Microeconomic Theory: Basic Principles and Extensions

Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 This is an electronic version of the print textbook Due to electronic rights restrictions some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing previous editions changes to current editions and alternate formats please visit wwwcengagecomhighered to search by ISBN author title or keyword for materials in your areas of interest Important notice Media content referenced within the product description or the product text may not be available in the eBook version Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 MICROECONOMIC THEORY Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Microeconomic Theory Basic Principles and Extensions TWELFTH EDITION WALTER NICHOLSON Amherst College CHRISTOPHER SNYDER Dartmouth College Australia Brazil Mexico Singapore United Kingdom United States Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 2017 2012 Cengage Learning WCN 01100101 ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced or distributed in any form or by any means except as permitted by US copyright law without the prior written permission of the copyright owner Library of Congress Control Number 2016941569 ISBN13 9781305505797 Cengage Learning 20 Channel Center Street Boston MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with employees residing in nearly 40 different countries and sales in more than 125 countries around the world Find your local representative at wwwcengagecom Cengage Learning products are represented in Canada by Nelson Education Ltd To learn more about Cengage Learning Solutions visit wwwcengagecom Purchase any of our products at your local college store or at our preferred online store wwwcengagebraincom Microeconomic Theory Basic Principles and Extensions Twelfth Edition Walter Nicholson Christopher Snyder Vice President General Manager Social Science Qualitative Business Erin Joyner Executive product Director Mike Schenk Product Director and Product Manager Jason Fremder Content Developer Anita Verma Product Assistant Emily Lehmann Marketing Director Kristen Hurd Marketing Manager Katie Jergens Marketing Coordinator Casey Binder Art and Cover Direction Production Management and Composition Lumina Datamatics Inc Intellectual 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duplicated in whole or in part WCN 02200203 vii Walter Nicholson is the Ward H Patton Professor of Economics Emeritus at Amherst College He received a BA in mathematics from Williams College and a PhD in econom ics from the Massachusetts Institute of Technology MIT Professor Nicholsons primary research interests are in the econometric analyses of labor market problems including wel fare unemployment and the impact of international trade For many years he has been Senior Fellow at Mathematica Inc and has served as an advisor to the US and Canadian governments He and his wife Susan live in Naples Florida and Montague Massachusetts Christopher M Snyder is the Joel and Susan Hyatt Professor of Economics at Dartmouth College currently serving as Chair of the Economics Department He received his BA in economics and mathematics from Fordham University and his PhD in eco nomics from MIT He is Research Associate in the National Bureau of Economic Research SecretaryTreasurer of the Industrial Organization Society and Associate Editor of the Review of Industrial Organization His research covers various theoretical and empirical topics in industrial organization contract theory and law and economics Professor Snyder and his wife Maura Doyle who also teaches economics at Dartmouth live within walking distance of campus in Hanover New Hampshire with their three daughters Professors Nicholson and Snyder are also the authors of Intermediate Microeconom ics and Its Application Cengage Learning 2015 an intuitive treatment of intermediate microeconomics emphasizing concepts and realworld applications over mathematical derivations About the Authors Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 ix Preface xix PART ONE Introduction 1 CHAPTER 1 Economic Models 3 CHAPTER 2 Mathematics for Microeconomics 21 PART TWO Choice and Demand 87 CHAPTER 3 Preferences and Utility 89 CHAPTER 4 Utility Maximization and Choice 115 CHAPTER 5 Income and Substitution Effects 141 CHAPTER 6 Demand Relationships among Goods 183 PART ThrEE Uncertainty and Strategy 205 CHAPTER 7 Uncertainty 207 CHAPTER 8 Game Theory 247 PART FOUr Production and Supply 295 CHAPTER 9 Production Functions 297 CHAPTER 10 Cost Functions 325 CHAPTER 11 Profit Maximization 363 PART FivE Competitive Markets 399 CHAPTER 12 The Partial Equilibrium Competitive Model 401 CHAPTER 13 General Equilibrium and Welfare 449 PART SiX Market Power 489 CHAPTER 14 Monopoly 491 CHAPTER 15 Imperfect Competition 525 PART SEvEN Pricing in Input Markets 573 CHAPTER 16 Labor Markets 575 CHAPTER 17 Capital and Time 599 PART EiGhT Market Failure 631 CHAPTER 18 Asymmetric Information 633 CHAPTER 19 Externalities and Public Goods 683 Brief Answers to Queries 717 Solutions to OddNumbered Problems 727 Glossary of Frequently Used Terms 741 Index 749 Brief Contents Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xi Preface xix PART ONE Introduction CHAPTER 1 Economic Models 3 Theoretical Models 3 Verification of Economic Models 4 General Features of Economic Models 5 Structure of Economic Models 6 Development of the Economic Theory of Value 9 Modern Developments 18 Summary 19 Suggestions for Further Reading 20 CHAPTER 2 Mathematics for Microeconomics21 Maximization of a Function of One Variable 21 Functions of Several Variables 26 Maximization of Functions of Several Variables 34 The Envelope Theorem 36 Constrained Maximization 40 Envelope Theorem in Constrained Maximization Problems 45 Inequality Constraints 46 SecondOrder Conditions and Curvature 48 Homogeneous Functions 55 Integration 58 Dynamic Optimization 63 Mathematical Statistics 67 Summary 76 Problems 77 Contents Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xii Contents Suggestions for Further Reading 81 Extensions SecondOrder Conditions and Matrix Algebra 82 PART TWO Choice and Demand CHAPTER 3 Preferences and Utility 89 Axioms of Rational Choice 89 Utility 90 Trades and Substitution 92 The Mathematics of Indifference Curves 99 Utility Functions for Specific Preferences 102 The ManyGood Case 106 Summary 107 Problems 107 Suggestions for Further Reading 110 Extensions Special Preferences 111 CHAPTER 4 Utility Maximization and Choice 115 An Initial Survey 116 The TwoGood Case A Graphical Analysis 117 The nGood Case 120 Indirect Utility Function 126 The Lump Sum Principle 127 Expenditure Minimization 129 Properties of Expenditure Functions 132 Summary 134 Problems 134 Suggestions for Further Reading 137 Extensions Budget Shares 138 CHAPTER 5 Income and Substitution Effects 141 Demand Functions 141 Changes in Income 143 Changes in a Goods Price 145 The Individuals Demand Curve 148 Compensated Hicksian Demand Curves and Functions 151 A Mathematical Development of Response to Price Changes 156 Demand Elasticities 159 Consumer Surplus 166 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Contents xiii Revealed Preference and the Substitution Effect 171 Summary 173 Problems 173 Suggestions for Further Reading 176 Extensions Demand Concepts and the Evaluation of Price Indices 178 CHAPTER 6 Demand Relationships among Goods 183 The TwoGood Case 183 Substitutes and Complements 186 Net Hicksian Substitutes and Complements 188 Substitutability with Many Goods 189 Composite Commodities 190 Home Production Attributes of Goods and Implicit Prices 193 Summary 196 Problems 197 Suggestions for Further Reading 200 Extensions Simplifying Demand and TwoStage Budgeting 202 PART ThrEE Uncertainty and Strategy CHAPTER 7 Uncertainty 207 Mathematical Statistics 207 Fair Gambles and the Expected Utility Hypothesis 208 Expected Utility 209 The Von NeumannMorgenstern Theorem 210 Risk Aversion 212 Measuring Risk Aversion 216 Methods for Reducing Uncertainty and Risk 221 Insurance 221 Diversification 222 Flexibility 223 Information 230 The StatePreference Approach to Choice Under Uncertainty 231 Asymmetry of Information 237 Summary 237 Problems 238 Suggestions for Further Reading 241 Extensions The Portfolio Problem 242 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xiv Contents CHAPTER 8 Game Theory 247 Basic Concepts 247 Prisoners Dilemma 248 Nash Equilibrium 250 Mixed Strategies 256 Existence of Equilibrium 260 Continuum of Actions 261 Sequential Games 263 Repeated Games 270 Incomplete Information 273 Simultaneous Bayesian Games 273 Signaling Games 278 Experimental Games 284 Evolutionary Games and Learning 286 Summary 287 Problems 287 Suggestions for Further Reading 290 Extensions Existence of Nash Equilibrium 291 PART FOUr Production and Supply CHAPTER 9 Production Functions 297 Marginal Productivity 297 Isoquant Maps and the Rate of Technical Substitution 300 Returns to Scale 304 The Elasticity of Substitution 307 Four Simple Production Functions 310 Technical Progress 314 Summary 318 Problems 319 Suggestions for Further Reading 321 Extensions ManyInput Production Functions 322 CHAPTER 10 Cost Functions 325 Definitions of Costs 325 Relationship between Profit Maximization and Cost Minimization 327 CostMinimizing Input Choices 328 Cost Functions 333 Shifts in Cost Curves 337 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Contents xv ShortRun LongRun Distinction 348 Summary 356 Problems 357 Suggestions for Further Reading 359 Extensions The Translog Cost Function 360 CHAPTER 11 Profit Maximization 363 The Nature and Behavior of Firms 363 Profit Maximization 365 Marginal Revenue 367 ShortRun Supply by a PriceTaking Firm 372 Profit Functions 376 Profit Maximization and Input Demand 381 Summary 388 Problems 388 Suggestions for Further Reading 392 Extensions Boundaries of the Firm 393 PART FivE Competitive Markets CHAPTER 12 The Partial Equilibrium Competitive Model 401 Market Demand 401 Timing of the Supply Response 405 Pricing in the Very Short Run 405 ShortRun Price Determination 407 Shifts in Supply and Demand Curves A Graphical Analysis 412 A Comparative Statics Model of Market Equilibrium 414 LongRun Analysis 418 LongRun Equilibrium Constant Cost Case 419 Shape of the LongRun Supply Curve 421 LongRun Elasticity of Supply 424 Comparative Statics Analysis of LongRun Equilibrium 424 Producer Surplus in the Long Run 428 Economic Efficiency and Applied Welfare Analysis 431 Price Controls and Shortages 434 Tax Incidence Analysis 435 Summary 440 Problems 440 Suggestions for Further Reading 444 Extensions Demand Aggregation and Estimation 445 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xvi Contents CHAPTER 13 General Equilibrium and Welfare 449 Perfectly Competitive Price System 449 A Graphical Model of General Equilibrium with Two Goods 450 Comparative Statics Analysis 460 General Equilibrium Modeling and Factor Prices 462 A Mathematical Model of Exchange 464 A Mathematical Model of Production and Exchange 475 Computable General Equilibrium Models 478 Summary 482 Problems 483 Suggestions for Further Reading 486 Extensions Computable General Equilibrium Models 487 PART SiX Market Power CHAPTER 14 Monopoly 491 Barriers to Entry 491 Profit Maximization and Output Choice 493 Misallocated Resources under Monopoly 498 Comparative Statics Analysis of Monopoly 501 Monopoly Product Quality 502 Price Discrimination 504 Price Discrimination through NonUniform Schedules 510 Regulation of Monopoly 513 Dynamic Views of Monopoly 516 Summary 518 Problems 518 Suggestions for Further Reading 522 Extensions Optimal Linear TwoPart Tariffs 523 CHAPTER 15 Imperfect Competition 525 ShortRun Decisions Pricing and Output 525 Bertrand Model 527 Cournot Model 528 Capacity Constraints 534 Product Differentiation 535 Tacit Collusion 541 LongerRun Decisions Investment Entry and Exit 545 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Contents xvii Strategic Entry Deterrence 550 Signaling 552 How Many Firms Enter 555 Innovation 559 Summary 561 Problems 562 Suggestions for Further Reading 565 Extensions Strategic Substitutes and Complements 567 PART SEvEN Pricing in Input Markets CHAPTER 16 Labor Markets 575 Allocation of Time 575 A Mathematical Analysis of Labor Supply 578 Market Supply Curve for Labor 582 Labor Market Equilibrium 583 Wage Variation 585 Monopsony in the Labor Market 589 Labor Unions 592 Summary 595 Problems 595 Suggestions for Further Reading 598 CHAPTER 17 Capital and Time 599 Capital and the Rate of Return 599 Determining the Rate of Return 601 Pricing of Risky Assets 608 The Firms Demand for Capital 610 Present Discounted Value Criterion 613 Natural Resource Pricing 617 Summary 620 Problems 620 Suggestions for Further Reading 623 APPENDIX The Mathematics of Compound Interest 625 Present Discounted Value 625 Continuous Time 627 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xviii Contents PART EiGhT Market Failure CHAPTER 18 Asymmetric Information 633 Complex Contracts as a Response to Asymmetric Information 633 PrincipalAgent Model 635 Hidden Actions 637 OwnerManager Relationship 638 Moral Hazard in Insurance 642 Hidden Types 647 Nonlinear Pricing 648 Adverse Selection in Insurance 658 Market Signaling 665 Auctions 667 Summary 671 Problems 671 Suggestions for Further Reading 674 Extensions Using Experiments to Measure AsymmetricInformation Problems 675 CHAPTER 19 Externalities and Public Goods 683 Defining Externalities 683 Externalities and Allocative Inefficiency 685 PartialEquilibrium Model of Externalities 689 Solutions to Negative Externality Problems 691 Attributes of Public Goods 695 Public Goods and Resource Allocation 697 Lindahl Pricing of Public Goods 701 Voting and Resource Allocation 704 A Simple Political Model 706 Voting Mechanisms 709 Summary 710 Problems 711 Suggestions for Further Reading 713 Extensions Pollution Abatement 715 Brief Answers to Queries 717 Solutions to OddNumbered Problems727 Glossary of Frequently Used Terms 741 Index 749 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xix The 12th edition of Microeconomic Theory Basic Principles and Extensions continues a suc cessful collaboration between the authors starting with the 10th edition This edition rep resents our efforts to continue refining and modernizing our treatment of microeconomic theory Despite the significant changes appearing in virtually every chapter the text retains all of the elements that have made it successful for so many editions The basic approach is to focus on building intuition about economic models while providing students with the mathematical tools needed to go further in their studies The text also seeks to facilitate that linkage by providing many numerical examples advanced problems and extended discussions of empirical implementationall of which are intended to show students how microeconomic theory is used today New developments continue to keep the field excit ing and we hope this edition manages to capture that excitement NEW TO THE TWELFTH EDITION We took a fresh look at every chapter to make sure that they continue to provide clear and uptodate coverage of all of the topics examined The major revisions include the following Many of the topics in our introductory chapter on mathematics Chapter 2 have been further revised to conform more closely to methods encountered in the recent econom ics literature Significant new material has been added on comparative statics analysis including the use of Cramers rule and on the interpretation of the envelope theorem New figures have been added to illustrate the most basic concepts risk aversion certainty equivalence in Chapter 7 on uncertainty and the notation streamlined throughout For all the figures exhibiting the gametheory examples in Chapter 8 detailed captions have been added providing synopses and further analytical points We tightened the exposition by removing several extraneous examples Passages have been added to Chapter 10 to help clear up perennial sources of student confusion regarding different categories of costseconomic versus accounting fixed versus sunk and so forthillustrating with examples from realworld industries Our discussion of the comparative statics of the competitive model in Chapter 12 has been extensively updated and expanded using the new mathematical material provided in Chapter 2 Preface Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 xx Preface Chapter 14 on monopoly has been extensively revised A passage has been added mak ing basic points about the monopoly problem connecting it to general profit maximiza tion from Chapter 11 Our revamped approach to comparative statics is now featured in several places in this chapter We cover recent advances in price discrimination tracta ble functional forms and innovation A significant amount of new material has been added to Chapter 17 on capital by look ing at savings decisions under uncertainty The concept of the stochastic discount factor is introduced and used to describe a number of issues in modern finance theory Coverage of behavioral economics has been further expanded with a number of added references throughout the relevant chapters One or more new behavioral econom ics problems have been added to most chapters covering topics such as decision util ity spurious product differentiation and the role of competition and advertising in unshrouding information about prices to consumers These appear at the end of the list of problems highlighted by the icon of the head with psychological gears turning Many new problems have been added with the goal of sharpening the focus on ones that will help students to develop their analytical skills SUPPLEMENTS TO THE TExT The thoroughly revised ancillaries for this edition include the following The Solutions Manual and Test Bank by the text authors The Solutions Manual con tains comments and solutions to all problems and the Test Bank has been revised to include additional questions Both are available to all adopting instructors in electronic version on Instructors companion site PowerPoint Lecture Presentation Slides PowerPoint slides for each chapter of the text provide a thorough set of outlines for classroom use or for students as a study aid The slides are available on Instructors companion site MindTap for Microeconomic Theory Basic Principles Extensions 12th Edition is a digital learning solution allowing instructors to chart paths of dynamic assignments and applications personalized for their own courses MindTap also includes realtime course analytics and an accessible reader to help engage students and encourage their high level thinking rather than memorization Cengage Learning Testing powered by Cognero is a flexible online system that allows instructors to import edit and manipulate content from the texts test bank or else where including the instructors own favorite test questions create multiple test ver sions in an instant and deliver tests from learning management system used for the course the classroom or wherever the instructor wants ONLINE RESOURCES Cengage Learning provides instructors with a set of valuable online resources that are an effective complement to this text Each new copy of the book comes with a registration card that provides access to Economic Applications and InfoTrac College Edition Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Preface xxi ACkNOWLEDgMENTS We are indebted to the team at Cengage most importantly Anita Verma for keeping all of the moving parts of this new edition moving and on schedule The copyeditors at Lumina Datamatics Inc did a great job of making sense of our messy manuscripts Joseph Malcolm coordinated the copyediting and supervised the production of page proofs deal ing expertly with many of the technical problems that arise in going from text to print equations We very much appreciate his attention to the complexities of this process and are grateful for his professionalism and hard work We thank our colleagues at Amherst and Dartmouth College for valuable conversa tions and understanding Several colleagues who used the book for their courses offered us detailed suggestions for revision We have also benefitted from the reactions of gen erations of students to the use of the book in our own microeconomics classes Over the years Amherst students Eric Budish Adrian Dillon David Macoy Tatyana Mamut Anoop Menon katie Merrill Jordan Milev and Doug Norton and Dartmouth students Wills Begor Paulina karpis glynnis kearny and Henry Senkfor worked with us revising various chapters Walter again gives special thanks to his wife Susan after providing muchneeded sup port through twentyfour editions of his microeconomics texts she is happy for the success but continues to wonder about his sanity Walters children kate David Tory and Paul still seem to be living happy and productive lives despite a severe lack of microeconomic education Perhaps this will be remedied as the next generation grows older At least he hopes they will wonder what the books dedicated to them are all about He is offering a prize for the first to read the entire text Chris gives special thanks to his familyhis wife Maura Doyle and their daughters Clare Tess and Megfor their patience during the revision process Maura has extensive experience using the book in her popular microeconomics courses at Dartmouth College and has been a rich source of suggestions reflected in this revision Perhaps our greatest debt is to instructors who adopt the text who share a similar view of how microeconomics should be taught We are grateful for the suggestions that teachers and students have shared with us over the years Special mention in this regard is due genevieve Briand Ramez guirguis Ron Harstad Bradley Ruffle and Adriaan Soetevent who provided pages of detailed perceptive comments on the previous edi tion We encourage teachers and students to continue to email us with any comments on the text wenicholsonamherstedu or chrissnyderdartmouthedu Walter Nicholson Amherst Massachusetts Christopher Snyder Hanover New Hampshire June 2016 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 1 Introduction Chapter 1 Economic Models Chapter 2 Mathematics for Microeconomics This part contains two chapters Chapter 1 examines the general philosophy of how economists build models of economic behavior Chapter 2 then reviews some of the mathematical tools used in the construction of these models The mathematical tools from Chapter 2 will be used throughout the remainder of this book PART ONE Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 3 CHAPTER ONE Economic Models The main goal of this book is to introduce you to the models that economists use to explain the behavior of consumers firms and markets These models are central to the study of all areas of economics Therefore it is essential to understand both the need for such models and the basic framework used to develop them This chapter begins by outlining some of the ways in which economists study practically every question that interests them 11 THEORETICAL MODELS A modern economy is a complicated place Thousands of firms engage in producing mil lions of different goods Many millions of people work in all sorts of occupations and make decisions about which of these goods to buy Lets use peanuts as an example Pea nuts must be harvested at the right time and shipped to processors who turn them into peanut butter peanut oil peanut brittle and numerous other peanut delicacies These processors in turn must make certain that their products arrive at thousands of retail outlets in the proper quantities to meet demand Because it would be impossible to describe the features of even these peanut markets in complete detail economists must abstract from the complexities of the real world and develop rather simple models that capture the essentials Just as a road map is helpful even though it does not record every house or every store the economic models of say the market for peanuts are also useful even though they do not record every minute fea ture of the peanut economy In this book we will study the most widely used economic models We will see that even though these models often make significant abstractions from the complexities of the real world they nonetheless capture the essential features that are common to all economic activities The use of models is widespread in the physical and social sciences In physics the notion of a perfect vacuum or an ideal gas is an abstraction that permits scientists to study realworld phenomena in simplified settings In chemistry the idea of an atom or a molecule is actually a simplified model of the structure of matter Architects use mockup models to plan buildings Television repairers refer to wiring diagrams to locate problems Economists models perform similar functions They provide simplified portraits of the way individuals make decisions the way firms behave and the way in which these two groups interact to establish markets Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 4 Part 1 Introduction 12 VERIFICATION OF ECONOMIC MODELS Of course not all models prove to be good For example the earthcentered model of planetary motion devised by Ptolemy was eventually discarded because it proved incapa ble of accurately explaining how the planets move around the sun An important purpose of scientific investigation is to sort out the bad models from the good models Two general methods have been used to verify economic models 1 a direct approach which seeks to establish the validity of the basic assumptions on which a model is based and 2 an indirect approach which attempts to confirm validity by showing that a simplified model correctly predicts realworld events To illustrate the basic differences between the two approaches lets briefly examine a model that we will use extensively in later chapters of this bookthe model of a firm that seeks to maximize profits 121 The profitmaximization model The model of a firm seeking to maximize profits is obviously a simplification of reality It ignores the personal motivations of the firms managers and does not consider conflicts among them It assumes that profits are the only relevant goal of the firm other possi ble goals such as obtaining power or prestige are treated as unimportant The model also assumes that the firm has sufficient information about its costs and the nature of the market to which it sells to discover its profitmaximizing options Most realworld firms of course do not have this information available at least not at zero cost Yet such shortcomings in the model are not necessarily serious No model can exactly describe reality The real question is whether this simple model has any claim to being a good one 122 Testing assumptions One test of the model of a profitmaximizing firm investigates its basic assumption Do firms really seek maximum profits Some economists have examined this question by sending questionnaires to executives asking them to specify the goals they pursue The results of such studies have been varied Businesspeople often mention goals other than profits or claim they only do the best they can to increase profits given their limited infor mation On the other hand most respondents also mention a strong interest in profits and express the view that profit maximization is an appropriate goal Therefore testing the profitmaximizing model by testing its assumptions has provided inconclusive results 123 Testing predictions Some economists most notably Milton Friedman deny that a model can be tested by inquir ing into the reality of its assumptions1 They argue that all theoretical models are based on unrealistic assumptions the very nature of theorizing demands that we make certain abstrac tions These economists conclude that the only way to determine the validity of a model is to see whether it is capable of predicting and explaining realworld events The ultimate test of an economic model comes when it is confronted with data from the economy itself Friedman provides an important illustration of that principle He asks what kind of the ory one should use to explain the shots expert pool players will make He argues that the laws of velocity momentum and angles from theoretical physics would be a suitable model 1See M Friedman Essays in Positive Economics Chicago University of Chicago Press 1953 chap 1 For an alternative view stressing the importance of using realistic assumptions see H A Simon Rational Decision Making in Business Organizations American Economic Review 69 no 4 September 1979 493513 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 5 Pool players shoot shots as if they follow these laws But most players asked whether they pre cisely understand the physical principles behind the game of pool will undoubtedly answer that they do not Nonetheless Friedman argues the physical laws provide accurate predic tions and therefore should be accepted as appropriate theoretical models of how experts play pool Thus a test of the profitmaximization model would be provided by predicting the behavior of realworld firms by assuming that these firms behave as if they were maximiz ing profits See Example 11 later in this chapter If these predictions are reasonably in accord with reality we may accept the profitmaximization hypothesis However we would reject the model if realworld data seem inconsistent with it Hence the ultimate test of any theory is its ability to predict realworld events 124 Importance of empirical analysis The primary concern of this book is the construction of theoretical models But the goal of such models is always to learn something about the real world Although the inclusion of a lengthy set of applied examples would needlessly expand an already bulky book2 the Extensions included at the end of many chapters are intended to provide a transition between the theory presented here and the ways that theory is applied in empirical studies 13 GENERAL FEATURES OF ECONOMIC MODELS The number of economic models in current use is of course large Specific assumptions used and the degree of detail provided vary greatly depending on the problem being addressed The models used to explain the overall level of economic activity in the United States for example must be considerably more aggregated and complex than those that seek to interpret the pricing of Arizona strawberries Despite this variety practically all economic models incorporate three common elements 1 the ceteris paribus other things the same assumption 2 the supposition that economic decisionmakers seek to optimize something and 3 a careful distinction between positive and normative questions Because we will encounter these elements throughout this book it may be helpful at the outset to describe the philosophy behind each of them 131 The ceteris paribus assumption As in most sciences models used in economics attempt to portray relatively simple relation ships A model of the wheat market for example might seek to explain wheat prices with a small number of quantifiable variables such as wages of farmworkers rainfall and consumer incomes This parsimony in model specification permits the study of wheat pricing in a sim plified setting in which it is possible to understand how the specific forces operate Although any researcher will recognize that many outside forces eg presence of wheat diseases changes in the prices of fertilizers or of tractors or shifts in consumer attitudes about eating bread affect the price of wheat these other forces are held constant in the construction of the model It is important to recognize that economists are not assuming that other factors do not affect wheat prices rather such other variables are assumed to be unchanged during the period of study In this way the effect of only a few forces can be studied in a simplified set ting Such ceteris paribus other things equal assumptions are used in all economic modeling 2For an intermediatelevel text containing an extensive set of realworld applications see W Nicholson and C Snyder Intermediate Microeconomics and Its Application 12th ed Mason OH ThomsonSouthwestern 2015 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 6 Part 1 Introduction Use of the ceteris paribus assumption does pose some difficulties for the verification of economic models from realworld data In other sciences the problems may not be so severe because of the ability to conduct controlled experiments For example a physicist who wishes to test a model of the force of gravity probably would not do so by dropping objects from the Empire State Building Experiments conducted in that way would be sub ject to too many extraneous forces eg wind currents particles in the air variations in temperature to permit a precise test of the theory Rather the physicist would conduct experiments in a laboratory using a partial vacuum in which most other forces could be controlled or eliminated In this way the theory could be verified in a simple setting with out considering all the other forces that affect falling bodies in the real world With a few notable exceptions economists have not been able to conduct controlled experiments to test their models Instead they have been forced to rely on various statisti cal methods to control for other forces when testing their theories Although these statis tical methods are as valid in principle as the controlled experiment methods used by other scientists in practice they raise a number of thorny issues For that reason the limitations and precise meaning of the ceteris paribus assumption in economics are subject to greater controversy than in the laboratory sciences 14 STRUCTURE OF ECONOMIC MODELS Most of the economic models you will encounter in this book have a mathematical struc ture They highlight the relationships between factors that affect the decisions of house holds and firms and the results of those decisions Economists tend to use different names for these two types of factors or in mathematical terms variables Variables that are out side of a decisionmakers control are called exogenous variables Such variables are inputs into economic models For example in consumer theory we will usually treat individuals as pricetakers The prices of goods are determined outside of our models of consumer behavior and we wish to study how consumers adjust to them The results of such deci sions eg the quantities of each good that a consumer buys are endogenous variables These variables are determined within our models This distinction is pictured schemati cally in Figure 11 Although the actual models developed by economists may be compli cated they all have this basic structure A good way to start studying a particular model is to identify precisely how it fits into this framework This distinction between exogenous and endogenous variables will become clearer as we explore a variety of economic models Keeping straight which variables are determined out side a particular model and which variables are determined within a model can be confus ing therefore we will try to remind you about this as we go along The distinction between exogenous and endogenous variables is also helpful in understanding the way in which the ceteris paribus assumption is incorporated into economic models In most cases we will want to study how the results of our models change when one of the exogenous variables such as a price or a persons income changes It is possible even likely that the change in such a single variable will change all the results calculated from the model For example as we will see it is likely that the change in the price of a single good will cause an individual to change the quantities of practically every good he or she buys Examining all such responses is precisely why economists build models The ceteris paribus assumption is enforced by changing only one exogenous variable holding all others constant If we wish to study the effects of a change in the price of gasoline on a households purchases we change that price in our model but we do not change the prices of other goods and in some cases we do not change the individuals income either Holding the other prices constant is what is meant by studying the ceteris paribus effect of an increase in the price of gasoline Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 7 141 Optimization assumptions Many economic models start from the assumption that the economic actors being studied are rationally pursuing some goal We briefly discussed such an assumption when investi gating the notion of firms maximizing profits Example 11 shows how that model can be used to make testable predictions Other examples we will encounter in this book include consumers maximizing their own wellbeing utility firms minimizing costs and gov ernment regulators attempting to maximize public welfare As we will show even though all these assumptions are unrealistic all have won widespread acceptance as good starting places for developing economic models There seem to be two reasons for this acceptance First the optimization assumptions are useful for generating precise solvable models pri marily because such models can draw on a variety of mathematical techniques suitable for optimization problems Many of these techniques together with the logic behind them are reviewed in Chapter 2 A second reason for the popularity of optimization models concerns their apparent empirical validity As some of our Extensions show such models seem to be fairly good at explaining reality In all then optimization models have come to occupy a prominent position in modern economic theory Values for exogenous variables are inputs into most economic models Model outputs results are values for the endogenous variables EXOGENOUS VARIABLES ENDOGENOUS VARIABLES ECONOMIC MODEL Households Prices of goods Firms Prices of inputs and output Households Utility maximization Firms Profit maximization Households Quantities bought Firms Output produced inputs hired FIGURE 11 Structure of a Typical Microeconomic Model EXAMPLE 11 Profit Maximization The profitmaximization hypothesis provides a good illustration of how optimization assumptions can be used to generate empirically testable propositions about economic behavior Suppose that a firm can sell all the output that it wishes at an exogenously determined price of p per unit and that the total costs of production C depend on the amount produced q Then profits are given by profits 5 π 5 pq 2 C1q2 11 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 8 Part 1 Introduction Maximization of profits consists of finding that value of q that maximizes the profit expression in Equation 11 This is a simple problem in calculus Differentiation of Equation 11 and setting that derivative equal to 0 give the following firstorder condition for a maximum dπ dq 5 p 2 Cr1q2 5 0 or p 5 Cr1q2 12 In words the profitmaximizing output level q is found by selecting that output level for which price is equal to marginal cost Cr 1q2 This result should be familiar to you from your introduc tory economics course Notice that in this derivation the price for the firms output is treated as a constant because the firm is a pricetaker That is price is an exogenous variable in this model Equation 12 is only the firstorder condition for a maximum Taking account of the sec ondorder condition can help us to derive a testable implication of this model The secondorder condition for a maximum is that at q it must be the case that d2π dq2 5 2Cs 1q2 0 or Cs 1q2 0 13 That is marginal cost must be increasing at q for this to be a true point of maximum profits Our model can now be used to predict how a firm will react to a change in price To do so we differentiate Equation 12 with respect to price p assuming that the firm continues to choose a profitmaximizing level of q d3p 2 Cr 1q2 5 04 dp 5 1 2 Cs 1q2 dq dp 5 0 14 Rearranging terms a bit gives dq dp 5 1 Cs 1q2 0 15 Here the final inequality again reflects the fact that marginal cost must be increasing at q if this point is to be a true maximum This then is one of the testable propositions of the profit maximization hypothesisif other things do not change a pricetaking firm should respond to an increase in price by increasing output On the other hand if firms respond to increases in price by reducing output there must be something wrong with our model Although this is a simple model it reflects the way we will proceed throughout much of this book Specifically the fact that the primary implication of the model is derived by calculus and consists of showing what sign a derivative should have is the kind of result we will see many times Notice that in this model there is only one endogenous variableq the quantity the firm chooses to produce There is also only one exogenous variablep the price of the product which the firm takes as a given Our model makes a specific prediction about how changes in this exog enous variable affect the firms output choice QUERY In general terms how would the implications of this model be changed if the price a firm obtains for its output were a function of how much it sold That is how would the model work if the pricetaking assumptions were abandoned 142 Positivenormative distinction A final feature of most economic models is the attempt to differentiate carefully between positive and normative questions Thus far we have been concerned primarily with positive economic theories Such theories take the real world as an object to be studied attempting to explain those economic phenomena that are observed Positive economics Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 9 seeks to determine how resources are in fact allocated in an economy A somewhat dif ferent use of economic theory is normative analysis taking a definite stance about what should be done Under the heading of normative analysis economists have a great deal to say about how resources should be allocated For example an economist engaged in posi tive analysis might investigate how prices are determined in the US healthcare economy The economist also might want to measure the costs and benefits of devoting even more resources to health care by for example offering governmentsubsidized health insurance But when he or she specifically advocates that such an insurance plan should be adopted the analysis becomes normative Some economists believe that the only proper economic analysis is positive analysis Draw ing an analogy with the physical sciences they argue that scientific economics should con cern itself only with the description and possibly prediction of realworld economic events To take political positions and to plead for special interests are considered to be outside the competence of an economist acting as such Of course an economist like any other citizen is free to express his or her views on political matters But when doing so he or she is acting as a citizen not an economist For other economists however the positivenormative distinction seems artificial They believe that the study of economics necessarily involves the researchers own views about ethics morality and fairness According to these economists searching for scientific objectivity in such circumstances is hopeless Despite some ambiguity this book tries to adopt a positivist tone leaving normative concerns for you to decide for yourself 15 DEVELOPMENT OF THE ECONOMIC THEORY OF VALUE Because economic activity has been a central feature of all societies it is surprising that these activities were not studied in any detail until fairly recently For the most part eco nomic phenomena were treated as a basic aspect of human behavior that was not suffi ciently interesting to deserve specific attention It is of course true that individuals have always studied economic activities with a view toward making some kind of personal gain Roman traders were not above making profits on their transactions But investigations into the basic nature of these activities did not begin in any depth until the eighteenth century3 Because this book is about economic theory as it stands today rather than the history of economic thought our discussion of the evolution of economic theory will be brief Only one area of economic study will be examined in its historical setting the theory of value 151 Early economic thoughts on value The theory of value not surprisingly concerns the determinants of the value of a com modity This subject is at the center of modern microeconomic theory and is closely intertwined with the fundamental economic problem of allocating scarce resources to alternative uses The logical place to start is with a definition of the word value Unfor tunately the meaning of this term has not been unambiguous throughout the history of economics Today we regard value as being synonymous with the price of a commodi ty4 Much of the early writings about economics however sought to establish the idea of a just price for some items and examine how actual market prices conformed to this ideal In many cases these discussions touched on modern concerns For example early 3For a detailed treatment of early economic thought see the classic work by J A Schumpeter History of Economic Analysis New York Oxford University Press 1954 pt II chaps 13 4This is not completely true when externalities are involved and a distinction must be made between private and social value see Chapter 19 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 10 Part 1 Introduction philosophereconomists were well aware of the effect that monopolies had on prices and readily condemned situations where such prices vastly exceeded production costs In other cases however these early thinkers adopted philosophical understandings of what a price should be and sometimes that price differed greatly from what was observed in actual markets This distinction was most clearly illustrated by arguments about whether inter est payments on loans were just throughout the fourteenth and fifteenth centuries The discussion focused primarily on whether lenders actually incurred any true costs in mak ing loans and if not how the charging of interest amounted to usury Similar arguments continue to this day not only with respect to interest on loans but also with respect to such topics as fair rental rates for housing or just wages for lowwage workers 152 The founding of modern economics During the latter part of the eighteenth century philosophers began to take a scientific approach to economic questions by focusing more explicitly on the mechanisms by which prices are determined The 1776 publication of The Wealth of Nations by Adam Smith 17231790 is generally considered the beginning of modern economics In his vast allencompassing work Smith laid the foundation for thinking about market forces in an ordered and systematic way Still Smith and his immediate successors such as David Ricardo 17721823 continued to struggle in finding a way to describe the relationship between value and price To Smith for example the value of a commodity often meant its value in use whereas the price represented its value in exchange The distinction between these two concepts was illustrated by the famous waterdiamond paradox Water which obviously has great value in use has little value in exchange it has a low price diamonds are of little practical use but have a great value in exchange The paradox with which early economists struggled derives from the observation that some useful items have low prices whereas certain nonessential items have high prices 153 Labor theory of exchange value Neither Smith nor Ricardo ever satisfactorily resolved the waterdiamond paradox The concept of value in use was left for philosophers to debate while economists turned their attention to explaining the determinants of value in exchange ie to explaining relative prices One obvious possible explanation is that exchange values of goods are determined by what it costs to produce them Costs of production are primarily influenced by labor costsat least this was so in the time of Smith and Ricardoand therefore it was a short step to embrace a labor theory of value For example to paraphrase an example from Smith if catching a deer takes twice the number of labor hours as catching a beaver then one deer should exchange for two beavers In other words the price of a deer should be twice that of a beaver Similarly diamonds are relatively costly because their production requires substantial labor input whereas water is freely available To students with even a passing knowledge of what we now call the law of supply and demand Smiths and Ricardos explanation must seem incomplete Did they not recognize the effects of demand on price The answer to this question is both yes and no They did observe periods of rapidly rising and falling relative prices and attributed such changes to demand shifts However they regarded these changes as abnormalities that produced only a temporary divergence of market price from labor value Because they had not really developed a theory of value in use ie demand they were unwilling to assign demand any more than a transient role in determining relative prices Rather longrun exchange values were assumed to be determined solely by labor costs of production Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 11 154 The marginalist revolution Between 1850 and 1880 economists became increasingly aware that to construct an adequate alternative to the labor theory of value they had to devise a theory of value in use During the 1870s several economists discovered that it is not the total usefulness of a commodity that helps to determine its exchange value but rather the usefulness of the last unit consumed For example water is certainly usefulit is necessary for all life However because water is relatively plentiful consuming one more pint ceteris paribus has a rela tively low value to people These marginalists redefined the concept of value in use from an idea of overall usefulness to one of marginal or incremental usefulnessthe usefulness of an additional unit of a commodity The concept of the demand for an incremental unit of output was now contrasted with Smiths and Ricardos analysis of production costs to derive a comprehensive picture of price determination5 155 Marshallian supplydemand synthesis The clearest statement of these marginal principles was presented by the English economist Alfred Marshall 18421924 in his Principles of Economics published in 1890 Marshall showed that demand and supply simultaneously operate to determine price As Marshall noted just as you cannot tell which blade of a scissors does the cutting so too you cannot say that either demand or supply alone determines price That analysis is illustrated by the famous Marshallian cross shown in Figure 12 In the diagram the quantity of a good pur chased per period is shown on the horizontal axis and its price appears on the vertical axis The curve DD represents the quantity of the good demanded per period at each possible price The curve is negatively sloped to reflect the marginalist principle that as quantity 5Ricardo had earlier provided an important first step in marginal analysis in his discussion of rent Ricardo theorized that as the production of corn increased land of inferior quality would be used and this would cause the price of corn to increase In his argument Ricardo recognized that it is the marginal costthe cost of producing an additional unitthat is relevant to pricing Notice that Ricardo implicitly held other inputs constant when discussing decreasing land productivity that is he used one version of the ceteris paribus assumption Marshall theorized that demand and supply interact to determine the equilibrium price p and the quantity q that will be traded in the market He concluded that it is not possible to say that either demand or supply alone determines price or therefore that either costs or usefulness to buyers alone determines exchange value Quantity per period Price S S D D q p FIGURE 12 The Marshallian SupplyDemand Cross Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 12 Part 1 Introduction increases people are willing to pay less for the last unit purchased It is the value of this last unit that sets the price for all units purchased The curve SS shows how marginal production costs increase as more output is produced This reflects the increasing cost of producing one more unit as total output expands In other words the upward slope of the SS curve reflects increasing marginal costs just as the downward slope of the DD curve reflects decreasing marginal value The two curves intersect at p q This is an equilibrium pointboth buyers and sellers are content with the quantity being traded and the price at which it is traded If one of the curves should shift the equilibrium point would shift to a new location Thus price and quantity are simultaneously determined by the joint opera tion of supply and demand EXAMPLE 12 SupplyDemand Equilibrium Although graphical presentations are adequate for some purposes economists often use algebraic representations of their models both to clarify their arguments and to make them more precise As an elementary example suppose we wished to study the peanut market and based on the statistical analysis of historical data concluded that the quantity of peanuts demanded each week q measured in bushels depended on the price of peanuts p measured in dollars per bushel according to the equation quantity demanded 5 qD 5 1000 2 100p 16 Because this equation for qD contains only the single independent variable p we are implicitly holding constant all other factors that might affect the demand for peanuts Equation 16 indicates that if other things do not change at a price of 5 per bushel people will demand 500 bushels of peanuts whereas at a price of 4 per bushel they will demand 600 bushels The negative coefficient for p in Equation 16 reflects the marginalist principle that a lower price will cause people to buy more peanuts To complete this simple model of pricing suppose that the quantity of peanuts supplied also depends on price quantity supplied 5 qS 5 2125 1 125p 17 Here the positive coefficient of price also reflects the marginal principle that a higher price will call forth increased supplyprimarily because as we saw in Example 11 it permits firms to incur higher marginal costs of production without incurring losses on the additional units produced Equilibrium price determination Therefore Equations 16 and 17 reflect our model of price determination in the peanut market An equilibrium price can be found by setting quantity demanded equal to quantity supplied qD 5 qS 18 or 1000 2 100p 5 2125 1 125p 19 or 225p 5 1125 110 Thus p 5 5 111 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 13 At a price of 5 per bushel this market is in equilibrium At this price people want to purchase 500 bushels and that is exactly what peanut producers are willing to supply This equilibrium is pictured graphically as the intersection of D and S in Figure 13 A more general model To illustrate how this supplydemand model might be used lets adopt a more general notation Suppose now that the demand and supply functions are given by qD 5 a 1 bp and qS 5 c 1 dp 112 where a and c are constants that can be used to shift the demand and supply curves respectively and b102 and d102 represent demanders and suppliers reactions to price Equilibrium in this market requires qD 5 qS or a 1 bp 5 c 1 dp 113 Thus equilibrium price is given by6 p 5 a 2 c d 2 b 114 The initial supplydemand equilibrium is illustrated by the intersection of D and S 1p 5 5 q 5 5002 When demand shifts to qDr 51450 2100p denoted as Dr the equilibrium shifts to p 5 7 q 5 750 0 Quantity per period bushels Price S S D D D D 145 10 7 5 500 750 1000 1450 FIGURE 13 Changing SupplyDemand Equilibria 6Equation 114 is sometimes called the reduced form for the supplydemand structural model of Equations 112 and 113 It shows that the equilibrium value for the endogenous variable p ultimately depends only on the exogenous factors in the model a and c and on the behavioral parameters b and d A similar equation can be calculated for equilibrium quantity Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 14 Part 1 Introduction Notice that in our previous example a 5 1000 b 5 2100 c 5 2125 and d 5 125 therefore p 5 1000 1 125 125 1 100 5 1125 225 5 5 115 With this more general formulation however we can pose questions about how the equilibrium price might change if either the demand or supply curve shifted For example differentiation of Equation 114 shows that dp da 5 1 d 2 b 0 dp dc 5 21 d 2 b 0 116 That is an increase in demand an increase in a increases equilibrium price whereas an increase in supply an increase in c reduces price This is exactly what a graphical analysis of supply and demand curves would show For example Figure 13 shows that when the constant term a in the demand equation increases from 1000 to 1450 equilibrium price increases to p 5 7 3 5 11450 1 12522254 QUERY How might you use Equation 116 to predict how each unit increase in the exogenous constant a affects the endogenous variable p Does this equation correctly predict the increase in p when the constant a increases from 1000 to 1450 156 Paradox resolved Marshalls model resolves the waterdiamond paradox Prices reflect both the marginal evaluation that demanders place on goods and the marginal costs of producing the goods Viewed in this way there is no paradox Water is low in price because it has both a low marginal value and a low marginal cost of production On the other hand diamonds are high in price because they have both a high marginal value because people are willing to pay quite a bit for one more and a high marginal cost of production This basic model of supply and demand lies behind much of the analysis presented in this book 157 General equilibrium models Although the Marshallian model is an extremely useful and versatile tool it is a partial equilibrium model looking at only one market at a time For some questions this narrowing of perspective gives valuable insights and analytical simplicity For other broader questions such a narrow viewpoint may prevent the discovery of important rela tionships among markets To answer more general questions we must have a model of the whole economy that suitably mirrors the connections among various markets and economic agents The French economist Leon Walras 18311910 building on a long Continental tradition in such analysis created the basis for modern investigations into those broad questions His method of representing the economy by a large number of simultaneous equations forms the basis for understanding the interrelationships implicit in general equilibrium analysis Walras recognized that one cannot talk about a single mar ket in isolation what is needed is a model that permits the effects of a change in one mar ket to be followed through other markets For example suppose that the demand for peanuts were to increase This would cause the price of peanuts to increase Marshallian analysis would seek to understand the size of Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 15 this increase by looking at conditions of supply and demand in the peanut market General equilibrium analysis would look not only at that market but also at repercussions in other markets An increase in the price of peanuts would increase costs for peanut butter makers which would in turn affect the supply curve for peanut butter Similarly the increasing price of peanuts might mean higher land prices for peanut farmers which would affect the demand curves for all products that they buy The demand curves for automobiles furni ture and trips to Europe would all shift out and that might create additional incomes for the providers of those products Consequently the effects of the initial increase in demand for peanuts eventually would spread throughout the economy General equilibrium anal ysis attempts to develop models that permit us to examine such effects in a simplified set ting Several models of this type are described in Chapter 13 158 Production possibility frontier Here we briefly introduce some general equilibrium ideas by using another graph you should remember from introductory economicsthe production possibility frontier This graph shows the various amounts of two goods that an economy can produce using its available resources during some period say one week Because the production possibility frontier shows two goods rather than the single good in Marshalls model it is used as a basic building block for general equilibrium models Figure 14 shows the production possibility frontier for two goods food and clothing The graph illustrates the supply of these goods by showing the combinations that can be produced with this economys resources For example 10 pounds of food and 3 units of clothing or 4 pounds of food and 12 units of clothing could be produced Many other combinations of food and clothing could also be produced The production possibility frontier shows all of them Combinations of food and clothing outside the frontier cannot be produced because not enough resources are available The production possibility fron tier reminds us of the basic economic fact that resources are scarcethere are not enough resources available to produce all we might want of every good This scarcity means that we must choose how much of each good to produce Figure 14 makes clear that each choice has its costs For example if this economy produces 10 pounds of food and 3 units of clothing at point A producing 1 more unit of clothing would cost ½ pound of foodincreasing the output of clothing by 1 unit means the production of food would have to decrease by ½ pound Thus the opportunity cost of 1 unit of clothing at point A is ½ pound of food On the other hand if the economy initially produces 4 pounds of food and 12 units of clothing at point B it would cost 2 pounds of food to produce 1 more unit of clothing The opportunity cost of 1 more unit of clothing at point B has increased to 2 pounds of food Because more units of clothing are produced at point B than at point A both Ricardos and Marshalls ideas of increasing incremental costs suggest that the opportunity cost of an additional unit of clothing will be higher at point B than at point A This effect is shown by Figure 14 The production possibility frontier provides two general equilibrium insights that are not clear in Marshalls supply and demand model of a single market First the graph shows that producing more of one good means producing less of another good because resources are scarce Economists often perhaps too often use the expression there is no such thing as a free lunch to explain that every economic action has opportunity costs Second the production possibility frontier shows that opportunity costs depend on how much of each good is produced The frontier is like a supply curve for two goods It shows the opportu nity cost of producing more of one good as the decrease in the amount of the second good Therefore the production possibility frontier is a particularly useful tool for studying sev eral markets at the same time Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 16 Part 1 Introduction The production possibility frontier shows the different combinations of two goods that can be produced from a certain amount of scarce resources It also shows the opportunity cost of producing more of one good as the amount of the other good that cannot then be produced The opportunity cost at two differ ent levels of clothing production can be seen by comparing points A and B Quantity of food per week B A 0 2 4 95 10 3 4 12 13 Quantity of clothing per week Opportunity cost of clothing 2 pounds of food Opportunity cost of clothing pound of food 1 2 FIGURE 14 Production Possibility Frontier EXAMPLE 13 The Production Possibility Frontier and Economic Inefficiency General equilibrium models are good tools for evaluating the efficiency of various economic arrangements As we will see in Chapter 13 such models have been used to assess a wide variety of policies such as trade agreements tax structures and environmental regulations In this simple example we explore the idea of efficiency in its most elementary form Suppose that an economy produces two goods x and y using labor as the only input The production function for good x is x 5 l05 x where lx is the quantity of labor used in x produc tion and the production function for good y is y 5 2l 05 y Total labor available is constrained by lx 1 ly 200 Construction of the production possibility frontier in this economy is extremely simple lx 1 ly 5 x2 1 025y2 200 117 where the equality holds exactly if the economy is to be producing as much as possible which after all is why it is called a frontier Equation 117 shows that the frontier here has the shape of a quarter ellipseits concavity derives from the diminishing returns exhibited by each produc tion function Opportunity cost Assuming this economy is on the frontier the opportunity cost of good y in terms of good x can be derived by solving for y as y2 5 800 2 4x2 or y 5 800 2 4x2 5 3800 2 4x24 05 118 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 17 159 Welfare economics In addition to using economic models to examine positive questions about how the econ omy operates the tools used in general equilibrium analysis have also been applied to the study of normative questions about the welfare properties of various economic arrange ments Although such questions were a major focus of the great eighteenth and nine teenthcentury economists eg Smith Ricardo Marx and Marshall perhaps the most significant advances in their study were made by the British economist Francis Y Edge worth 18481926 and the Italian economist Vilfredo Pareto 18481923 in the early years of the twentieth century These economists helped to provide a precise definition for the concept of economic efficiency and to demonstrate the conditions under which And then differentiating this expression dy dx 5 05 3800 2 4x24 205 128x2 5 24x y 119 Suppose for example labor is equally allocated between the two goods Then x 5 10 y 5 20 and dydx 5 24 110220 5 22 With this allocation of labor each unit increase in x output would require a reduction in y of 2 units This can be verified by considering a slightly differ ent allocation lx 5 101 and ly 5 99 Now production is x 5 1005 and y 5 199 Moving to this alternative allocation would have Dy Dx 5 1199 2 202 11005 2 102 5 201 005 522 which is precisely what was derived from the calculus approach Concavity Equation 119 clearly illustrates the concavity of the production possibility frontier The slope of the frontier becomes steeper more negative as x output increases and y output decreases For example if labor is allocated so that lx 5 144 and ly 5 56 then outputs are x 5 12 and y 15 and so dydx 5 24 112215 5 232 With expanded x production the opportunity cost of one more unit of x increases from 2 to 32 units of y Inefficiency If an economy operates inside its production possibility frontier it is operating inef ficiently Moving outward to the frontier could increase the output of both goods In this book we will explore many reasons for such inefficiency These usually derive from a failure of some mar ket to perform correctly For the purposes of this illustration lets assume that the labor market in this economy does not work well and that 20 workers are permanently unemployed Now the production possibility frontier becomes x2 1 025y2 5 180 120 and the output combinations we described previously are no longer feasible For example if x 5 10 then y output is now y 179 The loss of approximately 21 units of y is a measure of the cost of the labor market inefficiency Alternatively if the labor supply of 180 were allocated evenly between the production of the two goods then we would have x 95 and y 19 and the inef ficiency would show up in both goods productionmore of both goods could be produced if the labor market inefficiency were resolved QUERY How would the inefficiency cost of labor market imperfections be measured solely in terms of x production in this model How would it be measured solely in terms of y production What would you need to know to assign a single number to the efficiency cost of the imperfection when labor is equally allocated to the two goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 18 Part 1 Introduction markets will be able to achieve that goal By clarifying the relationship between the alloca tion and pricing of resources they provided some support for the idea first enunciated by Adam Smith that properly functioning markets provide an invisible hand that helps allo cate resources efficiently Later sections of this book focus on some of these welfare issues 16 MODERN DEVELOPMENTS Research activity in economics expanded rapidly in the years following World War II A major purpose of this book is to summarize much of this research By illustrating how economists have tried to develop models to explain increasingly complex aspects of eco nomic behavior this book provides an overall foundation for your study of these models 161 The mathematical foundations of economic models A major postwar development in microeconomic theory was the clarification and for malization of the basic assumptions that are made about individuals and firms The first landmark in this development was the 1947 publication of Paul Samuelsons Foundations of Economic Analysis in which the author the first American Nobel Prize winner in eco nomics laid out a number of models of optimizing behavior7 Samuelson demonstrated the importance of basing behavioral models on wellspecified mathematical postulates so that various optimization techniques from mathematics could be applied The power of his approach made it inescapably clear that mathematics had become an integral part of mod ern economics In Chapter 2 of this book we review some of the mathematical concepts most often used in microeconomics 162 New tools for studying markets A second feature that has been incorporated into this book is the presentation of a number of new tools for explaining market equilibria These include techniques for describing pric ing in single markets such as increasingly sophisticated models of monopolistic pricing or models of the strategic relationships among firms that use game theory They also include general equilibrium tools for simultaneously exploring relationships among many markets As we shall see all these new techniques help to provide a more complete and realistic picture of how markets operate 163 The economics of uncertainty and information A third major theoretical advance during the postwar period was the incorporation of uncertainty and imperfect information into economic models Some of the basic assump tions used to study behavior in uncertain situations were originally developed in the 1940s in connection with the theory of games Later developments showed how these ideas could be used to explain why individuals tend to be averse to risks and how they might gather information to reduce the uncertainties they face In this book problems of uncertainty and information enter the analysis on many occasions 164 Behavioral Economics A final theoretical advance in recent years is reflected in attempts to make economic mod els more realistic in terms of how they describe the decisions economic actors make By 7Paul A Samuelson Foundations of Economic Analysis Cambridge MA Harvard University Press 1947 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 1 Economic Models 19 drawing on insights from psychology and other social sciences these models seek to illus trate how imperfect information or various systematic biases can be used to explain why decisions may not always conform to the rational assumptions that underlie most tradi tional economic models In this book we examine such insights primarily through a series of analytical problems that show how behavioral elements can be incorporated into many of the traditional models that we examine 165 Computers and empirical analysis One other aspect of the postwar development of microeconomics should be mentioned the increasing use of computers to analyze economic data and build economic models As computers have become able to handle larger amounts of information and carry out com plex mathematical manipulations economists ability to test their theories has dramatically improved Whereas previous generations had to be content with rudimentary tabular or graphical analyses of realworld data todays economists have available a wide variety of sophisticated techniques together with extensive microeconomic data with which to test their models To examine these techniques and some of their limitations would be beyond the scope and purpose of this book However the Extensions at the end of most chapters are intended to help you start reading about some of these applications Summary This chapter provided a background on how economists approach the study of the allocation of resources Much of the material discussed here should be familiar to you from introductory economics In many respects the study of eco nomics represents acquiring increasingly sophisticated tools for addressing the same basic problems The purpose of this book and indeed of most upperlevel books on economics is to provide you with more of these tools As a beginning this chapter reminded you of the following points Economics is the study of how scarce resources are allo cated among alternative uses Economists seek to develop simple models to help understand that process Many of these models have a mathematical basis because the use of mathematics offers a precise shorthand for stating the models and exploring their consequences The most commonly used economic model is the supplydemand model first thoroughly developed by Alfred Marshall in the latter part of the nineteenth cen tury This model shows how observed prices can be taken to represent an equilibrium balancing of the production costs incurred by firms and the willingness of demanders to pay for those costs Marshalls model of equilibrium is only partialthat is it looks only at one market at a time To look at many markets together requires an expanded set of general equilibrium tools Testing the validity of an economic model is perhaps the most difficult task economists face Occasionally a mod els validity can be appraised by asking whether it is based on reasonable assumptions More often however mod els are judged by how well they can explain economic events in the real world Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 20 Part 1 Introduction Suggestions for Further Reading On Methodology Blaug Mark and John Pencavel The Methodology of Economics Or How Economists Explain 2nd ed Cambridge UK Cambridge University Press 1992 A revised and expanded version of a classic study on economic methodology Ties the discussion to more general issues in the phi losophy of science Boland Lawrence E A Critique of Friedmans Critics Jour nal of Economic Literature June 1979 50322 Good summary of criticisms of positive approaches to economics and of the role of empirical verification of assumptions Friedman Milton The Methodology of Positive Economics In Essays in Positive Economics pp 343 Chicago University of Chicago Press 1953 Basic statement of Friedmans positivist views Harrod Roy F Scope and Method in Economics Economic Journal 48 1938 383412 Classic statement of appropriate role for economic modeling Hausman David M and Michael S McPherson Economic Analysis Moral Philosophy and Public Policy 2nd ed Cam bridge UK Cambridge University Press 2006 The authors stress their belief that consideration of issues in moral philosophy can improve economic analysis McCloskey Donald N If Youre So Smart The Narrative of Economic Expertise Chicago University of Chicago Press 1990 Discussion of McCloskeys view that economic persuasion depends on rhetoric as much as on science For an interchange on this topic see also the articles in the Journal of Economic Literature June 1995 Sen Amartya On Ethics and Economics Oxford Blackwell Reprints 1989 The author seeks to bridge the gap between economics and ethical studies This is a reprint of a classic study on this topic Primary Sources on the History of Economics Edgeworth F Y Mathematical Psychics London Kegan Paul 1881 Initial investigations of welfare economics including rudimentary notions of economic efficiency and the contract curve Marshall A Principles of Economics 8th ed London Macmil lan Co 1920 Complete summary of neoclassical view A longrunning popular text Detailed mathematical appendix Marx K Capital New York Modern Library 1906 Full development of labor theory of value Discussion of trans formation problem provides a perhaps faulty start for general equilibrium analysis Presents fundamental criticisms of institu tion of private property Ricardo D Principles of Political Economy and Taxation Lon don J M Dent Sons 1911 Very analytical tightly written work Pioneer in developing care ful analysis of policy questions especially traderelated issues Dis cusses first basic notions of marginalism Smith A The Wealth of Nations New York Modern Library 1937 First great economics classic Long and detailed but Smith had the first word on practically every economic matter This edition has helpful marginal notes Walras L Elements of Pure Economics Translated by W Jaffe Homewood IL Richard D Irwin 1954 Beginnings of general equilibrium theory Rather difficult reading Secondary Sources on the History of Economics Backhouse Roger E The Ordinary Business of Life The His tory of Economics from the Ancient World to the 21st Century Princeton NJ Princeton University Press 2002 An iconoclastic history Good although brief on the earliest eco nomic ideas but some blind spots on recent uses of mathematics and econometrics Blaug Mark Economic Theory in Retrospect 5th ed Cam bridge UK Cambridge University Press 1997 Complete summary stressing analytical issues Excellent Readers Guides to the classics in each chapter Heilbroner Robert L The Worldly Philosophers 7th ed New York Simon Schuster 1999 Fascinating easytoread biographies of leading economists Chapters on Utopian Socialists and Thorstein Veblen highly recommended Keynes John M Essays in Biography New York W W Nor ton 1963 Essays on many famous persons Lloyd George Winston Chur chill Leon Trotsky and on several economists Malthus Mar shall Edgeworth F P Ramsey and Jevons Shows the true gift of Keynes as a writer Schumpeter J A History of Economic Analysis New York Oxford University Press 1954 Encyclopedic treatment Covers all the famous and many notso famous economists Also briefly summarizes concurrent develop ments in other branches of the social sciences Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 21 CHAPTER TWO Mathematics for Microeconomics Microeconomic models are constructed using a wide variety of mathematical techniques In this chapter we provide a brief summary of some of the most important of these that you will encounter in this book A major portion of the chapter concerns mathematical proce dures for finding the optimal value of some function Because we will frequently adopt the assumption that an economic actor seeks to maximize or minimize some function we will encounter these procedures most of which are based on calculus many times After our detailed discussion of the calculus of optimization we look into four topics that are covered more briefly First we look at a few special types of functions that arise in economics Knowledge of properties of these functions can often be helpful in solving problems Next we provide a brief summary of integral calculus Although integration is used in this book far less frequently than is differentiation we will nevertheless encounter situations where we will want to use integrals to measure areas that are important to eco nomic theory or to add up outcomes that occur over time or across many individuals One particular use of integration is to examine problems in which the objective is to maximize a stream of outcomes over time Our third added topic focuses on techniques to be used for such problems in dynamic optimization Finally the chapter concludes with a brief summary of mathematical statistics which will be particularly useful in our study of eco nomic behavior in uncertain situations 21 MAXIMIZATION OF A FUNCTION OF ONE VARIABLE We can motivate our study of optimization with a simple example Suppose that a man ager of a firm desires to maximize1 the profits received from selling a particular good Suppose also that the profits π received depend only on the quantity q of the good sold Mathematically π 5 f 1q2 21 Figure 21 shows a possible relationship between π and q Clearly to achieve maximum profits the manager should produce output q which yields profits π If a graph such as that of Figure 21 were available this would seem to be a simple matter to be accomplished with a ruler 1Here we will generally explore maximization problems A virtually identical approach would be taken to study minimization problems because maximization of f 1x2 is equivalent to minimizing 2f 1x2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 22 Part 1 Introduction Suppose however as is more likely the manager does not have such an accurate picture of the market He or she may then try varying q to see where a maximum profit is obtained For example by starting at q1 profits from sales would be π1 Next the manager may try output q2 observing that profits have increased to π2 The com monsense idea that profits have increased in response to an increase in q can be stated formally as π2 2 π1 q2 2 q1 0 or Dπ Dq 0 22 where the Δ notation is used to mean the change in π or q As long as DπDq is positive profits are increasing and the manager will continue to increase output For increases in output to the right of q however DπDq will be negative and the manager will realize that a mistake has been made 211 Derivatives As you probably know the limit of DπDq for small changes in q is called the derivative of the function π 5 f 1q2 and is denoted by dπdq or dfdq or fr1q2 More formally the derivative of a function π 5 f 1q2 at the point q1 is defined as dπ dq 5 df dq 5 lim hS0 f 1q1 1 h2 2 f 1q12 h 23 Notice that the value of this ratio obviously depends on the point q1 that is chosen The derivative of a function may not always exist or it may be undefined at certain points Most of the functions studied in this book are fully differentiable however 212 Value of the derivative at a point A notational convention should be mentioned Sometimes we wish to note explicitly the point at which the derivative is to be evaluated For example the evaluation of the deriva tive at the point q 5 q1 could be denoted by dπ dq q5q1 24 If a manager wishes to produce the level of output that maximizes profits then q should be produced Notice that at q dπdq 5 0 π fq π Quantity q1 q2 q q3 π π2 π3 π1 FIGURE 21 Hypothetical Relationship between Quantity Produced and Profits Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 23 At other times we are interested in the value of dπdq for all possible values of q and no explicit mention of a particular point of evaluation is made In the example of Figure 21 dπ dq q5q1 0 whereas dπ dq q5q3 0 What is the value of dπdq at q It would seem to be 0 because the value is positive for values of q less than q and negative for values of q greater than q The derivative is the slope of the curve in question this slope is positive to the left of q and negative to the right of q At the point q the slope of f 1q2 is 0 213 Firstorder condition for a maximum This result is general For a function of one variable to attain its maximum value at some point the derivative at that point if it exists must be 0 Hence if a manager could estimate the function f 1q2 from some sort of realworld data it would theoretically be possible to find the point where dfdq 5 0 At this optimal point say q df dq q5q 5 0 25 214 Secondorder conditions An unsuspecting manager could be tricked however by a naive application of this firstderivative rule alone For example suppose that the profit function looks like that shown in either Figure 22a or 22b If the profit function is that shown in Figure 22a the manager by producing where dπdq 5 0 will choose point q a This point in fact yields minimum not maximum profits for the manager Similarly if the profit function is that shown in Figure 22b the manager will choose point q b which although yields a profit greater than that for any output lower than q b is certainly inferior to any output greater than q b These situations illustrate the mathematical fact that dπdq 5 0 is a necessary con dition for a maximum but not a sufficient condition To ensure that the chosen point is indeed a maximum point a second condition must be imposed Intuitively this additional condition is clear The profit available by producing either a bit more or a bit less than q must be smaller than that available from q If this is not true the manager can do better than q Mathematically this means that dπdq must be greater than 0 for q q and must be less than 0 for q q Therefore at q dπdq must be decreasing Another way of saying this is that the derivative of dπdq must be negative at q 215 Second derivatives The derivative of a derivative is called a second derivative and is denoted by d2π dq2 or d2f dq2 or fs 1q2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 24 Part 1 Introduction The additional condition for q to represent a local maximum is therefore d 2π dq2 q5q 5 fs1q2 q5q 0 26 where the notation is again a reminder that this second derivative is to be evaluated at q Hence although Equation 25 1dπdq 5 02 is a necessary condition for a maximum that equation must be combined with Equation 26 1d2πdq2 02 to ensure that the point is a local maximum for the function Therefore Equations 25 and 26 together are sufficient conditions for such a maximum Of course it is possible that by a series of trials the manager may be able to decide on q by relying on market information rather than on mathematical reasoning remember Friedmans poolplayer analogy In this book we shall be less interested in how the point is discovered than in its properties and how the point changes when con ditions change A mathematical development will be helpful in answering these questions 216 Rules for finding derivatives Here are a few familiar rules for taking derivatives of a function of a single variable We will use these at many places in this book 1 If a is a constant then da dx 5 0 2 If a is a constant then d3af1x2 4 dx 5 afr 1x2 3 If a is a constant then dx a dx 5 axa21 4 d ln x dx 5 1 x where ln signifies the logarithm to the base e 15 2718282 In a the application of the first derivative rule would result in point q a being chosen This point is in fact a point of minimum profits Similarly in b output level q b would be recommended by the first derivative rule but this point is inferior to all outputs greater than q b This demonstrates graphically that finding a point at which the derivative is equal to 0 is a necessary but not a sufficient condition for a function to attain its maximum value qa πb πa qb π Quantity a b π Quantity FIGURE 22 Two Profit Functions That Give Misleading Results If the First Derivative Rule Is Applied Uncritically Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 25 25 dax dx 5 ax ln a for any constant a A particular case of this rule is dexdx 5 ex Now suppose that f 1x2 and g x are two functions of x and that fr 1x2 and gr 1x2 exist then 6 d3 f1x2 1 g 1x2 4 dx 5 fr 1x2 1 gr 1x2 7 d3 f1x2 g 1x2 4 dx 5 f 1x2gr 1x2 1 fr 1x2g 1x2 8 d3 f1x2g 1x2 4 dx 5 fr 1x2g 1x2 2 f 1x2gr 1x2 3g 1x2 4 2 provided that g 1x2 2 0 Finally if y 5 f 1x2 and x 5 g 1z2 and if both fr 1x2 and gr 1z2 exist then 9 dy dz 5 dy dx dx dz 5 df dx dg dz This result is called the chain rule It provides a convenient way to study how one variable z affects another variable y solely through its influence on some intermedi ate variable x Some examples are 10 deax dx 5 deax d1ax2 d1ax2 dx 5 eax a 5 aeax 11 d3 ln 1ax2 4 dx 5 d3 ln 1ax2 4 d1ax2 d1ax2 dx 5 1 ax a 5 1 x 12 d3 ln 1x22 4 dx 5 d3 ln 1x22 4 d1x22 d1x22 dx 5 1 x2 2x 5 2 x EXAMPLE 21 Profit Maximization Suppose that the relationship between profits π and quantity produced q is given by π1q2 5 1000q 2 5q2 27 A graph of this function would resemble the parabola shown in Figure 21 The value of q that maximizes profits can be found by differentiation dπ dq 5 1000 2 10q 5 0 28 Thus q 5 100 29 At q 5 100 Equation 27 shows that profits are 50000the largest value possible If for example the firm opted to produce q 5 50 profits would be 37500 At q 5 200 profits are precisely 0 That q 5 100 is a global maximum can be shown by noting that the second derivative of the profit function is 10 see Equation 28 Hence the rate of increase in profits is always decreasingup to q 5 100 this rate of increase is still positive but beyond that point it becomes negative In this example q 5 100 is the only local maximum value for the function π With more complex functions however there may be several such maxima QUERY Suppose that a firms output q is determined by the amount of labor l it hires according to the function q 5 2l Suppose also that the firm can hire all the labor it wants at 10 per unit and sells its output at 50 per unit Therefore profits are a function of l given by π1l2 5 100l 2 10l How much labor should this firm hire to maximize profits and what will those profits be Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 26 Part 1 Introduction 22 FUNCTIONS OF SEVERAL VARIABLES Economic problems seldom involve functions of only a single variable Most goals of inter est to economic agents depend on several variables and tradeoffs must be made among these variables For example the utility an individual receives from activities as a con sumer depends on the amount of each good consumed For a firms production function the amount produced depends on the quantity of labor capital and land devoted to pro duction In these circumstances this dependence of one variable y on a series of other variables 1x1 x2 c xn2 is denoted by y 5 f 1x1 x2 c xn2 210 221 Partial derivatives We are interested in the point at which y reaches a maximum and in the tradeoffs that must be made to reach that point It is again convenient to picture the agent as changing the variables at his or her disposal the x s to locate a maximum Unfortunately for a func tion of several variables the idea of the derivative is not well defined Just as the steepness of ascent when climbing a mountain depends on which direction you go so does the slope or derivative of the function depend on the direction in which it is taken Usually the only directional slopes of interest are those that are obtained by increasing one of the xs while holding all the other variables constant the analogy for mountain climbing might be to measure slopes only in a northsouth or eastwest direction These directional slopes are called partial derivatives The partial derivative of y with respect to ie in the direction of x1 is denoted by y x1 or f x1 or fx1 or f1 It is understood that in calculating this derivative all the other xs are held constant Again it should be emphasized that the numerical value of this slope depends on the value of x1 and on the preassigned and constant values of x2 c xn A somewhat more formal definition of the partial derivative is f x1 x2c xn 5 lim hS0 f 1x1 1 h x2 c xn2 2 f 1x1 x2 c xn2 h 211 where the notation is intended to indicate that x2 c xn are all held constant at the preas signed values x2 c xn so the effect of changing x1 only can be studied Partial derivatives with respect to the other variables 1x2 c xn2 would be calculated in a similar way 222 Calculating partial derivatives It is easy to calculate partial derivatives The calculation proceeds as for the usual deriva tive by treating x2 c xn as constants which indeed they are in the definition of a partial derivative Consider the following examples 1 If y 5 f 1x1 x22 5 ax2 1 1 bx1x2 1 cx2 2 then f x1 5 f1 5 2ax1 1 bx2 and f x2 5 f2 5 bx1 1 2cx2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 27 Notice that fx1 is in general a function of both x1 and x2 therefore its value will depend on the particular values assigned to these variables It also depends on the parameters a b and c which do not change as x1 and x2 change 2 If y 5 f 1x1 x22 5 eax11bx2 then f x1 5 f1 5 aeax11bx2 and f x2 5 f2 5 beax11bx2 3 If y 5 f 1x1 x22 5 a ln x1 1 b ln x2 then f x1 5 f1 5 a x1 and f x2 5 f2 5 b x2 Notice here that the treatment of x2 as a constant in the derivation of fx1 causes the term b ln x2 to disappear on differentiation because it does not change when x1 changes In this case unlike our previous examples the size of the effect of x1 on y is independent of the value of x2 In other cases the effect of x1 on y will depend on the level of x2 223 Partial derivatives and the ceteris paribus assumption In Chapter 1 we described the way in which economists use the ceteris paribus assumption in their models to hold constant a variety of outside influences so the particular relationship being studied can be explored in a simplified setting Partial derivatives are a precise math ematical way of representing this approach that is they show how changes in one variable affect some outcome when other influences are held constantexactly what economists need for their models For example Marshalls demand curve shows the relationship between price p and quantity q demanded when other factors are held constant Using partial derivatives we could represent the slope of this curve by qp to indicate the ceteris paribus assumptions that are in effect The fundamental law of demandthat price and quantity move in oppo site directions when other factors do not changeis therefore reflected by the mathematical statement qp 0 Again the use of a partial derivative serves as a reminder of the ceteris paribus assumptions that surround the law of demandthat is the law of demand only holds when the other factors that affect demand such as income or other prices are held constant 224 Partial derivatives and units of measurement In mathematics relatively little attention is paid to how variables are measured In fact most often no explicit mention is made of the issue However the variables used in economics usually refer to realworld magnitudes therefore we must be concerned with how they are measured Perhaps the most important consequence of choosing units of measurement is that the partial derivatives often used to summarize economic behavior will reflect these units For example if q represents the quantity of gasoline demanded by all US consumers during a given year measured in billions of gallons and p represents the price in dollars per gallon then qp will measure the change in demand in billions of gallons per year for a dollar per gallon change in price The numerical size of this derivative depends on Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 28 Part 1 Introduction how q and p are measured A decision to measure consumption in millions of gallons per year would multiply the size of the derivative by 1000 whereas a decision to measure price in cents per gallon would reduce it by a factor of 100 The dependence of the numerical size of partial derivatives on the chosen units of mea surement poses problems for economists Although many economic theories make predic tions about the sign direction of partial derivatives any predictions about the numerical magnitude of such derivatives would be contingent on how authors chose to measure their variables Making comparisons among studies could prove practically impossible espe cially given the wide variety of measuring systems in use around the world For this reason economists have chosen to adopt a different unitfree way to measure quantitative impacts 225 Elasticitya general definition Economists use elasticities to summarize virtually all the quantitative impacts that are of interest to them Because such measures focus on the proportional effect of a change in one variable on another they are unitfreethe units cancel out when the elasticity is calcu lated For example suppose that y is a function of x which we can denote by y1x2 Then the elasticity of y with respect to x which we will denote by eyx is defined as eyx 5 Dy y Dx x 5 Dy Dx x y 5 dy1x2 dx x y 212 If the variable y depends on several variables in addition to x as will often be the case the derivative in Equation 212 would be replaced by a partial derivative In either case the units in which y and x are measured cancel out in the definition of elasticity the result is a figure that is a pure number with no dimensions This makes it possible for economists to compare elasticities across different countries or across rather different goods You should already be familiar with the price elasticities of demand and supply usually encountered in a first eco nomics course Throughout this book you will encounter many more such concepts EXAMPLE 22 Elasticity and Functional Form The definition in Equation 212 makes clear that elasticity should be evaluated at a specific point on a function In general the value of this parameter would be expected to vary across different ranges of the function This observation is most clearly shown in the case where y is a linear func tion of x of the form y 5 a 1 bx 1 other terms In this case eyx 5 dy dx x y 5 b x y 5 b x a 1 bx 1 213 which makes clear that eyx is not constant Hence for linear functions it is especially important to note the point at which elasticity is to be computed If the functional relationship between y and x is of the exponential form y 5 axb then the elasticity is a constant independent of where it is measured eyx 5 dy dx x y 5 abxb21 x axb 5 b Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 29 226 Secondorder partial derivatives The partial derivative of a partial derivative is directly analogous to the second derivative of a function of one variable and is called a secondorder partial derivative This may be written as 1fxi2 xj or more simply as 2f xjxi 5 fij 215 For the examples discussed previously 1 y 5 f 1x1 x22 5 ax2 1 1 bx1x2 1 cx2 2 f11 5 2a f12 5 b f21 5 b f22 5 2c 2 y 5 f 1x1 x22 5 eax11bx2 f11 5 a2eax11bx2 f12 5 abeax11bx2 f21 5 abeax11bx2 f22 5 b2eax11bx2 3 y 5 a lnx1 1 b lnx2 f11 5 2ax22 1 f12 5 0 f21 5 0 f22 5 2bx22 2 A logarithmic transformation of this equation also provides a convenient alternative definition of elasticity Because ln y 5 ln a 1 b ln x we have eyx 5 b 5 d ln y d ln x 214 Hence elasticities can be calculated through logarithmic differentiation As we shall see this is frequently the easiest way to proceed in making such calculations QUERY Are there any functional forms in addition to the exponential that have a constant elasticity at least over some range Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 30 Part 1 Introduction 227 Youngs theorem These examples illustrate the mathematical result that under general conditions the order in which partial differentiation is conducted to evaluate secondorder partial derivatives does not matter That is fij 5 fji 216 for any pair of variables xi xj This result is sometimes called Youngs theorem For an intu itive explanation of the theorem we can return to our mountainclimbing analogy In this example the theorem states that the gain in elevation a hiker experiences depends on the directions and distances traveled but not on the order in which these occur That is the gain in altitude is independent of the actual path taken as long as the hiker proceeds from one set of map coordinates to another He or she may for example go one mile north then one mile east or proceed in the opposite order by first going one mile east then one mile north In either case the gain in elevation is the same because in both cases the hiker is moving from one specific place to another In later chapters we will make good use of this result because it provides a convenient way of showing some of the predictions that eco nomic models make about behavior2 228 Uses of secondorder partials Secondorder partial derivatives will play an important role in many of the economic theo ries that are developed throughout this book Probably the most important examples relate to the own secondorder partial fii This function shows how the marginal influence of xi on y ie yxi changes as the value of xi increases A negative value for fii is the mathematical way of indicating the economic idea of diminishing marginal effectiveness Similarly the crosspartial fij indicates how the marginal effectiveness of xi changes as xj increases The sign of this effect could be either positive or negative Youngs theorem indi cates that in general such crosseffects are symmetric More generally the secondorder partial derivatives of a function provide information about the curvature of the function Later in this chapter we will see how such information plays an important role in deter mining whether various secondorder conditions for a maximum are satisfied They also play an important role in determining the signs of many important derivatives in economic theory 229 The chain rule with many variables Calculating partial derivatives can be rather complicated in cases where some variables depend on other variables As we will see in many economic problems it can be hard to tell exactly how to proceed in differentiating complex functions In this section we illustrate a few simple cases that should help you to get the general idea We start with looking at how the chain rule discussed earlier in a singlevariable context can be generalized to many variables Specifically suppose that y is a function of three variables y 5 f 1x1 x2 x32 Sup pose further that each of these xs is itself a function of a single parameter say a Hence we can write y 5 f 3x1 1a2 x2 1a2 x3 1a2 4 Now we can ask how a change in a affects the value of y using the chain rule dy da 5 f x1 dx1 da 1 f x2 dx2 da 1 f x3 dx3 da 217 2Youngs theorem implies that the matrix of the secondorder partial derivatives of a function is symmetric This symmetry offers a number of economic insights For a brief introduction to the matrix concepts used in economics see the Extensions to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 31 In words changes in a affect each of the xs and then these changes in the xs affect the final value of y Of course some of the terms in this expression may be zero That would be the case if one of the xs is not affected by a or if a particular x had no effect on y in which case it should not be in the function But this version of the chain rule shows that a can influence y through many routes3 In our economic models we will want to be sure that all those routes are taken into account One special case of this chain rule might be explicitly mentioned here Suppose x3 1a2 5 a That is suppose that the parameter a enters directly into the determination of y 5 f 3x1 1a2 x2 1a2 a4 In this case the effect of a on y can be written as4 dy da 5 f x1 dx1 da 1 f x2 dx2 da 1 f a 220 3If the xs in Equation 217 depended on several parameters all the derivatives in the equation would be partial derivatives to indicate that the chain rule looks at the effect of only one parameter at a time holding the others constant 4The expression in Equation 220 is sometimes called the total derivative or full derivative of the function f although this usage is not consistent across various fields of applied mathematics EXAMPLE 23 Using the Chain Rule As a simple and probably unappetizing example suppose that each week a pizza fanatic con sumes three kinds of pizza denoted by x1 x2 and x3 Type 1 pizza is a simple cheese pizza costing p per pie Type 2 pizza adds two toppings and costs 2p Type 3 pizza is the house special which includes five toppings and costs 3p To ensure a modestly diversified menu this fanatic decides to allocate 30 each week to each type of pizza Here we wish to examine how the total number of pizzas purchased is affected by the underlying price p Notice that this problem includes a single exogenous variable p which is set by the pizza shop The quantities of each pizza purchased and total purchases are the endogenous variables in the model Because of the way this fanatic budgets his pizza purchases the quantity purchased of each type depends only on the price p Specifically x1 5 30p x2 5 302p x3 5 303p Now total pizza purchases y are given by y 5 f 3x1 1p2 x2 1p2 x3 1p2 4 5 x1 1p2 1 x2 1p2 1 x3 1p2 218 Applying the chain rule from Equation 217 to this function yields dy dp 5 f1 dx1 dp 1 f2 dx2 dp 1 f3 dx3 dp 5 230p22 2 15p22 2 10p22 5 255p22 219 We can interpret this with a numerical illustration Suppose that initially p 5 5 With this price total pizza purchases will be 11 pies Equation 219 implies that each unit price increase would reduce purchases by 22 1555252 pies but such a change is too large for calculus which assumes small changes to work correctly Therefore instead lets assume p increases by 5 cents to p 5 505 Equation 219 now predicts that total pizza purchases will decrease by 011 pies 1005 3 55252 If we calculate pie purchases directly we get x1 5 594 x2 5 297 x3 5 198 Hence total pies purchased are 1089a reduction of 011 from the original level just what was predicted by Equation 219 QUERY It should be obvious that a far easier way to solve this problem would be to define total pie purchases y directly as a function of p Provide a proof using this approach and then describe some reasons why this simpler approach may not always be possible to implement Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 32 Part 1 Introduction This shows that the effect of a on y can be decomposed into two different kinds of effects 1 a direct effect which is given by fa and 2 an indirect effect that operates only through the ways in which a affects the xs In many economic problems analyzing these two effects separately can provide a number of important insights 2210 Implicit functions If the value of a function is held constant an implicit relationship is created among the independent variables that enter into the function That is the independent variables can no longer take on any values but must instead take on only that set of values that result in the functions retaining the required value Examining these implicit relationships can often provide another analytical tool for drawing conclusions from economic models Probably the most useful result provided by this approach is in the ability to quantify the tradeoffs inherent in most economic models Here we will look at a simple case Consider the function y 5 f 1x1 x22 If we hold the value of y constant we have created an implicit relationship between the xs showing how changes in them must be related to keep the value of the function constant In fact under fairly general conditions5 the most import ant of which is that f2 2 0 it can be shown that holding y constant allows the creation of an implicit function of the form x2 5 g 1x12 Although computing this function may some times be difficult the derivative of the function g is related in a specific way to the partial derivatives of the original function f To show this first set the original function equal to a constant say zero and write the function as y 5 0 5 f 1x1 x22 5 f 1x1 g 1x12 2 221 Using the chain rule to differentiate this relationship with respect to x1 yields 0 5 f1 1 f2 dg 1x12 dx1 222 Rearranging terms gives the final result that dg 1x12 dx1 5 dx2 dx1 5 2 f1 f2 223 Thus we have shown6 that the partial derivatives of the function f can be used to derive an explicit expression for the tradeoffs between x1 and x2 The next example shows how this can make computations much easier in certain situations 5For a detailed discussion of this implicit function theorem and of how it can be extended to many variables see Carl P Simon and Lawrence Blume Mathematics for Economists New York WW Norton 1994 chapter 15 6An alternative approach to proving this result uses the total differential of f dy 5 f1 dx1 1 f2 dx2 Setting dy 5 0 and rearranging terms gives the same result assuming one can make the mathematically questionable move of dividing by dx1 EXAMPLE 24 A Production Possibility FrontierAgain In Example 13 we examined a production possibility frontier for two goods of the form x2 1 025y2 5 200 224 Because this function is set equal to a constant we can study the relationship between the vari ables by using the implicit function result Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 33 2211 A Special CaseComparative Statics Analysis One important application of the implicit function theorem arises when one of the vari ables is an exogenous variable defined outside the model such as a price and the other variable is an endogenous variable depending on that exogenous variable such as quantity supplied If we let this exogenous variable be represented by a then the twovariable func tion in Equation 221 can be written in implicit form as f1a x 1a2 2 5 0 and applying the implicit function theorem would yield dx1a2 da 5 2 f1 f2 5 2 f a f x 226 This shows how changes in the exogenous variable a affect the endogenous variable x directly That is this version of the implicit function theorem often provides a direct route to exploring the comparative statics of an economic model We will use this approach in two general situations in this book depending on the origin of the function f First the function might represent a firstorder condition for an optimization problem In this case the implicit function theorem can be used to examine how the optimal value of x changes when some exogenous variable changes A second use occurs when the function f represents an equilib rium condition such as a supplydemand equilibrium In this case the implicit function theorem can be used to show how the equilibrium value of x changes when the parameter a changes Perhaps the most useful aspect of this approach to such problems is that the result in Equation 226 can be readily generalized to include multiple exogenous variables or multiple endogenous variables We look briefly at the latter case in the Extensions to this chapter since dealing with multiple endogenous variables will usually require the use of matrix algebra dy dx 5 2fx fy 5 22x 05y 5 24x y 225 which is precisely the result we obtained earlier with considerably less work QUERY Why does the tradeoff between x and y here depend only on the ratio of x to y and not on the size of the labor force as reflected by the 200 constant EXAMPLE 25 Comparative Statics of a PriceTaking Firm In Example 11 we showed that the firstorder condition for a profit firm that takes market price as given was f1p q1p22 5 p 2 Cr 1q 1p2 2 5 0 Applying the implicit function theorem to this expression yields dq 1p2 dp 5 2 f p f q 5 2 1 12Cr 1q22q 5 1 Cs 1q2 0 227 which is precisely the result we obtained earlier In later chapters we will find it quite useful to follow this approach to study the comparative static implications of the equilibrium conditions in some of our models QUERY In elementary economics we usually assume that a pricetaking firm has an upward sloping supply curve Is the argument used to show that the result here is the same as the one used in that course Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 34 Part 1 Introduction 23 MAXIMIZATION OF FUNCTIONS OF SEVERAL VARIABLES Using partial derivatives allows us to find the maximum or minimum value for a func tion of several variables To understand the mathematics used in solving this problem an analogy to the onevariable case is helpful In this onevariable case we can picture an agent varying x by a small amount dx and observing the change in y dy This change is given by dy 5 fr 1x2dx The identity in Equation 226 records the fact that the change in y is equal to the change in x times the slope of the function This formula is equivalent to the pointslope formula used for linear equations in basic algebra As before the necessary condition for a maximum is dy 5 0 for small changes in x around the optimal point Otherwise y could be increased by suitable changes in x But because dx does not necessarily equal 0 in Equation 226 dy 5 0 must imply that at the desired point fr 1x2 5 0 This is another way of obtaining the firstorder condition for a maximum that we already derived Using this analogy lets look at the decisions made by an economic agent who must choose the levels of several variables Suppose that this agent wishes to find a set of xs that will maximize the value of y 5 f 1x1 x2 c xn2 The agent might consider changing only one of the xs say x1 while holding all the others constant The change in y ie dy that would result from this change in x1 is given by dy 5 f x1 dx1 5 f1dx1 228 This says that the change in y is equal to the change in x1 times the slope measured in the x1 direction Using the mountain analogy again the gain in altitude a climber heading north would achieve is given by the distance northward traveled times the slope of the mountain measured in a northward direction 231 Firstorder conditions for a maximum For a specific point to provide a local maximum value to the function f it must be the case that no small movement in any direction can increase its value Hence all the directional terms similar to Equation 228 must not increase y and the only way this can happen is if all the directional partial derivatives are zero remember the term dx1 in Equation 228 could be either positive or negative That is a necessary condition for a point to be a local maximum is that at this point f1 5 f2 5 c5 fn 5 0 229 Technically a point at which Equation 229 holds is called a critical point of the function It is not necessarily a maximum point unless certain secondorder conditions to be discussed later hold In most of our economic examples however these conditions will hold thus applying the firstorder conditions will allow us to find a maximum The necessary conditions for a maximum described by Equation 229 also have an important economic interpretation They say that for a function to reach its maximal value any input to the function must be increased up to the point at which its marginal or Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 35 incremental value to the function is zero If say f1 were positive at a point this could not be a true maximum because an increase in x1 holding all other variables constant would increase f EXAMPLE 26 Finding a Maximum Suppose that y is a function of x1 and x2 given by y 5 21x1 2 12 2 2 1x2 2 22 2 1 10 or y 5 2x2 1 1 2x1 2 x2 2 1 4x2 1 5 For example y might represent an individuals health measured on a scale of 0 to 10 and x1 and x2 might be daily dosages of two healthenhancing drugs We wish to find values for x1 and x2 that make y as large as possible Taking the partial derivatives of y with respect to x1 and x2 and apply ing the necessary conditions yields y x1 5 22x1 1 2 5 0 y x2 5 22x2 1 4 5 0 230 or x 1 5 1 x 2 5 2 Therefore the function is at a critical point when x1 5 1 x2 5 2 At that point y 5 10 is the best health status possible A bit of experimentation provides convincing evidence that this is the greatest value y can have For example if x1 5 x2 5 0 then y 5 5 or if x1 5 x2 5 1 then y 5 9 Values of x1 and x2 larger than 1 and 2 respectively reduce y because the negative quadratic terms become large Consequently the point found by applying the necessary conditions is in fact a local and global maximum7 QUERY Suppose y took on a fixed value say 5 What would the relationship implied between x1 and x2 look like How about for y 5 7 Or y 5 10 These graphs are contour lines of the func tion and will be examined in more detail in several later chapters See also Problem 21 232 Secondorder conditions Again however the conditions of Equation 229 are not sufficient to ensure a maxi mum This can be illustrated by returning to an already overworked analogy All hill tops are more or less flat but not every flat place is a hilltop A secondorder condition is needed to ensure that the point found by applying the firstorder conditions is a local maximum Intuitively for a local maximum y should be decreasing for any small changes in the xs away from the critical point As in the singlevariable case this involves looking at the curvature of the function around the critical point to be sure that the value of the 7More formally the point x1 5 1 x2 5 2 is a global maximum because the function is concave see our discussion later in this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 36 Part 1 Introduction function really does decrease for movements in every direction To do this we must look at the second partial derivatives of the function A first condition that draws in obvious ways from the singlevariable case is that the own second partial derivative for any vari able 1 fii2 must be negative If we confine our attention only to movements in a single direction a true maximum must be characterized by a pattern in which the slope of the function goes from positive up to zero flat to negative down That is what the math ematical condition fii 0 means Unfortunately the conditions that assure the value of f decreases for movements in any arbitrary direction involve all the second partial deriva tives A twovariable example is discussed later in this chapter but the general case is best discussed with matrix algebra see the Extensions to this chapter For economic theory however the fact that the own second partial derivatives must be negative for a maximum is often the most important fact 24 THE ENVELOPE THEOREM One major application related to the idea of implicit functions which will be used many times in this book is called the envelope theorem it concerns how an optimized function changes when a parameter of the function changes Because many of the economic prob lems we will be studying concern the effects of changing a parameter eg the effects that changing the market price of a commodity will have on an individuals purchases this is a type of calculation we will frequently make The envelope theorem often provides a nice shortcut to solving the problem 241 A specific example Perhaps the easiest way to understand the envelope theorem is through an example Sup pose y is a function of a single variable x and an exogenous parameter a given by y 5 2x2 1 ax 231 For different values of the parameter a this function represents a family of inverted parabolas If a is assigned a specific value Equation 231 is a function of x only and the value of x that maximizes y can be calculated For example if a 5 1 then x 5 1 2 and for these values of x and a y 5 1 4 its maximal value Similarly if a 5 2 then x 5 1 and y 5 1 Hence an increase of 1 in the value of the parameter a has increased the maximum value of y by 3 4 In Table 21 integral values of a between 0 and 6 are used to calculate the optimal values for x and the associated values of the objective y Notice that as a increases the maximal value for y also increases This is also illustrated in Figure 23 which shows that the relationship between a and y is quadratic Now we wish to calculate explicitly how y changes as the parameter a changes TABLE 21 OPTIMAL VALUES OF y AND x FOR ALTERNATIVE VALUES OF a LN y 5 2x2 1 ax Value of a Value of x Value of y 0 0 0 1 1 2 1 4 2 1 1 3 3 2 9 4 4 2 4 5 5 2 25 4 6 3 9 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 37 242 A direct timeconsuming approach The envelope theorem states that there are two equivalent ways we can make this calcu lation First we can calculate the slope of the function in Figure 23 directly To do so we must solve Equation 232 for the optimal value of x for any value of a dy dx 5 22x 1 a 5 0 hence x 5 a 2 Substituting this value of x in Equation 231 gives y 5 21x2 2 1 a 1x2 5 2aa 2b 2 1 aaa 2b 232 5 2a2 4 1 a2 2 5 a2 4 and this is precisely the relationship shown in Figure 23 From the previous equation it is easy to see that The envelope theorem states that the slope of the relationship between y the maximum value of y and the parameter a can be found by calculating the slope of the auxiliary relationship found by substituting the respective optimal values for x into the objective function and calculating ya a y 0 6 3 1 5 2 4 1 2 3 4 5 6 7 8 9 10 y fa FIGURE 23 Illustration of the Envelope Theorem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 38 Part 1 Introduction dy da 5 2a 4 5 a 2 233 and for example at a 5 2 dyda 5 1 That is near a 5 2 the marginal impact of increas ing a is to increase y by the same amount Near a 5 6 any small increase in a will increase y by three times this change Table 21 illustrates this result 243 The envelope shortcut Arriving at this conclusion was a bit complicated We had to find the optimal value of x for each value of a and then substitute this value for x into the equation for y In more gen eral cases this may be burdensome because it requires repeatedly maximizing the objective function The envelope theorem providing an alternative approach states that for small changes in a dyda can be computed by holding x at its optimal value and simply calculat ing ya from the objective function directly Proceeding in this way gives dy da 5 y a x5x1a2 5 12x2 1 ax2 a x5x1a2 5 x1a2 234 The notation here is a reminder that the partial derivative used in the envelope theorem must be evaluated at the value of x which is optimal for the particular parameter value for a In Equation 232 we showed that for any value of a x1a2 5 a2 Substitution into Equation 234 now yields dy da 5 x1a2 5 a 2 235 This is precisely the result obtained earlier The reason that the two approaches yield iden tical results is illustrated in Figure 23 The tangents shown in the figure report values of y for a fixed x The tangents slopes are ya Clearly at y this slope gives the value we seek This result is general and we will use it at several places in this book to simplify our analysis To summarize the envelope theorem states that the change in the value of an optimized function with respect to a parameter of that function can be found by partially differentiating the objective function while holding x at its optimal value That is dy da 5 y a 5x 5 x1a2 6 236 where the notation again provides a reminder that ya must be computed at that value of x that is optimal for the specific value of the parameter a being examined 244 Manyvariable case An analogous envelope theorem holds for the case where y is a function of several vari ables Suppose that y depends on a set of xs 1x1 c xn2 and on a particular parameter of interest say a y 5 f 1x1 c xn a2 237 Finding an optimal value for y would consist of solving n firstorder equations of the form y xi 5 0 1i 5 1 c n2 238 and a solution to this process would yield optimal values for these xs x 1 x 2 c x n that would implicitly depend on the parameter a Assuming the secondorder conditions are Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 39 met the implicit function theorem would apply in this case and ensure that we could solve each x i as a function of the parameter a x 1 5 x 1 1a2 x 2 5 x 2 1a2 239 x n 5 x n 1a2 Substituting these functions into our original objective Equation 237 yields an expres sion in which the optimal value of y say y depends on the parameter a both directly and indirectly through the effect of a on the xs y 5 f 3x 1 1a2 x 2 1a2 c x n 1a2 a4 This function which we will encounter at many places in this book is sometimes called a value function because it shows how the optimized value of the function depends on its parameters Totally differentiating this function with respect to a yields dy da 5 f x1 dx1 da 1 f x2 dx2 da 1 c1 f xn dxn da 1 f a 240 But because of the firstorder conditions all these terms except the last are equal to 0 if the xs are at their optimal values Hence we have the envelope result dy da 5 f a xi5x i 1a2 for all xi 241 Notice again that the partial derivative on the right side of this equation is to be evaluated at the optimal values of all of the xs The fact that these endogenous variables are at their optimal values is what makes the envelope theorem so useful because we can often use it to study the characteristics of these optimal values without actually having to compute them EXAMPLE 27 A PriceTaking Firms Supply Function Suppose that a pricetaking firm has a cost function given by C1q2 5 5q2 A direct way of finding its supply function is to use the firstorder condition p 5 Cr 1q2 5 10q to get q 5 01p An alterna tive and seemingly roundabout way to get this result is to calculate the firms profit function Since profits are given by π1p q2 5 pq 2 C1q2 we can calculate the optimal value of the firms profits as π1p2 5 pq 2 C1q2 5 p101p2 2 5 101p22 5 05p2 242 Notice how we have substituted the optimal value for q as a function of p into the expression for profits to obtain a value function in which the firms optimal profits ultimately depend only on the price of its product Now the envelope theorem states that dπ1p2 dp 5 01p 5 π1p q2 p q5q 5 q0 q5q 5 q 243 Hence in this case simple differentiation of the value function of the firm with respect to out put price yields the firms supply functiona quite general result Although use of the envelope theorem is certainly overkill for this example later we will find that this type of derivation often provides results more easily than does brute force application of the firstorder conditions This Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 40 Part 1 Introduction 25 CONSTRAINED MAXIMIZATION Thus far we have focused our attention on finding the maximum value of a function without restricting the choices of the xs available In most economic problems however not all values for the xs are feasible In many situations for example it is required that all the xs be positive This would be true for the problem faced by the manager choosing output to maximize prof its a negative output would have no meaning In other instances the xs may be constrained by economic considerations For example in choosing the items to consume an individual is not able to choose any quantities desired Rather choices are constrained by the amount of purchasing power available that is by this persons budget constraint Such constraints may lower the maximum value for the function being maximized Because we are not able to choose freely among all the xs y may not be as large as it could be The constraints would be nonbinding if we could obtain the same level of y with or without imposing the constraint 251 Lagrange multiplier method One method for solving constrained maximization problems is the Lagrange multiplier method which involves a clever mathematical trick that also turns out to have a useful economic interpretation The rationale of this method is simple although no rigorous pre sentation will be attempted here8 In a previous section the necessary conditions for a local maximum were discussed We showed that at the optimal point all the partial derivatives of f must be 0 Therefore there are n equations 1 fi 5 0 for i 5 1 c n2 in n unknowns the xs Generally these equations can be solved for the optimal xs When the xs are constrained however there is at least one additional equation the constraint but no addi tional variables Therefore the set of equations is overdetermined The Lagrangian tech nique introduces an additional variable the Lagrange multiplier which not only helps to solve the problem at hand because there are now n 1 1 equations in n 1 1 unknowns but also has an interpretation that is useful in a variety of economic circumstances 252 The formal problem More specifically suppose that we wish to find the values of x1 x2 c xn that maximize y 5 f 1x1 x2 c xn2 244 subject to a constraint that permits only certain values of the xs to be used A general way of writing that constraint is g 1x1 x2 c xn2 5 0 245 where the function9 g represents the relationship that must hold among all the xs 8For a detailed presentation see A K Dixit Optimization in Economic Theory 2nd ed Oxford Oxford University Press 1990 chapter 2 9As we pointed out earlier any function of x1 x2 c xn can be written in this implicit way For example the constraint x1 1 x2 5 10 could be written 10 2 x1 2 x2 5 0 In later chapters we will usually follow this procedure in dealing with constraints Often the constraints we examine will be linear is especially true if a firms profit function has been estimated from some sort of market data that provides an accurate picture of the ceteris paribus relationship between price and profits QUERY Why does the application of the envelope theorem in Equation 243 involve a total derivative on the lefthand side of the equation but a partial derivative on the righthand side Why is the value for this partial derivative q Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 41 253 Firstorder conditions The Lagrange multiplier method starts with setting up the Lagrangian expression 5 f 1x1 x2 c xn2 1 λg 1x1 x2 c xn2 246 where λ is an additional variable called the Lagrange multiplier Later we will interpret this new variable First however notice that when the constraint holds and f have the same value because g 1x1 x2 c xn2 5 0 Consequently if we restrict our attention only to values of the xs that satisfy the constraint finding the constrained maximum value of f is equivalent to finding a critical value of Lets proceed then to do so treating λ also as a variable in addition to the xs From Equation 246 the conditions for a critical point are x1 5 f1 1 λg1 5 0 x2 5 f2 1 λg2 5 0 247 xn 5 fn 1 λgn 5 0 λ 5 g 1x1 x2 c xn2 5 0 The equations comprised by Equation 247 are then the conditions for a critical point for the function Notice that there are n 1 1 equations one for each x and a final one for λ in n 1 1 unknowns The equations can generally be solved for x1 x2 c xn and λ Such a solution will have two properties 1 The xs will obey the constraint because the last line in Equation 247 imposes that condition and 2 among all those values of xs that satisfy the constraint those that also solve Equation 247 will make and hence f as large as possible assuming secondorder conditions are met Therefore the Lagrange multiplier method provides a way to find a solution to the constrained maximization problem we posed at the outset10 The solution to Equation 247 will usually differ from that in the unconstrained case see Equation 229 Rather than proceeding to the point where the marginal contribution of each x is 0 Equation 247 requires us to stop short because of the constraint Only if the constraint were ineffective in which case as we show below λ would be 0 would the con strained and unconstrained equations and their respective solutions agree These revised marginal conditions have economic interpretations in many different situations 254 Interpretation of the Lagrange multiplier Thus far we have used the Lagrange multiplier λ only as a mathematical trick to arrive at the solution we wanted In fact that variable also has an important economic interpre tation which will be central to our analysis at many points in this book To develop this interpretation rewrite the first n equations of Equation 247 as f1 2g1 5 f2 2g2 5 c5 fn 2gn 5 λ 248 10Strictly speaking these are the necessary conditions for an interior local maximum In some economic problems it is necessary to amend these conditions in fairly obvious ways to take account of the possibility that some of the xs may be on the boundary of the region of permissible xs For example if all the xs are required to be nonnegative it may be that the conditions of Equation 247 will not hold exactly because these may require negative xs We look at this situation later in this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 42 Part 1 Introduction In other words at the maximum point the ratio of fi to gi is the same for every xi The numerators in Equation 248 are the marginal contributions of each x to the function f They show the marginal benefit that one more unit of xi will have for the function that is being maximized ie for f A complete interpretation of the denominators in Equation 248 is probably best left until we encounter these ratios in actual economic applications There we will see that these usually have a marginal cost interpretation That is they reflect the added burden on the constraint of using slightly more xi As a simple illustration suppose the constraint required that total spending on x1 and x2 be given by a fixed dollar amount F Hence the constraint would be p1x1 1 p2x2 5 F where pi is the per unit cost of xi Using our present terminology this constraint would be written in implicit form as g 1x1 x22 5 F 2 p1x1 2 p2x2 5 0 249 In this situation then 2gi 5 pi 250 and the derivative 2gi does indeed reflect the per unit marginal cost of using xi Practically all the optimization problems we will encounter in later chapters have a similar interpretation 255 Lagrange multiplier as a benefitcost ratio Now we can give Equation 248 an intuitive interpretation The equation indicates that at the optimal choices for the xs the ratio of the marginal benefit of increasing xi to the marginal cost of increasing xi should be the same for every x To see that this is an obvi ous condition for a maximum suppose that it were not true Suppose that the benefit cost ratio were higher for x1 than for x2 In this case slightly more x1 should be used to achieve a maximum Consider using more x1 but giving up just enough x2 to keep g the constraint constant Hence the marginal cost of the additional x1 used would equal the cost saved by using less x2 But because the benefitcost ratio the amount of benefit per unit of cost is greater for x1 than for x2 the additional benefits from using more x1 would exceed the loss in benefits from using less x2 The use of more x1 and appropriately less x2 would then increase y because x1 provides more bang for your buck Only if the marginal benefitmarginal cost ratios are equal for all the xs will there be a local maximum one in which no small changes in the xs can increase the objective Concrete applications of this basic principle are developed in many places in this book The result is fundamental for the microeconomic theory of optimizing behavior The Lagrange multiplier 1λ2 can also be interpreted in light of this discussion λ is the common benefitcost ratio for all the xs That is λ 5 marginal benefit of xi marginal cost of xi 251 for every xi If the constraint were relaxed slightly it would not matter exactly which x is changed indeed all the xs could be altered because at the margin each promises the same ratio of benefits to costs The Lagrange multiplier then provides a measure of how such an overall relaxation of the constraint would affect the value of y In essence λ assigns a shadow price to the constraint A high λ indicates that y could be increased substan tially by relaxing the constraint because each x has a high benefitcost ratio A low value of λ on the other hand indicates that there is not much to be gained by relaxing the con straint If the constraint is not binding λ will have a value of 0 thereby indicating that the Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 43 constraint is not restricting the value of y In such a case finding the maximum value of y subject to the constraint would be identical to finding an unconstrained maximum The shadow price of the constraint is 0 This interpretation of λ can also be shown using the envelope theorem as described later in this chapter11 256 Duality This discussion shows that there is a clear relationship between the problem of maximiz ing a function subject to constraints and the problem of assigning values to constraints This reflects what is called the mathematical principle of duality Any constrained maxi mization problem has an associated dual problem in constrained minimization that focuses attention on the constraints in the original primal problem For example to jump a bit ahead of our story economists assume that individuals maximize their utility subject to a budget constraint This is the consumers primal problem The dual problem for the con sumer is to minimize the expenditure needed to achieve a given level of utility Or a firms primal problem may be to minimize the total cost of inputs used to produce a given level of output whereas the dual problem is to maximize output for a given total cost of inputs purchased Many similar examples will be developed in later chapters Each illustrates that there are always two ways to look at any constrained optimization problem Sometimes taking a frontal attack by analyzing the primal problem can lead to greater insights In other instances the back door approach of examining the dual problem may be more instructive Whichever route is taken the results will generally although not always be identical thus the choice made will mainly be a matter of convenience 11The discussion in the text concerns problems involving a single constraint In general one can handle m constraints m n by simply introducing m new variables Lagrange multipliers and proceeding in an analogous way to that discussed above EXAMPLE 28 Optimal Fences and Constrained Maximization Suppose a farmer had a certain length of fence P and wished to enclose the largest possible rect angular area What shape area should the farmer choose This is clearly a problem in constrained maximization To solve it let x be the length of one side of the rectangle and y be the length of the other side The problem then is to choose x and y so as to maximize the area of the field given by A 5 x y subject to the constraint that the perimeter is fixed at P 5 2x 1 2y Setting up the Lagrangian expression gives 5 x y 1 λ1P 2 2x 2 2y2 252 where λ is an unknown Lagrange multiplier The firstorder conditions for a maximum are x 5 y 2 2λ 5 0 y 5 x 2 2λ 5 0 λ 5 P 2 2x 2 2y 5 0 253 These three equations must be solved simultaneously for x y and λ The first two equations say that y2 5 x2 5 λ showing that x must be equal to y the field should be square They also imply that x and y should be chosen so that the ratio of marginal benefits to marginal cost is the same for both variables The benefit in terms of area of one more unit of x is given by y area is increased by 1 y and the marginal cost in terms of perimeter is 2 the available perimeter is reduced by 2 for each unit that the length of side x is increased The maximum conditions state that this ratio should be equal for each of the variables Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 44 Part 1 Introduction Because we have shown that x 5 y we can use the constraint to show that x 5 y 5 P 4 254 and because y 5 2λ λ 5 P 8 255 Interpretation of the Lagrange multiplier If the farmer were interested in knowing how much more field could be fenced by adding an extra yard of fence the Lagrange multiplier sug gests that he or she could find out by dividing the present perimeter by 8 Some specific numbers might make this clear Suppose that the field currently has a perimeter of 400 yards If the farmer has planned optimally the field will be a square with 100 yards 15P42 on a side The enclosed area will be 10000 square yards Suppose now that the perimeter ie the available fence were enlarged by one yard Equation 255 would then predict that the total area would be increased by approximately 50 15P82 square yards That this is indeed the case can be shown as follows Because the perimeter is now 401 yards each side of the square will be 4014 yards Therefore the total area of the field is 140142 2 which according to the authors calculator works out to be 1005006 square yards Hence the prediction of a 50squareyard increase that is provided by the Lagrange multiplier proves to be remarkably close As in all constrained maximization problems here the Lagrange multiplier provides useful information about the implicit value of the constraint Duality The dual of this constrained maximization problem is that for a given area of a rectan gular field the farmer wishes to minimize the fence required to surround it Mathematically the problem is to minimize P 5 2x 1 2y 256 subject to the constraint A 5 x y 257 Setting up the Lagrangian expression D 5 2x 1 2y 1 λD 1A 2 x y2 258 where the D denotes the dual concept yields the following firstorder conditions for a minimum D x 5 2 2 λD y 5 0 D y 5 2 2 λD x 5 0 D λD 5 A 2 x y 5 0 259 Solving these equations as before yields the result x 5 y 5 A 260 Again the field should be square if the length of fence is to be minimized The value of the Lagrange multiplier in this problem is λD 5 2 y 5 2 x 5 2 A 261 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 45 26 ENVELOPE THEOREM IN CONSTRAINED MAXIMIZATION PROBLEMS The envelope theorem which we discussed previously in connection with unconstrained maximization problems also has important applications in constrained maximization problems Here we will provide only a brief presentation of the theorem In later chapters we will look at a number of applications Suppose we seek the maximum value of y 5 f 1x1 c xn a2 262 subject to the constraint g 1x1 c xn a2 5 0 263 where we have made explicit the dependence of the functions f and g on some parameter a As we have shown one way to solve this problem is to set up the Lagrangian expression 5 f 1x1 c xn a2 1 λg 1x1 c xn a2 264 and solve the firstorder conditions see Equation 259 for the optimal constrained values x 1 c x n These values which will depend on the parameter a can then be substituted back into the original function f to yield a value function for the problem For this value function the envelope theorem states that dy da 5 a 1x 1 c x n a2 265 That is the change in the maximal value of y that results when the parameter a changes and all the xs are recalculated to new optimal values can be found by partially differenti ating the Lagrangian expression Equation 264 and evaluating the resultant partial deriv ative at the optimal values of the xs Hence the Lagrangian expression plays the same role in applying the envelope theorem to constrained problems as does the objective function alone in unconstrained problems The next example shows this for the optimal fencing problem A sketch of the proof of the envelope theorem in constrained problems is pro vided in Problem 212 As before this Lagrange multiplier indicates the relationship between the objective min imizing fence and the constraint needing to surround the field If the field were 10000 square yards as we saw before 400 yards of fence would be needed Increasing the field by one square yard would require about 002 more yards of fence 12A 5 21002 The reader may wish to fire up his or her calculator to show this is indeed the casea fence 100005 yards on each side will exactly enclose 10001 square yards Here as in most duality prob lems the value of the Lagrange multiplier in the dual is the reciprocal of the value for the Lagrange multiplier in the primal problem Both provide the same information although in a somewhat different form QUERY How would the answers to this problem change if one side of the field required a dou ble fence Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 46 Part 1 Introduction EXAMPLE 29 Optimal Fences and the Envelope Theorem In the fencing problem in Example 28 the value function shows the area of the field as a function of the perimeter fencing available the only exogenous variable in the problem A 5 x y 5 P 4 P 4 5 P 2 16 266 Now applying the envelope theorem yieldsremember that the Lagrangian expression for this problem is 5 xy 1 λ1P 2 2x 2 2y2 dA dP 5 P 8 5 P 5 λ 267 In this case as we already know the Lagrange multiplier shows how the optimized area of the field would be affected by a small increase in the available fencing More generally this prob lem illustrates the fact that the Lagrangian multiplier in a constrained maximization problem will often show the marginal gain in the objective function that can be obtained from a slight relax ation of the constraint QUERY How would you apply the envelope theorem to the dual problem of minimizing the fencing needed to enclose a certain field area 27 INEQUALITY CONSTRAINTS In some economic problems the constraints need not hold exactly For example an indi viduals budget constraint requires that he or she spend no more than a certain amount per period but it is at least possible to spend less than this amount Inequality constraints also arise in the values permitted for some variables in economic problems Usually for example economic variables must be nonnegative although they can take on the value of zero In this section we will show how the Lagrangian technique can be adapted to such circumstances Although we will encounter only a few problems later in the text that require this mathematics development here will illustrate a few general principles that are consistent with economic intuition 271 A twovariable example To avoid much cumbersome notation we will explore inequality constraints only for the simple case involving two choice variables The results derived are readily generalized Suppose that we seek to maximize y 5 f1x1 x22 subject to three inequality constraints 1 g 1x1 x22 0 2 x1 0 and 3 x2 0 268 Hence we are allowing for the possibility that the constraint we introduced before need not hold exactly a person need not spend all his or her income and for the fact that both of the xs must be nonnegative as in most economic problems 272 Slack variables One way to solve this optimization problem is to introduce three new variables a b and c that convert the inequality constraints into equalities To ensure that the inequalities Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 47 continue to hold we will square these new variables ensuring that the resulting values are positive Using this procedure the inequality constraints become 1 g 1x1 x22 2 a2 5 0 2 x1 2 b2 5 0 and 3 x2 2 c2 5 0 269 Any solution that obeys these three equality constraints will also obey the inequality con straints It will also turn out that the optimal values for a b and c will provide several insights into the nature of the solutions to a problem of this type 273 Solution using Lagrange multipliers By converting the original problem involving inequalities into one involving equalities we are now in a position to use Lagrangian methods to solve it Because there are three con straints we must introduce three Lagrange multipliers λ1 λ2 and λ3 The full Lagrangian expression is 5 f 1x1 x22 1 λ1 3g 1x1 x22 2 a24 1 λ2 1x1 2 b22 1 λ3 1x2 2 c22 270 We wish to find the values of x1 x2 a b c λ1 λ2 and λ3 that constitute a critical point for this expression This will necessitate eight firstorder conditions x1 5 f1 1 λ1g1 1 λ2 5 0 x2 5 f2 1 λ1g2 1 λ3 5 0 a 5 22aλ1 5 0 b 5 22bλ2 5 0 c 5 22cλ3 5 0 λ1 5 g 1x1 x22 2 a2 5 0 λ2 5 x1 2 b2 5 0 λ3 5 x2 2 c2 5 0 271 In many ways these conditions resemble those that we derived earlier for the case of a single equality constraint For example the final three conditions merely repeat the three revised constraints This ensures that any solution will obey these conditions The first two equations also resemble the optimal conditions developed earlier If λ2 and λ3 were 0 the conditions would in fact be identical But the presence of the additional Lagrange multipliers in the expressions shows that the customary optimality conditions may not hold exactly here Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 48 Part 1 Introduction 274 Complementary slackness The three equations involving the variables a b and c provide the most important insights into the nature of solutions to problems involving inequality constraints For example the third line in Equation 271 implies that in the optimal solution either λ1 or a must be 012 In the second case 1a 5 02 the constraint g 1x1 x22 5 0 holds exactly and the calculated value of λ1 indicates its relative importance to the objective function f On the other hand if a 2 0 then λ1 5 0 and this shows that the availability of some slackness in the con straint implies that its marginal value to the objective is 0 In the consumer context this means that if a person does not spend all his or her income even more income would do nothing to raise his or her wellbeing Similar complementary slackness relationships also hold for the choice variables x1 and x2 For example the fourth line in Equation 271 requires that the optimal solution have either b or λ2 as 0 If λ2 5 0 then the optimal solution has x1 0 and this choice variable meets the precise benefitcost test that f1 1 λ1g1 5 0 Alternatively solutions where b 5 0 have x1 5 0 and also require that λ2 0 Thus such solutions do not involve any use of x1 because that variable does not meet the benefitcost test as shown by the first line of Equation 271 which implies that f1 1 λ1g1 0 An identical result holds for the choice variable x2 These results which are sometimes called KuhnTucker conditions after their discover ers show that the solutions to optimization problems involving inequality constraints will differ from similar problems involving equality constraints in rather simple ways Hence we cannot go far wrong by working primarily with constraints involving equalities and assuming that we can rely on intuition to state what would happen if the problems involved inequalities That is the general approach we will take in this book13 28 SECONDORDER CONDITIONS AND CURVATURE Thus far our discussion of optimization has focused primarily on necessary firstorder conditions for finding a maximum That is indeed the practice we will follow throughout much of this book because as we shall see most economic problems involve functions for which the secondorder conditions for a maximum are also satisfied This is because these functions have the right curvature properties to ensure that the necessary conditions for an optimum are also sufficient In this section we provide a general treatment of these curva ture conditions and their relationship to secondorder conditions The economic implica tions of these curvature conditions will be discussed throughout the text 281 Functions of one variable First consider the case in which the objective y is a function of only a single variable x That is y 5 f1x2 272 A necessary condition for this function to attain its maximum value at some point is that dy dx 5 fr1x2 5 0 273 12We will not examine the degenerate case where both of these variables are 0 13The situation can become much more complex when calculus cannot be relied on to give a solution perhaps because some of the functions in a problem are not differentiable For a discussion see Avinash K Dixit Optimization in Economic Theory 2nd ed Oxford Oxford University Press 1990 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 49 at that point To ensure that the point is indeed a maximum we must have y decreasing for movements away from it We already know that for small changes in x the value of y does not change what we need to check is whether y is increasing before that plateau is reached and decreasing thereafter We have already derived an expression for the change in y dy dy 5 fr1x2dx 274 What we now require is that dy be decreasing for small increases in the value of x The dif ferential of Equation 274 is given by d1dy2 5 d2y 5 d3 fr1x2dx4 dx dx 5 fs1x2dx dx 5 fs1x2dx2 275 But d2y 0 implies that fs1x2dx2 0 276 and because dx2 must be positive because anything squared is positive we have fs1x2 0 277 as the required secondorder condition In words this condition requires that the function f have a concave shape at the critical point contrast Figures 21 and 22 The curvature conditions we will encounter in this book represent generalizations of this simple idea 282 Functions of two variables As a second case we consider y as a function of two independent variables y 5 f1x1 x22 278 A necessary condition for such a function to attain its maximum value is that its partial derivatives in both the x1 and the x2 directions be 0 That is y x1 5 f1 5 0 y x2 5 f2 5 0 279 A point that satisfies these conditions will be a flat spot on the function a point where dy 5 0 and therefore will be a candidate for a maximum To ensure that the point is a local maximum y must diminish for movements in any direction away from the critical point In pictorial terms there is only one way to leave a true mountaintop and that is to go down 283 An intuitive argument Earlier we described why a simple generalization of the singlevariable case shows that both own second partial derivatives f11 and f22 must be negative for a local maximum In our mountain analogy if attention is confined only to northsouth or eastwest Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 50 Part 1 Introduction movements the slope of the mountain must be diminishing as we cross its summit the slope must change from positive to negative The particular complexity that arises in the twovariable case involves movements through the optimal point that are not solely in the x1 or x2 directions say movements from northeast to southwest In such cases the secondorder partial derivatives do not provide complete information about how the slope is changing near the critical point Conditions must also be placed on the crosspartial derivative 1 f12 5 f212 to ensure that dy is decreasing for movements through the critical point in any direction As we shall see those conditions amount to requiring that the own secondorder partial derivatives be sufficiently negative so as to counterbalance any possible perverse crosspartial derivatives that may exist Intuitively if the mountain falls away steeply enough in the northsouth and eastwest directions relatively minor failures to do so in other directions can be compensated for 284 A formal analysis We now proceed to make these points more formally What we wish to discover are the conditions that must be placed on the second partial derivatives of the function f to ensure that d2y is negative for movements in any direction through the critical point Recall first that the total differential of the function is given by dy 5 f1dx1 1 f2dx2 280 The differential of that function is given by d2y 5 1 f11dx1 1 f12dx22dx1 1 1 f21dx1 1 f22dx22dx2 or d2y 5 f11dx2 1 1 f12dx2dx1 1 f21dx1dx2 1 f22dx2 2 Because by Youngs theorem f12 5 f21 we can arrange terms to get d2y 5 f11dx2 1 1 2f12dx1dx2 1 f22dx2 2 281 For this equation to be unambiguously negative for any change in the xs ie for any choices of dx1 and dx2 it is obviously necessary that f11 and f22 be negative If for example dx2 5 0 then d2y 5 f11dx2 1 282 and d2y 0 implies f11 0 283 An identical argument can be made for f22 by setting dx1 5 0 If neither dx1 nor dx2 is 0 we then must consider the crosspartial f12 in deciding whether d2y is unambiguously nega tive Relatively simple algebra can be used to show that the required condition is14 f11 f22 2 f 2 12 0 284 14The proof proceeds by adding and subtracting the term 1 f12 dx222f11 to Equation 281 and factoring But this approach is only applicable to this special case A more easily generalized approach that uses matrix algebra recognizes that Equation 281 is a Quadratic Form in dx1 and dx2 and that Equations 283 and 284 amount to requiring that the Hessian matrix cf11 f12 f21 f22 d be negative definite In particular Equation 284 requires that the determinant of this Hessian matrix be positive For a discussion see the Extensions to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 51 285 Concave functions Intuitively what Equation 284 requires is that the own second partial derivatives f11 and f22 be sufficiently negative so that their product which is positive will outweigh any pos sible perverse effects from the crosspartial derivatives 1 f12 5 f212 Functions that obey such a condition are called concave functions In three dimensions such functions resem ble inverted teacups for an illustration see Example 211 This image makes it clear that a flat spot on such a function is indeed a true maximum because the function always slopes downward from such a spot More generally concave functions have the property that they always lie below any plane that is tangent to themthe plane defined by the maximum value of the function is simply a special case of this property EXAMPLE 210 SecondOrder Conditions Health Status for the Last Time In Example 26 we considered the health status function y 5 f 1x1 x22 5 2x2 1 1 2x1 2 x2 2 1 4x2 1 5 285 The firstorder conditions for a maximum are f1 5 22x1 1 2 5 0 f2 5 22x2 1 4 5 0 286 or x 1 5 1 x 2 5 2 287 The secondorder partial derivatives for Equation 285 are f11 5 22 f22 5 22 f12 5 0 288 These derivatives clearly obey Equations 283 and 284 so both necessary and sufficient conditions for a local maximum are satisfied15 QUERY Describe the concave shape of the health status function and indicate why it has only a single global maximum value 15Notice that Equation 288 obeys the sufficient conditions not only at the critical point but also for all possible choices of x1 and x2 That is the function is concave In more complex examples this need not be the case The secondorder conditions need be satisfied only at the critical point for a local maximum to occur 286 Constrained maximization As another illustration of secondorder conditions consider the problem of choosing x1 and x2 to maximize y 5 f 1x1 x22 289 subject to the linear constraint c 2 b1x1 2 b2x2 5 0 290 where c b1 and b2 are constant parameters in the problem This problem is of the type that will be frequently encountered in this book and is a special case of the constrained Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 52 Part 1 Introduction maximum problems that we examined earlier There we showed that the firstorder condi tions for a maximum may be derived by setting up the Lagrangian expression 5 f 1x1 x22 1 λ 1c 2 b1x1 2 b2x22 291 Partial differentiation with respect to x1 x2 and λ yields the familiar results f1 2 λb1 5 0 f2 2 λb2 5 0 292 c 2 b1x1 2 b2x2 5 0 These equations can in general be solved for the optimal values of x1 x2 and λ To ensure that the point derived in that way is a local maximum we must again examine movements away from the critical points by using the second total differential d2y 5 f11dx2 1 1 2f12dx1dx2 1 f22dx2 2 293 In this case however not all possible small changes in the xs are permissible Only those values of x1 and x2 that continue to satisfy the constraint can be considered valid alterna tives to the critical point To examine such changes we must calculate the total differential of the constraint 2b1dx1 2 b2dx2 5 0 294 or dx2 5 2b1 b2 dx1 295 This equation shows the relative changes in x1 and x2 that are allowable in considering movements from the critical point To proceed further on this problem we need to use the firstorder conditions The first two of these imply f1 f2 5 b1 b2 296 and combining this result with Equation 295 yields dx2 5 2 f1 f2 dx1 297 We now substitute this expression for dx2 in Equation 293 to demonstrate the conditions that must hold for d2y to be negative d2y 5 f11dx2 1 1 2f12dx1a2 f1 f2 dx1b 1 f22a2 f1 f2 dx1b 2 5 f11dx2 1 2 2f12 f1 f2 dx2 1 1 f22 f 2 1 f 2 2 dx2 1 298 Combining terms and putting each over a common denominator gives d2y 5 1 f11 f 2 2 2 2f12 f1 f2 1 f22 f 2 12 dx2 1 f 2 2 299 Consequently for d2y 0 it must be the case that f11 f 2 2 2 2f12 f1 f2 1 f22 f 2 1 0 2100 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 53 287 Quasiconcave functions Although Equation 2100 appears to be little more than an inordinately complex mass of mathematical symbols the condition is an important one It characterizes a set of functions termed quasiconcave functions These functions have the property that the set of all points for which such a function takes on a value greater than any specific constant is a convex set ie any two points in the set can be joined by a line contained completely within the set Many economic models are characterized by such functions and as we will see in considerable detail in Chapter 3 in these cases the condition for quasiconcavity has a relatively simple economic interpretation Problems 29 and 210 examine two specific quasiconcave functions that we will frequently encounter in this book Example 211 shows the relationship between concave and quasiconcave functions EXAMPLE 211 Concave and QuasiConcave Functions The differences between concave and quasiconcave functions can be illustrated with the function16 y 5 f 1x1 x22 5 1x1 x22 k 2101 where the xs take on only positive values and the parameter k can take on a variety of positive values No matter what value k takes this function is quasiconcave One way to show this is to look at the level curves of the function by setting y equal to a specific value say c In this case y 5 c 5 1x1x22 k or x1x2 5 c1k 5 cr 2102 But this is just the equation of a standard rectangular hyperbola Clearly the set of points for which y takes on values larger than c is convex because it is bounded by this hyperbola A more mathematical way to show quasiconcavity would apply Equation 299 to this func tion Although the algebra of doing this is a bit messy it may be worth the struggle The various components of the equation are f1 5 kxk21 1 xk 2 f2 5 kxk 1xk21 2 f11 5 k1k 2 12xk22 1 xk 2 2103 f22 5 k1k 2 12xk 1xk22 2 f12 5 k2xk21 1 xk21 2 Thus f11 f 2 2 2 2f12 f1 f2 1 f2 2 f 2 1 5 k3 1k 2 12x3k22 1 x3k22 2 2 2k4x3k22 1 x3k22 2 1 k3 1k 2 12x3k22 1 x3k22 2 2104 5 2k3x3k22 1 x3k22 2 1212 which is clearly negative as is required for quasiconcavity Whether the function f is concave depends on the value of k If k 05 the function is indeed concave An intuitive way to see this is to consider only points where x1 5 x2 For these points 16This function is a special case of the CobbDouglas function See also Problem 210 and the Extensions to this chapter for more details on this function Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 54 Part 1 Introduction y 5 1x2 12 k 5 x2k 1 2105 which for k 05 is concave Alternatively for k 05 this function is convex A more definitive proof makes use of the partial derivatives of this function In this case the condition for concavity can be expressed as f11 f22 2 f 2 12 5 k2 1k 2 12 2x2k22 1 x2k22 2 2 k4x2k22 1 x2k22 2 5 x2k22 1 x2k22 2 3k2 1k 2 12 2 2 k44 2106 5 x2k21 1 x2k21 2 3k2 122k 1 12 4 In all three cases these functions are quasiconcave For a fixed y their level curves are convex But only for k 5 02 is the function strictly concave The case k 5 10 clearly shows nonconcavity because the function is not below its tangent plane a k 02 b k 05 c k 10 FIGURE 24 Concave and QuasiConcave Functions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 55 29 HOMOGENEOUS FUNCTIONS Many of the functions that arise naturally out of economic theory have additional mathematical properties One particularly important set of properties relates to how the functions behave when all or most of their arguments are increased proportionally Such situations arise when we ask questions such as what would happen if all prices increased by 10 percent or how would a firms output change if it doubled all the inputs that it uses Thinking about these questions leads naturally to the concept of homoge neous functions Specifically a function f 1x1 x2 c xn2 is said to be homogeneous of degree k if f 1tx1 tx2 c txn2 5 tkf 1x1 x2 c xn2 2107 The most important examples of homogeneous functions are those for which k 5 1 or k 5 0 In words when a function is homogeneous of degree 1 a doubling of all its argu ments doubles the value of the function itself For functions that are homogeneous of degree 0 a doubling of all its arguments leaves the value of the function unchanged Func tions may also be homogeneous for changes in only certain subsets of their arguments that is a doubling of some of the xs may double the value of the function if the other arguments of the function are held constant Usually however homogeneity applies to changes in all the arguments in a function and this expression is positive as is required for concavity for 122k 1 12 0 or k 05 On the other hand the function is convex for k 05 A graphic illustration Figure 24 provides threedimensional illustrations of three specific examples of this function for k 5 02 k 5 05 and k 5 1 Notice that in all three cases the level curves of the function have hyperbolic convex shapes That is for any fixed value of y the functions are similar This shows the quasiconcavity of the function The primary differ ences among the functions are illustrated by the way in which the value of y increases as both xs increase together In Figure 24a when k 5 02 the increase in y slows as the xs increase This gives the function a rounded teacuplike shape that indicates its concavity For k 5 05 y appears to increase linearly with increases in both of the xs This is the borderline between concavity and convexity Finally when k 5 1 as in Figure 24c simultaneous increases in the values of both of the xs increase y rapidly The spine of the function looks convex to reflect such increasing returns More formally the function is above its tangent plane whereas it should be below that plane for concavity A careful look at Figure 24a suggests that any function that is concave will also be quasi concave You are asked to prove that this is indeed the case in Problem 29 This example shows that the converse of this statement is not truequasiconcave functions need not necessarily be concave Most functions we will encounter in this book will also illustrate this fact most will be quasiconcave but not necessarily concave QUERY Explain why the functions illustrated both in Figures 24a and 24c would have maxi mum values if the xs were subject to a linear constraint but only the graph in Figure 24a would have an unconstrained maximum Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 56 Part 1 Introduction 291 Homogeneity and derivatives If a function is homogeneous of degree k and can be differentiated the partial derivatives of the function will be homogeneous of degree k 2 1 A proof of this follows directly from the definition of homogeneity For example differentiating Equation 2107 with respect to x1 f1tx1 c txn2 x1 5 f1tx1 c txn2 tx1 tx1 x1 5 f1 1tx1 c txn2 t 5 tk f1x1 c xn2 x1 or f1 1tx1 c txn2 5 tk21f1 1x1 c xn2 2108 which shows that f1 meets the definition for homogeneity of degree k 2 1 Because mar ginal ideas are so prevalent in microeconomic theory this property shows that some important properties of marginal effects can be inferred from the properties of the under lying function itself 292 Eulers theorem Another useful feature of homogeneous functions can be shown by differentiating the defi nition for homogeneity with respect to the proportionality factor t In this case we differ entiate the right side of Equation 2107 first then the left side ktk21f 1x1 c xn2 5 x1 f1 1tx1 c txn2 1 c1 xn fn1tx1 c txn2 If we let t 5 1 this equation becomes kf 1x1 c xn2 5 x1f1 1x1 c xn2 1 c1 xn fn1x1 c xn2 2109 This equation is termed Eulers theorem after the mathematician who also discovered the constant e for homogeneous functions It shows that for a homogeneous function there is a definite relationship between the values of the function and the values of its partial derivatives Several important economic relationships among functions are based on this observation 293 Homothetic functions A homothetic function is one that is formed by taking a monotonic transformation of a homogeneous function17 Monotonic transformations by definition preserve the order of the relationship between the arguments of a function and the value of that function If certain sets of xs yield larger values for f they will also yield larger values for a mono tonic transformation of f Because monotonic transformations may take many forms however they would not be expected to preserve an exact mathematical relationship such as that embodied in homogeneous functions Consider for example the function y 5 f 1x1 x22 5 x1x2 Clearly this function is homogeneous of degree 2a doubling of its two arguments will multiply the value of the function by 4 However the monotonic trans formation that simply adds 10 to f ie F1 f2 5 f 1 1 5 x1x2 1 1 is not homogeneous at all Thus except in special cases homothetic functions do not possess the homogene ity properties of their underlying functions Homothetic functions however do preserve one nice feature of homogeneous functionsthat the implicit tradeoffs implied by the function depend only on the ratio of the two variables being traded not on their abso lute levels To show this remember the implicit function theorem which showed that for 17Because a limiting case of a monotonic transformation is to leave the function unchanged all homogeneous functions are also homothetic Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 57 a twovariable function of the form y 5 f 1x1 x22 the implicit tradeoff between the two variables required to keep the value of the function constant is given by dx2 dx1 5 2 f1 f2 If we assume that f is homogeneous of degree k its partial derivatives will be homogeneous of degree k 2 1 therefore we can write this tradeoff as dx2 dx1 5 2 tk21f1 1x1 x22 tk21f2 1x1 x22 5 2 f1 1tx1 tx22 f2 1tx1 tx22 2110 Now let t 5 1x2 so Equation 2110 becomes dx2 dx1 5 2 f1 1x1x212 f2 1x1x212 2111 which shows that the tradeoffs implicit in f depend only on the ratio of x1 to x2 If we apply any monotonic transformation F with Fr 0 to the original homogeneous function f the tradeoffs implied by the new homothetic function F3 f1x1 x22 4 are unchanged dx2 dx1 5 2 Frf1 1x1x2 12 Frf2 1x1x2 12 5 2 f1 1x1x2 12 f2 1x1x2 12 2112 At many places in this book we will find it instructive to discuss some theoretical results with twodimensional graphs and Equation 2112 can be used to focus our attention on the ratios of the key variables rather than on their absolute levels EXAMPLE 212 Cardinal and Ordinal Properties In applied economics it is sometimes important to know the exact numerical relationship among variables For example in the study of production one might wish to know precisely how much extra output would be produced by hiring another worker This is a question about the cardinal ie numerical properties of the production function In other cases one may only care about the order in which various points are ranked In the theory of utility for example we assume that people can rank bundles of goods and will choose the bundle with the highest ranking but that there are no unique numerical values assigned to these rankings Mathematically ordinal proper ties of functions are preserved by any monotonic transformation because by definition a mono tonic transformation preserves order Usually however cardinal properties are not preserved by arbitrary monotonic transformations These distinctions are illustrated by the functions we examined in Example 211 There we studied monotonic transformations of the function f 1x1 x22 5 1x1x22 k 2113 by considering various values of the parameter k We showed that quasiconcavity an ordinal property was preserved for all values of k Hence when approaching problems that focus on max imizing or minimizing such a function subject to linear constraints we need not worry about pre cisely which transformation is used On the other hand the function in Equation 2113 is concave a cardinal property only for a narrow range of values of k Many monotonic transformations destroy the concavity of f The function in Equation 2113 can also be used to illustrate the difference between homoge neous and homothetic functions A proportional increase in the two arguments of f would yield Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 58 Part 1 Introduction f1tx1 tx22 5 t2kx1x2 5 t2kf 1x1 x22 2114 Hence the degree of homogeneity for this function depends on kthat is the degree of homoge neity is not preserved independently of which monotonic transformation is used Alternatively the function in Equation 2113 is homothetic because dx2 dx1 5 2 f1 f2 5 2kxk21 1 xk 2 kxk 1 xk21 2 5 2x2 x1 2115 That is the tradeoff between x2 and x1 depends only on the ratio of these two variables and is unaffected by the value of k Hence homotheticity is an ordinal property As we shall see this property is convenient when developing graphical arguments about economic propositions involving situations where the ratios of certain variables do not change often because they are determined by unchanging prices QUERY How would the discussion in this example be changed if we considered monotonic transformations of the form f 1x1 x2 k2 5 x1x2 1 k for various values of k 210 INTEGRATION Integration is another of the tools of calculus that finds a number of applications in micro economic theory The technique is used both to calculate areas that measure various eco nomic outcomes and more generally to provide a way of summing up outcomes that occur over time or across individuals Our treatment of the topic here necessarily must be brief therefore readers desiring a more complete background should consult the references at the end of this chapter 2101 Antiderivatives Formally integration is the inverse of differentiation When you are asked to calculate the integral of a function f1x2 you are being asked to find a function that has f1x2 as its deriv ative If we call this antiderivative F1x2 this function is supposed to have the property that dF1x2 dx 5 Fr 1x2 5 f 1x2 2116 If such a function exists then we denote it as F1x2 5 3f1x2 dx 2117 The precise reason for this notation will be described in detail later First lets look at a few examples If f 1x2 5 x then F1x2 5 3f1x2 dx 5 3x dx 5 x2 2 1 C 2118 where C is an arbitrary constant of integration that disappears on differentiation The correctness of this result can be easily verified Fr 1x2 5 d1x22 1 C2 dx 5 x 1 0 5 x 2119 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 59 2102 Calculating antiderivatives Calculation of antiderivatives can be extremely simple difficult agonizing or impossible depending on the particular f 1x2 specified Here we will look at three simple methods for making such calculations but as you might expect these will not always work 1 Creative guesswork Probably the most common way of finding integrals antideriva tives is to work backward by asking What function will yield f 1x2 as its derivative Here are a few obvious examples F1x2 5 3x2 dx 5 x3 3 1 C F1x2 5 3xn dx 5 xn11 n 1 1 1 C F1x2 5 3 1ax2 1 bx 1 c2 dx 5 ax3 3 1 bx2 2 1 cx 1 C F1x2 5 3ex dx 5 ex 1 C 2120 F1x2 5 3ax dx 5 ax ln a 1 C F1x2 5 3 a1 xb dx 5 ln 1 0x02 1 C F1x2 5 3 1 ln x2 dx 5 x ln x 2 x 1 C You should use differentiation to check that all these obey the property that Fr 1x2 5 f1x2 Notice that in every case the integral includes a constant of integration because antider ivatives are unique only up to an additive constant which would become zero on differ entiation For many purposes the results in Equation 2120 or trivial generalizations of them will be sufficient for our purposes in this book Nevertheless here are two more methods that may work when intuition fails 2 Change of variable A clever redefinition of variables may sometimes make a func tion much easier to integrate For example it is not at all obvious what the integral of 2x 11 1 x22 is But if we let y 5 1 1 x2 then dy 5 2xdx and 3 2x 1 1 x2 dx 5 3 1 y dy 5 ln 1 0y02 5 ln 1 01 1 x202 2121 The key to this procedure is in breaking the original function into a term in y and a term in dy It takes a lot of practice to see patterns for which this will work 3 Integration by parts A similar method for finding integrals makes use of the identity duv 5 udv 1 vdu for any two functions u and v Integration of this differential yields 3duv 5 uv 5 3u dv 1 3v du or 3u dv 5 uv 2 3v du 2122 Here the strategy is to define functions u and v in a way that the unknown integral on the left can be calculated by the difference between the two known expressions on the right For example you probably have no idea what the integral of xex is But we can define u 5 x thus du 5 dx and dv 5 exdx thus v 5 ex Hence we now have 3xex dx 5 3u dv 5 uv 2 3v du 5 xex 2 3ex dx 5 1x 2 12ex 1 C 2123 Again only practice can suggest useful patterns in the ways in which u and v can be defined Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 60 Part 1 Introduction 2103 Definite integrals The integrals we have been discussing thus far are indefinite integralsthey provide only a general function that is the antiderivative of another function A somewhat different although related approach uses integration to sum up the area under a graph of a function over some defined interval Figure 25 illustrates this process We wish to know the area under the function f 1x2 from x 5 a to x 5 b One way to do this would be to partition the interval into narrow slivers of x 1Dx2 and sum up the areas of the rectangles shown in the figure That is area under f1x2 a i f 1xi2Dxi 2124 where the notation is intended to indicate that the height of each rectangle is approximated by the value of f1x2 for a value of x in the interval Taking this process to the limit by shrinking the size of the Dx intervals yields an exact measure of the area we want and is denoted by area under f1x2 5 3 x5b x5a f1x2 dx 2125 This then explains the origin of the oddly shaped integral signit is a stylized S indicating sum As we shall see integrating is a general way of summing the values of a continuous function over some interval 2104 Fundamental theorem of calculus Evaluating the integral in Equation 2125 is simple if we know the antiderivative of f1x2 say F1x2 In this case we have Definite integrals measure the area under a curve by summing rectangular areas as shown in the graph The dimension of each rectangle is f 1x2dx fx fx a b x FIGURE 25 Definite Integrals Show the Areas under the Graph of a Function Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 61 area under f 1x2 5 3 x5b x5a f 1x2 dx 5 F1b2 2 F1a2 2126 That is all we need do is calculate the antiderivative of f 1x2 and subtract the value of this func tion at the lower limit of integration from its value at the upper limit of integration This result is sometimes termed the fundamental theorem of calculus because it directly ties together the two principal tools of calculusderivatives and integrals In Example 213 we show that this result is much more general than simply a way to measure areas It can be used to illustrate one of the primary conceptual principles of economicsthe distinction between stocks and flows EXAMPLE 213 Stocks and Flows The definite integral provides a useful way for summing up any function that is providing a con tinuous flow over time For example suppose that net population increase births minus deaths for a country can be approximated by the function f1t2 5 1000e002t Hence the net population change is growing at the rate of 2 percent per yearit is 1000 new people in year 0 1020 new people in the first year 1041 in the second year and so forth Suppose we wish to know how much in total the population will increase over a 50year period This might be a tedious calcu lation without calculus but using the fundamental theorem of calculus provides an easy answer increase in population 5 3 t550 t50 f1t2 dt 5 3 t550 t50 1000e002tdt 5 F1t2 50 0 5 1000e002t 002 50 0 5 1000e 002 2 50000 5 85914 2127 where the notation 0 b a indicates that the expression is to be evaluated as F1b2 2 F1a24 Hence the conclusion is that the population will grow by nearly 86000 people over the next 50 years Notice how the fundamental theorem of calculus ties together a flow concept net population increase which is measured as an amount per year with a stock concept total population which is measured at a specific date and does not have a time dimension Note also that the 86000 calcu lation refers only to the total increase between year 0 and year 50 To know the actual total pop ulation at any date we would have to add the number of people in the population at year 0 That would be similar to choosing a constant of integration in this specific problem Now consider an application with more economic content Suppose that total costs for a particular firm are given by C1q2 5 01q2 1 500 where q represents output during some period Here the term 01q2 represents variable costs costs that vary with output whereas the 500 figure represents fixed costs Marginal costs for this production process can be found through differentiationMC 5 dC1q2dq 5 02qhence marginal costs are increasing with q and fixed costs drop out on differentiation What are the total costs associated with produc ing say q 5 100 One way to answer this question is to use the total cost function directly C11002 5 01 110022 1 500 5 1500 An alternative way would be to integrate marginal cost over the range 0 to 100 to get total variable cost variable cost 5 3 q5100 q50 02q dq 5 01q2 100 0 5 1000 2 0 5 1000 2128 to which we would have to add fixed costs of 500 the constant of integration in this problem to get total costs Of course this method of arriving at total cost is much more cumbersome than just using the equation for total cost directly But the derivation does show that total variable cost between any two output levels can be found through integration as the area below the marginal cost curvea conclusion that we will find useful in some graphical applications QUERY How would you calculate the total variable cost associated with expanding output from 100 to 110 Explain why fixed costs do not enter into this calculation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 62 Part 1 Introduction 2105 Differentiating a definite integral Occasionally we will wish to differentiate a definite integralusually in the context of seeking to maximize the value of this integral Although performing such differentiations can sometimes be rather complex there are a few rules that should make the process easier 1 Differentiation with respect to the variable of integration This is a trick question but instructive nonetheless A definite integral has a constant value hence its derivative is zero That is d3 b a f 1x2 dx dx 5 0 2129 The summing process required for integration has already been accomplished once we write down a definite integral It does not matter whether the variable of integration is x or t or anything else The value of this integrated sum will not change when the variable x changes no matter what x is but see rule 3 below 2 Differentiation with respect to the upper bound of integration Changing the upper bound of integration will obviously change the value of a definite integral In this case we must make a distinction between the variable determining the upper bound of inte gration say x and the variable of integration say t The result then is a simple appli cation of the fundamental theorem of calculus For example de x a f1t2dt dx 5 d3F1x2 2 F1a2 4 dx 5 f1x2 2 0 5 f1x2 2130 where F1x2 is the antiderivative of f1x2 By referring back to Figure 25 we can see why this conclusion makes sensewe are asking how the value of the definite inte gral changes if x increases slightly Obviously the answer is that the value of the inte gral increases by the height of f1x2 notice that this value will ultimately depend on the specified value of x If the upper bound of integration is a function of x this result can be generalized using the chain rule de g1x2 a f1t2 dt dx 5 d3F1g 1x2 2 2 F1a2 4 dx 5 d3F1g 1x2 2 4 dx 5 f dg 1x2 dx 5 f1g 1x2 2gr 1x2 2131 where again the specific value for this derivative would depend on the value of x assumed Finally notice that differentiation with respect to a lower bound of integration just changes the sign of this expression de b g1x2f1t2 dt dx 5 d3F1b2 2 F1g 1x2 2 4 dx 5 2 dF1g 1x2 2 dx 5 2f1g 1x2 2gr 1x2 2132 3 Differentiation with respect to another relevant variable In some cases we may wish to integrate an expression that is a function of several variables In general this can involve multiple integrals and differentiation can become complicated But there is one simple case that should be mentioned Suppose that we have a function of two variables f1x y2 and that we wish to integrate this function with respect to the variable x The specific value for this integral will obviously depend on the value of y and we might Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 63 even ask how that value changes when y changes In this case it is possible to differen tiate through the integral sign to obtain a result That is de b af1xy2 dx dy 5 3 b a fy 1x y2 dx 2133 This expression shows that we can first partially differentiate f1x y2 with respect to y before proceeding to compute the value of the definite integral Of course the result ing value may still depend on the specific value that is assigned to y but often it will yield more economic insights than the original problem does Some further examples of using definite integrals are found in Problem 28 211 DYNAMIC OPTIMIZATION Some optimization problems that arise in microeconomics involve multiple periods18 We are interested in finding the optimal time path for a variable or set of variables that suc ceeds in optimizing some goal For example an individual may wish to choose a path of lifetime consumptions that maximizes his or her utility Or a firm may seek a path for input and output choices that maximizes the present value of all future profits The particular feature of such problems that makes them difficult is that decisions made in one period affect outcomes in later periods Hence one must explicitly take account of this interrela tionship in choosing optimal paths If decisions in one period did not affect later periods the problem would not have a dynamic structureone could just proceed to optimize decisions in each period without regard for what comes next Here however we wish to explicitly allow for dynamic considerations 2111 The optimal control problem Mathematicians and economists have developed many techniques for solving problems in dynamic optimization The references at the end of this chapter provide broad intro ductions to these methods Here however we will be concerned with only one such method that has many similarities to the optimization techniques discussed earlier in this chapterthe optimal control problem The framework of the problem is relatively simple A decisionmaker wishes to find the optimal time path for some variable x 1t2 over a speci fied time interval 3t0 t14 Changes in x are governed by a differential equation dx 1t2 dt 5 g 3x 1t2 c 1t2 t4 2134 where the variable c 1t2 is used to control the change in x 1t2 In each period the deci sionmaker derives value from x and c according to the function f 3x 1t2 c 1t2 t4 and his or her goal to optimize e t1 t0 f 3x 1t2 c 1t2 t4 dt Often this problem will also be subject to end point constraints on the variable x These might be written as x 1t02 5 x0 and x 1t12 5 x1 Notice how this problem is dynamic Any decision about how much to change x this period will affect not only the future value of x but it will also affect future values of the outcome function f The problem then is how to keep x 1t2 on its optimal path 18Throughout this section we treat dynamic optimization problems as occurring over time In other contexts the same techniques can be used to solve optimization problems that occur across a continuum of firms or individuals when the optimal choices for one agent affect what is optimal for others The material in this section will be used in only a few places in the text but is provided here as a convenient reference Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 64 Part 1 Introduction Economic intuition can help to solve this problem Suppose that we just focused on the function f and chose x and c to maximize it at each instant of time There are two diffi culties with this myopic approach First we are not really free to choose x at any time Rather the value of x will be determined by its initial value x0 and by its history of changes as given by Equation 2134 A second problem with this myopic approach is that it disre gards the dynamic nature of the problem by forgetting to ask how this periods decisions affect the future We need some way to reflect the dynamics of this problem in a single periods decisions Assigning the correct value price to x at each instant of time will do just that Because this implicit price will have many similarities to the Lagrange multipliers studied earlier in this chapter we will call it λ 1t2 The value of λ is treated as a function of time because the importance of x can obviously change over time 2112 The maximum principle Now lets look at the decisionmakers problem at a single point in time He or she must be concerned with both the current value of the objective function f 3x 1t2 c 1t2 t4 and with the implied change in the value of x 1t2 Because the current value of x 1t2 is given by λ 1t2x 1t2 the instantaneous rate of change of this value is given by d3λ 1t2x 1t2 4 dt 5 λ 1t2 dx 1t2 dt 1 x 1t2 dλ 1t2 dt 2135 and so at any time t a comprehensive measure of the value of concern19 to the decisionmaker is H 5 f 3x 1t2 c 1t2 t4 1 λ 1t2g 3x 1t2 c 1t2 t4 1 x 1t2 dλ 1t2 dt 2136 This comprehensive value represents both the current benefits being received and the instantaneous change in the value of x Now we can ask what conditions must hold for x 1t2 and c 1t2 to optimize this expression20 That is H c 5 fc 1 λgc 5 0 or fc 5 2λgc H x 5 fx 1 λgx 1 dλ 1t2 dt 5 0 or fx 1 λgx 5 2dλ 1t2 dt 2137 These are then the two optimality conditions for this dynamic problem They are usually referred to as the maximum principle This solution to the optimal control problem was first proposed by the Russian mathematician L S Pontryagin and his colleagues in the early 1960s Although the logic of the maximum principle can best be illustrated by the economic applications we will encounter later in this book a brief summary of the intuition behind them may be helpful The first condition asks about the optimal choice of c It suggests that at the margin the gain from increasing c in terms of the function f must be balanced by the losses from increasing c in terms of the way in which such a change would affect the change in x where that change is valued by the timevarying Lagrangian multiplier That is present gains must be weighed against future costs 19We denote this current value expression by H to suggest its similarity to the Hamiltonian expression used in formal dynamic optimization theory Usually the Hamiltonian expression does not have the final term in Equation 2136 however 20Notice that the variable x is not really a choice variable hereits value is determined by history Differentiation with respect to x can be regarded as implicitly asking the question If x 1t2 were optimal what characteristics would it have Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 65 The second condition relates to the characteristics that an optimal time path of x 1t2 should have It implies that at the margin any net gains from more current x either in terms of f or in terms of the accompanying value of changes in x must be balanced by changes in the implied value of x itself That is the net current gain from more x must be weighed against the declining future value of x EXAMPLE 214 Allocating a Fixed Supply As an extremely simple illustration of the maximum principle assume that someone has inher ited 1000 bottles of wine from a rich uncle He or she intends to drink these bottles over the next 20 years How should this be done to maximize the utility from doing so Suppose that this persons utility function for wine is given by u 3c 1t24 5 ln c 1t2 Hence the utility from wine drinking exhibits diminishing marginal utility 1ur 0 us 02 This persons goal is to maximize 3 20 0 u 3c 1t24 dt 5 3 20 0 ln c 1t2 dt 2138 Let x 1t2 represent the number of bottles of wine remaining at time t This series is constrained by x 102 5 1000 and x 1202 5 0 The differential equation determining the evolution of x 1t2 takes the simple form21 dx 1t2 dt 5 2c 1t2 2139 That is each instants consumption just reduces the stock of remaining bottles by the amount consumed The current value Hamiltonian expression for this problem is H 5 ln c 1t2 1 λ32c 1t24 1 x 1t2 dλ dt 2140 and the firstorder conditions for a maximum are H c 5 1 c 2 λ 5 0 H x 5 d λ dt 5 0 2141 The second of these conditions requires that λ the implicit value of wine be constant over time This makes intuitive sense Because consuming a bottle of wine always reduces the available stock by one bottle any solution where the value of wine differed over time would provide an incentive to change behavior by drinking more wine when it is cheap and less when it is expen sive Combining this second condition for a maximum with the first condition implies that c 1t2 itself must be constant over time If c 1t2 5 k the number of bottles remaining at any time will be x 1t2 5 1000 2 kt If k 5 50 the system will obey the endpoint constraints x 102 5 1000 and x 1202 5 0 Of course in this problem you could probably guess that the optimum plan would be to drink the wine at the rate of 50 bottles per year for 20 years because diminishing marginal utility suggests one does not want to drink excessively in any period The maximum principle confirms this intuition 21The simple form of this differential equation where dxdt depends only on the value of the control variable c means that this problem is identical to the one explored using the calculus of variations approach to dynamic optimization In such a case one can substitute dxdt into the function f and the firstorder conditions for a maximum can be compressed into the single equation fx 5 dfdxdtdt which is termed the Euler equation In Chapter 17 we will encounter many Euler equations Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 66 Part 1 Introduction More complicated utility Now lets take a more complicated utility function that may yield more interesting results Suppose that the utility of consuming wine at any date t is given by u 3c 1t2 4 5 e 3c 1t2 γγ if γ 2 0 γ 1 ln c 1t2 if γ 5 0 2142 Assume also that the consumer discounts future consumption at the rate δ Hence this persons goal is to maximize 3 20 0 u 3c 1t2 4 dt 5 3 20 0 e2δt 3c 1t24 γ γ dt 2143 subject to the following constraints dx 1t2 dt 5 2c 1t2 x 102 5 1000 x 1202 5 0 2144 Setting up the current value Hamiltonian expression yields H 5 e2δt 3c 1t24 γ γ 1 λ12c2 1 x 1t2 dλ1t2 dt 2145 and the maximum principle requires that H c 5 e2δt3c 1t24 γ21 2 λ 5 0 and H x 5 0 1 0 1 dλ dt 5 0 2146 Hence we can again conclude that the implicit value of the wine stock λ should be constant over time call this constant k and that e2δt3c 1t24 γ21 5 k or c 1t2 5 k1 1γ212eδt 1γ212 2147 Thus optimal wine consumption should fall over time because the coefficient of t in the expo nent of e is negative to compensate for the fact that future consumption is being discounted in the consumers mind If for example we let δ 5 01 and γ 5 21 reasonable values as we will show in later chapters then c 1t2 5 k205e2005t 2148 Now we must do a bit more work in choosing k to satisfy the endpoint constraints We want 3 20 0 c 1t2 dt 5 3 20 0 k205e2005t dt 5 220k205e2005t 20 0 5 220k205 1e21 2 12 5 1264k205 5 1000 2149 Finally then we have the optimal consumption plan as c 1t2 79e2005t 2150 This consumption plan requires that wine consumption start out fairly high and decrease at a continuous rate of 5 percent per year Because consumption is continuously decreasing we must use integration to calculate wine consumption in any particular year x as follows Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 67 consumption in year x 3 x x21 c 1t2 dt 5 3 x x21 79e2005tdt 5 21580e2005t x x21 5 1580 1e20051x212 2 e2005x2 2151 If x 5 1 consumption is approximately 77 bottles in this first year Consumption then decreases smoothly ending with approximately 30 bottles being consumed in the 20th year QUERY Our first illustration was just an example of the second in which δ 5 γ 5 0 Explain how alternative values of these parameters will affect the path of optimal wine consumption Explain your results intuitively for more on optimal consumption over time see Chapter 17 212 MATHEMATICAL STATISTICS In recent years microeconomic theory has increasingly focused on issues raised by uncer tainty and imperfect information To understand much of this literature it is important to have a good background in mathematical statistics Therefore the purpose of this section is to summarize a few of the statistical principles that we will encounter at various places in this book 2121 Random variables and probability density functions A random variable describes in numerical form the outcomes from an experiment that is subject to chance For example we might flip a coin and observe whether it lands heads or tails If we call this random variable x we can denote the possible outcomes realizations of the variable as x 5 e1 if coin is heads 0 if coin is tails Notice that before the flip of the coin x can be either 1 or 0 Only after the uncertainty is resolved ie after the coin is flipped do we know what the value of x is22 2122 Discrete and continuous random variables The outcomes from a random experiment may be either a finite number of possibilities or a continuum of possibilities For example recording the number that comes up on a single die is a random variable with six outcomes With two dice we could either record the sum of the faces in which case there are 12 outcomes some of which are more likely than others or we could record a twodigit number one for the value of each die in which case there would be 36 equally likely outcomes These are examples of discrete random variables Alternatively a continuous random variable may take on any value in a given range of real numbers For example we could view the outdoor temperature tomorrow as a 22Sometimes random variables are denoted by x to make a distinction between variables whose outcome is subject to random chance and nonrandom algebraic variables This notational device can be useful for keeping track of what is random and what is not in a particular problem and we will use it in some cases When there is no ambiguity however we will not use this special notation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 68 Part 1 Introduction continuous variable assuming temperatures can be measured finely ranging from say 250C to 150C Of course some of these temperatures would be unlikely to occur but in principle the precisely measured temperature could be anywhere between these two bounds Similarly we could view tomorrows percentage change in the value of a particular stock index as taking on all values between 2100 and say 11000 Again of course percentage changes around 0 would be considerably more likely to occur than would the extreme values 2123 Probability density functions For any random variable its probability density function PDF shows the probability that each specific outcome will occur For a discrete random variable defining such a function poses no particular difficulties In the coin flip case for example the PDF denoted by f1x2 would be given by f1x 5 12 5 05 f1x 5 02 5 05 2152 For the roll of a single die the PDF would be f1x 5 12 5 16 f1x 5 22 5 16 f1x 5 32 5 16 f1x 5 42 5 16 f1x 5 52 5 16 f1x 5 62 5 16 2153 Notice that in both these cases the probabilities specified by the PDF sum to 10 This is because by definition one of the outcomes of the random experiment must occur More generally if we denote all the outcomes for a discrete random variable by xi for i 5 1 c n then we must have a n i51 f 1xi2 5 1 2154 For a continuous random variable we must be careful in defining the PDF concept Because such a random variable takes on a continuum of values if we were to assign any nonzero value as the probability for a specific outcome ie a temperature of 12553470C we could quickly have sums of probabilities that are infinitely large Hence for a continuous random variable we define the PDF f 1x2 as a function with the property that the proba bility that x falls in a particular small interval dx is given by the area of f 1x2dx Using this convention the property that the probabilities from a random experiment must sum to 10 is stated as follows 3 1q 2q f1x2 dx 5 10 2155 2124 A few important PDFs Most any function will do as a PDF provided that f 1x2 0 and the function sums or integrates to 10 The trick of course is to find functions that mirror random experiments that occur in the real world Here we look at four such functions that we will find useful in various places in this book Graphs for all four of these functions are shown in Figure 26 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 69 1 Binomial distribution This is the most basic discrete distribution Usually x is assumed to take on only two values 1 and 0 The PDF for the binomial is given by f 1x 5 12 5 p f 1x 5 02 5 1 2 p 2156 where 0 p 1 The coin flip example is obviously a special case of the binomial where p 5 05 2 Uniform distribution This is the simplest continuous PDF It assumes that the possible val ues of the variable x occur in a defined interval and that each value is equally likely That is f1x2 5 1 b 2 a for a x b 2157 f1x2 5 0 for x a or x b Notice that here the probabilities integrate to 10 3 1q 2q f1x2 dx 5 3 b a 1 b 2 adx 5 x b 2 a b a 5 b b 2 a 2 a b 2 a 5 b 2 a b 2 a 5 10 2158 3 Exponential distribution This is a continuous distribution for which the probabilities decrease at a smooth exponential rate as x increases Formally f1x2 5 eλe2λx if x 0 0 if x 0 2159 where λ is a positive constant Again it is easy to show that this function integrates to 10 3 1q 2q f1x2 dx 5 3 q 0 λe2λxdx 5 2e2λx q 0 5 0 2 1212 5 10 2160 4 Normal distribution The Normal or Gaussian distribution is the most important in mathematical statistics Its importance stems largely from the central limit theo rem which states that the distribution of any sum of independent random variables will increasingly approximate the Normal distribution as the number of such variables increases Because sample averages can be regarded as sums of independent random variables this theorem says that any sample average will have a Normal distribution no matter what the distribution of the population from which the sample is selected Hence it may often be appropriate to assume a random variable has a Normal distribution if it can be thought of as some sort of average The mathematical form for the Normal PDF is f1x2 5 1 2π e2x22 2161 and this is defined for all real values of x Although the function may look complicated a few of its properties can be easily described First the function is symmetric around zero because of the x2 term Second the function is asymptotic to zero as x becomes large or small Third the function reaches its maximal value at x 5 0 This value is 12π 04 Finally the graph of this function has a general bell shape a shape used throughout the study of statistics Integration of this function is relatively tricky although easy in polar coordinates The presence of the constant 12π is needed if the function is to integrate to 10 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 70 Part 1 Introduction 2125 Expected value The expected value of a random variable is the numerical value that the random variable might be expected to have on average23 It is the center of gravity of the PDF For a discrete random variable that takes on the values x1 x2 c xn the expected value is defined as E1x2 5 a n i51 xi f 1xi2 2162 That is each outcome is weighted by the probability that it will occur and the result is summed over all possible outcomes For a continuous random variable Equation 2162 is readily generalized as E1x2 5 3 1q 2q xf 1x2 dx 2163 Again in this integration each value of x is weighted by the probability that this value will occur The concept of expected value can be generalized to include the expected value of any function of a random variable say g 1x2 In the continuous case for example we would write E3g 1x2 4 5 3 1q 2q g 1x2f 1x2 dx 2164 As a special case consider a linear function y 5 ax 1 b Then E1y2 5 E1ax 1 b2 5 3 1q 2q 1ax 1 b2f1x2 dx 5 a 3 1q 2q xf1x2 dx 1 b 3 1q 2q f1x2 dx 5 aE1x2 1 b 2165 Sometimes expected values are phrased in terms of the cumulative distribution function CDF F1x2 defined as F1x2 5 3 x 2q f 1t2 dt 2166 That is F1x2 represents the probability that the random variable t is less than or equal to x Using this notation the expected value of x can be written as E1x2 5 3 1q 2q xdF1x2 2167 Because of the fundamental theorem of calculus Equation 2167 and Equation 2163 mean exactly the same thing 23The expected value of a random variable is sometimes referred to as the mean of that variable In the study of sampling this can sometimes lead to confusion between the expected value of a random variable and the separate concept of the sample arithmetic average Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 71 EXAMPLE 215 Expected Values of a Few Random Variables The expected values of each of the random variables with the simple PDFs introduced earlier are easy to calculate All these expected values are indicated on the graphs of the functions PDFs in Figure 26 1 Binomial In this case E 1x2 5 1 f 1x 5 12 1 0 f 1x 5 02 5 1 p 1 0 11 2 p2 5 p 2168 Random variables that have these PDFs are widely used Each graph indicates the expected value of the PDF shown fx fx Ex Ex x fx 1 0 a Binomial c Exponential b Uniform Ex p a b 1 x x b a a b 2 λ 1λ 1 fx 1 2π Ex 0 d Normal x 1 p p FIGURE 26 Four Common Probability Density Functions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 72 Part 1 Introduction 2126 Variance and standard deviation The expected value of a random variable is a measure of central tendency On the other hand the variance of a random variable denoted by σ2 x or Var 1x2 is a measure of disper sion Specifically the variance is defined as the expected squared deviation of a random variable from its expected value Formally Var1x2 5 σ2 x 5 E3 1x 2 E1x2 2 24 5 3 1q 2q 1x 2 E1x2 2 2f1x2 dx 2172 Somewhat imprecisely the variance measures the typical squared deviation from the central value of a random variable In making the calculation deviations from the expected value are squared so that positive and negative deviations from the expected value will both contribute to this measure of dispersion After the calculation is made the squar ing process can be reversed to yield a measure of dispersion that is in the original units in which the random variable was measured This square root of the variance is called the standard deviation and is denoted as σx15σ2 x2 The wording of the term effectively For the coin flip case where p 5 05 this says that E1x2 5 p 5 05the expected value of this random variable is as you might have guessed one half 2 Uniform For this continuous random variable E 1x2 5 3 b a x b 2 adx 5 x2 2 1b 2 a2 b a 5 b2 2 1b 2 a2 2 a2 2 1b 2 a2 5 b 1 a 2 2169 Again as you might have guessed the expected value of the uniform distribution is precisely halfway between a and b 3 Exponential For this case of declining probabilities E 1x2 5 3 q 0 xλe2λxdx 5 2xe2λx 2 1 λe2λx q 0 5 1 λ 2170 where the integration follows from the integration by parts example shown earlier in this chapter Notice here that the faster the probabilities decline the lower is the expected value of x For example if λ 5 05 then E 1x2 5 2 whereas if λ 5 005 then E 1x2 5 20 4 Normal Because the Normal PDF is symmetric around zero it seems clear that E1x2 5 0 A formal proof uses a change of variable integration by letting u 5 x22 1du 5 xdx2 3 1q 2q 1 2π xe2x22dx 5 1 2π 3 1q 2q e2udu 5 1 2π 32e2x224 1q 2q 5 1 2π 30 2 04 5 0 2171 Of course the expected value of a normally distributed random variable or of any random variable may be altered by a linear transformation as shown in Equation 2165 QUERY A linear transformation changes a random variables expected value in a predictable wayif y 5 ax 1 b then E 1y2 5 aE 1x2 1 b Hence for this transformation say h 1x2 we have E 3h 1x2 4 5 h 3E 1x2 4 Suppose instead that x were transformed by a concave function say gx with gr 0 and gs 0 How would E 3g1x2 4 compare with g3E 1x24 NOTE This is an illustration of Jensens inequality a concept we will pursue in detail in Chapter 7 See also Problem 214 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 73 conveys its meaning σx is indeed the typical standard deviation of a random variable from its expected value When a random variable is subject to a linear transformation its variance and standard deviation will be changed in a fairly obvious way If y 5 ax 1 b then σ2 y 5 3 1q 2q 3ax 1 b 2 E1ax 1 b2 4 2f 1x2 dx 5 3 1q 2q a2 3x 2 E1x2 4 2f 1x2dx 5 a2σ2 x 2173 Hence addition of a constant to a random variable does not change its variance whereas multiplication by a constant multiplies the variance by the square of the constant There fore it is clear that multiplying a variable by a constant multiplies its standard deviation by that constant σax 5 aσx EXAMPLE 216 Variances and Standard Deviations for Simple Random Variables Knowing the variances and standard deviations of the four simple random variables we have been looking at can sometimes be useful in economic applications 1 Binomial The variance of the binomial can be calculated by applying the definition in its discrete analog σ2 x 5 a n i51 1xi 2 E 1x22 2f1xi2 5 11 2 p2 2 p 1 10 2 p2 2 11 2 p2 5 11 2 p2 1p 2 p2 1 p22 5 p11 2 p2 2174 Hence σx 5 p11 2 p2 One implication of this result is that a binomial variable has the largest variance and standard deviation when p 5 05 in which case σ2 x 5 025 and σx 5 05 Because of the relatively flat parabolic shape of p11 2 p2 modest deviations of p from 05 do not change this variance substantially 2 Uniform Calculating the variance of the uniform distribution yields a mildly interesting result σ2 x 5 3 b a ax 2 a 1 b 2 b 2 1 b 2 adx 5 ax 2 a 1 b 2 b 3 1 3 1b 2 a2 b a 5 1 3 1b 2 a2 c 1b 2 a2 3 8 2 1a 2 b2 3 8 d 5 1b 2 a2 2 12 2175 This is one of the few places where the number 12 has any use in mathematics other than in measuring quantities of oranges or doughnuts 3 Exponential Integrating the variance formula for the exponential is relatively labori ous Fortunately the result is simple for the exponential it turns out that σ2 x 5 1λ2 and σx 5 1λ Hence the expected value and standard deviation are the same for the exponential distributionit is a oneparameter distribution 4 Normal The integration can also be burdensome in this case But again the result is sim ple For the Normal distribution σ2 x 5 σx 5 1 Areas below the Normal curve can be read ily calculated and tables of these are available in any statistics text Two useful facts about the Normal PDF are 3 11 21 f1x2 dx 068 and 3 12 22 f1x2 dx 095 2176 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 74 Part 1 Introduction That is the probability is approximately two thirds that a Normal variable will be within 61 standard deviation of the expected value and most of the time ie with probability 095 it will be within 62 standard deviations Standardizing the Normal If the random variable x has a standard Normal PDF it will have an expected value of 0 and a standard deviation of 1 However a simple linear transformation can be used to give this random variable any desired expected value μ and standard deviation σ Consider the transformation y 5 σx 1 μ Now E 1y2 5 σE 1x2 1 μ 5 μ and Var1y2 5 σ2 y 5 σ2Var1x2 5 σ2 2177 Reversing this process can be used to standardize any Normally distributed random variable y with an arbitrary expected value μ and standard deviation σ this is sometimes denoted as y N1μ σ2 by using z 5 1y 2 μ2σ For example SAT scores y are distributed Normally with an expected value of 500 points and a standard deviation of 100 points ie y N1500 1002 Hence z 5 1y 2 5002100 has a standard Normal distribution with expected value 0 and stan dard deviation 1 Equation 2176 shows that approximately 68 percent of all scores lie between 400 and 600 points and 95 percent of all scores lie between 300 and 700 points QUERY Suppose that the random variable x is distributed uniformly along the interval 0 12 What are the mean and standard deviation of x What fraction of the x distribution is within 61 standard deviation of the mean What fraction of the distribution is within 62 standard devia tions of the expected value Explain why this differs from the fractions computed for the Normal distribution 2127 Covariance Some economic problems involve two or more random variables For example an investor may consider allocating his or her wealth among several assets the returns on which are taken to be random Although the concepts of expected value variance and so forth carry over more or less directly when looking at a single random variable in such cases it is also necessary to consider the relationship between the variables to get a complete picture The concept of covariance is used to quantify this relationship Before providing a definition however we will need to develop some background Consider a case with two continuous random variables x and y The PDF for these two variables denoted by f 1x y2 has the property that the probability associated with a set of outcomes in a small area with dimensions dxdy is given by f 1x y2dxdy To be a proper PDF it must be the case that f 1x y2 0 and 3 1q 2q 3 1q 2q f1x y2 dx dy 5 1 2178 The singlevariable measures we have already introduced can be developed in this twovariable context by integrating out the other variable That is E1x2 5 3 1q 2q 3 1q 2q xf1x y2 dy dx and Var1x2 5 3 1q 2q 3 1q 2q 3x 2 E1x2 4 2f 1x y2 dy dx 2179 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 75 In this way the parameters describing the random variable x are measured over all possible outcomes for y after taking into account the likelihood of those various outcomes In this context the covariance between x and y seeks to measure the direction of asso ciation between the variables Specifically the covariance between x and y denoted as Cov1x y2 is defined as Cov1x y2 5 3 1q 2q 3 1q 2q 3x 2 E1x2 4 3y 2 E1y2 4 f1x y2 dx dy 2180 The covariance between two random variables may be positive negative or zero If values of x that are greater than E1x2 tend to occur relatively frequently with values of y that are greater than E1y2 and similarly if low values of x tend to occur together with low values of y then the covariance will be positive In this case values of x and y tend to move in the same direction Alternatively if high values of x tend to be associated with low values for y and vice versa the covariance will be negative Two random variables are defined to be independent if the probability of any par ticular value of say x is not affected by the particular value of y that might occur and vice versa24 In mathematical terms this means that the PDF must have the property that f1x y2 5 g 1x2h 1y2that is the joint PDF can be expressed as the product of two singlevariable PDFs If x and y are independent their covariance will be zero Cov1x y2 5 3 1q 2q 3 1q 2q 3x 2 E1x2 4 3y 2 E1y2 4g 1x2h 1y2 dx dy 5 3 1q 2q 3x 2 E1x2 4g 1x2 dx 3 1q 2q 3y 2 E1y2 4h 1y2 dy 5 0 0 5 0 2181 The converse of this statement is not necessarily true however A zero covariance does not necessarily imply statistical independence Finally the covariance concept is crucial for understanding the variance of sums or dif ferences of random variables Although the expected value of a sum of two random vari ables is as one might guess the sum of their expected values E1x 1 y2 5 3 1q 2q 3 1q 2q 1x 1 y2f1x y2 dx dy 5 3 1q 2q xf1x y2 dy dx 1 3 1q 2q yf1x y2 dx dy 5 E1x2 1 E1y2 2182 the relationship for the variance of such a sum is more complicated Using the definitions we have developed yields 24A formal definition relies on the concept of conditional probability The conditional probability of an event B given that A has occurred written P 1B0A2 is defined as P 1B0A2 5 P 1A and B2P 1A2 B and A are defined to be independent if P 1B0A2 5 P 1B2 In this case P 1A and B2 5 P 1A2 P 1B2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 76 Part 1 Introduction Var1x 1 y2 5 3 1q 2q 3 1q 2q 3x 1 y 2 E1x 1 y2 4 2f1x y2 dx dy 5 3 1q 2q 3 1q 2q 3x 2 E1x2 1 y 2 E1y2 4 2f 1x y2 dx dy 5 3 1q 2q 3 1q 2q 3x 2 E1x2 4 2 1 3 y 2 E1 y2 4 2 1 2 3x 2 E1x2 4 3y 2 E1y2 4 f 1x y2 dx dy 5 Var1x2 1 Var 1y2 1 2Cov1x y2 2183 Hence if x and y are independent then Var 1x 1 y2 5 Var 1x2 1 Var 1y2 The variance of the sum will be greater than the sum of the variances if the two random variables have a positive covariance and will be less than the sum of the variances if they have a negative covariance Problems 214216 provide further details on some of the statistical results that are used in microeconomic theory Summary Despite the formidable appearance of some parts of this chap ter this is not a book on mathematics Rather the intention here was to gather together a variety of tools that will be used to develop economic models throughout the remainder of the text Material in this chapter will then be useful as a handy reference One way to summarize the mathematical tools introduced in this chapter is by stressing again the economic lessons that these tools illustrate Using mathematics provides a convenient shorthand way for economists to develop their models Implications of various economic assumptions can be studied in a simplified setting through the use of such mathematical tools The mathematical concept of the derivatives of a func tion is widely used in economic models because econ omists are often interested in how marginal changes in one variable affect another variable Partial derivatives are especially useful for this purpose because they are defined to represent such marginal changes when all other factors are held constant The mathematics of optimization is an important tool for the development of models that assume that economic agents rationally pursue some goal In the unconstrained case the firstorder conditions state that any activity that contributes to the agents goal should be expanded up to the point at which the marginal contribution of further expansion is zero In mathematical terms the firstorder condition for an optimum requires that all partial deriv atives be zero Most economic optimization problems involve con straints on the choices agents can make In this case the firstorder conditions for a maximum suggest that each activity be operated at a level at which the ratio of the marginal benefit of the activity to its marginal cost is the same for all activities actually used This common marginal benefitmarginal cost ratio is also equal to the Lagrange multiplier which is often introduced to help solve constrained optimization problems The Lagrange multiplier can also be interpreted as the implicit value or shadow price of the constraint The implicit function theorem is a useful mathematical device for illustrating the dependence of the choices that result from an optimization problem on the parameters of that problem eg market prices The envelope the orem is useful for examining how these optimal choices change when the problems parameters prices change Some optimization problems may involve constraints that are inequalities rather than equalities Solutions to these problems often illustrate complementary slack ness That is either the constraints hold with equality and their related Lagrange multipliers are nonzero or the constraints are strict inequalities and their related Lagrange multipliers are zero Again this illustrates how the Lagrange multiplier implies something about the importance of constraints Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 77 The firstorder conditions shown in this chapter are only the necessary conditions for a local maximum or mini mum One must also check secondorder conditions that require that certain curvature conditions be met Certain types of functions occur in many economic problems Quasiconcave functions those functions for which the level curves form convex sets obey the secondorder conditions of constrained maximum or minimum problems when the constraints are linear Homothetic functions have the useful property that implicit tradeoffs among the variables of the function depend only on the ratios of these variables Integral calculus is often used in economics both as a way of describing areas below graphs and as a way of sum ming results over time Techniques that involve various ways of differentiating integrals play an important role in the theory of optimizing behavior Many economic problems are dynamic in that decisions at one date affect decisions and outcomes at later dates The mathematics for solving such dynamic optimization problems is often a straightforward generalization of Lagrangian methods Concepts from mathematical statistics are often used in studying the economics of uncertainty and information The most fundamental concept is the notion of a random variable and its associated PDF Parameters of this dis tribution such as its expected value or its variance also play important roles in many economic models Problems 21 Suppose f 1x y2 5 4x2 1 3y2 a Calculate the partial derivatives of f b Suppose f 1x y2 5 16 Use the implicit function theorem to calculate dydx c What is the value of dydx if x 5 1 y 5 2 d Graph your results and use it to interpret the results in parts b and c of this problem 22 Suppose a firms total revenues depend on the amount pro duced q according to the function R 5 70q 2 q2 Total costs also depend on q C 5 q2 1 30q 1 100 a What level of output should the firm produce to maxi mize profits 1R 2 C2 What will profits be b Show that the secondorder conditions for a maximum are satisfied at the output level found in part a c Does the solution calculated here obey the marginal revenue equals marginal cost rule Explain 23 Suppose that f1x y2 5 xy Find the maximum value for f if x and y are constrained to sum to 1 Solve this problem in two ways by substitution and by using the Lagrange multiplier method 24 The dual problem to the one described in Problem 23 is minimize x 1 y subject to xy 5 025 Solve this problem using the Lagrangian technique Then compare the value you get for the Lagrange multiplier with the value you got in Problem 23 Explain the relationship between the two solutions 25 The height of a ball that is thrown straight up with a certain force is a function of the time t from which it is released given by f 1t2 5 205gt2 1 40t where g is a constant deter mined by gravity a How does the value of t at which the height of the ball is at a maximum depend on the parameter g b Use your answer to part a to describe how maximum height changes as the parameter g changes c Use the envelope theorem to answer part b directly d On the Earth g 5 32 but this value varies somewhat around the globe If two locations had gravitational con stants that differed by 01 what would be the difference in the maximum height of a ball tossed in the two places 26 A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3x feet long Now cut a smaller square that is t feet on Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 78 Part 1 Introduction a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a traylike structure with no top a Show that the volume of oil that can be held by this tray is given by V 5 t1x 2 2t2 13x 2 2t2 5 3tx2 2 8t2x 1 4t3 b How should t be chosen to maximize V for any given value of x c Is there a value of x that maximizes the volume of oil that can be carried d Suppose that a shipbuilder is constrained to use only 1000000 square feet of steel sheet to construct an oil tanker This constraint can be represented by the equa tion 3x2 2 4t2 5 1000000 because the builder can return the cutout squares for credit How does the solution to this constrained maximum problem com pare with the solutions described in parts b and c 27 Consider the following constrained maximization problem maximize y 5 x1 1 5 ln x2 subject to k 2 x1 2 x2 5 0 where k is a constant that can be assigned any specific value a Show that if k 5 10 this problem can be solved as one involving only equality constraints b Show that solving this problem for k 5 4 requires that x1 5 21 c If the xs in this problem must be nonnegative what is the optimal solution when k 5 4 This problem may be solved either intuitively or using the methods outlined in the chapter d What is the solution for this problem when k 5 20 What do you conclude by comparing this solution with the solution for part a Note This problem involves what is called a quasilinear function Such functions provide important examples of some types of behavior in consumer theoryas we shall see 28 Suppose that a firm has a marginal cost function given by MC1q2 5 q 1 1 a What is this firms total cost function Explain why total costs are known only up to a constant of integration which represents fixed costs b As you may know from an earlier economics course if a firm takes price p as given in its decisions then it will produce that output for which p 5 MC1q2 If the firm follows this profitmaximizing rule how much will it produce when p 5 15 Assuming that the firm is just breaking even at this price what are fixed costs c How much will profits for this firm increase if price increases to 20 d Show that if we continue to assume profit maximization then this firms profits can be expressed solely as a func tion of the price it receives for its output e Show that the increase in profits from p 5 15 to p 5 20 can be calculated in two ways i directly from the equa tion derived in part d and ii by integrating the inverse marginal cost function 3MC21 1p2 5 p 2 14 from p 5 15 to p 5 20 Explain this result intuitively using the envelope theorem Analytical Problems 29 Concave and quasiconcave functions Show that if f1x1 x22 is a concave function then it is also a quasiconcave function Do this by comparing Equation 2100 defining quasiconcavity with Equation 284 defining con cavity Can you give an intuitive reason for this result Is the converse of the statement true Are quasiconcave functions necessarily concave If not give a counterexample 210 The CobbDouglas function One of the most important functions we will encounter in this book is the CobbDouglas function y 5 1x12 α 1x22 β where α and β are positive constants that are each less than 1 a Show that this function is quasiconcave using a brute force method by applying Equation 2100 b Show that the CobbDouglas function is quasi concave by showing that any contour line of the form y 5 c where c is any positive constant is convex and there fore that the set of points for which y c is a convex set c Show that if α 1 β 1 then the CobbDouglas func tion is not concave thereby illustrating again that not all quasiconcave functions are concave Note The CobbDouglas function is discussed further in the Extensions to this chapter 211 The power function Another function we will encounter often in this book is the power function y 5 xδ where 0 δ 1 at times we will also examine this function for cases where δ can be negative too in which case we will use the form y 5 xδδ to ensure that the derivatives have the proper sign Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 79 a Show that this function is concave and therefore also by the result of Problem 29 quasiconcave Notice that the δ 5 1 is a special case and that the function is strictly concave only for δ 1 b Show that the multivariate form of the power function y 5 f1x1 x22 5 1x12 δ 1 1x22 δ is also concave and quasiconcave Explain why in this case the fact that f12 5 f21 5 0 makes the determination of concavity especially simple c One way to incorporate scale effects into the func tion described in part b is to use the monotonic transformation g 1x1 x22 5 yγ 5 3 1x12 δ 1 1x22 δ4 γ where g is a positive constant Does this transfor mation preserve the concavity of the function Is g quasiconcave 212 Proof of the envelope theorem in constrained optimization problems Because we use the envelope theorem in constrained opti mization problems often in the text proving this theorem in a simple case may help develop some intuition Thus sup pose we wish to maximize a function of two variables and that the value of this function also depends on a parameter a f1x1 x2 a2 This maximization problem is subject to a con straint that can be written as g1x1 x2 a2 5 0 a Write out the Lagrangian expression and the firstorder conditions for this problem b Sum the two firstorder conditions involving the xs c Now differentiate the above sum with respect to athis shows how the xs must change as a changes while requir ing that the firstorder conditions continue to hold d As we showed in the chapter both the objective function and the constraint in this problem can be stated as func tions of a f1x1 1a2 x2 1a2 a2 g 1x1 1a2 x2 1a2 a2 5 0 Dif ferentiate the first of these with respect to a This shows how the value of the objective changes as a changes while keeping the xs at their optimal values You should have terms that involve the xs and a single term in fa e Now differentiate the constraint as formulated in part d with respect to a You should have terms in the xs and a single term in ga f Multiply the results from part e by λ the Lagrange multiplier and use this together with the firstorder conditions from part c to substitute into the derivative from part d You should be able to show that df1x1 1a2 x2 1a2 a2 da 5 f a 1 λ g a which is just the partial derivative of the Lagrangian expression when all the xs are at their optimal values This proves the envelope theorem Explain intuitively how the various parts of this proof impose the condition that the xs are constantly being adjusted to be at their optimal values g Return to Example 28 and explain how the envelope the orem can be applied to changes in the fence perimeter Pthat is how do changes in P affect the size of the area that can be fenced Show that in this case the envelope theorem illustrates how the Lagrange multiplier puts a value on the constraint 213 Taylor approximations Taylors theorem shows that any function can be approxi mated in the vicinity of any convenient point by a series of terms involving the function and its derivatives Here we look at some applications of the theorem for functions of one and two variables a Any continuous and differentiable function of a single variable f1x2 can be approximated near the point a by the formula f1x2 5 f1a2 1 fr 1a2 1x 2 a2 1 05fs 1a2 1x 2 a22 1 1 terms in f t f tr c Using only the first three of these terms results in a quadratic Taylor approximation Use this approximation together with the definition of concavity to show that any concave function must lie on or below the tangent to the function at point a b The quadratic Taylor approximation for any function of two variables f1x y2 near the point a b is given by f1x y2 5 f1a b2 1 f1 1a b2 1x 2 a2 1 f2 1a b2 1y 2 b2 1 05 3 f11 1a b2 1x 2 a2 2 1 2f12 1a b2 1x 2 a2 1y 2 b2 1 f22 1y 2 b2 24 Use this approximation to show that any concave function as defined by Equation 284 must lie on or below its tangent plane at a b 214 More on expected value Because the expected value concept plays an important role in many economic theories it may be useful to summarize a few more properties of this statistical measure Throughout this problem x is assumed to be a continuous random variable with PDF f 1x2 a Jensens inequality Suppose that g 1x2 is a concave func tion Show that E 3g 1x24 g 3E 1x24 Hint Construct the tangent to g 1x2 at the point E 1x2 This tangent will have the form c 1 dx g 1x2 for all values of x and c 1 dE 1x2 5 g 3E 1x2 4 where c and d are constants b Use the procedure from part a to show that if g 1x2 is a convex function then E 3g 1x24 g 3E 1x2 4 c Suppose x takes on only nonnegative valuesthat is 0 x q Use integration by parts to show that E 1x2 5 3 q 0 31 2 F1x24dx Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 80 Part 1 Introduction where F1x2 is the cumulative distribution function for x ie F1x2 5 e x 0f1t2dt d Markovs inequality Show that if x takes on only posi tive values then the following inequality holds P 1x t2 E 1x2 t Hint E 1x2 5 e q 0 xf 1x2 dx 5 e t 0xf 1x2 dx 1 e q t xf 1x2 dx e Consider the PDF f1x2 5 2x23 for x 1 1 Show that this is a proper PDF 2 Calculate F1x2 for this PDF 3 Use the results of part c to calculate E 1x2 for this PDF 4 Show that Markovs inequality holds for this function f The concept of conditional expected value is useful in some economic problems We denote the expected value of x conditional on the occurrence of some event A as E 1x0A2 To compute this value we need to know the PDF for x given that A has occurred denoted by f1x0A2 With this notation E 1x0A2 5 e 1q 2qxf 1x0A2dx Perhaps the easiest way to understand these relationships is with an example Let f 1x2 5 x2 3 for 21 x 2 1 Show that this is a proper PDF 2 Calculate E 1x2 3 Calculate the probability that 21 x 0 4 Consider the event 0 x 2 and call this event A What is f 1x0A2 5 Calculate E 1x0A2 6 Explain your results intuitively 215 More on variances The definition of the variance of a random variable can be used to show a number of additional results a Show that Var1x2 5 E 1x22 2 3E 1x2 4 2 b Use Markovs inequality Problem 214d to show that if x can take on only nonnegative values P 3 1x 2 μx2 k4 σ2 x k2 This result shows that there are limits on how often a random variable can be far from its expected value If k 5 hσ this result also says that P 3 1x 2 μx2 hσ4 1 h2 Therefore for example the probability that a random variable can be more than 2 standard deviations from its expected value is always less than 025 The theoretical result is called Chebyshevs inequality c Equation 2182 showed that if 2 or more random variables are independent the variance of their sum is equal to the sum of their variances Use this result to show that the sum of n independent random variables each of which has expected value m and variance σ2 has expected value nμ and variance nσ2 Show also that the average of these n random variables which is also a ran dom variable will have expected value m and variance σ2n This is sometimes called the law of large numbers that is the variance of an average shrinks down as more independent variables are included d Use the result from part c to show that if x1 and x2 are independent random variables each with the same expected value and variance the variance of a weighted average of the two X 5 kx1 1 11 2 k2x2 0 k 1 is minimized when k 5 05 How much is the variance of this sum reduced by setting k properly relative to other possible values of k e How would the result from part d change if the two variables had unequal variances 216 More on covariances Here are a few useful relationships related to the covariance of two random variables x1 and x2 a S h o w t h a t Cov1x1 x22 5 E 1x1x22 2 E 1x12E 1x22 An important implication of this is that if Cov1x1 x22 5 0 E 1x1x22 5 E 1x12E 1x22 That is the expected value of a product of two random variables is the product of these variables expected values b Show that Var1ax1 1 bx22 5 a2Var1x12 1 b2Var1x22 1 2abCov1x1 x22 c In Problem 215d we looked at the variance of X 5 kx1 1 11 2 k2x2 0 k 1 Is the conclusion that this variance is minimized for k 5 05 changed by con sidering cases where Cov1x1 x22 2 0 d The correlation coefficient between two random variables is defined as Corr1x1 x22 5 Cov1x1 x22 Var1x12Var1x22 Explain why 21 Corr1x1 x22 1 and provide some intuition for this result e Suppose that the random variable y is related to the random variable x by the linear equation y 5 α 1 βx Show that β 5 Cov1y x2 Var1x2 Here β is sometimes called the theoretical regression coefficient of y on x With actual data the sample analog of this expression is the ordinary least squares OLS regression coefficient Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 81 Suggestions for Further Reading Dadkhan Kamran Foundations of Mathematical and Com putational Economics Mason OH ThomsonSouthWestern 2007 This is a good introduction to many calculus techniques The book shows how many mathematical questions can be approached using popular software programs such as Matlab or Excel Dixit A K Optimization in Economic Theory 2nd ed New York Oxford University Press 1990 A complete and modern treatment of optimization techniques Uses relatively advanced analytical methods Hoy Michael John Livernois Chris McKenna Ray Rees and Thanasis Stengos Mathematics for Economists 2nd ed Cam bridge MA MIT Press 2001 A complete introduction to most of the mathematics covered in microeconomics courses The strength of the book is its presenta tion of many workedout examples most of which are based on microeconomic theory Luenberger David G Microeconomic Theory New York McGraw Hill Inc 1995 This is an advanced text with a variety of novel microeconomic concepts The book also has five brief but useful mathematical appendices MasColell Andreu Michael D Whinston and Jerry R Green Microeconomic Theory New York Oxford University Press 1995 Encyclopedic treatment of mathematical microeconomics Exten sive mathematical appendices cover relatively highlevel topics in analysis Samuelson Paul A Foundations of Economic Analysis Cam bridge MA Harvard University Press 1947 Mathematical Appendix A A basic reference Mathematical Appendix A provides an advanced treatment of necessary and sufficient conditions for a maximum Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 A mathematical microeconomics text that stresses the observable predictions of economic theory The text makes extensive use of the envelope theorem Simon Carl P and Lawrence Blume Mathematics for Econo mists New York W W Norton 1994 A useful text covering most areas of mathematics relevant to econ omists Treatment is at a relatively high level Two topics discussed better here than elsewhere are differential equations and basic pointset topology Sydsaeter K A Strom and P Berck Economists Mathemat ical Manual 4th ed Berlin Germany SpringerVerlag 2005 An indispensable tool for mathematical review Contains 35 chap ters covering most of the mathematical tools that economists use Discussions are brief so this is not the place to encounter new con cepts for the first time Taylor Angus E and W Robert Mann Advanced Calculus 3rd ed New York John Wiley 1983 pp 18395 A comprehensive calculus text with a good discussion of the Lagrangian technique Thomas George B and Ross L Finney Calculus and Analytic Geometry 8th ed Reading MA AddisonWesley 1992 Basic calculus text with excellent coverage of differentiation techniques Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 7 See Poet i ia a ea a Se y 2 a LU Many mathematical results can be written in compact ways by first leading principal minor is a and the second is using matrix algebra In this Extension we look briefly at that A Ay Ay4 Ay notation We return to this notation at a few other places in 6 Ann X n square matrix A is positive definite if all its later chapters leading principal minors are positive The matrix is nega tive definite if its principal minors alternate in sign start Matrix algebra background ing aa ving The th seats presente here assume some genre familiar 7 A particularly useful symmetric matrix is the Hessian ity wit matrix algebra A succinct reminder of these princi matrix formed by all the secondorder partial derivatives ples might include of a function If fis a continuous and twice differentiable 1 Ann X k matrix A is a rectangular array of terms of the function of n variables then its Hessian is given by form fr fio 0 fin a a Le nu 42 1k Hf fa fin tt fin Gy Gyn f A a Jn San Sun Any On2 te Onk E21 Concave and convex functions Here i 1nj 1k Matrices can be added sub tracted or multiplied providing their dimensions are A concave fi unction is one that is always below or on any tan conformable gent to it Alternatively a convex function is always above or on any tangent The concavity or convexity of any function is 2 Ifn k then A is a square matrix A square matrix is y tangen yO y y a determined by its second derivatives For a function of a sin symmetric if a a The identity matrix Iisann X n J 4 pe pe gle variable fx the requirement is straightforward Using square matrix where a lifi janda Oifi j ae J the Taylor approximation at any point x 3 The determinant of a square matrix denoted by A is a scalar ie a single term found by suitably multiplying dx Xo dx fx f xodx f x9 together all the terms in the matrix If Ais 2 X 2 Fl flo F f 2 IA aay anap higherorder terms 1 3 Assuming that the higherorder terms are 0 we have Example A then P 5 Flay dx flo f xdx ww Al 215 18 if fs9 Oand f xo dx fx f xodx 4 The inverse of an a X n square matrix A is another ig f xo 0 Because the expressions on the right of these n X n matrix A such that inequalities are in fact the equation of the tangent to the func AAT1I tion at xp it is clear that the function is locally concave if f xo 0 and locally convex if f xp 0 Not ever y square matrix has an inverse A necessary Extending this intuitive idea to many dimensions is cum and sufficient condition for the existence of A is that bersome in terms of functional notation but relatively simple Al 0 when matrix algebra is used Concavity requires that the Hessian 5 The leading principal minors of ann X n square matrix A matrix be negative definite whereas convexity requires that this are the series of determinants of the first p rows and matrix be positive definite As in the single variable case these columns of A where p 1 n If A is 2 X 2 then the If some of the determinants in this definition are 0 then the matrix is said to be 82 oe oan a positive semidefinite or negative semidefinite Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 83 conditions amount to requiring that the function move consis tently away from any tangent to it no matter what direction is taken2 If f1x1 x22 is a function of two variables the Hessian is given by H 5 c f11 f12 f21 f22 d This is negative definite if f11 0 and f11 f12 2 f21 f12 0 which is precisely the condition described in Equation 284 Generalizations to functions of three or more variables follow the same matrix pattern Example 1 For the health status function in Example 26 the Hessian is given by H 5 c22 0 0 22d and the first and second leading principal minors are H1 5 22 0 and H2 5 1222 1222 2 0 5 4 0 Hence the function is concave Example 2 The CobbDouglas function xayb where a b 10 12 is used to illustrate utility functions and production functions in many places in this text The first and secondorder deriva tives of the function are fx 5 axa21yb fy 5 bxayb21 fxx 5 a 1a 2 12xa22yb fyy 5 b1b 2 12xayb22 Hence the Hessian for this function is H 5 ca 1a 2 12xa22yb abxa21yb21 abxa21yb21 b1b 2 12xayb22d The first leading principal minor of this Hessian is H1 5 a 1a 2 12xa22yb 0 and so the function will be concave providing H2 5 a 1a 2 12 1b2 1b 2 12x2a22y2b22 2 a2b2x2a22y2b22 5 ab11 2 a 2 b2x2a22y2b22 0 This condition clearly holds if a 1 b 1 That is in pro duction function terminology the function must exhibit diminishing returns to scale to be concave Geometrically the function must turn downward as both inputs are increased together E22 Maximization As we saw in Chapter 2 the firstorder conditions for an unconstrained maximum of a function of many variables require finding a point at which the partial derivatives are zero If the function is concave it will be below its tangent plane at this point therefore the point will be a true maxi mum3 Because the health status function is concave for example the firstorder conditions for a maximum are also sufficient E23 Constrained maxima When the xs in a maximization or minimization problem are subject to constraints these constraints have to be taken into account in stating secondorder conditions Again matrix algebra provides a compact if not intuitive way of denoting these conditions The notation involves adding rows and col umns of the Hessian matrix for the unconstrained problem and then checking the properties of this augmented matrix Specifically we wish to maximize f1x1 c xn2 subject to the constraint4 g 1x1 c xn2 5 0 We saw in Chapter 2 that the firstorder conditions for a max imum are of the form fi 1 λgi 5 0 where λ is the Lagrange multiplier for this problem Secondorder conditions for a maximum are based on the augmented bordered Hessian5 Hb 5 E 0 g1 g2 c gn g1 f11 f12 f1n g2 f21 f22 f2n gn fn1 fn2 c fnn U 2A proof using the multivariable version of Taylors approximation is provided in Simon and Blume 1994 chapter 21 3This will be a local maximum if the function is concave only in a region or global if the function is concave everywhere 4Here we look only at the case of a single constraint Generalization to many constraints is conceptually straightforward but notationally complex For a concise statement see Sydsaeter Strom and Berck 2005 p 103 5Notice that if gij 5 0 for all i and j then Hb can be regarded as the simple Hessian associated with the Lagrangian expression given in Equation 246 which is a function of the n 1 1 variables λ x1 xn Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 84 Part 1 Introduction For a maximum 1212Hb must be negative definitethat is the leading principal minors of Hb must follow the pat tern 2 1 21 2 and so forth starting with the second such minor6 The secondorder conditions for minimum require that 1212Hb be positive definitethat is all the leading principal minors of Hb except the first should be negative Example In the optimal fence problem Example 28 the bordered Hessian is Hb 5 C 0 22 22 22 0 1 22 1 0 S and Hb2 5 24 Hb3 5 8 thus again the leading principal minors have the sign pattern required for a maximum E24 Quasiconcavity If the constraint g is linear then the secondorder conditions explored in Extension 23 can be related solely to the shape of the function to be optimized f In this case the constraint can be written as g1x1 c xn2 5 c 2 b1x1 2 b2x2 2 c 2bnxn 5 0 and the firstorder conditions for a maximum are fi 5 λbi i 5 1 c n Using the conditions it is clear that the bordered Hessian Hb and the matrix Hr 5 E 0 f1 f2 c fn f1 f11 f12 f1n f2 f21 f22 f2n fn fn1 fn2 c fnn U have the same leading principal minors except for a positive constant of proportionality7 The conditions for a maximum of f subject to a linear constraint will be satisfied provided Hr follows the same sign conventions as Hbthat is 1212Hr must be negative definite A function f for which Hr does fol low this pattern is called quasiconcave As we shall see f has the property that the set of points x for which f1x2 c where c is any constant is convex For such a function the necessary conditions for a maximum are also sufficient Example For the fences problem f 1x y2 5 xy and Hr is given by Hr 5 C 0 y x y 0 1 x 1 0 S Thus Hr2 5 2y2 0 Hr3 5 2xy 0 and the function is quasiconcave8 Example More generally if f is a function of only two variables then quasiconcavity requires that Hr2 5 21 f12 2 0 and Hr3 5 2f11 f 2 2 2 f22 f 2 1 1 2f1 f2 f12 0 which is precisely the condition stated in Equation 2100 Hence we have a fairly simple way of determining quasiconcavity E25 Comparative Statics with two Endogenous Variables Often economists are concerned with models that have mul tiple endogenous variables For example simple supply and demand models typically have two endogenous variables price and quantity together with exogenous variables that may shift either the demand or supply curves Often matrix algebra provides a useful tool for devising comparative static results from such models Consider a situation with two endogenous variables 1x1 and x22 and a single exogenous parameter a It takes two equations to determine the equilibrium values of these two endogenous variables and the values taken by these variables will depend on the exogenous parameter a Write these two equations in implicit form as f 1 3x1 1a2 x2 1a2 a4 5 0 f 2 3x1 1a2 x2 1a2 a4 5 0 The first of these might represent a demand curve and the second a supply curve The above equations indicate that solving these equations will allow us to state the equilibrium 6Notice that the first leading principal minor of Hb is 0 7This can be shown by noting that multiplying a row or a column of a matrix by a constant multiplies the determinant by that constant 8Because f 1x y2 5 xy is a form of a CobbDouglas function that is not concave this shows that not every quasiconcave function is concave Notice that a monotonic function of f such as f 13 could be concave however Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 2 Mathematics for Microeconomics 85 values of each of the endogenous variables as functions of the parameter a Differentiation of these equilibrium equations with respect to a yields f 1 1 dx 1 da 1 f 1 2 dx 2 da 1 f 1 a 5 0 f 2 1 dx 1 da 1 f 2 2 dx 2 da 1 f 2 a 5 0 We wish to solve these simultaneous equations for the com parative static values of the derivatives that show how the equilibrium values change when a changes If we move the partial derivatives in a to the righthand side of the equations we can write them in matrix notation as cf 1 1 f 1 2 f 2 1 f 2 2 d D dx 1 da dx 2 da T 5 c2f 1 a 2f 2 a d and this can be solved as D dx 1 da dx 2 da T 5 cf 1 1 f 1 2 f 2 1 f 2 2 d 21 c2f 1 a 2f 2 a d This is the two variable analogue of the single endogenous vari able comparative statics problem illustrated in Equation 226 Extensions to more endogenous variables follow a similar process and can be readily solved using matrix algebra Cramers Rule Although the matrix formulation of this approach is the most general sometimes comparative statics problems are solved using Cramers rulea clever shortcut that does not require matrix inversion and is often more illuminating Specifically Cramers rule shows that each of the comparative static deriv atives can be solved as the ratio of two determinants see Syd saeter Strom and Berck 2005 p 144 dx1 da 5 2f 1 a f 1 2 2f 2 a f 2 2 f 1 1 f 1 2 f 2 1 f 2 2 dx 2 da 5 f 1 1 2f 1 a f 2 1 2f 2 a f 1 1 f 1 2 f 2 1 f 2 2 All of the derivatives in these equations should be calculated at the equilibrium values of the variables When the underly ing equations are linear this computation is especially easy as the next supply and demand example shows Example Suppose that the demand and supply functions for a product are given by q 5 cp 1 a or q 2 cp 2 a 5 0 1demand c 02 q 5 dp or q 2 dp 5 0 1supply d 02 Note that in this example only the demand curve is shifted by the parameter a Nevertheless a change in a affects the equi librium values of both of the endogenous variables q and p Now we can use the results of the previous section to derive dq da 5 1 2c 0 2d 1 2c 1 2d 5 2d c 2 d 5 d d 2 c 0 dp da 5 1 1 1 0 1 2c 1 2d 5 21 c 2 d 5 1 d 2 c 0 As is usually the case a shift outward in the demand curve increases both quantity and price Of course this could have been shown more easily by direct substitution but the use of Cramers rule will often make comparative statics problems easier to solve when the functions are messier Because the denominators to both derivatives are the same in Cramers rule often we need only to compute the numerators in order to discover whether the derivatives of the endogenous vari ables differ in sign References Simon C P and L Blume Mathematics for Economists New York W W Norton 1994 Sydsaeter R A Strom and P Berck Economists Mathemat ical Manual 4th ed Berlin Germany SpringerVerlag 2005 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 87 PART TWO Choice and Demand Chapter 3 Preferences and Utility Chapter 4 Utility Maximization and Choice Chapter 5 Income and Substitution Effects Chapter 6 Demand Relationships among Goods In Part 2 we will investigate the economic theory of choice One goal of this examination is to develop the notion of demand in a formal way so that it can be used in later sections of the text when we turn to the study of markets A more general goal of this part is to illustrate the approach economists use for explaining how individuals make choices in a wide variety of contexts Part 2 begins with a description of the way economists model individual preferences which are usually referred to by the formal term utility Chapter 3 shows how economists are able to conceptualize utility in a mathematical way This permits an examination of the various exchanges that individuals are willing to make voluntarily The utility concept is used in Chapter 4 to illustrate the theory of choice The fundamental hypothesis of the chapter is that people faced with limited incomes will make economic choices in such a way as to achieve as much utility as possible Chapter 4 uses mathematical and intuitive analyses to indicate the insights that this hypothesis provides about economic behavior Chapters 5 and 6 use the model of utility maximization to investigate how individuals will respond to changes in their circumstances Chapter 5 is primarily concerned with responses to changes in the price of a commodity an analysis that leads directly to the demand curve concept Chapter 6 applies this type of analysis to developing an understanding of demand relationships among different goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 89 CHAPTER THREE Preferences and Utility In this chapter we look at the way in which economists characterize individuals preferences We begin with a fairly abstract discussion of the preference relation but quickly turn to the economists primary tool for studying individual choicesthe utility function We look at some general characteristics of that function and a few simple examples of specific utility functions we will encounter throughout this book 31 AXIOMS OF RATIONAL CHOICE One way to begin an analysis of individuals choices is to state a basic set of postulates or axioms that characterize rational behavior These begin with the concept of preference An individual who reports that A is preferred to B is taken to mean that all things considered he or she feels better off under situation A than under situation B The preference relation is assumed to have three basic properties as follows I Completeness If A and B are any two situations the individual can always specify exactly one of the following three possibilities 1 A is preferred to B 2 B is preferred to A or 3 A and B are equally attractive Consequently people are assumed not to be paralyzed by indecision They completely understand and can always make up their minds about the desirability of any two alternatives The assumption also rules out the possibility that an individual can report both that A is preferred to B and that B is preferred to A II Transitivity If an individual reports that A is preferred to B and B is preferred to C then he or she must also report that A is preferred to C This assumption states that the individuals choices are internally consistent Such an assumption can be subjected to empirical study Generally such studies conclude that a persons choices are indeed transitive but this conclusion must be modified in cases where the individual may not fully understand the consequences of the choices he or she is making Because for the most part we will assume choices are fully informed but see the discussion of uncertainty in Chapter 7 and our problems in behavioral economics which are scattered throughout the book the transitivity property seems to be an appropriate assumption to make about preferences III Continuity If an individual reports A is preferred to B then situations suitably close to A must also be preferred to B This rather technical assumption is required if we wish to analyze individu als responses to relatively small changes in income and prices The purpose of the assumption is to rule out certain kinds of discontinuous knifeedge preferences that Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 90 Part 2 Choice and Demand pose problems for a mathematical development of the theory of choice Assuming continuity does not seem to risk missing types of economic behavior that are import ant in the real world but see Problem 314 for some counterexamples 32 UTILITY Given the assumptions of completeness transitivity and continuity it is possible to show formally that people are able to rank all possible situations from the least desirable to the most desirable ones1 Following the terminology introduced by the nineteenthcentury political theorist Jeremy Bentham economists call this ranking utility2 We also will follow Bentham by saying that more desirable situations offer more utility than do less desirable ones That is if a person prefers situation A to situation B we would say that the utility assigned to option A denoted by U1A2 exceeds the utility assigned to B U1B2 321 Nonuniqueness of utility measures We might even attach numbers to these utility rankings however these numbers will not be unique Any set of numbers we arbitrarily assign that accurately reflects the original pref erence ordering will imply the same set of choices It makes no difference whether we say that U1A2 5 5 and U1B2 5 4 or that U1A2 5 1000000 and U1B2 5 05 In both cases the numbers imply that A is preferred to B In technical terms our notion of utility is defined only up to an orderpreserving monotonic transformation3 Any set of numbers that accurately reflects a persons preference ordering will do Consequently it makes no sense to ask how much more is A preferred than B because that question has no unique answer Surveys that ask people to rank their happiness on a scale of 1 to 10 could just as well use a scale of 7 to 1000000 We can only hope that a person who reports he or she is a 6 on the scale one day and a 7 on the next day is indeed happier on the second day Therefore utility rankings are like the ordinal rankings of restaurants or movies using one two three or four stars They simply record the relative desirability of commodity bundles This lack of uniqueness in the assignment of utility numbers also implies that it is not possible to compare utilities of different people If one person reports that a steak dinner provides a utility of 5 and another person reports that the same dinner offers a utility of 100 we cannot say which individual values the dinner more because they could be using different scales Similarly we have no way of measuring whether a move from situation A to situation B provides more utility to one person or another Nonetheless as we will see economists can say quite a bit about utility rankings by examining what people voluntarily choose to do 322 The ceteris paribus assumption Because utility refers to overall satisfaction such a measure clearly is affected by a variety of factors A persons utility is affected not only by his or her consumption of physical commodities but also by psychological attitudes peer group pressures personal experiences and the general cultural environment Although economists do have a general interest in examining such influences a narrowing of focus is usually necessary 1These properties and their connection to representation of preferences by a utility function are discussed in detail in Andreu MasColell Michael D Whinston and Jerry R Green Microeconomic Theory New York Oxford University Press 1995 2J Bentham Introduction to the Principles of Morals and Legislation London Hafner 1848 3We can denote this idea mathematically by saying that any numerical utility ranking U can be transformed into another set of numbers by the function F providing that F 1U2 is order preserving This can be ensured if Fr 1U2 0 For example the transformation F 1U2 5 U2 is order preserving as is the transformation F 1U2 5 ln U At some places in the text and problems we will find it convenient to make such transformations to make a particular utility ranking easier to analyze Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 91 Consequently a common practice is to devote attention exclusively to choices among quantifiable options eg the relative quantities of food and shelter bought the number of hours worked per week or the votes among specific taxing formulas while holding constant the other things that affect behavior This ceteris paribus other things being equal assumption is invoked in all economic analyses of utilitymaximizing choices so as to make the analysis of choices manageable within a simplified setting 323 Utility from consumption of goods As an important example of the ceteris paribus assumption consider an individuals problem of choosing at a single point in time among n consumption goods x1 x2 xn We shall assume that the individuals ranking of these goods can be represented by a utility function of the form utility 5 U1x1 x2 xn other things2 31 where the xs refer to the quantities of the goods that might be chosen and the other things notation is used as a reminder that many aspects of individual welfare are being held constant in the analysis Often it is easier to write Equation 31 as utility 5 U1x1 x2 xn2 32 Or if only two goods are being considered as utility 5 U1x y2 32 where it is clear that everything is being held constant ie outside the frame of analysis except the goods actually referred to in the utility function It would be tedious to remind you at each step what is being held constant in the analysis but it should be remembered that some form of the ceteris paribus assumption will always be in effect 324 Arguments of utility functions The utility function notation is used to indicate how an individual ranks the particular arguments of the function being considered In the most common case the utility function Equation 32 will be used to represent how an individual ranks certain bundles of goods that might be purchased at one point in time On occasion we will use other arguments in the utility function and it is best to clear up certain conventions at the outset For example it may be useful to talk about the utility an individual receives from real wealth W Therefore we shall use the notation utility 5 U1W2 33 Unless the individual is a rather peculiar Scroogetype person wealth in its own right gives no direct utility Rather it is only when wealth is spent on consumption goods that any utility results For this reason Equation 33 will be taken to mean that the utility from wealth is in fact derived by spending that wealth in such a way as to yield as much utility as possible Two other arguments of utility functions will be used in later chapters In Chapter 16 we will be concerned with the individuals laborleisure choice and will therefore have to consider the presence of leisure in the utility function A function of the form utility 5 U1c h2 34 will be used Here c represents consumption and h represents hours of nonwork time ie leisure during a particular period Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 92 Part 2 Choice and Demand In Chapter 17 we will be interested in the individuals consumption decisions in different periods In that chapter we will use a utility function of the form utility 5 U1c1 c22 35 where c1 is consumption in this period and c2 is consumption in the next period By changing the arguments of the utility function therefore we will be able to focus on specific aspects of an individuals choices in a variety of simplified settings In summary we start our examination of individual behavior with the following definition D E F I N I T I O N Utility Individuals preferences are assumed to be represented by a utility function of the form U1x1 x2 xn2 36 where x1 x2 c xn are the quantities of each of n goods that might be consumed in a period This function is unique only up to an orderpreserving transformation 325 Economic goods In this representation the variables are taken to be goods that is whatever economic quantities they represent we assume that more of any particular xi during some period is preferred to less We assume this is true of every good be it a simple consumption item such as a hot dog or a complex aggregate such as wealth or leisure We have pictured this convention for a twogood utility function in Figure 31 There all consumption bundles in the shaded area are preferred to the bundle x y because any bundle in the shaded area provides more of at least one of the goods By our definition of goods bundles of goods in the shaded area are ranked higher than x y Similarly bundles in the area marked worse are clearly inferior to x y because they contain less of at least one of the goods and no more of the other Bundles in the two areas indicated by question marks are difficult to compare with x y because they contain more of one of the goods and less of the other Movements into these areas involve tradeoffs between the two goods 33 TRADES AND SUBSTITUTION Most economic activity involves voluntary trading between individuals When someone buys say a loaf of bread he or she is voluntarily giving up one thing money for something else bread that is of greater value to that individual To examine this kind of voluntary transaction we need to develop a formal apparatus for illustrating trades in the utility function context We first motivate our discussion with a graphical presentation and then turn to some more formal mathematics 331 Indifference curves and the marginal rate of substitution Voluntary trades can best be studied using the graphical device of an indifference curve In Figure 32 the curve U1 represents all the alternative combinations of x and y for which an individual is equally well off remember again that all other arguments of the utility function are held constant This person is equally happy consuming for example either the combination of goods x1 y1 or the combination x2 y2 This curve representing all the consumption bundles that the individual ranks equally is called an indifference curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 93 The slope of the indifference curve in Figure 32 is negative showing that if the individual is forced to give up some y he or she must be compensated by an additional amount of x to remain indifferent between the two bundles of goods The curve is also drawn so that the slope increases as x increases ie the slope starts at negative infinity and increases toward zero This is a graphical representation of the assumption that people become progressively less willing to trade away y to get more x In mathematical terms the absolute value of this slope diminishes as x increases Hence we have the following definition The shaded area represents those combinations of x and y that are unambiguously preferred to the combination x y Ceteris paribus individuals prefer more of any good rather than less Combinations identified by involve ambiguous changes in welfare because they contain more of one good and less of the other FIGURE 31 More of a Good Is Preferred to Less Quantity of x Quantity of y Preferred to x y Worse than x y y x D E F I N I T I O N Indifference curve An indifference curve or in many dimensions an indifference surface shows a set of consumption bundles about which the individual is indifferent That is the bundles all provide the same level of utility D E F I N I T I O N Marginal rate of substitution The negative of the slope of an indifference curve 1U12 at some point is termed the marginal rate of substitution MRS at that point That is MRS 5 2dy dx U5U1 37 where the notation indicates that the slope is to be calculated along the U1 indifference curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 94 Part 2 Choice and Demand Therefore the slope of U1 and the MRS tell us something about the trades this person will voluntarily make At a point such as x1 y1 the person has a lot of y and is willing to trade away a significant amount to get one more x Therefore the indifference curve at x1 y1 is rather steep This is a situation where the person has say many hamburgers y and little to drink with them x This person would gladly give up a few burgers say 5 to quench his or her thirst with one more drink At x2 y2 on the other hand the indifference curve is flatter Here this person has a few drinks and is willing to give up relatively few burgers say 1 to get another soft drink Consequently the MRS diminishes between x1 y1 and x2 y2 The changing slope of U1 shows how the particular consumption bundle available influences the trades this person will freely make 332 Indifference curve map In Figure 32 only one indifference curve was drawn The x y quadrant however is densely packed with such curves each corresponding to a different level of utility Because every bundle of goods can be ranked and yields some level of utility each point in Figure 32 must have an indifference curve passing through it Indifference curves are similar to contour lines on a map in that they represent lines of equal altitude of utility In Figure 33 several indifference curves are shown to indicate that there are infinitely many in the plane The level of utility represented by these curves increases as we move in a northeast direction the utility of curve U1 is less than that of U2 which is less than that of U3 This is because of the assumption made in Figure 31 More of a good is preferred to less As was discussed earlier there is no unique way to assign numbers to these utility levels The The curve U1 represents those combinations of x and y from which the individual derives the same utility The slope of this curve represents the rate at which the individual is willing to trade x for y while remaining equally well off This slope or more properly the negative of the slope is termed the marginal rate of substitution In the figure the indifference curve is drawn on the assumption of a diminishing marginal rate of substitution FIGURE 32 A Single Indifference Curve Quantity of x Quantity of y x2 x1 y1 U1 U1 y2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 95 curves only show that the combinations of goods on U3 are preferred to those on U2 which are preferred to those on U1 333 Indifference curves and transitivity As an exercise in examining the relationship between consistent preferences and the representation of preferences by utility functions consider the following question Can any two of an individuals indifference curves intersect Two such intersecting curves are shown in Figure 34 We wish to know if they violate our basic axioms of rationality Using our map analogy there would seem to be something wrong at point E where altitude is equal to two different numbers U1 and U2 But no point can be both 100 and 200 feet above sea level To proceed formally let us analyze the bundles of goods represented by points A B C and D By the assumption of nonsatiation ie more of a good always increases utility A is preferred to B and C is preferred to D But this person is equally satisfied with B and C they lie on the same indifference curve so the axiom of transitivity implies that A must be preferred to D But that cannot be true because A and D are on the same indifference curve and are by definition regarded as equally desirable This contradiction shows that indifference curves cannot intersect Therefore we should always draw indifference curve maps as they appear in Figure 33 334 Convexity of indifference curves An alternative way of stating the principle of a diminishing marginal rate of substitution uses the mathematical notion of a convex set A set of points is said to be convex if any two points within the set can be joined by a straight line that is contained completely within the There is an indifference curve passing through each point in the xy plane Each of these curves records combinations of x and y from which the individual receives a certain level of satisfaction Movements in a northeast direction represent movements to higher levels of satisfaction FIGURE 33 There Are Infinitely Many Indifference Curves in the xy Plane Quantity of x Quantity of y Increasing utility U1 U1 U2 U3 U2 U3 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 96 Part 2 Choice and Demand set The assumption of a diminishing MRS is equivalent to the assumption that all combi nations of x and y that are preferred or indifferent to a particular combination x y form a convex set4 This is illustrated in Figure 35a where all combinations preferred or indiffer ent to x y are in the shaded area Any two of these combinationssay x1 y1 and x2 y2 can be joined by a straight line also contained in the shaded area In Figure 35b this is not true A line joining x1 y1 and x2 y2 passes outside the shaded area Therefore the indiffer ence curve through x y in Figure 35b does not obey the assumption of a diminishing MRS because the set of points preferred or indifferent to x y is not convex 335 Convexity and balance in consumption By using the notion of convexity we can show that individuals prefer some balance in their consumption Suppose that an individual is indifferent between the combina tions x1 y1 and x2 y2 If the indifference curve is strictly convex then the combination 1x1 1 x222 1y1 1 y222 will be preferred to either of the initial combinations5 Intuitively wellbalanced bundles of commodities are preferred to bundles that are heavily weighted toward one commodity This is illustrated in Figure 36 Because the indifference curve is assumed to be convex all points on the straight line joining 1x1 y12 and 1x2 y22 are preferred to these initial points Therefore this will be true of the point 1x1 1 x222 1y1 1 y222 which lies at the midpoint of such a line Indeed any proportional combination of the two Combinations A and D lie on the same indifference curve and therefore are equally desirable But the axiom of transitivity can be used to show that A is preferred to D Hence intersecting indifference curves are not consistent with rational preferences That is point E cannot represent two different levels of utility FIGURE 34 Intersecting Indifference Curves Imply Inconsistent Preferences Quantity of x Quantity of y D C E A B U2 U1 4This definition is equivalent to assuming that the utility function is quasiconcave Such functions were discussed in Chapter 2 and we shall return to examine them in the next section Sometimes the term strict quasiconcavity is used to rule out the possibility of indifference curves having linear segments We generally will assume strict quasiconcavity but in a few places we will illustrate the complications posed by linear portions of indifference curves 5In the case in which the indifference curve has a linear segment the individual will be indifferent among all three combinations Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 97 In a the indifference curve is convex any line joining two points above U1 is also above U1 In b this is not the case and the curve shown here does not everywhere have a diminishing MRS FIGURE 35 The Notion of Convexity as an Alternative Definition of a Diminishing MRS If indifference curves are convex if they obey the assumption of a diminishing MRS then the line joining any two points that are indifferent will contain points preferred to either of the initial combinations Intuitively balanced bundles are preferred to unbalanced ones FIGURE 36 Balanced Bundles of Goods Are Preferred to Extreme Bundles Quantity of x Quantity of x Quantity of y Quantity of y b a U1 U1 U1 U1 y1 y2 y2 x1 x1 x2 x x2 x y y1 y Quantity of x Quantity of y 2 x1 x2 2 y1 y2 U1 U1 y1 x1 x2 y2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 98 Part 2 Choice and Demand indifferent bundles of goods will be preferred to the initial bundles because it will represent a more balanced combination Thus strict convexity is equivalent to the assumption of a diminishing MRS Both assumptions rule out the possibility of an indifference curve being straight over any portion of its length EXAMPLE 31 Utility and the MRS Suppose a persons ranking of hamburgers y and soft drinks x could be represented by the utility function utility 5 x y 38 An indifference curve for this function is found by identifying that set of combinations of x and y for which utility has the same value Suppose we arbitrarily set utility equal to 10 Then the equation for this indifference curve is utility 5 10 5 x y 39 Because squaring this function is order preserving the indifference curve is also represented by 100 5 x y 310 which is easier to graph In Figure 37 we show this indifference curve it is a familiar rectangular hyperbola One way to calculate the MRS is to solve Equation 310 for y y 5 100 x 311 This indifference curve illustrates the function 10 5 U 5 x y At point A 15 202 the MRS is 4 implying that this person is willing to trade 4y for an additional x At point B 120 52 however the MRS is 025 implying a greatly reduced willingness to trade FIGURE 37 Indifference Curve for Utility 5 x y Quantity of x Quantity of y 20 125 5 20 0 5 125 A C B U 10 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 99 34 THE MATHEMATICS OF INDIFFERENCE CURVES A mathematical derivation of the indifference curve concept provides additional insights about the nature of preferences In this section we look at a twogood example that ties directly to the graphical treatment provided previously Later in the chapter we look at the manygood case but conclude that this more complicated case adds only a few additional insights 341 The marginal rate of substitution Suppose an individual receives utility from consuming two goods whose quantities are given by x and y This persons ranking of bundles of these goods can be represented by a utility function of the form U1x y2 Those combinations of the two goods that yield a spe cific level of utility say k are represented by solutions to the implicit equation U1x y2 5 k In Chapter 2 see Equation 223 we showed that the tradeoffs implied by such an equa tion are given by dy dx U1x y25k 5 2Ux Uy 316 That is the rate at which x can be traded for y is given by the negative of the ratio of the marginal utility of good x to that of good y Assuming additional amounts of both And then use the definition Equation 37 MRS 5 2dy dx 1along U12 5 100 x2 312 Clearly this MRS decreases as x increases At a point such as A on the indifference curve with a lot of hamburgers say x 5 5 y 5 20 the slope is steep so the MRS is high MRS at 15 202 5 100 x2 5 100 25 5 4 313 Here the person is willing to give up 4 hamburgers to get 1 more soft drink On the other hand at B where there are relatively few hamburgers here x 5 20 y 5 5 the slope is flat and the MRS is low MRS at 120 52 5 100 x2 5 100 400 5 025 314 Now he or she will only give up one quarter of a hamburger for another soft drink Notice also how convexity of the indifference curve U1 is illustrated by this numerical example Point C is midway between points A and B at C this person has 125 hamburgers and 125 soft drinks Here utility is given by utility 5 x y 5 11252 2 5 125 315 which clearly exceeds the utility along U1 which was assumed to be 10 QUERY From our derivation here it appears that the MRS depends only on the quantity of x consumed Why is this misleading How does the quantity of y implicitly enter into Equations 313 and 314 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 100 Part 2 Choice and Demand goods provide added utility this tradeoff rate will be negative implying that increases in the quantity of good x must be met by decreases in the quantity of good y to keep utility constant Earlier we defined the marginal rate of substitution as the negative or absolute value of this tradeoff so now we have MRS 5 2 dy dx U1xy25k 5 Ux Uy 317 This derivation helps in understanding why the MRS does not depend specifically on how utility is measured Because the MRS is a ratio of two utility measures the units drop out in the computation For example suppose good x represents food and that we have chosen a utility function for which an extra unit of food yields 6 extra units of utility sometimes these units are called utils Suppose also that y represents clothing and with this utility function each extra unit of clothing provides 2 extra units of utility In this case it is clear that this person is willing to give up 3 units of clothing thereby losing 6 utils in exchange for 1 extra unit of food thereby gaining 6 utils MRS 5 2 dy dx 5 Ux Uy 5 6 utils per unit x 2 utils per unit y 5 3 units y per unit x 318 Notice that the utility measure used here utils drops out in making this computation and what remains is purely in terms of the units of the two goods This shows that the MRS at a particular combination of goods will be unchanged no matter what specific utility ranking is used6 342 Convexity of Indifference Curves In Chapter 1 we described how economists were able to resolve the waterdiamond paradox by proposing that the price of water is low because one more gallon provides relatively little in terms of increased utility Water is for the most part plentiful therefore its marginal utility is low Of course in a desert water would be scarce and its marginal utility and price could be high Thus one might conclude that the marginal utility associated with water consumption decreases as more water is consumedin formal terms the second partial derivative of the utility function ie Uxx 5 2Ux2 should be negative Intuitively it seems that this commonsense idea should also explain why indifference curves are convex The fact that people are increasingly less willing to part with good y to get more x while holding utility constant seems to refer to the same phenomenon that people do not want too much of any one good Unfortunately the precise connection between diminishing marginal utility and a diminishing MRS is complex even in the two good case As we showed in Chapter 2 a function will by definition have convex indiffer ence curves providing it is quasiconcave But the conditions required for quasiconcavity are messy and the assumption of diminishing marginal utility ie negative secondorder partial derivatives will not ensure that they hold7 Still as we shall see there are good rea sons for assuming that utility functions and many other functions used in microeconomics are quasiconcave thus we will not be too concerned with situations in which they are not 6More formally let F 3U1x y24 be any monotonic transformation of the utility function with Fr 1U2 0 With this new utility ranking the MRS is given by MRS 5 Fx Fy 5 Fr 1U2Ux Fr 1U2Uy 5 Ux Uy which is the same as the MRS for the original utility function 7Specifically for the function U1x y2 to be quasiconcave the following condition must hold see Equation 2114 UxxU2 y 2 2 UxyUxUy 1 UyyU2 x 0 The assumptions that Uxx Uyy 0 will not ensure this One must also be concerned with the sign of the cross partial derivative Uxy Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 101 EXAMPLE 32 Showing Convexity of Indifference Curves Calculation of the MRS for specific utility functions is frequently a good shortcut for showing convexity of indifference curves In particular the process can be much simpler than applying the definition of quasiconcavity although it is more difficult to generalize to more than two goods Here we look at how Equation 317 can be used for three different utility functions for more practice see Problem 31 1 U1x y2 5 x y This example just repeats the case illustrated in Example 31 One shortcut to applying Equation 317 that can simplify the algebra is to take the logarithm of this utility function Because taking logs is order preserving this will not alter the MRS to be calculated Thus let U1x y2 5 ln 3U1x y2 4 5 05 ln x 1 05 ln y 319 Applying Equation 317 yields MRS 5 Ux Uy 5 05x 05y 5 y x 320 which seems to be a much simpler approach than we used previously8 Clearly this MRS is diminishing as x increases and y decreases Therefore the indifference curves are convex 2 U1x y2 5 x 1 xy 1 y In this case there is no advantage to transforming this utility function Applying Equation 317 yields MRS 5 Ux Uy 5 1 1 y 1 1 x 321 Again this ratio clearly decreases as x increases and y decreases thus the indifference curves for this function are convex 3 U1x y2 5 x2 1 y2 For this example it is easier to use the transformation U1x y2 5 3U1x y24 2 5 x2 1 y2 322 Because this is the equation for a quartercircle we should begin to suspect that there might be some problems with the indifference curves for this utility function These suspi cions are confirmed by again applying the definition of the MRS to yield MRS 5 U x U y 5 2x 2y 5 x y 323 For this function it is clear that as x increases and y decreases the MRS increases Hence the indifference curves are concave not convex and this is clearly not a quasiconcave function QUERY Does a doubling of x and y change the MRS in each of these three examples That is does the MRS depend only on the ratio of x to y not on the absolute scale of purchases See also Example 33 8In Example 31 we looked at the U 5 10 indifference curve Thus for that curve y 5 100x and the MRS in Equation 320 would be MRS 5 100x2 as calculated before Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 102 Part 2 Choice and Demand 35 UTILITY FUNCTIONS FOR SPECIFIC PREFERENCES Individuals rankings of commodity bundles and the utility functions implied by these rankings are unobservable All we can learn about peoples preferences must come from the behavior we observe when they respond to changes in income prices and other factors Nevertheless it is useful to examine a few of the forms particular utility functions might take Such an examination may offer insights into observed behavior and more to the point understanding the properties of such functions can be of some help in solving problems Here we will examine four specific examples of utility functions for two goods Indifference curve maps for these functions are illustrated in the four panels of Figure 38 As should be visually apparent these cover a few possible shapes Even greater variety is possible once we move to functions for three or more goods and some of these possibilities are mentioned in later chapters The four indifference curve maps illustrate alternative degrees of substitutability of x for y The CobbDouglas and constant elasticity of substitution CES functions drawn here for relatively low substitutability fall between the extremes of perfect substitution b and no substitution c FIGURE 38 Examples of Utility Functions Quantity of x a CobbDouglas Quantity of y Quantity of y Quantity of y Quantity of y Quantity of x b Perfect substitutes Quantity of x c Perfect complements Quantity of x d CES U2 U2 U2 U2 U1 U0 U1 U0 U1 U1 U0 U0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 103 351 CobbDouglas utility Figure 38a shows the familiar shape of an indifference curve One commonly used utility function that generates such curves has the form U1x y2 5 xαyβ 324 where α and β are positive constants each less than 10 In Examples 31 and 32 we studied a particular case of this function for which α 5 β 5 05 The more general case presented in Equation 324 is termed a CobbDouglas utility function after two researchers who used such a function for their detailed study of production relationships in the US economy see Chapter 9 In general the relative sizes of α and β indicate the relative importance of the two goods to this individual Because utility is unique only up to a monotonic transformation it is often convenient to normalize these parameters so that α 1 β 5 1 In this case utility would be given by U1x y2 5 xδy12δ 325 where δ 5 α 1α 1 β2 1 2 δ 5 β 1α 1 β2 For example a CobbDouglas utility function with α 5 09 and β 5 03 would imply the same behavior as a function with δ 5 075 and 1 2 δ 5 025 352 Perfect substitutes The linear indifference curves in Figure 38b are generated by a utility function of the form U1x y2 5 αx 1 βy 326 where again α and β are positive constants That the indifference curves for this function are straight lines should be readily apparent Any particular level curve can be calculated by setting U1x y2 equal to a constant This would be the equation for a straight line The linear nature of these indifference curves gave rise to the term perfect substitutes to describe the implied relationship between x and y Because the MRS is constant and equal to αβ along the entire indifference curve our previous notions of a diminishing MRS do not apply in this case A person with these preferences would be willing to give up the same amount of y to get one more x no matter how much x was being consumed Such a situation might describe the rela tionship between different brands of what is essentially the same product For example many people including the author do not care where they buy gasoline A gallon of gas is a gallon of gas despite the best efforts of the Exxon and Shell advertising departments to convince me otherwise Given this fact I am always willing to give up 10 gallons of Exxon in exchange for 10 gallons of Shell because it does not matter to me which I use or where I got my last tankful Indeed as we will see in the next chapter one implication of such a relationship is that I will buy all my gas from the least expensive seller Because I do not experience a diminishing MRS of Exxon for Shell I have no reason to seek a balance among the gasoline types I use 353 Perfect complements A situation directly opposite to the case of perfect substitutes is illustrated by the Lshaped indifference curves in Figure 38c These preferences would apply to goods that go together coffee and cream peanut butter and jelly and cream cheese and lox are familiar examples The indifference curves shown in Figure 38c imply that these pairs of goods will be used in the fixed proportional relationship represented by the vertices of the curves A person who prefers 1 ounce of cream with 8 ounces of coffee will want 2 ounces of cream with 16 ounces of coffee Extra coffee without cream is of no value to this person just as extra cream would be of no value without coffee Only by choosing the goods together can utility be increased Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 104 Part 2 Choice and Demand These concepts can be formalized by examining the mathematical form of the utility function that generates these Lshaped indifference curves U1x y2 5 min1α x βy2 327 Here α and β are positive parameters and the operator min means that utility is given by the smaller of the two terms in the parentheses In the coffeecream example if we let ounces of coffee be represented by x and ounces of cream by y utility would be given by U1x y2 5 min1x 8y2 328 Now 8 ounces of coffee and 1 ounce of cream provide 8 units of utility But 16 ounces of coffee and 1 ounce of cream still provide only 8 units of utility because min116 82 5 8 The extra coffee without cream is of no value as shown by the horizontal section of the indifference curves for movement away from a vertex utility does not increase when only x increases with y constant Only if coffee and cream are both doubled to 16 and 2 respectively will utility increase to 16 More generally neither of the two goods specified in the utility function given by Equation 327 will be consumed in superfluous amounts if αx 5 βy In this case the ratio of the quantity of good y consumed to that of good x will be a constant given by y x 5 α β 329 Consumption will occur at the vertices of the indifference curves shown in Figure 38c 354 CES utility One problem with all of the simple utility functions illustrated so far is that they assume that the indifference curve map takes a predefined shape A function that permits a variety of shapes to be shown is the Constant Elasticity of Substitution CES function The customary form for this function is U1x y2 5 3xδ 1 yδ4 1 δ 330 where δ 1 δ 2 0 This function incorporates all three of the utility function described previously depending on the value of δ For δ 5 1 the correspondence to the case of perfect substitutes is obvious As δ approaches zero the function approaches the Cobb Douglas And as δ approaches 2q the function approaches the case of perfect complements Both of these results can be shown by using a limiting argument Often in our analysis we will simplify the calculations required for this function by using the monotonic transformation U 5 Uδδ which yields the more tractable form U1x y2 5 xδ δ 1 yδ δ 331 This form can be generalized a bit by providing differing weights for each of the goods see Problem 312 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 105 The use of the term elasticity of substitution for this function derives from the notion that the possibilities illustrated in Figure 38 correspond to various values for the substitution parameter σ which for this function is given by σ 5 1 11 2 δ2 For perfect substitutes then σ 5 q and the fixed proportions case has σ 5 09 Because the CES function allows us to explore all these cases and many intermediate cases such as the Cobb Douglas for which σ 5 1 it will prove useful for illustrating the degree of substitutability present in various economic relationships The specific shape of the CES function illustrated in Figure 38a is for the case δ 5 21 That is U1x y2 5 2x21 2 y21 5 21 x 2 1 y 332 For this situation σ 5 1 11 2 δ2 5 12 and as the graph shows these sharply curved indifference curves apparently fall between the CobbDouglas and fixed proportion cases The negative signs in this utility function may seem strange but the marginal utilities of both x and y are positive and diminishing as would be expected This explains why δ must appear in the denominators in Equation 330 In the particular case of Equation 332 utility increases from 2q when x 5 y 5 0 toward 0 as x and y increase This is an odd utility scale perhaps but perfectly acceptable and often useful EXAMPLE 33 Homothetic Preferences All the utility functions described in Figure 38 are homothetic see Chapter 2 That is the marginal rate of substitution for these functions depends only on the ratio of the amounts of the two goods not on the total quantities of the goods This fact is obvious for the case of the perfect substitutes when the MRS is the same at every point and the case of perfect complements where the MRS is infinite for yx αβ undefined when yx 5 αβ and zero when yx αβ For the general CobbDouglas function the MRS can be found as MRS 5 Ux Uy 5 αx α21y β βx αy β21 5 α β y x 333 which clearly depends only on the ratio yx Showing that the CES function is also homothetic is left as an exercise see Problem 312 The importance of homothetic functions is that one indifference curve is much like another Slopes of the curves depend only on the ratio yx not on how far the curve is from the origin Indifference curves for higher utility are simple copies of those for lower utility Hence we can study the behavior of an individual who has homothetic preferences by looking only at one indifference curve or at a few nearby curves without fearing that our results would change dramatically at different levels of utility QUERY How might you define homothetic functions geometrically What would the locus of all points with a particular MRS look like on an individuals indifference curve map 9The elasticity of substitution concept is discussed in more detail in connection with production functions in Chapter 9 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 106 Part 2 Choice and Demand 36 THE MANYGOOD CASE All the concepts we have studied thus far for the case of two goods can be generalized to situations where utility is a function of arbitrarily many goods In this section we will briefly explore those generalizations Although this examination will not add much to what we have already shown considering peoples preferences for many goods can be important in applied economics as we will see in later chapters If utility is a function of n goods of the form U1x1 x2 c xn2 then the equation U1x1 x2 xn2 5 k 336 defines an indifference surface in n dimensions This surface shows all those combinations of the n goods that yield the same level of utility Although it is probably impossible to picture what such a surface would look like we will continue to assume that it is convex That is balanced bundles of goods will be preferred to unbalanced ones Hence the utility function even in many dimensions will be assumed to be quasiconcave 361 The MRS with many goods We can study the trades that a person might voluntarily make between any two of these goods say x1 and x2 by again using the implicit function theorem MRS 5 2dx2 dx1 U1x1 x2 xn25k 5 Ux1 1x1 x2 xn2 Ux2 1x1 x2 xn2 337 The notation here makes the important point that an individuals willingness to trade x1 for x2 will depend not only on the quantities of these two goods but also on the quantities of all the other goods An individuals willingness to trade food for clothing will depend not only on the quantities of food and clothing he or she has but also on how much shelter he or she has In general it would be expected that changes in the quantities of any of these other goods would affect the tradeoff represented by Equation 337 It is this EXAMPLE 34 Nonhomothetic Preferences Although all the indifference curve maps in Figure 38 exhibit homothetic preferences not all proper utility functions do Consider the quasilinear utility function U1x y2 5 x 1 ln y 334 For this function good y exhibits diminishing marginal utility but good x does not The MRS can be computed as MRS 5 Ux Uy 5 1 1y 5 y 335 The MRS diminishes as the chosen quantity of y decreases but it is independent of the quantity of x consumed Because x has a constant marginal utility a persons willingness to give up y to get one more unit of x depends only on how much y he or she has Contrary to the homothetic case a doubling of both x and y doubles the MRS rather than leaving it unchanged QUERY What does the indifference curve map for the utility function in Equation 334 look like Why might this approximate a situation where y is a specific good and x represents every thing else Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 107 possibility that can sometimes make it difficult to generalize the findings of simple two good models to the manygood case One must be careful to specify what is being assumed about the quantities of the other goods In later chapters we will occasionally look at such complexities However for the most part the twogood model will be good enough for developing intuition about economic relationships Summary In this chapter we have described the way in which economists formalize individuals preferences about the goods they choose We drew several conclusions about such preferences that will play a central role in our analysis of the theory of choice in the following chapters If individuals obey certain basic behavioral postulates in their preferences among goods they will be able to rank all commodity bundles and that ranking can be represented by a utility function In making choices individuals will behave as though they were maximizing this function Utility functions for two goods can be illustrated by an indifference curve map Each indifference curve contour on this map shows all the commodity bundles that yield a given level of utility The negative of the slope of an indifference curve is defined as the marginal rate of substitution MRS This shows the rate at which an individual would willingly give up an amount of one good y if he or she were compensated by receiving one more unit of another good x The assumption that the MRS decreases as x is substituted for y in consumption is consistent with the notion that individuals prefer some balance in their consumption choices If the MRS is always decreasing individuals will have strictly convex indifference curves That is their utility function will be strictly quasiconcave A few simple functional forms can capture important differences in individuals preferences for two or more goods Here we examined the CobbDouglas function the linear function perfect substitutes the fixed proportions function perfect complements and the CES function which includes the other three as special cases It is a simple matter mathematically to generalize from twogood examples to many goods And as we shall see studying peoples choices among many goods can yield many insights But the mathematics of many goods is not especially intuitive therefore we will primarily rely on twogood cases to build such intuition Problems 31 Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves ie whether the MRS declines as x increases a U1x y2 5 3x 1 y b U1x y2 5 x y c U1x y2 5 x 1 y d U1x y2 5 x2 2 y2 e U1x y2 5 xy x 1 y 32 In footnote 7 we showed that for a utility function for two goods to have a strictly diminishing MRS ie to be strictly quasiconcave the following condition must hold UxxU2 y 2 2UxyUxUy 1 UyyU2 x 0 Use this condition to check the convexity of the indifference curves for each of the utility functions in Problem 31 Describe the precise relationship between diminishing marginal utility and quasiconcavity for each case 33 Consider the following utility functions a U1x y2 5 xy b U1x y2 5 x2y2 c U1x y2 5 ln x 1 ln y Show that each of these has a diminishing MRS but that they exhibit constant increasing and decreasing marginal utility respectively What do you conclude Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 108 Part 2 Choice and Demand 34 As we saw in Figure 35 one way to show convexity of indif ference curves is to show that for any two points 1x1 y12 and 1x2 y22 on an indifference curve that promises U 5 k the utility associated with the point a x1 1 x2 2 y1 1 y2 2 b is at least as great as k Use this approach to discuss the convexity of the indifference curves for the following three functions Be sure to graph your results a U1x y2 5 min 1x y2 b U1x y2 5 max1x y2 c U1x y2 5 x 1 y 35 The Phillie Phanatic PP always eats his ballpark franks in a special way he uses a footlong hot dog together with precisely half a bun 1 ounce of mustard and 2 ounces of pickle relish His utility is a function only of these four items and any extra amount of a single item without the other constituents is worthless a What form does PPs utility function for these four goods have b How might we simplify matters by considering PPs utility to be a function of only one good What is that good c Suppose footlong hot dogs cost 100 each buns cost 050 each mustard costs 005 per ounce and pickle relish costs 015 per ounce How much does the good defined in part b cost d If the price of footlong hot dogs increases by 50 percent to 150 each what is the percentage increase in the price of the good e How would a 50 percent increase in the price of a bun affect the price of the good Why is your answer different from part d f If the government wanted to raise 100 by taxing the goods that PP buys how should it spread this tax over the four goods so as to minimize the utility cost to PP 36 Many advertising slogans seem to be asserting something about peoples preferences How would you capture the following slogans with a mathematical utility function a Promise margarine is just as good as butter b Things go better with Coke c You cant eat just one Pringles potato chip d Krispy Kreme glazed doughnuts are just better than Dunkin Donuts e Miller Brewing advises us to drink beer responsibly What would irresponsible drinking be 37 a A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y She is indifferent between bundle 6 5 and bundle 12 3 What is the utility function for goods x and y Hint What is the shape of the indifference curve b A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle 8 1 She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle 4 4 She is indifferent between these two bundles Assuming that the utility function is CobbDouglas of the form U1x y2 5 x αy β where α and β are positive constants what is the utility function for this consumer c Was there a redundancy of information in part b If yes how much is the minimum amount of information required in that question to derive the utility function 38 Find utility functions given each of the following indifference curves defined by U 1 2 5 k a z 5 k1δ xαδy βδ b y 5 05x 2 2 4 1x2 2 k2 2 05x c z 5 y 4 2 4x 1x 2y 2 k2 2x 2 y 2 2x Analytical Problems 39 Initial endowments Suppose that a person has initial amounts of the two goods that provide utility to him or her These initial amounts are given by x and y a Graph these initial amounts on this persons indifference curve map b If this person can trade x for y or vice versa with other people what kinds of trades would he or she voluntarily make What kinds would not be made How do these trades relate to this persons MRS at the point 1x y2 c Suppose this person is relatively happy with the initial amounts in his or her possession and will only consider trades that increase utility by at least amount k How would you illustrate this on the indifference curve map Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 109 310 CobbDouglas utility Example 33 shows that the MRS for the CobbDouglas function U1x y2 5 x αy β is given by MRS 5 α β a y xb a Does this result depend on whether α 1 β 5 1 Does this sum have any relevance to the theory of choice b For commodity bundles for which y 5 x how does the MRS depend on the values of α and β Develop an intuitive explanation of why if α β MRS 1 Illustrate your argument with a graph c Suppose an individual obtains utility only from amounts of x and y that exceed minimal subsistence levels given by x0 y0 In this case U1x y2 5 1x 2 x02 α 1y 2 y02 β Is this function homothetic For a further discussion see the Extensions to Chapter 4 311 Independent marginal utilities Two goods have independent marginal utilities if 2U yx 5 2U xy 5 0 Show that if we assume diminishing marginal utility for each good then any utility function with independent mar ginal utilities will have a diminishing MRS Provide an exam ple to show that the converse of this statement is not true 312 CES utility with weights a Show that the CES function αx δ δ 1 β y δ δ is homothetic How does the MRS depend on the ratio yx b Show that your results from part a agree with our discussion of the cases δ 5 1 perfect substitutes and δ 5 0 CobbDouglas c Show that the MRS is strictly diminishing for all values of δ 1 d Show that if x 5 y the MRS for this function depends only on the relative sizes of α and β e Calculate the MRS for this function when yx 5 09 and yx 5 11 for the two cases δ 5 05 and δ 5 21 What do you conclude about the extent to which the MRS changes in the vicinity of x 5 y How would you interpret this geometrically 313 The quasilinear function Consider the function U1x y2 5 x 1 ln y This is a function that is used relatively frequently in economic modeling as it has some useful properties a Find the MRS of the function Now interpret the result b Confirm that the function is quasiconcave c Find the equation for an indifference curve for this function d Compare the marginal utility of x and y How do you interpret these functions How might consumers choose between x and y as they try to increase their utility by for example consuming more when their income increases We will look at this income effect in detail in the Chapter 5 problems e Considering how the utility changes as the quantities of the two goods increase describe some situations where this function might be useful 314 Preference relations The formal study of preferences uses a general vector nota tion A bundle of n commodities is denoted by the vec tor x 5 1x1 x2 c xn2 and a preference relation 1 s 2 is defined over all potential bundles The statement x1 s x2 means that bundle x1 is preferred to bundle x2 Indifference between two such bundles is denoted by x1 x2 The preference relation is complete if for any two bun dles the individual is able to state either x1 s x2 x2 s x1 or x1 x2 The relation is transitive if x1 s x2 and x2 s x3 implies that x1 s x3 Finally a preference relation is contin uous if for any bundle y such that y s x any bundle suitably close to y will also be preferred to x Using these definitions discuss whether each of the following preference relations is complete transitive and continuous a Summation preferences This preference relation assumes one can indeed add apples and oranges Specifically x1 s x2 if and only if a n i51 x1 i a n i51 x 2 i If a n i51 x1 i 5 a n i51 x 2 i x1 x2 b Lexicographic preferences In this case the preference relation is organized as a dictionary If x1 1 x2 1 x1 s x2 regardless of the amounts of the other n 2 1 goods If x1 1 5 x2 1 and x1 2 x2 2 x1 s x2 regardless of the amounts of the other n 2 2 goods The lexicographic preference relation then continues in this way throughout the entire list of goods c Preferences with satiation For this preference relation there is assumed to be a consumption bundle 1x2 that provides complete bliss The ranking of all other bundles is determined by how close they are to x That is x1 s x2 if and only if 0x1 2 x0 0x2 2 x0 where 0xi 2 x0 5 1x i 1 2 x 12 2 1 1x i 2 2 x 22 2 1 c1 1x i n 2 x n2 2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 110 Part 2 Choice and Demand 315 The benefit function In a 1992 article David G Luenberger introduced what he termed the benefit function as a way of incorporating some degree of cardinal measurement into utility theory10 The author asks us to specify a certain elementary consumption bundle and then measure how many replications of this bun dle would need to be provided to an individual to raise his or her utility level to a particular target Suppose there are only two goods and that the utility target is given by U 1x y2 Sup pose also that the elementary consumption bundle is given by 1x0 y02 Then the value of the benefit function b1U 2 is that value of α for which U1αx0 αy02 5 U a Suppose utility is given by U1x y2 5 xβy12β Calculate the benefit function for x0 5 y0 5 1 b Using the utility function from part a calculate the benefit function for x0 5 1 y0 5 0 Explain why your results differ from those in part a c The benefit function can also be defined when an indi vidual has initial endowments of the two goods If these initial endowments are given by x y then b1U x y2 is given by that value of α which satisfies the equation U1x 1 αx0 y 1 αy02 5 U In this situation the bene fit can be either positive when U1x y2 U or negative when U1x y2 U Develop a graphical description of these two possibilities and explain how the nature of the elementary bundle may affect the benefit calculation d Consider two possible initial endowments x1 y1 and x2 y2 Explain both graphically and intuitively why baU x1 1 x2 2 y1 1 y2 2 b 05b1U x1 y12 1 05b1U x2 y22 Note This shows that the benefit func tion is concave in the initial endowments 10Luenberger David G Benefit Functions and Duality Journal of Mathematical Economics 21 46181 The presentation here has been simplified considerably from that originally presented by the author mainly by changing the direction in which benefits are measured Suggestions for Further Reading Aleskerov Fuad and Bernard Monjardet Utility Maximiza tion Choice and Preference Berlin SpringerVerlag 2002 A complete study of preference theory Covers a variety of thresh old models and models of contextdependent decision making Jehle G R and P J Reny Advanced Microeconomic Theory 2nd ed Boston Addison WesleyLongman 2001 Chapter 2 has a good proof of the existence of utility functions when basic axioms of rationality hold Kreps David M A Course in Microeconomic Theory Prince ton NJ Princeton University Press 1990 Chapter 1 covers preference theory in some detail Good discussion of quasiconcavity Kreps David M Notes on the Theory of Choice London West view Press 1988 Good discussion of the foundations of preference theory Most of the focus of the book is on utility in uncertain situations MasColell Andrea Michael D Whinston and Jerry R Green Microeconomic Theory New York Oxford University Press 1995 Chapters 2 and 3 provide a detailed development of preference relations and their representation by utility functions Stigler G The Development of Utility Theory Journal of Political Economy 59 pts 12 AugustOctober 1950 30727 37396 A lucid and complete survey of the history of utility theory Has many interesting insights and asides Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 111 EXTENSIONS SPeCial PreferenCeS The utility function concept is a general one that can be adapted to a large number of special circumstances Discovery of ingenious functional forms that reflect the essential aspects of some problem can provide a number of insights that would not be readily apparent with a more literary approach Here we look at four aspects of preferences that economists have tried to model 1 threshold effects 2 quality 3 habits and addiction and 4 secondparty preferences In Chapters 7 and 17 we illustrate a number of additional ways of capturing aspects of preferences E31 Threshold effects The model of utility that we developed in this chapter implies an individual will always prefer commodity bundle A to bundle B provided U1A2 U1B2 There may be events that will cause people to shift quickly from consuming bun dle A to consuming B In many cases however such a light ningquick response seems unlikely People may in fact be set in their ways and may require a rather large change in circumstances to change what they do For example indi viduals may not have especially strong opinions about what precise brand of toothpaste they choose and may stick with what they know despite a proliferation of new and perhaps better brands Similarly people may stick with an old favor ite TV show even though it has declined in quality One way to capture such behavior is to assume individuals make deci sions as though they faced thresholds of preference In such a situation commodity bundle A might be chosen over B only when U1A2 U1B2 1 P i where P is the threshold that must be overcome With this specification indifference curves then may be rather thick and even fuzzy rather than the distinct contour lines shown in this chapter Threshold models of this type are used extensively in marketing The theory behind such models is presented in detail in Aleskerov and Monjardet 2002 There the authors consider a number of ways of specifying the threshold so that it might depend on the characteris tics of the bundles being considered or on other contextual variables Alternative fuels Vedenov Duffield and Wetzstein 2006 use the threshold idea to examine the conditions under which individuals will shift from gasoline to other fuels primarily ethanol for pow ering their cars The authors point out that the main disadvan tage of using gasoline in recent years has been the excessive price volatility of the product relative to other fuels They con clude that switching to ethanol blends is efficient especially during periods of increased gasoline price volatility provided that the blends do not decrease fuel efficiency E32 Quality Because many consumption items differ widely in quality economists have an interest in incorporating such differences into models of choice One approach is simply to regard items of different quality as totally separate goods that are rela tively close substitutes But this approach can be unwieldy because of the large number of goods involved An alternative approach focuses on quality as a direct item of choice Utility might in this case be reflected by utility 5 U1q Q2 ii where q is the quantity consumed and Q is the quality of that consumption Although this approach permits some exam ination of qualityquantity tradeoffs it encounters difficulty when the quantity consumed of a commodity eg wine con sists of a variety of qualities Quality might then be defined as an average see Theil1 1952 but that approach may not be appropriate when the quality of new goods is changing rapidly eg as in the case of personal computers A more general approach originally suggested by Lancaster 1971 focuses on a welldefined set of attributes of goods and assumes that those attributes provide utility If a good q provides two such attributes a1 and a2 then utility might be written as utility 5 U3q a1 1q2 a2 1q2 4 iii and utility improvements might arise either because this indi vidual chooses a larger quantity of the good or because a given quantity yields a higher level of valuable attributes 1Theil also suggests measuring quality by looking at correlations between changes in consumption and the income elasticities of various goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 112 Part 2 Choice and Demand Personal computers This is the practice followed by economists who study demand in such rapidly changing industries as personal computers In this case it would clearly be incorrect to focus only on the quantity of personal computers purchased each year because new machines are much better than old ones and presumably provide more utility For example Berndt Griliches and Rappaport 1995 find that personal computer quality has been increasing about 30 percent per year over a relatively long period primarily because of improved attri butes such as faster processors or better hard drives A person who spends say 2000 for a personal computer today buys much more utility than did a similar consumer 5 years ago E33 Habits and addiction Because consumption occurs over time there is the possibil ity that decisions made in one period will affect utility in later periods Habits are formed when individuals discover they enjoy using a commodity in one period and this increases their consumption in subsequent periods An extreme case is addiction be it to drugs cigarettes or Marx Brothers mov ies where past consumption significantly increases the util ity of present consumption One way to portray these ideas mathematically is to assume that utility in period t depends on consumption in period t and the total of all previous con sumption of the habitforming good say x utility 5 Ut1xt yt st2 iv where st 5 a q i51 xt2i In empirical applications however data on all past levels of consumption usually do not exist Therefore it is common to model habits using only data on current consumption 1xt2 and on consumption in the previous period 1xt212 A com mon way to proceed is to assume that utility is given by utility 5 Ut1x t yt2 v where x t is some simple function of xt and xt21 such as x t 5 xt 2 xt21 or x t 5 xtxt21 Such functions imply that ceteris paribus the higher the value of xt21 the more xt will be chosen in the current period Modeling habits These approaches to modeling habits have been applied to a wide variety of topics Stigler and Becker 1977 use such models to explain why people develop a taste for going to operas or playing golf Becker Grossman and Murphy 1994 adapt the models to studying cigarette smoking and other addictive behavior They show that reductions in smoking early in life can have large effects on eventual cigarette con sumption because of the dynamics in individuals utility functions Whether addictive behavior is rational has been extensively studied by economists For example Gruber and Koszegi 2001 show that smoking can be approached as a rational although timeinconsistent2 choice E34 Secondparty preferences Individuals clearly care about the wellbeing of other individ uals Phenomena such as making charitable contributions or making bequests to children cannot be understood without recognizing the interdependence that exists among people Secondparty preferences can be incorporated into the utility function of person i say by utility 5 Ui 1xi yi Uj2 vi where Uj is the utility of someone else If UiUj 0 then this person will engage in altruistic behavior whereas if UiUj 0 then he or she will demon strate the malevolent behavior associated with envy The usual case of UiUj 5 0 is then simply a middle ground between these alternative preference types Gary Becker was been a pioneer in the study of these possibilities and explored a vari ety of topics including the general theory of social interac tions 1976 and the importance of altruism in the theory of the family 1981 Evolutionary biology and genetics Biologists have suggested a particular form for the utility function in Equation vi drawn from the theory of genetics In this case utility 5 Ui 1xi yi2 1 a j rjUj vii where rj measures closeness of the genetic relationship between person i and person j For parents and children for example rj 5 05 whereas for cousins rj 5 0125 Bergstrom 1996 describes a few of the conclusions about evolutionary behavior that biologists have drawn from this particular func tional form References Aleskerov Fuad and Bernard Monjardet Utility Maximiza tion Choice and Preference Berlin SpringerVerlag 2002 Becker Gary S The Economic Approach to Human Behavior Chicago University of Chicago Press 1976 2For more on time inconsistency see Chapter 17 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 3 Preferences and Utility 113 A Treatise on the Family Cambridge MA Harvard University Press 1981 Becker Gary S Michael Grossman and Kevin M Murphy An Empirical Analysis of Cigarette Addiction American Economic Review June 1994 396418 Bergstrom Theodore C Economics in a Family Way Journal of Economic Literature December 1996 190334 Berndt Ernst R Zvi Griliches and Neal J Rappaport Econo metric Estimates of Price Indexes for Personal Computers in the 1990s Journal of Econometrics July 1995 24368 Gruber Jonathan and Botond Koszegi Is Addiction Ratio nal Theory and Evidence Quarterly Journal of Econom ics November 2001 1261303 Lancaster Kelvin J Consumer Demand A New Approach New York Columbia University Press 1971 Stigler George J and Gary S Becker De Gustibus Non Est Disputandum American Economic Review March 1977 7690 Theil Henri Qualities Prices and Budget Enquiries Review of Economic Studies April 1952 12947 Vedenov Dmitry V James A Duffield and Michael E Wetzstein Entry of Alternative Fuels in a Volatile US Gasoline Market Journal of Agricultural and Resource Economics April 2006 113 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 115 CHAPTER FOUR Utility Maximization and Choice In this chapter we examine the basic model of choice that economists use to explain individuals behavior That model assumes that individuals who are constrained by limited incomes will behave as though they are using their purchasing power in such a way as to achieve the highest utility possible That is individuals are assumed to behave as though they maximize utility subject to a budget constraint Although the specific applications of this model are varied as we will show all are based on the same fundamental mathematical model and all arrive at the same general conclusion To maximize utility individuals will choose bundles of commodities for which the rate of tradeoff between any two goods the MRS is equal to the ratio of the goods market prices Market prices convey information about opportunity costs to individuals and this information plays an important role in affecting the choices actually made Utility maximization and lightning calculations Before starting a formal study of the theory of choice it may be appropriate to dispose of two complaints noneconomists often make about the approach we will take First is the charge that no real person can make the kinds of lightning calculations required for utility maximization According to this complaint when moving down a supermarket aisle people just grab what is available with no real pattern or purpose to their actions Economists are not persuaded by this complaint They doubt that people behave randomly everyone after all is bound by some sort of budget constraint and they view the lightning calculation charge as misplaced Recall again Friedmans pool player from Chapter 1 The pool player also cannot make the lightning calculations required to plan a shot according to the laws of physics but those laws still predict the players behavior So too as we shall see the utilitymaximization model predicts many aspects of behavior even though no one carries around a computer with his or her utility function programmed into it To be precise economists assume that people behave as if they made such calculations thus the complaint that the calculations cannot possibly be made is largely irrelevant Still in recent times economists have increasingly tried to model some of the behavioral complications that arise in the actual decisions people make We look at some of these complications in a variety of problems throughout this book Altruism and selfishness A second complaint against our model of choice is that it appears to be extremely selfish no one according to this complaint has such solely selfcentered goals Although economists Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 116 Part 2 Choice and Demand are probably more ready to accept selfinterest as a motivating force than are other more Utopian thinkers Adam Smith observed We are not ready to suspect any person of being deficient in selfishness1 this charge is also misplaced Nothing in the utilitymaximization model prevents individuals from deriving satisfaction from philanthropy or generally doing good These activities also can be assumed to provide utility Indeed economists have used the utilitymaximization model extensively to study such issues as donating time and money to charity leaving bequests to children or even giving blood One need not take a position on whether such activities are selfish or selfless because economists doubt people would under take them if they were against their own best interests broadly conceived For an example of how altruism can be incorporated into the utility maximization framework see Problem 414 41 AN INITIAL SURVEY The general results of our examination of utility maximization can be stated succinctly as follows That spending all ones income is required for utility maximization is obvious Because extra goods provide extra utility there is no satiation and because there is no other use for income to leave any unspent would be to fail to maximize utility Throwing money away is not a utilitymaximizing activity The condition specifying equality of tradeoff rates requires a bit more explanation Because the rate at which one good can be traded for another in the market is given by the ratio of their prices this result can be restated to say that the individual will equate the MRS of x for y to the ratio of the price of x to the price of y 1pxpy2 This equating of a personal tradeoff rate to a marketdetermined tradeoff rate is a result common to all individual utilitymaximization problems and to many other types of maximization problems It will occur again and again throughout this text 411 A numerical illustration To see the intuitive reasoning behind this result assume that it were not true that an individual had equated the MRS to the ratio of the prices of goods Specifically suppose that the individuals MRS is equal to 1 and that he or she is willing to trade 1 unit of x for 1 unit of y and remain equally well off Assume also that the price of x is 2 per unit and of y is 1 per unit It is easy to show that this person can be made better off Suppose this person reduces x consumption by 1 unit and trades it in the market for 2 units of y Only 1 extra unit of y was needed to keep this person as happy as before the tradethe second unit of y is a net addition to wellbeing Therefore the individuals spending could not have been allocated optimally in the first place A similar method of reasoning can be used whenever the MRS and the price ratio pxpy differ The condition for maximum utility must be the equality of these two magnitudes 1Adam Smith The Theory of Moral Sentiments 1759 reprint New Rochelle NY Arlington House 1969 p 446 O P T I M I Z AT I O N P R I N C I P L E Utility maximization To maximize utility given a fixed amount of income to spend an indi vidual will buy those quantities of goods that exhaust his or her total income and for which the psychic rate of tradeoff between any two goods the MRS is equal to the rate at which the goods can be traded one for the other in the marketplace Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 117 42 THE TWOGOOD CASE A GRAPHICAL ANALYSIS This discussion seems eminently reasonable but it can hardly be called a proof Rather we must now show the result in a rigorous manner and at the same time illustrate several other important attributes of the maximization process First we take a graphic analysis then we take a more mathematical approach 421 Budget constraint Assume that the individual has I dollars to allocate between good x and good y If px is the price of good x and py is the price of good y then the individual is constrained by pxx 1 pyy I 41 That is no more than I can be spent on the two goods in question This budget constraint is shown graphically in Figure 41 This person can afford to choose only combinations of x and y in the shaded triangle of the figure If all of I is spent on good x it will buy Ipx units of x Similarly if all is spent on y it will buy Ipy units of y The slope of the constraint is easily seen to be 2pxpy This slope shows how y can be traded for x in the market If px 5 2 and py 5 1 then 2 units of y will trade for 1 unit of x Those combinations of x and y that the individual can afford are shown in the shaded triangle If as we usually assume the individual prefers more rather than less of every good the outer boundary of this triangle is the relevant constraint where all the available funds are spent either on x or on y The slope of this straightline boundary is given by 2pxpy FIGURE 41 The Individuals Budget Constraint for Two Goods Quantity of x 0 Quantity of y py I px I I pxx pyy Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 118 Part 2 Choice and Demand 422 Firstorder conditions for a maximum This budget constraint can be imposed on this persons indifference curve map to show the utilitymaximization process Figure 42 illustrates this procedure The individual would be irrational to choose a point such as A he or she can get to a higher utility level just by spend ing more of his or her income The assumption of nonsatiation implies that a person should spend all of his or her income to receive maximum utility Similarly by reallocating expen ditures the individual can do better than point B Point D is out of the question because income is not large enough to purchase D It is clear that the position of maximum utility is at point C where the combination x y is chosen This is the only point on indifference curve U2 that can be bought with I dollars no higher utility level can be bought C is a point of tangency between the budget constraint and the indifference curve Therefore at C we have slope of budget constraint 5 2px py 5 slope of indifference curve 5 dy dx U5constant 42 or px py 5 2 dy dx U5constant 5 MRS1of x for y2 43 Point C represents the highest utility level that can be reached by the individual given the budget constraint Therefore the combination x y is the rational way for the individual to allocate purchasing power Only for this combination of goods will two conditions hold All available funds will be spent and the individuals psychic rate of tradeoff MRS will be equal to the rate at which the goods can be traded in the market 1pxpy2 Quantity of x Quantity of y U1 U1 U2 U3 U2 U3 0 I pxx pyy B D C A y x FIGURE 42 A Graphical Demon stration of Utility Maximization Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 119 Our intuitive result is proved For a utility maximum all income should be spent and the MRS should equal the ratio of the prices of the goods It is obvious from the diagram that if this condition is not fulfilled the individual could be made better off by reallocating expenditures 423 Secondorder conditions for a maximum The tangency rule is only a necessary condition for a maximum To see that it is not a suf ficient condition consider the indifference curve map shown in Figure 43 Here a point of tangency C is inferior to a point of nontangency B Indeed the true maximum is at another point of tangency A The failure of the tangency condition to produce an unam biguous maximum can be attributed to the shape of the indifference curves in Figure 43 If the indifference curves are shaped like those in Figure 42 no such problem can arise But we have already shown that normally shaped indifference curves result from the assump tion of a diminishing MRS Therefore if the MRS is assumed to be always diminishing the condition of tangency is both a necessary and sufficient condition for a maximum2 With out this assumption one would have to be careful in applying the tangency rule 2As we saw in Chapters 2 and 3 this is equivalent to assuming that the utility function is quasiconcave Because we will usually assume quasiconcavity the necessary conditions for a constrained utility maximum will also be sufficient If indifference curves do not obey the assumption of a diminishing MRS not all points of tangency points for which MRS 2 pxpy may truly be points of maximum utility In this example tangency point C is inferior to many other points that can also be purchased with the available funds In order that the necessary conditions for a maximum ie the tangency conditions also be sufficient one usually assumes that the MRS is diminishing that is the utility function is strictly quasiconcave Quantity of x Quantity of y U1 U3 U2 U1 U2 U3 A C B I pxx pyy FIGURE 43 Example of an Indiffer ence Curve Map for Which the Tangency Con dition Does Not Ensure a Maximum Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 120 Part 2 Choice and Demand 424 Corner solutions The utilitymaximization problem illustrated in Figure 42 resulted in an interior maximum in which positive amounts of both goods were consumed In some situations individuals preferences may be such that they can obtain maximum utility by choosing to consume no amount of one of the goods If someone does not like hamburgers there is no reason to allocate any income to their purchase This possibility is reflected in Figure 44 There utility is maximized at E where x 5 x and y 5 0 thus any point on the budget constraint where positive amounts of y are consumed yields a lower utility than does point E Notice that at E the budget constraint is not precisely tangent to the indifference curve U2 Instead at the optimal point the budget constraint is flatter than U2 indicating that the rate at which x can be traded for y in the market is lower than the individuals psychic tradeoff rate the MRS At prevailing market prices the individual is more than willing to trade away y to get extra x Because it is impossible in this problem to consume negative amounts of y however the physical limit for this process is the Xaxis along which purchases of y are 0 Hence as this discussion makes clear it is necessary to amend the firstorder conditions for a utility maximum a bit to allow for corner solutions of the type shown in Figure 44 Following our discussion of the general ngood case we will use the mathematics from Chapter 2 to show how this can be accomplished 43 THE nGOOD CASE The results derived graphically in the case of two goods carry over directly to the case of n goods Again it can be shown that for an interior utility maximum the MRS between any two goods must equal the ratio of the prices of these goods To study this more general case however it is best to use some mathematics With the preferences represented by this set of indifference curves utility maximization occurs at E where 0 amounts of good y are consumed The firstorder conditions for a maximum must be modified somewhat to accommodate this possibility Quantity of x Quantity of y U3 U1 E x U2 FIGURE 44 Corner Solution for Utility Maximization Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 121 431 Firstorder conditions With n goods the individuals objective is to maximize utility from these n goods utility 5 U1x1 x2 xn2 44 subject to the budget constraint3 I 5 p1x1 1 p2x2 1 c1 pnxn 45 or I 2 p1x1 2 p2x2 2 c2 pnxn 5 0 46 Following the techniques developed in Chapter 2 for maximizing a function subject to a constraint we set up the Lagrangian expression 5 U1x1 x2 xn2 1 λ 1I 2 p1x1 2 p2x2 2 c2 pnxn2 47 Setting the partial derivatives of with respect to x1 x2 xn and λ equal to 0 yields n 1 1 equations representing the necessary conditions for an interior maximum x1 5 U x1 2 λp1 5 0 x2 5 U x2 2 λp2 5 0 48 xn 5 U xn 2 λpn 5 0 λ 5 I 2 p1x1 2 p2x2 2 c2 pnxn 5 0 These n 1 1 equations can in principle be solved for the optimal x1 x2 xn and for λ see Examples 41 and 42 to be convinced that such a solution is possible Equations 48 are necessary but not sufficient for a maximum The secondorder conditions that ensure a maximum are relatively complex and must be stated in matrix terms see the Extensions to Chapter 2 However the assumption of strict quasiconcavity a diminishing MRS in the twogood case along with the assumption that the budget constraint is linear is sufficient to ensure that any point obeying Equation 48 is in fact a true maximum 432 Implications of firstorder conditions The firstorder conditions represented by Equation 48 can be rewritten in a variety of instructive ways For example for any two goods xi and xj we have Uxi Uxj 5 pi pj 49 3Again the budget constraint has been written as an equality because given the assumption of nonsatiation it is clear that the individual will spend all available income Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 122 Part 2 Choice and Demand In Chapter 3 we showed that the ratio of the marginal utilities of two goods is equal to the marginal rate of substitution between them Therefore the conditions for an optimal allo cation of income become MRS1xi for xj2 5 pi pj 410 This is exactly the result derived graphically earlier in this chapter to maximize utility the individual should equate the psychic rate of tradeoff to the market tradeoff rate 433 Interpreting the Lagrange multiplier Another result can be derived by solving Equations 48 for λ λ 5 Ux1 p1 5 Ux2 p2 5 c5 Uxn pn 411 These equations state that at the utilitymaximizing point each good purchased should yield the same marginal utility per dollar spent on that good Therefore each good should have an identical marginal benefittomarginalcost ratio If this were not true one good would promise more marginal enjoyment per dollar than some other good and funds would not be optimally allocated Although the reader is again warned against talking confidently about marginal utility what Equation 411 says is that an extra dollar should yield the same additional utility no matter which good it is spent on The common value for this extra utility is given by the Lagrange multiplier for the consumers budget constraint ie by λ Consequently λ can be regarded as the marginal utility of an extra dollar of consumption expenditure the marginal utility of income One final way to rewrite the necessary conditions for a maximum is pi 5 Uxi λ 412 for every good i that is bought To interpret this expression remember from Equation 411 that the Lagrange multiplier λ represents the marginal utility value of an extra dollar of income no matter where it is spent Therefore the ratio in Equation 412 com pares the extra utility value of one more unit of good i to this common value of a mar ginal dollar in spending To be purchased the utility value of an extra unit of a good must be worth in dollar terms the price the person must pay for it For example a high price for good i can only be justified if it also provides a great deal of extra utility At the margin therefore the price of a good reflects an individuals willingness to pay for one more unit This is a result of considerable importance in applied welfare economics because willingness to pay can be inferred from market reactions to prices In Chapter 5 we will see how this insight can be used to evaluate the welfare effects of price changes and in later chapters we will use this idea to discuss a variety of questions about the effi ciency of resource allocation 434 Corner solutions The firstorder conditions of Equations 48 hold exactly only for interior maxima for which some positive amount of each good is purchased As discussed in Chapter 2 when corner Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 123 solutions such as those illustrated in Figure 44 arise the conditions must be modified slightly4 In this case Equations 48 become xi 5 U xi 2 λpi 0 1i 5 1 n2 413 and if xi 5 U xi 2 λpi 0 414 then xi 5 0 415 To interpret these conditions we can rewrite Equation 414 as pi Uxi λ 416 Hence the optimal conditions are as before except that any good whose price 1pi2 exceeds its marginal value to the consumer will not be purchased 1xi 5 02 Thus the mathematical results conform to the commonsense idea that individuals will not purchase goods that they believe are not worth the money Although corner solutions do not provide a major focus for our analysis in this book the reader should keep in mind the possibilities for such solutions arising and the economic interpretation that can be attached to the optimal conditions in such cases 4Formally these conditions are called the KuhnTucker conditions for nonlinear programming EXAMPLE 41 CobbDouglas Demand Functions As we showed in Chapter 3 the CobbDouglas utility function is given by U1x y2 5 xαyβ 417 where for convenience5 we assume α 1 β 5 1 We can now solve for the utilitymaximizing values of x and y for any prices 1px py2 and income I Setting up the Lagrangian expression 5 xαyβ 1 λ1I 2 pxx 2 pyy2 418 yields the firstorder conditions x 5 αxα21 yβ 2 λpx 5 0 y 5 βxαyβ21 2 λpy 5 0 λ 5 I 2 pxx 2 pyy 5 0 419 Taking the ratio of the first two terms shows that 5As we discussed in Chapter 3 the exponents in the CobbDouglas utility function can always be normalized to sum to 1 because U11α1β2 is a monotonic transformation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 124 Part 2 Choice and Demand αy βx 5 px py 420 or pyy 5 β α pxx 5 1 2 α α pxx 421 where the final equation follows because α 1 β 5 1 Substitution of this firstorder condition in Equation 421 into the budget constraint gives I 5 pxx 1 pyy 5 pxx 1 1 2 α α pxx 5 pxxa1 1 1 2 α α b 5 1 αpxx 422 solving for x yields x 5 αI px 423 and a similar set of manipulations would give y 5 βI py 424 These results show that an individual whose utility function is given by Equation 417 will always choose to allocate α proportion of his or her income to buying good x ie px xI 5 α and β propor tion to buying good y 1pyyI 5 β2 Although this feature of the CobbDouglas function often makes it easy to work out simple problems it does suggest that the function has limits in its ability to explain actual consumption behavior Because the share of income devoted to particular goods often changes significantly in response to changing economic conditions a more general functional form may pro vide insights not provided by the CobbDouglas function We illustrate a few possibilities in Example 42 and the general topic of budget shares is taken up in more detail in the Extensions to this chapter Numerical example First however lets look at a specific numerical example for the Cobb Douglas case Suppose that x sells for 1 and y sells for 4 and that total income is 8 Succinctly then assume that px 5 1 py 5 4 I 5 8 Suppose also that α 5 β 5 05 so that this individual splits his or her income equally between these two goods Now the demand Equations 423 and 424 imply x 5 αI px 5 05I px 5 05 182 1 5 4 y 5 βI py 5 05I py 5 05 182 4 5 1 425 and at these optimal choices utility 5 x05y05 5 142 05 112 05 5 2 426 We can compute the value for the Lagrange multiplier associated with this income allocation by using Equation 419 λ 5 αxα21y β px 5 05 142 205 112 05 1 5 025 427 This value implies that each small change in income will increase utility by approximately one fourth of that amount Suppose for example that this person had 1 percent more income 808 In this case he or she would choose x 5 404 and y 5 101 and utility would be 40405 10105 5 202 Hence a 008 increase in income increased utility by 002 as predicted by the fact that λ 5 025 QUERY Would a change in py affect the quantity of x demanded in Equation 423 Explain your answer mathematically Also develop an intuitive explanation based on the notion that the share of income devoted to good y is given by the parameter of the utility function β Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 125 EXAMPLE 42 CES Demand To illustrate cases in which budget shares are responsive to relative prices lets look at three spe cific examples of the CES function Case 1 δ 05 In this case utility is U1x y2 5 x05 1 y05 428 Setting up the Lagrangian expression 5 x05 1 y05 1 λ1I 2 pxx 2 pyy2 429 yields the following firstorder conditions for a maximum x 5 05x205 2 λpx 5 0 y 5 05y205 2 λpy 5 0 430 λ 5 I 2 pxx 2 pyy 5 0 Division of the first two of these shows that a y xb 05 5 px py 431 By substituting this into the budget constraint and doing some messy algebraic manipulation we can derive the demand functions associated with this utility function x 5 I px31 1 1pxpy2 4 432 y 5 I py31 1 1pypx24 433 Price responsiveness In these demand functions notice that the share of income spent on say good xthat is pxxI 5 1 31 1 1pxpy2 4is not a constant it depends on the price ratio pxpy The higher the relative price of x the smaller the share of income spent on that good In other words the demand for x is so responsive to its own price that an increase in the price reduces total spending on x That the demand for x is price responsive can also be illustrated by comparing the implied exponent on px in the demand function given by Equation 432 1222 to that from Equation 423 1212 In Chapter 5 we will discuss this observation more fully when we examine the elasticity concept in detail Case 2 δ 1 Alternatively lets look at a demand function with less substitutability6 than the CobbDouglas If δ 5 21 the utility function is given by U1x y2 5 2x21 2 y21 434 and it is easy to show that the firstorder conditions for a maximum require y x 5 a px py b 05 435 6One way to measure substitutability is by the elasticity of substitution which for the CES function is given by σ 5 1 11 2 δ2 Here δ 5 05 implies σ 5 2 δ 5 0 the CobbDouglas implies σ 5 1 and δ 5 21 implies σ 5 05 See also the discussion of the CES function in connection with the theory of production in Chapter 9 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 126 Part 2 Choice and Demand Again substitution of this condition into the budget constraint together with some messy alge bra yields the demand functions x 5 I px31 1 1pypx2 054 y 5 I py31 1 1pxpy2 054 436 That these demand functions are less price responsive can be seen in two ways First now the share of income spent on good xthat is pxxI 5 1 31 1 1pypx2 054responds positively to increases in px As the price of x increases this individual cuts back only modestly on good x thus total spending on that good increases That the demand functions in Equation 436 are less price responsive than the CobbDouglas is also illustrated by the relatively small implied expo nents of each goods own price 12052 Case 3 δ This is the important case in which x and y must be consumed in fixed propor tions Suppose for example that each unit of y must be consumed together with exactly 4 units of x The utility function that represents this situation is U1x y2 5 min 1x 4y2 437 In this situation a utilitymaximizing person will choose only combinations of the two goods for which x 5 4y that is utility maximization implies that this person will choose to be at a vertex of his or her Lshaped indifference curves Because of the shape of these indifference curves calcu lus cannot be used to solve this problem Instead one can adopt the simple procedure of substi tuting the utilitymaximizing condition directly into the budget constraint I 5 pxx 1 pyy 5 pxx 1 py x 4 5 1px 1 025py2x 438 Hence x 5 I px 1 025py 439 and similar substitutions yield y 5 I 4px 1 py 440 In this case the share of a persons budget devoted to say good x rises rapidly as the price of x increases because x and y must be consumed in fixed proportions For example if we use the values assumed in Example 41 1px 5 1 py 5 4 I 5 82 Equations 439 and 440 would predict x 5 4 y 5 1 and as before half of the individuals income would be spent on each good If we instead use px 5 2 py 5 4 and I 5 8 then x 5 83 y 5 23 and this person spends two thirds 3pxxI 5 12 8328 5 234 of his or her income on good x Trying a few other numbers suggests that the share of income devoted to good x approaches 1 as the price of x increases7 QUERY Do changes in income affect expenditure shares in any of the CES functions discussed here How is the behavior of expenditure shares related to the homothetic nature of this function 44 INDIRECT UTILITY FUNCTION Examples 41 and 42 illustrate the principle that it is often possible to manipulate the first order conditions for a constrained utilitymaximization problem to solve for the optimal values of x1 x2 xn These optimal values in general will depend on the prices of all the goods and on the individuals income That is 7These relationships for the CES function are pursued in more detail in Problem 49 and in Extension E43 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 127 x 1 5 x1 1p1 p2 pn I2 x 2 5 x2 1p1 p2 pn I2 x n 5 xn1p1 p2 pn I2 441 In the next chapter we will analyze in more detail this set of demand functions which show the dependence of the quantity of each xi demanded on p1 p2 pn and I Here we use the optimal values of the xs from Equation 442 to substitute in the original utility function to yield maximum utility 5 U3x 1 1p1 pn I2 x 2 1p1 pn I2 x n 1p1 pn I2 4 442 5 V1p1 p2 pn I2 443 In words because of the individuals desire to maximize utility given a budget constraint the optimal level of utility obtainable will depend indirectly on the prices of the goods being bought and the individuals income This dependence is reflected by the indirect utility function V If either prices or income were to change the level of utility that could be attained would also be affected The indirect utility function is the first example of a value function that we encounter in this book As described in Chapter 2 such a function solves out all of the endogenous variables in an optimization problem leaving the optimal value obtainable as a function only of exogenous variables usually prices Such an approach can provide a convenient shortcut to exploring how changes in the exogenous variables affect the bottom line outcome without having to redo the original optimization problem The envelope theorem see Chapter 2 can also be applied to such a value function often providing surprising insights Unfortunately applying the envelope theorem to the indirect utility function yields relatively minor returns in terms of significant insights The main result Roys identity is discussed briefly in the Extensions to Chapter 5 However we will encounter many more examples of value functions later and applying the envelope theorem to these will usually yield significant rewards 45 THE LUMP SUM PRINCIPLE Many economic insights stem from the recognition that utility ultimately depends on the income of individuals and on the prices they face One of the most important of these is the socalled lump sum principle that illustrates the superiority of taxes on a persons general purchasing power to taxes on specific goods A related insight is that general income grants to lowincome people will raise utility more than will a similar amount of money spent subsidizing specific goods The intuition behind these results derives directly from the utilitymaximization hypothesis an income tax or subsidy leaves the individual free to decide how to allocate whatever final income he or she has On the other hand taxes or subsidies on specific goods both change a persons purchasing power and distort his or her choices because of the artificial prices incorporated in such schemes Hence general income taxes and subsidies are to be preferred if efficiency is an important criterion in social policy The lump sum principle as it applies to taxation is illustrated in Figure 45 Initially this person has an income of I and is choosing to consume the combination x y A tax on good x would raise its price and the utilitymaximizing choice would shift to combination x1 y1 Tax collections would be t x1 where t is the tax rate imposed on good x Alternatively Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 128 Part 2 Choice and Demand an income tax that shifted the budget constraint inward to I would also collect this same amount of revenue8 But the utility provided by the income tax 1U22 exceeds that provided by the tax on x alone 1U12 Hence we have shown that the utility burden of the income tax is smaller A similar argument can be used to illustrate the superiority of income grants to subsidies on specific goods 8Because I 5 1px 1 t2x1 1 pyy1 we have Ir 5 I 2 tx1 5 pxx1 1 pyy1 which shows that the budget constraint with an equalsize income tax also passes through the point x1 y1 EXAMPLE 43 Indirect Utility and the Lump Sum Principle In this example we use the notion of an indirect utility function to illustrate the lump sum principle as it applies to taxation First we have to derive indirect utility functions for two illustrative cases Case 1 CobbDouglas In Example 41 we showed that for the CobbDouglas utility function with α 5 β 5 05 optimal purchases are x 5 I 2px y 5 I 2py 444 A tax on good x would shift the utilitymaximizing choice from x y to x1 y1 An income tax that collected the same amount would shift the budget constraint to I Utility would be higher 1U22 with the income tax than with the tax on x alone 1U12 Quantity of x Quantity of y y1 U1 U2 U3 I I x1 x2 x y2 y FIGURE 45 The Lump Sum Principle of Taxation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 129 Thus the indirect utility function in this case is V 1px py I2 5 U1x y2 5 1x2 05 1y2 05 5 I 2p05 x p05 y 445 Notice that when px 5 1 py 5 4 and I 5 8 we have V 5 8 12 1 22 5 2 which is the utility that we calculated before for this situation Case 2 Fixed proportions In the third case of Example 42 we found that x 5 I px 1 025py y 5 I 4px 1 py 446 Thus in this case indirect utility is given by V 1px py I2 5 min 1x 4y2 5 x 5 I px 1 025py 5 4y 5 4 4px 1 py 5 I px 1 025py 447 with px 5 1 py 5 4 and I 5 8 indirect utility is given by V 5 4 which is what we calculated before The lump sum principle Consider first using the CobbDouglas case to illustrate the lump sum principle Suppose that a tax of 1 were imposed on good x Equation 445 shows that indirect utility in this case would fall from 2 to 141 35 8 12 205 22 4 Because this person chooses x 5 2 with the tax total tax collections will be 2 Therefore an equalrevenue income tax would reduce net income to 6 and indirect utility would be 15 35 6 12 1 22 4 Thus the income tax is a clear improvement in utility over the case where x alone is taxed The tax on good x reduces utility for two reasons It reduces a persons purchasing power and it biases his or her choices away from good x With income taxation only the first effect is felt and so the tax is more efficient9 The fixedproportions case supports this intuition In that case a 1 tax on good x would reduce indirect utility from 4 to 83 35 8 12 1 12 4 In this case x 5 83 and tax collections would be 83 An income tax that collected 83 would leave this consumer with 163 in net income and that income would yield an indirect utility of V 5 83 35 11632 11 1 12 4 Hence aftertax utility is the same under both the excise and income taxes The reason the lump sum principle does not hold in this case is that with fixedproportions utility the excise tax does not distort choices because this persons preferences require the goods to be consumed in fixed pro portions and the tax will not bias choices away from this outcome QUERY Both indirect utility functions illustrated here show that a doubling of income and all prices would leave indirect utility unchanged Explain why you would expect this to be a property of all indirect utility functions That is explain why the indirect utility function is homogeneous of degree zero in all prices and income 46 EXPENDITURE MINIMIZATION In Chapter 2 we pointed out that many constrained maximum problems have associated dual constrained minimum problems For the case of utility maximization the associated dual minimization problem concerns allocating income in such a way as to achieve a given utility level with the minimal expenditure This problem is clearly analogous to the primary utilitymaximization problem but the goals and constraints of the problems have 9This discussion assumes that there are no incentive effects of income taxationprobably not a good assumption Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 130 Part 2 Choice and Demand been reversed Figure 46 illustrates this dual expenditureminimization problem There the individual must attain utility level U2 this is now the constraint in the problem Three possible expenditure amounts 1E1 E2 and E3 are shown as three budget constraint lines in the figure Expenditure level E1 is clearly too small to achieve U2 hence it cannot solve the dual problem With expenditures given by E3 the individual can reach U2 at either of the two points B or C but this is not the minimal expenditure level required Rather E2 clearly provides just enough total expenditures to reach U2 at point A and this is in fact the solution to the dual problem By comparing Figures 42 and 46 it is obvious that both the primary utilitymaximization approach and the dual expenditureminimization approach yield the same solution 1x y2 they are simply alternative ways of viewing the same process Often the expenditureminimization approach is more useful however because expenditures are directly observable whereas utility is not 461 A mathematical statement More formally the individuals dual expenditureminimization problem is to choose x1 x2 xn to minimize total expenditures 5 E 5 p1x1 1 p2x2 1 c1 pnxn 448 subject to the constraint utility 5 U 5 U 1x1 x2 xn2 449 The dual of the utilitymaximization problem is to attain a given utility level 1U22 with minimal expenditures An expenditure level of E1 does not permit U2 to be reached whereas E3 provides more spending power than is strictly necessary With expenditure E2 this person can just reach U2 by consuming x and y Quantity of x Quantity of y B E3 E2 U2 E1 C A x y FIGURE 46 The Dual Expenditure Minimization Problem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 131 The optimal amounts of x1 x2 xn chosen in this problem will depend on the prices of the various goods 1p1 p2 pn2 and on the required utility level U If any of the prices were to change or if the individual had a different utility target then another commodity bundle would be optimal This dependence can be summarized by an expenditure function D E F I N I T I O N Expenditure function The individuals expenditure function shows the minimal expenditures necessary to achieve a given utility level for a particular set of prices That is minimal expenditures 5 E 1p1 p2 pn U2 450 This definition shows that the expenditure function and the indirect utility function are inverse functions of one another compare Equations 443 and 450 Both depend on market prices but involve different constraints income or utility The expenditure function is the second value function encountered in this book and as we shall see in Chapters 5 and 6 it is far more useful than the indirect utility function This is primarily because application of the envelope theorem to the expenditure function provides a direct route to showing practically all of the key elements of demand theory Before providing a detailed listing of the general properties of expenditure functions lets first look at a few examples EXAMPLE 44 Two Expenditure Functions There are two ways one might compute an expenditure function The first most straightforward method would be to state the expenditureminimization problem directly and apply the Lagrang ian technique Some of the problems at the end of this chapter ask you to do precisely that Here however we will adopt a more streamlined procedure by taking advantage of the relationship between expenditure functions and indirect utility functions Because these two functions are inverses of each other calculation of one greatly facilitates the calculation of the other We have already calculated indirect utility functions for two important cases in Example 43 Retrieving the related expenditure functions is simple algebra Case 1 CobbDouglas utility Equation 445 shows that the indirect utility function in the twogood CobbDouglas case is V 1px py I2 5 I 2p05 x p05 y 451 If we now interchange the role of utility which we will now treat as the utility target denoted by U and income which we will now term expenditures E and treat as a function of the parame ters of this problem then we have the expenditure function E 1px py U2 5 2p05 x p05 y U 452 Checking this against our former results now we use a utility target of U 5 2 with again px 5 1 and py 5 4 With these parameters Equation 452 shows that the required minimal expenditures are 8 15 2 105 405 22 Not surprisingly both the primal utilitymaximization problem and the dual expenditureminimization problem are formally identical Case 2 Fixed proportions For the fixedproportions case Equation 447 gave the indirect utility function as V 1px py I2 5 I px 1 025py 453 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 132 Part 2 Choice and Demand If we again switch the role of utility and expenditures we quickly derive the expenditure function E 1px py U2 5 1px 1 025py2U 454 A check of the hypothetical values used in Example 43 1px 5 1 py 5 4 U 5 42 again shows that it would cost 8 35 11 1 025 42 44 to reach the utility target of 4 Compensating for a price change These expenditure functions allow us to investigate how a person might be compensated for a price change Specifically suppose that the price of good y were to increase from 4 to 5 This would clearly reduce a persons utility so we might ask what amount of monetary compensation would mitigate the harm Because the expenditure function requires utility to be held constant it provides a direct estimate of this amount Specifically in the CobbDouglas case expenditures would have to be increased from 8 to 894 15 2 1 505 22 to provide enough extra purchasing power to precisely compensate for this price increase With fixed proportions expenditures would have to be increased from 8 to 9 to compensate for the price increase Hence the compensations are about the same in these simple cases There is one important difference between the two examples however In the fixed proportions case the 1 of extra compensation simply permits this person to return to his or her previous consumption bundle 1x 5 4 y 5 12 That is the only way to restore utility to U 5 4 for this rigid person In the CobbDouglas case however this person will not use the extra com pensation to revert to his or her previous consumption bundle Instead utility maximization will require that the 894 be allocated so that x 5 447 and y 5 0894 This will still provide a utility level of U 5 2 but this person will economize on the now more expensive good y In the next chapter we will pursue this analysis of the welfare effects of price changes in much greater detail QUERY How should a person be compensated for a price decrease What sort of compensation would be required if the price of good y fell from 4 to 3 47 PROPERTIES OF EXPENDITURE FUNCTIONS Because expenditure functions are widely used in both theoretical and applied economics it is important to understand a few of the properties shared by all such functions Here we look at three properties All these follow directly from the fact that expenditure functions are based on individual expenditure minimization 1 Homogeneity For both of the functions illustrated in Example 44 a doubling of all prices will precisely double the value of required expenditures Technically these expen diture functions are homogeneous of degree one in all prices10 This is a general prop erty of expenditure functions Because the individuals budget constraint is linear in prices any proportional increase in all prices will require a similar increase in expen ditures in order to permit the person to buy the same utilitymaximizing commodity bundle that was chosen before the price increase In Chapter 5 we will see that for this reason demand functions are homogeneous of degree zero in all prices and income 2 Expenditure functions are nondecreasing in prices This property can be succinctly sum marized by the mathematical statement E pi 0 for every good i 455 10As described in Chapter 2 the function f 1x1 x2 xn2 is said to be homogeneous of degree k if f 1tx1 tx2 txn2 5 tkf 1x1 x2 xn2 In this case k 5 1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 133 This seems intuitively obvious Because the expenditure function reports the mini mum expenditure necessary to reach a given utility level an increase in any price must increase this minimum More formally suppose the price of one good increases and that all other prices stay the same Let A represent the bundle of goods purchased before the price increase and B the bundle purchased after the price increase Clearly bundle B costs more after the price increase than it did before the increase because one of the goods in that bundle has had an increase in price and the prices of all of the other goods have stayed the same Now compare the cost of bundle B before the price increase to the cost of bundle A Bundle A must have cost less because of the expenditure minimization assumptionthat is A was the cost minimizing way to achieve the utility target Hence we have the following string of logicbundle A costs less than bundle B before the price increase which costs less than bundle B after the price increase So the bundle chosen after the price increase B must cost more than the one chosen before the increase A A similar chain of logic could be used to show that a decrease in price should cause expenditures to decrease or possibly stay the same 3 Expenditure functions are concave in prices In Chapter 2 we discussed concave func tions which are defined as functions that always lie below tangents to them Although the technical mathematical conditions that describe such functions are complicated it is relatively simple to show how the concept applies to expenditure functions by con sidering the variation in a single price Figure 47 shows an individuals expenditures as a function of the single price p1 At the initial price p 1 this persons expenditures are given by E1p 1 2 Now consider prices higher or lower than p 1 If this person At p 1 this person spends E 1p 1 2 If he or she continues to buy the same set of goods as p1 changes then expenditures would be given by Epseudo Because his or her consumption patterns will likely change as p1 changes actual expenditures will be less than this Ep1 Ep1 p1 Ep1 E pseudo Ep1 FIGURE 47 Expenditure Functions Are Concave in Prices Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 134 Part 2 Choice and Demand continued to buy the same bundle of goods expenditures would increase or decrease linearly as this price changed This would give rise to the pseudoexpenditure function Epseudo in the figure This line shows a level of expenditures that would allow this person to buy the original bundle of goods despite the changing value of p1 If as seems more likely this person adjusted his or her purchases as p1 changed we know because of expenditure minimization that actual expenditures would be less than these pseudo amounts Hence the actual expenditure function E will lie everywhere below Epseudo and the function will be concave11 The concavity of the expenditure function is a useful property for a number of applications especially those related to the substitution effect from price changes see Chapter 5 Summary In this chapter we explored the basic economic model of util ity maximization subject to a budget constraint Although we approached this problem in a variety of ways all these approaches led to the same basic result To reach a constrained maximum an individual should spend all available income and should choose a com modity bundle such that the MRS between any two goods is equal to the ratio of those goods market prices This basic tangency will result in the individual equating the ratios of the marginal utility to market price for every good that is actually consumed Such a result is common to most constrained optimization problems The tangency conditions are only the firstorder condi tions for a unique constrained maximum however To ensure that these conditions are also sufficient the indi viduals indifference curve map must exhibit a diminish ing MRS In formal terms the utility function must be strictly quasiconcave The tangency conditions must also be modified to allow for corner solutions in which the optimal level of consumption of some goods is zero In this case the ratio of marginal utility to price for such a good will be below the common marginal benefitmarginal cost ratio for goods actually bought A consequence of the assumption of constrained utility maximization is that the individuals optimal choices will depend implicitly on the parameters of his or her budget constraint That is the choices observed will be implicit functions of all prices and income Therefore utility will also be an indirect function of these parameters The dual to the constrained utilitymaximization prob lem is to minimize the expenditure required to reach a given utility target Although this dual approach yields the same optimal solution as the primal constrained maximum problem it also yields additional insight into the theory of choice Specifically this approach leads to expenditure functions in which the spending required to reach a given utility target depends on goods market prices Therefore expenditure functions are in principle measurable Problems 41 Each day Paul who is in third grade eats lunch at school He likes only Twinkies t and soda s and these provide him a utility of utility 5 U1t s2 5 ts a If Twinkies cost 010 each and soda costs 025 per cup how should Paul spend the 1 his mother gives him to maximize his utility b If the school tries to discourage Twinkie consumption by increasing the price to 040 by how much will Pauls mother have to increase his lunch allowance to provide him with the same level of utility he received in part a 42 a A young connoisseur has 600 to spend to build a small wine cellar She enjoys two vintages in particular a 2001 French Bordeaux 1wF2 at 40 per bottle and a less expen sive 2005 California varietal wine 1wC2 priced at 8 If her utility is 11One result of concavity is that fii 5 2Ep2 i 0 This is precisely what Figure 47 shows Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 135 U1wF wC2 5 w23 F w13 C then how much of each wine should she purchase b When she arrived at the wine store this young oenologist discovered that the price of the French Bordeaux had fallen to 20 a bottle because of a decrease in the value of the euro If the price of the California wine remains stable at 8 per bottle how much of each wine should our friend purchase to maximize utility under these altered conditions c Explain why this wine fancier is better off in part b than in part a How would you put a monetary value on this utility increase 43 a On a given evening J P enjoys the consumption of cigars c and brandy b according to the function U1c b2 5 20c 2 c2 1 18b 2 3b2 How many cigars and glasses of brandy does he consume during an evening Cost is no object to J P b Lately however J P has been advised by his doctors that he should limit the sum of glasses of brandy and cigars consumed to 5 How many glasses of brandy and cigars will he consume under these circumstances 44 a Mr Odde Ball enjoys commodities x and y according to the utility function U1x y2 5 x2 1 y2 Maximize Mr Balls utility if px 5 3 py 5 4 and he has 50 to spend Hint It may be easier here to maximize U2 rather than U Why will this not alter your results b Graph Mr Balls indifference curve and its point of tangency with his budget constraint What does the graph say about Mr Balls behavior Have you found a true maximum 45 Mr A derives utility from martinis m in proportion to the number he drinks U1m2 5 m Mr A is particular about his martinis however He only enjoys them made in the exact proportion of two parts gin g to one part vermouth v Hence we can rewrite Mr As utility function as U1m2 5 U1g v2 5 min a g 2 vb a Graph Mr As indifference curve in terms of g and v for various levels of utility Show that regardless of the prices of the two ingredients Mr A will never alter the way he mixes martinis b Calculate the demand functions for g and v c Using the results from part b what is Mr As indirect utility function d Calculate Mr As expenditure function for each level of utility show spending as a function of pg and pv Hint Because this problem involves a fixedproportions utility function you cannot solve for utilitymaximizing decisions by using calculus 46 Suppose that a fastfood junkie derives utility from three goodssoft drinks x hamburgers y and ice cream sun daes zaccording to the CobbDouglas utility function U1x y z2 5 x05y05 11 1 z2 05 Suppose also that the prices for these goods are given by px 5 1 py 5 4 and pz 5 8 and that this consumers income is given by I 5 8 a Show that for z 5 0 maximization of utility results in the same optimal choices as in Example 41 Show also that any choice that results in z 0 even for a fractional z reduces utility from this optimum b How do you explain the fact that z 5 0 is optimal here c How high would this individuals income have to be for any z to be purchased 47 The lump sum principle illustrated in Figure 45 applies to transfer policy and taxation This problem examines this application of the principle a Use a graph similar to Figure 45 to show that an income grant to a person provides more utility than does a subsidy on good x that costs the same amount to the government b Use the CobbDouglas expenditure function presented in Equation 452 to calculate the extra purchasing power needed to increase this persons utility from U 5 2 to U 5 3 c Use Equation 452 again to estimate the degree to which good x must be subsidized to increase this persons utility from U 5 2 to U 5 3 How much would this subsidy cost the government How would this cost compare with the cost calculated in part b d Problem 410 asks you to compute an expenditure function for a more general CobbDouglas utility function than the one used in Example 44 Use that expenditure function to resolve parts b and c here Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 136 Part 2 Choice and Demand for the case α 5 03 a figure close to the fraction of income that lowincome people spend on food e How would your calculations in this problem have changed if we had used the expenditure function for the fixedproportions case Equation 454 instead 48 Two of the simplest utility functions are 1 Fixed proportions U1x y2 5 min 3x y4 2 Perfect substitutes U1x y2 5 x 1 y a For each of these utility functions compute the following Demand functions for x and y Indirect utility function Expenditure function b Discuss the particular forms of these functions you calculatedwhy do they take the specific forms they do 49 Suppose that we have a utility function involving two goods that is linear of the form U1x y2 5 ax 1 by Calculate the expenditure function for this utility function Hint The expenditure function will have kinks at various price ratios Analytical Problems 410 CobbDouglas utility In Example 41 we looked at the CobbDouglas utility func tion U1x y2 5 xα y12α where 0 α 1 This problem illus trates a few more attributes of that function a Calculate the indirect utility function for this Cobb Douglas case b Calculate the expenditure function for this case c Show explicitly how the compensation required to offset the effect of an increase in the price of x is related to the size of the exponent α 411 CES utility The CES utility function we have used in this chapter is given by U1x y2 5 xδ δ 1 yδ δ a Show that the firstorder conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion x y 5 a px py b 11δ212 b Show that the result in part a implies that individuals will allocate their funds equally between x and y for the CobbDouglas case 1δ 5 02 as we have shown before in several problems c How does the ratio pxxpyy depend on the value of δ Explain your results intuitively For further details on this function see Extension E43 d Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated 412 StoneGeary utility Suppose individuals require a certain level of food x to remain alive Let this amount be given by x0 Once x0 is purchased indi viduals obtain utility from food and other goods y of the form U1x y2 5 1x 2 x02 αyβ where α 1 β 5 1 a Show that if I pxx0 then the individual will maximize utility by spending α 1I 2 pxx02 1 pxx0 on good x and β1I 2 pxx02 on good y Interpret this result b How do the ratios pxxI and pyyI change as income increases in this problem See also Extension E42 for more on this utility function 413 CES indirect utility and expenditure functions In this problem we will use a more standard form of the CES utility function to derive indirect utility and expenditure functions Suppose utility is given by U1x y2 5 1xδ 1 yδ2 1δ in this function the elasticity of substitution σ 5 1 11 2 δ2 a Show that the indirect utility function for the utility function just given is V 5 I1pr x 1 pr y2 21r where r 5 δ 1δ 2 12 5 1 2 σ b Show that the function derived in part a is homo geneous of degree zero in prices and income c Show that this function is strictly increasing in income d Show that this function is strictly decreasing in any price e Show that the expenditure function for this case of CES utility is given by E 5 V 1 pr x 1 pr y2 1r f Show that the function derived in part e is homogeneous of degree one in the goods prices g Show that this expenditure function is increasing in each of the prices h Show that the function is concave in each price Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 137 Behavioral Problem 414 Altruism Michele who has a relatively high income I has altruis tic feelings toward Sofia who lives in such poverty that she essentially has no income Suppose Micheles preferences are represented by the utility function U1 1c1 c22 5 c12a 1 ca 2 where c1 and c2 are Michele and Sofias consumption levels appearing as goods in a standard CobbDouglas utility func tion Assume that Michele can spend her income either on her own or Sofias consumption through charitable donations and that 1 buys a unit of consumption for either thus the prices of consumption are p1 5 p2 5 1 a Argue that the exponent a can be taken as a measure of the degree of Micheles altruism by providing an interpretation of extremes values a 5 0 and a 5 1 What value would make her a perfect altruist regarding others the same as oneself b Solve for Micheles optimal choices and demonstrate how they change with a c Solve for Micheles optimal choices under an income tax at rate t How do her choices change if there is a charitable deduction so income spent on charitable deductions is not taxed Does the charitable deduction have a bigger incentive effect on more or less altruistic people d Return to the case without taxes for simplicity Now suppose that Micheles altruism is represented by the utility function U1 1c1 U22 5 c12a 1 Ua 2 which is similar to the representation of altruism in Extension E34 in the previous chapter According to this specification Michele cares directly about Sofias utility level and only indirectly about Sofias consumption level 1 Solve for Micheles optimal choices if Sofias utility function is symmetric to Micheles U2 1c2 U12 5 c12a 2 Ua 1 Compare your answer with part b Is Michele more or less charitable under the new specification Explain 2 Repeat the previous analysis assuming Sofias utility function is U2 1c22 5 c2 Suggestions for Further Reading Barten A P and Volker Böhm Consumer Theory In K J Arrow and M D Intriligator Eds Handbook of Mathematical Economics vol II Amsterdam NorthHolland 1982 Sections 10 and 11 have compact summaries of many of the con cepts covered in this chapter Deaton A and J Muelbauer Economics and Consumer Behavior Cambridge UK Cambridge University Press 1980 Section 25 provides a nice geometric treatment of duality concepts Dixit A K Optimization in Economic Theory Oxford UK Oxford University Press 1990 Chapter 2 provides several Lagrangian analyses focusing on the CobbDouglas utility function Hicks J R Value and Capital Oxford UK Clarendon Press 1946 Chapter II and the Mathematical Appendix provide some early suggestions of the importance of the expenditure function Luenberger D G Microeconomic Theory New York McGraw Hill 1992 In Chapter 4 the author shows several interesting relationships between his benefit function see Problem 315 and the more standard expenditure function This chapter also offers insights on a number of unusual preference structures MasColell A M D Whinston and J R Green Microeco nomic Theory Oxford UK Oxford University Press 1995 Chapter 3 contains a thorough analysis of utility and expenditure functions Samuelson Paul A Foundations of Economic Analysis Cam bridge MA Harvard University Press 1947 Chapter V and Appendix A provide a succinct analysis of the firstorder conditions for a utility maximum The appendix pro vides good coverage of secondorder conditions Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 A useful although fairly difficult treatment of duality in con sumer theory Theil H Theory and Measurement of Consumer Demand Amsterdam NorthHolland 1975 Good summary of basic theory of demand together with implica tions for empirical estimation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 138 EXTENSIONS BUdget ShareS The nineteenthcentury economist Ernst Engel was one of the first social scientists to intensively study peoples actual spending patterns He focused specifically on food consumption His find ing that the fraction of income spent on food decreases as income increases has come to be known as Engels law and has been con firmed in many studies Engels law is such an empirical regularity that some economists have suggested measuring poverty by the fraction of income spent on food Two other interesting applica tions are 1 the study by Hayashi 1995 showing that the share of income devoted to foods favored by the elderly is much higher in twogeneration households than in onegeneration house holds and 2 findings by Behrman 1989 from lessdeveloped countries showing that peoples desires for a more varied diet as their incomes increase may in fact result in reducing the fraction of income spent on particular nutrients In the remainder of this extension we look at some evidence on budget shares denoted by si 5 pixiI2 together with a bit more theory on the topic E41 The variability of budget shares Table E41 shows some recent budget share data from the United States Engels law is clearly visible in the table As income increases families spend a smaller proportion of their funds on food Other important variations in the table include the declining share of income spent on healthcare needs and the much larger share of income devoted to retirement plans by higherincome people Interestingly the shares of income devoted to shelter and transportation are relatively constant over the range of income shown in the table apparently highincome people buy bigger houses and cars The variable income shares in Table E41 illustrate why the CobbDouglas utility function is not useful for detailed empirical studies of household behavior When utility is given by U 1x y2 5 xαyβ where α 1 β 5 1 the implied demand equations are x 5 αIpx and y 5 βIpy Therefore sx 5 pxx I 5 α and sy 5 pyy I 5 β i and budget shares are constant for all observed income levels and relative prices Because of this shortcoming economists have investigated a number of other possible forms for the utility function that permit more flexibility TABLE E41 BUDGET SHARES OF US HOUSEHOLDS 2008 Annual Income 10000 2 14999 40000 2 49999 Over 70000 Expenditure Item Food 157 134 118 Shelter 231 212 193 Utilities fuel and public services 112 86 58 Transportation 141 178 168 Health insurance 53 40 26 Other healthcare expenses 26 28 23 Entertainment including alcohol 46 52 58 Education 23 12 26 Insurance and pensions 22 85 146 Other apparel personal care other housing expenses and misc 189 173 184 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 4 Utility Maximization and Choice 139 E42 Linear expenditure system A generalization of the CobbDouglas function that incor porates the idea that certain minimal amounts of each good must be bought by an individual 1x0 y02 is the utility function U1x y2 5 1x 2 x02 α 1y 2 y02 β ii for x x0 and y y0 where again α 1 β 5 1 Demand functions can be derived from this utility func tion in a way analogous to the CobbDouglas case by intro ducing the concept of supernumerary income 1I2 which represents the amount of purchasing power remaining after purchasing the minimum bundle I 5 I 2 pxx0 2 pyy0 iii Using this notation the demand functions are x 5 1pxx0 1 αI2 px y 5 1pyy0 1 βI2 py iv In this case the individual then spends a constant fraction of supernumerary income on each good once the minimum bundle has been purchased Manipulation of Equation iv yields the share equations sx 5 α 1 1βpxx0 2 αpyy02 I sy 5 β 1 1αpyy0 2 βpxx02 I v which show that this demand system is not homothetic Inspection of Equation v shows the unsurprising result that the budget share of a good is positively related to the mini mal amount of that good needed and negatively related to the minimal amount of the other good required Because the notion of necessary purchases seems to accord well with realworld observation this linear expenditure system LES which was first developed by Stone 1954 is widely used in empirical studies The utility function in Equation ii is also called a StoneGeary utility function Traditional purchases One of the most interesting uses of the LES is to examine how its notion of necessary purchases changes as conditions change For example Oczkowski and Philip 1994 study how access to modern consumer goods may affect the share of income that individuals in transitional economies devote to traditional local items They show that villagers of Papua New Guinea reduce such shares significantly as outside goods become increasingly accessible Hence such improvements as better roads for moving goods provide one of the primary routes by which traditional cultural practices are undermined E43 CES utility In Chapter 3 we introduced the CES utility function U1x y2 5 xδ δ 1 yδ δ vi for δ 1 δ 2 0 The primary use of this function is to illus trate alternative substitution possibilities as reflected in the value of the parameter δ Budget shares implied by this utility function provide a number of such insights Manipulation of the firstorder conditions for a constrained utility maximum with the CES function yields the share equations sx 5 1 31 1 1pypx2 K4 sy 5 1 31 1 1pxpy2 K4 vii where K 5 δ 1δ 2 12 The homothetic nature of the CES function is shown by the fact that these share expressions depend only on the price ratio pxpy Behavior of the shares in response to changes in rel ative prices depends on the value of the parameter K For the CobbDouglas case δ 5 0 and so K 5 0 and sx 5 sy 5 12 When δ 0 substitution possibilities are great and K 0 In this case Equation vii shows that sx and pxpy move in oppo site directions If pxpy increases the individual substitutes y for x to such an extent that sx decreases Alternatively if δ 0 then substitution possibilities are limited K 0 and sx and pxpy move in the same direction In this case an increase in pxpy causes only minor substitution of y for x and sx actually increases because of the relatively higher price of good x North American free trade CES demand functions are most often used in largescale computer models of general equilibrium see Chapter 13 that economists use to evaluate the impact of major economic changes Because the CES model stresses that shares respond to changes in relative prices it is particularly appropriate for looking at innovations such as changes in tax policy or in international trade restrictions where changes in relative prices are likely One important area of such research has been on the impact of the North American Free Trade Agreement for Canada Mexico and the United States In general these models find that all the countries involved might be expected to gain from the agreement but that Mexicos gains may be the greatest because it is experiencing the greatest change in relative prices Kehoe and Kehoe 1995 present a number of computable equilibrium models that economists have used in these examinations1 1The research on the North American Free Trade Agreement is discussed in more detail in the Extensions to Chapter 13 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 140 Part 2 Choice and Demand E44 The almost ideal demand system An alternative way to study budget shares is to start from a specific expenditure function This approach is especially convenient because the envelope theorem shows that budget shares can be derived directly from expenditure functions through logarithmic differentiation for more details see Chapter 5 ln E 1px py V2 ln px 5 1 E 1px py V2 E px px ln px 5 xpx E 5 sx viii Deaton and Muellbauer 1980 make extensive use of this relationship to study the characteristics of a particular class of expenditure functions that they term as an almost ideal demand system AIDS Their expenditure function takes the form ln E 1px py V2 5 a0 1 a1 ln px 1 a2 ln py 1 05b1 1 ln px2 2 1 b2 ln px ln py 1 05b3 1 ln py2 2 1 Vc0 pc1x pc2y ix This form approximates any expenditure function For the function to be homogeneous of degree one in the prices the parameters of the function must obey the constraints a1 1 a2 5 1 b1 1 b2 5 0 b2 1 b3 5 0 a n d c1 1 c2 5 0 Using the results of Equation viii shows that for this function sx 5 a1 1 b1 ln px 1 b2 ln py 1 c1Vc0 pc1x pc2y sy 5 a2 1 b2 ln px 1 b3 ln py 1 c2Vc0 pc1x pc2y x Notice that given the parameter restrictions sx 1 sy 5 1 Making use of the inverse relationship between indirect util ity and expenditure functions and some additional algebraic manipulation will put these budget share equations into a sim ple form suitable for econometric estimation sx 5 a1 1 b1 ln px 1 b2 ln py 1 c1 ln 1Ep2 sy 5 a2 1 b2 ln px 1 b3 ln py 1 c2 ln 1Ep2 xi where p is an index of prices defined by ln p 5 a0 1 a1 ln px 1 a2 ln py 1 05b1 1 ln px2 2 1 b2 ln px ln py 1 05b3 1 ln py2 2 xii In other words the AIDS share equations state that budget shares are linear in the logarithms of prices and in total real expenditures In practice simpler price indices are often sub stituted for the rather complex index given by Equation xii although there is some controversy about this practice see the Extensions to Chapter 5 British expenditure patterns Deaton and Muellbauer apply this demand system to the study of British expenditure patterns between 1954 and 1974 They find that food and housing have negative coefficients of real expenditures implying that the share of income devoted to these items decreases at least in Britain as people get richer The authors also find significant relative price effects in many of their share equations and prices have especially large effects in explaining the share of expenditures devoted to transportation and communication In applying the AIDS model to realworld data the authors also encounter a vari ety of econometric difficulties the most important of which is that many of the equations do not appear to obey the restric tions necessary for homogeneity Addressing such issues has been a major topic for further research on this demand system References Behrman Jere R Is Variety the Spice of Life Implications for Caloric Intake Review of Economics and Statistics November 1989 66672 Deaton Angus and John Muellbauer An Almost Ideal Demand System American Economic Review June 1980 31226 Hyashi Fumio Is the Japanese Extended Family Altruisti cally Linked A Test Based on Engel Curves Journal of Political Economy June 1995 66174 Kehoe Patrick J and Timothy J Kehoe Modeling North American Economic Integration London Kluwer Aca demic Publishers 1995 Oczkowski E and N E Philip Household Expenditure Patterns and Access to Consumer Goods in a Transitional Economy Journal of Economic Development June 1994 16583 Stone R Linear Expenditure Systems and Demand Analy sis Economic Journal September 1954 51127 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 141 CHAPTER FIVE Income and Substitution Effects In this chapter we will use the utilitymaximization model to study how the quantity of a good that an individual chooses is affected by a change in that goods price This examination allows us to construct the individuals demand curve for the good In the process we will provide a number of insights into the nature of this price response and into the kinds of assumptions that lie behind most analyses of demand 51 DEMAND FUNCTIONS As we pointed out in Chapter 4 in principle it will usually be possible to solve the necessary conditions of a utility maximum for the optimal levels of x1 x2 xn and λ the Lagrange multiplier as functions of all prices and income Mathematically this can be expressed as n demand functions1 of the form x 1 5 x1 1 p1 p2 pn I2 x 2 5 x2 1 p1 p2 pn I2 x n 5 xn1 p1 p2 pn I2 51 If there are only two goods x and y the case we will usually be concerned with this notation can be simplified a bit as x 5 x 1 px py I2 y 5 y 1 px py I2 52 Once we know the form of these demand functions and the values of all prices and income we can predict how much of each good this person will choose to buy The notation stresses that prices and income are exogenous to this process that is these are parameters over which the individual has no control at this stage of the analysis Changes in the parameters will of course shift the budget constraint and cause this person to make different choices That question is the focus of this chapter and the next Specifically in 1Sometimes the demand functions in Equation 51 are referred to as Marshallian demand functions after Alfred Marshall to differentiate them from the Hicksian demand functions named for John Hicks we will encounter later in this chapter The difference between the two concepts derives from whether income or utility enters the functions For simplicity throughout this text the term demand functions or demand curves will refer to the Marshallian concept whereas references to Hicksian or compensated demand functions and demand curves will be explicitly noted Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 142 Part 2 Choice and Demand this chapter we will be looking at the partial derivatives xI and xpx for any arbitrary good x Chapter 6 will carry the discussion further by looking at crossprice effects of the form xpy for any arbitrary pair of goods x and y 511 Homogeneity A first property of demand functions requires little mathematics If we were to double all prices and income indeed if we were to multiply them all by any positive constant then the optimal quantities demanded would remain unchanged Doubling all prices and income changes only the units by which we count not the real quantity of goods demanded This result can be seen in a number of ways although perhaps the easiest is through a graphic approach Referring back to Figures 41 and 42 it is clear that doubling px py and I does not affect the graph of the budget constraint Hence x y will still be the combination that is chosen In algebraic terms pxx 1 pyy 5 I is the same constraint as 2pxx 1 2pyy 5 2I Somewhat more technically we can write this result as saying that for any good xi x i 5 xi 1 p1 p2 pn I2 5 xi 1tp1 tp2 tpn tI2 53 for any t 0 Functions that obey the property illustrated in Equation 53 are said to be homogeneous of degree 02 Hence we have shown that individual demand functions are homogeneous of degree 0 in all prices and income Changing all prices and income in the same proportions will not affect the physical quantities of goods demanded This result shows that in theory individuals demands will not be affected by a pure inflation during which all prices and incomes increase proportionally They will continue to demand the same bundle of goods Of course if an inflation were not pure ie if some prices increased more rapidly than others this would not be the case 2More generally as we saw in Chapters 2 and 4 a function f 1x1 x2 xn2 is said to be homogeneous of degree k if f 1tx1 tx2 txn2 5 tkf 1x1 x2 xn2 for any t 0 The most common cases of homogeneous functions are k 5 0 and k 5 1 If f is homogeneous of degree 0 then doubling all its arguments leaves f unchanged in value If f is homogeneous of degree 1 then doubling all its arguments will double the value of f EXAMPLE 51 Homogeneity Homogeneity of demand is a direct result of the utilitymaximization assumption Demand functions derived from utility maximization will be homogeneous and conversely demand functions that are not homogeneous cannot reflect utility maximization unless prices enter directly into the utility function itself as they might for goods with snob appeal If for example an individuals utility for food x and housing y is given by utility 5 U1x y2 5 x03y07 54 then it is a simple matter following the procedure used in Example 41 to derive the demand functions x 5 03I px y 5 07I py 55 These functions obviously exhibit homogeneity because a doubling of all prices and income would leave x and y unaffected If the individuals preferences for x and y were reflected instead by the CES function U1x y2 5 x05 1 y05 56 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 143 then as shown in Example 42 the demand functions are given by x 5 a 1 1 1 pxpy b I px y 5 a 1 1 1 pypx b I py 57 As before both of these demand functions are homogeneous of degree 0 a doubling of px py and I would leave x and y unaffected QUERY Do the demand functions derived in this example ensure that total spending on x and y will exhaust the individuals income for any combination of px py and I Can you prove that this is the case 52 CHANGES IN INCOME As a persons purchasing power increases it is natural to expect that the quantity of each good purchased will also increase This situation is illustrated in Figure 51 As expenditures increase from I1 to I2 to I3 the quantity of x demanded increases from x1 to x2 to x3 Also the quantity of y increases from y1 to y2 to y3 Notice that the budget lines I1 I2 and I3 As income increases from I1 to I2 to I3 the optimal utilitymaximizing choices of x and y are shown by the successively higher points of tangency Observe that the budget constraint shifts in a parallel way because its slope given by 2pxpy does not change Quantity of x Quantity of y y3 U1 U2 U3 U3 I3 U2 I2 U1 I1 y2 y1 x1 x2 x3 FIGURE 51 Effect of an Increase in Income on the Quantities of x and y Chosen Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 144 Part 2 Choice and Demand In this diagram good z is inferior because the quantity purchased decreases as income increases Here y is a normal good as it must be if there are only two goods available and purchases of y increase as total expenditures increase Quantity of z Quantity of y y3 U3 I3 U2 U1 I2 I1 y2 y1 z3 z2 z1 FIGURE 52 An Indifference Curve Map Exhibiting Inferiority are all parallel reflecting that only income is changing not the relative prices of x and y Because the ratio pxpy stays constant the utilitymaximizing conditions also require that the MRS stay constant as the individual moves to higher levels of satisfaction Therefore the MRS is the same at point 1x3 y32 as at 1x1 y12 521 Normal and inferior goods In Figure 51 both x and y increase as income increasesboth xI and yI are positive This might be considered the usual situation and goods that have this property are called normal goods over the range of income change being observed For some goods however the quantity chosen may decrease as income increases in some ranges Examples of such goods are rotgut whiskey potatoes and secondhand clothing A good z for which zI is negative is called an inferior good This phenomenon is illustrated in Figure 52 In this diagram the good z is inferior because for increases in income in the range shown less of z is chosen Notice that indifference curves do not have to be oddly shaped to exhibit inferiority the curves corresponding to goods y and z in Figure 52 continue to obey the assumption of a diminishing MRS Good z is inferior because of the way it relates to the other goods available good y here not because of a peculiarity unique to it Hence we have developed the following definitions D E F I N I T I O N Inferior and normal goods A good xi for which xiI 0 over some range of income changes is an inferior good in that range If xiI 0 over some range of income variation then the good is a normal or noninferior good in that range Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 145 53 CHANGES IN A GOODS PRICE The effect of a price change on the quantity of a good demanded is more complex to analyze than is the effect of a change in income Geometrically this is because changing a price involves changing not only one of the intercepts of the budget constraint but also its slope Consequently moving to the new utilitymaximizing choice entails not only moving to another indifference curve but also changing the MRS Therefore when a price changes two analytically different effects come into play One of these is a substitution effect Even if the individual were to stay on the same indifference curve consumption patterns would be allocated so as to equate the MRS to the new price ratio A second effect the income effect arises because a price change necessarily changes an individuals real income The individual cannot stay on the initial indifference curve and must move to a new one We begin by analyzing these effects graphically Then we will provide a mathematical development 531 Graphical analysis of a decrease in price Income and substitution effects are illustrated in Figure 53 This individual is initially maximizing utility subject to total expenditures I by consuming the combination x y The initial budget constraint is I 5 p1 xx 1 pyy Now suppose that the price of x decreases to p2 x The new budget constraint is given by the equation I 5 p2 xx 1 pyy in Figure 53 It is clear that the new position of maximum utility is at x y where the new budget line is tangent to the indifference curve U2 The movement to this new point can be viewed as being composed of two effects First the change in the slope of the budget constraint would have motivated a move to point B even if choices had been confined to those on the original indifference curve U1 The dashed line in Figure 53 has the same slope as the new budget constraint 1I 5 p2 xx 1 pyy2 but is drawn to be tangent to U1 because we are concep tually holding real income ie utility constant A relatively lower price for x causes a move from x y to B if we do not allow this individual to be made better off as a result of the lower price This movement is a graphic demonstration of the substitution effect The additional move from B to the optimal point x y is analytically identical to the kind of change exhibited earlier for changes in income Because the price of x has decreased this person has a greater real income and can afford a utility level 1U22 that is greater than that which could previously be attained If x is a normal good more of it will be chosen in response to this increase in purchasing power This observation explains the origin of the term income effect for the movement Overall then the result of the price decrease is to cause more x to be demanded It is important to recognize that this person does not actually make a series of choices from x y to B and then to x y We never observe point B only the two optimal posi tions are reflected in observed behavior However the notion of income and substitution effects is analytically valuable because it shows that a price change affects the quantity of x that is demanded in two conceptually different ways We will see how this separation offers major insights in the theory of demand 532 Graphical analysis of an increase in price If the price of good x were to increase a similar analysis would be used In Figure 54 the budget line has been shifted inward because of an increase in the price of x from p1 x to p2 x The movement from the initial point of utility maximization 1x y2 to the new point 1x y2 can be decomposed into two effects First even if this person could stay on the initial indifference curve 1U22 there would still be an incentive to substitute y for x and move along U2 to point B However because purchasing power has been reduced by the Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 146 Part 2 Choice and Demand When the price of x decreases from p1 x to p2 x the utilitymaximizing choice shifts from x y to x y This movement can be broken down into two analytically different effects first the substitution effect involving a movement along the initial indifference curve to point B where the MRS is equal to the new price ratio and second the income effect entailing a movement to a higher level of utility because real income has increased In the diagram both the substitution and income effects cause more x to be bought when its price decreases Notice that point Ipy is the same as before the price change this is because py has not changed Therefore point Ipy appears on both the old and new budget constraints Quantity of x Quantity of y U2 U2 U1 U1 x y xB x y B I px 1x pyy I p2 xx pyy I py Substitution efect Income efect Total increase in x FIGURE 53 Demonstration of the Income and Substitution Effects of a Decrease in the Price of x increase in the price of x he or she must move to a lower level of utility This movement is again called the income effect Notice in Figure 54 that both the income and substitution effects work in the same direction and cause the quantity of x demanded to be reduced in response to an increase in its price 533 Effects of price changes for inferior goods Thus far we have shown that substitution and income effects tend to reinforce one another For a price decrease both cause more of the good to be demanded whereas for a price increase both cause less to be demanded Although this analysis is accurate for the case of normal noninferior goods the possibility of inferior goods complicates the story In this case income and substitution effects work in opposite directions and the combined result of a price change is indeterminate A decrease in price for example will always cause an Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 147 individual to tend to consume more of a good because of the substitution effect But if the good is inferior the increase in purchasing power caused by the price decrease may cause less of the good to be bought Therefore the result is indeterminate The substitution effect tends to increase the quantity of the inferior good bought whereas the perverse income effect tends to reduce this quantity Unlike the situation for normal goods it is not possible here to predict even the direction of the effect of a change in px on the quantity of x consumed 534 Giffens paradox If the income effect of a price change is strong enough the change in price and the resulting change in the quantity demanded could actually move in the same direction Legend has it that the English economist Robert Giffen observed this paradox in nineteenthcentury When the price of x increases the budget constraint shifts inward The movement from the initial util itymaximizing point 1x y2 to the new point 1x y2 can be analyzed as two separate effects The substitution effect would be depicted as a movement to point B on the initial indifference curve 1U22 The price increase however would create a loss of purchasing power and a consequent movement to a lower indifference curve This is the income effect In the diagram both the income and substitution effects cause the quantity of x to decrease as a result of the increase in its price Again the point Ipy is not affected by the change in the price of x Total reduction in x Substitution efect Income efect Quantity of x Quantity of y xB x x y y I py U2 U2 U1 U1 B I px 2x pyy I px 1x pyy FIGURE 54 Demonstration of the Income and Substitution Effects of an Increase in the Price of x Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 148 Part 2 Choice and Demand Ireland When the price of potatoes rose people reportedly consumed more of them This peculiar result can be explained by looking at the size of the income effect of a change in the price of potatoes Potatoes were not only inferior goods but they also used up a large portion of the Irish peoples income Therefore an increase in the price of potatoes reduced real income substantially The Irish were forced to cut back on other luxury food consumption to buy more potatoes Even though this rendering of events is historically implausible the possibility of an increase in the quantity demanded in response to an increase in the price of a good has come to be known as Giffens paradox3 Later we will provide a mathematical analysis of how Giffens paradox can occur 535 A summary Hence our graphical analysis leads to the following conclusions 54 THE INDIVIDUALS DEMAND CURVE Economists frequently wish to graph demand functions It will come as no surprise to you that these graphs are called demand curves Understanding how such widely used curves relate to underlying demand functions provides additional insights to even the most fundamental of economic arguments To simplify the development assume there are only two goods and that as before the demand function for good x is given by x 5 x 1 px py I 2 The demand curve derived from this function looks at the relationship between x and px while holding py I and preferences constant That is it shows the relationship x 5 x 1px py I 2 58 where the bars over py and I indicate that these determinants of demand are being held constant This construction is shown in Figure 55 The graph shows utilitymaximizing choices of x and y as this individual is presented with successively lower prices of good x while holding py and I constant We assume that the quantities of x chosen increase from xr to xrr to xt as that goods price decreases from prx to prr to ptx Such an assumption is in accord with our general conclusion that except in the unusual case of Giffens paradox xpx is negative 3A major problem with this explanation is that it disregards Marshalls observation that both supply and demand factors must be taken into account when analyzing price changes If potato prices increased because of the potato blight in Ireland then supply should have become smaller therefore how could more potatoes possibly have been consumed Also because many Irish people were potato farmers the potato price increase should have increased real income for them For a detailed discussion of these and other fascinating bits of potato lore see G P Dwyer and C M Lindsey Robert Giffen and the Irish Potato American Economic Review March 1984 18892 O P T I M I Z AT I O N P R I N C I P L E Substitution and income effects The utilitymaximization hypothesis suggests that for normal goods a decrease in the price of a good leads to an increase in quantity purchased because 1 the substitution effect causes more to be purchased as the individual moves along an indiffer ence curve and 2 the income effect causes more to be purchased because the price decrease has increased purchasing power thereby permitting movement to a higher indifference curve When the price of a normal good increases similar reasoning predicts a decrease in the quantity pur chased For inferior goods substitution and income effects work in opposite directions and no definite predictions can be made Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 149 In a the individuals utilitymaximizing choices of x and y are shown for three different prices of x1prx psx and ptx 2 In b this relationship between px and x is used to construct the demand curve for x The demand curve is drawn on the assumption that py I and preferences remain constant as px varies Quantity of x per period Quantity of y per period I px x pyy I px x pyy I px x pyy x x x U2 U3 U1 I py a Individuals indiference curve map Quantity of x per period px xpx py I b Demand curve x x x px px px FIGURE 55 Construction of an Indi viduals Demand Curve In Figure 55b information about the utilitymaximizing choices of good x is transferred to a demand curve with px on the vertical axis and sharing the same horizontal axis as Figure 55a The negative slope of the curve again reflects the assumption that xpx is negative Hence we may define an individual demand curve as follows Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 150 Part 2 Choice and Demand The demand curve illustrated in Figure 55 stays in a fixed position only so long as all other determinants of demand remain unchanged If one of these other factors were to change then the curve might shift to a new position as we now describe 541 Shifts in the demand curve Three factors were held constant in deriving this demand curve 1 income 2 prices of other goods say py and 3 the individuals preferences If any of these were to change the entire demand curve might shift to a new position For example if I were to increase the curve would shift outward provided that xI 0 ie provided the good is a normal good over this income range More x would be demanded at each price If another price say py were to change then the curve would shift inward or outward depending precisely on how x and y are related In the next chapter we will examine that relationship in detail Finally the curve would shift if the individuals preferences for good x were to change A sudden advertising blitz by the McDonalds Corporation might shift the demand for hamburgers outward for example As this discussion makes clear one must remember that the demand curve is only a twodimensional representation of the true demand function Equation 58 and that it is stable only if other things do stay constant It is important to keep clearly in mind the difference between a movement along a given demand curve caused by a change in px and a shift in the entire curve caused by a change in income in one of the other prices or in preferences Traditionally the term an increase in demand is reserved for an outward shift in the demand curve whereas the term an increase in the quantity demanded refers to a movement along a given curve caused by a fall in px D E F I N I T I O N Individual demand curve An individual demand curve shows the relationship between the price of a good and the quantity of that good purchased by an individual assuming that all other deter minants of demand are held constant EXAMPLE 52 Demand Functions and Demand Curves To be able to graph a demand curve from a given demand function we must assume that the preferences that generated the function remain stable and that we know the values of income and other relevant prices In the first case studied in Example 51 we found that x 5 03I px 59 and y 5 07I py If preferences do not change and if this individuals income is 100 these functions become x 5 30 px y 5 70 py 510 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 151 55 COMPENSATED HICKSIAN DEMAND CURVES AND FUNCTIONS In Figure 55 the level of utility this person gets varies along the demand curve As px decreases he or she is made increasingly better off as shown by the increase in utility from U1 to U2 to U3 The reason this happens is that the demand curve is drawn on the assumption that nominal income and other prices are held constant hence a decline in px makes this person better off by increasing his or her real purchasing power Although this is the most common way to impose the ceteris paribus assumption in developing a demand curve it is not the only way An alternative approach holds real income or utility constant while examining reactions to changes in px The derivation or pxx 5 30 pyy 5 70 which makes clear that the demand curves for these two goods are simple hyperbolas An increase in income would shift both of the demand curves outward Notice also in this case that the demand curve for x is not shifted by changes in py and vice versa For the second case examined in Example 51 the analysis is more complex For good x we know that x 5 a 1 1 1 pxpy b I px 511 so to graph this in the px 2 x plane we must know both I and py If we again assume I 5 100 and let py 5 1 then Equation 511 becomes x 5 100 p2 x 1 px 512 which when graphed would also show a general hyperbolic relationship between price and quantity consumed In this case the curve would be relatively flatter because substitution effects are larger than in the CobbDouglas case From Equation 511 we also know that x I 5 a 1 1 1 pxpy b 1 px 0 513 and x py 5 I 1px 1 py2 2 0 thus increases in I or py would shift the demand curve for good x outward QUERY How would the demand functions in Equations 510 change if this person spent half of his or her income on each good Show that these demand functions predict the same x consump tion at the point px 5 1 py 5 1 I 5 100 as does Equation 511 Use a numerical example to show that the CES demand function is more responsive to an increase in px than is the CobbDouglas demand function Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 152 Part 2 Choice and Demand is illustrated in Figure 56 where we hold utility constant at U2 while successively reducing px As px decreases the individuals nominal income is effectively reduced thus preventing any increase in utility In other words the effects of the price change on purchasing power are compensated to constrain the individual to remain on U2 Reactions to changing prices include only substitution effects If we were instead to examine effects of increases in px income compensation would be positive This individuals income would have to be increased to permit him or her to stay on the U2 indifference curve in response to the price increases We can summarize these results as follows The curve xc shows how the quantity of x demanded changes when px changes holding py and utility constant That is the individuals income is compensated to keep utility constant Hence xc reflects only substitution effects of changing prices FIGURE 56 Construction of a Com pensated Demand Curve px px x x Quantity of x Quantity of x Quantity of y py Slope py Slope px py Slope U2 px xcpx py U a Individuals indiference curve map b Compensated demand curve x px px px x x x Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 153 551 Shephards lemma Many facts about compensated demand functions can be easily proven by using a remarkable result from duality theory called Shephards lemma named for R W Shephard who pioneered the use of duality theory in production and cost functionssee Chapters 9 and 10 Consider the dual expenditure minimization problem discussed in Chapter 4 The Lagrangian expression for this problem was 5 pxx 1 pyy 1 λ 3U1x y2 2 U4 515 The solution to this problem yields the expenditure function E1px py U2 Because this is a value function the envelope theorem applies This means that we can interpret derivatives of the expenditure function by differentiating the original Lagrangian expression that produced it Differentiation with respect to the price of good x for example yields dE1px py U2 dpx 5 px 5 xc 1px py U2 516 That is the compensated demand function for a good can be found from the expenditure function by differentiation with respect to that goods price To see intuitively why such a derivative is a compensated demand function notice first that both the expenditure function and the compensated demand function depend on the same variables 1px py and U2the value of any derivative will always depend on the same variables that enter into the original function Second because we are differentiating a value function we are assured that any change in prices will be met by a series of adjustments in quantities bought that will continue to minimize the expenditures needed to reach a given utility level Finally changes in the price of a good will affect expenditures roughly in proportion to the quantity of that good being boughtthat is precisely what Equation 516 says One of the many insights that can be derived from Shephards lemma concerns the slope of the compensated demand curve In Chapter 4 we showed that the expenditure function must be concave in prices In mathematical terms 2E1px py V2p2 x 0 Taking account of Shephards lemma however implies that 2E1px py V2 p2 x 5 3E1px py V2px4 px 5 xc 1px py V2 px 0 517 Hence the compensated demand curve must have a negative slope The ambiguity that arises when substitution and income effects work in opposite directions for Marshallian demand curves does not arise in the case of compensated demand curves because they D E F I N I T I O N Compensated demand curve A compensated demand curve shows the relationship between the price of a good and the quantity purchased on the assumption that other prices and utility are held constant Therefore the curve which is sometimes termed a Hicksian demand curve after the British economist John Hicks illustrates only substitution effects Mathematically the curve is a twodimensional representation of the compensated demand function xc 5 xc 1px py U2 514 Notice that the only difference between the compensated demand function in Equation 514 and the uncompensated demand functions in Equations 51 or 52 is whether utility or income enters the functions Hence the major difference between compensated and uncompensated demand curves is whether utility or income is held constant in constructing the curves Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 154 Part 2 Choice and Demand involve only substitution effects and the assumption of quasiconcavity ensures that these will be negative 552 Relationship between compensated and uncompensated demand curves This relationship between the two demand curve concepts is illustrated in Figure 57 At ps x the curves intersect because at that price the individuals income is just sufficient to attain utility level U2 compare Figures 55 and Figure 56 Hence xs is demanded under either demand concept For prices below ps x however the individual suffers a compensat ing reduction in income on the curve xc that prevents an increase in utility arising from the lower price Assuming x is a normal good it follows that less x is demanded at ps x along xc than along the uncompensated curve x Alternatively for a price above ps x such as psx income compensation is positive because the individual needs some help to remain on U2 Again assuming x is a normal good at prx more x is demanded along xc than along x In general then for a normal good the compensated demand curve is somewhat less responsive to price changes than is the uncompensated curve This is because the latter reflects both substitution and income effects of price changes whereas the compensated curve reflects only substitution effects The choice between using compensated or uncompensated demand curves in economic analysis is largely a matter of convenience In most empirical work uncompensated or Marshallian demand curves are used because the data on prices and nominal incomes The compensated 1xc2 and uncompensated x demand curves intersect at psx because xs is demanded under each concept For prices above psx the individuals purchasing power must be increased with the compensated demand curve thus more x is demanded than with the uncompensated curve For prices below psx purchasing power must be reduced for the compensated curve therefore less x is demanded than with the uncompensated curve The standard demand curve is more priceresponsive because it incorporates both substitution and income effects whereas the curve xc reflects only substitution effects FIGURE 57 Comparison of Compensated and Uncompensated Demand Curves xpx py I xcpx py U Quantity of x px x x x px px px x x Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 155 needed to estimate them are readily available In the Extensions to Chapter 12 we will describe some of these estimates and show how they might be used for practical policy purposes For some theoretical purposes however compensated demand curves are a more appropriate concept because the ability to hold utility constant offers some advantages Our discussion of consumer surplus later in this chapter offers one illustration of these advantages EXAMPLE 53 Compensated Demand Functions In Example 31 we assumed that the utility function for hamburgers y and soft drinks x was given by utility 5 U1x y2 5 x05y05 518 and in Example 41 we showed that we can calculate the Marshallian demand functions for such utility functions as x 1px py I2 5 05I px y1px py I2 5 05I py 519 In Example 44 we found that the expenditure function in this case is given by E 1px py U2 5 2p05 x p05 y U Thus we can now use Shephards lemma to calculate the compensated demand func tions as xc 1px py U2 5 E 1px py U2 px 5 p205 x p05 y U yc 1px py U2 5 E 1px py U2 py 5 p05 x p205 y U 520 Sometimes indirect utility V is used in these compensated demand functions rather than U but this does not change the meaning of the expressionsthese demand functions show how an individual reacts to changes in prices while holding utility constant Although py did not enter into the uncompensated demand function for good x it does enter into the compensated function Increases in py shift the compensated demand curve for x out ward The two demand concepts agree at the assumed initial point px 5 1 py 5 4 I 5 8 and U 5 2 Equations 519 predict x 5 4 y 5 1 at this point as do Equations 520 For px 1 or px 1 the demands differ under the two concepts however If say px 5 4 then the uncompen sated functions predict x 5 1 y 5 1 whereas the compensated functions predict x 5 2 y 5 2 The reduction in x resulting from the increase in its price is smaller with the compensated demand function than it is with the uncompensated function because the former concept adjusts for the negative effect on purchasing power that comes about from the price increase This example makes clear the different ceteris paribus assumptions inherent in the two demand concepts With uncompensated demand expenditures are held constant at I 5 2 and so the increase in px from 1 to 4 results in a loss of utility in this case utility decreases from 2 to 1 In the compensated demand case utility is held constant at U 5 2 To keep utility constant expenditures must increase to E 5 4 122 1 4 122 5 16 to offset the effects of the price increase QUERY Are the compensated demand functions given in Equations 520 homogeneous of degree 0 in px and py if utility is held constant Why would you expect that to be true for all com pensated demand functions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 156 Part 2 Choice and Demand 56 A MATHEMATICAL DEVELOPMENT OF RESPONSE TO PRICE CHANGES Up to this point we have largely relied on graphical devices to describe how individuals respond to price changes Additional insights are provided by a more mathematical approach Our basic goal is to examine the partial derivative xpxthat is how a change in the price of a good affects its purchase ceteris paribus for the usual Marshallian demand curve In the next chapter we take up the question of how changes in the price of one commodity affect purchases of another commodity 561 Direct approach Our goal is to use the utilitymaximization model to learn something about how the demand for good x changes when px changes that is we wish to calculate xpx Following the procedures outlined in the Extensions to Chapter 2 we could approach this problem using comparative static methods by differentiating the three firstorder conditions for a maximum with respect to px This would yield three new equations containing the partial derivative we seek xpx These could then be solved using matrix algebra and Cramers rule4 Unfortunately obtaining this solution is cumbersome and the steps required yield little in the way of economic insights Hence we will instead adopt an indirect approach that relies on the concept of duality In the end both approaches yield the same conclusion but the indirect approach is much richer in terms of the economics it contains 562 Indirect approach To begin our indirect approach5 we will assume as before there are only two goods x and y and focus on the compensated demand function xc 1px py U2 and its relationship to the ordinary demand function x 1px py I2 By definition we know that xc 1px py U2 5 x 3px py E1px py U2 4 521 This conclusion was already introduced in connection with Figure 57 which showed that the quantity demanded is identical for the compensated and uncompensated demand functions when income is exactly what is needed to attain the required utility level Equation 521 is obtained by inserting that expenditure level into the demand function x 1px py I2 Now we can proceed by partially differentiating Equation 521 with respect to px and recognizing that this variable enters into the ordinary demand function in two places Hence xc px 5 x px 1 x E E px 522 and rearranging terms yields x px 5 xc px 2 x E E px 523 4See for example Paul A Samuelson Foundations of Economic Analysis Cambridge MA Harvard University Press 1947 pp 1013 5The following proof was first made popular by Phillip J Cook in A One Line Proof of the Slutsky Equation American Economic Review 62 March 1972 139 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 157 563 The substitution effect Consequently the derivative we seek has two terms Interpretation of the first term is straightforward It is the slope of the compensated demand curve But that slope represents movement along a single indifference curve it is in fact what we called the substitution effect earlier The first term on the right of Equation 523 is a mathematical representation of that effect 564 The income effect The second term in Equation 523 reflects the way in which changes in px affect the demand for x through changes in purchasing power Therefore this term reflects the income effect The negative sign in Equation 523 reflects the inverse relationship between changes in prices and changes in purchasing power For example an increase in px increases the expenditure level that would have been needed to keep utility constant mathematically Epx 0 But because nominal income is held constant in Marshallian demand these extra expenditures are not available Hence expenditures on x must be reduced to meet this shortfall The extent of the reduction in x is given by xE On the other hand if px decreases the expenditure level required to attain a given utility decreases But nominal income is constant in the Marshallian concept of demand hence there is an increase in purchasing power and therefore an increase in spending on good x 565 The Slutsky equation The relationships embodied in Equation 523 were first discovered by the Russian economist Eugen Slutsky in the late nineteenth century A slight change in notation is required to state the result the way Slutsky did First we write the substitution effect as substitution effect 5 xc px 5 x px U5constant 524 to indicate movement along a single indifference curve For the income effect we have income effect 5 2x E E px 5 2x I E px 525 because changes in income or expenditures amount to the same thing in the function x 1px py I2 The second term of the income effect can be interpreted using Shephards lemma That is Epx 5 xc Consequently the entire income effect is given by income effect 5 2xcx I 526 566 Final form of the Slutsky equation Bringing together Equations 524526 allows us to assemble the Slutsky equation in the form in which it was originally derived x 1px py I2 px 5 substitution effect 1 income effect 5 x px U5constant 2 xx I 527 where we have made use of the fact that x 1px py I2 5 xc 1px py V2 at the utilitymaximizing point Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 158 Part 2 Choice and Demand This equation allows a more definitive treatment of the direction and size of substitution and income effects than was possible with a graphic analysis First as we have just shown the substitution effect and the slope of the compensated demand curve is always negative This result derives both from the quasiconcavity of utility functions a diminishing MRS and from the concavity of the expenditure function We will show the negativity of the substitution effect in a somewhat different way in the final section of this chapter The sign of the income effect 12xxI2 depends on the sign of xI If x is a normal good then xI is positive and the entire income effect like the substitution effect is negative Thus for normal goods price and quantity always move in opposite directions For example a decrease in px increases real income and because x is a normal good purchases of x increase Similarly an increase in px reduces real income and so purchases of x decrease Overall then as we described previously using a graphic analysis substitution and income effects work in the same direction to yield a negatively sloped demand curve In the case of an inferior good xI 0 and the two terms in Equation 527 have different signs Hence the overall impact of a change in the price of a good is ambiguousit all depends on the relative sizes of the effects It is at least theoretically possible that in the inferior good case the second term could dominate the first leading to Giffens paradox 1xpx 02 EXAMPLE 54 A Slutsky Decomposition The decomposition of a price effect that was first discovered by Slutsky can be nicely illustrated with the CobbDouglas example studied previously In Example 53 we found that the Marshallian demand function for good x was x 1px py I2 5 05I px 528 and that the compensated demand function for this good was xc 1px py U2 5 p205 x p05 y U 529 Hence the total effect of a price change on Marshallian demand can be found by differentiating Equation 528 x 1px py I2 px 5 205I p2 x 530 We wish to show that this is the sum of the two effects that Slutsky identified To derive the substitution effect we must first differentiate the compensated demand function from Equation 529 substitution effect 5 xc 1px py U2 px 5 205p215 x p05 y U 531 Now in place of U we use indirect utility V 1px py I2 5 05Ip205 x p205 y substitution effect 5 205p215 x p05 y V 5 2025p22 x I 532 Calculation of the income effect in this example is considerably easier Applying the results from Equation 527 we have income effect 5 2xx I 5 2 c 05I px d 05 px 5 2025I p2 x 533 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 159 57 DEMAND ELASTICITIES Thus far in this chapter we have been examining how individuals respond to changes in prices and income by looking at the derivatives of the demand function For many analytical questions this is a good way to proceed because calculus methods can be directly applied However as we pointed out in Chapter 2 focusing on derivatives has one major disadvantage for empirical work The sizes of derivatives depend directly on how variables are measured That can make comparisons among goods or across countries and time periods difficult For this reason most empirical work in microeconomics uses some form of elasticity measure In this section we introduce the three most common types of demand elasticities and explore some of the mathematical relations among them Again for simplicity we will look at a situation where the individual chooses between only two goods although these ideas can be easily generalized 571 Marshallian demand elasticities Most of the commonly used demand elasticities are derived from the Marshallian demand function x 1px py I2 Specifically the following definitions are used A comparison of Equation 530 with Equations 532 and 533 shows that we have indeed decomposed the price derivative of this demand function into substitution and income components Interestingly the substitution and income effects are of precisely the same size This as we will see in later examples is one of the reasons that the CobbDouglas is a special case The wellworn numerical example we have been using also demonstrates this decomposition When the price of x increases from 1 to 4 the uncompensated demand for x decreases from x 5 4 to x 5 1 but the compensated demand for x decreases only from x 5 4 to x 5 2 That decline of 50 percent is the substitution effect The further 50 percent decrease from x 5 2 to x 5 1 represents reactions to the decline in purchasing power incorporated in the Marshallian demand function This income effect does not occur when the compensated demand notion is used QUERY In this example the individual spends half of his or her income on good x and half on good y How would the relative sizes of the substitution and income effects be altered if the expo nents of the CobbDouglas utility function were not equal D E F I N I T I O N 1 Price elasticity of demand 1ex px2 This measures the proportionate change in quantity demanded in response to a proportionate change in a goods own price Mathematically ex px 5 Dxx Dpxpx 5 Dx Dpx px x 5 x 1px py I2 px px x 534 2 Income elasticity of demand 1ex I2 This measures the proportionate change in quantity demanded in response to a proportionate change in income In mathematical terms ex I 5 Dxx DII 5 Dx DI I x 5 x 1px py I2 I I x 535 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 160 Part 2 Choice and Demand Notice that all these definitions use partial derivatives which signifies that all other determinants of demand are to be held constant when examining the impact of a specific variable In the remainder of this section we will explore the ownprice elasticity definition in some detail Examining the crossprice elasticity of demand is the primary topic of Chapter 6 572 Price elasticity of demand The own price elasticity of demand is probably the most important elasticity concept in all of microeconomics Not only does it provide a convenient way of summarizing how people respond to price changes for a wide variety of economic goods but it is also a central concept in the theory of how firms react to the demand curves facing them As you probably already learned in earlier economics courses a distinction is usually made between cases of elastic demand where price affects quantity significantly and inelastic demand where the effect of price is small One mathematical complication in making these ideas precise is that the price elasticity of demand itself is negative6 because except in the unlikely case of Giffens paradox xpx is negative The dividing line between large and small responses is generally set at 21 If ex px 5 21 changes in x and px are of the same proportionate size That is a 1 percent increase in price leads to a decrease of 1 per cent in quantity demanded In this case demand is said to be unitelastic Alternatively if ex px 21 then quantity changes are proportionately larger than price changes and we say that demand is elastic For example if ex px 5 23 each 1 percent increase in price leads to a decrease of 3 percent in quantity demanded Finally if ex px 21 then demand is inelastic and quantity changes are proportionately smaller than price changes A value of ex px 5 203 for example means that a 1 percent increase in price leads to a decrease in quantity demanded of 03 percent In Chapter 12 we will see how aggregate data are used to estimate the typical individuals price elasticity of demand for a good and how such esti mates are used in a variety of questions in applied microeconomics 573 Price elasticity and total spending The price elasticity of demand determines how a change in price ceteris paribus affects total spending on a good The connection is most easily shown with calculus 1px x2 px 5 px x px 1 x 5 x 1ex px 1 12 537 6Sometimes economists use the absolute value of the price elasticity of demand in their discussions Although this is mathematically incorrect such usage is common For example a study that finds that ex px 5 212 may sometimes report the price elasticity of demand as 12 We will not do so here however 3 Crossprice elasticity of demand 1ex py2 This measures the proportionate change in the quantity of x demanded in response to a proportionate change in the price of some other good y ex py 5 Dxx Dpypy 5 Dx Dpy py x 5 x 1px py I2 py py x 536 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 161 Thus the sign of this derivative depends on whether ex px is larger or smaller than 21 If demand is inelastic 10 ex px 212 the derivative is positive and price and total spend ing move in the same direction Intuitively if price does not affect quantity demanded very much then quantity stays relatively constant as price changes and total spending reflects mainly those price movements This is the case for example with the demand for most agricultural products Weatherinduced changes in price for specific crops usually cause total spending on those crops to move in the same direction On the other hand if demand is elastic 1ex px 212 reactions to a price change are so large that the effect on total spending is reversed An increase in price causes total spending to decrease because quantity decreases a lot and a decrease in price causes total spending to increase quantity increases significantly For the unitelastic case 1ex px 5 212 total spending is constant no matter how price changes 574 Compensated price elasticities Because some microeconomic analyses focus on the compensated demand function it is also useful to define elasticities based on that concept Such definitions follow directly from their Marshallian counterparts D E F I N I T I O N Let the compensated demand function be given by xc 1px py U2 Then we have the following definitions 1 Compensated ownprice elasticity of demand 1exc px2 This elasticity measures the proportion ate compensated change in quantity demanded in response to a proportionate change in a goods own price exc px 5 Dx cx c Dpxpx 5 Dxc Dpx px x c 5 x c 1px py U2 px px xc 538 2 Compensated crossprice elasticity of demand 1exc px2 This measures the proportionate com pensated change in quantity demanded in response to a proportionate change in the price of another good exc py 5 Dx cx c Dpypy 5 Dx c Dpy py x c 5 xc 1px py U2 py py x c 539 Whether these price elasticities differ much from their Marshallian counterparts depends on the importance of income effects in the overall demand for good x The precise connection between the two can be shown by multiplying the Slutsky result from Equation 527 by the factor pxx to yield the Slutsky equation in elasticity form px x x px 5 ex px 5 px x xc px 2 px x x x I 5 exc px 2 sxex I 540 where sx 5 pxxI is the share of total income devoted to the purchase of good x Equation 540 shows that compensated and uncompensated ownprice elasticities of demand will be similar if either of two conditions hold 1 The share of income devoted to good x 1sx2 is small or 2 the income elasticity of demand for good x 1ex I2 is small Either of these conditions serves to reduce the importance of the income effect as a component of the Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 162 Part 2 Choice and Demand Marshallian price elasticity If good x constitutes a small share of a persons expenditures his or her purchasing power will not be affected very much by a price change Even if a good has a large budget share if demand is not very responsive to changes in purchasing power the income effect will have a relatively small influence on demand elasticity Hence there will be many circumstances where one can use the two price elasticity concepts more or less interchangeably Put another way there are many economic circumstances in which substitution effects constitute the most important component of price responses 575 Relationships among demand elasticities There are a number of relationships among the elasticity concepts that have been developed in this section All these are derived from the underlying model of utility maximization Here we look at three such relationships that provide further insight on the nature of individual demand Homogeneity The homogeneity of demand functions can also be expressed in elas ticity terms Because any proportional increase in all prices and income leaves quantity demanded unchanged the net sum of all price elasticities together with the income elastic ity for a particular good must sum to zero A formal proof of this property relies on Eulers theorem see Chapter 2 Applying that theorem to the demand function x 1px py I2 and remembering that this function is homogeneous of degree 0 yields 0 5 px x px 1 py x py 1 I x I 541 If we divide Equation 541 by x then we obtain 0 5 ex px 1 ex py 1 ex I 542 as intuition suggests This result shows that the elasticities of demand for any good cannot follow a completely flexible pattern They must exhibit a sort of internal consistency that reflects the basic utilitymaximizing approach on which the theory of demand is based Engel aggregation In the Extensions to Chapter 4 we discussed the empirical analysis of market shares and took special note of Engels law that the share of income devoted to food decreases as income increases From an elasticity perspective Engels law is a statement of the empirical regularity that the income elasticity of demand for food is generally found to be considerably less than 1 Because of this it must be the case that the income elasticity of all nonfood items must be greater than 1 If an individual experiences an increase in his or her income then we would expect food expenditures to increase by a smaller proportional amount but the income must be spent somewhere In the aggregate these other expenditures must increase proportionally faster than income A formal statement of this property of income elasticities can be derived by differentiating the individuals budget constraint 1I 5 pxx 1 pyy2 with respect to income while treating the prices as constants 1 5 px x I 1 py y I 543 A bit of algebraic manipulation of this expression yields 1 5 px x I xI xI 1 py y I yI yI 5 sxex I 1 syey I 544 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 163 here as before si represents the share of income spent on good i Equation 544 shows that the weighted average on income elasticities for all goods that a person buys must be 1 If we knew say that a person spent a quarter of his or her income on food and the income elasticity of demand for food were 05 then the income elasticity of demand for everything else must be approximately 117 3511 2 025 0520754 Because food is an important necessity everything else is in some sense a luxury Cournot aggregation The eighteenthcentury French economist Antoine Cournot provided one of the first mathematical analyses of price changes using calculus His most important discovery was the concept of marginal revenue a concept central to the profitmaximization hypothesis for firms Cournot was also concerned with how the change in a single price might affect the demand for all goods Our final relationship shows that there are indeed connections among all of the reactions to the change in a single price We begin by differentiating the budget constraint again this time with respect to px I px 5 0 5 px x px 1 x 1 py y px Multiplication of this equation by pxI yields 0 5 px x px px I x x 1 x px I 1 py y px px I y y 0 5 sxex px 1 sx 1 syey px 545 so the final Cournot result is sxex px 1 syey px 5 2sx 546 Because the share coefficients are positive in this expression it shows that the budget constraint imposes some limits on the degree to which the crossprice elasticity 1ey px2 can be positive This is the first of many connections among the demands for goods that we will study more intensively in the next chapter Generalizations Although we have shown these aggregation results only for the case of two goods they are easily generalized to the case of many goods You are asked to do just that in Problem 511 A more difficult issue is whether these results should be expected to hold for typical economic data in which the demands of many people are combined Often economists treat aggregate demand relationships as describing the behavior of a typical per son and these relationships should in fact hold for such a person But the situation may not be that simple as we will show when discussing aggregation later in the book EXAMPLE 55 Demand Elasticities The Importance of Substitution Effects In this example we calculate the demand elasticities implied by three of the utility functions we have been using Although the possibilities incorporated in these functions are too simple to reflect how economists study demand empirically they do show how elasticities ultimately reflect peoples preferences One especially important lesson is to show why most of the variation in demand elasticities among goods probably arises because of differences in the size of substitution effects Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 164 Part 2 Choice and Demand Case 1 CobbDouglas 1σ 5 12 U1x y2 5 x α y β where α 1 β 5 1 The demand functions derived from this utility function are x 1px py I2 5 αI px y1px py I2 5 βI py 5 11 2 α2I py Application of the elasticity definitions shows that ex px 5 x px px x 5 2αI p2 x px αIpx 5 21 ex py 5 x py py x 5 0 py x 5 0 ex I 5 x I I x 5 α px I αIpx 5 1 547 The elasticities for good y take on analogous values Hence the elasticities associated with the CobbDouglas utility function are constant over all ranges of prices and income and take on especially simple values That these obey the three relationships shown in the previous section can be easily demonstrated using the fact that here sx 5 α and sy 5 β Homogeneity ex px 1 ex py 1 ex I 5 21 1 0 1 1 5 0 Engel aggregation sxex I 1 syey I 5 α 1 1 β 1 5 α 1 β 5 1 Cournot aggregation sxex px 1 syey px 5 α 1212 1 β 0 5 2α 5 2sx We can also use the Slutsky equation in elasticity form Equation 540 to derive the compen sated price elasticity in this example exc px 5 ex px 1 sxex I 5 21 1 α 112 5 α 2 1 5 2β 548 Here the compensated price elasticity for x depends on how important other goods y are in the utility function Case 2 CES 1σ 5 2 δ 5 052 U1x y2 5 x05 1 y05 In Example 42 we showed that the demand functions that can be derived from this utility function are x 1px py I2 5 I px11 1 pxp21 y 2 y1px py I2 5 I py11 1 p21 x py2 As you might imagine calculating elasticities directly from these functions can take some time Here we focus only on the ownprice elasticity and make use of the result from Problem 59 that the share elasticity of any good is given by esx px 5 sx px px sx 5 1 1 ex px 549 In this case sx 5 pxx I 5 1 1 1 px p21 y Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 165 so the share elasticity is more easily calculated and is given by esx px 5 sx px px sx 5 2p21 y 11 1 px p21 y 2 2 px 11 1 px p21 y 2 21 5 2px p21 y 1 1 px p21 y 550 Because the units in which goods are measured are rather arbitrary in utility theory we might as well define them so that initially px 5 py in which case7 we get ex px 5 esx px 2 1 5 21 1 1 1 2 1 5 215 551 Hence demand is more elastic in this case than in the CobbDouglas example The reason for this is that the substitution effect is larger for this version of the CES utility function This can be shown by again applying the Slutsky equation and using the facts that ex I 5 1 and sx 5 05 exc px 5 ex px 1 sxex I 5 215 1 05 112 5 21 552 which is twice the size of the elasticity for a CobbDouglas with equal shares Case 3 CES 1σ 5 05 δ 5 212 U1x y2 5 2x21 2 y21 Referring back to Example 42 we can see that the share of good x implied by this utility function is given by sx 5 1 1 1 p05 y p205 x so the share elasticity is given by esx px 5 sx px px sx 5 05p05 y p215 x 11 1 p05 y p205 x 2 2 px 11 1 p05 y p205 x 2 21 5 05p05 y p205 x 1 1 p05 y p205 x 553 If we again adopt the simplification of equal prices we can compute the ownprice elasticity as ex px 5 esx px 2 1 5 05 2 2 1 5 2075 554 and the compensated price elasticity as exc px 5 ex px 1 sxex I 5 2075 1 05 112 5 2025 555 Thus for this version of the CES utility function the ownprice elasticity is smaller than in Case 1 and Case 2 because the substitution effect is smaller Hence the main variation among the cases is indeed caused by differences in the size of the substitution effect If you never want to work out this kind of elasticity again it may be helpful to make use of the general result that exc px 5 211 2 sx2σ 556 You may wish to check out that this formula works in these three examples with sx 5 05 and σ 5 1 2 05 respectively and Problem 59 asks you to show that this result is generally true Because all these cases based on the CES utility function have a unitary income elasticity the ownprice elasticity can be computed from the compensated price elasticity by simply adding 2sx to the figure computed in Equation 556 QUERY Why is it that the budget share for goods other than x ie 1 2 sx enters into the com pensated ownprice elasticities in this example 7Notice that this substitution must be made after differentiation because the definition of elasticity requires that we change only px while holding py constant Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 166 Part 2 Choice and Demand 58 CONSUMER SURPLUS An important problem in applied welfare economics is to devise a monetary measure of the utility gains and losses that individuals experience when prices change One use for such a measure is to place a dollar value on the welfare loss that people experience when a market is monopolized with prices exceeding marginal costs Another application concerns mea suring the welfare gains that people experience when technical progress reduces the prices they pay for goods Related applications occur in environmental economics measuring the welfare costs of incorrectly priced resources law and economics evaluating the welfare costs of excess protections taken in fear of lawsuits and public economics measuring the excess burden of a tax To make such calculations economists use empirical data from studies of market demand in combination with the theory that underlies that demand In this section we will examine the primary tools used in that process 581 Consumer welfare and the expenditure function The expenditure function provides the first component for the study of the pricewelfare connection Suppose that we wished to measure the change in welfare that an individual experiences if the price of good x increases from p0 x to p1 x Initially this person requires expenditures of E1p0 x py U02 to reach a utility of U0 To achieve the same utility once the price of x increases he or she would require spending of at least E1p1 x py U02 Therefore to compensate for the price increase this person would require a compensation formally called a compensating variation8 or CV of CV 5 E1p1 x py U02 2 E1p0 x py U02 557 This situation is shown graphically in the top panel of Figure 58 This figure shows the quantity of the good whose price has changed on the horizontal axis and spending on all other goods in dollars on the vertical axis Initially this person consumes the combination x0 y0 and obtains utility of U0 When the price of x increases he or she would be forced to move to combination x2 y2 and suffer a loss in utility If he or she were compensated with extra purchasing power of amount CV he or she could afford to remain on the U0 indifference curve despite the price increase by choosing combination x1 y1 The distance CV therefore provides a monetary measure of how much this person needs to be compensated for the price increase 582 Using the compensated demand curve to show CV Unfortunately individuals utility functions and their associated indifference curve maps are not directly observable But we can make some headway on empirical measurement by determining how the CV amount can be shown on the compensated demand curve in the bottom panel of Figure 58 Shephards lemma shows that the compensated demand function for a good can be found directly from the expenditure function by differentiation 8Some authors define compensating variation as the amount of income that must be given to a person to increase utility from U1 to U0 given the new price of good x Under this definition CV 5 E1p1 x py U02 2 E1p1 x py U12 Rather than focusing on the added expenditures necessary to retain a given initial utility level as price changes this definition focuses on the added expenditures necessary to return to the prior utility level at the new price As suggested by the complex wording however these definitions are equivalent because of the way in which U1 is defined it is the utility obtainable with the old level of expenditures given the new price That is E1p0 x py U02 E1p1 x py U12 So it is clear that the two definitions are algebraically equivalent Some authors also look at CV from the point of view of a social planner who must make the required compensations for price changes In that case the positive CV illustrated here would be said to be negative Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 167 If the price of x increases from p0 x to p1 x this person needs extra expenditures of CV to remain on the U0 indifference curve Integration shows that CV can also be represented by the shaded area below the compensated demand curve in panel b Quantity of x Spending on other goods CV y1 U0 U1 y2 y0 x2 x1 x0 Epx 0 U0 Epx 0 U0 Epx 1 U0 Epx 1 U0 Quantity of x Price px 0 x1 x0 xcpx U0 B A px 1 px 2 a Indiference curve map b Compensated demand curve FIGURE 58 Showing Compensating Variation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 168 Part 2 Choice and Demand xc 1px py U2 5 E1px py U2 px 558 Hence the compensation described in Equation 557 can be found by integrating across a sequence of small increments to price from p0 x to p1 x CV 5 3 p1 x p0 x E1 px py U02 px dpx 5 3 p1 x p0 x xc 1 px py U02dpx 559 while holding py and utility constant The integral defined in Equation 559 has a geometric interpretation which is shown in the lower panel of Figure 58 It is the shaded area to the left of the compensated demand curve and bounded by p0 x and p1 x Thus the welfare cost of this price increase can also be illustrated using changes in the area below the compensated demand curve 583 The consumer surplus concept There is another way to look at this issue We can ask how much this person would be willing to pay for the right to consume all this good that he or she wanted at the market price of p0 x rather than doing without the good completely The compensated demand curve in the bottom panel of Figure 58 shows that if the price of x increased to p2 x this persons consumption would decrease to zero and he or she would require an amount of compensation equal to area p2 xAp0 x to accept the change voluntarily Therefore the right to consume x0 at a price of p0 x is worth this amount to this individual It is the extra benefit that this person receives by being able to make market transactions at the prevailing market price This value given by the area below the compensated demand curve and above the market price is termed consumer surplus Looked at in this way the welfare problem caused by an increase in the price of x can be described as a loss in consumer surplus When the price increases from p0 x to p1 x the consumer surplus triangle decreases in size from p2 xAp0 x to p2 xBp1 x As the figure makes clear that is simply another way of describing the welfare loss represented in Equation 559 584 Welfare changes and the Marshallian demand curve Thus far our analysis of the welfare effects of price changes has focused on the compensated demand curve This is in some ways unfortunate because most empirical work on demand actually estimates ordinary Marshallian demand curves In this section we will show that studying changes in the area below a Marshallian demand curve may in fact be a good way to measure welfare losses Consider the Marshallian demand curve x 1 px 2 illustrated in Figure 59 Initially this consumer faces the price p0 x and chooses to consume x0 This consumption yields a utility level of U0 and the initial compensated demand curve for x ie xc 1px py U02 also passes through the point x0 p0 x which we have labeled point A When price increases to p1 x the Marshallian demand for good x decreases to x1 point C on the demand curve and this persons utility also decreases to say U1 There is another compensated demand curve associated with this lower level of utility and it also is shown in Figure 59 Both the Marshallian demand curve and this new compensated demand curve pass through point C The presence of a second compensated demand curve in Figure 59 raises an intriguing conceptual question Should we measure the welfare loss from the price Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 169 increase as we did in Figure 58 using the compensating variation CV associated with the initial compensated demand curve area p1 xBAp0 x or should we perhaps use this new compensated demand curve and measure the welfare loss as area p1 xCDp0 x A potential rationale for using the area under the new curve would be to focus on the individuals situation after the price increase with utility level U1 We might ask how much he or she would now be willing to pay to see the price return to its old lower levelsa notion termed the equivalent variation EV9 The answer to this would be given by area p1 xCDp0 x Therefore the choice between which compensated demand curve to use boils down to choosing which level of utility one regards as the appropriate target for the analysis Luckily the Marshallian demand curve provides a convenient compromise between these two measures Because the size of the area between the two prices and below the Marshallian curve area p1 xCAp0 x is smaller than that below the compensated demand curve based on U0 but larger than that below the curve based on U1 it does seem an attractive middle ground Hence this is the measure of welfare losses we will primarily use throughout this book 9More formally EV 5 E1p1 x py U12 2 E1p0 x py U12 Again some authors use a different definition of EV as being the income necessary to restore utility given the old prices that is EV 5 E1p0 x py U02 2 E1p0 x py U12 But because E1p0 x py U02 5 E1p1 x py U12 these definitions are equivalent The usual Marshallian nominal income constant demand curve for good x is x1px 2 Further xc1 U02 and xc1 U12 denote the compensated demand curves associated with the utility levels experienced when p0 x and p1 x respectively prevail The area to the left of x1px 2 between p0 x and p1 x is bounded by the similar areas to the left of the compensated demand curves Hence for small changes in price the area to the left of the Marshallian demand curve is a good measure of welfare loss px px 0 px 1 x1 x0 Quantity of x per period A B C D xpx xc U0 xc U1 FIGURE 59 Welfare Effects of Price Changes and the Marshallian Demand Curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 170 Part 2 Choice and Demand We should point out that some economists use either CV or EV to compute the welfare effects of price changes Indeed economists are often not clear about which measure of welfare change they are using Our discussion in the previous section shows that if income effects are small it really does not make much difference in any case D E F I N I T I O N Consumer surplus Consumer surplus is the area below the Marshallian demand curve and above market price It shows what an individual would pay for the right to make voluntary trans actions at this price Changes in consumer surplus can be used to measure the welfare effects of price changes EXAMPLE 56 Welfare Loss from a Price Increase These ideas can be illustrated numerically by returning to our old hamburgersoft drink example Lets look at the welfare consequences of an unconscionable price increase for soft drinks good x from 1 to 4 In Example 53 we found that the compensated demand for good x was given by xc 1px py V2 5 Vp05 y p05 x 560 Hence the welfare cost of the price increase is given by CV 5 3 4 1 xc 1px py V2dpx 5 3 4 1 Vp05 y p205 x dpx 5 2Vp05 y p05 x px54 px51 561 If we use the values we have been assuming throughout this gastronomic feast 1V 5 2 py 5 42 then CV 5 2 2 2 142 05 2 2 2 2 112 05 5 8 562 This figure would be cut in half to 4 if we believed that the utility level after the price increase 1V 5 12 were the more appropriate utility target for measuring compensation If instead we had used the Marshallian demand function x 1px py I2 5 05Ip21 x 563 the loss would be calculated as loss 5 3 4 1 x 1px py I2dpx 5 3 4 1 05Ip21 x dpx 5 05I ln px 4 1 564 Thus with I 8 this loss is loss 5 4 ln 142 2 4 ln 112 5 4 ln 142 5 4 11392 5 555 565 which seems a reasonable compromise between the two alternative measures based on the compensated demand functions QUERY In this problem none of the demand curves has a finite price at which demand goes to precisely zero How does this affect the computation of total consumer surplus Does this affect the types of welfare calculations made here Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 171 59 REVEALED PREFERENCE AND THE SUBSTITUTION EFFECT The principal unambiguous prediction that can be derived from the utilitymaximization model is that the slope or price elasticity of the compensated demand curve is negative We have shown this result in two related ways The first proof was based on the quasiconcavity of utility functions that is because any indifference curve must exhibit a diminishing MRS any change in a price will induce a quantity change in the opposite direction when moving along that indifference curve A second proof derives from Shephards lemmabecause the expenditure function is concave in prices the compensated demand function which is the derivative of the expenditure function must have a negative slope Again utility is held con stant in this calculation as one argument in the expenditure function To some economists the reliance on a hypothesis about an unobservable utility function represented a weak foun dation on which to base a theory of demand An alternative approach which leads to the same result was first proposed by Paul Samuelson in the late 1940s10 This approach which Samuelson termed the theory of revealed preference defines a principle of rationality that is based on observed reactions to differing budget constraints and then uses this principle to approximate an individuals utility function In this sense a person who follows Samuelsons principle of rationality behaves as if he or she were maximizing a proper utility function and exhibits a negative substitution effect Because Samuelsons approach provides additional insights into our model of consumer choice we will briefly examine it here 10Paul A Samuelson Foundations of Economic Analysis Cambridge MA Harvard University Press 1947 With income I1 the individual can afford both points A and B If A is selected then A is revealed preferred to B It would be irrational for B to be revealed preferred to A in some other priceincome configuration Quantity of x Quantity of y ya I1 I3 I2 yb xa xb A B C FIGURE 510 Demonstration of the Principle of Rationality in the Theory of Revealed Preference Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 172 Part 2 Choice and Demand 591 Graphical approach The principle of rationality in the theory of revealed preference is as follows Consider two bundles of goods A and B If at some prices and income level the individual can afford both A and B but chooses A we say that A has been revealed preferred to B The principle of rationality states that under any different priceincome arrangement B can never be revealed preferred to A If B is in fact chosen at another priceincome configuration it must be because the individual could not afford A The principle is illustrated in Figure 510 Suppose that when the budget constraint is given by I1 point A is chosen even though B also could have been purchased Then A has been revealed preferred to B If for some other budget constraint B is in fact chosen then it must be a case such as that represented by I2 where A could not have been bought If B were chosen when the budget constraint is I3 this would be a violation of the principle of rationality because with I3 both A and B can be bought With budget constraint I3 it is likely that some point other than either A or B say C will be bought Notice how this principle uses observable reactions to alternative budget constraints to rank commodity bundles rather than assuming the existence of a utility function itself Also notice how the principle offers a glimpse of why indifference curves are convex Now we turn to a formal proof 592 Revealed preference and the negativity of the substitution effect Suppose that an individual is indifferent between two bundles C composed of xC and yC and D composed of xD and yD Let pC x pC y be the prices at which bundle C is chosen and pD x pD y the prices at which bundle D is chosen Because the individual is indifferent between C and D it must be the case that when C was chosen D cost at least as much as C pC xxC 1 pC yyC pC xxD 1 pC yyD 566 A similar statement holds when D is chosen pD x xD 1 pD y yD pD x xC 1 pD y yC 567 Rewriting these equations gives pC x 1xC 2 xD2 1 pC y 1yC 2 yD2 0 568 pD x 1xD 2 xC2 1 pD y 1yD 2 yC2 0 569 Adding these together yields 1pC x 2 pD x 2 1xC 2 xD2 1 1pC y 2 pD y 2 1yC 2 yD2 0 570 Now suppose that only the price of x changes assume that pC y 5 pD y Then 1pC x 2 pD x 2 1xC 2 xD2 0 571 But Equation 571 says that price and quantity move in the opposite direction when utility is held constant remember bundles C and D are equally attractive This is precisely a statement about the nonpositive nature of the substitution effect xc 1px py V2 px 5 x px U5constant 0 572 We have arrived at the result by an approach that does not require the existence of a quasi concave utility function Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 173 Summary In this chapter we used the utilitymaximization model to study how the quantity of a good that an individual chooses responds to changes in income or to changes in that goods price The final result of this examination is the derivation of the familiar downwardsloping demand curve In arriving at that result however we have drawn a wide variety of insights from the general economic theory of choice Proportional changes in all prices and income do not shift the individuals budget constraint and therefore do not change the quantities of goods chosen In formal terms demand functions are homogeneous of degree 0 in all prices and income When purchasing power changes ie when income increases with prices remaining unchanged budget con straints shift and individuals will choose new commodity bundles For normal goods an increase in purchasing power causes more to be chosen In the case of inferior goods however an increase in purchasing power causes less to be purchased Hence the sign of xiI could be either positive or negative although xiI 0 is the most common case A decrease in the price of a good causes substitution and income effects that for a normal good cause more of the good to be purchased For inferior goods however sub stitution and income effects work in opposite directions and no unambiguous prediction is possible Similarly an increase in price induces both substitution and income effects that in the normal case cause less to be demanded For inferior goods the net result is again ambiguous Marshallian demand curves represent twodimensional depictions of demand functions for which only the own price variesother prices and income are held constant Changes in these other variables will usually shift the position of the demand curve The sign of the slope of the Marshallian demand curve a x 1px py I2 px b is theoretically ambiguous because substitution and income effects may work in opposite directions The Slutsky equation permits a formal study of this ambiguity Compensated or Hicksian demand functions show how quantities demanded are functions of all prices and utility The compensated demand function for a good can be gen erated by partially differentiating the expenditure function with respect to that goods price Shephards lemma Compensated or Hicksian demand curves represent twodimensional depictions of compensated demand functions for which only the ownprice variesother prices and utility are held constant The sign of the slope of the compensated demand curve a xc 1px py U2 px b is unambiguously negative because of the quasiconcavity of utility functions or the related concavity of the expenditure function Demand elasticities are often used in empirical work to summarize how individuals react to changes in prices and income The most important such elasticity is the own price elasticity of demand ex px This measures the proportionate change in quantity in response to a 1 percent change in price A similar elasticity can be defined for movements along the compensated demand curve There are many relationships among demand elasticities Some of the more important ones are 1 ownprice elasticities determine how a price change affects total spending on a good 2 substitution and income effects can be summarized by the Slutsky equation in elasticity form and 3 various aggregation relations hold among elasticitiesthese show how the demands for different goods are related Welfare effects of price changes can be measured by changing areas below either compensated or Marshallian demand curves Such changes affect the size of the consumer surplus that individuals receive from being able to make market transactions The negativity of the substitution effect is the most basic conclusion from demand theory This result can be shown using revealed preference theory and so does not require assuming the existence of a utility function Problems 51 Thirsty Ed drinks only pure spring water but he can purchase it in two differentsized containers 075 liter and 2 liter Because the water itself is identical he regards these two goods as perfect substitutes a Assuming Eds utility depends only on the quantity of water consumed and that the containers themselves yield no utility express this utility function in terms of quantities of 075liter containers x and 2liter containers y Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 174 Part 2 Choice and Demand b State Eds demand function for x in terms of px py and I c Graph the demand curve for x holding I and py constant d How do changes in I and py shift the demand curve for x e What would the compensated demand curve for x look like in this situation 52 David gets 3 per week as an allowance to spend any way he pleases Because he likes only peanut butter and jelly sand wiches he spends the entire amount on peanut butter at 005 per ounce and jelly at 010 per ounce Bread is pro vided free of charge by a concerned neighbor David is a par ticular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter He is set in his ways and will never change these proportions a How much peanut butter and jelly will David buy with his 3 allowance in a week b Suppose the price of jelly were to increase to 015 an ounce How much of each commodity would be bought c By how much should Davids allowance be increased to compensate for the increase in the price of jelly in part b d Graph your results in parts a to c e In what sense does this problem involve only a single commodity peanut butter and jelly sandwiches Graph the demand curve for this single commodity f Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly 53 As defined in Chapter 3 a utility function is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope The MRS depends on the ratio yx a Prove that in this case xI is constant b Prove that if an individuals tastes can be represented by a homothetic indifference map then price and quantity must move in opposite directions that is prove that Giffens paradox cannot occur 54 As in Example 51 assume that utility is given by utility 5 U1x y2 5 x03y07 a Use the uncompensated demand functions given in Example 51 to compute the indirect utility function and the expenditure function for this case b Use the expenditure function calculated in part a together with Shephards lemma to compute the compensated demand function for good x c Use the results from part b together with the uncompensated demand function for good x to show that the Slutsky equation holds for this case 55 Suppose the utility function for goods x and y is given by utility 5 U1x y2 5 xy 1 y a Calculate the uncompensated Marshallian demand functions for x and y and describe how the demand curves for x and y are shifted by changes in I or the price of the other good b Calculate the expenditure function for x and y c Use the expenditure function calculated in part b to compute the compensated demand functions for goods x and y Describe how the compensated demand curves for x and y are shifted by changes in income or by changes in the price of the other good 56 Over a 3year period an individual exhibits the following consumption behavior px py x y Year 1 3 3 7 4 Year 2 4 2 6 6 Year 3 5 1 7 3 Is this behavior consistent with the principles of revealed preference theory 57 Suppose that a person regards ham and cheese as pure complementshe or she will always use one slice of ham in combination with one slice of cheese to make a ham and cheese sandwich Suppose also that ham and cheese are the only goods that this person buys and that bread is free a If the price of ham is equal to the price of cheese show that the ownprice elasticity of demand for ham is 205 and that the crossprice elasticity of demand for ham with respect to the price of cheese is also 205 b Explain why the results from part a reflect only income effects not substitution effects What are the compensated price elasticities in this problem c Use the results from part b to show how your answers to part a would change if a slice of ham cost twice the price of a slice of cheese d Explain how this problem could be solved intuitively by assuming this person consumes only one gooda ham and cheese sandwich 58 Show that the share of income spent on a good x is sx 5 d ln E d ln px where E is total expenditure Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 175 Analytical Problems 59 Share elasticities In the Extensions to Chapter 4 we showed that most empirical work in demand theory focuses on income shares For any good x the income share is defined as sx 5 pxxI In this problem we show that most demand elasticities can be derived from corresponding share elasticities a Show that the elasticity of a goods budget share with respect to income 1esx I 5 sxI Isx2 is equal to exI 2 1 Interpret this conclusion with a few numerical examples b Show that the elasticity of a goods budget share with respect to its own price 1esx px 5 sxpx pxsx2 is equal to ex px 1 1 Again interpret this finding with a few numerical examples c Use your results from part b to show that the expenditure elasticity of good x with respect to its own price 3epx x px 5 1px x2px 1x4 is also equal to ex px 1 1 d Show that the elasticity of a goods budget share with respect to a change in the price of some other good 1esx py 5 sxpy pysx2 is equal to ex py e In the Extensions to Chapter 4 we showed that with a CES utility function the share of income devoted to good x is given by sx 5 1 11 1 pk yp2k x 2 where k 5 δ 1δ 2 12 5 1 2 σ Use this share equation to prove Equation 556 exc px 5 211 2 sx2σ 510 More on elasticities Part e of Problem 59 has a number of useful applications because it shows how price responses depend ultimately on the underlying parameters of the utility function Specifically use that result together with the Slutsky equation in elasticity terms to show a In the CobbDouglas case 1σ 5 12 the following relationship holds between the ownprice elasticities of x and y ex px 1 ey py 5 22 b If σ 1 then ex px 1 ey py 22 and if σ 1 then ex px 1 ey py 22 Provide an intuitive explanation for this result c How would you generalize this result to cases of more than two goods Discuss whether such a generalization would be especially meaningful 511 Aggregation of elasticities for many goods The three aggregation relationships presented in this chapter can be generalized to any number of goods This problem asks you to do so We assume that there are n goods and that the share of income devoted to good i is denoted by si We also define the following elasticities ei I 5 xi I I xi ei j 5 xi pj pj xi Use this notation to show a Homogeneity g n j51ei j 1 ei I 5 0 b Engel aggregation g n i51siei I 5 1 c Cournot aggregation g n i51siei j 5 2sj 512 Quasilinear utility revisited Consider a simple quasilinear utility function of the form U1x y2 5 x 1 ln y a Calculate the income effect for each good Also calculate the income elasticity of demand for each good b Calculate the substitution effect for each good Also calculate the compensated ownprice elasticity of demand for each good c Show that the Slutsky equation applies to this function d Show that the elasticity form of the Slutsky equation also applies to this function Describe any special features you observe e A modest generalization of this quasilinear utility function is given by U1x y2 5 x 1 f 1y2 where f r 0 f s 0 How if at all would the results from parts ad differ if this general function were used instead of ln y in the utility function 513 The almost ideal demand system The general form for the expenditure function of the almost ideal demand system AIDS is given by ln E 1p1 pn U2 5 a0 1 a n i51 αi ln pi 1 1 2 a n i51 a n j51 γij ln pi ln pj 1 Uβ0 q k i51 pβk k For analytical ease assume that the following restrictions apply γij 5 γji a n i51 αi 5 1 and a n j51 γij 5 a n k51 βk 5 0 a Derive the AIDS functional form for a twogoods case b Given the previous restrictions show that this expenditure function is homogeneous of degree 1 in all prices This along with the fact that this function resembles closely the actual data makes it an ideal function c Using the fact that sx 5 d ln E d ln px see Problem 58 calculate the income share of each of the two goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 176 Part 2 Choice and Demand 514 Price indifference curves Price indifference curves are isoutility curves with the prices of two goods on the X and Yaxes respectively Thus they have the following general form 1p1 p22 0 v 1p1 p2 I2 5 v0 a Derive the formula for the price indifference curves for the CobbDouglas case with α 5 β 5 05 Sketch one of them b What does the slope of the curve show c What is the direction of increasing utility in your graph Behavioral Problems 515 The multiself model Many of the topics in behavioral economics can be approached using a simple model that pictures economic decision makers as having multiple selves each with a different utility function Here we examine two versions of this model In each we assume that this persons choices are dictated by one of two possible quasilinear utility functions 1 U1 1x y2 5 x 1 2 ln y 2 U2 1x y2 5 x 1 3 ln y a Decision utility In this model we make a distinction between the utility function that the person uses to make decisionsfunction 1and the function that determines the utility he or she actually experiences function 2 These functions may differ for a variety of reasons such as lack of information about good y or in a two period setting an unwillingness to change from past behavior Whatever the cause the divergence between the two concepts can lead to welfare losses To see this assume that px 5 py 5 1 and I 5 10 i What consumption choices will this person make using his or her decision utility function ii What will be the loss of experienced utility if this person makes the choice specified in part i iii How much of a subsidy would have to be given to good y purchases if this person is to be encour aged to consume commodity bundle that actually maximizes experienced utility remember this person still maximizes decision utility in his or her decision making iv We know from the lump sum principle that an income transfer could achieve the utility level specified in part iii at a cost lower than subsidizing good y Show this and then discuss whether this might not be a socially preferred solution to the problem b Preference uncertainty In this version of the multiself model the individual recognizes that he or she might experience either of the two utility functions in the future but does not know which will prevail One possible solution to this problem is to assume either is equally likely so make consumption choices that maximize U1x y2 5 x 1 25 ln y i What commodity bundle will this person choose ii Given the choice in part i what utility losses will be experienced once this person discovers his or her true preferences iii How much would this person pay to gather infor mation about his or her future preferences before making the consumption choices Note The distinction between decision utility and experienced utility is examined extensively in Chetty 2015 Suggestions for Further Reading Chetty Raj Behavioral Economics and Public Policy A Pragmatic Perspective American Economic Review May 2015 133 Provides a consistent theoretical framework for much of the lit erature on behavioral economics Also describes several empirical examples that make use of modern big data techniques Cook P J A One Line Proof of the Slutsky Equation Amer ican Economic Review 62 March 1972 139 Clever use of duality to derive the Slutsky equation uses the same method as in Chapter 5 but with rather complex notation Fisher F M and K Shell The Economic Theory of Price Indi ces New York Academic Press 1972 Complete technical discussion of the economic properties of vari ous price indexes describes ideal indexes based on utilitymaxi mizing models in detail Luenberger D G Microeconomic Theory New York McGraw Hill 1992 Pages 147151 provide a concise summary of how to state the Slutsky equations in matrix notation MasColell Andreu Michael D Whinston and Jerry R Green Microeconomic Theory New York Oxford University Press 1995 Chapter 3 covers much of the material in this chapter at a some what higher level Section I on measurement of the welfare effects of price changes is especially recommended Samuelson Paul A Foundations of Economic Analysis Cam bridge MA Harvard University Press 1947 chapter 5 Provides a complete analysis of substitution and income effects Also develops the revealed preference notion Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 177 Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 Provides an extensive derivation of the Slutsky equation and a lengthy presentation of elasticity concepts Sydsaetter K A Strom and P Berck Economists Mathemati cal Manual Berlin Germany SpringerVerlag 2003 Provides a compact summary of elasticity concepts The coverage of elasticity of substitution notions is especially complete Varian H Microeconomic Analysis 3rd ed New York W W Norton 1992 Formal development of preference notions Extensive use of expenditure functions and their relationship to the Slutsky equa tion Also contains a nice proof of Roys identity Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 178 EXTENSIONS DEmanD ConCEptS anD thE EvaluatIon of prICE InDICES In Chapters 4 and 5 we introduced a number of related demand concepts all of which were derived from the underly ing model of utility maximization Relationships among these various concepts are summarized in Figure E51 We have already looked at most of the links in the table formally We have not yet discussed the mathematical relationship between indirect utility functions and Marshallian demand functions Roys identity and we will do that below All the entries in the table make clear that there are many ways to learn some thing about the relationship between individuals welfare and the prices they face In this extension we will explore some of these approaches Specifically we will look at how the con cepts can shed light on the accuracy of the consumer price index CPI the primary measure of inflation in the United States We will also look at a few other price index concepts The CPI is a market basket index of the cost of living Researchers measure the amounts that people consume of a set of goods in some base period in the twogood case these baseperiod consumption levels might be denoted by x0 and y0 and then use current price data to compute the changing price of this market basket Using this procedure the cost of the market basket initially would be I0 5 p0 xx0 1 p0 yy0 and the cost in period 1 would be I1 5 p1 xx0 1 p1 yy0 The change in the cost of living between these two periods would then be measured by I1I0 Although this procedure is an intuitively plausible way of measuring inflation and market basket price indices are widely used such indices have many shortcomings FIGURE E51 Relationships among Demand Concepts Primal Dual Inverses Shephards lemma Roys identity Maximize Ux y st I pxx pyy Indirect utility function U Vpx py I Minimize Ex y st U Ux y xpx py I Marshallian demand xcpx py U E px Compensated demand Expenditure function E Epx py U px I V V Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 179 E51 Expenditure functions and substitution bias Market basket price indices suffer from substitution bias Because the indices do not permit individuals to make substitutions in the market basket in response to changes in relative prices they will tend to overstate the welfare losses that people incur from increasing prices This exaggeration is illustrated in Figure E52 To achieve the utility level U0 initially requires expenditures of E0 resulting in a purchase of the basket x0 y0 If pxpy decrease the initial utility level can now be obtained with expenditures of E1 by altering the consumption bundle to x1 y1 Computing the expenditure level needed to continue consuming x0 y0 exaggerates how much extra purchasing power this person needs to restore his or her level of wellbeing Economists have extensively studied the extent of this substitution bias Aizcorbe and Jackman 1993 for example find that this difficulty with a market basket index may exaggerate the level of inflation shown by the CPI by approximately 02 percent per year E52 Roys identity and new goods bias When new goods are introduced it takes some time for them to be integrated into the CPI For example Hausman 1999 2003 states that it took more than 15 years for cell phones to appear in the index The problem with this delay is that market basket indices will fail to reflect the welfare gains that people experience from using new goods To measure these costs Hausman sought to measure a virtual price 1p2 at which the demand for say cell phones would be zero and then argued that the introduction of the good at its market price represented a change in consumer surplus that could be measured Hence the author was faced with the problem of how to get from the Marshallian demand function for cell phones which he estimated econometrically to the expenditure function To do so he used Roys identity see Roy 1942 Remember that the consumers utilitymaximizing problem can be represented by the Lagrangian expression 5 U1x y2 1 λ1I 2 pxx 2 pyy2 The indirect utility function arising from this maximization problem is given by V 1px py I2 Applying the envelope theorem to this value function yields dV 1px py I2 dpx 5 px 5 2λx 1px py I2 dV 1px py I2 dI 5 I 5 λ i Initially expenditures are given by E0 and this individual buys x0 y0 If pxpy decreases utility level U0 can be reached most cheaply by consuming x1 y1 and spending E1 Purchasing x0 y0 at the new prices would cost more than E1 Hence holding the consumption bundle constant imparts an upward bias to CPItype computations E0 U0 U0 U E1 x0 y0 x1 Quantity of ty of ty x Quantity of y FIGURE E52 Substitution Bias in the CPI Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 180 Part 2 Choice and Demand These equations allow us to extract the Marshall demand function as x 1px py I2 5 2Vpx VI ii This expression is called Roys identity Using his estimates of the Marshallian demand function Hausman integrated Equation ii to obtain the implied indirect utility function and then calculated its inverse the expendi ture function check Figure E51 to see the logic of the pro cess Although this certainly is a roundabout scheme it did yield large estimates for the gain in consumer welfare from cell phonesa present value in 1999 of more than 100 bil lion Delays in the inclusion of such goods into the CPI can therefore result in a misleading measure of consumer welfare E53 Other complaints about the CPI Researchers have found several other faults with the CPI as currently constructed Most of these focus on the consequences of using incorrect prices to compute the index For example when the quality of a good improves people are made better off although this may not show up in the goods price Throughout the 1970s and 1980s the reliability of color television sets improved dramatically but the price of a set did not change much A market basket that included one color television set would miss this source of improved welfare Similarly the opening of big box retailers such as Costco and Home Depot during the 1990s undoubtedly reduced the prices that consumers paid for various goods But including these new retail outlets into the sample scheme for the CPI took several years so the index misrepresented what people were actually paying Assessing the magnitude of error introduced by these cases where incorrect prices are used in the CPI can also be accomplished by using the various demand concepts in Figure E51 For a summary of this research see Moulton 1996 E54 Exact price indices In principle it is possible that some of the shortcomings of price indices such as the CPI might be ameliorated by more careful attention to demand theory If the expenditure function for the representative consumer were known for example it would be possible to construct an exact index for changes in purchasing power that would take commodity substitution into account To illustrate this suppose there are only two goods and we wish to know how purchasing power has changed between period 1 and period 2 If the expenditure function is given by E 1px py U2 then the ratio I1 2 5 E 1p2 x p2 y U 2 E 1p1 x p1 y U 2 iii shows how the cost of attaining the target utility level U has changed between the two periods If for example I1 2 5 104 then we would say that the cost of attaining the utility target had increased by 4 percent Of course this answer is only a conceptual one Without knowing the representa tive persons utility function we would not know the specific form of the expenditure function But in some cases Equa tion iii may suggest how to proceed in index construction Suppose for example that the typical persons preferences could be represented by the CobbDouglas utility function U1x y2 5 xαy12α In this case it is easy to show that the expendi ture function is a generalization of the one given in Example 44 E 1px py U2 5 pα xp12α y Uαα 11 2 α2 12α 5 kpα xp12α y U Inserting this function into Equation iii yields I1 2 5 k1p2 x2 α 1p2 y2 12αU k1p1 x2 α 1p1 y2 12αU 5 1p2 x2 α 1p2 y2 12α 1p1 x2 α 1p1 y2 12α iv Thus in this case the exact price index is a relatively simple function of the observed prices The particularly useful feature of this example is that the utility target cancels out in the construction of the costofliving index as it will anytime the expenditure function is homogeneous in utility Notice also that the expenditure shares α and 1 2 α play an important role in the indexthe larger a goods share the more important will changes be in that goods price in the final index E55 Development of exact price indices The CobbDouglas utility function is of course a simple one Much recent research on price indices has focused on more general types of utility functions and on the discovery of the exact price indices they imply For example Feenstra and Reinsdorf 2000 show that the almost ideal demand system described in the Extensions to Chapter 4 implies an exact price index I that takes a Divisia form ln 1I2 5 a n i51 wiD ln pi v here the wi are weights to be attached to the change in the logarithm of each goods price Often the weights in Equation v are taken to be the budget shares of the goods Interestingly this is precisely the price index implied by the CobbDouglas utility function in Equation iv because ln 1I1 22 5 α ln p2 x 1 11 2 α2 ln p2 y 2 α ln p1 x 2 11 2 α2 ln p1 y vi 5 αD ln px 1 11 2 α2D ln py In actual applications the weights would change from period to period to reflect changing budget shares Similarly changes over several periods would be chained together from a number of singleperiod price change indices Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 5 Income and Substitution Effects 181 Changing demands for food in China China has one of the fastest growing economies in the world Its GDP per capita is currently growing at a rate of approx imately 8 percent per year Chinese consumers also spend a large fraction of their incomes on foodapproximately 38 percent of total expenditures in recent survey data One implication of the rapid growth in Chinese incomes however is that patterns of food consumption are changing rapidly Purchases of staples such as rice or wheat are declining in relative importance whereas purchases of poultry fish and processed foods are growing rapidly An article by Gould and Villarreal 2006 studies these patterns in detail using the AIDS model They identify a variety of substitution effects across specific food categories in response to changing relative prices Such changing patterns imply that a fixed market bas ket price index such as the US Consumer Price Index would be particularly inappropriate for measuring changes in the cost of living in China and that some alternative approaches should be examined References Aizcorbe Ana M and Patrick C Jackman The Commodity Substitution Effect in CPI Data 198291 Monthly Labor Review December 1993 2533 Feenstra Robert C and Marshall B Reinsdorf An Exact Price Index for the Almost Ideal Demand System Eco nomics Letters February 2000 15962 Gould Brain W and Hector J Villarreal An Assessment of the Current Structure of Food Demand in Urban China Agricultural Economics January 2006 116 Hausman Jerry Cellular Telephone New Products and the CPI Journal of Business and Economic Statistics April 1999 18894 Hausman Jerry Sources of Bias and Solutions to Bias in the Consumer Price Index Journal of Economic Perspectives Winter 2003 2344 Moulton Brent R Bias in the Consumer Price Index What Is the Evidence Journal of Economic Perspectives Fall 1996 15977 Roy R De Iutilité contribution á la théorie des choix Paris Hermann 1942 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 183 CHAPTER SIX Demand Relationships among Goods In Chapter 5 we examined how changes in the price of a particular good say good x affect the quantity of that good chosen Throughout the discussion we held the prices of all other goods constant It should be clear however that a change in one of these other prices could also affect the quantity of x chosen For example if x were taken to repre sent the quantity of automobile miles that an individual drives this quantity might be expected to decrease when the price of gasoline increases or increase when air and bus fares increase In this chapter we will use the utilitymaximization model to study such relationships 61 THE TWOGOOD CASE We begin our study of the demand relationship among goods with the twogood case Unfortunately this case proves to be rather uninteresting because the types of relation ships that can occur when there are only two goods are limited Still the twogood case is useful because it can be illustrated with twodimensional graphs Figure 61 starts our examination by showing two examples of how the quantity of x chosen might be affected by a change in the price of y In both panels of the figure py has decreased This has the result of shifting the budget constraint outward from I0 to I1 In both cases the quantity of good y chosen has also increased from y0 to y1 as a result of the decrease in py as would be expected if y is a normal good For good x however the results shown in the two panels differ In a the indifference curves are nearly Lshaped implying a fairly small substitu tion effect A decrease in py does not induce a large move along U0 as y is substituted for x That is x drops relatively little as a result of the substitution The income effect however reflects the greater purchasing power now available and this causes the total quantity of x chosen to increase Hence xpy is negative x and py move in opposite directions In Figure 61b this situation is reversed xpy is positive The relatively flat indiffer ence curves in Figure 61a result in a large substitution effect from the fall in py The quan tity of x decreases sharply as y is substituted for x along U0 As in Figure 61a the increased Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 184 Part 2 Choice and Demand purchasing power from the decrease in py causes more x to be bought but now the sub stitution effect dominates and the quantity of x decreases to x1 In this case x and py then move in the same direction 611 A mathematical treatment The ambiguity in the effect of changes in py can be further illustrated by a Slutskytype equation By using procedures similar to those in Chapter 5 it is fairly simple to show that x 1 px py I2 py 5 substitution effect 1 income effect 5 x py U5constant 2 y x I 61 or in elasticity terms ex py 5 exc py 2 syex I 62 Notice that the size of the income effect is determined by the share of good y in this per sons purchases The impact of a change in py on purchasing power is determined by how important y is to this person For the twogood case the terms on the right side of Equations 61 and 62 have different signs Assuming that indifference curves are convex the substitution effect xpy0 U5constant is positive If we confine ourselves to moves along one indifference curve increases in py In both panels the price of y has decreased In a substitution effects are small therefore the quantity of x consumed increases along with y Because xpy 0 x and y are gross complements In b substitu tion effects are large therefore the quantity of x chosen decreases Because xpy 0 x and y would be termed gross substitutes Quantity of x Quantity of x Quantity of y Quantity of y a Gross complements b Gross substitutes x0 y0 y1 x1 y0 y1 x1 x0 I0 I0 I1 I1 U0 U1 U0 U1 FIGURE 61 Differing Directions of CrossPrice Effects Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 185 increase x and decreases in py decrease the quantity of x chosen However assuming x is a normal good the income effect 2yxI or 2syex I is clearly negative Hence the combined effect is ambiguous xpy could be either positive or negative Even in the two good case the demand relationship between the demand for x and py is rather complex EXAMPLE 61 Another Slutsky Decomposition for CrossPrice Effects In Example 54 we examined the Slutsky decomposition for the effect of a change in the price of x Now lets look at the crossprice effect of a change in y prices on x purchases Remember that the uncompensated and compensated demand functions for x are given by x 1 px py I2 5 05I px 63 and xc 1 px py V2 5 Vp05 y p205 x 64 As we have pointed out before the Marshallian demand function in this case yields xpy 5 0 that is changes in the price of y do not affect x purchases Now we show that this occurs because the substitution and income effects of a price change are precisely counterbalancing The substi tution effect in this case is given by x py U5constant 5 xc py 5 05Vp205 y p205 x 65 Substituting for V from the indirect utility function 1V 5 05Ip205 y p205 x 2 gives a final statement for the substitution effect x py U5constant 5 025Ip21 y p21 x 66 Returning to the Marshallian demand function for y1y 5 05Ip21 y 2 to calculate the income effect yields 2y x I 5 2305Ip21 y 4 305p21 x 4 5 2025Ip21 y p21 x 67 and combining Equations 66 and 67 gives the total effect of the change in the price of y as x py 5 025Ip21 y p21 x 2 025Ip21 y p21 x 5 0 68 This makes clear that the reason that changes in the price of y have no effect on x purchases in the CobbDouglas case is that the substitution and income effects from such a change are precisely offsetting neither of the effects alone however is zero Returning to our numerical example 1 px 5 1 py 5 4 I 5 8 V 5 22 suppose now that py falls to 2 This should have no effect on the Marshallian demand for good x The compensated demand function in Equation 64 shows that the price change would cause the quantity of x demanded to decrease from 4 to 283 15 222 as y is substituted for x with utility unchanged However the increased purchasing power arising from the price decrease precisely reverses this effect QUERY Why would it be incorrect to argue that if xpy 5 0 then x and y have no substitu tion possibilitiesthat is they must be consumed in fixed proportions Is there any case in which such a conclusion could be drawn Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 186 Part 2 Choice and Demand 62 SUBSTITUTES AND COMPLEMENTS With many goods there is much more room for interesting relations among goods It is relatively easy to generalize the Slutsky equation for any two goods xi xj as xi 1p1 pn I2 pj 5 xi pj U5constant 2 xj xi I 69 and again this can be readily translated into an elasticity relation ei j 5 ec i j 2 sjei I 610 This says that the change in the price of any good here good j induces income and sub stitution effects that may change the quantity of every good demanded Equations 69 and 610 can be used to discuss the idea of substitutes and complements Intuitively these ideas are rather simple Two goods are substitutes if one good may as a result of changed condi tions replace the other in use Some examples are tea and coffee hamburgers and hot dogs and butter and margarine Complements on the other hand are goods that go together such as coffee and cream fish and chips or brandy and cigars In some sense substitutes substitute for one another in the utility function whereas complements complement each other There are two different ways to make these intuitive ideas precise One of these focuses on the gross effects of price changes by including both income and substitution effects the other looks at substitution effects alone Because both definitions are used we will examine each in detail 621 Gross Marshallian substitutes and complements Whether two goods are substitutes or complements can be established by referring to observed price reactions as follows That is two goods are gross substitutes if an increase in the price of one good causes more of the other good to be bought The goods are gross complements if an increase in the price of one good causes less of the other good to be purchased For example if the price of coffee increases the demand for tea might be expected to increase they are substitutes whereas the demand for cream might decrease coffee and cream are complements Equa tion 69 makes it clear that this definition is a gross definition in that it includes both income and substitution effects that arise from price changes Because these effects are in fact combined in any realworld observation we can make it might be reasonable always to speak only of gross substitutes and gross complements D E F I N I T I O N Gross substitutes and complements Two goods xi and xj are said to be gross substitutes if xi pj 0 611 and gross complements if xi pj 0 612 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 187 622 Asymmetry of the gross definitions There are however several things that are undesirable about the gross definitions of sub stitutes and complements The most important of these is that the definitions are not sym metric It is possible by the definitions for x1 to be a substitute for x2 and at the same time for x2 to be a complement of x1 The presence of income effects can produce paradoxical results Lets look at a specific example EXAMPLE 62 Asymmetry in CrossPrice Effects Suppose the utility function for two goods x and y has the quasilinear form U1x y2 5 ln x 1 y 613 Setting up the Lagrangian expression 5 ln x 1 y 1 λ1I 2 pxx 2 pyy2 614 yields the following firstorder conditions x 5 1 x 2 λpx 5 0 y 5 1 2 λpy 5 0 λ 5 I 2 pxx 2 pyy 5 0 615 Moving the terms in λ to the right and dividing the first equation by the second yields 1 x 5 px py 616 pxx 5 py 617 Substitution into the budget constraint now permits us to solve for the Marshallian demand function for y I 5 px x 1 pyy 5 py 1 pyy 618 Hence y 5 I 2 py py 619 This equation shows that an increase in py must decrease spending on good y ie py y There fore because px and I are unchanged spending on x must increase Thus x py 0 620 and we would term x and y gross substitutes On the other hand Equation 619 shows that spend ing on y is independent of px Consequently y px 5 0 621 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 188 Part 2 Choice and Demand 63 NET HICKSIAN SUBSTITUTES AND COMPLEMENTS Because of the possible asymmetries involved in the definition of gross substitutes and complements an alternative definition that focuses only on substitution effects is often used and looked at in this way x and y would be said to be independent of each other they are neither gross substitutes nor gross complements Relying on gross responses to price changes to define the relationship between x and y would therefore run into ambiguity QUERY In Example 34 we showed that a utility function of the form given by Equation 613 is not homothetic The MRS does not depend only on the ratio of x to y Can asymmetry arise in the homothetic case D E F I N I T I O N Net substitutes and complements Goods xi and xj are said to be net substitutes if xi pj U5constant 0 622 and net complements if xi pj U5constant 0 623 These definitions1 then look only at the substitution terms to determine whether two goods are substitutes or complements This definition is both intuitively appealing because it looks only at the shape of an indifference curve and theoretically desirable because it is unambiguous Once xi and xj have been discovered to be substitutes they stay substitutes no matter in which direction the definition is applied As a matter of fact the definitions are symmetric xi pj U5constant 5 xj pi U5constant 624 1These are sometimes called Hicksian substitutes and complements named after the British economist John Hicks who originally developed the definitions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 189 The substitution effect of a change in pi on good xj is identical to the substitution effect of a change in pj on the quantity of xi chosen This symmetry is important in both theoretical and empirical work2 The difference between Hicks and Marshalls definitions of substitutes and complements is readily apparent in Figure 61 In that figure the two goods are always Hicks substitutes Because of the convexity of the indifference curves a decrease in py must cause the quantity of x chosen to fall when utility is held constant That is xcpy 0 However Marshalls definition also involves income effects which are always negative assuming both goods are normal and that can cause ambiguity In Figure 61a this negative income effect domi nates the positive substitution effect so that xpy 0 and the two goods would be termed Marshall complements In Figure 61b however the positive substitution effect outweighs the negative income effect In this case xpy 0 and the goods would be called Marshall substitutes Of course with many goods much more complex patterns are possible but the Hicks definition simplifies the situation considerably by eliminating the ambiguities 64 SUBSTITUTABILITY WITH MANY GOODS Once the utilitymaximizing model is extended to many goods a wide variety of demand patterns become possible Whether a particular pair of goods are net substitutes or net com plements is basically a question of a persons preferences thus one might observe all sorts of relationships A major theoretical question that has concerned economists is whether substitutability or complementarity is more prevalent In most discussions we tend to regard goods as substitutes a price increase in one market tends to increase demand in most other markets It would be nice to know whether this intuition is justified The British economist John Hicks studied this issue in some detail more than 75 years ago and reached the conclusion that most goods must be substitutes The result is sum marized in what has come to be called Hicks second law of demand3 A modern proof starts with the compensated demand function for a particular good xc i 1p1 pn V2 This func tion is homogeneous of degree 0 in all prices if utility is held constant and prices dou ble quantities demanded do not change because the utilitymaximizing tangencies do not change Applying Eulers theorem to the function yields p1 xc i p1 1 p2 xc i p2 1 c1 pn xc i pn 5 0 625 2This symmetry is easily shown using Shephards lemma Compensated demand functions can be calculated from expenditure functions by differentiation xc i 1p1 pn V2 5 E1p1 pn V2 pi Hence the substitution effect is given by xi pj U5constant 5 xc i pj 5 2E pjpi 5 Eij But now we can apply Youngs theorem to the expenditure function Eij 5 Eji 5 xc j pi 5 xj pi U5constant which proves the symmetry 3See John Hicks Value and Capital Oxford UK Oxford University Press 1939 mathematical appendices There is some debate about whether this result should be called Hicks second or third law In fact two other laws that we have already seen are listed by Hicks 1 xc ipi 0 negativity of the ownsubstitution effect and 2 xc ipj 5 xc jpi symmetry of crosssubstitution effects But he refers explicitly only to two properties in his written summary of his results Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 190 Part 2 Choice and Demand We can put this result into elasticity terms by dividing Equation 625 by xi ec i1 1 ec i2 1 c1 ec in 5 0 626 But we know that ec ii 0 because of the negativity of the ownsubstitution effect Hence it must be the case that a j2i ec ij 0 627 In words the sum of all the compensated crossprice elasticities for a particular good must be positive or zero This is the sense that most goods are substitutes Empirical evidence seems generally consistent with this theoretical finding Instances of net complementarity between goods are encountered relatively infrequently in empirical studies of demand 65 COMPOSITE COMMODITIES Our discussion in the previous section showed that the demand relationships among goods can be complicated In the most general case an individual who consumes n goods will have demand functions that reflect n1n 1 122 different substitution effects4 When n is large as it surely is for all the specific goods that individuals actually consume this general case can be unmanageable It is often far more convenient to group goods into larger aggregates such as food clothing shelter and so forth At the most extreme level of aggregates we might wish to examine one specific good say gasoline which we might call x and its relationship to all other goods which we might call y This is the procedure we have been using in some of our twodimensional graphs and we will continue to do so at many other places in this book In this section we show the conditions under which this procedure can be defended In the Extensions to this chapter we explore more general issues involved in aggregating goods into larger groupings 651 Composite commodity theorem Suppose consumers choose among n goods but that we are only interested specifically in one of themsay x1 In general the demand for x1 will depend on the individual prices of the other n 2 1 commodities But the analysis can be simplified greatly if we are willing to assume that the prices of all of these other n 2 1 goods move proportionally together In this case the composite commodity theorem states that we can define a single composite of all of these goods say y so that the individuals utility maximization problem can be compressed into a simpler problem of choosing only between x1 and y Formally if we let p0 2 p0 n represent the initial prices of these goods then we assume that these prices can only vary together They might all double or all decrease by 50 percent but the relative prices of x2 xn would not change Now we define the composite commodity y to be total expenditures on x2 xn using the initial prices p0 2 p0 n y 5 p0 2x2 1 p0 3x3 1 c1 p0 nxn 628 This persons initial budget constraint is given by I 5 p1x1 1 p0 2x2 1 c1 p0 nxn 5 p1x1 1 y 629 4To see this notice that all substitution effects sij could be recorded in an n 3 n matrix However symmetry of the effects 1sij 5 sji2 implies that only those terms on and below the principal diagonal of this matrix may be distinctly different from each other This amounts to half the terms in the matrix 1n222 plus the remaining half of the terms on the main diagonal of the matrix 1n22 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 191 By assumption all the prices p2 pn change in unison Assume all these prices change by a factor of t 1t 02 Now the budget constraint is I 5 p1x1 1 tp0 2x2 1 c1 tp0 nxn 5 p1x1 1 ty 630 Consequently the factor of proportionality t plays the same role in this persons budget constraint as did the price of y 1py2 in our earlier twogood analysis Changes in p1 or t induce the same kinds of substitution effects we have been analyzing As long as p2 pn move together we can therefore confine our examination of demand to choices between buying x1 or buying everything else5 Simple graphs that show only these two goods are therefore consistent with more general principles of utility maximization as long as the conditions of the theorem that all other prices move together are satisfied Notice how ever that the theorem makes no predictions about how choices of x2 xn behave they need not move in unison The theorem focuses only on total spending on x2 xn not on how that spending is allocated among specific items although this allocation is assumed to be done in a utilitymaximizing way 652 Generalizations and limitations The composite commodity theorem applies to any group of commodities whose relative prices all move together It is possible to have more than one such commodity if there are several groupings that obey the theorem ie expenditures on food clothing and so forth Hence we have developed the following definition This definition and the related theorem are powerful results They help simplify many problems that would otherwise be intractable Still one must be rather careful in applying the theorem to the real world because its conditions are stringent Finding a set of commodities whose prices move together is rare Slight departures from strict proportionality may negate the composite commodity theorem if crosssubstitution effects are large In the Extensions to this chapter we look at ways to simplify situations where prices move independently 5This definition of a composite commodity was made popular by JR Hicks in Value and Capital Oxford Oxford University Press 1939 The difficult part of his composite commodity theorem is in showing that choices made between x1 and y are precisely the same ones as would be made if the full utility maximization process were used Hicks proof of the theorem relied on relatively complex matrix algebra More recent proofs make use of duality and the envelope theorem For two examples see Problem 613 D E F I N I T I O N Composite commodity A composite commodity is a group of goods for which all prices move together These goods can be treated as a single commodity in that the individual behaves as though he or she were choosing between other goods and total spending on the entire composite group EXAMPLE 63 Housing Costs as a Composite Commodity Suppose that an individual receives utility from three goods food x housing services y mea sured in hundreds of square feet and household operations z as measured by electricity use If the individuals utility is given by the threegood CES function utility 5 U1x y z2 5 21 x 2 1 y 2 1 z 631 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 192 Part 2 Choice and Demand then the Lagrangian technique can be used to calculate Marshallian demand functions for these goods as x 5 I px 1 px py 1 px pz y 5 I py 1 py px 1 py pz z 5 I pz 1 pz px 1 pz py 632 If initially I 5 100 px 5 1 py 5 4 and pz 5 1 then the demand functions predict x 5 25 y 5 125 z 5 25 633 Hence 25 is spent on food and a total of 75 is spent on housingrelated needs If we assume that housing service prices 1 py2 and household operation prices 1 pz2 always move together then we can use their initial prices to define the composite commodity housing h as h 5 4y 1 1z 634 Here we also arbitrarily define the initial price of housing 1 ph2 to be 1 The initial quantity of housing is simply total dollars spent on h h 5 4 11252 1 1 1252 5 75 635 Furthermore because py and pz always move together ph will always be related to these prices by ph 5 pz 5 025py 636 Using this information we can recalculate the demand function for x as a function of I px and ph x 5 I px 1 4px ph 1 px ph 5 I py 1 3px ph 637 As before initially I 5 100 px 5 1 and ph 5 1 thus x 5 25 Total spending on housing can be computed from the budget constraint phh 5 I 2 pxx 5 100 2 25 5 75 Or since ph 5 1 h 5 75 An increase in housing costs If the prices of y and z were to increase proportionally to py 5 16 pz 5 4 with px remaining at 1 then ph would also increase to 4 Equation 637 now predicts that the demand for x would decrease to x 5 100 1 1 34 5 100 7 638 and that housing purchases would be given by phh 5 100 2 100 7 5 600 7 639 or because ph 5 4 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 193 66 HOME PRODUCTION ATTRIBUTES OF GOODS AND IMPLICIT PRICES Thus far in this chapter we have focused on what economists can learn about the relation ships among goods by observing individuals changing consumption of these goods in reac tion to changes in market prices In some ways this analysis skirts the central question of why coffee and cream go together or why fish and chicken may substitute for each other in a persons diet To develop a deeper understanding of such questions economists have sought to explore activities within individuals households That is they have devised models of nonmarket types of activities such as parental child care meal preparation or doityourself construction to understand how such activities ultimately result in demands for goods in the market In this section we briefly review some of these models Our primary goal is to illustrate some of the implications of this approach for the traditional theory of choice 661 Household production model The starting point for most models of household production is to assume that individuals do not receive utility directly from goods they purchase in the market as we have been assuming thus far Instead it is only when market goods are combined with time inputs by the individual that utilityproviding outputs are produced In this view raw beef and uncooked potatoes then yield no utility until they are cooked together to produce stew Sim ilarly market purchases of beef and potatoes can be understood only by examining the indi viduals preferences for stew and the underlying technology through which it is produced h 5 150 7 640 Notice that this is precisely the level of housing purchases predicted by the original demand func tions for three goods in Equation 632 With I 5 100 px 5 1 py 5 16 and pz 5 4 these equa tions can be solved as x 5 100 7 y 5 100 28 z 5 100 14 641 and so the total amount of the composite good housing consumed according to Equation 634 is given by h 5 4y 1 1z 5 150 7 642 Hence we obtained the same responses to price changes regardless of whether we chose to examine demands for the three goods x y and z or to look only at choices between x and the composite good h QUERY How do we know that the demand function for x in Equation 637 continues to ensure utility maximization Why is the Lagrangian for the constrained maximization problem unchanged by making the substitutions represented by Equation 636 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 194 Part 2 Choice and Demand In formal terms assume as before that there are three goods that a person might pur chase in the market x y and z Purchasing these goods provides no direct utility but the goods can be combined by the individual to produce either of two homeproduced goods a1 or a2 The technology of this household production can be represented by the produc tion functions f1 and f2 see Chapter 9 for a more complete discussion of the production function concept Therefore a1 5 f1 1x y z2 a2 5 f2 1x y z2 643 and utility 5 U1a1 a22 644 The individuals goal is to choose x y z so as to maximize utility subject to the production constraints and to a financial budget constraint6 pxx 1 pyy 1 pzz 5 I 645 Although we will not examine in detail the results that can be derived from this gen eral model two insights that can be drawn from it might be mentioned First the model may help clarify the nature of market relationships among goods Because the production functions in Equations 643 are in principle measurable using detailed data on household operations households can be treated as multiproduct firms and studied using many of the techniques economists use to study production A second insight provided by the household production approach is the notion of the implicit or shadow prices associated with the homeproduced goods a1 and a2 Because consuming more a1 say requires the use of more of the ingredients x y and z this activ ity obviously has an opportunity cost in terms of the quantity of a2 that can be produced To produce more bread say a person must not only divert some flour milk and eggs from using them to make cupcakes but may also have to alter the relative quantities of these goods purchased because he or she is bound by an overall budget constraint Hence bread will have an implicit price in terms of the number of cupcakes that must be forgone to be able to consume one more loaf That implicit price will reflect not only the market prices of bread ingredients but also the available household production technology and in more complex models the relative time inputs required to produce the two goods As a starting point however the notion of implicit prices can be best illustrated with a simple model 662 The linear attributes model A particularly simple form of the household production model was first developed by K J Lancaster to examine the underlying attributes of goods7 In this model it is the attributes of goods that provide utility to individuals and each specific good contains a fixed set of attributes If for example we focus only on the calories 1a12 and vitamins 1a22 that various foods provide Lancasters model assumes that utility is a function of these attributes and that individuals purchase various foods only for the purpose of obtaining the calories and vitamins they offer In mathematical terms the model assumes that the production equations have the simple form a1 5 a1 xx 1 a1 yy 1 a1 zz a2 5 a2 xx 1 a2 yy 1 a2 zz 646 6Often household production theory also focuses on the individuals allocation of time to producing a1 and a2 or to working in the market In Chapter 16 we look at a few simple models of this type 7See K J Lancaster A New Approach to Consumer Theory Journal of Political Economy 74 April 1966 13257 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 195 where a1 x represents the number of calories per unit of food x a2 x represents the number of vitamins per unit of food x and so forth In this form of the model there is no actual pro duction in the home Rather the decision problem is how to choose a diet that provides the optimal mix of calories and vitamins given the available food budget 663 Illustrating the budget constraints To begin our examination of the theory of choice under the attributes model we first illus trate the budget constraint In Figure 62 the ray 0x records the various combinations of a1 and a2 available from successively larger amounts of good x Because of the linear produc tion technology assumed in the attributes model these combinations of a1 and a2 lie along such a straight line although in more complex models of home production that might not be the case Similarly rays of 0y and 0z show the quantities of the attributes a1 and a2 pro vided by various amounts of goods y and z that might be purchased If this person spends all his or her income on good x then the budget constraint Equa tion 645 allows the purchase of x 5 I px 647 and that will yield a 1 5 a1 xx 5 a1 xI px a 2 5 a2 xx 5 a2 xI px 648 This point is recorded as point x on the 0x ray in Figure 62 Similarly the points y and z represent the combinations of a1 and a2 that would be obtained if all income were spent on good y or good z respectively Bundles of a1 and a2 that are obtainable by purchasing both x and y with a fixed bud get are represented by the line joining x and y in Figure 628 Similarly the line xz represents the combinations of a1 and a2 available from x and z and the line yz shows combinations available from mixing y and z All possible combinations from mixing the three market goods are represented by the shaded triangular area xyz 664 Corner solutions One fact is immediately apparent from Figure 62 A utilitymaximizing individual would never consume positive quantities of all three of these goods Only the northeast perimeter of the xyz triangle represents the maximal amounts of a1 and a2 available to this person given his or her income and the prices of the market goods Individuals with a preference toward a1 will have indifference curves similar to U0 and will maximize utility by choosing a point such as E The combination of a1 and a2 specified by that point can be obtained by consum ing only goods y and z Similarly a person with preferences represented by the indifference 8Mathematically suppose a fraction α of the budget is spent on x and 11 2 α2 on y then a1 5 αa1 xx 1 11 2 α2a1 yy a2 5 αa2 xx 1 11 2 α2a2 yy The line xy is traced out by allowing α to vary between 0 and 1 The lines xz and yz are traced out in a similar way as is the triangular area xyz Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 196 Part 2 Choice and Demand curve Ur0 will choose point Er and consume only goods x and y Therefore the attributes model predicts that corner solutions at which individuals consume zero amounts of some commodities will be relatively common especially in cases where individuals attach value to fewer attributes here two than there are market goods to choose from three If income prices or preferences change then consumption patterns may also change abruptly Goods that were previously consumed may cease to be bought and goods previously neglected may experience a significant increase in purchases This is a direct result of the linear assump tions inherent in the production functions assumed here In household production models with greater substitutability assumptions such discontinuous reactions are less likely Summary In this chapter we used the utilitymaximizing model of choice to examine relationships among consumer goods Although these relationships may be complex the analysis presented here provided a number of ways of categorizing and simplifying them When there are only two goods the income and substi tution effects from the change in the price of one good say py on the demand for another good x usually work in opposite directions Therefore the sign of xpy is ambiguous Its substitution effect is positive but its income effect is negative In cases of more than two goods demand relationships can be specified in two ways Two goods 1xi and xj2 are gross substitutes if xipj 0 and gross comple ments if xipj 0 Unfortunately because these price effects include income effects they need not be The points x y and z show the amounts of attributes a1 and a2 that can be purchased by buying only x y or z respectively The shaded area shows all combinations that can be bought with mixed bundles Some individuals may maximize utility at E others at Er a2 a2 a1 0 U0 U0 x y z E z E y x a1 FIGURE 62 Utility Maximization in the Attributes Model Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 197 symmetric That is xipj does not necessarily equal xjpi Focusing only on the substitution effects from price changes eliminates this ambiguity because substitution effects are symmetric that is xc ipj 5 xc jpi Now two goods are defined as net or Hicksian substitutes if xc ipj 0 and net complements if xc ipj 0 Hicks second law of demand shows that net substitutes are more prevalent If a group of goods has prices that always move in uni son then expenditures on these goods can be treated as a composite commodity whose price is given by the size of the proportional change in the composite goods prices An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utilityproviding attributes This may provide additional insights into relationships among goods Problems 61 Heidi receives utility from two goods goats milk m and strudel s according to the utility function U1m s2 5 m s a Show that increases in the price of goats milk will not affect the quantity of strudel Heidi buys that is show that spm 5 0 b Show also that mps 5 0 c Use the Slutsky equation and the symmetry of net sub stitution effects to prove that the income effects involved with the derivatives in parts a and b are identical d Prove part c explicitly using the Marshallian demand functions for m and s 62 Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him For Burt rotgut whiskey is an inferior good that exhibits Giffens paradox although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense Develop an intuitive explanation to suggest why an increase in the price of rotgut whiskey must cause fewer jelly donuts to be bought That is the goods must also be gross complements 63 Donald a frugal graduate student consumes only coffee c and buttered toast bt He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast a In this problem buttered toast can be treated as a com posite commodity What is its price in terms of the prices of butter 1 pb2 and toast 1 pt2 b Explain why cpbt 5 0 c Is it also true here that cpb and cpt are equal to 0 64 Ms Sarah Traveler does not own a car and travels only by bus train or plane Her utility function is given by utility 5 b t p where each letter stands for miles traveled by a specific mode Suppose that the ratio of the price of train travel to that of bus travel 1 ptpb2 never changes a How might one define a composite commodity for ground transportation b Phrase Sarahs optimization problem as one of choosing between ground g and air p transportation c What are Sarahs demand functions for g and p d Once Sarah decides how much to spend on g how will she allocate those expenditures between b and t 65 Suppose that an individual consumes three goods x1 x2 and x3 and that x2 and x3 are similar commodities ie cheap and expensive restaurant meals with p2 5 kp3 where k 1 that is the goods prices have a constant relationship to one another a Show that x2 and x3 can be treated as a composite commodity b Suppose both x2 and x3 are subject to a transaction cost of t per unit for some examples see Problem 66 How will this transaction cost affect the price of x2 relative to that of x3 How will this effect vary with the value of t c Can you predict how an incomecompensated increase in t will affect expenditures on the composite commod ity x2 and x3 Does the composite commodity theorem strictly apply to this case d How will an incomecompensated increase in t affect how total spending on the composite commodity is allo cated between x2 and x3 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 198 Part 2 Choice and Demand 66 Apply the results of Problem 65 to explain the following observations a It is difficult to find highquality apples to buy in Washington State or good fresh oranges in Florida see also Problem 612 b People with significant babysitting expenses are more likely to have meals out at expensive rather than cheap restaurants than are those without such expenses c Individuals with a high value of time are more likely to fly the Concorde than those with a lower value of time d Individuals are more likely to search for bargains for expensive items than for cheap ones Note Observations b and d form the bases for perhaps the only two mur der mysteries in which an economist solves the crime see Marshall Jevons Murder at the Margin and The Fatal Equilibrium 67 In general uncompensated crossprice effects are not equal That is xi pj 2 xj pi Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices This is a generalization of Problem 61 68 Example 63 computes the demand functions implied by the threegood CES utility function U1x y z2 5 21 x 2 1 y 2 1 z a Use the demand function for x in Equation 632 to deter mine whether x and y or x and z are gross substitutes or gross complements b How would you determine whether x and y or x and z are net substitutes or net complements Analytical Problems 69 Consumer surplus with many goods In Chapter 5 we showed how the welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves This problem asks you to gen eralize this to price changes in two or many goods a Suppose that an individual consumes n goods and that the prices of two of those goods say p1 and p2 increase How would you use the expenditure function to measure the compensating variation CV for this person of such a price increase b A way to show these welfare costs graphically would be to use the compensated demand curves for goods x1 and x2 by assuming that one price increased before the other Illustrate this approach c In your answer to part b would it matter in which order you considered the price changes Explain d In general would you think that the CV for a price increase of these two goods would be greater if the goods were net substitutes or net complements Or would the relationship between the goods have no bearing on the welfare costs 610 Separable utility A utility function is called separable if it can be written as U1x y2 5 U1 1x2 1 U2 1y2 where Uri 0 Usi 0 and U1 U2 need not be the same function a What does separability assume about the crosspartial derivative Uxy Give an intuitive discussion of what word this condition means and in what situations it might be plausible b Show that if utility is separable then neither good can be inferior c Does the assumption of separability allow you to con clude definitively whether x and y are gross substitutes or gross complements Explain d Use the CobbDouglas utility function to show that separability is not invariant with respect to monotonic transformations Note Separable functions are examined in more detail in the Extensions to this chapter 611 Graphing complements Graphing complements is complicated because a complemen tary relationship between goods under Hicks definition cannot occur with only two goods Rather complementarity necessarily involves the demand relationships among three or more goods In his review of complementarity Samuel son provides a way of illustrating the concept with a twodi mensional indifference curve diagram see the Suggested Readings To examine this construction assume there are three goods that a consumer might choose The quantities of these are denoted by x1 x2 and x3 Now proceed as follows a Draw an indifference curve for x2 and x3 holding the quantity of x1 constant at x0 1 This indifference curve will have the customary convex shape Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 199 b Now draw a second indifference curve for x2 x3 that pro vides the same level of utility as in part a holding x1 constant at x0 1 2 h this new indifference curve will be to the right of the old one For this new indifference curve show the amount of extra x2 that would compensate this person for the loss of x1 call this amount j Similarly show that amount of extra x3 that would compensate for the loss of x1 and call this amount k c Suppose now that an individual is given both amounts j and k thereby permitting him or her to move to an even higher x2 x3 indifference curve Show this move on your graph and draw this new indifference curve d Samuelson now suggests the following definitions If the new indifference curve corresponds to the indif ference curve when x1 5 x0 1 2 2h goods 2 and 3 are independent If the new indifference curve provides more utility than when x1 5 x0 1 2 2h goods 2 and 3 are complements If the new indifference curve provides less utility than when x1 5 x0 1 2 2h goods 2 and 3 are substitutes Show that these graphical definitions are symmetric e Discuss how these graphical definitions correspond to Hicks more mathematical definitions given in the text f Looking at your final graph do you think that this approach fully explains the types of relationships that might exist between x2 and x3 612 Shipping the good apples out Details of the analysis suggested in Problems 65 and 66 were originally worked out by Borcherding and Silberberg see the Suggested Readings based on a supposition first proposed by Alchian and Allen These authors look at how a transaction charge affects the relative demand for two closely substitutable items Assume that goods x2 and x3 are close substitutes and are subject to a transaction charge of t per unit Suppose also that good 2 is the more expensive of the two goods ie good apples as opposed to cooking apples Hence the transac tion charge lowers the relative price of the more expensive good ie 1p2 1 t2 1p3 1 t2 decreases as t increases This will increase the relative demand for the expensive good if 1x c 2x c 32t 0 where we use compensated demand func tions to eliminate pesky income effects Borcherding and Silberberg show this result will probably hold using the fol lowing steps a Use the derivative of a quotient rule to expand 1x c 2x c 32t b Use your result from part a together with the fact that in this problem xc it 5 xc ip2 1 xc ip3 for i 5 2 3 to show that the derivative we seek can be writ ten as 1xc 2xc 32 t 5 xc 2 xc 3 c s22 x2 1 s23 x2 2 s32 x3 2 s33 x3 d where sij 5 xc ipj c Rewrite the result from part b in terms of compensated price elasticities ec ij 5 xc i pj pj xc i d Use Hicks third law Equation 626 to show that the term in brackets in parts b and c can now be written as 3 1e22 2 e232 11p2 2 1p32 1 1e21 2 e312p34 e Develop an intuitive argument about why the expression in part d is likely to be positive under the conditions of this problem Hints Why is the first product in the brackets positive Why is the second term in brackets likely to be small f Return to Problem 66 and provide more complete expla nations for these various findings 613 Proof of the Composite Commodity Theorem Proving the composite commodity theorem consists of show ing that choices made when we use a composite commodity are identical to those that would be made if we specified the complete utilitymaximization problem This problem asks you to show this using two different approaches For both of these we assume there are only three goods x1 x2 and x3 and that the prices of x2 and x3 always move togetherthat is p2 5 tp0 2 and p3 5 tp0 3 where p0 2 and p0 3 are the initial prices of these two goods With this notation the composite commod ity y is defined as y 5 p0 2x2 1 p0 3x3 a Proof using duality Let the expenditure function for the original threegood problem be given by E 1p1 p2 p3 U2 and consider the alternative expenditure minimization problem Minimize p1x1 1 ty st U1x1 x2 x32 5 U This problem will also yield an expenditure function of the form E1p1 t U2 i Use the envelope theorem to show that E t 5 E t 5 y This shows that the demand for the composite good y is the same under either approach ii Explain why the demand for x1 is also the same under either approach This proof is taken from Deaton and Muellbauer 1980 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 200 Part 2 Choice and Demand b Proof using twostage maximization Now consider this problem from the perspective of utility maximiza tion The problem can be simplified by adopting the nor malization that p2 1that is all purchasing power is measured in units of x2 and the prices p1 and p3 are now treated as prices relative top2 Under the assumptions of the composite commodity theorem p1 can vary but p3 is a fixed value The original utility maximization problem is Maximize U1x1 x2 x32 st p1x1 1 x2 1 p3x3 5 M 1where M 5 Ip22 and the firstorder conditions for a maximum are Ui 5 λpi i 5 1 3 where λ is the Lagrange multiplier The alternative twostage approach to the problem is Stage 1 Maximize U1x1 x2 x32 st x2 1 p3x3 5 m where m is the portion of M devoted to purchasing the composite good This maximization problem treats x1 as an exogenous parameter in the maximization pro cess so it becomes an element of the value function in this problem The firstorder conditions for this problem are Ui 5 μpi for i 2 3 where μ is the Lagrange mul tiplier for this stage of the problem Let the value indi rect utility function for this stage 1 problem be given by V 1x1 m2 The final part of this twostage problem is then Stage 2 Maximize V 1x1 m2 st p1x1 1 m 5 M This will have firstorder conditions of the form Vx1 5 δp1 and Vm 5 δ where δ is the Lag range multiplier for stage 2 Given this setup answer the following questions i Explain why the value function in stage 1 depends only on x1 and m Hint This is where the fact that p3 is constant plays a key role ii Show that the two approaches to this maximiza tion problem yield the same result by showing that λ 5 μ 5 δ What do you have to assume to ensure the results are equivalent This problem is adapted from Carter 1995 Behavioral Problem 614 Spurious Product Differentiation As we shall see in Chapter 15 a firm may sometimes seek to differentiate its product from those of its competitors in order to increase profits In this problem we examine the possibility that such differentiation may be spurious that is more apparent than real and that such a possibil ity may reduce the buyers utility To do so assume that a consumer sets out to buy a flat screen television y Two brands are available Utility provided by brand 1 is given by U1x y12 5 x 1 500 ln 11 1 y12 where x represents all other goods This person believes brand 2 is a bit better and therefore provides utility of U1x y22 5 x 1 600 ln 11 1 y22 Because this person only intends to buy one television his or her purchase decision will determine which utility function prevails a Suppose px 5 1 and I 5 1000 what is the maxi mum price that this person will pay for each brand of television based on his or her beliefs about qual ity Hint When this person purchases a TV either y1 5 1 y2 5 0 or y1 5 0 y2 5 1 b If this person does have to pay the prices calculated in part a which TV will he or she purchase c Suppose that the presumed superiority of brand 2 is spuriousperhaps the belief that it is better has been created by some clever advertising why would firm 2 pay for such advertising How would you calculate the utility loss associated with the purchase of a brand 2 TV d What kinds of actions might this consumer take to avoid the utility loss experienced in part c How much would he or she be willing to spend on such actions Suggestions for Further Reading Borcherding T E and E Silberberg Shipping the Good Apples OutThe AlchianAllen Theorem Reconsidered Journal of Political Economy February 1978 13138 Good discussion of the relationships among three goods in demand theory See also Problems 65 and 66 Carter M An expository note on the composite commodity theorem Economic Theory March 1995 175179 A nice graphical interpretation of the composite commodity theorem The mathematical proof provided may be incomplete however Deaton A and J Muellbauer Economics and Consumer Behavior Cambridge UK Cambridge University Press 1980 Uses duality theory to prove the composite commodity the orem and many other results from consumer theory Also offers some details on the almost ideal demand system Hicks J R Value and Capital 2nd ed Oxford UK Oxford University Press 1946 See Chapters IIII and related appendices Proof of the composite commodity theorem Also has one of the first treatments of net substitutes and complements Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 201 MasColell A M D Whinston and J R Green Microeco nomic Theory New York Oxford University Press 1995 Explores the consequences of the symmetry of compensated cross price effects for various aspects of demand theory Rosen S Hedonic Prices and Implicit Markets Journal of Political Economy JanuaryFebruary 1974 3455 Nice graphical and mathematical treatment of the attribute approach to consumer theory and of the concept of markets for attributes Samuelson P A ComplementarityAn Essay on the 40th Anniversary of the HicksAllen Revolution in Demand Theory Journal of Economic Literature December 1977 125589 Reviews a number of definitions of complementarity and shows the connections among them Contains an intuitive graphical dis cussion and a detailed mathematical appendix Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 Good discussion of expenditure functions and the use of indirect utility functions to illustrate the composite commodity theorem and other results Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 202 In Chapter 6 we saw that the theory of utility maximization in its full generality imposes rather few restrictions on what might happen Other than the fact that net crosssubstitu tion effects are symmetric practically any type of relationship among goods is consistent with the underlying theory This situation poses problems for economists who wish to study consumption behavior in the real worldtheory just does not provide much guidance when there are many thousands of goods potentially available for study There are two general ways in which simplifications are made The first uses the composite commodity theorem from Chapter 6 to aggregate goods into categories within which rel ative prices move together For situations where economists are specifically interested in changes in relative prices within a category of spending such as changes in the relative prices of various forms of energy however this process will not do An alternative is to assume that consumers engage in a two stage process in their consumption decisions First they allo cate income to various broad groupings of goods eg food clothing and then given these expenditure constraints they maximize utility within each of the subcategories of goods using only information about those goods relative prices In that way decisions can be studied in a simplified setting by looking only at one category at a time This process is called twostage budgeting In these Extensions we first look at the general theory of twostage budgeting and then turn to exam ine some empirical examples E61 Theory of twostage budgeting The issue that arises in twostage budgeting can be stated succinctly Does there exist a partition of goods into m non overlapping groups denoted by r 5 1 m and a separate budget 1lr2 devoted to each category such that the demand functions for the goods within any one category depend only on the prices of goods within the category and on the catego rys budget allocation That is can we partition goods so that demand is given by xi 1 p1 pn I2 5 xir1 pir Ir2 i for r 5 1 m That it might be possible to do this is suggested by comparing the following twostage maximization problem V1p1 pn I1 Im2 5 max x1 xn cU1x1 xn2 st a ir pixi Ir r 5 1 md ii and max I1 ImV st a M r51 Ir 5 I to the utilitymaximization problem we have been studying max xi U1x1 xn2 st a n i51 pixi I iii Without any further restrictions these two maximization processes will yield the same result that is Equation ii is just a more complicated way of stating Equation iii Thus some restrictions have to be placed on the utility function to ensure that the demand functions that result from solving the two stage process will be of the form specified in Equation i Intu itively it seems that such a categorization of goods should work providing that changes in the price of a good in one category do not affect the allocation of spending for goods in any category other than its own In Problem 69 we showed a case where this is true for an additively separable utility function Unfortunately this proves to be a special case The more general mathematical restrictions that must be placed on the utility function to justify twostage budgeting have been derived see Blackorby Primont and Russell 1978 but these are not especially intuitive Of course economists who wish to study decentralized decisions by consumers or per haps more importantly by firms that operate many divisions must do something to simplify matters Now we look at a few applied examples E62 Relation to the composition commodity theorem Unfortunately neither of the two available theoretical approaches to demand simplification is completely satisfying The composite commodity theorem requires that the relative prices for goods within one group remain constant over time an assumption that has been rejected during many different historical periods EXTENSIONS SimplifyinG DemanD anD TwoSTaGe BuDGeTinG Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 6 Demand Relationships among Goods 203 On the other hand the kind of separability and twostage budgeting indicated by the utility function in Equation i also requires strong assumptions about how changes in prices for a good in one group affect spending on goods in any other group These assumptions appear to be rejected by the data see Diewert and Wales 1995 Economists have tried to devise even more elaborate hybrid methods of aggregation among goods For example Lewbel 1996 shows how the composite commodity theo rem might be generalized to cases where withingroup relative prices exhibit considerable variability He uses this general ization for aggregating US consumer expenditures into six large groups ie food clothing household operation med ical care transportation and recreation Using these aggre gates he concludes that his procedure is much more accurate than assuming twostage budgeting among these expenditure categories E63 Homothetic functions and energy demand One way to simplify the study of demand when there are many commodities is to assume that utility for certain sub categories of goods is homothetic and may be separated from the demand for other commodities This procedure was followed by Jorgenson Slesnick and Stoker 1997 in their study of energy demand by US consumers By assuming that demand functions for specific types of energy are propor tional to total spending on energy the authors were able to concentrate their empirical study on the topic that is of most interest to them estimating the price elasticities of demand for various types of energy They conclude that most types of energy ie electricity natural gas gasoline have fairly elas tic demand functions Demand appears to be most respon sive to price for electricity References Blackorby Charles Daniel Primont and R Robert Russell Duality Separability and Functional Structure Theory and Economic Applications New York North Holland 1978 Diewert W Erwin and Terrence J Wales Flexible Func tional Forms and Tests of Homogeneous Separability Journal of Econometrics June 1995 259302 Jorgenson Dale W Daniel T Slesnick and Thomas M Stoker TwoStage Budgeting and Consumer Demand for Energy In Dale W Jorgenson Ed Welfare vol 1 Aggre gate Consumer Behavior pp 475510 Cambridge MA MIT Press 1997 Lewbel Arthur Aggregation without Separability A Stan dardized Composite Commodity Theorem American Economic Review June 1996 52443 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 205 PART three Uncertainty and Strategy Chapter 7 Uncertainty Chapter 8 Game Theory This part extends the analysis of individual choice to more complicated settings In Chapter 7 we look at individual behavior in uncertain situations A decision is no longer associated with a single outcome but a number of more or less likely ones We describe why people generally dislike the risk involved in such situations We seek to understand the steps they take to mitigate risk including buy ing insurance acquiring more information and preserving options Chapter 8 looks at decisions made in strategic situations in which a persons wellbeing depends not just on his or her own actions but also on the actions of others and vice versa This leads to a certain circularity in analyzing strategic decisions which we will resolve using the tools of game theory The equilibrium notions we develop in studying such situations are widely used throughout economics Although this part can be regarded as the natural extension of the analysis of consumer choice from Part 2 to more complicated settings it applies to a much broader set of decisionmakers including firms other organizations even whole countries For example game theory will provide the framework to study imperfect competition among few firms in Chapter 15 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 207 CHAPTER SeVeN Uncertainty In this chapter we explore some of the basic elements of the theory of individual behavior in uncertain situations We discuss why individuals do not like risk and the various meth ods buying insurance acquiring more information and preserving options they may adopt to reduce it More generally the chapter is intended to provide a brief introduction to issues raised by the possibility that information may be imperfect when individuals make utilitymaximizing decisions The Extensions section provides a detailed application of the concepts in this chapter to the portfolio problem a central problem in financial eco nomics Whether a wellinformed person can take advantage of a poorly informed person in a market transaction asymmetric information is a question put off until Chapter 18 71 MATHEMATICAL STATISTICS Many of the formal tools for modeling uncertainty in economic situations were originally developed in the field of mathematical statistics Some of these tools were reviewed in Chapter 2 and in this chapter we will make a great deal of use of the concepts introduced there Specifically four statistical ideas will recur throughout this chapter Random variable A random variable is a variable that records in numerical form the possible outcomes from some random event1 Probability density function PDF A function f 1x2 that shows the probabilities asso ciated with the possible outcomes from a random variable Expected value of a random variable The outcome of a random variable that will occur on average The expected value is denoted by E1x2 If x is a discrete random variable with n outcomes then E1x2 5 g n i51 xi f 1xi2 If x is a continuous random vari able then E1x2 5 e 1q 2q xf 1x2 dx Variance and standard deviation of a random variable These concepts measure the dispersion of a random variable about its expected value In the discrete case Var1x2 5 σ2 x 5 g n i51 3xi 2 E1x2 4 2f 1xi2 in the continuous case Var1x2 5 σ2 x 5 e 1q 2q 3x 2 E1x2 4 2f 1x2 dx The standard deviation is the square root of the variance As we shall see all these concepts will come into play when we begin looking at the decisionmaking process of a person faced with a number of uncertain outcomes that can be conceptually represented by a random variable 1When it is necessary to distinguish between random variables and nonrandom variables we will use the notation x to denote the fact that the variable x is random in that it takes on a number of potential randomly determined outcomes Often however it will not be necessary to make the distinction because randomness will be clear from the context of the problem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 208 Part 3 Uncertainty and Strategy 72 FAIR GAMBLES AND THE EXPECTED UTILITY HYPOTHESIS A fair gamble is a specified set of prizes and associated probabilities that has an expected value of zero For example if you flip a coin with a friend for a dollar the expected value of this gamble is zero because E1x2 5 05 1112 1 05 1212 5 0 71 where wins are recorded with a plus sign and losses with a minus sign Similarly a game that promised to pay you 10 if a coin came up heads but would cost you only 1 if it came up tails would be unfair because E1x2 5 05 11102 1 05 1212 5 450 72 This game can easily be converted into a fair game however simply by charging you an entry fee of 450 for the right to play It has long been recognized that most people would prefer not to take fair gambles2 Although people may wager a few dollars on a coin flip for entertainment purposes they would generally balk at playing a similar game whose outcome was 11 million or 21 million One of the first mathematicians to study the reasons for this unwillingness to engage in fair bets was Daniel Bernoulli in the eighteenth century3 His examination of the famous St Petersburg paradox provided the starting point for virtually all studies of the behavior of individuals in uncertain situations 721 St Petersburg paradox In the St Petersburg paradox the following gamble is proposed A coin is flipped until a head appears If a head first appears on the nth flip the player is paid 2n This gamble has an infinite number of outcomes a coin might be flipped from now until doomsday and never come up a head although the likelihood of this is small but the first few can easily be written down If xi represents the prize awarded when the first head appears on the ith trial then x1 5 2 x2 5 4 x3 5 8 xn 5 2n 73 The probability of getting a head for the first time on the ith trial is 11 22 i it is the probabil ity of getting 1i 2 12 tails and then a head Hence the probabilities of the prizes given in Equation 73 are π1 5 1 2 π2 5 1 4 π3 5 1 8 πn 5 1 2n 74 Therefore the expected value of the gamble is infinite E1x2 5 a q i51 πixi 5 a q i51 112i22i 5 1 1 1 1 1 1 c1 1 1 c5 q 75 2The gambles discussed here are assumed to yield no utility in their play other than the prizes hence the observation that many individuals gamble at unfair odds is not necessarily a refutation of this statement Rather such individuals can reasonably be assumed to be deriving some utility from the circumstances associated with the play of the game Therefore it is possible to differentiate the consumption aspect of gambling from the pure risk aspect 3The paradox is named after the city where Bernoullis original manuscript was published The article has been reprinted as D Bernoulli Exposition of a New Theory on the Measurement of Risk Econometrica 22 January 1954 2336 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 209 Some introspection however should convince anyone that no player would pay very much much less than infinity to take this bet If we charged 1 billion to play the game we would surely have no takers despite the fact that 1 billion is still considerably less than the expected value of the game This then is the paradox Bernoullis gamble is in some sense not worth its infinite expected dollar value 73 EXPECTED UTILITY Bernoullis solution to this paradox was to argue that individuals do not care directly about the dollar prizes of a gamble rather they respond to the utility these dollars provide If we assume that the marginal utility of wealth decreases as wealth increases the St Petersburg gamble may converge to a finite expected utility value even though its expected monetary value is infinite Because the gamble only provides a finite expected utility individuals would only be willing to pay a finite amount to play it Example 71 looks at some issues related to Bernoullis solution EXAMPLE 71 Bernoullis Solution to the Paradox and Its Shortcomings Suppose as did Bernoulli that the utility of each prize in the St Petersburg paradox is given by U1xi2 5 ln1xi2 76 This logarithmic utility function exhibits diminishing marginal utility ie Ur 0 but Us 0 and the expected utility value of this game converges to a finite number expected utility 5 a q i51 πiU1xi2 5 a q i51 1 2i ln 12i2 77 Some manipulation of this expression yields4 the result that the expected utility from this gam ble is 139 Therefore an individual with this type of utility function might be willing to invest resources that otherwise yield up to 139 units of utility a certain wealth of approximately 4 pro vides this utility in purchasing the right to play this game Thus assuming that the large prizes promised by the St Petersburg paradox encounter diminishing marginal utility permitted Bernoulli to offer a solution to the paradox Unbounded utility Unfortunately Bernoullis solution to the St Petersburg paradox does not completely solve the problem As long as there is no upper bound to the utility function the para dox can be regenerated by redefining the gambles prizes For example with the logarithmic utility function prizes can be set as xi 5 e2i in which case U1xi2 5 ln 1e2i2 5 2i 78 and the expected utility from the gamble would again be infinite Of course the prizes in this redefined gamble are large For example if a head first appears on the fifth flip a person would 4Proof expected utility 5 a q i51 i 2i ln 2 5 ln 2 a q i51 i 2i But the value of this final infinite series can be shown to equal 2 Hence expected utility 5 2 ln 2 139 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 210 Part 3 Uncertainty and Strategy 74 THE VON NEUMANNMORGENSTERN THEOREM Among many contributions relevant to Part 3 of our text in their book The Theory of Games and Economic Behavior John von Neumann and Oscar Morgenstern developed a mathematical foundation for Bernoullis solution to the St Petersburg paradox5 In partic ular they laid out basic axioms of rationality and showed that any person who is rational in this way would make choices under uncertainty as though he or she had a utility function over money U1x2 and maximized the expected value of U1x2 rather than the expected value of the monetary payoff x itself Although most of these axioms seem eminently rea sonable at first glance many important questions about their tenability have been raised6 We will not pursue these questions here however 741 The von NeumannMorgenstern utility index To begin suppose that there are n possible prizes that an individual might win by partici pating in a lottery Let these prizes be denoted by x1 x2 xn and assume that these have been arranged in order of ascending desirability Therefore x1 is the least preferred prize for the individual and xn is the most preferred prize Now assign arbitrary utility numbers to these two extreme prizes For example it is convenient to assign U1x12 5 0 U1xn2 5 1 79 but any other pair of numbers would do equally well7 Using these two values of utility the point of the von NeumannMorgenstern theorem is to show that a reasonable way exists to assign specific utility numbers to the other prizes available Suppose that we choose any other prize say xi Consider the following experiment Ask the individual to state the probability say πi at which he or she would be indifferent between xi with certainty win e25 5 79 trillion although the probability of winning this would be only 125 5 0031 The idea that people would pay a great deal say trillions of dollars to play games with small proba bilities of such large prizes seems to many observers to be unlikely Hence in many respects the St Petersburg game remains a paradox QUERY Here are two alternative solutions to the St Petersburg paradox For each calculate the expected value of the original game 1 Suppose individuals assume that any probability less than 001 is in fact zero 2 Suppose that the utility from the St Petersburg prizes is given by U1xi2 5 bxi if xi 1000000 1000000 if xi 1000000 5J von Neumann and O Morgenstern The Theory of Games and Economic Behavior Princeton NJ Princeton University Press 1944 The axioms of rationality in uncertain situations are discussed in the books appendix 6For a discussion of some of the issues raised in the debate over the von NeumannMorgenstern axioms especially the assumption of independence see C Gollier The Economics of Risk and Time Cambridge MA MIT Press 2001 chap 1 7Technically a von NeumannMorgenstern utility index is unique only up to a choice of scale and originthat is only up to a linear transformation This requirement is more stringent than the requirement that a utility function be unique up to a monotonic transformation Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 211 and a gamble offering prizes of xn with probability πi and x1 with probability 1 2 πi It seems reasonable although this is the most problematic assumption in the von Neumann Morgenstern approach that such a probability will exist The individual will always be indifferent between a gamble and a sure thing provided that a high enough probability of winning the best prize is offered It also seems likely that πi will be higher the more desir able xi is the better xi is the better the chance of winning xn must be to get the individual to gamble Therefore the probability πi measures how desirable the prize xi is In fact the von NeumannMorgenstern technique defines the utility of xi as the expected utility of the gamble that the individual considers equally desirable to xi U1xi2 5 πiU1xn2 1 11 2 πi2U1x12 710 Because of our choice of scale in Equation 79 we have U1xi2 5 πi 1 1 11 2 πi2 0 5 πi 711 By judiciously choosing the utility numbers to be assigned to the best and worst prizes we have been able to devise a scale under which the utility index attached to any other prize is simply the probability of winning the top prize in a gamble the individual regards as equivalent to the prize in question This choice of utility indices is arbitrary Any other two numbers could have been used to construct this utility scale but our initial choice Equation 79 is a particularly convenient one 742 Expected utility maximization In line with the choice of scale and origin represented by Equation 79 suppose that a util ity index πi has been assigned to every prize xi Notice in particular that π1 5 0 πn 5 1 and that the other utility indices range between these extremes Using these utility indi ces we can show that a rational individual will choose among gambles based on their expected utilities ie based on the expected value of these von NeumannMorgenstern utility index numbers As an example consider two gambles Gamble A offers x2 with probability a and x3 with probability 1 2 a Gamble B offers x4 with probability b and x5 with probability 1 2 b We want to show that this person will choose gamble A if and only if the expected utility of gamble A exceeds that of gamble B Now for the gambles expected utility of A 5 EA3U1x2 4 5 aU1x22 1 11 2 a2U1x32 expected utility of B 5 EB 3U1x2 4 5 bU1x42 1 11 2 b2U1x52 712 The notation for the expected values of these gambles uses a subscript to indicate the gamble that is being evaluated in each case This is the same convention used statistics to subscript the expectations operator E when one needs to be clear about which probability density function is meant when several are being discussed What are called gambles in this chapter are nothing other than different probability density functions over x Substi tuting the utility index numbers ie π2 is the utility of x2 and so forth gives EA3U1x2 4 5 aπ2 1 11 2 a2π3 EB 3U1x2 4 5 bπ4 1 11 2 b2π5 713 We wish to show that the individual will prefer gamble A to gamble B if and only if EA3U1x2 4 EB 3U1x2 4 714 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 212 Part 3 Uncertainty and Strategy To show this recall the definitions of the utility index The individual is indifferent between x2 and a gamble promising x1 with probability 1 2 π2 and xn with probability π2 We can use this fact to substitute gambles involving only x1 and xn for all utilities in Equation 713 even though the individual is indifferent between these the assumption that this substitution can be made implicitly assumes that people can see through complex lottery combinations After a bit of messy algebra we can conclude that gamble A is equivalent to a gamble promising xn with prob ability aπ2 1 11 2 a2π3 and gamble B is equivalent to a gamble promising xn with probability bπ4 1 11 2 b2π5 The individual will presumably prefer the gamble with the higher probabil ity of winning the best prize Consequently he or she will choose gamble A if and only if aπ2 1 11 2 a2π3 bπ4 1 11 2 b2π5 715 But this is precisely what we wanted to show in Equation 714 Consequently we have proved that an individual will choose the gamble that provides the highest level of expected von NeumannMorgenstern utility We now make considerable use of this result which can be summarized as follows O P T I M I Z AT I O N P R I N C I P L E Expected utility maximization If individuals obey the von NeumannMorgenstern axioms of behavior in uncertain situations they will act as though they choose the option that maximizes the expected value of their von NeumannMorgenstern utility 75 RISK AVERSION Economists have found that people tend to avoid risky situations even if the situation amounts to a fair gamble For example few people would choose to take a 10000 bet on the outcome of a coin flip even though the average payoff is 0 The reason is that the gambles money prizes do not completely reflect the utility provided by the prizes The utility that people obtain from an increase in prize money may increase less rapidly than the dollar value of these prizes A gamble that is fair in money terms may be unfair in utility terms and thus would be rejected In more technical terms extra money may provide people with decreasing marginal utility A simple example can help explain why An increase in income from say 40000 to 50000 may substantially increase a persons wellbeing ensuring he or she does not have to go without essentials such as food and housing A further increase from 50000 to 60000 allows for an even more comfortable lifestyle perhaps providing tastier food and a bigger house but the improvement will likely not be as great as the initial one Starting from a wealth of 50000 the individual would be reluctant to take a 10000 bet on a coin flip The 50 percent chance of the increased comforts that he or she could have with 60000 does not compensate for the 50 percent chance that he or she will end up with 40000 and perhaps have to forgo some essentials These effects are only magnified with riskier gambles that is gambles having even more variable outcomes8 The person with initial wealth of 50000 would likely be even more reluctant to take a 20000 bet on a coin flip because he or she would face the prospect of ending up with only 30000 if the flip turned out badly severely cutting into lifes essen tials The equal chance of ending up with 70000 is not adequate compensation On the other hand a bet of only 1 on a coin flip is relatively inconsequential Although the person may still decline the bet he or she would not try hard to do so because his or her ultimate wealth hardly varies with the outcome of the coin toss 8Often the statistical concepts of variance and standard deviation are used to measure We will do so at several places later in this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 213 751 Risk aversion and fair gambles These arguments are illustrated in Figures 71 and 72 Here W0 represents an individuals current wealth and U1W2 is a von NeumannMorgenstern utility function just called a utility function from now on which reflects how he or she feels about various levels of wealth9 In the figures U1W2 is drawn as a concave function to reflect the assumption of diminishing marginal utility of wealth Figure 71 shows how this person would evaluate the offer of fair gamble A which is a 5050 chance of winning or losing h dollars The utility of initial wealth W0 before any gamble is taken as U1W02 which is also the expected value of initial wealth because it is certain The expected utility if he or she participates in gamble A is EA3U1W2 4 5 1 2 U1W0 1 h2 1 1 2 U1W0 2 h2 716 halfway between the utilities from the unfavorable outcome W0 2 h and the favorable out come W0 1 h It is clear from the geometry of the figure that U1W02 EA3U1W2 4 717 If the utility function is concave in wealth ie exhibits diminishing marginal utility of wealth then this person will refuse fair bets such as gamble A which involves a 5050 chance of winning or losing h dollars The expected utility EA 3U1W24 from gamble A is less than the expected utility U1W02 from refusing the bet and keeping the original wealth W0 The person would be willing to trade gamble A for the certainty equivalent CEA which is considerably less wealth than W0 Utility UW UW0 EAUW UCEA W0 h W0 h CEA W0 Wealth W FIGURE 71 Utility of Wealth Facing a Fair Bet 9Technically U1W2 is an indirect utility function because it is the consumption enabled by wealth that provides direct utility In Chapter 17 we will take up the relationship between consumptionbased utility and their implied indirect utility of wealth functions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 214 Part 3 Uncertainty and Strategy By the way the result in Equation 717 is not special to Figure 71 it is a consequence of a general result from mathematical statistics called Jensens inequality that holds for any concave function10 According to Equation 717 this person will prefer to keep his or her current wealth rather than taking the fair gamble The intuitive reason is that winning a fair bet adds to enjoyment less than losing hurts Figure 72 compares gamble A to a new gamble B which is a 5050 chance of winning or losing 2h dollars The persons expected utility from gamble B equals EB 3U1W2 4 5 1 2U1W0 1 2h2 1 1 2U1W0 2 2h2 718 This expected utility is again halfway between the unfavorable and favorable outcome but because the outcomes are more variable in gamble B than A the expected utility of B is lower and so the person prefers A to B although he or she would prefer to keep initial wealth W0 than take either gamble Compare gamble A from Figure 71 to gamble B which involves a 5050 chance of winning and losing twice as much Both are fair gambles but B involves more variability and so is worse for the person Gam ble B has lower expected utility than A that is EB3U1W24 EA 3U1W24 and a lower certainty equivalent that is CEB CEA FIGURE 72 Comparing Two Fair Bets of Differing Variability Utility UW UW0 EAUW UCEA EBUW UCEB W0 2h W0 h W0 2h W0 h W0 CEA CEB Wealth W 10Jensens inequality states that if g 1x2 is a strictly concave function of random variable x then E3g 1x24 g 1E1x22 In the utility context this means that if utility is concave in a random variable measuring wealth ie if Us 1W2 0 then the expected utility of wealth will be less than the utility from receiving the expected value of wealth with certainty Applied to gamble A we have EA3U1W24 U3EA1W24 5 U1W02 because as a fair gamble expected wealth from A equals W0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 215 D E F I N I T I O N Risk aversion An individual who always refuses fair bets is said to be risk averse If individuals exhibit a diminishing marginal utility of wealth they will be risk averse As a consequence they will be willing to pay something to avoid taking fair bets 752 Risk aversion and insurance As a matter of fact this person might be willing to pay some amount to avoid participating in any gamble at all Notice that a certain wealth of CEA provides the same expected utility as does participating in gamble A CEA is referred to as the certainty equivalent of gamble A The individual would be willing to pay up to W0 2 CEA to avoid participating in the gamble This explains why people buy insurance They are giving up a small cer tain amount the insurance premium to avoid the risky outcome they are being insured against The premium a person pays for automobile collision insurance for example pro vides a policy that agrees to repair his or her car should an accident occur The widespread use of insurance would seem to imply that aversion to risk is prevalent In fact the person in Figure 72 would pay even more to avoid taking the larger gamble B As an exercise try to identify the certainty equivalent CEB of gamble B and the amount the person would pay to avoid gamble B on the figure The analysis in this section can be summarized by the following definition EXAMPLE 72 Willingness to Pay for Insurance To illustrate the connection between risk aversion and insurance consider a person with a cur rent wealth of 100000 who faces the prospect of a 25 percent chance of losing his or her 20000 automobile through theft during the next year Suppose also that this persons von Neumann Morgenstern utility function is logarithmic that is U1W2 5 ln 1W2 If this person faces next year without insurance expected utility will be Eno 3U1W24 5 075U11000002 1 025U1800002 5 075 ln 100000 1 025 ln 80000 5 1145714 719 In this situation a fair insurance premium would be 5000 25 percent of 20000 assuming that the insurance company has only claim costs and that administrative costs are 0 Consequently if this person completely insures the car his or her wealth will be 95000 regardless of whether the car is stolen In this case then the expected utility of fair insurance is Efair 3U1W2 4 5 U1950002 5 ln 1950002 5 1146163 720 This person is made better off by purchasing fair insurance Indeed he or she would be willing to pay more than the fair premium for insurance We can determine the maximum insurance pre mium 1x2 by setting the expected utility from a policy charging this premium to solve Ewtp 3U1W24 5 U1100000 2 x2 5 ln 1100000 2 x2 5 1145714 721 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 216 Part 3 Uncertainty and Strategy Solving this equation for x yields 100000 2 x 5 e1145714 722 or x 5 5426 723 This person would be willing to pay up to 426 in administrative costs to an insurance company in addition to the 5000 premium to cover the expected value of the loss Even when these costs are paid this person is as well off as he or she would be when facing the world uninsured QUERY Suppose utility had been linear in wealth Would this person be willing to pay anything more than the actuarially fair amount for insurance How about the case where utility is a convex function of wealth 76 MEASURING RISK AVERSION In the study of economic choices in risky situations it is sometimes convenient to have a quantitative measure of how averse to risk a person is The most commonly used measure of risk aversion was initially developed by J W Pratt in the 1960s11 This risk aversion mea sure rW is defined as r1W2 5 2Us 1W2 Ur 1W2 724 Because the distinguishing feature of riskaverse individuals is a diminishing marginal utility of wealth 3Us1W2 04 Pratts measure is positive in such cases The measure is invariant with respect to linear transformations of the utility function and therefore not affected by which particular von NeumannMorgenstern ordering is used 761 Risk aversion and insurance premiums A useful feature of the Pratt measure of risk aversion is that it is proportional to the amount an individual will pay for insurance against taking a fair bet Suppose the winnings from such a fair bet are denoted by the random variable h which takes on both positive and negative values Because the bet is fair E1h2 5 0 Now let p be the size of the insurance premium that would make the individual exactly indifferent between taking the fair bet h and paying p with certainty to avoid the gamble E3U1W 1 h2 4 5 U1W 2 p2 725 where W is the individuals current wealth We now expand both sides of Equation 725 using Taylors series12 Because p is a fixed amount a linear approximation to the right side of the equation will suffice U1W 2 p2 5 U1W2 2 pUr 1W2 1 higherorder terms 726 11J W Pratt Risk Aversion in the Small and in the Large Econometrica JanuaryApril 1964 12236 12Taylors series provides a way of approximating any differentiable function around some point If f 1x2 has derivatives of all orders it can be shown that f 1x 1 h2 5 f 1x2 1 hfr 1x2 1 1h222fs 1x2 1 higherorder terms The pointslope formula in algebra is a simple example of Taylors series Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 217 13In this case the factor of proportionality is also proportional to the variance of h because Var 1h2 5 E3h 2 E1h24 2 5 E1h22 For an illustration where this equation fits exactly see Example 73 For the left side we need a quadratic approximation to allow for the variability in the gamble h E3U1W 1 h2 4 5 EcU1W2 1 hUr 1W2 1 h2 2 Us 1W2 1 higherorder termsd 727 5 U1W2 1 E1h2Ur 1W2 1 E1h22 2 Us 1W2 1 higherorder terms 728 If we recall that E1h2 5 0 and then drop the higherorder terms and use the constant k to represent E1h222 we can equate Equations 726 and 728 as U1W2 2 pUr 1W2 U1W2 2 kUs 1W2 729 or p 2kUs 1W2 Ur 1W2 5 kr1W2 730 That is the amount that a riskaverse individual is willing to pay to avoid a fair bet is approximately proportional to Pratts risk aversion measure13 Because insurance premiums paid are observable in the real world these are often used to estimate individuals risk aver sion coefficients or to compare such coefficients among groups of individuals Therefore it is possible to use market information to learn a bit about attitudes toward risky situations 762 Risk aversion and wealth An important question is whether risk aversion increases or decreases with wealth Intui tively one might think that the willingness to pay to avoid a given fair bet would decrease as wealth increases because decreasing marginal utility would make potential losses less serious for highwealth individuals This intuitive answer is not necessarily correct how ever because decreasing marginal utility also makes the gains from winning gambles less attractive Thus the net result is indeterminate it all depends on the precise shape of the utility function Indeed if utility is quadratic in wealth U1W2 5 a 1 bW 1 cW 2 731 where b 0 and c 0 then Pratts risk aversion measure is r1W2 5 2Us 1W2 Ur 1W2 5 22c b 1 2cW 732 which contrary to intuition increases as wealth increases On the other hand if utility is logarithmic in wealth U1W2 5 ln1W2 733 then we have r1W2 5 2Us 1W2 Ur 1W2 5 1 W 734 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 218 Part 3 Uncertainty and Strategy which does indeed decrease as wealth increases The exponential utility function U1W2 5 2e2AW 735 where A is a positive constant exhibits constant absolute risk aversion over all ranges of wealth because now r1W2 5 2Us 1W2 Ur 1W2 5 A2e2AW Ae2AW 5 A 736 This feature of the exponential utility function14 can be used to provide some numerical estimates of the willingness to pay to avoid gambles as the next example shows EXAMPLE 73 Constant Risk Aversion Suppose an individual whose initial wealth is W0 and whose utility function exhibits constant absolute risk aversion is facing a 5050 chance of winning or losing 1000 How much 1 f 2 would he or she pay to avoid the risk To find this value we set the utility of W0 2 f equal to the expected utility from the gamble 2e2A1W02f 2 5 21 2 e2A1W0110002 2 1 2 e2A1W0210002 737 Because the factor 2exp 12AW02 is contained in all the terms in Equation 737 this may be divided out thereby showing that for the exponential utility function the willingness to pay to avoid a given gamble is independent of initial wealth The remaining terms eAf 5 1 2 e21000A 1 1 2 e1000A 738 can now be used to solve for f for various values of A If A 5 00001 then f 5 499 a person with this degree of risk aversion would pay approximately 50 to avoid a fair bet of 1000 Alter natively if A 5 00003 this more riskaverse person would pay f 5 1478 to avoid the gamble Because intuition suggests that these values are not unreasonable values of the risk aversion parameter A in these ranges are sometimes used for empirical investigations Normally distributed risk The constant risk aversion utility function can be combined with the assumption that random shocks to a persons wealth follow a Normal distribution to arrive at a particularly simple result Before doing so we need to generalize the notion of a Normal dis tribution introduced in Chapter 2 There we provided the formula for the probability density function in the special case called a standard Normal in which the mean is 0 and variance is 1 More generally if wealth W is a Normal random variable with mean μ and variance σ2 its probability density function is f 1W2 5 112πσ22e21W2μ222σ2 If this persons utility for wealth is U1W2 5 2e2AW then his or her expected utility over risky wealth is E 3U1W2 4 5 3 q 2q U1W2f 1W2 dW 5 1 2πσ2 3 q 2q e2A1σz1μ2e2z22 σ dz 739 14Because the exponential utility function exhibits constant absolute risk aversion it is sometimes abbreviated by the term CARA utility Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 219 where the second equality follows from substituting for U1W2 and f 1W2 and making the change of variables z 5 1W 2 μ2σ This integral can be computed using a few tricks15 arriving at E 3U1W2 4 5 e2Aexp 1μ 2 Aσ222 But this is just a monotonic transformation of the argument inside the exp operator Hence the persons preferences can be simply represented by this argument μ 2 A 2 σ 2 5 CE 740 We have labeled this expression CE because it is the certainty equivalent of risky wealth The per son would be indifferent between his or her risky wealth Normally distributed with mean μ and variance σ2 and certain wealth with mean CE and no variance We can evaluate the persons pref erences over different distributions of wealth with different combinations of mean and variance simply by determining which provides the higher CE Equation 740 shows that CE is a linear function of mean and variance with the individuals riskaversion parameter A determining how detrimental variance is to the person For example suppose a person has invested so that wealth has a mean of 100000 and a stan dard deviation σ of 10000 To get a rough idea what these numbers mean with the Normal distribution he or she has about a 5 percent chance of having less wealth than 83500 and about the same chance of more wealth than 116500 With these parameters the certainty equivalent is given by CE 5 100000 2 A 1000022 If A 5 110000 then CE 5 95000 In that case this person would be indifferent between his or her risky wealth and certain wealth of 95000 A more riskaverse person might have A 5 310000 in this case the certainty equivalent of his or her wealth would be 85000 QUERY Suppose this person had two ways to invest his or her wealth allocation 1 with μ1 5 107000 and σ1 5 10000 or allocation 2 with μ2 5 102000 and σ2 5 2000 How would this persons attitude toward risk affect his or her choice between these allocations16 763 Relative risk aversion It seems unlikely that the willingness to pay to avoid a given gamble is independent of a persons wealth A more appealing assumption may be that such willingness to pay is inversely proportional to wealth and that the expression rr1W2 5 Wr1W2 5 2WUs 1W2 Ur 1W2 741 might be approximately constant Following the terminology proposed by J W Pratt17 the rrW function defined in Equation 741 is a measure of relative risk aversion The power utility function U1W R2 5 bWRR if R 1 R 2 0 ln W if R 5 0 742 15Canceling σ from numerator and denominator moving a constant exponential expression out of the integral and completing a square in the exponent of the integrand the righthand side of Equation 739 equals e2Aμ 2π 3 q 2q e21z1Aσ222 eA2σ22dz 5 exp12Aμ 1 A2σ222 c 1 2π 3 q 2q e21z1Aσ222dzd The factor in square brackets is the integral of the probability density function for a Normal random variable with mean 2Aσ and variance 1 and thus equals 1 Rearranging the resulting expression gives the expression for CE in Equation 740 16This numerical example roughly approximates historical data on real returns of stocks and bonds respectively although the calculations are illustrative only 17Pratt Risk Aversion Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 220 Part 3 Uncertainty and Strategy exhibits diminishing absolute risk aversion r1W2 5 2Us 1W2 Ur 1W2 5 2 1R 2 12WR22 WR21 5 1 2 R W 743 but constant relative risk aversion18 rr1W2 5 Wr1W2 5 1 2 R 744 Empirical evidence is generally consistent with values of R in the range of 3 to 1 Hence individuals seem to be somewhat more risk averse than is implied by the logarithmic utility function although in many applications that function provides a reasonable approximation It is useful to note that the constant relative risk aversion utility function in Equation 742 has the same form as the general CES utility function we first described in Chapter 3 This provides some geometric intuition about the nature of risk aversion that we will explore later in this chapter 18Some authors write the utility function in Equation 742 as U1W2 5 W12a 11 2 a2 and seek to measure a 5 1 2 R In this case a is the relative risk aversion measure The constant relative risk aversion function is sometimes abbreviated as CRRA utility EXAMPLE 74 Constant Relative Risk Aversion An individual whose behavior is characterized by a constant relative risk aversion utility function will be concerned about proportional gains or loss of wealth Therefore we can ask what fraction of initial wealth f such a person would be willing to give up to avoid a fair gamble of say 10 percent of initial wealth First we assume R 5 0 so the logarithmic utility function is appro priate Setting the utility of this individuals certain remaining wealth equal to the expected utility of the 10 percent gamble yields ln 311 2 f 2W04 5 05 ln111 W02 1 05 ln109 W02 745 Because each term contains ln W0 initial wealth can be eliminated from this expression ln11 2 f 2 5 05 3ln1112 1ln10924 5 ln1099205 hence 1 2 f 5 10992 05 5 0995 and f 5 0005 746 Thus this person will sacrifice up to 05 percent of wealth to avoid the 10 percent gamble A similar calculation can be used for the case R 5 22 to yield f 5 0015 747 Hence this more riskaverse person would be willing to give up 15 percent of his or her initial wealth to avoid a 10 percent gamble QUERY With the constant relative risk aversion function how does this persons willingness to pay to avoid a given absolute gamble say of 1000 depend on his or her initial wealth Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 221 77 METHODS FOR REDUCING UNCERTAINTY AND RISK We have seen that riskaverse people will avoid gambles and other risky situations if pos sible Often it is impossible to avoid risk entirely Walking across the street involves some risk of harm Burying ones wealth in the backyard is not a perfectly safe investment strat egy because there is still some risk of theft to say nothing of inflation Our analysis thus far implies that people would be willing to pay something to at least reduce these risks if they cannot be avoided entirely In the next four sections we will study each of four dif ferent methods that individuals can take to mitigate the problem of risk and uncertainty insurance diversification flexibility and information 78 INSURANCE We have already discussed one such strategy buying insurance Riskaverse people would pay a premium to have the insurance company cover the risk of loss Each year people in the United States spend more than half a trillion dollars on insurance of all types Most commonly they buy coverage for their own life for their home and cars and for their health care costs But insurance can be bought perhaps at a high price for practically any risk imaginable ranging from earthquake insurance for a house along a fault line to special coverage for a surgeon against a hand injury A riskaverse person would always want to buy fair insurance to cover any risk he or she faces No insurance company could afford to stay in business if it offered fair insurance in the sense that the premium exactly equals the expected payout for claims Besides cover ing claims insurance companies must also maintain records collect premiums investi gate fraud and perhaps return a profit to shareholders Hence an insurance customer can always expect to pay more than an actuarially fair premium If people are sufficiently risk averse they will even buy unfair insurance as shown in Example 72 the more risk averse they are the higher the premium they would be willing to pay Several factors make insurance difficult or impossible to provide Largescale disasters such as hurricanes and wars may result in such large losses that the insurance company would go bankrupt before it could pay all the claims Rare and unpredictable events eg war nuclear power plant accidents offer no reliable track record for insurance companies to establish premiums Two other reasons for absence of insurance coverage relate to the informational disadvantage the company may have relative to the customer In some cases the individual may know more about the likelihood that they will suffer a loss than the insurance company Only the worst customers those who expect larger or more likely losses may end up buying an insurance policy This adverse selection problem may unravel the whole insurance market unless the company can find a way to control who buys through some sort of screening or compulsion Another problem is that hav ing insurance may make customers less willing to take steps to avoid losses for example driving more recklessly with auto insurance or eating fatty foods and smoking with health insurance This socalled moral hazard problem again may impair the insurance market unless the insurance company can find a way to cheaply monitor customer behavior We will discuss the adverse selection and moral hazard problems in more detail in Chapter 18 and discuss ways the insurance company can combat these problems which besides the above strategies include offering only partial insurance and requiring the payment of deductibles and copayments Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 222 Part 3 Uncertainty and Strategy 79 DIVERSIFICATION A second way for riskaverse individuals to reduce risk is by diversifying This is the eco nomic principle behind the adage Dont put all your eggs in one basket By suitably spreading risk around it may be possible to reduce the variability of an outcome without lowering the expected payoff The most familiar setting in which diversification comes up is in investing Investors are routinely advised to diversify your portfolio To understand the wisdom behind this advice consider an example in which a person has wealth W to invest This money can be invested in two independent risky assets 1 and 2 which have equal expected values the mean returns are μ1 5 μ2 and equal variances the variances are σ2 1 5 σ2 2 A person whose undiversified portfolio UP includes just one of the assets putting all his or her eggs in that basket would earn an expected return of μUP 5 μ1 5 μ2 and would face a variance of σ2 UP 5 σ2 1 5 σ2 2 Suppose instead the individual chooses a diversified portfolio DP Let α1 be the frac tion invested in the first asset and α2 5 1 2 α1 in the second We will see that the person can do better than the undiversified portfolio in the sense of getting a lower variance with out changing the expected return The expected return on the diversified portfolio does not depend on the allocation across assets and is the same as for either asset alone μDP 5 α1μ1 1 11 2 α12μ2 5 μ1 5 μ2 748 To see this refer back to the rules for computed expected values from Chapter 2 The vari ance will depend on the allocation between the two assets σ2 DP 5 α2 1σ2 1 1 11 2 α12 2σ2 2 5 11 2 2α1 1 2α2 12σ2 1 749 This calculation again can be understood by reviewing the section on variances in Chapter 2 There you will be able to review the two facts used in this calculation First the variance of a constant times a random variable is that constant squared times the variance of a random variable second the variance of independent random variables because their covariance is 0 equals the sum of the variances Choosing α1 to minimize Equation 749 yields α1 5 1 2 and σ2 DP 5 σ2 1 2 Therefore the optimal portfolio spreads wealth equally between the two assets maintaining the same expected return as an undiversified portfolio but reducing variance by half Diversification works here because the assets returns are independent When one return is low there is a chance the other will be high and vice versa Thus the extreme returns are balanced out at least some of the time reducing the overall variance Diversification will work in this way as long as there is not perfect correlation in the asset returns so that they are not effectively the same asset The less correlated the assets are the better diversification will work to reduce the variance of the overall portfolio The example constructed to highlight the benefits of diversification as simply as possi ble has the artificial element that asset returns were assumed to be equal Diversification was a free lunch in that the variance of the portfolio could be reduced without reducing the expected return compared with an undiversified portfolio If the expected return from one of the assets say asset 1 is higher than the other then diversification into the other asset would no longer be a free lunch but would result in a lower expected return Still the benefits from risk reduction can be great enough that a riskaverse investor would be willing to put some share of wealth into the asset with the lower expected return A prac tical example of this idea is related to advice one would give to an employee of a firm with a stock purchase plan Even if the plan allows employees to buy shares of the companys stock at a generous discount compared with the market the employee may still be advised Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 223 not to invest all savings in that stock because otherwise the employees entire savings to say nothing of his or her salary and perhaps even house value to the extent house values depend on the strength of businesses in the local economy is tied to the fortunes of a sin gle company generating a tremendous amount of risk The Extensions provide a much more general analysis of the problem of choosing the optimal portfolio However the principle of diversification applies to a much broader range of situations than financial markets For example students who are uncertain about where their interests lie or about what skills will be useful on the job market are well advised to register for a diverse set of classes rather than exclusively technical or artistic ones 710 FLEXIBILITY Diversification is a useful method to reduce risk for a person who can divide up a deci sion by allocating small amounts of a larger sum among a number of different choices In some situations a decision cannot be divided it is all or nothing For example in shopping for a car a consumer cannot combine the attributes that he or she likes from one model say fuel efficiency with those of another say horsepower or power windows by buying half of each cars are sold as a unit With allornothing decisions the decisionmaker can obtain some of the benefit of diversification by making flexible decisions Flexibility allows the person to adjust the initial decision depending on how the future unfolds The more uncertain the future the more valuable this flexibility Flexibility keeps the decisionmaker from being tied to one course of action and instead provides a number of options The decisionmaker can choose the best option to suit later circumstances A good example of the value of flexibility comes from considering the fuels on which cars are designed to run Until now most cars were limited in how much biofuel such as ethanol made from crops could be combined with petroleum products such as gasoline or diesel in the fuel mix A purchaser of such a car would have difficulties if governments passed new regulations increasing the ratio of ethanol in car fuels or banning petroleum products entirely New cars have been designed that can burn ethanol exclusively but such cars are not useful if current conditions continue to prevail because most filling stations do not sell fuel with high concentrations of ethanol A third type of car has internal compo nents that can handle a variety of types of fuel both petroleumbased and ethanol and any proportions of the two Such cars are expensive to build because of the specialized compo nents involved but a consumer might pay the additional expense anyway because the car would be useful whether or not biofuels become more important over the life of the car19 7101 Types of options The ability of flexiblefuel cars to be able to burn any mix of petroleumbased fuels and biofuels is valuable because it provides the owner with more options relative to a car that can run on only one type of fuel Readers are probably familiar with the notion that options are valuable from another context where the term is frequently usedfinancial markets where one hears about stock options and other forms of options contracts There is a close connection between the option implicit in the flexiblefuel cars and these option con tracts that we will investigate in more detail Before discussing the similarities between the options arising in different contexts we introduce some terms to distinguish them 19While the current generation of flexiblefuel cars involve stateoftheart technology the first such car produced back in 1908 was Henry Fords ModelT one of the topselling cars of all time The availability of cheap gasoline may have swung the market toward competitors singlefuel cars spelling the demise of the ModelT For more on the history of this model see L Brooke Ford Model T The Car That Put the World on Wheels Minneapolis Motorbooks 2008 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 224 Part 3 Uncertainty and Strategy The flexiblefuel car can be viewed as an ordinary car combined with an additional real option to burn biofuels if those become more important in the future Financial option contracts come in a variety of forms some of which can be complex There are also many different types of real options and they arise in many different set tings sometimes making it difficult to determine exactly what sort of option is embedded in the situation Still all options share three fundamental attributes First they specify the underlying transaction whether it is a stock to be traded or a car or fuel to be purchased Second they specify a period over which the option may be exercised A stock option may specify a period of 1 year for example The option embedded in a flexiblefuel car preserves the owners option during the operating life of the car The longer the period over which the option extends the more valuable it is because the more uncertainty that can be resolved during this period Third the option contract specifies a price A stock option might sell for a price of 70 If this option is later traded on an exchange its price might vary from moment to moment as the markets move Real options do not tend to have explicit prices but sometimes implicit prices can be calculated For example if a flex iblefuel car costs 5000 more than an otherwise equivalent car that burns one type of fuel then this 5000 could be viewed as the option price 7102 Model of real options Let x embody all the uncertainty in the economic environment In the case of the flexi blefuel car x might reflect the price of fossil fuels relative to biofuels or the stringency of government regulation of fossil fuels In terms of the section on statistics in Chapter 2 x is a random variable sometimes referred to as the state of the world that can take on possibly many different values The individual has some number i 5 1 n of choices currently available Let Oi 1x2 be the payoffs provided by choice i where the argument 1x2 allows each choice to provide a different pattern of returns depending on how the future turns out Figure 73a illustrates the case of two choices The first choice provides a decreasing payoff as x increases indicated by the downward slope of O1 This might correspond to ownership of a car that runs only on fossil fuels as biofuels become more important than fossil fuels the value of a car burning only fossil fuels decreases The second choice pro vides an increasing payoff perhaps corresponding to ownership of a car that runs only on biofuels Figure 73b translates the payoffs into von NeumannMorgenstern utilities that the person obtains from the payoffs by graphing U1Oi2 rather than Oi The bend intro duced in moving from payoffs to utilities reflects the diminishing marginal utility from higher payoffs for a riskaverse person If the person does not have the flexibility provided by a real option he or she must make the choice before observing how the state x turns out The individual should choose the single alternative that is best on average His or her expected utility from this choice is max5E3U1O12 4 E3U1On2 46 750 D E F I N I T I O N Financial option contract A financial option contract offers the right but not the obligation to buy or sell an asset such as a share of stock during some future period at a certain price D E F I N I T I O N Real option A real option is an option arising in a setting outside of financial markets Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 225 Figure 73 does not provide enough information to judge which expected utility is higher because we do not know the likelihoods of the different xs but if the xs are about equally likely then it looks as though the individual would choose the second alternative provid ing higher utility over a larger range The individuals expected utility from this choice is E3U1O22 4 On the other hand if the real option can be preserved to make a choice that responds to which state of the world x has occurred the person will be better off In the car application the real option could correspond to buying a flexiblefuel car which does not lock the buyer into one fuel but allows the choice of whatever fuel turns out to be most common or inexpensive over the life of the car In Figure 73 rather than choosing a single alternative the person would choose the first option if x xr and the second option if x xr The utility provided by this strategy is given by the bold curve which is the upper envelope of the curves for the individual options With a general number 1n2 of choices expected utility from this upper envelope of individual options is E5max 3U1O12 U1On2 46 751 The expected utility in Equation 751 is higher than in 750 This may not be obvious at first glance because it seems that simply swapping the order of the expectations and max operators should not make a difference But indeed it does Whereas Equation 750 is the expected utility associated with the best single utility curve Equation 751 is the expected utility associated with the upper envelope of all the utility curves20 Panel a shows the payoffs and panel b shows the utilities provided by two alternatives across states of the world 1x2 If the decision has to be made upfront the individual chooses the single curve having the highest expected utility If the real option to make either decision can be preserved until later the indi vidual can obtain the expected utility of the upper envelope of the curves shown in bold Payof State x x O1 O2 Utility State x x UO2 UO1 a Payofs from alternatives b Utilities from alternatives FIGURE 73 The Nature of a Real Option 20The result can be proved formally using Jensens inequality introduced in footnote 10 The footnote discusses the implications of Jensens inequality for concave functions E3f 1x24 f 3E1x24 Jensens inequality has the reverse implication for convex functions E3f 1x24 f 3E1x24 In other words for convex functions the result is greater if the expectations operator is applied outside of the function than if the order of the two is reversed In the options context the max operator has the properties of a convex function This can be seen from Figure 73b where taking the upper envelope convexifies the individual curves turning them into more of a Vshape Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 226 Part 3 Uncertainty and Strategy 7103 More options are better typically Adding more options can never harm an individual decisionmaker as long as he or she is not charged for them because the extra options can always be ignored This is the essence of options They give the holder the rightbut not the obligationto choose them Figure 74 illustrates this point showing the effect of adding a third option to the two drawn in Figure 73 In the first panel the person strictly benefits from the third option because there are some states of the world the highest values of x in the figure for which it is better than any other alternative shifting the upper envelope of utilities the bold curve up The third option is worthless in the second panel Although the third option is not the worst option for many states of the world it is never the best and so does not improve the upper envelope of utilities relative to Figure 73 Still the addi tion of the third option is not harmful This insight may no longer hold in a strategic setting with multiple decisionmakers In a strategic setting economic actors may benefit from having some of their options cut off This may allow a player to commit to a narrower course of action that he or she would not have chosen otherwise and this commitment may affect the actions of other parties possibly to the benefit of the party making the commitment A famous illustra tion of this point is provided in one of the earliest treatises on military strategy by Sun Tzu a Chinese general writing in 400 BC It seems crazy for an army to destroy all means of retreat burning bridges behind itself and sinking its own ships among other mea sures Yet this is what Sun Tzu advocated as a military tactic If the second army observes that the first cannot retreat and will fight to the death it may retreat itself before engag ing the first We will analyze such strategic issues more formally in the next chapter on game theory The addition of a third alternative to the two drawn in Figure 73 is valuable in a because it shifts the upper envelope shown in bold of utilities up The new alternative is worthless in b because it does not shift the upper envelope but the individual is not worse off for having it Utility State x a Additional valuable option UO1 UO1 UO2 UO2 UO3 UO3 Utility State x b Additional worthless option FIGURE 74 More Options Cannot Make the Individual DecisionMaker Worse Off Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 227 7104 Computing option value We can push the analysis further to derive a mathematical expression for the value of a real option Let F be the fee that has to be paid for the ability to choose the best alternative after x has been realized instead of before The individual would be willing to pay the fee as long as E5max 3U1O1 1x2 2 F24 3U1On1x2 2 F2 46 max5E3U1O1 1x2 2 4 E3U1On1x2 2 46 752 The right side is the expected utility from making the choice beforehand repeated from Equation 750 The left side allows for the choice to be made after x has occurred a benefit but subtracts the fee for option from every payoff The fee is naturally assumed to be paid up front and thus reduces wealth by F whichever option is chosen later The real options value is the highest F for which Equation 752 is still satisfied which of course is the F for which the condition holds with equality EXAMPLE 75 Value of a FlexibleFuel Car Lets work out the option value provided by a flexiblefuel car in a numerical example Let O1 1x2 5 1 2 x be the payoff from a fossilfuelonly car and O2 1x2 5 x be the payoff from a biofuelonly car The state of the world x reflects the relative importance of biofuels compared with fossil fuels over the cars lifespan Assume x is a random variable that is uniformly distrib uted between 0 and 1 the simplest continuous random variable to work with here The sta tistics section in Chapter 2 provides some detail on the uniform distribution showing that the probability density function PDF is f 1x2 5 1 in the special case when the uniform random variable ranges between 0 and 1 Risk neutrality To make the calculations as easy as possible to start suppose first that the car buyer is risk neutral obtaining a utility level equal to the payoff level Suppose the buyer is forced to choose a biofuel car This provides an expected utility of E 1O22 5 3 1 0 O2 1x2f 1x2 dx 5 3 1 0 x dx 5 x2 2 x51 x50 5 1 2 753 where the integral simplifies because f 1x2 5 1 Similar calculations show that the expected util ity from buying a fossilfuel car is also 12 Therefore if only singlefuel cars are available the person is indifferent between them obtaining expected utility 12 from either Now suppose that a flexiblefuel car is available which allows the buyer to obtain either O1 1x2 or O2 1x2 whichever is higher under the latter circumstances The buyers expected utility from this car is E 3max1O1 O224 5 3 1 0 max11 2 x x2f 1x2dx 5 3 1 2 0 11 2 x2dx 1 3 1 1 2 x dx 5 23 1 1 2 x dx 5 x2 0 x51 x51 2 5 3 4 754 The second line in Equation 754 follows from the fact that the two integrals in the preceding expression are symmetric Because the buyers utility exactly equals the payoffs we can compute the option value of the flexiblefuel car directly by taking the difference between the expected payoffs in Equations 753 and 754 which equals 14 This is the maximum premium the person would pay for the flexiblefuel car over a singlefuel car Scaling payoffs to more realistic levels by multiplying by say 10000 the price premium and the option value of the flexiblefuel car would be 2500 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 228 Part 3 Uncertainty and Strategy This calculation demonstrates the general insight that options are a way of dealing with uncer tainty that have value even for riskneutral individuals The next part of the example investigates whether risk aversion makes options more or less valuable Risk aversion Now suppose the buyer is risk averse having von NeumannMorgenstern utility function U1x2 5 x The buyers expected utility from a biofuel car is E 3U1O224 5 3 1 0 O2 1x2f 1x2dx 5 3 1 0 x 1 2 dx 5 2 3x 3 2 x51 x50 5 2 3 755 which is the same as from a fossilfuel car as similar calculations show Therefore a singlefuel car of whatever type provides an expected utility of 23 The expected utility from a flexiblefuel car that costs F more than a singlefuel car is E5max3U1O1 1x2 2 F2 U1O2 1x2 2 F2 46 5 3 1 0 max11 2 x 2 F x 2 F2f1x2 dx 5 3 1 2 0 1 2 x 2 F dx 1 3 1 1 2 x 2 F dx 5 23 1 1 2 x 2 F dx 5 23 12F 1 22F u 1 2 du 5 4 3u 3 2 u512F u5 1 22F 5 4 3 c 11 2 F2 3 2 2 a1 2 2 Fb 3 2 d 756 To find the maximum premium F that the riskaverse buyer would be willing to pay for the flexiblefuel car we plot the expected utility from a singlefuel car from Equation 755 and from the flexiblefuel car from Equation 756 and see the value of F where the curves cross F Expected utility Single fuel Flexible fuel 000 00 01 02 03 04 05 06 07 08 09 10 010 020 030 040 050 FIGURE 75 Graphical Method for Computing the Premium for a FlexibleFuel Car Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 229 The calculations involved in Equation 756 are somewhat involved and thus require some discus sion The second line relies on the symmetry of the two integrals appearing there which allows us to collapse them into two times the value of one of them and we chose the simpler of the two for these purposes The third line uses the change of variables u 5 x 2 F to simplify the integral See Equation 2121 in Chapter 2 for another example of the changeofvariables trick and further discussion To find the maximum premium the buyer would pay for a flexiblefuel car we can set Equations 755 and 756 equal and solve for F Unfortunately the resulting equation is too compli cated to be solved analytically One simple approach is to graph the last line of Equation 756 for a range of values of F and eyeball where the graph hits the required value of 23 from Equation 755 This is done in Figure 75 where we see that this value of F is slightly less than 03 0294 to be more precise Therefore the riskaverse buyer is willing to pay a premium of 0294 for the flexi blefuel car which is also the option value of this type of car Scaling up by 10000 for more real istic monetary values the price premium would be 2940 This is 440 more than the riskneutral buyer was willing to pay Thus the option value is greater in this case for the riskaverse buyer QUERY Does risk aversion always increase option value If so explain why If not modify the example with different shapes to the payoff functions to provide an example where the risk neutral buyer would pay more 7105 Option value of delay Society seems to frown on procrastinators Do not put off to tomorrow what you can do today is a familiar maxim Yet the existence of real options suggests a possible value in procrastination There may be a value in delaying big decisionssuch as the purchase of a carthat are not easily reversed later Delaying these big decisions allows the decision maker to preserve option value and gather more information about the future To the outside observer who may not understand all the uncertainties involved in the situation it may appear that the decisionmaker is too inert failing to make what looks to be the right decision at the time In fact delaying may be exactly the right choice to make in the face of uncertainty Choosing one course of action rules out other courses later Delay preserves options If circumstances continue to be favorable or become even more so the action can still be taken later But if the future changes and the action is unsuitable the decision maker may have saved a lot of trouble by not making it The value of delay can be seen by returning to the car application Suppose for the sake of this example that only singlefuel cars of either type fossil fuel or biofuel are available on the market flexiblefuel cars have not yet been invented Even if circumstances start to favor the biofuel car with the number of filling stations appearing to tip toward offering biofuels the buyer may want to hold off buying a car until he or she is more sure This may be true even if the buyer is forgoing considerable consumer surplus from the use of a new car during the period of delay The problem is that if biofuels do not end up taking over the market the buyer may be left with a car that is hard to fuel up and hard to trade in for a car burning the other fuel type The buyer would be willing to experience delay costs up to F to preserve flexibility The value of delay hinges on the irreversibility of the underlying decision If in the car example the buyer manufacturer could recover close to the purchase price by selling it on the usedcar market there would be no reason to delay purchasing But it is well known that the value of a new car decreases precipitously once it is driven off the car lot we will discuss reasons for this including the lemons effect in Chapter 18 therefore it may not be so easy to reverse the purchase of a car Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 230 Part 3 Uncertainty and Strategy 7106 Implications for costbenefit analysis To an outside observer delay may seem like a symptom of irrationality or ignorance Why is the decisionmaker overlooking an opportunity to take a beneficial action The chapter has now provided several reasons why a rational decisionmaker might not want to pursue an action even though the expected benefits from the action outweigh the expected costs First a riskaverse individual might avoid a gamble even if it provided a positive expected monetary payoff because of the decreasing marginal utility from money And option value provides a further reason for the action not to be undertaken The decisionmaker might be delaying until he or she has more certainty about the potential results of the decision Many of us have come across the costbenefit rule which says that an action should be taken if anticipated costs are less than benefits This is generally a sensible rule providing the correct course of action in simple settings without uncertainty One must be more care ful in applying the rule in settings involving uncertainty The correct decision rule is more complicated because it should account for risk preferences by converting payoffs into util ities and for the option value of delay if present Failure to apply the simple costbenefit rule in settings with uncertainty may indicate sophistication rather than irrationality21 711 INFORMATION The fourth method of reducing the uncertainty involved in a situation is to acquire better information about the likely outcome that will arise We have already considered a ver sion of this in the previous section where we considered the strategy of preserving options while delaying a decision until better information is received Delay involved some costs which can be thought of as a sort of purchase price for the information acquired Here we will be more direct in considering information as a good that can be purchased directly and analyze in greater detail why and how much individuals are willing to pay for it 7111 Information as a good By now it should be clear to the reader that information is a valuable economic resource We have seen an example already A buyer can make a better decision about which type of car to buy if he or she has better information about the sort of fuels that will be readily available during the life of the car But the examples do not end there Shoppers who know where to buy highquality goods cheaply can make their budgets stretch further than those who do not doctors can provide better medical care if they are up to date on the latest sci entific research The study of information economics has become one of the major areas in current research Several challenges are involved Unlike the consumer goods we have been study ing thus far information is difficult to quantify Even if it could be quantified information has some technical properties that make it an unusual sort of good Most information is durable and retains value after it has been used Unlike a hot dog which is consumed only once knowledge of a special sale can be used not only by the person who discovers it but also by anyone else with whom the information is shared The friends then may gain from this information even though they do not have to spend anything to obtain it Indeed in a 21Economists are puzzled by consumers reluctance to install efficient appliances even though the savings on energy bills are likely to defray the appliances purchase price before long An explanation from behavioral economics is that consumers are too ignorant to perform the costbenefit calculations or are too impatient to wait for the energy savings to accumulate K Hassett and G Metcalf in Energy Conservation Investment Do Consumers Discount the Future Correctly Energy Policy June 1993 71016 suggest that consumer inertia may be rational delay in the face of fluctuating energy prices See Problem 710 for a related numerical example Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 231 special case of this situation information has the characteristic of a pure public good see Chapter 19 That is the information is both nonrival in that others may use it at zero cost and nonexclusive in that no individual can prevent others from using the information The classic example of these properties is a new scientific discovery When some prehistoric people invented the wheel others could use it without detracting from the value of the discovery and everyone who saw the wheel could copy it freely Information is also difficult to sell because the act of describing the good that is being offered to a potential consumer gives it away to them These technical properties of information imply that market mechanisms may often operate imperfectly in allocating resources to information provision and acquisition After all why invest in the production of information when one can just acquire it from others at no cost Therefore standard models of supply and demand may be of relatively lim ited use in understanding such activities At a minimum models have to be developed that accurately reflect the properties being assumed about the informational environment Throughout the latter portions of this book we will describe some of the situations in which such models are called for Here however we will pay relatively little attention to supplydemand equilibria and will instead focus on an example that illustrates the value of information in helping individuals make choices under uncertainty 7112 Quantifying the value of information We already have all the tools needed to quantify the value of information from the section on option values Suppose again that the individual is uncertain about what the state of the world 1x2 will be in the future He or she needs to make one of n choices today this allows us to put aside the option value of delay and other issues we have already studied As before Oi 1x2 represents the payoffs provided by choice i Now reinterpret F as the fee charged to be told the exact value that x will take on in the future perhaps this is the salary of the economist hired to make such forecasts The same calculations from the option section can be used here to show that the persons willingness to pay is again the value of F such that Equation 752 holds with equality Just as this was the value of the real option in that section here it is the value of information The value of information would be lower than this F if the forecast of future conditions were imperfect rather than perfect as assumed here Other factors affecting an individuals value of information include the extent of uncertainty before acquiring the information the number of options he or she can choose between and his or her risk preferences The more uncertainty resolved by the new information the more valuable it is of course If the individual does not have much scope to respond to the information because of having only a limited range of choices to make the information will not be valuable The degree of risk aversion has ambiguous effects on the value of information answering the Query in Example 75 will provide you with some idea why 712 THE STATEPREFERENCE APPROACH TO CHOICE UNDER UNCERTAINTY Although our analysis in this chapter has offered insights on a number of issues it seems rather different from the approach we took in other chapters The basic model of utility maximization subject to a budget constraint seems to have been lost To make further progress in the study of behavior under uncertainty we will develop some new techniques that will permit us to bring the discussion of such behavior back into the standard choice theoretic framework Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 232 Part 3 Uncertainty and Strategy 7121 States of the world and contingent commodities We start by pushing a bit further on an idea already mentioned thinking about an uncer tain future in term of states of the world We cannot predict exactly what will happen say tomorrow but we assume that it is possible to categorize all the possible things that might happen into a fixed number of welldefined states For example we might make the crude approximation of saying that the world will be in only one of two possible states tomorrow It will be either good times or bad times One could make a much finer gradation of states of the world involving even millions of possible states but most of the essentials of the theory can be developed using only two states A conceptual idea that can be developed concurrently with the notion of states of the world is that of contingent commodities These are goods delivered only if a particular state of the world occurs As an example 1 in good times is a contingent commodity that promises the individual 1 in good times but nothing should tomorrow turn out to be bad times It is even possible by stretching ones intuitive ability somewhat to conceive of being able to purchase this commodity I might be able to buy from someone the promise of 1 if tomorrow turns out to be good times Because tomorrow could be bad this good will probably sell for less than 1 If someone were also willing to sell me the contingent commodity 1 in bad times then I could assure myself of having 1 tomorrow by buying the two contingent commodities 1 in good times and 1 in bad times 7122 Utility analysis Examining utilitymaximizing choices among contingent commodities proceeds formally in much the same way we analyzed choices previously The principal difference is that after the fact a person will have obtained only one contingent good depending on whether it turns out to be good or bad times Before the uncertainty is resolved however the indi vidual has two contingent goods from which to choose and will probably buy some of each because he or she does not know which state will occur We denote these two contingent goods by Wg wealth in good times and Wb wealth in bad times Assuming that utility is independent of which state occurs22 and that this individual believes that bad times will occur with probability π the expected utility associated with these two contingent goods is E3U1W2 4 5 11 2 π2U1Wg2 1 πU1Wb2 757 This is the magnitude this individual seeks to maximize given his or her initial wealth W0 7123 Prices of contingent commodities Assuming that this person can purchase 1 of wealth in good times for pg and 1 of wealth in bad times for pb his or her budget constraint is then W0 5 pgWg 1 pbWb 758 The price ratio pgpb shows how this person can trade dollars of wealth in good times for dollars in bad times If for example pg 5 080 and pb 5 020 the sacrifice of 1 of wealth in good times would permit this person to buy contingent claims yielding 4 of wealth should times turn out to be bad Whether such a trade would improve utility will of course depend on the specifics of the situation But looking at problems involving uncer tainty as situations in which various contingent claims are traded is the key insight offered by the statepreference model 22This assumption is untenable in circumstances where utility of wealth depends on the state of the world For example the utility provided by a given level of wealth may differ depending on whether an individual is sick or healthy We will not pursue such complications here however For most of our analysis utility is assumed to be concave in wealth Ur 1W2 0 Us 1W2 0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 233 7124 Fair markets for contingent goods If markets for contingent wealth claims are well developed and there is general agreement about the likelihood of bad times 1π2 then prices for these claims will be actuarially fair that is they will equal the underlying probabilities pg 5 1 2 π pb 5 π 759 Hence the price ratio pgpb will simply reflect the odds in favor of good times pg pb 5 1 2 π π 760 In our previous example if pg 5 1 2 π 5 08 and pb 5 π 5 02 then 11 2 π2π 5 4 In this case the odds in favor of good times would be stated as 4 to 1 Fair markets for contingent claims such as insurance markets will also reflect these odds An analogy is provided by the odds quoted in horse races These odds are fair when they reflect the true probabilities that various horses will win 7125 Risk aversion We are now in a position to show how risk aversion is manifested in the stateprefer ence model Specifically we can show that if contingent claims markets are fair then a utilitymaximizing individual will opt for a situation in which Wg 5 Wb that is he or she will arrange matters so that the wealth ultimately obtained is the same no matter what state occurs As in previous chapters maximization of utility subject to a budget constraint requires that this individual set the MRS of Wg for Wb equal to the ratio of these goods prices MRS 5 E3U1W2 4Wg E3U1W2 4Wb 5 11 2 π2Ur 1Wg2 πUr 1Wb2 5 pg pb 761 In view of the assumption that markets for contingent claims are fair Equation 760 this firstorder condition reduces to Ur 1Wg2 Ur 1Wb2 5 1 or23 Wg 5 Wb 762 Hence this individual when faced with fair markets in contingent claims on wealth will be risk averse and will choose to ensure that he or she has the same level of wealth regardless of which state occurs 7126 A graphic analysis Figure 76 illustrates risk aversion with a graph This individuals budget constraint 1I2 is shown to be tangent to the U1 indifference curve where Wg 5 Wba point on the cer tainty line where wealth 1W 2 is independent of which state of the world occurs At W the slope of the indifference curve 3 11 2 π2π4 is precisely equal to the price ratio pgpb 23This step requires that utility be state independent and that Ur 1W2 0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 234 Part 3 Uncertainty and Strategy If the market for contingent wealth claims were not fair utility maximization might not occur on the certainty line Suppose for example that 11 2 π2π 5 4 but that pgpb 5 2 because ensuring wealth in bad times proves costly In this case the budget constraint would resemble line Ir in Figure 76 and utility maximization would occur below the cer tainty line24 In this case this individual would gamble a bit by opting for Wg Wb because claims on Wb are relatively costly Example 76 shows the usefulness of this approach in evaluating some of the alternatives that might be available The line I represents the individuals budget constraint for contingent wealth claims W0 5 pgWg 1 pbWb If the market for contingent claims is actuarially fair 3pgpb 5 11 2 π2π4 then utility maximization will occur on the certainty line where Wg 5 Wb 5 W If prices are not actuarially fair the budget constraint may resemble Ir and utility maximization will occur at a point where Wg Wb FIGURE 76 Risk Aversions in the StatePreference Model Certainty line Wb Wb W Wg Wg W I I W W U1 EXAMPLE 76 Insurance in the StatePreferences Model We can illustrate the statepreference approach by recasting the auto insurance illustration from Example 72 as a problem involving the two contingent commodities wealth with no theft 1Wg2 and wealth with a theft 1Wb2 If as before we assume logarithmic utility and that the probabil ity of a theft is π 5 025 then expected utility is E 3U1W2 4 5 075U1Wg2 1 025U1Wb2 5 075 lnWg 1 025 lnWb 763 24Because as Equation 761 shows the MRS on the certainty line is always 11 2 π2π tangencies with a flatter slope than this must occur below the line Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 235 If the individual takes no action then utility is determined by the initial wealth endowment W0g 5 100000 and W0b 5 80000 so Eno 3U1W24 5 075 ln 100000 1 025 ln 80000 5 1145714 764 To study trades away from these initial endowments we write the budget constraint in terms of the prices of the contingent commodities pg and pb pgW0g 1 pbW0b 5 pgWg 1 pbWb 765 Assuming that these prices equal the probabilities of the two states 1pg 5 075 pb 5 0252 this constraint can be written as 075 11000002 1 025 1800002 5 95000 5 075Wg 1 025Wb 766 that is the expected value of wealth is 95000 and this person can allocate this amount between Wg and Wb Now maximization of utility with respect to this budget constraint yields Wg 5 Wb 5 95000 Consequently the individual will move to the certainty line and receive an expected utility of EA 3U1W2 4 5 ln 95000 5 1146163 767 a clear improvement over doing nothing To obtain this improvement this person must be able to transfer 5000 of wealth in good times no theft into 15000 of extra wealth in bad times theft A fair insurance contract call it contract A would allow this because it would cost 5000 but return 20000 should a theft occur but nothing should no theft occur Notice here that the wealth changes promised by insurancedWbdWg 5 1500025000 5 23exactly equal the negative of the odds ratio 211 2 π2π 5 2075025 5 23 A policy with a deductible provision A number of other insurance contracts might be utility improving in this situation although not all of them would lead to choices that lie on the certainty line For example a policy B that cost 5200 and returned 20000 in case of a theft would permit this person to reach the certainty line with Wg 5 Wb 5 94800 and expected utility EB 3U1W24 5 ln 94800 5 1145953 768 which also exceeds the utility obtainable from the initial endowment A policy that costs 4900 and requires the individual to incur the first 1000 of a loss from theft would yield Wg 5 100000 2 4900 5 95100 Wb 5 80000 2 4900 1 19000 5 94100 769 the expected utility from this policy label it C equals EC3U1W2 4 5 075 ln 95100 1 025 ln 94100 5 1146004 770 Although this policy does not permit this person to reach the certainty line it is utility improving Insurance need not be complete to offer the promise of higher utility QUERY What is the maximum amount an individual would be willing to pay for an insurance policy under which he or she had to absorb the first 1000 of loss Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 236 Part 3 Uncertainty and Strategy 7127 Risk aversion and risk premiums The statepreference model is also especially useful for analyzing the relationship between risk aversion and individuals willingness to pay for risk Consider two people each of whom starts with a certain wealth W0 Each person seeks to maximize an expected utility function of the form E3U1W2 4 5 11 2 π2 WR g R 1 πWR b R 771 Here the utility function exhibits constant relative risk aversion see Example 74 Notice also that the function closely resembles the CES utility function we examined in Chapter 3 and elsewhere The parameter R determines both the degree of risk aversion and the degree of curvature of indifference curves implied by the function A riskaverse individual will have a large negative value for R and have sharply curved indifference curves such as U1 shown in Figure 77 A person with more tolerance for risk will have a higher value of R and flatter indifference curves such as U225 Suppose now these individuals are faced with the prospect of losing h dollars of wealth in bad times Such a risk would be acceptable to individual 2 if wealth in good times were to increase from W0 to W2 For the riskaverse individual 1 however wealth would have to increase to W1 to make the risk acceptable Therefore the difference between W1 and W2 Indifference curve U1 represents the preferences of a riskaverse person whereas the person with prefer ences represented by U2 is willing to assume more risk When faced with the risk of losing h in bad times person 2 will require compensation of W2 2 W0 in good times whereas person 1 will require a larger amount given by W1 2 W0 FIGURE 77 Risk Aversion and Risk Premiums Certainty line Wb Wb W Wg Wg W W0 W0 W W0 W0 W h W0 W0 W W2 W1 W2 W U1 U2 U2 U 25Tangency of U1 and U2 at W0 is ensured because the MRS along the certainty line is given by 11 2 π2π regardless of the value of R Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 237 indicates the effect of risk aversion on willingness to assume risk Some of the problems in this chapter make use of this graphic device for showing the connection between prefer ences as reflected by the utility function in Equation 771 and behavior in risky situations 713 ASYMMETRY OF INFORMATION One obvious implication of the study of information acquisition is that the level of infor mation that an individual buys will depend on the perunit price of information messages Unlike the market price for most goods which we usually assume to be the same for every one there are many reasons to believe that information costs may differ significantly among individuals Some individuals may possess specific skills relevant to information acquisition eg they may be trained mechanics whereas others may not possess such skills Some individuals may have other types of experience that yield valuable informa tion whereas others may lack that experience For example the seller of a product will usually know more about its limitations than will a buyer because the seller will know pre cisely how the good was made and where possible problems might arise Similarly large scale repeat buyers of a good may have greater access to information about it than would firsttime buyers Finally some individuals may have invested in some types of informa tion services eg by having a computer link to a brokerage firm or by subscribing to Con sumer Reports that make the marginal cost of obtaining additional information lower than for someone without such an investment All these factors suggest that the level of information will sometimes differ among the participants in market transactions Of course in many instances information costs may be low and such differences may be minor Most people can appraise the quality of fresh vegetables fairly well just by looking at them for example But when information costs are high and variable across individuals we would expect them to find it advantageous to acquire different amounts of information We will postpone a detailed study of such situa tions until Chapter 18 Summary The goal of this chapter was to provide some basic material for the study of individual behavior in uncertain situations The key concepts covered are listed as follows The most common way to model behavior under uncer tainty is to assume that individuals seek to maximize the expected utility of their actions Individuals who exhibit a diminishing marginal utility of wealth are risk averse That is they generally refuse fair bets Riskaverse individuals will wish to insure themselves completely against uncertain events if insurance premi ums are actuarially fair They may be willing to pay more than actuarially fair premiums to avoid taking risks Two utility functions have been extensively used in the study of behavior under uncertainty the constant abso lute risk aversion CARA function and the constant relative risk aversion CRRA function Neither is com pletely satisfactory on theoretical grounds Methods for reducing the risk involved in a situation include transferring risk to those who can bear it more effectively through insurance spreading risk across several activities through diversification preserving options for dealing with the various outcomes that arise and acquiring information to determine which outcomes are more likely One of the most extensively studied issues in the eco nomics of uncertainty is the portfolio problem which asks how an investor will split his or her wealth among available assets A simple version of the problem is used to illustrate the value of diversification in the text the Extensions provide a detailed analysis Information is valuable because it permits individuals to make better decisions in uncertain situations Informa tion can be most valuable when individuals have some flexibility in their decision making The statepreference approach allows decision mak ing under uncertainty to be approached in a familiar choicetheoretic framework Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 238 Part 3 Uncertainty and Strategy Problems 71 George is seen to place an evenmoney 100000 bet on the Bulls to win the NBA Finals If George has a logarith mic utilityofwealth function and if his current wealth is 1000000 what must he believe is the minimum probability that the Bulls will win 72 Show that if an individuals utilityofwealth function is con vex then he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles Do you believe this sort of risktaking behavior is common What factors might tend to limit its occurrence 73 An individual purchases a dozen eggs and must take them home Although making trips home is costless there is a 50 percent chance that all the eggs carried on any one trip will be broken during the trip The individual considers two strat egies 1 take all 12 eggs in one trip or 2 take two trips with six eggs in each trip a List the possible outcomes of each strategy and the prob abilities of these outcomes Show that on average six eggs will remain unbroken after the trip home under either strategy b Develop a graph to show the utility obtainable under each strategy Which strategy will be preferable c Could utility be improved further by taking more than two trips How would this possibility be affected if addi tional trips were costly 74 Suppose there is a 5050 chance that a riskaverse individual with a current wealth of 20000 will contract a debilitating disease and suffer a loss of 10000 a Calculate the cost of actuarially fair insurance in this sit uation and use a utilityofwealth graph such as shown in Figure 71 to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured b Suppose two types of insurance policies were available 1 a fair policy covering the complete loss and 2 a fair policy covering only half of any loss incurred Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the first 75 Ms Fogg is planning an aroundtheworld trip on which she plans to spend 10000 The utility from the trip is a function of how much she actually spends on it 1Y2 given by U1Y2 5 ln Y a If there is a 25 percent probability that Ms Fogg will lose 1000 of her cash on the trip what is the trips expected utility b Suppose that Ms Fogg can buy insurance against los ing the 1000 say by purchasing travelers checks at an actuarially fair premium of 250 Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the 1000 without insurance c What is the maximum amount that Ms Fogg would be willing to pay to insure her 1000 76 In deciding to park in an illegal place any individual knows that the probability of getting a ticket is p and that the fine for receiving the ticket is f Suppose that all individuals are risk averse ie Us 1W2 0 where W is the individuals wealth Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking Hint Use the Taylor series approx imation U1W 2 f 2 5 U1W2 2 fUr 1W2 1 1 f 222Us 1W2 77 In Equation 730 we showed that the amount an individ ual is willing to pay to avoid a fair gamble 1h2 is given by p 5 05E 1h22r1W2 where r1W2 is the measure of absolute risk aversion at this persons initial level of wealth In this problem we look at the size of this payment as a function of the size of the risk faced and this persons level of wealth a Consider a fair gamble 1v2 of winning or losing 1 For this gamble what is E 1v22 b Now consider varying the gamble in part a by multiply ing each prize by a positive constant k Let h 5 kv What is the value of E 1h22 c Suppose this person has a logarithmic utility function U1W2 5 ln W What is a general expression for r1W2 d Compute the risk premium 1p2 for k 5 05 1 and 2 and for W 5 10 and 100 What do you conclude by compar ing the six values Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 239 78 A farmer believes there is a 5050 chance that the next growing season will be abnormally rainy His expected utility function has the form E 3U1Y2 4 5 1 2 ln YNR 1 1 2 ln YR where YNR and YR represent the farmers income in the states of normal rain and rainy respectively a Suppose the farmer must choose between two crops that promise the following income prospects Crop YNR YR Wheat 28000 10000 Corn 19000 15000 Which of the crops will he plant b Suppose the farmer can plant half his field with each crop Would he choose to do so Explain your result c What mix of wheat and corn would provide maximum expected utility to this farmer d Would wheat crop insurancewhich is available to farm ers who grow only wheat and which costs 4000 and pays off 8000 in the event of a rainy growing seasoncause this farmer to change what he plants 79 Maria has 1 she can invest in two assets A and B A dollar invested in A has a 5050 chance of returning 16 or nothing and in B has a 5050 chance of returning 9 or nothing Marias utility over wealth is given by the function U1W2 5 W a Suppose the assets returns are independent 1 Despite the fact that A has a much higher expected return than B show that Maria would prefer to invest half of her money in B rather than investing everything in A 2 Let a be the fraction of the dollar she invests in A What value would Maria choose if she could pick any a between 0 and 1 Hint Write down her expected utility as a function of a and then either graph this function and look for the peak or compute this function over the grid of values a 5 0 01 02 etc b Now suppose the assets returns are perfectly negatively correlated When A has a positive return B returns noth ing and vice versa 1 Show that Maria is better off investing half her money in each asset now than when the assets returns were independent 2 If she can choose how much to invest in each show that she would choose to invest a greater fraction in B than when assets returns were independent 710 Return to Example 75 in which we computed the value of the real option provided by a flexiblefuel car Continue to assume that the payoff from a fossilfuelburning car is O1 1x2 5 1 2 x Now assume that the payoff from the biofuel car is higher O2 1x2 5 2x As before x is a random variable uniformly distrib uted between 0 and 1 capturing the relative availability of biofuels versus fossil fuels on the market over the future lifespan of the car a Assume the buyer is risk neutral with von Neumann Morgenstern utility function U1x2 5 x Compute the option value of a flexiblefuel car that allows the buyer to reproduce the payoff from either singlefuel car b Repeat the option value calculation for a riskaverse buyer with utility function U1x2 5 x c Compare your answers with Example 75 Discuss how the increase in the value of the biofuel car affects the option value provided by the flexiblefuel car Analytical Problems 711 HARA utility The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic abso lute risk aversion HARA functions The general form for this function is U1W2 5 θ 1μ 1 Wγ2 12γ where the various parameters obey the following restrictions γ 1 μ 1 Wγ 0 θ 3 11 2 γ2γ4 0 The reasons for the first two restrictions are obvious the third is required so that Ur 0 a Calculate r1W2 for this function Show that the reciprocal of this expression is linear in W This is the origin of the term harmonic in the functions name b Show that when μ 5 0 and θ 5 3 11 2 γ2γ4 γ21 this function reduces to the CRRA function given in Chapter 7 see footnote 17 c Use your result from part a to show that if γ S q then r1W2 is a constant for this function d Let the constant found in part c be represented by A Show that the implied form for the utility function in this case is the CARA function given in Equation 735 e Finally show that a quadratic utility function can be gener ated from the HARA function simply by setting γ 5 21 f Despite the seeming generality of the HARA func tion it still exhibits several limitations for the study of behavior in uncertain situations Describe some of these shortcomings Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 240 Part 3 Uncertainty and Strategy 712 More on the CRRA function For the CRRA utility function Equation 742 we showed that the degree of risk aversion is measured by 1 2 R In Chapter 3 we showed that the elasticity of substitution for the same func tion is given by 1 11 2 R2 Hence the measures are reciprocals of each other Using this result discuss the following questions a Why is risk aversion related to an individuals willingness to substitute wealth between states of the world What phenomenon is being captured by both concepts b How would you interpret the polar cases R 5 1 and R 5 2q in both the riskaversion and substitution frameworks c A rise in the price of contingent claims in bad times 1pb2 will induce substitution and income effects into the demands for Wg and Wb If the individual has a fixed bud get to devote to these two goods how will choices among them be affected Why might Wg rise or fall depending on the degree of risk aversion exhibited by the individual d Suppose that empirical data suggest an individual requires an average return of 05 percent before being tempted to invest in an investment that has a 5050 chance of gain ing or losing 5 percent That is this person gets the same utility from W0 as from an even bet on 1055 W0 and 0955 W0 1 What value of R is consistent with this behavior 2 How much average return would this person require to accept a 5050 chance of gaining or losing 10 percent Note This part requires solving nonlinear equations so approximate solutions will suffice The comparison of the riskreward tradeoff illustrates what is called the equity premium puzzle in that risky investments seem actually to earn much more than is consistent with the degree of risk aversion suggested by other data See N R Kocherlakota The Equity Premium Its Still a Puzzle Journal of Economic Literature March 1996 4271 713 Graphing risky investments Investment in risky assets can be examined in the statepref erence framework by assuming that W0 dollars invested in an asset with a certain return r will yield W0 11 1 r2 in both states of the world whereas investment in a risky asset will yield W0 1l 1 rg2 in good times and W0 1l 1 rb2 in bad times where rg r rb a Graph the outcomes from the two investments b Show how a mixed portfolio containing both riskfree and risky assets could be illustrated in your graph How would you show the fraction of wealth invested in the risky asset c Show how individuals attitudes toward risk will deter mine the mix of riskfree and risky assets they will hold In what case would a person hold no risky assets d If an individuals utility takes the constant relative risk aversion form Equation 742 explain why this person will not change the fraction of risky assets held as his or her wealth increases26 714 The portfolio problem with a Normally distributed risky asset In Example 73 we showed that a person with a CARA util ity function who faces a Normally distributed risk will have expected utility of the form E 3U1W2 4 5 μW 2 1A22σ2 W where μW is the expected value of wealth and σ2 W is its variance Use this fact to solve for the optimal portfolio allocation for a person with a CARA utility function who must invest k of his or her wealth in a Normally distributed risky asset whose expected return is μr and variance in return is σ2 r your answer should depend on A Explain your results intuitively Behavioral Problem 715 Prospect theory Two pioneers of the field of behavioral economics Dan iel Kahneman winner of the Nobel Prize in economics and author of bestselling book Thinking Fast and Slow and Amos Tversky deceased before the prize was awarded conducted an experiment in which they presented different groups of sub jects with one of the following two scenarios Scenario 1 In addition to 1000 up front the subject must choose between two gambles Gamble A offers an even chance of winning 1000 or nothing Gamble B pro vides 500 with certainty Scenario 2 In addition to 2000 given up front the sub ject must choose between two gambles Gamble C offers an even chance of losing 1000 or nothing Gamble D results in the loss of 500 with certainty a Suppose Standard Stan makes choices under uncer tainty according to expected utility theory If Stan is risk neutral what choice would he make in each scenario b What choice would Stan make if he is risk averse c Kahneman and Tversky found 16 percent of subjects chose A in the first scenario and 68 percent chose C in the second scenario Based on your preceding answers explain why these findings are hard to reconcile with expected utility theory d Kahneman and Tversky proposed an alternative to expected utility theory called prospect theory to explain the experimental results The theory is that 26This problem is based on J E Stiglitz The Effects of Income Wealth and Capital Gains Taxation in Risk Taking Quarterly Journal of Economics May 1969 26383 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 241 peoples current income level functions as an anchor point for them They are risk averse over gains beyond this point but sensitive to small losses below this point This sensitivity to small losses is the opposite of risk aversion A riskaverse person suffers disproportionately more from a large than a small loss 1 Prospect Pete makes choices under uncertainty according to prospect theory What choices would he make in Kahneman and Tverskys experiment Explain 2 Draw a schematic diagram of a utility curve over money for Prospect Pete in the first scenario Draw a utility curve for him in the second scenario Can the same curve suffice for both scenarios or must it shift How do Petes utility curves differ from the ones we are used to drawing for people like Standard Stan Suggestions for Further Reading Arrow K J The Role of Securities in the Optimal Allocation of Risk Bearing Review of Economic Studies 31 1963 9196 Introduces the statepreference concept and interprets securities as claims on contingent commodities Uncertainty and the Welfare Economics of Medical Care American Economic Review 53 1963 94173 Excellent discussion of the welfare implications of insurance Has a clear concise mathematical appendix Should be read in con junction with Paulys article on moral hazard see Chapter 18 Bernoulli D Exposition of a New Theory on the Measure ment of Risk Econometrica 22 1954 2336 Reprint of the classic analysis of the St Petersburg paradox Dixit A K and R S Pindyck Investment under Uncertainty Princeton NJ Princeton University Press 1994 Focuses mainly on the investment decision by firms but has good coverage of option concepts Friedman M and L J Savage The Utility Analysis of Choice Journal of Political Economy 56 1948 279304 Analyzes why individuals may both gamble and buy insurance Very readable Gollier Christian The Economics of Risk and Time Cambridge MA MIT Press 2001 Contains a complete treatment of many of the issues discussed in this chapter Especially good on the relationship between alloca tion under uncertainty and allocation over time Kahneman D Thinking Fast and Slow New York Farrar Straus and Giroux 2011 A bestselling book synthesizing the authors enormous body of pathbreaking research on behavioral psychology and its implica tions for economics Chapter 26 covers Prospect Theory MasColell Andreu Michael D Whinston and Jerry R Green Microeconomic Theory New York Oxford University Press 1995 chap 6 Provides a good summary of the foundations of expected utility theory Also examines the state independence assumption in detail and shows that some notions of risk aversion carry over into cases of state dependence Pratt J W Risk Aversion in the Small and in the Large Econometrica 32 1964 12236 Theoretical development of riskaversion measures Fairly techni cal treatment but readable Rothschild M and J E Stiglitz Increasing Risk 1 A Defini tion Journal of Economic Theory 2 1970 22543 Develops an economic definition of what it means for one gamble to be riskier than another A sequel article in the Journal of Eco nomic Theory provides economic illustrations Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 Chapter 13 provides a nice introduction to the relationship between statistical concepts and expected utility maximization Also shows in detail the integration mentioned in Example 73 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 EXTENSIONS THE PORTFOLIO PROBLEM One of the classic problems in the theory of behavior under uncertainty is the issue of how much of his or her wealth a riskaverse investor should invest in a risky asset Intuitively it seems that the fraction invested in risky assets should be smaller for more riskaverse investors and one goal of our analysis in these Extensions will be to show that formally We will then see how to generalize the model to consider port folios with many such assets finally working up to the Capital Asset Pricing model a staple of financial economics courses E71 Basic model with one risky asset To get started assume that an investor has a certain amount of wealth W0 to invest in one of two assets The first asset yields a certain return of rf whereas the second assets return is a random variable r If we let the amount invested in the risky asset be denoted by k then this persons wealth at the end of one period will be W 5 1W0 2 k2 11 1 rf2 1 k11 1 r2 5 W0 11 1 rf2 1 k1r 2 rf2 i Notice three things about this endofperiod wealth First W is a random variable because its value depends on r Second k can be either positive or negative here depending on whether this person buys the risky asset or sells it short As we shall see how ever in the usual case E 1r 2 rf2 0 and this will imply k 0 Finally notice also that Equation i allows for a solution in which k W0 In this case this investor would leverage his or her investment in the risky asset by borrowing at the riskfree rate rf If we let U1W2 represent this investors utility function then the von NeumannMorgenstern theorem states that he or she will choose k to maximize E 3U1W2 4 The firstorder condition for such a maximum is E 3U1W24 k 5 E 3U1W0 11 1 rf2 1 k1r 2 rf22 4 k 5 E 3Ur 1r 2 rf2 4 5 0 ii In calculating this firstorder condition we can differenti ate through the expected value operator E See Chapter 2 for a discussion of differentiating integrals of which an expected value operator is an example Equation ii involves the expected value of the product of marginal utility and the term r 2 rf Both of these terms are random Whether r 2 rf is positive or negative will depend on how well the risky assets perform over the next period But the return on this risky asset will also affect this investors endofperiod wealth and thus will affect his or her marginal utility If the investment does well W will be large and marginal utility will be relatively low because of diminishing marginal utility If the investment does poorly wealth will be relatively low and marginal utility will be relatively high Hence in the expected value calculation in Equation ii negative outcomes for r 2 rf will be weighted more heavily than positive outcomes to take the utility consequences of these outcomes into account If the expected value in Equation ii were positive a person could increase his or her expected utility by investing more in the risky asset If the expected value were negative he or she could increase expected utility by reducing the amount of the risky asset held Only when the firstorder condition holds will this person have an optimal portfolio Two other conclusions can be drawn from Equation ii First as long as E 1r 2 rf2 0 an investor will choose posi tive amounts of the risky asset To see why notice that meet ing Equation ii will require that fairly large values of Ur be attached to situations where r 2 rf turns out to be negative That can only happen if the investor owns positive amounts of the risky asset so that endofperiod wealth is low in such situations A second conclusion from Equation ii is that investors who are more risk averse will hold smaller amounts of the risky asset Again the reason relates to the shape of the Ur function For riskaverse investors marginal utility rises rapidly as wealth falls Hence they need relatively little exposure to potential negative outcomes from holding the risky asset to satisfy Equation ii E72 CARA utility To make further progress on the portfolio problem requires that we make some specific assumptions about the inves tors utility function Suppose it is given by the CARA form U1W2 5 2e2AW 5 2exp 12AW2 Then the marginal utility function is given by Ur 1W2 5 A exp 12AW2 substituting for endofperiod wealth we have Ur 1W2 5 A exp 32A1W0 11 1 rf2 1 k1r 2 rf2 24 5 A exp 32AW0 11 1 rf2 4exp 32Ak1r 2 rf24 iii EXTENSIONS The PorTfolio Problem 242 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 243 That is the marginal utility function can be separated into a random part and a nonrandom part both initial wealth and the riskfree rate are nonrandom Hence the optimality con dition from Equation ii can be written as E 3Ur 1r 2 rf24 5 A exp 32AW0 11 1 rf2 4 E 3exp 12Ak1r 2 rf2 2 1r 2 rf2 4 5 0 iv Now we can divide by the exponential function of initial wealth leaving an optimality condition that involves only terms in k A and r 2 rf Solving this condition for the optimal level of k can in general be difficult but see Problem 714 Regard less of the specific solution however Equation iv shows that this optimal investment amount will be a constant regardless of the level of initial wealth Hence the CARA function implies that the fraction of wealth that an investor holds in risky assets should decrease as wealth increasesa conclusion that seems precisely contrary to empirical data which tend to show the fraction of wealth held in risky assets increasing with wealth If we instead assumed utility took the CRRA rather than the CARA form we could show with some patience that all individuals with the same risk tolerance will hold the same fraction of wealth in risky assets regardless of their absolute levels of wealth Although this conclusion is slightly more in accord with the facts than is the conclusion from the CARA function it still falls short of explaining why the fraction of wealth held in risky assets tends to increase with wealth E73 Portfolios of many risky assets Additional insight can be gained if the model is generalized to allow for many risky assets Let the return on each of n risky assets be the random variable ri 1i 5 1 n2 The expected values and variances of these assets returns are denoted by E 1ri2 5 μi and Var1ri2 5 σ2 i respectively An investor who invests a portion of his or her wealth in a portfolio of these assets will obtain a random return 1rp2 given by rp 5 a n i51 αiri v where αi 0 is the fraction of the risky portfolio held in asset i and where g n i51αi 5 1 In this situation the expected return on this portfolio will be E 1rp2 5 μp 5 a n i51 αiμi vi If the returns of each asset are independent then the variance of the portfolios return will be Var1rp2 5 σ2 p 5 a n i51 α2 iσ2 i vii If the returns are not independent Equation vii would have to be modified to take covariances among the returns into account Using this general notation we now proceed to look at some aspects of this portfolio allocation problem E74 Optimal portfolios With many risky assets the optimal portfolio problem can be divided into two steps The first step is to consider portfolios of just the risky assets The second step is to add in the risk less one To solve for the optimal portfolio of just the risky assets one can proceed as in the text where in the section on diver sification we looked at the optimal investment weights across just two risky assets Here we will choose a general set of asset weightings the αi to minimize the variance or standard deviation of the portfolio for each potential expected return The solution to this problem yields an efficiency frontier for risky asset portfolios such as that represented by the line EE in Figure E71 Portfolios that lie below this frontier are infe rior to those on the frontier because they offer lower expected returns for any degree of risk Portfolio returns above the frontier are unattainable Sharpe 1970 discusses the mathe matics associated with constructing the EE frontier Now add a riskfree asset with expected return μf and σf 5 0 shown as point R in Figure E71 Optimal portfolios will now consist of mixtures of this asset with risky ones All such port folios will lie along the line RP in the figure because this shows the maximum return attainable for each value of σ for various portfolio allocations These allocations will contain only one spe cific set of risky assets the set represented by point M In equilib rium this will be the market portfolio consisting of all capital assets held in proportion to their market valuations This market portfolio will provide an expected return of μM and a standard deviation of that return of σM The equation for the line RP that represents any mixed portfolio is given by the linear equation μp 5 μf 1 μM 2 μf σM σp viii This shows that the market line RP permits individual investors to purchase returns in excess of the riskfree return 1μM 2 μf2 by taking on proportionally more risk 1σPσM2 For choices on RP to the left of the market point M σPσM 1 and μf μP μM Highrisk points to the right of Mwhich can be obtained by borrowing to produce a leveraged portfoliowill have σPσM 1 and will promise an expected return in excess of what is provided by the market portfolio 1μP μM2 Tobin 1958 was one of the first economists to recognize the role that riskfree assets play in identifying the market portfolio and in setting the terms on which investors can obtain returns above riskfree levels E75 Individual choices Figure E72 illustrates the portfolio choices of various investors facing the options offered by the line RP This figure illustrates the type of portfolio choice model previously described in this chapter Individuals with low tolerance for risk I will opt for portfolios that are heavily weighted toward the riskfree asset Investors willing to assume a modest degree of risk II Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 244 Part 3 Uncertainty and Strategy The frontier EE represents optimal mixtures of risky assets that minimize the standard deviation of the portfolio σP for each expected return μP A riskfree asset with return μf offers investors the opportu nity to hold mixed portfolios along RP that mix this riskfree asset with the market portfolio M E E R P M P μ μ μ σ σ M M P f FIGURE E71 Efficient Portfolios Given the market options RP investors can choose how much risk they wish to assume Very riskaverse investors 1UI2 will hold mainly riskfree assets whereas risk takers 1UIII2 will opt for leveraged portfolios M P UII UIII UI R P f P μ μ σ FIGURE E72 Investor Behavior and Risk Aversion Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 7 Uncertainty 245 will opt for portfolios close to the market portfolio Highrisk investors III may opt for leveraged portfolios Notice that all investors face the same price of risk 1μM 2 μf2 with their expected returns being determined by how much relative risk 1σPσM2 they are willing to incur Notice also that the risk associated with an investors portfolio depends only on the fraction of the portfolio invested in the market portfolio 1α2 because σ2 P 5 α2σ2 M 1 11 2 α2 2 0 Hence σPσM 5 α and so the investors choice of portfolio is equivalent to his or her choice of risk Mutual funds The notion of portfolio efficiency has been widely applied to the study of mutual funds In general mutual funds are a good answer to small investors diversification needs Because such funds pool the funds of many individuals they are able to achieve economies of scale in transactions and management costs This permits fund owners to share in the fortunes of a much wider variety of equities than would be possible if each acted alone But mutual fund managers have incentives of their own therefore the portfolios they hold may not always be perfect representations of the risk attitudes of their clients For example Scharfstein and Stein 1990 developed a model that shows why mutual fund managers have incentives to fol low the herd in their investment picks Other studies such as the classic investigation by Jensen 1968 find that mutual fund managers are seldom able to attain extra returns large enough to offset the expenses they charge investors In recent years this has led many mutual fund buyers to favor index funds that seek simply to duplicate the market average as represented say by the Standard and Poors 500 stock index Such funds have low expenses and therefore permit investors to achieve diversification at minimal cost E76 Capital asset pricing model Although the analysis of E75 shows how a portfolio that mixes a riskfree asset with the market portfolio will be priced it does not describe the riskreturn tradeoff for a single asset Because assuming transactions are costless an investor can always avoid risk unrelated to the overall market by choosing to diversify with a market portfolio such unsystematic risk will not warrant any excess return An asset will however earn an excess return to the extent that it contributes to over all market risk An asset that does not yield such extra returns would not be held in the market portfolio so it would not be held at all This is the fundamental insight of the capital asset pricing model CAPM To examine these results formally consider a portfolio that combines a small amount α of an asset with a random return of x with the market portfolio which has a random return of M The return on this portfolio z would be given by z 5 αx 1 11 2 α2M ix The expected return is μz 5 αμx 1 11 2 α2μM x with variance σ2 z 5 α2σ2 x 1 11 2 α2 2σ2 M 1 2α 11 2 α2σxM xi where σx M is the covariance between the return on x and the return on the market But our previous analysis shows μz 5 μf 1 1μM 2 μf2 σz σM xii Setting Equation x equal to xii and differentiating with respect to α yields μz α 5 μx 2 μM 5 μM 2 μf σM σz α xiii By calculating σzα from Equation xi and taking the limit as α approaches zero we get μx 2 μM 5 μM 2 μf σM a σxM 2 σ2 M σM b xiv or rearranging terms μx 5 μf 1 1μM 2 μf2 σxM σ2 M xv Again risk has a reward of μM 2 μf but now the quantity of risk is measured by σxMσ2 M This ratio of the covariance between the return x and the market to the variance of the market return is referred to as the beta coefficient for the asset Estimated beta coefficients for financial assets are reported in many publications Studies of the CAPM This version of the CAPM carries strong implications about the determinants of any assets expected rate of return Because of this simplicity the model has been subject to a large number of empirical tests In general these find that the models measure of systemic risk beta is indeed correlated with expected returns whereas simpler measures of risk eg the standard deviation of past returns are not Perhaps the most influential early empirical test that reached such a conclusion was that of Fama and MacBeth 1973 But the CAPM itself explains only a small fraction of differences in the returns of various assets And contrary to the CAPM a number of authors have found that many other economic factors significantly affect expected returns Indeed a prom inent challenge to the CAPM comes from one of its original founderssee Fama and French 1992 References Fama E F and K R French The Cross Section of Expected Stock Returns Journal of Finance 47 1992 42766 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 246 Part 3 Uncertainty and Strategy Fama E F and J MacBeth Risk Return and Equilibrium Journal of Political Economy 8 1973 60736 Jensen M The Performance of Mutual Funds in the Period 19451964 Journal of Finance May 1968 386416 Scharfstein D S and J Stein Herd Behavior and Invest ment American Economic Review June 1990 46589 Sharpe W F Portfolio Theory and Capital Markets New York McGrawHill 1970 Tobin J Liquidity Preference as Behavior towards Risk Review of Economic Studies February 1958 6586 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 247 CHAPTER eight Game Theory This chapter provides an introduction to noncooperative game theory a tool used to understand the strategic interactions among two or more agents The range of applica tions of game theory has been growing constantly including all areas of economics from labor economics to macroeconomics and other fields such as political science and biol ogy Game theory is particularly useful in understanding the interaction between firms in an oligopoly so the concepts learned here will be used extensively in Chapter 15 We begin with the central concept of Nash equilibrium and study its application in simple games We then go on to study refinements of Nash equilibrium that are used in games with more complicated timing and information structures 81 BASIC CONCEPTS Thus far in Part 3 of this text we have studied individual decisions made in isolation In this chapter we study decision making in a more complicated strategic setting In a strategic setting a person may no longer have an obvious choice that is best for him or her What is best for one decisionmaker may depend on what the other is doing and vice versa For example consider the strategic interaction between drivers and the police Whether drivers prefer to speed may depend on whether the police set up speed traps Whether the police find speed traps valuable depends on how much drivers speed This confusing circularity would seem to make it difficult to make much headway in analyzing strategic behavior In fact the tools of game theory will allow us to push the analysis nearly as far for example as our analysis of consumer utility maximization in Chapter 4 There are two major tasks involved when using game theory to analyze an economic situation The first is to distill the situation into a simple game Because the analysis involved in strategic settings quickly grows more complicated than in simple decision problems it is important to simplify the setting as much as possible by retaining only a few essential elements There is a certain art to distilling games from situations that is hard to teach The examples in the text and problems in this chapter can serve as models that may help in approaching new situations The second task is to solve the given game which results in a prediction about what will happen To solve a game one takes an equilibrium concept eg Nash equilibrium and runs through the calculations required to apply it to the given game Much of the chapter will be devoted to learning the most widely used equilibrium concepts and to practicing the calculations necessary to apply them to particular games A game is an abstract model of a strategic situation Even the most basic games have three essential elements players strategies and payoffs In complicated settings it is sometimes also necessary to specify additional elements such as the sequence of moves Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 248 Part 3 Uncertainty and Strategy and the information that players have when they move who knows what when to describe the game fully 811 Players Each decisionmaker in a game is called a player These players may be individuals as in poker games firms as in markets with few firms or entire nations as in military con flicts A player is characterized as having the ability to choose from among a set of possible actions Usually the number of players is fixed throughout the play of the game Games are sometimes characterized by the number of players involved twoplayer threeplayer or nplayer games As does much of the economic literature this chapter often focuses on twoplayer games because this is the simplest strategic setting We will label the players with numbers thus in a twoplayer game we will have players 1 and 2 In an nplayer game we will have players 1 2 n with the generic player labeled i 812 Strategies Each course of action open to a player during the game is called a strategy Depending on the game being examined a strategy may be a simple action drive over the speed limit or not or a complex plan of action that may be contingent on earlier play in the game say speeding only if the driver has observed speed traps less than a quarter of the time in past drives Many aspects of game theory can be illustrated in games in which players choose between just two possible actions Let S1 denote the set of strategies open to player 1 S2 the set open to player 2 and more generally Si the set open to player i Let s1 S1 be a particular strategy chosen by player 1 from the set of possibilities s2 S2 the particular strategy chosen by player 2 and si Si for player i A strategy profile will refer to a listing of particular strategies chosen by each of a group of players 813 Payoffs The final return to each player at the conclusion of a game is called a payoff Payoffs are measured in levels of utility obtained by the players For simplicity monetary payoffs say profits for firms are often used More generally payoffs can incorporate nonmonetary fac tors such as prestige emotion risk preferences and so forth In a twoplayer game U1 1s1 s22 denotes player 1s payoff given that he or she chooses s1 and the other player chooses s2 and similarly U2 1s2 s12 denotes player 2s payoff1 The fact that player 1s payoff may depend on player 2s strategy and vice versa is where the strategic interdependence shows up In an nplayer game we can write the payoff of a generic player i as Ui 1si s2i2 which depends on player is own strategy si and the profile s2i 5 1s1 si21 si11 sn2 of the strategies of all players other than i 82 PRISONERS DILEMMA The Prisoners Dilemma introduced by A W Tucker in the 1940s is one of the most famous games studied in game theory and will serve here as a nice example to illustrate all the nota tion just introduced The title stems from the following situation Two suspects are arrested for a crime The district attorney has little evidence in the case and is eager to extract a con fession She separates the suspects and tells each If you fink on your companion but your companion doesnt fink on you I can promise you a reduced 1year sentence whereas 1Technically these are the von NeumannMorgenstern utility functions from the previous chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 249 your companion will get 4 years If you both fink on each other you will each get a 3year sentence Each suspect also knows that if neither of them finks then the lack of evidence will result in being tried for a lesser crime for which the punishment is a 2year sentence Boiled down to its essence the Prisoners Dilemma has two strategic players the sus pects labeled 1 and 2 There is also a district attorney but because her actions have already been fully specified there is no reason to complicate the game and include her in the specification Each player has two possible strategies open to him fink or remain silent Therefore we write their strategy sets as S1 5 S2 5 5fink silent6 To avoid negative numbers we will specify payoffs as the years of freedom over the next 4 years For example if suspect 1 finks and suspect 2 does not suspect 1 will enjoy 3 years of freedom and sus pect 2 none that is U1 1fink silent2 5 3 and U2 1silent fink2 5 0 821 Normal form The Prisoners Dilemma and games like it can be summarized by the matrix shown in Figure 81 called the normal form of the game Each of the four boxes represents a different combination of strategies and shows the players payoffs for that combination The usual convention is to have player 1s strategies in the row headings and player 2s in the column headings and to list the payoffs in order of player 1 then player 2 in each box 822 Thinking strategically about the Prisoners Dilemma Although we have not discussed how to solve games yet it is worth thinking about what we might predict will happen in the Prisoners Dilemma Studying Figure 81 on first thought one might predict that both will be silent This gives the most total years of freedom for both four compared with any other outcome Thinking a bit deeper this may not be the best prediction in the game Imagine ourselves in player 1s position for a moment We do not know what player 2 will do yet because we have not solved out the game so lets investigate each possibility Suppose player 2 chose to fink By finking ourselves we would earn 1 year of freedom versus none if we remained silent so finking is better for us Suppose player 2 chose to remain silent Finking is still better for us than remaining silent because we get three rather than 2 years of freedom Regardless of what the other player does finking is better for us than being silent because it results in an extra year of free dom Because players are symmetric the same reasoning holds if we imagine ourselves in Two suspects simultaneously choose to fink or be silent Rows refer to player 1s actions and columns refer to player 2s Each box is an outcome the first entry is player 1s payoff and the second is player 2s in that outcome Suspect 2 Fink Silent Suspect 1 Fink U1 5 1 U2 5 1 U1 5 0 U2 5 3 U1 5 3 U2 5 0 U1 5 2 U2 5 2 Silent FIGURE 81 Normal Form for the Prisoners Dilemma Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 250 Part 3 Uncertainty and Strategy player 2s position Therefore the best prediction in the Prisoners Dilemma is that both will fink When we formally introduce the main solution conceptNash equilibrium we will indeed find that both finking is a Nash equilibrium The prediction has a paradoxical property By both finking the suspects only enjoy 1 year of freedom but if they were both silent they would both do better enjoying 2 years of freedom The paradox should not be taken to imply that players are stupid or that our pre diction is wrong Rather it reveals a central insight from game theory that pitting players against each other in strategic situations sometimes leads to outcomes that are inefficient for the players2 The suspects might try to avoid the extra prison time by coming to an agreement beforehand to remain silent perhaps reinforced by threats to retaliate afterward if one or the other finks Introducing agreements and threats leads to a game that differs from the basic Prisoners Dilemma a game that should be analyzed on its own terms using the tools we will develop shortly Solving the Prisoners Dilemma was easy because there were only two players and two strategies and because the strategic calculations involved were fairly straightforward It would be useful to have a systematic way of solving this as well as more complicated games Nash equilibrium provides us with such a systematic solution 83 NASH EQUILIBRIUM In the economic theory of markets the concept of equilibrium is developed to indicate a situation in which both suppliers and demanders are content with the market outcome Given the equilibrium price and quantity no market participant has an incentive to change his or her behavior In the strategic setting of game theory we will adopt a related notion of equilibrium formalized by John Nash in the 1950s called Nash equilibrium3 Nash equi librium involves strategic choices that once made provide no incentives for the players to alter their behavior further A Nash equilibrium is a strategy for each player that is the best choice for each player given the others equilibrium strategies The next several sections provide a formal definition of Nash equilibrium apply the concept to the Prisoners Dilemma and then demonstrate a shortcut involving under lining payoffs for picking Nash equilibria out of the normal form As at other points in the chapter the reader who wants to avoid wading through a lot of math can skip over the notation and definitions and jump right to the applications without losing too much of the basic insight behind game theory 831 A formal definition Nash equilibrium can be defined simply in terms of best responses In an nplayer game strategy si is a best response to rivals strategies s2i if player i cannot obtain a strictly higher payoff with any other possible strategy sri Si given that rivals are playing s2i 2When we say the outcome is inefficient we are focusing just on the suspects utilities if the focus were shifted to society at large then both finking might be a good outcome for the criminal justice systempresumably the motivation behind the district attorneys offer 3John Nash Equilibrium Points in nPerson Games Proceedings of the National Academy of Sciences 36 1950 4849 Nash is the principal figure in the 2001 film A Beautiful Mind see Problem 85 for a gametheory example from the film and cowinner of the 1994 Nobel Prize in economics D E F I N I T I O N Best response si is a best response for player i to rivals strategies s2i denoted si BRi 1s2i2 if Ui 1si s2i2 Ui 1sri s2i2 for all sri Si 81 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 251 A technicality embedded in the definition is that there may be a set of best responses rather than a unique one that is why we used the set inclusion notation si BRi 1s2i2 There may be a tie for the best response in which case the set BRi 1s2i2 will contain more than one element If there is not a tie then there will be a single best response si and we can simply write si 5 BRi 1s2i2 We can now define a Nash equilibrium in an nplayer game as follows D E F I N I T I O N Nash equilibrium A Nash equilibrium is a strategy profile 1 s 1 s 2 s n2 such that for each player i 5 1 2 n s i is a best response to the other players equilibrium strategies s 2i That is s i BRi 1s 2i These definitions involve a lot of notation The notation is a bit simpler in a twoplayer game In a twoplayer game 1s 1 s 22 is a Nash equilibrium if s 1 and s 2 are mutual best responses against each other U1 1s 1 s 22 U1 1s1 s 22 for all s1 S1 82 and U2 1s 1 s 22 U2 1s2 s 12 for all s2 S2 83 A Nash equilibrium is stable in that even if all players revealed their strategies to each other no player would have an incentive to deviate from his or her equilibrium strategy and choose something else Nonequilibrium strategies are not stable in this way If an outcome is not a Nash equilibrium then at least one player must benefit from deviating Hyperrational players could be expected to solve the inference problem and deduce that all would play a Nash equilibrium especially if there is a unique Nash equilibrium Even if players are not hyperrational over the long run we can expect their play to converge to a Nash equilibrium as they abandon strategies that are not mutual best responses Besides this stability property another reason Nash equilibrium is used so widely in eco nomics is that it is guaranteed to exist for all games we will study allowing for mixed strategies to be defined below Nash equilibria in pure strategies do not have to exist The mathematics behind this existence result are discussed at length in the Extensions to this chapter Nash equilibrium has some drawbacks There may be multiple Nash equilibria making it hard to come up with a unique prediction Also the definition of Nash equilibrium leaves unclear how a player can choose a bestresponse strategy before knowing how rivals will play 832 Nash equilibrium in the Prisoners Dilemma Lets apply the concepts of best response and Nash equilibrium to the example of the Prison ers Dilemma Our educated guess was that both players will end up finking We will show that both finking is a Nash equilibrium of the game To do this we need to show that fink ing is a best response to the other players finking Refer to the payoff matrix in Figure 81 If player 2 finks we are in the first column of the matrix If player 1 also finks his payoff is 1 if he is silent his payoff is 0 Because he earns the most from finking given player 2 finks finking is player 1s best response to player 2s finking Because players are symmetric the same logic implies that player 2s finking is a best response to player 1s finking Therefore both finking is indeed a Nash equilibrium We can show more that both finking is the only Nash equilibrium To do so we need to rule out the other three outcomes Consider the outcome in which player 1 finks and player 2 is silent abbreviated fink silent the upper right corner of the matrix This is not a Nash equilibrium Given that player 1 finks as we have already said player 2s best response is Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 252 Part 3 Uncertainty and Strategy to fink not to be silent Symmetrically the outcome in which player 1 is silent and player 2 finks in the lower left corner of the matrix is not a Nash equilibrium That leaves the out come in which both are silent Given that player 2 is silent we focus our attention on the second column of the matrix The two rows in that column show that player 1s payoff is 2 from being silent and 3 from finking Therefore silent is not a best response to fink thus both being silent cannot be a Nash equilibrium To rule out a Nash equilibrium it is enough to find just one player who is not playing a best response and thus would want to deviate to some other strategy Considering the outcome fink silent although player 1 would not deviate from this outcome he earns 3 which is the most possible player 2 would prefer to deviate from silent to fink Symmet rically considering the outcome silent fink although player 2 does not want to deviate player 1 prefers to deviate from silent to fink so this is not a Nash equilibrium Consid ering the outcome silent silent both players prefer to deviate to another strategy more than enough to rule out this outcome as a Nash equilibrium 833 Underlining bestresponse payoffs A quick way to find the Nash equilibria of a game is to underline bestresponse payoffs in the matrix The underlining procedure is demonstrated for the Prisoners Dilemma in Figure 82 The first step is to underline the payoffs corresponding to player 1s best responses Player 1s best response is to fink if player 2 finks so we underline U1 5 1 in the upper left box and to fink if player 2 is silent so we underline U1 5 3 in the upper left box Next we move to underlining the payoffs corresponding to player 2s best responses Player 2s best response is to fink if player 1 finks so we underline U2 5 1 in the upper left box and to fink if player 1 is silent so we underline U2 5 3 in the lower left box Now that the bestresponse payoffs have been underlined we look for boxes in which every players payoff is underlined These boxes correspond to Nash equilibria There may be additional Nash equilibria involving mixed strategies defined later in the chapter In Figure 82 only in the upper left box are both payoffs underlined verifying that fink finkand none of the other outcomesis a Nash equilibrium The first step is to underline player 1s best responses Player 1 prefers to fink if 2 finks so we underline U1 5 1 in the upper left box Player 1 prefers to fink if 2 is silent so we underline U1 5 3 in the upper right box The next step is to underline player 2s best responses Player 2 prefers to fink if 1 finks so we underline U2 5 1 in the upper left box Player 2 prefers to fink if 1 is silent so we underline U2 5 3 in the lower left box The final step is to circle any box with both payoffs underlined here showing the Nash equilibrium involves both finking Suspect 2 Fink Silent Suspect 1 Fink U1 5 0 U2 5 3 U1 5 3 U2 5 0 U1 5 2 U2 5 2 Silent U1 5 1 U2 5 1 FIGURE 82 Underlining Procedure in the Prisoners Dilemma Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 253 834 Dominant strategies Fink fink is a Nash equilibrium in the Prisoners Dilemma because finking is a best response to the other players finking We can say more Finking is the best response to all the other players strategies fink and silent This can be seen among other ways from the underlining procedure shown in Figure 82 All player 1s payoffs are underlined in the row in which he plays fink and all player 2s payoffs are underlined in the column in which he plays fink A strategy that is a best response to any strategy the other players might choose is called a dominant strategy Players do not always have dominant strategies but when they do there is strong reason to believe they will play that way Complicated strategic consider ations do not matter when a player has a dominant strategy because what is best for that player is independent of what others are doing D E F I N I T I O N Dominant strategy A dominant strategy is a strategy s i for player i that is a best response to all strategy profiles of other players That is s i BRi 1s2i2 for all s2i Note the difference between a Nash equilibrium strategy and a dominant strategy A strat egy that is part of a Nash equilibrium need only be a best response to one strategy profile of other playersnamely their equilibrium strategies A dominant strategy must be a best response not just to the Nash equilibrium strategies of other players but to all the strategies of those players If all players in a game have a dominant strategy then we say the game has a dominant strategy equilibrium As well as being the Nash equilibrium of the Prisoners Dilemma fink fink is a dominant strategy equilibrium It is generally true for all games that a dominant strategy equilibrium if it exists is also a Nash equilibrium and is the unique such equilibrium 835 Battle of the Sexes The famous Battle of the Sexes game is another example that illustrates the concepts of best response and Nash equilibrium The story goes that a wife player 1 and husband player 2 would like to meet each other for an evening out They can go either to the ballet or to a boxing match Both prefer to spend time together than apart Conditional on being together the wife prefers to go to the ballet and the husband to the boxing match The nor mal form of the game is presented in Figure 83 For brevity we dispense with the u1 and u2 The wife and husband simultaneously choose which event ballet or boxing to show up to They obtain 0 payoff if they do not end up coordinating Player 2 Husband Ballet Boxing Player 1 Wife Ballet 2 1 0 0 0 0 1 2 Boxing FIGURE 83 Normal Form for the Battle of the Sexes Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 254 Part 3 Uncertainty and Strategy labels on the payoffs and simply reemphasize the convention that the first payoff is player 1s and the second is player 2s We will examine the four boxes in Figure 83 and determine which are Nash equilib ria and which are not Start with the outcome in which both players choose ballet writ ten ballet ballet the upper left corner of the payoff matrix Given that the husband plays ballet the wifes best response is to play ballet this gives her highest payoff in the matrix of 2 Using notation ballet 5 BR1 1ballet2 We do not need the fancy setinclu sion symbol as in ballet BR1 1ballet2 because the husband has only one best response to the wifes choosing ballet Given that the wife plays ballet the husbands best response is to play ballet If he deviated to boxing then he would earn 0 rather than 1 because they would end up not coordinating Using notation ballet 5 BR2 1ballet2 Thus ballet ballet is indeed a Nash equilibrium Symmetrically boxing boxing is a Nash equilibrium Consider the outcome ballet boxing in the upper left corner of the matrix Given the husband chooses boxing the wife earns 0 from choosing ballet but 1 from choosing box ing therefore ballet is not a best response for the wife to the husbands choosing boxing In notation ballet o BR1 1boxing2 Hence ballet boxing cannot be a Nash equilibrium The husbands strategy of boxing is not a best response to the wifes playing ballet either thus both players would prefer to deviate from ballet boxing although we only need to find one player who would want to deviate to rule out an outcome as a Nash equilibrium Symmetrically boxing ballet is not a Nash equilibrium either The Battle of the Sexes is an example of a game with more than one Nash equilibrium in fact it has threea third in mixed strategies as we will see It is hard to say which of the two we have found thus far is more plausible because they are symmetric Therefore it is difficult to make a firm prediction in this game The Battle of the Sexes is also an exam ple of a game with no dominant strategies A player prefers to play ballet if the other plays ballet and boxing if the other plays boxing Figure 84 applies the underlining procedure used to find Nash equilibria quickly to the Battle of the Sexes The procedure verifies that the two outcomes in which the players succeed in coordinating are Nash equilibria and the two outcomes in which they do not coordinate are not Example 81 provides additional practice in finding Nash equilibria in a more compli cated setting with three strategies for each player The underlining procedure yields two Nash equilibria in pure strategies both go to ballet and both go to boxing Player 2 Husband Ballet Boxing Player 1 Wife Ballet 2 1 0 0 0 0 1 2 Boxing FIGURE 84 Underlining Procedure in the Battle of the Sexes Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 255 EXAMPLE 81 Rock Paper Scissors Rock Paper Scissors is a childrens game in which the two players simultaneously display one of three hand symbols Figure 85 presents the normal form The zero payoffs along the diagonal show that if players adopt the same strategy then no payments are made In other cases the pay offs indicate a 1 payment from loser to winner under the usual hierarchy rock breaks scissors scissors cut paper paper covers rock As anyone who has played this game knows and as the underlining procedure reveals none of the nine boxes represents a Nash equilibrium Any strategy pair is unstable because it offers at least one of the players an incentive to deviate For example scissors scissors provides an incentive for either player 1 or 2 to choose rock paper rock provides an incentive for player 2 to choose scissors Rock Paper Scissors involves three strategies for each player The underlining procedure shows that it has no Nash equilibrium in pure strategies Player 2 Rock Paper Player 1 Rock 0 0 1 1 1 1 0 0 Paper Scissors 1 1 1 1 1 1 1 1 Scissors 0 0 Player 2 Rock Paper Player 1 Rock 0 0 1 1 1 1 0 0 Paper Scissors 1 1 1 1 1 1 1 1 Scissors 0 0 a Normal form b Underlining procedure FIGURE 85 Rock Paper Scissors The game does have a Nash equilibriumnot any of the nine boxes in the figure but in mixed strategies defined in the next section QUERY Does any player have a dominant strategy Why is paper scissors not a Nash equilibrium Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 256 Part 3 Uncertainty and Strategy 84 MIxED STRATEGIES Players strategies can be more complicated than simply choosing an action with certainty In this section we study mixed strategies which have the player randomly select from sev eral possible actions By contrast the strategies considered in the examples thus far have a player choose one action or another with certainty these are called pure strategies For example in the Battle of the Sexes we have considered the pure strategies of choosing either ballet or boxing for sure A possible mixed strategy in this game would be to flip a coin and then attend the ballet if and only if the coin comes up heads yielding a 5050 chance of showing up at either event Although at first glance it may seem bizarre to have players flipping coins to deter mine how they will play there are good reasons for studying mixed strategies First some games such as Rock Paper Scissors have no Nash equilibria in pure strategies As we will see in the section on existence such games will always have a Nash equilib rium in mixed strategies therefore allowing for mixed strategies will enable us to make predictions in such games where it is impossible to do so otherwise Second strategies involving randomization are natural and familiar in certain settings Students are famil iar with the setting of class exams Class time is usually too limited for the professor to examine students on every topic taught in class but it may be sufficient to test students on a subset of topics to induce them to study all the material If students knew which topics were on the test then they might be inclined to study only those and not the others therefore the professor must choose the topics at random to get the students to study everything Random strategies are also familiar in sports the same soccer player sometimes shoots to the right of the net and sometimes to the left on penalty kicks and in card games the poker player sometimes folds and sometimes bluffs with a similarly poor hand at different times4 841 Formal definitions To be more formal suppose that player i has a set of M possible actions Ai 5 5a1 i am i aM i 6 where the subscript refers to the player and the superscript to the different choices A mixed strategy is a probability distribution over the M actions si 5 1σ1 i σm i σM i 2 where σm i is a number between 0 and 1 that indicates the probability of player i playing action am i The probabilities in si must sum to unity σ1 i 1 c1 σm i 1 c1 σM i 5 1 In the Battle of the Sexes for example both players have the same two actions of ballet and boxing so we can write A1 5 A2 5 5ballet boxing6 We can write a mixed strategy as a pair of probabilities 1σ 1 2 σ2 where σ is the probability that the player chooses ballet The probabilities must sum to unity and so with two actions once the probability of one action is specified the probability of the other is determined Mixed strategy 13 23 means that the player plays ballet with probability 13 and boxing with probability 23 12 12 means that the player is equally likely to play ballet or boxing 1 0 means that the player chooses ballet with certainty and 0 1 means that the player chooses boxing with certainty 4A third reason is that mixed strategies can be purified by specifying a more complicated game in which one or the other action is better for the player for privately known reasons and where that action is played with certainty For example a history professor might decide to ask an exam question about World War I because unbeknownst to the students she recently read an interesting journal article about it See John Harsanyi Games with Randomly Disturbed Payoffs A New Rationale for Mixed Strategy Equilibrium Points International Journal of Game Theory 2 1973 123 Harsanyi was a cowinner along with Nash of the 1994 Nobel Prize in economics Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 257 In our terminology a mixed strategy is a general category that includes the special case of a pure strategy A pure strategy is the special case in which only one action is played with positive probability Mixed strategies that involve two or more actions being played with positive probability are called strictly mixed strategies Returning to the examples from the previous paragraph of mixed strategies in the Battle of the Sexes all four strategies 13 23 12 12 1 0 and 0 1 are mixed strategies The first two are strictly mixed and the second two are pure strategies With this notation for actions and mixed strategies behind us we do not need new defi nitions for best response Nash equilibrium and dominant strategy The definitions intro duced when si was taken to be a pure strategy also apply to the case in which si is taken to be a mixed strategy The only change is that the payoff function Ui 1si s2i2 rather than being a certain payoff must be reinterpreted as the expected value of a random payoff with probabilities given by the strategies si and s2i Example 82 provides some practice in computing expected payoffs in the Battle of the Sexes EXAMPLE 82 Expected Payoffs in the Battle of the Sexes Lets compute players expected payoffs if the wife chooses the mixed strategy 19 89 and the husband 45 15 in the Battle of the Sexes The wifes expected payoff is U1aa1 9 8 9b a4 5 1 5bb 5 a1 9ba4 5bU1 1ballet ballet2 1 a1 9ba1 5bU1 1ballet boxing2 1 a8 9ba4 5bU1 1boxing ballet2 1 a8 9ba1 5bU1 1boxing boxing2 84 5 a1 9ba4 5b 122 1 a1 9ba1 5b 102 1 a8 9ba4 5b 102 1 a8 9ba1 5b 112 5 16 45 To understand Equation 84 it is helpful to review the concept of expected value from Chapter 2 The expected value of a random variable equals the sum over all outcomes of the probability of the outcome multiplied by the value of the random variable in that outcome In the Battle of the Sexes there are four outcomes corresponding to the four boxes in Figure 83 Because play ers randomize independently the probability of reaching a particular box equals the product of the probabilities that each player plays the strategy leading to that box Thus for example the probability boxing balletthat is the wife plays boxing and the husband plays balletequals 1892 3 1452 The probabilities of the four outcomes are multiplied by the value of the relevant random variable in this case players 1s payoff in each outcome Next we compute the wifes expected payoff if she plays the pure strategy of going to ballet the same as the mixed strategy 1 0 and the husband continues to play the mixed strategy 45 15 Now there are only two relevant outcomes given by the two boxes in the row in which the wife plays ballet The probabilities of the two outcomes are given by the probabilities in the husbands mixed strategy Therefore U1aballet a4 5 1 5bb 5 a4 5bU1 1ballet ballet2 1 a1 5bU1 1ballet boxing2 5 a4 5b 122 1 a1 5b 102 5 8 5 85 Finally we will compute the general expression for the wifes expected payoff when she plays mixed strategy 1w 1 2 w2 and the husband plays 1h 1 2 h2 If the wife plays ballet with proba bility w and the husband with probability h then Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 258 Part 3 Uncertainty and Strategy U1 1 1w 1 2 w2 1h 1 2 h2 2 5 1w2 1h2U1 1ballet ballet2 1 1w2 11 2 h2U1 1ballet boxing2 1 11 2 w2 1h2U1 1boxing ballet2 1 11 2 w2 11 2 h2U1 1boxing boxing2 5 1w2 1h2 122 1 1w2 11 2 h2 102 1 11 2 w2 1h2 102 1 11 2 w2 11 2 h2 112 5 1 2 h 2 w 1 3hw 86 QUERY What is the husbands expected payoff in each case Show that his expected payoff is 2 2 2h 2 2w 1 3hw in the general case Given the husband plays the mixed strategy 45 15 what strategy provides the wife with the highest payoff 842 Computing mixedstrategy equilibria Computing Nash equilibria of a game when strictly mixed strategies are involved is a bit more complicated than when pure strategies are involved Before wading in we can save a lot of work by asking whether the game even has a Nash equilibrium in strictly mixed strategies If it does not having found all the purestrategy Nash equilibria then one has finished analyzing the game The key to guessing whether a game has a Nash equilibrium in strictly mixed strategies is the surprising result that almost all games have an odd num ber of Nash equilibria5 Lets apply this insight to some of the examples considered thus far We found an odd number one of purestrategy Nash equilibria in the Prisoners Dilemma suggesting we need not look further for one in strictly mixed strategies In the Battle of the Sexes we found an even number two of purestrategy Nash equilibria suggesting the existence of a third one in strictly mixed strategies Example 81Rock Paper Scissorshas no purestrategy Nash equilibria To arrive at an odd number of Nash equilibria we would expect to find one Nash equilibrium in strictly mixed strategies 5John Harsanyi Oddness of the Number of Equilibrium Points A New Proof International Journal of Game Theory 2 1973 23550 Games in which there are ties between payoffs may have an even or infinite number of Nash equilibria EXAMPLE 83 MixedStrategy Nash Equilibrium in the Battle of the Sexes A general mixed strategy for the wife in the Battle of the Sexes is 1w 1 2 w2 and for the hus band is 1h 1 2 h2 where w and h are the probabilities of playing ballet for the wife and husband respectively We will compute values of w and h that make up Nash equilibria Both players have a continuum of possible strategies between 0 and 1 Therefore we cannot write these strategies in the rows and columns of a matrix and underline bestresponse payoffs to find the Nash equilibria Instead we will use graphical methods to solve for the Nash equilibria Given players general mixed strategies we saw in Example 82 that the wifes expected payoff is U1 1 1w 1 2 w2 1h 1 2 h2 2 5 1 2 h 2 w 1 3hw 87 As Equation 87 shows the wifes best response depends on h If h 13 she wants to set w as low as possible w 5 0 If h 13 her best response is to set w as high as possible w 5 1 When h 5 13 her expected payoff equals 23 regardless of what w she chooses In this case there is a tie for the best response including any w from 0 to 1 In Example 82 we stated that the husbands expected payoff is U2 1 1h 1 2 h2 1w 1 2 w22 5 2 2 2h 2 2w 1 3hw 88 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 259 When w 23 his expected payoff is maximized by h 5 0 when w 23 his expected payoff is maximized by h 5 1 and when w 5 23 he is indifferent among all values of h obtaining an expected payoff of 23 regardless The best responses are graphed in Figure 86 The Nash equilibria are given by the intersection points between the best responses At these intersection points both players are best responding to each other which is what is required for the outcome to be a Nash equilibrium There are three Nash equilibria The points E1 and E2 are the purestrategy Nash equilibria we found before with E1 corresponding to the purestrategy Nash equilibrium in which both play boxing and E2 to that in which both play ballet Point E3 is the strictly mixedstrategy Nash equilibrium which can be spelled out as the wife plays ballet with probability 23 and boxing with probability 13 and the husband plays ballet with probability 13 and boxing with probability 23 More succinctly hav ing defined w and h we may write the equilibrium as w 5 23 and h 5 13 Ballet is chosen by the wife with probability w and by the husband with probability h Players best responses are graphed on the same set of axes The three intersection points E1 E2 and E3 are Nash equilibria The Nash equilibrium in strictly mixed strategies E3 is w 5 23 and h 5 13 Husbands best response BR2 1 23 13 13 23 1 0 Wifes best response BR1 E2 E3 E1 h w FIGURE 86 Nash Equilibria in Mixed Strategies in the Battle of the Sexes QUERY What is a players expected payoff in the Nash equilibrium in strictly mixed strategies How does this payoff compare with those in the purestrategy Nash equilibria What arguments might be offered that one or another of the three Nash equilibria might be the best prediction in this game Example 83 runs through the lengthy calculations involved in finding all the Nash equilibria of the Battle of the Sexes those in pure strategies and those in strictly mixed strategies A shortcut to finding the Nash equilibrium in strictly mixed strategies is based on the insight that a player will be willing to randomize between two actions in equilib rium only if he or she gets the same expected payoff from playing either action or in other words is indifferent between the two actions in equilibrium Otherwise one of the two Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 260 Part 3 Uncertainty and Strategy actions would provide a higher expected payoff and the player would prefer to play that action with certainty Suppose the husband is playing mixed strategy 1h 1 2 h2 that is playing ballet with probability h and boxing with probability 1 2 h The wifes expected payoff from playing ballet is U1 1ballet 1h 1 2 h2 2 5 1h2 122 1 11 2 h2 102 5 2h 89 Her expected payoff from playing boxing is U1 1boxing 1h 1 2 h2 2 5 1h2 102 1 11 2 h2 112 5 1 2 h 810 For the wife to be indifferent between ballet and boxing in equilibrium Equations 89 and 810 must be equal 2h 5 1 2 h implying h 5 13 Similar calculations based on the hus bands indifference between playing ballet and boxing in equilibrium show that the wifes probability of playing ballet in the strictly mixedstrategy Nash equilibrium is w 5 23 Work through these calculations as an exercise Notice that the wifes indifference condition does not pin down her equilibrium mixed strategy The wifes indifference condition cannot pin down her own equilibrium mixed strategy because given that she is indifferent between the two actions in equilibrium her overall expected payoff is the same no matter what probability distribution she plays over the two actions Rather the wifes indifference condition pins down the other playersthe husbandsmixed strategy There is a unique probability distribution he can use to play ballet and boxing that makes her indifferent between the two actions and thus makes her willing to randomize Given any probability of his playing ballet and boxing other than 13 23 it would not be a stable outcome for her to randomize Thus two principles should be kept in mind when seeking Nash equilibria in strictly mixed strategies One is that a player randomizes over only those actions among which he or she is indifferent given other players equilibrium mixed strategies The second is that one players indifference condition pins down the other players mixed strategy 85 ExISTENCE OF EQUILIBRIUM One of the reasons Nash equilibrium is so widely used is that a Nash equilibrium is guaranteed to exist in a wide class of games This is not true for some other equilib rium concepts Consider the dominant strategy equilibrium concept The Prisoners Dilemma has a dominant strategy equilibrium both suspects fink but most games do not Indeed there are many gamesincluding for example the Battle of the Sexesin which no player has a dominant strategy let alone all the players In such games we cannot make predictions using dominant strategy equilibrium but we can using Nash equilibrium The Extensions section at the end of this chapter will provide the technical details behind John Nashs proof of the existence of his equilibrium in all finite games games with a finite number of players and a finite number of actions The existence theo rem does not guarantee the existence of a purestrategy Nash equilibrium We already saw an example Rock Paper Scissors in Example 81 However if a finite game does not have a purestrategy Nash equilibrium the theorem guarantees that it will have a mixedstrategy Nash equilibrium The proof of Nashs theorem is similar to the proof in Chapter 13 of the existence of prices leading to a general competitive equilibrium The Extensions section includes an existence theorem for games with a continuum of actions as studied in the next section Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 261 86 CONTINUUM OF ACTIONS Most of the insight from economic situations can often be gained by distilling the situation down to a few or even two actions as with all the games studied thus far Other times additional insight can be gained by allowing a continuum of actions To be clear we have already encountered a continuum of strategiesin our discussion of mixed strategiesbut still the probability distributions in mixed strategies were over a finite number of actions In this section we focus on continuum of actions Some settings are more realistically modeled via a continuous range of actions In Chapter 15 for example we will study competition between strategic firms In one model Bertrand firms set prices in another Cournot firms set quantities It is natural to allow firms to choose any nonnegative price or quantity rather than artificially restricting them to just two prices say 2 or 5 or two quantities say 100 or 1000 units Continuous actions have several other advantages The familiar methods from calculus can often be used to solve for Nash equilibria It is also possible to analyze how the equilibrium actions vary with changes in underlying parameters With the Cournot model for example we might want to know how equilibrium quantities change with a small increase in a firms marginal costs or a demand parameter 861 Tragedy of the Commons Example 84 illustrates how to solve for the Nash equilibrium when the game in this case the Tragedy of the Commons involves a continuum of actions The first step is to write down the payoff for each player as a function of all players actions The next step is to compute the firstorder condition associated with each players payoff maximum This will give an equation that can be rearranged into the best response of each player as a function of all other players actions There will be one equation for each player With n players the system of n equations for the n unknown equilibrium actions can be solved simultaneously by either algebraic or graphical methods EXAMPLE 84 Tragedy of the Commons The term Tragedy of the Commons has come to signify environmental problems of overuse that arise when scarce resources are treated as common property6 A gametheoretic illustration of this issue can be developed by assuming that two herders decide how many sheep to graze on the village commons The problem is that the commons is small and can rapidly succumb to overgrazing To add some mathematical structure to the problem let qi be the number of sheep that herder i 5 1 2 grazes on the commons and suppose that the persheep value of grazing on the com mons in terms of wool and sheepmilk cheese is v 1q1 q22 5 120 2 1q1 1 q22 811 This function implies that the value of grazing a given number of sheep is lower when more sheep are around competing for grass We cannot use a matrix to represent the normal form of this game of continuous actions Instead the normal form is simply a listing of the herders payoff functions U1 1q1 q22 5 q1v 1q1 q22 5 q1 1120 2 q1 2 q22 U2 1q1 q22 5 q2v 1q1 q22 5 q2 1120 2 q1 2 q22 812 To find the Nash equilibrium we solve herder 1s valuemaximization problem max q1 5q1 1120 2 q1 2 q22 6 813 6This term was popularized by G Hardin The Tragedy of the Commons Science 162 1968 124348 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 262 Part 3 Uncertainty and Strategy The firstorder condition for a maximum is 120 2 2q1 2 q2 5 0 814 or rearranging q1 5 60 2 q2 2 5 BR1 1q22 815 Similar steps show that herder 2s best response is q2 5 60 2 q1 2 5 BR2 1q12 816 The Nash equilibrium is given by the pair 1q 1 q 22 that satisfies Equations 815 and 816 simulta neously Taking an algebraic approach to the simultaneous solution Equation 816 can be substi tuted into Equation 815 which yields q1 5 60 2 1 2a60 2 q1 2 b 817 on rearranging this implies q 1 5 40 Substituting q 1 5 40 into Equation 817 implies q 2 5 40 as well Thus each herder will graze 40 sheep on the common Each earns a payoff of 1600 as can be seen by substituting q 1 5 q 2 5 40 into the payoff function in Equation 813 Equations 815 and 816 can also be solved simultaneously using graphical methods Figure 87 plots the two best responses on a graph with player 1s action on the horizontal axis and player 2s on the vertical axis These best responses are simply lines and thus are easy to graph in this exam ple To be consistent with the axis labels the inverse of Equation 815 is actually what is graphed The two best responses intersect at the Nash equilibrium E1 The intersection E1 between the two herders best responses is the Nash equilibrium An increase in the persheep value of grazing in the Tragedy of the Commons shifts out herder 1s best response resulting in a Nash equilibrium E2 in which herder 1 grazes more sheep and herder 2 fewer sheep than in the original Nash equilibrium 120 60 120 40 0 40 60 E1 E2 BR2q1 BR1q2 q2 q1 FIGURE 87 BestResponse Diagram for the Tragedy of the Commons Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 263 The graphical method is useful for showing how the Nash equilibrium shifts with changes in the parameters of the problem Suppose the persheep value of grazing increases for the first herder while the second remains as in Equation 811 perhaps because the first herder starts rais ing merino sheep with more valuable wool This change would shift the best response out for herder 1 while leaving herder 2s the same The new intersection point E2 in the figure which is the new Nash equilibrium involves more sheep for 1 and fewer for 2 The Nash equilibrium is not the best use of the commons In the original problem both herd ers persheep value of grazing is given by Equation 811 If both grazed only 30 sheep then each would earn a payoff of 1800 as can be seen by substituting q1 5 q2 5 30 into Equation 813 Indeed the joint payoff maximization problem max q1 q2 5 1q1 1 q22v 1q1 q226 5 max q1 q2 5 1q1 1 q22 1120 2 q1 2 q22 6 818 is solved by q1 5 q2 5 30 or more generally by any q1 and q2 that sum to 60 QUERY How would the Nash equilibrium shift if both herders benefits increased by the same amount What about a decrease in only herder 2s benefit from grazing As Example 84 shows graphical methods are particularly convenient for quickly determining how the equilibrium shifts with changes in the underlying parameters The example shifted the benefit of grazing to one of herders This exercise nicely illustrates the nature of strategic interaction Herder 2s payoff function has not changed only herder 1s has yet his equilibrium action changes The second herder observes the firsts higher benefit anticipates that the first will increase the number of sheep he grazes and reduces his own grazing in response The Tragedy of the Commons shares with the Prisoners Dilemma the feature that the Nash equilibrium is less efficient for all players than some other outcome In the Prisoners Dilemma both fink in equilibrium when it would be more efficient for both to be silent In the Tragedy of the Commons the herders graze more sheep in equilibrium than is effi cient This insight may explain why ocean fishing grounds and other common resources can end up being overused even to the point of exhaustion if their use is left unregulated More detail on such problemsinvolving what we will call negative externalitiesis pro vided in Chapter 19 87 SEQUENTIAL GAMES In some games the order of moves matters For example in a bicycle race with a staggered start it may help to go last and thus know the time to beat On the other hand competition to establish a new highdefinition video format may be won by the first firm to market its technology thereby capturing an installed base of consumers Sequential games differ from the simultaneous games we have considered thus far in that a player who moves later in the game can observe how others have played up to that moment The player can use this information to form more sophisticated strategies than simply choosing an action the players strategy can be a contingent plan with the action played depending on what the other players have done To illustrate the new concepts raised by sequential gamesand in particular to make a stark contrast between sequential and simultaneous gameswe take a simultaneous game we have discussed already the Battle of the Sexes and turn it into a sequential game Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 264 Part 3 Uncertainty and Strategy 871 Sequential Battle of the Sexes Consider the Battle of the Sexes game analyzed previously with all the same actions and payoffs but now change the timing of moves Rather than the wife and husband making a simultaneous choice the wife moves first choosing ballet or boxing the husband observes this choice say the wife calls him from her chosen location and then the husband makes his choice The wifes possible strategies have not changed She can choose the simple actions ballet or boxing or perhaps a mixed strategy involving both actions although this will not be a relevant consideration in the sequential game The husbands set of possi ble strategies has expanded For each of the wifes two actions he can choose one of two actions therefore he has four possible strategies which are listed in Table 81 TABLE 81 HUSBANDS CONTINGENT STRATEGIES Contingent Strategy Written in Conditional Format Always go to the ballet ballet ballet ballet boxing Follow his wife ballet ballet boxing boxing Do the opposite boxing ballet ballet boxing Always go to boxing boxing ballet boxing boxing The vertical bar in the husbands strategies means conditional on and thus for example boxing ballet should be read as the husband chooses boxing conditional on the wifes choosing ballet Given that the husband has four pure strategies rather than just two the normal form given in Figure 88 must now be expanded to eight boxes Roughly speaking the normal form is twice as complicated as that for the simultaneous version of the game in Figure 82 This motivates a new way to represent games called the extensive form which is especially convenient for sequential games The column player husband has more complicated contingent strategies in the sequential Battle of the Sexes The normal form expands to reflect his expanded strategy space FIGURE 88 Normal Form for the Sequential Battle of the Sexes Husband Ballet Ballet Ballet Boxing Ballet Ballet Boxing Boxing Boxing Ballet Boxing Boxing Boxing Ballet Ballet Boxing Wife Ballet 2 1 2 1 0 0 1 2 Boxing 0 0 0 0 0 0 1 2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 265 872 Extensive form The extensive form of a game shows the order of moves as branches of a tree rather than collapsing everything down into a matrix The extensive form for the sequential Battle of the Sexes is shown in Figure 89a The action proceeds from left to right Each node shown as a dot on the tree represents a decision point for the player indicated there The first move belongs to the wife After any action she might take the husband gets to move Payoffs are listed at the end of the tree in the same order player 1s player 2s as in the normal form Contrast Figure 89a with Figure 89b which shows the extensive form for the simulta neous version of the game It is hard to harmonize an extensive form in which moves hap pen in progression with a simultaneous game in which everything happens at the same time The trick is to pick one of the two players to occupy the role of the second mover but then highlight that he or she is not really the second mover by connecting his or her deci sion nodes in the same information set the dotted oval around the nodes The dotted oval in Figure 89b indicates that the husband does not know his wifes move when he chooses his action It does not matter which player is picked for first and second mover in a simul taneous game we picked the husband in the figure to make the extensive form in Figure 89b look as much like Figure 89a as possible The similarity between the two extensive forms illustrates the point that that form does not grow in complexity for sequential games the way the normal form does We next will draw on both normal and extensive forms in our analysis of the sequential Battle of the Sexes In the sequential version a the husband moves second after observing his wifes move In the simulta neous version b he does not know her choice when he moves so his decision nodes must be connected in one information set FIGURE 89 Extensive Form for the Battle of the Sexes 2 1 Ballet Ballet Boxing Boxing 2 1 0 0 2 Ballet Boxing 0 0 1 2 2 1 Ballet Ballet Boxing Boxing 2 1 0 0 2 Ballet Boxing 0 0 1 2 a Sequential version b Simultaneous version Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 266 Part 3 Uncertainty and Strategy 873 Nash equilibria To solve for the Nash equilibria return to the normal form in Figure 88 Applying the method of underlining bestresponse payoffsbeing careful to underline both payoffs in cases of ties for the best responsereveals three purestrategy Nash equilibria 1 wife plays ballet husband plays ballet ballet ballet boxing 2 wife plays ballet husband plays ballet ballet boxing boxing 3 wife plays boxing husband plays boxing ballet boxing boxing As with the simultaneous version of the Battle of the Sexes here again we have multiple equilibria Yet now game theory offers a good way to select among the equilibria Con sider the third Nash equilibrium The husbands strategy boxing ballet boxing boxing involves the implicit threat that he will choose boxing even if his wife chooses ballet This threat is sufficient to deter her from choosing ballet Given that she chooses boxing in equilibrium his strategy earns him 2 which is the best he can do in any outcome Thus the outcome is a Nash equilibrium But the husbands threat is not crediblethat is it is an empty threat If the wife really were to choose ballet first then he would give up a payoff of 1 by choosing boxing rather than ballet It is clear why he would want to threaten to choose boxing but it is not clear that such a threat should be believed Similarly the husbands strategy ballet ballet ballet boxing in the first Nash equilibrium also involves an empty threat that he will choose ballet if his wife chooses boxing This is an odd threat to make because he does not gain from making it but it is an empty threat nonetheless Another way to understand empty versus credible threats is by using the concept of the equilibrium path the connected path through the extensive form implied by equilibrium strategies In Figure 810 which reproduces the extensive form of the sequential Battle of In the third of the Nash equilibria listed for the sequential Battle of the Sexes the wife plays boxing and the husband plays boxing ballet boxing boxing tracing out the branches indicated with thick lines both solid and dashed The dashed line is the equilibrium path the rest of the tree is referred to as being off the equilibrium path FIGURE 810 Equilibrium Path Ballet 2 0 0 1 2 0 0 2 1 2 1 Ballet Ballet Boxing Boxing Boxing Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 267 the Sexes from Figure 89 a dotted line is used to identify the equilibrium path for the third of the listed Nash equilibria The third outcome is a Nash equilibrium because the strategies are rational along the equilibrium path However following the wifes choosing balletan event that is off the equilibrium paththe husbands strategy is irrational The concept of subgameperfect equilibrium in the next section will rule out irrational play both on and off the equilibrium path 874 Subgameperfect equilibrium Game theory offers a formal way of selecting the reasonable Nash equilibria in sequential games using the concept of subgameperfect equilibrium Subgameperfect equilibrium is a refinement that rules out empty threats by requiring strategies to be rational even for contingencies that do not arise in equilibrium Before defining subgameperfect equilibrium formally we need a few preliminary defi nitions A subgame is a part of the extensive form beginning with a decision node and including everything that branches out to the right of it A proper subgame is a subgame that starts at a decision node not connected to another in an information set Conceptu ally this means that the player who moves first in a proper subgame knows the actions played by others that have led up to that point It is easier to see what a proper subgame is than to define it in words Figure 811 shows the extensive forms from the simultane ous and sequential versions of the Battle of the Sexes with boxes drawn around the proper subgames in each The sequential version a has three proper subgames the game itself and two lower subgames starting with decision nodes where the husband gets to move The simultaneous version b has only one decision nodethe topmost nodenot con nected to another in an information set Hence this version has only one subgame the whole game itself D E F I N I T I O N Subgameperfect equilibrium A subgameperfect equilibrium is a strategy profile 1s 1 s 2 s n2 that is a Nash equilibrium on every proper subgame A subgameperfect equilibrium is always a Nash equilibrium This is true because the whole game is a proper subgame of itself thus a subgameperfect equilibrium must be a Nash equilibrium for the whole game In the simultaneous version of the Battle of the Sexes there is nothing more to say because there are no subgames other than the whole game itself In the sequential version subgameperfect equilibrium has more bite Strategies must not only form a Nash equilibrium on the whole game itself but they must also form Nash equilibria on the two proper subgames starting with the decision points at which the hus band moves These subgames are simple decision problems so it is easy to compute the corresponding Nash equilibria For subgame B beginning with the husbands decision node following his wifes choosing ballet he has a simple decision between ballet which earns him a payoff of 1 and boxing which earns him a payoff of 0 The Nash equilib rium in this simple decision subgame is for the husband to choose ballet For the other subgame C he has a simple decision between ballet which earns him 0 and boxing which earns him 2 The Nash equilibrium in this simple decision subgame is for him to choose boxing Therefore the husband has only one strategy that can be part of a sub gameperfect equilibrium ballet ballet boxing boxing Any other strategy has him Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 268 Part 3 Uncertainty and Strategy playing something that is not a Nash equilibrium for some proper subgame Returning to the three enumerated Nash equilibria only the second is subgame perfect the first and the third are not For example the third equilibrium in which the husband always goes to boxing is ruled out as a subgameperfect equilibrium because the husbands strategy boxing boxing is not a Nash equilibrium in proper subgame B Thus subgameperfect equilibrium rules out the empty threat of always going to boxing that we were uncom fortable with earlier More generally subgameperfect equilibrium rules out any sort of empty threat in a sequential game In effect Nash equilibrium requires behavior to be rational only on the equilibrium path Players can choose potentially irrational actions on other parts of the extensive form In particular one player can threaten to damage both to scare the other from choosing certain actions Subgameperfect equilibrium requires ratio nal behavior both on and off the equilibrium path Threats to play irrationallythat is threats to choose something other than ones best responseare ruled out as being empty 875 Backward induction Our approach to solving for the equilibrium in the sequential Battle of the Sexes was to find all the Nash equilibria using the normal form and then to seek among those for the sub gameperfect equilibrium A shortcut for finding the subgameperfect equilibrium directly is to use backward induction the process of solving for equilibrium by working backward The sequential version in a has three proper subgames labeled A B and C The simultaneous version in b has only one proper subgame the whole game itself labeled D FIGURE 811 Proper Subgames in the Battle of the Sexes a Sequential 1 2 2 Boxing Ballet D Boxing 1 2 0 0 0 0 2 1 Boxing Ballet Ballet 1 2 2 Boxing Ballet A B C Boxing 1 2 0 0 0 0 2 1 Boxing Ballet Ballet b Simultaneous Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 269 from the end of the game to the beginning Backward induction works as follows Identify all the subgames at the bottom of the extensive form Find the Nash equilibria on these subgames Replace the potentially complicated subgames with the actions and payoffs resulting from Nash equilibrium play on these subgames Then move up to the next level of subgames and repeat the procedure Figure 812 illustrates the use of backward induction in the sequential Battle of the Sexes First we compute the Nash equilibria of the bottommost subgames at the husbands decision nodes In the subgame following his wifes choosing ballet he would choose ballet giving payoffs 2 for her and 1 for him In the subgame following his wifes choosing boxing he would choose boxing giving payoffs 1 for her and 2 for him Next substitute the hus bands equilibrium strategies for the subgames themselves The resulting game is a simple decision problem for the wife drawn in the lower panel of the figure a choice between ballet which would give her a payoff of 2 and boxing which would give her a payoff of 1 The Nash equilibrium of this game is for her to choose the action with the higher payoff ballet In sum backward induction allows us to jump straight to the subgameperfect equi librium in which the wife chooses ballet and the husband chooses ballet ballet boxing boxing bypassing the other Nash equilibria Backward induction is particularly useful in games that feature many rounds of sequential play As rounds are added it quickly becomes too hard to solve for all the Nash equilibria and then to sort through which are subgameperfect With backward induc tion an additional round is simply accommodated by adding another iteration of the procedure The last subgames where player 2 moves are replaced by the Nash equilibria on these subgames The simple game that results at right can be solved for player 1s equilibrium action FIGURE 812 Applying Backward Induction 1 2 2 Boxing Ballet Boxing 1 2 0 0 0 0 2 1 Boxing Ballet Ballet Boxing Ballet 1 2 plays boxing boxing payoff 1 2 2 plays ballet ballet payoff 2 1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 270 Part 3 Uncertainty and Strategy 88 REPEATED GAMES In the games examined thus far each player makes one choice and the game ends In many realworld settings players play the same game over and over again For example the players in the Prisoners Dilemma may anticipate committing future crimes and thus playing future Prisoners Dilemmas together Gasoline stations located across the street from each other when they set their prices each morning effectively play a new pric ing game every day The simple constituent game eg the Prisoners Dilemma or the gasolinepricing game that is played repeatedly is called the stage game As we saw with the Prisoners Dilemma the equilibrium in one play of the stage game may be worse for all players than some other more cooperative outcome Repeated play of the stage game opens up the possibility of cooperation in equilibrium Players can adopt trigger strate gies whereby they continue to cooperate as long as all have cooperated up to that point but revert to playing the Nash equilibrium if anyone deviates from cooperation We will investigate the conditions under which trigger strategies work to increase players payoffs As is standard in game theory we will focus on subgameperfect equilibria of the repeated games 881 Finitely repeated games For many stage games repeating them a known finite number of times does not increase the possibility for cooperation To see this point concretely suppose the Prisoners Dilemma were played repeatedly for T periods Use backward induction to solve for the subgameperfect equilibrium The lowest subgame is the Prisoners Dilemma stage game played in period T Regardless of what happened before the Nash equilibrium on this subgame is for both to fink Folding the game back to period T 2 1 trigger strategies that condition period T play on what happens in period T 2 1 are ruled out Although a player might like to promise to play cooperatively in period T and thus reward the other for playing cooperatively in period T 2 1 we have just seen that nothing that happens in period T 2 1 affects what happens subsequently because players both fink in period T regardless It is as though period T 2 1 were the last and the Nash equilibrium of this subgame is again for both to fink Working backward in this way we see that players will fink each period that is players will simply repeat the Nash equilibrium of the stage game T times Reinhard Selten winner of the Nobel Prize in economics for his contributions to game theory showed that this logic is general For any stage game with a unique Nash equilib rium the unique subgameperfect equilibrium of the finitely repeated game involves play ing the Nash equilibrium every period7 If the stage game has multiple Nash equilibria it may be possible to achieve some coop eration in a finitely repeated game Players can use trigger strategies sustaining coop eration in early periods on an outcome that is not an equilibrium of the stage game by threatening to play in later periods the Nash equilibrium that yields a worse outcome for the player who deviates from cooperation8 Rather than delving into the details of finitely repeated games we will instead turn to infinitely repeated games which greatly expand the possibility of cooperation 7R Selten A Simple Model of Imperfect Competition Where 4 Are Few and 6 Are Many International Journal of Game Theory 2 1973 141201 8J P Benoit and V Krishna Finitely Repeated Games Econometrica 53 1985 890940 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 271 882 Infinitely repeated games With finitely repeated games the folk theorem applies only if the stage game has multiple equilibria If like the Prisoners Dilemma the stage game has only one Nash equilibrium then Seltens result tells us that the finitely repeated game has only one subgameperfect equilibrium repeating the stagegame Nash equilibrium each period Backward induction starting from the last period T unravels any other outcomes With infinitely repeated games however there is no definite ending period T from which to start backward induction Outcomes involving cooperation do not necessarily end up unraveling Under some conditions the opposite may be the case with essentially anything being possible in equilibrium of the infinitely repeated game This result is some times called the folk theorem because it was part of the folk wisdom of game theory before anyone bothered to prove it formally One difficulty with infinitely repeated games involves adding up payoffs across periods An infinite stream of low payoffs sums to infinity just as an infinite stream of high payoffs How can the two streams be ranked We will circumvent this problem with the aid of dis counting Let δ be the discount factor discussed in the Chapter 17 Appendix measuring how much a payoff unit is worth if received one period in the future rather than today In Chapter 17 we show that δ is inversely related to the interest rate9 If the interest rate is high then a person would much rather receive payment today than next period because invest ing todays payment would provide a return of principal plus a large interest payment next period Besides the interest rate δ can also incorporate uncertainty about whether the game continues in future periods The higher the probability that the game ends after the current period the lower the expected return from stage games that might not actually be played Factoring in a probability that the repeated game ends after each period makes the set ting of an infinitely repeated game more believable The crucial issue with an infinitely repeated game is not that it goes on forever but that its end is indeterminate Interpreted in this way there is a sense in which infinitely repeated games are more realistic than finitely repeated games with large T Suppose we expect two neighboring gasoline stations to play a pricing game each day until electric cars replace gasolinepowered ones It is unlikely the gasoline stations would know that electric cars were coming in exactly T 5 2000 days More realistically the gasoline stations will be uncertain about the end of gasoline powered cars thus the end of their pricing game is indeterminate Players can try to sustain cooperation using trigger strategies Trigger strategies have them continuing to cooperate as long as no one has deviated deviation triggers some sort of punishment The key question in determining whether trigger strategies work is whether the punishment can be severe enough to deter the deviation in the first place Suppose both players use the following specific trigger strategy in the Prisoners Dilemma Continue being silent if no one has deviated fink forever afterward if anyone has deviated to fink in the past To show that this trigger strategy forms a subgameperfect equilibrium we need to check that a player cannot gain from deviating Along the equilib rium path both players are silent every period this provides each with a payoff of 2 every period for a present discounted value of 9Beware of the subtle difference between the formulas for the present value of an annuity stream used here and in Chapter 17 Appendix There the payments came at the end of the period rather than at the beginning as assumed here So here the present value of 1 payment per period from now on is 1 1 1 δ 1 1 δ2 1 1 δ3 1 5 1 1 2 δ Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 272 Part 3 Uncertainty and Strategy V eq 5 2 1 2δ 1 2δ2 1 2δ3 1 c 5 2 11 1 δ 1 δ2 1 δ3 1 c 2 5 2 1 2 δ 819 A player who deviates by finking earns 3 in that period but then both players fink every period from then oneach earning 1 per period for a total presented discounted payoff of V dev 5 3 1 112 1δ2 1 112 1δ22 1 112 1δ32 1 c 5 3 1 δ11 1 δ 1 δ2 1 c 2 5 3 1 δ 1 2 δ 820 The trigger strategies form a subgameperfect equilibrium if Veq Vdev implying that 2 1 2 δ 3 1 δ 1 2 δ 821 After multiplying through by 1 2 δ and rearranging we obtain δ 12 In other words players will find continued cooperative play desirable provided they do not discount future gains from such cooperation too highly If δ 12 then no cooperation is possible in the infinitely repeated Prisoners Dilemma the only subgameperfect equilibrium involves finking every period The trigger strategy we considered has players revert to the stagegame Nash equilib rium of finking each period forever This strategy which involves the harshest possible punishment for deviation is called the grim strategy Less harsh punishments include the socalled titfortat strategy which involves only one round of punishment for cheating Because the grim strategy involves the harshest punishment possible it elicits cooperation for the largest range of cases the lowest value of δ of any strategy Harsh punishments work well because if players succeed in cooperating they never experience the losses from the punishment in equilibrium10 The discount factor δ is crucial in determining whether trigger strategies can sustain cooperation in the Prisoners Dilemma or indeed in any stage game As δ approaches 1 grimstrategy punishments become infinitely harsh because they involve an unending stream of undiscounted losses Infinite punishments can be used to sustain a wide range of possible outcomes This is the logic behind the folk theorem for infinitely repeated games Take any stagegame payoff for a player between Nash equilibrium one and the highest one that appears anywhere in the payoff matrix Let V be the present discounted value of the infinite stream of this payoff The folk theorem says that the player can earn V in some subgameperfect equilibrium for δ close enough to 111 10Nobel Prizewinning economist Gary Becker introduced a related point the maximal punishment principle for crime The principle says that even minor crimes should receive draconian punishments which can deter crime with minimal expenditure on policing The punishments are costless to society because no crimes are committed in equilibrium so punishments never have to be carried out See G Becker Crime and Punishment An Economic Approach Journal of Political Economy 76 1968 169 217 Less harsh punishments may be suitable in settings involving uncertainty For example citizens may not be certain about the penal code police may not be certain they have arrested the guilty party 11A more powerful version of the folk theorem was proved by D Fudenberg and Nobel laureate E Maskin The Folk Theorem in Repeated Games with Discounting or with Incomplete Information Econometrica 54 1986 53356 Payoffs below even the Nash equilibrium ones can be generated by some subgameperfect equilibrium payoffs all the way down to players minmax level the lowest level a player can be reduced to by all other players working against him or her Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 273 89 INCOMPLETE INFORMATION In the games studied thus far players knew everything there was to know about the setup of the game including each others strategy sets and payoffs Matters become more complicated and potentially more interesting if some players have information about the game that others do not Poker would be different if all hands were played face up The fun of playing poker comes from knowing what is in your hand but not others Incom plete information arises in many other realworld contexts besides parlor games A sports team may try to hide the injury of a star player from future opponents to prevent them from exploiting this weakness Firms production technologies may be trade secrets and thus firms may not know whether they face efficient or weak competitors This section and the next two will introduce the tools needed to analyze games of incomplete infor mation The analysis integrates the material on game theory developed thus far in this chapter with the material on uncertainty and information from the previous chapter Games of incomplete information can quickly become complicated Players who lack full information about the game will try to use what they do know to make inferences about what they do not The inference process can be involved In poker for example knowing what is in your hand can tell you something about what is in others A player who holds two aces knows that others are less likely to hold aces because two of the four aces are not available Information on others hands can also come from the size of their bets or from their facial expressions of course a big bet may be a bluff and a facial expression may be faked Probability theory provides a formula called Bayes rule for making inferences about hidden information We will encounter Bayes rule in a later section The relevance of Bayes rule in games of incomplete information has led them to be called Bayesian games To limit the complexity of the analysis we will focus on the simplest possible setting throughout We will focus on twoplayer games in which one of the players player 1 has private information and the other player 2 does not The analysis of games of incomplete information is divided into two sections The next section begins with the simple case in which the players move simultaneously The subsequent section then analyzes games in which the informed player 1 moves first Such games called signaling games are more complicated than simultaneous games because player 1s action may signal something about his or her private information to the uninformed player 2 We will introduce Bayes rule at that point to help analyze player 2s inference about player 1s hidden information based on observations of player 1s action 810 SIMULTANEOUS BAYESIAN GAMES In this section we study a twoplayer simultaneousmove game in which player 1 has pri vate information but player 2 does not We will use he for player 1 and she for player 2 to facilitate the exposition We begin by studying how to model private information 8101 Player types and beliefs John Harsanyi who received the Nobel Prize in economics for his work on games with incomplete information provided a simple way to model private information by introduc ing player characteristics or types12 Player 1 can be one of a number of possible such types 12J Harsanyi Games with Incomplete Information Played by Bayesian Players Management Science 14 196768 15982 32034 486502 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 274 Part 3 Uncertainty and Strategy denoted t Player 1 knows his own type Player 2 is uncertain about t and must decide on her strategy based on beliefs about t Formally the game begins at an initial node called a chance node at which a par ticular value tk is randomly drawn for player 1s type t from a set of possible types T 5 5t1 tk tK6 Let Pr 1tk2 be the probability of drawing the particular type tk Player 1 sees which type is drawn Player 2 does not see the draw and only knows the prob abilities using them to form her beliefs about player 1s type Thus the probability that player 2 places on player 1s being of type tk is Pr 1tk2 Because player 1 observes his type t before moving his strategy can be conditioned on t Conditioning on this information may be a big benefit to a player In poker for exam ple the stronger a players hand the more likely the player is to win the pot and the more aggressively the player may want to bid Let s1 1t2 be player 1s strategy contingent on his type Because player 2 does not observe t her strategy is simply the unconditional one s2 As with games of complete information players payoffs depend on strategies In Bayes ian games payoffs may also depend on types Therefore we write player 1s payoff as U1 1s1 1t2 s2 t2 and player 2s as U2 1s2 s1 1t2 t2 Note that t appears in two places in player 2s payoff function Player 1s type may have a direct effect on player 2s payoffs Player 1s type also has an indirect effect through its effect on player 1s strategy s1 1t2 which in turn affects player 2s payoffs Because player 2s payoffs depend on t in these two ways her beliefs about t will be crucial in the calculation of her optimal strategy Figure 813 provides a simple example of a simultaneous Bayesian game Each player chooses one of two actions All payoffs are known except for player 1s payoff when 1 chooses up and 2 chooses left Player 1s payoff in outcome up left is identified as his type t There are two possible values for player 1s type t 5 6 and t 5 0 each occurring with equal probability Player 1 knows his type before moving Player 2s beliefs are that each type has probability 12 The extensive form is drawn in Figure 814 8102 BayesianNash equilibrium Extending Nash equilibrium to Bayesian games requires two small matters of interpreta tion First recall that player 1 may play a different action for each of his types Equilibrium In this game all payoffs are known to both players except for t in the upper left Player 2 only knows the distribution an equal chance that t 5 0 or t 5 6 Player 1 knows the realized value of t equivalent to knowing his or her type FIGURE 813 Simple Game of Incom plete Information Player 2 Lef Right Player 1 Up t 2 0 0 2 0 2 4 Down Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 275 requires that player 1s strategy be a best response for each and every one of his types Second recall that player 2 is uncertain about player 1s type Equilibrium requires that player 2s strategy maximize an expected payoff where the expectation is taken with respect to her beliefs about player 1s type We encountered expected payoffs in our discus sion of mixed strategies The calculations involved in computing the best response to the pure strategies of different types of rivals in a game of incomplete information are similar to the calculations involved in computing the best response to a rivals mixed strategy in a game of complete information Interpreted in this way Nash equilibrium in the setting of a Bayesian game is called BayesianNash equilibrium Next we provide a formal definition of the concept for refer ence Given that the notation is fairly dense it may be easier to first skip to Examples 85 and 86 which provide a blueprint on how to solve for equilibria in Bayesian games you might come across This figure translates Figure 813 into an extensiveform game The initial chance node is indicated by an open circle Player 2s decision nodes are in the same information set because she does not observe player 1s type or action before moving FIGURE 814 Extensive Form for Simple Game of Incomplete Information Down Down Up Lef Lef Lef Lef Right Right Right Right Up 1 2 2 6 2 0 0 0 2 0 0 Pr 12 t 6 Pr 12 t 0 2 0 2 4 2 4 2 0 2 2 1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 276 Part 3 Uncertainty and Strategy Because the difference between Nash equilibrium and BayesianNash equilibrium is only a matter of interpretation all our previous results for Nash equilibrium including the existence proof apply to BayesianNash equilibrium as well D E F I N I T I O N BayesianNash equilibrium In a twoplayer simultaneousmove game in which player 1 has private information a BayesianNash equilibrium is a strategy profile 1s 1 1t2 s 22 such that s 1 1t2 is a best response to s 2 for each type t T of player 1 U1 1s 1 1t2 s 2 t2 U1 1sr1 s 2 t2 for all sr1 S1 822 and such that s 2 is a best response to s 1 1t2 given player 2s beliefs Pr 1tk2 about player 1s types a tkT Pr 1tk2U2 1s 2 s 1 1tk2 tk2 a tkT Pr 1tk2U2 1sr2 s 1 1tk2 tk2 for all sr2 S2 823 EXAMPLE 85 BayesianNash Equilibrium of Game in Figure 814 To solve for the BayesianNash equilibrium of the game in Figure 814 first solve for the informed players player 1s best responses for each of his types If player 1 is of type t 5 0 then he would choose down rather than up because he earns 0 by playing up and 2 by playing down regardless of what player 2 does If player 1 is of type t 5 6 then his best response is up to player 2s playing left and down to her playing right This leaves only two possible candidates for an equilibrium in pure strategies 1 plays up t 5 6 down t 5 0 and 2 plays left 1 plays down t 5 6 down t 5 0 and 2 plays right The first candidate cannot be an equilibrium because given that player 1 plays up t 5 6 down t 5 0 player 2 earns an expected payoff of 1 from playing left Player 2 would gain by deviating to right earning an expected payoff of 2 The second candidate is a BayesianNash equilibrium Given that player 2 plays right player 1s best response is to play down providing a payoff of 2 rather than 0 regardless of his type Given that both types of player 1 play down player 2s best response is to play right providing a payoff of 4 rather than 0 QUERY If the probability that player 1 is of type t 5 6 is high enough can the first candidate be a BayesianNash equilibrium If so compute the threshold probability EXAMPLE 86 Tragedy of the Commons as a Bayesian Game For an example of a Bayesian game with continuous actions consider the Tragedy of the Com mons in Example 84 but now suppose that herder 1 has private information regarding his value of grazing per sheep v1 1q1 q2 t2 5 t 2 1q1 1 q22 824 where herder 1s type is t 5 130 the high type with probability 23 and t 5 100 the low type with probability 13 Herder 2s value remains the same as in Equation 811 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 277 To solve for the BayesianNash equilibrium we first solve for the informed players herder 1s best responses for each of his types For any type t and rivals strategy q2 herder 1s value maximization problem is max q1 5q1v1 1q1 q2 t26 5 max q1 5q1 1t 2 q1 2 q226 825 The firstorder condition for a maximum is t 2 2q1 2 q2 5 0 826 Rearranging and then substituting the values t 5 130 and t 5 100 we obtain q1H 5 65 2 q2 2 and q1L 5 50 2 q2 2 827 where q1H is the quantity for the high type of herder 1 ie the t 5 130 type and q1L for the low type the t 5 100 type Next we solve for herder 2s best response Herder 2s expected payoff is 2 3 3q2 1120 2 q1H 2 q224 1 1 3 3q2 1120 2 q1L 2 q224 5 q2 1120 2 q1 2 q22 828 where q1 5 2 3 q1H 1 1 3 q1L 829 Rearranging the firstorder condition from the maximization of Equation 828 with respect to q2 gives q2 5 60 2 q1 2 830 Substituting for q1H and q1L from Equation 827 into Equation 829 and then substituting the resulting expression for q1 into Equation 830 yields q2 5 30 1 q2 4 831 implying that q 2 5 40 Substituting q 2 5 40 back into Equation 827 implies q 1H 5 45 and q 1L 5 30 Figure 815 depicts the BayesianNash equilibrium graphically Herder 2 imagines playing against an average type of herder 1 whose average best response is given by the thick dashed line The intersection of this best response and herder 2s at point B determines herder 2s equilibrium quantity q 2 5 40 The best response of the low resp high type of herder 1 to q 2 5 40 is given by point A resp point C For comparison the fullinformation Nash equilibria are drawn when herder 1 is known to be the low type point Ar or the high type point Cr QUERY Suppose herder 1 is the high type How does the number of sheep each herder grazes change as the game moves from incomplete to full information moving from point Cr to C What if herder 1 is the low type Which type prefers full information and thus would like to sig nal its type Which type prefers incomplete information and thus would like to hide its type We will study the possibility player 1 can signal his type in the next section Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 278 Part 3 Uncertainty and Strategy 811 SIGNALING GAMES In this section we move from simultaneousmove games of private information to sequen tial games in which the informed player player 1 takes an action that is observable to player 2 before player 2 moves Player 1s action provides information a signal that player 2 can use to update her beliefs about player 1s type perhaps altering the way player 2 would play in the absence of such information In poker for example player 2 may take a big raise by player 1 as a signal that he has a good hand perhaps leading player 2 to fold A firm considering whether to enter a market may take the incumbent firms low price as a signal that the incumbent is a lowcost producer and thus a tough competitor perhaps keeping the entrant out of the market A prestigious college degree may signal that a job applicant is highly skilled The analysis of signaling games is more complicated than simultaneous games because we need to model how player 2 processes the information in player 1s signal and then updates her beliefs about player 1s type To fix ideas we will focus on a concrete applica tion a version of Michael Spences model of jobmarket signaling for which he won the Nobel Prize in economics13 13M Spence JobMarket Signaling Quarterly Journal of Economics 87 1973 35574 Best responses for herder 2 and both types of herder 1 are drawn as thick solid lines the expected best response as perceived by 2 is drawn as the thick dashed line The BayesianNash equilibrium of the incompleteinformation game is given by points A and C Nash equilibria of the corresponding fullinformation games are given by points Ar and Cr High types best response 2s best response Low types best response q1 q2 A B C C A 0 40 30 45 40 FIGURE 815 Equilibrium of the Bayesian Tragedy of the Commons Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 279 8111 Jobmarket signaling Player 1 is a worker who can be one of two types highskilled 1t 5 H2 or lowskilled 1t 5 L2 Player 2 is a firm that considers hiring the applicant A lowskilled worker is com pletely unproductive and generates no revenue for the firm a highskilled worker gener ates revenue π If the applicant is hired the firm must pay the worker w think of this wage as being fixed by government regulation Assume π w 0 Therefore the firm wishes to hire the applicant if and only if he or she is highskilled But the firm cannot observe the applicants skill it can observe only the applicants prior education Let cH be the high types cost of obtaining an education and cL the low types cost Assume cH cL implying that education requires less effort for the highskilled applicant than the lowskilled one We make the extreme assumption that education does not increase the workers productivity directly The applicant may still decide to obtain an education because of its value as a signal of ability to future employers Figure 816 shows the extensive form Player 1 observes his or her type at the start player 2 observes only player 1s action education signal before moving Let PrH and PrL be player 2s beliefs before observing player 1s education signal that player 1 is high or lowskilled These are called player 2s prior beliefs Observing player 1s action will lead player 2 to revise his or her beliefs to form what are called posterior beliefs Player 1 worker observes his or her own type Then player 1 chooses to become educated E or not NE After observing player 1s action player 2 firm decides to make him or her a job offer J or not NJ The nodes in player 2s information sets are labeled n1 c n4 for reference NE NE J J J J NJ NJ NJ NJ E E 1 2 2 n3 n2 n1 n4 0 0 PrL PrH cL 0 cH 0 0 0 w w w π w w cH π w w cL w 2 2 1 FIGURE 816 JobMarket Signaling Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 280 Part 3 Uncertainty and Strategy For example the probability that the worker is highskilled is conditional on the workers having obtained an education PrHE and conditional on no education PrHNE Player 2s posterior beliefs are used to compute his or her best response to player 1s education decision Suppose player 2 sees player 1 choose E Then player 2s expected pay off from playing J is Pr1H0E2 1π 2 w2 1 Pr 1L0E2 12w2 5 Pr 1H0E2π 2 w 832 where the right side of this equation follows from the fact that because L and H are the only types Pr1L0E2 5 1 2 Pr1H0E2 Player 2s payoff from playing NJ is 0 To determine the best response to E player 2 compares the expected payoff in Equation 832 to 0 Player 2s best response is J if and only if Pr 1H0E2 wπ The question remains of how to compute posterior beliefs such as Pr 1H0E2 Rational players use a statistical formula called Bayes rule to revise their prior beliefs to form pos terior beliefs based on the observation of a signal 8112 Bayes rule Bayes rule gives the following formula for computing player 2s posterior belief Pr 1H0E214 Pr 1H0E2 5 Pr 1E0H2 Pr 1H2 Pr 1E0H2 Pr 1H2 1 Pr 1E0L2 Pr 1L2 833 Similarly Pr 1H0NE2 is given by Pr 1H0NE2 5 Pr 1NE0H2 Pr 1H2 Pr 1NE0H2 Pr 1H2 1 Pr 1NE0L2 Pr 1L2 834 Two sorts of probabilities appear on the left side of Equations 833 and 834 the prior beliefs Pr1H2 and Pr 1L2 the conditional probabilities Pr 1E0H2 Pr 1NE0L2 and so forth The prior beliefs are given in the specification of the game by the probabilities of the differ ent branches from the initial chance node The conditional probabilities Pr 1E0H2 Pr 1NE0L2 and so forth are given by player 1s equilibrium strategy For example Pr 1E0H2 is the prob ability that player 1 plays E if he or she is of type H Pr 1NE0L2 is the probability that player 1 plays NE if he or she is of type L and so forth As the schematic diagram in Figure 817 summarizes Bayes rule can be thought of as a black box that takes prior beliefs and strategies as inputs and gives as outputs the beliefs we must know to solve for an equilib rium of the game player 2s posterior beliefs 14Equation 833 can be derived from the definition of conditional probability in footnote 25 of Chapter 2 Equation 834 can be derived similarly By definition Pr 1H0E2 5 Pr 1H and E2 Pr 1E2 Reversing the order of the two events in the conditional probability yields Pr 1E0H2 5 Pr 1H and E2 Pr 1H2 or after rearranging Pr 1H and E2 5 Pr 1E0H2 Pr 1H2 Substituting the preceding equation into the first displayed equation of this footnote gives the numerator of Equation 833 The denominator follows because the events of player 1s being of type H or L are mutually exclusive and jointly exhaustive so Pr 1E2 5 Pr 1E and H2 1 Pr 1E and L2 5 Pr 1E0H2 Pr 1H2 1 Pr 1E0L2 Pr 1L2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 281 When player 1 plays a pure strategy Bayes rule often gives a simple result Suppose for example that Pr1E0H2 5 1 and Pr 1E0L2 5 0 or in other words that player 1 obtains an education if and only if he or she is highskilled Then Equation 833 implies Pr 1H0E2 5 1 Pr 1H2 1 Pr 1H2 1 0 Pr 1L2 5 1 835 That is player 2 believes that player 1 must be highskilled if it sees player 1 choose E On the other hand suppose that Pr 1E0H2 5 Pr 1E0L2 5 1that is suppose player 1 obtains an education regardless of his or her type Then Equation 833 implies Pr 1H0E2 5 1 Pr 1H2 1 Pr 1H2 1 1 Pr 1L2 5 Pr 1H2 836 because Pr1H2 1 Pr 1L2 5 1 That is seeing player 1 play E provides no information about player 1s type so player 2s posterior belief is the same as his or her prior one More gen erally if player 1 plays the mixed strategy Pr 1E0H2 5 p and Pr 1E0L2 5 q then Bayes rule implies that Pr 1H0E2 5 p Pr 1H2 p Pr 1H2 1 q Pr 1L2 837 8113 Perfect Bayesian equilibrium With games of complete information we moved from Nash equilibrium to the refinement of subgameperfect equilibrium to rule out noncredible threats in sequential games For the same reason with games of incomplete information we move from BayesianNash equilibrium to the refinement of perfect Bayesian equilibrium Bayes rule is a formula for computing player 2s posterior beliefs from other pieces of information in the game Inputs Output Bayes rule Player 2s posterior beliefs Player 2s prior beliefs Player 1s strategy FIGURE 817 Bayes Rule as a Black Box D E F I N I T I O N Perfect Bayesian equilibrium A perfect Bayesian equilibrium consists of a strategy profile and a set of beliefs such that at each information set the strategy of the player moving there maximizes his or her expected payoff where the expectation is taken with respect to his or her beliefs and at each information set where possible the beliefs of the player moving there are formed using Bayes rule based on prior beliefs and other players strategies Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 282 Part 3 Uncertainty and Strategy The requirement that players play rationally at each information set is similar to the requirement from subgameperfect equilibrium that play on every subgame form a Nash equilibrium The requirement that players use Bayes rule to update beliefs ensures that players incorporate the information from observing others play in a rational way The remaining wrinkle in the definition of perfect Bayesian equilibrium is that Bayes rule need only be used where possible Bayes rule is useless following a completely unex pected eventin the context of a signaling model an action that is not played in equilibrium by any type of player 1 For example if neither H nor L type chooses E in the job market signaling game then the denominator of Equation 833 equals zero and the fraction is undefined If Bayes rule gives an undefined answer then perfect Bayesian equilibrium puts no restrictions on player 2s posterior beliefs and thus we can assume any beliefs we like As we saw with games of complete information signaling games may have multiple equilibria The freedom to specify any beliefs when Bayes rule gives an undefined answer may support additional perfect Bayesian equilibria A systematic analysis of multiple equi libria starts by dividing the equilibria into three classesseparating pooling and hybrid Then we look for perfect Bayesian equilibria within each class In a separating equilibrium each type of player 1 chooses a different action Therefore player 2 learns player 1s type with certainty after observing player 1s action The posterior beliefs that come from Bayes rule are all zeros and ones In a pooling equilibrium differ ent types of player 1 choose the same action Observing player 1s action provides player 2 with no information about player 1s type Pooling equilibria arise when one of player 1s types chooses an action that would otherwise be suboptimal to hide his or her private information In a hybrid equilibrium one type of player 1 plays a strictly mixed strategy it is called a hybrid equilibrium because the mixed strategy sometimes results in the types being separated and sometimes pooled Player 2 learns a little about player 1s type Bayes rule refines player 2s beliefs a bit but does not learn player 1s type with certainty Player 2 may respond to the uncertainty by playing a mixed strategy itself The next three examples solve for the three different classes of equilibrium in the jobmarket signaling game EXAMPLE 87 Separating Equilibrium in the JobMarket Signaling Game A good guess for a separating equilibrium is that the highskilled worker signals his or her type by getting an education and the lowskilled worker does not Given these strategies player 2s beliefs must be Pr1H0E2 5 Pr1L0NE2 5 1 and Pr1H0NE2 5 Pr1L0E2 5 0 according to Bayes rule Condi tional on these beliefs if player 2 observes that player 1 obtains an education then player 2 knows it must be at node n1 rather than n2 in Figure 817 Its best response is to offer a job J given the payoff of π 2 w 0 If player 2 observes that player 1 does not obtain an education then player 2 knows it must be at node n4 rather than n3 and its best response is not to offer a job NJ because 0 2w The last step is to go back and check that player 1 would not want to deviate from the separating strategy 1E0H NE0L2 given that player 2 plays 1J0E NJ0NE2 Type H of player 1 earns w 2 cH by obtaining an education in equilibrium If type H deviates and does not obtain an education then he or she earns 0 because player 2 believes that player 1 is type L and does not offer a job For type H not to prefer to deviate it must be that w 2 cH 0 Next turn to type L of player 1 Type L earns 0 by not obtaining an education in equilibrium If type L deviates and obtains an education then he or she earns w 2 cL because player 2 believes that player 1 is type H and offers a job For type L not to prefer to deviate we must have w 2 cL 0 Putting these conditions together there is sepa rating equilibrium in which the worker obtains an education if and only if he or she is highskilled and in which the firm offers a job only to applicants with an education if and only if cH w cL Another possible separating equilibrium is for player 1 to obtain an education if and only if he or she is lowskilled This is a bizarre outcomebecause we expect education to be a signal of Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 283 high rather than low skilland fortunately we can rule it out as a perfect Bayesian equilibrium Player 2s best response would be to offer a job if and only if player 1 did not obtain an education Type L would earn 2cL from playing E and w from playing NE so it would deviate to NE QUERY Why does the worker sometimes obtain an education even though it does not raise his or her skill level Would the separating equilibrium exist if a lowskilled worker could obtain an education more easily than a highskilled one EXAMPLE 88 Pooling Equilibria in the JobMarket Signaling Game Lets investigate a possible pooling equilibrium in which both types of player 1 choose E For player 1 not to deviate from choosing E player 2s strategy must be to offer a job if and only if the worker is educatedthat is 1J0E NJ0NE2 If player 2 does not offer jobs to educated workers then player 1 might as well save the cost of obtaining an education and choose NE If player 2 offers jobs to uneducated workers then player 1 will again choose NE because he or she saves the cost of obtaining an education and still earns the wage from the job offer Next we investigate when 1J0E NJ0NE2 is a best response for player 2 Player 2s posterior beliefs after seeing E are the same as his or her prior beliefs in this pooling equilibrium Player 2s expected payoff from choosing J is Pr1H0E2 1π 2 w2 1 Pr1L0E2 12w2 5 Pr1H2 1π 2 w2 1 Pr1L2 12w2 5 Pr1H2π 2 w 838 For J to be a best response to E Equation 838 must exceed player 2s zero payoff from choos ing NJ which on rearranging implies that Pr1H2 wπ Player 2s posterior beliefs at nodes n3 and n4 are not pinned down by Bayes rule because NE is never played in equilibrium and so seeing player 1 play NE is a completely unexpected event Perfect Bayesian equilibrium allows us to specify any probability distribution we like for the posterior beliefs Pr1H0NE2 at node n3 and Pr1L0NE2 at node n4 Player 2s payoff from choosing NJ is 0 For NJ to be a best response to NE 0 must exceed player 2s expected payoff from playing J 0 Pr1H0NE2 1π 2 w2 1 Pr1L0NE2 12w2 5 Pr1H0NE2π 2 w 839 where the right side follows because Pr1H0NE2 1 Pr1L0NE2 5 1 Rearranging yields Pr1H0NE2 wπ In sum for there to be a pooling equilibrium in which both types of player 1 obtain an edu cation we need Pr1H0NE2 wπ Pr1H2 The firm has to be optimistic about the proportion of skilled workers in the populationPr1H2 must be sufficiently highand pessimistic about the skill level of uneducated workersPr1H0NE2 must be sufficiently low In this equilibrium type L pools with type H to prevent player 2 from learning anything about the workers skill from the education signal The other possibility for a pooling equilibrium is for both types of player 1 to choose NE There are a number of such equilibria depending on what is assumed about player 2s posterior beliefs out of equilibrium ie player 2s beliefs after he or she observes player 1 choosing E Per fect Bayesian equilibrium does not place any restrictions on these posterior beliefs Problem 810 asks you to search for various of these equilibria and introduces a further refinement of perfect Bayesian equilibrium the intuitive criterion that helps rule out unreasonable outofequilibrium beliefs and thus implausible equilibria QUERY Return to the pooling outcome in which both types of player 1 obtain an education Consider player 2s posterior beliefs following the unexpected event that a worker shows up with Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 284 Part 3 Uncertainty and Strategy no education Perfect Bayesian equilibrium leaves us free to assume anything we want about these posterior beliefs Suppose we assume that the firm obtains no information from the no educa tion signal and so maintains its prior beliefs Is the proposed pooling outcome an equilibrium What if we assume that the firm takes no education as a bad signal of skill believing that player 1s type is L for certain EXAMPLE 89 Hybrid Equilibria in the JobMarket Signaling Game One possible hybrid equilibrium is for type H always to obtain an education and for type L to ran domize sometimes pretending to be a high type by obtaining an education Type L randomizes between playing E and NE with probabilities e and 1 2 e Player 2s strategy is to offer a job to an educated applicant with probability j and not to offer a job to an uneducated applicant We need to solve for the equilibrium values of the mixed strategies e and j and the posterior beliefs Pr1H0E2 and Pr1H0NE2 that are consistent with perfect Bayesian equilibrium The poste rior beliefs are computed using Bayes rule Pr1H0E2 5 Pr1H2 Pr1H2 1 ePr1L2 5 Pr1H2 Pr1H2 1 e31 2 Pr1H2 4 840 and Pr1H0NE2 5 0 For type L of player 1 to be willing to play a strictly mixed strategy he or she must get the same expected payoff from playing Ewhich equals jw 2 cL given player 2s mixed strategyas from playing NEwhich equals 0 given that player 2 does not offer a job to uneducated applicants Hence jw 2 cL 5 0 or solving for j j 5 cLw Player 2 will play a strictly mixed strategy conditional on observing E only if he or she gets the same expected payoff from playing J which equals Pr1H0E2 1π 2 w2 1 Pr1L0E2 12w2 5 Pr1H0E2π 2 w 841 as from playing NJ which equals 0 Setting Equation 841 equal to 0 substituting for Pr1H0E2 from Equation 840 and then solving for e gives e 5 1π 2 w2Pr1H2 w31 2 Pr1H24 842 QUERY To complete our analysis In this equilibrium type H of player 1 cannot prefer to devi ate from E Is this true If so can you show it How does the probability of type L trying to pool with the high type by obtaining an education vary with player 2s prior belief that player 1 is the high type 812 ExPERIMENTAL GAMES Experimental economics is a recent branch of research that explores how well economic theory matches the behavior of experimental subjects in laboratory settings The methods are similar to those used in experimental psychologyoften conducted on campus using undergraduates as subjectsalthough experiments in economics tend to involve incen tives in the form of explicit monetary payments paid to subjects The importance of exper imental economics was highlighted in 2002 when Vernon Smith received the Nobel Prize in economics for his pioneering work in the field An important area in this field is the use of experimental methods to test game theory Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 285 8121 Experiments with the Prisoners Dilemma There have been hundreds of tests of whether players fink in the Prisoners Dilemma as predicted by Nash equilibrium or whether they play the cooperative outcome of Silent In one experiment subjects played the game 20 times with each player being matched with a different anonymous opponent to avoid repeatedgame effects Play converged to the Nash equilibrium as subjects gained experience with the game Players played the cooper ative action 43 percent of the time in the first five rounds falling to only 20 percent of the time in the last five rounds15 As is typical with experiments subjects behavior tended to be noisy Although 80 per cent of the decisions were consistent with Nash equilibrium play by the end of the exper iment 20 percent of them still were anomalous Even when experimental play is roughly consistent with the predictions of theory it is rarely entirely consistent 8122 Experiments with the Ultimatum Game Experimental economics has also tested to see whether subgameperfect equilibrium is a good predictor of behavior in sequential games In one widely studied sequential game the Ultimatum Game the experimenter provides a pot of money to two players The first mover Proposer proposes a split of this pot to the second mover The second mover Responder then decides whether to accept the offer in which case players are given the amount of money indicated or reject the offer in which case both players get nothing In the subgameperfect equilibrium the Proposer offers a minimal share of the pot and this is accepted by the Responder One can see this by applying backward induction The Responder should accept any positive division no matter how small knowing this the Pro poser should offer the Responder only a minimal share In experiments the division tends to be much more even than in the subgameperfect equilibrium16 The most common offer is a 5050 split Responders tend to reject offers giving them less than 30 percent of the pot This result is observed even when the pot is as high as 100 so that rejecting a 30 percent offer means turning down 30 Some econ omists have suggested that the money players receive may not be a true measure of their payoffs They may care about other factors such as fairness and thus obtain a benefit from a more equal division of the pot Even if a Proposer does not care directly about fairness the fear that the Responder may care about fairness and thus might reject an uneven offer out of spite may lead the Proposer to propose an even split The departure of experimental behavior from the predictions of game theory was too systematic in the Ultimatum Game to be attributed to noisy play leading some game theo rists to rethink the theory and add an explicit consideration for fairness17 8123 Experiments with the Dictator Game To test whether players care directly about fairness or act out of fear of the other players spite researchers experimented with a related game the Dictator Game In the Dictator Game the Proposer chooses a split of the pot and this split is implemented without input from the Responder Proposers tend to offer a lesseven split than in the Ultimatum Game 15R Cooper D V DeJong R Forsythe and T W Ross Cooperation Without Reputation Experimental Evidence from Prisoners Dilemma Games Games and Economic Behavior February 1996 187218 16For a review of Ultimatum Game experiments and a textbook treatment of experimental economics more generally see D D Davis and C A Holt Experimental Economics Princeton NJ Princeton University Press 1993 17See for example E Fehr and KM Schmidt A Theory of Fairness Competition and Cooperation Quarterly Journal of Economics August 1999 817868 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 286 Part 3 Uncertainty and Strategy but still offer the Responder some of the pot suggesting that Proposers have some residual concern for fairness The details of the experimental design are crucial however as one ingenious experiment showed18 The experiment was designed so that the experimenter would never learn which Proposers had made which offers With this element of anonym ity Proposers almost never gave an equal split to Responders and indeed took the whole pot for themselves two thirds of the time Proposers seem to care more about appearing fair to the experimenter than truly being fair 813 EVOLUTIONARY GAMES AND LEARNING The frontier of gametheory research regards whether and how players come to play a Nash equilibrium Hyperrational players may deduce each others strategies and instantly settle on the Nash equilibrium How can they instantly coordinate on a single outcome when there are multiple Nash equilibria What outcome would realworld players for whom hyperrational deductions may be too complex settle on Game theorists have tried to model the dynamic process by which an equilibrium emerges over the long run from the play of a large population of agents who meet others at random and play a pairwise game Game theorists analyze whether play converges to Nash equilibrium or some other outcome which Nash equilibrium if any is converged to if there are multiple equilibria and how long such convergence takes Two models which make varying assumptions about the level of players rationality have been most widely studied an evolutionary model and a learning model In the evolutionary model players do not make rational decisions instead they play the way they are genetically programmed The more successful a players strategy in the population the more fit is the player and the more likely will the player survive to pass his or her genes on to future generations and thus the more likely the strategy spreads in the population Evolutionary models were initially developed by John Maynard Smith and other biol ogists to explain the evolution of such animal behavior as how hard a lion fights to win a mate or an ant fights to defend its colony Although it may be more of a stretch to apply evolutionary models to humans evolutionary models provide a convenient way of analyz ing population dynamics and may have some direct bearing on how social conventions are passed down perhaps through culture In a learning model players are again matched at random with others from a large pop ulation Players use their experiences of payoffs from past play to teach them how others are playing and how they themselves can best respond Players usually are assumed to have a degree of rationality in that they can choose a static best response given their beliefs may do some experimenting and will update their beliefs according to some reasonable rule Players are not fully rational in that they do not distort their strategies to affect others learning and thus future play Game theorists have investigated whether more or lesssophisticated learning strategies converge more or less quickly to a Nash equilibrium Current research seeks to integrate theory with experimental study trying to identify the specific algorithms that realworld subjects use when they learn to play games 18E Hoffman K McCabe K Shachat and V Smith Preferences Property Rights and Anonymity in Bargaining Games Games and Economic Behavior November 1994 34680 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 287 SUMMARY This chapter provided a structured way to think about stra tegic situations We focused on the most important solution concept used in game theory Nash equilibrium We then progressed to several more refined solution concepts that are in standard use in game theory in more complicated settings with sequential moves and incomplete information Some of the principal results are as follows All games have the same basic components players strategies payoffs and an information structure Games can be written down in normal form providing a payoff matrix or payoff functions or extensive form providing a game tree Strategies can be simple actions more complicated plans contingent on others actions or even probability distri butions over simple actions mixed strategies A Nash equilibrium is a set of strategies one for each player that are mutual best responses In other words a players strategy in a Nash equilibrium is optimal given that all others play their equilibrium strategies A Nash equilibrium always exists in finite games in mixed if not pure strategies Subgameperfect equilibrium is a refinement of Nash equilibrium that helps to rule out equilibria in sequential games involving noncredible threats Repeating a stage game a large number of times intro duces the possibility of using punishment strategies to attain higher payoffs than if the stage game is played once If players are sufficiently patient in an infinitely repeated game then a folk theorem holds implying that essentially any payoffs are possible in the repeated game In games of private information one player knows more about his or her type than another Players maximize their expected payoffs given knowledge of their own type and beliefs about the others In a perfect Bayesian equilibrium of a signaling game the second mover uses Bayes rule to update his or her beliefs about the first movers type after observing the first mov ers action The frontier of gametheory research combines theory with experiments to determine whether players who may not be hyperrational come to play a Nash equilibrium which particular equilibrium if there are more than one and what path leads to the equilibrium Problems 81 Consider the following game Player 2 D E Player 1 A 7 6 5 8 5 8 7 6 B F 0 0 1 1 0 0 1 1 C 4 4 a Find the purestrategy Nash equilibria if any b Find the mixedstrategy Nash equilibrium in which each player randomizes over just the first two actions c Compute players expected payoffs in the equilibria found in parts a and b d Draw the extensive form for this game 82 The mixedstrategy Nash equilibrium in the Battle of the Sexes in Figure 83 may depend on the numerical values for the payoffs To generalize this solution assume that the payoff matrix for the game is given by Player 2 Husband Ballet Boxing Player 1 Wife Ballet K 1 0 0 0 0 1 K Boxing where K 1 Show how the mixedstrategy Nash equilibrium depends on the value of K 83 The game of Chicken is played by two macho teens who speed toward each other on a singlelane road The first to veer off is branded the chicken whereas the one who does not veer gains Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 288 Part 3 Uncertainty and Strategy peergroup esteem Of course if neither veers both die in the resulting crash Payoffs to the Chicken game are provided in the following table Teen 2 Veer Does not veer Teen 1 Veer 2 2 1 3 3 1 0 0 Does not veer a Draw the extensive form b Find the purestrategy Nash equilibrium or equilibria c Compute the mixedstrategy Nash equilibrium As part of your answer draw the bestresponse function diagram for the mixed strategies d Suppose the game is played sequentially with teen 1 moving first and committing to this action by throwing away the steering wheel What are teen 2s contingent strategies Write down the normal and extensive forms for the sequential version of the game e Using the normal form for the sequential version of the game solve for the Nash equilibria f Identify the proper subgames in the extensive form for the sequential version of the game Use backward induction to solve for the subgameperfect equilibrium Explain why the other Nash equilibria of the sequential game are unreasonable 84 Two neighboring homeowners i 5 1 2 simultaneously choose how many hours li to spend maintaining a beautiful lawn The average benefit per hour is 10 2 li 1 lj 2 and the opportunity cost per hour for each is 4 Home owner is average benefit is increasing in the hours neighbor j spends on his own lawn because the appearance of ones property depends in part on the beauty of the surrounding neighborhood a Compute the Nash equilibrium b Graph the bestresponse functions and indicate the Nash equilibrium on the graph c On the graph show how the equilibrium would change if the intercept of one of the neighbors average benefit functions fell from 10 to some smaller number 85 The Academy Awardwinning movie A Beautiful Mind about the life of John Nash dramatizes Nashs scholarly contribution in a single scene His equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students They notice several women one blond and the rest brunette and agree that the blond is more desirable than the brunettes The Nash character views the situation as a game among the male graduate students along the following lines Suppose there are n males who simultaneously approach either the blond or one of the brunettes If male i alone approaches the blond then he is successful in getting a date with her and earns payoff a If one or more other males approach the blond along with i the competition causes them all to lose her and i as well as the others who approached her earns a payoff of zero On the other hand male i earns a payoff of b 0 from approaching a brunette because there are more brunettes than males therefore i is certain to get a date with a brunette The desirability of the blond implies a b a Argue that this game does not have a symmetric purestrategy Nash equilibrium b Solve for the symmetric mixedstrategy equilibrium That is letting p be the probability that a male approaches the blond find p c Show that the more males there are the less likely it is in the equilibrium from part b that the blond is approached by at least one of them Note This paradox ical result was noted by S Anderson and M Engers in Participation Games Market Entry Coordination and the Beautiful Blond Journal of Economic Behavior Organization 63 2007 12037 86 The following game is a version of the Prisoners Dilemma but the payoffs are slightly different than in Figure 81 Suspect 2 Fink Silent Suspect 1 Fink 0 0 3 1 1 3 1 1 Silent a Verify that the Nash equilibrium is the usual one for the Prisoners Dilemma and that both players have dominant strategies b Suppose the stage game is repeated infinitely many times Compute the discount factor required for their suspects to be able to cooperate on silent each period Outline the trigger strategies you are considering for them Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 289 87 Return to the game with two neighbors in Problem 84 Con tinue to suppose that player is average benefit per hour of work on landscaping is 10 2 li 1 lj 2 Continue to suppose that player 2s opportunity cost of an hour of landscaping work is 4 Suppose that player 1s oppor tunity cost is either 3 or 5 with equal probability and that this cost is player 1s private information a Solve for the BayesianNash equilibrium b Indicate the BayesianNash equilibrium on a best response function diagram c Which type of player 1 would like to send a truthful sig nal to player 2 if it could Which type would like to hide his or her private information 88 In Blind Texan Poker player 2 draws a card from a standard deck and places it against her forehead without looking at it but so player 1 can see it Player 1 moves first deciding whether to stay or fold If player 1 folds he must pay player 2 50 If player 1 stays the action goes to player 2 Player 2 can fold or call If player 2 folds she must pay player 1 50 If player 2 calls the card is examined If it is a low card 28 player 2 pays player 1 100 If it is a high card 9 10 jack queen king or ace player 1 pays player 2 100 a Draw the extensive form for the game b Solve for the hybrid equilibrium c Compute the players expected payoffs Analytical Problems 89 Alternatives to Grim Strategy Suppose that the Prisoners Dilemma stage game see Figure 81 is repeated for infinitely many periods a Can players support the cooperative outcome by using titfortat strategies punishing deviation in a past period by reverting to the stagegame Nash equilibrium for just one period and then returning to cooperation Are two periods of punishment enough b Suppose players use strategies that punish deviation from cooperation by reverting to the stagegame Nash equi librium for 10 periods before returning to cooperation Compute the threshold discount factor above which cooperation is possible on the outcome that maximizes the joint payoffs 810 Refinements of perfect Bayesian equilibrium Recall the jobmarket signaling game in Example 89 a Find the conditions under which there is a pooling equilib rium where both types of worker choose not to obtain an education NE and where the firm offers an uneducated worker a job Be sure to specify beliefs as well as strategies b Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education NE and where the firm does not offer an uneducated worker a job What is the lowest posterior belief that the worker is lowskilled condi tional on obtaining an education consistent with this pooling equilibrium Why is it more natural to think that a lowskilled worker would never deviate to E and thus an educated worker must be highskilled Cho and Krepss intuitive criterion is one of a series of complicated refinements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identified in this part see I K Cho and D M Kreps Signalling Games and Stable Equilibria Quarterly Jour nal of Economics 102 1987 179221 Behavioral Problems 811 Fairness in the Ultimatum Game Consider a simple version of the Ultimatum Game discussed in the text The first mover proposes a division of 1 Let r be the share received by the other player in this proposal so the first mover keeps 1 2 r where 0 r 12 Then the other player moves responding by accepting or rejecting the proposal If the responder accepts the proposal the players are paid their shares if the responder rejects it both players receive nothing Assume that if the responder is indifferent between accepting or rejecting a proposal he or she accepts it a Suppose that players only care about monetary payoffs Verify that the outcome mentioned in the text in fact occurs in the unique subgameperfect equilibrium of the Ultimatum Game b Compare the outcome in the Ultimatum Game with the outcome in the Dictator Game also discussed in the text in which the proposers surplus division is imple mented regardless of whether the second mover accepts or rejects so it is not much of a strategic game c Now suppose that players care about fairness as well as money Following the article by Fehr and Schmidt cited in the text suppose these preferences are represented by the utility function U1 1x1 x22 5 x1 2 a0x1 2 x20 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 290 Part 3 Uncertainty and Strategy where x1 is player 1s payoff and x2 is player 2s a symmetric function holds for player 2 The first term reflects the usual desire for more money The second term reflects the desire for fairness that the players payoffs not be too unequal The parameter a measures how intense the preference for fairness is relative to the desire for more money Assume a 12 1 Solve for the responders equilibrium strategy in the Ultimatum Game 2 Taking into account how the second mover will respond solve for the proposers equilibrium strategy r in the Ultimatum Game Hint r will be a corner solution which depends on the value of a 3 Continuing with the fairness preferences compare the outcome in the Ultimatum Game with that in the Dicta tor Game Find cases that match the experimental results described in the text in particular in which the split of the pot of money is more even in the Ultimatum Game than in the Dictator Game Is there a limit to how even the split can be in the Ultimatum Game 812 Rotten Kid Theorem In A Treatise on the Family Cambridge MA Harvard Uni versity Press 1981 Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child player 1 and the childs parent player 2 The child moves first choosing an action r that affects both his own income Y1 1r2 and the income of his par ent Y2 1r2 where Yr1 1r2 0 and Yr2 1r2 0 Later the parent moves leaving a monetary bequest L to the child The child cares only for his own utility U1 1Y1 1 L2 but the parent max imizes U2 1Y2 2 L2 1 αU1 where α 0 reflects the parents altruism toward the child Prove that in a subgameperfect equilibrium the child will opt for the value of r that maxi mizes Y1 1 Y2 even though he has no altruistic intentions Hint Apply backward induction to the parents problem first which will give a firstorder condition that implicitly deter mines L although an explicit solution for L cannot be found the derivative of L with respect to rrequired in the childs firststage optimization problemcan be found using the implicit function rule Suggestions for Further Reading Fudenberg D and J Tirole Game Theory Cambridge MA MIT Press 1991 A comprehensive survey of game theory at the graduate student level although selected sections are accessible to advanced undergraduates Holt C A Markets Games Strategic Behavior Boston Pearson 2007 An undergraduate text with emphasis on experimental games Rasmusen E Games and Information 4th ed Malden MA Blackwell 2007 An advanced undergraduate text with many realworld applications Watson Joel Strategy An Introduction to Game Theory New York Norton 2002 An undergraduate text that balances rigor with simple examples often 2 2 games Emphasis on bargaining and contracting examples Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 291 EXTENSIONS ExisTEncE of nash Equilibrium This section will sketch John Nashs original proof that all finite games have at least one Nash equilibrium in mixed if not in pure strategies We will provide some of the details of the proof here the original proof is in Nash 1950 and a clear textbook presentation of the full proof is provided in Fuden berg and Tirole 1991 The section concludes by mentioning a related existence theorem for games with continuous actions Nashs proof is similar to the proof of the existence of a general competitive equilibrium in Chapter 13 Both proofs rely on a fixed point theorem The proof of the existence of Nash equilibrium requires a slightly more powerful theorem Instead of Brouwers fixed point theorem which applies to functions Nashs proof relies on Kakutanis fixed point theo rem which applies to correspondencesmore general map pings than functions E81 Correspondences versus functions A function maps each point in a first set to a single point in a second set A correspondence maps a single point in the first set to possibly many points in the second set Figure E81 illustrates the difference An example of a correspondence that we have already seen is the best response BRi 1s2i2 The best response need not map other players strategies si into a single strategy that is a best response for player i There may be ties among several best responses As shown in Figure 84 in the Bat tle of the Sexes the husbands best response to the wifes playing the mixed strategy of going to ballet with proba bility 23 and boxing with probability 13 or just w 5 23 for short is not just a single point but the whole interval of possible mixed strategies Both the husbands and the wifes best responses in this figure are correspondences not functions The reason Nash needed a fixed point theorem involv ing correspondences rather than just functions is precisely because his proof works with players best responses to prove existence The function graphed in a looks like a familiar curve Each value of x is mapped into a single value of y With the correspondence graphed in b each value of x may be mapped into many values of y Thus correspondences can have bulges as shown by the shaded regions in b a Function b Correspondence y x y x FIGURE E81 Comparison of Functions and Correspondences Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 292 Part 3 Uncertainty and Strategy E82 Kakutanis fixed point theorem Here is the statement of Kakutanis fixed point theorem Any convex uppersemicontinuous correspondence 3f 1x24 from a closed bounded convex set into itself has at least one fixed point 1x2 such that x f 1x2 Comparing the statement of Kakutanis fixed point theorem with Brouwers in Chapter 13 they are similar except for the substitution of correspondence for function and for the conditions on the correspondence Brouwers theorem requires the function to be continuous Kakutanis theorem requires the correspondence to be convex and upper semicontinuous These properties which are related to continuity are less familiar and worth spending a moment to understand Figure E82 provides examples of correspondences violating a convexity and b upper semicontinuity The figure shows why the two properties are needed to guarantee a fixed point Without both properties the correspondence can jump across the 45 line and thus fail to have a fixed pointthat is a point for which x 5 f 1x2 E83 Nashs proof We use R1s2 to denote the correspondence that underlies Nashs existence proof This correspondence takes any profile of players strategies s 5 1s1 s2 sn2 possibly mixed and maps it into another mixed strategy profile the profile of best responses R1s2 5 1BR1 1s212 BR2 1s222 BRn 1s2n22 i A fixed point of the correspondence is a strategy for which s R1s2 this is a Nash equilibrium because each players strategy is a best response to others strategies The proof checks that all the conditions involved in Kaku tanis fixed point theorem are satisfied by the bestresponse correspondence R1s2 First we need to show that the set of mixedstrategy profiles is closed bounded and convex Because a strategy profile is just a list of individual strategies the set of strategy profiles will be closed bounded and convex if each players strategy set Si has these properties individually As Figure E83 shows for the case of two and three actions the set of mixed strategies over actions has a simple shape1 The set is closed contains its boundary bounded does not go off to infinity in any direction and convex the segment between any two points in the set is also in the set We then need to check that the bestresponse correspon dence R1s2 is convex Individual best responses cannot look like Figure E82a because if any two mixed strategies such as A and B are best responses to others strategies then mixed strategies between them must also be best responses For example in the Battle of the Sexes if 13 23 and 23 13 are best responses for the husband against his wifes playing 23 13 where in each pair the first number is the probability of playing ballet and the second of playing boxing then mixed strategies between the two such as 12 12 must also be best responses for him Figure 84 showed that in fact all possible mixed strategies for the husband are best responses to the wifes playing 23 13 The correspondence in a is not convex because the dashed vertical segment between A and B is not inside the correspondence The correspondence in b is not upper semicontinuous because there is a path C inside the correspondence leading to a point D that as indicated by the open circle is not inside the correspondence Both a and b fail to have fixed points b Correspondence that is not upper semicontinuous 1 D 45 a Correspondence that is not convex fx 1 1 x fx x B A 45 C FIGURE E82 Kakutanis Conditions on Correspondence 1Mathematicians study them so frequently that they have a special name for such a set a simplex Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 8 Game Theory 293 Finally we need to check that R1s2 is upper semicontin uous Individual best responses cannot look like in Figure E82b They cannot have holes like point D punched out of them because payoff functions Ui 1si s2i2 are continuous Recall that payoffs when written as functions of mixed strat egies are actually expected values with probabilities given by the strategies si and s2i As Equation 2176 showed expected values are linear functions of the underlying probabilities Linear functions are of course continuous E84 Games with continuous actions Nashs existence theorem applies to finite gamesthat is games with a finite number of players and actions per player Nashs theorem does not apply to games that feature continuous actions such as the Tragedy of the Commons in Example 85 Is a Nash equilibrium guaranteed to exist for these games too Glicksberg 1952 proved that the answer is yes as long as payoff functions are continuous References Fudenberg D and J Tirole Game Theory Cambridge MA MIT Press 1991 sec 13 Glicksberg I L A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilib rium Points Proceedings of the National Academy of Sci ences 38 1952 17074 Nash John Equilibrium Points in nPerson Games Proceed ings of the National Academy of Sciences 36 1950 4849 Player 1s set of possible mixed strategies over two actions is given by the diagonal line segment in a The set for three actions is given by the shaded triangle on the threedimensional graph in b a Two actions b Tree actions 1 0 1 1 1 1 p1 1 p1 1 p1 2 p1 2 p1 3 0 FIGURE E83 Set of Mixed Strategies for an Individual Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 295 PART FOUR Production and Supply Chapter 9 Production Functions Chapter 10 Cost Functions Chapter 11 Profit Maximization In this part we examine the production and supply of economic goods Institutions that coordinate the transformation of inputs into outputs are called firms They may be large institutions such as Google Sony or the US Department of Defense or small ones such as Mom and Pop stores or selfem ployed individuals Although they may pursue different goals Google may seek maximum profits whereas an Israeli kibbutz may try to make members of the kibbutz as well off as possible all firms must make certain basic choices in the production process The purpose of Part 4 is to develop some tools for analyzing those choices In Chapter 9 we examine ways of modeling the physical relationship between inputs and outputs We introduce the concept of a production function a useful abstraction from the complexities of realworld production processes Two measurable aspects of the production function are stressed its returns to scale ie how output expands when all inputs are increased and its elasticity of substitution ie how easily one input may be replaced by another while maintaining the same level of output We also briefly describe how technical improvements are reflected in produc tion functions The production function concept is then used in Chapter 10 to discuss costs of production We assume that all firms seek to produce their output at the lowest possible cost an assumption that permits the development of cost functions for the firm Chapter 10 also focuses on how costs may differ between the short run and the long run In Chapter 11 we investigate the firms supply decision To do so we assume that the firms manager will make input and output choices to maximize profits The chapter concludes with the fundamental model of supply behavior by profitmaximizing firms that we will use in many subsequent chapters Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 CHAPTER NINE Production Functions The principal activity of any firm is to turn inputs into outputs Because economists are interested in the choices the firm makes in accomplishing this goal but wish to avoid discussing many of the engineering intricacies involved they have chosen to construct an abstract model of production In this model the relationship between inputs and outputs is formalized by a production function of the form q 5 f 1k l m c2 91 where q represents the firms output of a particular good during a period1 k represents the machine ie capital usage during the period l represents hours of labor input m represents raw materials used2 and the notation indicates the possibility of other variables affecting the production process Equation 91 is assumed to provide for any conceivable set of inputs the engineers solution to the problem of how best to combine those inputs to get output 91 MARGINAL PRODUCTIVITY In this section we look at the change in output brought about by a change in one of the productive inputs For the purposes of this examination and indeed for most of the pur poses of this book it will be more convenient to use a simplified production function defined as follows D E F I N IT ION Production function The firms production function for a par ticular good q q 5 f 1k l2 92 shows the maximum amount of the good that can be produced using alternative combinations of capital k and labor l 1Here we use a lowercase q to represent one firms output We reserve the uppercase Q to represent total output in a market Generally we assume that a firm produces only one output Issues that arise in multiproduct firms are discussed in a few footnotes and problems 2In empirical work raw material inputs often are disregarded and output q is measured in terms of value added Of course most of our analysis will hold for any two inputs to the production process we might wish to examine The terms capital and labor are used only for convenience Similarly it would be a simple matter to generalize our discussion to cases involving more than two inputs occasionally we will do so For the most part however limiting the discussion to two inputs will be helpful because we can show these inputs on two dimensional graphs 297 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 298 Part 4 Production and Supply 911 Marginal physical product To study variation in a single input we define marginal physical product as follows D E F I N I T I O N Marginal physical product The marginal physical product of an input is the additional output that can be produced by using one more unit of that input while holding all other inputs constant Mathematically marginal physical product of capital 5 MPk 5 q k 5 fk marginal physical product of labor 5 MPl 5 q l 5 fl 93 Notice that the mathematical definitions of marginal product use partial derivatives thereby properly reflecting the fact that all other input usage is held constant while the input of interest is being varied For example consider a farmer hiring one more laborer to harvest the crop but holding all other inputs constant The extra output this laborer produces is that farmhands marginal physical product measured in physical quantities such as bushels of wheat crates of oranges or heads of lettuce We might observe for example that 50 workers on a farm are able to produce 100 bushels of wheat per year whereas 51 workers with the same land and equipment can produce 102 bushels The marginal physical product of the 51st worker is then 2 bushels per year 912 Diminishing marginal productivity We might expect that the marginal physical product of an input depends on how much of that input is used Labor for example cannot be added indefinitely to a given field while keeping the amount of equipment fertilizer and so forth fixed without eventually exhibit ing some deterioration in its productivity Mathematically the assumption of diminishing marginal physical productivity is an assumption about how secondorder partial deriva tives of the production function behave in the limit MPk k 5 2f k2 5 fkk 0 for high enough k MPl l 5 2f l2 5 fll 0 for high enough l 94 The assumption of diminishing marginal productivity was originally proposed by the nineteenthcentury economist Thomas Malthus who worried that rapid increases in pop ulation would result in lower labor productivity His gloomy predictions for the future of humanity led economics to be called the dismal science But the mathematics of the production function suggests that such gloom may be misplaced Changes in the mar ginal productivity of labor over time depend not only on how labor input is growing but also on changes in other inputs such as capital That is we must also be concerned with MPlk 5 flk In most cases flk 0 thus declining labor productivity as both l and k increase is not a foregone conclusion Indeed it appears that labor productivity has risen significantly since Malthus time primarily because increases in capital inputs along with technical improvements have offset the impact of decreasing marginal productivity alone Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 299 913 Average physical productivity In common usage the term labor productivity often means average productivity When it is said that a certain industry has experienced productivity increases this is taken to mean that output per unit of labor input has increased Although the concept of aver age productivity is not nearly as important in theoretical economic discussions as mar ginal productivity is it receives a great deal of attention in empirical discussions Because average productivity is easily measured say as so many bushels of wheat per laborhour input it is often used as a measure of efficiency We define the average product of labor 1APl2 to be APl 5 output labor input 5 q l 5 f1k l2 l 95 Notice that APl also depends on the level of capital used This observation will prove to be important when we examine the measurement of technical change at the end of this chapter EXAMPLE 91 A TwoInput Production Function Suppose the production function for flyswatters during a particular period can be represented by q 5 f1k l2 5 600k2l2 2 k3l3 96 To construct the marginal and average productivity functions of labor l for this function we must assume a particular value for the other input capital k Suppose k 5 10 Then the produc tion function is given by q 5 60000l2 2 1000l3 97 Marginal product The marginal productivity function when k 10 is given by MPl 5 q l 5 120000l 2 3000l2 98 which diminishes as l increases eventually becoming negative This implies that q reaches a max imum value Setting MPl equal to 0 120000l 2 3000l 2 5 0 99 yields 40l 5 l2 or l 5 40 as the point at which q reaches its maximum value Labor input beyond 40 units per period actually reduces total output For example when l 5 40 Equation 97 shows that q 5 32 million flyswatters whereas when l 5 50 production of flyswatters amounts to only 25 million Average product To find the average productivity of labor in flyswatter production we divide q by l still holding k 5 10 APl 5 q l 5 60000l 2 1000l2 910 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 300 Part 4 Production and Supply 92 ISOQUANT MAPS AND THE RATE OF TECHNICAL SUBSTITUTION To illustrate possible substitution of one input for another in a production function we use its isoquant map Again we study a production function of the form q 5 f 1k l2 with the understanding that capital and labor are simply convenient examples of any two inputs that might happen to be of interest An isoquant from iso meaning equal records those combinations of k and l that are able to produce a given quantity of output For example all those combinations of k and l that fall on the curve labeled q 5 10 in Figure 91 are capable of producing 10 units of output per period This isoquant then records the fact that there are many alternative ways of producing 10 units of output One way might be rep resented by point A We would use lA and kA to produce 10 units of output Alternatively we might prefer to use relatively less capital and more labor and therefore would choose a point such as B Hence we may define an isoquant as follows Again this is an inverted parabola that reaches its maximum value when APl l 5 60000 2 2000l 5 0 911 which occurs when l 5 30 At this value for labor input Equation 910 shows that APl 5 900000 and Equation 98 shows that MPl is also 900000 When APl is at a maximum average and mar ginal productivities of labor are equal3 Notice the relationship between total output and average productivity that is illustrated by this example Even though total production of flyswatters is greater with 40 workers 32 million than with 30 workers 27 million output per worker is higher in the second case With 40 work ers each worker produces 800000 flyswatters per period whereas with 30 workers each worker produces 900000 Because capital input flyswatter presses is held constant in this definition of productivity the diminishing marginal productivity of labor eventually results in a declining level of output per worker QUERY How would an increase in k from 10 to 11 affect the MPl and APl functions here Explain your results intuitively 3This result is general Because APl l 5 l MPl 2 q l2 at a maximum l MPl 5 q or MPl 5 APl D E F I N I T I O N Isoquant An isoquant shows those combinations of k and l that can produce a given level of output say q0 Mathematically an isoquant records the set of k and l that satisfies f1k l2 5 q0 912 As was the case for indifference curves there are infinitely many isoquants in the k 2 l plane Each isoquant represents a different level of output Isoquants record successively higher levels of output as we move in a northeasterly direction Presumably using more Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 301 of each of the inputs will permit output to increase Two other isoquants for q 5 20 and q 5 30 are shown in Figure 91 You will notice the similarity between an isoquant map and the individuals indifference curve map discussed in Part 2 They are indeed similar concepts because both represent contour maps of a particular function For isoquants however the labeling of the curves is measurablean output of 10 units per period has a quantifiable meaning Therefore economists are more interested in studying the shape of production functions than in examining the exact shape of utility functions 921 The marginal rate of technical substitution RTS The slope of an isoquant shows how one input can be traded for another while holding output constant Examining the slope provides information about the technical possibility of substituting labor for capital A formal definition follows Isoquants record the alternative combinations of inputs that can be used to produce a given level of out put The slope of these curves shows the rate at which l can be substituted for k while keeping output con stant The negative of this slope is called the marginal rate of technical substitution RTS In the figure the RTS is positive and diminishing for increasing inputs of labor FIGURE 91 An Isoquant Map k per period l per period kA lA lB kB A B q 30 q 20 q 10 D E F I N I T I O N Marginal rate of technical substitution The marginal rate of technical substitution RTS shows the rate at which having added a unit of labor capital can be decreased while holding output constant along an isoquant In mathematical terms RTS 1l for k2 5 2dk dl q5q0 913 In this definition the notation is intended as a reminder that output is to be held constant as l is substituted for k The particular value of this tradeoff rate will depend not only on the level of output but also on the quantities of capital and labor being used Its value depends on the point on the isoquant map at which the slope is to be measured Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 302 Part 4 Production and Supply 922 RTS and marginal productivities To examine the shape of production function isoquants it is useful to prove the following result The RTS of l for k is equal to the ratio of the marginal physical productivity of labor 1MPl2 to the marginal physical productivity of capital 1MPk2 Imagine using Equa tion 912 to graph the q0 isoquant We would substitute a sequence of increasing values of l and see how k would have to adjust to keep output constant at q0 The graph of the iso quant is really the graph of the implicit function k1l2 satisfying q0 5 f 1k1l2 l2 914 Just as we did in the section on implicit functions in Chapter 2 see in particular Equation 222 we can use the chain rule to differentiate Equation 914 giving 0 5 fk dk dl 1 fl 5 MPk dk dl 1 MPl 915 where the initial 0 appears because q0 is being held constant therefore the derivative of the left side of Equation 914 with respect to l equals 0 Rearranging Equation 915 gives RTS 1l for k2 5 2dk dl q5q0 5 MPl MPk 916 Hence the RTS is given by the ratio of the inputs marginal productivities Equation 916 shows that those isoquants that we actually observe must be negatively sloped Because both MPl and MPk will be nonnegative no firm would choose to use a costly input that reduced output the RTS also will be positive or perhaps zero Because the slope of an isoquant is the negative of the RTS any firm we observe will not be oper ating on the positively sloped portion of an isoquant Although it is mathematically possi ble to devise production functions whose isoquants have positive slopes at some points it would not make economic sense for a firm to opt for such input choices 923 Reasons for a diminishing RTS The isoquants in Figure 91 are drawn not only with a negative slope as they should be but also as convex curves Along any one of the curves the RTS is diminishing For high ratios of k to l the RTS is a large positive number indicating that a great deal of capital can be given up if one more unit of labor becomes available On the other hand when a lot of labor is already being used the RTS is low signifying that only a small amount of capital can be traded for an additional unit of labor if output is to be held constant This assump tion would seem to have some relationship to the assumption of diminishing marginal productivity A hasty use of Equation 916 might lead one to conclude that an increase in l accompanied by a decrease in k would result in a decrease in MPl an increase in MPk and therefore a decrease in the RTS The problem with this quick proof is that the marginal productivity of an input depends on the level of both inputschanges in l affect MPk and vice versa It is not possible to derive a diminishing RTS from the assumption of diminish ing marginal productivity alone To see why this is so mathematically assume that q 5 f1k l2 and that fk and fl are pos itive ie the marginal productivities are positive Assume also that fkk 0 and fll 0 that the marginal productivities are diminishing To show that isoquants are convex we would like to show that d1RTS2dl 0 Because RTS 5 flfk we have dRTS dl 5 d1 flfk2 dl 917 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 303 Because fl and fk are functions of both k and l we must be careful in taking the derivative of this expression dRTS dl 5 fk 1 fll 1 flk dkdl2 2 fl1 fkl 1 fkk dkdl2 1 fk2 2 918 Using the fact that dkdl 5 2flfk along an isoquant and Youngs theorem 1 fkl 5 flk2 we have dRTS dl 5 f 2 k fll 2 2fk flfkl 1 f 2 l fkk 1 fk2 3 919 Because we have assumed fk 0 the denominator of this function is positive Hence the whole fraction will be negative if the numerator is negative Because fll and fkk are both assumed to be negative the numerator definitely will be negative if fkl is positive If we can assume this we have shown that dRTSdl 0 that the isoquants are convex4 924 Importance of crossproductivity effects Intuitively it seems reasonable that the crosspartial derivative fkl 5 flk should be positive If workers had more capital they would have higher marginal productivities Although this is probably the most prevalent case it does not necessarily have to be so Some production functions have fkl 0 at least for a range of input values When we assume a diminishing RTS as we will throughout most of our discussion we are therefore making a stronger assumption than simply diminishing marginal productivities for each inputspecifically we are assuming that marginal productivities diminish rapidly enough to compensate for any possible negative crossproductivity effects Of course as we shall see later with three or more inputs things become even more complicated EXAMPLE 92 A Diminishing RTS In Example 91 the production function for flyswatters was given by q 5 f 1k l2 5 600k2l2 2 k3l3 920 General marginal productivity functions for this production function are MPl 5 fl 5 q l 5 1200k2l 2 3k3l2 MPk 5 fk 5 q k 5 1200kl2 2 3k2l3 921 Notice that each of these depends on the values of both inputs Simple factoring shows that these marginal productivities will be positive for values of k and l for which kl 400 Because fll 5 1200k2 2 6k3l fkk 5 1200l2 2 6kl3 922 it is clear that this function exhibits diminishing marginal productivities for sufficiently large values of k and l Indeed again by factoring each expression it is easy to show that fll fkk 0 if kl 200 However even within the range 200 kl 400 where the marginal productivity 4As we pointed out in Chapter 2 functions for which the numerator in Equation 919 is negative are called strictly quasiconcave functions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 304 Part 4 Production and Supply 93 RETURNS TO SCALE We now proceed to characterize production functions A first question that might be asked about them is how output responds to increases in all inputs together For example suppose that all inputs were doubled Would output double or would the relationship not be so simple This is a question of the returns to scale exhibited by the production func tion that has been of interest to economists ever since Adam Smith intensively studied the production of pins Smith identified two forces that came into operation when the conceptual experiment of doubling all inputs was performed First a doubling of scale permits a greater division of labor and specialization of function Hence there is some presumption that efficiency might increaseproduction might more than double Sec ond doubling of the inputs also entails some loss in efficiency because managerial over seeing may become more difficult given the larger scale of the firm Which of these two tendencies will have a greater effect is an important empirical question These concepts can be defined technically as follows relations for this function behave normally this production function may not necessarily have a diminishing RTS Crossdifferentiation of either of the marginal productivity functions Equation 921 yields fkl 5 flk 5 2400kl 2 9k2l 2 923 which is positive only for kl 266 Therefore the numerator of Equation 919 will definitely be negative for 200 kl 266 but for largerscale flyswatter factories the case is not so clear because fkl is negative When fkl is neg ative increases in labor input reduce the marginal productivity of capital Hence the intuitive argument that the assumption of diminishing marginal productivities yields an unambiguous prediction about what will happen to the RTS 15flfk2 as l increases and k decreases is incor rect It all depends on the relative effects on marginal productivities of diminishing marginal pro ductivities which tend to reduce fl and increase fk and the contrary effects of crossmarginal productivities which tend to increase fl and reduce fk Still for this flyswatter case it is true that the RTS is diminishing throughout the range of k and l where marginal productivities are positive For cases where 266 kl 400 the diminishing marginal productivities exhibited by the function are sufficient to overcome the influence of a negative value for fkl on the convexity of isoquants QUERY For cases where k 5 l what can be said about the marginal productivities of this production function How would this simplify the numerator for Equation 919 How does this permit you to more easily evaluate this expression for some larger values of k and l D E F I N I T I O N Returns to scale If the production function is given by q 5 f1k l2 and if all inputs are multi plied by the same positive constant t where t 1 then we classify the returns to scale of the production function by Effect on Output Returns to Scale f 1tk tl2 5 tf 1k l2 5 tq Constant f 1tk tl2 tf 1k l2 5 tq Decreasing f 1tk tl2 tf 1k l2 5 tq Increasing Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 305 In intuitive terms if a proportionate increase in inputs increases output by the same propor tion the production function exhibits constant returns to scale If output increases less than proportionately the function exhibits diminishing returns to scale And if output increases more than proportionately there are increasing returns to scale As we shall see it is theoreti cally possible for a function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels5 Often however economists refer to returns to scale of a production function with the implicit understanding that only a fairly narrow range of variation in input usage and the related level of output is being considered 931 Constant returns to scale There are economic reasons why a firms production function might exhibit constant returns to scale If the firm operates many identical plants it may increase or decrease production simply by varying the number of them in current operation That is the firm can double output by doubling the number of plants it operates and that will require it to employ precisely twice as many inputs Empirical studies of production functions often find that returns to scale are roughly constant for the firms studied at least around for outputs close to the firms established operating levelsthe firms may exhibit increasing returns to scale as they expand to their established size For all these reasons the constant returnstoscale case seems worth examining in somewhat more detail When a production function exhibits constant returns to scale it meets the definition of homogeneity that we introduced in Chapter 2 That is the production is homogeneous of degree 1 in its inputs because f1tk tl2 5 t1f1k l2 5 tq 924 In Chapter 2 we showed that if a function is homogeneous of degree k its derivatives are homogeneous of degree k 2 1 In this context this implies that the marginal productivity functions derived from a constant returnstoscale production function are homogeneous of degree 0 That is MPk 5 f1k l2 k 5 f1tk tl2 k MPl 5 f1k l2 l 5 f1tk tl2 l 925 for any t 0 In particular we can let t 5 1l in Equations 925 and get MPk 5 f1kl 12 k MPl 5 f1kl 12 l 926 That is the marginal productivity of any input depends only on the ratio of capital to labor input not on the absolute levels of these inputs This fact is especially important for exam ple in explaining differences in productivity among industries or across countries 5A local measure of returns to scale is provided by the scale elasticity defined as eq t 5 f 1tk tl2 t t f 1tk tl2 where this expression is to be evaluated at t 5 1 This parameter can in principle take on different values depending on the level of input usage For some examples using this concept see Problem 99 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 306 Part 4 Production and Supply 932 Homothetic production functions One consequence of Equations 926 is that the RTS 15MPlMPk2 for any constant returns toscale production function will depend only on the ratio of the inputs not on their abso lute levels That is such a function will be homothetic see Chapter 2its isoquants will be radial expansions of one another This situation is shown in Figure 92 Along any ray through the origin where the ratio kl does not change the slopes of successively higher isoquants are identical This property of the isoquant map will be useful to us on several occasions A simple numerical example may provide some intuition about this result Suppose a large bread order consisting of say 200 loaves can be filled in one day by three bakers working with three ovens or by two bakers working with four ovens Therefore the RTS of ovens for bakers is one for oneone extra oven can be substituted for one baker If this production process exhibits constant returns to scale two large bread orders totaling 400 loaves can be filled in one day either by six bakers with six ovens or by four bakers with eight ovens In the latter case two ovens are substituted for two bakers so again the RTS is one for one In constant returnstoscale cases expanding the level of production does not alter tradeoffs among inputs thus production functions are homothetic A production function can have a homothetic indifference curve map even if it does not exhibit constant returns to scale As we showed in Chapter 2 this property of homo theticity is retained by any monotonic transformation of a homogeneous function Hence increasing or decreasing returns to scale can be incorporated into a constant returnsto scale function through an appropriate transformation Perhaps the most common such Because a constant returnstoscale production function is homothetic the RTS depends only on the ratio of k to l not on the scale of production Consequently along any ray through the origin a ray of constant kl the RTS will be the same on all isoquants An additional feature is that the isoquant labels increase proportionately with the inputs FIGURE 92 Isoquant Map for a Constant ReturnstoScale Production Function k per period l per period q 1 q 2 q 3 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 307 transformation is exponential Thus if f1k l2 is a constant returnstoscale production function we can let F1k l2 5 3 f1k l2 4 γ 927 where γ is any positive exponent If γ 1 then F1tk tl2 5 3f1tk tl2 4 γ 5 3tf1k l2 4 γ 5 tγ 3f1k l2 4 γ 5 tγF1k l2 tF1k l2 928 for any t 1 Hence this transformed production function exhibits increasing returns to scale The exponent γ captures the degree of the increasing returns to scale A doubling of inputs would lead to a fourfold increase in output if γ 5 2 but an eightfold increase if γ 5 3 An identical proof shows that the function F exhibits decreasing returns to scale for γ 1 Because this function remains homothetic through all such transformations we have shown that there are important cases where the issue of returns to scale can be sepa rated from issues involving the shape of an isoquant In these cases changes in the returns to scale will just change the labels on the isoquants rather than their shapes In the next section we will look at how shapes of isoquants can be described 933 The ninput case The definition of returns to scale can be easily generalized to a production function with n inputs If that production function is given by q 5 f 1x1 x2 c xn2 929 and if all inputs are multiplied by t 1 we have f1tx1 tx2 c txn2 5 t kf1x1 x2 c xn2 5 tkq 930 for some constant k If k 5 1 the production function exhibits constant returns to scale Decreasing and increasing returns to scale correspond to the cases k 1 and k 1 respectively The crucial part of this mathematical definition is the requirement that all inputs be increased by the same proportion t In many realworld production processes this provision may make little economic sense For example a firm may have only one boss and that number would not necessarily be doubled even if all other inputs were Or the output of a farm may depend on the fertility of the soil It may not be literally possible to double the acres planted while maintaining fertility because the new land may not be as good as that already under cultivation Hence some inputs may have to be fixed or at least imperfectly variable for most practical purposes In such cases some degree of diminishing productivity a result of increasing employment of variable inputs seems likely although this cannot properly be called diminishing returns to scale because of the presence of inputs that are held fixed 94 THE ELASTICITY OF SUBSTITUTION Another important characteristic of the production function is how easy it is to substitute one input for another This is a question about the shape of a single isoquant rather than about the whole isoquant map Along one isoquant the rate of technical substitution will decrease as the capitallabor ratio decreases ie as kl decreases now we wish to define some parameter that measures this degree of responsiveness If the RTS does not change at all for changes in kl we might say that substitution is easy because the ratio of the mar ginal productivities of the two inputs does not change as the input mix changes Alterna tively if the RTS changes rapidly for small changes in kl we would say that substitution Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 308 Part 4 Production and Supply is difficult because minor variations in the input mix will have a substantial effect on the inputs relative productivities A scalefree measure of this responsiveness is provided by the elasticity of substitution a concept we encountered informally in our discussion of CES utility functions Here we will work on providing a more formal definition For discrete changes the elasticity of substitution is given by σ 5 percent D1kl2 percent DRTS 5 D1kl2 kl 4 DRTS RTS 5 D1kl2 DRTS RTS kl 931 More often we will be interested in considering small changes therefore a modification of Equation 931 will be of more interest σ 5 d1kl2 d RTS RTS kl 5 d ln 1kl2 d ln RTS 932 The logarithmic expression follows from mathematical derivations along the lines of Example 22 from Chapter 2 All these equations can be collected in the following formal definition 6The elasticity of substitution can be phrased directly in terms of the production function and its derivatives in the constant returnstoscale case as σ 5 fk fl f fkl But this form is cumbersome Hence usually the logarithmic definition in Equation 933 is easiest to apply For a concise summary see P Berck and K Sydsaeter Economists Mathematical Manual Berlin Germany SpringerVerlag 1999 chap 5 D E F I N I T I O N Elasticity of substitution For the production function q 5 f 1k l2 the elasticity of substitution σ measures the proportionate change in kl relative to the proportionate change in the RTS along an isoquant That is σ 5 percent D 1kl2 percent DRTS 5 d1kl2 d RTS RTS kl 5 d ln 1kl2 d ln RTS 5 d ln 1kl2 d ln 1 flfk2 933 Because along an isoquant kl and RTS move in the same direction the value of σ is always positive Graphically this concept is illustrated in Figure 93 as a movement from point A to point B on an isoquant In this movement both the RTS and the ratio kl will change we are interested in the relative magnitude of these changes If σ is high then the RTS will not change much relative to kl and the isoquant will be close to linear On the other hand a low value of σ implies a rather sharply curved isoquant the RTS will change by a substantial amount as kl changes In general it is possible that the elasticity of sub stitution will vary as one moves along an isoquant and as the scale of production changes Often however it is convenient to assume that σ is constant along an isoquant If the pro duction function is also homothetic thenbecause all the isoquants are merely radial blowupsσ will be the same along all isoquants We will encounter such functions later in this chapter and in many of the end of chapter problems6 941 The ninput case Generalizing the elasticity of substitution to the manyinput case raises several complica tions One approach is to adopt a definition analogous to Equation 933 that is to define Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 309 the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in the RTS between them while holding out put constant7 To make this definition complete it is necessary to require that all inputs other than the two being examined be held constant However this latter requirement which is not relevant when there are only two inputs restricts the value of this poten tial definition In realworld production processes it is likely that any change in the ratio of two inputs will also be accompanied by changes in the levels of other inputs Some of these other inputs may be complementary with the ones being changed whereas others may be substitutes and to hold them constant creates a rather artificial restriction For this reason an alternative definition of the elasticity of substitution that permits such comple mentarity and substitutability in the firms cost function is generally used in the ngood case Because this concept is usually measured using cost functions we will describe it in the next chapter In moving from point A to point B on the q0 isoquant both the capitallabor ratio 1kl2 and the RTS will change The elasticity of substitution σ is defined to be the ratio of these proportional changes it is a measure of how curved the isoquant is FIGURE 93 Graphic Description of the Elasticity of Substitution k per period l per period q0 A B k l A k l B RTSA RTSB 7That is the elasticity of substitution between input i and input j might be defined as σij 5 ln 1xixj2 ln 1fjfi2 for movements along f 1x1 x2 c xn2 5 q0 Notice that the use of partial derivatives in this definition effectively requires that all inputs other than i and j be held constant when considering movements along the q0 isoquant Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 310 Part 4 Production and Supply 95 FOUR SIMPLE PRODUCTION FUNCTIONS In this section we illustrate four simple production functions each characterized by a dif ferent elasticity of substitution These are shown only for the case of two inputs but gener alization to many inputs is easily accomplished see the Extensions for this chapter 951 Case 1 Linear 1σ 5 2 Suppose that the production function is given by q 5 f 1k l2 5 αk 1 βl 934 It is easy to show that this production function exhibits constant returns to scale For any t 1 f1tk tl2 5 αtk 1 βtl 5 t1αk 1 βl2 5 tf1k l2 935 All isoquants for this production function are parallel straight lines with slope 2βα Such an isoquant map is pictured in Figure 94a Because the RTS is constant along any straight line isoquant the denominator in the definition of σ Equation 933 is equal to 0 and hence σ is infinite Although this linear production function is a useful example it is rarely encountered in practice because few production processes are characterized by such ease of substitution Indeed in this case capital and labor can be thought of as perfect substi tutes for each other An industry characterized by such a production function could use only capital or only labor depending on these inputs prices It is hard to envision such a production process Every machine needs someone to press its buttons and every laborer requires some capital equipment however modest 952 Case 2 Fixed proportions 1σ 5 02 Production functions characterized by σ 5 0 have Lshaped isoquants as depicted in Figure 94b At the corner of an Lshaped isoquant a negligible increase in kl causes an infinite increase in RTS because the isoquant changes suddenly from horizontal to vertical there Substituting 0 for the change in kl in the numerator of the formula for σ in Equation 931 and infinity for the change in RTS in the denominator implies σ 5 0 A firm would always operate at the corner of an isoquant Operating anywhere else is inefficient because the same output could be produced with fewer inputs by moving along the isoquant toward the corner As drawn in Figure 94 the corners of the isoquants all lie along the same ray from the origin This illustrates the important special case of a fixedproportions production function Because the firm always operates at the corner of some isoquant and all isoquants line up along the same ray it must be the case that the firm uses inputs in the fixed proportions given by the slope of this ray regardless of how much it produces8 The inputs are perfect complements in that starting from the fixed proportion an increase in one input is useless unless the other is increased as well The mathematical form of the fixedproportions production function is given by q 5 min 1αk βl2 α β 0 936 where the operator min means that q is given by the smaller of the two values in paren theses For example suppose that αk βl then q 5 αk and we would say that capital is 8Production functions with σ 5 0 need not be fixed proportions The other possibility is that the corners of the isoquants lie along a nonlinear curve from the origin rather than lining up along a ray Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 311 the binding constraint in this production process The employment of more labor would not increase output and hence the marginal product of labor is zero additional labor is superfluous in this case Similarly if αk βl then labor is the binding constraint on out put and additional capital is superfluous When αk 5 βl both inputs are fully utilized When this happens kl 5 βα and production takes place at a vertex on the isoquant map If both inputs are costly this is the only costminimizing place to operate The locus of all such vertices is a straight line through the origin with a slope given by βα9 Three possible values for the elasticity of substitution are illustrated in these figures In a capital and labor are perfect substitutes In this case the RTS will not change as the capitallabor ratio changes In b the fixedproportions case no substitution is possible The capitallabor ratio is fixed at βα A case of intermediate substitutability is illustrated in c FIGURE 94 Isoquant Maps for Simple Production Functions with Various Values for σ k per period k per period k per period l per period l per period l per period q3 q2 q1 q3 q2 q1 q3 q2 q1 a σ b σ 0 c σ 1 Slope q3 q3 9With the form reflected by Equation 936 the fixedproportions production function exhibits constant returns to scale because f 1tk tl2 5 min 1αtk βtl2 5 t min 1αk βl2 5 tf 1k l2 for any t 1 As before increasing or decreasing returns can be easily incorporated into the functions by using a nonlinear transformation of this functional formsuch as 3 f 1k l24 γ where γ may be greater than or less than 1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 312 Part 4 Production and Supply The fixedproportions production function has a wide range of applications Many machines for example require a certain number of people to run them but any excess labor is superfluous Consider combining capital a lawn mower and labor to mow a lawn It will always take one person to run the mower and either input without the other is not able to produce any output at all It may be that many machines are of this type and require a fixed complement of workers per machine10 953 Case 3 CobbDouglas 1σ 5 12 The production function for which σ 5 1 called a CobbDouglas production function11 provides a middle ground between the two polar cases previously discussed Isoquants for the CobbDouglas case have the normal convex shape and are shown in Figure 94c The mathematical form of the CobbDouglas production function is given by q 5 f 1k l2 5 Akαlβ 937 where A α and β are all positive constants The CobbDouglas function can exhibit any degree of returns to scale depending on the values of α and β Suppose all inputs were increased by a factor of t Then f1tk tl2 5 A 1tk2 α1tl2 β 5 Atα1βkαlβ 5 tα1β 1k l2 938 Hence if α 1 β 5 1 the CobbDouglas function exhibits constant returns to scale because output also increases by a factor of t If α 1 β 1 then the function exhibits increasing returns to scale whereas α 1 β 1 corresponds to the decreasing returnstoscale case It is a simple matter to show that the elasticity of substitution is 1 for the CobbDouglas function12 This fact has led researchers to use the constant returnstoscale version of the function for a general description of aggregate production relationships in many countries The CobbDouglas function has also proved to be useful in many applications because it is linear in logarithms ln q 5 ln A 1 α ln k 1 β ln l 939 The constant α is then the elasticity of output with respect to capital input and β is the elas ticity of output with respect to labor input 13 These constants can sometimes be estimated 10The lawn mower example points up another possibility however Presumably there is some leeway in choosing what size of lawn mower to buy Hence before the actual purchase the capitallabor ratio in lawn mowing can be considered variable Any device from a pair of clippers to a gang mower might be chosen Once the mower is purchased however the capitallabor ratio becomes fixed 11Named after C W Cobb and P H Douglas See P H Douglas The Theory of Wages New York Macmillan Co 1934 pp 13235 12For the CobbDouglas function RTS 5 fl fk 5 βAkαlβ21 αAkα21l β 5 β α k l or ln RTS 5 ln a β αb 1 ln ak l b Hence σ 5 ln 1kl2 ln RTS 5 1 13See Problem 95 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 313 from actual data and such estimates may be used to measure returns to scale by examin ing the sum α 1 β and for other purposes 954 Case 4 CES production function A functional form that incorporates all three previous cases and allows σ to take on other values as well is the constant elasticity of substitution CES production function first introduced by Arrow et al in 196114 This function is given by q 5 f 1k l2 5 1k ρ 1 l ρ2 γρ 940 for ρ 1 ρ 2 0 and γ 0 This function closely resembles the CES utility function discussed in Chapter 3 although now we have added the exponent γρ to permit explicit introduction of returnstoscale factors For γ 1 the function exhibits increasing returns to scale whereas for γ 1 it exhibits decreasing returns Direct application of the definition of σ to this function15 gives the important result that σ 5 1 1 2 ρ 941 Hence the linear fixedproportions and CobbDouglas cases correspond to ρ 5 1 ρ 5 2q and ρ 5 20 respectively Proof of this result for the fixedproportions and CobbDouglas cases requires a limit argument Often the CES function is used with a distributional weight α 10 α 12 to indicate the relative significance of the inputs q 5 f1k l2 5 3αkρ 1 11 2 α2lρ4 γρ 942 With constant returns to scale and ρ 5 0 this function converges to the CobbDouglas form q 5 f 1k l2 5 kαl 12α 943 14K J Arrow H B Chenery B S Minhas and R M Solow CapitalLabor Substitution and Economic Efficiency Review of Economics and Statistics August 1961 22550 15For the CES function we have RTS 5 fl fk 5 1γρ2 q1γ2ρ2γ ρlρ21 1γρ2 q1γ2ρ2γ ρkρ21 5 a l kb ρ21 5 ak l b 12ρ Applying the definition of the elasticity of substitution then yields σ 5 ln 1kl2 ln RTS 5 1 1 2 ρ Notice in this computation that the factor ρ cancels out of the marginal productivity functions thereby ensuring that these marginal productivities are positive even when ρ is negative as it is in many cases This explains why ρ appears in two different places in the definition of the CES function EXAMPLE 93 A Generalized Leontief Production Function Suppose that the production function for a good is given by q 5 f1k l2 5 k 1 l 1 2kl 944 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 314 Part 4 Production and Supply 96 TECHNICAL PROGRESS Methods of production improve over time and it is important to be able to capture these improvements with the production function concept A simplified view of such progress is provided by Figure 95 Initially isoquant IQr records those combinations of capital and labor that can be used to produce an output level of q0 Following the development of supe rior production techniques this isoquant shifts to IQs Now the same level of output can be This function is a special case of a class of functions named for the RussianAmerican economist Wassily Leontief16 The function clearly exhibits constant returns to scale because f1tk tl2 5 tk 1 tl 1 2tkl 5 tf1k l2 945 Marginal productivities for the Leontief function are fk 5 1 1 1kl2 205 fl 5 1 1 1kl2 05 946 Hence marginal productivities are positive and diminishing As would be expected because this function exhibits constant returns to scale the RTS here depends only on the ratio of the two inputs RTS 5 fl fk 5 1 1 1kl2 05 1 1 1kl2 205 947 This RTS diminishes as kl falls so the isoquants have the usual convex shape There are two ways you might calculate the elasticity of substitution for this production func tion First you might notice that in this special case the function can be factored as q 5 k 1 l 1 2kl 5 1k 1 l2 2 5 1k05 1 l052 2 948 which makes clear that this function has a CES form with ρ 5 05 and g 5 1 Hence the elasticity of substitution here is σ 5 1 11 2 ρ2 5 2 Of course in most cases it is not possible to do such a simple factorization A more exhaus tive approach is to apply the definition of the elasticity of substitution given in footnote 6 of this chapter σ 5 fk fl f fkl 5 31 1 1kl2 054 31 1 1kl2 2054 q 105kl2 5 2 1 1kl2 05 1 1kl2 205 1 1 05 1kl2 05 1 05 1kl2 205 5 2 949 Notice that in this calculation the input ratio 1kl2 drops out leaving a simple result In other applications one might doubt that such a fortuitous result would occur and hence doubt that the elasticity of substitution is constant along an isoquant see Problem 97 But here the result that σ 5 2 is intuitively reasonable because that value represents a compromise between the elastic ity of substitution for this production functions linear part 1q 5 k 1 l σ 5 q2 and its Cobb Douglas part 1q 5 2k05l05 σ 5 12 QUERY What can you learn about this production function by graphing the q 5 4 isoquant Why does this function generalize the fixedproportions case 16Leontief was a pioneer in the development of inputoutput analysis In inputoutput analysis production is assumed to take place with a fixedproportions technology The Leontief production function generalizes the fixedproportions case For more details see the discussion of Leontief production functions in the Extensions to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 315 produced with fewer inputs One way to measure this improvement is by noting that with a level of capital input of say k1 it previously took l2 units of labor to produce q0 whereas now it takes only l1 Output per worker has risen from q0l2 to q0l1 But one must be care ful in this type of calculation An increase in capital input to k2 would also have permitted a reduction in labor input to l1 along the original q0 isoquant In this case output per worker would also increase although there would have been no true technical progress Use of the production function concept can help to differentiate between these two concepts and therefore allow economists to obtain an accurate estimate of the rate of technical change 961 Measuring technical progress The first observation to be made about technical progress is that historically the rate of growth of output over time has exceeded the growth rate that can be attributed to the growth in conventionally defined inputs Suppose that we let q 5 A 1t2f1k l2 950 be the production function for some good or perhaps for societys output as a whole The term A 1t2 in the function represents all the influences that go into determining q other than k machinehours and l laborhours Changes in A over time represent technical progress For this reason A is shown as a function of time Presumably dAdt 0 partic ular levels of input of labor and capital become more productive over time Technical progress shifts the q0 isoquant labeled IQr toward the origin The new q0 isoquant IQs shows that a given level of output can now be produced with less input For example with k1 units of capital it now only takes l1 units of labor to produce q0 whereas before the technical advance it took l2 units of labor FIGURE 95 Technical Progress k per period l per period k1 k2 l2 l1 IQ IQ Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 316 Part 4 Production and Supply 962 Growth accounting Differentiating Equation 950 with respect to time gives dq dt 5 dA dt f1k l2 1 A df1k l2 dt 5 dA dt q A 1 q f1k l2 a f k dk dt 1 f l dl dtb 951 Dividing by q gives dqdt q 5 dAdt A 1 fk f1k l2 dk dt 1 fl f1k l2 dl dt 952 or dqdt q 5 dAdt A 1 f k k f1k l2 dkdt k 1 f l l f1k l2 dldt l 953 Now for any variable x 1dxdt2x is the proportional rate of growth of x per unit of time We shall denote this by Gx17 Hence Equation 953 can be written in terms of growth rates as Gq 5 GA 1 f k k f1k l2 Gk 1 f l l f1k l2 Gt 954 But f k k f1k l2 5 q k k q 5 elasticity of output with respect to capital 5 eq k 955 and f l l f1k l2 5 q l l q 5 elasticity of output with respect to labor 5 eq l 956 Therefore our growth equation finally becomes Gq 5 GA 1 eqkGk 1 eqlGl 957 This shows that the rate of growth in output can be broken down into the sum of two com ponents growth attributed to changes in inputs k and l and other residual growth ie changes in A that represents technical progress Equation 957 provides a way of estimating the relative importance of technical progress 1GA2 in determining the growth of output For example in a pioneering study of the entire US economy between the years 1909 and 1949 R M Solow recorded the following values for the terms in the equation18 Gq 5 275 percent per year Gl 5 100 percent per year Gk 5 175 percent per year eq l 5 065 eq k 5 035 958 17Two useful features of this definition are first Gx y 5 Gx 1 Gy that is the growth rate of a product of two variables is the sum of each ones growth rate and second Gxy 5 Gx 2 Gy 18R M Solow Technical Progress and the Aggregate Production Function Review of Economics and Statistics 39 August 1957 31220 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 317 Consequently GA 5 Gq 2 eq lGl 2 eq kGk 5 275 2 065 11002 2 035 11752 5 150 959 The conclusion Solow reached then was that technology advanced at a rate of 15 percent per year from 1909 to 1949 More than half of the growth in real output could be attributed to technical change rather than to growth in the physical quantities of the factors of pro duction More recent evidence has tended to confirm Solows conclusions about the rela tive importance of technical change Considerable uncertainty remains however about the precise causes of such change EXAMPLE 94 Technical Progress in the CobbDouglas Production Function The CobbDouglas production function provides an especially easy avenue for illustrating tech nical progress Assuming constant returns to scale such a production function with technical progress might be represented by q 5 A1t2f1k l2 5 A1t2kαl12α 960 If we also assume that technical progress occurs at a constant exponential θ then we can write A1t2 5 Aeθt and the production function becomes q 5 Aeθtkαl12α 961 A particularly easy way to study the properties of this type of function over time is to use logarithmic differentiation ln q t 5 ln q q q t 5 qt q 5 Gq 5 3 ln A 1 θt 1 α ln k 1 11 2 α2 ln l4 t 5 θ 1 α ln k t 1 11 2 α2 ln l t 5 θ 1 αGk 1 11 2 α2Gl 962 Thus this derivation just repeats Equation 957 for the CobbDouglas case Here the technical change factor is explicitly modeled and the output elasticities are given by the values of the expo nents in the CobbDouglas The importance of technical progress can be illustrated numerically with this function Sup pose A 5 10 θ 5 003 α 5 05 and that a firm uses an input mix of k 5 l 5 4 Then at t 5 0 output is 40 15 10 405 4052 After 20 years 1t 5 202 the production function becomes q 5 10e00320k05l05 5 10 11822k05l05 5 182k05l05 963 In year 20 the original input mix now yields q 5 728 Of course one could also have produced q 5 728 in year 0 but it would have taken a lot more inputs For example with k 5 1325 and l 5 4 output is indeed 728 but much more capital is used Output per unit of labor input would increase from 10 1ql 5 4042 to 182 15 72842 in either circumstance but only the first case would have been true technical progress Inputaugmenting technical progress It is tempting to attribute the increase in the average productivity of labor in this example to say improved worker skills but that would be mislead ing in the CobbDouglas case One might just as well have said that output per unit of capital increased from 10 to 182 over the 20 years and attribute this increase to improved machinery Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 318 Part 4 Production and Supply Summary A plausible approach to modeling improvements in labor and capital separately is to assume that the production function is q 5 A1eφtk2 α 1eεtl2 12α 964 where φ represents the annual rate of improvement in capital input and ε represents the annual rate of improvement in labor input But because of the exponential nature of the CobbDouglas function this would be indistinguishable from our original example q 5 Ae3α φ1112α2ε4 tkαl12α 5 Aeθtkαl12α 965 where θ 5 αφ 1 11 2 α2ε Hence to study technical progress in individual inputs it is neces sary either to adopt a more complex way of measuring inputs that allows for improving quality or what amounts to the same thing to use a multiinput production function QUERY Actual studies of production using the CobbDouglas tend to find α 03 Use this finding together with Equation 965 to discuss the relative importance of improving capital and labor quality to the overall rate of technical progress In this chapter we illustrated the ways in which economists conceptualize the production process of turning inputs into outputs The fundamental tool is the production func tion whichin its simplest formassumes that output per period q is a simple function of capital and labor inputs during that period q 5 f1k l2 Using this starting point we developed several basic results for the theory of production If all but one of the inputs are held constant a relation ship between the singlevariable input and output can be derived From this relationship one can derive the mar ginal physical productivity MP of the input as the change in output resulting from a oneunit increase in the use of the input The marginal physical productivity of an input is assumed to decrease as use of the input increases The entire production function can be illustrated by its isoquant map The negative of the slope of an isoquant is termed the marginal rate of technical substitution RTS because it shows how one input can be substituted for another while holding output constant The RTS is the ratio of the marginal physical productivities of the two inputs Isoquants are usually assumed to be convexthey obey the assumption of a diminishing RTS This assumption cannot be derived exclusively from the assumption of diminishing marginal physical produc tivities One must also be concerned with the effect of changes in one input on the marginal productivity of other inputs The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs If output increases proportionately with input use there are constant returns to scale If there are greater than proportionate increases in output there are increasing returns to scale whereas if there are less than proportionate increases in output there are decreasing returns to scale The elasticity of substitution σ provides a measure of how easy it is to substitute one input for another in production A high σ implies nearly linear isoquants whereas a low σ implies that isoquants are nearly Lshaped Technical progress shifts the entire production func tion and its related isoquant map Technical improve ments may arise from the use of improved more productive inputs or from better methods of economic organization Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 319 Problems 91 Power Goat Lawn Company uses two sizes of mowers to cut lawns The smaller mowers have a 22inch deck The larger ones combine two of the 22inch decks in a single mower For each size of mower Power Goat has a different production function given by the rows of the following table Output per Hour square feet Capital Input of 22s mowers Labor Input Small mowers 5000 1 1 Large mowers 8000 2 1 a Graph the q 5 40000 square feet isoquant for the first production function How much k and l would be used if these factors were combined without waste b Answer part a for the second function c How much k and l would be used without waste if half of the 40000squarefoot lawn were cut by the method of the first production function and half by the method of the second How much k and l would be used if one fourth of the lawn were cut by the first method and three fourths by the second What does it mean to speak of fractions of k and l d Based on your observations in part c draw a q 5 40000 isoquant for the combined production functions 92 Suppose the production function for widgets is given by q 5 kl 2 08k2 2 02l2 where q represents the annual quantity of widgets produced k represents annual capital input and l represents annual labor input a Suppose k 5 10 graph the total and average productiv ity of labor curves At what level of labor input does this average productivity reach a maximum How many wid gets are produced at that point b Again assuming that k 5 10 graph the MPl curve At what level of labor input does MPl 5 0 c Suppose capital inputs were increased to k 5 20 How would your answers to parts a and b change d Does the widget production function exhibit constant increasing or decreasing returns to scale 93 Sam Malone is considering renovating the bar stools at Cheers The production function for new bar stools is given by q 5 01k 02l 08 where q is the number of bar stools produced during the ren ovation week k represents the number of hours of bar stool lathes used during the week and l represents the number of worker hours employed during the period Sam would like to provide 10 new bar stools and he has allocated a budget of 10000 for the project a Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount 50 per hour he might as well hire these two inputs in equal amounts If Sam proceeds in this way how much of each input will he hire and how much will the renovation project cost b Norm who knows something about bar stools argues that once again Sam has forgotten his microeconomics He asserts that Sam should choose quantities of inputs so that their marginal not average productivities are equal If Sam opts for this plan instead how much of each input will he hire and how much will the renovation project cost c On hearing that Norms plan will save money Cliff argues that Sam should put the savings into more bar stools to provide seating for more of his USPS colleagues How many more bar stools can Sam get for his budget if he follows Cliffs plan d Carla worries that Cliffs suggestion will just mean more work for her in delivering food to bar patrons How might she convince Sam to stick to his original 10bar stool plan 94 Suppose that the production of crayons q is conducted at two locations and uses only labor as an input The production function in location 1 is given by q1 5 10l 05 1 and in location 2 by q2 5 50l 05 2 a If a single firm produces crayons in both locations then it will obviously want to get as large an output as possi ble given the labor input it uses How should it allocate labor between the locations to do so Explain precisely the relationship between l1 and l2 b Assuming that the firm operates in the efficient manner described in part a how does total output q depend on the total amount of labor hired l 95 As we have seen in many places the general CobbDouglas production function for two inputs is given by q 5 f1k l2 5 Akαl β where 0 α 1 and 0 β 1 For this production function a Show that fk 0 f1 0 fkk 0 fll 0 and fkl 5 flk 0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 320 Part 4 Production and Supply b Show that eq k 5 α and eq l 5 β c In footnote 5 we defined the scale elasticity as eq t 5 f 1tk tl2 t t f 1tk tl2 where the expression is to be evaluated at t 5 1 Show that for this CobbDouglas function eq t 5 α 1 β Hence in this case the scale elasticity and the returns to scale of the production function agree for more on this concept see Problem 99 d Show that this function is quasiconcave e Show that the function is concave for α 1 β 1 but not concave for α 1 β 1 96 Suppose we are given the constant returnstoscale CES pro duction function q 5 1k ρ 1 l ρ2 1ρ a Show that MPk 5 1qk2 12ρ and MPl 5 1ql2 12ρ b Show that RTS 5 1kl2 12ρ use this to show that σ 5 1 11 2 ρ2 c Determine the output elasticities for k and l and show that their sum equals 1 d Prove that q l 5 a q l b σ and hence that ln a q l b 5 σ ln a q l b Note The latter equality is useful in empirical work because we may approximate ql by the competitively determined wage rate Hence σ can be estimated from a regression of ln 1ql2 on ln w 97 Consider a generalization of the production function in Example 93 q 5 β0 1 β1kl 1 β2k 1 β3l where 0 βi 1 i 5 0 c 3 a If this function is to exhibit constant returns to scale what restrictions should be placed on the parameters β0 c β3 b Show that in the constant returnstoscale case this function exhibits diminishing marginal productivities and that the marginal productivity functions are homo geneous of degree 0 c Calculate σ in this case Although σ is not in general constant for what values of the βs does σ 5 0 1 or 98 Show that Eulers theorem implies that for a constant returns toscale production function q 5 f 1k l2 q 5 fkk 1 fll Use this result to show that for such a production function if MPl APl then MPk must be negative What does this imply about where production must take place Can a firm ever pro duce at a point where APl is increasing Analytical Problems 99 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity eqt 5 f 1tk tl2t tq evaluated at t 5 1 a Show that if the production function exhibits constant returns to scale then eqt 5 1 b We can define the output elasticities of the inputs k and l as eq k 5 f1k l2 k k q eq l 5 f1k l2 l l q Show that eq t 5 eq k 1 eq l c A function that exhibits variable scale elasticity is q 5 11 1 k21l212 21 Show that for this function eq t 1 for q 05 and that eq t 1 for q 05 d Explain your results from part c intuitively Hint Does q have an upper bound for this production function 910 Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale often that assumption is not neces sary This problem illustrates some of these cases a In footnote 6 we pointed out that in the constant returnstoscale case the elasticity of substitution for a twoinput production function is given by σ 5 fk fl f fkl Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 321 Suppose now that we define the homothetic production function F as F1k l2 5 3 f 1k l2 4 γ where f1k l2 is a constant returnstoscale production function and γ is a positive exponent Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function f b Show how this result can be applied to both the Cobb Douglas and CES production functions 911 More on Eulers theorem Suppose that a production function f1x1 x2 c xn2 is homogeneous of degree k Eulers theorem shows that g ixi fi 5 kf and this fact can be used to show that the partial derivatives of f are homogeneous of degree k 2 1 a Prove that g n i51g n j51xi xj f ij 5 k1k 2 12f b In the case of n 5 2 and k 5 1 what kind of restrictions does the result of part a impose on the secondorder partial derivative f12 How do your conclusions change when k 1 or k 1 c How would the results of part b be generalized to a pro duction function with any number of inputs d What are the implications of this problem for the param eters of the multivariable CobbDouglas production function f1x1 x2 c xn2 5 w n i51xαi i for αi 0 Suggestions for Further Reading Clark J M Diminishing Returns In Encyclopaedia of the Social Sciences vol 5 New York CrowellCollier and Macmillan 1931 pp 14446 Lucid discussion of the historical development of the diminishing returns concept Douglas P H Are There Laws of Production American Economic Review 38 March 1948 141 A nice methodological analysis of the uses and misuses of produc tion functions Ferguson C E The Neoclassical Theory of Production and Dis tribution New York Cambridge University Press 1969 A thorough discussion of production function theory as of 1970 Good use of threedimensional graphs Fuss M and D McFadden Production Economics A Dual Approach to Theory and Application Amsterdam North Holland 1980 An approach with a heavy emphasis on the use of duality MasCollell A M D Whinston and J R Green Microeconomic Theory New York Oxford University Press 1995 Chapter 5 provides a sophisticated if somewhat spare review of production theory The use of the profit function see Chapter 11 is sophisticated and illuminating Shephard R W Theory of Cost and Production Functions Princeton NJ Princeton University Press 1978 Extended analysis of the dual relationship between production and cost functions Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 Thorough analysis of the duality between production functions and cost curves Provides a proof that the elasticity of substitution can be derived as shown in footnote 6 of this chapter Stigler G J The Division of Labor Is Limited by the Extent of the Market Journal of Political Economy 59 June 1951 18593 Careful tracing of the evolution of Smiths ideas about economies of scale Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 322 EXTENSIONS ManyInPut ProductIon FunctIonS Most of the production functions illustrated in Chapter 9 can be easily generalized to manyinput cases Here we show this for the CobbDouglas and CES cases and then examine two flexible forms that such production functions might take In all these examples the αs are nonnegative parameters and the n inputs are represented by x1 c xn E91 CobbDouglas The manyinput CobbDouglas production function is given by q 5 q n i51 xαi i i a This function exhibits constant returns to scale if a n i51 αi 5 1 ii b In the constantreturnstoscale CobbDouglas func tion αi is the elasticity of q with respect to input xi Because 0 αi 1 each input exhibits diminishing marginal productivity c Any degree of increasing returns to scale can be incorpo rated into this function depending on ε 5 a n i51 αi iii d The elasticity of substitution between any two inputs in this production function is 1 This can be shown by using the definition given in footnote 7 of this chapter σij 5 ln 1xixj2 ln 1 fjfi2 Here fj fi 5 αi xαj21 j qi2jxαi i αi xαi21 i qj2ixαj j 5 αj αi xi xj Hence ln a fj fi b 5 ln a αj αi b 1 ln a xi xj b and σij 5 1 Because this parameter is so constrained in the CobbDouglas function the function is generally not used in econometric analyses of microeconomic data on firms However the function has a variety of general uses in macroeconomics as the next example illustrates The Solow growth model The manyinput CobbDouglas production function is a primary feature of many models of economic growth For example Solows 1956 pioneering model of equilibrium growth can be most easily derived using a twoinput constant returnstoscale CobbDouglas function of the form q 5 Akαl12α iv where A is a technical change factor that can be represented by exponential growth of the form A 5 eat v Dividing both sides of Equation iv by l yields q 5 eatkα vi where q 5 ql and k 5 kl Solow shows that economies will evolve toward an equilib rium value of k the capitallabor ratio Hence crosscountry differences in growth rates can be accounted for only by dif ferences in the technical change factor a Two features of Equation vi argue for including more inputs in the Solow model First the equation as it stands is incapable of explaining the large differences in per capita output 1q2 that are observed around the world Assuming α 5 03 say a figure consistent with many empirical stud ies it would take crosscountry differences in kl of as much as 4000000 to 1 to explain the 100to1 differences in per capita income observeda clearly unreasonable magnitude By introducing additional inputs such as human capital these differences become more explainable A second shortcoming of the simple CobbDouglas for mulation of the Solow model is that it offers no explanation of the technical change parameter aits value is determined exogenously By adding additional factors it becomes easier to understand how the parameter a may respond to economic Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 9 Production Functions 323 incentives This is the key insight of literature on endoge nous growth theory for a summary see Romer 1996 E92 CES The manyinput constant elasticity of substitution CES pro duction function is given by q 5 C aαix ρ i D γρ ρ 1 vii a By substituting txi for each output it is easy to show that this function exhibits constant returns to scale for γ 5 1 For γ 1 the function exhibits increasing returns to scale b The production function exhibits diminishing marginal productivities for each input when γ 1 c As in the twoinput case the elasticity of substitution here is given by σ 5 1 1 2 ρ viii and this elasticity applies to substitution between any two of the inputs Checking the CobbDouglas in the Soviet Union One way in which the multiinput CES function is used is to determine whether the estimated substitution parameter ρ is consistent with the value implied by the CobbDouglas 1ρ 5 0 σ 5 12 For example in a study of five major indus tries in the former Soviet Union E Bairam 1991 finds that the CobbDouglas provides a relatively good explanation of changes in output in most major manufacturing sectors Only for food processing does a lower value for σ seem appropriate The next three examples illustrate flexibleform produc tion functions that may approximate any general function of n inputs In the Chapter 10 extensions we examine the cost function analogs to some of these functions which are more widely used than the production functions themselves E93 Nested production functions In some applications CobbDouglas and CES production functions are combined into a nested single function To accomplish this the original n primary inputs are categorized into say m general classes of inputs The specific inputs in each of these categories are then aggregated into a single com posite input and the final production function is a function of these m composites For example assume there are three primary inputs x1 x2 x3 Suppose however that x1 and x2 are relatively closely related in their use by firms eg capital and energy whereas the third input labor is relatively distinct Then one might want to use a CES aggregator function to construct a composite input for capital services of the form x4 5 3γx ρ 1 1 11 2 γ2x ρ 24 1ρ ix Then the final production function might take a CobbDouglas form q 5 x α 3x β 4 x This structure allows the elasticity of substitution between x1 and x2 to take on any value 3σ 5 1 11 2 ρ2 4 but constrains the elasticity of substitution between x3 and x4 to be one A variety of other options are available depending on how pre cisely the embedded functions are specified The dynamics of capitalenergy substitutability Nested production functions have been widely used in studies that seek to measure the precise nature of the substitutability between capital and energy inputs For example Atkeson and Kehoe 1999 use a model rather close to the one specified in Equations ix and x to try to reconcile two facts about the way in which energy prices affect the economy 1 Over time use of energy in production seems rather unresponsive to price at least in the short run and 2 across countries energy prices seem to have a large influence over how much energy is used By using a capital service equation of the form given in Equa tion ix with a low degree of substitutability 1ρ 5 2232 along with a CobbDouglas production function that combines labor with capital servicesthey are able to replicate the facts about energy prices fairly well They conclude however that this model implies a much more negative effect of higher energy prices on economic growth than seems actually to have been the case Hence they ultimately opt for a more complex way of modeling production that stresses differences in energy use among capital investments made at different dates E94 Generalized Leontief q 5 a n i51 a n j51 αijxixj where αij 5 αji a The function considered in Problem 97 is a simple case of this function for the case n 5 2 For n 5 3 the function would have linear terms in the three inputs along with three radical terms representing all possible crossproducts of the inputs b The function exhibits constant returns to scale as can be shown by using txi Increasing returns to scale can be incor porated into the function by using the transformation qr 5 qε ε 1 c Because each input appears both linearly and under the radical the function exhibits diminishing marginal pro ductivities to all inputs d The restriction αij 5 αji is used to ensure symmetry of the secondorder partial derivatives Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 324 Part 4 Production and Supply E95 Translog ln q 5 α0 1 a n i51 αi ln xi 1 05 a n i51 a n j51 αij ln xi ln xj αij 5 αji a Note that the CobbDouglas function is a special case of this function where α0 5 αij 5 0 for all i j b As for the CobbDouglas this function may assume any degree of returns to scale If a n i51 αi 5 1 and a n j51 αij 5 0 for all i then this function exhibits constant returns to scale The proof requires some care in dealing with the double summation c Again the condition αij 5 αji is required to ensure equality of the crosspartial derivatives Immigration Because the translog production function incorporates a large number of substitution possibilities among various inputs it has been widely used to study the ways in which newly arrived workers may substitute for existing workers Of particular interest is the way in which the skill level of immigrants may lead to differing reactions in the demand for skilled and unskilled workers in the domestic economy Studies of the United States and many other countries eg Canada Germany and France have suggested that the overall size of such effects is modest especially given rela tively small immigration flows But there is some evidence that unskilled immigrant workers may act as substitutes for unskilled domestic workers but as complements to skilled domestic workers Hence increased immigration flows may exacerbate trends toward increasing wage differentials For a summary see Borjas 1994 References Atkeson Andrew and Patrick J Kehoe Models of Energy Use PuttyPutty versus PuttyClay American Economic Review September 1999 102843 Bairam Erkin Elasticity of Substitution Technical Prog ress and Returns to Scale in Branches of Soviet Industry A New CES Production Function Approach Journal of Applied Economics JanuaryMarch 1991 9196 Borjas G J The Economics of Immigration Journal of Economic Literature December 1994 1667717 Romer David Advanced Macroeconomics New York McGrawHill 1996 Solow R M A Contribution to the Theory of Economic Growth Quarterly Journal of Economics February 1956 6594 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 325 CHAPTER TEN Cost Functions This chapter analyzes the costs incurred by a firm for the inputs it needs to produce a given output The next chapter pursues this same topic further by combining costs with revenues to show how firms make profitmaximizing input and output decisions 101 DEFINITIONS OF COSTS Before we can discuss the theory of costs some difficulties about the proper definition of costs must be cleared up Specifically we must distinguish between 1 accounting cost and 2 economic cost The accountants view of cost stresses outofpocket expenses his torical costs depreciation and other bookkeeping entries The economists definition of cost which in obvious ways draws on the fundamental opportunitycost notion is that the cost of any input is given by the size of the payment necessary to keep the resource in its present employment Alternatively the economic cost of using an input is what that input would be paid in its next best use One way to distinguish between these two views is to consider how the costs of various inputs labor capital and entrepreneurial services are defined under each system 1011 Labor costs Economists and accountants regard labor costs in much the same way To accountants expenditures on labor are current expenses and hence costs of production For economists labor is an explicit cost Labor services laborhours are contracted at some hourly wage rate w and it is usually assumed that this is also what the labor services would earn in their best alternative employment The hourly wage of course includes costs of fringe ben efits provided to employees 1012 Capital costs The two cost concepts diverge more in the case of capital services machinehours In cal culating capital costs accountants use the historical price of the particular machine under investigation and apply some moreorless arbitrary depreciation rule to determine how much of that machines original price to charge to current costs Economists regard the historical price of a machine as a sunk cost which is irrelevant to output decisions They instead regard the implicit cost of the machine to be what someone else would be willing to pay for its use Thus the cost of one machinehour is the rental rate for that machine in its best alternative use By continuing to use the machine itself the firm is implicitly forgoing Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 326 Part 4 Production and Supply what someone else would be willing to pay to use it This rental rate for one machinehour will be denoted by v1 Suppose a company buys a computer for 2000 An accountant applying a straight line depreciation method over 5 years would regard the computer as having a cost of 400 a year An economist would look at the market value of the computer The availability of much faster computers in subsequent years can cause the secondhand price of the original computer to decrease precipitously If the secondhand price decreases all the way to for example 200 after the first year the economic cost will be related to this 200 the original 2000 price will no longer be relevant All these yearly costs can easily be converted into computerhour costs of course The distinction between accounting and economic costs of capital largely disappears if the company rents it at a price of v each period rather than purchasing Then v reflects a current company expenditure that shows up directly as an accounting cost it also reflects the market value of one periods use of the capital and thus is an opportunityeconomic cost 1013 Costs of entrepreneurial services The owner of a firm is a residual claimant who is entitled to whatever extra revenues or losses are left after paying other input costs To an accountant these would be called profits which might be either positive or negative Economists however ask whether owners or entrepre neurs also encounter opportunity costs by working at a particular firm or devoting some of their funds to its operation If so these services should be considered an input and some cost should be imputed to them For example suppose a highly skilled computer programmer starts a software firm with the idea of keeping any accounting profits that might be gener ated The programmers time is clearly an input to the firm and a cost should be attributed to it Perhaps the wage that the programmer might command if he or she worked for someone else could be used for that purpose Hence some part of the accounting profits generated by the firm would be categorized as entrepreneurial costs by economists Economic profits would be smaller than accounting profits and might be negative if the programmers oppor tunity costs exceeded the accounting profits being earned by the business Similar arguments apply to the capital that an entrepreneur provides to the firm 1014 Economic costs In this book not surprisingly we use economists definition of cost Our focus on economic definitions of cost does not mean that we regard accounting as a useless endeavor Accounting data are often readily available whereas the corresponding economic concepts may be more difficult to measure For example returning to the pre ceding case of a computer the firm can easily keep track of the 2000 the firm had paid for it to determine its accounting cost but may not bother undertaking a study needed for a precise measure of the economic cost of what it could rent the obsolescing unit for given that in fact the firm is planning not to rent it If the accounting measure is not too far D E F I N I T I O N Economic cost The economic cost of any input is the payment required to keep that input in its present employment Equivalently the economic cost of an input is the remuneration the input would receive in its best alternative employment 1Sometimes the symbol r is chosen to represent the rental rate on capital Because this variable is often confused with the related but distinct concept of the market interest rate an alternative symbol was chosen here The exact relationship between v and the interest rate is examined in Chapter 17 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 327 from its economic counterpart the accounting measure may be good enough for many practical purposes Furthermore there exists a whole branch of accounting managerial accounting devoted to developing measures that help guide the economic decisions faced by the manager of a firm which can end up resembling many of the economic concepts studied here We put measurement issues aside and use the decisionrelevant concepts economic coststhroughout the analysis 1015 Simplifying assumptions As a start we will make two simplifications about the inputs a firm uses First we assume that there are only two inputs homogeneous labor l measured in laborhours and homo geneous capital k measured in machinehours Entrepreneurial costs are included in capital costs That is we assume that the primary opportunity costs faced by a firms owner are those associated with the capital that the owner provides Second we assume that inputs are hired in perfectly competitive markets Firms can buy or sell all the labor or capital services they want at the prevailing rental rates w and v In graphic terms the supply curve for these resources is horizontal at the prevailing factor prices Both w and v are treated as parameters in the firms decisions there is nothing the firm can do to affect them These conditions will be relaxed in later chapters notably Chapter 16 but for the moment the pricetaker assumption is a convenient and useful one to make Therefore with these simplifications total cost C for the firm during the period is given by total cost 5 C 5 wl 1 vk 101 where l and k represent input usage during the period 102 RELATIONSHIP BETWEEN PROFIT MAxIMIzATION AND COST MINIMIzATION Lets look ahead to the next chapter on profit maximization and compare the analysis here with the analysis in that chapter We will define economic profits π as the differ ence between the firms total revenues R and its total costs C Suppose the firm takes the market price p for its total output q as given and that its production function is q 5 f 1k l2 Then its profit can be written π 5 R 2 C 5 pq 2 wl 2 vk 5 pf 1k l2 2 wl 2 vk 102 Equation 102 shows that the economic profits obtained by this firm are a function of the amount of capital and labor employed If as we will assume in many places in this book this firm seeks maximum profits then we might study its behavior by examining how k and l are chosen to maximize Equation 102 This would in turn lead to a theory of supply and to a theory of the derived demand for capital and labor inputs In the next chapter we will take up those subjects in detail Here however we wish to develop a theory of costs that is somewhat more general apply ing not only to firms that are pricetakers on their output markets perfect competitors but also to those whose output choice affects the market price monopolies and oligopolies The more general theory will even apply to nonprofits as long as they are interested in operating efficiently The other advantage of looking at cost minimization separately from profit max imization is that it is simpler to analyze this small piece in isolation and only later add the insights obtained into the overall puzzle of the firms operations The conditions derived for costminimizing input choices in this chapter will emerge again as a byproduct of the analysis of the maximization of profits as specified in Equation 102 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 328 Part 4 Production and Supply Hence we begin the study of costs by finessing for the moment a discussion of output choice That is we assume that for some reason the firm has decided to produce a partic ular output level say q0 The firm will of course earn some revenue R from this output choice but we will ignore revenue for now We will focus solely on the question of how the firm can produce q0 at minimal cost 103 COSTMINIMIzING INPUT CHOICES Mathematically we will end up solving a constrained minimization problem But before pro ceeding with a rigorous solution it is useful to state the result to be derived with an intuitive argument To minimize the cost of producing a given level of output a firm should choose that point on the q0 isoquant at which the rate of technical substitution RTS of l for k is equal to the ratio wv It should equate the rate at which k can be traded for l in production to the rate at which they can be traded in the marketplace Suppose that this were not true In particular suppose that the firm were producing output level q0 using k 5 10 l 5 10 and assume that the RTS were 2 at this point Assume also that w 5 1 v 5 1 and hence that wv 5 1 which is not equal to 2 At this input combination the cost of producing q0 is 20 It is easy to show this is not the minimal input cost For example q0 can also be pro duced using k 5 8 and l 5 11 we can give up two units of k and keep output constant at q0 by adding one unit of l But at this input combination the cost of producing q0 is 19 and hence the initial input combination was not optimal A contradiction similar to this one can be demonstrated whenever the RTS and the ratio of the input costs differ 1031 Mathematical analysis Mathematically we seek to minimize total costs given q 5 f 1k l2 5 q0 Setting up the Lagrangian 5 wl 1 vk 1 λ 3q0 2 f 1k l2 4 103 the firstorder conditions for a constrained minimum are l 5 w 2 λ f l 5 0 k 5 v 2 λ f k 5 0 104 λ 5 q0 2 f1k l2 5 0 or dividing the first two equations w v 5 fl fk 5 RTS 1of l for k2 105 This says that the costminimizing firm should equate the RTS for the two inputs to the ratio of their prices 1032 Further interpretations These firstorder conditions for minimal costs can be manipulated in several different ways to yield interesting results For example crossmultiplying Equation 105 gives fk v 5 fl w 106 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 329 That is for costs to be minimized the marginal productivity per dollar spent should be the same for all inputs If increasing one input promised to increase output by a greater amount per dollar spent than did another input costs would not be minimalthe firm should hire more of the input that promises a bigger bang per buck and less of the more costly in terms of productivity input Any input that cannot meet the common benefit cost ratio defined in Equation 106 should not be hired at all Equation 106 can of course also be directly derived from Equation 104 as can the following useful reciprocal relationship w fl 5 v fk 5 λ 107 This equation reports the extra cost of obtaining an extra unit of output by hiring either added labor or added capital input Because of cost minimization this marginal cost is the same no matter which input is hired This common marginal cost is also measured by the Lagrange multiplier from the costminimization problem As is the case for all constrained optimization problems here the Lagrange multiplier shows how much in extra costs would be incurred by increasing the output constraint slightly Because marginal cost plays an important role in a firms supply decisions we will return to this feature of cost minimization frequently 1033 Graphical analysis Cost minimization is shown graphically in Figure 101 Given the output isoquant q0 we wish to find the least costly point on the isoquant Lines showing equal cost are parallel straight lines with slopes 2wv Three lines of equal total cost are shown in Figure 101 A firm is assumed to choose k and l to minimize total costs The condition for this minimization is that the rate at which k and l can be traded technically while keeping q 5 q0 should be equal to the rate at which these inputs can be traded in the market In other words the RTS of l for k should be set equal to the price ratio wv This tangency is shown in the figure costs are minimized at C1 by choosing inputs kc and l c FIgurE 101 Minimization of Costs of Producing q0 l per period lc kc k per period C1 q0 C2 C3 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 330 Part 4 Production and Supply C1 C2 C3 It is clear from the figure that the minimum total cost for producing q0 is given by C1 where the total cost curve is just tangent to the isoquant The associated inputs are l c and k c where the superscripts emphasize that these input levels are a solution to a costminimization problem This combination will be a true minimum if the isoquant is convex if the RTS diminishes for decreases in kl The mathematical and graphic analyses arrive at the same conclusion as follows O P T I M I Z AT I O N P r I N C I P L E Cost minimization To minimize the cost of any given level of output 1q02 the firm should pro duce at that point on the q0 isoquant for which the RTS of l for k is equal to the ratio of the inputs rental prices 1wv2 1034 Contingent demand for inputs Figure 101 exhibits the formal similarity between the firms costminimization problem and the individuals expenditureminimization problem studied in Chapter 4 see Figure 46 In both problems the economic actor seeks to achieve his or her target output or utility at minimal cost In Chapter 5 we showed how this process is used to construct a theory of compensated demand for a good In the present case cost minimization leads to a demand for capital and labor input that is contingent on the level of output being produced Therefore this is not the complete story of a firms demand for the inputs it uses because it does not address the issue of output choice But studying the contingent demand for inputs provides an important building block for analyzing the firms overall demand for inputs and we will take up this topic in more detail later in this chapter 1035 Firms expansion path A firm can follow the costminimization process for each level of output For each q it finds the input choice that minimizes the cost of producing it If input costs w and v remain constant for all amounts the firm may demand we can easily trace this locus of costminimizing choices This procedure is shown in Figure 102 The curve 0E records the costminimizing tangencies for successively higher levels of output For example the min imum cost for producing output level q1 is given by C1 and inputs k1 and l1 are used Other tangencies in the figure can be interpreted in a similar way The locus of these tangencies is called the firms expansion path because it records how input expands as output expands while holding the prices of the inputs constant As Figure 102 shows the expansion path need not be a straight line The use of some inputs may increase faster than others as output expands Which inputs expand more rap idly will depend on the shape of the production isoquants Because cost minimization requires that the RTS always be set equal to the ratio wv and because the wv ratio is assumed to be constant the shape of the expansion path will be determined by where a particular RTS occurs on successively higher isoquants If the production function exhibits constant returns to scale or more generally if it is homothetic then the expansion path will be a straight line because in that case the RTS depends only on the ratio of k to l That ratio would be constant along such a linear expansion path It would seem reasonable to assume that the expansion path will be positively sloped that is successively higher output levels will require more of both inputs This need not be the case however as Figure 103 illustrates Increases of output beyond q2 cause the quantity of labor used to decrease In this range labor would be said to be an inferior input The occurrence of inferior inputs is then a theoretical possibility that may happen even when isoquants have their usual convex shape Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 331 The firms expansion path is the locus of costminimizing tangencies Assuming fixed input prices the curve shows how inputs increase as output increases FIgurE 102 Firms Expansion Path l per period l1 k1 0 C3 C2 C1 q3 q2 q1 k per period With this particular set of isoquants labor is an inferior input because less l is chosen as output expands beyond q2 FIgurE 103 Input Inferiority l per period 0 E q4 q3 q2 q1 k per period Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 332 Part 4 Production and Supply Much theoretical discussion has centered on the analysis of factor inferiority Whether inferiority is likely to occur in realworld production functions is a difficult empirical question to answer It seems unlikely that such comprehensive magnitudes as capital and labor could be inferior but a finer classification of inputs may bring inferiority to light For example the use of shovels may decrease as production of building foundations and the use of backhoes increases In this book we shall not be particularly concerned with the analytical issues raised by this possibility although complications raised by inferior inputs will be mentioned in a few places EXAMPLE 101 Cost Minimization The costminimization process can be readily illustrated with two of the production functions we encountered in the last chapter 1 CobbDouglas q 5 f1k l2 5 k αl β For this case the relevant Lagrangian for minimizing the cost of producing say q0 is 5 vk 1 wl 1 λ1q0 2 kαl β2 108 The firstorder conditions for a minimum are k 5 v 2 λαkα21l β 5 0 l 5 w 2 λβkαl β21 5 0 λ 5 q0 2 kαl β 5 0 109 Dividing the second of these by the first yields w v 5 βkαlβ21 αkα21lβ 5 β α k l 1010 which again shows that costs are minimized when the ratio of the inputs prices is equal to the RTS Because the CobbDouglas function is homothetic the RTS depends only on the ratio of the two inputs If the ratio of input costs does not change the firms will use the same input ratio no matter how much it producesthat is the expansion path will be a straight line through the origin As a numerical example suppose α 5 β 5 05 w 5 12 v 5 3 and that the firm wishes to produce q0 5 40 The firstorder condition for a minimum requires that k 5 4l Inserting that into the production function the final requirement in Equation 109 we have q0 5 40 5 k05l 05 5 2l Thus the costminimizing input combination is l 5 20 and k 5 80 and total costs are given by vk 1 wl 5 3 80 1 12 20 5 480 That this is a true cost minimum is suggested by looking at a few other input combinations that also are capable of producing 40 units of output k 5 40 l 5 40 C 5 600 k 5 10 l 5 160 C 5 2220 k 5 160 l 5 10 C 5 600 1011 Any other input combination able to produce 40 units of output will also cost more than 480 Cost minimization is also suggested by considering marginal productivities At the optimal point Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 333 MPk 5 fk 5 05k205l 05 5 05a20 80b 05 5 025 MPl 5 fl 5 05k05l205 5 05a80 20b 05 5 10 1012 Hence at the margin labor is four times as productive as capital and this extra productivity precisely compensates for the higher unit price of labor input 2 CES q 5 f 1k l2 5 1k ρ 1 l ρ2 γ ρ Again we set up the Lagrangian 5 vk 1 wl 1 λ3q0 2 1kρ 1 l ρ2 γ ρ4 1013 The firstorder conditions for a minimum are k 5 v 2 λ1γρ2 1k ρ 1 l ρ2 1γ2ρ2ρ1ρ2k ρ21 5 0 l 5 w 2 λ1γρ2 1k ρ 1 l ρ2 1γ2ρ2ρ1ρ2l ρ21 5 0 λ 5 q0 2 1k ρ 1 l ρ2 1γ2ρ2 5 0 1014 Dividing the first two of these equations causes a lot of this mass of symbols to drop out leaving w v 5 a l kb ρ21 5 ak l b 12ρ 5 ak l b 1σ or k l 5 aw v b σ 1015 where σ 5 1 11 2 ρ2 is the elasticity of substitution Because the CES function is also homothetic the costminimizing input ratio is independent of the absolute level of pro duction The result in Equation 1015 is a simple generalization of the CobbDouglas result when σ 5 1 With the CobbDouglas the costminimizing capitallabor ratio changes directly in proportion to changes in the ratio of wages to capital rental rates In cases with greater substitutability 1σ 12 changes in the ratio of wages to rental rates cause a greater than proportional increase in the costminimizing capitallabor ratio With less substitut ability 1σ 12 the response is proportionally smaller QUERY In the CobbDouglas numerical example with wv 5 4 we found that the cost minimizing input ratio for producing 40 units of output was kl 5 8020 5 4 How would this value change for σ 5 2 or σ 5 05 What actual input combinations would be used What would total costs be 104 COST FUNCTIONS We are now in a position to examine the firms overall cost structure To do so it will be convenient to use the expansion path solutions to derive the total cost function D E F I N I T I O N Total cost function The total cost function shows that for any set of input costs and for any out put level the minimum total cost incurred by the firm is C 5 C1v w q2 1016 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 334 Part 4 Production and Supply Figure 102 makes clear that total costs increase as output q increases We will begin by analyzing this relationship between total cost and output while holding input prices fixed Then we will consider how a change in an input price shifts the expansion path and its related cost functions 1041 Average and marginal cost functions Although the total cost function provides complete information about the outputcost relationship it is often convenient to analyze costs on a perunit of output basis because that approach corresponds more closely to the analysis of demand which focused on the price per unit of a commodity Two different unit cost measures are widely used in eco nomics 1 average cost which is the cost per unit of output and 2 marginal cost which is the cost of one more unit of output D E F I N I T I O N Average and marginal cost functions The average cost function AC is found by computing total costs per unit of output average cost 5 AC1v w q2 5 C1v w q2 q 1017 The marginal cost function MC is found by computing the change in total costs for a change in output produced marginal cost 5 MC1v w q2 5 C1v w q2 q 1018 Notice that in these definitions average and marginal costs depend both on the level of out put being produced and on the prices of inputs In many places throughout this book we will graph simple twodimensional relationships between costs and output As the definitions make clear all such graphs are drawn on the assumption that the prices of inputs remain con stant and that technology does not change If input prices change or if technology advances cost curves generally will shift to new positions Later in this chapter we will explore the likely direction and size of such shifts when we study the entire cost function in detail 1042 Graphical analysis of total costs Figures 104a and 105a illustrate two possible shapes for the relationship between total cost and the level of the firms output In Figure 104a total cost is simply proportional to output Such a situation would arise if the underlying production function exhibits con stant returns to scale In that case suppose k1 units of capital input and l1 units of labor input are required to produce one unit of output Then C1v w 12 5 vk1 1 wl1 1019 To produce m units of output mk1 units of capital and ml1 units of labor are required because of the constant returnstoscale assumption2 Hence C1v w m2 5 vmk1 1 wml1 5 m 1vk1 1 wl12 5 mC1v w 12 1020 and the proportionality between output and cost is established 2The input combination ml1 mk1 minimizes the cost of producing m units of output because the ratio of the inputs is still k1l1 and the RTS for a constant returnstoscale production function depends only on that ratio Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 335 The situation in Figure 105a is more complicated There it is assumed that initially the total cost curve is concave although initially costs increase rapidly for increases in output that rate of increase slows as output expands into the midrange of output Beyond this mid dle range however the total cost curve becomes convex and costs begin to increase pro gressively more rapidly One possible reason for such a shape for the total cost curve is that there is some third factor of production say the services of an entrepreneur that is fixed as capital and labor usage expands In this case the initial concave section of the curve might be explained by the increasingly optimal usage of the entrepreneurs serviceshe or she needs a moderate level of production to use his or her skills fully Beyond the point of inflection however the entrepreneur becomes overworked in attempting to coordinate production and diminishing returns set in Hence total costs increase rapidly A variety of other explanations have been offered for the cubictype total cost curve in Figure 105a but we will not examine them here Ultimately the shape of the total cost curve is an empirical question that can be determined only by examining realworld data In the Extensions to this chapter we review some of the literature on cost functions In a total costs are proportional to output level Average and marginal costs as shown in b are equal and constant for all output levels FIGURE 104 Cost Curves in the Constant ReturnstoScale Case Total costs Average and marginal costs Output per period Output per period C AC MC a b Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 336 Part 4 Production and Supply 1043 Graphical analysis of average and marginal costs Information from the total cost curves can be used to construct the average and mar ginal cost curves shown in Figures 104b and 105b For the constant returnstoscale case Figure 104 this is simple Because total costs are proportional to output average and marginal costs are constant and equal for all levels of output3 These costs are shown by the horizontal line AC 5 MC in Figure 104b For the cubic total cost curve case Figure 105 computation of the average and mar ginal cost curves requires some geometric intuition As the definition in Equation 1018 If the total cost curve has the cubic shape shown in panel a then the average and marginal cost curves shown in panel b will be Ushaped The marginal cost curve passes through the low point of the average cost curve at output level q This same q has the property in panel a that a chord from the origin to the total cost curve is tangent to the curve at this output level FIGURE 105 Total Average and Marginal Cost Curves for the Cubic Total Cost Curve Case Total costs Average and marginal costs Output per period Output per period C AC q q a b Chord from origin MC 3Mathematically because C aq where a is the cost of one unit of output AC 5 C q 5 a 5 C q 5 MC Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 337 makes clear marginal cost is simply the slope of the total cost curve Hence because of the assumed shape of the curve the MC curve is Ushaped with MC falling over the concave portion of the total cost curve and rising beyond the point of inflection Because the slope is always positive however MC is always greater than 0 Average costs AC start out being equal to marginal costs for the first unit of output4 As output expands however AC exceeds MC because AC reflects both the marginal cost of the last unit produced and the higher marginal costs of the previously produced units As long as AC MC average costs must be decreasing Because the lower costs of the newly produced units are below average cost they continue to pull average costs downward Marginal costs increase however and eventually at q equal average cost Beyond this point MC AC and average costs will increase because they are pulled upward by increasingly higher marginal costs Consequently we have shown that the AC curve also has a Ushape and that it reaches a low point at q where AC and MC intersect5 In empirical studies of cost functions there is considerable interest in this point of min imum average cost It reflects the minimum efficient scale MES for the particular pro duction process being examined The point is also theoretically important because of the role it plays in perfectly competitive price determination in the long run see Chapter 12 Although we identified q using properties of the AC and MC curves in Figure 105b it can also be identified in Figure 105a as the output level at which a chord from the origin to the total cost curve is tangent to that curve6 105 SHIFTS IN COST CURvES The cost curves illustrated in Figures 104 and 105 show the relationship between costs and quantity produced on the assumption that all other factors are held constant Specif ically construction of the curves assumes that input prices and the level of technology do not change7 If these factors do change the cost curves will shift In this section we delve further into the mathematics of cost functions as a way of studying these shifts We begin with some examples 4Technically AC and MC approach each other in the limit as q approaches 0 This can be shown by lHôpitals rule which states that if f 1a2 5 g 1a2 5 0 then lim xSa f 1x2 g 1x2 5 lim xSa f r1x2 gr1x2 Applying the rule to unit cost functions because C 5 0 at q 5 0 we have lim qS0 AC 5 lim qS0 C q 5 lim qS0 Cq qq 5 lim qS0 MC 5Mathematically we can find the minimum AC by setting its derivative equal to 0 AC q 5 1Cq2 q 5 q 1Cq2 2 C 1 q2 5 q MC 2 C q2 5 0 implying q MC 2 C 5 0 or MC 5 Cq 5 AC Thus MC AC when AC is minimized 6 To understand why we need to be able to read marginal and average costs of panel a of Figure 105 Marginal cost can be read of panel a simply as the slope of C Reading average cost off panel a requires the device of a chord from the origin to C Since this chord starts at 00 it is easy to compute its slope slope of chord from origin 5 rise run 5 C 2 0 q 2 0 5 AC We saw from panel b that MC 5 AC at q Therefore at output level q the slope of C must equal the slope of a chord from the origin to C implying that the chord must be tangent to C 7For multiproduct firms an additional complication must be considered For such firms it is possible that the costs associated with producing one output say q1 are also affected by the amount of some other output being produced q2 In this case the firm is said to exhibit economies of scope and the total cost function will be of the form Cv w q1 q2 Hence q2 must also be held constant in constructing the q1 cost curves Presumably increases in q2 shift the q1 cost curves downward Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 338 Part 4 Production and Supply EXAMPLE 102 Some Illustrative Cost Functions In this example we calculate the cost functions associated with three different production functions Later we will use these examples to illustrate some of the general properties of cost functions 1 Fixed Proportions q 5 f 1k l2 5 min 1αk βl2 The calculation of cost functions from their underlying production functions is one of the more frustrating tasks for economics students Thus lets start with a simple example What we wish to do is show how total costs depend on input costs and on quantity produced In the fixedproportions case we know that production will occur at a vertex of the Lshaped isoquants where q 5 αk 5 βl Hence total costs are C1v w q2 5 vk 1 wl 5 va q αb 1 wa q βb 5 qa v α 1 w β b 1021 This is indeed the sort of function we want because it states total costs as a function of v w and q only together with some parameters of the underlying production function Because of the constant returnstoscale nature of this production function it takes the special form C1v w q2 5 qC1v w 12 1022 That is total costs are given by output times the cost of producing one unit Increases in input prices clearly increase total costs with this function and technical improvements that take the form of increasing the parameters α and β reduce costs 2 CobbDouglas q 5 f1k l2 5 kαl β This is our first example of burdensome computation but we can clarify the process by recognizing that the final goal is to use the results of cost minimization to replace the inputs in the production function with costs From Example 101 we know that cost minimization requires that w v 5 β α k l 1023 and so k 5 α β w v l 1024 Substitution into the production function permits a solution for labor input in terms of q v and w as q 5 kαl β 5 aα β w v b α l α1β 1025 or l c 1v w q2 5 q1α1βa β αb α1α1β2 w2α1α1β2vα1α1β2 1026 A similar set of manipulations gives kc 1v w q2 5 q1α1βaα βb β1α1β2 wβ1α1β2v2β1α1β2 1027 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 339 Now we are ready to derive total costs as C1v w q2 5 vkc 1 wl c 5 q11α1β2Bvα1α1β2wβ1α1β2 1028 where B 5 1α 1 β2α2α1α1β2β2β1α1β2a constant that involves only the parameters α and β Although this derivation was a bit messy several interesting aspects of this CobbDouglas cost function are readily apparent First whether the function is a convex linear or concave function of output depends on whether there are decreasing returns to scale 1α 1 β 12 constant returns to scale 1α 1 β 5 12 or increasing returns to scale 1α 1 β 12 Second an increase in any input price increases costs with the extent of the increase being deter mined by the relative importance of the input as reflected by the size of its exponent in the production function Finally the cost function is homogeneous of degree 1 in the input pricesa general feature of all cost functions as we shall show shortly 3 CES q 5 f 1k l2 5 1k ρ 1 l ρ2 γρ For this case your authors will mercifully spare you the algebra To derive the total cost function we use the costminimization condition specified in Equation 1015 solve for each input individually and eventually get C1v w q2 5 vk 1 wl 5 q1γ 1v ρ1 ρ212 1 w ρ1 ρ2122 1 ρ212ρ 5 q1g 1v12σ 1 w12σ2 1112σ2 1029 where the elasticity of substitution is given by σ 5 1 11 2 ρ2 Once again the shape of the total cost is determined by the scale parameter γ for this production function and the cost function increases in both of the input prices The function is also homogeneous of degree 1 in those prices One limiting feature of this form of the CES function is that the inputs are given equal weightshence their prices are equally important in the cost func tion This feature of the CES is easily generalized however see Problem 109 QUERY How are the various substitution possibilities inherent in the CES function reflected in the CES cost function in Equation 1029 1051 Properties of cost functions These examples illustrate some properties of total cost functions that are general 1 Homogeneity The total cost functions in Example 102 are all homogeneous of degree 1 in the input prices That is a doubling of input prices will precisely double the cost of producing any given output level you might check this out for yourself This is a property of all proper cost functions When all input prices double or are increased by any uniform proportion the ratio of any two input prices will not change Because cost minimization requires that the ratio of input prices be set equal to the RTS along a given isoquant the costminimizing input combination also will not change Hence the firm will buy exactly the same set of inputs and pay precisely twice as much for them One implication of this result is that a pure uniform inflation in all input costs will not change a firms input decisions Its cost curves will shift upward in precise correspon dence to the rate of inflation 2 Total cost functions are nondecreasing in q v and w This property seems obvious but it is worth dwelling on it a bit Because cost functions are derived from a cost minimization process any decrease in costs from an increase in one of the functions arguments would lead to a contradiction For example if an increase in output from q1 to q2 caused total costs to decrease it must be the case that the firm was not minimizing costs in the first place It should have produced q2 and thrown away an output of q2 2 q1 thereby Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 340 Part 4 Production and Supply producing q1 at a lower cost Similarly if an increase in the price of an input ever reduced total cost the firm could not have been minimizing its costs in the first place To see this suppose the firm was using the input combination 1l1 k12 and that w increases Clearly that will increase the cost of the initial input combination But if changes in input choices caused total costs to decrease that must imply that there was a lowercost input mix than 1l1 k12 initially Hence we have a contradiction and this property of cost func tions is established8 3 Total cost functions are concave in input prices It is probably easiest to illustrate this property with a graph Figure 106 shows total costs for various values of an input price say w holding q and v constant Suppose that initially input prices wr and vr prevail and 8A formal proof could also be based on the envelope theorem as applied to constrained minimization problems Consider the Lagrangian in Equation 103 As was pointed out in Chapter 2 we can calculate the change in the objective in such an expression here total cost with respect to a change in a variable by differentiating the Lagrangian Performing this differentiation yields MC 5 C q 5 q 5 λ 0 C v 5 v 5 k c 0 C w 5 w 5 l c 0 Not only do these envelope results prove this property of cost functions but they also are useful in their own right as we will show later in this chapter With input prices wr and vr total costs of producing q0 are C1vr wr q02 If the firm does not change its input mix costs of producing q0 would follow the straight line CPSEUDO With input substitution actual costs C1vr w q02 will fall below this line and hence the cost function is concave in w FIGURE 106 Cost Functions Are Concave in Input Prices Costs Cv w q0 Cv w q0 w w CPSEUDO Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 341 that total output q0 is produced at total cost C1vr wr q02 using costminimizing inputs lr and kr If the firm did not change its input mix in response to changes in wages then its total cost curve would be linear as reflected by the line CPSEUDO 1vr w q02 5 vrkr 1 wlr in the figure But a costminimizing firm probably would change the input mix it uses to produce q0 when wages change and these actual costs C1vr w q02 would fall below the pseudo costs Hence the total cost function must have the concave shape shown in Figure 106 One implication of this finding is that costs will be lower when a firm faces input prices that fluctuate around a given level than when they remain constant at that level With fluctuating input prices the firm can adapt its input mix to take advantage of such fluctuations by using a lot of say labor when its price is low and economizing on that input when its price is high 4 Properties carrying over to average and marginal costs Some but not all of these properties of total cost functions carry over to their related average and marginal cost functions Homogeneity is one property that carries over directly Because C1tv tw q2 5 tC1v w q2 we have AC1tv tw q2 5 C1tv tw q2 q 5 tC1v w q2 q 5 tAC1v w q2 1030 and9 MC1tv tw q2 5 C1tv tw q2 q 5 tC1v w q2 q 5 tMC1v w q2 1031 The effects of changes in q v and w on average and marginal costs are sometimes ambiguous however We have already shown that average and marginal cost curves may have negatively sloped segments so neither AC nor MC is nondecreasing in q Because total costs must not decrease when an input price increases it is clear that average cost is increasing in w and v But the case of marginal cost is more complex The main com plication arises because of the possibility of input inferiority In that admittedly rare case an increase in an inferior inputs price will actually cause marginal cost to decrease Although the proof of this is relatively straightforward10 an intuitive explanation for it is elusive Still in most cases it seems clear that the increase in the price of an input will increase marginal cost as well 1052 Input substitution A change in the price of an input will cause the firm to alter its input mix Hence a full study of how cost curves shift when input prices change must also include an examination of sub stitution among inputs The previous chapter provided a concept measuring how substitut able inputs arethe elasticity of substitution Here we will modify the definition using some results from cost minimization so that it is expressed only in terms of readily observable variables The modified definition will turn out to be more useful for empirical work 10The proof follows the envelope theorem results presented in footnote 8 Because the MC function can be derived by differentiation from the Lagrangian for cost minimization we can use Youngs theorem to show MC v 5 1q2 v 5 2 vq 5 2 qv 5 k q Hence if capital is a normal input an increase in v will raise MC whereas if capital is inferior an increase in v will actually reduce MC 9This result does not violate the theorem that the derivative of a function that is homogeneous of degree k is homogeneous of degree k 1 because we are differentiating with respect to q and total costs are homogeneous with respect to input prices only Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 342 Part 4 Production and Supply Recall the formula for the elasticity of substitution from Chapter 9 repeated here σ 5 d1kl2 d RTS RTS kl 5 d ln 1kl2 d ln RTS 1032 But the costminimization principle says that RTS 1of l for k2 5 wv at an optimum Sub stituting gives a new version of the elasticity of substitution11 s 5 d1kl2 d1wv2 wv kl 5 d ln 1kl2 d ln 1wv2 1033 distinguished by changing the label from σ to s The elasticities differ in two respects Whereas σ applies to any point on any isoquant s applies only to a single point on a single isoquant the equilibrium point where there is a tangency between the isoquant and an equal total cost line Although this would seem to be a drawback of s the big advantage of focusing on the equilibrium point is that s then involves only easily observable variables input amounts and prices By contrast σ involves the RTS the slope of an isoquant Knowl edge of the RTS would require detailed knowledge of the production process that even the firms engineers may not have let alone an outside observer In the twoinput case s must be nonnegative an increase in wv will be met by an increase in kl or in the limiting fixedproportions case kl will stay constant Large values of s indicate that firms change their input proportions significantly in response to changes in relative input prices whereas low values indicate that changes in input prices have relatively little effect 1053 Substitution with many inputs Instead of just the two inputs k and l now suppose there are many inputs to the produc tion process 1x1 x2 c xn2 that can be hired at competitive rental rates 1w1 w2 c wn2 Then the elasticity of substitution between any two inputs 1sij2 is defined as follows 11This definition is usually attributed to R G D Allen who developed it in an alternative form in his Mathematical Analysis for Economists New York St Martins Press 1938 pp 5049 12This definition is attributed to the Japanese economist M Morishima and these elasticities are sometimes referred to as Morishima elasticities In this version the elasticity of substitution for substitute inputs is positive Some authors reverse the order of subscripts in the denominator of Equation 1034 and in this usage the elasticity of substitution for substitute inputs is negative D E F I N I T I O N Elasticity of substitution The elasticity of substitution between inputs xi and xj is given by sij 5 1xixj2 1wjwi2 wjwi xixj 5 ln 1xixj2 ln 1wjwi2 1034 where output and all other input prices are held constant A subtle point that did not arise in the twoinput case regards what is assumed about the firms usage of the other inputs besides i and j Should we perform the thought experiment of holding them fixed as are other input prices and output Or should we take into account the adjustment of these other inputs to achieve cost minimization The latter assumption has proved to be more useful in economic analysis therefore that is the one we will take to be embodied in Equation 103412 For example a major topic in the theory of firms input choices is to describe the relationship between capital and energy inputs The definition Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 343 in Equation 1034 would permit a researcher to study how the ratio of energy to capital input changes when relative energy prices increase while permitting the firm to make any adjustments to labor input whose price has not changed that would be required for cost minimization Hence this would give a realistic picture of how firms behave with regard to whether energy and capital are more like substitutes or complements Later in this chapter we will look at this definition in a bit more detail because it is widely used in empirical studies of production 1054 Quantitative size of shifts in cost curves We have already shown that increases in an input price will raise total average and except in the inferior input case marginal costs We are now in a position to judge the extent of such increases First and most obviously the increase in costs will be influenced importantly by the relative significance of the input in the production process If an input constitutes a large fraction of total costs an increase in its price will raise costs signifi cantly An increase in the wage rate would sharply increase homebuilders costs because labor is a major input in construction On the other hand a price increase for a relatively minor input will have a small cost impact An increase in nail prices will not raise home costs much A less obvious determinant of the extent of cost increases is input substitutability If firms can easily substitute another input for the one that has increased in price there may be little increase in costs Increases in copper prices in the late 1960s for example had little impact on electric utilities costs of distributing electricity because they found they could easily substitute aluminum for copper cables Alternatively if the firm finds it difficult or impossible to substitute for the input that has become more costly then costs may increase rapidly The cost of gold jewelry along with the price of gold rose rapidly during the early 1970s because there was simply no substitute for the raw input It is possible to give a precise mathematical statement of the quantitative sizes of all these effects by using the elasticity of substitution To do so however would risk further cluttering the book with symbols13 For our purposes it is sufficient to rely on the previ ous intuitive discussion This should serve as a reminder that changes in the price of an input will have the effect of shifting firms cost curves with the size of the shift depend ing on the relative importance of the input and on the substitution possibilities that are available 1055 Technical change Technical improvements allow the firm to produce a given output with fewer inputs Such improvements obviously shift total costs downward if input prices stay constant Although the actual way in which technical change affects the mathematical form of the total cost curve can be complex there are cases where one may draw simple conclusions Suppose for example that the production function exhibits constant returns to scale and that technical change enters that function as described in Chapter 9 ie q 5 A 1t2f 1k l2 where A 102 5 1 In this case total costs in the initial period are given by C0 1v w q2 5 qC0 1v w 12 1035 13For a complete statement see C Ferguson Neoclassical Theory of Production and Distribution Cambridge UK Cambridge University Press 1969 pp 15460 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 344 Part 4 Production and Supply Because the same inputs that produced one unit of output in period 0 are also the cost minimizing way of producing A 1t2 units of output in period t we know that C0 1v w 12 5 Ct1v w A 1t2 2 5 A 1t2Ct1v w 12 1036 Therefore we can compute the total cost function in period t as Ct1v w q2 5 qCt1v w 12 5 qC0 1v w 12 A 1t2 5 C0 1v w q2 A 1t2 1037 Hence total costs decrease over time at the rate of technical change14 Note that in this case technical change is neutral in that it does not affect the firms input choices as long as input prices stay constant This neutrality result might not hold in cases where technical progress takes a more complex form or where there are variable returns to scale Even in these more complex cases however technical improvements will cause total costs to decrease 14To see that the indicated rates of change are the same note first that the rate of change of technical progress is r1t2 5 Ar 1t2 A 1t2 while the rate of change in total cost is Ct t 1 Ct 5 C0Ar 1t2 A 1t22 1 Ct 5 Ar 1t2 A 1t2 5 r1t2 using Equation 1037 EXAMPLE 103 Shifting the CobbDouglas Cost Function In Example 102 we computed the CobbDouglas cost function as C1v w q2 5 q11α1β2Bvα1α1β2wβ1α1β2 1038 where B 5 1α 1 β2α2α1α1β2β2β1α1β2 As in the numerical illustration in Example 101 lets assume that α 5 β 5 05 in which case the total cost function is greatly simplified C1v w q2 5 2qv05w05 1039 This function will yield a total cost curve relating total costs and output if we specify particular values for the input prices If as before we assume v 5 3 and w 5 12 then the relationship is C13 12 q2 5 2q36 5 12q 1040 and as in Example 101 it costs 480 to produce 40 units of output Here average and marginal costs are easily computed as AC 5 C q 5 12 MC 5 C q 5 12 1041 As expected average and marginal costs are constant and equal to each other for this constant returnstoscale production function Changes in input prices If either input price were to change all these costs would change also For example if wages were to increase to 27 an easy number with which to work costs would become Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 345 C13 27 q2 5 2q81 5 18q AC 5 18 MC 5 18 1042 Notice that an increase in wages of 125 percent increased costs by only 50 percent here both because labor represents only 50 percent of all costs and because the change in input prices encouraged the firm to substitute capital for labor The total cost function because it is derived from the costminimization assumption accomplishes this substitution behind the scenes reporting only the final impact on total costs Technical progress Lets look now at the impact that technical progress can have on costs Specifi cally assume that the CobbDouglas production function is q 5 A1t2k05l 05 5 e 03tk05l 05 1043 That is we assume that technical change takes an exponential form and that the rate of technical change is 3 percent per year Using the results of the previous section Equation 1037 yields Ct1v w q2 5 C0 1v w q2 A1t2 5 2qv 05w 05e203t 1044 Thus if input prices remain the same then total costs decrease at the rate of technical improvementthat is at 3 percent per year After say 20 years costs will be with v 5 3 w 5 12 C20 13 12 q2 5 2q36 e260 5 12q 10552 5 66q AC20 5 66 MC20 5 66 1045 Consequently costs will have decreased by nearly 50 percent as a result of the technical change This would for example more than have offset the wage increase illustrated previously QUERY In this example what are the elasticities of total costs with respect to changes in input costs Is the size of these elasticities affected by technical change 1056 Contingent demand for inputs and Shephards lemma As we described earlier the process of cost minimization creates an implicit demand for inputs Because that process holds quantity produced constant this demand for inputs will also be contingent on the quantity being produced This relationship is fully reflected in the firms total cost function and perhaps surprisingly contingent demand functions for all the firms inputs can be easily derived from that function The process involves what has come to be called Shephards lemma15 which states that the contingent demand func tion for any input is given by the partial derivative of the total cost function with respect to that inputs price Because Shephards lemma is widely used in many areas of economic research we will provide a relatively detailed examination of it 15Named for R W Shephard who highlighted the important relationship between cost functions and input demand functions in his Cost and Production Functions Princeton NJ Princeton University Press 1970 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 346 Part 4 Production and Supply The intuition behind Shephards lemma is straightforward Suppose that the price of labor w were to increase slightly How would this affect total costs If nothing else changed it seems that costs would increase by approximately the amount of labor l that the firm was currently hiring Roughly speaking then Cw 5 l and that is what Shephards lemma claims Figure 106 makes roughly the same point graphically Along the pseudo cost function all inputs are held constant therefore an increase in the wage increases costs in direct proportion to the amount of labor used Because the true cost function is tangent to the pseudofunction at the current wage its slope ie its partial derivative also will show the current amount of labor input demanded Technically Shephards lemma is one result of the envelope theorem that was first dis cussed in Chapter 2 There we showed that the change in the optimal value in a constrained optimization problem with respect to one of the parameters of the problem can be found by differentiating the Lagrangian for that optimization problem with respect to this chang ing parameter In the costminimization case the Lagrangian is 5 vk 1 wl 1 λ 3q 2 f 1k l2 4 1046 and the envelope theorem applied to either input is C1v w q2 v 5 1v w q λ2 v 5 k c 1v w q2 C1v w q2 w 5 1v w q λ2 w 5 l c 1v w q2 1047 where the notation is intended to make clear that the resulting demand functions for capital and labor input depend on v w and q Because quantity produced enters these functions input demand is indeed contingent on that variable This feature of the demand functions is also reflected by the c in the notation16 Hence the demand relations in Equation 1047 do not represent a complete picture of input demand because they still depend on a vari able that is under the firms control In the next chapter we will complete the study of input demand by showing how the assumption of profit maximization allows us to effectively replace q in the input demand relationships with the market price of the firms output p 16The notation mirrors the one used for compensated demand curves in Chapter 5 which were derived from the expenditure function In that case such demand functions were contingent on the utility target assumed EXAMPLE 104 Contingent Input Demand Functions In this example we will show how the total cost functions derived in Example 102 can be used to derive contingent demand functions for the inputs capital and labor 1 Fixed Proportions C1v w q2 5 q 1vα 1 wβ2 For this cost function contingent demand functions are simple kc 1v w q2 5 C1v w q2 v 5 q α lc 1v w q2 5 C1v w q2 w 5 q β 1048 To produce any particular output with a fixed proportions production function at minimal cost the firm must produce at the vertex of its isoquants no matter what the inputs prices are Hence the demand for inputs depends only on the level of output and v and w do not enter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 347 the contingent input demand functions Input prices may however affect total input demands in the fixed proportions case because they may affect how much the firm decides to sell 2 CobbDouglas C1v w q2 5 q11α1β2Bvα1α1β2wβ1α1β2 In this case the derivation is mess ier but also more instructive k c 1v w q2 5 C v 5 α α 1 β q11α1β2Bv2β1α1β2w β1α1β2 5 α α 1 β q11α1β2Baw v b β1α1β2 l c 1v w q2 5 C w 5 β α 1 β q11α1β2Bvα1α1β2w2α1α1β2 5 β α 1 β q11α1β2Baw v b 2α1α1β2 1049 Consequently the contingent demands for inputs depend on both inputs prices If we assume α 5 β 5 05 so B 5 2 these reduce to k c 1v w q2 5 05 q 2 aw v b 05 5 qaw v b 05 l c 1v w q2 5 05 q 2 aw v b 205 5 qaw v b 205 1050 With v 5 3 w 5 12 and q 5 40 Equations 1050 yield the result we obtained previously that the firm should choose the input combination k 5 80 l 5 20 to minimize the cost of producing 40 units of output If the wage were to increase to say 27 the firm would choose the input combination k 5 120 l 5 403 to produce 40 units of output Total costs would increase from 480 to 520 but the ability of the firm to substitute capital for the now more expensive labor does save considerably For example the initial input combination would now cost 780 3 CES C1v w q2 5 q1g 1v12σ 1 w12σ2 1112σ2 The importance of input substitution is shown even more clearly with the contingent demand functions derived from the CES function For that function k c 1v w q2 5 C v 5 1 1 2 σ q1γ 1v12σ 1 w12σ2 σ112σ2 11 2 σ2v2σ 5 q1g 1v12σ 1 w12σ2 σ112σ2v2σ l c 1v w q2 5 C w 5 1 1 2 σ q1γ 1v12σ 1 w12σ2 σ112σ2 11 2 σ2w2σ 5 q1γ 1v 12σ 1 w 12σ2 σ112σ2w2σ 1051 These functions collapse when σ 5 1 the CobbDouglas case but we can study exam ples with either more 1σ 5 22 or less 1σ 5 052 substitutability and use CobbDouglas as the middle ground If we assume constant returns to scale 1g 5 12 and v 5 3 w 5 12 and q 5 40 then contingent demands for the inputs when σ 5 2 are k c 13 12 402 5 40 1321 1 12212 22 322 5 256 l c 13 12 402 5 40 1321 1 12212 22 1222 5 16 1052 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 348 Part 4 Production and Supply That is the level of capital input is 16 times the amount of labor input With less substitut ability 1σ 5 052 contingent input demands are k c 13 12 402 5 40 1305 1 12052 1 3205 5 120 l c 13 12 402 5 40 1305 1 12052 1 12205 5 60 1053 Thus in this case capital input is only twice as large as labor input Although these various cases cannot be compared directly because different values for σ scale output differently we can as an example look at the consequence of an increase in w to 27 in the low substitutability case With w 5 27 the firm will choose k 5 160 l 5 533 In this case the cost savings from substitution can be calculated by comparing total costs when using the initial input combination 15 3 120 1 27 60 5 19802 to total costs with the optimal combination 15 3 160 1 27 533 5 19192 Hence moving to the optimal input combination reduces total costs by only about 3 percent In the CobbDouglas case cost savings are over 20 percent QUERY How would total costs change if w increased from 12 to 27 and the production function took the simple linear form q 5 k 1 4l What light does this result shed on the other cases in this example 1057 Shephards lemma and the elasticity of substitution One especially nice feature of Shephards lemma is that it can be used to show how to derive information about input substitution directly from the total cost function through differentiation Using the definition in Equation 1034 yields sij 5 ln 1xixj2 ln 1wjwi2 5 ln 1CiCj2 ln 1wjwi2 1054 where Ci and Cj are the partial derivatives of the total cost function with respect to the input prices Once the total cost function is known perhaps through econometric estima tion information about substitutability among inputs can thus be readily obtained from it In the Extensions to this chapter we describe some of the results that have been obtained in this way Problems 1011 and 1012 provide some additional details about ways in which substitutability among inputs can be measured 106 SHORTRUN LONGRUN DISTINCTION It is traditional in economics to make a distinction between the short run and the long run Although no precise temporal definition can be provided for these terms the general purpose of the distinction is to differentiate between a short period during which eco nomic actors have only limited flexibility in their actions and a longer period that provides greater freedom One area of study in which this distinction is important is in the theory of the firm and its costs because economists are interested in examining supply reactions over differing time intervals In the remainder of this chapter we will examine the implications of such differential response To illustrate why shortrun and longrun reactions might differ assume that capital input is held fixed at a level of k1 and that in the short run the firm is free to vary only its labor input17 The idea is that the firm has inherited k1 a level of capital that was suited to Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 349 the longrun equilibrium under previous conditions but that may or may not suit present conditions The idea is that the firm has inherited k1 a level of capital that was suited to the longrun equilibrium under previous conditions but that may or may not suit present conditions Implicitly we are assuming that alterations to this level of capital are infinitely costly in the short run As a result of this assumption the shortrun production function is q 5 f 1k1 l2 1055 where this notation explicitly shows that capital inputs may not vary Of course the level of output still may be changed if the firm alters its use of labor 1061 Shortrun total costs Total cost for the firm continues to be defined as C 5 vk 1 wl 1056 for our shortrun analysis but now capital input is fixed at k1 To denote this fact we will write SC 5 vk1 1 wl 1057 where the S indicates that we are analyzing shortrun costs with the level of capital input fixed Throughout our analysis we will use this method to indicate shortrun costs Usu ally we will not denote the level of capital input explicitly but it is understood that this input is fixed The cost concepts introduced earlierC AC MCare in fact longrun con cepts because in their definitions all inputs were allowed to vary freely Their longrun nature is indicated by the absence of a leading S18 1062 Fixed and variable costs The two types of input costs in Equation 1057 are given special names The term vk1 is referred to as shortrun fixed costs because k1 is constant these costs will not change in the short run The term wl is referred to as shortrun variable costslabor input can indeed be varied in the short run Hence we have the following definitions 17Of course this approach is for illustrative purposes only In many actual situations labor input may be less flexible in the short run than is capital input 18The astute reader may worry that since capital k1 is locked in the firm and thus cannot be rented out for alternative uses short run fixed cost vk1 is an accounting cost and not an economic cost and thus should not figure into the shortrun cost function at all While there is some merit to this perspective the standard convention is to include vk1 as part of shortrun costs The reason for the convention is that it allows an applestoapples comparison to the theoretical case in which the firm is free to choose capital rather than inheriting a given level k1 Certainly vk is an economic cost when the firm can flexibly choose capital so it is important to include the analogous capital expenditure vk1 as a cost when capital is inflexible Otherwise we would erroneously conclude that inflexibility somehow helps to reduce a firms costs D E F I N I T I O N Shortrun fixed and variable costs Shortrun fixed costs are costs associated with inputs that cannot be varied in the short run Shortrun variable costs are costs of those inputs that can be varied to change the firms output level While these definitions emphasize the firms choice of inputs they have implications for how different costs categories vary with the firms output choice Shortrun variable costs can be reduced or increased by producing more or less and can be avoided entirely by pro ducing nothing By contrast shortrun fixed costs must be paid regardless of the output level chosen even zero Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 350 Part 4 Production and Supply 1063 Nonoptimality of shortrun costs It is important to understand that total shortrun costs are not the minimal costs for pro ducing the various output levels Because we are holding capital fixed in the short run the firm does not have the flexibility of input choice that we assumed when we discussed cost minimization earlier in this chapter Rather to vary its output level in the short run the firm will be forced to use nonoptimal input combinations The RTS will not necessarily be equal to the ratio of the input prices This is shown in Figure 107 In the short run the firm is constrained to use k1 units of capital To produce output level q0 it will use l0 units of labor Similarly it will use l1 units of labor to produce q1 and l2 units to produce q2 The total costs of these input combinations are given by SC0 SC1 and SC2 respectively Only for the input combination k1 l1 is output being produced at minimal cost Only at that point is the RTS equal to the ratio of the input prices From Figure 107 it is clear that q0 is being produced with too much capital in this shortrun situation Cost minimization should suggest a southeasterly movement along the q0 isoquant indicating a substitution of labor for capital in production Similarly q2 is being produced with too little capital and costs could be reduced by substituting capital for labor Neither of these substitutions is possible in the short run Over a longer period however the firm will be able to change its level of capital input and will adjust its input usage to the costminimizing combinations We have already discussed this flexible case earlier in this chapter and shall return to it to illustrate the connection between longrun and shortrun cost curves Because capital input is fixed at k in the short run the firm cannot bring its RTS into equality with the ratio of input prices Given the input prices q0 should be produced with more labor and less capital than it will be in the short run whereas q2 should be produced with more capital and less labor than it will be FIgurE 107 Nonoptimal Input Choices Must Be Made in the Short Run SC0 l2 l1 k1 l0 q2 q1 q0 SC1 C SC2 k per period l per period Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 351 1064 Shortrun marginal and average costs Frequently it is more useful to analyze shortrun costs on a perunit of output basis rather than on a total basis The two most important perunit concepts that can be derived from the shortrun total cost function are the shortrun average total cost function SAC and the shortrun marginal cost function SMC These concepts are defined as SAC 5 total costs total output 5 SC q SMC 5 change in total costs change in output 5 SC q 1058 where again these are defined for a specified level of capital input These definitions for average and marginal costs are identical to those developed previously for the longrun fully flexible case and the derivation of cost curves from the total cost function proceeds in exactly the same way Because the shortrun total cost curve has the same general type of cubic shape as did the total cost curve in Figure 105 these shortrun average and marginal cost curves will also be Ushaped 1065 Relationship between shortrun and longrun cost curves It is easy to demonstrate the relationship between the shortrun costs and the fully flexible longrun costs that were derived previously in this chapter Figure 108 shows this relationship for both the constant returnstoscale and cubic total cost curve cases Shortrun total costs for three levels of capital input are shown although of course it would be possible to show many more such shortrun curves The figures show that longrun total costs C are always less than shortrun total costs except at that output level for which the assumed fixed capital input is appropriate to longrun cost minimi zation For example as in Figure 107 with capital input of k1 the firm can obtain full cost minimization when q1 is produced Hence shortrun and longrun total costs are equal at this point For output levels other than q1 however SC C as was the case in Figure 107 Technically the longrun total cost curves in Figure 108 are said to be an envelope of their respective shortrun curves These shortrun total cost curves can be represented parametrically by shortrun total cost 5 SC1v w q k2 1059 and the family of shortrun total cost curves is generated by allowing k to vary while hold ing v and w constant The longrun total cost curve C must obey the shortrun relationship in Equation 1059 and the further condition that k be cost minimizing for any level of out put A firstorder condition for this minimization is that SC1v w q k2 k 5 0 1060 Solving Equations 1059 and 1060 simultaneously then generates the longrun total cost function Although this is a different approach to deriving the total cost function it should give precisely the same results derived earlier in this chapteras the next example illustrates Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 352 Part 4 Production and Supply By considering all possible levels of capital input the longrun total cost curve C can be traced In a the underlying production function exhibits constant returns to scale In the long run although not in the short run total costs are proportional to output In b the longrun total cost curve has a cubic shape as do the shortrun curves Diminishing returns set in more sharply for the shortrun curves however because of the assumed fixed level of capital input FIgurE 108 Two Possible Shapes for LongRun Total Cost Curves Total costs Total costs a Constant returns to scale b Cubic total cost curve case Output per period Output per period SCk0 SCk0 q0 q1 q2 q0 q1 q2 C C SCk1 SCk1 SCk2 SCk2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 353 EXAMPLE 105 Envelope relations and CobbDouglas Cost Functions Again we start with the CobbDouglas production function q 5 kαβ but now we hold capital input constant at k1 Thus in the short run q 5 kα 1l β or l 5 q1βk2αβ 1 1061 and total costs are given by SC1v w q k12 5 vk1 1 wl 5 vk1 1 wq1βk2αβ 1 1062 Notice that the fixed level of capital enters into this shortrun total cost function in two ways 1 k1 determines fixed costs and 2 k1 also in part determines variable costs because it deter mines how much of the variable input labor is required to produce various levels of output To derive longrun costs we require that k be chosen to minimize total costs SC1v w q k2 k 5 v 1 2α β wq1βk21α1β2β 5 0 1063 Although the algebra is messy this equation can be solved for k and substituted into Equation 1062 to return us to the CobbDouglas cost function C1v w q2 5 Bq11α1β2v α1α1β2w β1α1β2 1064 Numerical example If we again let α 5 β 5 05 v 5 3 and w 5 12 then the shortrun cost func tion is SC13 12 q k12 5 3k1 1 12q2k21 1 1065 In Example 101 we found that the costminimizing level of capital input for q 5 40 was k 5 80 Equation 1065 shows that shortrun total costs for producing 40 units of output with k1 5 80 is SC13 12 q 802 5 3 80 1 12 q2 1 80 5 240 1 3q2 20 5 240 1 240 5 480 1066 which is just what we found before We can also use Equation 1065 to show how costs differ in the short and long run Table 101 shows that for output levels other than q 5 40 shortrun costs are larger than longrun costs and that this difference is proportionally larger the farther one gets from the output level for which k 5 80 is optimal TABLE 101 DIFFErENCE BETWEEN SHOrTruN AND LONgruN TOTAL COST k 5 80 q C 5 12q SC 5 240 1 3q220 10 120 255 20 240 300 30 360 375 40 480 480 50 600 615 60 720 780 70 840 975 80 960 1200 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 354 Part 4 Production and Supply TABLE 102 uNIT COSTS IN THE LONg ruN AND THE SHOrT ruN k 5 80 q AC MC SAC SMC 10 12 12 255 3 20 12 12 150 6 30 12 12 125 9 40 12 12 120 12 50 12 12 123 15 60 12 12 130 18 70 12 12 139 21 80 12 12 150 24 It is also instructive to study differences between the longrun and shortrun perunit costs in this situation Here AC 5 MC 5 12 We can compute the shortrun equivalents when k 5 80 as SAC 5 SC q 5 240 q 1 3q 20 SMC 5 SC q 5 6q 20 1067 Both of these shortrun unit costs are equal to 12 when q 5 40 However as Table 102 shows shortrun unit costs can differ significantly from this figure depending on the output level that the firm produces Notice in particular that shortrun marginal cost increases rapidly as output expands beyond q 5 40 because of diminishing returns to the variable input labor This conclu sion plays an important role in the theory of shortrun price determination QUERY Explain why an increase in w will increase both shortrun average cost and shortrun marginal cost in this illustration but an increase in v affects only shortrun average cost 1066 Graphs of perunit cost curves The envelope total cost curve relationships exhibited in Figure 108 can be used to show geometric connections between shortrun and longrun average and marginal cost curves These are presented in Figure 109 for the cubic total cost curve case In the figure short run and longrun average costs are equal at that output for which the fixed capital input is appropriate At q1 for example SAC1k12 5 AC because k1 is used in producing q1 at minimal costs For movements away from q1 shortrun average costs exceed longrun average costs thus reflecting the costminimizing nature of the longrun total cost curve Because the minimum point of the longrun average cost curve AC plays a major role in the theory of longrun price determination it is important to note the various curves that pass through this point in Figure 109 First as is always true for average and marginal cost curves the MC curve passes through the low point of the AC curve At q1 longrun average and marginal costs are equal Associated with q1 is a certain level of capital input say k1 the shortrun average cost curve for this level of capital input is tangent to the AC curve at its minimum point The SAC curve also reaches its minimum at output level q1 For movements away from q1 the AC curve is much flatter than the SAC curve and this reflects the greater flexibility open to firms in the long run Shortrun costs increase rapidly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 355 because capital inputs are fixed In the long run such inputs are not fixed and diminishing marginal productivities do not occur so abruptly Finally because the SAC curve reaches its minimum at q1 the shortrun marginal cost curve SMC also passes through this point Therefore the minimum point of the AC curve brings together the four most important perunit costs At this point AC 5 MC 5 SAC 5 SMC 1068 For this reason as we shall show in Chapter 12 the output level q1 is an important equilib rium point for a competitive firm in the long run 1067 Practical examples of fixed costs The model we have studied associates fixed costs with inputs that are inflexible in the short run though flexible in the long run This model covers a broad range of industries and is particularly relevant for manufacturing For example a luxurycar manufacturer may have built a large factory and hired a large workforce when gas prices were expected to be low The manufacturer may have difficulty scaling back these inputs if it turns out that gasoline prices unexpectedly rise reducing demand for its luxury cars which guzzle more gas than other models However the passage of several years may be enough time to downsize the factory and renegotiate labor contracts so that a lower output can be efficiently produced The astute reader may worry that since capital k1 is locked in the firm and thus cannot be rented out for alternative uses shortrun fixed cost vk1 is an accounting not an economic cost and thus should not figure into the shortrun cost function at all While there is some merit to this perspective the standard convention is to include vk1 as part of shortrun costs The reason for the convention is that it allows an applestoapples comparison to the theoretical case in which the firm is free to choose capital rather than inheriting a given level k1 Certainly vk is an economic cost when the firm can flexibly choose capital so it is important to include the analogous capital expenditure vk1 as a cost when capital is inflexible Otherwise we would erroneously conclude that inflexibility somehow helps to reduce a firms costs Costs Output per period q0 q1 q2 SMCk0 SMCk1 SMCk2 MC SACk0 SACk1 AC SACk2 FIgurE 109 Average and Marginal Cost Curves for the Cubic Cost Curve Case Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 356 Part 4 Production and Supply Fixed costs arise for reasons other than input inflexibility in other settings especially hightech industries For example in media marketsbooks movies musicmuch of the production cost lies in the initial creation of the work the socalled first copy costs The unit cost of distributing the work to consumers afterward may be quite lowessentially zero in the case of digital media Firstcopy costs are fixed in the sense of being indepen dent of how many consumers receive the work after it is created The term vk1 used earlier for expenditures on inflexible inputs can be reinterpreted as the firstcopy cost The first copy can be improved by devoting more inputs k1 to it in the case of a movie for example using bigger movie sets or more or higher quality actors We noticed a related sidebene fit of higher k1 in the previous analysis In the numerical calculations in Example 105 we found that an increase in k1 while of course raising shortrun fixed costs had the benefit of lowering the shortrun variable cost of producing a given output Fixed costs arise in a broader set of hightech markets besides digital media Any prod uct that must be inventedranging from a new drug to a faster flying dronemay require substantial investment in research and development Because this investment is expended before output is produced it is of necessity independent of the subsequent output level and in that sense is a fixed cost The vk1 term used earlier for expenditures on inflexi ble inputs can be reinterpreted as the research and development investment The more inputs k1 devoted to research and development the better the resulting product better can mean that the product can subsequently be manufactured at lower variable cost or that it is higher quality generating higher demand Network markets such as electricity natural gas and fixedline telephone utilities also involve a large upfront cost to connect to consumers homes The cost of connecting the consumer to the distribution network is fixed in that it is independent of the consum ers subsequent usage Solving for the optimal upfront investment in media hightech network or other markets raises complex issues that are not addressed until Chapter 17 where we develop a detailed theory of investment so we are content with just mentioning these cases here19 19 One approach is to treat the upfront investment as a sunk cost that can be ignored in the analysis of the subsequent output choice There is merit to this approach if the output choice is the only decision of interest and in essence is the approach adopted in this chapter However if one is interested in analyzing the upfront investment decision at that point it is not sunk and so is involves real economic costs In that analysis the upfront investment would need to be treated as a fixed cost since it will not vary with the number of units subsequently sold Summary In this chapter we examined the relationship between the level of output a firm produces and the input costs associated with that level of production The resulting cost curves should gen erally be familiar to you because they are widely used in most courses in introductory economics Here we have shown how such curves reflect the firms underlying production function and the firms desire to minimize costs By developing cost curves from these basic foundations we were able to illustrate a number of important findings A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution RTS is equal to the ratio of the inputs rental prices Repeated application of this minimization procedure yields the firms expansion path Because the expansion path shows how input usage expands with the level of output it also shows the relationship between output level and total cost That relationship is summarized by the total cost function C1v w q2 which shows production costs as a function of output levels and input prices The firms average cost 1AC 5 Cq2 and marginal cost 1MC 5 Cq2 functions can be derived directly from the total cost function If the total cost curve has a gen eral cubic shape then the AC and MC curves will be Ushaped Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 357 All cost curves are drawn on the assumption that the input prices are held constant When input prices change cost curves will shift to new positions The extent of the shifts will be determined by the overall importance of the input whose price has changed and by the ease with which the firm may substitute one input for another Technical progress will also shift cost curves Input demand functions can be derived from the firms total cost function through partial differentiation These input demand functions will depend on the quantity of output that the firm chooses to produce and are therefore called contingent demand functions In the short run the firm may not be able to vary some inputs It can then alter its level of production only by changing its employment of variable inputs In so doing it may have to use nonoptimal highercost input combinations than it would choose if it were possible to vary all inputs Problems 101 Suppose that a firm produces two different outputs the quantities of which are represented by q1 and q2 In general the firms total costs can be represented by C1q1 q22 This function exhibits economies of scope if C1q1 02 1 C10 q22 C1q1 q22 for all output levels of either good a Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct firm than in two singleproduct firms producing each good separately b If the two outputs are actually the same good we can define total output as q 5 q1 1 q2 Suppose that in this case average cost 15 Cq2 decreases as q increases Show that this firm also enjoys economies of scope under the definition provided here 102 Professor Smith and Professor Jones are going to produce a new introductory textbook As true scientists they have laid out the production function for the book as q 5 S12J12 where q is the number of pages in the finished book S is the number of working hours spent by Smith and J is the number of hours spent working by Jones After having spent 900 hours preparing the first draft time which he valued at 3 per working hour Smith has to move on to other things and cannot contribute any more to the book Jones whose labor is valued at 12 per working hour will revise Smiths draft to complete the book a How many hours will Jones have to spend to produce a finished book of 150 pages Of 300 pages Of 450 pages b What is the marginal cost of the 150th page of the finished book Of the 300th page Of the 450th page 103 Suppose that a firms fixed proportion production function is given by q 5 min 15k 10l2 a Calculate the firms longrun total average and marginal cost functions b Suppose that k is fixed at 10 in the short run Calculate the firms shortrun total average and marginal cost functions c Suppose v 5 1 and w 5 3 Calculate this firms longrun and shortrun average and marginal cost curves 104 A firm producing hockey sticks has a production function given by q 5 2kl In the short run the firms amount of capital equipment is fixed at k 5 100 The rental rate for k is v 5 1 and the wage rate for l is w 5 4 a Calculate the firms shortrun total cost curve Calculate the shortrun average cost curve b What is the firms shortrun marginal cost function What are the SC SAC and SMC for the firm if it produces 25 hockey sticks Fifty hockey sticks One hundred hockey sticks Two hundred hockey sticks c Graph the SAC and the SMC curves for the firm Indicate the points found in part b d Where does the SMC curve intersect the SAC curve Explain why the SMC curve will always intersect the SAC curve at its lowest point Suppose now that capital used for producing hockey sticks is fixed at k1 in the short run e Calculate the firms total costs as a function of q w v and k1 f Given q w and v how should the capital stock be chosen to minimize total cost g Use your results from part f to calculate the longrun total cost of hockey stick production h For w 5 4 v 5 1 graph the longrun total cost curve for hockey stick production Show that this is an enve lope for the shortrun curves computed in part e by examining values of k1 of 100 200 and 400 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 358 Part 4 Production and Supply 105 An enterprising entrepreneur purchases two factories to produce widgets Each factory produces identical products and each has a production function given by qi 5 kili i 5 1 2 The factories differ however in the amount of capital equipment each has In particular factory 1 has k1 5 25 whereas factory 2 has k2 5 100 Rental rates for k and l are given by w 5 v 5 1 a If the entrepreneur wishes to minimize shortrun total costs of widget production how should output be allocated between the two factories b Given that output is optimally allocated between the two factories calculate the shortrun total average and marginal cost curves What is the marginal cost of the 100th widget The 125th widget The 200th widget c How should the entrepreneur allocate widget production between the two factories in the long run Calculate the longrun total average and marginal cost curves for widget production d How would your answer to part c change if both factories exhibited diminishing returns to scale 106 Suppose the totalcost function for a firm is given by C 5 qw 23v13 a Use Shephards lemma to compute the constant output demand functions for inputs l and k b Use your results from part a to calculate the underlying production function for q 107 Suppose the totalcost function for a firm is given by C 5 q1v 1 2vw 1 w2 a Use Shephards lemma to compute the constant output demand function for each input k and l b Use the results from part a to compute the underlying production function for q c You can check the result by using results from Exam ple 102 to show that the CES cost function with σ 5 05 ρ 5 21 generates this totalcost function 108 In a famous article J viner Cost Curves and Supply Curves Zeitschrift fur Nationalokonomie 3 September 1931 2346 viner criticized his draftsman who could not draw a family of SAC curves whose points of tangency with the Ushaped AC curve were also the minimum points on each SAC curve The draftsman protested that such a drawing was impossible to construct Whom would you support in this debate Analytical Problems 109 Generalizing the CES cost function The CES production function can be generalized to permit weighting of the inputs In the twoinput case this function is q 5 f1k l2 5 3 1αk2 ρ 1 1βl2 ρ4 γ ρ a What is the totalcost function for a firm with this pro duction function Hint You can of course work this out from scratch easier perhaps is to use the results from Example 102 and reason that the price for a unit of capi tal input in this production function is vα and for a unit of labor input is wβ b If γ 5 1 and α 1 β 5 1 it can be shown that this pro duction function converges to the CobbDouglas form q 5 kαl β as ρ S 0 What is the total cost function for this particular version of the CES function c The relative labor cost share for a twoinput production function is given by wlvk Show that this share is con stant for the CobbDouglas function in part b How is the relative labor share affected by the parameters α and b d Calculate the relative labor cost share for the general CES function introduced above How is that share affected by changes in wv How is the direction of this effect determined by the elasticity of substitution σ How is it affected by the sizes of the parameters α and β 1010 Input demand elasticities The ownprice elasticities of contingent input demand for labor and capital are defined as el cw 5 l c w w l c ek cv 5 k c v v k c a Calculate el c w and ek c v for each of the cost functions shown in Example 102 b Show that in general el c w 1 ek c v 5 0 c Show that the crossprice derivatives of contin gent demand functions are equalthat is show that l cv 5 k cw Use this fact to show that slel c v 5 skek c w where sl sk are respectively the share of labor in total cost 1wlC2 and of capital in total cost 1vkC2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 359 d Use the results from parts b and c to show that slel c w 1 skekc w 5 0 e Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input demand 1011 The elasticity of substitution and input demand elasticities The definition of the Morishima elasticity of substitution sij in Equation 1054 can be recast in terms of input demand elas ticities This illustrates the basic asymmetry in the definition a Show that if only wj changes sij 5 ex c i wj 2 ex c j wj b Show that if only wi changes sji 5 ex c j wi 2 ex c i wi c Show that if the production function takes the general CES form q 5 1 g n i51 x ρ i 2 γρ for ρ 2 0 then all of the Morishima elasticities are the same sij 5 1 11 2 ρ2 5 σ This is the only case in which the Morishima definition is symmetric 1012 The Allen elasticity of substitution Many empirical studies of costs report an alternative defini tion of the elasticity of substitution between inputs This alter native definition was first proposed by R G D Allen in the 1930s and further clarified by H Uzawa in the 1960s This definition builds directly on the production functionbased elasticity of substitution defined in footnote 6 of Chapter 9 Aij 5 CijCCiCj where the subscripts indicate partial differen tiation with respect to various input prices Clearly the Allen definition is symmetric a Show that Aij 5 exc i wjsj where sj is the share of input j in total cost b Show that the elasticity of si with respect to the price of input j is related to the Allen elasticity by esi pj 5 sj1Aij 2 12 c Show that with only two inputs Akl 5 1 for the Cobb Douglas case and Akl 5 σ for the CES case d Read Blackorby and Russell 1989 Will the Real Elas ticity of Substitution Please Stand Up to see why the Morishima definition is preferred for most purposes Suggestions for Further Reading Allen R G D Mathematical Analysis for Economists New York St Martins Press 1938 various pagessee index Complete though dated mathematical analysis of substitution possibilities and cost functions Notation somewhat difficult Blackorby C and R R Russell Will the Real Elasticity of Substitution Please Stand Up A Comparison of the Allen Uzawa and Morishima Elasticities American Economic Review September 1989 88288 A nice clarification of the proper way to measure substitutability among many inputs in production Argues that the AllenUzawa definition is largely useless and that the Morishima definition is by far the best Ferguson C E The Neoclassical Theory of Production and Distribution Cambridge Cambridge University Press 1969 Chap 6 Nice development of cost curves especially strong on graphic analysis Fuss M and D McFadden Production Economics A Dual Approach to Theory and Applications Amsterdam North Holland 1978 Difficult and quite complete treatment of the dual relationship between production and cost functions Some discussion of empir ical issues Knight H H Cost of Production and Price over Long and Short Periods Journal of Political Economics 29 April 1921 30435 Classic treatment of the shortrun longrun distinction Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 Chapters 79 have a great deal of material on cost functions Especially recommended are the authors discussions of reci procity effects and their treatment of the shortrun long run distinction as an application of the Le Chatelier principle from physics Sydsaeter K A Strom and P Berck Economists Mathematical Manual 3rd ed Berlin Springerverlag 2000 Chapter 25 provides a succinct summary of the mathematical con cepts in this chapter A nice summary of many input cost func tions but beware of typos Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 360 The two cost functions studied in Chapter 10 the Cobb Douglas and the CES are very restrictive in the substitu tion possibilities they permit The CobbDouglas implicitly assumes that σ 5 1 between any two inputs The CES permits σ to take any value but it requires that the elasticity of substi tution be the same between any two inputs Because empirical economists would prefer to let the data show what the actual substitution possibilities among inputs are they have tried to find more flexible functional forms One especially popular such form is the translog cost function first made popular by Fuss and McFadden 1978 In this extension we will look at this function E101 The translog with two inputs In Example 102 we calculated the CobbDouglas cost multi function in the twoinput case as C1v w q2 5 Bq11α1β2 3 v α1α1β2w β1α1β2 If we take the natural logarithm of this we have ln C1v w q2 5 ln B 1 31 1α 1 β2 4 ln q 1 3α 1α 1 β2 4 ln v 1 3β 1α 1 β2 4 ln w i That is the log of total costs is linear in the logs of output and the input prices The translog function generalizes this by per mitting secondorder terms in input prices ln C1v w q2 5 ln q 1 a0 1 a1 ln v 1 a2 ln w 1 a3 1 ln v2 2 1 a4 1 ln w2 2 1 a5 ln v ln w ii where this function implicitly assumes constant returns to scale because the coefficient of ln q is 10although that need not be the case Some of the properties of this function are For the function to be homogeneous of degree 1 in input prices it must be the case that a1 1 a2 5 1 and a3 1 a4 1 a5 5 0 This function includes the CobbDouglas as the spe cial case a3 5 a4 5 a5 5 0 Hence the function can be used to test statistically whether the CobbDouglas is appropriate Input shares for the translog function are especially easy to compute using the result that si 5 1 ln C2 1 ln wi2 In the twoinput case this yields sk 5 ln C ln v 5 a1 1 2a3 ln v 1 a5 ln w sl 5 ln C ln w 5 a2 1 2a4 ln w 1 a5 ln v iii In the CobbDouglas case 1a3 5 a4 5 a5 5 02 these shares are constant but with the general translog function they are not Calculating the elasticity of substitution in the translog case proceeds by using the result given in Problem 1011 that sk l 5 ek c w 2 elc w Making this calculation is straightforward provided one keeps track of how to use logarithms ekcw 5 ln Cv ln w 5 ln 1C v ln C ln v2 ln w 5 3ln C 2 ln v 1 ln 1 ln C ln v24 ln w 5 sl 2 0 1 ln sk sk 2 ln C vw 5 sl 1 a5 sk iv Observe that in the CobbDouglas case 1a5 5 02 the contingent price elasticity of demand for k with respect to the wage has a simple form ek cw 5 sl A similar set of manipulations yields el cw 5 2sk 1 2a4sl and in the CobbDouglas case el cw 5 2sk Bringing these two elas ticities together yields skl 5 ekcw 2 el cw 5 sl 1 sk 1 a5 sk 2 2a4 sl 5 1 1 sl a5 2 2ska4 sksl v EXTENSIONS The Translog CosT FunCTion Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 10 Cost Functions 361 Again in the CobbDouglas case we have skl 5 1 as should have been expected The Allen elasticity of substitution see Problem 1012 for the translog function is Akl 5 1 1 a5sk sl This function can also be used to calculate that the contingent crossprice elasticity of demand is ekc w 5 slAkl 5 sl 1 a5sk as was shown previously Here again Akl 5 1 in the CobbDouglas case In general however the Allen and Morishima definitions will differ even with just two inputs E102 The manyinput translog cost function Most empirical studies include more than two inputs The translog cost function is especially easy to generalize to these situations If we assume there are n inputs each with a price of wi 1i 5 1 c n2 then this function is C1w1 wn q2 5 ln q 1 a0 1 a n i51 ai ln wi 1 05 a n i51 a n j51 aij ln wi ln wj vi where we have once again assumed constant returns to scale This function requires aij 5 aji so each term for which i 2 j appears twice in the final double sum which explains the presence of the 05 in the expression For this function to be homogeneous of degree 1 in the input prices it must be the case that g i51 n ai 5 1 and g i51 n aij 5 0 Two useful properties of this function are Input shares take the linear form si 5 ai 1 a n j51 aij ln wj vii Again this shows why the translog is usually estimated in a share form Sometimes a term in ln q is also added to the share equations to allow for scale effects on the shares see Sydsæter Strøm and Berck 2000 The elasticity of substitution between any two inputs in the translog function is given by sij 5 1 1 sj aij 2 siajj sisj viii Hence substitutability can again be judged directly from the parameters estimated for the translog function E103 Some applications The translog cost function has become the main choice for empirical studies of production Two factors account for this popularity First the function allows a fairly complete charac terization of substitution patterns among inputsit does not require that the data fit any prespecified pattern Second the functions format incorporates input prices in a flexible way so that one can be reasonably sure that he or she has controlled for such prices in regression analysis When such control is assured measures of other aspects of the cost function such as its returns to scale will be more reliable One example of using the translog function to study input substitution is the study by Westbrook and Buckley 1990 of the responses that shippers made to changing relative prices of moving goods that resulted from deregulation of the railroad and trucking industries in the United States The authors look specifically at the shipping of fruits and vegeta bles from the western states to Chicago and New York They find relatively high substitution elasticities among shipping options and so conclude that deregulation had significant welfare benefits Doucouliagos and Hone 2000 provide a similar analysis of deregulation of dairy prices in Australia They show that changes in the price of raw milk caused dairy processing firms to undertake significant changes in input usage They also show that the industry adopted significant new technologies in response to the price change An interesting study that uses the translog primarily to judge returns to scale is Latzkos 1999 analysis of the US mutual fund industry He finds that the elasticity of total costs with respect to the total assets managed by the fund is less than 1 for all but the largest funds those with more than 4 billion in assets Hence the author concludes that money management exhibits substantial returns to scale A number of other studies that use the translog to estimate economies of scale focus on municipal services For example Garcia and Thomas 2001 look at water supply systems in local French communities They conclude that there are significant operat ing economies of scale in such systems and that some merging of systems would make sense Yatchew 2000 reaches a simi lar conclusion about electricity distribution in small commu nities in Ontario Canada He finds that there are economies of scale for electricity distribution systems serving up to about 20000 customers Again some efficiencies might be obtained from merging systems that are much smaller than this size References Doucouliagos H and P Hone Deregulation and Sub equilibrium in the Australian Dairy Processing Industry Economic Record June 2000 15262 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 362 Part 4 Production and Supply Fuss M and D McFadden Eds Production Economics A Dual Approach to Theory and Applications Amsterdam North Holland 1978 Garcia S and A Thomas The Structure of Municipal Water Supply Costs Application to a Panel of French Local Com munities Journal of Productivity Analysis July 2001 529 Latzko D Economies of Scale in Mutual Fund Administra tion Journal of Financial Research Fall 1999 33139 Sydsæter K A Strøm and P Berck Economists Mathemati cal Manual 3rd ed Berlin Springerverlag 2000 Westbrook M D and P A Buckley Flexible Functional Forms and Regularity Assessing the Competitive Rela tionship between Truck and Rail Transportation Review of Economics and Statistics November 1990 62330 Yatchew A Scale Economies in Electricity Distribution A Semiparametric Analysis Journal of Applied Econometrics MarchApril 2000 187210 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 363 CHAPTER ELEVEN Profit Maximization In Chapter 10 we examined the way in which firms minimize costs for any level of output they choose In this chapter we focus on how the level of output is chosen by profit maximizing firms Before investigating that decision however it is appropriate to discuss briefly the nature of firms and the ways in which their choices should be analyzed 111 THE NATURE AND BEHAVIOR OF FIRMS In this chapter we delve deeper into the analysis of decisions made by suppliers in the market The analysis of the supplyfirm side of the market raises questions that did not come up in our previous analysis of the demandconsumer side Whereas consumers are easy to identify as single individuals firms come in all shapes and sizes ranging from a corner mom and pop grocery store to a vast modern corporation supplying hundreds of different products produced in factories operating across the globe Economists have long puzzled over what determines the size of firms how their management is structured what sort of financial instruments should be used to fund needed investment and so forth The issues involved turn out to be rather deep and philosophical To make progress in this chapter we will continue to analyze the standard neoclassical model of the firm which brushes most of these deeper issues aside We will provide only a hint of the deeper issues involved returning to a fuller discussion in the Extensions to this chapter 1111 Simple model of a firm Throughout Part 4 we have been examining a simple model of the firm without being explicit about the assumptions involved It is worth being a bit more explicit here The firm has a technology given by the production function say f k l The firm is run by an entre preneur who makes all the decisions and receives all the profits and losses from the firms operations The combination of these elementsproduction technology entrepreneur and inputs used labor l capital k and otherstogether constitutes what we will call the firm The entrepreneur acts in his or her own selfinterest typically leading to decisions that maximize the firms profits as we will see 1112 Complicating factors Before pushing ahead further with the analysis of the simple model of the firm which will occupy most of this chapter we will hint at some complicating factors In the simple model just described a single partythe entrepreneurmakes all the decisions and receives all Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 364 Part 4 Production and Supply the returns from the firms operations With most large corporations decisions and returns are separated among many parties Shareholders are really the owners of the corporation receiving returns in the form of dividends and stock returns But shareholders do not run the firm the average shareholder may own hundreds of different firms stock through mutual funds and other holdings and could not possibly have the time or expertise to run all these firms The firm is run on shareholders behalf usually by the chief executive officer CEO and his or her management team The CEO does not make all the decisions but del egates most to managers at one of any number of levels in a complicated hierarchy The fact that firms are often not run by the owner leads to another complication Whereas the shareholders may like profits to be maximized the manager may act in his or her own interest rather than the interests of the shareholders The manager may pre fer the prestige from expanding the business empire beyond what makes economic sense may seek to acquire expensive perks and may shy away from profitable but uncomfort able actions such as firing redundant workers Different mechanisms may help align the managers interests with those of the shareholder Managerial compensation in the form of stock and stock options may provide incentives for profit maximization as might the threat of firing if a poorly performing firm goes bankrupt or is taken over by a corporate raider But there is no telling that such mechanisms will work perfectly Even a concept as simple as the size of the firm is open to question The simple defi nition of the firm includes all the inputs it uses to produce its output for example all the machines and factories involved If part of this production process is outsourced to another firm using its machines and factories then several firms rather than one are responsible for supply A classic example is provided by the automaker General Motors GM1 Ini tially GM purchased the car bodies from another firm Fisher Body who designed and made these to order GM was only responsible for final assembly of the body with the other auto parts After experiencing a sequence of supply disruptions over several decades GM decided to acquire Fisher Body in 1926 Overnight much more of the productionthe construction of the body and final assemblywas concentrated in a single firm What then should we say about the size of a firm in the automaking business Is the combination of GM and Fisher Body after the acquisition or the smaller GM beforehand a better definition of the firm in this case Should we expect the acquisition of Fisher Body to make any real economic difference to the auto market say reducing input supply disruptions or is it a mere name change These are deep questions we will touch on in the Extensions to this chapter For now we will take the size and nature of the firm as given specified by the production function f k l 1113 Relationship to consumer theory Part 2 of this book was devoted to understanding the decisions of consumers on the demand side of the market this Part 4 is devoted to understanding firms on the supply side As we have already seen there are many common elements between the two analyses and much of the same mathematical methods can be used in both There are two essential differences that merit all the additional space devoted to the study of firms First as just discussed firms are not individuals but can be much more complicated organizations We will mostly finesse this difference by assuming that the firm is represented by the entre preneur as an individual decisionmaker dealing with the complications in more detail in the Extensions 1GMs acquisition of Fisher Body has been extensively analyzed by economists See for example B Klein Vertical Integration as Organization Ownership the FisherBodyGeneral Motors Relationship Revisited Journal of Law Economics and Organization Spring 1988 199213 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 365 Another difference between firms and consumers is that we can be more concrete about the firms objectives than a consumers With consumers there is no accounting for taste There is no telling why one consumer likes hot dogs more than hamburgers and another consumer the opposite By contrast it is usually assumed that firms do not have an inher ent preference regarding the production of hot dogs or hamburgers the natural assump tion is that it produces the product or makes any number of other decisions earning the most profit There are certainly a number of caveats with the profitmaximization assump tion but if we are willing to make it we can push the analysis farther than we did with consumer theory 112 PROFIT MAXIMIZATION Most models of supply assume that the firm and its manager pursue the goal of achieving the largest economic profits possible The following definition embodies this assumption and also reminds the reader of the definition of economic profits D E F I N I T I O N Profitmaximizing firm The firm chooses both its inputs and its outputs with the sole goal of maximizing economic profits the difference between its total revenues and its total economic costs This assumptionthat firms seek maximum economic profitshas a long history in economic literature It has much to recommend it It is plausible because firm owners may indeed seek to make their asset as valuable as possible and because competitive markets may punish firms that do not maximize profits This assumption comes with caveats We already noted in the previous section that if the manager is not the owner of the firm he or she may act in a selfinterested way and not try to maximize owner wealth Even if the manager is also the owner he or she may have other concerns besides wealth say reducing pollution at a power plant or curing illness in developing countries in a pharmaceutical lab We will put such other objectives aside for now not because they are unrealistic but rather because it is hard to say exactly which of the broad set of additional goals are most important to people and how much they mat ter relative to wealth The social goals may be addressed more efficiently by maximizing the firms profit and then letting the owners use their greater wealth to fund other goals directly through taxes or charitable contributions In any event a rich set of theoretical results explaining actual firms decisions can be derived using the profitmaximization assumption thus we will push ahead with it for most of the rest of the chapter 1121 Profit maximization and marginalism If firms are strict profit maximizers they will make decisions in a marginal way The entrepreneur will perform the conceptual experiment of adjusting those variables that can be controlled until it is impossible to increase profits further This involves say look ing at the incremental or marginal profit obtainable from producing one more unit of output or at the additional profit available from hiring one more laborer As long as this incremental profit is positive the extra output will be produced or the extra laborer will Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 366 Part 4 Production and Supply be hired When the incremental profit of an activity becomes zero the entrepreneur has pushed that activity far enough and it would not be profitable to go further In this chapter we will explore the consequences of this assumption by using increasingly sophisticated mathematics 1122 Output choice First we examine a topic that should be familiar what output level a firm will produce to obtain maximum profits A firm sells some level of output q at a market price of p per unit Total revenues R are given by R 1q2 5 p 1q2 q 111 where we have allowed for the possibility that the selling price the firm receives might be affected by how much it sells In the production of q certain economic costs are incurred and as in Chapter 10 we will denote these by Cq The difference between revenues and costs is called economic profits 1π2 We will recap this definition here for reference D E F I N I T I O N Economic profit A firms economic profits are the difference between its revenues and costs economic profits 5 π1q2 5 R1q2 2 C1q2 112 D E F I N I T I O N Marginal revenue Marginal revenue is the change in total revenue R resulting from a change in output q marginal revenue 5 MR 5 dR dq 115 Because both revenues and costs depend on the quantity produced economic profits will also depend on it The necessary condition for choosing the value of q that maximizes profits is found by setting the derivative of Equation 112 with respect to q equal to 02 dπ dq 5 πr 1q2 5 dR dq 2 dC dq 5 0 113 so the firstorder condition for a maximum is that dR dq 5 dC dq 114 In the previous chapter the derivative dCdq was defined to be marginal cost MC The other derivative dRdq can be defined analogously as follows 2Notice that this is an unconstrained maximization problem the constraints in the problem are implicit in the revenue and cost functions Specifically the demand curve facing the firm determines the revenue function and the firms production function together with input prices determines its costs Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 367 With the definitions of MR and MC in hand we can see that Equation 114 is a mathe matical statement of the marginal revenue equals marginal cost rule usually studied in introductory economics courses The rule is important enough to be highlighted as an optimization principle O P T I M I Z AT I O N P R I N C I P L E Profit maximization To maximize economic profits the firm should choose output q at which marginal revenue is equal to marginal cost That is MR1q2 5 MC1q2 116 1123 Secondorder conditions Equation 114 or 115 is only a necessary condition for a profit maximum For sufficiency it is also required that d 2π dq2 q5q 5 dπr 1q2 dq q5q 0 117 or that marginal profit must decrease at the optimal level of output q For q less than q profit must increase 3πr 1q2 04 for q greater than q profit must decrease 3πr 1q2 04 Only if this condition holds has a true maximum been achieved Clearly the condition holds if marginal revenue decreases or remains constant in q and marginal cost increases in q 1124 Graphical analysis These relationships are illustrated in Figure 111 where the top panel depicts typical cost and revenue functions For low levels of output costs exceed revenues thus economic profits are negative In the middle ranges of output revenues exceed costs this means that profits are positive Finally at high levels of output costs rise sharply and again exceed rev enues The vertical distance between the revenue and cost curves ie profits is shown in Figure 111b Here profits reach a maximum at q At this level of output it is also true that the slope of the revenue curve marginal revenue is equal to the slope of the cost curve marginal cost It is clear from the figure that the sufficient conditions for a maximum are also satisfied at this point because profits are increasing to the left of q and decreasing to the right of q Therefore output level q is a true profit maximum This is not so for output level q Although marginal revenue is equal to marginal cost at this output profits are in fact at a local minimum there 113 MARGINAL REVENUE Marginal revenue is simple to compute when a firm can sell all it wishes without having any effect on market price The extra revenue obtained from selling one more unit is just this market price A firm may not always be able to sell all it wants at the prevailing market price however If it faces a downwardsloping demand curve for its product then more output can be sold only by reducing the goods price In this case the revenue obtained from selling one more unit will be less than the price of that unit because to get consumers to take the extra unit the price of all other units must be lowered Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 368 Part 4 Production and Supply This result can be easily demonstrated As before total revenue R is the product of the quantity sold q times the price at which it is sold p which may also depend on q Using the product rule to compute the derivative marginal revenue is MR 1q2 5 dR dq 5 d3 p 1q2 q4 dq 5 p 1 q dp dq 118 Notice that the marginal revenue is a function of output In general MR will be different for different levels of q From Equation 118 it is easy to see that if price does not change Profits defined as revenues R minus costs C reach a maximum when the slope of the revenue func tion marginal revenue is equal to the slope of the cost function marginal cost This equality is only a necessary condition for a maximum as may be seen by comparing points q a true maximum and q a local minimum points at which marginal revenue equals marginal cost Revenues costs Profts Losses Output per period Output per period a b q q q C R 0 FIGURE 111 Marginal Revenue Must Equal Marginal Cost for Profit Maximization Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 369 as quantity increases 1dpdq 5 02 marginal revenue will be equal to price In this case we say that the firm is a pricetaker because its output decisions do not influence the price it receives On the other hand if price decreases as quantity increases 1dpdq 02 marginal revenue will be less than price A profitmaximizing manager must know how increases in output will affect the price received before making an optimal output decision If increases in q cause market price to decrease this must be taken into account EXAMPLE 111 Marginal Revenue from a Linear Demand Function Suppose a shop selling sub sandwiches also called grinders torpedoes or in Philadelphia hoa gies faces a linear demand curve for its daily output over period q of the form q 5 100 2 10p 119 Solving for the price the shop receives we have p 5 2q 10 1 10 1110 and total revenues as a function of q are given by R 5 pq 5 2q2 10 1 10q 1111 The sub firms marginal revenue function is MR 5 dR dq 5 2q 5 1 10 1112 and in this case MR p for all values of q If for example the firm produces 40 subs per day Equation 1110 shows that it will receive a price of 6 per sandwich But at this level of output Equation 1112 shows that MR is only 2 If the firm produces 40 subs per day then total rev enue will be 240 15 6 3 402 whereas if it produced 39 subs then total revenue would be 238 15 61 3 392 because price will increase slightly when less is produced Hence the mar ginal revenue from the 40th sub sold is considerably less than its price Indeed for q 5 50 marginal revenue is zero total revenues are a maximum at 250 5 5 3 50 and any further expansion in daily sub output will result in a reduction in total revenue to the firm To determine the profitmaximizing level of sub output we must know the firms marginal costs If subs can be produced at a constant average and marginal cost of 4 then Equation 1112 shows that MR 5 MC at a daily output of 30 subs With this level of output each sub will sell for 7 and profits are 90 35 17 2 42 304 Although price exceeds average and marginal cost here by a substantial margin it would not be in the firms interest to expand output With q 5 35 for example price will decrease to 650 and profits will decrease to 8750 35 1650 2 4002 354 Marginal revenue not price is the primary determinant of profitmaximizing behavior QUERY How would an increase in the marginal cost of sub production to 5 affect the output decision of this firm How would it affect the firms profits 1131 Marginal revenue and elasticity The concept of marginal revenue is directly related to the elasticity of the demand curve facing the firm Remember that the elasticity of demand 1eq p2 is defined as the percentage change in quantity demanded that results from a 1 percent change in price eq p 5 dqq dpp 5 dq dp p q Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 370 Part 4 Production and Supply TABLE 111 RELATIONSHIP BETWEEN ELASTICITY AND MARGINAL REVENUE eq p 21 MR 0 eq p 5 21 MR 5 0 eq p 21 MR 0 Now this definition can be combined with Equation 118 to give MR 5 p 1 q dp dq 5 pa1 1 q p dp dqb 5 pa1 1 1 eq p b 1113 As long as the demand curve facing the firm is negatively sloped eq p 0 and marginal revenue will be less than price as we have already shown If demand is elastic 1eq p 212 then marginal revenue will be positive If demand is elastic the sale of one more unit will not affect price very much and hence more revenue will be yielded by the sale In fact if demand facing the firm is infinitely elastic 1eq p 5 2q2 marginal revenue will equal price The firm is in this case a pricetaker However if demand is inelastic 1eq p 212 marginal revenue will be negative Increases in q can be obtained only through large decreases in market price and these decreases will cause total revenue to decrease The relationship between marginal revenue and elasticity is summarized by Table 111 1132 Pricemarginal cost markup If we assume the firm wishes to maximize profits this analysis can be extended to illustrate the connection between price and marginal cost Setting MR 5 MC in Equation 1113 yields MC 5 pa1 1 1 eq p b or after rearranging p 2 MC p 5 1 2eq p 5 1 0eq p0 1114 where the last equality holds if demand is downward sloping and thus eq p 0 This formula for the percentage markup of price over marginal cost is sometimes called the Lerner index after the economist Abba Lerner who first proposed it in the 1930s The markup depends in a specific way on the elasticity of demand facing the firm First notice that this demand must be elastic 1eq p 212 for this formula to make any sense If demand were inelastic the ratio in Equation 1114 would be greater than 1 which is impossible if a positive MC is subtracted from a positive p in the numerator This sim ply reflects that when demand is inelastic marginal revenue is negative and cannot be equated to a positive marginal cost It is important to stress that it is the demand facing the firm that must be elastic This may be consistent with an inelastic market demand for the product in question if the firm faces competition from other firms producing the same good Equation 1114 implies that the percentage markup over marginal cost will be higher the closer eq p is to 21 If the demand facing the firm is infinitely elastic perhaps because there are many other firms producing the same good then eq p 5 2q and there is no Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 371 markup 1 p 5 MC2 On the other hand with an elasticity of demand of say eq p 5 22 the markup over marginal cost will be 50 percent of price that is 1 p 2 MC2p 5 12 1133 Marginal revenue curve Any demand curve has a marginal revenue curve associated with it If as we sometimes assume the firm must sell all its output at one price it is convenient to think of the demand curve facing the firm as an average revenue curve That is the demand curve shows the revenue per unit in other words the price yielded by alternative output choices The marginal revenue curve on the other hand shows the extra revenue provided by the last unit sold In the usual case of a downwardsloping demand curve the marginal revenue curve will lie below the demand curve because according to Equation 118 MR p In Figure 112 we have drawn such a curve together with the demand curve from which it was derived Notice that for output levels greater than q1 marginal revenue is negative As output increases from 0 to q1 total revenues 1 p q2 increase However at q1 total revenues 1 p1 q12 are as large as possible beyond this output level price decreases proportionately faster than output increases In Part 2 we talked in detail about the possibility of a demand curves shifting because of changes in income prices of other goods or preferences Whenever a demand curve does shift its associated marginal revenue curve shifts with it This should be obvious because a marginal revenue curve cannot be calculated without referring to a specific demand curve Because the demand curve is negatively sloped the marginal revenue curve will fall below the demand average revenue curve For output levels beyond q1 MR is negative At q1 total revenues 1 p1 q12 are a maximum beyond this point additional increases in q cause total revenues to decrease because of the concomitant decreases in price FIGURE 112 Market Demand Curve and Associated Marginal Revenue Curve Price Quantity per period D average revenue p1 q1 MR 0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 372 Part 4 Production and Supply 114 SHORTRUN SUPPLY BY A PRICETAKING FIRM We are now ready to study the supply decision of a profitmaximizing firm In this chapter we will examine only the case in which the firm is a pricetaker In Part 6 we will look at other cases in considerably more detail Also we will focus only on supply decisions in the short run here Longrun questions concern entry and exit by firms and are the primary focus of the next chapter Therefore the firms set of shortrun cost curves is the appropri ate model for our analysis 1141 Profitmaximizing decision Figure 113 shows the firms shortrun decision The market price3 is given by P There fore the demand curve facing the firm is a horizontal line through P This line is labeled P 5 MR as a reminder that an extra unit can always be sold by this pricetaking firm without affecting the price it receives Output level q provides maximum profits because at q price is equal to shortrun marginal cost The fact that profits are positive can be seen by noting that price at q exceeds average costs The firm earns a profit on each unit sold If price were below average cost as is the case for P the firm would have a loss 3We will usually use an uppercase italic P to denote market price here and in later chapters When notation is complex however we will sometimes revert to using a lowercase p EXAMPLE 112 The Constant Elasticity Case In Chapter 5 we showed that a demand function of the form q 5 apb 1115 has a constant price elasticity of demand equal to 2b To compute the marginal revenue function for this function first solve for p p 5 a1 ab 1b q1b 5 kq1b 1116 where k 5 11a2 1b Hence R 5 pq 5 kq111b2b and MR 5 dRdq 5 11b b kq1b 5 11b b p 1117 For this particular function MR is proportional to price If for example eq p 5 b 5 22 then MR 5 05p For a more elastic case suppose b 5 210 then MR 5 09p The MR curve approaches the demand curve as demand becomes more elastic Again if b 5 2q then MR 5 p that is in the case of infinitely elastic demand the firm is a pricetaker For inelastic demand on the other hand MR is negative and profit maximization would be impossible QUERY Suppose demand depended on other factors in addition to p How would this change the analysis of this example How would a change in one of these other factors shift the demand curve and its marginal revenue curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 373 on each unit sold If price and average cost were equal profits would be zero Notice that at q the marginal cost curve has a positive slope This is required if profits are to be a true maximum If P 5 MC on a negatively sloped section of the marginal cost curve then this would not be a point of maximum profits because increasing output would yield more in revenues price times the amount produced than this production would cost marginal cost would decrease if the MC curve has a negative slope Consequently profit maximiza tion requires both that P 5 MC and that marginal cost increase at this point4 1142 The firms shortrun supply curve The positively sloped portion of the shortrun marginal cost curve is the shortrun supply curve for this pricetaking firm That curve shows how much the firm will produce for every possible market price For example as Figure 113 shows at a higher price of P the firm will produce q because it is in its interest to incur the higher marginal costs entailed by q With a price of P on the other hand the firm opts to produce less 1q2 because 4Mathematically because π1q2 5 Pq 2 C1q2 profit maximization requires the firstorder condition πr 1q2 5 P 2 MC1q2 5 0 and the secondorder condition πs 1q2 5 2MCr 1q2 0 Hence it is required that MCr 1q2 0 marginal cost must be increasing In the short run a pricetaking firm will produce the level of output for which SMC 5 P At P for example the firm will produce q The SMC curve also shows what will be produced at other prices For prices below SAVC however the firm will choose to produce no output The heavy lines in the figure represent the firms shortrun supply curve Market price Quantity per period SMC SAVC SAC P P q q q P MR Ps 0 FIGURE 113 ShortRun Supply Curve for a Price Taking Firm Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 374 Part 4 Production and Supply only a lower output level will result in lower marginal costs to meet this lower price By considering all possible prices the firm might face we can see by the marginal cost curve how much output the firm should supply at each price 1143 The shutdown decision For low prices we must be careful about this conclusion Should market price fall below Ps the shutdown price the profitmaximizing decision would be to produce nothing As Figure 113 shows prices less than Ps do not cover average variable costs There will be a loss on each unit produced in addition to the loss of all fixed costs By shutting down produc tion the firm must still pay fixed costs but avoids the losses incurred on each unit produced Because in the short run the firm cannot close down and avoid all costs its best decision is to produce no output On the other hand a price only slightly above Ps means the firm should produce some output Although profits may be negative which they will be if price falls below shortrun average total costs the case at P the profitmaximizing decision is to continue production as long as variable costs are covered Fixed costs must be paid in any case and any price that covers variable costs will provide revenue as an offset to the fixed costs5 Hence we have a complete description of this firms supply decisions in response to alternative prices for its output These are summarized in the following definition Of course any factor that shifts the firms shortrun marginal cost curve such as changes in input prices or changes in the level of fixed inputs used will also shift the short run supply curve In Chapter 12 we will make extensive use of this type of analysis to study the operations of perfectly competitive markets 5Some algebra may clarify matters We know that total costs equal the sum of fixed and variable costs SC 5 SFC 1 SVC and that profits are given by π 5 R 2 SC 5 P q 2 SFC 2 SVC If q 5 0 then variable costs and revenues are 0 and thus π 5 2SFC The firm will produce something only if π 2SFC But that means that p q SVC or p SVCq D E F I N I T I O N Shortrun supply curve The firms shortrun supply curve shows how much it will produce at various possible output prices For a profitmaximizing firm that takes the price of its output as given this curve consists of the positively sloped segment of the firms shortrun marginal cost above the point of minimum average variable cost For prices below this level the firms profitmaximizing decision is to shut down and produce no output EXAMPLE 113 ShortRun Supply In Example 105 we calculated the shortrun totalcost function for the CobbDouglas produc tion function as SC1v w q k12 5 vk1 1 wq1βk1 2αβ 1118 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 375 where k1 is the level of capital input that is held constant in the short run6 Shortrun marginal cost is easily computed as SMC1v w q k12 5 SC q 5 w β q112β2βk2αβ 1 1119 Notice that shortrun marginal cost increases in output for all values of q Shortrun profit max imization for a pricetaking firm requires that output be chosen so that market price P is equal to shortrun marginal cost SMC 5 w β q112β2βk2αβ 1 5 P 1120 and we can solve for quantity supplied as q 5 aw β b 2β112β2 kα112β2 1 P β112β2 1121 This supply function provides a number of insights that should be familiar from earlier econom ics courses 1 The supply curve is positively slopedincreases in P cause the firm to produce more because it is willing to incur a higher marginal cost7 2 the supply curve is shifted to the left by increases in the wage rate wthat is for any given output price less is supplied with a higher wage 3 the supply curve is shifted outward by increases in capital input k1with more capital in the short run the firm incurs a given level of shortrun marginal cost at a higher output level and 4 the rental rate of capital v is irrelevant to shortrun supply decisions because it is only a component of fixed costs Numerical example We can pursue once more the numerical example from Example 105 where α 5 β 5 05 v 5 3 w 5 12 and k1 5 80 For these specific parameters the supply function is q 5 a w 05b 21 1k12 1 p1 5 40 P w 5 40P 12 5 10P 3 1122 That this computation is correct can be checked by comparing the quantity supplied at various prices with the computation of shortrun marginal cost in Table 102 For example if P 5 12 then the supply function predicts that q 5 40 will be supplied and Table 102 shows that this will agree with the P 5 SMC rule If price were to double to P 5 24 an output level of 80 would be supplied and again Table 102 shows that when q 5 80 SMC 5 24 A lower price say P 5 6 would cause less to be produced 1q 5 202 Before adopting Equation 1122 as the supply curve in this situation we should also check the firms shutdown decision Is there a price where it would be more profitable to produce q 5 0 than to follow the P 5 SMC rule From Equation 1118 we know that shortrun variable costs are given by SVC 5 wq1βk2αβ 1 1123 and so SVC q 5 wq112β2βk2αβ 1 1124 A comparison of Equation 1124 with Equation 1119 shows that SVCq SMC for all values of q provided that β 1 Thus in this problem there is no price low enough such that by following the P 5 SMC rule the firm would lose more than if it produced nothing 6Because capital input is held constant the shortrun cost function exhibits increasing marginal cost and will therefore yield a unique profitmaximizing output level If we had used a constant returnstoscale production function in the long run there would have been no such unique output level We discuss this point later in this chapter and in Chapter 12 7In fact the shortrun elasticity of supply can be read directly from Equation 1121 as β 11 2 β2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 376 Part 4 Production and Supply 115 PROFIT FUNCTIONS Additional insights into the profitmaximization process for a pricetaking firm8 can be obtained by looking at the profit function This function shows the firms maximized profits as depending only on the prices that the firm faces To understand the logic of its construction remember that economic profits are defined as π 5 Pq 2 C 5 Pf 1k l2 2 vk 2 wl 1125 Only the variables k and l and also q 5 f 1k l2 are under the firms control in this expres sion The firm chooses levels of these inputs to maximize profits treating the three prices P v and w as fixed parameters in its decision Looked at in this way the firms maximum profits ultimately depend only on these three exogenous prices together with the form of the production function We summarize this dependence by the profit function In this definition we use an upper case P to indicate that the value given by the function is the maximum profits obtainable given the prices This function implicitly incorporates the form of the firms production functiona process we will illustrate in Example 114 The profit function can refer to either longrun or shortrun profit maximization but in the latter case we would need also to specify the levels of any inputs that are fixed in the short run 1151 Properties of the profit function As for the other optimized functions we have already looked at the profit function has a number of properties that are useful for economic analysis 8Much of the analysis here would also apply to a firm that had some market power over the price it received for its product but we will delay a discussion of that possibility until Part 5 In our numerical example consider the case P 5 3 With such a low price the firm would opt for q 5 10 Total revenue would be R 5 30 and total shortrun costs would be SC 5 255 see Table 101 Hence profits would be π 5 R 2 SC 5 2225 Although the situation is dismal for the firm it is better than opting for q 5 0 If it produces nothing it avoids all variable labor costs but still loses 240 in fixed costs of capital By producing 10 units of output its revenues cover variable costs 1R 2 SVC 5 30 2 15 5 152 and contribute 15 to offset slightly the loss of fixed costs QUERY How would you graph the shortrun supply curve in Equation 1122 How would the curve be shifted if w rose to 15 How would it be shifted if capital input increased to k1 5 100 How would the shortrun supply curve be shifted if v fell to 2 Would any of these changes alter the firms determination to avoid shutting down in the short run D E F I N I T I O N Profit function The firms profit function shows its maximal profits as a function of the prices that the firm faces P1P v w2 5 max k l π1k l2 5 max k l 3Pf 1k l2 2 vk 2 wl4 1126 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 377 1 Homogeneity A doubling of all the prices in the profit function will precisely double profitsthat is the profit function is homogeneous of degree 1 in all prices We have already shown that marginal costs are homogeneous of degree 1 in input prices hence a doubling of input prices and a doubling of the market price of a firms output will not change the profitmaximizing quantity it decides to produce However because both revenues and costs have doubled profits will double This shows that with pure infla tion where all prices rise together firms will not change their production plans and the levels of their profits will just keep up with that inflation 2 Profit functions are nondecreasing in output price P This result seems obviousa firm could always respond to an increase in the price of its output by not changing its input or output plans Given the definition of profits they must increase Hence if the firm changes its plans it must be doing so to make even more profits If profits were to decrease the firm would not be maximizing profits 3 Profit functions are nonincreasing in input prices v and w Again this feature of the profit function seems obvious A proof is similar to that used above in our discussion of output prices 4 Profit functions are convex in output prices This important feature of profit functions says that the profits obtainable by averaging those available from two different output prices will be at least as large as those obtainable from the average9 of the two prices Mathematically P 1P1 v w2 1 P 1P2 v w2 2 PaP1 1 P2 2 v wb 1127 The intuitive reason for this convexity is that when firms can freely adapt their decisions to two different prices better results are possible than when they can make only one set of choices in response to the single average price More formally let P3 5 1P1 1 P222 and let qi ki li represent the profitmaximizing output and input choices for these various prices Then P 1P3 v w2 P3q3 2 vk3 2 wl3 5 P1q3 2 vk3 2 wl3 2 1 P2q3 2 vk3 2 wl3 2 P1q1 2 vk1 2 wl1 2 1 P2q2 2 vk2 2 wl2 2 P 1P1 v w2 1 P 1P2 v w2 2 1128 which proves Equation 1127 The key step is Equation 1128 Because 1q1 k1 l12 is the profitmaximizing combination of output and inputs when the market price is P1 it must generate as much profit as any other choice including 1q3 k3 l32 By similar reasoning the profit from 1q2 k2 l22 is at least as much as that from 1q3 k3 l32 when the market price is P2 The convexity of the profit function has many applications to topics such as price stabilization 1152 Envelope results Because the profit function reflects an underlying process of unconstrained maximization we may also apply the envelope theorem to see how profits respond to changes in output 9Although we only discuss a simple averaging of prices here it is clear that with convexity a condition similar to Equation 1127 holds for any weighted average price P 5 tP1 1 11 2 t2P2 where 0 t 1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 378 Part 4 Production and Supply and input prices This application of the theorem yields a variety of useful results Specifi cally using the definition of profits shows that P 1P v w2 P 5 q 1P v w2 1129 P 1P v w2 v 5 2k1P v w2 1130 P 1P v w2 w 5 2l 1P v w2 1131 Again these equations make intuitive sense A small increase in output price will increase profits in proportion to how much the firm is producing whereas a small increase in the price of an input will reduce profits in proportion to the amount of that input being used The first of these equations says that the firms supply function can be calculated from its profit function by partial differentiation with respect to the output price10 The second and third equations show that input demand functions11 can also be derived from the profit functions Because the profit function itself is homogeneous of degree 1 all the functions described in Equations 11291131 are homogeneous of degree 0 That is a doubling of both output and input prices will not change the input levels that the firm chooses nor will this change the firms profitmaximizing output level All these findings also have short run analogs as will be shown later with a specific example 1153 Producer surplus in the short run In Chapter 5 we discussed the concept of consumer surplus and showed how areas below the demand curve can be used to measure the welfare costs to consumers of price changes We also showed how such changes in welfare could be captured in the individuals expen diture function The process of measuring the welfare effects of price changes for firms is similar in shortrun analysis and this is the topic we pursue here However as we show in the next chapter measuring the welfare impact of price changes for producers in the long run requires a different approach because most such longterm effects are felt not by firms themselves but rather by their input suppliers In general it is this longrun approach that will prove more useful for our subsequent study of the welfare impacts of price changes Because the profit function is nondecreasing in output prices we know that if P2 P1 then P 1P2 2 P 1P1 2 and it would be natural to measure the welfare gain to the firm from the price change as welfare gain 5 P 1P2 2 2 P 1P1 2 1132 Figure 114 shows how this value can be measured graphically as the area bounded by the two prices and above the shortrun supply curve Intuitively the supply curve shows the minimum price that the firm will accept for producing its output Hence when market price increases from P1 to P2 the firm is able to sell its prior output level 1q12 at a higher price and also opts to sell additional output 1q2 2 q12 for which at the margin it likewise 10This relationship is sometimes referred to as Hotellings lemmaafter the economist Harold Hotelling who discovered it in the 1930s 11Unlike the input demand functions derived in Chapter 10 these input demand functions are not conditional on output levels Rather the firms profitmaximizing output decision has already been taken into account in the functions Therefore this demand concept is more general than the one we introduced in Chapter 10 and we will have much more to say about it in the next section Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 379 earns added profits on all but the final unit Hence the total gain in the firms profits is given by area P2 ABP1 Mathematically we can make use of the envelope results from the previous section to derive welfare gain 5 P 1P2 2 2 P 1P1 2 5 3 P2 P1 P P dP 5 3 P2 P1 q 1P2dP 1133 Thus the geometric and mathematical measures of the welfare change agree Using this approach we can also measure how much the firm values the right to pro duce at the prevailing market price relative to a situation where it would produce no out put If we denote the shortrun shutdown price as PS which may or may not be a price of zero then the extra profits available from facing a price of P1 are defined to be producer surplus producer surplus 5 P 1P1 2 2 P 1PS 2 5 3 P1 PS q 1P2dP 1134 This is shown as area P1BCPs in Figure 114 Hence we have the following formal definition If price increases from P1 to P2 then the increase in the firms profits is given by area P2 ABP1 At a price of P1 the firm earns shortrun producer surplus given by area PsCBP1 This measures the increase in shortrun profits for the firm when it produces q1 rather than shutting down when price is Ps or below Market price P2 q q1 SMC q2 P1 Ps A B C FIGURE 114 Changes in ShortRun Producer Surplus Measure Firm Profits Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 380 Part 4 Production and Supply In this definition we have made no distinction between the short run and the long run although our development thus far has involved only shortrun analysis In the next chap ter we will see that the same definition can serve dual duty by describing producer surplus in the long run so using this generic definition works for both concepts Of course as we will show the meaning of longrun producer surplus is different from what we have stud ied here One more aspect of shortrun producer surplus should be pointed out Because the firm produces no output at its shutdown price we know that P 1PS 2 5 2vk1 that is profits at the shutdown price are solely made up of losses of all fixed costs Therefore producer surplus 5 P 1P1 2 2 P 1PS 2 5 P 1P1 2 2 12vk12 5 P 1P1 2 1 vk1 1135 That is producer surplus is given by current profits being earned plus shortrun fixed costs Further manipulation shows that magnitude can also be expressed as producer surplus 5 P 1P1 2 2 P 1PS 2 5 P1q1 2 vk1 2 wl1 1 vk1 5 P1q1 2 wl1 1136 In words a firms shortrun producer surplus is given by the extent to which its revenues exceed its variable coststhis is indeed what the firm gains by producing in the short run rather than shutting down and producing nothing D E F I N I T I O N Producer surplus Producer surplus is the extra return that producers earn by making transac tions at the market price over and above what they would earn if nothing were produced It is illustrated by the size of the area below the market price and above the supply curve EXAMPLE 114 A ShortRun Profit Function These various uses of the profit function can be illustrated with the CobbDouglas production function we have been using Because q 5 k αl β and because we treat capital as fixed at k1 in the short run it follows that profits are π 5 Pk α 1l β 2 vk1 2 wl 1137 To find the profit function we use the firstorder conditions for a maximum to eliminate l from this expression π l 5 βPk α 1l β21 2 w 5 0 1138 so l 5 a w βPkα 1 b 11β212 1139 We can simplify the process of substituting this back into the profit equation by letting A 5 1wβPkα 1 2 Making use of this shortcut we have P1P v w k12 5 Pkα 1Aβ1β212 2 vk1 2 wA11β212 5 wA11β212aPkα 1 A w 2 1b 2 vk1 5 1 2 β ββ1β212 w β1β212P1112β2kα112β2 1 2 vk1 1140 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 381 Though admittedly messy this solution is what was promisedthe firms maximal profits are expressed as a function of only the prices it faces and its technology Notice that the firms fixed costs 1vk12 enter this expression in a simple linear way The prices the firm faces determine the extent to which revenues exceed variable costs then fixed costs are subtracted to obtain the final profit number Because it is always wise to check that ones algebra is correct lets try out the numerical exam ple we have been using With α 5 β 5 05 v 5 3 w 5 12 and k1 5 80 we know that at a price of P 5 12 the firm will produce 40 units of output and use labor input of l 5 20 Hence profits will be π 5 R 2 C 5 12 40 2 3 80 2 12 20 5 0 The firm will just break even at a price of P 5 12 Using the profit function yields P1P v w k12 5 P112 3 12 802 5 025 1221 122 80 2 3 80 5 0 1141 Thus at a price of 12 the firm earns 240 in profits on its variable costs and these are precisely offset by fixed costs in arriving at the final total With a higher price for its output the firm earns positive profits If the price falls below 12 however the firm incurs shortrun losses12 Hotellings lemma We can use the profit function in Equation 1140 together with the enve lope theorem to derive this firms shortrun supply function q 1P v w k12 5 P P 5 aw β b β1β212 kα112β2 1 P β112β2 1142 which is precisely the shortrun supply function that we calculated in Example 113 see Equation 1121 Producer surplus We can also use the supply function to calculate the firms shortrun pro ducer surplus To do so we again return to our numerical example α 5 β 5 05 v 5 3 w 5 12 and k1 5 80 With these parameters the shortrun supply relationship is q 5 10P3 and the shut down price is zero Hence at a price of P 5 12 producer surplus is producer surplus 5 3 12 0 10P 3 dP 5 10P12 6 12 0 5 240 1143 This precisely equals shortrun profits at a price of 12 1π 5 02 plus shortrun fixed costs 15 vk1 5 3 80 5 2402 If price were to rise to say 15 then producer surplus would increase to 375 which would still consist of 240 in fixed costs plus total profits at the higher price 1P 5 1352 QUERY How is the amount of shortrun producer surplus here affected by changes in the rental rate for capital v How is it affected by changes in the wage w 116 PROFIT MAXIMIZATION AND INPUT DEMAND Thus far we have treated the firms decision problem as one of choosing a profit maximizing level of output But our discussion throughout has made clear that the firms output is in fact determined by the inputs it chooses to use a relationship that is summa rized by the production function q 5 f 1k l2 Consequently the firms economic profits can also be expressed as a function of only the inputs it uses π 1k l2 5 Pq 2 C1q2 5 Pf 1k l2 2 1vk 1 wl2 1144 12In Table 102 we showed that if q 5 40 then SAC 5 12 Hence zero profits are also indicated by P 5 12 5 SAC Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 382 Part 4 Production and Supply Viewed in this way the profitmaximizing firms decision problem becomes one of choos ing the appropriate levels of capital and labor input13 The firstorder conditions for a max imum are π k 5 P f k 2 v 5 0 1145 π l 5 P f l 2 w 5 0 1146 These conditions make the intuitively appealing point that a profitmaximizing firm should hire any input up to the point at which the inputs marginal contribution to reve nue is equal to the marginal cost of hiring the input Because the firm is assumed to be a pricetaker in its hiring the marginal cost of hiring any input is equal to its market price The inputs marginal contribution to revenue is given by the extra output it produces the marginal product times that goods market price This demand concept is given a special name as follows Marginal revenue product The marginal revenue product is the extra revenue a firm receives when it uses one more unit of an input In the pricetaking14 case MRPl 5 Pfl and MRPk 5 Pfk Hence profit maximization requires that the firm hire each input up to the point at which its marginal revenue product is equal to its market price Notice also that the profitmaximizing Equations 1145 and 1146 also imply cost minimization because RTS 5 flfk 5 wv 1161 Secondorder conditions Because the profit function in Equation 1144 depends on two variables k and l the secondorder conditions for a profit maximum are somewhat more complex than in the singlevariable case we examined earlier In Chapter 2 we showed that to ensure a true maximum the profit function must be concave That is πkk 5 f kk 0 πll 5 f ll 0 1147 and πkkπll 2 π2 kl 5 fkk fll 2 f 2 kl 0 1148 Therefore concavity of the profit relationship amounts to requiring that the production function itself be concave Notice that diminishing marginal productivity for each input is not sufficient to ensure increasing marginal costs Expanding output usually requires the firm to use more capital and more labor Thus we must also ensure that increases in capital input do not raise the marginal productivity of labor and thereby reduce marginal cost by a large enough amount to reverse the effect of diminishing marginal productivity of labor itself Therefore Equation 1147 requires that such crossproductivity effects be rela tively smallthat they be dominated by diminishing marginal productivities of the inputs 13Throughout our discussion in this section we assume that the firm is a pricetaker thus the prices of its output and its inputs can be treated as fixed parameters Results can be generalized fairly easily in the case where prices depend on quantity 14If the firm is not a pricetaker in the output market then this definition is generalized by using marginal revenue in place of price That is MRPl 5 Rl 5 Rq ql 5 MR MPl A similar derivation holds for capital input D E F I N I T I O N Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 383 If these conditions are satisfied then marginal costs will increase at the profitmaximizing choices for k and l and the firstorder conditions will represent a local maximum 1162 Input demand functions In principle the firstorder conditions for hiring inputs in a profitmaximizing way can be manipulated to yield input demand functions that show how hiring depends on the prices that the firm faces We will denote these demand functions by capital demand 5 k1P v w2 labor demand 5 l 1P v w2 1149 Notice that contrary to the input demand concepts discussed in Chapter 10 these demand functions are unconditionalthat is they implicitly permit the firm to adjust its output to changing prices Hence these demand functions provide a more complete picture of how prices affect input demand than did the contingent demand functions introduced in Chap ter 10 We have already shown that these input demand functions can also be derived from the profit function through differentiation in Example 115 we show that process explicitly First however we will explore how changes in the price of an input might be expected to affect the demand for it To simplify matters we look only at labor demand but the analy sis of the demand for any other input would be the same In general we conclude that the direction of this effect is unambiguous in all casesthat is lw 0 no matter how many inputs there are To develop some intuition for this result we begin with some simple cases 1163 Singleinput case One reason for expecting lw to be negative is based on the presumption that the marginal physical product of labor decreases as the quantity of labor employed increases A decrease in w means that more labor must be hired to bring about the equality w 5 P MPl A decrease in w must be met by a decrease in MPl because P is fixed as required by the ceteris paribus assumption and this can be brought about by increasing l That this argument is strictly cor rect for the case of one input can be shown as follows With one input Equation 1144 is the sole firstorder condition for profit maximization rewritten here in a slightly different form Pfl 2 w 5 F1l w P2 5 0 1150 where F is just a shorthand we will use to refer to the left side of Equation 1150 If w changes the optimal value of l must adjust so that this condition continues to hold which defines l as an implicit function of w Applying the rule for finding the derivative of an implicit function in Chapter 2 Equation 223 in particular gives dl dw 5 2Fw Fl 5 w Pfll 0 1151 where the final inequality holds because the marginal productivity of labor is assumed to be diminishing 1 fll 02 Hence we have shown that at least in the singleinput case a ceteris paribus increase in the wage will cause less labor to be hired 1164 Twoinput case For the case of two or more inputs the story is more complex The assumption of a diminishing marginal physical product of labor can be misleading here If w falls there Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 384 Part 4 Production and Supply will not only be a change in l but also a change in k as a new costminimizing combination of inputs is chosen When k changes the entire fl function changes labor now has a dif ferent amount of capital to work with and the simple argument used previously cannot be made First we will use a graphic approach to suggest why even in the twoinput case lw must be negative A more precise mathematical analysis is presented in the next section 1165 Substitution effect In some ways analyzing the twoinput case is similar to the analysis of the individuals response to a change in the price of a good that was presented in Chapter 5 When w falls we can decompose the total effect on the quantity of l hired into two components The first of these components is called the substitution effect If q is held constant at q1 then there will be a tendency to substitute l for k in the production process This effect is illustrated in Figure 115a Because the condition for minimizing the cost of producing q1 requires that RTS 5 wv a fall in w will necessitate a movement from input combination A to combina tion B And because the isoquants exhibit a diminishing RTS it is clear from the diagram that this substitution effect must be negative A decrease in w will cause an increase in labor hired if output is held constant 1166 Output effect It is not correct however to hold output constant It is when we consider a change in q the output effect that the analogy to the individuals utilitymaximization problem breaks down Consumers have budget constraints but firms do not Firms produce as much as the When the price of labor falls two analytically different effects come into play One of these the substi tution effect would cause more labor to be purchased if output were held constant This is shown as a movement from point A to point B in a At point B the costminimizing condition 1RTS 5 wv2 is satisfied for the new lower w This change in wv will also shift the firms expansion path and its marginal cost curve A normal situation might be for the MC curve to shift downward in response to a decrease in w as shown in b With this new curve 1MCr2 a higher level of output 1q22 will be chosen Conse quently the hiring of labor will increase to l2 also from this output effect Price k1 k2 P q1 q2 A B C l1 l2 l per period k per period q1 q2 Output per period a Te isoquant map b Te output decision MC MC FIGURE 115 The Substitution and Output Effects of a Decrease in the Price of a Factor Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 385 available demand allows To investigate what happens to the quantity of output produced we must investigate the firms profitmaximizing output decision A change in w because it changes relative input costs will shift the firms expansion path Consequently all the firms cost curves will be shifted and probably some output level other than q1 will be chosen Fig ure 115b shows what might be considered the normal case There the fall in w causes MC to shift downward to MCr Consequently the profitmaximizing level of output rises from q1 to q2 The profitmaximizing condition 1P 5 MC2 is now satisfied at a higher level of out put Returning to Figure 115a this increase in output will cause even more l to be demanded as long as l is not an inferior input inferior inputs will be discussed in more detail in the mathematical development below The result of both the substitution and output effects will be to move the input choice to point C on the firms isoquant map Both effects work to increase the quantity of labor hired in response to a decrease in the real wage The analysis provided in Figure 115 assumed that the market price or marginal rev enue if this does not equal price of the good being produced remained constant This would be an appropriate assumption if only one firm in an industry experienced a fall in unit labor costs However if the decline were industry wide then a slightly different anal ysis would be required In that case all firms marginal cost curves would shift outward and hence the industry supply curve which as we will see in the next chapter is the sum of firms individual supply curves would shift also Assuming that output demand is down ward sloping this will lead to a decline in product price Output for the industry and for the typical firm will still increase and as before more labor will be hired but the precise cause of the output effect is different see Problem 1111 1167 Crossprice effects We have shown that at least in simple cases lw is unambiguously negative substitution and output effects cause more labor to be hired when the wage rate falls From Figure 115 it should be clear that no definite statement can be made about how capital usage responds to the wage change That is the sign of kw is indeterminate In the simple twoinput case a fall in the wage will cause a substitution away from capital that is less capital will be used to produce a given output level However the output effect will cause more capital to be demanded as part of the firms increased production plan Thus substitution and output effects in this case work in opposite directions and no definite conclusion about the sign of kw is possible 1168 A summary of substitution and output effects The results of this discussion can be summarized by the following principle O P T I M I Z AT I O N P R I N C I P L E Substitution and output effects in input demand When the price of an input falls two effects cause the quantity demanded of that input to rise 1 the substitution effect causes any given output level to be produced using more of the input and 2 the fall in costs causes more of the good to be sold thereby creating an additional output effect that increases demand for the input Conversely when the price of an input rises both substitution and output effects cause the quantity demanded of the input to decline We now provide a more precise development of these concepts using a mathematical approach to the analysis Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 386 Part 4 Production and Supply 1169 A mathematical development Our mathematical development of the substitution and output effects that arise from the change in an input price follows the method we used to study the effect of price changes in consumer theory The final result is a Slutskystyle equation that resembles the one we derived in Chapter 5 However the ambiguity stemming from Giffens paradox in the the ory of consumption demand does not occur here We start with a reminder that we have two concepts of demand for any input say labor 1 the conditional demand for labor denoted by l c 1v w q2 and 2 the uncon ditional demand for labor which is denoted by lP v w At the profitmaximizing choice for labor input these two concepts agree about the amount of labor hired The two concepts also agree on the level of output produced which is a function of all the prices l 1P v w2 5 l c 1v w q 1P v w2 2 1152 Differentiation of this expression with respect to the wage and holding the other prices constant yields l 1P v w2 w 5 l c 1v w q2 w 1 l c 1v w q2 q q 1P v w2 w 1153 Thus the effect of a change in the wage on the demand for labor is the sum of two com ponents a substitution effect in which output is held constant and an output effect in which the wage change has its effect through changing the quantity of output that the firm opts to produce The first of these effects is clearly negativebecause the production function is quasiconcave ie it has convex isoquants the outputcontingent demand for labor must be negatively sloped Figure 115b provides an intuitive illustration of why the output effect in Equation 1153 is negative but it can hardly be called a proof The particular complicating factor is the possibility that the input under consideration here labor may be inferior Perhaps oddly inferior inputs also have negative output effects but for rather arcane reasons that are best relegated to a footnote15 The bottom line however is that Giffens paradox cannot occur in the theory of the firms demand for inputs Input demand functions are unambiguously downward sloping In this case the theory of profit maximization imposes more restrictions on what might happen than does the theory of utility maximization In Example 115 we show how decomposing input demand into its substitution and output components can yield useful insights into how changes in input prices affect firms 15In other words an increase in the price of an inferior reduces marginal cost and thereby increases output But when output increases less of the inferior input is hired Hence the end result is a decrease in quantity demanded in response to an increase in price A formal proof makes extensive use of envelope relationships The output effect equals l c q q w 5 l c q 2P w P 5 l c q a2 l Pb 5 2al c q b 2 q P 5 2al c q b 2 2P P 2 where the first step holds by Equation 1152 the second by Equation 1129 the third by Youngs theorem and Equation 1131 the fourth by Equation 1152 and the last by Equation 1129 But the convexity of the profit function in output prices implies the last factor is positive so the whole expression is clearly negative Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 387 EXAMPLE 115 Decomposing Input Demand into Substitution and Output Components To study input demand we need to start with a production function that has two features 1 The function must permit capitallabor substitution because substitution is an important part of the story and 2 the production function must exhibit increasing marginal costs so that the secondorder conditions for profit maximization are satisfied One function that satisfies these conditions is a threeinput CobbDouglas function when one of the inputs is held fixed Thus let q 5 f 1k l g2 5 k025l 025g 05 where k and l are the familiar capital and labor inputs and g is a third input size of the factory that is held fixed at g 5 16 square meters for all our analysis Therefore the shortrun production function is q 5 4k 025l 025 We assume that the factory can be rented at a cost of r per square meter per period To study the demand for say labor input we need both the total cost function and the profit function implied by this production function Mercifully your author has computed these functions for you as C1v w r q2 5 q2v05w05 8 1 16r 1154 and P1P v w r2 5 2P 2v205w205 2 16r 1155 As expected the costs of the fixed input g enter as a constant in these equations and these costs will play little role in our analysis Envelope results Labordemand relationships can be derived from both of these functions through differentiation l c 1v w r q2 5 C w 5 q2v05w205 16 1156 and l 1P v w r2 5 P w 5 P 2v205w215 1157 These functions already suggest that a change in the wage has a larger effect on total labor demand than it does on contingent labor demand because the exponent of w is more negative in the total demand equation That is the output effect must also play a role here To see that directly we turn to some numbers Numerical example Lets start again with the assumed values that we have been using in sev eral previous examples v 5 3 w 5 12 and P 5 60 Lets first calculate what output the firm will choose in this situation To do so we need its supply function q 1P v w r2 5 P P 5 4Pv205w205 1158 With this function and the prices we have chosen the firms profitmaximizing output level is surprise q 5 40 With these prices and an output level of 40 both of the demand functions predict that the firm will hire l 5 50 Because the RTS here is given by kl we also know that kl 5 wv therefore at these prices k 5 200 Suppose now that the wage rate rises to w 5 27 but that the other prices remain unchanged The firms supply function Equation 1158 shows that it will now produce q 5 2667 The rise in the wage shifts the firms marginal cost curve upward and with a constant output price this causes the firm to produce less To produce this output either of the labordemand functions can be used to show that the firm will hire l 5 148 Hiring of capital will also fall to k 5 1333 because of the large reduction in output Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 388 Part 4 Production and Supply We can decompose the fall in labor hiring from l 5 50 to l 5 148 into substitution and out put effects by using the contingent demand function If the firm had continued to produce q 5 40 even though the wage rose Equation 1156 shows that it would have used l 5 3333 Capital input would have increased to k 5 300 Because we are holding output constant at its initial level of q 5 40 these changes represent the firms substitution effects in response to the higher wage The decline in output needed to restore profit maximization causes the firm to cut back on its output In doing so it substantially reduces its use of both inputs Notice in particular that in this example the rise in the wage not only caused labor usage to decline sharply but also caused capital usage to fall because of the large output effect QUERY How would the calculations in this problem be affected if all firms had experienced the rise in wages Would the decline in labor and capital demand be greater or smaller than found here Summary In this chapter we studied the supply decision of a profit maximizing firm Our general goal was to show how such a firm responds to price signals from the marketplace In address ing that question we developed a number of analytical results To maximize profits the firm should choose to produce that output level for which marginal revenue the reve nue from selling one more unit is equal to marginal cost the cost of producing one more unit If a firm is a pricetaker then its output decisions do not affect the price of its output thus marginal revenue is given by this price If the firm faces a downwardslop ing demand for its output however then it can sell more only at a lower price In this case marginal revenue will be less than price and may even be negative Marginal revenue and the price elasticity of demand are related by the formula MR 5 P a1 1 1 eq p b where P is the market price of the firms output and eqp is the price elasticity of demand for its product The supply curve for a pricetaking profit maximizing firm is given by the positively sloped portion of its mar ginal cost curve above the point of minimum average variable cost AVC If price falls below minimum AVC the firms profitmaximizing choice is to shut down and produce nothing The firms reactions to changes in the various prices it faces can be studied through use of its profit function P1P v w2 That function shows the maximum profits that the firm can achieve given the price for its output the prices of its input and its production technology The profit function yields particularly useful envelope results Differentiation with respect to market price yields the supply function whereas differentiation with respect to any input price yields the negative of the demand func tion for that input Shortrun changes in market price result in changes to the firms shortrun profitability These can be measured graphically by changes in the size of producer surplus The profit function can also be used to calculate changes in producer surplus Profit maximization provides a theory of the firms derived demand for inputs The firm will hire any input up to the point at which its marginal revenue product is just equal to its perunit market price Increases in the price of an input will induce substi tution and output effects that cause the firm to reduce hiring of that input Problems 111 Johns Lawn Mowing Service is a small business that acts as a pricetaker ie MR 5 P The prevailing market price of lawn mowing is 20 per acre Johns costs are given by total cost 5 01q2 1 10q 1 50 where q 5 the number of acres John chooses to cut a day a How many acres should John choose to cut to maximize profit b Calculate Johns maximum daily profit c Graph these results and label Johns supply curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 389 112 Universal Widget produces highquality widgets at its plant in Gulch Nevada for sale throughout the world The cost func tion for total widget production q is given by total cost 5 025q2 Widgets are demanded only in Australia where the demand curve is given by qA 5 100 2 2PA and Lapland where the demand curve is given by qL 5 100 2 4PL thus total demand equals q 5 qA 1 qL If Universal Widget can control the quantities supplied to each market how many should it sell in each location to maximize total profits What price will be charged in each location 113 The production function for a firm in the business of calcula tor assembly is given by q 5 2l where q denotes finished calculator output and l denotes hours of labor input The firm is a pricetaker both for calcu lators which sell for P and for workers which can be hired at a wage rate of w per hour a What is the total cost function for this firm b What is the profit function for this firm c What is the supply function for assembled calculators qP w d What is this firms demand for labor function lP w e Describe intuitively why these functions have the form they do 114 The market for highquality caviar is dependent on the weather If the weather is good there are many fancy parties and caviar sells for 30 per pound In bad weather it sells for only 20 per pound Caviar produced one week will not keep until the next week A small caviar producer has a cost func tion given by C 5 05q2 1 5q 1 100 where q is the weekly caviar production Production decisions must be made before the weather and the price of caviar is known but it is known that good weather and bad weather each occur with a probability of 05 a How much caviar should this firm produce if it wishes to maximize the expected value of its profits b Suppose the owner of this firm has a utility function of the form utility 5 π where π is weekly profits What is the expected utility associated with the output strategy defined in part a c Can this firm owner obtain a higher utility of profits by producing some output other than that specified in parts a and b Explain d Suppose this firm could predict next weeks price but could not influence that price What strategy would maximize expected profits in this case What would expected profits be 115 The Acme Heavy Equipment School teaches students how to drive construction machinery The number of students that the school can educate per week is given by q 5 10 min 1k l2 r where k is the number of backhoes the firm rents per week l is the number of instructors hired each week and γ is a parame ter indicating the returns to scale in this production function a Explain why development of a profitmaximizing model here requires 0 γ 1 b Supposing γ 5 05 calculate the firms total cost function and profit function c If v 5 1000 w 5 500 and P 5 600 how many students will Acme serve and what are its profits d If the price students are willing to pay rises to P 5 900 how much will profits change e Graph Acmes supply curve for student slots and show that the increase in profits calculated in part d can be plotted on that graph 116 Would a lumpsum profits tax affect the profitmaximizing quantity of output How about a proportional tax on profits How about a tax assessed on each unit of output How about a tax on labor input 117 This problem concerns the relationship between demand and marginal revenue curves for a few functional forms a Show that for a linear demand curve the marginal rev enue curve bisects the distance between the vertical axis and the demand curve for any price b Show that for any linear demand curve the vertical dis tance between the demand and marginal revenue curves is 21b q where b102 is the slope of the demand curve c Show that for a constant elasticity demand curve of the form q 5 aP b the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve with this constant depend ing on the price elasticity of demand d Show that for any downwardsloping demand curve the vertical distance between the demand and marginal rev enue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part b e Graph the results of parts ad of this problem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 390 Part 4 Production and Supply 118 How would you expect an increase in output price P to affect the demand for capital and labor inputs a Explain graphically why if neither input is inferior it seems clear that a rise in P must not reduce the demand for either factor b Show that the graphical presumption from part a is demonstrated by the input demand functions that can be derived in the CobbDouglas case c Use the profit function to show how the presence of infe rior inputs would lead to ambiguity in the effect of P on input demand Analytical Problems 119 A CES profit function With a CES production function of the form q 5 1k ρ 1 l ρ2 γρ a whole lot of algebra is needed to compute the profit func tion as P1P v w2 5 KP1 112γ2 1v12σ 1 w12σ2 γ112σ21γ212 where σ 5 1 11 2 ρ2 and K is a constant a If you are a glutton for punishment or if your instructor is prove that the profit function takes this form Per haps the easiest way to do so is to start from the CES cost function in Example 102 b Explain why this profit function provides a reasonable representation of a firms behavior only for 0 γ 1 c Explain the role of the elasticity of substitution 1σ2 in this profit function d What is the supply function in this case How does σ determine the extent to which that function shifts when input prices change e Derive the input demand functions in this case How are these functions affected by the size of σ 1110 Some envelope results Youngs theorem can be used in combination with the enve lope results in this chapter to derive some useful results a Show that l 1P v w2v 5 k1P v w2w Interpret this result using substitution and output effects b Use the result from part a to show how a unit tax on labor would be expected to affect capital input c Show that qw 5 2lP Interpret this result d Use the result from part c to discuss how a unit tax on labor input would affect quantity supplied 1111 Le Châteliers Principle Because firms have greater flexibility in the long run their reactions to price changes may be greater in the long run than in the short run Paul Samuelson was perhaps the first economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Châte liers Principle The basic idea of the principle is that any dis turbance to an equilibrium such as that caused by a price change will not only have a direct effect but may also set off feedback effects that enhance the response In this prob lem we look at a few examples Consider a pricetaking firm that chooses its inputs to maximize a profit function of the form P1P v w2 5 Pf1k l 2 2 wl 2 vk This maximiza tion process will yield optimal solutions of the general form q1P v w2 l 1P v w2 and k1P v w2 If we constrain capital input to be fixed at k in the short run this firms shortrun responses can be represented by qs 1P w k 2 and ls 1P w k 2 a Using the definitional relation q1P v w2 5 qs 1P w k1P v w2 2 show that q P 5 qs P 1 2ak P b 2 k v Do this in three steps First differentiate the definitional relation with respect to P using the chain rule Next differentiate the definitional relation with respect to v again using the chain rule and use the result to substi tute for qsk in the initial derivative Finally substitute a result analogous to part c of Problem 1110 to give the displayed equation b Use the result from part a to argue that qP qsP This establishes Le Châteliers Principle for sup ply Longrun supply responses are larger than con strained shortrun supply responses c Using similar methods as in parts a and b prove that Le Châteliers Principle applies to the effect of the wage on labor demand That is starting from the definitional relation l 1P v w2 5 l s 1P w k1P v w22 show that l w l sw implying that longrun labor demand falls more when wage goes up than shortrun labor demand note that both of these derivatives are negative d Develop your own analysis of the difference between the short and longrun responses of the firms cost function Cv w q to a change in the wage w 1112 More on the derived demand with two inputs The demand for any input depends ultimately on the demand for the goods that input produces This can be shown most explicitly by deriving an entire industrys demand for inputs To do so we assume that an industry produces a homoge neous good Q under constant returns to scale using only capital and labor The demand function for Q is given by Q 5 D 1P2 where P is the market price of the good being pro duced Because of the constant returnstoscale assumption P 5 MC 5 AC Throughout this problem let Cv w 1 be the firms unit cost function Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 391 a Explain why the total industry demands for capital and labor are given by k 5 QCv and l 5 QCw b Show that k v 5 QCvv 1 DrC 2 v and l w 5 QCww 1 DrC2 w c Prove that Cvv 5 2w v Cvw and Cww 5 2v w Cvw d Use the results from parts b and c together with the elasticity of substitution defined σ 5 CCvwCvCw to show that k v 5 wl Q σk vC 1 Drk2 Q2 and l w 5 vk Q σl wC 1 Drl2 Q2 e Convert the derivatives in part d into elasticities to show that ekv 5 2sl σ 1 sk eQP and el w 5 2sk σ 1 sl eQP where eQP is the price elasticity of demand for the prod uct being produced f Discuss the importance of the results in part e using the notions of substitution and output effects from Chapter 11 Note The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was first suggested by Alfred Marshall The proof given here follows that in D Hamermesh Labor Demand Princeton NJ Princeton University Press 1993 1113 Crossprice effects in input demand With two inputs crossprice effects on input demand can be easily calculated using the procedure outlined in Problem 1112 a Use steps b d and e from Problem 1112 to show that ekw 5 sl 1σ 1 eQP2 and el v 5 sk 1σ 1 eQP2 b Describe intuitively why input shares appear somewhat differently in the demand elasticities in part e of Prob lem 1112 than they do in part a of this problem c The expression computed in part a can be easily gen eralized to the manyinput case as exiwj 5 sj1Aij 1 eQP2 where Aij is the Allen elasticity of substitution defined in Problem 1012 For reasons described in Problems 1011 and 1012 this approach to input demand in the multiinput case is generally inferior to using Mor ishima elasticities One oddity might be mentioned however For the case i 5 j this expression seems to say that el w 5 sl 1Al l 1 eQ P2 and if we jumped to the conclusion that Al l 5 σ in the twoinput case then this would contradict the result from Problem 1112 You can resolve this paradox by using the definitions from Problem 1012 to show that with two inputs Al l 5 12sksl2 Akl 5 12sksl2 σ and so there is no disagreement 1114 Profit functions and technical change Suppose that a firms production function exhibits technical improvements over time and that the form of the function is q 5 f 1k l t2 In this case we can measure the proportional rate of technical change as ln q t 5 ft f compare this with the treatment in Chapter 9 Show that this rate of change can also be measured using the profit function as ln q t 5 P1P v w t2 Pq ln P t That is rather than using the production function directly technical change can be measured by knowing the share of profits in total revenue and the proportionate change in prof its over time holding all prices constant This approach to measuring technical change may be preferable when data on actual input levels do not exist 1115 Property rights theory of the firm This problem has you work through some of the calculations associated with the numerical example in the Extensions Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors GM who we imag ine are deciding between remaining as separate firms or hav ing GM acquire Fisher Body and thus become one larger firm Let the total surplus that the units generate together be S 1xF xG2 5 x12 F 1 ax12 G where xF and xG are the investments undertaken by the managers of the two units before negotiat ing and where a unit of investment costs 1 The parameter a measures the importance of GMs managers investment Show that according to the property rights model worked out in the Extensions it is efficient for GM to acquire Fisher Body if and only if GMs managers investment is important enough in particular if a 3 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 392 Part 4 Production and Supply Suggestions for Further Reading Hart O Firms Contracts and Financial Structure Oxford UK Oxford University Press 1995 Discusses the philosophical issues addressed by alternative theories of the firm Derives further results for the property rights theory discussed in the Extensions Hicks J R Value and Capital 2nd ed Oxford UK Oxford University Press 1947 The Appendix looks in detail at the notion of factor complementarity MasColell A M D Whinston and J R Green Micro economic Theory New York Oxford University Press 1995 Provides an elegant introduction to the theory of production using vector and matrix notation This allows for an arbitrary number of inputs and outputs Samuelson P A Foundations of Economic Analysis Cam bridge MA Harvard University Press 1947 Early development of the profit function idea together with a nice discussion of the consequences of constant returns to scale for market equilibrium Pages 3646 have extensive applications of Le Châteliers Principle see Problem 1111 Sydsaeter K A Strom and P Berck Economists Mathemati cal Manual 3rd ed Berlin SpringerVerlag 2000 Chapter 25 offers formulas for a number of profit and factor demand functions Varian H R Microeconomic Analysis 3rd ed New York W W Norton 1992 Includes an entire chapter on the profit function Varian offers a novel approach for comparing short and longrun responses using Le Châteliers Principle Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 393 Chapter 11 provided fairly straightforward answers to the questions of what determines the boundaries of a firm and its objectives The firm is identified by the production function f k l it uses to produce its output and the firm makes its input and output decisions to maximize profit Ronald Coase winner of the Nobel Prize in economics in 1991 was the first to point out back in the 1930s that the nature of the firm is a bit more subtle than that The firm is one way to organize the economic transactions necessary for output to be produced and sold transactions including the purchase of inputs financing of investment advertising management and so forth But these transactions could also be conducted in other ways Parties could sign long term contracts or even just trade on a spot market see Coase 1937 There is a sense in which firms and spot markets are not just different ways of organizing transactions but polar opposites Moving a transaction within a firm is tantamount to insulating the transaction from shortterm market forces eliminating price signals by placing it inside a more durable institution This presents a puzzle Economists are supposed to love marketswhy are they then so willing to take the existence of firms for granted On the other hand if firms are so great why is there not just one huge firm that con trols the whole economy removing all transactions from the market Clearly a theory is needed to explain why there are firms of intermediate sizes and why these sizes vary across different industries and even across different firms in the same industry To make the ideas in the Extensions concrete we will couch the discussion in terms of the classic case of Fisher Body and General Motors GM mentioned at the begin ning of Chapter 11 Recall that Fisher Body was the main supplier of auto bodies to GM which GM would assemble with other auto parts into a car that it then sold to consum ers At first the firms operated separately but GM acquired Fisher Body in 1926 after a series of supply disruptions We will narrow the broad question of where firm boundaries should be set down to the question of whether it made eco nomic sense for GM and Fisher Body to merge into a single firm E111 Common features of alternative theories A considerable amount of theoretical and empirical research continues to be directed toward the fundamental question of the nature of the firm but it is fair to say that it has not pro vided a final answer Reflecting this uncertainty the Exten sions presents two different theories that have been proposed as alternatives to the neoclassical model studied in Chapter 11 The first is the property rights theory associated with the work of Sanford Grossman Oliver Hart and John Moore The second is the transactions cost theory associated with the work of Oliver Williamson cowinner of the Nobel Prize in economics in 20091 The theories share some features Both acknowledge that if all markets looked like the supplydemand model encoun tered in principles courseswhere a large number of suppli ers and buyers trade a commodity anonymouslythat would be the most efficient way to organize transactions leaving no role for firms However it is unrealistic to assume that all transactions work that way Three factorsuncertainty complexity and specializationcan lead a transaction to look more like haggling among a few participants rather than an impersonal sale on a large exchange We can see how these three factors would have operated in the GMFisher Body example The presence of uncer tainty and complexity would have made it difficult for GM to sign contracts years in advance for auto bodies Such con tracts would have to specify how the auto bodies should be designed but successful design depends on the vagaries of consumer taste which are difficult to predict after all large tail fins were popular at one point in history and hard to specify in writing The best way to cope with uncertainty and complexity may be for GM to negotiate the purchase of auto bodies at the time they are needed for assembly rather than years in advance at the signing of a longterm contract The third factor specialization leads to obvious advantages Auto EXTENSIONS Boundaries of the firM 1Seminal articles on the property rights theory are Grossman and Hart 1986 and Hart and Moore 1990 See Williamson 1979 for a comprehensive treatment of the transactions cost theory Gibbons 2005 provides a good summary of these and other alternatives to the neoclassical model Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 394 Part 4 Production and Supply bodies that are tailored to GMs styling and other technical requirements would be more valuable than generic ones But specialization has the drawback of limiting GM to a small set of suppliers rather than buying auto bodies as it would an input on a competitive commodity market Markets exhibiting these three factorsuncertainty complexity and specializationwill not involve the sale of perfect longterm contracts in a competitive equilibrium with large numbers of suppliers and demanders Rather they will often involve few parties perhaps just two negotiating often not far in advance of when the input is required This makes the alternative theories of the firm interesting If the alternative theories merely compared firms to perfectly com petitive markets markets would always end up winning in the comparison Instead firms are compared to negotiated sales a more subtle comparison without an obvious win ner We will explore the subtle comparisons offered by the two different theories next E112 Propertyrights theory To make the analysis of this alternative theory as stark as possible suppose that there are just two owner managers one who runs Fisher Body and one who runs GM Let S 1xF xG2 be the total surplus generated by the transaction between Fisher Body and GM the sum of both firms profits Fisher Body from its sale of auto bodies to GM and GM from its sale of cars to consumers Instead of being a func tion of capital and labor or input and output prices we now put those factors aside and just write surplus as a function of two new variables the investments made by Fisher Body 1xF2 and GM 1xG2 The surplus function subtracts all pro duction costs just as the producer surplus concept from Chapter 11 did but does not subtract the cost of the invest ments xF and xG The investments are sunk before negoti ations between them over the transfer of the auto bodies take place The investments include for example any effort made by Fisher Bodys manager to improve the precision of its metalcutting dies and to refine the shapes to GMs spec ifications as well as the effort expended by GMs manager in designing and marketing the car and tailoring its assem bly process to use the bodies Both result in a better car model that can be sold at a higher price and that generates more profit not including the investment effort For sim plicity assume one unit of investment costs a manager 1 implying that investment level xF costs Fisher Bodys man ager xF dollars and that the marginal cost of investment for both parties is 1 Before computing the equilibrium investment levels under various ownership structures as a benchmark we will com pute the efficient investment levels The efficient levels maxi mize total surplus minus investment costs S 1xF xG2 2 xF 2 xG i The firstorder conditions for maximization of this objective are S xF 5 S xG 5 1 ii The efficient investment levels equalize the total marginal benefit with the marginal cost Next lets compute equilibrium investment levels under various ownership structures Assume the investments are too complicated to specify in a contract before they are under taken So too is the specification of the auto bodies them selves Instead starting with the case in which Fisher Body and GM are separate firms they must bargain over the terms of trade of the auto bodies prices quantities nature of the product when they are needed There is a large body of liter ature on how to model bargaining we will touch on this a bit more in Chapter 13 when we introduce Edgeworth boxes and contract curves To make the analysis as simple as possible we will not solve for all the terms of the bargain but will just assume that they come to an agreement to split any gains from the transaction equally2 Because cars cannot be produced without auto bodies no surplus is generated if parties do not consummate a deal Therefore the gain from bargaining is the whole surplus S 1xF xG2 The investment expenditures are not part of the negotiation because they were sunk before Fisher Body and GM each end up with S 1xF xG22 in equilibrium from bargaining To solve for equilibrium investments subtract Fisher Bodys cost of investment from its share of the bargaining gains yielding the objective function 1 2 S 1xF xG2 2 xF iii Taking the firstorder condition with respect to xF and rear ranging yields the condition 1 2a S xF b 5 1 iv The left side of Equation iv is the marginal benefit to Fisher Body from additional investment Fisher Body receives its bargaining share half of the surplus The right side is the marginal cost which is 1 because investment xF is measured in dollar terms As usual the optimal choice here invest ment equalizes marginal benefit and marginal cost A similar condition characterizes GMs investment decision 1 2a S xG b 5 1 v In sum if Fisher Body and GM are separate firms invest ments are given by Equations iv and v 2This is a special case of socalled Nash bargaining an influential bargaining theory developed by the same John Nash behind Nash equilibrium Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 395 If instead GM acquires Fisher Body so they become one firm the manager of the auto body subsidiary is now in a worse bargaining position He or she can no longer extract half of the bargaining surplus by threatening not to use Fisher Bodys assets to produce bodies for GM the assets are all under GMs control To make the point as clear as possible assume that Fisher Bodys manager obtains no bargaining sur plus GM obtains all of it Without the prospect of a return the manager will not undertake any investment implying xF 5 0 On the other hand because GMs manager now obtains the whole surplus S 1xF xG2 the objective function determining his or her investment is now 1 2S 1xF xG2 2 xG vi yielding firstorder condition S xG 5 1 vii When both parties were in separate firms each had less than efficient investment incentives compare the firstorder conditions in the efficient outcome in Equation ii with Equa tions iv and v because they only obtain half the bargain ing surplus Combining the two units under GMs ownership further dilutes Fisher Bodys investment incentives reducing its investment all the way down to xF 5 0 but boosts GMs so that GMs firstorder condition resembles the efficient one Intuitively asset ownership gives parties more bargaining power and this bargaining power in turn protects the party from having the returns from their investment appropriated by the other party in bargaining3 Of course there is only so much bargaining power to go around A shift of assets from one party to another will increase ones bargaining power at the expense of the other Therefore a tradeoff is involved in merging two units into one the merger only makes economic sense under certain conditions If GMs investment is much more important for surplus then it will be efficient to allocate ownership over all the assets to GM If both units investments are roughly equally important then maintaining both parties bargaining power by apportioning some of the assets to each might be a good idea If Fisher Bodys investment is the most important then having Fisher Body acquire GM may produce the most efficient structure More specific recommendations would depend on functional forms as will be illustrated in the following numerical example E113 Numerical example For a simple numerical example of the property rights theory let S 1xF xG2 5 x12 F 1 x12 G The firstorder condition for the efficient level of Fisher Bodys investment is 1 2 x 212 F 5 1 implying x F 5 14 Likewise x G 5 14 Total surplus sub tracting the investment costs is 12 If Fisher Body and GM remain separate firms half the surplus from each partys investment is held up by the other party Fisher Bodys firstorder condition is 1 4 x212 F 5 1 implying xF 5 116 Likewise xG 5 116 Thus parties are underinvesting relative to the efficient outcome Total surplus subtracting investment costs is only 38 If GM acquires Fisher Body the manager of the auto body unit does not invest 1xF 5 02 because he or she obtains no bar gaining surplus The manager of the integrated firm obtains all the bargaining surplus and invests at the efficient level x G 5 14 Overall total surplus subtracting investment costs is 14 Combining the firms decreases Fisher Bodys invest ment and increases GMs but the net effect is to make them jointly worse off therefore the firms should remain separate If GMs investment were more important than Fisher Bodys merging them could be efficient Let S 1xF xG2 5 x12 F 1 ax12 G where a allows the impact of GMs investment on surplus to vary One of the problems at the end of this chapter asks you to show that having GMs manager own all assets is more effi cient than keeping the firms separate for high enough a in particular a 3 E114 Transaction cost theory Next turn to the second alternative theory of the firmthe transaction cost theory As discussed previously it shares many common elements with the property rights theory but there are subtle differences With the property rights theory the main benefit of restructuring the firm was to get the right incentives for investments made before bargaining With the transaction cost theory the main benefit is to reduce haggling costs at the time of bargaining Let hF be a costly action undertaken by Fisher Body at the time of bargaining that increases its bargaining power at the expense of GM We loosely interpret this action as haggling but more concretely it could be a costly signal such as was seen in the Spence education signaling game in Chapter 8 or it could represent bargaining delay or an input supplier strike GM can take a similar haggling action hG Rather than fixing the bargaining shares at 12 each we now assume α 1hF hG2 is the share accruing to Fisher Body and 1 2 α 1hF hG2 is the share accruing to GM where α is between 0 and 1 and is increasing in hF and decreasing in hG For simplicity assume that the marginal cost for one unit of the haggling action is 1 implying a haggling level of hF costs Fisher Body hF dol lars and of hG costs GM hG dollars To abstract from some of the bargaining issues in the previous theory assume that 3The appropriation of the returns from one partys investment by the other party in bargaining is called the holdup problem referring to the colorful image of a bandit holding up a citizen at gunpoint Nothing illegal is happening here hold up is just a feature of bargaining Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 396 Part 4 Production and Supply investments are made at the time of bargaining rather than beforehand so that in principle they can be set at the efficient levels x F and x G satisfying Equation ii The efficient outcome is for investments to be set at x F and x G and for parties not to undertake any haggling actions hF 5 hG 5 0 Haggling does not generate any more total surplus but rather reallocates it from one party to another If Fisher Body and GM are separate firms they will undertake some of these actions much like the prisoners were led to fink on each other in equilibrium of the Prisoners Dilemma in Chapter 8 when it would have been better for the two of them to remain silent Fisher Bodys objective function determining its equilibrium level of haggling is α 1hF hG2 3S 1x F x G2 2 x F 2 x G4 2 hF viii where it is assumed the parties naturally would agree on the investments maximizing their joint surplus Fisher Bodys firstorder condition is after rearranging α xF 3S 1x F x G2 2 x F 2 x G4 5 1 ix Similarly GM will have firstorder condition α xG 3S 1x F x G2 2 x F 2 x G4 5 1 x The main point to take away from these somewhat compli cated conditions is that both parties will engage in some wasteful haggling if they remain separate If instead GM acquires Fisher Body and they become one firm assume this enables GM to authorize what investment levels should be undertaken without having to resort to bar gaining This rules out haggling therefore hF 5 hG 5 0 a savings with this organizational structure In many accounts of the transactions cost theory that is the end of the story Combining separate units together in the same firm reduces haggling and thus firms are always more efficient than mar kets when haggling costs are significant The trouble with stopping there with the model is that there is no tradeoff associated with firms In theory one large firm should oper ate the entire economy which is certainly an unrealistic outcome One way to generate a tradeoff is to assume that there is drawback to having one party here GM make a unilateral decision One natural drawback is that GM may not choose the efficient investment levels either because it lacks valuable information to which the manager of the auto body unit is privy or because the manager of the merged firm makes the investment for his or her own benefit rather than to maxi mize joint surplus Letting x F and x G be the investment levels authorized by the manager of the merged firm total surplus as a result of the merger is S 1x F x G2 2 x F 2 x G xi compared with total surplus when the firms remain separate S 1x F x G2 2 x F 2 x G 2 hF 2 hG xii The tradeoffs involved in different firm structures are appar ent from a comparison of these equations Giving GM the unilateral authority to make the investment decision avoids any haggling costs but may result in inefficient investment levels Whether it is more efficient to keep the firms separate or to merge the two units together and have one manager con trol them depends on the significance of the investment dis tortion relative to the haggling costs which in turn depends on functional forms E115 Classic empirical studies Early empirical studies of these alternative theories of the firm were not designed to distinguish between these spe cific theories or additional alternatives The focus was instead on seeing whether the conditions pushing input markets away from perfect competition toward negotiated salesuncertainty complexity and specialization leading to few bargaining partiescould help explain the deci sion to have a transaction occur within the boundaries of a firm rather than having it occur between separate parties Monteverde and Teece 1982 surveyed engineers at US auto manufacturers about more than 100 parts assembled together to make cars asking them how much engineer ing effort was required to design the part and whether the part was specialized to a single manufacturer The authors found that these variables had a significant positive effect on the decision of the manufacturer to produce the part in house rather than purchasing from a separate supplier Mas ten 1984 found similar results in the aerospace industry Anderson and Schmittlein 1984 found that proxies for complexity and specialization could help explain why some electronic components were sold by sales representatives employed by the manufacturers themselves and some by independent operators References Anderson E and D C Schmittlein Integration of the Sales Force An Empirical Examination Rand Journal of Eco nomics Autumn 1984 38595 Coase R H The Nature of the Firm Economica November 1937 386405 Gibbons R Four Formalizable Theories of the Firm Journal of Economic Behavior and Organization October 2005 20045 Grossman S J and O D Hart The Costs and Benefits of Ownership A Theory of Vertical and Lateral Integration Journal of Political Economy August 1986 691719 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 11 Profit Maximization 397 Hart O D and J Moore Property Rights and the Nature of the Firm Journal of Political Economy December 1990 11191158 Masten S E The Organization of Production Evidence from the Aerospace Industry Journal of Law and Econom ics October 1984 40317 Monteverde K and D J Teece Supplier Switching Costs and Vertical Integration in the Automobile Industry Bell Journal of Economics Spring 1982 20613 Williamson O Transaction Cost Economics The Gover nance of Contractual Relations Journal of Law and Eco nomics October 1979 23361 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 399 Competitive Markets Chapter 12 The Partial Equilibrium Competitive Model Chapter 13 General Equilibrium and Welfare In Parts 2 and 4 we developed models to explain the demand for goods by utilitymaximizing individ uals and the supply of goods by profitmaximizing firms In the next two parts we will bring together these strands of analysis to discuss how prices are determined in the marketplace The discussion in this part concerns competitive markets The principal characteristic of such markets is that firms as well as individual demanders behave as pricetakers That is firms are assumed to respond to market prices but they believe they have no control over these prices The primary reason for such a belief is that competitive markets are characterized by many suppliers therefore the decisions of any one of them indeed has little effect on prices In Part 6 we will relax this assumption by looking at markets with only a few suppliers perhaps only one For these cases the assumption of pricetak ing behavior is untenable thus the likelihood that firms actions can affect prices must be taken into account Chapter 12 develops the familiar partial equilibrium model of price determination in competi tive markets The principal result is the Marshallian cross diagram of supply and demand that we first discussed in Chapter 1 This model illustrates a partial equilibrium view of price determination because it focuses on only a single market We look at the comparative statics analysis of this model in considerable detail because it is one of the key building blocks of microeconomics In the concluding sections of the chapter we show some of the ways in which these models are applied A specific focus is on illustrating how the competitive model can be used to judge the wel fare consequences for market participants of changes in market equilibria brought about by taxes Although the partial equilibrium competitive model is useful for studying a single market in detail it is inappropriate for examining relationships among markets To capture such crossmarket effects requires the development of general equilibrium modelsa topic we take up in Chapter 13 There we show how an entire economy can be viewed as a system of interconnected competitive markets that determine all prices simultaneously We also examine how welfare consequences of various economic questions can be studied in this model PART five Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 401 CHAPTER TWeLve The Partial Equilibrium Competitive Model In this chapter we describe the familiar model of price determination under perfect com petition that was originally developed by Alfred Marshall in the late nineteenth century That is we provide a fairly complete analysis of the supplydemand mechanism as it applies to a single market This is perhaps the most widely used model for the study of price determination 121 MARKET DEMAND In Part 2 we showed how to construct individual demand functions that illustrate changes in the quantity of a good that a utilitymaximizing individual chooses as the market price and other factors change With only two goods x and y we concluded that an individuals Marshallian demand function can be summarized as quantity of x demanded 5 x 1px py I2 121 Now we wish to show how these demand functions can be added up to reflect the demand of all individuals in a marketplace Using a subscript i 1i 5 1 n2 to represent each persons demand function for good x we can define the total demand in the market as market demand for X 5 a n i51 xi 1px py Ii2 122 Notice three things about this summation First we assume that everyone in this market place faces the same prices for both goods That is px and py enter Equation 122 without personspecific subscripts On the other hand each persons income enters into his or her own specific demand function Market demand depends not only on the total income of all market participants but also on how that income is distributed among consumers Finally observe that we have used an uppercase X to refer to market demanda notation we will soon modify Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 402 Part 5 Competitive Markets 1211 The market demand curve Equation 122 makes clear that the total quantity of a good demanded depends not only on its own price but also on the prices of other goods and on the income of each person To construct the market demand curve for good X we allow px to vary while holding py and the income of each person constant Figure 121 shows this construction for the case where there are only two consumers in the market For each potential price of x the point on the market demand curve for X is found by adding up the quantities demanded by each person For example at a price of p x person 1 demands x 1 and person 2 demands x 2 The total quantity demanded in this twoperson market is the sum of these two amounts 1X 5 x 1 1 x 22 Therefore the point p x X is one point on the market demand curve for X Other points on the curve are derived in a similar way Thus the market demand curve is a horizontal sum of each individuals demand curve1 1212 Shifts in the market demand curve The market demand curve summarizes the ceteris paribus relationship between X and px It is important to keep in mind that the curve is in reality a twodimensional representa tion of a manyvariable function Changes in px result in movements along this curve but changes in any of the other determinants of the demand for X cause the curve to shift to a new position A general increase in incomes would for example cause the demand curve to shift outward assuming X is a normal good because each individual would choose to buy more X at every price Similarly an increase in py would shift the demand curve to X outward if individuals regarded X and Y as substitutes but it would shift the demand curve for X inward if the goods were regarded as complements Accounting for all such shifts may sometimes require returning to examine the individual demand functions that constitute the market relationship especially when examining situations in which the dis tribution of income changes and thereby raises some incomes while reducing others To 1Compensated market demand curves can be constructed in exactly the same way by summing each individuals compensated demand Such a compensated market demand curve would hold each persons utility constant A market demand curve is the horizontal sum of each individuals demand curve At each price the quantity demanded in the market is the sum of the amounts each individual demands For example at p x the demand in the market is x 1 1 x 2 5 X px px px x1 X x2 x2 x1 px X X a Individual 1 b Individual 2 c Market demand x1 x2 FIGURE 121 Construction of a Market Demand Curve from Indi vidual Demand Curves Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 403 keep matters straight economists usually reserve the term change in quantity demanded for a movement along a fixed demand curve in response to a change in px Alternatively any shift in the position of the demand curve is referred to as a change in demand EXAMPLE 121 Shifts in Market Demand These ideas can be illustrated with a simple set of linear demand functions Suppose individual 1s demand for oranges x measured in dozens per year is given by2 x1 5 10 2 2px 1 01I1 1 05py 123 where px 5 price of oranges dollars per dozen I1 5 individual 1s income in thousands of dollars py 5 price of grapefruit a gross substitute for orangesdollars per dozen Individual 2s demand for oranges is given by x2 5 17 2 px 1 005I2 1 05py 124 Hence the market demand function is X1px py I1 I22 5 x1 1 x2 5 27 2 3px 1 01I1 1 005I2 1 py 125 Here the coefficient for the price of oranges represents the sum of the two individuals coeffi cients as does the coefficient for grapefruit prices This reflects the assumption that orange and grapefruit markets are characterized by the law of one price Because the individuals have dif fering coefficients for income however the demand function depends on each persons income To graph Equation 125 as a market demand curve we must assume values for I1 I2 and py because the demand curve reflects only the twodimensional relationship between x and px If I1 5 40 I2 5 20 and py 5 4 then the market demand curve is given by X 5 27 2 3px 1 4 1 1 1 4 5 36 2 3px 126 which is a simple linear demand curve If the price of grapefruit were to increase to py 5 6 then the curve would assuming incomes remain unchanged shift outward to X 5 27 2 3px 1 4 1 1 1 6 5 38 2 3px 127 whereas an income tax that took 10 thousand dollars from individual 1 and transferred it to individual 2 would shift the demand curve inward to X 5 27 2 3px 1 3 1 15 1 4 5 355 2 3px 128 because individual 1 has a larger marginal effect of income changes on orange purchases All these changes shift the demand curve in a parallel way because in this linear case none of them affects either individuals coefficient for px In all cases an increase in px of 010 ten cents would cause X to decrease by 030 dozen per year QUERY For this linear case when would it be possible to express market demand as a linear function of total income 1I1 1 I22 Alternatively suppose the individuals had differing coeffi cients for py Would that change the analysis in any fundamental way 2This linear form is used to illustrate some issues in aggregation It is difficult to defend this form theoretically however For example it is not homogeneous of degree 0 in all prices and income Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 404 Part 5 Competitive Markets 1213 Generalizations Although our construction concerns only two goods and two individuals it is easily gener alized Suppose there are n goods denoted by xi i 5 1 n with prices pi i 5 1 n Assume also that there are m individuals in society Then the jth individuals demand for the ith good will depend on all prices and on Ij the income of this person This can be denoted by xi j 5 xi j 1 p1 pn Ij2 129 where i 5 1 n and j 5 1 m Using these individual demand functions market demand concepts are provided by the following definition D E F I N I T I O N Market demand The market demand function for a particular good Xi is the sum of each individuals demand for that good Xi 1 p1 pn I1 Im2 5 a m j51 xi j1p1 pn Ij2 1210 The market demand curve for Xi is constructed from the demand function by varying pi while holding all other determinants of Xi constant Assuming that each individuals demand curve is downward sloping this market demand curve will also be downward sloping Of course this definition is just a generalization of our previous discussion but three features warrant repetition First the functional representation of Equation 1210 makes clear that the demand for Xi depends not only on pi but also on the prices of all other goods Therefore a change in one of those other prices would be expected to shift the demand curve to a new position Second the functional notation indicates that the demand for Xi depends on the entire distribution of individuals incomes Although in many eco nomic discussions it is customary to refer to the effect of changes in aggregate total pur chasing power on the demand for a good this approach may be a misleading simplification because the actual effect of such a change on total demand will depend on precisely how the income changes are distributed among individuals Finally although they are obscured somewhat by the notation we have been using the role of changes in preferences should be mentioned We have constructed individuals demand functions with the assumption that preferences as represented by indifference curve maps remain fixed If preferences were to change so would individual and market demand functions Hence market demand curves can clearly be shifted by changes in preferences In many economic analyses how ever it is assumed that these changes occur so slowly that they may be implicitly held con stant without misrepresenting the situation 1214 A simplified notation Often in this book we look at only one market To simplify the notation in these cases we use QD or sometimes just D to refer to the quantity of the particular good demanded in this market and P to denote its market price As always when we draw a demand curve in the QP plane the ceteris paribus assumption is in effect If any of the factors mentioned in the previous section eg other prices individuals incomes or preferences should Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 405 change the QP demand curve will shift and we should keep that possibility in mind When we turn to consider relationships among two or more goods however we will return to the notation we have been using up until now ie denoting goods by x and y or by xi 1215 Elasticity of market demand When we use this simplified notation for market demand we will also use a compact nota tion for various elasticity concepts Specifically if the market demand function is repre sented by QD 5 D 1P Pr I2 then we define Price Elasticity of Market Demand 5 eD P 5 D 1P Pr I2 P P QD Cross Price Elasticity of Market Demand 5 eD Pr 5 D 1P Pr I2 Pr Pr QD Income Elasticity of Market Demand 5 eD I 5 D 1P Pr I2 I I QD 1211 The most important of these concepts is the own price elasticity of demand eD P which as we shall see plays a large role in the comparative statics of supply and demand models As in Chapter 5 we also characterize market demand as being elastic 1eD P 212 inelastic 10 eD P 212 or unit elastic eD P 5 21 Many of the relationships among elasticities discussed in Chapter 5 also apply to these marketwide concepts3 122 TIMING OF THE SUPPLY RESPONSE In the analysis of competitive pricing it is important to decide the length of time to be allowed for a supply response to changing demand conditions The establishment of equi librium prices will be different if we are talking about a short period during which most inputs are fixed than if we are envisioning a longrun process in which it is possible for new firms to enter an industry For this reason it has been traditional in economics to discuss pricing in three different time periods 1 very short run 2 short run and 3 long run Although it is not possible to give these terms an exact chronological definition the essen tial distinction being made concerns the nature of the supply response that is assumed to be possible In the very short run there is no supply response The quantity supplied is fixed and does not respond to changes in demand In the short run existing firms may change the quantity they are supplying but no new firms can enter the industry In the long run new firms may enter an industry thereby producing a flexible supply response In this chapter we will discuss each of these possibilities 123 PRICING IN THE VERY SHORT RUN In the very short run or the market period there is no supply response The goods are already in the marketplace and must be sold for whatever the market will bear In this situation price acts only as a device for rationing demand Price will adjust to clear the market of the quantity that must be sold during the period Although the market price may 3In many applications market demand is modeled in per capita terms and the demand relationship is said to apply to the typical person Whether such aggregation across individuals can be justified on theoretical grounds is discussed briefly in the Extensions to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 406 Part 5 Competitive Markets act as a signal to producers in future periods it does not perform such a function in the current period because currentperiod output is fixed Figure 122 depicts this situation Market demand is represented by the curve D Supply is fixed at Q and the price that clears the market is P1 At P1 individuals are willing to take all that is offered in the market Sellers want to dispose of Q without regard to price suppose that the good in question is perishable and will be worthless if it is not sold in the very short run Hence P1 Q is an equilibrium pricequantity combination If demand should shift to Dr then the equi librium price would increase to P2 but Q would stay fixed because no supply response is possible The supply curve in this situation is a vertical straight line at output Q The analysis of the very short run is not particularly useful for many markets Such a theory may adequately represent some situations in which goods are perishable or must be sold on a given day as is the case in auctions Indeed the study of auctions provides a number of insights about the informational problems involved in arriving at equilibrium prices which we take up in Chapter 18 But auctions are unusual in that supply is fixed The far more usual case involves some degree of supply response to changing demand It is presumed that an increase in price will bring additional quantity into the market In the remainder of this chapter we will examine this process Before beginning our analysis we should note that increases in quantity supplied need not come only from increased production In a world in which some goods are durable ie last longer than a single period current owners of these goods may supply them in increasing amounts to the market as price increases For example even though the sup ply of Rembrandts is fixed we would not want to draw the market supply curve for these paintings as a vertical line such as that shown in Figure 122 As the price of Rembrandts When quantity is fixed in the very short run price acts only as a device to ration demand With quantity fixed at Q price P1 will prevail in the marketplace if D is the market demand curve at this price indi viduals are willing to consume exactly that quantity available If demand should shift upward to Dr the equilibrium market price would increase to P2 Price Quantity per period Q P1 P2 S S D D D D FIGURE 122 Pricing in the Very Short Run Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 407 increases individuals and museums will become increasingly willing to part with them From a market point of view therefore the supply curve for Rembrandts will have an upward slope even though no new production takes place A similar analysis would follow for many types of durable goods such as antiques used cars vintage baseball cards or cor porate shares all of which are in nominally fixed supply Because we are more interested in examining how demand and production are related we will not be especially concerned with such cases here 124 SHORTRUN PRICE DETERMINATION In shortrun analysis the number of firms in an industry is fixed These firms are able to adjust the quantity they produce in response to changing conditions They will do this by altering levels of usage for those inputs that can be varied in the short run and we shall investigate this supply decision here Before beginning the analysis we should perhaps state explicitly the assumptions of this perfectly competitive model D E F I N I T I O N Perfect competition A perfectly competitive market is one that obeys the following assumptions 1 There are a large number of firms each producing the same homogeneous product 2 Each firm attempts to maximize profits 3 Each firm is a pricetaker It assumes that its actions have no effect on market price 4 Prices are assumed to be known by all market participantsinformation is perfect 5 Transactions are costless Buyers and sellers incur no costs in making exchanges for more on this and the previous assumption see Chapter 18 Throughout our discussion we continue to assume that the market is characterized by a large number of demanders each of whom operates as a pricetaker in his or her consump tion decisions 1241 Shortrun market supply curve In Chapter 11 we showed how to construct the shortrun supply curve for a single profitmaximizing firm To construct a market supply curve we start by recognizing that the quantity of output supplied to the entire market in the short run is the sum of the quantities supplied by each firm Because each firm uses the same market price to deter mine how much to produce the total amount supplied to the market by all firms will obvi ously depend on this price The relationship between price and quantity supplied is called a shortrun market supply curve Figure 123 illustrates the construction of the curve For simplicity assume there are only two firms A and B The shortrun supply ie marginal cost curves for firms A and B are shown in Figures 123a and 123b The market supply curve shown in Figure 123c is the horizontal sum of these two curves For example at a price of P1 firm A is willing to supply qA 1 and firm B is willing to supply qB 1 Therefore at this price the total supply in the market is given by Q1 which is equal to qA 1 1 qB 1 The other points on the curve are constructed in an identical way Because each firms supply curve has a positive slope the market supply curve will also have a positive slope The positive slope reflects the fact that shortrun marginal costs increase as firms attempt to increase their outputs Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 408 Part 5 Competitive Markets 1242 Shortrun market supply More generally if we let qi 1P v w2 represent the shortrun supply function for each of the n firms in the industry we can define the shortrun market supply function as follows The supply marginal cost curves of two firms are shown in a and b The market supply curve c is the horizontal sum of these curves For example at P1 firm A supplies qA 1 firm B supplies qB 1 and total market supply is given by Q1 5 qA 1 1 qB 1 a Firm A b Firm B c Te market Total output per period P P P P1 S SA SB Q1 qA qB 1 qA 1 qB FIGURE 123 ShortRun Market Supply Curve Notice that the firms in the industry are assumed to face the same market price and the same prices for inputs4 The shortrun market supply curve shows the twodimensional relationship between Q and P holding v and w and each firms underlying technology constant The notation makes clear that if v w or technology were to change the supply curve would shift to a new location 1243 Shortrun supply elasticity One way of summarizing the responsiveness of the output of firms in an industry to higher prices is by the shortrun supply elasticity This measure shows how proportional changes in market price are met by changes in total output Consistent with the elasticity concepts developed in Chapter 5 this is defined as follows 4Several assumptions that are implicit in writing Equation 1212 should be highlighted First the only one output price P enters the supply functionimplicitly firms are assumed to produce only a single output The supply function for multiproduct firms would also depend on the prices of the other goods these firms might produce Second the notation implies that input prices v and w can be held constant in examining firms reactions to changes in the price of their output That is firms are assumed to be pricetakers for inputstheir hiring decisions do not affect these input prices Finally the notation implicitly assumes the absence of externalitiesthe production activities of any one firm do not affect the production possibilities for other firms Models that relax these assumptions will be examined at many places later in this book D E F I N I T I O N Shortrun market supply function The shortrun market supply function shows total quantity supplied by each firm to a market QS 1P v w2 5 SP v w2 5 a n i51 qi 1P v w2 1212 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 409 Because quantity supplied is an increasing function of price 1QSP 02 the supply elasticity is positive High values for eS P imply that small increases in market price lead to a relatively large supply response by firms because marginal costs do not increase steeply and input price interaction effects are small Alternatively a low value for eS P implies that it takes relatively large changes in price to induce firms to change their output levels because marginal costs increase rapidly Notice that as for all elasticity notions computation of eS P requires that input prices and technology be held constant To make sense as a market response the concept also requires that all firms face the same price for their output If firms sold their output at different prices we would need to define a supply elasticity for each firm D E F I N I T I O N Shortrun elasticity of supply eS P eS P 5 percentage change in Q supplied percentage change in P 5 QS P P QS 1213 EXAMPLE 122 A ShortRun Supply Function In Example 113 we calculated the general shortrun supply function for any single firm with a twoinput CobbDouglas production function as qi 1P v w k12 5 aw β b 2β112β2 kα112β2 1 P β112β2 1214 If we let α 5 β 5 05 v 5 3 w 5 12 and k1 5 80 then this yields the simple singlefirm supply function qi 1P v w 5 12 k1 5 802 5 10P 3 1215 Now assume that there are 100 identical such firms and that each firm faces the same market prices for both its output and its input hiring Given these assumptions the shortrun market supply function is given by S 1P v w 5 12 k1 5 802 5 a 100 i51 qi 5 a 100 i51 10P 3 5 1000P 3 1216 Thus at a price of say P 5 12 total market supply will be 4000 with each of the 100 firms sup plying 40 units We can compute the shortrun elasticity of supply in this situation as eS P 5 S 1P v w 5 12 k1 5 802 P P S 5 1000 3 P 1000P3 5 1 1217 this might have been expected given the unitary exponent of P in the supply function Effect of an increase in w If all the firms in this marketplace experienced an increase in the wage they must pay for their labor input then the shortrun supply curve would shift to a new position To calculate the shift we must return to the single firms supply function Equation 1214 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 410 Part 5 Competitive Markets 1244 Equilibrium price determination We can now combine demand and supply curves to demonstrate the establishment of equi librium prices in the market Figure 124 shows this process Looking first at Figure 124b we see the market demand curve D ignore Dr for the moment and the shortrun supply curve S The two curves intersect at a price of P1 and a quantity of Q1 This pricequantity combination represents an equilibrium between the demands of individuals and the costs of firms The equilibrium price P1 serves two important functions First this price acts as a signal to producers by providing them with information about how much should be pro duced To maximize profits firms will produce that output level for which marginal costs are equal to P1 In the aggregate production will be Q1 A second function of the price is to ration demand Given the market price P1 utilitymaximizing individuals will decide how much of their limited incomes to devote to buying the particular good At a price of P1 total quantity demanded will be Q1 and this is precisely the amount that will be produced Hence we define equilibrium price as follows and now use a new wage say w 5 15 If none of the other parameters of the problem have changed the firms production function and the level of capital input it has in the short run the supply function becomes qi 1P v w 5 15 k1 5 802 5 8P 3 1218 and the market supply function is S 1P v w 5 15 k1 5 802 5 a 100 i51 8P 3 5 800P 3 1219 Thus at a price of P 5 12 now this industry will supply only QS 5 3200 with each firm produc ing qi 5 32 In other words the supply curve has shifted upward because of the increase in the wage Notice however that the price elasticity of supply has not changedit remains eS P 5 1 QUERY How would the results of this example change by assuming different values for the weight of labor in the production function ie for α and β D E F I N I T I O N Equilibrium price An equilibrium price is one at which quantity demanded is equal to quantity supplied At such a price neither demanders nor suppliers have an incentive to alter their eco nomic decisions Mathematically an equilibrium price P solves the equation D 1P Pr I2 5 S 1P v w2 1220 or more compactly D 1P 2 5 S 1P 2 1221 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 411 The definition given in Equation 1221 makes clear that an equilibrium price depends on the values of many exogenous factors such as incomes or prices of other goods and of firms inputs As we will see in the next section changes in any of these factors will likely result in a change in the equilibrium price required to equate quantity supplied to quantity demanded The implications of the equilibrium price P1 for a typical firm and a typical individual are shown in Figures 124a and 124c respectively For the typical firm the price P1 will cause an output level of q1 to be produced The firm earns a small profit at this particular price because shortrun average total costs are covered The demand curve d ignore dr for the moment for a typical individual is shown in Figure 124c At a price of P1 this indi vidual demands q1 By adding up the quantities that each individual demands at P1 and the quantities that each firm supplies we can see that the market is in equilibrium The market supply and demand curves provide a convenient way of making such a summation 1245 Market reaction to a shift in demand The three panels in Figure 124 can be used to show two important facts about shortrun market equilibrium the individuals impotence in the market and the nature of short run supply response First suppose that a single individuals demand curve were to shift outward to dr as shown in Figure 124c Because the competitive model assumes there are many demanders this shift will have practically no effect on the market demand curve Consequently market price will be unaffected by the shift to dr that is price will remain at P1 Of course at this price the person for whom the demand curve has shifted will con sume slightly more 1qr12 as shown in Figure 124c But this amount is a tiny part of the market Market demand curves and market supply curves are each the horizontal sum of numerous components These market curves are shown in b Once price is determined in the market each firm and each indi vidual treat this price as a fixed parameter in their decisions Although individual firms and persons are important in determining price their interaction as a whole is the sole determinant of price This is illus trated by a shift in an individuals demand curve to dr If only one individual reacts in this way market price will not be affected However if everyone exhibits an increased demand market demand will shift to Dr in the short run price will increase to P2 Price Price Price Output per period Total output per period Quantity demanded per period a A typical firm b Te market c A typical individual P 1 P 2 q 1 q 2 Q 1 Q 2 D D D S D d d d d SMC SAC q1 q 2 q 1 FIGURE 124 Interactions of Many Individuals and Firms Determine Market Price in the Short Run Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 412 Part 5 Competitive Markets If many individuals experience outward shifts in their demand curves the entire market demand curve may shift Figure 124b shows the new demand curve Dr The new equilib rium point will be at P2 Q2 at this point supplydemand balance is reestablished Price has increased from P1 to P2 in response to the demand shift Notice also that the quantity traded in the market has increased from Q1 to Q2 The increase in price has served two functions First as in our previous analysis of the very short run it has acted to ration demand Whereas at P1 a typical individual demanded qr1 at P2 only qr2 is demanded The increase in price has also acted as a signal to the typical firm to increase production In Fig ure 124a the firms profitmaximizing output level has increased from q1 to q2 in response to the price increase That is what we mean by a shortrun supply response An increase in market price acts as an inducement to increase production Firms are willing to increase production and to incur higher marginal costs because the price has increased If market price had not been permitted to increase suppose that government price controls were in effect then firms would not have increased their outputs At P1 there would now be an excess unfilled demand for the good in question If market price is allowed to increase a supplydemand equilibrium can be reestablished so that what firms produce is again equal to what individuals demand at the prevailing market price Notice also that at the new price P2 the typical firm has increased its profits This increasing profitability in the short run will be important to our discussion of longrun pricing later in this chapter 125 SHIFTS IN SUPPLY AND DEMAND CURVES A GRAPHICAL ANALYSIS In previous chapters we established many reasons why either a demand curve or a supply curve might shift These reasons are briefly summarized in Table 121 Although most of these merit little additional explanation it is important to note that a change in the number of firms will shift the shortrun market supply curve because the sum in Equation 1212 will be over a different number of firms This observation allows us to tie together short run and longrun analysis It seems likely that the types of changes described in Table 121 are constantly occurring in realworld markets When either a supply curve or a demand curve does shift equilib rium price and quantity will change In this section we investigate graphically the relative magnitudes of such changes In the next section we show the results mathematically 1251 Shifts in supply curves Importance of the shape of the demand curve Consider first a shift inward in the shortrun supply curve for a good As in Example 122 such a shift might have resulted from an increase in the prices of inputs used by firms to produce the good Whatever the cause of the shift it is important to recognize that the TABLE 121 REASONS FOR SHIFTS IN DEMAND OR SUPPLY CURVES Demand Curves Shift Because Supply Curves Shift Because Incomes change Input prices change Prices of substitutes or complements change Technology changes Preferences change Number of producers changes Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 413 effect of the shift on the equilibrium level of P and Q will depend on the shape of the demand curve for the product Figure 125 illustrates two possible situations The demand curve in Figure 125a is relatively price elastic that is a change in price substantially affects quantity demanded For this case a shift in the supply curve from S to Sr will cause equilib rium price to increase only moderately from P to Pr whereas quantity decreases sharply from Q to Qr Rather than being passed on in higher prices the increase in the firms input costs is met primarily by a decrease in quantity a movement down each firms mar ginal cost curve and only a slight increase in price This situation is reversed when the market demand curve is inelastic In Figure 125b a shift in the supply curve causes equilibrium price to increase substantially while quantity is little changed The reason for this is that individuals do not reduce their demands much if prices increase Consequently the shift upward in the supply curve is almost entirely passed on to demanders in the form of higher prices 1252 Shifts in demand curves Importance of the shape of the supply curve Similarly a shift in a market demand curve will have different implications for P and Q depending on the shape of the shortrun supply curve Two illustrations are shown in Figure 126 In Figure 126a the supply curve for the good in question is inelastic In this situation a shift outward in the market demand curve will cause price to increase sub stantially On the other hand the quantity traded increases only slightly Intuitively what has happened is that the increase in demand and in Q has caused firms to move up their steeply sloped marginal cost curves The concomitant large increase in price serves to ration demand In a the shift upward in the supply curve causes price to increase only slightly while quantity decreases sharply This results from the elastic shape of the demand curve In b the demand curve is inelastic price increases substantially with only a slight decrease in quantity FIGURE 125 Effect of a Shift in the ShortRun Supply Curve Depends on the Shape of the Demand Curve Price Price a Elastic demand b Inelastic demand S S S P P Q per period Q Q Q Q per period Q P P S S S D D Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 414 Part 5 Competitive Markets Figure 126b shows a relatively elastic shortrun supply curve Such a curve would occur for an industry in which marginal costs do not increase steeply in response to out put increases For this case an increase in demand produces a substantial increase in Q However because of the nature of the supply curve this increase is not met by great cost increases Consequently price increases only moderately These examples again demonstrate Marshalls observation that demand and supply simultaneously determine price and quantity Recall his analogy from Chapter 1 Just as it is impossible to say which blade of a scissors does the cutting so too is it impossible to attribute price solely to demand or to supply characteristics Rather the effect of shifts in either a demand curve or a supply curve will depend on the shapes of both curves 126 A COMPARATIVE STATICS MODEL OF MARKET EQUILIBRIUM All of the graphical analysis provided in the previous section can be succinctly developed using the comparative statics methods illustrated in Chapter 2 Because this is perhaps the most important way in which comparative statics methods are applied to examining changing equilibria here we will offer a rather extended analysis To do so we assume that the demand function is given by QD 5 D 1P α2 where α is an exogenous variable that shifts the demand function such as income or the price of another good Similarly the shortrun5 supply function is given by QS 5 S1P β2 where β is an exogenous variable that 5Most of the comparative statics analysis developed here for the short run would apply to longrun analysis also by simply substituting the longrun supply function In that case one would also like to model the equilibrium number of firms in an industrya topic we will take up later in this chapter In a supply is inelastic a shift in demand causes price to increase greatly with only a small concomitant increase in quantity In b on the other hand supply is elastic price increases only slightly in response to a demand shift FIGURE 126 Effect of a Shift in the Demand Curve Depends on the Shape of the Short Run Supply Curve Price Price b Elastic supply a Inelastic supply Q per period Q per period D D D D Q Q P P S S S S D D D D P P Q Q Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 415 shifts the supply function such as input prices or technical progress With this notation market equilibrium values for price 1P2 and quantity 1Q2 are determined by QD 5 QS 5 Q 5 D 1P α2 5 S1P β2 1222 To show how these equilibrium values change when one of the exogenous variables changes we write the equilibrium conditions as D 1P α2 2 Q 5 0 S1P β2 2 Q 5 0 1223 and note that these two equations are to be solved simultaneously to determine the equi librium values Now consider a shift in the demand6 function shown by a change in α Differentiation of Equations 1223 with respect to α yields DP dP dα 1 Dα 2 dQ dα 5 0 or DP dP dα 2 dQ dα 5 2Dα SP dP dα 2 dQ dα 5 0 1224 These equations show how the equilibrium values of price and quantity change when the demand curve shifts We could solve the equations for these derivatives by substitution but using the matrix algebra introduced in the Extensions to Chapter 2 gives an approach that can be more readily generalized Equations 1224 can be written in matrix notation as cDP 21 SP 21d D dP dα dQ dα T 5 c2Dα 0 d 1225 Applying Cramers rule to solve these equations for the change in equilibrium price and quantity yields dP dα 5 2Dα 21 0 21 DP 21 SP 21 5 Dα SP 2 DP 1226 dQ dα 5 DP 2Dα SP 0 DP 21 SP 21 5 Dα SP SP 2 DP 1227 Because SP 0 DP 0 the denominators of these expressions will be positive Hence the overall sign of dPdα and dQdα will both have the same sign as that of Dα If α rep resents an exogenous variable such as income or the price of a substitute an increase in this variable will shift the demand curve outward and increase both equilibrium price and quantity On the other hand if α is a variable such as the price of a complement for which an increase shifts the demand curve inward such an increase will both reduce equilib rium price and equilibrium quantity The extent of these changes is given by Equations 1226 and 1227 where all the derivatives are to be evaluated at the market equilibrium 6A similar approach can be used to calculate an expression for changes in equilibrium price and quantity brought about by a shift in the supply curve see Problem 1213 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 416 Part 5 Competitive Markets 1261 An elasticity interpretation Some algebraic manipulation of Equation 1226 and 1227 can be used to derive these rela tionships in elasticity forma form that is often more useful for empirical analysis Multi plying Equation 1226 by αP gives eP α 5 dP dα α P 5 Dα SP 2 DP α P 5 Dα SP 2 DP αQ PQ 5 eD α eS P 2 eD P 1228 Similarly multiplying Equation 1227 by αQ yields eQ α 5 dQ dα α Q 5 DαSP SP 2 DP 1αQ2 1PQ2 PQ 5 eD αeS P eS P 2 eD P 1229 Because all of the elasticities in Equations 1228 and 1229 may be available from prior empirical studies it can be used to make rough estimates of the effect of various events on equilibrium market prices and quantities As an example suppose that α represents con sumer income and we are interested in predicting how an increase in income might affect automobile output and pricing Suppose that empirical data estimate that eD P 5 212 and eD I 5 3 these data are taken from Table 123 in the Extensions to this chapter Because the auto market is a complex one it is difficult to specify a clear price elasticity of supply so we might as well assume simply that eS P 5 1 Inserting these values into Equation 1228 yields eP I 5 eD I eS P 2 eD P 5 30 10 2 12122 5 30 22 5 136 1230 Making similar insertions into Equation 1229 gives eQ I 5 1302 1102 10 2 12122 5 30 22 5 136 1231 The empirical data therefore suggest that each one percentage point increase in consumer income will increase both the equilibrium price and quantity of autos by 136 percentage points The identical values for price and quantity change here arise because the price elas ticity of supply is assumed to be 10 Hence shifts out in demand increase price and quan tity in the same proportion Equation 1229 shows how the results would differ depending on the price elasticity of supply If supply were price elastic 1eS P 12 the proportional increase in equilibrium quantity would exceed the proportional increase in price With an inelastic supply 1eS P 12 the situation would be reversed So this simple compara tive statics model confirms many of the things one learns in introductory economics Of course in the real world many other factors will undoubtedly affect equilibrium outcomes in the auto market but this simple model gives researchers a start on the issue EXAMPLE 123 Equilibria with Constant Elasticity Functions An even more complete analysis of supplydemand equilibrium can be provided if we use spe cific functional forms Constant elasticity functions are especially useful for this purpose Sup pose the demand for automobiles is given by D 1P I2 5 01P212I 3 1232 here price P is measured in dollars as is real family income I The supply function for auto mobiles is S 1P w2 5 6400Pw205 1233 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 417 where w is the hourly wage of automobile workers Notice that the elasticities assumed here are those used previously in the text eD P 5 212 eD I 5 30 and eS P 5 1 If the values for the exogenous variables I and w are 20000 and 25 respectively then demandsupply equilibrium requires D 1P I2 5 01P212I3 5 18 3 10112P212 5 S 1P w2 5 6400Pw205 5 1280P 1234 or P 22 5 18 3 101121280 5 625 3 108 or P 5 9957 Q 5 1280 P 5 12745000 1235 Hence the initial equilibrium in the automobile market has a price of nearly 10000 with approx imately 13 million cars being sold A shift in demand A 10 percent increase in real family income all other factors remaining constant would shift the demand function to D 1P2 5 1106 3 10122P12 1236 and proceeding as before P 22 5 1106 3 101221280 5 832 3 108 1237 or P 5 11339 Q 5 14514000 1238 As we predicted earlier the 10 percent increase in real income made car prices increase by nearly 14 percent In the process quantity sold increased by approximately 177 million automobiles again about a 14 percent increase A shift in supply An exogenous shift in automobile supply as a result say of changing auto workers wages would also affect market equilibrium If wages were to increase from 25 to 30 per hour the supply function would shift to S 1P2 5 6400P 1302205 5 1168P 1239 returning to our original demand function with I 5 20000 then yields P 22 5 18 3 101121168 5 685 3 108 1240 or P 5 10381 Q 5 12125000 1241 Therefore the 20 percent increase in wages led to a 43 percent increase in auto prices and to a decrease in sales of more than 600000 units Changing equilibria in many types of markets can be approxi mated by using this general approach together with empirical estimates of the relevant elasticities QUERY Do the results of changing auto workers wages agree with what might have been pre dicted using an equation similar to Equation 1230 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 418 Part 5 Competitive Markets 127 LONGRUN ANALYSIS We saw in Chapter 10 that in the long run a firm may adapt all its inputs to fit market conditions For longrun analysis we should use the firms longrun cost curves A profitmaximizing firm that is a pricetaker will produce the output level for which price is equal to longrun marginal cost MC However we must consider a second and ulti mately more important influence on price in the long run the entry of entirely new firms into the industry or the exit of existing firms from that industry In mathematical terms we must allow the number of firms n to vary in response to economic incentives The per fectly competitive model assumes that there are no special costs of entering or exiting from an industry Consequently new firms will be lured into any market in which economic profits are positive Similarly firms will leave any industry in which profits are negative The entry of new firms will cause the shortrun industry supply curve to shift outward because there are now more firms producing than there were previously Such a shift will cause market price and industry profits to decrease The process will continue until no firm contemplating entry would be able to earn a profit in the industry7 At that point entry will cease and the industry will have an equilibrium number of firms A similar argu ment can be made for the case in which some of the firms are suffering shortrun losses Some firms will choose to leave the industry and this will cause the supply curve to shift to the left Market price will increase thus restoring profitability to those firms remaining in the industry 1271 Equilibrium conditions To begin with we will assume that all the firms in an industry have identical cost functions that is no firm controls any special resources or technologies8 Because all firms are iden tical the equilibrium longrun position requires that each firm earn exactly zero economic profits In graphic terms the longrun equilibrium price must settle at the low point of each firms longrun average total cost curve Only at this point do the two equilibrium conditions P 5 MC which is required for profit maximization and P 5 AC which is required for zero profit hold It is important to emphasize however that these two equi librium conditions have rather different origins Profit maximization is a goal of firms Therefore the P 5 MC rule derives from the behavioral assumptions we have made about firms and is similar to the output decision rule used in the short run The zeroprofit con dition is not a goal for firms firms obviously would prefer to have large positive profits The longrun operation of the market however forces all firms to accept a level of zero economic profits 1P 5 AC2 because of the willingness of firms to enter and to leave an industry in response to the possibility of making supranormal returns Although the firms in a perfectly competitive industry may earn either positive or negative profits in the short run in the long run only a level of zero profits will prevail Hence we can summarize this analysis by the following definition 7Remember that we are using the economists definition of profits here These profits represent a return to the owner of a business in excess of that which is strictly necessary to stay in the business 8If firms have different costs then lowcost firms can earn positive longrun profits and such extra profits will be reflected in the price of the resource that accounts for the firms low costs In this sense the assumption of identical costs is not restrictive because an active market for the firms inputs will ensure that average costs which include opportunity costs are the same for all firms See also the discussion of Ricardian rent later in this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 419 128 LONGRUN EQUILIBRIUM CONSTANT COST CASE To discuss longrun pricing in detail we must make an assumption about how the entry of new firms into an industry affects the prices of firms inputs The simplest assumption we might make is that entry has no effect on the prices of those inputsperhaps because the industry is a relatively small hirer in its various input markets Under this assumption no matter how many firms enter or leave this market each firm will retain the same set of cost curves with which it started This assumption of constant input prices may not be ten able in many important cases which we will look at in the next section For the moment however we wish to examine the equilibrium conditions for a constant cost industry 1281 Initial equilibrium Figure 127 demonstrates longrun equilibrium in this situation For the market as a whole Figure 127b the demand curve is given by D and the shortrun supply curve by SS Therefore the shortrun equilibrium price is P1 The typical firm Figure 127a will D E F I N I T I O N Longrun competitive equilibrium A perfectly competitive market is in longrun equilibrium if there are no incentives for profitmaximizing firms to enter or to leave the market This will occur when a the number of firms is such that P 5 MC 5 AC and b each firm operates at the low point of its longrun average cost curve An increase in demand from D to Dr will cause price to increase from P1 to P2 in the short run This higher price will create profits in the industry and new firms will be drawn into the market If it is assumed that the entry of these new firms has no effect on the cost curves of the firms in the industry then new firms will continue to enter until price is pushed back down to P1 At this price economic prof its are zero Therefore the longrun supply curve LS will be a horizontal line at P1 Along LS output is increased by increasing the number of firms each producing q1 b Total market Q1 Q2 Q3 D SS SS SS SS D D D Price Price Total quantity per period Quantity per period a A typical firm P 1 P 2 q2 q1 SMC MC AC LS FIGURE 127 LongRun Equilibrium for a Perfectly Competitive Industry Constant Cost Case Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 420 Part 5 Competitive Markets produce output level q1 because at this level of output price is equal to shortrun mar ginal cost SMC In addition with a market price of P1 output level q1 is also a longrun equilibrium position for the firm The firm is maximizing profits because price is equal to longrun marginal costs MC Figure 127a also implies our second longrun equilibrium property Price is equal to longrun average costs AC Consequently economic profits are zero and there is no incentive for firms either to enter or to leave the industry Therefore the market depicted in Figure 127 is in both shortrun and longrun equilibrium Firms are in equilibrium because they are maximizing profits and the number of firms is stable because economic profits are zero This equilibrium will tend to persist until either supply or demand conditions change 1282 Responses to an increase in demand Suppose now that the market demand curve in Figure 127b shifts outward to Dr If SS is the relevant shortrun supply curve for the industry then in the short run price will increase to P2 The typical firm in the short run will choose to produce q2 and will earn profits on this level of output In the long run these profits will attract new firms into the market Because of the constant cost assumption this entry of new firms will have no effect on input prices New firms will continue to enter the market until price is forced down to the level at which there are again no pure economic profits Therefore the entry of new firms will shift the shortrun supply curve to SSr where the equilibrium price P1 is rees tablished At this new longrun equilibrium the pricequantity combination P1 Q3 will prevail in the market The typical firm will again produce at output level q1 although now there will be more firms than in the initial situation 1283 Infinitely elastic supply We have shown that the longrun supply curve for the constant cost industry will be a hor izontal straight line at price P1 This curve is labeled LS in Figure 127b No matter what happens to demand the twin equilibrium conditions of zero longrun profits because free entry is assumed and profit maximization will ensure that no price other than P1 can pre vail in the long run9 For this reason P1 might be regarded as the normal price for this commodity If the constant cost assumption is abandoned however the longrun supply curve need not have this infinitely elastic shape as we will show in the next section 9These equilibrium conditions also point out what seems to be somewhat imprecisely an efficient aspect of the longrun equilibrium in perfectly competitive markets The good under investigation will be produced at minimum average cost We will have much more to say about efficiency in the next chapter EXAMPLE 124 Infinitely Elastic LongRun Supply Handmade bicycle frames are produced by a number of identically sized firms Total longrun monthly costs for a typical firm are given by C1q2 5 q3 2 20q2 1 100q 1 8000 1242 where q is the number of frames produced per month Demand for handmade bicycle frames is given by QD 5 D 1P2 5 2500 2 3P 1243 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 421 where QD is the quantity demanded per month and P is the price per frame To determine the longrun equilibrium in this market we must find the low point of the typical firms average cost curve Because AC 5 C1q2 q 5 q2 2 20q 1 100 1 8000 q 1244 and MC 5 C1q2 q 5 3q2 2 40q 1 100 1245 and because we know this minimum occurs where AC 5 MC we can solve for this output level q2 2 20q 1 100 1 8000 q 5 3q2 1 40q 1 100 or 2q2 2 20q 5 8000 q 1246 which has a convenient solution of q 5 20 With a monthly output of 20 frames each producer has a longrun average and marginal cost of 500 This is the longrun equilibrium price of bicy cle frames handmade frames cost a bundle as any cyclist can attest With P 5 500 Equation 1243 shows QD 5 1000 Therefore the equilibrium number of firms is 50 When each of these 50 firms produces 20 frames per month supply will precisely balance what is demanded at a price of 500 If demand in this problem were to increase to QD 5 D 1P2 5 3000 2 3P 1247 then we would expect longrun output and the number of frames to increase Assuming that entry into the frame market is free and that such entry does not alter costs for the typical bicycle maker the longrun equilibrium price will remain at 500 and a total of 1500 frames per month will be demanded That will require 75 frame makers so 25 new firms will enter the market in response to the increase in demand QUERY Presumably the entry of frame makers in the long run is motivated by the shortrun profitability of the industry in response to the increase in demand Suppose each firms shortrun costs were given by SC 5 50q2 2 1500q 1 20000 Show that shortrun profits are zero when the industry is in longterm equilibrium What are the industrys shortrun profits as a result of the increase in demand when the number of firms stays at 50 129 SHAPE OF THE LONGRUN SUPPLY CURVE Contrary to the shortrun situation longrun analysis has little to do with the shape of the longrun marginal cost curve Rather the zeroprofit condition centers attention on the low point of the longrun average cost curve as the factor most relevant to longrun price determination In the constant cost case the position of this low point does not change as new firms enter the industry Consequently if input prices do not change then only one price can prevail in the long run regardless of how demand shiftsthe longrun sup ply curve is horizontal at this price Once the constant cost assumption is abandoned this need not be the case If the entry of new firms causes average costs to rise the longrun supply curve will have an upward slope On the other hand if entry causes average costs to decline it is even possible for the longrun supply curve to be negatively sloped We shall now discuss these possibilities Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 422 Part 5 Competitive Markets 1291 Increasing cost industry The entry of new firms into an industry may cause the average costs of all firms to increase for several reasons New and existing firms may compete for scarce inputs thus driving up their prices New firms may impose external costs on existing firms and on themselves in the form of air or water pollution They may increase the demand for taxfinanced ser vices eg police forces sewage treatment plants and the required taxes may show up as increased costs for all firms Figure 128 demonstrates two market equilibria in such an increasing cost industry The initial equilibrium price is P1 At this price the typical firm produces q1 and total industry output is Q1 Suppose now that the demand curve for the industry shifts outward to Dr In the short run price will rise to P2 because this is where Dr and the industrys shortrun supply curve SS intersect At this price the typical firm will produce q2 and will earn a substantial profit This profit then attracts new entrants into the market and shifts the shortrun supply curve outward Suppose that this entry of new firms causes the cost curves of all firms to increase The new firms may compete for scarce inputs thereby driving up the prices of these inputs A typical firms new higher set of cost curves is shown in Figure 128b The new long run equilibrium price for the industry is P3 here P3 5 MC 5 AC and at this price Q3 is demanded We now have two points P1 Q1 and P3 Q3 on the longrun supply curve All other points on the curve can be found in an analogous way by considering all possi ble shifts in the demand curve These shifts will trace out the longrun supply curve LS Here LS has a positive slope because of the increasing cost nature of the industry Observe that the LS curve is flatter more elastic than the shortrun supply curves This indicates the greater flexibility in supply response that is possible in the long run Still the curve is upward sloping so price increases with increasing demand This situation is probably common we will have more to say about it in later sections Initially the market is in equilibrium at P1 Q1 An increase in demand to Dr causes price to increase to P2 in the short run and the typical firm produces q2 at a profit This profit attracts new firms into the industry The entry of these new firms causes costs for a typical firm to increase to the levels shown in b With this new set of curves equilibrium is reestablished in the market at P3 Q3 By considering many possible demand shifts and connecting all the resulting equilibrium points the longrun supply curve LS is traced out SS D D Price Price Price P 1 P 1 P 2 P 2 P 3 P 3 q 2 q 3 Q 3 Q 1 Q 2 q 1 SMC SMC MC MC AC AC LS Output per period Output per period Output per period a Typical firm before entry b Typical firm afer entry c Te market SS D D FIGURE 128 An Increasing Cost Industry Has a Positively Sloped LongRun Supply Curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 423 1292 Decreasing cost industry Not all industries exhibit constant or increasing costs In some cases the entry of new firms may reduce the costs of firms in an industry For example the entry of new firms may pro vide a larger pool of trained labor from which to draw than was previously available thus reducing the costs associated with the hiring of new workers Similarly the entry of new firms may provide a critical mass of industrialization which permits the development of more efficient transportation and communications networks Whatever the exact reason for the cost reductions the final result is illustrated in the three panels of Figure 129 The ini tial market equilibrium is shown by the pricequantity combination P1 Q1 in Figure 129c At this price the typical firm produces q1 and earns exactly zero in economic profits Now suppose that market demand shifts outward to Dr In the short run price will increase to P2 and the typical firm will produce q2 At this price level positive profits are being earned These profits cause new entrants to come into the market If this entry causes costs to decline a new set of cost curves for the typical firm might resemble those shown in Figure 129b Now the new equilibrium price is P3 at this price Q3 is demanded By considering all possible shifts in demand the longrun supply curve LS can be traced out This curve has a negative slope because of the decreasing cost nature of the industry Therefore as output expands price falls This possibility has been used as the justification for protective tariffs to shield new industries from foreign competition It is assumed only occasionally correctly that the protection of the infant industry will permit it to grow and ultimately to compete at lower world prices 1293 Classification of longrun supply curves Thus we have shown that the longrun supply curve for a perfectly competitive industry may assume a variety of shapes The principal determinant of the shape is the way in which the entry of firms into the industry affects all firms costs The following definitions cover the various possibilities In c the market is in equilibrium at P1 Q1 An increase in demand to Dr causes price to increase to P2 in the short run and the typical firm produces q2 at a profit This profit attracts new firms to the indus try If the entry of these new firms causes costs for the typical firm to decrease a set of new cost curves might look like those in b With this new set of curves market equilibrium is reestablished at P3 Q3 By connecting such points of equilibrium a negatively sloped longrun supply curve LS is traced out SS D D Price Price Price P1 P1 P2 P2 P3 P3 Q3 Q2 Q1 q1 q2 q3 LS LS Output per period Output per period Output per period c Te market SMC SMC MC MC AC AC a Typical firm before entry b Typical firm afer entry D D SS FIGURE 129 A Decreasing Cost Industry Has a Negatively Sloped LongRun Supply Curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 424 Part 5 Competitive Markets Now we show how the shape of the longrun supply curve can be further quantified 1210 LONGRUN ELASTICITY OF SUPPLY The longrun supply curve for an industry incorporates information on internal firm adjustments to changing prices and changes in the number of firms and input costs in response to profit opportunities All these supply responses are summarized in the follow ing elasticity concept The value of this elasticity may be positive or negative depending on whether the industry exhibits increasing or decreasing costs As we have seen eLS P is infinite in the constant cost case because industry expansions or contractions can occur without having any effect on product prices 12101 Empirical estimates It is obviously important to have good empirical estimates of longrun supply elasticities These indicate whether production can be expanded with only a slight increase in relative price ie supply is price elastic or whether expansions in output can occur only if rela tive prices increase sharply ie supply is price inelastic Such information can be used to assess the likely effect of shifts in demand on longrun prices and to evaluate alternative policy proposals intended to increase supply Table 122 presents several longrun supply elasticity estimates These relate primarily although not exclusively to natural resources because economists have devoted considerable attention to the implications of increasing demand for the prices of such resources As the table makes clear these estimates vary widely depending on the spatial and geological properties of the particular resources involved All the estimates however suggest that supply does respond positively to price 1211 COMPARATIVE STATICS ANALYSIS OF LONGRUN EQUILIBRIUM Earlier in this chapter we showed how to develop a simple comparative statics analysis of changing shortrun equilibria in competitive markets By using estimates of the longrun D E F I N I T I O N Constant increasing and decreasing cost industries An industry supply curve exhibits one of three shapes Constant cost Entry does not affect input costs the longrun supply curve is horizontal at the longrun equilibrium price Increasing cost Entry increases input costs the longrun supply curve is positively sloped Decreasing cost Entry reduces input costs the longrun supply curve is negatively sloped D E F I N I T I O N Longrun elasticity of supply The longrun elasticity of supply 1eLS P2 records the proportionate change in longrun industry output in response to a proportionate change in product price Mathematically eLS P 5 percentage change in Q percentage change in P 5 QLS P P QLS 1248 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 425 elasticities of demand and supply exactly the same sort of analysis can be conducted for the long run as well For example the hypothetical auto market model in Example 123 might serve equally well for longrun analysis although some differences in interpretation might be required Indeed in applied models of supply and demand it is often not clear whether the author intends his or her results to reflect the short run or the long run and some care must be taken to understand how the issue of entry is being handled Sources Agricultural acreageM Nerlove Estimates of the Elasticities of Supply of Selected Agricultural Commodities Journal of Farm Economics 38 May 1956 496509 Aluminum and chromiumestimated from US Department of Interior Critical Materials Commodity Action Analysis Washington DC US Government Printing Office 1975 Coalestimated from M B Zimmerman The Supply of Coal in the Long Run The Case of Eastern Deep Coal MIT Energy Laboratory Report No MITEL 75021 Sep tember 1975 Natural gasbased on estimate for oil see text and J D Khazzoom The FPC Staffs Econometric Model of Natural Gas Supply in the United States The Bell Journal of Economics and Management Science Spring 1971 10317 OilE W Erickson S W Millsaps and R M Spann Oil Supply and Tax Incentives Brookings Papers on Economic Activity 2 1974 44978 Urban housingB A Smith The Supply of Urban Housing Journal of Political Economy 40 August 1976 389405 12111 Industry structure One aspect of the changing longrun equilibria in a perfectly competitive market that is obscured by using a simple supplydemand analysis is how the number of firms varies as market equilibria change Becauseas we will see in Part 6the functioning of mar kets may in some cases be affected by the number of firms and because there may be direct public policy interest in entry and exit from an industry some additional analysis is required In this section we will examine in detail determinants of the number of firms in the constant cost case Brief reference will also be made to the increasing cost case and some of the problems for this chapter examine that case in more detail 12112 Shifts in demand Because the longrun supply curve for a constant cost industry is infinitely elastic analyzing shifts in market demand is particularly easy If the initial equilibrium industry output is Q0 and if q represents the output level for which the typical firms longrun average cost is minimized then the initial equilibrium number of firms n0 is given by n0 5 Q0 q 1249 TABLE 122 SELECTED ESTIMATES OF LONGRUN SUPPLY ELASTICITIES Agricultural acreage Corn 018 Cotton 067 Wheat 093 Aluminum Nearly infinite Chromium 030 Coal eastern reserves 150300 Natural gas US reserves 020 Oil US reserves 076 Urban housing Density 53 Quality 38 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 426 Part 5 Competitive Markets A shift in demand that changes equilibrium output to Q1 will in the long run change the equilibrium number of firms to n1 5 Q1 q 1250 and the change in the number of firms is given by n1 2 n0 5 Q1 2 Q0 q 1251 That is the change in the equilibrium number of firms is completely determined by the extent of the demand shift and by the optimal output level for the typical firm 12113 Changes in input costs Even in the simple constant cost industry case analyzing the effect of an increase in an input price and hence an upward shift in the infinitely elastic longrun supply curve is relatively complicated First to calculate the decrease in industry output it is necessary to know both the extent to which minimum average cost is increased by the input price increase and how such an increase in the longrun equilibrium price affects total quan tity demanded Knowledge of the typical firms average cost function and of the price elas ticity of demand permits such a calculation to be made in a straightforward way But an increase in an input price may also change the minimum average cost output level for the typical firm Such a possibility is illustrated in Figure 1210 Both the average and marginal costs have been shifted upward by the input price increase but because average cost has An increase in the price of an input will shift average and marginal cost curves upward The precise effect of these shifts on the typical firms optimal output level q will depend on the relative magnitudes of the shifts Average and marginal costs Output per period q1 q0 MC1 AC1 MC0 AC0 FIGURE 1210 An Increase in an Input Price May Change Long Run Equilibrium Output for the Typical Firm Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 427 shifted up by a relatively greater extent than the marginal cost the typical firms optimal output level has increased from q 0 to q 1 If the relative sizes of the shifts in cost curves were reversed however the typical firms optimal output level would have decreased10 Taking account of this change in optimal scale Equation 1251 becomes n1 2 n0 5 Q1 q 1 2 Q0 q 0 1252 and a number of possibilities arise If q 1 q 0 the decrease in quantity brought about by the increase in market price will definitely cause the number of firms to decrease However if q 1 q 0 then the result will be indeterminate Industry output will decrease but optimal firm size also will decrease thus the ultimate effect on the number of firms depends on the relative magnitude of these changes A decrease in the number of firms still seems the most likely outcome when an input price increase causes industry output to decrease but an increase in n is at least a theoretical possibility 10A mathematical proof proceeds as follows Optimal output q is defined such that AC1v w q2 5 MC1v w q2 Differentiating both sides of this expression by say v yields AC v 1 AC q q v 5 MC v 1 MC q q v but ACq 5 0 because average costs are minimized Manipulating terms we obtain q v 5 aMC q b 1 aAC v 2 MC v b Because MCq 0 at the minimum AC it follows that qv will be positive or negative depending on the sizes of the relative shifts in the AC and MC curves EXAMPLE 125 Increasing Input Costs and Industry Structure An increase in costs for bicycle frame makers will alter the equilibrium described in Example 124 but the precise effect on market structure will depend on how costs increase The effects of an increase in fixed costs are fairly clear The longrun equilibrium price will increase and the size of the typical firm will also increase This latter effect occurs because an increase in fixed costs increases AC but not MC To ensure that the equilibrium condition for AC 5 MC holds output and MC must also increase For example if an increase in shop rents causes the typical frame makers costs to increase to C1q2 5 q3 2 20q2 1 100q 1 11616 1253 it is an easy matter to show that MC 5 AC when q 5 22 Therefore the increase in rent has increased the efficient scale of bicycle frame operations by 2 bicycle frames per month At q 5 22 the longrun average cost and the marginal cost are both 672 and that will be the longrun equi librium price for frames At this price QD 5 D 1P2 5 2500 2 3P 5 484 1254 so there will be room in the market now for only 22 15 484 4 222 firms The increase in fixed costs resulted not only in an increase in price but also in a significant reduction in the number of frame makers from 50 to 22 Increases in other types of input costs may however have more complex effects Although a complete analysis would require an examination of frame makers production functions and their related input choices we can provide a simple illustration by assuming that an increase in some variable input prices causes the typical firms total cost function to become C1q2 5 q3 2 8q2 1 100q 1 4950 1255 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 428 Part 5 Competitive Markets Now MC 5 3q2 2 16q 1 100 and AC 5 q2 2 8q 1 100 1 4950 q 1256 Setting MC 5 AC yields 2q2 2 8q 5 4950 q 1257 which has a solution of q 5 15 Therefore this particular change in the total cost function has significantly reduced the optimal size for frame shops With q 5 15 Equations 1256 show AC 5 MC 5 535 and with this new longrun equilibrium price we have QD 5 D 1P2 5 2500 2 3P 5 895 1258 These 895 frames will in equilibrium be produced by about 60 firms 895 4 15 5 5967 problems do not always work out evenly Even though the increase in costs results in a higher price the equilibrium number of frame makers expands from 50 to 60 because the optimal size of each shop is now smaller QUERY How do the total marginal and average functions derived from Equation 1255 differ from those in Example 124 Are costs always greater for all levels of q for the former cost curve Why is longrun equilibrium price higher with the former curves See footnote 10 for a formal discussion 1212 PRODUCER SURPLUS IN THE LONG RUN In Chapter 11 we described the concept of shortrun producer surplus which represents the return to a firms owners in excess of what would be earned if output were zero We showed that this consisted of the sum of shortrun profits plus shortrun fixed costs In longrun equilibrium profits are zero and there are no fixed costs therefore all such short run surplus is eliminated Owners of firms are indifferent about whether they are in a par ticular market because they could earn identical returns on their investments elsewhere Suppliers of firms inputs may not be indifferent about the level of production in a partic ular industry however In the constant cost case of course input prices are assumed to be independent of the level of production on the presumption that inputs can earn the same amount in alternative occupations But in the increasing cost case entry will bid up some input prices and suppliers of these inputs will be made better off Consideration of these price effects leads to the following alternative notion of producer surplus D E F I N I T I O N Producer surplus Producer surplus is the extra return that producers make by making trans actions at the market price over and above what they would earn if nothing were produced It is illustrated by the size of the area below the market price and above the supply curve Although this is the same definition we introduced in Chapter 11 the context is now different Now the extra returns that producers make should be interpreted as meaning the higher prices that productive inputs receive For shortrun producer surplus the gainers from market transactions are firms that are able to cover fixed costs and possibly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 429 earn profits over their variable costs For longrun producer surplus we must penetrate back into the chain of production to identify who the ultimate gainers from market transactions are It is perhaps surprising that longrun producer surplus can be shown graphically in much the same way as shortrun producer surplus The former is given by the area above the longrun supply curve and below equilibrium market price In the constant cost case longrun supply is infinitely elastic and this area will be zero showing that returns to inputs are independent of the level of production With increasing costs however longrun supply will be positively sloped and input prices will be bid up as industry output expands Because this notion of longrun producer surplus is widely used in applied analysis as we show later in this chapter we will provide a formal development 12121 Ricardian rent Longrun producer surplus can be most easily illustrated with a situation first described by David Ricardo in the early part of the nineteenth century11 Assume there are many parcels of land on which a particular crop might be grown These range from fertile land low costs of production to poor dry land high costs The longrun supply curve for the crop is constructed as follows At low prices only the best land is used As output increases highercost plots of land are brought into production because higher prices make it profit able to use this land The longrun supply curve is positively sloped because of the increas ing costs associated with using less fertile land Market equilibrium in this situation is illustrated in Figure 1211 At an equilibrium price of P owners of both the lowcost and the mediumcost firms earn longrun prof its The marginal firm earns exactly zero economic profits Firms with even higher costs stay out of the market because they would incur losses at a price of P Profits earned by the intramarginal firms can persist in the long run however because they reflect a return to a unique resourcelowcost land Free entry cannot erode these profits even over the long term The sum of these longrun profits constitutes longrun producer surplus as given by area PEB in Figure 1211d Equivalence of these areas can be shown by recognizing that each point in the supply curve in Figure 1211d represents minimum average cost for some firm For each such firm P 2 AC represents profits per unit of output Total long run profits can then be computed by summing over all units of output12 11See David Ricardo The Principles of Political Economy and Taxation 1817 reprinted London J M Dent and Son 1965 chap 2 and chap 32 12More formally suppose that firms are indexed by i 1i 5 1 n2 from lowest to highest cost and that each firm produces q In the longrun equilibrium Q 5 nq where n is the equilibrium number of firms and Q is total industry output Suppose also the inverse of the supply function competitive price as a function of quantity supplied is given by P 5 P 1Q2 Because of the indexing of firms price is determined by the highest cost firm in the market P 5 P 1iq2 5 ACi and P 5 P 1Q2 5 P 1nq2 Now in longrun equilibrium profits for firm i are given by πi 5 1P 2 ACi2q and total profits are given by π 5 3 n 0 πi di 5 3 n 0 1P 2 ACi2q di 5 3 n 0 pq di 2 3 n 0 ACiq di 5 pnq 2 3 n 0 P 1iq2q di 5 PQ 2 3 Q 0 P 1Q2 dQ which is the shaded area in Figure 1211d Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 430 Part 5 Competitive Markets 12122 Capitalization of rents The longrun profits for the lowcost firms in Figure 1211 will often be reflected in prices for the unique resources owned by those firms In Ricardos initial analysis for example one might expect fertile land to sell for more than an untillable rock pile Because such prices will reflect the present value of all future profits these profits are said to be capi talized into inputs prices Examples of capitalization include such disparate phenomena as the higher prices of nice houses with convenient access for commuters the high value of rock and sport stars contracts and the lower value of land near toxic waste sites Notice that in all these cases it is market demand that determines rentsthese rents are not tradi tional input costs that indicate forgone opportunities 12123 Input supply and longrun producer surplus It is the scarcity of lowcost inputs that creates the possibility of Ricardian rent If lowcost farmland were available at infinitely elastic supply there would be no such rent More gen erally any input that is scarce in the sense that it has a positively sloped supply curve will obtain rents in the form of earning a higher return than would be obtained if industry output were zero In such cases increases in output not only raise firms costs and thereby the price for which the output will sell but also generate rents for inputs The sum of all Owners of lowcost and mediumcost land can earn longrun profits Longrun producers surplus rep resents the sum of all these rentsarea PEB in d Usually Ricardian rents will be capitalized into input prices Price Price Price Price MC MC MC AC AC AC S E D B Quantity per period Quantity per period Quantity per period Quantity per period a Lowcost frm b Mediumcost frm c Marginal frm d Te market P P P P q q q Q FIGURE 1211 Ricardian Rent Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 431 such rents is again measured by the area above the longrun supply curve and below equi librium price Changes in the size of this area of longrun producer surplus indicate chang ing rents earned by inputs to the industry Notice that although longrun producer surplus is measured using the market supply curve it is inputs to the industry that receive this surplus Empirical measurements of changes in longrun producer surplus are widely used in applied welfare analysis to indicate how suppliers of various inputs fare as conditions change The final sections of this chapter illustrate several such analyses 1213 ECONOMIC EFFICIENCY AND APPLIED WELFARE ANALYSIS Longrun competitive equilibria may have the desirable property of allocating resources efficiently Although we will have far more to say about this concept in a general equi librium context in Chapter 13 here we can offer a partial equilibrium description of why the result might hold Remember from Chapter 5 that the area below a demand curve and above market price represents consumer surplusthe extra utility consumers receive from choosing to purchase a good voluntarily rather than being forced to do without it Sim ilarly as we saw in the previous section producer surplus is measured as the area below market price and above the longrun supply curve which represents the extra return that productive inputs receive rather than having no transactions in the good Overall then the area between the demand curve and the supply curve represents the sum of consumer and producer surplus It measures the total additional value obtained by market participants by being able to make market transactions in this good It seems clear that this total area is maximized at the competitive market equilibrium 12131 A graphic proof Figure 1212 shows a simplified proof Given the demand curve D and the longrun supply curve S the sum of consumer and producer surplus is given by distance AB for the first unit produced Total surplus continues to increase as additional output is producedup to the competitive equilibrium level Q This level of production will be achieved when price is at the competitive level P Total consumer surplus is represented by the light shaded area in the figure and total producer surplus is noted by the darker shaded area Clearly for output levels less than Q say Q1 total surplus would be reduced One sign of this misallocation is that at Q1 demanders would value an additional unit of output at P1 whereas average and marginal costs would be given by P2 Because P1 P2 total welfare would clearly increase by producing one more unit of output A transaction that involved trading this extra unit at any price between P1 and P2 would be mutually beneficial Both parties would gain The total welfare loss that occurs at output level Q1 is given by area FEG The distribu tion of surplus at output level Q1 will depend on the precise nonequilibrium price that prevails in the market At a price of P1 consumer surplus would be reduced substantially to area AFP1 whereas producers might gain because producer surplus is now P1 FGB At a low price such as P2 the situation would be reversed with producers being much worse off than they were initially Hence the distribution of the welfare losses from producing less than Q will depend on the price at which transactions are conducted However the size of the total loss is given by FEG regardless of the price settled upon13 13Increases in output beyond Q also clearly reduce welfare Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 432 Part 5 Competitive Markets 12132 A mathematical proof Mathematically welfare maximization consists of the problem of how to choose Q to maximize consumer surplus 1 producer surplus 5 3U1Q2 2 PQ4 1 cPQ 2 3 Q 0 P 1Q2dQd 5 U1Q2 2 3 Q 0 P 1Q2dQ 1259 where UQ is the utility function of the representative consumer and PQ is the longrun supply relation In longrun equilibria along the longrun supply curve P 1Q2 5 AC 5 MC Maximization of Equation 1259 with respect to Q yields Ur 1Q2 5 P 1Q2 5 AC 5 MC 1260 so maximization occurs where the marginal value of Q to the representative consumer is equal to market price But this is precisely the competitive supplydemand equilibrium because the demand curve represents consumers marginal valuations whereas the supply curve reflects marginal and in longterm equilibrium average cost 12133 Applied welfare analysis The conclusion that the competitive equilibrium maximizes the sum of consumer and producer surplus mirrors a series of more general economic efficiency theorems we will At the competitive equilibrium Q the sum of consumer surplus shaded lighter gray and producer surplus shaded darker is maximized For an output level Q1 Q there is a deadweight loss of con sumer and producer surplus that is given by area FEG Price Quantity per period 0 A B P1 Q1 P2 P Q F E G S D FIGURE 1212 Competitive Equilibrium and ConsumerProducer Surplus Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 433 examine in Chapter 13 Describing the major caveats that attach to these theorems is best delayed until that more extended discussion Here we are more interested in showing how the competitive model is used to examine the consequences of changing economic condi tions on the welfare of market participants Usually such welfare changes are measured by looking at changes in consumer and producer surplus In the final sections of this chapter we look at two examples EXAMPLE 126 Welfare Loss Computations Use of consumer and producer surplus notions makes possible the explicit calculation of wel fare losses from restrictions on voluntary transactions In the case of linear demand and supply curves this computation is especially simple because the areas of loss are frequently triangular For example if demand is given by Q D 5 D 1P2 5 10 2 P 1261 and supply by QS 5 S 1P2 5 P 2 2 1262 then market equilibrium occurs at the point P 5 6 Q 5 4 Restriction of output to Q 5 3 would create a gap between what demanders are willing to pay 1PD 5 10 2 Q 5 72 and what suppliers require 1PS 5 2 1 Q 5 52 The welfare loss from restricting transactions is given by a triangle with a base of 2 15 PD 2 PS 5 7 2 52 and a height of 1 the difference between Q and Q Hence the welfare loss is 1 if P is measured in dollars per unit and Q is measured in units More generally the loss will be measured in the units in which P Q is measured Computations with constant elasticity curves More realistic results can usually be obtained by using constant elasticity demand and supply curves based on econometric studies In Example 123 we examined such a model of the US automobile market We can simplify that example a bit by assuming that P is measured in thousands of dollars and Q in millions of auto mobiles and that demand is given by QD 5 D 1P2 5 200P12 1263 and supply by QS 5 S 1P2 5 13P 1264 Equilibrium in the market is given by P 5 987 Q 5 128 Suppose now that government pol icy restricts automobile sales to 11 million to control emissions of pollutants An approxima tion to the direct welfare loss from such a policy can be found by the triangular method used earlier With Q 5 11 we have PD 5 1112002 083 5 111 and PS 5 1113 5 846 Hence the wel fare loss triangle is given by 05 1PD 2 PS2 1Q 2 Q 2 5 05 1111 2 8462 1128 2 112 5 238 Here the units are those of P times Q billions of dollars Therefore the approximate14 value of the welfare loss is 24 billion which might be weighed against the expected gain from emissions control Distribution of loss In the automobile case the welfare loss is shared about equally by consumers and producers An approximation for consumers losses is given by 05 1PD 2 P2 1Q 2 Q 2 5 05 1111 2 9872 1128 2 112 5 111 and for producers by 05 1987 2 8462 1128 2 112 5 127 Because the price elasticity of demand is somewhat greater in absolute value than the price elasticity of supply consumers incur less than half the 14A more precise estimate of the loss can be obtained by integrating PD 2 PS over the range Q 5 11 to Q 5 128 With exponential demand and supply curves this integration is often easy In the present case the technique yields an estimated welfare loss of 228 showing that the triangular approximation is not too bad even for relatively large price changes Hence we will primarily use such approximations in later analysis Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 434 Part 5 Competitive Markets loss and producers somewhat more than half With a more price elastic demand curve consum ers would incur a smaller share of the loss QUERY How does the size of the total welfare loss from a quantity restriction depend on the elasticities of supply and demand What determines how the loss will be shared 1214 PRICE CONTROLS AND SHORTAGES Sometimes governments may seek to control prices at below equilibrium levels Although adoption of such policies may be based on noble motives the controls deter longrun supply responses and create welfare losses for both consumers and producers A simple analysis of this possibility is provided by Figure 1213 Initially the market is in longrun equilibrium at P1 Q1 point E An increase in demand from D to Dr would cause the price to rise to P2 in the short run and encourage entry by new firms Assuming this market is characterized by increasing costs as reflected by the positively sloped longrun supply curve LS price would decrease somewhat as a result of this entry ultimately settling at P3 If these price changes were regarded as undesirable then the government could in prin ciple prevent them by imposing a legally enforceable ceiling price of P1 This would cause firms to continue to supply their previous output Q1 but because at P1 demanders now want to purchase Q4 there will be a shortage given by Q4 2 Q1 A shift in demand from D to Dr would increase price to P2 in the short run Entry over the long run would yield a final equilibrium of P3 Q3 Controlling the price at P1 would prevent these actions and yield a shortage of Q4 2 Q1 Relative to the uncontrolled situation the price control yields a transfer from producers to consumers area P3CEP1 and a deadweight loss of forgone transactions given by the two areas AErC and CErE FIGURE 1213 Price Controls and Shortages Price Quantity per period P 1 P 2 P 3 Q 1 Q 4 Q 3 E D A SS C E D LS Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 435 12141 Welfare evaluation The welfare consequences of this pricecontrol policy can be evaluated by comparing con sumer and producer surplus measures prevailing under this policy with those that would have prevailed in the absence of controls First the buyers of Q1 gain the consumer surplus given by area P3CEP1 because they can buy this good at a lower price than would exist in an uncontrolled market This gain reflects a pure transfer from producers out of the amount of producer surplus that would exist without controls What current consumers have gained from the lower price producers have lost Although this transfer does not represent a loss of overall welfare it does clearly affect the relative wellbeing of the market participants Second the area AErC represents the value of additional consumer surplus that would have been attained without controls Similarly the area CErE reflects additional producer surplus available in the uncontrolled situation Together these two areas ie area AErE represent the total value of mutually beneficial transactions that are prevented by the government policy of controlling price This is therefore a measure of the pure welfare costs of that policy 12142 Disequilibrium behavior The welfare analysis depicted in Figure 1213 also suggests some of the types of behavior that might be expected as a result of the pricecontrol policy Assuming that observed mar ket outcomes are generated by Q 1P12 5 min3QD 1P12 QS 1P12 4 1265 suppliers will be content with this outcome but demanders will not because they will be forced to accept a situation of excess demand They have an incentive to signal their dissatis faction to suppliers through increasing price offers Such offers may not only tempt existing suppliers to make illegal transactions at higher than allowed prices but may also encourage new entrants to make such transactions It is this kind of activity that leads to the prevalence of black markets in most instances of price control Modeling the resulting transactions is dif ficult for two reasons First these may involve nonpricetaking behavior because the price of each transaction must be individually negotiated rather than set by the market Second nonequilibrium transactions will often involve imperfect information Any pair of market participants will usually not know what other transactors are doing although such actions may affect their welfare by changing the options available Some progress has been made in modeling such disequilibrium behavior using game theory techniques see Chapter 18 However other than the obvious prediction that transactions will occur at prices above the price ceiling no general results have been obtained The types of blackmarket transactions undertaken will depend on the specific institutional details of the situation 1215 TAX INCIDENCE ANALYSIS The partial equilibrium model of competitive markets has also been widely used to study the impact of taxes Although as we will point out these applications are necessarily lim ited by their inability to analyze tax effects that spread through many markets they do provide important insights on a number of issues 12151 A comparative statics model of tax incidence The comparative statics methods we used earlier in this chapter to examine the effects of shifts in supply and demand curves can also be used to examine the issue of tax Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 436 Part 5 Competitive Markets incidencethat is to look at who actually pays a tax To do so we will look at a simple perunit tax of amount t imposed on a good produced in a competitive industry Imposi tion of such a tax creates a wedge between what demanders pay for the good which we will denote by P and what suppliers receive P 2 t As before we let Q represent equi librium output in this market and this equilibrium requires that D 1P2 2 Q 5 0 S1P 2 t2 2 Q 5 0 1266 Differentiation of these equilibrium conditions with respect to t yields DP dP dt 2 dQ dt 5 0 SP dP dt 2 SP 2 dQ dt 5 0 or SP dP dt 2 dQ dt 5 SP 1267 Writing these two equations in matrix notation cDP 21 SP 21d D dP dt dQ dt T 5 c0 SP d 1268 allows us to use Cramers rule to solve for the desired derivatives dP dt 5 0 21 SP 21 DP 21 SP 21 5 SP SP 2 DP dQ dt 5 DP 0 SP SP DP 21 SP 21 5 DP SP SP 2 DP 1269 As before we can make more sense out of these results by stating them in elasticity terms dP dt 5 SP SP 2 DP PQ PQ 5 eS P eS P 2 eD P dQ dt 5 DP dP dt PQ PQ 5 eD P eS P eS P 2 eD P Q P 1270 Here we explore the consequences of the first of these derivatives but will take up the second shortly Because both the denominator and numerator of dPdt are positive imposition of the tax will likely increase the price paid by demanders Only in the case where eS P 5 0 would this price not rise If supply were perfectly inelastic suppliers would incur all of the tax in terms of a reduced price for its output When eS P 0 demand ers will incur at least part of the tax In the extreme case of completely inelastic demand 1eD P 5 02 the first of Equations 1270 would show that dPdt 5 1that is the entire tax is paid by demanders of the product To study intermediate cases we can compare the Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 437 change in price paid by demanders 1dPdt2 to the change in price received by suppliers 1d1P 2 t2dt 5 dPdt 2 12 dPdt dPdt 2 1 5 eS P 1eS P 2 eD P2 eS P 1eS P 2 eD P2 2 1eS P 2 eD P2 1eS P 2 eD P2 5 eS P eD P 1271 This ratio is negative because demanders experience a price increase and suppliers expe rience a decrease But the ratio shows that the extent of these price changes will be in an inverse relationship to the elasticities involved If the elasticity of supply is greater than the absolute value of the elasticity of demand demanders will pay a greater portion of the tax Alternatively if the absolute value of the elasticity of demand exceeds the elasticity of sup ply suppliers will pay the greater portion of the tax One way to remember this is to view price elasticity as reflecting the capacity of economic actors to escape a tax The actors with the greater elasticity will be more able to escape 12152 A welfare analysis A simplified welfare analysis of the tax incidence issue looks only at the single market in which the tax is imposed It therefore avoids the general equilibrium effects of such a tax which may spread through many markets In this simplified view imposition of the unit tax t creates a vertical wedge between the supply and demand curves Suppliers now receive a lower price for their output and demanders pay more Total quantity traded in the market place declines As a result of the tax demanders incur a loss of consumer surplus of which a portion is transferred to the government as part of total tax revenues Producers also incur a loss of producer surplus and again a portion of this is transferred to the gov ernment in the form of tax revenue In general the reduction in combined consumer and producer surplus exceeds total tax revenues collected This represents a deadweight loss that arises because some mutually beneficial transactions are discouraged by the tax In public finance this deadweight loss is referred to the excess burden of the tax It reflects a loss in consumer and producer surplus that is not garnered in tax revenues Hence payers of the tax may suffer a loss in welfare even if the tax finances goods or services that are as beneficial to them as what could have been purchased with the tax revenues In general the sizes of all these effects will be related to the price elasticities involved To determine the final incidence of the producers share of the tax would also require an explicit analysis of input marketsthe burden of the tax would be reflected in reduced rents for those inputs characterized by relatively inelastic supply More generally a com plete analysis of the incidence question requires a full general equilibrium model that can treat many markets simultaneously We will discuss such models in the next chapter 12153 Deadweight loss and elasticity All nonlumpsum taxes involve deadweight losses because they alter the behavior of eco nomic actors The size of such losses will depend in a rather complex ways on the elas ticities of demand and supply in many markets In a single market model the size of the deadweight loss from a tax can be approximated by the area of a triangle whose base is given by the size of the tax t and whose height is given by the reduction in quantity brought about by the tax Hence the deadweight loss from such a tax is given by DW 5 205t dQ dt t 5 205t2 dQ dt 1272 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 438 Part 5 Competitive Markets Here the negative sign is needed because dQdt 0 and we wish our deadweight loss fig ure to be positive Now we can use the results of the previous section to phrase this dead weight loss in elasticity terms DW 5 205t2 dQ dt 5 205t2 eD P eS P eS P 2 eD P Q P 5 205a t Pb 2 eD P eS P eS P 2 eD P P Q 1273 This rather complicated equation shows that the deadweight loss from a tax is proportional to total spending on the good This proportion rises with the square of the tax as a pro portion of the products price One implication of this is that at the margin excess burden of a tax increases as the tax rate increases Excess burden also depends on the elasticities of supply and demand for this product The lower are these elasticities the smaller is the excess burden Indeed if either eD P 5 0 or eS P 5 0 there is no excess burdenbecause the tax does not affect the quantity transacted These observations suggest that supply and demand elasticities might play an important role in developing a tax system that sought to keep excess burden to a minimum For an illustration see Problem 1211 12154 Transaction costs Although we have developed this discussion in terms of tax incidence theory models incorporating a wedge between buyers and sellers prices have a number of other appli cations in economics Perhaps the most important of these involve costs associated with making market transactions In some cases these costs may be explicit Most real estate transactions for example take place through a thirdparty broker who charges a fee for the service of bringing buyer and seller together Similar explicit transaction fees occur in the trading of stocks and bonds boats and airplanes and practically everything that is sold at auction In all these instances buyers and sellers are willing to pay an explicit fee to an agent or broker who facilitates the transaction In other cases transaction costs may be largely implicit Individuals trying to purchase a used car for example will spend consid erable time and effort reading classified advertisements and examining vehicles and these activities amount to an implicit cost of making the transaction EXAMPLE 127 The Excess Burden of a Tax In Example 126 we examined the loss of consumer and producer surplus that would occur if automobile sales were cut from their equilibrium level of 128 million to 11 million An auto tax of 2640 ie 264 thousand dollars would accomplish this reduction because it would intro duce exactly the wedge between demand and supply price that was calculated previously Because we have assumed eD P 5 212 and eS P 5 10 in Example 126 and because initial spending on automobiles is approximately 126 billion Equation 1273 predicts that the excess burden from the auto tax would be DW 5 05a264 987b 2 a12 22b126 5 246 1274 This loss of 246 billion dollars is approximately the same as the loss from emissions control cal culated in Example 126 It might be contrasted to total tax collections which in this case amount to 29 billion 2640 per automobile times 11 million automobiles in the posttax equilibrium Here the deadweight loss equals approximately 8 percent of total tax revenues collected Marginal burden An incremental increase in the auto tax would be relatively more costly in terms of excess burden Suppose the government decided to round the auto tax upward to Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 439 a flat 3000 per car In this case car sales would drop to approximately 107 million Tax collections would amount to 321 billion an increase of 31 billion over what was computed previously Equation 1273 can be used to show that deadweight losses now amount to 317 billionan increase of 071 billion above the losses experienced with the lower tax At the margin additional deadweight losses amount to approximately 23 percent 07231 of addi tional revenues collected Hence marginal and average excess burden computations may differ significantly QUERY Can you explain intuitively why the marginal burden of a tax exceeds its average bur den Under what conditions would the marginal excess burden of a tax exceed additional tax revenues collected To the extent that transaction costs are on a perunit basis as they are in the real estate securities and auction examples our previous taxation example applies exactly From the point of view of the buyers and sellers it makes little difference whether t represents a perunit tax or a perunit transaction fee because the analysis of the fees effect on the mar ket will be the same That is the fee will be shared between buyers and sellers depending on the specific elasticities involved Trading volume will be lower than in the absence of such fees15 A somewhat different analysis would hold however if transaction costs were a lumpsum amount per transaction In that case individuals would seek to reduce the number of transactions made but the existence of the charge would not affect the supply demand equilibrium itself For example the cost of driving to the supermarket is mainly a lumpsum transaction cost on shopping for groceries The existence of such a charge may not significantly affect the price of food items or the amount of food consumed unless it tempts people to grow their own but the charge will cause individuals to shop less fre quently to buy larger quantities on each trip and to hold larger inventories of food in their homes than would be the case in the absence of such a cost 12155 Effects on the attributes of transactions More generally taxes or transaction costs may affect some attributes of transactions more than others In our formal model we assumed that such costs were based only on the phys ical quantity of goods sold Therefore the desire of suppliers and demanders to minimize costs led them to reduce quantity traded When transactions involve several dimensions such as quality risk or timing taxes or transaction costs may affect some or all of these dimensionsdepending on the precise basis on which the costs are assessed For exam ple a tax on quantity may cause firms to upgrade product quality or informationbased transaction costs may encourage firms to produce less risky standardized commodities Similarly a pertransaction cost travel costs of getting to the store may cause individuals to make fewer but larger transactions and to hold larger inventories The possibilities for these various substitutions will obviously depend on the particular circumstances of the transaction We will examine several examples of costinduced changes in attributes of transactions in later chapters16 15This analysis does not consider possible benefits obtained from brokers To the extent that these services are valuable to the parties in the transaction demand and supply curves will shift outward to reflect this value Hence trading volume may expand with the availability of services that facilitate transactions although the costs of such services will continue to create a wedge between sellers and buyers prices 16For the classic treatment of this topic see Y Barzel An Alternative Approach to the Analysis of Taxation Journal of Political Economy December 1976 117797 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 440 Part 5 Competitive Markets Summary In this chapter we developed a detailed model of how the equilibrium price is determined in a single competitive mar ket This model is basically the one first fully articulated by Alfred Marshall in the latter part of the nineteenth century It remains the single most important component of all of micro economics Some of the properties of this model we examined may be listed as follows Shortrun equilibrium prices are determined by the interaction of what demanders are willing to pay demand and what existing firms are willing to produce supply Both demanders and suppliers act as pricetak ers in making their respective decisions In the long run the number of firms may vary in response to profit opportunities If free entry is assumed then firms will earn zero economic profits over the long run Therefore because firms also maximize profits the longrun equilibrium condition is P 5 MC 5 AC The shape of the longrun supply curve depends on how the entry of new firms affects input prices If entry has no impact on input prices the longrun supply curve will be horizontal infinitely elastic If entry increases input prices the longrun supply curve will have a positive slope If shifts in longrun equilibrium affect input prices this will also affect the welfare of input suppliers Such wel fare changes can be measured by changes in longrun producer surplus The twin concepts of consumer and producer surplus provide useful ways of measuring the welfare impact on market participants of various economic changes Changes in consumer surplus represent the monetary value of changes in consumer utility Changes in pro ducer surplus represent changes in the monetary returns that inputs receive The competitive model can be used to study the impact of various economic policies For example it can be used to illustrate the transfers and welfare losses associated with price controls The competitive model can also be applied to study tax ation The model illustrates both tax incidence ie who bears the actual burden of a tax and the welfare losses associated with taxation the excess burden Similar conclusions can be derived by using the competitive model to study transaction costs Problems 121 Suppose there are 100 identical firms in a perfectly competi tive industry Each firm has a shortrun total cost function of the form C1q2 5 1 300 q3 1 02q2 1 4q 1 10 a Calculate the firms shortrun supply curve with q as a function of market price P b On the assumption that firms output decisions do not affect their costs calculate the shortrun industry supply curve c Suppose market demand is given by Q 5 200P 1 8000 What will be the shortrun equilibrium pricequantity combination 122 Suppose there are 1000 identical firms producing diamonds Let the total cost function for each firm be given by C1q w2 5 q2 1 wq where q is the firms output level and w is the wage rate of dia mond cutters a If w 5 10 what will be the firms shortrun supply curve What is the industrys supply curve How many diamonds will be produced at a price of 20 each How many more diamonds would be produced at a price of 21 b Suppose the wages of diamond cutters depend on the total quantity of diamonds produced and suppose the form of this relationship is given by w 5 0002Q here Q represents total industry output which is 1000 times the output of the typical firm In this situation show that the firms marginal cost and shortrun supply curve depends on Q What is the industry supply curve How much will be produced at a price of 20 How much more will be produced at a price of 21 What do you conclude about the shape of the shortrun supply curve 123 Suppose that the demand function for a good has the linear form Q 5 D 1P I2 5 a 1 bP 1 cI and the supply function is also of the linear form Q 5 S 1P2 5 d 1 gP Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 441 a Calculate equilibrium price and quantity for this mar ket as a function of the parameters a b c d and g and of I income the exogenous shift term for the demand function b Use your results from part a to calculate the compara tive statics derivative dPdI c Now calculate the same derivative using the comparative statics analysis of supply and demand presented in this chapter You should be able to show that you get the same results in each case d Specify some assumed values for the various parameters of this problem and describe why the derivative dPdI takes the form it does here 124 A perfectly competitive industry has a large number of poten tial entrants Each firm has an identical cost structure such that longrun average cost is minimized at an output of 20 units 1qi 5 202 The minimum average cost is 10 per unit Total market demand is given by Q 5 D 1P2 5 1500 2 50P a What is the industrys longrun supply schedule b What is the longrun equilibrium price 1P2 The total industry output 1Q2 The output of each firm 1q2 The number of firms The profits of each firm c The shortrun total cost function associated with each firms longrun equilibrium output is given by C1q2 5 05q2 2 10q 1 200 Calculate the shortrun average and marginal cost func tion At what output level does shortrun average cost reach a minimum d Calculate the shortrun supply function for each firm and the industry shortrun supply function e Suppose now that the market demand function shifts upward to Q 5 D 1P2 5 2000 2 50P Using this new demand curve answer part b for the very short run when firms cannot change their outputs f In the short run use the industry shortrun supply func tion to recalculate the answers to b g What is the new longrun equilibrium for the industry 125 Suppose that the demand for stilts is given by Q 5 D 1P2 5 1500 2 50P and that the longrun total operating costs of each stilt making firm in a competitive industry are given by C1q2 5 05q2 2 10q Entrepreneurial talent for stilt making is scarce The supply curve for entrepreneurs is given by QS 5 025w where w is the annual wage paid Suppose also that each stiltmaking firm requires one and only one entrepreneur hence the quantity of entrepreneurs hired is equal to the number of firms Longrun total costs for each firm are then given by C1q w2 5 05q2 2 10q 1 w a What is the longrun equilibrium quantity of stilts pro duced How many stilts are produced by each firm What is the longrun equilibrium price of stilts How many firms will there be How many entrepreneurs will be hired and what is their wage b Suppose that the demand for stilts shifts outward to Q 5 D 1P2 5 2428 2 50P How would you now answer the questions posed in part a c Because stiltmaking entrepreneurs are the cause of the upwardsloping longrun supply curve in this problem they will receive all rents generated as industry output expands Calculate the increase in rents between parts a and b Show that this value is identical to the change in longrun producer surplus as measured along the stilt supply curve 126 The handmade snuffbox industry is composed of 100 identi cal firms each having shortrun total costs given by STC 5 05q2 1 10q 1 5 and shortrun marginal costs given by SMC 5 q 1 10 where q is the output of snuffboxes per day a What is the shortrun supply curve for each snuffbox maker What is the shortrun supply curve for the mar ket as a whole b Suppose the demand for total snuffbox production is given by Q 5 D 1P2 5 1100 2 50P What will be the equilibrium in this marketplace What will each firms total shortrun profits be c Graph the market equilibrium and compute total short run producer surplus in this case d Show that the total producer surplus you calculated in part c is equal to total industry profits plus industry shortrun fixed costs Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 442 Part 5 Competitive Markets e Suppose the government imposed a 3 tax on snuffboxes How would this tax change the market equilibrium f How would the burden of this tax be shared between snuffbox buyers and sellers g Calculate the total loss of producer surplus as a result of the taxation of snuffboxes Show that this loss equals the change in total shortrun profits in the snuffbox indus try Why do fixed costs not enter into this computation of the change in shortrun producer surplus 127 The perfectly competitive videotapecopying industry is com posed of many firms that can copy five tapes per day at an average cost of 10 per tape Each firm must also pay a royalty to film studios and the perfilm royalty rate r is an increas ing function of total industry output Q r 5 0002Q Demand is given by Q 5 D 1P2 5 1050 2 50P a Assuming the industry is in longrun equilibrium what will be the equilibrium price and quantity of copied tapes How many tape firms will there be What will the perfilm royalty rate be b Suppose that demand for copied tapes increases to Q 5 D 1P2 5 1600 2 50P In this case what is the longrun equilibrium price and quantity for copied tapes How many tape firms are there What is the perfilm royalty rate c Graph these longrun equilibria in the tape market and calculate the increase in producer surplus between the situations described in parts a and b d Show that the increase in producer surplus is precisely equal to the increase in royalties paid as Q expands incrementally from its level in part b to its level in part c e Suppose that the government institutes a 550 perfilm tax on the filmcopying industry Assuming that the demand for copied films is that given in part a how will this tax affect the market equilibrium f How will the burden of this tax be allocated between consumers and producers What will be the loss of con sumer and producer surplus g Show that the loss of producer surplus as a result of this tax is borne completely by the film studios Explain your result intuitively 128 The domestic demand for portable radios is given by Q 5 D 1P2 5 5000 2 100P where price P is measured in dollars and quantity Q is measured in thousands of radios per year The domestic sup ply curve for radios is given by Q 5 S 1P2 5 150P a What is the domestic equilibrium in the portable radio market b Suppose portable radios can be imported at a world price of 10 per radio If trade were unencumbered what would the new market equilibrium be How many porta ble radios would be imported c If domestic portable radio producers succeeded in hav ing a 5 tariff implemented how would this change the market equilibrium How much would be collected in tariff revenues How much consumer surplus would be transferred to domestic producers What would the deadweight loss from the tariff be d How would your results from part c be changed if the government reached an agreement with foreign suppli ers to voluntarily limit the portable radios they export to 1250000 per year Explain how this differs from the case of a tariff 129 Suppose that the market demand for a product is given by QD 5 D 1P2 5 A 2 BP Suppose also that the typical firms cost function is given by C1q2 5 k 1 aq 1 bq2 a Compute the longrun equilibrium output and price for the typical firm in this market b Calculate the equilibrium number of firms in this market as a function of all the parameters in this problem c Describe how changes in the demand parameters A and B affect the equilibrium number of firms in this market Explain your results intuitively d Describe how the parameters of the typical firms cost function affect the longrun equilibrium number of firms in this example Explain your results intuitively Analytical Problems 1210 Ad valorem taxes Throughout this chapter our analysis of taxes has assumed that they are imposed on a perunit basis Many taxes such as sales taxes are proportional based on the price of the item In this problem you are asked to show that assuming the tax rate is reasonably small the market consequences of such a tax are quite similar to those already analyzed To do so we now assume that the price received by suppliers is given by P and the price paid by demanders is P 11 1 t2 where t is the ad valorem tax rate ie with a tax rate of 5 percent t 5 005 the Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 443 price paid by demanders is 105P In this problem then the supply function is given by Q 5 S 1P2 and the demand func tion by Q 5 D 3 11 1 t2P4 a Show that for such a tax d ln P dt 5 eD P eS P 2 eD P Hint Remember that d ln Pdt 5 1 P dP dt and that here we are assuming t 0 b Show that the excess burden of such a small ad valorem tax is given by DW 5 05 eD P eS P eS P 2 eD P t2P Q c Compare these results to those derived in this chapter for a perunit tax Can you make any statements about which tax would be superior in various circumstances 1211 The Ramsey formula for optimal taxation The development of optimal tax policy has been a major topic in public finance for centuries17 Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey who conceptualized the problem as how to structure a tax system that would collect a given amount of revenues with the minimal deadweight loss18 Spe cifically suppose there are n goods xi with prices pi to be taxed with a sequence of ad valorem taxes see Problem 1210 whose rates are given by ti 1i 5 1 n2 Therefore total tax rev enue is given by T 5 g n i51ti pi xi Ramseys problem is for a fixed T to choose tax rates that will minimize total deadweight loss DW 5 g n i51DW1ti2 a Use the Lagrange multiplier method to show that the solu tion to Ramseys problem requires ti 5 λ11eS 2 1eD2 where λ is the Lagrange multiplier for the tax constraint b Interpret the Ramsey result intuitively c Describe some shortcomings of the Ramsey approach to optimal taxation 1212 Cobweb models One way to generate disequilibrium prices in a simple model of supply and demand is to incorporate a lag into produc ers supply response To examine this possibility assume that quantity demanded in period t depends on price in that period 1QD t 5 a 2 bPt2 but that quantity supplied depends 17The seventeenthcentury French finance minister JeanBaptiste Colbert captured the essence of the problem with his memorable statement that the art of taxation consists in so plucking the goose as to obtain the largest possible amount of feathers with the smallest amount of hissing 18See F Ramsey A Contribution to the Theory of Taxation Economic Journal March 1927 4761 on the previous periods priceperhaps because farmers refer to that price in planting a crop 1QS t 5 c 1 dPt212 a What is the equilibrium price in this model 1P 5 Pt 5 Pt212 for all periods t b If P0 represents an initial price for this good to which suppliers respond what will the value of P1 be c By repeated substitution develop a formula for any arbi trary Pt as a function of P0 and t d Use your results from part a to restate the value of Pt as a function of P0 P and t e Under what conditions will Pt converge to P as t S q f Graph your results for the case a 5 4 b 5 2 c 5 1 d 5 1 and P0 5 0 Use your graph to discuss the origin of the term cobweb model 1213 More on the comparative statics of supply and demand The supply and demand model presented earlier in this chap ter can be used to look at many other comparative statics questions In this problem you are asked to explore three of them In all of these quantity demanded is given by D 1P α2 and quantity supplied by S 1P β2 a Shifts in supply In Chapter 12 we analyzed the case of a shift in demand by looking at a comparative statics analysis of how changes in α affect equilibrium price and quantity For this problem you are to make a similar set of computations for a shift in a parameter of the supply function β That is calculate dPdβ and dQdβ Be sure to calculate your results in both derivative and elas ticity terms Also describe with some simple graphs why the results here differ from those shown in the body of Chapter 12 b A quantity wedge In our analysis of the imposition of a unit tax we showed how such a tax wedge can affect equilibrium price and quantity A similar analysis can be done for a quantity wedge for which in equilibrium the quantity supplied may exceed the quantity demanded Such a situation might arise for example if some portion of production were lost through spoilage or if some por tion of production were demanded by the government as a payment for the right to do business Formally let Q be the amount of the good lost In this case equilibrium requires D 1P2 5 Q and S 1P2 5 Q 1 Q Use the com parative statics methods developed in this chapter to cal culate dPdQ and dQdQ In many cases it might be more reasonable to assume Q 5 δQ where δ is a small decimal value Without making any explicit calcula tions how do you think this case would differ from the one you explicitly analyzed c The identification problem An important issue in the empirical study of competitive markets is to decide whether observed pricequantity data points represent Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 444 Part 5 Competitive Markets demand curves supply curves or some combination of the two Explain the following conclusions using the comparative statics results we have obtained i If only the demand parameter α takes on chang ing values data on changing equilibrium values of price and quantity can be used to estimate the price elasticity of supply ii If only the supply parameter β takes on chang ing values data on changing equilibrium values of price and quantity can be used to evaluate the price elasticity of supply to answer this you must have done part a of this problem iii If demand and supply curves are both only shifted by the same parameter ie the demand and sup ply functions are D 1P α2 and S 1P α2 neither of the price elasticities can be evaluated 1214 The Le Chatelier principle Our analysis of supply response in this chapter focused on the fact that firms have greater flexibility in the long run both in their hiring of inputs and in their entry decisions For this rea son price increases resulting from an increase in demand may be large in the short run but price will tend to return toward its initial equilibrium value over the longer term Paul Samu elson noted that this tendency resembled a similar principle in chemistry in which an initial disturbance to an equilibrium tends to be moderated over the longer term He therefore introduced the term used in chemistry the Le Chatelier prin ciple to economics To examine this principle we now write the supply function as S 1P t2 where t represents time and our discussion in this chapter shows why SP t 0that is the effect of a price increase on quantity supplied becomes greater over time a Using this new supply function differentiate Equations 1224 with respect to t This results in two equations in the two secondorder crossderivatives d 2P dαdt and d 2Q dαdt These derivatives show how equilibrium price and quan tity react to a given shift in demand over time b Solve these two equations for the secondorder cross partial derivatives identified in the previous part Show that d 2P dαdt has the opposite sign from dP dα This is the Le Chatelier resultthe initial change in equilibrium price is moderated over time c Show that d 2Q dαdt has the same sign as dP dα This is a situation therefore in which the Le Chatelier moderat ing result is not reflected in all of the equilibrium values of all outcomes d Describe how your mathematical results mirror the graphical analysis presented in this chapter SUGGESTIONS FOR FURTHER READING Arnott R Time for Revision on Rent Control Journal of Economic Perspectives Winter 1995 99120 Provides an assessment of actual soft rentcontrol policies and provides a rationale for them Knight F H Risk Uncertainty and Profit Boston Houghton Mifflin 1921 chaps 5 and 6 Classic treatment of the role of economic events in motivating industry behavior in the long run Marshall A Principles of Economics 8th ed New York Crowell Collier and Macmillan 1920 book 5 chaps 1 2 and 3 Classic development of the supplydemand mechanism MasColell A M D Whinston and J R Green Micro economic Theory New York Oxford University Press 1995 chap 10 Provides a compact analysis at a high level of theoretical precision There is a good discussion of situations where competitive markets may not reach an equilibrium Reynolds L G CutThroat Competition American Eco nomic Review 30 December 1940 73647 Critique of the notion that there can be too much competition in an industry Robinson J What Is Perfect Competition Quarterly Jour nal of Economics 49 1934 10420 Critical discussion of the perfectly competitive assumptions Salanie B The Economics of Taxation Cambridge MA MIT Press 2003 This provides a compact study of many issues in taxation Describes a few simple models of incidence and develops some general equilibrium models of taxation Stigler G J Perfect Competition Historically Contem plated Journal of Political Economy 65 1957 117 Fascinating discussion of the historical development of the com petitive model Varian H R Microeconomic Analysis 3rd ed New York W W Norton 1992 chap 13 Terse but instructive coverage of many of the topics in this chapter The importance of entry is stressed although the precise nature of the longrun supply curve is a bit obscure Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 445 In Chapters 46 we showed that the assumption of util ity maximization implies several properties for individual demand functions the functions are continuous the functions are homogeneous of degree 0 in all prices and income incomecompensated substitution effects are negative and crossprice substitution effects are symmetric In this extension we will examine the extent to which these properties would be expected to hold for aggregated market demand functions and what if any restrictions should be placed on such functions In addition we illustrate some other issues that arise in estimating these aggregate functions and some results from such estimates E121 Continuity The continuity of individual demand functions clearly implies the continuity of market demand functions But there are situ ations in which market demand functions may be continuous whereas individual functions are not Consider the case where goodssuch as an automobilemust be bought in large dis crete units Here individual demand is discontinuous but the aggregated demands of many people are nearly continuous E122 Homogeneity and income aggregation Because each individuals demand function is homogeneous of degree 0 in all prices and income market demand functions are also homogeneous of degree 0 in all prices and individual incomes However market demand functions are not neces sarily homogeneous of degree 0 in all prices and total income To see when demand might depend just on total income suppose individual is demand for X is given by xi 5 ai 1P2 1 b1P2yi i 5 1 n i where P is the vector of all market prices ai 1P2 is a set of individualspecific price effects and bP is a marginal pro pensitytospend function that is the same across all individ uals although the value of this parameter may depend on market prices In this case the market demand functions will depend on P and on total income y 5 a n i51 yi ii This shows that market demand reflects the behavior of a single typical consumer Gorman 1959 shows that this is the most general form of demand function that can represent such a typical consumer E123 Crossequation constraints Suppose a typical individual buys k items and that expendi tures on each are given by pj xj 5 a k i51 aij pi 1 bj y j 5 1 k iii If expenditures on these k items exhaust total income that is a k j51 pj xj 5 y iv then summing over all goods shows that a k j51 aij 5 0 for all i v and that a k j51 bj 5 1 vi for each person This implies that researchers are generally not able to estimate expenditure functions for k goods inde pendently Rather some account must be taken of relation ships between the expenditure functions for different goods E124 Econometric practice The degree to which these theoretical concerns are reflected in the actual practices of econometricians varies widely At the least sophisticated level an equation similar to Equation iii might be estimated directly using ordinary least squares OLS with little attention to the ways in which the assump tions might be violated Various elasticities could be calcu lated directly from this equationalthough because of the EXTENSIONS DEMAND AGGREGATION AND ESTIMATION Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 446 Part 5 Competitive Markets linear form used these would not be constant for changes in pi or y A constant elasticity formulation of Equation iii would be ln 1 pj xj2 5 a k i51 aij ln 1 pi2 1 bj ln y j 5 1 k vii where price and income elasticities would be given directly by exj pj 5 aj j 2 1 exj pi 5 ai j 1i 2 j2 viii exj y 5 bj Notice here however that no specific attention is paid to biases introduced by the use of aggregate income or by the disregard of possible crossequation restrictions such as those in Equations v and vi Further restrictions are also implied by the homogeneity of each of the demand functions 1g k i51 aij 1 bj 5 12 although this restriction too is often disregarded in the development of simple econometric estimates More sophisticated studies of aggregated demand equa tions seek to remedy these problems by explicitly considering potential income distribution effects and by estimating entire systems of demand equations Theil 1971 1975 provides a good introduction to some of the procedures used Econometric results Table 123 reports a number of economic estimates of repre sentative price and income elasticities drawn from a variety of sources The original sources for these estimates should be consulted to determine the extent to which the authors have been attentive to the theoretical restrictions outlined previously Overall these estimates accord fairly well with TABLE 123 REPRESENTATIVE PRICE AND INCOME ELASTICITIES OF DEMAND Price Elasticity Income Elasticity Food 021 1028 Medical services 018 1022 Housing Rental 018 1100 Owner occupied 120 1120 Electricity 114 1061 Automobiles 120 1300 Gasoline 055 1160 Beer 026 1038 Wine 088 1097 Marijuana 150 000 Cigarettes 035 1050 Abortions 081 1079 Transatlantic air travel 130 1140 Imports 058 1273 Money 040 1100 Note Price elasticity refers to interest rate elasticity Sources Food H Wold and L Jureen Demand Analysis New York John Wiley Sons 1953 203 Medical services income elasticity from R Andersen and L Ben ham Factors Affecting the Relationship between Family Income and Medical Care Consumption in Herbert Klarman Ed Empirical Studies in Health Economics Baltimore Johns Hopkins University Press 1970 price elasticity from W C Manning et al Health Insurance and the Demand for Medical Care Evidence from a Randomized Experiment American Economic Review June 1987 25177 Housing income elasticities from F de Leeuw The Demand for Housing Review for Economics and Statistics February1971 price elasticities from H S Houthakker and L D Taylor Consumer Demand in the United States Cambridge MA Harvard University Press 1970 16667 Electricity R F Halvorsen Residential Demand for Electricity unpublished PhD dissertation Harvard University December 1972 Automobiles Gregory C Chow Demand for Automobiles in the United States Amsterdam North Holland 1957 Gasoline C Dahl Gasoline Demand Survey Energy Journal 7 1986 6782 Beer and wine J A Johnson E H Oksanen M R Veall and D Fritz ShortRun and LongRun Elasticities for Canadian Consump tion of Alcoholic Beverages Review of Economics and Statistics February 1992 6474 Marijuana T C Misket and F Vakil Some Estimate of Price and Expendi ture Elasticities among UCLA Students Review of Economics and Statistics November 1972 47475 Cigarettes F Chalemaker Rational Addictive Behavior and Cigarette Smoking Journal of Political Economy August 1991 72242 Abortions M H Medoff An Economic Analysis of the Demand for Abortions Economic Inquiry April 1988 25359 Transatlantic air travel J M Cigliano Price and Income Elasticities for Airline Travel Business Economics September 1980 1721 Imports M D Chinn Beware of Econometricians Bearing Estimates Journal of Policy Analysis and Management Fall 1991 54667 Money D L Hoffman and R H Rasche LongRun Income and Interest Elasticities of Money Demand in the United States Review of Economics and Statistics November 1991 66574 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 12 The Partial Equilibrium Competitive Model 447 intuitionthe demand for transatlantic air travel is more price elastic than is the demand for medical care for example Perhaps somewhat surprising are the high price and income elasticities for owneroccupied housing because shelter is often regarded in everyday discussion as a necessity The high estimated income elasticity of demand for automobiles prob ably conflates the measurement of both quantity and quality demanded But it does suggest why the automobile industry is so sensitive to the business cycle References Gorman W M Separable Utility and Aggregation Econo metrica November 1959 46981 Theil H Principles of Econometrics New York John Wiley Sons 1971 pp 32646 Theory and Measurement of Consumer Demand vol 1 Amsterdam North Holland 1975 chaps 5 and 6 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 449 CHAPTER THIRTEEN General Equilibrium and Welfare The partial equilibrium models of perfect competition that were introduced in Chapter 12 are clearly inadequate for describing all the effects that occur when changes in one market have repercussions in other markets Therefore they are also inadequate for making gen eral welfare statements about how well market economies perform Instead what is needed is an economic model that permits us to view many markets simultaneously In this chap ter we will develop a few simple versions of such models The Extensions to the chapter show how general equilibrium models are used in empirical applications to the real world 131 PERFECTLY COMPETITIVE PRICE SYSTEM The model we will develop in this chapter is primarily an elaboration of the supply demand mechanism presented in Chapter 12 Here we will assume that all markets are of the type described in that chapter and refer to such a set of markets as a perfectly com petitive price system The assumption is that there is some large number of homogeneous goods in this simple economy Included in this list of goods are not only consumption items but also factors of production Each of these goods has an equilibrium price estab lished by the action of supply and demand1 At this set of prices every market is cleared in the sense that suppliers are willing to supply the quantity that is demanded and consumers will demand the quantity that is supplied We also assume that there are no transaction or transportation charges and that both individuals and firms have perfect knowledge of prevailing market prices 1311 The law of one price Because we assume zero transaction cost and perfect information each good obeys the law of one price A homogeneous good trades at the same price no matter who buys it or which firm sells it If one good traded at two different prices demanders would rush to buy the good where it was cheaper and firms would try to sell all their output where the good was more expensive These actions in themselves would tend to equalize the price of the good 1One aspect of this market interaction should be made clear from the outset The perfectly competitive market determines only relative not absolute prices In this chapter we speak only of relative prices It makes no difference whether the prices of apples and oranges are 010 and 020 respectively or 10 and 20 The important point in either case is that two apples can be exchanged for one orange in the market The absolute level of prices is determined mainly by monetary factorsa topic usually covered in macroeconomics Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 450 Part 5 Competitive Markets In the perfectly competitive market each good must have only one price This is why we may speak unambiguously of the price of a good 1312 Behavioral assumptions The perfectly competitive model assumes that people and firms react to prices in specific ways 1 There are assumed to be a large number of people buying any one good Each person takes all prices as given and adjusts his or her behavior to maximize utility given the prices and his or her budget constraint People may also be suppliers of productive ser vices eg labor and in such decisions they also regard prices as given2 2 There are assumed to be a large number of firms producing each good and each firm produces only a small share of the output of any one good In making input and output choices firms are assumed to operate to maximize profits The firms treat all prices as given when making these profitmaximizing decisions These various assumptions should be familiar because we have been making them throughout this book Our purpose here is to show how an entire economic system oper ates when all markets work in this way 132 A GRAPHICAL MODEL OF GENERAL EQUILIBRIUM WITH TWO GOODS We begin our analysis with a graphical model of general equilibrium involving only two goods which we will call x and y This model will prove useful because it incorporates many of the features of far more complex general equilibrium representations of the economy 1321 General equilibrium demand Ultimately demand patterns in an economy are determined by individuals preferences For our simple model we will assume that all individuals have identical preferences which can be represented by an indifference curve map3 defined over quantities of the two goods x and y The benefit of this approach for our purposes is that this indifference curve map which is identical to the ones used in Chapters 36 shows how individuals rank con sumption bundles containing both goods These rankings are precisely what we mean by demand in a general equilibrium context Of course we cannot illustrate which bundles of commodities will be chosen until we know the budget constraints that demanders face Because incomes are generated as individuals supply labor capital and other resources to the production process we must delay introducing budget constraints until we have exam ined the forces of production and supply in our model 2Hence unlike our partial equilibrium models incomes are endogenously determined in general equilibrium models 3There are some technical problems in using a single indifference curve map to represent the preferences of an entire community of individuals In this case the marginal rate of substitution ie the slope of the community indifference curve will depend on how the available goods are distributed among individuals The increase in total y required to compensate for a oneunit reduction in x will depend on which specific individuals the x is taken from Although we will not discuss this issue in detail here it has been widely examined in the international trade literature Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 451 1322 General equilibrium supply Developing a notion of general equilibrium supply in this twogood model is a somewhat more complex process than describing the demand side of the market because we have not thus far illustrated production and supply of two goods simultaneously Our approach is to use the familiar production possibility curve see Chapter 1 for this purpose By detailing the way in which this curve is constructed we can illustrate in a simple context the ways in which markets for outputs and inputs are related 1323 Edgeworth box diagram for production Construction of the production possibility curve for two outputs x and y begins with the assumption that there are fixed amounts of capital and labor inputs that must be allo cated to the production of the two goods The possible allocations of these inputs can be illustrated with an Edgeworth box diagram with dimensions given by the total amounts of capital and labor available In Figure 131 the length of the box represents total laborhours and the height of the box represents total capitalhours The lower left corner of the box represents the origin for measuring capital and labor devoted to production of good x The upper right corner of the box represents the origin for resources devoted to y Using these conventions any point in the box can be regarded as a fully employed allocation of the available resources between goods x and y Point A for example represents an allocation in which the indi cated number of labor hours are devoted to x production together with a specified number The dimensions of this diagram are given by the total quantities of labor and capital available Quantities of these resources devoted to x production are measured from origin Ox quantities devoted to y are mea sured from Oy Any point in the box represents a fully employed allocation of the available resources to the two goods A Ox Total labor Labor for x Labor in y production Labor for y Total capital Capital for y Capital in y production Labor in x production Capital in x production O y Capital for x FIGURE 131 Construction of an Edgeworth Box Diagram for Production Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 452 Part 5 Competitive Markets of hours of capital Production of good y uses whatever labor and capital are left over Point A in Figure 131 for example also shows the exact amount of labor and capital used in the production of good y Any other point in the box has a similar interpretation Thus the Edgeworth box shows every possible way the existing capital and labor might be used to produce x and y 1324 Efficient allocations Many of the allocations shown in Figure 131 are technically inefficient in that it is possi ble to produce both more x and more y by shifting capital and labor around a bit In our model we assume that competitive markets will not exhibit such inefficient input choices for reasons we will explore in more detail later in the chapter Hence we wish to discover the efficient allocations in Figure 131 because these illustrate the production outcomes in this model To do so we introduce isoquant maps for good x using Ox as the origin and good y using Oy as the origin as shown in Figure 132 In this figure it is clear that the arbitrarily chosen allocation A is inefficient By reallocating capital and labor one can pro duce both more x than x2 and more y than y2 The efficient allocations in Figure 132 are those such as P1 P2 P3 and P4 where the isoquants are tangent to one another At any other points in the box diagram the two goods isoquants will intersect and we can show inefficiency as we did for point A At the points of tangency however this kind of unambiguous improvement cannot be made In going from P2 to P3 for example more x is being produced but at the cost of less y being produced therefore P3 is not more efficient than P2both of the points are efficient Tangency of the isoquants for good x and good y implies that their slopes are equal That is This diagram adds production isoquants for x and y to Figure 131 It then shows technically efficient ways to allocate the fixed amounts of k and l between the production of the two outputs The line joining Ox and Oy is the locus of these efficient points Along this line the RTS of l for k in the production of good x is equal to the RTS in the production of y Total l P1 Total k Ox O y P2 P3 P4 x1 x2 x3 x4 y1 y2 y3 y4 A FIGURE 132 Edgeworth Box Diagram of Efficiency in Production Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 453 the RTS of capital for labor is equal in x and y production We will show this result mathe matically shortly As we will also show competitive input markets will help to bring about this equality The curve joining Ox and Oy that includes all these points of tangency shows all the efficient allocations of capital and labor Points off this curve are inefficient in that unam biguous increases in output can be obtained by reshuffling inputs between the two goods Points on the curve OxOy are all efficient allocations however because more x can be pro duced only by cutting back on y production and vice versa 1325 Production possibility frontier The efficiency locus in Figure 132 shows the maximum output of y that can be produced for any preassigned output of x We can use this information to construct a production pos sibility frontier which shows the alternative outputs of x and y that can be produced with the fixed capital and labor inputs In Figure 133 the OxOy locus has been taken from Figure 132 and transferred onto a graph with x and y outputs on the axes At Ox for example no resources are devoted to x production consequently y output is as large as is possible with the existing resources Similarly at Oy the output of x is as large as possible The other points on the production possibility frontier say P1 P2 P3 and P4 are derived from the efficiency locus in an identical way Hence we have derived the following definition The production possibility frontier shows the alternative combinations of x and y that can be efficiently produced by a firm with fixed resources The curve can be derived from Figure 132 by varying inputs between the production of x and y while maintaining the conditions for efficiency The negative of the slope of the production possibility curve is called the rate of product transformation RPT O y x4 A P1 P2 P3 P4 x3 x2 x1 y1 y2 y3 y4 Ox Quantity of x Quantity of y FIGURE 133 Production Possibility Frontier Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 454 Part 5 Competitive Markets 1326 Rate of product transformation The slope of the production possibility frontier shows how x output can be substituted for y output when total resources are held constant For example for points near Ox on the production possibility frontier the slope is a small negative numbersay 14 this implies that by reducing y output by 1 unit x output could be increased by 4 Near Oy on the other hand the slope is a large negative number say 5 implying that y output must be reduced by 5 units to permit the production of one more x The slope of the production possibility frontier clearly shows the possibilities that exist for trading y for x in produc tion The negative of this slope is called the rate of product transformation RPT D E F I N I T I O N Production possibility frontier The production possibility frontier shows the alternative com binations of two outputs that can be produced with fixed quantities of inputs if those inputs are employed efficiently D E F I N I T I O N Rate of product transformation The rate of product transformation RPT between two outputs is the negative of the slope of the production possibility frontier for those outputs Mathematically RPT 1of x for y2 5 3slope of production possibility frontier4 5 2 dy dx 1along OxOy2 131 The RPT records how x can be technically traded for y while continuing to keep the avail able productive inputs efficiently employed 1327 A Mathematical Derivation Showing the mathematics of how the production possibility frontier is constructed can help prove formally many of the graphical points we have made so far and provide some added insights as well Technically the production possibility frontier results from a con strained maximization problemthat is for any given level of x output say x we wish to maximize y output when our choices are constrained by the total amounts of capital and labor available denoted by k and l respectively Because there are three constraints in this problem we will need three Lagrangian multipliers 1λ1 λ2 and λ32 We use subscripts to indicate the quantities of capital and labor devoted to x and y production and assume that the production functions for these two goods are given by f x1kx lx2 and f y 1ky ly2 The Lagrangian expression for the production possibility frontier is therefore 1ly ky lx kx2 5 f y 1ky ly2 1 λ1 3x 2 f x1kx lx2 4 1 λ2 1k 2 kx 2 ky2 132 1 λ3 1l 2 lx 2 ly2 and the firstorder conditions in addition to the three constraints for a maximum are ly 5 f y l 2 λ3 5 0 ky 5 f y k 2 λ2 5 0 lx 5 λ1 f x l 2 λ3 5 0 kx 5 λ1 f x k 2 λ2 5 0 133 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 455 If we move the terms in λ2 and λ3 to the right it is obvious that the ratio of the top two equations must equal the ratio of the bottom two λ3 λ2 5 f y l f y k 5 f x l f x k 134 This equation simply repeats our graphical finding from Figure 132that for an efficient allocation of resources the RTS between the two inputs in y production must equal the RTS in x production 13271 Rate of product transformation We can also derive the Rate of Product Transformation from this optimization problem To do so first recognize that the value function that results from this optimization yields optimal y output as a function of the three constraints in the problem Let this value function be given by y 1x k l2 Apply ing the envelope theorem we get RPT 1x for y2 5 dy dx 5 x 5 λ1 135 As might have been expected the RPT is given by the absolute value of the first of the Lagrange multipliers in our optimization problem Some further manipulation of the firstorder conditions sheds a bit more light on this concept RPT 5 λ1 5 λ3 f x l 5 f y l f x l 5 λ2 f x k 5 f y k f x k 136 In words the RPT is given by the ratios of the marginal productivities of both labor and capital in the production of y and x It is this result that can help to explain why the pro duction possibility frontier has a concave shape an increasing RPT There are three plau sible explanations 13272 Diminishing returns If each input experiences diminishing returns it seems likely that the ratios shown in Equation 136 will increase as more x and less y is pro duced With diminishing returns such a change in production should decrease the mar ginal productivity of say labor in x production and increase the marginal productivity of labor in y production According to Equation 136 therefore the RPT should increase A similar argument can be made by recalling from Chapter 10 that cost minimization requires that MCx 5 wf x l and MCy 5 wf y l where w is the wage paid for a unit of labor4 Because the law of one price ensures that the price of labor is the same in the production of each output equation 136 implies that RPT 5 f y l f x l 5 MCx MCy 137 That is the RPT is also given by the ratio of the two outputs marginal costs With diminish ing returns to production of these goods we would expect the marginal cost of x to increase as its output increases and the marginal cost of y to decrease as its output decreases Over all then the RPT will increase as production is reallocated toward greater x output 13273 Factor intensities Diminishing returns are not necessary for the pro duction possibility frontier to take a concave shape Even when both goods exhibit constant 4A similar set of manipulations could also be done for capital input choice given the rental rate on capital v This would also show that with competitively determined input prices cost minimization in the production of the two goods will result in their having the same Rates of Technical Substitution as is required for productive efficiency Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 456 Part 5 Competitive Markets returns to scale if efficiency requires that they use the inputs with differing intensities the frontier will still be concave This is best illustrated by returning to the construction of the frontier from Figure 132 In that figure the efficiency locus took a general bowed shape above the Ox Oy diagonal of the Edgeworth Box This shows that when the inputs are efficiently allocated production of good x will be capital intensivethat is the ratio of capital to labor input will be higher in the production of good x than it is in the production of good y you can check this out by noticing that the overall ratio of capital to labor in this economy is klthe fact that the efficiency locus is above the Ox Oy diagonal of the box implies that x production will have a capitallabor ratio that always exceeds this average To show why this implies that with constant returns to scale the production possibility frontier will be concave consider input allocations along the diagonal All of these have the average capitallabor ratio With constant returns to scale production of good x would increase proportionally to the increase in inputs along the diagonal as more resources are allocated to its production Similarly production of good y would fall proportionally as these resources are taken away Hence the production possibilities generated by moving along Ox Oy could be represented by a negatively sloped straight line But we know that input allocations along Ox Oy are inefficientthat is more of both goods could be pro duced by moving from the diagonal to the efficiency locus Hence the true production possibility frontier must bulge out beyond linearthat is it must be concave The intuitive reason for this is that as resource allocations move from Ox toward Oy the capitallabor ratio must fall for both goods check this out for yourself This has the effect of raising the relative marginal cost of producing the capital intensive good x so the RPT must increase Example 131 provides a numerical illustration of this phenomenon when production is characterized by CobbDouglas production functions More generally the fact that Figure 132 can be used to examine how the capitallabor ratio changes as one moves along the production possibility frontier has important implications for input pricing and lies at the heart of traditional theorems about the effect of international trade on such prices see Figure 136 and Problem 1310 If production functions were to exhibit increasing returns to scale the production pos sibility frontier no longer needs to be concave Of course the factor intensity arguments presented above still generally hold so there are some forces tending toward concavity But with significant enough increasing returns to scale given increases in the production of good x can be achieved with progressively fewer inputs Consequently it is possible that the opportunity costs in terms of reduced y output could fallthat is the RPT could fall making the production possibility frontier convex For some numerical illustrations see Problem 139 13274 Nonhomogeneous inputs In the abstract world of economic the ory labor and capital inputs are treated as homogeneouseach input can be used equally well in the production of either good Of course in the real world that may not be the case Some inputs may be quite efficient in producing some goods but not in producing others a skilled mechanic may be good at making jet engines but not so good at milking cows This possibility provides another reason for believing that production possibility frontiers are concave When x output is low it can use inputs that are especially good at producing that good But as x output increases it must increasingly use inputs that while good at producing y are not much good at producing x Hence the relative marginal cost of producing good x would tend to rise and that of producing y would tend to fall Again Equation 137 shows that the RPT would rise and the production possibility frontier would be concave Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 457 1328 Opportunity cost and supply In conclusion then the production possibility curve demonstrates that there are many pos sible efficient combinations of the two goods and that producing more of one good necessi tates cutting back on the production of some other good This is precisely what economists mean by the term opportunity cost The cost of producing more x can be most readily mea sured by the reduction in y output that this entails Therefore the cost of one more unit of x is best measured as the RPT of x for y at the prevailing point on the production possi bility frontier The fact that this cost increases as more x is produced represents the general formulation of supply in a general equilibrium context EXAMPLE 131 Concavity of the Production Possibility Frontier In this example we look at two characteristics of production functions that may cause the produc tion possibility frontier to be concave Diminishing returns Suppose that the production of both x and y depends only on labor input and that the production functions for these goods are x 5 f 1lx2 5 l 05 x y 5 f 1ly2 5 l 05 y 138 Hence production of each of these goods exhibits diminishing returns to scale If total labor sup ply is limited by lx 1 ly 5 100 139 then simple substitution shows that the production possibility frontier is given by x 2 1 y 2 5 100 for x y 0 1310 In this case the frontier is a quartercircle and is concave The RPT can now be computed directly from the equation for the production possibility frontier written in implicit form as f1x y2 5 x2 1 y2 2 100 5 02 RPT 5 dy dx 5 a fx fy b 5 2x 2y 5 x y 1311 and this slope increases as x output increases A numerical illustration of concavity starts by not ing that the points 10 0 and 0 10 both lie on the frontier A straight line joining these two points would also include the point 5 5 but that point lies below the frontier If equal amounts of labor are devoted to both goods then production is x 5 y 5 50 which yields more of both goods than this midpoint Factor intensity To show how differing factor intensities yield a concave production possibility frontier suppose this the two goods are produced under constant returns to scale but with differ ent CobbDouglas production functions x 5 f 1k l2 5 k 05 x l 05 x y 5 g 1k l2 5 k 025 y l 075 y 1312 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 458 Part 5 Competitive Markets Suppose also that total capital and labor are constrained by kx 1 ky 5 100 lx 1 ly 5 100 1313 It is easy to show that RTSx 5 kx lx 5 κx RTSy 5 3ky ly 5 3κy 1314 where κi 5 kili Being located on the production possibility frontier requires RTSx 5 RTSy or κx 5 3κy That is no matter how total resources are allocated to production being on the pro duction possibility frontier requires that x be the capitalintensive good because capital is more productive in x production than in y production The capitallabor ratios in the production of the two goods are also constrained by the available resources kx 1 ky lx 1 ly 5 kx lx 1 ly 1 ky lx 1 ly 5 ακx 1 11 2 α2κy 5 100 100 5 1 1315 where α 5 lx 1lx 1 ly2that is α is the share of total labor devoted to x production Using the condition that κx 5 3κy we can find the input ratios of the two goods in terms of the overall allocation of labor ky 5 1 1 1 2α kx 5 3 1 1 2α 1316 Now we are in a position to phrase the production possibility frontier in terms of the share of labor devoted to x production x 5 κ05 x lx 5 κ05 x α 11002 5 100αa 3 1 1 2αb 05 y 5 κ025 y ly 5 κ025 y 11 2 α2 11002 5 100 11 2 α2 a 1 1 1 2αb 025 1317 We could push this algebra even further to eliminate α from these two equations to get an explicit functional form for the production possibility frontier that involves only x and y but we can show concavity with what we already have First notice that if α 5 0 x production gets no labor or cap ital inputs then x 5 0 y 5 100 With α 5 1 we have x 5 100 y 5 0 Hence a linear production possibility frontier would include the point 50 50 But if α takes on a middle value say 039 then x 5 100αa 3 1 1 2αb 05 5 39a 3 178b 05 5 506 y 5 100 11 2 α2 a 1 1 1 2αb 025 5 61a 1 178b 025 5 528 1318 which shows that the actual frontier is bowed outward beyond a linear frontier It is worth repeat ing that both of the goods in this example are produced under constant returns to scale and that the two inputs are fully homogeneous It is only the differing input intensities involved in the production of the two goods that yields the concave production possibility frontier QUERY How would an increase in the total amount of labor available shift the production possibility frontiers in these examples 1329 Determination of equilibrium prices Given these notions of demand and supply in our simple twogood economy we can now illustrate how equilibrium prices are determined Figure 134 shows PP the production Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 459 possibility frontier for the economy and the set of indifference curves represents individ uals preferences for these goods First consider the price ratio pxpy At this price ratio firms will choose to produce the output combination x1 y1 Profitmaximizing firms will choose the more profitable point on PP At x1 y1 the ratio of the two goods prices 1pxpy2 is equal to the ratio of the goods marginal costs the RPT thus profits are maximized there On the other hand given this budget constraint line C5 individuals will demand xr1 yr1 Consequently with these prices there is an excess demand for good x individuals demand more than is being produced but an excess supply of good y The workings of the marketplace will cause px to increase and py to decrease The price ratio pxpy will increase the price line will take on a steeper slope Firms will respond to these price changes by moving clockwise along the production possibility frontier that is they will increase their production of good x and decrease their production of good y Similarly individuals will respond to the changing prices by substituting y for x in their consumption choices These actions of both firms and individuals serve to eliminate the excess demand for x and the excess supply of y as market prices change 5It is important to recognize why the budget constraint has this location Because px and py are given the value of total production is px x1 1 py y1 This is the value of GDP in the simple economy pictured in Figure 134 It is also therefore the total income accruing to people in society Societys budget constraint therefore passes through x1 y1 and has a slope of pxpy This is precisely the budget constraint labeled C in the figure With a price ratio given by px py firms will produce x1 y1 societys budget constraint will be given by line C With this budget constraint individuals demand xr1 and yr1 that is there is an excess demand for good x and an excess supply of good y The workings of the market will move these prices toward their equilibrium levels p x p y At those prices societys budget constraint will be given by line C and supply and demand will be in equilibrium The combination x y of goods will be chosen FIGURE 134 Determination of Equilibrium Prices Quantity of y Quantity of x px P py U1 C C y x P C C U2 y1 x1 U3 Slope E y1 x1 px Slope py Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 460 Part 5 Competitive Markets Equilibrium is reached at x y with a price ratio of p xp y With this price ratio6 supply and demand are equilibrated for both good x and good y Given px and py firms will pro duce x and y in maximizing their profits Similarly with a budget constraint given by C individuals will demand x and y The operation of the price system has cleared the mar kets for both x and y simultaneously Therefore this figure provides a general equilibrium view of the supplydemand process for two markets working together For this reason we will make considerable use of this figure in our subsequent analysis 133 COMPARATIVE STATICS ANALYSIS As in our partial equilibrium analysis the equilibrium price ratio p xp y illustrated in Figure 134 will tend to persist until either preferences or production technologies change This competitively determined price ratio reflects these two basic economic forces If pref erences were to shift say toward good x then pxpy would increase and a new equilibrium would be established by a clockwise move along the production possibility frontier More x and less y would be produced to meet these changed preferences Similarly technical prog ress in the production of good x would shift the production possibility frontier outward as illustrated in Figure 135 This would tend to decrease the relative price of x and increase the quantity of x consumed assuming x is a normal good In the figure the quantity of y 6Notice again that competitive markets determine only equilibrium relative prices Determination of the absolute price level requires the introduction of money into this barter model Technical advances that lower marginal costs of x production will shift the production possibility fron tier This will generally create income and substitution effects that cause the quantity of x produced to increase assuming x is a normal good Effects on the production of y are ambiguous because income and substitution effects work in opposite directions FIGURE 135 Effects of Technical Progress in x Production Quantity of y Quantity of x x1 x0 y1 y0 E1 E0 U0 U1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 461 consumed also increases as a result of the income effect arising from the technical advance however a slightly different drawing of the figure could have reversed that result if the sub stitution effect had been dominant Example 132 looks at a few such effects EXAMPLE 132 Comparative Statics in a General Equilibrium Model To explore how general equilibrium models work lets start with a simple example based on the production possibility frontier in Example 131 In that example we assumed that production of both goods was characterized by decreasing returns x 5 l 05 x and y 5 l 05 y and also that total labor available was given by lx 1 ly 5 100 The resulting production possibility frontier was given by x2 1 y2 5 100 and RPT 5 xy To complete this model we assume that the typical individuals utility function is given by U1x y2 5 x05y05 so the demand functions for the two goods are x 5 x 1px py I2 5 05I px y 5 y1px py I2 5 05I py 1319 Basecase equilibrium Profit maximization by firms requires that pxpy 5 MCxMCy 5 RPT 5 xy and utilitymaximizing demand requires that pxpy 5 yx Thus equilibrium requires that xy 5 yx or x 5 y Inserting this result into the equation for the production possi bility frontier shows that x 5 y 5 50 5 707 and px py 5 1 1320 This is the equilibrium for our base case with this model The budget constraint The budget constraint that faces individuals is not especially transparent in this illustration therefore it may be useful to discuss it explicitly To bring some degree of absolute pricing into the model lets consider all prices in terms of the wage rate w Because total labor supply is 100 it follows that total labor income is 100w However because of the diminish ing returns assumed for production each firm also earns profits For firm x say the total cost function is C1w x2 5 wlx 5 wx2 so px 5 MCx 5 2wx 5 2w50 Therefore the profits for firm x are πx 5 1 px 2 ACx2x 5 1 px 2 wx2x 5 wx2 5 50w A similar computation shows that profits for firm y are also given by 50w Because general equilibrium models must obey the national income identity we assume that consumers are also shareholders in the two firms and treat these profits also as part of their spendable incomes Hence total consumer income is total income 5 labor income 1 profits 5 100w 1 2 150w2 5 200w 1321 This income will just permit consumers to spend 100w on each good by buying 50 units at a price of 2w50 so the model is internally consistent A shift in supply There are only two ways in which this basecase equilibrium can be disturbed 1 by changes in supplythat is by changes in the underlying technology of this economy or 2 by changes in demandthat is by changes in preferences Lets first consider changes in technology Suppose that there is technical improvement in x production so that the production function is x 5 2l 05 x Now the production possibility frontier is given by x24 1 y2 5 100 and RPT 5 x4y Proceeding as before to find the equilibrium in this model px py 5 x 4y 1supply2 px py 5 y x 1demand2 1322 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 462 Part 5 Competitive Markets 134 GENERAL EQUILIBRIUM MODELING AND FACTOR PRICES This simple general equilibrium model reinforces Marshalls observations about the impor tance of both supply and demand forces in the price determination process By providing an explicit connection between the markets for all goods the general equilibrium model makes it possible to examine more complex questions about market relationships than is possible by looking at only one market at a time General equilibrium modeling also per mits an examination of the connections between goods and factor markets we can illus trate that with an important historical case 1341 The Corn Laws debate High tariffs on grain imports were imposed by the British government following the Napo leonic wars Debate over the effects of these Corn Laws dominated the analytical efforts of economists between the years 1829 and 1845 A principal focus of the debate concerned so x2 5 4y 2 and the equilibrium is x 5 250 y 5 50 and px py 5 1 2 1323 Technical improvements in x production have caused its relative price to decrease and the consumption of this good to increase As in many examples with CobbDouglas util ity the income and substitution effects of this price decrease on y demand are precisely off setting Technical improvements clearly make consumers better off however Whereas utility was previously given by U1x y2 5 x05y05 5 50 5 707 now it has increased to U1x y2 5 x05y05 5 1250 2 05 150 2 05 5 2 50 5 10 Technical change has increased consumer welfare substantially A shift in demand If consumer preferences were to switch to favor good y as U1x y2 5 x01y 09 then demand functions would be given by x 5 01Ipx and y 5 09Ipy and demand equilibrium would require px py 5 y9x Returning to the original production possibility frontier to arrive at an overall equilibrium we have px py 5 x y 1supply2 px py 5 y 9x 1demand2 1324 so 9x2 5 y2 and the equilibrium is given by x 5 10 y 5 310 and px py 5 1 3 1325 Hence the decrease in demand for x has significantly reduced its relative price Observe that in this case however we cannot make a welfare comparison to the previous cases because the utility function has changed QUERY What are the budget constraints in these two alternative scenarios How is income dis tributed between wages and profits in each case Explain the differences intuitively Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 463 the effect that elimination of the tariffs would have on factor pricesa question that con tinues to have relevance today as we will see The production possibility frontier in Figure 136 shows those combinations of grain x and manufactured goods y that could be produced by British factors of production Assuming somewhat contrary to actuality that the Corn Laws completely prevented trade market equilibrium would be at E with the domestic price ratio given by p xp y Removal of the tariffs would reduce this price ratio to prx pry Given that new ratio Britain would produce combination A and consume combination B Grain imports would amount to xB xA and these would be financed by export of manufactured goods equal to yA yB Overall utility for the typical British consumer would be increased by the opening of trade Therefore use of the production possibility diagram demonstrates the implications that relaxing the tariffs would have for the production of both goods 1342 Trade and factor prices We can also analyze the effect of tariff reductions on factor prices by using our previous discussion of Figure 132 The movement from point E to point A in Figure 136 is similar to a movement from P3 to P1 in Figure 132 where production of x is decreased and pro duction of y is increased Reduction of tariff barriers on grain would cause production to be reallocated from point E to point A consumption would be reallocated from E to B If grain production is relatively capital intensive the rela tive price of capital would decrease as a result of these reallocations FIGURE 136 Analysis of the Corn Laws Debate U2 xA xE yE yA yB P xBP U1 Slope pʹxpʹy Slope px py A E B Output of grain x Output of manufactured goods y Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 464 Part 5 Competitive Markets This figure also records the reallocation of capital and labor made necessary by such a move If we assume that grain production is relatively capital intensive then the move ment from P3 to P1 causes the ratio of k to l to increase in both industries7 This in turn will cause the relative price of capital to decrease and the relative price of labor to increase Hence we conclude that repeal of the Corn Laws would be harmful to capital owners ie landlords and helpful to laborers It is not surprising that landed interests fought repeal of the laws 1343 Political support for trade policies The possibility that trade policies may affect the relative incomes of various factors of production continues to exert a major influence on political debates about such policies In the United States for example exports tend to be intensive in their use of skilled labor whereas imports tend to be intensive in unskilled labor input By analogy to our discussion of the Corn Laws it might thus be expected that further movements toward free trade policies would result in increasing relative wages for skilled workers and in decreasing relative wages for unskilled workers Therefore it is not surprising that unions representing skilled workers the machinists or aircraft workers tend to favor free trade whereas unions of unskilled workers those in textiles shoes and related businesses tend to oppose it8 135 A MATHEMATICAL MODEL OF EXCHANGE Although the previous graphical model of general equilibrium with two goods is fairly instructive it cannot reflect all the features of general equilibrium modeling with an arbi trary number of goods and productive inputs In the remainder of this chapter we will illustrate how such a more general model can be constructed and we will look at some of the insights that such a model can provide For most of our presentation we will look only at a model of exchangequantities of various goods already exist and are merely traded among individuals In such a model there is no production Later in the chapter we will look briefly at how production can be incorporated into the general model we have constructed 1351 Vector notation Most general equilibrium modeling is conducted using vector notation This provides great flexibility in specifying an arbitrary number of goods or individuals in the models Consequently this seems to be a good place to offer a brief introduction to such notation A vector is simply an ordered array of variables which each may take on specific values Here we will usually adopt the convention that the vectors we use are column vectors Hence we will write an n 3 1 column vector as x 5 D x1 x2 xn T 1326 7In the Corn Laws debate attention centered on the factors of land and labor 8The finding that the opening of trade will raise the relative price of the abundant factor is called the StolperSamuelson theorem after the economists who rigorously proved it in the 1950s Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 465 where each xi is a variable that can take on any value If x and y are two n 3 1 column vectors then the vector sum of them is defined as x 1 y 5 D x1 x2 xn T 1 D y1 y2 yn T 5 D x1 1 y1 x2 1 y2 xn 1 yn T 1327 Notice that this sum only is defined if the two vectors are of equal length In fact checking the length of vectors is one good way of deciding whether one has written a meaningful vector equation The dot product of two vectors is defined as the sum of the componentby component product of the elements in the two vectors That is xy 5 x1 y1 1 x2 y2 1 c1 xn yn 1328 Notice again that this operation is only defined if the vectors are of the same length With these few concepts we are now ready to illustrate the general equilibrium model of exchange 1352 Utility initial endowments and budget constraints In our model of exchange there are assumed to be n goods and m individuals Each indi vidual gains utility from the vector of goods he or she consumes ui 1xi2 where i 5 1 m Individuals also possess initial endowments of the goods given by x i Individuals are free to exchange their initial endowments with other individuals or to keep some or all the endow ment for themselves In their trading individuals are assumed to be pricetakersthat is they face a price vector p that specifies the market price for each of the n goods Each individual seeks to maximize utility and is bound by a budget constraint that requires that the total amount spent on consumption equals the total value of his or her endowment pxi 5 px i 1329 Although this budget constraint has a simple form it may be worth contemplating it for a minute The right side of Equation 1329 is the market value of this individuals endow ment sometimes referred to as his or her full income He or she could afford to consume this endowment and only this endowment if he or she wished to be selfsufficient But the endowment can also be spent on some other consumption bundle which presumably provides more utility Because consuming items in ones own endowment has an oppor tunity cost the terms on the left of Equation 1329 consider the costs of all items that enter into the final consumption bundle including endowment goods that are retained 1353 Demand functions and homogeneity The utility maximization problem outlined in the previous section is identical to the one we studied in detail in Part 2 of this book As we showed in Chapter 4 one outcome of this process is a set of n individual demand functions one for each good in which quantities demanded depend on all prices and income Here we can denote these in vector form as xi 1p px i2 These demand functions are continuous and as we showed in Chapter 4 they are homogeneous of degree 0 in all prices and income This latter property can be indi cated in vector notation by xi 1tp tpx i2 5 xi 1p px i2 1330 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 466 Part 5 Competitive Markets for any t 0 This property will be useful because it will permit us to adopt a convenient normalization scheme for prices which because it does not alter relative prices leaves quantities demanded unchanged 1354 Equilibrium and Walras law Equilibrium in this simple model of exchange requires that the total quantities of each good demanded be equal to the total endowment of each good available remember there is no production in this model Because the model used is similar to the one originally developed by Leon Walras9 this equilibrium concept is customarily attributed to him The n equations in Equation 1331 state that in equilibrium demand equals supply in each market This is the multimarket analog of the single market equilibria examined in the previous chapter Because there are n prices to be determined a simple counting of equations and unknowns might suggest that the existence of such a set of prices is guar anteed by the simultaneous equation solution procedures studied in elementary algebra Such a supposition would be incorrect for two reasons First the algebraic theorem about simultaneous equation systems applies only to linear equations Nothing suggests that the demand equations in this problem will be linearin fact most examples of demand equa tions we encountered in Part 2 were definitely nonlinear A second problem with Equation 1331 is that the equations are not independent of one anotherthey are related by what is known as Walras law Because each individual in this exchange economy is bound by a budget constraint of the form given in Equation 1329 we can sum over all individuals to obtain a m i51 pxi 5 a m i51 px i or a m i51 p1xi 2 x i2 5 0 1332 In words Walras law states that the value of all quantities demanded must equal the value of all endowments This result holds for any set of prices not just for equilibrium prices10 The general lesson is that the logic of individual budget constraints necessarily creates a relationship among the prices in any economy It is this connection that helps to ensure that a demandsupply equilibrium exists as we now show 1355 Existence of equilibrium in the exchange model The question of whether all markets can reach equilibrium together has fascinated econ omists for nearly 200 years Although intuitive evidence from the real world suggests that this must indeed be possible market prices do not tend to fluctuate wildly from one day 9The concept is named for the nineteenth century FrenchSwiss economist Leon Walras who pioneered the development of general equilibrium models Models of the type discussed in this chapter are often referred to as models of Walrasian equilibrium primarily because of the pricetaking assumptions inherent in them 10Walras law holds trivially for equilibrium prices as multiplication of Equation 1331 by p shows D E F I N I T I O N Walrasian equilibrium Walrasian equilibrium is an allocation of resources and an associated price vector p such that a m i51 xi 1p px i2 5 a m i51 x i 1331 where the summation is taken over the m individuals in this exchange economy Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 467 to the next proving the result mathematically proved to be rather difficult Walras him self thought he had a good proof that relied on evidence from the market to adjust prices toward equilibrium The price would increase for any good for which demand exceeded supply and decrease when supply exceeded demand Walras believed that if this process continued long enough a full set of equilibrium prices would eventually be found Unfor tunately the pure mathematics of Walras solution were difficult to state and ultimately there was no guarantee that a solution would be found But Walras idea of adjusting prices toward equilibrium using market forces provided a starting point for the modern proofs which were largely developed during the 1950s A key aspect of the modern proofs of the existence of equilibrium prices is the choice of a good normalization rule Homogeneity of demand functions makes it possible to use any absolute scale for prices providing that relative prices are unaffected by this choice Such an especially convenient scale is to normalize prices so that they sum to one Consider an arbitrary set of n nonnegative prices p1 p2 pn We can normalize11 these to form a new set of prices pri 5 pi a n k51 pk 1333 These new prices will have the properties that g n k51 prk 5 1 and that relative price ratios are maintained pri prj 5 pi a pk pj a pk 5 pi pj 1334 Because this sort of mathematical process can always be done we will assume without loss of generality that the price vectors we use p have all been normalized in this way Therefore proving the existence of equilibrium prices in our model of exchange amounts to showing that there will always exist a price vector p that achieves equilibrium in all markets That is a m i51 xi 1p px i 2 5 a m i51 x i or a m i51 xi 1p px i 2 2 a m i51 x i 5 0 or z 1p2 5 0 1335 where we use zp as a shorthand way of recording the excess demands for goods at a particular set of prices In equilibrium excess demand is zero in all markets12 Now consider the following way of implementing Walras idea that goods in excess demand should have their prices increased whereas those in excess supply should have their prices reduced13 Starting from any arbitrary set of prices p0 we define a new set p1 as p1 5 f 1p02 5 p0 1 kz 1p02 1336 where k is a small positive constant This function will be continuous because demand functions are continuous and it will map one set of normalized prices into another 11 This is possible only if at least one of the prices is nonzero Throughout our discussion we will assume that not all equilibrium prices can be zero 12 Goods that are in excess supply at equilibrium will have a zero price We will not be concerned with such free goods here 13 What follows is an extremely simplified version of the proof of the existence of equilibrium prices In particular problems of free goods and appropriate normalizations have been largely assumed away For a mathematically correct proof see for example G Debreu Theory of Value New York John Wiley Sons 1959 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 468 Part 5 Competitive Markets because of our assumption that all prices are normalized Hence it will meet the condi tions of the Brouwers fixed point theorem which states that any continuous function from a closed compact set onto itself in the present case from the unit simplex onto itself will have a fixed point such that x 5 f 1x2 The theorem is illustrated for a single dimen sion in Figure 137 There no matter what shape the function f 1x2 takes as long as it is continuous it must somewhere cross the 45 line and at that point x 5 f 1x2 If we let p represent the fixed point identified by Brouwers theorem for Equation 1336 we have p 5 f 1p2 5 p 1 kz 1p2 1337 Hence at this point z 1p2 5 0 thus p is an equilibrium price vector The proof that Walras sought is easily accomplished using an important mathematical result developed a few years after his death The elegance of the proof may obscure the fact that it uses a number of assumptions about economic behavior such as 1 pricetaking by all parties 2 homogeneity of demand functions 3 continuity of demand functions and 4 presence of budget constraints and Walras law All these play important roles in showing that a system of simple markets can indeed achieve a multimarket equilibrium 1356 First theorem of welfare economics Given that the forces of supply and demand can establish equilibrium prices in the general equilibrium model of exchange we have developed it is natural to ask what are the welfare consequences of this finding Adam Smith14 hypothesized that market forces provide an invisible hand that leads each market participant to promote an end social welfare which was no part of his intention Modern welfare economics seeks to understand the extent to which Smith was correct 14Adam Smith The Wealth of Nations New York Modern Library 1937 p 423 Because any continuous function must cross the 45 line somewhere in the unit square this function must have a point for which f 1x2 5 x This point is called a fixed point 1 0 1 x x fx Fixed point fx fx 45 FIGURE 137 A Graphical Illustration of Brouwers Fixed Point Theorem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 469 Perhaps the most important welfare result that can be derived from the exchange model is that the resulting Walrasian equilibrium is efficient in the sense that it is not possible to devise some alternative allocation of resources in which at least some people are better off and no one is worse off This definition of efficiency was originally developed by Italian economist Vilfredo Pareto in the early 1900s Understanding the definition is easiest if we consider what an inefficient allocation might be The total quantities of goods included in initial endowments would be allocated inefficiently if it were possible by shifting goods around among individuals to make at least one person better off ie receive a higher util ity and no one worse off Clearly if individuals preferences are to count such a situation would be undesirable Hence we have a formal definition D E F I N I T I O N Pareto efficient allocation An allocation of the available goods in an exchange economy is effi cient if it is not possible to devise an alternative allocation in which at least one person is better off and no one is worse off D E F I N I T I O N First theorem of welfare economics Every Walrasian equilibrium is Pareto efficient A proof that all Walrasian equilibria are Pareto efficient proceeds indirectly Suppose that p generates a Walrasian equilibrium in which the quantity of goods consumed by each person is denoted by xk 1k 5 1 m2 Now assume that there is some alternative allocation of the available goods rxk 1k 5 1 m2 such that for at least one person say person i it is that case that rxi is preferred to xi For this person it must be the case that prxi p xi 1338 because otherwise this person would have bought the preferred bundle in the first place If all other individuals are to be equally well off under this new proposed allocation it must be the case for them that prxk 5 p xk k 5 1 m k 2 i 1339 If the new bundle were less expensive such individuals could not have been minimiz ing expenditures at p Finally to be feasible the new allocation must obey the quantity constraints a m i51 rxi 5 a m i51 x i 1340 Multiplying Equation 1340 by p yields a m i51 prxi 5 a m i51 px i 1341 but Equations 1338 and 1339 together with Walras law applied to the original equilib rium imply that a m i51 prxi a m i51 p xi 5 a m i51 px i 1342 Hence we have a contradiction and must conclude that no such alternative allocation can exist Therefore we can summarize our analysis with the following definition Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 470 Part 5 Competitive Markets The significance of this theorem should not be overstated The theorem does not say that every Walrasian equilibrium is in some sense socially desirable Walrasian equilibria can for example exhibit vast inequalities among individuals arising in part from inequal ities in their initial endowments see the discussion in the next section The theorem also assumes pricetaking behavior and full information about pricesassumptions that need not hold in other models Finally the theorem does not consider possible effects of one individuals consumption on another In the presence of such externalities even a perfect competitive price system may not yield Pareto optimal results see Chapter 19 Still the theorem does show that Smiths invisible hand conjecture has some valid ity The simple markets in this exchange world can find equilibrium prices and at those equilibrium prices the resulting allocation of resources will be efficient in the Pareto sense Developing this proof is one of the key achievements of welfare economics 1357 A graphic illustration of the first theorem In Figure 138 we again use the Edgeworth box diagram this time to illustrate an exchange economy In this economy there are only two goods x and y and two individuals A and B The total dimensions of the Edgeworth box are determined by the total quantities of the two goods available x and y Goods allocated to individual A are recorded using 0A as an origin Individual B gets those quantities of the two goods that are left over and can be measured using 0B as an origin Individual As indifference curve map is drawn in the usual way whereas individual Bs map is drawn from the perspective of 0B Point E in the Edgeworth box represents the initial endowments of these two individuals Individual A starts with x A and y A Individual B starts with x B 5 x 2 x A and y B 5 y 2 y A The initial endowments provide a utility level of U 2 A for person A and U 2 B for person B These levels are clearly inefficient in the Pareto sense For example we could by real locating the available goods15 increase person Bs utility to U 3 B while holding person As utility constant at U 2 A point B Or we could increase person As utility to U 3 A while keep ing person B on the U 2 B indifference curve point A Allocations A and B are Pareto effi cient however because at these allocations it is not possible to make either person better off without making the other worse off There are many other efficient allocations in the Edgeworth box diagram These are identified by the tangencies of the two individuals indifference curves The set of all such efficient points is shown by the line joining OA to OB This line is sometimes called the contract curve because it represents all the Pare toefficient contracts that might be reached by these two individuals Notice however that assuming that no individual would voluntarily opt for a contract that made him or her worse off only contracts between points B and A are viable with initial endowments given by point E The line PP in Figure 138 shows the competitively established price ratio that is guaran teed by our earlier existence proof The line passes through the initial endowments E and shows the terms at which these two individuals can trade away from these initial positions Notice that such trading is beneficial to both partiesthat is it allows each of them to get a higher utility level than is provided by their initial endowments Such trading will continue until all such mutual beneficial trades have been completed That will occur at allocation E on the contract curve Because the individuals indifference curves are tangent at this point no further trading would yield gains to both parties Therefore the competitive allo cation E meets the Pareto criterion for efficiency as we showed mathematically earlier 15This point could in principle be found by solving the following constrained optimization problem Maximize UB1xB yB2 subject to the constraint UA1xA yA2 5 U2 A See Example 133 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 471 1358 Second theorem of welfare economics The first theorem of welfare economics shows that a Walrasian equilibrium is Pareto effi cient but the social welfare consequences of this result are limited because of the role played by initial endowments in the demonstration The location of the Walrasian equilibrium at E in Figure 138 was significantly influenced by the designation of E as the starting point for trading Points on the contract curve outside the range of AB are not attainable through vol untary transactions even though these may in fact be more socially desirable than E per haps because utilities are more equal The second theorem of welfare economics addresses this issue It states that for any Pareto optimal allocation of resources there exists a set of initial endowments and a related price vector such that this allocation is also a Walrasian equilibrium Phrased another way any Pareto optimal allocation of resources can also be a Walrasian equilibrium providing that initial endowments are adjusted accordingly A graphical proof of the second theorem should suffice Figure 139 repeats the key aspects of the exchange economy pictures in Figure 138 Given the initial endowments at point E all voluntary Walrasian equilibrium must lie between points A and B on the contract curve Suppose however that these allocations were thought to be undesirableperhaps because they involve too much inequality of utility Assume that the Pareto optimal alloca tion Q is believed to be socially preferable but it is not attainable from the initial endow ments at point E The second theorem states that one can draw a price line through Q that is tangent to both individuals respective indifference curves This line is denoted by PrPr in Figure 139 Because the slope of this line shows potential trades these individuals are willing to make any point on the line can serve as an initial endowment from which trades lead to Q One such point is denoted by Q If a benevolent government wished to ensure that Q would emerge as a Walrasian equilibrium it would have to transfer initial endowments of the goods from E to Q making person A better off and person B worse off in the process With initial endowments at point E individuals trade along the price line PP until they reach point E This equilibrium is Pareto efficient O B P A B P U 3 B U 3 A U 2 B U 2 A E E y O A y A y B x A x B x FIGURE 138 The First Theorem of Welfare Economics Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 472 Part 5 Competitive Markets If allocation Q is regarded as socially optimal this allocation can be supported by any initial endow ments on the price line PrPr To move from E to say Q would require transfers of initial endowments O A O B Total y Total x B E A E Pʹ Q Q Pʹ FIGURE 139 The Second Theorem of Welfare Economics EXAMPLE 133 A TwoPerson Exchange Economy To illustrate these various principles consider a simple twoperson twogood exchange economy Suppose that total quantities of the goods are fixed at x 5 y 5 1000 Person As utility takes the CobbDouglas form UA 1xA yA2 5 x 23 A y 13 A 1343 and person Bs preferences are given by UB 1xB yB2 5 x 13 B y 23 B 1344 Notice that person A has a relative preference for good x and person B has a relative preference for good y Hence you might expect that the Paretoefficient allocations in this model would have the property that person A would consume relatively more x and person B would consume relatively more y To find these allocations explicitly we need to find a way of dividing the available goods in such a way that the utility of person A is maximized for any preassigned utility level for person B Setting up the Lagrangian expression for this problem we have 1xA yA2 5 UA 1xA yA2 1 λ3UB 11000 2 xA 1000 2 yA2 2 UB4 1345 Substituting for the explicit utility functions assumed here yields 1xA yA2 5 x 23 A y 13 A 1 λ3 11000 2 xA2 13 11000 2 yA2 23 2 UB4 1346 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 473 and the firstorder conditions for a maximum are xA 5 2 3a yA xA b 13 2 λ 3 a 1000 2 yA 1000 2 xA b 23 5 0 yA 5 1 3a xA yA b 23 2 2λ 3 a 1000 2 xA 1000 2 yA b 13 5 0 1347 Moving the terms in λ to the right and dividing the top equation by the bottom gives 2a yA xA b 5 1 2a 1000 2 yA 1000 2 xA b or 1348 xA 1000 2 xA 5 4yA 1000 2 yA This equation allows us to identify all the Pareto optimal allocations in this exchange economy For example if we were to arbitrarily choose xA 5 xB 5 500 Equation 1348 would become 4yA 1000 2 yA 5 1 so yA 5 200 yB 5 800 1349 This allocation is relatively favorable to person B At this point on the contract curve UA 5 5002320013 5 369 UB 5 5001380023 5 683 Notice that although the available quantity of x is divided evenly by assumption most of good y goes to person B as efficiency requires Equilibrium price ratio To calculate the equilibrium price ratio at this point on the contract curve we need to know the two individuals marginal rates of substitution For person A MRS 5 UAxA UAyA 5 2 yA xA 5 2 200 500 5 08 1350 and for person B MRS 5 UBxB UByB 5 05 yA xA 5 05 800 500 5 08 1351 Hence the marginal rates of substitution are indeed equal as they should be and they imply a price ratio of px py 5 08 Initial endowments Because this equilibrium price ratio will permit these individ uals to trade 8 units of y for each 10 units of x it is a simple matter to devise initial endow ments consistent with this Pareto optimum Consider for example the endowment xA 5 350 yA 5 320 xB 5 650 yB 5 680 If px 5 08 py 5 1 the value of person As initial endowment is 600 If he or she spends two thirds of this amount on good x it is possible to purchase 500 units of good x and 200 units of good y This would increase utility from its initial level of UA 5 35023 32013 5 340 to 369 Similarly the value of person Bs endowment is 1200 If he or she spends one third of this on good x 500 units can be bought With the remaining two thirds of the value of the endowment being spent on good y 800 units can be bought In the process Bs utility increases from 670 to 683 Thus trading from the proposed initial endowment to the contract curve is indeed mutually beneficial as shown in Figure 138 QUERY Why did starting with the assumption that good x would be divided equally on the contract curve result in a situation favoring person B throughout this problem What point on the contract curve would provide equal utility to persons A and B What would the price ratio of the two goods be at this point Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 474 Part 5 Competitive Markets 1359 Social welfare functions Figure 139 shows that there are many Paretoefficient allocations of the available goods in an exchange economy We are assured by the second theorem of welfare economics that any of these can be supported by a Walrasian system of competitively determined prices providing that initial endowments are adjusted accordingly A major question for welfare economics is how if at all to develop criteria for choosing among all these allocations In this section we look briefly at one strand of this large topicthe study of social welfare functions Simply put a social welfare function is a hypothetical scheme for ranking poten tial allocations of resources based on the utility they provide to individuals In mathemat ical terms Social Welfare 5 SW3U1 1x12 U2 1x22 Um 1xm2 4 1352 The social planners goal then is to choose allocations of goods among the m individuals in the economy in a way that maximizes SW Of course this exercise is a purely conceptual onein reality there are no clearly articulated social welfare functions in any economy and there are serious doubts about whether such a function could ever arise from some type of democratic process16 Still assuming the existence of such a function can help to illuminate many of the thorniest problems in welfare economics A first observation that might be made about the social welfare function in Equation 1352 is that any welfare maximum must also be Pareto efficient If we assume that every individuals utility is to count it seems clear that any allocation that permits further Pareto improvements that make one person better off and no one else worse off cannot be a welfare maximum Hence achieving a welfare maximum is a problem in choosing among Paretoefficient allocations and their related Walrasian price systems We can make further progress in examining the idea of social welfare maximization by considering the precise functional form that SW might take Specifically if we assume util ity is measurable using the CES form can be particularly instructive SW1U1 U2 Um2 5 U R 1 R 1 U R 2 R 1 c1 U R m R R 1 1353 Because we have used this functional form many times before in this book its properties should by now be familiar Specifically if R 5 1 the function becomes SW1U1 U2 Um2 5 U1 1 U2 1 c1 Um 1354 Thus utility is a simple sum of the utility of every person in the economy Such a social welfare function is sometimes called a utilitarian function With such a function social welfare is judged by the aggregate sum of utility or perhaps even income with no regard for how utility income is distributed among the members of society At the other extreme consider the case R 5 q In this case social welfare has a fixed proportions character and as we have seen in many other applications SW1U1 U2 Um2 5 Min3U1 U2 Um4 1355 Therefore this function focuses on the worseoff person in any allocation and chooses that allocation for which this person has the highest utility Such a social welfare func tion is called a maximin function It was made popular by the philosopher John Rawls who argued that if individuals did not know which position they would ultimately have in 16The impossibility of developing a social welfare function from the underlying preferences of people in society was first studied by K Arrow in Social Choice and Individual Values 2nd ed New York Wiley 1963 There is a large body of literature stemming from Arrows initial discovery Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 475 society ie they operate under a veil of ignorance they would opt for this sort of social welfare function to guard against being the worseoff person17 Our analysis in Chapter 7 suggests that people may not be this risk averse in choosing social arrangements However Rawls focus on the bottom of the utility distribution is probably a good antidote to think ing about social welfare in purely utilitarian terms It is possible to explore many other potential functional forms for a hypothetical wel fare function Problem 1314 looks at some connections between social welfare functions and the income distribution for example But such illustrations largely miss a crucial point if they focus only on an exchange economy Because the quantities of goods in such an economy are fixed issues related to production incentives do not arise when evaluat ing social welfare alternatives In actuality however any attempt to redistribute income or utility through taxes and transfers will necessarily affect production incentives and affect the size of the Edgeworth box Therefore assessing social welfare will involve study ing the tradeoff between achieving distributional goals and maintaining levels of pro duction To examine such possibilities we must introduce production into our general equilibrium framework 136 A MATHEMATICAL MODEL OF PRODUCTION AND EXCHANGE Adding production to the model of exchange developed in the previous section is a rel atively simple process First the notion of a good needs to be expanded to include fac tors of production Therefore we will assume that our list of n goods now includes inputs whose prices also will be determined within the general equilibrium model Some inputs for one firm in a general equilibrium model are produced by other firms Some of these goods may also be consumed by individuals cars are used by both firms and final consum ers and some of these may be used only as intermediate goods steel sheets are used only to make cars and are not bought by consumers Other inputs may be part of individuals initial endowments Most importantly this is the way labor supply is treated in general equilibrium models Individuals are endowed with a certain number of potential labor hours They may sell these to firms by taking jobs at competitively determined wages or they may choose to consume the hours themselves in the form of leisure In making such choices we continue to assume that individuals maximize utility18 We will assume that there are r firms involved in production Each of these firms is bound by a production function that describes the physical constraints on the ways the firm can turn inputs into outputs By convention outputs of the firm take a positive sign whereas inputs take a negative sign Using this convention each firms production plan can be described by an n 3 1 column vector y j1 j 5 1 r2 which contains both posi tive and negative entries The only vectors that the firm may consider are those that are feasible given the current state of technology Sometimes it is convenient to assume each firm produces only one output But that is not necessary for a more general treatment of production Firms are assumed to maximize profits Production functions are assumed to be suffi ciently convex to ensure a unique profit maximum for any set of output and input prices This rules out both increasing returns to scale technologies and constant returns because neither yields a unique maxima Many general equilibrium models can handle such 17J Rawls A Theory of Justice Cambridge MA Harvard University Press 1971 18A detailed study of labor supply theory is presented in Chapter 16 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 476 Part 5 Competitive Markets possibilities but there is no need to introduce such complexities here Given these assump tions the profits for any firm can be written as πj1p2 5 py j if πj1p2 0 and y j 5 0 if πj1p2 0 1356 Hence this model has a long run orientation in which firms that lose money at a partic ular price configuration hire no inputs and produce no output Notice how the convention that outputs have a positive sign and inputs a negative sign makes it possible to phrase prof its in a compact way19 1361 Budget constraints and Walras law In an exchange model individuals purchasing power is determined by the values of their initial endowments Once firms are introduced we must also consider the income stream that may flow from ownership of these firms To do so we adopt the simplifying assumption that each individual owns a predefined share si where a m i51 si 5 1 of the profits of all firms That is each person owns an index fund that can claim a proportionate share of all firms profits We can now rewrite each individuals budget constraint from Equation 1329 as pxi 5 si a r j51 py j 1 px i i 5 1 m 1357 Of course if all firms were in longrun equilibrium in perfectly competitive industries all profits would be zero and the budget constraint in Equation 1357 would revert to that in Equation 1329 But allowing for longterm profits does not greatly complicate our model therefore we might as well consider the possibility As in the exchange model the existence of these m budget constraints implies a con straint of the prices that are possiblea generalization of Walras law Summing the budget constraints in Equation 1357 over all individuals yields pa m i51 xi 1p2 5 pa r j51 y j1p2 1 pa m i51 x i 1358 and letting x1p2 5 gxi 1p2 y 1p2 5 gy j1p2 x 5 gx i provides a simple statement of Wal ras law px1p2 5 py 1p2 1 px 1359 Notice again that Walras law holds for any set of prices because it is based on individuals budget constraints 1362 Walrasian equilibrium As before we define a Walrasian equilibrium price vector 1p2 as a set of prices at which demand equals supply in all markets simultaneously In mathematical terms this means that x1p2 5 y 1p2 1 x 1360 19As we saw in Chapter 11 profit functions are homogeneous of degree 1 in all prices Hence both output supply functions and input demand functions are homogeneous of degree 0 in all prices because they are derivatives of the profit function Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 477 Initial endowments continue to play an important role in this equilibrium For example it is individuals endowments of potential labor time that provide the most important input for firms production processes Therefore determination of equilibrium wage rates is a major output of general equilibrium models operating under Walrasian conditions Exam ining changes in wage rates that result from changes in exogenous influences is perhaps the most important practical use of such models As in the study of an exchange economy it is possible to use some form of fixed point theorem20 to show that there exists a set of equilibrium prices that satisfy the n equations in Equation 1360 Because of the constraint of Walras law such an equilibrium price vector will be unique only up to a scalar multiplethat is any absolute price level that preserves relative prices can also achieve equilibrium in all markets Technically excess demand functions z 1p2 5 x1p2 2 y 1p2 2 x 1361 are homogeneous of degree 0 in prices therefore any price vector for which z 1p2 5 0 will also have the property that z 1tp2 5 0 and t 0 Frequently it is convenient to normalize prices so that they sum to one But many other normalization rules can also be used In macroeconomic versions of general equilibrium models it is usually the case that the abso lute level of prices is determined by monetary factors 1363 Welfare economics in the Walrasian model with production Adding production to the model of an exchange economy greatly expands the number of feasible allocations of resources One way to visualize this is shown in Figure 1310 There PP represents that production possibility frontier for a twogood economy with a fixed endowment of primary factors of production Any point on this frontier is fea sible Consider one such allocation say allocation A If this economy were to produce xA and yA we could use these amounts for the dimensions of the Edgeworth exchange box shown inside the frontier Any point within this box would also be a feasible allo cation of the available goods between the two people whose preferences are shown Clearly a similar argument could be made for any other point on the production pos sibility frontier Despite these complications the first theorem of welfare economics continues to hold in a general equilibrium model with production At a Walrasian equilibrium there are no further market opportunities either by producing something else or by reallocating the available goods among individuals that would make one individual or group of individuals better off without making other individuals worse off Adam Smiths invisible hand continues to exert its logic to ensure that all such mutually beneficial opportunities are exploited in part because transaction costs are assumed to be zero Again the general social welfare implications of the first theorem of welfare economics are far from clear There is of course a second theorem which shows that practically any Walrasian equilibrium can be supported by suitable changes in initial endowments One also could hypothesize a social welfare function to choose among these But most such exercises are rather uninformative about actual policy issues 20For some illustrative proofs see K J Arrow and F H Hahn General Competitive Analysis San Francisco CA HoldenDay 1971 chap 5 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 478 Part 5 Competitive Markets More interesting is the use of the Walrasian mechanism to judge the hypothetical impact of various tax and transfer policies that seek to achieve specific social welfare criteria In this case as we shall see the fact that Walrasian models stress interconnections among markets especially among product and input markets can yield important and often sur prising results In the next section we look at a few of these 137 COMPUTABLE GENERAL EQUILIBRIUM MODELS Two advances have spurred the rapid development of general equilibrium models in recent years First the theory of general equilibrium itself has been expanded to include many features of realworld markets such as imperfect competition environmental externalities and complex tax systems Models that involve uncertainty and that have a dynamic struc ture also have been devised most importantly in the field of macroeconomics A second related trend has been the rapid development of computer power and the associated soft ware for solving general equilibrium models This has made it possible to study models with virtually any number of goods and types of households In this section we will briefly Any point on the production possibility frontier PP can serve as the dimensions of an Edgeworth exchange box FIGURE 1310 Production Increases the Number of Feasible Allocations y A x A P P A Quantity of x Quantity of y Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 479 explore some conceptual aspects of these models21 The Extensions to the chapter describe a few important applications 1371 Structure of general equilibrium models Specification of any general equilibrium model begins by defining the number of goods to be included in the model These goods include not only consumption goods but also intermediate goods that are used in the production of other goods eg capital equipment productive inputs such as labor or natural resources and goods that are to be produced by the government public goods The goal of the model is then to solve for equilibrium prices for all these goods and to study how these prices change when conditions change Some of the goods in a general equilibrium model are produced by firms The technol ogy of this production must be specified by production functions The most common such specification is to use the types of CES production functions that we studied in Chapters 9 and 10 because these can yield some important insights about the ways in which inputs are substituted in the face of changing prices In general firms are assumed to maximize their profits given their production functions and the input and output prices they face Demand is specified in general equilibrium models by defining utility functions for var ious types of households Utility is treated as a function both of goods that are consumed and of inputs that are not supplied to the marketplace eg available labor that is not sup plied to the market is consumed as leisure Households are assumed to maximize utility Their incomes are determined by the amounts of inputs they sell in the market and by the net result of any taxes they pay or transfers they receive Finally a full general equilibrium model must specify how the government operates If there are taxes in the model how those taxes are to be spent on transfers or on public goods which provide utility to consumers must be modeled If government borrowing is allowed the bond market must be explicitly modeled In short the model must fully specify the flow of both sources and uses of income that characterize the economy being modeled 1372 Solving general equilibrium models Once technology supply and preferences demand have been specified a general equi librium model must be solved for equilibrium prices and quantities The proof earlier in this chapter shows that such a model will generally have such a solution but actually find ing that solution can sometimes be difficultespecially when the number of goods and households is large General equilibrium models are usually solved on computers via mod ifications of an algorithm originally developed by Herbert Scarf in the 1970s22 This algo rithm or more modern versions of it searches for market equilibria by mimicking the way markets work That is an initial solution is specified and then prices are raised in markets with excess demand and lowered in markets with excess supply until an equilibrium is found in which all excess demands are zero Sometimes multiple equilibria will occur but usually economic models have sufficient curvature in the underlying production and util ity functions that the equilibrium found by the Scarf algorithm will be unique 21For more detail on the issues discussed here see W Nicholson and F Westhoff General Equilibrium Models Improving the Microeconomics Classroom Journal of Economic Education Summer 2009 297314 22Herbert Scarf with Terje Hansen On the Computation of Economic Equilibria New Haven CT Yale University Press 1973 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 480 Part 5 Competitive Markets 1373 Economic insights from general equilibrium models General equilibrium models provide a number of insights about how economies operate that cannot be obtained from the types of partial equilibrium models studied in Chapter 12 Some of the most important of these are the following All prices are endogenous in these models The exogenous elements of general equilib rium models are preferences and productive technologies All firms and productive inputs are owned by households All income ultimately accrues to households Any model with a government sector is incomplete if it does not specify how tax receipts are used The bottom line in any policy evaluation is the utility of households Firms and gov ernments are only intermediaries in getting to this final accounting All taxes distort economic decisions along some dimension The welfare costs of such distortions must always be weighed against the benefits of such taxes in terms of public good production or equityenhancing transfers Some of these insights are illustrated in the next two examples In later chapters we will return to general equilibrium modeling whenever such a perspective seems necessary to gain a more complete understanding of the topic being covered EXAMPLE 134 A Simple General Equilibrium Model Lets look at a simple general equilibrium model with only two households two consumer goods x and y and two inputs capital k and labor l Each household has an endowment of capital and labor that it can choose to retain or sell in the market These endowments are denoted by k1 l 1 and k2 l 2 respectively Households obtain utility from the amounts of the consumer goods they purchase and from the amount of labor they do not sell into the market ie leisure 5 l i 2 li The households have simple CobbDouglas utility functions U1 5 x 05 1 y 03 1 1l 1 2 l12 02 U2 5 x 04 2 y 04 2 1l 2 2 l22 02 1362 Hence household 1 has a relatively greater preference for good x than does household 2 Notice that capital does not enter into these utility functions directly Consequently each household will provide its entire endowment of capital to the marketplace Households will retain some labor however because leisure provides utility directly Production of goods x and y is characterized by simple CobbDouglas technologies x 5 k 02 x l 08 x y 5 k 08 y l 02 y 1363 Thus in this example production of x is relatively labor intensive whereas production of y is relatively capital intensive To complete this model we must specify initial endowments of capital and labor Here we assume that k1 5 40 l 1 5 24 and k2 5 10 l 2 5 24 1364 Although the households have equal labor endowments ie 24 hours household 1 has signifi cantly more capital than does household 2 Basecase simulation Equations 13621364 specify our complete general equilibrium model in the absence of a government A solution to this model will consist of four equilibrium prices Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 481 23Because firms production functions are characterized by constant returns to scale in equilibrium each earns zero profits therefore there is no need to specify firm ownership in this model for x y k and l at which households maximize utility and firms maximize profits23 Because any general equilibrium model can compute only relative prices we are free to impose a pricenor malization scheme Here we assume that the prices will always sum to unity That is px 1 py 1 pk 1 pl 5 1 1365 Solving24 for these prices yields px 5 0363 py 5 0253 pk 5 0136 pl 5 0248 1366 At these prices total production of x is 237 and production of y is 251 The utilitymaximizing choices for household 1 are x1 5 157 y1 5 81 l 1 2 l1 5 24 2 148 5 92 U1 5 135 1367 for household 2 these choices are x2 5 81 y2 5 116 l 2 2 l2 5 24 2 181 5 59 U2 5 875 1368 Observe that household 1 consumes quite a bit of good x but provides less in labor supply than does household 2 This reflects the greater capital endowment of household 1 in this basecase simulation QUERY How would you show that each household obeys its budget constraint in this simula tion Does the budgetary allocation of each household exhibit the budget shares that are implied by the form of its utility function 24For details of these solutions together with a link to the program that generated them see W Nicholson and F Westhoff General Equilibrium Models Improving the Microeconomics Classroom Journal of Economic Education Summer 2009 297314 EXAMPLE 135 The Excess Burden of a Tax In Chapter 12 we showed that taxation may impose an excess burden in addition to the tax rev enues collected because of the incentive effects of the tax With a general equilibrium model we can show much more about this effect Specifically assume that the government in the econ omy of Example 134 imposes an ad valorem tax of 04 on good x This introduces a wedge between what demanders pay for this good x 1 px2 and what suppliers receive for the good 1 prx 5 11 2 t2px 5 06px2 To complete the model we must specify what happens to the revenues generated by this tax For simplicity we assume that these revenues are rebated to the households in a 5050 split In all other respects the economy remains as described in Example 134 Solving for the new equilibrium prices in this model yields px 5 0472 py 5 0218 pk 5 0121 pl 5 0188 1369 At these prices total production of x is 179 and total production of y is 288 Hence the alloca tion of resources has shifted significantly toward y production Even though the relative price of x experienced by consumers 15 px py 5 04720218 5 2172 has increased significantly from its value of 143 in Example 134 the price ratio experienced by firms 106px py 5 1302 has decreased somewhat from this prior value Therefore one might expect based on a partial equi librium analysis that consumers would demand less of good x and likewise that firms would Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 482 Part 5 Competitive Markets similarly produce less of that good Partial equilibrium analysis would not however allow us to predict the increased production of y which comes about because the relative price of y has decreased for consumers but has increased for firms nor the reduction in relative input prices because there is less being produced overall A more complete picture of all these effects can be obtained by looking at the final equilibrium positions of the two households The posttax alloca tion for household 1 is x1 5 116 y1 5 152 l 1 2 l1 5 118 U1 5 127 1370 for household 2 x2 5 63 y2 5 136 l 2 2 l2 5 79 U2 5 896 1371 Hence imposition of the tax has made household 1 considerably worse off Utility decreases from 135 to 127 Household 2 is made slightly better off by this tax and transfer scheme primarily because it receives a relatively large share of the tax proceeds that come mainly from household 1 Although total utility has decreased as predicted by the simple partial equilibrium analysis of excess burden general equilibrium analysis gives a more complete picture of the distributional conse quences of the tax Notice also that the total amount of labor supplied decreases as a result of the tax Total leisure increases from 151 hours to 197 Therefore imposition of a tax on good x has had a relatively substantial labor supply effect that is completely invisible in a partial equilibrium model QUERY Would it be possible to make both households better off relative to Example 134 in this taxation scenario by changing how the tax revenues are redistributed Summary This chapter has provided a general exploration of Adam Smiths conjectures about the efficiency properties of compet itive markets We began with a description of how to model many competitive markets simultaneously and then used that model to make a few statements about welfare Some high lights of this chapter are listed here Preferences and production technologies provide the build ing blocks on which all general equilibrium models are based One particularly simple version of such a model uses individual preferences for two goods together with a con cave production possibility frontier for those two goods Competitive markets can establish equilibrium prices by making marginal adjustments in prices in response to information about the demand and supply for individual goods Walras law ties markets together so that such a solution is assured in most cases General equilibrium models can usually be solved by using computer algorithms The resulting solutions yield many insights about the economy that are not obtainable from partial equilibrium analysis of single markets Competitive prices will result in a Paretoefficient allo cation of resources This is the first theorem of welfare economics Factors that interfere with competitive markets abilities to achieve efficiency include 1 market power 2 exter nalities 3 existence of public goods and 4 imperfect information We explore all these issues in detail in later chapters Competitive markets need not yield equitable distribu tions of resources especially when initial endowments are highly skewed In theory any desired distribution can be attained through competitive markets accompanied by appropriate transfers of initial endowments the sec ond theorem of welfare economics But there are many practical problems in implementing such transfers Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 483 Problems 131 Suppose the production possibility frontier for guns x and butter y is given by x2 1 2y2 5 900 a Graph this frontier b If individuals always prefer consumption bundles in which y 5 2x how much x and y will be produced c At the point described in part b what will be the RPT and hence what price ratio will cause production to take place at that point This slope should be approximated by considering small changes in x and y around the opti mal point d Show your solution on the figure from part a 132 Suppose two individuals Smith and Jones each have 10 hours of labor to devote to producing either ice cream x or chicken soup y Smiths utility function is given by US 5 x03y07 whereas Jones is given by UJ 5 x05y05 The individuals do not care whether they produce x or y and the production function for each good is given by x 5 2l and y 5 3l where l is the total labor devoted to production of each good a What must the price ratio pxpy be b Given this price ratio how much x and y will Smith and Jones demand Hint Set the wage equal to 1 here c How should labor be allocated between x and y to satisfy the demand calculated in part b 133 Consider an economy with just one technique available for the production of each good Good Food Cloth Labor per unit output 1 1 Land per unit output 2 1 a Suppose land is unlimited but labor equals 100 Write and sketch the production possibility frontier b Suppose labor is unlimited but land equals 150 Write and sketch the production possibility frontier c Suppose labor equals 100 and land equals 150 Write and sketch the production possibility frontier Hint What are the intercepts of the production possibility frontier When is land fully employed Labor Both d Explain why the production possibility frontier of part c is concave e Sketch the relative price of food as a function of its out put in part c f If consumers insist on trading 4 units of food for 5 units of cloth what is the relative price of food Why g Explain why production is exactly the same at a price ratio of pFpC 5 11 as at pFpC 5 19 h Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 08 and 09 respectively There are 100 units of capital available What is the pro duction possibility curve in this case Answer part e for this case 134 Suppose that Robinson Crusoe produces and consumes fish F and coconuts C Assume that during a certain period he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts Robinsons production for fish is given by F 5 lF and for coconuts by C 5 lC where lF and lC are the number of hours spent fishing or gath ering coconuts Consequently lC 1 lF 5 200 Robinson Crusoes utility for fish and coconuts is given by utility 5 F C a If Robinson cannot trade with the rest of the world how will he choose to allocate his labor What will the opti mal levels of F and C be What will his utility be What will be the RPT of fish for coconuts b Suppose now that trade is opened and Robinson can trade fish and coconuts at a price ratio of pFpC 5 21 If Rob inson continues to produce the quantities of F and C from part a what will he choose to consume once given the opportunity to trade What will his new level of utility be c How would your answer to part b change if Robinson adjusts his production to take advantage of the world prices d Graph your results for parts a b and c 135 Smith and Jones are stranded on a desert island Each has in his possession some slices of ham H and cheese C Smith is a choosy eater and will eat ham and cheese only in the fixed Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 484 Part 5 Competitive Markets proportions of 2 slices of cheese to 1 slice of ham His utility function is given by US 5 min 1H C22 Jones is more flexible in his dietary tastes and has a utility function given by UJ 5 4H 1 3C Total endowments are 100 slices of ham and 200 slices of cheese a Draw the Edgeworth box diagram that represents the possibilities for exchange in this situation What is the only exchange ratio that can prevail in any equilibrium b Suppose Smith initially had 40H and 80C What would the equilibrium position be c Suppose Smith initially had 60H and 80C What would the equilibrium position be d Suppose Smith much the stronger of the two decides not to play by the rules of the game Then what could the final equilibrium position be 136 In the country of Ruritania there are two regions A and B Two goods x and y are produced in both regions Production functions for region A are given by xA 5 lx yA 5 ly here lx and ly are the quantities of labor devoted to x and y production respectively Total labor available in region A is 100 units that is lx 1 ly 5 100 Using a similar notation for region B production func tions are given by xB 5 1 2lx yB 5 1 2ly There are also 100 units of labor available in region B lx 1 ly 5 100 a Calculate the production possibility curves for regions A and B b What condition must hold if production in Ruritania is to be allocated efficiently between regions A and B assum ing labor cannot move from one region to the other c Calculate the production possibility curve for Ruritania again assuming labor is immobile between regions How much total y can Ruritania produce if total x output is 12 Hint A graphical analysis may be of some help here 137 Use the computer algorithm discussed in footnote 24 to exam ine the consequences of the following changes to the model in Example 134 For each change describe the final results of the modeling and offer some intuition about why the results worked as they did a Change the preferences of household 1 to U1 5 x 06 1 y 02 1 1l1 2 l12 02 b Reverse the production functions in Equation 1358 so that x becomes the capitalintensive good c Increase the importance of leisure in each households utility function Analytical Problems 138 Tax equivalence theorem Use the computer algorithm discussed in the reference given in footnotes 21 and 24 to show that a uniform ad valorem tax of both goods yields the same equilibrium as does a uniform tax on both inputs that collects the same revenue Note This tax equivalence theorem from the theory of public finance shows that taxation may be done on either the output or input sides of the economy with identical results 139 Returns to scale and the production possibility frontier The purpose of this problem is to examine the relationships among returns to scale factor intensity and the shape of the production possibility frontier Suppose there are fixed supplies of capital and labor to be allocated between the production of good x and good y The production functions for x and y are given respectively by x 5 kαl β and y 5 kγl δ where the parameters α β γ and δ will take on different val ues throughout this problem Using either intuition a computer or a formal mathemat ical approach derive the production possibility frontier for x and y in the following cases a α 5 β 5 γ 5 δ 5 12 b α 5 β 5 12 γ 5 13 δ 5 23 c α 5 β 5 12 γ 5 δ 5 23 d α 5 β 5 γ 5 δ 5 23 e α 5 β 5 06 γ 5 02 δ 5 10 f α 5 β 5 07 γ 5 06 δ 5 08 Do increasing returns to scale always lead to a convex produc tion possibility frontier Explain 1310 The trade theorems The construction of the production possibility curve shown in Figures 132 and 133 can be used to illustrate three import ant theorems in international trade theory To get started notice in Figure 132 that the efficiency line Ox Oy is bowed above the main diagonal of the Edgeworth box This shows that the production of good x is always capital intensive relative to the production of good y That is when produc tion is efficient 1k l 2 x 1k l 2 y no matter how much of the goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 13 General Equilibrium and Welfare 485 are produced Demonstration of the trade theorems assumes that the price ratio p 5 pxpy is determined in international marketsthe domestic economy must adjust to this ratio in trade jargon the country under examination is assumed to be a small country in a large world a Factor price equalization theorem Use Figure 134 to show how the international price ratio p determines the point in the Edgeworth box at which domestic produc tion will take place Show how this determines the fac tor price ratio wv If production functions are the same throughout the world what will this imply about relative factor prices throughout the world b StolperSamuelson theorem An increase in p will cause the production to move clockwise along the pro duction possibility frontierx production will increase and y production will decrease Use the Edgeworth box diagram to show that such a move will decrease kl in the production of both goods Explain why this will cause wv to decrease What are the implications of this for the opening of trade relations which typically increases the price of the good produced intensively with a countrys most abundant input c Rybczynski theorem Suppose again that p is set by external markets and does not change Show that an increase in k will increase the output of x the capital intensive good and reduce the output of y the labor intensive good 1311 An example of Walras law Suppose there are only three goods 1x1 x2 x32 in an econ omy and that the excess demand functions for x2 and x3 are given by ED2 5 2 3p2 p1 1 2p3 p1 2 1 ED3 5 2 4p2 p1 2 2p3 p1 2 2 a Show that these functions are homogeneous of degree 0 in p1 p2 and p3 b Use Walras law to show that if ED2 5 ED3 5 0 then ED1 must also be 0 Can you also use Walras law to cal culate ED1 c Solve this system of equations for the equilibrium rela tive prices p2p1 and p3p1 What is the equilibrium value for p3p2 1312 Productive efficiency with calculus In Example 133 we showed how a Pareto efficiency exchange equilibrium can be described as the solution to a constrained maximum problem In this problem we provide a similar illustration for an economy involving production Suppose that there is only one person in a twogood economy and that his or her utility function is given by U1x y2 Suppose also that this economys production possibility frontier can be written in implicit form as T 1x y2 5 0 a What is the constrained optimization problem that this economy will seek to solve if it wishes to make the best use of its available resources b What are the firstorder conditions for a maximum in this situation c How would the efficient situation described in part b be brought about by a perfectly competitive system in which this individual maximizes utility and the firms underlying the production possibility frontier maximize profits d Under what situations might the firstorder conditions described in part b not yield a utility maximum 1313 Initial endowments equilibrium prices and the first theorem of welfare economics In Example 133 we computed an efficient allocation of the available goods and then found the price ratio consistent with this allocation That then allowed us to find initial endow ments that would support this equilibrium In that way the example demonstrates the second theorem of welfare eco nomics We can use the same approach to illustrate the first theorem Assume again that the utility functions for persons A and B are those given in the example a For each individual show how his or her demand for x and y depends on the relative prices of these two goods and on the initial endowment that each person has To simplify the notation here set py 5 1 and let p represent the price of x relative to that of y Hence the value of say As initial endowment can be written as pxA 1 yA b Use the equilibrium conditions that total quantity demanded of goods x and y must equal the total quan tities of these two goods available assumed to be 1000 units each to solve for the equilibrium price ratio as a function of the initial endowments of the goods held by each person remember that initial endowments must also total 1000 for each good c For the case xA 5 yA 5 500 compute the resulting mar ket equilibrium and show that it is Pareto efficient d Describe in general terms how changes in the initial endowments would affect the resulting equilibrium prices in this model Illustrate your conclusions with a few numerical examples 1314 Social welfare functions and income taxation The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 486 Part 5 Competitive Markets one in welfare economics In this problem we look at a few elements of this theory Throughout we assume that there are m individuals in the economy and that each individ ual is characterized by a skill level ai which indicates his or her ability to earn income Without loss of generality sup pose also that individuals are ordered by increasing ability Pretax income itself is determined by skill level and effort ci which may or may not be sensitive to taxation That is Ii 5 I1ai ci2 Suppose also that the utility cost of effort is given by ψ1c2 ψr 0 ψs 0 ψ102 5 0 Finally the government wishes to choose a schedule of income taxes and transfers T 1I2 which maximizes social welfare subject to a government budget constraint satisfying g m i51T 1Ii2 5 R where R is the amount needed to finance public goods a Suppose that each individuals income is unaffected by effort and that each persons utility is given by ui 5 ui 3Ii 2 T 1Ii2 2 ψ1c2 4 Show that maximization of a CES social welfare function requires perfect equality of income no matter what the precise form of that function Note For some individuals T 1Ii2 may be negative b Suppose now that individuals incomes are affected by effort Show that the results of part a still hold if the gov ernment based income taxation on ai rather than on Ii c In general show that if income taxation is based on observed income this will affect the level of effort indi viduals undertake d Characterization of the optimal tax structure when income is affected by effort is difficult and often coun terintuitive Diamond25 shows that the optimal marginal rate schedule may be Ushaped with the highest rates for both low and highincome people He shows that the optimal top rate marginal rate is given by Tr 1I max 2 5 11 1 eLw2 11 2 ki2 2eLw 1 11 1 eLw2 11 2 ki2 where ki 10 ki 12 is the top income persons relative weight in the social welfare function and eL w is the elas ticity of labor supply with respect to the aftertax wage rate Try a few simulations of possible values for these two parameters and describe what the top marginal rate should be Give an intuitive discussion of these results 25P Diamond Optimal Income Taxation An Example with a UShaped Pattern of Optimal Marginal Tax Rates American Economic Review March 1998 8393 Suggestions for Further Reading Arrow K J and F H Hahn General Competitive Analysis Amsterdam NorthHolland 1978 chaps 1 2 and 4 Sophisticated mathematical treatment of general equilibrium analysis Each chapter has a good literary introduction Debreu G Theory of Value New York John Wiley Sons 1959 Basic reference difficult mathematics Does have a good introduc tory chapter on the mathematical tools used Debreu G Existence of Competitive Equilibrium In K J Arrow and M D Intriligator Eds Handbook of Mathemati cal Economics vol 2 Amsterdam NorthHolland 1982 pp 697743 Fairly difficult survey of existence proofs based on fixed point the orems Contains a comprehensive set of references Ginsburgh V and M Keyzer The Structure of Applied Gen eral Equilibrium Models Cambridge MA MIT Press 1997 Detailed discussions of the problems in implementing computable general equilibrium models Some useful references to the empir ical literature Harberger A The Incidence of the Corporate Income Tax Journal of Political Economy JanuaryFebruary 1962 21540 Nice use of a twosector general equilibrium model to examine the final burden of a tax on capital MasColell A M D Whinston and J R Green Microeco nomic Theory Oxford UK Oxford University Press 1995 Part Four is devoted to general equilibrium analysis Chapters 17 existence and 18 connections to game theory are especially use ful Chapters 19 and 20 pursue several of the topics in the Exten sions to this chapter Salanie B Microeconomic Models of Market Failure Cam bridge MA MIT Press 2000 Nice summary of the theorems of welfare economics along with detailed analyses of externalities public goods and imperfect competition Sen A K Collective Choice and Social Welfare San Francisco CA HoldenDay 1970 chaps 1 and 2 Basic reference on social choice theory Early chapters have a good discussion of the meaning and limitations of the Pareto efficiency concept Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 487 As discussed briefly in Chapter 13 recent improvements in computer technology have made it feasible to develop com putable general equilibrium CGE models of considerable detail These may involve literally hundreds of industries and individuals each with somewhat different technologies or preferences The general methodology used with these models is to assume various forms for production and utility functions and then choose particular parameters of those functions based on empirical evidence Numeri cal general equilibrium solutions are then generated by the models and compared with realworld data After calibrat ing the models to reflect reality various policy elements in the models are varied as a way of providing general equilibrium estimates of the overall impact of those policy changes In this extension we briefly review a few of these types of applications E131 Trade models One of the first uses for applied general equilibrium mod els was to the study of the impact of trade barriers Because much of the debate over the effects of such barriers or of their reduction focuses on impacts on real wages such general equilibrium models are especially appropriate for the task Two unusual features tend to characterize such mod els First because the models often have an explicit focus on domestic versus foreign production of specific goods it is necessary to introduce a large degree of product differentia tion into individuals utility functions That is US textiles are treated as being different from Mexican textiles even though in most trade theories textiles might be treated as homogeneous goods Modelers have found they must allow for only limited substitutability among such goods if their models are to replicate actual trade patterns A second feature of CGE models of trade is the interest in incorporating increasing returnstoscale technologies into their production sectors This permits the models to capture one of the primary advantages of trade for smaller economies Unfortunately introduction of the increasing returnsto scale assumption also requires that the models depart from perfectly competitive pricetaking assumptions Often some type of markup pricing together with Cournottype imperfect competition see Chapter 15 is used for this purpose North American free trade Some of the most extensive CGE modeling efforts have been devoted to analyzing the impact of the North American Free Trade Agreement NAFTA Virtually all these models find that the agreement offered welfare gains to all the countries involved Gains for Mexico accrued primarily because of reduced US trade barriers on Mexican textiles and steel Gains to Canada came primarily from an increased ability to benefit from economies of scale in certain key industries Brown 1992 surveys a number of CGE models of North American free trade and concludes that gains on the order of 23 percent of GDP might be experienced by both countries For the United States gains from NAFTA might be considerably smaller but even in this case significant welfare gains were found to be associated with the increased competitiveness of domestic markets E132 Tax and transfer models A second major use of CGE models is to evaluate poten tial changes in a nations tax and transfer policies For these applications considerable care must be taken in modeling the factor supply side of the models For example at the mar gin the effects of rates of income taxation either positive or negative can have important labor supply effects that only a general equilibrium approach can model properly Similarly taxtransfer policy can also affect savings and investment decisions and for these too it may be necessary to adopt more detailed modeling procedures eg differentiating individuals by age to examine effects of retirement programs The Dutch MIMIC model Probably the most elaborate taxtransfer CGE model is that developed by the Dutch Central Planning Bureauthe Micro Macro Model to Analyze the Institutional Context MIMIC This model puts emphasis on social welfare programs and on some of the problems they seek to ameliorate most notably unemployment which is missing from many other CGE mod els Gelauff and Graaflund 1994 summarize the main fea tures of the MIMIC model They also use it to analyze such policy proposals as the 1990s tax reform in the Netherlands and potential changes to the generous unemployment and disability benefits in that country EXTENSIONS ComputablE GEnEral Equilibrium modEls Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 488 Part 5 Competitive Markets E133 Environmental models CGE models are also appropriate for understanding the ways in which environmental policies may affect the economy In such applications the production of pollutants is considered as a major side effect of the other economic activities in the model By specifying environmental goals in terms of a given reduction in these pollutants it is possible to use these models to study the economic costs of various strategies for achieving these goals One advantage of the CGE approach is to provide some evidence on the impact of environmental policies on income distributiona topic largely omitted from more nar row industrybased modeling efforts Assessing CO2 reduction strategies Concern over the possibility that CO2 emissions in various energyusing activities may be contributing to global warm ing has led to a number of plans for reducing these emis sions Because the repercussions of such reductions may be widespread and varied CGE modeling is one of the pre ferred assessment methods Perhaps the most elaborate such model is that developed by the Organisation for Economic Cooperation and Development OECDthe General Equilibrium Environmental GREEN model The basic structure of this model is described by Burniaux Nicoletti and OlivieraMartins 1992 The model has been used to simulate various policy options that might be adopted by European nations to reduce CO2 emissions such as insti tution of a carbon tax or increasingly stringent emissions regulations for automobiles and power plants In general these simulations suggest that economic costs of these poli cies would be relatively modest given the level of restrictions currently anticipated But most of the policies would have adverse distributional effects that may require further atten tion through government transfer policy E134 Regional and urban models A final way in which CGE models can be used is to examine economic issues that have important spatial dimensions Con struction of such models requires careful attention to issues of transportation costs for goods and moving costs associated with labor mobility because particular interest is focused on where transactions occur Incorporation of these costs into CGE models is in many ways equivalent to adding extra lev els of product differentiation because these affect the relative prices of otherwise homogeneous goods Calculation of equi libria in regional markets can be especially sensitive to how transport costs are specified Changing government procurement CGE regional models have been widely used to examine the local impact of major changes in government spending policies For example Hoffmann Robinson and Subramanian 1996 use a CGE model to evaluate the regional impact of reduced defense expenditures on the California economy They find that the size of the effects depends importantly on the assumed costs of migration for skilled workers A similar finding is reported by Bernat and Hanson 1995 who examine possible reduc tions in US pricesupport payments to farms Although such reductions would offer overall efficiency gains to the economy they could have significant negative impacts on rural areas References Bernat G A and K Hanson Regional Impacts of Farm Pro grams A TopDown CGE Analysis Review of Regional Studies Winter 1995 33150 Brown D K The Impact of North American Free Trade Area Applied General Equilibrium Models In N Lustig B P Bosworth and R Z Lawrence Eds North American Free Trade Assessing the Impact Washington DC Brook ings Institution 1992 pp 2668 Burniaux J M G Nicoletti and J OlivieraMartins GREEN A Global Model for Quantifying the Costs of Policies to Curb CO2 Emissions OECD Economic Studies Winter 1992 4992 Gelauff G M M and J J Graaflund Modeling Welfare State Reform Amsterdam North Holland 1994 Hoffmann S S Robinson and S Subramanian The Role of Defense Cuts in the California Recession Computable General Equilibrium Models and Interstate Fair Mobility Journal of Regional Science November 1996 57195 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 489 PART SIX Market Power Chapter 14 Monopoly Chapter 15 Imperfect Competition In this part we examine the consequences of relaxing the assumption that firms are pricetakers When firms have some power to set prices they will no longer treat them as fixed parameters in their decisions but will instead treat price setting as one part of the profitmaximization process Usually this will mean prices no longer accurately reflect marginal costs and the efficiency theorems that apply to competitive markets no longer hold Chapter 14 looks at the relatively simple case where there is only a single monopoly supplier of a good This supplier can choose to operate at any point on the demand curve for its product that it finds most profitable Its activities are constrained only by this demand curve not by the behavior of rival producers As we shall see this offers the firm a number of avenues for increasing profits such as using novel pricing schemes or adapting the characteristics of its product Although such decisions will indeed provide more profits for the monopoly in general they will also result in welfare losses for consumers relative to perfect competition In Chapter 15 we consider markets with few producers Models of such markets are consid erably more complicated than are markets of monopoly or of perfect competition for that matter because the demand curve faced by any one firm will depend in an important way on what its rivals choose to do Studying the possibilities will usually require gametheoretic ideas to capture accu rately the strategic possibilities involved Hence you should review the basic game theory material in Chapter 8 before plunging into Chapter 15 whose general conclusion is that outcomes in markets with few firms will depend crucially on the details of how the game is played In many cases the same sort of inefficiencies that occur in monopoly markets appear in imperfectly competitive markets as well Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 491 CHAPTER FOURTEEN Monopoly A monopoly is a single firm that serves an entire market This single firm faces the market demand curve for its output Using its knowledge of this demand curve the monopoly makes a decision on how much to produce Unlike the perfectly competitive firms output decision which has no effect on market price the monopolys output decision will in fact determine the goods price In this sense monopoly markets and markets character ized by perfect competition are polaropposite cases D E F I N I T I O N Monopoly A monopoly is a single supplier to a market This firm may choose to produce at any point on the market demand curve At times it is more convenient to treat monopolies as having the power to set prices Tech nically a monopoly can choose that point on the market demand curve at which it prefers to operate It may choose either market price or quantity but not both In this chapter we will usually assume that monopolies choose the quantity of output that maximizes profits and then settle for the market price that the chosen output level yields It would be a simple matter to rephrase the discussion in terms of price setting and in some places we shall do so 141 BARRIERS TO ENTRY The reason a monopoly exists is that other firms find it unprofitable or impossible to enter the market Therefore barriers to entry are the source of all monopoly power If other firms could enter a market then the firm would by definition no longer be a monopoly There are two general types of barriers to entry technical barriers and legal barriers 1411 Technical barriers A primary technical barrier is that the production of the good in question may exhibit decreasing marginal and average costs over a wide range of output levels The technology of production is such that relatively largescale firms are lowcost producers In this situa tion which is sometimes referred to as natural monopoly one firm may find it profitable to drive others out of the industry by cutting prices Similarly once a monopoly has been established entry will be difficult because any new firm must produce at relatively low lev els of output and therefore at relatively high average costs It is important to stress that the range of declining costs need only be large relative to the market in question Declining costs on some absolute scale are not necessary For example the production and delivery of concrete does not exhibit declining marginal costs over a broad range of output when Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 492 Part 6 Market Power compared with the total US market However in any particular small town declining marginal costs may permit a monopoly to be established The high costs of transportation in this industry tend to isolate one market from another Another technical basis of monopoly is special knowledge of a lowcost productive technique The monopoly has an incentive to keep its technology secret but unless this technology is protected by a patent see next paragraph this may be extremely difficult Ownership of unique resourcessuch as mineral deposits or land locations or the posses sion of unique managerial talentsmay also be a lasting basis for maintaining a monopoly 1412 Legal barriers Many pure monopolies are created as a matter of law rather than as a matter of economic conditions One important example of a governmentgranted monopoly position is the legal protection of a product by a patent or copyright Prescription drugs computer chips and Disney animated movies are examples of profitable products that are shielded for a time from direct competition by potential imitators Because the basic technology for these products is uniquely assigned to one firm a monopoly position is established The defense made of such a governmentally granted monopoly is that the patent and copy right system makes innovation more profitable and therefore acts as an incentive Whether the benefits of such innovative behavior exceed the costs of having monopolies is an open question that has been much debated by economists A second example of a legally created monopoly is the awarding of an exclusive fran chise to serve a market These franchises are awarded in cases of public utility gas and electric service communications services the post office some television and radio sta tion markets and a variety of other situations Usually the restriction of entry is combined with a regulatory cap on the price the franchised monopolist is allowed to charge The argument usually put forward in favor of creating these franchised monopolies is that the industry in question is a natural monopoly average cost is diminishing over a broad range of output levels and minimum average cost can be achieved only by organizing the indus try as a monopoly The public utility and communications industries are often considered good examples Certainly that does appear to be the case for local electricity and telephone service where a given network probably exhibits declining average cost up to the point of universal coverage But recent deregulation in telephone services and electricity genera tion show that even for these industries the natural monopoly rationale may not be all inclusive In other cases franchises may be based largely on political rationales This seems to be true for the postal service in the United States and for a number of nationalized industries airlines radio and television banking in other countries 1413 Barriers erected by the monopolist Although some barriers to entry may be independent of the monopolists own activities other barriers may result directly from those activities For example firms may develop unique products or technologies and take extraordinary steps to keep these from being copied by competitors Or firms may buy up unique resources to prevent potential entry The De Beers cartel for example controls a large fraction of the worlds diamond mines Finally a wouldbe monopolist may enlist government aid in devising barriers to entry It may lobby for legislation that restricts new entrants to maintain an orderly market or for health and safety regulations that raise potential entrants costs Because the monopolist has both special knowledge of its business and significant incentives to pursue these goals it may have considerable success in creating such barriers to entry Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 493 The attempt by a monopolist to erect barriers to entry may involve real resource costs Maintaining secrecy buying unique resources and engaging in political lobbying are all costly activities A full analysis of monopoly should involve not only questions of cost minimization and output choice as under perfect competition but also an analysis of profitmaximizing creation of entry barriers However we will not provide a detailed investigation of such questions here1 Instead we will take the presence of a single supplier on the market and this single firms cost function as given 142 PROFIT MAXIMIZATION AND OuTPuT CHOICe The monopolist chooses quantity Q to maximize its profits π 1Q2 5 R 1Q2 2 C1Q2 5 P 1Q2Q 2 C1Q2 141 just as we saw in Chapter 11 on profit maximization The new issue here is one of inter pretation Here the monopolists output Q constitutes the entire market output and the monopolists inverse demand P 1Q2 represents the entire markets demand whereas the output q and price p 1q2 of the generic firm studied in Chapter 11 may have only repre sented a fraction of the market The monopolists firstorder condition for profit maximi zation is πr 1Q2 5 dR dQ 2 dC dQ 5 MR 1Q2 2 MC1Q2 5 0 142 To maximize profit the monopolist produces that output level at which marginal revenue MR 1Q2 equals marginal cost MC1Q2 The monopoly in contrast to a perfectly competitive firm faces a negatively sloped market demand curve implying Pr 1Q2 0 Thus marginal revenue will be less than the market price MR 1Q2 5 P 1Q2 1 QPr 1Q2 P 1Q2 143 To sell an additional unit the monopoly must lower its price on all units to be sold if it is to generate the extra demand necessary to absorb this marginal unit Figure 141 illustrates monopoly profit maximization The profitmaximizing output level is denoted Qm at the intersection of marginal revenue and marginal cost Given the monopolys decision to produce Qm the inverse demand curve P 1Q2 indi cates that a market price of Pm will prevail This is the price that demanders as a group are willing to pay for the output of the monopoly In the market an equilibrium pricequantity combination of Pm Qm will be observed Assuming Pm ACm this output level will be profitable and the monopolist will have no incentive to alter output levels unless demand or cost conditions change Hence we have the following principle 1For a simple treatment see R A Posner The Social Costs of Monopoly and Regulation Journal of Political Economy 83 August 1975 80727 O P T I M I Z AT I O N P R I N C I P L E Monopolists output A monopolist will choose to produce that output for which marginal reve nue equals marginal cost Because the monopolist faces a downwardsloping demand curve mar ket price will exceed marginal revenue and the firms marginal cost at this output level Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 494 Part 6 Market Power 1421 The inverse elasticity rule again In Chapter 11 we showed that the assumption of profit maximization implies that the gap between a price of a firms output and its marginal cost in percentage terms called the Lerner index is inversely related to the price elasticity of the demand curve faced by the firm Applying Equation 1114 to the case of monopoly yields Pm 2 MC Pm 5 2 1 eD P 144 where now we use the elasticity of demand for the entire market 1eD P2 because the monop oly is the sole supplier of the good in question This observation leads to two general con clusions about monopoly pricing First a monopoly will choose to operate only in regions in which the market demand curve is elastic 1eD P 212 If demand were inelastic then marginal revenue would be negative and thus could not be equated to marginal cost which presumably is positive Equation 144 also shows that eD P 21 implies an implausible negative marginal cost A second implication of Equation 144 is that the firms markup over marginal cost measured as a fraction of price depends inversely on the elasticity of market demand For example if eD P 5 22 then Equation 144 shows that Pm 5 2MC whereas if eD P 5 210 then Pm 5 111MC Notice also that if the elasticity of demand were con stant along the entire demand curve the proportional markup over marginal cost would A profitmaximizing monopolist produces that quantity for which marginal revenue is equal to marginal cost In the diagram this quantity is given by Qm which will yield a price of Pm in the market Monopoly profits can be read as the rectangle of PmABACm Price costs B C Output per period Q PQ A MRQ ACQ ACm MCQ Qm P m FIGURE 141 Profit Maximization and Price Determination for a Monopoly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 495 remain unchanged in response to changes in input costs In that case therefore market price moves proportionally to marginal cost Increases in marginal cost will prompt the monopoly to increase its price proportionally and decreases in marginal cost will cause the monopoly to reduce its price proportionally Even if elasticity is not constant along the demand curve it seems clear from Figure 141 that increases in marginal cost will increase price although not necessarily in the same proportion As long as the demand curve fac ing the monopoly is downward sloping upward shifts in MC will prompt the monopoly to reduce output and thereby obtain a higher price We will examine all these relationships mathematically in more detail below 1422 Monopoly profits Total profits earned by the monopolist can be read directly from Figure 141 These are shown by the rectangle PmEACm and again represent the profit per unit price minus aver age cost times the number of units sold These profits will be positive if market price exceeds average total cost If Pm ACm however then the monopolist can operate only at a longterm loss and will decline to serve the market Because by assumption no entry is possible into a monopoly market the monopo lists positive profits can exist even in the long run For this reason some authors refer to the profits that a monopoly earns in the long run as monopoly rents These profits can be regarded as a return to that factor that forms the basis of the monopoly eg a patent a favorable location or a dynamic entrepreneur hence another possible owner might be willing to pay that amount in rent for the right to the monopoly The potential for profits is the reason why some firms pay other firms for the right to use a patent and why conces sioners at sporting events and on some highways are willing to pay for the right to the concession To the extent that monopoly rights are given away at less than their true mar ket value as in radio and television licensing the wealth of the recipients of those rights is increased Although a monopoly may earn positive profits in the long run2 the size of such profits will depend on the relationship between the monopolists average costs and the demand for its product Figure 142 illustrates two situations in which the demand mar ginal revenue and marginal cost curves are rather similar As Equation 141 suggests the pricemarginal cost markup is about the same in these two cases But average costs in Figure 142a are considerably lower than in Figure 142b Although the profitmaximizing decisions are similar in the two cases the level of profits ends up being different In Figure 142a the monopolists price 1Pm2 exceeds the average cost of producing Qm labeled ACm by a large extent and significant profits are obtained In Figure 142b however Pm 5 ACm and the monopoly earns zero economic profits the largest amount possible in this case Hence large profits from a monopoly are not inevitable and the actual extent of economic profits may not always be a good guide to the significance of monopolistic influences in a market 1423 There is no monopoly supply curve In the theory of perfectly competitive markets presented in Part 4 it was possible to speak of an industry supply curve We constructed the longrun supply curve by allowing the 2As in the competitive case the profitmaximizing monopolist would be willing to produce at a loss in the short run as long as market price exceeds average variable cost Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 496 Part 6 Market Power market demand curve to shift and observing the supply curve that was traced out by the series of equilibrium pricequantity combinations This type of construction is not possi ble for monopolistic markets With a fixed market demand curve the supply curve for a monopoly will be only one pointnamely that pricequantity combination for which MR 5 MC If the demand curve should shift then the marginal revenue curve would also shift and a new profitmaximizing output would be chosen However connecting the resulting series of equilibrium points on the market demand curves would have little meaning This locus might have a strange shape depending on how the market demand curves elasticity and its associated MR curve changes as the curve is shifted In this sense the monopoly firm has no welldefined supply curve Each demand curve is a unique profitmaximizing opportunity for a monopolist Both monopolies in this figure are equally strong if by this we mean they produce similar divergences between market price and marginal cost However because of the location of the demand and average cost curves it turns out that the monopoly in a earns high profits whereas that in b earns no profits Consequently the size of profits is not a measure of the strength of a monopoly Quantity per period Price Price a Monopoly with large profts Q m Q m Pm D D D D MR MR MC MC AC AC b Zeroproft monopoly Quantity per period ACm Pm ACm FIGURE 142 Monopoly Profits Depend on the Relationship between the Demand and Average Cost Curves EXAMPLE 141 Calculating Monopoly Output Suppose the market for Olympicquality Frisbees Q measured in Frisbees bought per year has a linear demand curve of the form Q 5 2000 2 20P 145 or P 5 100 2 Q 20 146 and let the costs of a monopoly Frisbee producer be given by C1Q2 5 005Q2 1 10000 147 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 497 EXAMPLE 142 Monopoly with Simple Demand Curves We can derive a few simple facts about monopoly pricing in cases where the demand curve fac ing the monopoly takes a simple algebraic form and the firm has constant marginal costs ie C1Q2 5 cQ and MC 5 c Linear demand Suppose that the inverse demand function facing the monopoly is of the linear form P 5 a 2 bQ In this case revenue is PQ 5 aQ 2 bQ2 and MR 5 a 2 2bQ Hence profit maximization requires that MR 5 a 2 2bQ 5 MC 5 c or Qm 5 a 2 c 2b 1413 Inserting this solution for the profitmaximizing output level back into the inverse demand func tion yields a direct relationship between price and marginal cost Pm 5 a 2 bQm 5 a 2 a 2 c 2 5 a 1 c 2 1414 To maximize profits this producer chooses that output level for which MR 5 MC To solve this problem we must phrase both MR and MC as functions of Q alone Toward this end write total revenue as P Q 5 100Q 2 Q2 20 148 Consequently MR 5 100 2 Q 10 5 MC 5 01Q 149 and Qm 5 500 Pm 5 75 1410 At the monopolys preferred output level C1Q2 5 005 150022 1 10000 5 22500 ACm 5 22500 500 5 45 1411 Using this information we can calculate profits as πm 5 1Pm 2 ACm2Qm 5 175 2 452 500 5 15000 1412 Observe that at this equilibrium there is a large markup between price 75 and marginal cost 1MC 5 01Q 5 502 Yet as long as entry barriers prevent a new firm from producing Olym picquality Frisbees this gap and positive economic profits can persist indefinitely QUERY How would an increase in fixed costs from 10000 to 12500 affect the monopolys output plans How would profits be affected Suppose total costs shifted to C1Q2 5 0075Q2 1 10000 How would the equilibrium change Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 498 Part 6 Market Power An interesting implication is that in this linear case Pmc 5 12 That is only half of the amount of any increase in marginal cost will show up in the market price of the monopoly product3 Constant elasticity demand If the demand curve facing the monopoly takes the con stant elasticity form Q 5 aP e implying e is the price elasticity of demand then we know MR 5 P 11 1 1e2 and thus profit maximization requires Pma1 1 1 eb 5 c or Pm 5 c a e 1 1 eb 1415 Because it must be the case that e 21 for profit maximization price will clearly exceed mar ginal cost and this gap will be larger the closer e is to 21 Notice also that Pmc 5 e 11 1 e2 and so any given increase in marginal cost will increase price by more than this amount Of course as we have already pointed out the proportional increase in marginal cost and price will be the same That is ePm c 5 1Pmc2 1cPm2 5 1 QUERY The demand function in both cases is shifted by the parameter a Discuss the effects of such a shift for both linear and constant elasticity demand Explain your results intuitively 143 MISALLOCATED RESOURCES UNDER MONOPOLY In Chapter 13 we briefly mentioned why the presence of monopoly distorts the allocation of resources Because the monopoly produces a level of output for which MC 5 MR P the market price of its good no longer conveys accurate information about production costs Hence consumers decisions will no longer reflect true opportunity costs of produc tion and resources will be misallocated In this section we explore this misallocation in some detail in a partialequilibrium context 1431 Basis of comparison To evaluate the allocational effect of a monopoly we need a precisely defined basis of com parison A particularly useful comparison is provided by a perfectly competitive industry It is convenient to think of a monopoly as arising from the capture of such a competitive industry and to treat the individual firms that constituted the competitive industry as now being single plants in the monopolists empire A prototype case would be John D Rocke fellers purchase of most of the US petroleum refineries in the late nineteenth century and his decision to operate them as part of the Standard Oil trust We can then compare the performance of this monopoly with the performance of the previously competitive indus try to arrive at a statement about the welfare consequences of monopoly 1432 A graphical analysis Figure 143 provides a graphical analysis of the welfare effects of monopoly If this mar ket were competitive output would be Qcthat is production would occur where price is equal to longrun average and marginal cost Under a simple singleprice monop oly output would be Qm because this is the level of production for which marginal 3Notice that when c 5 0 we have Pm 5 a2 That is price should be halfway between zero and the price intercept of the demand curve Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 499 revenue is equal to marginal cost The restriction in output from Qc to Qm represents the misallocation brought about through monopolization The total value of resources released by this output restriction is shown in Figure 143 as area FEQcQm Essentially the monopoly closes down some of the plants that were operating in the competitive case These transferred inputs can be productively used elsewhere thus area FEQcQm is not a social loss The restriction in output from Qc to Qm involves a total loss in consumer surplus of PmBEPc Part of this loss PmBCPc is transferred to the monopoly as increased profit Another part of the consumers loss BEC is not transferred to anyone but is a pure dead weight loss in the market A second source of deadweight loss is given by area CEF This is an area of lost producer surplus that does not get transferred to another source4 The total deadweight loss from both sources is area BEF sometimes called the deadweight loss tri angle because of its roughly triangular shape The gain PmBCPc in monopoly profit from 4More precisely region CEF represents lost producer surplus equivalently lost profit if output were reduced holding prices constant at Pc To understand how to measure producer surplus on a graph review the section on producer surplus in Chapter 11 especially Figure 114 Monopolization of this previously competitive market would cause output to be reduced from Qc to Qm Productive inputs worth FEQcQm are reallocated to the production of other goods Consumer surplus equal to PmBCPc is transferred into monopoly profits Deadweight loss is given by BEF Price Quantity per period PQ MRQ MCQ Qc Qm Pm Pc A B C E F G Transfer from consumers to firm Dead weight loss Value of trans ferred inputs FIGURE 143 Allocational and Distributional Effects of Monopoly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 500 Part 6 Market Power an increased price more than compensates for its loss of producer surplus CEF from the output reduction so that overall the monopolist finds reducing output from Qc to Qm to be profitable To illustrate the nature of this deadweight loss consider Example 141 in which we calculated an equilibrium price of 75 and a marginal cost of 50 This gap between price and marginal cost is an indication of the efficiencyimproving trades that are for gone through monopolization Undoubtedly there is a wouldbe buyer who is willing to pay say 60 for an Olympic Frisbee but not 75 A price of 60 would more than cover all the resource costs involved in Frisbee production but the presence of the monopoly prevents such a mutually beneficial transaction between Frisbee users and the providers of Frisbeemaking resources For this reason the monopoly equilibrium is not Pareto optimalan alternative allocation of resources would make all parties better off Economists have made many attempts to estimate deadweight loss in actual industries The estimates have varied wildly depending on sometimes heroic assumptions needed to fill in for variables that cannot be directly measured5 5The classic study is A Harberger Monopoly and Resource Allocation American Economic Review May 1954 7787 Using data from fairly broadly defined industries Harberger estimates that deadweight losses constitute a tiny fraction about 01 percent of gross national product GNP Using more detailed firmlevel data Cowling and Mueller estimate much larger deadweight losses ranging from 4 to 13 percent of GNP See K Cowling and D C Mueller The Social Cost of Monopoly Power Economic Journal December 1978 72748 EXAMPLE 143 Welfare Losses and Elasticity The allocational effects of monopoly can be characterized fairly completely in the case of constant marginal costs and a constant price elasticity demand curve To do so assume again that constant marginal and average costs for a monopolist are given by c and that the demand curve has a constant elasticity form of Q 5 P e 1416 where e is the price elasticity of demand 1e 212 We know the competitive price in this market will be Pc 5 c 1417 and the monopoly price is given by Pm 5 c 1 1 1e 1418 The consumer surplus associated with any price 1P02 can be computed as CS 5 3 q P0 Q 1P2dP 5 3 q P0 P edP 5 P e11 e 1 1 q P0 5 2 P e11 0 e 1 1 1419 Hence under perfect competition we have CSc 5 2 ce11 e 1 1 1420 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 501 and under monopoly CSm 5 2 a c 1 1 1eb e11 e 1 1 1421 Taking the ratio of these two surplus measures yields CSm CSc 5 a 1 1 1 1eb e11 1422 If e 5 22 for example then this ratio is 12 Consumer surplus under monopoly is half what it is under perfect competition For more elastic cases this figure decreases a bit because output restrictions under monopoly are more significant For elasticities closer to 21 the ratio increases Profits The transfer from consumer surplus into monopoly profits can also be computed fairly easily in this case Monopoly profits are given by πm 5 PmQm 2 cQm 5 a c 1 1 1e 2 cbQm 5 a 2ce 1 1 1eb a c 1 1 1eb e 5 2a c 1 1 1eb e11 1 e 1423 Dividing this expression by Equation 1417 yields πm CSc 5 ae 1 1 e ba 1 1 1 1eb e11 5 a e 1 1 eb e 1424 For e 5 22 this ratio is 14 Hence one fourth of the consumer surplus enjoyed under perfect competition is transferred into monopoly profits Therefore the deadweight loss from monopoly in this case is also a fourth of the level of consumer surplus under perfect competition QUERY Suppose e 5 215 What fraction of consumer surplus is lost through monopolization How much is transferred into monopoly profits Why do these results differ from the case e 5 22 144 COMPARATIVE STATICS ANALYSIS OF MONOPOLY The techniques of comparative statics analysis introduced in Chapter 2 and applied for example to study shifts in demand and supply in Chapter 12 can be applied to provide rigorous results about monopoly behavior For the sake of illustration we will prove that a monopolist reduces its output in response to an upward shift in its marginal cost curve This is not a revolutionary new insightwe already discussed the intuition for it previ ously in the chapterbut proving it helps to expand the range of settings in which students are comfortable applying the tools of comparative statics analysis Let marginal cost be given by MC1Q γ2 where γ is some factor shifting the curve up that is MCγ 0 The firstorder condition for the profitmaximizing choice of output from Equation 142 becomes MR 1Q2 2 MC1Q γ2 5 0 1425 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 502 Part 6 Market Power Totally differentiating with respect to γ MRr 1Q2 dQm dγ 2 MC Q dQm dγ 2 MC γ 5 0 1426 Solving for the derivative of interest dQm dγ 5 MCγ MRr 1Q2 2 MCQ 1427 The denominator of the previous equation is nothing other than the second derivative of profits with respect to quantity If the secondorder condition for profitmaximiza tion given in Equation 117 holds the denominator of Equation 1427 must be negative Because the numerator is positive we know dQmdγ 0 Hence an increase in the mar ginal cost shifter reduces monopoly output as was to be shown Using similar methods we could introduce a shifter α into demand as P 1Q α2 and try to see how Qm varies with α assuming say that Pα 0 Unfortunately the compar ative statics predictions from this exercise are ambiguous As Equation 1425 might sug gest what matters for the result is not how P 1Q α2 varies with the parameter but how MR 1Q α2 5 P 1Q α2 1 Q PQ does If the increase in α makes the inverse demand curve steeper ie makes PQ more negative at the same it shifts the curve P 1Q α2 out α will have an ambiguous effect on MR 1Q α2 and thus an ambiguous effect on Qm We could derive unequivocal comparative statics results based on direct assumptions on how α shifts MR 1Q α2 but these results would not be very useful given we usually lack strong intuition about how factors such as α should shift marginal revenue 145 MONOPOLY PRODUCT QUALITY The market power enjoyed by a monopoly may be exercised along dimensions other than the market price of its product If the monopoly has some leeway in the type quality or diversity of the goods it produces then it would not be surprising for the firms decisions to differ from those that might prevail under a competitive organization of the market Whether a monopoly will produce higherquality or lowerquality goods than would be produced under competition is unclear however It all depends on the firms costs and the nature of consumer demand 1451 A formal treatment of quality Suppose consumers willingness to pay for a good of quality X is given by the inverse demand function P 1Q X2 where PQ 5 PQ 0 inverse demand is downward slop ing as usual and PX 5 PX 0 consumers desire quality Let C1Q X2 be the cost of producing Q units of quality X with CQ 5 CQ 0 and CX 5 CX 0 more and better output is costlier to produce First consider the monopolys decisions The monopolist chooses Q and X to maximize π 5 P 1Q X2Q 2 C1Q X2 1428 The optimal output Qm and quality Xm for the monopolist can be found by solving the system of firstorder conditions πQ 5 0 and πX 5 0 To avoid the complications of solving this system of two equations for the two unknown choice variables and to help Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 503 focus on the quality choice imagine we are told the value of Qm and asked just to solve for Xm We can do this by solving the single firstorder condition for X π X 5 PX1Qm X2Qm 2 CX1Qm X2 50 1429 This equation says that just like optimal monopoly quantity optimal monopoly quality satisfies a marginal revenue equals marginal cost condition only here the marginal reve nue PX1Qm X2Qm is that associated with a quality increase and the marginal cost CX1Qm X2 is likewise that associated with a quality increase The marginal revenue from a quality increase is the product of two factors the price increase PX1Qm X2 that can be extracted from the marginal demander is multiplied by the number of units sold Qm because the price increase can be charged to all these demanders To compare the monopolys to the efficient choice imagine now that quality is chosen by a social planner who maximizes social welfare SW the sum of profit π and consumer surplus CS To ensure an applestoapples comparison suppose the social planner leaves Qm unchanged and focuses only on setting X A bit of algebra can help clarify the social planners objective SW 5 π 1 CS 5 PmQm 2 C1Qm X2 1 3 Qm 0 3P 1Q X2 2 Pm4dQ 5 3 Qm 0 P 1Q X2dQ 2 C1Qm X2 1430 To understand the integral expression for CS this is Marshallian consumer surplus which can be seen in Figure 143 as the area of the roughly triangular region ABPm which can be computed by integrating the difference between two curves inverse demand P 1Q X2 and the horizontal line of height Pm Canceling PmQm from outside and inside the integral this is revenue which cancels out because it is just a transfer of surplus from consumers to the producer leaves the final equality Differentiation of Equation 1430 with respect to X yields the firstorder condition for a maximum SW X 5 3 Qm 0 PX1Q X2dQ 2 CX1Qm X2 5 0 1431 The monopolists choice of quality in Equation 1429 targets the marginal consumer The monopolist cares about the marginal consumers valuation of quality because increas ing the attractiveness of the product to the marginal consumer is how it increases sales By contrast the efficient quality chosen by the social planner maximizes consumer surplus across all buyers which given output is kept constant at Qm is equivalent to maximizing the average consumer surplus across buyers We can now see that whether the monop olist sets the quality level too high or too low is ambiguous If the marginal consumer is more responsive to quality than the average consumer the monopolist will choose an inef ficiently high quality If the marginal consumer cares less about quality than the average consumer the monopolist will choose an inefficiently low quality Only by knowing specif ics about the market is it possible to predict the direction for example see Problem 149 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 504 Part 6 Market Power 1452 Product durability Much of the research on the effect of monopolization on quality has focused on durable goods These are goods such as automobiles houses or refrigerators that provide services to their owners over several periods rather than being completely consumed soon after they are bought The element of time that enters into the theory of durable goods leads to many interesting problems and paradoxes Initial interest in the topic started with the question of whether monopolies would produce goods that lasted as long as similar goods produced under perfect competition The intuitive notion that monopolies would under produce durability just as they choose an output below the competitive level was soon shown to be incorrect by the Australian economist Peter Swan in the early 1970s6 Swans insight was to view the demand for durable goods as the demand for a flow of services ie automobile transportation over several periods He argued that both a monopoly and a competitive market would seek to minimize the cost of providing this flow to consumers The monopoly would of course choose an output level that restricted the flow of services to maximize profits butassuming constant returns to scale in productionthere is no reason that durability per se would be affected by mar ket structure This result is sometimes referred to as Swans independence assumption Output decisions can be treated independently from decisions about product durability Subsequent research on the Swan result has focused on showing how it can be under mined by different assumptions about the nature of a particular durable good or by relaxing the implicit assumption that all demanders are the same For example the result depends critically on how durable goods deteriorate The simplest type of deterioration is illus trated by a durable good such as a lightbulb that provides a constant stream of services until it becomes worthless With this type of good Equations 1429 and 1431 are identical so Swans independence result holds Even when goods deteriorate smoothly the indepen dence result continues to hold if a constant flow of services can be maintained by simply replacing what has been usedthis requires that new goods and old goods be perfect substi tutes and infinitely divisible Outdoor house paint may more or less meet this requirement On the other hand most goods clearly do not It is just not possible to replace a rundown refrigerator with say half of a new one Once such more complex forms of deterioration are considered Swans result may not hold because we can no longer fall back on the notion of providing a given flow of services at minimal cost over time In these more complex cases however it is not always the case that a monopoly will produce less durability than will a competitive marketit all depends on the nature of the demand for durability 146 PRICE DISCRIMINATION In some circumstances a monopoly may be able to increase profits by departing from a sin gleprice policy for its output The possibility of selling identical goods at different prices is called price discrimination7 6P L Swan Durability of Consumption Goods American Economic Review December 1970 88494 7A monopoly may also be able to sell differentiated products at differential pricecost margins Here however we treat price discrimination only for a monopoly that produces a single homogeneous product Price discrimination is an issue in other imperfectly competitive markets besides monopoly but is easiest to study in the simple case of a single firm D E F I N I T I O N Price discrimination A monopoly engages in price discrimination if it is able to sell otherwise identical units of output at different prices Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 505 Examples of price discrimination include senior citizen discounts for restaurant meals which could instead be viewed as a price premium for younger customers coffee sold at a lower price per ounce when bought in larger cup sizes and different net tuition charged to different college students after subtracting their more or less generous financial aid awards A nonexample of price discrimination might be higher auto insurance premiums charged to younger drivers It might be clearer to think of the insurance policies sold to younger and older drivers as being different products to the extent that younger drivers are riskier and result in many more claims having to be paid Whether a price discrimination strategy is feasible depends crucially on the inability of buyers of the good to practice arbitrage In the absence of transactions or information costs the law of one price implies that a homogeneous good must sell everywhere for the same price Consequently price discrimination schemes are doomed to failure because demanders who can buy from the monopoly at lower prices will be more attractive sources of the goodfor those who must pay high pricesthan is the monopoly itself Profit seeking middlemen will destroy any discriminatory pricing scheme However when resale is costly or can be prevented entirely then price discrimination becomes possible 1461 Perfect price discrimination If each buyer can be separately identified by a monopolist then it may be possible to charge each the maximum price he or she would willingly pay for the good This strategy of per fect price discrimination sometimes called firstdegree price discrimination would then extract all available consumer surplus leaving demanders as a group indifferent between buying the monopolists good or doing without it The strategy is illustrated in Figure 144 Under perfect price discrimination the monopoly charges a different price to each buyer It sells Q1 units at P1 Q2 2 Q1 units at P2 and so forth In this case the firm will produce Q and total revenues will approach AEQ0 Price Quantity per period A P1 P2 Q1 Q2 0 PQ E Q MC FIGURE 144 Perfect Price Discrimination Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 506 Part 6 Market Power The figure assumes that buyers are arranged in descending order of willingness to pay The first buyer is willing to pay up to P1 for Q1 units of output therefore the monopolist charges P1 and obtains total revenues of P1Q1 as indicated by the lightly shaded rectan gle A second buyer is willing to pay up to P2 for Q2 2 Q1 units of output therefore the monopolist obtains total revenue of P2 1Q2 2 Q12 from this buyer Notice that this strategy cannot succeed unless the second buyer is unable to resell the output he or she buys at P2 to the first buyer who pays P1 P2 The monopolist will proceed in this way up to the marginal buyer the last buyer who is willing to pay at least the goods marginal cost labeled MC in Figure 144 Hence total quantity produced will be Q Total revenues collected will be given by the area AEQ0 All consumer surplus has been extracted by the monopolist and there is no deadweight loss in this situation Compare Figures 143 and 144 Therefore the allocation of resources under perfect price discrimination is efficient although it does entail a large transfer from consumer surplus into monopoly profits EXAMPLE 144 Perfect Price Discrimination Consider again the Frisbee monopolist in Example 141 Because there are relatively few highquality Frisbees sold the monopolist may find it possible to discriminate perfectly among a few worldclass flippers In this case it will choose to produce that quantity for which the mar ginal buyer pays exactly the marginal cost of a Frisbee P 5 100 2 Q 20 5 MC 5 01Q 1432 Hence Q 5 666 and at the margin price and marginal cost are given by P 5 MC 5 666 1433 Now we can compute total revenues by integration R 5 3 Q 0 P 1Q2dQ 5 a100Q 2 Q2 40b Q5666 Q50 5 55511 1434 Total costs are C1Q2 5 005Q2 1 10000 5 32178 1435 total profits are given by π 5 R 2 C 5 23333 1436 which represents a substantial increase over the singleprice policy examined in Example 141 which yielded 15000 QUERY What is the maximum price any Frisbee buyer pays in this case Use this to obtain a geometric definition of profits Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 507 1462 Price discrimination across segmented markets Perfect price discrimination poses a considerable information burden for the monopoly it must know the demand function for each potential buyer A less stringent requirement would be to assume the monopoly can segment its buyers into relatively few identifiable markets such as ruralurban domesticforeign or primetimeoffprime and pur sue a separate monopoly pricing policy in each market This pricing strategy is sometimes called thirddegree price discrimination following an historical classification scheme Knowledge of the price elasticities of demand in these markets is sufficient to pursue such a policy The monopoly then sets a price in each market according to the inverse elasticity rule Assuming that marginal cost is the same in all markets the result is a pricing policy in which Pi a1 1 1 ei b 5 Pj a1 1 1 ej b 1437 or Pi Pj 5 11 1 1ej2 11 1 1ei2 1438 where Pi and Pj are the prices charged in markets i and j which have price elasticities of demand given by ei and ej An immediate consequence of this pricing policy is that the profitmaximizing price will be higher in markets in which demand is less elastic If for example ei 5 22 and ej 5 23 then Equation 1438 shows that PiPj 5 43prices will be one third higher in market i the less elastic market Figure 145 illustrates this result for two markets that the monopoly can serve at con stant marginal cost MC Demand is less elastic in market 1 than in market 2 thus the gap between price and marginal revenue is larger in the former market Profit maximi zation requires that the firm produce Q 1 in market 1 and Q 2 in market 2 resulting in a higher price in the less elastic market As long as arbitrage between the two markets can be prevented this price difference can persist The twoprice discriminatory policy is clearly more profitable for the monopoly than a singleprice policy would be because the firm can always opt for the latter policy should market conditions warrant The welfare consequences of price discrimination across segmented markets are in principle ambiguous If the total amount sold in the two markets is the same under price discrimination as under a single price then the single price will generally lead to higher wel fare This is because a single price does a better job allocating output to the consumers who value it the most than two prices There will always be some consumers who are denied the good in the highprice market who value it more than some consumers who end up pur chasing in the lowprice market Welfare could be raised by reallocating the good between such consumers Under a single price there is never a need for such reallocation because all consumers who end up with the good value it more than those who do not A possible offsetting effect is that price discrimination can in some cases increase total output sold across the markets Example 145 provides such a case If forced to charge a sin gle price in this example the monopolist sets it so high that no one in market 2 is served When price discrimination is allowed the monopolist charges the monopoly price in this market which leads to higher welfare than when it is excluded entirely This is only a pos sible effectnot guaranteedbecause price discrimination is not guaranteed to increase output only doing so in some cases Taking these potentially offsetting effects together what can we conclude about the wel fare effects of price discrimination across segmented markets One can be sure that welfare Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 508 Part 6 Market Power falls under price discrimination if total output does not increase compared to a single price If price discrimination increases total output then one cannot be sure about welfare without further detailed computations The conclusions for consumer surplus are similar The fundamental ambiguity of the welfare and consumersurplus effects of price discrimina tion was highlighted by an important recent article8 Rather than taking two segmented markets as in Example 145 and asking what happens to welfare when the markets are combined and a single price charged the article takes the opposite perspective starting with a single market and examining different ways to divide those consumers into market segments across which the monopolist can then price discriminate The results on perfect price discrimination should already suggest how extreme segmentations can generate large swings in surplus Imagine seg menting each consumer value into its own market The monopolist could then approximate perfect price discrimination by charging the suitable price to each of these numerous tiny mar kets Compared to a single price charged to the combined market price discrimination across this segmentation increases welfare approximating the efficient level under perfect competition while eliminating all consumer surplus The article demonstrates many other possibilities For any initial demand curve there is a segmentation that as perfect price discrimination eliminates all consumer surplus however instead of shifting to the monopolist the consumer surplus is destroyed so the monopolist earns no more than it did under a single price No segmentation can reduce its profit below the singleprice level because the monopolist can always recover that profit by charging the optimal single price on all segmented markets Another segmentation maintains the same profit as under a single price but increases social welfare all the way up to the 8D Bergemann B Brooks and S Morris The Limits of Price Discrimination American Economic Review March 2015 92157 If two markets are separate then a monopolist can maximize profits by selling its product at different prices in the two markets This would entail choosing that output for which MC 5 MR in each of the markets The diagram shows that the market with a less elastic demand curve will be charged the higher price by the price discriminator Price 0 P1Q1 P 1 P 2 Q 2 Q 1 Quantity in market 2 Quantity in market 1 MR1Q1 MC MR2Q2 P2Q2 FIGURE 145 Price Discrimination across Segmented Markets Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 509 perfectly competitive efficient level so that consumers receive all the surplus gains For every surplus allocation between these extremes there is a segmentation that can support it as an out come Problem 1412 guides you through the construction of some of these segmentations In real markets a monopolist is not free to segment consumers any way it likes seg mentation is determined by the nature of geography or other identifiable characteristics However this discussion shows that there is nothing about a given set of consumer values in a market that precludes price discrimination from having a wide range of effects on the level of welfare and its distribution between producer and consumers EXAMPLE 145 Price Discrimination across Segmented Markets Suppose that a monopoly producer of widgets has a constant marginal cost of c 5 6 and sells its products in two separated markets whose inverse demand functions are P1 5 24 2 Q1 and P2 5 12 2 05Q2 1439 Notice that consumers in market 1 are more eager to buy than are consumers in market 2 in the sense that the former are willing to pay more for any given quantity Using the results for linear demand curves from Example 142 shows that the profitmaximizing pricequantity combina tions in these two markets are P 1 5 24 1 6 2 5 15 Q 1 5 9 P 2 5 12 1 6 2 5 9 Q 2 5 6 1440 With this pricing strategy profits are π 5 115 2 62 9 1 19 2 62 6 5 81 1 18 5 99 We can compute the deadweight losses in the two markets with the help of Figure 146 which shows This figure provides a graphical representation of thirddegree price discrimination as in Figure 145 but for the special case of the widget markets in the present numerical example The numbers along the axes can be used to compute the area of the shaded deadweightloss triangles FIGURE 146 Scale drawing of the two widget markets in numerical example Price 0 9 12 15 24 6 12 9 18 P1Q1 Quantity in market 2 Quantity in market 1 MR1Q1 MC MR2Q2 P2Q2 DW1 DW2 6 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 510 Part 6 Market Power 147 PRICE DISCRIMINATION THROUGH NONUNIFORM SCHEDULES The examples of price discrimination examined in the previous section require the monop oly to separate demanders into a number of identifiable categories and then choose a profitmaximizing price for each such category An alternative approach would be for the monopoly to choose a possibly rather complex price schedule that provides incentives for demanders to separate themselves depending on how much they wish to buy Such schemes include quantity discounts minimum purchase requirements or cover charges and tiein sales These plans would be adopted by a monopoly if they yielded greater profits than would a uniform price per unit after accounting for any possible costs of implement ing the price schedule Because the schedules will result in demanders paying different aver age prices per unit for identical goods this form of price discrimination sometimes called second degree price discrimination following an historical classification scheme is feasible only when there are no arbitrage possibilities Here we look at one simple case The Exten sions to this chapter and portions of Chapter 18 look at more complex nonuniform schemes 1471 Twopart tariffs One form of pricing schedule that has been extensively studied is a twopart tariff under which demanders must pay a fixed fee for the right to consume a good in addition to a uniform price for each unit consumed The prototype case first studied by Walter Oi is the dimensions of the shaded deadweightloss triangles whose areas can be computed using the usual formula DW 5 DW1 1 DW2 5 1 2 115 2 92 118 2 92 1 1 2 19 2 62 112 2 62 1441 5 405 1 9 5 495 A singleprice policy In this case constraining the monopoly to charge a single price would reduce welfare Under a singleprice policy the monopoly would simply cease serving market 2 because it can maximize profits by charging a price of 15 and at that price no widgets will be bought in market 2 because the maximum willingness to pay is 12 Therefore total deadweight loss in this situation is increased from its level in Equation 1441 because total potential consumer surplus in market 2 is now lost DW 5 DW1 1 DW2 5 405 1 1 2 112 2 62 112 2 02 5 405 1 36 5 765 1442 This illustrates a situation where price discrimination is welfare improving over a singleprice policywhen the discriminatory policy permits smaller markets to be served Whether such a situation is common is an important policy question eg consider the case of US pharmaceuti cal manufacturers charging higher prices at home than abroad QUERY Suppose you werent told that under a singleprice policy the monopolist maximizes its profits by serving only market 1 at a price of 15 A natural approach to solving the problem would be to combine the two linear demands to obtain a market demand substitute this demand into the profit function and solve the resulting firstorder condition What price and profit does this method yield Why doesnt this method work in finding the true solution a price of 15 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 511 an amusement park perhaps Disneyland that sets a basic entry fee coupled with a stated marginal price for each amusement used9 Mathematically this scheme can be represented by the tariff any demander must pay to purchase q units of a good T1q2 5 a 1 pq 1443 where a is the fixed fee and p is the marginal price to be paid The monopolists goal then is to choose a and p to maximize profits given the demand for this product Because the average price paid by any demander is given by p 5 T q 5 a q 1 p 1444 this tariff is feasible only when those who pay low average prices those for whom q is large cannot resell the good to those who must pay high average prices those for whom q is small One approach described by Oi for establishing the parameters of this linear tariff would be for the firm to set the marginal price p equal to MC and then set a to extract the max imum consumer surplus from a given set of buyers One might imagine buyers being arrayed according to willingness to pay The choice of p 5 MC would then maximize con sumer surplus for this group and a could be set equal to the surplus enjoyed by the least eager buyer He or she would then be indifferent about buying the good but all other buy ers would experience net gains from the purchase This feasible tariff might not be the most profitable however Consider the effects on profits of a small increase in p above MC This would result in no net change in the profits earned from the least willing buyer Quantity demanded would drop slightly at the margin where p 5 MC and some of what had previously been consumer surplus and therefore part of the fixed fee a would be converted into variable profits because now p MC For all other demanders profits would be increased by the price increase Although each will pay a bit less in fixed charges profits per unit bought will increase to a greater extent10 In some cases it is possible to make an explicit calculation of the optimal twopart tariff Example 146 provides an illustration More generally however optimal schedules will depend on a variety of contingencies Some of the possibilities are examined in the Extensions to this chapter To illustrate the mathematics of twopart tariffs lets return to the demand equations introduced in Example 145 but now assume that they apply to two specific demanders q1 5 24 2 p1 1445 q2 5 24 2 2p2 where now the ps refer to the marginal prices faced by these two buyers11 9W Y Oi A Disneyland Dilemma TwoPart Tariffs for a Mickey Mouse Monopoly Quarterly Journal of Economics February 1971 7790 Interestingly the Disney empire once used a twopart tariff but abandoned it because the costs of administering the payment schemes for individual rides became too high Like other amusement parks Disney moved to a singleadmissions price policy which still provided them with ample opportunities for price discrimination especially with the multiple parks at Disney World 10This follows because qi1MC2 q11MC2 where qi1MC2 is the quantity demanded when p 5 MC for all except the least willing buyer person 1 Hence the gain in profits from an increase in price above MC Dpqi1MC2 exceeds the loss in profits from a smaller fixed fee Dpq11MC2 11The theory of utility maximization that underlies these demand curves is that the quantity demanded is determined by the marginal price paid whereas the entry fee a determines whether q 5 0 might instead be optimal EXAMPLE 146 TwoPart Tariffs Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 512 Part 6 Market Power 1472 Dynamic price discrimination and the Coase conjecture One might think that interesting opportunities for dynamic price discrimination arise if the monopolist can target consumers with different values for the product by varying the path of prices over time Take the case of highdefinition televisions A monopolist in the next generation of yet higherdefinition televisions might consider selling at an extremely high price to rich or techsavvy people who want the latest gadgets Given televisions are durable once these consumers have purchased they will be taken off the television mar ket at least for a while leaving a creamskimmed market with lower demanders The monopolist can then target the highest demanders remaining in the market next period at a slightly lower price and so on effectively using time as a device to extract much of the surplus from each type of consumer Surprisingly the possibility of intertemporal price discrimination may not help the monopolist indeed the very opposite may be true Rather than having all their surplus extracted anticipating the price drop to come highvalue consumers can wait to buy at the lower prices reducing demand early on causing the monopolists plans to unravel Ronald Coase was the first economist to note the problem that a declining price path raises for a An Oi tariff Implementing the twopart tariff suggested by Oi would require the monopolist to set p1 5 p2 5 MC 5 6 Hence in this case q1 5 18 and q2 5 12 With this marginal price demander 2 the less eager of the two obtains consumer surplus of 36 35 05 112 2 62 124 That is the maximal entry fee that might be charged without causing this person to leave the mar ket Consequently the twopart tariff in this case would be T 1q2 5 36 1 6q If the monopolist opted for this pricing scheme its profits would be π 5 R 2 C 5 T 1q12 1 T 1q22 2 AC1q1 1 q22 1446 5 72 1 6 30 2 6 30 5 72 These fall short of those obtained in Example 145 The optimal tariff The optimal twopart tariff in this situation can be computed by noting that total profits with such a tariff are π 5 2a 1 1 p 2 MC2 1q1 1 q22 Here the entry fee a must equal the consumer surplus obtained by person 2 Inserting the specific parameters of this problem yields π 5 05 2q2 112 2 p2 1 1 p 2 62 1q1 1 q22 5 124 2 2p2 112 2 p2 1 1 p 2 62 148 2 3p2 1447 5 18p 2 p2 Hence maximum profits are obtained when p 5 9 and a 5 05 124 2 2p2 112 2 p2 5 9 There fore the optimal tariff is T 1q2 5 9 1 9q With this tariff q1 5 15 and q2 5 6 and the monop olists profits are 81 35 2 192 1 19 2 62 115 1 62 4 The monopolist might opt for this pricing scheme if it were under political pressure to have a uniform pricing policy and to agree not to price demander 2 out of the market The twopart tariff permits a degree of differential pricing 1 p1 5 960 p2 5 9752 but appears fair because all buyers face the same schedule QUERY Suppose a monopolist could choose a different entry fee for each demander What pric ing policy would be followed Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 513 monopolist selling a durable good12 Coase argued that its presence would severely undercut potential monopoly power as the monopolist in one period is effectively competing against its own lower prices in future periods His conjecture which was later formally proved13 is that in the limit as the time between periods becomes very short the outcome would converge to that under perfect competition with most consumers buying at a price near marginal cost Only if the monopoly can credibly commit either not to lower price or not to produce any more in the future can it hope to get back to the monopoly profit absent price discrimination Price protection guarantees rebating a consumer the difference if price falls after he or she buys capacity constraints limited issues rental rather than sales and so forth are strategies that can help the monopolist preserve some of its commitment power14 148 REGULATION OF MONOPOLY The regulation of natural monopolies is an important subject in applied economic analysis The utility communications and transportation industries are highly regulated in most countries and devising regulatory procedures that induce these industries to operate in a desirable way is an important practical problem Here we will examine a few aspects of the regulation of monopolies that relate to pricing policies 1481 Marginal cost pricing and the natural monopoly dilemma Many economists believe it is important for the prices charged by regulated monopolies to reflect marginal costs of production accurately In this way the deadweight loss may be minimized The principal problem raised by an enforced policy of marginal cost pricing is that it will require natural monopolies to operate at a loss Natural monopolies by defini tion exhibit decreasing average costs over a broad range of output levels The cost curves for such a firm might look like those shown in Figure 147 In the absence of regulation the monopoly would produce output level QA and receive a price of PA for its product Profits in this situation are given by the rectangle PAABC A regulatory agency might instead set a price of PR for the monopoly At this price QR is demanded and the marginal cost of pro ducing this output level is also PR Consequently marginal cost pricing has been achieved Unfortunately because of the negative slope of the firms average cost curve the price PR 15 marginal cost2 decreases below average costs With this regulated price the monop oly must operate at a loss of GFEPR Because no firm can operate indefinitely at a loss this poses a dilemma for the regulatory agency Either it must abandon its goal of marginal cost pricing or the government must subsidize the monopoly forever 1482 Twotier pricing systems One way out of the marginal cost pricing dilemma is the implementation of a multiprice system Under such a system the monopoly is permitted to charge some users a high price 12R Coase Durability and Monopoly Journal of Law and Economics April 1972 14349 Coases insight is not restricted to durable goods it also applies for example to an entertainment say novel or movie that a person typically enjoys once rather than repeatedly The monopolist may want to sell at a high price to diehard first and then a declining path of prices to lower demand consumers but consumers anticipation and reaction to this plan can unravel it 13It was proved by Nancy Stokey in Rational Expectations and Durable Goods Pricing Bell Journal of Economics Spring 1981 11228 14For a summary of more recent ideas on the durablegood monopoly problem see M Waldman Durable Goods Theory for Real World Markets Journal of Economic Perspectives Winter 2003 13154 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 514 Part 6 Market Power while maintaining a low price for marginal users In this way the demanders paying the high price in effect subsidize the losses of the lowprice customers Such a pricing scheme is shown in Figure 148 Here the regulatory commission has decided that some users will pay a relatively high price P1 At this price Q1 is demanded Other users presumably those who would not buy the good at the P1 price are offered a lower price P2 This lower price generates additional demand of Q2 2 Q1 Consequently a total output of Q2 is produced at an average cost of A With this pricing system the profits on the sales to highprice demanders given by the rectangle P1DBA balance the losses incurred on the lowpriced sales BFEC Furthermore for the marginal user the marginal cost pricing rule is being followed It is the inframarginal user who subsidizes the firm so it does not operate at a loss Although in practice it may not be so simple to establish pricing schemes that main tain marginal cost pricing and cover operating costs many regulatory commissions do use price schedules that intentionally discriminate against some users eg businesses to the advantage of others consumers 1483 Rate of return regulation Another approach followed in many regulatory situations is to permit the monopoly to charge a price above marginal cost that is sufficient to earn a fair rate of return on invest ment Much analytical effort is then devoted to defining the fair rate concept and to devel oping ways in which it might be measured From an economic point of view some of the Because natural monopolies exhibit decreasing average costs marginal costs decrease below average costs Consequently enforcing a policy of marginal cost pricing will entail operating at a loss A price of PR for example will achieve the goal of marginal cost pricing but will necessitate an operating loss of GFEPR Price Quantity per period F E B A MR AC MC D D PA PR QA QR C G FIGURE 147 Price Regulation for a Decreasing Cost Monopoly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 515 most interesting questions about this procedure concern how the regulatory activity affects the firms input choices If for example the rate of return allowed to firms exceeds what own ers might obtain on investment under competitive circumstances there will be an incentive to use relatively more capital input than would truly minimize costs And if regulators delay in making rate decisions this may give firms costminimizing incentives that would not otherwise exist We will now briefly examine a formal model of such possibilities15 1484 A formal model Suppose a regulated utility has a production function of the form q 5 f 1k l2 1448 This firms actual rate of return on capital is then defined as s 5 pf 1k l2 2 wl k 1449 where p is the price of the firms output which depends on q and w is the wage rate for labor input If s is constrained by regulation to be equal to say s then the firms problem is to maximize profits 15This model is based on H Averch and L L Johnson Behavior of the Firm under Regulatory Constraint American Economic Review December 1962 105269 By charging a high price 1P12 to some users and a low price 1P22 to others it may be possible for a regu latory commission to 1 enforce marginal cost pricing and 2 create a situation where the profits from one class of user 1P1DBA2 subsidize the losses of the other class BFEC Price Quantity per period F E B D C AC MC D P1 P2 Q1 Q2 A FIGURE 148 TwoTier Pricing Schedule Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 516 Part 6 Market Power π 5 pf 1k l2 2 wl 2 vk 1450 subject to this regulatory constraint The Lagrangian for this problem is 5 pf 1k l2 2 wl 2 vk 1 λ 3wl 1 sk 2 pf 1k l2 4 1451 Notice that if λ 5 0 regulation is ineffective and the monopoly behaves like any profitmaximizing firm If λ 5 1 Equation 1451 reduces to 5 1s 2 v2k 1452 which assuming s v which it must be if the firm is not to earn less than the prevailing rate of return on capital elsewhere means this monopoly will hire infinite amounts of capitalan implausible result Hence 0 λ 1 The firstorder conditions for a maximum are l 5 pfl 2 w 1 λ 1w 2 pf12 5 0 k 5 pfk 2 v 1 λ 1s 2 pfk2 5 0 1453 λ 5 wl 2 1 sk 2 pf 1k l2 5 0 The first of these conditions implies that the regulated monopoly will hire additional labor input up to the point at which pfl 5 w a result that holds for any profitmaximizing firm For capital input however the second condition implies that 11 2 λ2pfk 5 v 2 λs 1454 or pfk 5 v 2 λ s 1 2 λ 5 v 2 λ 1s 2 v2 1 2 λ 1455 Because s v and λ 1 Equation 1455 implies pfk v 1456 The firm will hire more capital and achieve a lower marginal productivity of capital than it would under unregulated conditions Therefore overcapitalization may be a reg ulatoryinduced misallocation of resources for some utilities Although we shall not do so here it is possible to examine other regulatory questions using this general analytical framework 149 DYNAMIC VIEWS OF MONOPOLY The static view that monopolistic practices distort the allocation of resources provides the principal economic rationale for favoring antimonopoly policies Not all economists believe that the static analysis should be definitive however Some authors most nota bly J A Schumpeter have stressed the beneficial role that monopoly profits can play in the process of economic development16 These authors place considerable emphasis on 16See for example J A Schumpeter Capitalism Socialism and Democracy 3rd ed New York Harper Row 1950 especially chap 8 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 517 innovation and the ability of particular types of firms to achieve technical advances In this context the profits that monopolistic firms earn provide funds that can be invested in research and development Whereas perfectly competitive firms must be content with a normal return on invested capital monopolies have surplus funds with which to under take the risky process of research More important perhaps the possibility of attaining a monopolistic positionor the desire to maintain such a positionprovides an import ant incentive to keep one step ahead of potential competitors Innovations in new prod ucts and costsaving production techniques may be integrally related to the possibility of monopolization To take an extreme case imagine that a firm needs to undertake an expensive research program to develop a new product Unless the firm has some way to prevent others from copying the product competitors enter the market at will competing away much of the innovators profit Without the promise of profit from its innovation the firm may lose much of its incentive to invest in the creation of the product in the first place The firm must have some way to prevent the idea behind its innovation from being easily copied In some cases a new product may be too complex for anyone to reverse engineer and the firm may maintain the design and production process as trade secrets Some products are by nature easily copied however for example pharmaceutical manufacturers are required by law to state the active ingredients in their products essentially telling potential competitors what makes their product work For such product some other protection must be found to preserve innovation incentives Certain inventions can secure a government patent which forbids competitors from using the idea in their own products for a period 20 years in the United States Other intellectualproperty protections include copyright protecting pub lished works such a song or this textbook and trademarks protecting brand names How long these intellectualproperty protections should last and what innovations they should cover raise complex issues The protections introduce a tradeoff between the dynamic incentives to create new products and the static monopoly pricing distortions they create by preventing competitive entry at least for the temporary period they are in force The dynamic investment incentives need not be socially efficient but can be insuffi cient or excessive depending on the circumstances The protection rewards an innovator with the promise of monopoly profit over the duration of the protection The reward can be measured in Figure 143 as the area of region PmBFG However the innovation also creates consumer surplus equal to the area of the roughly triangular region ABPm in the figure a benefit to society that the firm cannot capture The inability to capture the con sumer surplus is a factor that weighs on the side of insufficient innovation incentives On the other hand the race to obtain the protection may lead to excessive investment incen tives Having a new product one day sooner hardly matters to society in the grand scheme of things To the firms one day can mean the difference between winning and losing the patent and the associated monopoly profits Firms may race so hard for a patent that they dissipate much of the potential gains in the process Whether the benefits of monopoly purported by Schumpeter and others outweigh their allocational and distributional disadvantages whether patents and other intel lectualproperty protections are too short or too long whether there is too little or too much innovative activity in the overall economythese are all questions that cannot be answered by recourse to a priori arguments They are empirical questions requiring detailed investigation of realworld markets17 17An excellent example of excellent recent empirical work on innovation incentives is provided by E Budish B Roin and H Williams Do Firms Underinvest in LongTerm Research Evidence from Cancer Clinical Trials American Economic Review July 2015 204485 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Summary In this chapter we have examined models of markets in which there is only a single monopoly supplier Unlike the competi tive case investigated in Part 4 monopoly firms do not exhibit pricetaking behavior Instead the monopolist can choose the pricequantity combination on the market demand curve that is most profitable A number of consequences then follow from this market power The most profitable level of output for the monopolist is the one for which marginal revenue is equal to marginal cost At this output level price will exceed marginal cost The profitability of the monopolist will depend on the relationship between price and average cost Relative to perfect competition monopoly involves a loss of consumer surplus for demanders Some of this is transferred into monopoly profits whereas some of the loss in consumer supply represents a deadweight loss of overall economic welfare Monopolists may opt for higher or lower levels of quality than would perfectly competitive firms depending on the circumstances A monopoly may be able to increase its profits further through price discriminationthat is charging dif ferent prices to different buyers based in part on their valuations Various strategies can be used including seg menting markets based on identifiable characteristics or letting buyers sort themselves on a nonuniform price schedule The ability of the monopoly to practice price discrimination depends on its ability to prevent arbitrage among buyers Governments often choose to regulate natural monop olies firms with diminishing average costs over a broad range of output levels The type of regulatory mech anisms adopted can affect the behavior of the regulated firm The deadweight loss from high monopoly prices can be dwarfed in the long run by dynamic gains if monopolies can be shown to be more innovative than competitive firms still an open empirical question Problems 141 A monopolist can produce at constant average and marginal costs of AC 5 MC 5 5 The firm faces a market demand curve given by Q 5 53 2 P a Calculate the profitmaximizing pricequantity combi nation for the monopolist Also calculate the monopo lists profits b What output level would be produced by this industry under perfect competition 1where price 5 marginal cost2 c Calculate the consumer surplus obtained by consum ers in case b Show that this exceeds the sum of the monopolists profits and the consumer surplus received in case a What is the value of the deadweight loss from monopolization 142 A monopolist faces a market demand curve given by Q 5 70 2 p a If the monopolist can produce at constant average and marginal costs of AC 5 MC 5 6 what output level will the monopolist choose to maximize profits What is the price at this output level What are the monopolists profits b Assume instead that the monopolist has a cost structure where total costs are described by C1Q2 5 025Q2 2 5Q 1 300 With the monopolist facing the same market demand and marginal revenue what pricequantity combination will be chosen now to maximize profits What will prof its be c Assume now that a third cost structure explains the monopolists position with total costs given by C1Q2 5 00133Q3 2 5Q 1 250 Again calculate the monopolists pricequantity combi nation that maximizes profits What will profit be Hint Set MC 5 MR as usual and use the quadratic formula to solve the secondorder equation for Q d Graph the market demand curve the MR curve and the three marginal cost curves from parts a b and c Notice that the monopolists profitmaking ability is con strained by 1 the market demand curve along with its associated MR curve and 2 the cost structure underly ing production 518 Part 6 Market Power Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 143 A single firm monopolizes the entire market for widgets and can produce at constant average and marginal costs of AC 5 MC 5 10 Originally the firm faces a market demand curve given by Q 5 60 2 P a Calculate the profitmaximizing pricequantity combi nation for the firm What are the firms profits b Now assume that the market demand curve shifts out ward becoming steeper and is given by Q 5 45 2 05P What is the firms profitmaximizing pricequantity com bination now What are the firms profits c Instead of the assumptions of part b assume that the market demand curve shifts outward becoming flatter and is given by Q 5 100 2 2P What is the firms profitmaximizing pricequantity combination now What are the firms profits d Graph the three different situations of parts a b and c Using your results explain why there is no real sup ply curve for a monopoly 144 Suppose the market for Hula Hoops is monopolized by a sin gle firm a Draw the initial equilibrium for such a market b Now suppose the demand for Hula Hoops shifts out ward slightly Show that in general contrary to the competitive case it will not be possible to predict the effect of this shift in demand on the market price of Hula Hoops c Consider three possible ways in which the price elastic ity of demand might change as the demand curve shifts It might increase it might decrease or it might stay the same Consider also that marginal costs for the monop olist might be increasing decreasing or constant in the range where MR 5 MC Consequently there are nine different combinations of types of demand shifts and marginal cost slope configurations Analyze each of these to determine for which it is possible to make a definite prediction about the effect of the shift in demand on the price of Hula Hoops 145 Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price P but also on the amount of advertising the firm does A measured in dollars The specific form of this function is Q 5 120 2 P2 11 1 01A 2 001A22 The monopolistic firms cost function is given by C 5 10Q 1 15 1 A a Suppose there is no advertising 1A 5 02 What output will the profitmaximizing firm choose What mar ket price will this yield What will be the monopolys profits b Now let the firm also choose its optimal level of adver tising expenditure In this situation what output level will be chosen What price will this yield What will the level of advertising be What are the firms profits in this case Hint This can be worked out most easily by assum ing the monopoly chooses the profitmaximizing price rather than quantity 146 Suppose a monopoly can produce any level of output it wishes at a constant marginal and average cost of 5 per unit Assume the monopoly sells its goods in two different markets separated by some distance The demand curve in the first market is given by Q1 5 55 2 P1 and the demand curve in the second market is given by Q2 5 70 2 2P2 a If the monopolist can maintain the separation between the two markets what level of output should be pro duced in each market and what price will prevail in each market What are total profits in this situation b How would your answer change if it costs demanders only 4 to transport goods between the two markets What would be the monopolists new profit level in this situation c How would your answer change if transportation costs were zero and then the firm was forced to follow a sin gleprice policy d Now assume the two different markets 1 and 2 are just two individual consumers Suppose the firm could adopt a linear twopart tariff under which marginal prices charged to the two consumers must be equal but their lumpsum entry fees might vary What pricing policy should the firm follow 147 Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of 10 per unit Monopolized mar ginal costs increase to 12 per unit because 2 per unit must Chapter 14 Monopoly 519 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 520 Part 6 Market Power be paid to lobbyists to retain the widget producers favored position Suppose the market demand for widgets is given by QD 5 1000 2 50P a Calculate the perfectly competitive and monopoly out puts and prices b Calculate the total loss of consumer surplus from monopolization of widget production c Graph your results and explain how they differ from the usual analysis 148 Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of a subsidy a Why would a lumpsum subsidy not achieve the govern ments goal b Use a graphical proof to show how a perunitofoutput subsidy might achieve the governments goal c Suppose the government wants its subsidy to maximize the difference between the total value of the good to con sumers and the goods total cost Show that to achieve this goal the government should set t P 5 2 1 eD P where t is the perunit subsidy and P is the competitive price Explain your result intuitively 149 Suppose a monopolist produces alkaline batteries that may have various useful lifetimes X Suppose also that consum ers inverse demand depends on batteries lifetimes and quantity Q purchased according to the function P 1Q X2 5 g 1X Q2 where gr 0 That is consumers care only about the prod uct of quantity times lifetime They are willing to pay equally for many shortlived batteries or few longlived ones Assume also that battery costs are given by C1Q X2 5 C1X2Q where Cr 1X2 0 Show that in this case the monopoly will opt for the same level of X as does a competitive industry even though levels of output and prices may differ Explain your result Hint Treat XQ as a composite commodity Analytical Problems 1410 Taxation of a monopoly good The taxation of monopoly can sometimes produce results different from those that arise in the competitive case This problem looks at some of those cases Most of these can be analyzed by using the inverse elasticity rule Equa tion 141 a Consider first an ad valorem tax on the price of a monopolys good This tax reduces the net price received by the monopoly from P to P 112t2 where t is the pro portional tax rate Show that with a linear demand curve and constant marginal cost the imposition of such a tax causes price to increase by less than the full extent of the tax b Suppose that the demand curve in part a were a con stant elasticity curve Show that the price would now increase by precisely the full extent of the tax Explain the difference between these two cases c Describe a case where the imposition of an ad valorem tax on a monopoly would cause the price to increase by more than the tax d A specific tax is a fixed amount per unit of output If the tax rate is τ per unit total tax collections are τQ Show that the imposition of a specific tax on a monopoly will reduce output more and increase price more than will the imposition of an ad valorem tax that collects the same tax revenue 1411 Flexible functional forms In an important recent working paper M Fabinger and E G Weyl characterize tractable monopoly problems18 A tracta ble problem satisfies three conditions First it must be pos sible to move back and forth between explicit expressions for inverse and direct demand invertibility Second inverse demandwhich can also be interpreted as average revenue must have the same functional form as marginal revenue and average cost must have the same functional form as marginal cost form preservation Third the monopolists firstorder condition must be a linear equation linearity if not imme diately after differentiation then at least after suitable sub stitution The authors show that the broadest possible class of tractable problems has the following functional form for inverse demand and average cost P 1Q2 5 a0 1 a1Q2s AC1Q2 5 c0 1 c1Q2s where a0 a1 c0 c1 and s are nonnegative constants a Solve for the monopoly equilibrium quantity and price given these functional forms What substitution x 5 f1Q2 do you need to make the firstorder condition linear in x b Derive the solution in the special case with constant average and marginal cost 18M Fabinger and E G Weyl A Tractable Approach to PassThrough Patterns March 2015 SSRN working paper no 2194855 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 14 Monopoly 521 c If one is willing to relax tractability a bit to allow the monopolys firstorder condition to be a quadratic equa tion at least after suitable substitution the authors show that the broadest class of tractable problems then involves the following functional forms P 1Q2 5 a0 1 a1Q2s 1 a2Qs AC1Q2 5 c0 1 c1Q2s 1 c2Qs Solve for the monopoly equilibrium quantity and price What substitution x 5 f1Q2 is needed to make the firstorder condition quadratic in x d While slightly complicated the functional forms in part c have the advantage of being flexible enough to allow for Ushaped average cost curves such as drawn in Figure 142 in addition to constant increasing and decreasing Demonstrate this by graphing this average cost curve for wellchosen values of c0 c1 c2 to illustrate the various cases The flexible functional forms in part c also allow for realis tic demand shapes for example one that closely fits the US income distribution which implicitly takes income to proxy for consumers willingness to pay These realistic demand shapes can be used in calibrations to address important pol icy questions For example the text mentioned that in the ory the welfare effects of monopoly price discrimination can go either way either being higher or lower than under uni form pricing Calibrations involving the demand curves from part c invariably show that welfare is higher under price discrimination 1412 Welfare possibilities with different market segmentations The article by D Bergemann B Brooks and S Morris dis cussed in the text highlights the fundamental ambiguity of the welfare effects of price discrimination This question guides you through the construction of market segmentations that can achieve extreme welfare gains and losses relative to a sin gleprice policy Here we focus on the simple case of a market containing two consumer types but the results hold generally for any number of types and indeed for arbitrary continuous distributions of types Consider a market served by a monopolist in which q con sumers have value maximum willingness to pay v for the good and q consumers have value v where v v 0 Pro duction is costless a For comparison first solve for the socially efficient out put and welfare associated with the perfectly competitive outcome b Find a way to segment consumers into just two markets that allows the monopolist to recover the profit from perfect price discrimination Compute the associated profit consumer surplus and social welfare c The analysis in the rest of the problem is divided into two exhaustive cases First suppose q v 1q 1 q2v i Find the monopoly price quantity profit con sumer surplus and welfare when the monopolist charges a single price in the initial market before any segmentation ii Divide the single market into two segments by moving all of the lowvalue consumers and a fraction b of the highvalue ones into segment B leaving the remaining consumers in the ini tial market to constitute segment A and assume the monopolist engages in price discrimination across the segments Show that there exists b in the interval 10 12 such that the monopolist is indifferent between charging a high and low price in segment B Consider the equilibrium in which the monopolist charges the low price on a segment when indifferent Solve for the monop olists discriminatory prices across the segments Solve for profit consumer surplus and welfare in total across the two segments Compare this outcome to a singleprice monopoly showing that consumer surplus and welfare is created How do surpluses compare to those under perfect competition iii Plot the outcomes from parts i and ii on a graph with consumer surplus on the horizontal axis and monopoly profit on the vertical Also plot the perfect price discrimination from part b Connect the points as vertices of a triangle For a challenge think of ways to further segment the market to achieve the surplus divisions along the sides and in the interior of the triangle d Now suppose q v 1q 1 q2v i Find the monopoly price quantity profit con sumer surplus and welfare when the monopolist charges a single price before segmentation ii Divide the single market into two segments by moving all of the highvalue consumers and a fraction a of the lowvalue consumers into seg ment A leaving the remaining consumers in the initial market to constitute segment B and assume the monopolist engages in price discrimination across the segments Show that there exists a in the interval 10 12 such that the monopolist is indifferent between charging a high and low price in segment A Consider the equilibrium in which the monopolist charges the high price on a seg ment when indifferent Solve for the monopolists discriminatory prices across the segments Solve for profit consumer surplus and welfare in total across the two segments Compare this outcome to a singleprice monopoly showing that con sumer surplus and welfare is destroyed Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 522 Part 6 Market Power iii Show that plotting the consumer surplus and monopoly profit from the various pricing strate gies yields a similar triangle as in part c The analysis is similar when production involves a positive marginal cost c rather than being costless We just need to reinterpret consumer values above as being net of c Behavioral Problem 1413 Shrouded prices Some firms employ the marketing strategy of posting a low price for the good but then tack on hidden fees or high prices for addons that can add up to an allin price that is exor bitant compared to the posted price A television ad may blare that a perpetually sharp knife sells for 20 leaving the additional 10 handling chargeor worse that the 20 is just one of three installmentsfor the small print A laser printer printing photoquality color prints may seem like a bargain at 300 if one doesnt consider that the five toner cartridges must be replaced each year at 100 each If consumers under stand and account for these additional expenses we are firmly in a neoclassical model which can be analyzed using standard methods Behavioral economists worry about the possibility that unsophisticated consumers may underestimate or even ignore these shrouded prices and firms do their best to keep it that way This question introduces a model of shrouded prices and analyzes their efficiency consequences a Consumers demand for a good whose price they per ceive to be P is given by Q 5 10 2 P A monopolist produces the good at constant average and marginal cost equal to 6 Compute the monopoly price quantity profit consumer surplus and welfare the sum of con sumer surplus and profit assuming the perceived is the same as the actual price so there is no shrouding b Now assume that while the perceived price is still P the actual price charged by the monopolist is P 1 s where s is the shrouded part which goes unrecognized by con sumers Compute the monopoly price quantity and profit assuming the same demand and cost as in part a What amount of shrouding does the firm prefer c Compute the consumer surplus CS associated with the outcome in b This requires some care because con sumers spend more than they expect to Letting Ps and Qs be the equilibrium price and quantity charged by the monopoly with shrouded prices CS 5 3 Qs 0 P 1Q2dQ 2 PsQs This equals gross consumer surplus the area under inverse demand up to the quantity sold less actual rather than perceived expenditures d Compute welfare Find the welfaremaximizing level of shrouding Explain why this is positive rather than zero e Return to the case of no shrouding in part a but now assume the government offers a subsidy s Show that the welfaremaximizing subsidy equals welfaremaximizing level of shrouding found in part d Are the distribu tional consequences surplus going to consumers firm and government the same in the two cases Use the con nection between shrouding and a subsidy to argue infor mally that any amount of shrouding will be inefficient in a perfectly competitive market Suggestions for Further Reading Posner R A The Social Costs of Monopoly and Regulation Journal of Political Economy 83 1975 80727 An analysis of the probability that monopolies will spend resources on the creation of barriers to entry and thus have higher costs than perfectly competitive firms Schumpeter J A Capitalism Socialism and Democracy 3rd ed New York Harper Row 1950 Classic defense of the role of the entrepreneur and economic profits in the economic growth process Spence M Monopoly Quality and Regulation Bell Journal of Economics April 1975 41729 Develops the approach to product quality used in this text and provides a detailed analysis of the effects of monopoly Stigler G J The Theory of Economic Regulation Bell Jour nal of Economics and Management Science 2 Spring 1971 3 Early development of the capture hypothesis of regulatory behav iorthat the industry captures the agency supposed to regulate it and uses that agency to enforce entry barriers and further enhance profits Tirole J The Theory of Industrial Organization Cambridge MA MIT Press 1989 chaps 13 A complete analysis of the theory of monopoly pricing and product choice Varian H R Microeconomic Analysis 3rd ed New York W W Norton 1992 chap 14 Provides a succinct analysis of the role of incentive compatibility constraints in seconddegree price discrimination Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 523 EXTENSIONS OPTIMAL LINEAR TWOPART TARIFFS In Chapter 14 we examined a simple illustration of ways in which a monopoly may increase profits by practicing seconddegree price discriminationthat is by establishing price or outlay schedules that prompt buyers to separate themselves into distinct market segments Here we pursue the topic of linear tariff schedules a bit further Nonlinear pricing schedules are discussed in Chapter 18 E141 Structure of the problem To examine issues related to price schedules in a simple con text for each demander we define the valuation function as vi 1q2 5 pi 1q2 q 1 si i where pi 1q2 is the inverse demand function for individual i and si is consumer surplus Hence vi represents the total value to individual i of undertaking transactions of amount q which includes total spending on the good plus the value of consumer surplus obtained Here we will assume a there are only two demanders1 or homogeneous groups of demanders and b person 1 has stronger preferences for this good than person 2 in the sense that v1 1q2 v2 1q2 ii for all values of q The monopolist is assumed to have constant marginal costs denoted by c and chooses a tariff revenue schedule T 1q2 that maximizes profits given by π 5 T 1q12 1 T 1q22 2 c 1q1 1 q22 iii where qi represents the quantity chosen by person i In select ing a price schedule that successfully distinguishes among consumers the monopolist faces two constraints To ensure that the lowdemand person 2 is served it is necessary that v2 1q22 2 T 1q22 0 iv That is person 2 must derive a net benefit from her optimal choice q2 Person 1 the highdemand individual must also obtain a net gain from his chosen consumption level 1q12 and must prefer this choice to the output choice made by person 2 v1 1q12 2 T 1q12 v1 1q22 2 T 1q22 v If the monopolist does not recognize this incentive com patibility constraint it may find that person 1 opts for the portion of the price schedule intended for person 2 thereby destroying the goal of obtaining selfselected market separa tion Given this general structure we can proceed to illustrate a number of interesting features of the monopolists problem E142 Pareto superiority Permitting the monopolist to depart from a simple sin gleprice scheme offers the possibility of adopting Pareto superior tariff schedules under which all parties to the transaction are made better off For example suppose the monopolists profitmaximizing price is pM At this price person 2 consumes qM 2 and receives a net value from this consumption of v2 1q M 2 2 2 pMq M 2 vi A tariff schedule for which T 1q2 5 c pMq for q q M 2 a 1 pq for q q M 2 vii where a 0 and c p pM may yield increased profits for the monopolist as well as increased welfare for person 1 Spe cifically consider values of a and p such that a 1 pq M 1 5 pMq M 1 or a 5 1 pM 2 p2q M 1 viii where q M 1 represents consumption of person 1 under a single price policy In this case a and p are set so that person 1 can still afford to buy q M 1 under the new price schedule Because p pM however he will opt for q 1 q M 1 Because person 1 could have bought q M 1 but chose q 1 instead he must be better off under the new schedule The monopolys profits are now given by π 5 a 1 pq1 1 pMq M 2 2 c 1q1 1 q M 2 2 ix and π 2 πM 5 a 1 pq1 1 pMq M 1 2 c 1q1 2 q M 1 2 x 1Generalizations to many demanders are nontrivial For a discussion see Wilson 1993 chaps 25 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 524 Part 6 Market Power where πM is the monopolys singleprice profits 35 1pM 2 c2 3 1q M 1 1 q M 2 2 4 Substitution for a from Equa tion viii shows π 2 πM 5 1 p 2 c2 1q1 2 q M 1 2 0 xi Hence this new price schedule also provides more profits to the monopoly some of which might be shared with person 2 The price schedule is Pareto superior to a single monopoly price The notion that multipart schedules may be Pareto superior has been used not only in the study of price discrim ination but also in the design of optimal tax schemes and auc tion mechanisms see Willig 1978 Pricing a farmland reserve The potential Pareto superiority of complex tariff schedules is used by R B W Smith 1995 to estimate a leastcost method for the US government to finance a conservation reserve pro gram for farmland The specific plan the author studies would maintain a 34millionacre reserve out of production in any given year He calculates that use of carefully constructed nonlinear tariff schedules for such a program might cost only 1 billion annually E143 Tied sales Sometimes a monopoly will market two goods together This situation poses a number of possibilities for discriminatory pricing schemes Consider for example laser printers that are sold with toner cartridges or electronic game players sold with patented additional games Here the pricing situation is similar to that examined in Chapter 14usually consumers buy only one unit of the basic product the printer or camera and thereby pay the entry fee Then they consume a variable number of tied products toner and film Because our anal ysis in Chapter 14 suggests that the monopoly will choose a price for its tied product that exceeds marginal cost there will be a welfare loss relative to a situation in which the tied good is produced competitively Perhaps for this reason tied sales are prohibited by law in some cases Prohibition may not nec essarily increase welfare however if the monopoly declines to serve lowdemand consumers in the absence of such a prac tice Oi 1971 Automobiles and wine One way in which tied sales can be accomplished is through creation of a multiplicity of quality variants that appeal to different classes of buyers Automobile companies have been especially ingenious at devising quality variants of their basic models eg the Honda Accord comes in DX LX EX and SX configurations that act as tied goods in separating buyers into various market niches A 1992 study by J E Kwoka examines one specific US manufacturer Chrysler and shows how market segmentation is achieved through quality variation The author calculates that significant transfer from consumer surplus to firms occurs as a result of such segmentation Generally this sort of price discrimination in a tied good will be infeasible if that good is also produced under com petitive conditions In such a case the tied good will sell for marginal cost and the only possibility for discriminatory behavior open to the monopolist is in the pricing of its basic good ie by varying entry fees among demanders In some special cases however choosing to pay the entry fee will con fer monopoly power in the tied good on the monopolist even though it is otherwise reduced under competitive conditions For example Locay and Rodriguez 1992 examine the case of restaurants pricing of wine Here group decisions to patron ize a particular restaurant may confer monopoly power to the restaurant owner in the ability to practice wine price discrim ination among buyers with strong grape preferences Because the owner is constrained by the need to attract groups of cus tomers to the restaurant the power to price discriminate is less than under the pure monopoly scenario References Kwoka J E Market Segmentation by PriceQuality Sched ules Some Evidence from Automobiles Journal of Busi ness October 1992 61528 Locay L and A Rodriguez Price Discrimination in Com petitive Markets Journal of Political Economy October 1992 95468 Oi W Y A Disneyland Dilemma TwoPart Tariffs on a Mickey Mouse Monopoly Quarterly Journal of Economics February 1971 7790 Smith R B W The Conservation Reserve Program as a Least Cost Land Retirement Mechanism American Jour nal of Agricultural Economics February 1995 93105 Willig R Pareto Superior NonLinear Outlay Schedules Bell Journal of Economics January 1978 5669 Wilson W Nonlinear Pricing Oxford Oxford University Press 1993 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 525 CHAPTER FIFTEEN Imperfect Competition This chapter discusses oligopoly markets falling between the extremes of perfect compe tition and monopoly Oligopolies raise the possibility of strategic interaction among firms To analyze this strategic interaction rigorously we will apply the concepts from game theory that were introduced in Chapter 8 Our gametheoretic analysis will show that small changes in details concerning the variables firms choose the timing of their moves or their infor mation about market conditions or rival actions can have a dramatic effect on market outcomes The first half of the chapter deals with shortterm decisions such as pricing and output and the second half covers longerterm decisions such as investment advertising and entry 151 ShOrTrun DeCiSiOnS PriCing anD OuTPuT it is difficult to predict exactly the possible outcomes for price and output when there are few firms prices depend on how aggressively firms compete which in turn depends on which strategic variables firms choose how much information firms have about rivals and how often firms interact with each other in the market For example consider the Bertrand game studied in the next section The game involves two identical firms choosing prices simultaneously for their identical products in their one meeting in the market The Bertrand game has a nash equilibrium at point C in Figure 151 even though there may be only two firms in the market in this equilibrium they behave as though they were perfectly competitive setting price equal to marginal cost and earning zero profit We will discuss whether the Bertrand game is a realistic depiction of actual firm behavior but an analysis of the model shows that it is possible to think up rigorous gametheoretic models in which one extremethe competitive outcomecan emerge in concentrated markets with few firms at the other extreme as indicated by point M in Figure 151 firms as a group may act as a cartel recognizing that they can affect price and coordinate their decisions indeed D E F I N IT ION Oligopoly a market with relatively few firms but more than one Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 526 Part 6 Market Power they may be able to act as a perfect cartel and achieve the highest possible profitsnamely the profit a monopoly would earn in the market One way to maintain a cartel is to bind firms with explicit pricing rules Such explicit pricing rules are often prohibited by anti trust law But firms need not resort to explicit pricing rules if they interact on the market repeatedly they can collude tacitly high collusive prices can be maintained with the tacit threat of a price war if any firm undercuts We will analyze this game formally and discuss the difficulty of maintaining collusion The Bertrand and cartel models determine the outer limits between which actual prices in an imperfectly competitive market are set one such intermediate price is represented by point A in Figure 151 This band of outcomes may be wide and given the plethora of available models there may be a model for nearly every point within the band For exam ple in a later section we will show how the Cournot model in which firms set quantities rather than prices as in the Bertrand model leads to an outcome such as point A some where between C and M in Figure 151 it is important to know where the industry is on the line between points C and M because total welfare as measured by the sum of consumer surplus and firms profits see Chapter 12 depends on the location of this point at point C total welfare is as high as possible at point A total welfare is lower by the area of the shaded triangle 3 in Chapter 12 this shortfall in total welfare relative to the highest possible level was called deadweight loss at point M deadweight loss is even greater and is given by the area of shaded regions Market equilibrium under imperfect competition can occur at many points on the demand curve in the figure which assumes that marginal costs are constant over all output ranges the equilibrium of the Bertrand game occurs at point C also corresponding to the perfectly competitive outcome The perfect cartel outcome occurs at point M also corresponding to the monopoly outcome Many solutions may occur between points M and C depending on the specific assumptions made about how firms compete For example the equilibrium of the Cournot game might occur at a point such as A The deadweight loss given by the shaded triangle increases as one moves from point C to M Price PM PA PC QM QA QC MR MC C A M Quantity D 1 2 3 FIGURE 151 Pricing and Output under Imperfect Competition Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 527 1 2 and 3 The closer the imperfectly competitive outcome to C and the farther from M the higher is total welfare and the better off society will be1 152 BerTranD MODeL The Bertrand model is named after the economist who first proposed it2 The model is a game involving two identical firms labeled 1 and 2 producing identical products at a constant marginal cost and constant average cost c The firms choose prices p1 and p2 simultaneously in a single period of competition Because firms products are perfect sub stitutes all sales go to the firm with the lowest price Sales are split evenly if p1 5 p2 Let D 1 p2 be market demand We will look for the nash equilibrium The game has a continuum of actions as does example 84 the Tragedy of the Commons in Chapter 8 unlike example 84 we cannot use calculus to derive bestresponse functions because the profit functions are not differen tiable here Starting from equal prices if one firm lowers its price by the smallest amount then its sales and profit would essentially double We will proceed by first guessing what the nash equilibrium is and then spending some time to verify that our guess was in fact correct 1521 Nash equilibrium of the Bertrand game The only purestrategy nash equilibrium of the Bertrand game is p 1 5 p 2 5 c That is the nash equilibrium involves both firms charging marginal cost in saying that this is the only nash equilibrium we are making two statements that need to be verified This out come is a nash equilibrium and there is no other nash equilibrium To verify that this outcome is a nash equilibrium we need to show that both firms are playing a best response to each otheror in other words that neither firm has an incentive to deviate to some other strategy in equilibrium firms charge a price equal to marginal cost which in turn is equal to average cost But a price equal to average cost means firms earn zero profit in equilibrium Can a firm earn more than the zero it earns in equilibrium by deviating to some other price no if it deviates to a higher price then it will make no sales and therefore no profit not strictly more than in equilibrium if it deviates to a lower price then it will make sales but will be earning a negative margin on each unit sold because price would be below marginal cost Thus the firm would earn negative profit less than in equilibrium Because there is no possible profitable deviation for the firm we have succeeded in verifying that both firms charging marginal cost is a nash equilibrium it is clear that marginal cost pricing is the only purestrategy nash equilibrium if prices exceeded marginal cost the highprice firm would gain by undercutting the other slightly and capturing all the market demand More formally to verify that p 1 5 p 2 5 c is the only nash equilibrium we will go one by one through an exhaustive list of cases for various values of p1 p2 and c verifying that none besides p1 5 p2 5 c is a nash equilibrium To reduce the number of cases assume firm 1 is the lowprice firmthat is p1 p2 The same conclusions would be reached taking 2 to be the lowprice firm 1Because this section deals with shortrun decision variables price and quantity the discussion of total welfare in this paragraph focuses on shortrun considerations as discussed in a later section an imperfectly competitive market may produce considerably more deadweight loss than a perfectly competitive one in the short run yet provide more innovation incentives leading to lower production costs and new products and perhaps higher total welfare in the long run The patent system intentionally impairs competition by granting a monopoly right to improve innovation incentives 2J Bertrand Théorie Mathematique de la richess Sociale Journal de Savants 1883 499508 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 528 Part 6 Market Power There are three exhaustive cases i c p1 ii c p1 and iii c 5 p1 Case i cannot be a nash equilibrium Firm 1 earns a negative margin p1 2 c on every unit it sells and because it makes positive sales it must earn negative profit it could earn higher profit by deviating to a higher price For example firm 1 could guarantee itself zero profit by deviating to p1 5 c Case ii cannot be a nash equilibrium either at best firm 2 gets only half of market demand if p1 5 p2 and at worst gets no demand if p1 p2 Firm 2 could capture all the market demand by undercutting firm 1s price by a tiny amount ε This ε could be cho sen small enough that market price and total market profit are hardly affected if p1 5 p2 before the deviation the deviation would essentially double firm 2s profit if p1 p2 before the deviation the deviation would result in firm 2 moving from zero to positive profit in either case firm 2s deviation would be profitable Case iii includes the subcase of p1 5 p2 5 c which we saw is a nash equilibrium The only remaining subcase in which p1 p2 is c 5 p1 p2 This subcase cannot be a nash equilibrium Firm 1 earns zero profit here but could earn positive profit by deviating to a price slightly above c but still below p2 although the analysis focused on the game with two firms it is clear that the same outcome would arise for any number of firms n 2 The nash equilibrium of the nfirm Bertrand game is p 1 5 p 2 5 5 p n 5 c 1522 Bertrand paradox The nash equilibrium of the Bertrand model is the same as the perfectly competitive out come Price is set to marginal cost and firms earn zero profit This resultthat the nash equilibrium in the Bertrand model is the same as in perfect competition even though there may be only two firms in the marketis called the Bertrand paradox it is paradoxical that competition between as few as two firms would be so tough The Bertrand paradox is a general result in the sense that we did not specify the marginal cost c or the demand curve therefore the result holds for any c and any downwardsloping demand curve in another sense the Bertrand paradox is not general it can be undone by changing various of the models other assumptions each of the next several sections will present a different model generated by changing a different one of the Bertrand assumptions in the next section for example we will assume that firms choose quantity rather than price leading to what is called the Cournot game We will see that firms do not end up charging marginal cost and earning zero profit in the Cournot game in subsequent sections we will show that the Bertrand paradox can also be avoided if still other assumptions are changed if firms face capacity constraints rather than being able to produce an unlimited amount at cost c if products are slightly differentiated rather than being perfect substitutes or if firms engage in repeated interaction rather than one round of competition 153 COurnOT MODeL The Cournot model named after the economist who proposed it3 is similar to the Bertrand model except that firms are assumed to simultaneously choose quantities rather than prices as we will see this simple change in strategic variable will lead to a big change in implications Price will be above marginal cost and firms will earn positive profit in the nash equilibrium of the Cournot game it is somewhat surprising but nonetheless an important point to keep in mind that this simple change in choice variable matters in the 3a Cournot Researches into the Mathematical Principles of the Theory of Wealth trans n T Bacon new York Macmillan 1897 although the Cournot model appears after Bertrands in this chapter Cournots work originally published in 1838 predates Bertrands Cournots work is one of the first formal analyses of strategic behavior in oligopolies and his solution concept anticipated nash equilibrium Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 529 strategic setting of an oligopoly when it did not matter with a monopoly The monopolist obtained the same profitmaximizing outcome whether it chose prices or quantities We will start with a general version of the Cournot game with n firms indexed by i 5 1 c n each firm chooses its output qi of an identical product simultaneously The out puts are combined into a total industry output Q 5 q1 1 q2 1 c1 qn resulting in market price PQ Observe that PQ is the inverse demand curve corresponding to the market demand curve Q 5 D 1P2 assume market demand is downward sloping and so inverse demand is too that is Pr 1Q2 0 Firm is profit equals its total revenue P 1Q2qi minus its total cost Ci 1qi2 πi 5 P 1Q2qi 2 Ci 1qi2 151 1531 Nash equilibrium of the Cournot game unlike the Bertrand game the profit function 151 in the Cournot game is differentia ble hence we can proceed to solve for the nash equilibrium of this game just as we did in example 84 the Tragedy of the Commons That is we find each firm is best response by taking the firstorder condition of the objective function 151 with respect to qi πi qi 5 P 1Q2 1 Pr 1Q2qi 2 MR MC Cri 1qi2 5 0 152 equation 152 must hold for all i 5 1 c n in the nash equilibrium according to equation 152 the familiar condition for profit maximization from Chapter 11marginal revenue MR equals marginal cost MCholds for the Cournot firm as we will see from an analysis of the particular form that the marginal revenue term takes for the Cournot firm price is above the perfectly competitive level above marginal cost but below the level in a perfect cartel that maximizes firms joint profits in order for equation 152 to equal 0 price must exceed marginal cost by the magni tude of the wedge term Pr 1Q2qi if the Cournot firm produces another unit on top of its existing production of qi units then because demand is downward sloping the additional unit causes market price to decrease by Pr 1Q2 leading to a loss of revenue of Pr 1Q2qi the wedge term from firm is existing production To compare the Cournot outcome with the perfect cartel outcome note that the objec tive for the cartel is to maximize joint profit a n j51 πj 5 P 1Q2 a n j51 qj 2 a n j51 Cj1qj2 153 Taking the firstorder condition of equation 153 with respect to qi gives qi a a n j51 πjb 5 P 1Q2 1 Pr 1Q2 a n j51 qj 2 Cri 1qi2 5 0 MR MC 154 This firstorder condition is similar to equation 152 except that the wedge term Pr 1Q2 a n j51 qj 5 Pr 1Q2Q 155 is larger in magnitude with a perfect cartel than with Cournot firms in maximizing joint profits the cartel accounts for the fact that an additional unit of firm is output by reduc ing market price reduces the revenue earned on all firms existing output hence Pr 1Q2 is Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 530 Part 6 Market Power multiplied by total cartel output Q in equation 155 The Cournot firm accounts for the reduction in revenue only from its own existing output qi hence Cournot firms will end up overproducing relative to the joint profitmaximizing outcome That is the extra pro duction in the Cournot outcome relative to a perfect cartel will end up in lower joint profit for the firms What firms would regard as overproduction is good for society because it means that the Cournot outcome point A referring back to Figure 151 will involve more total welfare than the perfect cartel outcome point M in Figure 151 EXAMPLE 151 NaturalSpring Duopoly as a numerical example of some of these ideas we will consider a case with just two firms and simple demand and cost functions Following Cournots nineteenthcentury example of two natural springs we assume that each spring owner has a large supply of possibly healthful water and faces the prob lem of how much to provide the market a firms cost of pumping and bottling qi liters is Ci 1qi2 5 cqi implying that marginal costs are a constant c per liter inverse demand for spring water is P 1Q2 5 a 2 Q 156 where a is the demand intercept measuring the strength of spring water demand and Q 5 q1 1 q2 is total spring water output We will now examine various models of how this market might operate Bertrand model in the nash equilibrium of the Bertrand game the two firms set price equal to marginal cost hence market price is P 5 c total output is Q 5 a 2 c firm profit is π i 5 0 and total profit for all firms is P 5 0 For the Bertrand quantity to be positive we must have a c which we will assume throughout the problem Cournot model The solution for the nash equilibrium follows example 86 closely Profits for the two Cournot firms are π1 5 P 1Q2q1 2 cq1 5 1a 2 q1 2 q2 2 c2q1 π2 5 P 1Q2q2 2 cq2 5 1a 2 q1 2 q2 2 c2q2 157 using the firstorder conditions to solve for the bestresponse functions we obtain q1 5 a 2 q2 2 c 2 q2 5 a 2 q1 2 c 2 158 Solving equations 158 simultaneously yields the nash equilibrium q 1 5 q 2 5 a 2 c 3 159 Thus total output is Q 5 1232 1a 2 c2 Substituting total output into the inverse demand curve implies an equilibrium price of P 5 1a 1 2c23 Substituting price and outputs into the profit functions equations 157 implies π 1 5 π 2 5 1192 1a 2 c2 2 so total market profit equals P 5 π 1 1 π 2 5 1292 1a 2 c2 2 Perfect cartel The objective function for a perfect cartel involves joint profits π1 1 π2 5 1a 2 q1 2 q2 2 c2q1 1 1a 2 q1 2 q2 2 c2q2 1510 The two firstorder conditions for maximizing equation 1510 with respect to q1 and q2 are the same q1 1π1 1 π22 5 q2 1π1 1 π22 5 a 2 2q1 2 2q2 2 c 5 0 1511 as is evident from the firstorder conditions firms market shares are not pinned down in a per fect cartel because they produce identical products at constant marginal cost But equation 1511 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 531 EXAMPLE 152 Cournot BestResponse Diagrams Continuing with the naturalspring duopoly from example 151 it is instructive to solve for the nash equilibrium using graphical methods We will graph the bestresponse functions given in equation 158 the intersection between the best responses is the nash equilibrium as back ground you may want to review a similar diagram Figure 87 for the Tragedy of the Commons The linear bestresponse functions are most easily graphed by plotting their intercepts as shown in Figure 152 The bestresponse functions intersect at the point q 1 5 q 2 5 1a 2 c23 which was the nash equilibrium of the Cournot game computed using algebraic methods in example 151 does pin down total output q 1 1 q 2 5 Q 5 1122 1a 2 c2 Substituting total output into inverse demand implies that the cartel price is P 5 1122 1a 1 c2 Substituting price and quantities into equation 1510 implies a total cartel profit of P 5 1142 1a 2 c2 2 Comparison Moving from the Bertrand model to the Cournot model to a perfect cartel because a c we can show that quantity Q decreases from a 2 c to 1232 1a 2 c2 to 1122 1a 2 c2 it can also be shown that price P and industry profit P increase For example if a 5 120 and c 5 0 implying that inverse demand is P 1Q2 5 120 2 Q and that production is costless then market quantity is 120 with Bertrand competition 80 with Cournot competition and 60 with a perfect cartel Price increases from 0 to 40 to 60 across the cases and industry profit increases from 0 to 3200 to 3600 QUERY in a perfect cartel do firms play a best response to each others quantities if not in which direction would they like to change their outputs What does this say about the stability of cartels Firms best responses are drawn as thick lines their intersection E is the nash equilibrium of the Cournot game isoprofit curves for firm 1 increase until point M is reached which is the monopoly outcome for firm 1 a c a c a c 2 3 0 q2 q1 M E BR1q2 BR2q1 π1 100 π1 200 a c a c 2 a c 3 FIGURE 152 BestResponse Diagram for Cournot Duopoly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 532 Part 6 Market Power Figure 152 displays firms isoprofit curves an isoprofit curve for firm 1 is the locus of quan tity pairs providing it with the same profit level To compute the isoprofit curve associated with a profit level of say 100 we start by setting equation 157 equal to 100 π1 5 1a 2 q1 2 q2 2 c2q1 5 100 1512 Then we solve for q2 to facilitate graphing the isoprofit q2 5 a 2 c 2 q1 2 100 q1 1513 Several example isoprofits for firm 1 are shown in the figure as profit increases from 100 to 200 to yet higher levels the associated isoprofits shrink down to the monopoly point which is the highest isoprofit on the diagram To understand why the individual isoprofits are shaped like frowns refer back to equation 1513 as q1 approaches 0 the last term 12100q12 dominates causing the left side of the frown to turn down as q1 increases the 2q1 term in equation 1513 begins to dominate causing the right side of the frown to turn down Figure 153 shows how to use bestresponse diagrams to quickly tell how changes in such under lying parameters as the demand intercept a or marginal cost c would affect the equilibrium Figure 153a depicts an increase in both firms marginal cost c The best responses shift inward resulting in a new equilibrium that involves lower output for both although firms have the same marginal cost in this example one can imagine a model in which firms have different marginal cost parameters and so can be varied independently Figure 153b depicts an increase in just firm 1s marginal cost only firm 1s best response shifts The new equilibrium involves lower output for firm 1 and higher output for firm 2 although firm 2s best response does not shift it still increases its output as it anticipates a reduction in firm 1s output and best responds to this anticipated output reduction QUERY explain why firm 1s individual isoprofits reach a peak on its bestresponse function in Figure 152 What would firm 2s isoprofits look like in Figure 152 how would you represent an increase in demand intercept a in Figure 153 Firms initial best responses are drawn as solid lines resulting in a nash equilibrium at point Er Panel a depicts an increase in both firms marginal costs shifting their best responsesnow given by the dashed linesinward The new intersection point and thus the new equilibrium is point Es Panel b depicts an increase in just firm 1s marginal cost q2 q2 BR1q2 BR1q2 BR2q1 BR2q1 q1 q1 E E E E a Increase in both frms marginal costs b Increase in frm 1s marginal cost FIGURE 153 Shifting Cournot Best Responses Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 533 1532 Varying the number of Cournot firms The Cournot model is particularly useful for policy analysis because it can represent the whole range of outcomes from perfect competition to perfect cartelmonopoly ie the whole range of points between C and M in Figure 151 by varying the number of firms n from n 5 q to n 5 1 For simplicity consider the case of identical firms which here means the n firms sharing the same cost function C1qi2 in equilibrium firms will pro duce the same share of total output qi 5 Qn Substituting qi 5 Qn into equation 1512 the wedge term becomes Pr 1Q2Qn The wedge term disappears as n grows large firms become infinitesimally small an infinitesimally small firm effectively becomes a price taker because it produces so little that any decrease in market price from an increase in output hardly affects its revenue Price approaches marginal cost and the market outcome approaches the perfectly competitive one as n decreases to 1 the wedge term approaches that in equation 155 implying the Cournot outcome approaches that of a perfect cartel as the Cournot firms market share grows it internalizes the revenue loss from a decrease in market price to a greater extent EXAMPLE 153 NaturalSpring Oligopoly return to the natural springs in example 151 but now consider a variable number n of firms rather than just two The profit of one of them firm i is πi 5 P 1Q2qi 2 cqi 5 1a 2 Q 2 c2qi 5 1a 2 qi 2 Q2i 2 c2qi 1514 it is convenient to express total output as Q 5 qi 1 Q2i where Q2i 5 Q 2qi is the output of all firms except for i Taking the firstorder condition of equation 1514 with respect to qi we recog nize that firm i takes Q2i as a given and thus treats it as a constant in the differentiation πi qi 5 a 2 2qi 2 Q2i 2 c 5 0 1515 which holds for all i 5 1 2 n The key to solving the system of n equations for the n equilibrium quantities is to recognize that the nash equilibrium involves equal quantities because firms are symmetric Symmetry implies that Q 2i 5 Q 2 q i 5 nq i 2 q i 5 1n 2 12q i 1516 Substituting equation 1516 into 1515 yields a 2 2q i 2 1n 2 12q i 2 c 5 0 1517 or q i 5 1a 2 c2 1n 1 12 Total market output is Q 5 nq i 5 a n n 1 1b 1a 2 c2 1518 and market price is P 5 a 2 Q 5 a 1 n 1 1ba 1 a n n 1 1bc 1519 Substituting for q i Q and P into the firms profit equation 1514 we have that total profit for all firms is P 5 nπ i 5 n a a 2 c n 1 1b 2 1520 Setting n 5 1 in equations 15181520 gives the monopoly outcome which gives the same price total output and profit as in the perfect cartel case computed in example 151 Letting n grow Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 534 Part 6 Market Power without bound in equations 15181520 gives the perfectly competitive outcome the same out come computed in example 151 for the Bertrand case QUERY We used the trick of imposing symmetry after taking the firstorder condition for firm is quantity choice it might seem simpler to impose symmetry before taking the firstorder condi tion Why would this be a mistake how would the incorrect expressions for quantity price and profit compare with the correct ones here 1533 Prices or quantities Moving from price competition in the Bertrand model to quantity competition in the Cournot model changes the market outcome dramatically This change is surprising on first thought after all the monopoly outcome from Chapter 14 is the same whether we assume the monopolist sets price or quantity Further thought suggests why price and quantity are such different strategic variables Starting from equal prices a small reduction in one firms price allows it to steal all the market demand from its competitors This sharp benefit from undercutting makes price competition extremely tough Quantity compe tition is softer Starting from equal quantities a small increase in one firms quantity has only a marginal effect on the revenue that other firms receive from their existing output Firms have less of an incentive to outproduce each other with quantity competition than to undercut each other with price competition an advantage of the Cournot model is its realistic implication that the industry grows more competitive as the number n of firms entering the market increases from monopoly to perfect competition in the Bertrand model there is a discontinuous jump from monop oly to perfect competition if just two firms enter and additional entry beyond two has no additional effect on the market outcome an apparent disadvantage of the Cournot model is that firms in realworld markets tend to set prices rather than quantities contrary to the Cournot assumption that firms choose quantities For example grocers advertise prices for orange juice say 300 a con tainer in newspaper circulars rather than the number of containers it stocks as we will see in the next section the Cournot model applies even to the orange juice market if we reinterpret quantity to be the firms capacity defined as the most the firm can sell given the capital it has in place and other available inputs in the short run 154 CaPaCiTY COnSTrainTS For the Bertrand model to generate the Bertrand paradox the result that two firms essen tially behave as perfect competitors firms must have unlimited capacities Starting from equal prices if a firm lowers its price the slightest amount then its demand essentially dou bles The firm can satisfy this increased demand because it has no capacity constraints giving firms a big incentive to undercut if the undercutting firm could not serve all the demand at its lower price because of capacity constraints that would leave some residual demand for the higherpriced firm and would decrease the incentive to undercut Consider a twostage game in which firms build capacity in the first stage and firms choose prices p1 and p2 in the second stage4 Firms cannot sell more in the second stage 4 The model is due to D Kreps and J Scheinkman Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes Bell Journal of Economics autumn 1983 32637 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 535 than the capacity built in the first stage if the cost of building capacity is sufficiently high it turns out that the subgameperfect equilibrium of this sequential game leads to the same outcome as the nash equilibrium of the Cournot model To see this result we will analyze the game using backward induction Consider the secondstage pricing game supposing the firms have already built capacities q1 and q2 in the first stage Let p be the price that would prevail when production is at capacity for both firms a situation in which p1 5 p2 p 1521 is not a nash equilibrium at this price total quantity demanded exceeds total capacity therefore firm 1 could increase its profits by raising price slightly and continuing to sell q1 Similarly p1 5 p2 p 1522 is not a nash equilibrium because now total sales fall short of capacity at least one firm say firm 1 is selling less than its capacity By cutting price slightly firm 1 can increase its profits by selling up to its capacity q1 hence the nash equilibrium of this secondstage game is for firms to choose the price at which quantity demanded exactly equals the total capacity built in the first stage5 p1 5 p2 5 p 1523 anticipating that the price will be set such that firms sell all their capacity the firststage capacity choice game is essentially the same as the Cournot game Therefore the equilib rium quantities price and profits will be the same as in the Cournot game Thus even in markets such as orange juice sold in grocery stores where it looks like firms are setting prices the Cournot model may prove more realistic than it first seems 155 PrODuCT DiFFerenTiaTiOn another way to avoid the Bertrand paradox is to replace the assumption that the firms products are identical with the assumption that firms produce differentiated products Many if not most realworld markets exhibit product differentiation For example toothpaste brands vary somewhat from supplier to supplierdiffering in flavor fluoride content whitening agents endorsement from the american Dental association and so forth even if suppliers product attributes are similar suppliers may still be differ entiated in another dimension physical location Because demanders will be closer to some suppliers than to others they may prefer nearby sellers because buying from them involves less travel time 1551 Meaning of the market The possibility of product differentiation introduces some fuzziness into what we mean by the market for a good With identical products demanders were assumed to be indifferent about which firms output they bought hence they shop at the lowestprice firm leading to 5For completeness it should be noted that there is no purestrategy nash equilibrium of the secondstage game with unequal prices 1p1 2 p22 The lowprice firm would have an incentive to increase its price andor the highprice firm would have an incentive to lower its price For large capacities there may be a complicated mixedstrategy nash equilibrium but this can be ruled out by supposing the cost of building capacity is sufficiently high Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 536 Part 6 Market Power the law of one price The law of one price no longer holds if demanders strictly prefer one supplier to another at equal prices are greengel and whitepaste toothpastes in the same market or in two different ones is a pizza parlor at the outskirts of town in the same mar ket as one in the middle of town With differentiated products we will take the market to be a group of closely related products that are more substitutable among each other as measured by crossprice elastic ities than with goods outside the group We will be somewhat loose with this definition avoiding precise thresholds for how high the crossprice elasticity must be between goods within the group and how low with outside goods arguments about which goods should be included in a product group often dominate antitrust proceedings and we will try to avoid this contention here 1552 Bertrand competition with differentiated products return to the Bertrand model but now suppose there are n firms that simultaneously choose prices pi 1i 5 1 n2 for their differentiated products Product i has its own spe cific attributes ai possibly reflecting special options quality brand advertising or location a product may be endowed with the attribute orange juice is by definition made from oranges and cranberry juice from cranberries or the attribute may be the result of the firms choice and spending level the orange juice supplier can spend more and make its juice from fresh oranges rather than from frozen concentrate The various attributes serve to differentiate the products Firm is demand is qi 1 pi P2i ai A2i2 1524 where P2i is a list of all other firms prices besides is and A2i is a list of all other firms attri butes besides is Firm is total cost is Ci 1qi ai2 1525 and profit is thus πi 5 pi qi 2 Ci 1qi ai2 1526 With differentiated products the profit function equation 1526 is differentiable so we do not need to solve for the nash equilibrium on a casebycase basis as we did in the Bertrand model with identical products We can solve for the nash equilibrium as in the Cournot model solving for bestresponse functions by taking each firms firstorder con dition here with respect to price rather than quantity The firstorder condition from equation 1526 with respect to pi is πi pi 5 qi 1 pi qi pi 2 Ci qi qi pi 5 0 A B 1527 The first two terms labeled A on the right side of equation 1527 are a sort of marginal revenuenot the usual marginal revenue from an increase in quantity but rather the marginal revenue from an increase in price The increase in price increases revenue on existing sales of qi units but we must also consider the negative effect of the reduction in sales qipi multiplied by the price pi that would have been earned on these sales The last term labeled B is the cost savings associated with the reduced sales that accompany an increased price The nash equilibrium can be found by simultaneously solving the system of first order conditions in equation 1527 for all i 5 1 n if the attributes ai are also choice Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 537 variables rather than just endowments there will be another set of firstorder conditions to consider For firm i the firstorder condition with respect to ai has the form πi ai 5 pi qi ai 2 Ci ai 2 Ci qi qi ai 5 0 1528 The simultaneous solution of these firstorder conditions can be complex and they yield few definitive conclusions about the nature of market equilibrium Some insights from particular cases will be developed in the next two examples EXAMPLE 154 Toothpaste as a Differentiated Product Suppose that two firms produce toothpaste one a green gel and the other a white paste To sim plify the calculations suppose that production is costless Demand for product i is qi 5 ai 2 pi 1 pj 2 1529 The positive coefficient on pj the other goods price indicates that the goods are gross substitutes Firm is demand is increasing in the attribute ai which we will take to be demanders inherent prefer ence for the variety in question we will suppose that this is an endowment rather than a choice vari able for the firm and so will abstract from the role of advertising to promote preferences for a variety Algebraic solution Firm is profit is πi 5 piqi 2 Ci 1qi2 5 pi aai 2 pi 1 pj 2 b 1530 where Ci 1qi2 5 0 because is production is costless The firstorder condition for profit maximi zation with respect to pi is πi pi 5 ai 2 2pi 1 pj 2 5 0 1531 Solving for pi gives the following bestresponse functions for i 5 1 2 p1 5 1 2 aa1 1 p2 2 b p2 5 1 2 aa2 1 p1 2 b 1532 Solving equations 1532 simultaneously gives the nash equilibrium prices p i 5 8 15 ai 1 2 15 aj 1533 The associated profits are π i 5 a 8 15 ai 1 2 15 ajb 2 1534 Firm i s equilibrium price is not only increasing in its own attribute ai but also in the other products attribute aj an increase in aj causes firm j to increase its price which increases firm i s demand and thus the price i charges Graphical solution We could also have solved for equilibrium prices graphically as in Figure 154 The best responses in equation 1532 are upward sloping They intersect at the nash equilib rium point E The isoprofit curves for firm 1 are smileshaped To see this take the expression for firm 1s profit in equation 1530 set it equal to a certain profit level say 100 and solve for p2 to facilitate graphing it on the bestresponse diagram We have p2 5 100 p1 1 p1 2 a1 1535 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 538 Part 6 Market Power The smile turns up as p1 approaches 0 because the denominator of 100p1 approaches 0 The smile turns up as p1 grows large because then the second term on the right side of equation 1535 grows large isoprofit curves for firm 1 increase as one moves away from the origin along its bestresponse function QUERY how would a change in the demand intercepts be represented on the diagram Firm best responses are drawn as thick lines their intersection E is the nash equilibrium isoprofit curves for firm 1 increase moving out along firm 1s bestresponse function p2 p1 E p1 p2 a2 c 2 0 a1 c 2 BR1p2 BR2p1 π1 100 π1 200 FIGURE 154 Best Responses for Bertrand Model with Differentiated Products EXAMPLE 155 Hotellings Beach a simple model in which identical products are differentiated because of the location of their suppli ers spatial differentiation was provided by h hotelling in the 1920s6 as shown in Figure 155 two ice cream stands labeled A and B are located along a beach of length L The stands make identical ice cream cones which for simplicity are assumed to be costless to produce Let a and b represent the firms locations on the beach We will take the locations of the ice cream stands as given in a later example we will revisit firms equilibrium location choices assume that demanders are located uni formly along the beach one at each unit of length Carrying ice cream a distance d back to ones beach umbrella costs td 2 because ice cream melts more the higher the temperature t and the further one must walk7 Consistent with the Bertrand assumption firms choose prices pA and pB simultaneously Determining demands Let x be the location of the consumer who is indifferent between buy ing from the two ice cream stands The following condition must be satisfied by x pA 1 t1x 2 a2 2 5 pB 1 t1b 2 x2 2 1536 6h hotelling Stability in Competition Economic Journal 39 1929 4157 7The assumption of quadratic transportation costs turns out to simplify later work when we compute firms equilibrium locations in the model Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 539 The left side of equation 1536 is the generalized cost of buying from A including the price paid and the cost of transporting the ice cream the distance x 2 a Similarly the right side is the gen eralized cost of buying from B Solving equation 1536 for x yields x 5 b 1 a 2 1 pB 2 pA 2t1b 2 a2 1537 if prices are equal the indifferent consumer is located midway between a and b if As price is less than Bs then x shifts toward endpoint L This is the case shown in Figure 155 Because all demanders between 0 and x buy from A and because there is one consumer per unit distance it follows that As demand equals x qA 1 pA pB a b2 5 x 5 b 1 a 2 1 pB 2 pA 2t1b 2 a2 1538 The remaining L 2 x consumers constitute Bs demand qB 1 pB pA b a2 5 L 2 x 5 L 2 b 1 a 2 1 pA 2 pB 2t1b 2 a2 1539 Solving for Nash equilibrium The nash equilibrium is found in the same way as in example 154 except that for demands we use equations 1538 and 1539 in place of equation 1529 Skip ping the details of the calculations the nash equilibrium prices are p A 5 t 3 1b 2 a2 12L 1 a 1 b2 p B 5 t 3 1b 2 a2 14L 2 a 2 b2 1540 These prices will depend on the precise location of the two stands and will differ from each other For example if we assume that the beach is L 5 100 yards long a 5 40 yards b 5 70 yards and t 5 0001 one tenth of a penny then p A 5 310 and p B 5 290 These price differences arise only from the locational aspects of this problemthe cones themselves are identical and costless to produce Because A is somewhat more favorably located than B it can charge a higher price for its cones without losing too much business to B using equation 1538 shows that x 5 110 2 1 310 2 290 122 100012 11102 52 1541 so stand A sells 52 cones whereas B sells only 48 despite its lower price at point x the consumer is indifferent between walking the 12 yards to A and paying 310 or walking 18 yards to B and paying 290 The equilibrium is inefficient in that a consumer slightly to the right of x would incur a shorter walk by patronizing A but still chooses B because of As power to set higher prices ice cream stands A and B are located at points a and b along a beach of length L The consumer who is indifferent between buying from the two stands is located at x Consumers to the left of x buy from A and to the right buy from B FIGURE 155 Hotellings Beach As demand Bs demand a 0 x b L Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 540 Part 6 Market Power 1553 Consumer search and price dispersion hotellings model analyzed in example 155 suggests the possibility that competitors may have some ability to charge prices above marginal cost and earn positive profits even if the physical characteristics of the goods they sell are identical Firms various locations closer to some demanders and farther from othersmay lead to spatial differentiation The internet makes the physical location of stores less relevant to consumers especially if shipping charges are independent of distance or are not assessed even in this setting firms can avoid the Bertrand paradox if we drop the assumption that demanders know every firms price in the market instead we will assume that demanders face a small cost s called a search cost to visit the store or click to its website to find its price Peter Diamond winner of the nobel Prize in economics in 2010 developed a model in which demanders search by picking one of the n stores at random and learning its price Demanders know the equilibrium distribution of prices but not which store is charging which price Demanders get their first price search for free but then must pay s for addi tional searches They need at most one unit of the good and they all have the same gross surplus v for the one unit8 not only do stores manage to avoid the Bertrand paradox in this model they obtain the polar opposite outcome all charge the monopoly price v which extracts all consumer surplus This outcome holds no matter how small the search cost s isas long as s is posi tive say a penny it is easy to see that all stores charging v is an equilibrium if all charge the same price v then demanders may as well buy from the first store they search because additional searches are costly and do not end up revealing a lower price it can also be seen that this is the only equilibrium Consider any outcome in which at least one store charges less than v and consider the lowestprice store label it i in this outcome Store i could raise its price pi by as much as s and still make all the sales it did before The lowest price a demander could expect to pay elsewhere is no less than pt and the demander would have to pay the cost s to find this other price 8P Diamond a Model of Price adjustment Journal of Economic Theory 3 1971 15668 equilibrium profits are π A 5 t 18 1b 2 a2 12L 1 a 1 b2 2 π B 5 t 18 1b 2 a2 14L 2 a 2 b2 2 1542 Somewhat surprisingly the ice cream stands benefit from faster melting as measured here by the transportation cost t For example if we take L 5 100 a 5 40 b 5 70 and t 5 0001 as in the previous paragraph then π A 5 160 and π B 5 140 rounding to the nearest dollar if transportation costs doubled to t 5 0002 then profits would double to π A 5 320 and π B 5 280 The transportationmelting cost is the only source of differentiation in the model if t 5 0 then we can see from equation 1540 that prices equal 0 which is marginal cost given that pro duction is costless and from equation 1542 that profits equal 0in other words the Bertrand paradox results QUERY What happens to prices and profits if ice cream stands locate in the same spot if they locate at the opposite ends of the beach Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 541 Less extreme equilibria are found in models where consumers have different search costs9 For example suppose one group of consumers can search for free and another group has to pay s per search in equilibrium there will be some price dispersion across stores One set of stores serves the lowsearchcost demanders and the lucky highsearch cost consumers who happen to stumble on a bargain These bargain stores sell at marginal cost The other stores serve the highsearchcost demanders at a price that makes these demanders indifferent between buying immediately and taking a chance that the next price search will uncover a bargain store 156 TaCiT COLLuSiOn in Chapter 8 we showed that players may be able to earn higher payoffs in the subgameperfect equilibrium of an infinitely repeated game than from simply repeating the nash equilibrium from the singleperiod game indefinitely For example we saw that if players are patient enough they can cooperate on playing silent in the infinitely repeated version of the Prisoners Dilemma rather than finking on each other each period From the perspective of oligopoly theory the issue is whether firms must endure the Bertrand para dox marginal cost pricing and zero profits in each period of a repeated game or whether they might instead achieve more profitable outcomes through tacit collusion a distinction should be drawn between tacit collusion and the formation of an explicit cartel an explicit cartel involves legal agreements enforced with external sanctions if the agreements eg to sustain high prices or low outputs are violated Tacit collusion can only be enforced through punishments internal to the marketthat is only those that can be generated within a subgameperfect equilibrium of a repeated game antitrust laws gen erally forbid the formation of explicit cartels so tacit collusion is usually the only way for firms to raise prices above the static level 1561 Finitely repeated game Taking the Bertrand game to be the stage game Seltens theorem from Chapter 8 tells us that repeating the stage game any finite number of times T does not change the out come The only subgameperfect equilibrium of the finitely repeated Bertrand game is to repeat the stagegame nash equilibriummarginal cost pricingin each of the T periods The game unravels through backward induction in any subgame starting in period T the unique nash equilibrium will be played regardless of what happened before Because the out come in period T 2 1 does not affect the outcome in the next period it is as though period T 2 1 is the last period and the unique nash equilibrium must be played then too applying backward induction the game unravels in this manner all the way back to the first period 1562 Infinitely repeated game if the stage game is repeated infinitely many periods however the folk theorem applies The folk theorem indicates that any feasible and individually rational payoff can be sus tained each period in an infinitely repeated game as long as the discount factor δ is close enough to unity recall that the discount factor is the value in the present period of one dollar earned one period in the futurea measure roughly speaking of how patient play ers are Because the monopoly outcome with profits divided among the firms is a feasible and individually rational outcome the folk theorem implies that the monopoly outcome 9The following model is due to S Salop and J Stiglitz Bargains and ripoffs a Model of Monopolistically Competitive Price Dispersion Review of Economic Studies 44 1977 493510 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 542 Part 6 Market Power must be sustainable in a subgameperfect equilibrium for δ close enough to 1 Lets investi gate the threshold value of δ needed First suppose there are two firms competing in a Bertrand game each period Let PM denote the monopoly profit and PM the monopoly price in the stage game The firms may collude tacitly to sustain the monopoly pricewith each firm earning an equal share of the monopoly profitby using the grim trigger strategy of continuing to collude as long as no firm has undercut PM in the past but reverting to the stagegame nash equilibrium of marginal cost pricing every period from then on if any firm deviates by undercutting Successful tacit collusion provides the profit stream V collude 5 PM 2 1 δ PM 2 1 δ2 PM 2 1 c 5 PM 2 11 1 δ 1 δ2 1 c2 5 aPM 2 b a 1 1 2 δb 1543 refer to Chapter 8 for a discussion of adding up a series of discount factors 1 1 δ 1 δ2 1 c We need to check that a firm has no incentive to deviate By undercutting the collusive price PM slightly a firm can obtain essentially all the monopoly profit for itself in the current period This deviation would trigger the grim strategy punishment of marginal cost pricing in the second and all future periods so all firms would earn zero profit from there on hence the stream of profits from deviating is V deviate 5 PM For this deviation not to be profitable we must have V collude V deviate or on substituting aPM 2 b a 1 1 2 δb PM 1544 rearranging equation 1544 the condition reduces to δ 12 To prevent deviation firms must value the future enough that the threat of losing profits by reverting to the oneperiod nash equilibrium outweighs the benefit of undercutting and taking the whole monopoly profit in the present period EXAMPLE 156 Tacit Collusion in a Bertrand Model Bertrand duopoly Suppose only two firms produce a certain medical device used in surgery The medical device is produced at constant average and marginal cost of 10 and the demand for the device is given by Q 5 5000 2 100P 1545 if the Bertrand game is played in a single period then each firm will charge 10 and a total of 4000 devices will be sold Because the monopoly price in this market is 30 firms have a clear incentive to consider collusive strategies at the monopoly price total profits each period are 40000 and each firms share of total profits is 20000 according to equation 1544 collusion at the monopoly price is sustainable if 20000 a 1 1 2 δb 40000 1546 or if δ 12 as we saw is the condition δ 12 likely to be met in this market That depends on what factors we consider in computing δ including the interest rate and possible uncertainty about whether the game will continue Leave aside uncertainty for a moment and consider only the interest rate Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 543 if the period length is one year then it might be reasonable to assume an annual interest rate of r 5 10 as shown in the appendix to Chapter 17 δ 5 1 11 1 r2 therefore if r 5 10 then δ 5 091 This value of δ clearly exceeds the threshold of 12 needed to sustain collusion For δ to be less than the 12 threshold for collusion we must incorporate uncertainty into the discount factor There must be a significant chance that the market will not continue into the next period perhaps because a new surgical procedure is developed that renders the medical device obsolete We focused on the best possible collusive outcome the monopoly price of 30 Would collusion be easier to sustain at a lower price say 20 no at a price of 20 total profits each period are 30000 and each firms share is 15000 Substituting into equation 1544 collusion can be sustained if 15000 a 1 1 2 δb 30000 1547 again implying δ 12 Whatever collusive profit the firms try to sustain will cancel out from both sides of equation 1544 leaving the condition δ 12 Therefore we get a discrete jump in firms ability to collude as they become more patientthat is as δ increases from 0 to 110 For δ below 12 no collusion is possible For δ above 12 any price between marginal cost and the monopoly price can be sustained as a collusive outcome in the face of this multiplicity of sub gameperfect equilibria economists often focus on the one that is most profitable for the firms but the formal theory as to why firms would play one or another of the equilibria is still unsettled Bertrand oligopoly now suppose n firms produce the medical device The monopoly profit continues to be 40000 but each firms share is now only 40000n By undercutting the monop oly price slightly a firm can still obtain the whole monopoly profit for itself regardless of how many other firms there are replacing the collusive profit of 20000 in equation 1546 with 40000n we have that the n firms can successfully collude on the monopoly price if 40000 n a 1 1 2 δb 40000 1548 or δ 1 2 1 n 1549 Taking the reasonable discount factor of δ 5 091 used previously collusion is possible when 11 or fewer firms are in the market and impossible with 12 or more With 12 or more firms the only subgameperfect equilibrium involves marginal cost pricing and zero profits equation 1549 shows that tacit collusion is easier the more patient are firms as we saw before and the fewer of them there are One rationale used by antitrust authorities to challenge certain mergers is that a merger may reduce n to a level such that equation 1549 begins to be satisfied and collusion becomes possible resulting in higher prices and lower total welfare QUERY a period can be interpreted as the length of time it takes for firms to recognize and respond to undercutting by a rival What would be the relevant period for competing gasoline stations in a small town in what industries would a year be a reasonable period 10The discrete jump in firms ability to collude is a feature of the Bertrand model the ability to collude increases continuously with δ in the Cournot model of example 157 EXAMPLE 157 Tacit Collusion in a Cournot Model Suppose that there are again two firms producing medical devices but that each period they now engage in quantity Cournot rather than price Bertrand competition We will again investi gate the conditions under which firms can collude on the monopoly outcome To generate the Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 544 Part 6 Market Power monopoly outcome in a period firms need to produce 1000 each this leads to a price of 30 total profits of 40000 and firm profits of 20000 The present discounted value of the stream of these collusive profits is V collude 5 20000 a 1 1 2 δb 1550 Computing the present discounted value of the stream of profits from deviating is somewhat complicated The optimal deviation is not as simple as producing the whole monopoly output oneself and having the other firm produce nothing The other firms 1000 units would be pro vided to the market The optimal deviation by firm 1 say would be to best respond to firm 2s output of 1000 To compute this best response first note that if demand is given by equation 1545 then inverse demand is given by P 5 50 2 Q 100 1551 Firm 1s profit is π1 5 Pq1 2 cq1 5 q1 a40 2 q1 1 q2 100 b 1552 Taking the firstorder condition with respect to q1 and solving for q1 yields the bestresponse function q1 5 2000 2 q2 2 1553 Firm 1s optimal deviation when firm 2 produces 1000 units is to increase its output from 1000 to 1500 Substituting these quantities into equation 1552 implies that firm 1 earns 22500 in the period in which it deviates how much firm 1 earns in the second and later periods following a deviation depends on the trigger strategies firms use to punish deviation assume that firms use the grim strategy of revert ing to the nash equilibrium of the stage gamein this case the nash equilibrium of the Cournot gameevery period from then on in the nash equilibrium of the Cournot game each firm best responds to the other in accordance with the bestresponse function in equation 1553 switching subscripts in the case of firm 2 Solving these bestresponse equations simultaneously implies that the nash equilibrium outputs are q 1 5 q 2 5 40003 and that profits are π 1 5 π 2 5 17778 Firm 1s present discounted value of the stream of profits from deviation is V deviate 5 22500 1 17778 δ 1 17778 δ2 1 17778 δ3 1 c 5 22500 1 117778 δ2 11 1 δ 1 δ2 1 c2 5 22500 1 17778 a δ 1 2 δb 1554 We have V collude V deviate if 20000 a 1 1 2 δb 22500 1 17778 a δ 1 2 δb 1555 or after some algebra if δ 053 unlike with the Bertrand stage game with the Cournot stage game there is a possibility of some collusion for discount factors below 053 however the outcome would have to involve higher outputs and lower profits than monopoly QUERY The benefit to deviating is lower with the Cournot stage game than with the Bertrand stage game because the Cournot firm cannot steal all the monopoly profit with a small deviation Why then is a more stringent condition δ 053 rather than δ 05 needed to collude on the monopoly outcome in the Cournot duopoly compared with the Bertrand duopoly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 545 157 LOngerrun DeCiSiOnS inVeSTMenT enTrY anD exiT The chapter has so far focused on the most basic shortrun decisions regarding what price or quantity to set The scope for strategic interaction expands when we introduce longerrun decisions Take the case of the market for cars Longerrun decisions include whether to update the basic design of the car a process that might take up to 2 years to complete Longerrun decisions may also include investing in robotics to lower produc tion costs moving manufacturing plants closer to consumers and cheap inputs engaging in a new advertising campaign and entering or exiting certain product lines say ceasing the production of station wagons or starting production of hybrid cars in making such decisions an oligopolist must consider how rivals will respond to it Will competition with existing rivals become tougher or milder Will the decision lead to the exit of current rivals or encourage new ones to enter is it better to be the first to make such a decision or to wait until after rivals move 1571 Flexibility versus commitment Crucial to our analysis of longerrun decisions such as investment entry and exit is how easy it is to reverse a decision once it has been made On first thought it might seem that it is better for a firm to be able to easily reverse decisions because this would give the firm more flexibility in responding to changing circumstances For example a car manufac turer might be more willing to invest in developing a hybridelectric car if it could easily change the design back to a standard gasolinepowered one should the price of gasoline and the demand for hybrid cars along with it decrease unexpectedly absent strategic considerationsand so for the case of a monopolista firm would always value flexibility and reversibility The option value provided by flexibility is discussed in further detail in Chapter 7 Surprisingly the strategic considerations that arise in an oligopoly setting may lead a firm to prefer its decision be irreversible What the firm loses in terms of flexibility may be offset by the value of being able to commit to the decision We will see a number of instances of the value of commitment in the next several sections if a firm can commit to an action before others move the firm may gain a firstmover advantage a firm may use its firstmover advantage to stake out a claim to a market by making a commitment to serve it and in the process limit the kinds of actions its rivals find profitable Commitment is essential for a firstmover advantage if the first mover could secretly reverse its decision then its rival would anticipate the reversal and the firms would be back in the game with no firstmover advantage We already encountered a simple example of the value of commitment in the Battle of the Sexes game from Chapter 8 in the simultaneous version of the model there were three nash equilibria in one purestrategy equilibrium the wife obtains her highest payoff by attending her favorite event with her husband but she obtains lower payoffs in the other two equilibria a purestrategy equilibrium in which she attends her less favored event and a mixedstrategy equilibrium giving her the lowest payoff of all three in the sequential version of the game if a player were given the choice between being the first mover and having the ability to commit to attending an event or being the second mover and having the flexibility to be able to meet up with the first wherever he or she showed up a player would always choose the ability to commit The first mover can guarantee his or her pre ferred outcome as the unique subgameperfect equilibrium by committing to attend his or her favorite event Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 546 Part 6 Market Power 1572 Sunk costs expenditures on irreversible investments are called sunk costs Sunk costs include expenditures on unique types of equipment eg a newsprintmak ing machine or jobspecific training for workers developing the skills to use the newsprint machine There is sometimes confusion between sunk costs and what we have called fixed costs They are similar in that they do not vary with the firms output level in a production period and are incurred even if no output is produced in that period But instead of being incurred peri odically as are many fixed costs heat for the factory salaries for secretaries and other adminis trators sunk costs are incurred only once in connection with a single investment11 Some fixed costs may be avoided over a sufficiently long runsay by reselling the plant and equipment involvedbut sunk costs can never be recovered because the investments involved cannot be moved to a different use When the firm makes a sunk investment it has committed itself to that investment and this may have important consequences for its strategic behavior 1573 Firstmover advantage in the Stackelberg model The simplest setting to illustrate the firstmover advantage is in the Stackelberg model named after the economist who first analyzed it12 The model is similar to a duopoly ver sion of the Cournot model except thatrather than simultaneously choosing the quanti ties of their identical outputsfirms move sequentially with firm 1 the leader choosing its output first and then firm 2 the follower choosing after observing firm 1s output We use backward induction to solve for the subgameperfect equilibrium of this sequential game Begin with the followers output choice Firm 2 chooses the output q2 that maximizes its own profit taking firm 1s output as given in other words firm 2 best responds to firm 1s out put This results in the same bestresponse function for firm 2 as we computed in the Cournot game from the firstorder condition equation 152 Label this bestresponse function BR2 1q12 Turn then to the leaders output choice Firm 1 recognizes that it can influence the followers action because the follower best responds to 1s observed output Substituting BR2 1q12 into the profit function for firm 1 given by equation 151 we have π1 5 P 1q1 1 BR2 1q12 2q1 2 C1 1q12 1556 The firstorder condition with respect to q1 is π1 q1 5 P 1Q2 1 Pr 1Q2q1 1 Pr 1Q2BRr2 1q12q1 2 Cri 1qi2 5 0 S 1557 This is the same firstorder condition computed in the Cournot model see equation 152 except for the addition of the term S which accounts for the strategic effect of firm 1s output on firm 2s The strategic effect S will lead firm 1 to produce more than it would have in a Cournot 11Mathematically the notion of sunk costs can be integrated into the perperiod total cost function as Ct 1qt2 5 S 1 Ft 1 cqt where S is the perperiod amortization of sunk costs eg the interest paid for funds used to finance capital investments Ft is the perperiod fixed costs c is marginal cost and qt is perperiod output if qt 5 0 then Ct 5 S 1 Ft but if the production period is long enough then some or all of Ft may also be avoidable no portion of S is avoidable however 12h von Stackelberg The Theory of the Market Economy trans a T Peacock new York Oxford university Press 1952 D E F I N I T I O N Sunk cost a sunk cost is an expenditure on an investment that cannot be reversed and has no resale value Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 547 model By overproducing firm 1 leads firm 2 to reduce q2 by the amount BRr2 1q12 the fall in firm 2s output increases market price thus increasing the revenue that firm 1 earns on its existing sales We know that q2 decreases with an increase in q1 because bestresponse functions under quantity competition are generally downward sloping see Figure 152 for an illustration The strategic effect would be absent if the leaders output choice was unobservable to the follower or if the leader could reverse its output choice in secret The leader must be able to commit to an observable output choice or else firms are back in the Cournot game it is easy to see that the leader prefers the Stackelberg game to the Cournot game The leader could always reproduce the outcome from the Cournot game by choosing its Cournot output in the Stackelberg game The leader can do even better by producing more than its Cournot output thereby taking advantage of the strategic effect S EXAMPLE 158 Stackelberg Springs recall the two naturalspring owners from example 151 now rather than having them choose outputs simultaneously as in the Cournot game assume that they choose outputs sequentially as in the Stackelberg game with firm 1 being the leader and firm 2 the follower Firm 2s output We will solve for the subgameperfect equilibrium using backward induction starting with firm 2s output choice We already found firm 2s bestresponse function in equation 158 repeated here q2 5 a 2 q1 2 c 2 1558 Firm 1s output now fold the game back to solve for firm 1s output choice Substituting firm 2s best response from equation 1558 into firm 1s profit function from equation 1556 yields π1 5 ca 2 q1 2 a a 2 q1 2 c 2 b 2 cd q1 5 1 2 1a 2 q1 2 c2q1 1559 Taking the firstorder condition π1 q1 5 1 2 1a 2 2q1 2 c2 5 0 1560 and solving gives q 1 5 1a 2 c22 Substituting q 1 back into firm 2s bestresponse function gives q 2 5 1a 2 c24 Profits are π 1 5 1182 1a 2 c2 2 and π 2 5 11162 1a 2 c2 2 To provide a numerical example suppose a 5 120 and c 5 0 Then q 1 5 60 q 2 5 30 π 1 5 1800and π 2 5 900 Firm 1 produces twice as much and earns twice as much as firm 2 recall from the simultaneous Cournot game in example 151 that for these numerical val ues total market output was 80 and total industry profit was 3200 implying that each of the two firms produced 802 5 40 units and earned 32002 5 1600 Therefore when firm 1 is the first mover in a sequential game it produces 160 2 40240 5 50 more and earns 11800 2 160021600 5 125 more than in the simultaneous game Graphing the Stackelberg outcome Figure 156 illustrates the Stackelberg equilibrium on a bestresponse function diagram The leader realizes that the follower will always best respond so the resulting outcome will always be on the followers bestresponse function The leader effec tively picks the point on the followers bestresponse function that maximizes the leaders profit The highest isoprofit highest in terms of profit level but recall from Figure 152 that higher profit levels are reached as one moves down toward the horizontal axis is reached at the point S of tan gency between firm 1s isoprofit and firm 2s bestresponse function This is the Stackelberg equi librium Compared with the Cournot equilibrium at point C the Stackelberg equilibrium involves higher output and profit for firm 1 Firm 1s profit is higher because by committing to the high output level firm 2 is forced to respond by reducing its output Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 548 Part 6 Market Power Commitment is required for the outcome to stray from firm 1s bestresponse function as happens at point S if firm 1 could secretly reduce q1 perhaps because q1 is actual capacity that can be secretly reduced by reselling capital equipment for close to its purchase price to a manufacturer of another product that uses similar capital equipment then it would move back to its best response firm 2 would best respond to this lower quantity and so on following the dotted arrows from S back to C Bestresponse functions from the Cournot game are drawn as thick lines Frownshaped curves are firm 1s isoprofits Point C is the nash equilibrium of the Cournot game invoking simultaneous output choices The Stackelberg equilibrium is point S the point at which the highest isoprofit for firm 1 is reached on firm 2s bestresponse function at S firm 1s isoprofit is tangent to firm 2s bestresponse function if firm 1 cannot commit to its output then the outcome function unravels following the dot ted line from S back to C q2 q1 BR2q1 BR1q2 S C FIGURE 156 Stackelberg Game QUERY What would be the outcome if the identity of the first mover were not given and instead firms had to compete to be the first how would firms vie for this position Do these considerations help explain overinvestment in internet firms and telecommunications during the dotcom bubble 1574 Contrast with price leadership in the Stackelberg game the leader uses what has been called a top dog strategy13 aggressively overproducing to force the follower to scale back its production The leader earns more than in the associated simultaneous game Cournot whereas the follower earns less although it is gen erally true that the leader prefers the sequential game to the simultaneous game the leader can do at least as well and generally better by playing its nash equilibrium strategy from the simultane ous game it is not generally true that the leader harms the follower by behaving as a top dog Sometimes the leader benefits by behaving as a puppy dog as illustrated in example 159 13Top dog puppy dog and other colorful labels for strategies are due to D Fudenberg and J Tirole The Fat Cat effect the Puppy Dog Ploy and the Lean and hungry Look American Economic Review Papers and Proceedings 74 1984 36168 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 549 EXAMPLE 159 PriceLeadership Game return to example 154 in which two firms chose price for differentiated toothpaste brands simultaneously So that the following calculations do not become too tedious we make the sim plifying assumptions that a1 5 a2 5 1 and c 5 0 Substituting these parameters back into exam ple 154 shows that equilibrium prices are 23 0667 and profits are 49 0444 for each firm now consider the game in which firm 1 chooses price before firm 214 We will solve for the subgameperfect equilibrium using backward induction starting with firm 2s move Firm 2s best response to its rivals choice p1 is the same as computed in example 154which on substituting a2 5 1 and c 5 0 into equation 1532 is p2 5 1 2 1 p1 4 1561 Fold the game back to firm 1s move Substituting firm 2s best response into firm 1s profit func tion from equation 1530 gives π1 5 p1 c1 2 p1 1 1 2a1 2 1 p1 4 b d 5 p1 8 110 2 7p12 1562 Taking the firstorder condition and solving for the equilibrium price we obtain p 1 0714 Substituting into equation 1561 gives p 2 0679 equilibrium profits are π 1 0446 and π 2 0460 Both firms prices and profits are higher in this sequential game than in the simulta neous one but now the follower earns even more than the leader as illustrated in the bestresponse function diagram in Figure 157 firm 1 commits to a high price to induce firm 2 to raise its price also essentially softening the competition between them 14Sometimes this game is called the Stackelberg price game although technically the original Stackelberg game involved quantity competition Thick lines are bestresponse functions from the game in which firms choose prices for differentiated products ushaped curves are firm 1s isoprofits Point B is the nash equilibrium of the simultaneous game and L is the subgameperfect equilibrium of the sequential game in which firm 1 moves first at L firm 1s isoprofit is tangent to firm 2s best response p2 BR1p2 BR2p1 B L p1 FIGURE 157 PriceLeadership Game Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 550 Part 6 Market Power The leader needs a moderate price increase from 0667 to 0714 to induce the follower to increase its price slightly from 0667 to 0679 so the leaders profits do not increase as much as the followers QUERY What choice variable realistically is easier to commit to prices or quantities What business strategies do firms use to increase their commitment to their list prices We say that the first mover is playing a puppy dog strategy in example 159 because it increases its price relative to the simultaneousmove game when translated into outputs this means that the first mover ends up producing less than in the simultaneousmove game it is as though the first mover strikes a less aggressive posture in the market and so leads its rival to compete less aggressively a comparison of Figures 156 and 157 suggests the crucial difference between the games that leads the first mover to play a top dog strategy in the quantity game and a puppy dog strategy in the price game The bestresponse functions have different slopes The goal is to induce the follower to compete less aggressively The slopes of the best response functions determine whether the leader can best do that by playing aggressively itself or by softening its strategy The first mover plays a top dog strategy in the sequential quantity game or indeed any game in which best responses slope down When best responses slope down playing more aggressively induces a rival to respond by competing less aggressively Conversely the first mover plays a puppy dog strategy in the price game or any game in which best responses slope up When best responses slope up playing less aggressively induces a rival to respond by competing less aggressively Therefore knowing the slope of firms best responses provides considerable insight into the sort of strategies firms will choose if they have commitment power The exten sions at the end of this chapter provide further technical details including shortcuts for determining the slope of a firms bestresponse function just by looking at its profit function 158 STraTegiC enTrY DeTerrenCe We saw that by committing to an action a first mover may be able to manipulate the second mover into being a less aggressive competitor in this section we will see that the first mover may be able to prevent the entry of the second mover entirely leaving the first mover as the sole firm in the market in this case the firm may not behave as an unconstrained monopolist because it may have distorted its actions to fend off the rivals entry in deciding whether to deter the second movers entry the first mover must weigh the costs and benefits relative to accommodating entrythat is allowing entry to happen accommodating entry does not mean behaving nonstrategically The first mover would move off its bestresponse function to manipulate the second mover into being less com petitive as described in the previous section The cost of deterring entry is that the first mover would have to move off its bestresponse function even further than it would if it accommodates entry The benefit is that it operates alone in the market and has market demand to itself Deterring entry is relatively easy for the first mover if the second mover must pay a substantial sunk cost to enter the market Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 551 EXAMPLE 1510 Deterring Entry of a Natural Spring recall example 158 where two naturalspring owners choose outputs sequentially We now add an entry stage in particular after observing firm 1s initial quantity choice firm 2 decides whether to enter the market entry requires the expenditure of sunk cost K2 after which firm 2 can choose output Market demand and cost are as in example 158 To simplify the calculations we will take the specific numerical values a 5 120 and c 5 0 implying that inverse demand is P 1Q2 5 120 2 Q and that production is costless To further simplify we will abstract from firm 1s entry decision and assume that it has already sunk any cost needed to enter before the start of the game We will look for conditions under which firm 1 prefers to deter rather than accommo date firm 2s entry Accommodating entry Start by computing firm 1s profit if it accommodates firm 2s entry denoted π acc 1 This has already been done in example 158 in which there was no issue of deterring firm 2s entry There we found firm 1s equilibrium output to be 1a 2 c22 5 q acc 1 and its profit to be 1a 2 c2 28 5 π acc 1 Substituting the specific numerical values a 5 120 and c 5 0 we have q acc 1 5 60 and π acc 1 5 1120 2 02 28 5 1800 Deterring entry next compute firm 1s profit if it deters firm 2s entry denoted πdet 1 To deter entry firm 1 needs to produce an amount qdet 1 high enough that even if firm 2 best responds to qdet 1 it cannot earn enough profit to cover its sunk cost K2 We know from equation 1558 that firm 2s bestresponse function is q2 5 120 2 q1 2 1563 Substituting for q2 in firm 2s profit function equation 157 and simplifying gives π2 5 a 120 2 q det 1 2 b 2 2 K2 1564 Setting firm 2s profit in equation 1564 equal to 0 and solving yields q det 1 5 120 2 2K2 1565 q det 1 is the firm 1 output needed to keep firm 2 out of the market at this output level firm 1s profit is π det 1 5 2K2 1120 2 2K22 1566 which we found by substituting q det 1 a 5 120 and c 5 0 into firm 1s profit function from equa tion 157 We also set q2 5 0 because if firm 1 is successful in deterring entry it operates alone in the market Comparison The final step is to juxtapose π acc 1 and π det 1 to find the condition under which firm 1 prefers deterring to accommodating entry To simplify the algebra let x 5 2K2 Then π det 1 5 π acc 1 if x2 2 120x 1 1800 5 0 1567 applying the quadratic formula yields x 5 120 6 7200 2 1568 Taking the smaller root because we will be looking for a minimum threshold we have x 5 176 rounding to the nearest decimal Substituting x 5 176 into x 5 2K2 and solving for K2 yields K2 5 ax 2b 2 5 a176 2 b 2 77 1569 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 552 Part 6 Market Power if K2 5 77 then entry is so cheap for firm 2 that firm 1 would have to increase its output all the way to q det 1 5 102 in order to deter entry This is a significant distortion above what it would produce when accommodating entry q acc 1 5 60 if K2 77 then the output distortion needed to deter entry wastes so much profit that firm 1 prefers to accommodate entry if K2 77 output need not be distorted as much to deter entry thus firm 1 prefers to deter entry QUERY Suppose the first mover must pay the same entry cost as the second K1 5 K2 5 K Suppose further that K is high enough that the first mover prefers to deter rather than accommo date the second movers entry Would this sunk cost not be high enough to keep the first mover out of the market too Why or why not a realworld example of overproduction or overcapacity to deter entry is provided by the 1945 antitrust case against alcoa a uS aluminum manufacturer a uS federal court ruled that alcoa maintained much higher capacity than was needed to serve the market as a strategy to deter rivals entry and it held that alcoa was in violation of antitrust laws To recap what we have learned in the last two sections With quantity competition the first mover plays a top dog strategy regardless of whether it deters or accommodates the second movers entry True the entrydeterring strategy is more aggressive than the entryaccommodating one but this difference is one of degree rather than kind however with price competition as in example 159 the first movers entrydeterring strategy would differ in kind from its entryaccommodating strategy it would play a puppy dog strategy if it wished to accommodate entry because this is how it manipulates the second mover into playing less aggressively it plays a top dog strategy of lowering its price rel ative to the simultaneous game if it wants to deter entry Two general principles emerge entry deterrence is always accomplished by a top dog strategy whether competition is in quantities or prices or more generally whether bestresponse functions slope down or up The first mover simply wants to create an inhospitable environment for the second mover if firm 1 wants to accommodate entry whether it should play a puppy dog or top dog strategy depends on the nature of competitionin particular on the slope of the bestresponse functions 159 SignaLing The preceding sections have shown that the first movers ability to commit may afford it a big strategic advantage in this section we will analyze another possible firstmover advantage the ability to signal if the second mover has incomplete information about market conditions eg costs demand then it may try to learn about these conditions by observing how the first mover behaves The first mover may try to distort its actions to manipulate what the second learns The analysis in this section is closely tied to the material on signaling games in Chapter 8 and the reader may want to review that material before proceeding with this section The ability to signal may be a plausible benefit of being a first mover in some settings in which the benefit we studied earliercommitmentis implausible For example in indus tries where the capital equipment is readily adapted to manufacture other products costs are not very sunk thus capacity commitments may not be especially credible The first mover can reduce its capacity with little loss For another example the priceleadership game involved a commitment to price it is hard to see what sunk costs are involved in Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 553 setting a price and thus what commitment value it has15 Yet even in the absence of com mitment value prices may have strategic signaling value 1591 Entrydeterrence model Consider the incomplete information game in Figure 158 The game involves a first mover firm 1 and a second mover firm 2 that choose prices for their differentiated products Firm 1 has private information about its marginal cost which can take on one of two values high with probability PrH or low with probability Pr 1L2 5 1 2 Pr 1H2 in period 1 firm 1 serves the market alone at the end of the period firm 2 observes firm 1s price and decides whether to enter the market if it enters it sinks an entry cost K2 and learns the true level of firm 1s costs then firms compete as duopolists in the second period choosing prices for differentiated products as in example 154 or 155 We do not need to be specific about the 15The Query in example 159 asks you to consider reasons why a firm may be able to commit to a price The firm may gain commitment power by using contracts eg longterm supply contracts with customers or a mostfavored customer clause which ensures that if the firm lowers price in the future to other customers then the favored customer gets a rebate on the price difference The firm may advertise a price through an expensive national advertising campaign The firm may have established a valuable reputation as charging everyday low prices Firm 1 signals its private information about its cost high H or low L through the price it sets in the first period Firm 2 observes firm 1s price and then decides whether to enter if firm 2 enters the firms compete as duopolists otherwise firm 1 operates alone on the market again in the second period Firm 2 earns positive profit if and only if it enters against a high cost rival E E E NE NE NE 1 2 PrL PrH M1 H D1 H D2 H M1 H R D1 H D2 H M1 L D1 L D2 L 2M1 H R 0 2M1 L 0 2M1 H 0 p1 H p1 L p1 L 2 2 1 FIGURE 158 Signaling for Entry Deterrence Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 554 Part 6 Market Power exact form of demands if firm 2 does not enter it obtains a payoff of zero and firm 1 again operates alone in the market assume there is no discounting between periods Firm 2 draws inferences about firm 1s cost from the price that firm 1 charges in the first period Firm 2 earns more if it competes against the highcost type because the highcost types price will be higher and as we saw in examples 154 and 155 the higher the rivals price for a differentiated product the higher the firms own demand and profit Let D t i be the duopoly profit not including entry costs for firm i 51 26 if firm 1 is of type t 5L H6 To make the model interesting we will suppose D L 2 K2 D H 2 so that firm 2 earns more than its entry cost if it faces the highcost type but not if it faces the lowcost type Otherwise the information in firm 1s signal would be useless because firm 2 would always enter or always stay out regardless of firm 1s type To simplify the model we will suppose that the lowcost type only has one relevant action in the first periodnamely setting its monopoly price p L 1 The highcost type can choose one of two prices it can set the monopoly price associated with its type p H 1 or it can choose the same price as the low type p L 1 Presumably the optimal monopoly price is increasing in mar ginal cost thus p L 1 p H 1 Let M t 1 be firm 1s monopoly profit if it is of type t 5L H6 the profit if it is alone and charges its optimal monopoly price p H 1 if it is the high type and p L 1 if it is the low type Let R be the high types loss relative to the optimal monopoly profit in the first period if it charges p L 1 rather than its optimal monopoly price p H 1 Thus if the high type charges p H 1 in the first period then it earns M H 1 in that period but if it charges p L 1 it earns M H 1 2 R 1592 Separating equilibrium We will look for two kinds of perfect Bayesian equilibria separating and pooling in a sep arating equilibrium the different types of the first mover must choose different actions here there is only one such possibility for firm 1 The lowcost type chooses p L 1 and the highcost type chooses p H 1 Firm 2 learns firm 1s type from these actions perfectly and stays out on seeing p L 1 and enters on seeing p H 1 it remains to check whether the highcost type would prefer to deviate to p L 1 in equilibrium the high type earns a total profit of M H 1 1 D H 1 M H 1 in the first period because it charges its optimal monopoly price and D H 1 in the second because firm 2 enters and the firms compete as duopolists if the high type were to deviate to p L 1 then it would earn M H 1 2 R in the first period the loss R coming from charging a price other than its firstperiod optimum but firm 2 would think it is the low type and would not enter hence firm 1 would earn M H 1 in the second period for a total of 2M H 1 2 R across periods For deviation to be unprofitable we must have M H 1 1 D H 1 2M H 1 2 R 1570 or after rearranging R M H 1 2 D H 1 1571 That is the hightypes loss from distorting its price from its monopoly optimum in the first period exceeds its gain from deterring firm 2s entry in the second period if the condition in equation 1571 does not hold there still may be a separating equilib rium in an expanded game in which the low type can charge other prices besides p L 1 The high type could distort its price downward below p L 1 increasing the firstperiod loss the high type would suffer from pooling with the low type to such an extent that the high type would rather charge p H 1 even if this results in firm 2s entry 1593 Pooling equilibrium if the condition in equation 1571 does not hold then the high type would prefer to pool with the low type if pooling deters entry Pooling deters entry if firm 2s prior belief Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 555 that firm 1 is the high type PrHwhich is equal to its posterior belief in a pooling equilibriumis low enough that firm 2s expected payoff from entering Pr 1H2D H 2 1 31 2 Pr 1H2 4D L 2 2 K2 1572 is less than its payoff of zero from staying out of the market 1594 Predatory pricing The incompleteinformation model of entry deterrence has been used to explain why a ratio nal firm might want to engage in predatory pricing the practice of charging an artificially low price to prevent potential rivals from entering or to force existing rivals to exit The predatory firm sacrifices profits in the short run to gain a monopoly position in future periods Predatory pricing is prohibited by antitrust laws in the most famous antitrust case dating back to 1911 John D rockefellerowner of the Standard Oil Company that con trolled a substantial majority of refined oil in the united Stateswas accused of attempt ing to monopolize the oil market by cutting prices dramatically to drive rivals out and then raising prices after rivals had exited the market or sold out to Standard Oil Predatory pricing remains a controversial antitrust issue because of the difficulty in distinguishing between predatory conduct which authorities would like to prevent and competitive con duct which authorities would like to promote in addition economists initially had trouble developing gametheoretic models in which predatory pricing is rational and credible Suitably interpreted predatory pricing may emerge as a rational strategy in the incompleteinformation model of entry deterrence Predatory pricing can show up in a separating equilibriumin particular in the expanded model where the lowcost type can separate only by reducing price below its monopoly optimum Total welfare is actually higher in this separating equilibrium than it would be in its fullinformation counterpart Firm 2s entry decision is the same in both outcomes but the lowcost types price may be lower to signal its type in the predatory outcome Predatory pricing can also show up in a pooling equilibrium in this case it is the high cost type that charges an artificially low price pricing below its firstperiod optimum to keep firm 2 out of the market Whether social welfare is lower in the pooling equilibrium than in a fullinformation setting is unclear in the first period price is lower and total welfare presumably higher in the pooling equilibrium than in a fullinformation setting On the other hand deterring firm 2s entry results in higher secondperiod prices and lower welfare Weighing the firstperiod gain against the secondperiod loss would require detailed knowledge of demand curves discount factors and so forth The incompleteinformation model of entry deterrence is not the only model of predatory pricing that economists have developed another model involves frictions in the market for financial capital that stem perhaps from informational problems between borrowers and lend ers of the sort we will discuss in Chapter 18 With limits on borrowing firms may only have limited resources to make a go in a market a larger firm may force financially strapped rivals to endure losses until their resources are exhausted and they are forced to exit the market 1510 hOW ManY FirMS enTer To this point we have taken the number of firms in the market as given often assuming that there are at most two firms as in examples 151 153 and 1510 We did allow for a general number of firms n in some of our analysis as in examples 153 and 157 but were silent about how this number n was determined in this section we provide a gametheoretic anal ysis of the number of firms by introducing a first stage in which a large number of potential entrants can each choose whether to enter We will abstract from firstmover advantages Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 556 Part 6 Market Power entry deterrence and other strategic considerations by assuming that firms make their entry decisions simultaneously Strategic considerations are interesting and important but we have already developed some insights into strategic considerations from the previous sections andby abstracting from themwe can simplify the analysis here 15101 Barriers to entry For the market to be oligopolistic with a finite number of firms rather than perfectly com petitive with an infinite number of infinitesimal firms some factors called barriers to entry must eventually make entry unattractive or impossible We discussed many of these factors at length in the previous chapter on monopoly if a sunk cost is required to enter the market theneven if firms can freely choose whether to enteronly a limited number of firms will choose to enter in equilibrium because competition among more than that number would drive profits below the level needed to recoup the sunk entry cost govern ment intervention in the form of patents or licensing requirements may prevent firms from entering even if it would be profitable for them to do so Some of the new concepts discussed in this chapter may introduce additional barri ers to entry Search costs may prevent consumers from finding new entrants with lower prices andor higher quality than existing firms Product differentiation may raise entry barriers because of strong brand loyalty existing firms may bolster brand loyalty through expensive advertising campaigns and softening this brand loyalty may require entrants to conduct similarly expensive advertising campaigns existing firms may take other strategic measures to deter entry such as committing to a high capacity or output level engaging in predatory pricing or other measures discussed in previous sections 15102 Longrun equilibrium Consider the following gametheoretic model of entry in the long run a large number of sym metric firms are potential entrants into a market Firms make their entry decisions simultane ously entry requires the expenditure of sunk cost K Let n be the number of firms that decide to enter in the next stage the n firms engage in some form of competition over a sequence of periods during which they earn the present discounted value of some constant profit stream To simplify we will usually collapse the sequence of periods of competition into a single period Let g 1n2 be the profit earned by an individual firm in this competition subgame not including the sunk cost so g 1n2 is a gross profit Presumably the more firms in the market the more competitive the market is and the less an individual firm earns so gr 1n2 0 We will look for a subgameperfect equilibrium in pure strategies16 This will be the number of firms n satisfying two conditions First the n entering firms earn enough to cover their entry cost g 1n2 K Otherwise at least one of them would have preferred to have deviated to not entering Second an additional firm cannot earn enough to cover its entry cost g 1n 1 12 K Otherwise a firm that remained out of the market could have profitably deviated by entering given that gr 1n2 0 we can put these two conditions together and say that n is the greatest integer satisfying g 1n2 K This condition is reminiscent of the zeroprofit condition for longrun equilibrium under perfect competition The slight nuance here is that active firms are allowed to earn positive profits especially if K is large relative to the size of the market there may only be a few long run entrants thus the market looks like a canonical oligopoly earning well above what they need to cover their sunk costs yet an additional firm does not enter because its entry would depress individual profit enough that the entrant could not cover its large sunk cost 16a symmetric mixedstrategy equilibrium also exists in which sometimes more and sometimes fewer firms enter than can cover their sunk costs There are multiple purestrategy equilibria depending on the identity of the n entrants but n is uniquely identified Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 557 is the longrun equilibrium efficient Does the oligopoly involve too few or too many firms relative to what a benevolent social planner would choose for the market Suppose the social planner can choose the number of firms restricting entry through licenses and pro moting entry through subsidizing the entry cost but cannot regulate prices or other compet itive conduct of the firms once in the market The social planner would choose n to maximize CS1n2 1 ng 1n2 2 nK 1573 where CSn is equilibrium consumer surplus in an oligopoly with n firms ngn is total equilibrium profit gross of sunk entry costs across all firms and nK is the total expendi ture on sunk entry costs Let n be the social planners optimum in general the longrun equilibrium number of firms n may be greater or less than the social optimum n depending on two offsetting effects the appropriability effect and the businessstealing effect The social planner takes account of the benefit of increased consumer surplus from lower prices but firms do not appropriate consumer surplus and so do not take into account this benefit This appropriability effect would lead a social planner to choose more entry than in the longrun equilibrium n n Working in the opposite direction is that entry causes the profits of existing firms to decrease as indicated by the derivative gr 1n2 0 entry increases the competitiveness of the market destroying some of firms profits in addition the entrant steals some market share from existing firmshence the term businessstealing effect The marginal firm does not take other firms loss in profits when making its entry decision whereas the social planner would The businessstealing effect biases longrun equilibrium toward more entry than a social planner would choose n n Depending on the functional forms for demand and costs the appropriability effect dom inates in some cases and there is less entry in longrun equilibrium than is efficient in other cases the businessstealing dominates and there is more entry in longrun equilib rium than is efficient as in example 1511 EXAMPLE 1511 Cournot in the Long Run Longrun equilibrium return to example 153 of a Cournot oligopoly We will determine the longrun equilibrium number of firms in the market Let K be the sunk cost a firm must pay to enter the market in an initial entry stage Suppose there is one period of Cournot competition after entry To further simplify the calculations assume that a 5 1 and c 5 0 Substituting these values back into example 153 we have that an individual firms gross profit is g1n2 5 a 1 n 1 1b 2 1574 The longrun equilibrium number of firms is the greatest integer n satisfying g1n2 K ignor ing integer problems n satisfies n 5 1 K 2 1 1575 Social planners problem We first compute the individual terms in the social planners objec tive function equation 1573 Consumer surplus equals the area of the shaded triangle in Figure 159 which using the formula for the area of a triangle is CS 1n2 5 1 2 Q 1n2 3a 2 P 1n2 4 5 n2 2 1n 1 12 2 1576 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 558 Part 6 Market Power here the last equality comes from substituting for price and quantity from equations 1518 and 1519 Total profits for all firms gross of sunk costs equal the area of the shaded rectangle ng1n2 5 Q 1n2P 1n2 5 n 1n 1 12 2 1577 Substituting from equations 1576 and 1577 into the social planners objective function equa tion 1573 gives n2 2 1n 1 12 2 1 n 1n 1 12 2 2nK 1578 after removing positive constants the firstorder condition with respect to n is 1 2 K 1n 1 12 3 5 0 1579 implying that n 5 1 K13 2 1 1580 ignoring integer problems this is the optimal number of firms for a social planner Comparison if K 1 a condition required for there to be any entry then n n and so there is more entry in longrun equilibrium than a social planner would choose To take a particular numerical example let K 5 01 Then n 5 216 and n 5 115 implying that the market would be a duopoly in longrun equilibrium but a social planner would have preferred a monopoly QUERY if the social planner could set both the number of firms and the price in this example what choices would he or she make how would these compare to longrun equilibrium equilibrium for n firms drawn for the demand and cost assumptions in example 1511 Consumer surplus CSn is the area of the shaded triangle Total profits ngn for all firms gross of sunk costs is the area of the shaded rectangle Price 1 Qn Pn 1 CSn ngn c 0 Quantity Demand FIGURE 159 Profit and Consumer Surplus in Example 1511 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 559 15103 Feedback effect We found that certain factors decreased the stringency of competition and increased firms profits eg quantity rather than price competition product differentiation search costs discount factors sufficient to sustain collusion a feedback effect is that the more profit able the market is for a given number of firms the more firms will enter the market mak ing the market more competitive and less profitable than it would be if the number of firms were fixed To take an extreme example compare the Bertrand and Cournot games Taking as given that the market involves two identical producers we would say that the Bertrand game is much more competitive and less profitable than the Cournot game This conclu sion would be reversed if firms facing a sunk entry cost were allowed to make rational entry decisions Only one firm would choose to enter the Bertrand market a second firm would drive gross profit to zero and so its entry cost would not be covered The longrun equilibrium outcome would involve a monopolist and thus the highest prices and profits possible exactly the opposite of our conclusions when the number of firms was fixed On the other hand the Cournot market may have space for several entrants driving prices and profits below their monopoly levels in the Bertrand market The moderating effect of entry should lead economists to be careful when drawing con clusions about oligopoly outcomes Product differentiation search costs collusion and other factors may reduce competition and increase profits in the short run but they may also lead to increased entry and competition in the long run and thus have ambiguous effects overall on prices and profits Perhaps the only truly robust conclusions about prices and profits in the long run involve sunk costs greater sunk costs constrain entry even in the long run so we can confidently say that prices and profits will tend to be higher in industries requiring higher sunk costs as a percentage of sales to enter17 1511 innOVaTiOn at the end of the previous chapter we asked which market structuremonopoly or perfect competitionleads to more innovation in new products and costreducing processes if monopoly is more innovative will the longrun benefits of innovation offset the shortrun deadweight loss of monopoly The same questions can be asked in the context of oligopoly Do concentrated market structures with few firms perhaps charging high prices provide better incentives for innovation Which is more innovative a large or a small firm an established firm or an entrant answers to these questions can help inform policy toward mergers entry regulation and smallfirm subsidies as we will see with the aid of some simple models there is no definite answer as to what level of concentration is best for longrun total welfare We will derive some general trade offs but quantifying these tradeoffs to determine whether a particular market would be more innovative if it were concentrated or unconcentrated will depend on the nature of competition for innovation the nature of competition for consumers and the specification of demand and cost functions The same can be said for determining what firm size or age is most innovative The models we introduce here are of product innovations the invention of a product eg plasma televisions that did not exist before another class of innovations is that of process innovations which reduce the cost of producing existing productsfor example the use of robot technology in automobile manufacture 17For more on robust conclusions regarding industry structure and competitiveness see J Sutton Sunk Costs and Market Structure Cambridge Ma MiT Press 1991 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 560 Part 6 Market Power 15111 Monopoly on innovation Begin by supposing that only a single firm call it firm 1 has the capacity to innovate For example a pharmaceutical manufacturer may have an idea for a malaria vaccine that no other firm is aware of how much would the firm be willing to complete research and development for the vaccine and to test it with largescale clinical trials how does this willingness to spend which we will take as a measure of the innovativeness of the firm depend on concentration of firms in the market Suppose first that there is currently no other vaccine available for malaria if firm 1 suc cessfully develops the vaccine then it will be a monopolist Letting PM be the monopoly profit firm 1 would be willing to spend as much as PM to develop the vaccine next to examine the case of a less concentrated market suppose that another firm firm 2 already has a vaccine on the market for which firm 1s would be a therapeutic substitute if firm 1 also develops its vaccine the firms compete as duopolists Let πD be the duopoly profit in a Bertrand model with identical products πD 5 0 but πD 0 in other modelsfor example models involving quantity competition or collusion Firm 1 would be willing to spend as much as πD to develop the vaccine in this case Comparing the two cases because PM πD it follows that firm 1 would be willing to spend more and by this measure would be more innovative in a more concentrated market The general principle here can be labeled a dissipation effect Competition dissipates some of the profit from innova tion and thus reduces incentives to innovate The dissipation effect is part of the rationale behind the patent system a patent grants monopoly rights to an inventor intentionally restricting competition to ensure higher profits and greater innovation incentives another comparison that can be made is to see which firm 1 or 2 has more of an incentive to innovate given that it has a monopoly on the initial idea Firm 1 is initially out of the market and must develop the new vaccine to enter Firm 2 is already in the malaria market with its first vaccine but can consider developing a second one as well which we will continue to assume is a perfect substitute as shown in the previous paragraph firm 1 would be willing to pay up to πD for the innovation Firm 2 would not be willing to pay anything because it is currently a monopolist in the malaria vaccine market and would continue as a monopolist whether or not it developed the second medicine Crucial to this conclusion is that the firm with the initial idea can decline to develop it but still not worry that the other firm will take the idea we will change this assumption in the next subsec tion Therefore the potential competitor firm 1 is more innovative by our measure than the existing monopolist firm 2 The general principle here has been labeled a replacement effect Firms gain less incremental profit and thus have less incentive to innovate if the new product replaces an existing product already making profit than if the firm is a new entrant in the market The replacement effect can explain turnover in certain industries where old firms become increasingly conservative and are eventually displaced by innovative and quickly growing startups as Microsoft displaced iBM as the dominant company in the computer industry and as google now threatens to replace Microsoft 15112 Competition for innovation new firms are not always more innovative than existing firms The dissipation effect may counteract the replacement effect leading old firms to be more innovative To see this tradeoff requires yet another variant of the model Suppose now that more than one firm has an initial idea for a possible innovation and that they compete to see which can develop the idea into a viable product For example the idea for a new malaria vaccine may have occurred to scientists in two firms laboratories at about the same time and the firms may Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 561 engage in a race to see who can produce a viable vaccine from this initial idea Continue to assume that firm 2 already has a malaria vaccine on the market and that this new vaccine would be a perfect substitute for it The difference between the models in this and the previous section is that if firm 2 does not win the race to develop the idea then the idea does not simply fall by the wayside but rather is developed by the competitor firm 1 Firm 2 has an incentive to win the innovation competition to prevent firm 1 from becoming a competitor Formally if firm 1 wins the innovation competition then it enters the market and is a competitor with firm 2 earning duopoly profit πD as we have repeatedly seen this is the maximum that firm 1 would pay for the innovation Firm 2s profit is PM if it wins the competition for the innovation but πD if it loses and firm 1 wins Firm 2 would pay up to the difference PM 2 πD for the inno vation if PM 2πDthat is if industry profit under a monopoly is greater than under a duopoly which it is when some of the monopoly profit is dissipated by duopoly competition then PM 2 πD πD and firm 2 will have more incentive to innovate than firm 1 This model explains the puzzling phenomenon of dominant firms filing for sleeping patents patents that are never implemented Dominant firms have a substantial incentive as we have seen possibly greater than entrantsto file for patents to prevent entry and pre serve their dominant position Whereas the replacement effect may lead to turnover in the market and innovation by new firms the dissipation effect may help preserve the position of dominant firms and retard the pace of innovation Summary Many markets fall between the polar extremes of perfect com petition and monopoly in such imperfectly competitive mar kets determining market price and quantity is complicated because equilibrium involves strategic interaction among the firms in this chapter we used the tools of game theory developed in Chapter 8 to study strategic interaction in oli gopoly markets We first analyzed oligopoly firms shortrun choices such as prices and quantities and then went on to analyze firms longerrun decisions such as product location innovation entry and the deterrence of entry We found that seemingly small changes in modeling assumptions may lead to big changes in equilibrium outcomes Therefore predicting behavior in oligopoly markets may be difficult based on the ory alone and may require knowledge of particular industries and careful empirical analysis Still some general principles did emerge from our theoretical analysis that aid in under standing oligopoly markets One of the most basic oligopoly models the Bertrand model involves two identical firms that set prices simul taneously The equilibrium resulted in the Bertrand para dox even though the oligopoly is the most concentrated possible firms behave as perfect competitors pricing at marginal cost and earning zero profit The Bertrand paradox is not the inevitable outcome in an oligopoly but can be escaped by changing assumptions underlying the Bertrand modelfor example allowing for quantity competition differentiated products search costs capacity constraints or repeated play leading to collusion as in the Prisoners Dilemma firms could profit by coor dinating on a less competitive outcome but this outcome will be unstable unless firms can explicitly collude by forming a legal cartel or tacitly collude in a repeated game For tacit collusion to sustain supercompetitive profits firms must be patient enough that the loss from a price war in future periods to punish undercutting exceeds the benefit from undercutting in the current period Whereas a nonstrategic monopolist prefers flexibility to respond to changing market conditions a strategic oligopolist may prefer to commit to a single choice The firm can commit to the choice if it involves a sunk cost that cannot be recovered if the choice is later reversed a first mover can gain an advantage by committing to a different action from what it would choose in the nash equilibrium of the simultaneous game To deter entry the first mover should commit to reducing the entrants profits using an aggressive top dog strategy high output or low price if it does not deter entry the first mover should commit to a strategy leading its rival to compete less aggressively This is sometimes a top dog and sometimes a puppy dog strategy depending on the slope of firms best responses Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 562 Part 6 Market Power holding the number of firms in an oligopoly constant in the short run the introduction of a factor that soft ens competition eg product differentiation search costs collusion will increase firms profit but an offset ting effect in the long run is that entrywhich tends to reduce oligopoly profitwill be more attractive innovation may be even more important than low prices for total welfare in the long run Determining which oligopoly structure is the most innovative is difficult because offsetting effects dissipation and replacement are involved Problems 151 assume for simplicity that a monopolist has no costs of pro duction and faces a demand curve given by Q 5 150 2 P a Calculate the profitmaximizing pricequantity combi nation for this monopolist also calculate the monopo lists profit b Suppose instead that there are two firms in the market facing the demand and cost conditions just described for their identical products Firms choose quantities simul taneously as in the Cournot model Compute the outputs in the nash equilibrium also compute market output price and firm profits c Suppose the two firms choose prices simultaneously as in the Bertrand model Compute the prices in the nash equilibrium also compute firm output and profit as well as market output d graph the demand curve and indicate where the market pricequantity combinations from parts ac appear on the curve 152 Suppose that firms marginal and average costs are constant and equal to c and that inverse market demand is given by P 5 a 2 bQ where a b 0 a Calculate the profitmaximizing pricequantity combi nation for a monopolist also calculate the monopolists profit b Calculate the nash equilibrium quantities for Cournot duopolists which choose quantities for their identical products simultaneously also compute market output market price and firm and industry profits c Calculate the nash equilibrium prices for Bertrand duopolists which choose prices for their identical prod ucts simultaneously also compute firm and market out put as well as firm and industry profits d Suppose now that there are n identical firms in a Cournot model Compute the nash equilibrium quanti ties as functions of n also compute market output mar ket price and firm and industry profits e Show that the monopoly outcome from part a can be reproduced in part d by setting n 5 1 that the Cournot duopoly outcome from part b can be repro duced in part d by setting n 5 2 in part d and that letting n approach infinity yields the same market price output and industry profit as in part c 153 Let ci be the constant marginal and average cost for firm i so that firms may have different marginal costs Suppose demand is given by P 5 1 2 Q a Calculate the nash equilibrium quantities assuming there are two firms in a Cournot market also compute market output market price firm profits industry prof its consumer surplus and total welfare b represent the nash equilibrium on a bestresponse function diagram Show how a reduction in firm 1s cost would change the equilibrium Draw a representative isoprofit for firm 1 154 Suppose that firms 1 and 2 operate under conditions of con stant average and marginal cost but that firm 1s marginal cost is c1 5 10 and firm 2s is c2 5 8 Market demand is Q 5 500 2 20P a Suppose firms practice Bertrand competition that is setting prices for their identical products simultaneously Compute the nash equilibrium prices To avoid techni cal problems in this question assume that if firms charge equal prices then the lowcost firm makes all the sales b Compute firm output firm profit and market output c is total welfare maximized in the nash equilibrium if not suggest an outcome that would maximize total wel fare and compute the deadweight loss in the nash equi librium compared with your outcome 155 Consider the following Bertrand game involving two firms producing differentiated products Firms have no costs of production Firm 1s demand is q1 5 1 2 p1 1 bp2 where b 0 a symmetric equation holds for firm 2s demand Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 563 a Solve for the nash equilibrium of the simultaneous pricechoice game b Compute the firms outputs and profits c represent the equilibrium on a bestresponse function dia gram Show how an increase in b would change the equilib rium Draw a representative isoprofit curve for firm 1 156 recall example 156 which covers tacit collusion Suppose as in the example that a medical device is produced at constant average and marginal cost of 10 and that the demand for the device is given by Q 5 5000 2 100P The market meets each period for an infinite number of peri ods The discount factor is δ a Suppose that n firms engage in Bertrand competition each period Suppose it takes two periods to discover a devia tion because it takes two periods to observe rivals prices Compute the discount factor needed to sustain collusion in a subgameperfect equilibrium using grim strategies b now restore the assumption that as in example 157 deviations are detected after just one period next assume that n is not given but rather is determined by the number of firms that choose to enter the market in an initial stage in which entrants must sink a onetime cost K to participate in the market Find an upper bound on n Hint Two conditions are involved 157 assume as in Problem 151 that two firms with no production costs facing demand Q 5 150 2 P choose quantities q1 and q2 a Compute the subgameperfect equilibrium of the Stack elberg version of the game in which firm 1 chooses q1 first and then firm 2 chooses q2 b now add an entry stage after firm 1 chooses q1 in this stage firm 2 decides whether to enter if it enters then it must sink cost K2 after which it is allowed to choose q2 Compute the threshold value of K2 above which firm 1 prefers to deter firm 2s entry c represent the Cournot Stackelberg and entrydeterrence outcomes on a bestresponse function diagram 158 recall the hotelling model of competition on a linear beach from example 155 Suppose for simplicity that ice cream stands can locate only at the two ends of the line segment zoning prohibits commercial development in the middle of the beach This question asks you to analyze an entry deterring strategy involving product proliferation a Consider the subgame in which firm A has two ice cream stands one at each end of the beach and B locates along with A at the right endpoint What is the nash equilibrium of this subgame Hint Bertrand competition ensues at the right endpoint b if B must sink an entry cost KB would it choose to enter given that firm A is in both ends of the market and remains there after entry c is As product proliferation strategy credible Or would A exit the right end of the market after B enters To answer these questions compare As profits for the case in which it has a stand on the left side and both it and B have stands on the right to the case in which A has one stand on the left end and B has one stand on the right end so Bs entry has driven A out of the right side of the market Analytical Problems 159 Herfindahl index of market concentration One way of measuring market concentration is through the use of the herfindahl index which is defined as H 5a n i51 s2 i where st 5 qiQ is firm is market share The higher is H the more concentrated the industry is said to be intuitively more concentrated markets are thought to be less competitive because dominant firms in concentrated markets face little competitive pressure We will assess the validity of this intu ition using several models a if you have not already done so answer Problem 152d by computing the nash equilibrium of this nfirm Cournot game also compute market output market price con sumer surplus industry profit and total welfare Com pute the herfindahl index for this equilibrium b Suppose two of the n firms merge leaving the market with n 2 1 firms recalculate the nash equilibrium and the rest of the items requested in part a how does the merger affect price output profit consumer surplus total welfare and the herfindahl index c Put the model used in parts a and b aside and turn to a different setup that of Problem 153 where Cournot duopolists face different marginal costs use your answer to Problem 153a to compute equilibrium firm outputs market output price consumer surplus industry profit and total welfare substituting the particular cost param eters c1 5 c2 5 14 also compute the herfindahl index d repeat your calculations in part c while assuming that firm 1s marginal cost c1 falls to 0 but c2 stays at 14 how does the cost change affect price output profit con sumer surplus total welfare and the herfindahl index e given your results from parts ad can we draw any general conclusions about the relationship between Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 564 Part 6 Market Power market concentration on the one hand and price profit or total welfare on the other 1510 Inverse elasticity rule use the firstorder condition equation 152 for a Cournot firm to show that the usual inverse elasticity rule from Chap ter 11 holds under Cournot competition where the elasticity is associated with an individual firms residual demand the demand left after all rivals sell their output on the market Manipulate equation 152 in a different way to obtain an equivalent version of the inverse elasticity rule P 2 MC P 52 si eQP where si 5 qiQ is firm i s market share and eQP is the elasticity of market demand Compare this version of the inverse elastic ity rule with that for a monopolist from the previous chapter 1511 Competition on a circle hotellings model of competition on a linear beach is used widely in many applications but one application that is dif ficult to study in the model is free entry Free entry is easiest to study in a model with symmetric firms but more than two firms on a line cannot be symmetric because those located nearest the endpoints will have only one neighboring rival whereas those located nearer the middle will have two To avoid this problem Steven Salop introduced competi tion on a circle18 as in the hotelling model demanders are located at each point and each demands one unit of the good a consumers surplus equals v the value of consuming the good minus the price paid for the good as well as the cost of having to travel to buy from the firm Let this travel cost be td where t is a parameter measuring how burdensome travel is and d is the distance traveled note that we are here assuming a linear rather than a quadratic travelcost function in con trast to example 155 initially we take as given that there are n firms in the market and that each has the same cost function Ci 5 K 1 cqi where K is the sunk cost required to enter the market this will come into play in part e of the question where we consider free entry and c is the constant marginal cost of production For simplicity assume that the circumference of the circle equals 1 and that the n firms are located evenly around the circle at intervals of 1n The n firms choose prices pi simultaneously a each firm i is free to choose its own price 1 pi2 but is con strained by the price charged by its nearest neighbor to either side Let p be the price these firms set in a sym metric equilibrium explain why the extent of any firms market on either side x is given by the equation p 1 tx 5 p 1 t3 11n2 2 x4 18See S Salop Monopolistic Competition with Outside goods Bell Journal of Economics Spring 1979 14156 b given the pricing decision analyzed in part a firm i sells qi 5 2x because it has a market on both sides Cal culate the profitmaximizing price for this firm as a func tion of p c t and n c noting that in a symmetric equilibrium all firms prices will be equal to p show that pi 5 p 5 c 1 tn explain this result intuitively d Show that a firms profits are tn2 2 K in equilibrium e What will the number of firms n be in longrun equilib rium in which firms can freely choose to enter f Calculate the socially optimal level of differentiation in this model defined as the number of firms and prod ucts that minimizes the sum of production costs plus demander travel costs Show that this number is precisely half the number calculated in part e hence this model illustrates the possibility of overdifferentiation 1512 Signaling with entry accommodation This question will explore signaling when entry deterrence is impossible thus the signaling firm accommodates its rivals entry assume deterrence is impossible because the two firms do not pay a sunk cost to enter or remain in the market The setup of the model will follow example 154 so the calcula tions there will aid the solution of this problem in particular firm is demand is given by qi 5 ai 2 pi 1 pj 2 where ai is product i s attribute say quality Production is costless Firm 1s attribute can be one of two values either a1 5 1 in which case we say firm 1 is the low type or a1 5 2 in which case we say it is the high type assume there is no discounting across periods for simplicity a Compute the nash equilibrium of the game of complete information in which firm 1 is the high type and firm 2 knows that firm 1 is the high type b Compute the nash equilibrium of the game in which firm 1 is the low type and firm 2 knows that firm 1 is the low type c Solve for the Bayesiannash equilibrium of the game of incomplete information in which firm 1 can be either type with equal probability Firm 1 knows its type but firm 2 only knows the probabilities Because we did not spend time in this chapter on Bayesian games you may want to consult Chapter 8 especially example 86 d Which of firm 1s types gains from incomplete informa tion Which type would prefer complete information and thus would have an incentive to signal its type if possible Does firm 2 earn more profit on aver age under complete information or under incomplete information Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 565 e Consider a signaling variant of the model chat has two periods Firms 1 and 2 choose prices in the first period when firm 2 has incomplete information about firm 1s type Firm 2 observes firm 1s price in this period and uses the information to update its beliefs about firm 1s type Then firms engage in another period of price competi tion Show that there is a separating equilibrium in which each type of firm 1 charges the same prices as computed in part d You may assume that if firm 1 chooses an outofequilibrium price in the first period then firm 2 believes that firm 1 is the low type with probability 1 Hint To prove the existence of a separating equilibrium show that the loss to the low type from trying to pool in the first period exceeds the secondperiod gain from having con vinced firm 2 that it is the high type use your answers from parts ad where possible to aid in your solution Behavioral Problem 1513 Can competition unshroud prices in this problem we return to the question of shrouded prod uct attributes and prices introduced in Problem 614 and further analyzed in Problem 1413 here we will pursue the question of whether market forces can be counted on to atten uate consumer behavioral biases in particular whether com petition and advertising can serve to unshroud previously shrouded prices We will study a model inspired by xavier gabaix and David Laibsons influential article19 a population of consumers nor malize their mass to 1 have gross surplus v for a homogeneous good produced by duopoly firms at constant marginal and average cost c Firms i 5 1 2 simultaneously post prices pi in addition to these posted prices each firm i can add a shrouded fee si which are anticipated by some consumers but not others For example the fees could be for checked baggage associated with plane travel or for not making a minimum monthly pay ment on a credit card a fraction α are sophisticated consum ers who understand the equilibrium and anticipate equilibrium shrouded fees at a small inconvenience cost e they are able to avoid the shrouded fee packing only carryons in the airline example or being sure to make the minimum monthly payment in the creditcard example The remaining fraction 1 2 α of consumers are myopic They do not anticipate shrouded fees only considering posted prices in deciding from which firm to buy Their only way of avoiding the fee is void the entire transaction saving the total expenditure pi 1 si but forgoing surplus v Suppose firms choose posted prices simultaneously as in the Bertrand model a argue that in equilibrium p i 1 s i 5 v at least as long as e is sufficiently small that firms do not try to induce sophisticated consumers not to avoid the shrouded fee Compute the nash equilibrium posted prices p i Hint as in the standard Bertrand game an undercutting argument suggests that a zeroprofit condition is crucial in determining p i here too how do the posted prices compare to cost are they guaranteed to be positive how is surplus allocated across consumers b Can you give examples of realworld products that seem to be priced as in part a c Suppose that one of the firms say firm 2 can devi ate to an advertising strategy advertising has several effects First it converts myopic consumers into sophis ticated ones who rationally forecast shrouded fees and who can avoid them at cost e Second it allows firm 2 to make both p2 and s2 transparent to all types of con sumers Show that this deviation is unprofitable if e a1 2 α α b 1v 2 c2 d hence we have shown that even costless advertising need not result in unshrouding explain the forces leading advertising to be an unprofitable deviation e return to the case in part a with no advertising but now suppose firms cannot post negative prices One reason is that sophisticated consumers could exact huge losses by purchasing an untold number of units to earn the negative price which are simply disposed of Com pute the nash equilibrium how does it compare to part a Can firms earn positive profits Suggestions for Further Reading Carlton D W and J M Perloff Modern Industrial Organiza tion 4th ed Boston addisonWesley 2005 Classic undergraduate text on industrial organization that covers theoretical and empirical issues Kwoka J e Jr and L J White The Antitrust Revolution 4th ed new York Oxford university Press 2004 Summarizes economic arguments on both sides of a score of important recent antitrust cases Demonstrates the policy rele vance of the theory developed in this chapter Pepall L D J richards and g norman Industrial Organiza tion Contemporary Theory and Practice 2nd ed Cincinnati Oh Thomson SouthWestern 2002 19x gabaix and D Laibson Shrouded attributes Consumer Myopia and information Suppression in Competitive Markets Quarterly Journal of Economics May 2006 461504 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 566 Part 6 Market Power An undergraduate textbook providing a simple but thorough treatment of oligopoly theory Uses the Hotelling model in a vari ety of additional applications including advertising Sutton J Sunk Costs and Market Structure Cambridge Ma MiT Press 1991 Argues that the robust predictions of oligopoly theory regard the size and nature of sunk costs Second half provides detailed case studies of competition in various manufacturing industries Tirole J The Theory of Industrial Organization Cambridge Ma MiT Press 1988 A comprehensive survey of the topics discussed in this chapter and a host of others Standard text used in graduate courses but selected sections are accessible to advanced undergraduates Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 567 We saw in the chapter that one can often understand the nature of strategic interaction in a market simply from the slope of firms bestresponse functions For example we argued that a first mover that wished to accept rather than deter entry should commit to a strategy that leads its rival to behave less aggressively What sort of strategy this is depends on the slope of firms best responses if best responses slope downward as in a Cournot model then the first mover should play a top dog strategy and produce a large quan tity leading its rival to reduce its production if best responses slope upward as in a Bertrand model with price competition for differentiated products then the first mover should play a puppy dog strategy and charge a high price leading its rival to increase its price as well More generally we have seen repeatedly that best response function diagrams are often helpful in understand ing the nature of nash equilibrium how the nash equilibrium changes with parameters of the model how incomplete infor mation might affect the game and so forth Simply knowing the slope of the bestresponse function is often all one needs to draw a usable bestresponse function diagram By analogy to similar definitions from consumer and pro ducer theory game theorists define firms actions to be stra tegic substitutes if an increase in the level of the action eg output price investment by one firm is met by a decrease in that action by its rival On the other hand actions are strategic complements if an increase in an action by one firm is met by an increase in that action by its rival E151 Nash equilibrium To make these ideas precise suppose that firm 1s profit π1 1a1 a22 is a function of its action a1 and its rivals firm 2s action a2 here we have moved from subscripts to super scripts for indicating the firm to which the profits belong to make room for subscripts that will denote partial derivatives Firm 2s profit function is denoted similarly a nash equilib rium is a profile of actions for each firm 1a 1 a 22 such that each firms equilibrium action is a best response to the others Let BR1 1a22 be firm 1s bestresponse function and let BR2 1a12 be firm 2s then a nash equilibrium is given by a 1 5 BR1 1a 22 and a 2 5 BR2 1a 12 E152 Bestresponse functions in more detail The firstorder condition for firm 1s action choice is π1 1 1a1 a22 5 0 i where subscripts for π represent partial derivatives with respect to its various arguments a unique maximum and thus a unique best response is guaranteed if we assume that the profit function is concave π1 11 1a1 a22 0 ii given a rivals action a2 the solution to equation i for a maxi mum is firm 1s bestresponse function a1 5 BR1 1a22 iii Since the best response is unique BR1 1a22 is indeed a func tion rather than a correspondence see Chapter 8 for more on correspondences The strategic relationship between actions is determined by the slope of the bestresponse functions if best responses are downward sloping ie if BRr1 1a22 0 and BRr2 1a12 0 then a1 and a2 are strategic substitutes if best responses are upward sloping ie if BRr1 1a22 0 and BRr2 1a12 04 then a1 and a2 are strategic complements E153 Inferences from the profit function We just saw that a direct route for determining whether actions are strategic substitutes or complements is first to solve explic itly for bestresponse functions and then to differentiate them in some applications however it is difficult or impossible to find an explicit solution to equation i We can still determine whether actions are strategic substitutes or complements by drawing inferences directly from the profit function Substituting equation iii into the firstorder condition of equation i gives π1 1 1BR1 1a22 a22 5 0 iv Totally differentiating equation iv with respect to a2 yields after dropping the arguments of the functions for brevity π1 11BRr1 1 π1 12 5 0 v EXTENSIONS StrategIC SubStItuteS and ComplementS Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 568 Part 6 Market Power rearranging equation v gives the derivative of the best response function BRr152π1 12 π1 11 vi in view of the secondorder condition equation ii the denominator of equation vi is negative Thus the sign of BRr1 is the same as the sign of the numerator π1 12 That is π1 12 0 implies BRr1 0 and π1 12 0 implies BRr1 0 The strategic relationship between the actions can be inferred directly from the crosspartial derivative of the profit function E154 Cournot model in the Cournot model profits are given as a function of the two firms quantities π1 1q1 q22 5 q1P 1q1 q22 2 C1q12 vii The firstorder condition is π1 1 5 q1Pr 1q1 1 q22 1 P 1q1 1 q22 2 Cr 1q12 viii as we have already seen equation 152 The derivative of equation viii with respect to q2 is after dropping functions arguments for brevity π1 12 5 q1Ps 1 Pr ix Because Pr 0 the sign of π1 12 will depend on the sign of Psthat is the curvature of demand With linear demand Ps 5 0 and so π1 12 is clearly negative Quantities are strategic substitutes in the Cournot model with linear demand Figure 152 illustrates this general principle This figure is drawn for an example involving linear demand and indeed the best responses are downward sloping More generally quantities are strategic substitutes in the Cournot model unless the demand curve is very con vex ie unless Ps is positive and large enough to offset the last term in equation ix For a more detailed discussion see Bulow geanakoplous and Klemperer 1985 E155 Bertrand model with differentiated products in the Bertrand model with differentiated products demand can be written as q1 5 D1 1p1 p22 x See equation 1524 for a related expression using this nota tion profit can be written as π1 5 p1q1 2 C1q12 5 p1D1 1 p1 p22 2 C1D1 1 p1 p22 2 xi The firstorder condition with respect to p1 is π1 1 5 p1D1 1 1 p1 p22 1 D1 1 p1 p22 2Cr 1D1 1 p1 p22 2D1 1 1 p1 p22 xii The crosspartial derivative is after dropping functions argu ments for brevity π1 12 5 p1D1 12 1 D1 2 2 CrD1 12 2 CsD1 2D1 1 xiii interpreting this mass of symbols is no easy task in the special case of constant marginal cost 1Cs 5 02 and linear demand 1D1 12 5 02 the sign of π1 12 is given by the sign of D1 2 ie how a firms demand is affected by changes in the rivals price in the usual case when the two goods are themselves substitutes we have D1 2 0 and so π1 12 0 That is prices are strategic complements The terminology here can seem contradictory so the result bears repeating if the goods that the firms sell are substitutes then the variables the firms choose pricesare strategic complements Firms in such a duopoly would either raise or lower prices together see Tirole 1988 We saw an example of this in Figure 154 The figure was drawn for the case of linear demand and constant marginal cost and we saw that best responses are upward sloping E156 Entry accommodation in a sequential game Consider a sequential game in which firm 1 chooses a1 and then firm 2 chooses a2 Suppose firm 1 finds it more profitable to accommodate than to deter firm 2s entry Because firm 2 moves after firm 1 we can substitute firm 2s best response into firm 1s profit function to obtain π1 1a1 BR2 1a12 2 xiv Firm 1s firstorder condition is π1 1 1 π1 2BRr2 5 0 S xv By contrast the firstorder condition from the simultaneous game see equation i is simply π1 1 5 0 The firstorder con ditions from the sequential and simultaneous games differ in the term S This term captures the strategic effect of moving firstthat is whether the first mover would choose a higher or lower action in the sequential game than in the simultane ous game The sign of S is determined by the signs of the two factors in S We will argue in the next paragraph that these two fac tors will typically have the same sign both positive or both negative implying that S 0 and hence that the first mover will typically distort its action upward in the sequential game compared with the simultaneous game This result confirms the findings from several of the examples in the text in Figure 156 we see that the Stackelberg quantity is higher than the Cournot quantity in Figure 157 we see that the price leader distorts its price upward in the sequential game compared with the simultaneous one Section e153 showed that the sign of BRr2 is the same as the sign of π2 12 if there is some symmetry to the market then the sign of π2 12 will be the same as the sign of π1 12 Typically Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 569 π1 2 and π1 12 will have the same sign For example consider the case of Cournot competition By equation 151 firm 1s profit is π1 5 P 1q1 1 q22q1 2 C1q12 xvi Therefore π1 2 5 Pr 1q1 1 q22q1 xvii Because demand is downward sloping it follows that π1 2 0 Differentiating equation xvii with respect to q1 yields π1 12 5 Pr1q1Ps xviii This expression is also negative if demand is linear so Ps 5 0 or if demand is not too convex so the last term in equation xviii does not swamp the term Pr E157 Extension to general investments The model from the previous section can be extended to general investmentsthat is beyond a mere commitment to a quantity or price Let K1 be this general investment say advertising investment in lowercost manufacturing or product positioningsunk at the outset of the game The two firms then choose their productmarket actions a1 and a2 representing prices or quantities simultaneously in the second period Firms profits in this extended model are respectively π1 1a1 a2 K12 and π2 1a1 a22 xix The analysis is simplified by assuming that firm 2s profit is not directly a function of K1 although firm 2s profit will indirectly depend on K1 in equilibrium because equilibrium actions will depend on K1 Let a 1 1K12 and a 2 1K12 be firms actions in a subgameperfect equilibrium a 1 1K12 5 BR1 1a 2 1K12 K12 a 2 1K12 5 BR2 1a 1 1K12 2 xx Because firm 2s profit function does not depend directly on K1 in equation xix neither does its best response in equa tion xx The analysis here draws on Fudenberg and Tirole 1984 and Tirole 1988 Substituting from equation xx into equa tion xix the firms nash equilibrium profits in the subgame following firm 1s choice of K1 are π11K12 5 π1 1a 1 1K12 a 2 1K12 K12 π21K12 5 π2 1a 1 1K12 a 2 1K122 xxi Fold the game back to firm 1s firstperiod choice of K1 Because firm 1 wants to accommodate entry it chooses K1 to maximize π11K12 Totally differentiating π11K12 the first order condition is dπ1 dK1 5 π1 1 da 1 dK1 1 π1 2 da 2 dK1 1 π1 K1 S 5 π1 2 da 2 dK1 1 π1 K1 xxii The second equality in equation xxii holds by the envelope theorem The envelope theorem just says that π1 1 da 1dK1 disappears because a1 is chosen optimally in the second period so π1 1 5 0 by the firstorder condition for a1 The first of the remaining two terms in equation xxii S is the strategic effect of an increase in K1 on firm 1s profit through firm 2s action if firm 1 cannot make an observable commit ment to K1 then S disappears from equation xxii and only the last term the direct effect of K1 on firm 1s profit will be present The sign of S determines whether firm 1 strategically over or underinvests in K1 when it can make a strategic commit ment We have the following steps sign 1S2 5 sign aπ2 1 da 2 dK1 b 5 sign aπ 2 1BRr2 da 1 dK1 b 5 sign adπ2 dK1 BRr2b xxiii The first line of equation xxiii holds if there is some symme try to the market so that the sign of π1 2 equals the sign of π2 1 The second line follows from differentiating a 2 1K12 in equa tion xx The third line follows by totally differentiating π 2 in equation xxi dπ2 dK1 5 π2 1 da 1 dK1 1 π2 2 da 2 dK1 5 π2 1 da 1 dK1 xxiv where the second equality again follows from the envelope theorem By equation xxiii the sign of the strategic effect S is determined by the sign of two factors The first factor dπ2dK1 indicates the effect of K1 on firm 2s equilibrium profit in the subgame if dπ2dK1 0 then an increase in K1 harms firm 2 and we say that investment makes firm 1 tough if dπ2dK1 0 then an increase in K1 benefits firm 2 and we say that investment makes firm 1 soft The second factor BRr2 is the slope of firm 2s best response which depends on whether actions a1 and a2 are strategic substitutes or complements each of the two terms in S can have one of two signs for a total of four possible combina tions displayed in Table 151 if investment makes firm 1 tough then the strategic effect S leads firm 1 to reduce K1 if actions are strategic complements or to increase K1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 570 Part 6 Market Power if actions are strategic substitutes The opposite is true if investment makes firm 1 soft For example actions could be prices in a Bertrand model with differentiated products and thus would be strategic com plements investment K1 could be advertising that steals mar ket share from firm 2 Table 151 indicates that when K1 is observable firm 1 should strategically underinvest to induce less aggressive price competition from firm 2 E158 Mostfavored customer program The preceding analysis applies even if K1 is not a continu ous investment variable but instead a 01 choice For exam ple consider the decision by firm 1 of whether to start a mostfavored customer program studied in Cooper 1986 a mostfavored customer program rebates the price difference sometimes in addition to a premium to past customers if the firm lowers its price in the future Such a program makes firm 1 soft by reducing its incentive to cut price if firms compete in strategic complements say in a Bertrand model with dif ferentiated products then Table 151 says that firm 1 should overinvest in the mostfavored customer program meaning that it should be more willing to implement the program if doing so is observable to its rival The strategic effect leads to less aggressive price competition and thus to higher prices and profits Ones first thought might have been that such a most favored customer program should be beneficial to consumers and lead to lower prices because the clause promises payments back to them as we can see from this example strategic con siderations sometimes prove ones initial intuition wrong suggesting that caution is warranted when examining strate gic situations E159 Trade policy The analysis in Section e157 applies even if K1 is not a choice by firm 1 itself For example researchers in international trade sometimes take K1 to be a governments policy choice on behalf of its domestic firms Brander and Spencer 1985 stud ied a model of international trade in which exporting firms from country 1 engage in Cournot competition with domestic firms in country 2 The actions quantities are strategic sub stitutes The authors ask whether the government of country 1 would want to implement an export subsidy program a deci sion that plays the role of K1 in their model an export subsidy makes exporting firms tough because it effectively lowers their marginal costs increasing their exports to country 2 and reducing market price there according to Table 151 the government of country 1 should overinvest in the subsidy pol icy adopting the policy if it is observable to domestic firms in country 2 but not otherwise The model explains why coun tries unilaterally adopt export subsidies and other trade inter ventions when free trade would be globally efficient at least in this simple model Our analysis can be used to show that Brander and Spen cers rationalization of export subsidies may not hold up under alternative assumptions about competition if exporting firms and domestic firms were to compete in strategic complements say Bertrand competition in differentiated products rather than Cournot competition then an export subsidy would be a bad idea according to Table 151 Country 1 should then underinvest in the export subsidy ie not adopt it to avoid overly aggressive price competition E1510 Entry deterrence Continue with the model from Section e157 but now sup pose that firm 1 prefers to deter rather than accommodate entry Firm 1s objective is then to choose K1 to reduce firm 2s profit π2 to zero Whether firm 1 should distort K1 upward or downward to accomplish this depends only on the sign of dπ2dK1that is on whether investment makes firm 1 tough or softand not on whether actions are strate gic substitutes or complements if investment makes firm 1 tough it should overinvest to deter entry relative to the case in which it cannot observably commit to investment On the other hand if investment makes firm 1 soft it should under invest to deter entry For example if K1 is an investment in marginal cost reduc tion this likely makes firm 1 tough and so it should over invest to deter entry if K1 is an advertisement that increases demand for the whole product category more than its own brand advertisements for a particular battery brand involving an unstoppable batterypowered bunny may increase sales of all battery brands if consumers have difficulty remembering exactly which battery was in the bunny then this will likely make firm 1 soft so it should underinvest to deter entry TABLE 151 STRATEGIC EFFECT WHEN ACCOMMODATING ENTRY Firm 1s investment Tough 1dπ2dK1 02 Soft 1dπ2dK1 02 actions Strategic Complements 1BRr 02 underinvest 122 Overinvest 112 Strategic Substitutes 1BRr 02 Overinvest 112 underinvest 122 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 15 Imperfect Competition 571 References Brander J a and B J Spencer export Subsidies and international Market Share rivalry Journal of Interna tional Economics 18 February 1985 83100 Bulow J g geanakoplous and P Klemperer Multimarket Oligopoly Strategic Substitutes and Complements Jour nal of Political Economy June 1985 488511 Cooper T MostFavoredCustomer Pricing and Tacit Col lusion Rand Journal of Economics 17 autumn 1986 37788 Fudenberg D and J Tirole The Fat Cat effect the Puppy Dog Ploy and the Lean and hungry Look American Economic Review Papers and Proceedings 74 May 1984 36168 Tirole J The Theory of Industrial Organization Cambridge Ma MiT Press 1988 chap 8 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 573 PART SEVEN Pricing in Input Markets Chapter 16 Labor Markets Chapter 17 Capital and Time Our study of input demand in Chapter 11 was quite general in that it can be applied to any factor of production In Chapters 16 and 17 we take up several issues specifically related to pricing in the labor and capital markets Chapter 16 focuses mainly on labor supply Most of our analysis deals with various aspects of individual labor supply In successive sections we look at the supply of hours of work decisions related to the accumulation of human capital and modeling the job search process For each of these topics we show how the decisions of individuals affect labor market equilibria The final sections of Chapter 16 take up some aspects of imperfect competition in labor markets In Chapter 17 we examine the market for capital The central purpose of the chapter is to emphasize the connection between capital and the allocation of resources over time Some care is also taken to integrate the theory of capital into the models of firms behavior we developed in Part 4 A brief appendix to Chapter 17 presents some useful mathematical results about interest rates In The Principles of Political Economy and Taxation Ricardo wrote The produce of the earth is divided among three classes of the community namely the propri etor of the land the owner of the stock of capital necessary for its cultivation and the laborers by whose industry it is cultivated To determine the laws which regulate this distribution is the principal problem in Political Economy The purpose of Part 7 is to illustrate how the study of these laws has advanced since Ricardos time D Ricardo The Principles of Political Economy and Taxation 1817 reprinted London J M Dent and Son 1965 p 1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 CHAPTER SIXTEEN Labor Markets In this chapter we examine some aspects of input pricing that are related particularly to the labor market Because we have already discussed questions about the demand for labor or any other input in some detail in Chapter 11 here we will be concerned primar ily with analyzing the supply of labor We start by looking at a simple model of utility max imization that explains individuals supply of work hours to the labor market Subsequent sections then take up various generalizations of this model 161 ALLOCATION OF TIME In Part 2 we studied the way in which an individual chooses to allocate a fixed amount of income among a variety of available goods Individuals must make similar choices in deciding how they will spend their time The number of hours in a day or in a year is absolutely fixed and time must be used as it passes by Given this fixed amount of time any individual must decide how many hours to work how many hours to spend consuming a wide variety of goods ranging from cars and television sets to operas how many hours to devote to selfmaintenance and how many hours to sleep By examining how individuals choose to divide their time among these activities economists are able to understand the labor supply decision 1611 Simple twogood model For simplicity we start by assuming there are only two uses to which an individual may devote his or her timeeither engaging in market work at a real wage rate of w per hour or not working We shall refer to nonwork time as leisure but this word is not meant to carry any connotation of idleness Time not spent in market work can be devoted to work in the home to selfimprovement or to consumption it takes time to use a television set or a bowling ball1 All of those activities contribute to an individuals wellbeing and time will be allocated to them in what might be assumed to be a utilitymaximizing way More specifically assume that an individuals utility U during a typical day depends on consumption during that period c and on hours of leisure enjoyed h utility 5 U1c h2 161 Notice that in writing this utility function we have used two composite goods consump tion and leisure Of course utility is actually derived by devoting real income and time to 1Perhaps the first formal theoretical treatment of the allocation of time was given by G S Becker in A Theory of the Allocation of Time Economic Journal 75 September 1965 493517 575 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 576 Part 7 Pricing in Input Markets the consumption of a wide variety of goods and services2 In seeking to maximize utility an individual is bound by two constraints The first of these concerns the time that is avail able If we let l represent hours of work then l 1 h 5 24 162 That is the days time must be allocated either to work or to leisure nonwork A sec ond constraint records the fact that an individual can purchase consumption items only by working later in this chapter we will allow for the availability of nonlabor income If the real hourly market wage rate the individual can earn is given by w then the income constraint is given by c 5 wl 163 Combining the two constraints we have c 5 w 124 2 h2 164 or c 1 wh 5 24w 165 This combined constraint has an important interpretation Any person has a full income given by 24w That is an individual who worked all the time would have this much com mand over real consumption goods each day Individuals may spend their full income either by working for real income and consumption or by not working and thereby enjoying leisure Equation 165 shows that the opportunity cost of consuming leisure is w per hour it is equal to earnings forgone by not working 1612 Utility maximization The individuals problem then is to maximize utility subject to the full income constraint Given the Lagrangian expression 5 U1c h2 1 λ 124w 2 c 2 wh2 166 the firstorder conditions for a maximum are c 5 U c 2 λ 5 0 h 5 U h 2 wλ 5 0 167 Dividing the two lines in Equation 167 we obtain Uh Uc 5 w 5 MRS 1h for c2 168 Hence we have derived the following principle 2The production of goods in the home has received considerable attention especially since household time allocation diaries have become available For a survey see R Granau The Theory of Home Production The Past Ten Years in J T Addison Ed Recent Developments in Labor Economics Cheltenham UK Elgar Reference Collection 2007 vol 1 pp 23543 O P T I M I Z AT I O N P R I N C I P L E Utilitymaximizing labor supply decision To maximize utility given the real wage w the indi vidual should choose to work that number of hours for which the marginal rate of substitution of leisure for consumption is equal to w Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 577 Of course the result derived in Equation 168 is only a necessary condition for a maxi mum As in Chapter 4 this tangency will be a true maximum provided the MRS of leisure for consumption is diminishing 1613 Income and substitution effects of a change in w A change in the real wage rate w can be analyzed in a manner identical to that used in Chapter 5 When w increases the price of leisure becomes higher A person must give up more in lost wages for each hour of leisure consumed As a result the substitution effect of an increase in w on the hours of leisure will be negative As leisure becomes more expensive there is reason to consume less of it However the income effect will be positivebecause lei sure is a normal good the higher income resulting from a higher w will increase the demand for leisure Thus the income and substitution effects work in opposite directions It is impos sible to predict on a priori grounds whether an increase in w will increase or decrease the demand for leisure time Because leisure and work are mutually exclusive ways to spend ones time it is also impossible to predict what will happen to the number of hours worked The substitution effect tends to increase hours worked when w increases whereas the income effectbecause it increases the demand for leisure timetends to decrease the number of hours worked Which of these two effects is the stronger is an important empirical question3 1614 A graphical analysis The two possible reactions to a change in w are illustrated in Figure 161 In both graphs the initial wage is w0 and the initial optimal choices of c and h are given by 3If the family is taken to be the relevant decision unit then even more complex questions arise about the income and substitution effects that changes in the wages of one family member will have on the labor force behavior of other family members Because the individual is a supplier of labor the income and substitution effects of an increase in the real wage rate w work in opposite directions in their effects on the hours of leisure demanded or on hours of work In a the substitution effect movement to point S outweighs the income effect and a higher wage causes hours of leisure to decrease to h1 Therefore hours of work increase In b the income effect is stronger than the substitution effect and h increases to h1 In this case hours of work decrease Consumption Consumption Leisure Leisure a b c1 c1 c0 c0 h1 h0 h0 h1 S S U1 U1 U0 U0 c w124 h c w024 h c w124 h c w024 h FIGURE 161 Income and Substitution Effects of a Change in the Real Wage Rate w Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 578 Part 7 Pricing in Input Markets the point c0 h0 When the wage rate increases to w1 the optimal combination moves to point c1 h1 This movement can be considered the result of two effects The substitu tion effect is represented by the movement of the optimal point from c0 h0 to S and the income effect by the movement from S to c1 h1 In the two panels of Figure 161 these two effects combine to produce different results In panel a the substitution effect of an increase in w outweighs the income effect and the individual demands less leisure 1h1 h02 Another way of saying this is that the individual will work longer hours when w increases In panel b of Figure 161 the situation is reversed The income effect of an increase in w more than offsets the substitution effect and the demand for leisure increases 1h1 h02 The individual works shorter hours when w increases In the cases examined in Chapter 5 this would have been considered an unusual resultwhen the price of leisure increases the individual demands more of it For the case of normal consumption goods the income and substitution effects work in the same direction Only for inferior goods do they differ in sign In the case of leisure and labor however the income and substitution effects always work in opposite directions An increase in w makes an individual betteroff because he or she is a supplier of labor In the case of a consumption good individuals are made worseoff when a price increases because they are consumers of that good We can summarize this analysis as follows O P T I M I Z AT I O N P R I N C I P L E Income and substitution effects of a change in the real wage When the real wage rate increases a utilitymaximizing individual may increase or decrease hours worked The sub stitution effect will tend to increase hours worked as the individual substitutes earnings for leisure which is now relatively more costly On the other hand the income effect will tend to reduce hours worked as the individual uses his or her increased purchasing power to buy more leisure hours We now turn to examine a mathematical development of these responses that provides additional insights into the labor supply decision 162 A MATHEMATICAL ANALYSIS OF LABOR SUPPLY To derive a mathematical statement of labor supply decisions it is helpful first to amend the budget constraint slightly to allow for the presence of nonlabor income To do so we rewrite Equation 163 as c 5 wl 1 n 169 where n is real nonlabor income and may include such items as dividend and inter est income receipt of government transfer benefits or simply gifts from other persons Indeed n could stand for lumpsum taxes paid by this individual in which case its value would be negative Maximization of utility subject to this new budget constraint would yield results vir tually identical to those we have already derived That is the necessary condition for a maximum described in Equation 168 would continue to hold as long as the value of n is unaffected by the laborleisure choices being made that is so long as n is a lumpsum Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 579 receipt or loss of income4 the only effect of introducing nonlabor income into the analysis is to shift the budget constraints in Figure 161 outward or inward in a parallel manner without affecting the tradeoff rate between earnings and leisure This discussion suggests that we can write the individuals labor supply function as lw n to indicate that the number of hours worked will depend both on the real wage rate and on the amount of real nonlabor income received On the assumption that leisure is a normal good ln will be negative that is an increase in n will increase the demand for leisure and because there are only 24 hours in the day reduce l Before studying wage effects on labor supply 1lw2 we will find it helpful to consider the dual problem to the individuals primary utilitymaximization problem 1621 Dual to the labor supply problem As we showed in Chapter 5 the dual to the individuals utility maximization problem is to minimize the expenditures necessary to attain a given utility level This alternative approach to the problem yielded a variety of useful results and that is also the case for our model of labor supply In this case we wish to minimize the nonlabor income necessary to attain a given utility target That is the individuals optimization problem is minimize n 5 c 2 wl 5 c 2 w 124 2 h2 1610 subject to U1c h2 5 U 1611 The Lagrangian expression for this minimization problem is 1c h λ2 5 c 2 w 124 2 h2 1 λ 3 U 2 U1c h2 4 1612 The tangency condition for this minimization is identical to that shown in Equation 168 check this for yourself In this case the value function for the problem shows the min imal nonlabor income necessary to achieve a given utility level as a function of the wage and that target utility leveln 1w U2 Applying the envelope theorem to this value func tion shows that dn 1w U2 dw 5 w 5 2124 2 h2 52l c 1w U2 1613 where the notation l c 1w U2 conveys the idea that this derivative yields a compensated labor supply function which allows us to hold utility constant while examining the effect of changing wages on labor supply The equation makes intuitive sensefor a small increase in the wage the nonlabor income needed to reach a given utility target falls by the number of hours this person works times the change in that wage 1622 Slutsky equation for labor supply Now we can use this result to derive a Slutsky equation for the uncompensated labor sup ply function l 1w n2 In equilibrium hours of labor supplied are identical under both the compensated or uncompensated notions of labor supply l c 1w U2 5 l 1w n2 5 l 3w n1w U24 1614 4In many situations however n itself may depend on labor supply decisions For example the value of welfare or unemployment benefits a person can receive depends on his or her earnings as does the amount of income taxes paid In such cases the slope of the individuals budget constraint will no longer be reflected by the real wage but must instead reflect the net return to additional work after taking increased taxes and reductions in transfer payments into account For some examples see the problems at the end of this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 580 Part 7 Pricing in Input Markets Partial differentiation of this equation with respect to the wage w yields l c 1w U2 w 5 l 1w n2 w 1 l 1w n2 n n1w U2 w 5 l 1w n2 w 2l c 1w U2 l 1w n2 n 1615 Rearranging terms we have l 1w n2 w 5 l c 1w U2 w 1l c 1w U2 l 1w n2 n 1616 This is the Slutsky decomposition of the effect of a wage change on labor supply It shows why the sign of this effect is ambiguous The first term on the right of Equation 1616 is the substitution effectif we hold utility constant an increase in the wage must increase labor supply because of the convexity of the consumptionleisure indifference curve But the second term on the right of Equation 1615 is negative because l 1w n2n is negative This is the income effecta higher wage provides a higher real income and some of that income will be spent on increased leisure reduced labor supply As a general proposi tion then we cannot say whether an increase in the wage has a positive or a negative effect on the quantity of labor supplied Empirical evidence tends to suggest however that the effect of nonlabor income on labor supply is relatively small and therefore that the positive substitution effect in Equation 1616 dominates the negative income effect In most cases therefore we can assume that the labor supply curve is upward sloping though this will not be universally true EXAMPLE 161 Labor Supply Functions Individual labor supply functions can be constructed from underlying utility functions in much the same way that we constructed demand functions in Part 2 Here we will begin with a fairly extended treatment of a simple CobbDouglas case and then provide a shorter summary of labor supply with CES utility 1 CobbDouglas utility Suppose that an individuals utility function for consumption c and leisure h is given by U1c h2 5 cα hβ 1617 and assume for simplicity that α 1 β 5 1 This person is constrained by two equations 1 an income constraint that shows how consumption can be financed c 5 wl 1 n 1618 and 2 a total time constraint l 1 h 5 1 1619 where we have arbitrarily set the available time to be 1 By combining the financial and time con straints into a full income constraint we can arrive at the following Lagrangian expression for this utilitymaximization problem 5 U1c h2 1 λ1w 1 n 2 wh 2 c2 5 cα hβ 1 λ1w 1 n 2 wh 2 c2 1620 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 581 Firstorder conditions for a maximum are c 5 αc2βhβ 2 λ 5 0 h 5 βcαh2α 2 λw 5 0 1621 λ 5 w 1 n 2 wh 2 c 5 0 Dividing the first of these by the second yields αh βc 5 αh 11 2 α2c 5 1 w or wh 5 1 2 α α c 1622 Substitution into the full income constraint then yields the familiar results c 5 α 1w 1 n2 h 5 β1w 1 n2 w 1623 In words this person spends a fixed fraction α of his or her full income 1w 1 n2 on consump tion and the complementary fraction β 5 1 2 α on leisure which costs w per unit The labor supply function for this person is then given by l 1w n2 5 1 2 h 5 11 2 β2 2 βn w 1624 2 Properties of the CobbDouglas labor supply function This labor supply function shares many of the properties exhibited by consumer demand functions derived from CobbDouglas utility For example if n 5 0 then lw 5 0this person always devotes 12β proportion of his or her time to working no matter what the wage rate Income and substitution effects of a change in w are precisely offsetting in this case just as they are with crossprice effects in CobbDouglas demand functions On the other hand if n 0 then lw 0 When there is positive nonlabor income this person spends βn of it on leisure But leisure costs w per hour so an increase in the wage means that fewer hours of leisure can be bought Hence an increase in w increases labor supply Finally observe that ln 0 An increase in nonlabor income allows this person to buy more leisure so labor supply decreases One interpretation of this result is that transfer programs such as welfare benefits or unemployment compensation reduce labor supply Another inter pretation is that lumpsum taxation increases labor supply But actual tax and transfer programs are seldom lump sumusually they affect net wage rates as well Hence any precise prediction requires a detailed look at how such programs affect the budget constraint 3 CES labor supply In the Extensions to Chapter 4 we derived the general form for demand functions generated from a CES constant elasticity of substitution utility function We can apply that derivation directly here to study CES labor supply Specifically if utility is given by U1c h2 5 cδ δ 1 hδ δ 1625 then budget share equations are given by sc 5 c w 1 n 5 1 1 1 wκ sh 5 wh w 1 n 5 1 1 1 w2κ 1626 where κ 5 δ 1δ 2 12 Solving explicitly for leisure demand gives h 5 w 1 n w 1 w12κ 1627 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 582 Part 7 Pricing in Input Markets and l 1w n2 5 1 2 h 5 w12κ 2 n w 1 w12κ 1628 It is perhaps easiest to explore the properties of this function by taking some examples If δ 5 05 and κ 521 the labor supply function is l 1w n2 5 w 2 2 n w 1 w 2 5 1 2 nw 2 1 1 1w 1629 If n 5 0 then clearly lw 0 because of the relatively high degree of substitutability between consumption and leisure in this utility function the substitution effect of a higher wage outweighs the income effect On the other hand if δ 521 and κ 5 05 then the labor supply function is l 1w n2 5 w05 2 n w 1 w05 5 1 2 nw05 1 1 w05 1630 Now when n 5 0 lw 0 because there is a smaller degree of substitutability in the utility function the income effect outweighs the substitution effect in labor supply5 QUERY Why does the effect of nonlabor income in the CES case depend on the consumption leisure substitutability in the utility function 163 MARKET SUPPLY CURVE FOR LABOR We can plot a curve for market supply of labor based on individual labor supply decisions At each possible wage rate we add together the quantity of labor offered by each individual to arrive at a market total One particularly interesting aspect of this procedure is that as the wage rate increases more individuals may be induced to enter the labor force Figure 162 5In the CobbDouglas case 1δ 5 0 κ 5 02 the constantshare result for n 5 0 is given by l1w n2 5 1w 2 n22w 5 05 2 n2w FIGURE 162 Construction of the Market Supply Curve for Labor As the real wage increases there are two reasons why the supply of labor may increase First higher real wages may cause each person in the market to work more hours Second higher wages may induce more individuals for example individual 2 to enter the labor market Real wage Real wage Real wage Hours Hours Total labor supply a Individual 1 b Individual 2 c Te market w1 w2 w3 S1 S2 S Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 583 illustrates this possibility for the simple case of two people For a real wage below w1 neither individual chooses to work Consequently the market supply curve of labor Fig ure 162c shows that no labor is supplied at real wages below w1 A wage in excess of w1 causes individual 1 to enter the labor market However as long as wages fall short of w2 individual 2 will not work Only at a wage rate above w2 will both individuals participate in the labor market In general the possibility of the entry of new workers makes the market supply of labor somewhat more responsive to wagerate increases than would be the case if the number of workers were assumed to be fixed The most important example of higher real wage rates inducing increased labor force participation is the labor force behavior of married women in the United States in the postWorld War II period Since 1950 the percentage of working married women has increased from 32 percent to over 65 percent economists attribute this at least in part to the increasing wages that women are able to earn 164 LABOR MARKET EQUILIBRIUM Equilibrium in the labor market is established through the interaction of individuals labor supply decisions with firms decisions about how much labor to hire That process is illus trated by the familiar supplydemand diagram in Figure 163 At a real wage rate of w the quantity of labor demanded by firms is precisely matched by the quantity supplied by indi viduals A real wage higher than w would create a disequilibrium in which the quantity of labor supplied is greater than the quantity demanded There would be some involuntary unemployment at such a wage and this may create pressure for the real wage to decrease Similarly a real wage lower than w would result in disequilibrium behavior because firms would want to hire more workers than are available In the scramble to hire workers firms may bid up real wages to restore equilibrium FIGURE 163 Equilibrium in the Labor Market A real wage of w creates an equilibrium in the labor market with an employment level of l Real wage Quantity of labor S D l w Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 584 Part 7 Pricing in Input Markets Possible reasons for disequilibria in the labor market are a major topic in macroeco nomics especially in relationship to the business cycle Perceived failures of the market to adjust to changing equilibria have been blamed on sticky real wages inaccurate expecta tions by workers or firms about the price level the impact of government unemployment insurance programs labor market regulations and minimum wages and intertemporal work decisions by workers We will encounter a few of these applications later in this chap ter and in Chapters 17 and 19 Equilibrium models of the labor market can also be used to study a number of questions about taxation and regulatory policy For example the partial equilibrium tax incidence modeling illustrated in Chapter 12 can be readily adapted to the study of employment taxa tion One interesting possibility that arises in the study of labor markets is that a given pol icy intervention may shift both demand and supply functionsa possibility we examine in Example 162 EXAMPLE 162 Mandated Benefits A number of recent laws have mandated that employers provide special benefits to their workers such as health insurance paid time off or minimum severance packages The effect of such man dates on equilibrium in the labor market depends importantly on how the benefits are valued by workers Suppose that prior to implementation of a mandate the supply and demand for labor are given by lS 5 a 1 bw lD 5 c 2 dw 1631 Setting lS 5 lD yields an equilibrium wage of w 5 c 2 a b 1 d 1632 Now suppose that the government mandates that all firms provide a particular benefit to their workers and that this benefit costs t per unit of labor hired Therefore unit labor costs increase to w 1 t Suppose also that the new benefit has a monetary value to workers of k per unit of labor suppliedhence the net return from employment increases to w 1 k Equilibrium in the labor market then requires that a 1 b1w 1 k2 5 c 2 d1w 1 t2 1633 A bit of manipulation of this expression shows that the net wage is given by w 5 c 2 a b 1 d 2 bk 1 dt b 1 d 5 w 2 bk 1 dt b 1 d 1634 If workers derive no value from the mandated benefit 1k 5 02 then the mandate is just like a tax on employment Employees pay a share of the tax given by the ratio d 1b 1 d2 and the equilibrium quantity of labor hired decreases Qualitatively similar results will occur so long as k t On the other hand if workers value the benefit at precisely its cost 1k 5 t2 then the new wage decreases precisely by the amount of this cost 1w 5 w2 t2 and the equilibrium level of employment does not change Finally if workers value the benefit at more than it costs the firm to provide it k ta situation where one might wonder why the benefit was not already pro vided then the equilibrium wage will decrease by more than the benefit costs and equilibrium employment will increase QUERY How would you graph this analysis Would its conclusions depend on using linear sup ply and demand functions Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 585 165 WAGE VARIATION The labor market equilibrium illustrated in Figure 163 implies that there is a single marketclearing wage established by the supply decisions of households and the demands of firms The most cursory examination of labor markets would suggest that this view is far too simplistic Even in a single geographical area wages vary significantly among workers perhaps by a multiple of 10 or even 50 Of course such variation probably has some sort of supplydemand explanation but possible reasons for the differentials are obscured by thinking of wages as being determined in a single market In this section we look at three major causes of wage differences 1 human capital 2 compensating wage differentials and 3 job search uncertainty In the final sections of the chapter we look at a fourth set of causesimperfect competition in the labor market 1651 Human capital Workers vary significantly in the skills and other attributes they bring to a job Because firms pay wages commensurate with the values of workers productivities such differences can clearly lead to large differences in wages By drawing a direct analogy to the physical capital used by firms economists6 refer to such differences as differences in human cap ital Such capital can be accumulated in many ways by workers Elementary and second ary education often provides the foundation for human capitalthe basic skills learned in school make most other learning possible Formal education after high school can also provide a variety of jobrelated skills College and university courses offer many general skills and professional schools provide specific skills for entry into specific occupations Other types of formal education may also enhance human capital often by providing training in specific tasks Of course elementary and secondary education is compulsory in many countries but postsecondary education is often voluntary and thus attendance may be more amenable to economic analysis In particular the general methods to study a firms investment in physical capital see Chapter 17 have been widely applied to the study of individuals investments in human capital Workers may also acquire skills on the job As they gain experience their productivity will increase and presumably they will be paid more Skills accumulated on the job may sometimes be transferable to other possible employment Acquiring such skills is similar to acquiring formal education and hence is termed general human capital In other cases the skills acquired on a job may be quite specific to a particular job or employer These skills are termed specific human capital As Example 163 shows the economic consequences of these two types of investment in human capital can be quite different 6Widespread use of the term human capital is generally attributed to the American economist T W Schultz An important pioneering work in the field is G Becker Human Capital A Theoretical and Empirical Analysis with Special Reference to Education New York National Bureau of Economic Research 1964 EXAMPLE 163 General and Specific Human Capital Suppose that a firm and a worker are entering into a twoperiod employment relationship In the first period the firm must decide on how much to pay the worker 1w12 and how much to invest in general g and specific s human capital for this worker Suppose that the value of the workers marginal product is v1 in the first period In the second period the value of the workers marginal product is given by v2 1 g s2 5 v1 1 v g1 g2 1 v s 1s2 1635 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 586 Part 7 Pricing in Input Markets where v g and v s represent the increase in human capital as a result of the firms investments in period one Assume also that both investments are profitable in that v g1 g2 pg g and v s 1s2 ps s where pg and ps are the perunit prices of providing the different types of skills Profits7 for the firm are given by π1 5 v1 2 w1 2 pg g 2 ps s π2 5 v1 1 v g1 g2 1 v s 1s2 2 w2 1636 π 5 π1 1 π2 5 2v1 1 v g1 g2 2 pg g 1 v s 1s2 2 ps s 2 w1 2 w2 where w2 is the secondperiod wage paid to the worker In this contractual situation the worker wishes to maximize w1 1 w2 and the firm wishes to maximize twoperiod profits Competition in the labor market will play an important role in the contract chosen in this situation because the worker can always choose to work somewhere else If he or she is paid the marginal product in this alternative employment alternative wages must be w1 5 v1 and w2 5 v1 1 v g1 g2 Note that investments in general human capital increase the workers alterna tive wage rate but investments in specific human capital do not because by definition such skills are useless on other jobs If the firm sets wages equal to these alternatives profits are given by π 5 v s 1s2 2 pg g 2 ps s 1637 and the firms optimal choice is to set g 5 0 Intuitively if the firm cannot earn any return on its investment in general human capital its profitmaximizing decision is to refrain from such investing From the workers point of view however this decision would be nonoptimal He or she would command a higher wage with such added human capital Hence the worker may opt to pay for his or her own general human capital accumulation by taking a reduction in firstperiod wages Total wages are then given by w 5 w1 1 w2 5 2w1 1 v g1 g2 2 pg g 1638 and the firstorder condition for an optimal g for the worker is v g1 g2g 5 pg Note that this is the same optimality condition that would prevail if the firm could capture all the gains from its investment in general human capital Note also that the worker could not opt for this optimal contract if legal restrictions such as a minimum wage law prevented him or her from paying for the human capital investment with lower firstperiod wages The firms firstorder condition for a profitmaximizing choice of s immediately follows from Equation 1637v s 1s2s 5 ps Once the firm makes this investment however it must decide how if at all the increase in the value of the marginal product is to be shared with the worker This is ultimately a bargaining problem The worker can threaten to leave the firm unless he or she gets a share of the increased marginal product On the other hand the firm can threaten to invest little in specific human capital unless the worker promises to stay around A number of outcomes seem plausible depending on the success of the bargaining strategies employed by the two parties QUERY Suppose that the firm offered to provide a share of the increased marginal product given by αv s 1s2 with the worker where 0 α 1 How would this affect the firms investment in s How might this sharing affect wage bargaining in future periods One final type of investment in human capital might be mentionedinvestments in health Such investments can occur in many ways Individuals can purchase preventive care to guard against illness they may take other actions such as exercise with the same goal or they may purchase medical care to restore health if they have contracted an illness 7For simplicity we do not discount future profits here Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 587 All of these actions are intended to augment a persons health capital which is one com ponent of human capital There is ample evidence that such capital pays off in terms of increased productivity indeed firms themselves may wish to invest in such capital for the reasons outlined in Example 163 All components of human capital have certain characteristics that differentiate them from the types of physical capital also used in the production process First acquisition of human capital is often a timeconsuming process Attending school enrolling in a jobtraining program or even daily exercise can take many hours and these hours will usually have significant opportunity costs for individuals Hence human capital acquisi tion is often studied as part of the same time allocation process that we looked at earlier in this chapter Second human capital once obtained cannot be sold Unlike the owner of a piece of machinery the owner of human capital may only rent out that capital to others the owner cannot sell the capital outright Hence human capital is perhaps the most illiq uid way in which one can hold assets Finally human capital depreciates in an unusual way Workers may indeed lose skills as they get older or if they are unemployed for a long time However the death of a worker constitutes an abrupt loss of all human capital That together with their illiquidity makes human capital investments rather risky 1652 Compensating wage differentials Differences in working conditions are another reason why wages may differ among workers In general one might expect that jobs with pleasant surroundings would pay less for a given set of skills and jobs that are dirty or dangerous would pay more In this section we look at how such compensating wage differentials might arise in competitive labor markets Consider first a firms willingness to provide good working conditions Suppose that the firms output is a function of the labor it hires l and the amenities it provides to its work ers A Hence q 5 f 1l A2 We assume that amenities themselves are productive 1 fA 02 and exhibit diminishing marginal productivity 1 fAA 02 The firms profits are π 1l A2 5 pf 1l A2 2 wl 2 pAA 1639 where p w and pA are respectively the price of the firms output the wage rate paid and the price of amenities For a fixed wage the firm can choose profitmaximizing levels for its two inputs l and A The resulting equilibrium will have differing amenity levels among firms because these amenities will have different productivities in different applications happy workers may be important for retail sales but not for managing oil refineries In this case however wage levels will be determined independent of amenities Consider now the possibility that wage levels can change in response to amenities provided on the job Specifically assume that the wage actually paid by a firm is given by w 5 w0 2 k1A 2 A2 where k represents the implicit price of a unit of amenityan implicit price that will be determined in the marketplace as we shall show Given this possibility the firms profits are given by π 1l A2 5 pf 1l A2 2 3w0 2 k1A 2 A2 4l 2 pA A 1640 and the firstorder condition for a profitmaximizing choice of amenities is π A 5 p fA 1 kl 2 pA 5 0 or pfA 5 pA 2 kl 1641 Hence the firm will have an upward sloping supply curve for amenities in which a higher level of k causes the firm to choose to provide more amenities to its workers a fact derived from the assumed diminishing marginal productivity of amenities Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 588 Part 7 Pricing in Input Markets A workers valuation of amenities on the job is derived from his or her utility function Uw A The worker will choose among employment opportunities in a way that maxi mizes utility subject to the budget constraint w 5 w0 2 k1A 2 A2 As in other models of utility maximization the firstorder conditions for this constrained maximum problem can be manipulated to yield MRS 5 UA Uw 5 k 1642 That is the worker will choose a job that offers a combination of wages and amenities for which his or her MRS is precisely equal to the implicit price of amenities Therefore the utilitymaximizing process will generate a downward sloping demand curve for amenities as a function of k An equilibrium value of k can be determined in the marketplace by the interaction of the aggregate supply curve for firms and the aggregate demand curve for workers Given this value of k actual levels of amenities will differ among firms according to the specif ics of their production functions Individuals will also take note of the implicit price of amenities in sorting themselves among jobs Those with strong preferences for amenities will opt for jobs that provide them but they will also accept lower wages in the process Inferring the extent to which compensating such wage differentials explains wage vari ation in the real world is complicated by the many other factors that affect wages Most importantly linking amenities to wage differentials across individuals must also account for possible differences in human capital among these workers The simple observation that some unpleasant jobs do not seem to pay very well is not necessarily evidence against the theory of compensating wage differentials The presence or absence of such differen tials can be determined only by comparing workers with the same levels of human capital 1653 Job search Wage differences can also arise from differences in the success that workers have in finding good job matches The primary difficulty is that the job search process involves uncer tainty Workers new to the labor force may have little idea of how to find work Work ers who have been laid off from jobs face special problems in part because they lose the returns to the specific human capital they have accumulated unless they are able to find another job that uses these skills Therefore in this section we will look briefly at the ways economists have tried to model the job search process Suppose that the job search process proceeds as follows An individual samples the available jobs one at a time by calling a potential employer or perhaps obtaining an inter view The individual does not know what wage will be offered by the employer until he or she makes the contact the wage offered might also include the value of various fringe benefits or amenities on the job Before making the contact the job seeker does know that the labor market reflects a probability distribution of potential wages This probability density function see Chapter 2 of potential wages is given by f w The job seeker spends an amount c on each employer contact One way to approach the job seeker strategy is to argue that he or she should choose the number of employer contacts n for which the marginal benefit of further searching and thereby possibly finding a higher wage is equal to the marginal cost of the additional con tact Because search encounters diminish returns8 such an optimal n will generally exist 8The probability that a job seeker will encounter a specific high wage say w0 for the first time on the nth employer contact is given by 3F 1w024 n21 f 1w02 where Fw is the cumulative distribution of wages showing the probability that wages are less than or equal to a given level see Chapter 2 Hence the expected maximum wage after n contacts is wn max 5 e q 0 3F 1w24 n21 f 1w2wdw It is a fairly simple matter to show that w n11 max 2 w n max diminishes as n increases Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 589 although its value will depend on the precise shape of the wage distribution Therefore individuals with differing views of the distribution of potential wages may adopt differing search intensities and may ultimately settle for differing wage rates Setting the optimal search intensity on a priori grounds may not be the best strategy in this situation If a job seeker encountered an especially attractive job on say the third employer contact it would make little sense for him or her to continue looking An alter native strategy would be to set a reservation wage and take the first job that offered this wage An optimal reservation wage 1wr2 would be set so that the expected gain from one more employer contact should be equal to the cost of that contact That is wr should be chosen so that c 5 3 q wr 1w 2 wr2f1w2dw 1643 Equation 1643 makes clear that an increase in c will cause this person to reduce his or her reservation wage Hence people with high search costs may end the job search process with low wages Alternatively people with low search costs perhaps because the search is subsi dized by unemployment benefits will opt for higher reservation wages and possibly higher future wages although at the cost of a longer search Examining issues related to job search calls into question the definition of equilibrium in the labor market Figure 163 implies that labor markets will function smoothly settling at an equilibrium wage at which the quantity of labor supplied equals the quantity demanded In a dynamic context however it is clear that labor markets experience considerable flows into and out of employment and that there may be significant frictions involved in this pro cess Economists have developed a number of models that explore what equilibrium might look like in a market with search unemployment but we will not pursue these here9 166 MONOPSONY IN THE LABOR MARKET In many situations firms are not pricetakers for the inputs they buy That is the supply curve for say labor faced by the firm is not infinitely elastic at the prevailing wage rate It often may be necessary for the firm to offer a wage above that currently prevailing if it is to attract more employees In order to study such situations it is most convenient to examine the polar case of monopsony a single buyer in the labor market If there is only one buyer in the labor market then this firm faces the entire market supply curve To increase its hir ing of labor by one more unit it must move to a higher point on this supply curve This will involve paying not only a higher wage to the marginal worker but also additional wages to those workers already employed Therefore the marginal expense associated with hiring the extra unit of labor 1MEl 2 exceeds its wage rate We can show this result mathematically as follows The total cost of labor to the firm is wl Hence the change in those costs brought about by hiring an additional worker is MEl 5 wl l 5 w 1 l w l 1644 In the competitive case wl 5 0 and the marginal expense of hiring one more worker is simply the market wage w However if the firm faces a positively sloped labor supply curve then wl 0 and the marginal expense exceeds the wage These ideas are sum marized in the following definition 9For a pioneering example see P Diamond Wage Determination and Efficiency in Search Equilibrium Review of Economic Studies XLIX 1982 21727 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 590 Part 7 Pricing in Input Markets A profitmaximizing firm will hire any input up to the point at which its marginal revenue product is just equal to its marginal expense This result is a generalization of our previous discussion of marginalist choices to cover the case of monopsony power in the labor mar ket As before any departure from such choices will result in lower profits for the firm If for example MRPl MEl then the firm should hire more workers because such an action would increase revenues more than costs Alternatively if MRPl MEl then employment should be reduced because that would lower costs more rapidly than revenues 1661 Graphical analysis The monopsonists choice of labor input is illustrated in Figure 164 The firms demand curve for labor D is drawn negatively sloped as we have shown it must be10 Here 10Figure 164 is intended only as a pedagogic device and cannot be rigorously defended In particular the curve labeled D although it is supposed to represent the demand or marginal revenue product curve for labor has no precise meaning for the monopsonist buyer of labor because we cannot construct this curve by confronting the firm with a fixed wage rate Instead the firm views the entire supply curve S and uses the auxiliary curve MEl to choose the most favorable point on S In a strict sense there is no such thing as the monopsonists demand curve This is analogous to the case of a monopoly for which we could not speak of a monopolists supply curve D E F I N I T I O N Marginal input expense The marginal expense ME associated with any input is the increase in total costs of the input that results from hiring one more unit If the firm faces an upwardsloping supply curve for the input the marginal expense will exceed the market price of the input If a firm faces a positively sloped supply curve for labor S it will base its decisions on the marginal expense of additional hiring 1MEl2 Because S is positively sloped the MEl curve lies above S The curve S can be thought of as an average cost of labor curve and the MEl curve is marginal to S At l1 the equilib rium condition MEl 5 MRPl holds and this quantity will be hired at a market wage rate w1 Notice that the monopsonist buys less labor than would be bought if the labor market were perfectly competitive 1l 2 FIGURE 164 Pricing in a Monopsonistic Labor Market Wage Labor input per period S S D D MEl l w w1 l1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 591 also the MEl curve associated with the labor supply curve S is constructed in much the same way that the marginal revenue curve associated with a demand curve can be constructed Because S is positively sloped the MEl curve lies everywhere above S The profitmaximizing level of labor input for the monopsonist is given by l1 for at this level of input the profitmaximizing condition holds At l1 the wage rate in the market is given by w1 Notice that the quantity of labor demanded falls short of that which would be hired in a perfectly competitive labor market 1l 2 The firm has restricted input demand by virtue of its monopsonistic position in the market The formal similarities between this analysis and that of monopoly presented in Chapter 14 should be clear In partic ular the demand curve for a monopsonist consists of a single point given by l1 w1 The monopsonist has chosen this point as the most desirable of all points on the supply curve S A different point will not be chosen unless some external change such as a shift in the demand for the firms output or a change in technology affects labors mar ginal revenue product11 11A monopsony may also practice price discrimination in all of the ways described for a monopoly in Chapter 14 For a detailed discussion of the comparative statics analysis of factor demand in the monopoly and monopsony cases see W E Diewert Duality Approaches to Microeconomic Theory in K J Arrow and M D Intriligator Eds Handbook of Mathematical Economics Amsterdam NorthHolland 1982 vol 2 pp 58490 EXAMPLE 164 Monopsonistic Hiring To illustrate these concepts in a simple context suppose a coal mines workers can dig two tons of coal per hour and coal sells for 10 per ton Therefore the marginal revenue product of a coal miner is 20 per hour If the coal mine is the only hirer of miners in a local area and faces a labor supply curve of the form l 5 50w 1645 then this firm must recognize that its hiring decisions affect wages Expressing the total wage bill as a function of l wl 5 l2 50 1646 permits the mine operator perhaps only implicitly to calculate the marginal expense associated with hiring miners MEl 5 wl l 5 l 25 1647 Equating this to miners marginal revenue product of 20 implies that the mine operator should hire 500 workers per hour At this level of employment the wage will be 10 per houronly half the value of the workers marginal revenue product If the mine operator had been forced by market competition to pay 20 per hour regardless of the number of miners hired then market equilibrium would have been established with l 5 1000 rather than the 500 hired under monop sonistic conditions QUERY Suppose the price of coal increases to 15 per ton How would this affect the monop sonists hiring and the wages of coal miners Would the miners benefit fully from the increase in their MRP Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 592 Part 7 Pricing in Input Markets 167 LABOR UNIONS Workers may at times find it advantageous to join together in a labor union to pursue goals that can more effectively be accomplished by a group If association with a union were wholly voluntary we could assume that every union member derives a positive ben efit from belonging Compulsory membership the closed shop however is often used to maintain the viability of the union organization If all workers were left on their own to decide on membership their rational decision might be not to join the union thereby avoiding dues and other restrictions However they would benefit from the higher wages and better working conditions that have been won by the union What appears to be ratio nal from each individual workers point of view may prove to be irrational from a groups point of view because the union is undermined by free riders Therefore compulsory membership may be a necessary means of maintaining the union as an effective bargaining agent 1671 Unions goals A good starting place for our analysis of union behavior is to describe union goals A first assumption we might make is that the goals of a union are in some sense an adequate rep resentation of the goals of its members This assumption avoids the problem of union lead ership and disregards the personal aspirations of those leaders which may be in conflict with rankandfile goals Therefore union leaders are assumed to be conduits for express ing the desires of the membership12 In some respects unions can be analyzed in the same way as monopoly firms The union faces a demand curve for labor because it is the sole source of supply it can choose at which point on this curve it will operate The point actually chosen by the union will obviously depend on what particular goals it has decided to pursue Three possible choices are illustrated in Figure 165 For example the union may choose to offer that quantity of labor that maximizes the total wage bill w l If this is the case it will offer that quantity for which the marginal revenue from labor demand is equal to 0 This quantity is given by l1 in Figure 165 and the wage rate associated with this quantity is w1 Therefore the point E1 is the preferred wagequantity combination Notice that at wage rate w1 there may be an excess supply of labor and the union must somehow allocate available jobs to those workers who want them Another possible goal the union may pursue would be to choose the quantity of labor that would maximize the total economic rent that is wages less opportunity costs obtained by those members who are employed This would necessitate choosing that quan tity of labor for which the additional total wages obtained by having one more employed union member the marginal revenue are equal to the extra cost of luring that member into the market Therefore the union should choose that quantity l2 at which the marginal revenue curve crosses the supply curve13 The wage rate associated with this quantity is w2 and the desired wagequantity combination is labeled E2 in the diagram With the wage w2 many individuals who desire to work at the prevailing wage are left unemployed Perhaps the union may tax the large economic rent earned by those who do work to transfer income to those who dont 12Much recent analysis however revolves around whether potential union members have some voice in setting union goals and how union goals may affect the desires of workers with differing amounts of seniority on the job 13Mathematically the unions goal is to choose l so as to maximize wl 2 1area under S2 where S is the compensated supply curve for labor and reflects workers opportunity costs in terms of forgone leisure Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 593 A third possibility would be for the union to aim for maximum employment of its members This would involve choosing the point w3 l3 which is precisely the point that would result if the market were organized in a perfectly competitive way No employment greater than l3 could be achieved because the quantity of labor that union members supply would be reduced for wages less than w3 A union has a monopoly in the supply of labor so it may choose its most preferred point on the demand curve for labor Three such points are shown in the figure At point E1 total labor payments 1w l2 are maximized at E2 the economic rent that workers receive is maximized and at E3 the total amount of labor services supplied is maximized w1 w2 w3 S D D l 2 l 1 l 3 E2 E1 E3 Quantity of labor per period Real wage MR FIGURE 165 Three Possible Points on the Labor Demand Curve That a Monopolistic Union Might Choose EXAMPLE 165 Modeling a Union In Example 164 we examined a monopsonistic hirer of coal miners who faced a supply curve given by l 5 50w 1648 To study the possibilities for unionization to combat this monopsonist assume contrary to Example 164 that the monopsonist has a downwardsloping marginal revenue product for labor curve of the form MRP 5 70 2 01l 1649 By setting MRP 5 MEl it is easy to show that in the absence of an effective union a monopsonist would opt for the same wagehiring combination as in Example 164 500 workers would be hired at a wage of 10 If the union can establish control over the entire supply of labor to this employer it can try to achieve the various results shown in Figure 165 For example the union could press for the competitive solution 1E32 By solving Equations 1648 and 1649 together and assum ing that MRP 5 w this supplydemand equilibrium would result in a labor contract in which l 5 583 w 5 117 This union could opt for the other solutions shown in Figure 165 by calcu lating the marginal revenue curve associated with this firms demand curve for labor Since total wages along this demand curve are given by MRP l 5 70l 2 01l 2 this relationship is given by 1MRP l2l 5 70 2 02l The total wage bill is maximized when 70 2 02l 5 0 so l 5 350 w 5 35 1650 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 594 Part 7 Pricing in Input Markets This is the contract shown by point E1 in Figure 165 Finally the union could seek to maximize economic rent per worker This can be done by finding where the marginal revenue curve intersects the labor supply curve 70 2 02l 5 l 50 1651 which has a solution of 11l 5 3500 or l 5 318 w 5 382 1652 This contract is represented by point E2 in Figure 165 The fact that all three of the unions preferred contracts differ substantially from the monopsonists desired outcome suggests that the final outcome in this situation will be determined through some sort of bargaining between the two parties Notice that the variation in desired levels of hiring is quite a bit smaller than the variation in the desired wage level This suggests that the impact of bargaining power at least in this example will probably be most strongly reflected in the wage that is ultimately settled upon QUERY Which if any of the three wage contracts described in this example might represent an Nash equilibrium EXAMPLE 166 A Union Bargaining Model Game theory can be used to gain insights into the economics of unions As a simple illustration suppose a union and a firm engage in a twostage game In the first stage the union sets the wage rate its workers will accept Given this wage the firm then chooses its employment level This twostage game can be solved by backward induction Given the wage w specified by the union the firms secondstage problem is to maximize π 5 R1l2 2 wl 1653 where R is the total revenue function of the firm expressed as a function of employment The firstorder condition for a maximum here assuming that the wage is fixed is the familiar Rr 1l2 5 w 1654 Assuming l solves Equation 1654 the unions goal is to choose w to maximize utility U1w l2 5 U3w l 1w24 1655 and the firstorder condition for a maximum is U1 1 U2lr 5 0 1656 or U1U2 52lr 1657 In words the union should choose w so that its MRS is equal to the absolute value of the slope of the firms labor demand function The w l combination resulting from this game is clearly a Nash equilibrium Efficiency of the labor contract The labor contract w l is Pareto inefficient To see this notice that Equation 1657 implies that small movements along the firms labor demand curve l leave the union equally welloff But the envelope theorem implies that a decrease in w must increase profits to the firm Hence there must exist a contract w p l p where w p w and l p l with which both the firm and union are better off Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 595 The inefficiency of the labor contract in this twostage game is similar to the inefficiency of some of the repeated Nash equilibria we studied in Chapter 15 This suggests that with repeated rounds of contract negotiations trigger strategies might be developed that form a subgameperfect equilibrium and maintain Paretosuperior outcomes For a simple example see Problem 1610 QUERY Suppose the firms total revenue function differed depending on whether the economy was in an expansion or a recession What kinds of labor contracts might be Pareto optimal Summary In this chapter we examined some models that focus on pric ing in the labor market Because labor demand was already treated as being derived from the profitmaximization hypoth esis in Chapter 11 most of the new material here focused on labor supply Our primary findings were as follows A utilitymaximizing individual will choose to supply an amount of labor at which his or her marginal rate of sub stitution of leisure for consumption is equal to the real wage rate An increase in the real wage creates substitution and income effects that work in opposite directions in affect ing the quantity of labor supplied This result can be summarized by a Slutskytype equation much like the one already derived in consumer theory A competitive labor market will establish an equilibrium real wage at which the quantity of labor supplied by indi viduals is equal to the quantity demanded by firms Wages may vary among workers for a number of reasons Workers may have invested in different levels of skills and therefore have different productivities Jobs may dif fer in their characteristics thereby creating compensat ing wage differentials And individuals may experience differing degrees of jobfinding success Economists have developed models that address all of these features of the labor market Monopsony power by firms on the demand side of the labor market will reduce both the quantity of labor hired and the real wage As in the monopoly case there will also be a welfare loss Labor unions can be treated analytically as monopoly suppliers of labor The nature of labor market equilib rium in the presence of unions will depend importantly on the goals the union chooses to pursue Problems 161 Suppose there are 8000 hours in a year actually there are 8760 and that an individual has a potential market wage of 5 per hour a What is the individuals full income If he or she chooses to devote 75 percent of this income to leisure how many hours will be worked b Suppose a rich uncle dies and leaves the individual an annual income of 4000 per year If he or she continues to devote 75 percent of full income to leisure how many hours will be worked c How would your answer to part b change if the market wage were 10 per hour instead of 5 per hour d Graph the individuals supply of labor curve implied by parts b and c 162 As we saw in this chapter the elements of labor supply the ory can also be derived from an expenditureminimization approach Suppose a persons utility function for consumption and leisure takes the CobbDouglas form U1c h2 5 cαh12α Then the expenditureminimization problem is minimize c 2 w124 2 h2 st U1c h2 5 cαh12α 5 U a Use this approach to derive the expenditure function for this problem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 596 Part 7 Pricing in Input Markets b Use the envelope theorem to derive the compensated demand functions for consumption and leisure c Derive the compensated labor supply function Show that lcw 0 d Compare the compensated labor supply function from part c to the uncompensated labor supply function in Example 162 with n 5 0 Use the Slutsky equation to show why income and substitution effects of a change in the real wage are precisely offsetting in the uncompen sated CobbDouglas labor supply function 163 A welfare program for lowincome people offers a family a basic grant of 6000 per year This grant is reduced by 075 for each 1 of other income the family has a How much in welfare benefits does the family receive if it has no other income If the head of the family earns 2000 per year How about 4000 per year b At what level of earnings does the welfare grant become 0 c Assume the head of this family can earn 4 per hour and that the family has no other income What is the annual budget constraint for this family if it does not participate in the welfare program That is how are consumption c and hours of leisure h related d What is the budget constraint if the family opts to par ticipate in the welfare program Remember the welfare grant can only be positive e Graph your results from parts c and d f Suppose the government changes the rules of the welfare program to permit families to keep 50 percent of what they earn How would this change your answers to parts d and e g Using your results from part f can you predict whether the head of this family will work more or less under the new rules described in part f 164 Suppose demand for labor is given by l 5250w 1 450 and supply is given by l 5 100w where l represents the number of people employed and w is the real wage rate per hour a What will be the equilibrium levels for w and l in this market b Suppose the government wishes to increase the equi librium wage to 4 per hour by offering a subsidy to employers for each person hired How much will this subsidy have to be What will the new equilibrium level of employment be How much total subsidy will be paid c Suppose instead that the government declared a mini mum wage of 4 per hour How much labor would be demanded at this price How much unemployment would there be d Graph your results 165 Carl the clothier owns a large garment factory on an isolated island Carls factory is the only source of employment for most of the islanders and thus Carl acts as a monopsonist The supply curve for garment workers is given by l 5 80w where l is the number of workers hired and w is their hourly wage Assume also that Carls labor demand marginal reve nue product curve is given by l 5 400 2 40MRPl a How many workers will Carl hire to maximize his prof its and what wage will he pay b Assume now that the government implements a mini mum wage law covering all garment workers How many workers will Carl now hire and how much unemploy ment will there be if the minimum wage is set at 4 per hour c Graph your results d How does a minimum wage imposed under monop sony differ in results as compared with a minimum wage imposed under perfect competition Assume the mini mum wage is above the marketdetermined wage 166 The Ajax Coal Company is the only hirer of labor in its area It can hire any number of female workers or male workers it wishes The supply curve for women is given by lf 5 100wf and for men by lm 5 9w2 m where wf and wm are the hourly wage rates paid to female and male workers respectively Assume that Ajax sells its coal in a perfectly competitive market at 5 per ton and that each worker hired both men and women can mine 2 tons per hour If the firm wishes to maximize profits how many female and male workers should be hired and what will the wage rates be for these two groups How much will Ajax earn in profits per hour on its mine machinery How will that result compare to one in which Ajax was constrained say by market forces to pay all workers the same wage based on the value of their marginal products Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 16 Labor Markets 597 167 Universal Fur is located in Clyde Baffin Island and sells highquality fur bow ties throughout the world at a price of 5 each The production function for fur bow ties q is given by q 5 240x 2 2x2 where x is the quantity of pelts used each week Pelts are supplied only by Dans Trading Post which obtains them by hiring Eskimo trappers at a rate of 10 per day Dans weekly production function for pelts is given by x 5 l where l represents the number of days of Eskimo time used each week a For a quasicompetitive case in which both Universal Fur and Dans Trading Post act as pricetakers for pelts what will be the equilibrium price 1 px2 and how many pelts will be traded b Suppose Dan acts as a monopolist while Universal Fur continues to be a pricetaker What equilibrium will emerge in the pelt market c Suppose Universal Fur acts as a monopsonist but Dan acts as a pricetaker What will the equilibrium be d Graph your results and discuss the type of equilibrium that is likely to emerge in the bilateral monopoly bar gaining between Universal Fur and Dan 168 Following in the spirit of the labor market game described in Example 166 suppose the firms total revenue function is given by R 5 10l 2 l 2 and the unions utility is simply a function of the total wage bill U1w l2 5 wl a What is the Nash equilibrium wage contract in the two stage game described in Example 166 b Show that the alternative wage contract wr 5 lr 5 4 is Pareto superior to the contract identified in part a c Under what conditions would the contract described in part b be sustainable as a subgameperfect equilibrium Analytical Problems 169 Compensating wage differentials for risk An individual receives utility from daily income y given by U1 y2 5 100y 2 1 2 y 2 The only source of income is earnings Hence y 5 wl where w is the hourly wage and l is hours worked per day The indi vidual knows of a job that pays 5 per hour for a certain 8hour day What wage must be offered for a construction job where hours of work are randomwith a mean of 8 hours and a standard deviation of 6 hoursto get the individual to accept this more risky job Hint This problem makes use of the statistical identity E 1x22 5 Var x 1 E 1x22 1610 Family labor supply A family with two adult members seeks to maximize a utility function of the form U1c h1 h22 where c is family consumption and h1 and h2 are hours of lei sure of each family member Choices are constrained by c 5 w1 124 2 h12 1 w2 124 2 h22 1 n where w1 and w2 are the wages of each family member and n is nonlabor income a Without attempting a mathematical presentation use the notions of substitution and income effects to discuss the likely signs of the crosssubstitution effects h1w2 and h2w1 b Suppose that one family member say individual 1 can work in the home thereby converting leisure hours into consumption according to the function c1 5 f 1h12 where f r 0 and f s 0 How might this additional option affect the optimal division of work among family members 1611 A few results from demand theory The theory developed in this chapter treats labor supply as the mirror image of the demand for leisure Hence the entire body of demand theory developed in Part 2 of the text becomes relevant to the study of labor supply as well Here are three examples a Roys identity In the Extensions to Chapter 5 we showed how demand functions can be derived from indirect utility functions by using Roys identity Use a similar approach to show that the labor supply function associ ated with the utilitymaximization problem described in Equation 1620 can be derived from the indirect utility function by l 1w n2 5 V 1w n2w V 1w n2n Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 598 Part 7 Pricing in Input Markets Illustrate this result for the CobbDouglas case described in Example 161 b Substitutes and complements A change in the real wage will affect not only labor supply but also the demand for specific items in the preferred consumption bundle Develop a Slutskytype equation for the crossprice effect of a change in w on a particular consumption item and then use it to discuss whether leisure and the item are net or gross substitutes or complements Provide an example of each type of relationship c Labor supply and marginal expense Use a derivation similar to that used to calculate marginal revenue for a given demand curve to show that MEl 5 w11 1 1elw2 1612 Intertemporal labor supply It is relatively easy to extend the singleperiod model of labor supply presented in Chapter 16 to many periods Here we look at a simple example Suppose that an individ ual makes his or her labor supply and consumption deci sions over two periods14 Assume that this person begins period 1 with initial wealth W0 and that he or she has 1 unit of time to devote to work or leisure in each period Therefore the twoperiod budget constraint is given by W0 5 c1 1 c2 2 w1 11 2 h12 2 w2 11 2 h22 where the ws are the real wage rates prevailing in each period Here we treat w2 as uncertain so utility in period 2 will also be uncertain If we assume utility is additive across the two periods we have E 3U1c1 h1 c2 h22 4 5 U1c1 h12 1 E 3U1c2 h224 a Show that the firstorder conditions for utility maxi mization in period 1 are the same as those shown in Chapter 16 in particular show MRS 1c1 for h12 5 w1 Explain how changes in W0 will affect the actual choices of c1 and h1 b Explain why the indirect utility function for the second period can be written as V 1w2 W 2 where W 5 W0 1 w1 11 2 h12 2 c1 Note that because w2 is a random vari able V is also random c Use the envelope theorem to show that optimal choice of W requires that the Lagrange multipliers for the wealth constraint in the two periods obey the condi tion λ1 5 E 1λ22 1where λ1 is the Lagrange multiplier for the original problem and λ2 is the implied Lagrange multiplier for the period 2 utilitymaximization prob lem2 That is the expected marginal utility of wealth should be the same in the two periods Explain this result intuitively d Although the comparative statics of this model will depend on the specific form of the utility function dis cuss in general terms how a governmental policy that added k dollars to all period 2 wages might be expected to affect choices in both periods Suggestions for Further Reading Ashenfelter O C and D Card Handbook of Labor Econom ics 3 Amsterdam North Holland 1999 Contains a variety of highlevel essays on many labor market top ics Survey articles on labor supply and demand in volumes 1 and 2 1986 are also highly recommended Becker G A Theory of the Allocation of Time Economic Journal September 1965 493517 One of the most influential papers in microeconomics Beckers observations on both labor supply and consumption decisions were revolutionary Binger B R and E Hoffman Microeconomics with Calculus 2nd ed Reading MA AddisonWesley 1998 Chapter 17 has a thorough discussion of the labor supply model including some applications to household labor supply Hamermesh D S Labor Demand Princeton NJ Princeton University Press 1993 The author offers a complete coverage of both theoretical and empirical issues The book also has nice coverage of dynamic issues in labor demand theory Silberberg E and W Suen The Structure of Economics A Mathematical Analysis 3rd ed Boston IrwinMcGrawHill 2001 Provides a nice discussion of the dual approach to labor supply theory 14Here we assume that the individual does not discount utility in the second period and that the real interest rate between the two periods is zero Discounting in a multiperiod context is taken up in Chapter 17 The discussion in that chapter also generalizes the approach to studying changes in the Lagrange multiplier over time shown in part c Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 599 CHAPTER SEVENTEEN Capital and Time In this chapter we provide an introduction to the theory of capital In many ways that theory resembles our previous analysis of input pricing in generalthe principles of profitmaximizing input choice do not change But capital theory adds an import ant time dimension to economic decision making our goal here is to explore that extra dimension We begin with a broad characterization of the capital accumulation process and the notion of the rate of return Then we turn to more specific models of economic behavior over time 171 CAPITAL AND THE RATE OF RETURN When we speak of the capital stock of an economy we mean the sum total of machines buildings and other reproducible resources in existence at some point in time These assets represent some part of an economys past output that was not consumed but was instead set aside to be used for production in the future All societies from the most prim itive to the most complex engage in capital accumulation Hunters in a primitive society taking time off from hunting to make arrows individuals in a modern society using part of their incomes to buy houses or governments taxing citizens in order to purchase dams and post office buildings are all engaging in essentially the same sort of activity Some portion of current output is being set aside for use in producing output in future periods As we saw in the previous chapter this is also true for human capitalindividuals invest time and money in improving their skills so that they can earn more in the future Present sacrifice for future gain is the essential aspect of all capital accumulation 1711 Rate of return The process of capital accumulation is pictured schematically in Figure 171 In both pan els of the figure society is initially consuming level c0 and has been doing so for some time At time t1 a decision is made to withhold some output amount s from current consumption for one period Starting in period t2 this withheld consumption is in some way put to use producing future consumption An important concept connected with this process is the rate of return which is earned on that consumption that is put aside In panel a for example all of the withheld consumption is used to produce additional out put only in period t2 Consumption is increased by amount x in period t2 and then returns to the longrun level c0 Society has saved in 1 year in order to splurge in the next year The oneperiod rate of return from this activity is defined as follows Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 600 Part 7 Pricing in Input Markets In a society withdraws some current consumption s to gorge itself with x extra consumption in the next period The oneperiod rate of return would be measured by xs 21 The society in b takes a more longterm view and uses s to increase its consumption perpetually by y The perpetual rate of return would be given by ys Consumption Consumption Time Time a Oneperiod return b Perpetual return c0 t1 t2 t3 t1 t2 t3 c0 x s s y FIGURE 171 Two Views of Capital Accumulation If x s if more consumption comes out of this process than went into it we would say that the oneperiod rate of return to capital accumulation is positive For example if withholding 100 units from current consumption permitted society to consume an extra 110 units next year then the oneperiod rate of return would be 110 100 2 1 5 010 or 10 percent In panel b of Figure 171 society takes a more longterm view in its capital accumula tion Again an amount s is set aside at time t1 Now however this setaside consumption is used to increase the consumption level for all periods in the future If the permanent level of consumption is increased to c0 1 y we define the perpetual rate of return as follows D E F I N I T I O N Singleperiod rate of return The singleperiod rate of return 1r12 on an investment is the extra consumption provided in period 2 as a fraction of the consumption forgone in period 1 That is r1 5 x 2 s s 5 x s 2 1 171 D E F I N I T I O N Perpetual rate of return The perpetual rate of return 1rq2 is the permanent increment to future consumption expressed as a fraction of the initial consumption forgone That is rq 5 y s 172 If capital accumulation succeeds in raising c0 permanently then rq will be positive For example suppose that society set aside 100 units of output in period t1 to be devoted to capital accumulation If this capital would permit output to be increased by 10 units for every period in the future starting at time period t2 the perpetual rate of return would be 10 percent Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 601 When economists speak of the rate of return to capital accumulation they have in mind something between these two extremes Somewhat loosely we shall speak of the rate of return as being a measure of the terms at which consumption today may be turned into consumption tomorrow this will be made more explicit soon A natural question to ask is how the economys rate of return is determined Again the equilibrium arises from the supply and demand for present and future goods In the next section we present a simple twoperiod model in which this supplydemand interaction is demonstrated 172 DETERMINING THE RATE OF RETURN In this section we will describe how operation of supply and demand in the market for future goods establishes an equilibrium rate of return We begin by analyzing the connec tion between the rate of return and the price of future goods Then we show how indi viduals and firms are likely to react to this price Finally these actions are brought together as we have done for the analysis of other markets to demonstrate the determination of an equilibrium price of future goods and to examine some of the characteristics of that solution 1721 Rate of return and price of future goods For most of the analysis in this chapter we assume there are only two periods to be con sidered the current period denoted by the subscript 0 and the next period subscript 1 We will use r to denote the oneperiod rate of return between these two periods Hence as defined in the previous section r 5 Dc1 Dc0 2 1 173 where the D notation indicates the change in consumption during the two periods Note that throughout this discussion we are using the absolute values of the changes in consumption as in Equations 171 and 172 Rewriting Equation 173 yields Dc1 Dc0 5 1 1 r 174 or Dc0 Dc1 5 1 1 1 r 175 The term on the left of Equation 175 records how much c0 must be forgone if c1 is to be increased by 1 unit that is the expression represents the relative price of 1 unit of c1 in terms of c0 Thus we have defined the price of future goods1 We now proceed to develop a demandsupply analysis of the determination of p1 By so doing we also will have developed a theory of the determination of r the rate of return in this simple model 1This price is identical to the discount factor introduced in connection with repeated games in Chapter 8 D E F I N I T I O N Price of future goods The relative price of future goods 1p12 is the quantity of present goods that must be forgone to increase future consumption by 1 unit That is p1 5 Dc0 Dc1 5 1 1 1 r 176 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 602 Part 7 Pricing in Input Markets 1722 Demand for future goods The theory of the demand for future goods is one further application of the utility maximization model developed in Part 2 of this book Here the individuals utility depends on present and future consumption ie utility 5 U1c0 c12 and he or she must decide how much current wealth W to allocate to these two goods2 Wealth not spent on current consumption can be invested at the rate of return r to obtain consumption next period As before p1 reflects the present cost of future consumption and the individuals budget constraint is given by W 5 c0 1 p1c1 177 This constraint is illustrated in Figure 172 If the individual chooses to spend all of his or her wealth on c0 then total current consumption will be W with no consumption occurring in period 2 Alternatively if c0 5 0 then c1 will be given by Wp1 5 W11 1 r2 That is if all wealth is invested at the rate of return r current wealth will grow to W11 1 r2 in period 23 2For an analysis of the case where the individual has income in both periods see Problem 171 3This observation yields an alternative interpretation of the intertemporal budget constraint which can be written in terms of the rate of return as W 5 c0 1 c1 1 1 r This illustrates that it is the present value of c1 that enters into the individuals current budget constraint The concept of present value is discussed in more detail later in this chapter When faced with the intertemporal budget constraint W 5 c0 1 p1c1 the individual will maximize util ity by choosing to consume c 0 currently and c 1 in the next period A decrease in p1 an increase in the rate of return r will cause c1 to increase but the effect on c0 is indeterminate because substitution and income effects operate in opposite directions assuming that both c0 and c1 are normal goods Future consumption c1 Current consumption c0 W c0 p1c1 U2 U1 U0 Wp1 c1 c0 W FIGURE 172 Individuals Intertemporal Utility Maximization Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 603 1723 Utility maximization Imposing the individuals indifference curve map for c0 and c1 onto the budget constraint in Figure 172 illustrates utility maximization Here utility is maximized at the point c 0 c 1 The individual consumes c 0 currently and chooses to save W 2 c 0 to consume next period This future consumption can be found from the budget constraint as p1c 1 5 W 2 c 0 178 or c 1 5 1W 2 c 02 p1 179 5 1W 2 c 02 11 1 r2 1710 In words wealth that is not currently consumed 1W 2 c 02 is invested at the rate of return r and will grow to yield c 1 in the next period EXAMPLE 171 Intertemporal Impatience Individuals utilitymaximizing choices over time will obviously depend on how they feel about the relative merits of consuming currently or waiting to consume in the future One way of reflecting the possibility that people exhibit some impatience in their choices is to assume that the utility from future consumption is implicitly discounted in the individuals mind For example we might assume that the utility function for consumption Uc is the same in both periods with Ur 0 Us 0 but that period 1s utility is discounted in the individuals mind by a rate of time preference of 1 11 1 δ2 where δ 0 If the intertemporal utility function is also separa ble for more discussion of this concept see the Extensions to Chapter 6 we can write U1c0 c12 5 U1c02 1 1 1 1 δ U1c12 1711 Maximization of this function subject to the intertemporal budget constraint W 5 c0 1 c1 1 1 r 1712 yields the following Lagrangian expression 5 U1c0 c12 1 λcW 2 c0 2 c1 1 1 rd 1713 and the firstorder conditions for a maximum are c0 5 Ur 1c02 2 λ 5 0 c1 5 1 1 1 δ Ur 1c12 2 λ 1 1 r 5 0 1714 λ 5 W 2 c0 2 c1 1 1 r 5 0 Dividing the first and second of these and rearranging terms gives4 Ur 1c02 5 1 1 r 1 1 δ Ur 1c12 1715 4Equation 1715 is sometimes called the Euler equation for intertemporal utility maximization As we show once a specific utility function is defined the equation indicates how consumption changes over time Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 604 Part 7 Pricing in Input Markets Because the utility function for consumption is assumed to be the same in two periods we can conclude that c0 5 c1 if r 5 δ that c0 c1 if δ r to obtain Ur 1c02 Ur 1c12 requires c0 c1 and that c0 c1 for r δ Therefore whether this individuals consumption increases or decreases from period 0 to period 1 will depend on exactly how impatient he or she is Although a consumer may have a preference for present goods 1δ 02 he or she may still consume more in the future than in the present if the rate of return received on savings is high enough Consumption smoothing Because utility functions generally exhibit a diminishing marginal utility from consumption individuals will seek to equalize utility across periods The extent of such smoothing will depend on individuals willingness to substitute consumption over time that is illustrated by the curvature of the utility functions indifference curves For example if the util ity function takes the CES form U1c2 5 c12γ 1 2 γ for γ 0 γ 2 1 5 ln 1c2 for γ 5 1 1716 Greater values of γ will make the indifference map more sharply curved and this person will be less willing to substitute one periods consumption for anothers If this persons rate of time pref erence is δ 5 0 Equation 1715 becomes c2γ 0 5 11 1 r2c2γ 1 or c1 c0 5 11 1 r2 1γ 1717 If r 5 0 this person will equalize consumption across the two periods no matter what value γ takes A positive real interest rate will however encourage this person to have c1 c0 and the extent of this preference will depend on γ which we previously called the coefficient of relative risk aversion but in this context it is sometimes called the coefficient of fluctuation aversion For example if r 5 005 and γ 5 05 c1 c0 5 11052 2 5 11025 1718 and period 2 consumption will be about 10 percent larger than period 1 consumption On the other hand if r 5 005 and γ 5 3 c1 c0 5 11052 13 5 10162 1719 and period 2 consumption will be only 16 percent larger than period 1 consumption In this latter case a positive real interest rate provides a much smaller incentive to depart from equal consumption levels QUERY Empirical data show that per capita consumption has increased at an annual rate of approximately 2 percent in the US economy over the past 50 years What real interest rate would be needed to make this increase utility maximizing again assuming that δ 5 0 Note We will return to the relationship between consumption smoothing and the real interest rate in Example 172 Problem 1713 shows how intertemporal discount rates that follow a hyperbolic pattern can be used to explain why people may sometimes make decisions that seem shortsighted 1724 Effects of changes in r A comparative statics analysis of the equilibrium illustrated in Figure 172 is straightfor ward If p1 decreases that is if r increases then both income and substitution effects will cause more c1 to be demandedexcept in the unlikely event that c1 is an inferior good Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 605 Hence the demand curve for c1 will be downward sloping An increase in r effectively low ers the price of c1 and consumption of that good thereby increases This demand curve is labeled D in Figure 173 Before leaving our discussion of individuals intertemporal decisions we should point out that the analysis does not permit an unambiguous statement to be made about the sign of c0p1 In Figure 172 substitution and income effects work in opposite directions and thus no definite prediction is possible A decrease in p1 will cause the individual to sub stitute c1 for c0 in his or her consumption plans But the decrease in p1 increases the real value of wealth and this income effect causes both c0 and c1 to increase Phrased somewhat differently the model illustrated in Figure 172 does not permit a definite prediction about how changes in the rate of return affect currentperiod wealth accumulation saving A higher r produces substitution effects that favor more saving and income effects that favor less Ultimately then the direction of the effect is an empirical question 1725 Supply of future goods In one sense the analysis of the supply of future goods is quite simple We can argue that an increase in the relative price of future goods 1p12 will induce firms to produce more of them because the yield from doing so is now greater This reaction is reflected in the pos itively sloped supply curve S in Figure 173 It might be expected that as in our previous perfectly competitive analysis this supply curve reflects the increasing marginal costs or diminishing returns firms experience when attempting to turn present goods into future ones through capital accumulation Unfortunately by delving deeper into the nature of capital accumulation one runs into complications that have occupied economists for hundreds of years5 Basically all of these 5For a discussion of some of this debate see M Blaug Economic Theory in Retrospect rev ed Homewood IL Richard D Irwin 1978 chap 12 The point p 1 c 1 represents an equilibrium in the market for future goods The equilibrium price of future goods determines the rate of return via Equation 1716 Future consumption c1 Price p1 c1 p1 D D s s FIGURE 173 Determination of the Equilibrium Price of Future Goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 606 Part 7 Pricing in Input Markets derive from problems in developing a tractable model of the capital accumulation process For our model of individual behavior this problem did not arise because we could assume that the market quoted a rate of return to individuals so they could adapt their behavior to it We shall also follow this route when describing firms investment decisions later in the chapter But to develop an adequate model of capital accumulation by firms we must describe precisely how c0 is turned into c1 and doing so would take us too far afield into the intri cacies of capital theory Instead we will be content to draw the supply curve in Figure 173 with a positive slope on the presumption that such a shape is intuitively reasonable Much of the subsequent analysis in this chapter may serve to convince you that this is indeed the case 1726 Equilibrium price of future goods Equilibrium in the market shown in Figure 173 is at p 1 c 1 At that point individuals sup ply and demand for future goods are in balance and the required amount of current goods will be put into capital accumulation to produce c 1 in the future6 There are a number of reasons to expect that p1 will be less than 1 that is it will cost less than the sacrifice of one current good to buy one good in the future As we showed in Example 171 it might be argued that individuals require some reward for waiting Everyday adages a bird in the hand is worth two in the bush live for today and more substantial realities the uncertainty of the future and the finiteness of life suggest that individuals are generally impatient in their consumption decisions Hence capital accumulation such as that shown in Figure 173 will take place only if the current sacrifice is in some way worthwhile There are also supply reasons for believing p1 will be less than 1 All of these involve the idea that capital accumulation is productive Sacrificing one good today will yield more than one good in the future Some simple examples of the productivity of capital invest ment are provided by such pastoral activities as the growing of trees or the aging of wine and cheese Tree nursery owners and vineyard and dairy operators abstain from selling their wares in the belief that time will make them more valuable in the future Although it is obvious that capital accumulation in a modern industrial society is more complex than growing trees consider building a steel mill or an electric power system economists believe the two processes have certain similarities In both cases investing current goods makes the production process longer and more complex thereby increasing the contribu tion of other resources used in production 1727 The equilibrium rate of return Figure 173 shows how the equilibrium price of future goods 1p 12 is determined in the market for those goods Because present and future consumption consists of the same homogeneous good this will also determine the equilibrium rate of return according to the relationship p 1 5 1 1 1 r or r 5 1 2 p 1 p 1 1720 Because p 1 will be less than 1 this equilibrium rate of return will be positive For exam ple if p 1 5 095 then r 5 005095 005 and we would say that the rate of return is 5 percent By withholding 1 unit of consumption in year 0 an individual would be able to purchase 105 units of consumption in period 1 Hence the equilibrium rate of return shows the terms on which goods can be reallocated over time for both individuals and firms 6This is a much simplified form of an analysis originally presented by I Fisher The Rate of Interest New York Macmillan 1907 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 607 1728 Rate of return real interest rates and nominal interest rates The concept of the rate of return that we have been analyzing here is sometimes used syn onymously with the related concept of the real interest rate In this context both are taken to refer to the real return that is available from capital accumulation This concept must be differentiated from the nominal interest rate actually available in financial mar kets Specifically if overall prices are expected to increase by p e between two periods ie p e 5 010 for a 10 percent inflation rate then we would expect the nominal interest rate i to be given by the equation 1 1 i 5 11 1 r2 11 1 p e2 1721 because a wouldbe lender would expect to be compensated for both the opportunity cost of not investing in real capital r and for the general increase in prices 1 p e2 Expansion of Equation 1717 yields 1 1 i 5 1 1 r 1 p e 1 rp e 1722 and assuming r p e is small we have the simpler approximation i 5 r 1 p e 1723 If the real rate of return is 4 percent 004 and the expected rate of inflation is 10 percent 010 then the nominal interest rate would be approximately 14 percent 014 Therefore the difference between observed nominal interest rates and real interest rates may be sub stantial in inflationary environments EXAMPLE 172 Determination of the Real Interest Rate A simple model of real interest rate determination can be developed by assuming that consumption grows at some exogenous rate g For example suppose that the only consumption good is perishable fruit and that this fruit comes from trees that are growing at the rate g More realistically g might be determined by macroeconomic forces such as the rate of technical change in the Solow growth model see the Extensions to Chapter 9 No matter how the growth rate is determined the real interest rate must adjust so that consumers are willing to accept this rate of growth in consumption Optimal consumption The typical consumer wants his or her consumption pattern to maxi mize the utility received from this consumption over time That is the goal is to maximize utility 5 3 q 0 e2δtU1c 1t2 2dt 1724 where δ is the rate of pure time preference At each instant of time this person earns a wage w and earns interest r on his or her capital stock k Hence this persons capital evolves according to the equation dk dt 5 w 1 rk 2 c 1725 and is bound by the endpoint constraints k102 5 0 and k1q2 5 0 Setting up the augmented Hamiltonian for this dynamic optimization problem see Chapter 2 yields H 5 e2δtU1c2 1 λ1w 1 rk 2 c2 1 k d λ dt 1726 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 608 Part 7 Pricing in Input Markets Therefore the maximum principle requires Hc 5 e2δtUr 1c2 2 λ 5 0 Hk 5 r λ 1 d λ dt 5 0 or rλ 5 2d λ dt 1727 Solving the differential equation implied by the second of these conditions yields the conclusion that λ 5 e2rt and substituting this into the first of the conditions shows that Ur 1c2 5 e1δ2r2t 1728 Consistent with our results from Example 171 the marginal utility of consumption should increase or decrease over time depending on the relationship between the rate of time preference and the real interest rate When utility takes the CES form U1c2 5 c12γ 11 2 γ2 Ur 1c2 5 c2γ Equation 1728 yields the explicit solution c 1t2 5 exp e 1r 2 δ2 γ tf 1729 When r δ consumption will increase over time But this rate of increase will depend on how willing this person is to accept unequal consumption A high value of γ indicates an unwilling ness to substitute consumption over time so the rate of optimal consumption increase will be slower Real interest rate determination The only price in this simple fruit tree economy is the real interest rate r If the rate of increase in consumption is exogenously given as g the real inter est rate must adjust to make such a rate of increase desired by the typical person It must be the case therefore that g 5 1r 2 δ2 γ or r 5 δ 1 γg 1730 Real interest rate paradox Equation 1730 provides the basis for what is termed the real interest rate paradox Over time real per capita consumption grows at about 1 percent per year in the US economy Most empirical studies suggest that γ is about 3 Consequently even if the rate of time preference is zero the real interest rate should be around 3 percent With a more realistic value of δ 5 002 the real interest rate should be about 5 percent But the actual riskfree real interest rate in the United States over the past 75 years has been around 2 percent or less Either there is something wrong with this model or people are much more willing to accept unequal consumption than is generally believed QUERY How should the results of this example be augmented to allow for the possibility that g maybe subject to random fluctuations See also Problem 179 173 PRICING OF RISKY ASSETS The model of intertemporal consumption also provides insights on the pricing of risky assets In this section we briefly summarize a few of the basic results that can be obtained from using this approach We define a risky asset as a oneperiod investment made at period 0 that will yield an uncertain return of xi in period 1 The price of this asset in period 0 is given by pi and we wish to discover the determinants of this price Establishing the price will also establish the gross rate of return on this asset Ri 5 xipi Obviously this gross rate of return is also uncertain Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 609 In order to develop a theory of how this asset is priced we assume that the typical indi vidual chooses how much to invest in the risky asset in order to maximize a twoperiod utility function of the form U1c0 c12 5 U1c02 1 1 1 1 δ E3U1c12 4 1731 subject to c0 5 y0 2 npi and c1 5 y1 1 nxi 1732 where y0 and y1 are exogenous earnings in the two periods and n is the number of units of the risky asset purchased Because of the risks involved in this investing c1 is uncertainthereby explaining the use of the expected value operator in Equation 1731 Substituting the budget constraints into the utility function we get U1c0 c12 5 U1 y0 2 npi2 1 1 1 1 δE3U1y1 1 nxi2 4 1733 Differentiation of this expression with respect to n provides the firstorder condition for optimal investment in this risky asset U1c1 c22 n 5 2piUr 1c02 1 1 1 1 δE3xiUr 1c12 4 5 0 or pi 5 1 1 1 δ E xi Ur 1c12 Ur 1c02 1734 This is the fundamental equation for the pricing of risky assets derived from a consump tionbased model If we let m 5 Ur 1c12 11 1 δ2Ur 1c02 the equation can be simplified as pi 5 E1m xi2 1735 This shows that the price of the risky asset is given by the expected value of the product of two random variables The random term m in the expression serves to discount the risky return xi in the same way that the real interest rate serves to discount a oneperiod certain return of x as x 11 1 r2 For this reason m is sometimes called the stochastic discount factor7 This factor itself is random because the return on the asset affects consumption and the marginal utility thereof in period 1 1731 Riskfree rate of return When the return on an asset is certain Equation 1735 basically repeats what we have shown in Example 171 If we denote the period 1 value of this risk free asset as xf we get pf 5 E1m2xf or Rf 5 xf pf 5 1 E1m2 1736 where Rf is the gross return on the riskfree asset This is also the result given in Equation 1715 with a slightly different notation 1732 Systematic and idiosyncratic risk More generally the price of a risky asset will according to Equation 1735 depend on the product of two random variables To gain further insight on this relationship we can use a general result from mathematical statistics that for any two random variables x and y 7An extensive use of this concept is provided in J Cochrane Asset Pricing Revised Edition Princeton Princeton University Press 2005 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 610 Part 7 Pricing in Input Markets E1x y2 5 E1x2 E1y2 1 Cov 1x y2 That is the expected value of the product of two ran dom variables can be decomposed into two termsthe product of the expected values of the two variables and a term representing the covariance between the variables see also Problem 216 Applying this to Equation 1735 yields pi 5 E1m xi2 5 E1m2 E1xi2 1 Cov 1m xi2 5 E1xi2 Rf 1 Cov 1m xi2 1737 This decomposition provides one of the most important insights from the consump tionbased approach to the pricing of risky assets If the stochastic discount factor m and the random return from a risky asset are uncorrelated that is their covariance is zero then the period 0 price of the risky asset will be simply the expected value of the period 1 uncertain return discounted using the riskfree rate to period 0 Such an absence of cor relation between m and xi would occur if the variation in the return to the risky asset were noise related only to that asset itself and not to any other outcome relevant to the individ uals consumption planning In financial economics such risk is said to be idiosyncratic The conclusion then is that idiosyncratic risk does not affect the pricing of risky assets Possible correlations between m and xi are termed systematic risk To see how such risk affects asset pricing remember that the only random element in the stochastic dis count factor m is the marginal utility of consumption in period 1 that is Ur 1c12 Consider a risky assets whose payoff is positively correlated with good times in the economy as a whole When good times occur the asset will have a favorable payoff and consumption will also be high But when consumption is high the marginal utility of consumption is low Hence m and xi will be negatively correlated and the price of this asset will be lower than that of an otherwise similar risky asset that incorporates only idiosyncratic risk Alter natively an asset that pays off favorably in bad times will induce a positive correlation between m and xi Its price will exceed that of an otherwise similar asset that incorporates only idiosyncratic risk The conclusion then is that the pricing8 of risky assets will reflect individuals desire to mitigate fluctuations in consumption Assets that help to do that will be highly priced whereas those that exacerbate such fluctuations will be priced lower Of course our presentation of the consumptionbased approach to the pricing of risky assets is extremely simplistic The results shown here provide only the most elementary start to the vast subject of financial economics A few additional results are illustrated in Problem 1712 But all of these only scratch the surface of this rapidly expanding field 174 THE FIRMS DEMAND FOR CAPITAL Firms rent machines in accordance with the same principles of profit maximization we derived in Chapter 11 Specifically in a perfectly competitive market the firm will choose to hire that number of machines for which the marginal revenue product is precisely equal to their market rental rate In this section we first investigate the determinants of this mar ket rental rate and implicitly assume all machines are rented from other firms Later in the section we will see that this analysis is little changed when firms actually own the machines they use 8Often this result is stated in terms of expected rates of return E1Ri2 5 E1xi2pi Assets whose returns are negatively correlated with m will have higher expected rates of return than those whose returns are positively correlated with m For a somewhat different approach that reaches the same conclusion see the discussion of the beta approach to portfolio theory in the Extensions to Chapter 7 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 611 1741 Determinants of market rental rates Consider a firm in the business of renting machines to other firms Suppose the firm owns a machine say a car or a backhoe that has a current market price of p How much will the firm charge its clients for the use of the machine The owner of the machine faces two kinds of costs depreciation on the machine and the opportunity cost of having its funds tied up in a machine rather than in an investment earning the current available rate of return If it is assumed that depreciation costs per period are a constant percentage d of the machines market price and that the real interest rate is given by r then the total costs to the machine owner for one period are given by pd 1 pr 5 p 1r 1 d2 1738 If we assume that the machine rental market is perfectly competitive then no longrun profits can be earned by renting machines The workings of the market will ensure that the rental rate per period for the machine v is exactly equal to the costs of the machine owner Hence we have the basic result that v 5 p 1r 1 d2 1739 The competitive rental rate is the sum of forgone interest and depreciation costs the machines owner must pay For example suppose the real interest rate is 5 percent ie 005 and the physical depreciation rate is 15 percent 015 Suppose also that the cur rent market price of the machine is 10000 Then in this simple model the machine would have an annual rental rate of 2000 35 10000 3 1005 1 0152 4 per year 500 of this would represent the opportunity cost of the funds invested in the machine and the remaining 1500 would reflect the physical costs of deterioration 1742 Nondepreciating machines In the hypothetical case of a machine that does not depreciate 1d 5 02 Equation 1739 can be written as v p 5 r 1740 In equilibrium an infinitely longlived nondepreciating machine is equivalent to a per petual bond see the Appendix to this chapter and hence must yield the market rate of return The rental rate as a percentage of the machines price must be equal to r If vp r then everyone would rush out to buy machines because renting out machines would yield more than rates of return elsewhere Similarly if vp r then no one would be in the business of renting out machines because more could be made on alternative investments 1743 Ownership of machines Our analysis so far has assumed that firms rent all of the machines they use Although such rental does take place in the real world for example many firms are in the business of leas ing airplanes trucks freight cars and computers to other firms it is more common for firms to own the machines they use A firm will buy a machine and use it in combination with the labor it hires to produce output The ownership of machines makes the analysis of the demand for capital somewhat more complex than that of the demand for labor How ever by recognizing the important distinction between a stock and a flow we can show that these two demands are quite similar Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 612 Part 7 Pricing in Input Markets A firm uses capital services to produce output These services are a flow magnitude It is the number of machinehours that is relevant to the productive process just as it is laborhours not the number of machines per se Often however the assumption is made that the flow of capital services is proportional to the stock of machines 100 machines if fully employed for 1 hour can deliver 100 machinehours of service there fore these two different concepts are often used synonymously If during a period a firm desires a certain number of machinehours this is usually taken to mean that the firm desires a certain number of machines The firms demand for capital services is also a demand for capital9 A profitmaximizing firm in perfect competition will choose its level of inputs so that the marginal revenue product from an extra unit of any input is equal to its cost This result also holds for the demand for machinehours The cost of capital services is given by the rental rate v in Equation 1739 This cost is borne by the firm whether it rents the machine in the open market or owns the machine itself In the former case it is an explicit cost whereas in the latter case the firm is essentially in two businesses 1 producing output and 2 owning machines and renting them to itself In this second role the firms decisions would be the same as any other machine rental firm because it incurs the same costs The fact of ownership to a first approximation is irrelevant to the determination of cost Hence our prior analysis of capital demand applies to the owners by case as well 1744 Theory of investment If a firm obeys the profitmaximizing rule of Equation 1741 and finds that it desires more capital services than can be provided by its currently existing stock of machinery then it has two choices First it may hire the additional machines that it needs in the rental mar ket This would be formally identical to its decision to hire additional labor Second the firm can buy new machinery to meet its needs This second alternative is the one most often chosen we call the purchase of new equipment by the firm investment Investment demand is an important component of aggregate demand in macroeco nomic theory It is often assumed this demand for plant and equipment ie machines is inversely related to the real rate of interest or what we have called the rate of return Using the analysis developed in this part of the text we can demonstrate the links in this argu ment A decrease in the real interest rate r will ceteris paribus decrease the rental rate on capital Because forgone interest represents an implicit cost for the owner of a machine a decrease in r in effect reduces the price ie the rental rate of capital inputs This decrease in v implies that capital has become a relatively less expensive input this will prompt firms to increase their capital usage 9Firms decisions on how intensively to use a given capital stock during a period are often analyzed as part of the study of business cycles D E F I N I T I O N Demand for capital A profitmaximizing firm that faces a perfectly competitive rental market for capital will hire additional capital input up to the point at which its marginal revenue product 1MRPk2 is equal to the market rental rate v Under perfect competition the rental rate will reflect both depreciation costs and opportunity costs of alternative investments Thus we have MRPk 5 v 5 p1r 1 d2 1741 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 613 175 PRESENT DISCOUNTED VALUE CRITERION Often analysts will take a different approach to the theory of firms physical invest ments by focusing on the present value of the returns such an investment provides This approach arrives at many of the same conclusions we have already seen When a firm buys a machine it is saving itself a stream of net rentals that lasts as long as the machine is being used To decide whether to purchase the machine therefore the firm must com pute the present discounted value10 PDV of this stream of rentals and compare this to the present price of the machine In this way the firm will be taking into account the opportunity costs associated with the interest payments foregone by putting its funds into the piece of equipment If the price of the machine exceeds the PDV of the stream of rental rates this is not a good investment and the firm will decline to make it On the other hand the firm will buy any machine for which its price falls below the PDV of its rental stream and the firm will continue to buy additional machines until no further such gains can be made 1751 A Simple Case As a particularly simple application of this principle assume that a machines rental rate is a constant v in every period and that the machine will last forever With these sim plifying assumptions we may write the present discounted value from machine owner ship as PDV 5 v 11 1 r2 1 v 11 1 r2 2 1 c1 v 11 1 r2 n 1 c 5 v a 1 11 1 r2 1 1 11 1 r2 2 1 c1 1 11 1 r2 n 1 cb 5 v a 1 1 2 1 11 1 r2 2 1b 1742 5 v a1 1 r r 2 1b 5 v 1 r But in equilibrium p 5 PDV so p 5 v 1 r 1743 or v p 5 r 1744 as was already shown in Equation 1740 For this case the present discounted value crite rion gives results identical to those outlined in the previous section 10For a discussion of the logic of the present discounted value process see the Appendix to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 614 Part 7 Pricing in Input Markets 17511 General case Equation 1739 can also be derived for the more general case in which the rental rate on machines is not constant over time and in which there is some depreciation This analysis is most easily carried out by using continuous time Sup pose that the rental rate for a new machine at anytime s is given by vs Assume also that the machine depreciates exponentially at the rate of d11 Therefore the net rental rate and the marginal revenue product of a machine decreases over time as the machine gets older In year s the net rental rate of an old machine bought in a previous year t would be v 1s2e2d1s2t2 1745 because s 2 t is the number of years over which the machine has been decaying For exam ple suppose that a machine is bought new in 2011 Its net rental rate in 2016 then would be the rental rate earned by new machines in 2016 v2016 discounted by the e2 5d to account for the amount of depreciation that has taken place over the 5 years of the machines life If the firm is considering buying the machine when it is new in year t it should discount all of these net rental amounts back to that date Therefore the present value of the net rental in year s discounted back to year t is if r is real the interest rate e2r 1s2t2v 1s2e2d1s2t2 5 e1r1d2v 1s2e21r1d2s 1746 because again 1s 2 t2 years elapse from when the machine is bought until the net rental is received Therefore the present discounted value of a machine bought in year t is the sum integral of these present values This sum should be taken from year t when the machine is bought over all years into the future PDV1t2 5 3 q t e1r1d2tv 1s2e21r1d2sds 1747 Since in equilibrium the price of the machine at year t 3p 1t2 4 will be equal to this present value we have the following fundamental equation p 1t2 5 3 q t e1r1d2tv 1s2e21r1d2sds 1748 This rather formidable equation is simply a more complex version of Equation 1741 and can be used to derive Equation 1739 First rewrite the equation as p 1t2 5 e1r1d2t 3 q t v 1s2e21r1d2sds 1749 Now differentiate with respect to t using the rule for taking the derivative of a product dp 1t2 dt 5 1r 1 d2e1r1d2t 3 q t v 1s2e21r1d2sds 2 e1r1d2tv 1t2e21r1d2t 5 1r 1 d2p 1t2 2 v 1t2 1750 11In this view of depreciation machines are assumed to evaporate at a fixed rate per unit of time This model of decay is in many ways identical to the assumptions of radioactive decay made in physics There are other possible forms that physical depreciation might take this is just one that is mathematically tractable It is important to keep the concept of physical depreciation depreciation that affects a machines productivity distinct from accounting depreciation The latter concept is important only in that the method of accounting depreciation chosen may affect the rate of taxation on the profits from a machine From an economic point of view however the cost of a machine is a sunk cost any choice on how to write off this cost is to some extent arbitrary Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 615 Hence v 1t2 5 1r 1 d2p 1t2 2 dp 1t2 dt 1751 This is precisely the result shown earlier in Equation 1739 except that the term 2dp 1t2dt has been added The economic explanation for the presence of this added term is that it represents the capital gains accruing to the owner of the machine If the machines price can be expected to increase for example the owner may accept somewhat less than 1r 1 d2p for its rental12 On the other hand if the price of the machine is expected to decrease 3dp 1t2dt 04 the owner will require more in rent than is specified in Equation 1739 If the price of the machine is expected to remain constant over time then dp 1t2dt 5 0 and the equations are identical This analysis shows there is a definite relationship between the price of a machine at anytime the stream of future implicit rentals the machine promises and the current rental rate for the machine 12For example rental houses in suburbs with rapidly appreciating house prices will usually rent for less than the landlords actual costs because the landlord also gains from price appreciation EXAMPLE 173 Cutting Down a Tree As an example of the PDV criterion consider the case of a forester who must decide when to cut down a growing tree Suppose the value of the tree at any time t is given by f1t2 where f r 1t2 0 f s 1t2 0 and that l dollars were invested initially as payments to workers who planted the tree Assume also that the continuous market interest rate is given by r When the tree is planted the present discounted value of the tree owners profits is given by PDV 1t2 5 e2rtf1t2 2 l 1752 which is simply the difference between the present value of revenues and present costs The foresters decision then consists of choosing the harvest date t to maximize this value As always this value may be found by differentiation dPDV 1t2 dt 5 e2rtfr 1t2 2 re2rtf1t2 5 0 1753 or dividing both sides by e2rt f r 1t2 2 rf1t2 5 0 1754 Therefore r 5 f r 1t2 f1t2 1755 Two features of this optimal condition are worth noting First observe that the cost of the initial labor input drops out upon differentiation This cost is even in a literal sense a sunk cost that is irrelevant to the profitmaximizing decision Second Equation 1755 can be interpreted as saying the tree should be harvested when the rate of interest is equal to the proportional rate of growth of the tree This result makes intuitive sense If the tree is growing more rapidly than the prevailing interest rate then its owner should leave his or her funds invested in the tree because the tree provides the best return available On the other hand if the tree is growing less rapidly Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 616 Part 7 Pricing in Input Markets than the prevailing interest rate then the tree should be cut and the funds obtained from its sale should be invested elsewhere at the rate r Equation 1755 is only a necessary condition for a maximum By differentiating Equation 1754 again it is easy to see that it is also required that at the chosen value of t f s 1t2 2 rf r 1t2 0 1756 if the firstorder conditions are to represent a true maximum Because we assumed f r 1t2 0 the tree is always growing and f s 1t2 0 the growth slows over time it is clear that this con dition holds A numerical illustration Suppose trees grow according to the equation f 1t2 5 exp 504t 6 1757 This equation always exhibits a positive growth rate 3 f r 1t2 04 and because f r 1t2 f1t2 5 02 t 1758 the trees proportional growth rate diminishes over time If the real interest rate were say 004 then we could solve for the optimal harvesting age as r 5 004 5 f r 1t2 f1t2 5 02 t 1759 or t 5 02 004 5 5 so t 5 25 1760 Up to 25 years of age the volume of wood in the tree is increasing at a rate in excess of 4 percent per year so the optimal decision is to permit the tree to stand But for t 25 the annual growth rate decreases below 4 percent and thus the forester can find better investmentsperhaps plant ing new trees Comparative statics analysis The effect of a change in the real interest rate on tree harvesting can be shown in this example by applying the comparative statics methods introduced in Chapter 2 to the optimality condition given in Equation 1754 dt1r2 dr 5 2 2f1t2 f s 1t2 2 rf r 1t2 0 1761 where the final inequality derives from the second order condition for a maximum Equation 1756 As might have been expected a higher real interest rate will lead the firm to harvest the tree sooner before its growth rate drops too low For example if r 5 005 the optimal harvesting time declines from 25 years to 16 years QUERY Suppose all prices including those of trees were increasing at 10 percent per year How would this change the optimal harvesting results in this problem Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 617 176 NATURAL RESOURCE PRICING Pricing of natural resources has been a concern of economists at least since the time of Thomas Malthus A primary issue has been whether the market system can achieve a desir able allocation of such resources given their ultimately finite and exhaustible nature In this section we look at a simple model of resource pricing to illustrate some of the insights that economic analysis can provide 1761 Profitmaximizing pricing and output Suppose that a firm owns a finite stock of a particular resource Let the stock of the resource at any time be denoted by xt and current production from this stock by qt Hence the stock of this resource evolves according to the differential equation dx 1t2 dt 5 x 1t2 5 2q 1t2 1762 where we use the dot notation to denote a time derivative The stock of this resource is constrained by x 102 5 x and x 1q2 5 0 Extraction of this resource exhibits constant average and marginal cost for changes in output levels but this cost may change over time Hence the firms total costs at any point in time are C1t2 5 c 1t2q 1t2 The firms goal then is to maximize the present discounted value of profits subject to the constraint given in Equation 1762 If we let p 1t2 be the price of the resource at time t then the present value of future profits is given by π 5 3 q 0 3p 1t2q 1t2 2 c 1t2q 1t2 4e2rtdt 1763 where r is the real interest rate assumed to be constant throughout our analysis Setting up the augmented Hamiltonian for this dynamic optimization problem yields H 5 3p 1t2q 1t2 2 c 1t2q 1t2 4e2rt 1 λ 32q 1t2 4 1 x 1t2 d λ dt 1764 The maximum principle applied to this dynamic problem has two firstorder conditions for a maximum Hq 5 3p 1t2 2 c 1t2 4e2rt 2 λ 5 0 Hx 5 dλ dt 5 0 1765 The second of these conditions implies that the shadow price of the resource stock should remain constant over time Because producing a unit of the resource reduces the stock by precisely 1 unit no matter when it is produced any time path along which this shadow price changed would be nonoptimal If we now solve the firstorder condition for λ and differentiate with respect to time we get using the fact that dλdt 5 0 dλ 1t2 dt 5 0 5 λ 5 1 p 2 c 2e2rt 2 r1 p 2 c2e2rt 1766 Dividing by e2rt and rearranging terms provides an equation that explains how the price of the resource must change over time p 5 r1 p 2 c2 1 c 1767 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 618 Part 7 Pricing in Input Markets Notice that the price change has two components The second component shows that price changes must follow any changes in marginal extraction costs The first shows that even if extraction costs do not change there will be an upward trend in prices that reflects the scarcity value of the resource The firm will have an incentive to delay some resource pro duction only if so refraining will yield a return equivalent to the real interest rate Other wise it is better for the firm to sell all its resource assets and invest the funds elsewhere This result first noted13 by Harold Hotelling in the early 1930s can be further simplified by assuming that marginal extraction costs are always zero In this case Equation 1767 reduces to the simple differential equation p 5 rp 1768 whose solution is p 5 p0ert 1769 That is prices increase exponentially at the real rate of interest More generally suppose that marginal costs also follow an exponential trend given by c 1t2 5 c0eγt 1770 where γ may be either positive or negative In this case the solution to the differential Equation 1767 is p 1t2 5 1 p0 2 c02ert 1 c0eγt 1771 This makes it even clearer that the resource price is influenced by two trends an increas ing scarcity rent that reflects the asset value of the resource and the trend in marginal extraction costs 13H Hotelling The Economics of Exhaustible Resources Journal of Political Economy April 1931 13775 EXAMPLE 174 Can Resource Prices Decrease Although Hotellings original observation suggests that natural resource prices should increase at the real rate of interest Equation 1771 makes clear that this conclusion is not unambiguous If marginal extraction costs decrease because of technical advances ie if γ is negative then it is possible that the resource price will decrease The conditions that would lead to decreasing resource prices can be made more explicit by calculating the first and second time derivatives of price in Equation 1771 dp dt 5 r1 p0 2 c02er t 1 γc0eγ t d 2p dt 2 5 r 2 1 p0 2 c02er t 1 γ 2c0eγ t 0 1772 Because the second derivative is always positive we need to only examine the sign of the first derivative at t 5 0 to conclude when prices decrease At this initial date dp dt 5 r1 p0 2 c02 1 γc0 1773 Hence prices will decrease at least initially providing 2γ r p0 2 c0 c0 1774 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 619 1762 Generalizing the model The description of natural resource pricing given here provides only a brief glimpse of this important topic14 Some additional issues that have been considered by economists include social optimality substitution and renewable resources 17621 Social optimality Are the price trends described in Equation 1771 eco nomically efficient That is do they maximize consumer surplus in addition to maximiz ing the firms profits Our previous discussion of optimal consumption over time suggests that the marginal utility of consumption should change in certain prescribed ways if the consumer is to remain on his or her optimal path Because individuals will consume any resource up to the point at which its price is proportional to marginal utility it seems plau sible that the price trends calculated here might be consistent with optimal consumption But a more complete analysis would need to introduce the consumers rate of time prefer ence and his or her willingness to substitute for an increasingly highpriced resource so there is no clearcut answer Rather the optimality of the path indicated by Equation 1766 will depend on the specifics of the situation 17622 Substitution A related issue is how substitute resources should be inte grated into this analysis A relatively simple answer is provided by considering how the initial price 1p02 should be chosen in Equation 1771 If that price is such that the ini tial pricequantity combination is a market equilibrium thenassuming all other finite resource prices follow a similar time trendrelative resource prices will not change and with certain utility functions the pricequantity time paths for all of them may constitute an equilibrium An alternative approach would be to assume that a perfect substitute for the resource will be developed at some date in the future If this new resource is available in perfectly elastic supply then its availability would put a cap on the price or the original resource this also would have implications for p0 see Problem 177 But all of these solu tions to modeling substitutability are special cases To model the situation more generally requires a dynamic general equilibrium model capable of capturing interactions in many markets 17623 Renewable resources A final complication that might be added to the model of resource pricing presented here is the possibility that the resource in question is not finite it can be renewed through natural or economic actions This would be the 14For a sampling of dynamic optimization models applied to natural resource issues see J M Conrad and C W Clark Natural Resource Economics Notes and Problems Cambridge Cambridge University Press 2004 Clearly this condition cannot be met if marginal extraction costs are increasing over time 1γ 02 But if costs are decreasing a period of decreasing real price is possible For example if r 5 005 and γ 5 2002 then prices would decrease provided initial scarcity rents were less than 40 percent of extraction costs Although prices must eventually increase a fairly abundant resource that experienced significant decreases in extraction costs could have a relatively long period of decreasing prices This seems to have been the case for crude oil for example QUERY Is the firm studied in this section a pricetaker How would the analysis differ if the firm were a monopolist See also Problem 1710 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 620 Part 7 Pricing in Input Markets case for timber or fishing grounds where various types of renewal activities are possible The formal consideration of renewable resources requires a modification of the differential equation defining changes in the resource stock which no longer takes the simple form given in Equation 1762 Specification of profitmaximizing price trajectories in such cases can become quite complicated Summary In this chapter we examined several aspects of the theory of capital with particular emphasis on integrating it with the theory of resource allocation over time Some of the results were as follows Capital accumulation represents the sacrifice of present for future consumption The rate of return measures the terms at which this trade can be accomplished The rate of return is established through mechanisms much like those that establish any equilibrium price The equilibrium rate of return will be positive reflecting not only individuals relative preferences for present over future goods but also the positive physical productivity of capital accumulation The rate of return or real interest rate is an import ant element in the overall costs associated with capital ownership It is an important determinant of the market rental rate on capital v Future returns on capital investments must be dis counted at the prevailing real interest rate Use of such present value notions provides an alternative way to approach studying the firms investment decisions Individual wealth accumulation natural resource pric ing and other dynamic problems can be studied using the techniques of optimal control theory Often such models will yield competitivetype results Problems 171 An individual has a fixed wealth W to allocate between con sumption in two periods c1 and c2 The individuals utility function is given by U1c1 c22 and the budget constraint is W 5 c1 1 c2 1 1 r where r is the oneperiod interest rate a Show that in order to maximize utility given this budget constraint the individual should choose c1 and c2 such that the MRS of c1 for c2 is equal to 1 1 r b Show that c2r 0 but that the sign of c1r is ambig uous If c1r is negative what can you conclude about the price elasticity of demand for c2 c How would your conclusions from part b be amended if the individual received income in each period y1 and y2 such that the budget constraint is given by y1 2 c1 1 y2 2 c2 1 1 r 5 0 172 Assume that an individual expects to work for 40 years and then retire with a life expectancy of an additional 20 years Suppose also that the individuals earnings increase at a rate of 3 percent per year and that the interest rate is also 3 per cent the overall price level is constant in this problem What constant fraction of income must the individual save in each working year to be able to finance a level of retirement income equal to 60 percent of earnings in the year just prior to retirement 173 As scotch whiskey ages its value increases One dollar of scotch at year 0 is worth V 1t2 5 exp 52t 2 015t6 dollars at time t If the interest rate is 5 percent after how many years should a person sell scotch in order to maximize the PDV of this sale 174 As in Example 173 suppose trees are produced by applying 1 unit of labor at time 0 The value of the wood contained in a tree is given at any time t by ft If the market wage rate is w and the real interest rate is r what is the PDV of this Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 621 production process and how should t be chosen to maximize this PDV a If the optimal value of t is denoted by t show that the no pure profit condition of perfect competition will necessitate that w 5 e2rtf1t 2 Can you explain the meaning of this expression b A tree sold before t will not be cut down immediately Rather it still will make sense for the new owner to let the tree continue to mature until t Show that the price of a uyearold tree will be weru and that this price will exceed the value of the wood in the tree 3f1u24 for every value of u except u 5 t when these two values are equal c Suppose a landowner has a balanced woodlot with one tree of each age from 0 to t What is the value of this woodlot Hint It is the sum of the values of all trees in the lot d If the value of the woodlot is V show that the instanta neous interest on V that is r V is equal to the profits earned at each instant by the landowner where by prof its we mean the difference between the revenue obtained from selling a fully matured tree 3f1t2 4 and the cost of planting a new one w This result shows there is no pure profit in borrowing to buy a woodlot because one would have to pay in interest at each instant exactly what would be earned from cutting a fully matured tree 175 This problem focuses on the interaction of the corporate prof its tax with firms investment decisions a Suppose contrary to fact that profits were defined for tax purposes as what we have called pure economic prof its How would a tax on such profits affect investment decisions b In fact profits are defined for tax purposes as πr 5 pq 2 wl 2 depreciation where depreciation is determined by governmental and industry guidelines that seek to allocate a machines costs over its useful lifetime If depreciation were equal to actual physical deterioration and if a firm were in long run competitive equilibrium how would a tax on πr affect the firms choice of capital inputs c Given the conditions of part b describe how capi tal usage would be affected by adoption of accelerated depreciation policies which specify depreciation rates in excess of physical deterioration early in a machines life but much lower depreciation rates as the machine ages d Under the conditions of part c how might a decrease in the corporate profits tax affect capital usage 176 A highpressure life insurance salesman was heard to make the following argument At your age a 100000 whole life policy is a much better buy than a similar term policy Under a whole life policy youll have to pay 2000 per year for the first 4 years but nothing more for the rest of your life A term pol icy will cost you 400 per year essentially forever If you live 35 years youll pay only 8000 for the whole life policy but 14000 15 400 352 for the term policy Surely the whole life is a better deal Assuming the salesmans life expectancy assumption is correct how would you evaluate this argument Specifically calculate the present discounted value of the premium costs of the two policies assuming the interest rate is 10 percent 177 Suppose that a perfect substitute for crude oil will be dis covered in 15 years and that the price of this substitute will be the equivalent of an oil price of 125 per barrel Suppose the current marginal extraction cost for oil is 7 per barrel Assume also that the real interest rate is 5 percent and that real extraction costs decrease at a rate of 2 percent annu ally If crude oil prices follow the path described in Equation 1771 what should the current price of crude oil be Does your answer shed any light on actual pricing in the crude oil market Analytical Problems 178 Capital gains taxation Suppose an individual has W dollars to allocate between con sumption this period 1c02 and consumption next period 1c12 and that the interest rate is given by r a Graph the individuals initial equilibrium and indicate the total value of currentperiod savings 1W 2 c02 b Suppose that after the individual makes his or her savings decision by purchasing oneperiod bonds the interest rate decreases to rr How will this alter the individuals budget constraint Show the new utility maximizing position Discuss how the individuals improved position can be interpreted as resulting from a capital gain on his or her initial bond purchases c Suppose the tax authorities wish to impose an income tax based on the value of capital gains If all such gains are valued in terms of c0 as they are accrued show how those gains should be measured Call this value G1 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 622 Part 7 Pricing in Input Markets d Suppose instead that capital gains are measured as they are realizedthat is capital gains are defined to include only that portion of bonds that is cashed in to buy additional c0 Show how these realized gains can be measured Call this amount G2 e Develop a measure of the true increase in utility that results from the decrease in r measured in terms of c0 Call this true capital gain G3 Show that G3 G2 G1 What do you conclude about a tax policy that taxes only realized gains Note This problem is adapted from J Whalley Capital Gains Taxation and Interest Rate Changes National Tax Journal March 1979 8791 179 Precautionary saving and prudence The Query to Example 172 asks how uncertainty about the future might affect a persons savings decisions In this prob lem we explore this question more fully All of our analysis is based on the simple twoperiod model in Example 171 a To simplify matters assume that r 5 δ in Equation 1715 If consumption is certain this implies that ur 1c02 5 ur 1c12 or c0 5 c1 But suppose that consump tion in period 1 will be subject to a zeromean random shock so that c1 5 c p 1 1 x where c p 1 is planned period1 consumption and x is a random variable with an expected value of 0 Describe why in this context utility maximization requires ur 1c02 5 E 3ur 1c12 4 b Use Jensens inequality see Chapters 2 and 7 to show that this person will opt for c p 1 c0 if and only if ur is convexthat is if and only if urrr 0 c Kimball15 suggests using the term prudence to describe a person whose utility function is characterized by urrr 0 Describe why the results from part b show that such a definition is consistent with everyday usage d In Example 172 we showed that real interest rates in the US economy seem too low to reconcile actual con sumption growth rates with evidence on individuals willingness to experience consumption fluctuations If consumption growth rates were uncertain would this explain or exacerbate the paradox 1710 Monopoly and natural resource prices Suppose that a firm is the sole owner of a stock of a natural resource a How should the analysis of the maximization of the discounted profits from selling this resource Equation 1763 be modified to take this fact into account 15M S Kimball Precautionary Savings in the Small and in the Large Econometrica January 1990 5373 b Suppose that the demand for the resource in question had a constant elasticity form q 1t2 5 a 3p1t2 4b How would this change the price dynamics shown in Equation 1767 c How would the answer to Problem 177 be changed if the entire crude oil supply were owned by a single firm 1711 Renewable timber economics The calculations in Problem 174 assume there is no dif ference between the decisions to cut a single tree and to manage a woodlot But managing a woodlot also involves replanting which should be explicitly modeled To do so assume a lot owner is considering planting a single tree at a cost w harvesting the tree at t planting another and so forth forever The discounted stream of profits from this activity is then V 5 2w 1 e2rt3 f 1t2 2 w4 1 e2r2t 3f1t2 2 w4 1 c1 e2rnt3 f 1t2 2 w4 1 a Show that the total value of this planned harvesting activity is given by V 5 f 1t2 2 w e2rt 2 1 2 w b Find the value of t that maximizes V Show that this value solves the equation f r 1t 2 5 rf 1t 2 1 rV 1t 2 c Interpret the results of part b How do they reflect optimal usage of the input time Why is the value of t specified in part b different from that in Example 172 d Suppose tree growth measured in constant dollars fol lows the logistic function f1t2 5 50 11 1 e10201t2 What is the maximum value of the timber available from this tree e If tree growth is characterized by the equation given in part d what is the optimal rotation period if r 5 005 and w 5 0 Does this period produce a maximum sus tainable yield f How would the optimal period change if r decreased to 004 Note The equation derived in part b is known in for estry economics as Faustmanns equation 1712 More on the rate of return on a risky asset Many results from the theory of finance are framed in terms of the expected gross rate of return E 1Ri2 5 E 1xi2pi on a risky asset In this problem you are asked to derive a few of these results Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 623 a Use Equation 1737 to show that E 1Ri2 2 Rf 5 2Rf Cov 1m Ri2 b In mathematical statistics the CauchySchwarz inequality states that for any two random variables x and y 0Cov 1x y2 0 σx σy Use this result to show that 0E 1Ri2 2 Rf 0 Rf σm σRi c Sharpe ratio bound In finance the Sharpe ratio is defined as the excess expected return of a risky asset over the riskfree rate divided by the standard deviation of the return on that risky asset That is Sharpe ratio 5 3E 1Ri2 2 Rf4σRi Use the results of part b to show that the upper bound for the Sharpe ratio is σmE 1m2 Note The ratio of the standard deviation of a random variable to its mean is termed the coefficient of variation or CV This part shows that the upper bound of the Sharpe ratio is given by the CV of the stochastic discount rate d Approximating the CV of m The stochastic discount factor m is random because consumption growth is random Sometimes it is convenient to assume that consumption growth follows a lognormal distributionthat is the logarithm of consumption growth follows a Normal distribution Let the standard deviation of the logarithm consumption growth be given by σln Dc Given these assumptions it can be shown that CV 1m2 5 eγ2 σ2 ln Dc 2 1 Use this result to show that an approximation to the value of this radical can be expressed as CV 1m2 γσln Dc e Equity premium paradox Search the Internet for histori cal data on the average Sharpe ratio for a broad stock mar ket index over the past 50 years Use this result together with the rough estimate that σlnDc 01 to show that parts c and d of this problem imply a very high value for individuals relative risk aversion parameter γ That is the relatively high historical Sharpe ratio for stocks can only be justified by our theory if people are much more risk averse than is usually assumed This is termed the equity premium paradox What do you make of it Behavioral Problem 1713 Hyperbolic discounting The notion that people might be shortsighted was formal ized by David Laibson in Golden Eggs and Hyperbolic Dis counting Quarterly Journal of Economics May 1997 pp 44377 In this paper the author hypothesizes that individ uals maximize an intertemporal utility function of the form utility 5 U1ct2 1 βa τ5T τ51 δτU1ct1τ2 where 0 β 1 and 0 δ 1 The particular time pat tern of these discount factors leads to the possibility of shortsightedness a Laibson suggests hypothetical values of β 5 06 and δ 5 099 Show that for these values the factors by which future consumption is discounted follow a gen eral hyperbolic pattern That is show that the factors decrease significantly for period t 1 1 and then fol low a steady geometric rate of decrease for subsequent periods b Describe intuitively why this pattern of discount rates might lead to shortsighted behavior c More formally calculate the MRS between ct11 and ct12 at time t Compare this to the MRS between ct11 and ct12 at time t 1 1 Explain why with a constant real inter est rate this would imply dynamically inconsistent choices over time Specifically how would the relation ship between optimal ct11 and ct12 differ from these two perspectives d Laibson explains that the pattern described in part c will lead early selves to find ways to constrain future selves and so achieve full utility maximization Explain why such constraints are necessary e Describe a few of the ways in which people seek to con strain their future choices in the real world Suggestions For Further Reading Blaug M Economic Theory in Retrospect rev ed Home wood IL Richard D Irwin 1978 chap 12 Good review of Austrian capital theory and of attempts to concep tualize the capital accumulation process Conrad J M and C W Clark Natural Resource Economics Notes and Problems Cambridge Cambridge University Press 2004 Provides several illustrations of how optimal control theory can be applied to problems in natural resource pricing Dixit A K Optimization in Economic Theory 2nd ed New York Oxford University Press 1990 Extended treatment of optimal control theory in a fairly easyto follow format Dorfman R An Economic Interpretation of Optimal Con trol Theory American Economic Review 59 December 1969 81731 Uses the approach of this chapter to examine optimal capital accu mulation Excellent intuitive introduction Hotelling H The Economics of Exhaustible Resources Journal of Political Economy 39 April 1931 13775 Fundamental work on allocation of natural resources Analyzes both competitive and monopoly cases Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 624 Part 7 Pricing in Input Markets MasColell A M D Whinston and J R Green Microeco nomic Theory New York Oxford University Press 1995 Chapter 20 offers extensive coverage of issues in defining equilibrium over time The discussion of overlapping generations models is especially useful Ramsey F P A Mathematical Theory of Saving Economic Journal 38 December 1928 54259 One of the first uses of the calculus of variations to solve economic problems Solow R M Capital Theory and the Rate of Return Amster dam NorthHolland 1964 Lectures on the nature of capital Very readable Sydsaeter K A Strom and P Berck Economists Mathemati cal Manual 3rd ed Berlin SpringerVerlag 2000 Chapter 27 provides a variety of formulas that are valuable for finance and growth theory Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 APPENDIX TO CHAPTER SEVENTEEN The Mathematics of Compound Interest The purpose of this appendix is to gather some simple results concerning the mathematics of compound interest These results have applications in a wide variety of economic prob lems that range from macroeconomic policy to the optimal way of raising Christmas trees We assume there is a current prevailing market interest rate of i per periodsay of 1 year This interest rate is assumed to be both certain and constant over all future periods1 If 1 is invested at this rate i and if the interest is then compounded ie future interest is paid on post interest earned then at the end of one period 1 will be 1 3 11 1 i2 at the end of two periods 1 will be 1 3 11 1 i2 3 11 1 i2 5 1 3 11 1 i2 2 and at the end of n periods 1 will be 1 3 11 1 i2 n Similarly N grows like N 3 11 1 i2 n 17A1 PRESENT DISCOUNTED VALUE The present value of 1 payable one period from now is 1 1 1 i This is simply the amount an individual would be willing to pay now for the promise of 1 at the end of one period Similarly the present value of 1 payable n periods from now is 1 11 1 i2 n and the present value of N payable n periods from now is N 11 1 i2 n 1The assumption of a constant i is obviously unrealistic Because problems introduced by considering an interest rate that varies from period to period greatly complicate the notation without adding a commensurate degree of conceptual knowledge such an analysis is not undertaken here In many cases the generalization to a varying interest rate is merely a trivial application of the notion that any multiperiod interest rate can be regarded as resulting from compounding several singleperiod rates If we let rij be the interest rate prevailing between periods i and j where i j then 1 1 rij 5 11 1 ri i112 3 11 1 ri11 i122 3 c3 11 1 rj21 j2 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 626 Part 7 Pricing in Input Markets The present discounted value of a stream of payments N0 N1 N2 Nn where the subscripts indicate the period in which the payment is to be made is PDV 5 N0 1 N1 11 1 i2 1 N2 11 1 i2 2 1 c1 Nn 11 1 i2 n 17A1 The PDV is the amount an individual would be willing to pay in return for a promise to receive the stream N0 N1 N2 Nn It represents the amount that would have to be invested now if one wished to duplicate the payment stream 17A11 Annuities and perpetuities An annuity is a promise to pay N in each period for n periods starting next period The PDV of such a contract is PDV 5 N 1 1 i 1 N 11 1 i2 2 1 c1 N 11 1 i2 n 17A2 Let δ 5 1 11 1 i2 then PDV 5 N1δ 1 δ2 1 c1 δn2 5 Nδ11 1 δ 1 δ2 1 c1 δn212 5 Nδ a1 2 δn 1 2 δ b 17A3 Observe that lim nSq δn 5 0 Therefore for an annuity of infinite duration PDV of infinite annuity 5 lim nSqPDV 5 Nδ a 1 1 2 δb 17A4 by the definition of δ Nδ a 1 1 2 δb 5 N a 1 1 1 ib a 1 1 2 1 11 1 i2 b 5 N a 1 1 1 ib a1 1 i i b 5 N i 17A5 This case of an infiniteperiod annuity is sometimes called a perpetuity or a consol The formula simply says that the amount that must be invested if one is to obtain N per period forever is simply Ni because this amount of money would earn N in interest each period 1i Ni 5 N2 17A12 The special case of a bond An nperiod bond is a promise to pay N each period starting next period for n periods It also promises to return the principal face value of the bond at the end of n periods If the principal value of the bond is P usually 1000 in the US bond market then the present discounted value of such a promise is PDV 5 N 1 1 i 1 N 11 1 i2 2 1 c1 N 11 1 i2 n 1 P 11 1 i2 n 17A6 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 627 Again let δ 5 1 11 1 i2 then PDV 5 Nδ 1 Nδ2 1 c1 1N 1 P2δn 17A7 Equation 17A7 can be looked at in another way Suppose we knew the price say B at which the bond is currently trading Then we could ask what value of i gives the bond a PDV equal to B To find this i we set B 5 PDV 5 Nδ 1 Nδ2 1 c1 1N 1 P2δn 17A8 Because B N and P are known we can solve this equation for δ and hence for i2 The i that solves the equation is called the yield on the bond and is the best measure of the return actually available from the bond The yield of a bond represents the return available both from direct interest payments and from any price differential between the initial price B and the maturity price P Notice that as i increases PDV decreases This is a precise way of formulating the well known concept that bond prices PDVs and interest rates yields are inversely correlated 17A2 CONTINUOUS TIME Thus far our approach has dealt with discrete timethe analysis has been divided into periods Often it is more convenient to deal with continuous time In such a case the inter est on an investment is compounded instantaneously and growth over time is smooth This facilitates the analysis of maximization problems because exponential functions are more easily differentiated Many financial intermediaries for example savings banks have adopted nearly continuous interest formulas in recent years Suppose that i is given as the nominal interest rate per year but that half this nominal rate is compounded every 6 months Then at the end of 1 year the investment of 1 would have grown to 1 3 a1 1 i 2b 2 17A9 Observe that this is superior to investing for 1 year at the simple rate i because interest has been paid on interest that is a1 1 i 2b 2 11 1 i2 17A10 Consider the limit of this process For the nominal rate of i per period consider the amount that would be realized if i were in fact compounded n times during the period Letting n S q we have lim nSq 3 a1 1 i nb n 17A11 This limit exists and is simply ei where e is the base of natural logarithms the value of e is approximately 272 It is important to note that ei 11 1 i2it is much better to have continuous compounding over the period than to have simple interest 2Because this equation is an nthdegree polynomial there are in reality n solutions roots Only one of these solutions is the relevant one reported in bond tables or on calculators The other solutions are either imaginary or unreasonable In the present example there is only one real solution Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 628 Part 7 Pricing in Input Markets TABLE 17A1 EFFECTIVE ANNUAL INTEREST RATES FOR SELECTED CONTINUOUSLY COMPOUNDED RATES Continuously Compounded Rate Effective Annual Rate 30 305 40 408 50 513 55 565 60 618 65 672 70 725 80 833 90 942 100 1052 We can ask what continuous rate r yields the same amount at the end of one period as the simple rate i We are looking for the value of r that solves the equation er 5 11 1 i2 17A12 Hence r 5 ln11 1 i2 17A13 Using this formula it is a simple matter to translate from discrete interest rates into continuous ones If i is measured as a decimal yearly rate then r is a yearly continuous rate Table 17A1 shows the effective annual interest rate i associated with selected interest rates r that are continuously compounded3 Tables similar to 17A1 often appear in the windows of banks advertising the true yields on their accounts 17A21 Continuous growth One dollar invested at a continuous interest rate of r will become V 5 1 erT 17A14 after T years This growth formula is a convenient one to work with For example it is easy to show that the instantaneous relative rate of change in V is as would be expected simply given by r relative rate of change 5 dVdt V 5 rert ert 5 r 17A15 Continuous interest rates also are convenient for calculating present discounted values Suppose we wished to calculate the PDV of 1 to be paid T years from now This would be given by4 1 erT 5 1 3 e2rT 17A16 3To compute the figures in Table 17A1 interest rates are used in decimal rather than percent form that is a 5 percent interest rate is recorded as 005 for use in Equation 17A12 4In physics this formula occurs as an example of radioactive decay If 1 unit of a substance decays continuously at the rate δ then after T periods e2δT units will remain This amount never exactly reaches zero no matter how large T is Depreciation can be treated the same way in capital theory Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 17 Capital and Time 629 The logic of this calculation is exactly the same as that used in the discrete time analysis of this appendix Future dollars are worth less than present dollars 17A22 Payment streams One interesting application of continuous discounting occurs in calculating the PDV of 1 per period paid in small installments at each instant of time from today time 0 until time T Because there would be an infinite number of payments the mathematical tool of integra tion must be used to compute this result PDV 5 3 T 0 e2rtdt 17A17 What this expression means is that we are adding all the discounted dollars over the time period 0 to T The value of this definite integral is given by PDV 5 2e2rt r T 0 5 2e2rT r 1 1 r 17A18 As T approaches infinity this value becomes PDV 5 1 r 17A19 as was the case for the infinitely long annuity considered in the discrete case Continuous discounting is particularly convenient for calculating the PDV of an arbi trary stream of payments over time Suppose that ft records the number of dollars to be paid during period t Then the PDV of the payment at time t is e2rtf1t2 17A20 and the PDV of the entire stream from the present time year 0 until year T is given by PDV 5 3 T 0 f1t2e2rtdt 17A21 Often economic agents may seek to maximize an expression such as that given in Equa tion 17A21 Use of continuous time makes the analysis of such choices straightforward because standard calculus methods of maximization can be used 17A23 Duration The use of continuous time can also clarify a number of otherwise rather difficult financial concepts For example suppose we wished to know how long on average it takes for an individual to receive a payment from a given payment stream ft The present value of the stream is given by V 5 3 T 0 f1t2e2rtdt 17A22 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 630 Part 7 Pricing in Input Markets Differentiation of this value by the discount factor e2r yields V e2r 5 3 T 0 tf1t2e2r1t212dt 17A23 and the elasticity of this change is given by e 5 V e2r e2r V 5 e T 0 tf 1t2e2rtdt V 17A24 Hence the elasticity of the present value of this payment stream with respect to the annual discount factor which is similar to say the elasticity of bond prices with respect to changes in interest rates is given by the ratio of the present value of a timeweighted stream of payments to an unweighted stream Conceptually then this elasticity represents the average time an individual must wait to receive the typical payment In the financial press this concept is termed the duration of the payment stream This is an important measure of the volatility of the present value of such a stream with respect to interest rate changes5 5As an example a duration of 8 years would mean that the mean length of time that the individual must wait for the typical payment is 8 years It also means that the elasticity of the value of this stream with respect to the discount factor is 80 Because the elasticity of the discount factor itself with respect to the interest rate is simply 2r the elasticity of the value of the stream with respect to this interest rate is 28r If r 5 005 for example then the elasticity of the present value of this stream with respect to r is 2040 A more common way of stating this is that each percentage point change in the interest rate will change the price of the bond by D percent In this case each 01 change in r would change the bond price by 8 percent Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 631 Market Failure Chapter 18 Asymmetric Information Chapter 19 Externalities and Public Goods In this part we look more closely at some of the reasons why markets may perform poorly In allocating resources We will also examine some of the ways in which such market failures might be mitigated Chapter 18 focuses on situations where some market participants are better informed than oth ers In such cases of asymmetric information establishing efficient contracts between these parties can be quite complicated and may involve a variety of strategic choices We will see that in many situations the firstbest fully informed solution is not attainable Therefore secondbest solutions that may involve some efficiency losses must be considered Externalities are the principal topic of Chapter 19 The first part of the chapter is concerned with situations in which the actions of one economic actor directly affect the wellbeing of another actor We show that unless these costs or benefits can be internalized into the decision process resources will be misallocated In the second part of the chapter we turn to a particular type of externality that posed by public goods goods that are both nonexclusive and nonrival We show that markets will often underallocate resources to such goods so other ways of financing such as compulsory taxation should be considered Chapter 19 concludes with an examination of how voting may affect this process PART EIGHT Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 633 CHAPTER EIGHTEEN Asymmetric Information Markets may not be fully efficient when one side has information that the other side does not asymmetric information Contracts with more complex terms than simple perunit prices may be used to help solve problems raised by such asymmetric information The two important classes of asymmetric information problems studied in this chapter include moral hazard problems in which one partys actions during the term of the contract are unobservable to the other and adverse selection problems in which a party obtains asymmetric information about market conditions before signing the contract Carefully designed contracts may reduce such problems by providing incentives to reveal ones information and take appropriate actions But these contracts seldom eliminate the inef ficiencies entirely Surprisingly unbridled competition may worsen private information problems although a carefully designed auction can harness competitive forces to the auc tioneers advantage 181 COMPLEX CONTRACTS AS A RESPONSE TO ASYMMETRIC INFORMATION So far the transactions we have studied have involved simple contracts We assumed that firms bought inputs from suppliers at constant perunit prices and likewise sold output to consumers at constant perunit prices Many realworld transactions involve much more complicated contracts Rather than an hourly wage a corporate executives com pensation usually involves complex features such as the granting of stock stock options and bonuses Insurance policies may cap the insurers liability and may require the cus tomer to bear costs in the form of deductibles and copayments In this chapter we will show that such complex contracts may arise as a way for transacting parties to deal with the problem of asymmetric information 1811 Asymmetric information Transactions can involve a considerable amount of uncertainty The value of a snow shovel will depend on how much snow falls during the winter season The value of a hybrid car will depend on how much gasoline prices increase in the future Uncertainty need not lead to inefficiency when both sides of a transaction have the same limited knowledge Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 634 Part 8 Market Failure concerning the future but it can lead to inefficiency when one side has better information The side with better information is said to have private information or equivalently asym metric information There are several sources of asymmetric information Parties will often have inside information concerning themselves that the other side does not have Consider the case of health insurance A customer seeking insurance will often have private information about his or her own health status and family medical history that the insurance com pany does not Consumers in good health may not bother to purchase health insurance at the prevailing rates A consumer in poor health would have higher demand for insur ance wishing to shift the burden of large anticipated medical expenses to the insurer A medical examination may help the insurer learn about a customers health status but examinations are costly and may not reveal all of the customers private health infor mation The customer will be reluctant to report family medical history and genetic disease honestly if the insurer might use this information to deny coverage or increase premiums Other sources of asymmetric information arise when what is being bought is an agents service The buyer may not always be able to monitor how hard and well the agent is work ing The agent may have better information about the requirements of the project because of his or her expertise which is the reason the agent was hired in the first place For exam ple a repairer called to fix a kitchen appliance will know more about the true severity of the appliances mechanical problems than does the homeowner Asymmetric information can lead to inefficiencies Insurance companies may offer less insurance and charge higher premiums than if they could observe the health of poten tial clients and could require customers to obey strict health regimens The whole market may unravel as consumers who expect their health expenditures to be lower than the aver age insured consumers withdraw from the market in successive stages leaving only the few worst health risks as consumers With appliance repair the repairer may pad his or her bill by replacing parts that still function and may take longer than neededa waste of resources 1812 The value of contracts Contractual provisions can be added in order to circumvent some of these inefficiencies An insurance company can offer lower health insurance premiums to customers who sub mit to medical exams or who are willing to bear the cost of some fraction of their own medical services Lowerrisk consumers may be more willing than highrisk consumers to submit to medical exams and to bear a fraction of their medical expenses A homeowner may buy a service contract that stipulates a fixed fee for keeping the appliance running rather than a payment for each service call and part needed in the event the appliance breaks down Although contracts may help reduce the inefficiencies associated with asymmetric information rarely do they eliminate the inefficiencies altogether In the health insur ance example having some consumers undertake a medical exam requires the expendi ture of real resources Requiring lowrisk consumers to bear some of their own medical expenditures means that they are not fully insured which is a social loss to the extent that a riskneutral insurance company would be a more efficient risk bearer than a risk averse consumer A fixedfee contract to maintain an appliance may lead the repairer to supply too little effort overlooking potential problems in the hope that nothing breaks until after the service contract expires and so then the problems become the homeowners Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 635 182 PRINCIPALAGENT MODEL Models of asymmetric information can quickly become quite complicated and thus before considering a fullblown market model with many suppliers and demanders we will devote much of our analysis to a simpler modelcalled a principalagent modelin which there is only one party on each side of the market The party who proposes the con tract is called the principal The party who decides whether or not to accept the contract and then performs under the terms of the contract if accepted is called the agent The agent is typically the party with the asymmetric information We will use she for the prin cipal and he for the agent to facilitate the exposition 1821 Two leading models Two models of asymmetric information are studied most often In the first model the agents actions taken during the term of the contract affect the principal but the princi pal does not observe these actions directly The principal may observe outcomes that are correlated with the agents actions but not the actions themselves This first model is called a hiddenaction model For historical reasons stemming from the insurance context the hiddenaction model is also called a moral hazard model In the second model the agent has private information about the state of the world before signing the contract with the principal The agents private information is called his type consistent with our terminology from games of private information studied in Chapter 8 The second model is thus called a hiddentype model For historical reasons stemming from its application in the insurance context which we discuss later the hiddentype model is also called an adverse selection model As indicated by Table 181 the hiddentype and hiddenaction models cover a wide variety of applications Note that the same party might be a principal in one setting and an agent in another For example a companys CEO is the principal in dealings with the com panys employees but is the agent of the firms shareholders We will study several of the applications from Table 181 in detail throughout the remainder of this chapter 1822 First second and third best In a fullinformation environment the principal could propose a contract to the agent that maximizes their joint surplus and captures all of this surplus for herself leaving the agent with just enough surplus to make him indifferent between signing the contract or not This TABLE 181 APPLICATIONS OF THE PRINCIPALAGENT MODEL Agents Private Information Principal Agent Hidden Type Hidden Action Shareholders Manager Managerial skill Effort executive decisions Manager Employee Job skill Effort Homeowner Appliance repairer Skill severity of appliance malfunction Effort unnecessary repairs Student Tutor Subject knowledge Preparation patience Monopoly Customer Value for good Care to avoid breakage Health insurer Insurance purchaser Preexisting condition Risky activity Parent Child Moral fiber Delinquency Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 636 Part 8 Market Failure outcome is called the first best and the contract implementing this outcome is called the first best contract The first best is a theoretical benchmark that is unlikely to be achieved in practice because the principal is rarely fully informed The outcome that maximizes the principals sur plus subject to the constraint that the principal is less well informed than the agent is called the second best and the contract that implements this outcome is called the secondbest contract Adding further constraints to the principals problem besides the informational constraint for example restricting contracts to some simple form such as constant perunit pricesleads to the third best the fourth best and so on depending on how many constraints are added Since this chapter is in the part of the book that examines market failures we will be interested in determining how important a market failure is asymmetric information Comparing the first to the second best will allow us to quantify the reduction in total wel fare due to asymmetric information Also illuminating is a comparison of the second and third best This comparison will indicate how surpluses are affected when moving from simple contracts in the third best to potentially quite sophisticated contracts in the second best Of course the principals sur plus cannot decrease when she has access to a wider range of contracts with which to max imize her surplus However total welfarethe sum of the principals and agents surplus in a principalagent modelmay decrease Figure 181 suggests why In the example in The total welfare is the area of the circle pie the principals surplus is the area of the shaded region In panel a the complex contract increases total welfare and the principals surplus along with it because she obtains a constant share In panel b the principal offers the complex contracteven though this reduces total welfarebecause the complex contract allows her to appropriate a larger share FIGURE 181 The Contracting Pie a Complex contract increases parties joint surplus b Complex contract increases principals share of surplus Complex secondbest contract Simple thirdbest contract Complex secondbest contract Simple thirdbest contract Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 637 panel a of the figure the complex contract increases the total welfare pie that is divided between the principal and the agent The principal likes the complex contract because it allows her to obtain a roughly constant share of a bigger pie In panel b the principal likes the complex contract even though the total welfare pie is smaller with it than with the simple contract The complex contract allows her to appropriate a larger slice at the expense of reducing the pies total size The different cases in panels a and b will come up in the applications analyzed in subsequent sections 183 HIDDEN ACTIONS The first of the two important models of asymmetric information is the hiddenaction model also sometimes called the moral hazard model in insurance and other contexts The principal would like the agent to take an action that maximizes their joint surplus and given that the principal makes the contract offer she would like to appropriate most of the surplus for herself In the application to the ownermanager relationship that we will study the owner would like the manager whom she hires to show up during business hours and work diligently In the application to the accident insurance the insurance company would like the insured individual to avoid accidents The agents actions may be unobserv able to the principal Observing the action may require the principal to monitor the agent at all times and such monitoring may be prohibitively expensive If the agents action is unobservable then he will prefer to shirk choosing an action to suit himself rather than the principal In the ownermanager application shirking might mean showing up late for work and slacking off while on the job in the insurance example shirking might mean taking more risk than the insurance company would like Although contracts cannot prevent shirking directly by tying the agents compensation to his actionbecause his action is unobservablecontracts can mitigate shirking by tying compensation to observable outcomes In the ownermanager application the relevant observable outcome might be the firms profit The owner may be able to induce the man ager to work hard by tying the managers pay to the firms profit which depends on the managers effort The insurance company may be able to induce the individual to take care by having him bear some of the cost of any accident Often the principal is more concerned with the observable outcome than with the agents unobservable action anyway so it seems the principal should do just as well by condition ing the contract on outcomes as on actions The problem is that the outcome may depend in part on random factors outside of the agents control In the ownermanager application the firms profit may depend on consumer demand which may depend on unpredictable economic conditions In the insurance application whether an accident occurs depends in part on the care taken by the individual but also on a host of other factors including other peoples actions and acts of nature Tying the agents compensation to partially random out comes exposes him to risk If the agent is risk averse then this exposure causes disutility and requires the payment of a risk premium before he will accept the contract see Chapter 7 In many applications the principal is less risk averse and thus is a more efficient risk bearer than the agent In the ownermanager application the owner might be one of many shareholders who each hold only a small share of the firm in a diversified portfolio In the insurance appli cation the company may insure a large number of agents whose accidents are uncorrelated and thus face little aggregate risk If there were no issue of incentives then the agents com pensation should be independent of risky outcomes completely insuring him against risk and shifting the risk to the efficient bearer the principal The secondbest contract strikes the optimal balance between incentives and insurance but it does not provide as strong incen tives or as full insurance as the firstbest contract Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 638 Part 8 Market Failure In the following sections we will study two specific applications of the hiddenaction model First we will study employment contracts signed between a firms owners and a manager who runs the firm on behalf of the owners Second we will study contracts offered by an insurance company to insure an individual against accident risk 184 OWNERMANAGER RELATIONSHIP Modern corporations may be owned by millions of dispersed shareholders who each own a small percentage of the corporations stock The shareholderswho may have lit tle expertise in the line of business and who may own too little of the firm individually to devote much attention to itdelegate the operation of the firm to a managerial team consisting of the chief executive officer CEO and other officers We will simplify the setting and suppose that the firm has one representative owner and one manager The owner who plays the role of the principal in the model offers a contract to the manager who plays the role of the agent The manager decides whether to accept the employ ment contract and if so how much effort e 0 to exert An increase in e increases the firms gross profit not including payments to the manager but is personally costly to the manager1 Assume the firms gross profit πg takes the following simple form πg 5 e 1 ε 181 Gross profit is increasing in the managers effort e and also depends on a random variable ε which represents demand cost and other economic factors outside of the managers con trol Assume that ε is normally distributed with mean 0 and variance σ2 The managers personal disutility or cost of undertaking effort ce is increasing 3cr 1e2 04 and convex 3cs 1e2 04 Let s be the salarywhich may depend on effort andor gross profit depending on what the owner can observeoffered as part of the contract between the owner and man ager Because the owner represents individual shareholders who each own a small share of the firm as part of a diversified portfolio we will assume that she is risk neutral Letting net profit πn equal gross profit minus payments to the manager πn 5 πg 2 s 182 the riskneutral owner wants to maximize the expected value of her net profit E1πn2 5 E1e 1 ε 2 s2 5 e 2 E1s2 183 To introduce a tradeoff between incentives and risk we will assume the manager is risk averse in particular we assume the manager has a utility function with respect to salary whose constant absolute risk aversion parameter is A 0 We can use the results from Example 73 to show that his expected utility is E1U2 5 E1s2 2 A 2 Var1s2 2 c 1e2 184 We will examine the optimal salary contract that induces the manager to take appro priate effort e under different informational assumptions We will study the firstbest con tract when the owner can observe e perfectly and then the secondbest contract when there is asymmetric information about e 1Besides effort e could represent distasteful decisions such as firing unproductive workers Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 639 1841 First best fullinformation case With full information it is relatively easy to design an optimal salary contract The owner can pay the manager a fixed salary s if he exerts the firstbest level of effort e which we will compute shortly and nothing otherwise The managers expected utility from the contract can be found by substituting the expected value 3E1s2 5 s4 and variance 3Var 1s2 5 04 of the fixed salary as well as the effort e into Equation 184 For the manager to accept the contract this expected utility must exceed what he would obtain from his nextbest job offer E1U2 5 s 2 c 1e2 0 185 where we have assumed for simplicity that he obtains 0 from his nextbest job offer In principalagent models a condition like Equation 185 is called a participation constraint ensuring the agents participation in the contract The owner optimally pays the lowest salary satisfying Equation 185 s 5 c 1e2 The owners net profit then is E1πn2 5 e 2 E1s2 5 e 2 c 1e2 186 which is maximized for e satisfying the firstorder condition cr 1e2 5 1 187 At an optimum the marginal cost of effort cr 1e2 equals the marginal benefit 1 1842 Second best hiddenaction case If the owner can observe the managers effort then she can implement the first best by sim ply ordering the manager to exert the firstbest effort level If she cannot observe effort the contract cannot be conditioned on e However she can still induce the manager to exert some effort if the managers salary depends on the firms gross profit The manager is given performance pay The more the firm earns the more the manager is paid Suppose the owner offers a salary to the manager that is linear in gross profit s 1πg2 5 a 1 bπg 188 where a is the fixed component of salary and b measures the slope sometimes called the power of the incentive scheme If b 5 0 then the salary is constant and as we saw provides no effort incentives As b increases toward 1 the incentive scheme provides increasingly powerful incentives The fixed component a can be thought of as the managers base salary and b as the incentive pay in the form of stocks stock options and performance bonuses The ownermanager relationship can be viewed as a threestage game In the first stage the owner sets the salary which amounts to choosing a and b In the second stage the manager decides whether or not to accept the contract In the third stage the manager decides how much effort to exert conditional on accepting the contract We will solve for the subgameperfect equilibrium of this game by using backward induction starting with the managers choice of e in the last stage and taking as given that the manager was offered salary scheme a 1 bπg and accepted it Substituting from Equation 188 into Equation 184 the managers expected utility from the linear salary is E1a 1 bπg2 2 A 2 Var1a 1 bπg2 2 c 1e2 189 Reviewing a few facts about expectations and variances of a random variable will help us simplify Equation 189 First note that E1a 1 bπg2 5 E1a 1 be 1 bε2 5 a 1 be 1 bE1ε2 5 a 1 be 1810 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 640 Part 8 Market Failure see Equation 2165 Furthermore Var1a 1 bπg2 5 Var 1a 1 be 1 bε2 5 b2Var 1ε2 5 b2σ2 1811 see Equation 2173 Therefore Equation 189 reduces to managers expected utility 5 a 1 be 2 Ab2σ2 2 2 c 1e2 1812 The firstorder condition for the e maximizing the managers expected utility yields cr 1e2 5 b 1813 Because ce is convex the marginal cost of effort cr 1e2 is increasing in e Hence as shown in Figure 182 the higher is the power b of the incentive scheme the more effort e the manager exerts The managers effort depends only on the slope b and not on the fixed part a of his incentive scheme Now fold the game back to the managers secondstage choice of whether to accept the contract The manager accepts the contract if his expected utility in Equation 1812 is nonnegative or upon rearranging if a c 1e2 1 Ab2σ2 2 2 be 1814 The fixed part of the salary a must be high enough for the manager to accept the contract Next fold the game back to the owners firststage choice of the parameters a and b of the salary scheme The owners objective is to maximize her expected surplus which upon substituting from Equation 1810 into 183 is owners surplus 5 e 11 2 b2 2 a 1815 Because the managers marginal cost of effort cr 1e2 slopes upward an increase in the power of the incen tive scheme from b1 to b2 induces the manager to increase his effort from e1 to e2 ce e1 e2 e b2 b1 FIGURE 182 Managers Effort Responds to Increased Incentives Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 641 subject to two constraints The first constraint Equation 1814 is that the manager must accept the contract in the second stage As mentioned in the previous section this is called a participation constraint Although Equation 1814 is written as an inequality it is clear that the owner will keep lowering a until the condition holds with equality since a does not affect the managers effort and since the owner does not want to pay the manager any more than necessary to induce him to accept the contract The second constraint Equation 1813 is that the manager will choose e to suit himself rather than the owner who cannot observe e This is called the incentive compatibility constraint Substituting the constraints into Equation 1815 allows us to express the owners surplus as a function only of the managers effort e 2 c 1e2 2 Aσ2 3cr 1e2 4 2 2 1816 The secondbest effort e satisfies the firstorder condition cr 1e2 5 1 1 1 Aσ2cs 1e2 1817 The righthand side of Equation 1817 is also equal to the power b of the incentive scheme in the second best since cr 1e2 5 b by Equation 1813 Comparing Equation 1817 to 187 shows cr 1e2 1 5 cr 1e2 But the convexity of c 1e2 then implies e e The presence of asymmetric information leads to lower equilib rium effort If the owner cannot specify e in a contract then she can induce effort only by tying the managers pay to firm profit however doing so introduces variation into his pay for which the riskaverse manager must be paid a risk premium This risk premium the third term in Equation 1816 adds to the owners cost of inducing effort If effort incentives were not an issue then the riskneutral owner would be betteroff bearing all risk herself and insuring the riskaverse manager against any fluctuations in profit by offering a constant salary as we saw in the firstbest problem Yet if effort is unobservable then a constant salary will not provide any incentive to exert effort The secondbest contract trades off the owners desire to induce high effort which would come from setting b close to 1 against her desire to insure the riskaverse manager against varia tions in his salary which would come from setting b close to 0 Hence the resulting value of b falls somewhere between 0 and 1 In short the fundamental tradeoff in the ownermanager relationship is between incentives and insurance The more risk averse is the manager ie the higher is A the more important is insurance relative to incentives The owner insures the manager by reducing the dependence of his salary on fluctuating profit reducing b and therefore e For the same reason the more that profit varies owing to factors outside of the managers control ie the higher is σ2 the lower is b and e2 2A study has confirmed that CEOs and other top executives receive more powerful incentives if they work for firms with less volatile stock prices See R Aggarwal and A Samwick The Other Side of the Tradeoff The Impact of Risk on Executive Compensation Journal of Political Economy 107 1999 65105 EXAMPLE 181 OwnerManager Relationship As a numerical example of some of these ideas suppose the managers cost of effort has the simple form c 1e2 5 e22 and suppose σ2 5 1 First best The firstbest level of effort satisfies cr 1e2 5 e 5 1 A firstbest contract specifies that the manager exerts firstbest effort e 5 1 in return for a fixed salary of 12 which leaves Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 642 Part 8 Market Failure the manager indifferent between accepting the contract and pursuing his nextbest available job which we have assumed provides him with utility 0 The owners net profit equals 12 Second best The secondbest contract depends on the degree of the managers risk aversion measured by A Suppose first that A 5 13 Then by Equation 1817 the secondbest level of effort is e 5 12 and b 5 12 as well To compute the fixed part a of the managers salary recall that Equation 1814 holds as an equality in the second best and substitute the variables computed so far yielding a 5 0 The manager receives no fixed pay but does receive incentive pay equal to 50 cents for every dollar of gross profit Substituting the variables computed into Equation 1815 we see that the owners expected net profit is 14 Now suppose A 5 2 so that the manager is more risk averse The secondbest effort decreases to e 5 13 and b decreases to 13 as well The fixed part of the managers salary increases to a 5 118 The owners expected net profit decreases to 16 Empirical evidence In an influential study of performance pay Jensen and Murphy estimated that b 5 0003 for top executives in a sample of large US firms which is orders of magnitude smaller than the values of b we just computed4 The fact that realworld incentive schemes are less sensitive to performance than theory would indicate is a puzzle for future research to unravel QUERY How would the analysis change if the owners did not perfectly observe gross profit but instead depended on the manager for a selfreport Could this explain the puzzle that top execu tives incentives are unexpectedly lowpowered 1843 Comparison to standard model of the firm It is natural to ask how the results with hidden information about the managers action com pare to the standard model of a perfectly competitive market with no asymmetric informa tion First the presence of hidden information raises a possibility of shirking and inefficiency that is completely absent in the standard model The manager does not exert as much effort as he would if effort were observable Even if the owner does as well as she can in the pres ence of asymmetric information to provide incentives for effort she must balance the bene fits of incentives against the cost of exposing the manager to too much risk Second although the manager can be regarded as an input like any other capital labor materials and so forth in the standard model he becomes a unique sort of input when his actions are hidden information It is not enough to pay a fixed unit price for this input as a firm would the rental rate for capital or the market price for materials How productive the manager is depends on how his compensation is structured The same can be said for any sort of labor input Workers may shirk on the job unless monitored or given incentives not to shirk 185 MORAL HAZARD IN INSURANCE Another important context in which hidden actions lead to inefficiencies is the market for insurance Individuals can take a variety of actions that influence the probability that a risky event will occur Car owners can install alarms to deter theft consumers can eat healthier foods to prevent illness In these activities utilitymaximizing individuals will 3To make the calculations easier we have scaled A up from its more realistic values in Chapter 7 and have rescaled several other parameters as well 4M Jensen and K Murphy Performance Pay and TopManagement Incentives Journal of Political Economy 98 1990 564 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 643 pursue risk reduction up to the point at which marginal gains from additional precautions are equal to the marginal cost of these precautions In the presence of insurance coverage however this calculation may change If a person is fully insured against losses then he or she will have a reduced incentive to undertake costly precautions which may increase the likelihood of a loss occurring In the automobile insurance case for example a person who has a policy that covers theft may not bother to install a car alarm This behavioral response to insurance coverage is termed moral hazard The use of the term moral to describe this response is perhaps unfortunate There is nothing particularly immoral about the behavior being described since individuals are simply responding to the incentives they face In some applications this response might even be desirable For example people with medical insurance may be encouraged to seek early treatment because the insurance reduces their outofpocket cost of medical care But because insurance providers may find it costly to measure and evaluate such responses moral hazard may have important implications for the allocation of resources To examine these we need a model of utilitymaximizing behavior by insured individuals 1851 Mathematical model Suppose a riskaverse individual faces the possibility of incurring a loss l that will reduce his initial wealth 1W02 The probability of loss is π An individual can reduce π by spending more on preventive measures e5 Let UW be the individuals utility given wealth W An insurance company here playing the role of principal offers an insurance contract involving a payment x to the individual if a loss occurs The premium for this coverage is p If the individual takes the coverage then his wealth in state 1 no loss and state 2 loss are W1 5 W0 2 e 2 p W2 5 W0 2 e 2 p 2 l 1 x 1818 and his expected utility is 11 2 π2U1W12 1 πU1W22 1819 The riskneutral insurance companys objective is to maximize expected profit expected insurance profit 5 p 2 πx 1820 1852 Firstbest insurance contract In the firstbest case the insurance company can perfectly monitor the agents precaution ary effort e It sets e and the other terms of the insurance contract x and p to maximize its expected profit subject to the participation constraint that the individual accepts the contract 11 2 π2U1W12 1 πU1W22 U 1821 5For consistency we use the same variable e as we did for managerial effort In this context since e is subtracted from the individuals wealth e should be thought of as either a direct expenditure or the monetary equivalent of the disutility of effort D E F I N I T I O N Moral hazard The effect of insurance coverage on an individuals precautions which may change the likelihood or size of losses Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 644 Part 8 Market Failure where U is the highest utility the individual can attain in the absence of insurance It is clear that the insurance company will increase the premium until the participation con straint holds with equality Thus the firstbest insurance contract is the solution to a maxi mization problem subject to an equality constraint which we can use Lagrange methods to solve The associated Lagrangian is 5 p 2 πx 1 λ 3 11 2 π2U1W12 1 πU1W22 2 U 4 1822 The firstorder conditions are 0 5 p 5 1 2 λ 3 11 2 π2Ur 1W0 2 e 2 p2 1 πUr 1W0 2 e 2 p 2 l 1 x2 4 1823 0 5 x 5 2π 1 λπUr 1W0 2 e 2 p 2 l 1 x2 1824 0 5 e 5 2 π e x 2 λ5 11 2 π2Ur 1W0 2 e 2 p2 1 πUr 1W0 2 e 2 p 2 l 1 x2 1 π e 3U1W0 2 e 2 p2 2 U1W0 2 e 2 p 2 l 1 x2 46 1825 These conditions may seem complicated but they have simple implications Equations 1823 and 1824 together imply 1 λ 5 11 2 π2Ur 1W0 2 e 2 p2 1 πUr 1W0 2 e 2 p 2 l 1 x2 5 Ur 1W0 2 e 2 p 2 l 1 x2 1826 which in turn implies x 5 l This is the familiar result that the first best involves full insurance Substituting for λ from Equation 1826 into Equation 1825 and noting x 5 l we have 2π e l 5 1 1827 At an optimum the marginal social benefit of precaution the reduction in the probability of a loss multiplied by the amount of the loss equals the marginal social cost of precaution which here is just 1 In sum the firstbest insurance contract provides the individual with full insurance but requires him to choose the socially efficient level of precaution 1853 Secondbest insurance contract To obtain the first best the insurance company would need to monitor the insured individ ual to ensure that the person was constantly taking the firstbest level of precaution e In the case of insurance for automobile accidents the company would have to make sure that the driver never exceeds a certain speed always keeps alert and never drives while talking on his cell phone for example Even if a blackbox recorder could be installed to constantly track the cars speed it would still be impossible to monitor the drivers alertness Similarly for health insurance it would be impossible to watch everything the insured party eats to make sure he doesnt eat anything unhealthy Assume for simplicity that the insurance company cannot monitor precaution e at all so that e cannot be specified by the contract directly This secondbest problem is similar to the firstbest except that a new constraint must to be added an incentive compatibility constraint specifying that the agent is free to choose the level of precaution that suits him and maximizes his expected utility 11 2 π2U1W12 1 πU1W22 1828 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 645 Unlike the first best the secondbest contract will typically not involve full insurance Under full insurance x 5 l and as Equation 1818 shows W1 5 W2 But then the insured partys expected utility from Equation 1828 is U1W12 5 U1W0 2 e 2 p2 1829 which is maximized by choosing the lowest level of precaution possible e 5 0 To induce the agent to take precaution the company should provide him only partial insurance Exposing the individual to some risk induces him to take at least some precau tion The company will seek to offer just the right level of partial insurance not too much insurance else the agents precaution drops too low and not too little insurance else the agent would not be willing to pay much in premiums The principal faces the same trade off in this insurance example as in the ownermanager relationship studied previously incentives versus insurance The solution for the optimal secondbest contract is quite complicated given the gen eral functional forms for utility that we are using6 Example 182 provides some further practice on the moral hazard problem with specific functional forms 6For more analysis see S Shavell On Moral Hazard and Insurance Quarterly Journal of Economics November 1979 54162 EXAMPLE 182 Insurance and Precaution against Car Theft In Example 72 we examined the decision by a driver endowed with 100000 of wealth to purchase insurance against the theft of a 20000 car Here we reexamine the market for theft insurance when he can also take the precaution of installing a car alarm that costs 1750 and that reduces the probability of theft from 025 to 015 No insurance In the absence of insurance the individual can decide either not to install the alarm in which case as we saw from Example 72 his expected utility is 1145714 or to install the alarm in which case his expected utility is 085 ln 1100000 2 17502 1 015 ln 1100000 2 1750 2 200002 5 1146113 1830 He prefers to install the device First best The firstbest contract maximizes the insurance companys profit given that it requires the individual to install an alarm and can costlessly verify whether the individual has complied The firstbest contract provides full insurance paying the full 20000 if the car is stolen The highest premium p that the company can charge leaves the individual indifferent between accept ing the fullinsurance contract and going without insurance ln 1100000 2 1750 2 p2 5 1146113 1831 Solving for p yields 98250 2 p 5 e1146113 1832 implying that p 5 3298 Note that the e in Equation 1832 is the number 27818 not the individuals precaution The companys profit equals the premium minus the expected payout 3298 2 1015 3 200002 5 298 Second best If the company cannot monitor whether the individual has installed an alarm then it has two choices It can induce him to install the alarm by offering only partial insurance or it can disregard the alarm and provide him with full insurance Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 646 Part 8 Market Failure 1854 Competitive insurance market So far in this chapter we have studied insurance using the same principalagent framework as we used to study the ownermanager relationship In particular we have assumed that a monopoly insurance company principal makes a takeitorleaveit offer to the individual agent This is a different perspective than in Chapter 7 where we implicitly assumed that insurance is offered at fair ratesthat is at a premium that just covers the insurers expected payouts for losses Fair insurance would arise in a perfectly competitive insurance market With competitive insurers the first best maximizes the insurance customers expected utility given that the contract can specify his precaution level The second best maximizes the customers expected utility under the constraint that his precaution level must be induced by having the contract offer only partial insurance Our conclusions about the moral hazard problem remain essentially unchanged when moving from a monopoly insurer to perfect competition The first best still involves full insurance and a precaution level satisfying Equation 1827 The second best still involves partial insurance and a moderate level of precaution The main difference is in the dis tribution of surplus Insurance companies no longer earn positive profits since the extra surplus now accrues to the individual If the company offers full insurance then the individual will certainly save the 1750 by not installing the alarm The highest premium that the company can charge him solves ln 1100000 2 p2 5 1146113 1833 implying that p 5 5048 The companys profit is then 5048 2 1025 3 200002 5 48 On the other hand the company can induce the individual to install the alarm if it reduces the payment after theft from the full 20000 down to 3374 and lowers the premium to 602 These sec ondbest contractual terms were computed by the authors using numerical methods we will forgo the complicated computations and just take these terms as given Lets check that the individual would indeed want to install the alarm His expected utility if he accepts the contract and installs the alarm is 085 ln 1100000 2 1750 2 6022 1 015 ln 1100000 2 1750 2 602 2 20000 1 33742 5 1146113 1834 the same as if he accepts the contract and does not install the alarm 075 ln 1100000 2 6022 1 025 ln 1100000 2 602 2 20000 1 33742 5 1146113 1835 also the same as he obtains if he goes without insurance So he weakly prefers to accept the contract and install the alarm The insurance companys profit is 602 2 1015 3 33742 5 96 Thus par tial insurance is more profitable than full insurance when the company cannot observe precaution QUERY What is the most that the insurance company would be willing to spend in order to monitor whether the individual has installed an alarm EXAMPLE 183 Competitive Theft Insurance Return to Example 182 but now assume that car theft insurance is sold by perfectly competitive companies rather than by a monopolist First best If companies can costlessly verify whether or not the individual has installed an alarm then the firstbest contract requires him to install the alarm and fully insures him for a Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 647 premium of 3000 This is a fair insurance premium because it equals the expected payout for a loss 3000 5 015 3 20000 Firms earn zero profit at this fair premium The individuals expected utility increases to 1146426 from the 1146113 of Example 182 Second best Suppose now that insurance companies cannot observe whether the individual has installed an alarm The secondbest contract is similar to that computed in Example 182 except that the 96 earned by the monopoly insurer is effectively transferred back to the customer in the form of a reduced premium charged by competing insurers The equilibrium premium is p 5 506 and the payment for loss is x 5 3374 QUERY Which casemonopoly or perfect competitionbest describes the typical insurance market Which types of insurance car health life disability and which countries do you think have more competitive markets 186 HIDDEN TYPES Next we turn to the other leading variant of principalagent model the model of hidden types Whereas in the hiddenaction model the agent has private information about a choice he has made in the hiddentype model he has private information about an innate characteristic he cannot choose For example a students type may be his innate intelli gence as opposed to an action such as the effort he expends in studying for an exam At first glance it is not clear why there should be a fundamental economic differ ence between hidden types and hidden actions that requires us to construct a whole new model and devote a whole new section to it The fundamental economic difference is this In a hiddentype model the agent has private information before signing a con tract with the principal in a hiddenaction model the agent obtains private information afterward Having private information before signing the contract changes the game between the principal and the agent In the hiddenaction model the principal shares symmetric infor mation with the agent at the contracting stage and so can design a contract that extracts all of the agents surplus In the hiddentype model the agents private information at the time of contracting puts him in a better position There is no way for the principal to extract all the surplus from all types of agents A contract that extracts all the surplus from the high types those who benefit more from a given contract would provide the low types with negative surplus and they would refuse to sign it The principal will try to extract as much surplus as possible from agents through clever contract design She will even be willing to shrink the size of the contracting pie sacrificing some joint surplus in order to obtain a larger share for herself as in panel b of Figure 181 To extract as much surplus as possible from each type while ensuring that low types are not scared off the principal will offer a contract in the form of a cleverly designed menu that includes options targeted to each agent type The menu of options will be more profitable for the principal than a contract with a single option but the principal will still not be able to extract all the surplus from all agent types Since the agents type is hidden he cannot be forced to select the option targeted at his type but is free to select any of the options and this ability will ensure that the high types always end up with positive surplus To make these ideas more concrete we will study two applications of the hiddentype model that are important in economics First we will study the optimal nonlinear pricing problem and then we will study private information in insurance Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 648 Part 8 Market Failure 187 NONLINEAR PRICING In the first application of the hiddentype model we consider a monopolist the princi pal who sells to a consumer the agent with private information about his own valuation for the good Rather than allowing the consumer to purchase any amount he wants at a constant price per unit the monopolist offers the consumer a nonlinear price schedule The nonlinear price schedule is a menu of differentsized bundles at different prices from which the consumer makes his selection In such schedules the larger bundle generally sells for a higher total price but a lower perunit price than a smaller bundle Our approach builds on the analysis of seconddegree price discrimination in Chapter 14 Here we analyze general nonlinear pricing schedules the most general form of second degree price discrimination In the earlier chapter we limited our attention to a simpler form of seconddegree price discrimination involving twopart tariffs The linear twopart and general nonlinear pricing schedules are plotted in Figure 183 The figure graphs the total tariffthe total cost to the consumer of buying q units for the three different schedules Basic and intermediate economics courses focus on the case of a constant perunit price which is called a linear pricing schedule The linear pricing schedule is graphed as a straight line that intersects the origin because nothing needs to be paid if no units are purchased The twopart tariff is also a straight line but its interceptreflecting the fixed feeis above the origin The darkest curve is a general nonlinear pricing schedule Examples of nonlinear pricing schedules include a coffee shop selling three different sizessay a small 8ounce cup for 150 a medium 12ounce cup for 180 and a large The graph shows the shape of three different pricing schedules Thicker curves are more complicated pricing schedules and so represent more sophisticated forms of seconddegree price discrimination q Total tarif 0 Linear Twopart Nonlinear FIGURE 183 Shapes of Various Pricing Schedules Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 649 16ounce cup for 200 Although larger cups cost more in total they cost less per ounce 1875 cents per ounce for the small 15 for the medium and 125 for the large The con sumer does not have the choice of buying as much coffee as he wants at a given perounce price instead he must pick one of these three menu options each specifying a particular bundled quantity In other examples the q that is bundled in a menu item is the quality of a single unit of the product rather than the quantity or number of units For example an airline ticket involves a single unit ie a single flight whose quality varies depending on the class of the ticket which ranges from first class with fancy drinks and meals and plush seats offering plenty of leg room to coach class with peanuts for meals and small seats having little leg room 1871 Mathematical model To understand the economic principles involved in nonlinear pricing consider a formal model in which a single consumer obtains surplus U 5 θv 1q2 2 T 1836 from consuming a bundle of q units of a good for which he pays a total tariff of T The first term in the consumers utility function θv 1q2 reflects the consumers benefit from con sumption Assume vr 1q2 0 and vs 1q2 0 implying that the consumer prefers more of the good to less but that the marginal benefit of more units is decreasing The consumers type is given by θ which can be high 1θH2 with probability β and low 1θL2 with probability 1 2 β The high type enjoys consuming the good more than the low type 0 θL θH The total tariff T paid by the consumer for the bundle is subtracted from his benefit to compute his net surplus For simplicity we are assuming that there is a single consumer in the market The analysis would likewise apply to markets with many consumers a proportion β of which are high types and 1 2 β of which are low types The only complication in extending the model to many consumers is that we would need to assume that consumers can not divide bundles into smaller packages for resale among themselves Of course such repackaging would be impossible for a single unit of the good involving a bundle of qual ity and reselling may be impossible even for quantity bundles if the costs of reselling are prohibitive Suppose the monopolist has a constant marginal and average cost c of producing a unit of the good Then the monopolists profit from selling a bundle of q units for a total tariff of T is P 5 T 2 cq 1837 1872 Firstbest nonlinear pricing In the firstbest case the monopolist can observe the consumers type θ before offering him a contract The monopolist chooses the contract terms q and T to maximize her profit subject to Equation 1837 and subject to a participation constraint that the consumer accepts the contract Setting the consumers utility to 0 if he rejects the contract the partic ipation constraint may be written as θv 1q2 2 T 0 1838 The monopolist will choose the highest value of T satisfying the participation constraint T 5 θv 1q2 Substituting this value of T into the monopolists profit function yields θv 1q2 2 cq 1839 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 650 Part 8 Market Failure Taking the firstorder condition and rearranging provides a condition for the firstbest quantity θvr 1q2 5 c 1840 This equation is easily interpreted In the first best the marginal social benefit of increased quantity on the lefthand side the consumers marginal private benefit θvr 1q2 equals the marginal social cost on the righthand side the monopolists marginal cost c The firstbest quantity offered to the high type 1q H2 satisfies Equation 1840 for θ 5 θH and that offered to the low type 1q L2 satisfies the equation for θ 5 θL The tariffs are set so as to extract all the types surplus The first best for the monopolist is identical to what we termed firstdegree price discrimination in Chapter 14 It is instructive to derive the monopolists first best in a different way using methods simi lar to those used to solve the consumers utility maximization problem in Chapter 4 The con tract q T can be thought of as a bundle of two different goods over which the monopolist has preferences The monopolist regards T as a good more money is better than less and q as a bad higher quantity requires higher production costs Her indifference curve actually an isoprofit curve over q T combinations is a straight line with slope c To see this note that the slope of the monopolists indifference curve is her marginal rate of substitution MRS 5 2Pq PT 5 2 1c2 1 5 c 1841 The monopolists indifference curves are drawn as dashed lines in Figure 184 Because q is a bad for the monopolist her indifference curves are higher as one moves toward the upper left The consumers indifference curves over the bundle of contractual terms are drawn as solid lines the thicker one for the high type and thinner for the low type the monopolists isoprofits are drawn as dashed lines Point A is the firstbest contract option offered to the high type and point B is that offered to the low type A q T B 0 UL 0 UH 0 FIGURE 184 FirstBest Nonlinear Pricing Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 651 Figure 184 also draws indifference curves for the two consumer types the high types labeled U 0 H and the low types labeled U 0 L Because T is a bad for consumers higher indifference curves for both types of consumer are reached as one moves toward the lower right The U 0 H indifference curve for the high type is special because it intersects the origin implying that the high type gets the same surplus as if he didnt sign the contract at all The firstbest contract offered by the monopolist to the high type is point A at which the highest indifference curve for the monopolist still intersects the high types U 0 H indifference curve and thus still provides the high type with nonnegative surplus This is a point of tangency between the contracting parties indifference curvesthat is a point at which the indifference curves have the same slope The monopolists indifference curves have slope c everywhere as we saw in Equation 1841 The slope of type θs indifference curve is the marginal rate of substitution MRS 5 Uq UT 5 2θvr 1q2 21 5 θvr 1q2 1842 Equating the slopes gives the same condition for the first best as we found in Equation 1840 marginal social benefit equals marginal social cost of an additional unit The same arguments imply that point B is the firstbest contract offered to the low type and we can again verify that Equation 1840 is satisfied there To summarize the firstbest contract offered to each type specifies a quantity q H or q L respectively that maximizes social surplus given the type of consumer and a tariff T H or T L respectively that allows the monopolist to extract all of the types surplus 1873 Secondbest nonlinear pricing Now suppose that the monopolist does not observe the consumers type when offering him a contract but knows only the distribution θ 5 θH with probability β and θ 5 θL with probability 1 2 β As Figure 185 shows the firstbest contract would no longer work because the high type obtains more utility moving from the indifference curve labeled U 0 H to the one labeled U 2 H by choosing the bundle targeted to the low type B rather than the bundle targeted to him A In other words choosing A is no longer incentive compatible for the high type To keep the high type from choosing B the monopolist must reduce the high types tariff offering C instead of A The substantial reduction in the high types tariff indicated by the downwardpointing arrow puts a big dent in the monopolists expected profit The monopolist can do bet ter than offering the menu of contracts B C She can distort the low types bundle in order to make it less attractive to the high type Then the high types tariff need not be reduced as much to keep him from choosing the wrong bundle Figure 186 shows how this new contract would work The monopolist reduces the quantity in the low types bundle while reducing the tariff so that the low type stays on his U 0 L indifference curve and thus continues to accept the contract offering bundle D rather than B The high type obtains less utility from D than B as D reaches only his U 1 H indifference curve and is short of his U 2 H indifference curve To keep the high type from choosing D the monopolist need only lower the high types tariff by the amount given by the vertical distance between A and E rather than all the way down to C Relative to B C the secondbest menu of contracts D E trades off a distortion in the low types quantity moving from the firstbest quantity in B to the lower quantity in D and destroying some social surplus in the process against an increase in the tariff that can be extracted from the high type in moving from C to E An attentive student might wonder why the monopolist would want to make this tradeoff After all the monopolist Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 652 Part 8 Market Failure must reduce the low types tariff in moving from B to D or else the low type would refuse to accept the contract How can we be sure that this reduction in the low types tariff doesnt more than offset any increase in the high types tariff The reason is that a reduction in quantity harms the high type more than it does the low type As Equation 1842 shows the consumers marginal rate of substitution between contractual terms quantity and tariff depends on his type θ and is higher for the high type Since the high type values quantity more than does the low type the high type would pay more to avoid the decrease in quan tity in moving from B to D than would the low type Further insight can be gained from an algebraic characterization of the second best The secondbest contract is a menu that targets bundle 1qH TH2 at the high type and 1qL TL2 at the low type The contract maximizes the monopolists expected profit β 1TH 2 cqH2 1 11 2 β2 1TL 2 cqL2 1843 subject to four constraints θLv 1qL2 2 TL 0 1844 θHv 1qH2 2 TH 0 1845 θLv 1qL2 2 TL θLv 1qH2 2 TH 1846 θHv 1qH2 2 TH θHv 1qL2 2 TL 1847 The firstbest contract involving points A and B is not incentive compatible if the consumer has private information about his type The high type can reach a higher indifference curve by choosing the bundle B that is targeted at the low type To keep him from choosing B the monopolist must reduce the high types tariff by replacing bundle A with C q T A B 0 C Reduction in tarif UH 0 UH 2 UL 0 FIGURE 185 First Best Not Incentive Compatible Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 653 The first two are participation constraints for the low and high type of consumer ensuring that they accept the contract rather than forgoing the monopolists good The last two are incentive compatibility constraints ensuring that each type chooses the bundle targeted to him rather than the other types bundle As suggested by the graphical analysis in Figure 186 only two of these constraints play a role in the solution The most important constraint was to keep the high type from choosing the low types bundle this is Equation 1847 incentive compatibility constraint for the high type The other relevant constraint was to keep the low type on his U 0 L indifference curve to prevent him from rejecting the contract this is Equation 1844 participation constraint for the low type Hence Equations 1844 and 1847 hold with equality in the second best The other two constraints can be ignored as can be seen in Figure 186 The high types secondbest bundle E puts him on a higher indifference curve 1U 1 H2 than if he rejects the contract 1U 0 H2 so the high types participation constraint Equation 1845 can be safely ignored The low type would be on a lower indifference curve if he chose the high types bundle E rather than his own D so the low types incentive compatibility constraint Equation 1846 can also be safely ignored Treating Equations 1844 and 1847 as equalities and using them to solve for TL and TH yields TL 5 θLv 1qL2 1848 The secondbest contract is indicated by the circled points D and E Relative to the incentivecompatible contract found in Figure 185 points B and C the secondbest contract distorts the low types quantity indicated by the move from B to D in order to make the low types bundle less attractive to the high type This allows the principal to charge tariff to the high type indicated by the move from C to E q T A B 0 C D E qL q qH q qL q UH 0 UH 1 UH 2 UL 0 FIGURE 186 SecondBest Nonlinear Pricing Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 654 Part 8 Market Failure and TH 5 θH3v 1qH2 2 v 1qL2 4 1 TL 5 θH3v 1qH2 2 v 1qL2 4 1 θLv 1qL2 1849 By substituting these expressions for TL and TH into the monopolists objective func tion Equation 1839 we convert a complicated maximization problem with four inequality constraints into the simpler unconstrained problem of choosing qL and qH to maximize β5θH3v 1qH2 2 v 1qL2 4 1 θLv 1qL2 2 cqH6 1 11 2 β2 3θLv 1qL2 2 cqL4 1850 The low types quantity satisfies the firstorder condition with respect to qL which upon considerable rearranging yields θLvr 1q L 2 5 c 1 β 1θH 2 θL2vr 1q L 2 1 2 β 1851 The last term is clearly positive and thus the equation implies that θLvr 1q L 2 c whereas θLvr 1q L2 5 c in the first best Since vq is concave we see that the secondbest quantity is lower than the first best verifying the insight from our graphical analysis that the low types quantity is distorted downward in the second best to extract surplus from the high type The high types quantity satisfies the firstorder condition from the maximization of Equation 1843 with respect to qH upon rearranging this yields θHvr 1q H 2 5 c 1852 This condition is identical to the first best implying that there is no distortion of the high types quantity in the second best There is no reason to distort the high types quantity because there is no higher type from whom to extract surplus The result that the highest type is offered an efficient contract is often referred to as no distortion at the top Returning to the low types quantity how much the monopolist distorts this quantity downward depends on the probabilities of the two consumer types orequivalently in a model with many consumerson the relative proportions of the two types If there are many low types β is low then the monopolist would not be willing to distort the low types quantity very much because the loss from this distortion would be substantial and there would be few high types from whom additional surplus could be extracted The more the high types the higher is β the more the monopolist is willing to distort the low types quantity downward Indeed if there are enough high types the monopolist may decide not to serve the low types at all and just offer one bundle that would be purchased by the high types This would allow the monopolist to squeeze all the surplus from the high types because they would have no other option EXAMPLE 184 Monopoly Coffee Shop The college has a single coffee shop whose marginal cost is 5 cents per ounce of coffee The repre sentative customer is equally likely to be a coffee hound high type with θH 5 20 or a regular Joe low type with θL 5 15 Assume v 1q2 5 2q First best Substituting the functional form v 1q2 5 2q into the condition for firstbest quantities 3θvr 1q2 5 c4 and rearranging we have q 5 1θc2 2 Therefore q L 5 9 and q H 5 16 The tariff extracts all of each types surplus 3T 5 θv 1q2 4 here implying that T L 5 90 and T H 5 160 The shops expected profit is Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 655 1 2 1T H 2 cq H2 1 1 2 1T L 2 cq L2 5 625 1853 cents per customer The first best can be implemented by having the owner sell a 9ounce cup for 90 cents to the low type and a 16ounce cup for 160 to the high type Somehow the barista can discern the customers type just by looking at him as he walks in the door Incentive compatibility when types are hidden The first best is not incentive compati ble if the barista cannot observe the customers type The high type obtains no surplus from the 16ounce cup sold at 160 If he instead paid 90 cents for the 9ounce cup he would obtain a surplus of θHv 192 2 90 5 30 cents Keeping the same cup sizes as in the first best the price for the large cup would have to be reduced by 30 cents to 130 in order to keep the high type from buying the small cup The shops expected profit from this incentive compatible menu is 1 2 1130 2 5 162 1 1 2 190 2 5 92 5 475 1854 Second best The shop can do even better by reducing the size of the small cup to make it less attractive to high demanders The size of the small cup in the second best satisfies Equation 1851 which for the functional forms in this example implies that θLq212 L 5 c 1 1θH 2 θL2q212 L 1855 or rearranging q L 5 a 2θL 2 θH c b 2 5 a2 15 2 20 5 b 2 5 4 1856 The highest price that can be charged without losing the lowtype customers is T L 5 θLv 1q L 2 5 1152 1242 5 60 1857 The large cup is the same size as in the first best 16 ounces It can be sold for no more than 140 or else the coffee hound would buy the 4ounce cup instead Although the total tariff for the large cup is higher at 140 than for the small cup at 60 cents the unit price is lower 875 cents versus 15 cents per ounce Hence the large cup sells at a quantity discount The shops expected profit is 1 2 1140 2 5 162 1 1 2 160 2 5 42 5 50 1858 cents per consumer Reducing the size of the small cup from 9 to 4 ounces allows the shop to recapture some of the profit lost when the customers type cannot be observed QUERY In the firstbest menu the price per ounce is the same 10 cents for both the low and high types cup Can you explain why it is still appropriate to consider this a nonlinear pricing scheme 1874 Continuum of types Similar results for nonlinear pricing hold if we allow for a continuum of consumer types rather than just two The analysis requires more complicated mathematics in particular the techniques of optimal control introduced in Chapter 2 so the casual reader may want to skip this subsection7 7Besides drawing on Chapter 2 this subsection draws on Section 233 of P Bolton and M Dewatripont Contract Theory Cambridge MA MIT Press 2005 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 656 Part 8 Market Failure Suppose the consumers type θ is a continuous random variable in the interval between θL at the low end and θH at the high end Let φ1θ2 be the associated probability density function and F 1θ2 the cumulative distribution function These concepts can be reviewed in the section on probability and statistics from Chapter 2 As in the twotype case the consumer sees his or her type but the monopolist only knows the distribution from which θ is drawn The monopolist offers a menu of bundles one for each type θ where a bun dle is a specification of a quantity q 1θ2 and a total tariff for this quantity T1θ2 Where the menu only needed to include two options in the twotype case with a continuum of types the menu will be a continuous schedule and q 1θ2 and T1θ2 will be functions that vary continuously with θ The consumers utility function is U1θ2 5 θv 1q 1θ2 2 2 T1θ2 as before The monopolists profit from serving type θ is P 1θ2 5 T1θ2 2 cq 1θ2 where c is the constant marginal and average cost of production The first best for the monopolist assuming for the moment it has full information is easy to solve for Each type is offered the socially optimal quantity which satisfies the condition θvr 1q2 5 c Each type is charged the tariff that extracts all of his surplus T1θ2 5 θv 1q 1θ2 2 The monopolist earns profit θv 1q 1θ2 2 2 cq 1θ2 which is clearly all of social surplus The monopolists secondbest pricing scheme now treating the consumers type as pri vate information is the menu of bundles q 1θ2 and T1θ2 maximizing its expected profit 3 θH θL P 1θ2 φ1θ2 dθ 5 3 θH θL 3T1θ2 2 cq 1θ2 4 φ1θ2 dθ 1859 subject to participation and incentivecompatibility constraints for the consumer As in the twotype case participation is only a concern for the lowest type that the monopolist serves Then all types will participate as long as θL does The relevant participation con straint is thus8 θLv 1q 1θL2 2 2 T1θL2 0 1860 Incentive compatibility requires more detailed discussion Incentive compatibility requires that type θ prefers its bundle to any other types say q 1θ2 and T1θ2 In other words θv 1q 1θ2 2 2 T1θ2 is maximized at θ 5 θ Taking the firstorder condition with respect to θ we have that θvr 1q 1θ2 2qr 1θ2 2 Tr 1θ2 5 0 holds for θ 5 θ that is9 θvr 1q 1θ2 2qr 1θ2 2 Tr 1θ2 5 0 1861 This equation has too many derivatives to be able to apply the optimalcontrol methods from Chapter 2 directly The analogous equation in Chapter 2 Equation 2134 has only one derivative To get Equation 1861 into the right shape we will perform a clever change of variables Differentiating the utility function Ur 1θ2 5 v 1q 1θ2 2 1 θvr 1q 1θ2 2qr 1θ2 2 Tr 1θ2 5 v 1q 1θ2 2 1862 where the second equality uses the information from Equation 1861 Using Equation 1862 rather than 1861 as the incentivecompatibility constraint it is now expressed in a form with only one derivative as needed Since the differential equation Ur 1θ2 5 v 1q 1θ2 2 involves the derivative of U1θ2 rather than of T1θ2 we can make the substitution 8The fact that all types participate in the contract does not require the monopolist to serve them with a positive quantity The monopolist may choose to offer the null contract zero quantity and tariff to a range of types By reducing some types down to the null contract the monopolist can extract even more surplus from higher types 9This equation is necessary and sufficient for incentive compatibility under a set of conditions that hold in many examples but are too technical to discuss here Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 657 T1θ2 5 θv 1q 1θ2 2 2 U1θ2 everywhere in the maximization problem to get it in terms of q 1θ2 and U1θ2 rather than q 1θ2 and T1θ2 The reformulated problem is to maximize 3 θH θL 3θv 1q 1θ2 2 2 U1θ2 2 cq 1θ2 4 φ1θ2 dθ 1863 subject to the participation constraint Equation 1860 and the incentive compatibility constraint Equation 1862 Following Equation 2136 the Hamiltonian associated with the optimal control problem is H 5 3θv 1q 1θ2 2 2 U1θ2 2 cq 1θ2 4 φ1θ2 1 λ 1θ2v 1q 1θ2 2 1 U1θ2λr 1θ2 1864 To see how this Hamiltonian is constructed θ is here playing the role played by t in Chap ter 2 q 1θ2 is playing the role of control variable c 1t2 U1θ2 is playing the role of state vari able x 1t2 the integrand 3θv 1q 1θ2 2 2 U1θ2 2 cq 1θ2 4 φ1θ2 in Equation 1863 is playing the role of f and the incentivecompatibility condition Ur 1θ2 5 v 1q 1θ2 2 is playing the role of differential equation dx 1t2dt 5 g 1x 1t2 c 1t2 t2 Having set up the Hamiltonian we can proceed to solve this optimalcontrol problem Analogous to the conditions Hc 5 0 and Hx 5 0 from Equation 2137 here the conditions for the optimalcontrol solution are H q 5 3θvr 1q 1θ2 2 2 c4φ1θ2 1 λ 1θ2vr 1θ2 5 0 1865 H U 5 2φ1θ2 1 λr 1θ2 5 0 1866 We will solve this system of equations by first using Equation 1866 to get an expression for the Lagrange multiplier which can then be eliminated from the preceding equation Using the fun damental theorem of calculus and a bit of work10 one can show that Equation 1866 implies λ 1θ2 5 F 1θ2 2 1 Substituting for the Lagrange multiplier in Equation 1865 and rearranging θvr 1q 1θ2 2 5 c 1 1 2 F 1θ2 φ1θ2 vr 1q 1θ2 2 1867 This equation tells us a lot about the second best Since F 1θH2 5 1 for the highest type the equation reduces to θHvr 1q 1θH2 2 5 c the firstbest condition We again get no distortion at the top for the high type All other types face some downward distortion in q 1θ2 To see this note θvr 1q 1θ2 2 c for these implying q 1θ2 is less than the first best for all θ θH 10We have λ 1θH2 2 λ1θ2 5 3 θH θ λr 1s2 ds 5 3 θH θ φ 1s2 ds 5 F1θH2 2 F1θ2 5 1 2 F1θ2 where the first equality follows from the fundamental theorem of calculus discussed in Chapter 2 the second equality from Equation 1866 the third equality from the fact that a probability density function is the derivative of the cumulative distribution function and the last equality from F1θH2 5 1 true because F is a cumulative distribution function which equals 1 when evaluated at the greatest possible value of the random variable Therefore λ 1θ2 5 λ 1θH2 1 F1θ2 2 1 5 F1θ2 2 1 since λ1θH2 5 0 as there are no types above θH from whom to extract surplus and thus the value as measured by λ 1θH2 from distorting the contract offered to type θH is 0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 658 Part 8 Market Failure 188 ADVERSE SELECTION IN INSURANCE For the second application of the hiddentype model we will return to the insurance mar ket in which an individual with stateindependent preferences and initial income W0 faces the prospect of loss l Assume the individual can be one of two types a highrisk type with probability of loss πH or a lowrisk type with probability πL where πH πL We will first assume the insurance company is a monopolist later we will study the case of competi tive insurers The presence of hidden risk types in an insurance market is said to lead to adverse selection Insurance tends to attract more risky than safe consumers the selection in adverse selection because it is more valuable to risky types yet risky types are more expensive to serve the adverse in adverse selection D E F I N I T I O N Adverse selection The problem facing insurers that risky types are both more likely to accept an insurance policy and more expensive to serve As we will see if the insurance company is clever then it can mitigate the adverse selection problem by offering a menu of contracts The policy targeted to the safe type offers only partial insurance so that it is less attractive to the highrisk type 1881 First best In the first best the insurer can observe the individuals risk type and offer a different pol icy to each Our previous analysis of insurance makes it clear that the first best involves full insurance for each type so the insurance payment x in case of a loss equals the full amount of the loss l Different premiums are charged to each type and are set to extract all of the surplus that each type obtains from the insurance The solution is shown in Figure 187 the construction of this figure is discussed further in Chapter 7 Without insurance each type finds himself at point E Point A resp B is the first best policy offered to the highrisk resp lowrisk type Points A and B lie on the certainty line because both are fully insured Since the premiums extract each types surplus from insurance both types are on their indifference curves through the noinsurance point E The high types premium is higher so A is further down the certainty line toward the origin than is B11 1882 Second best If the monopoly insurer cannot observe the agents type then the firstbest contracts will not be incentive compatible The highrisk type would claim to be low risk and take full 11Mathematically A appears further down the certainty line than B in Figure 187 because the high types indifference curve through E is flatter than the low types To see this note that expected utility equals 11 2 π2U1W12 1 πU1W22 and so the MRS is given by 2 dW1 dW2 5 11 2 π2Ur 1W12 πUr 1W22 At a given 1W1 W22 combination on the graph the marginal rates of substitution differ only because the underlying probabilities of loss differ Since 1 2 πH πH 1 2 πL πL it follows that the highrisk types indifference curve will be flatter This proof follows the analysis presented in M Rothschild and J Stiglitz Equilibrium in Competitive Insurance Markets An Essay on the Economics of Imperfect Information Quarterly Journal of Economics November 1976 62950 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 659 insurance coverage at the lower premium As in the nonlinear pricing problem the second best will involve a menu of contracts Other principles from the nonlinear pricing problem also carry over here The high type continues to receive the firstbest quantity here full insurancethere is no distortion at the top The low types quantity is distorted downward from the first best so he receives only partial insurance Again we see that with hidden types the principal is willing to sacrifice some social surplus in order to extract some of the surplus the agent would otherwise derive from his private information Figure 188 depicts the second best If the insurer tried to offer a menu containing the firstbest contracts A and B then the highrisk type would choose B rather than A To maintain incentive compatibility the insurer distorts the low types policy from B along its indifference curve U 0 L down to D The low type is only partially insured and this allows the insurer to extract more surplus from the high type The high type continues to be fully insured but the increase in his premium shifts his policy down the certainty line to C In the first best the monopoly insurer offers policy A to the highrisk type and B to the lowrisk type Both types are fully insured The premiums are sufficiently high to keep each type on his indifference curve through the noinsurance point E A B 0 E W1 W2 Certainty line UH 0 UL 0 FIGURE 187 First Best for a Monopoly Insurer The analysis of automobile insurance in Example 182 which is based on Example 72 can be recast as an adverse selection problem Suppose that the probability of theft depends not on the act of installing an antitheft device but rather on the color of the car Because thieves prefer red to gray cars the probability of theft is higher for red cars 1πH 5 0252 than for gray cars 1πL 5 0152 EXAMPLE 185 Insuring the Little Red Corvette Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 660 Part 8 Market Failure Secondbest insurance policies are represented by the circled points C for the highrisk type and D for the lowrisk type A B 0 E C D W1 W2 Certainty line UH 0 UH 1 UL 0 FIGURE 188 Second Best for a Monopoly Insurer First best The monopoly insurer can observe the car color and offer different policies for dif ferent colors Both colors are fully insured for the 20000 loss of the car The premium is the maximum amount that each type would be willing to pay in lieu of going without insurance as computed in Example 72 this amount is 5426 for the high type red cars Similar calcula tions show that a graycar owners expected utility if he is not insured is 114795 and the max imum premium he would be willing to pay for full insurance is 3287 Although the insurer pays more claims for red cars the higher associated premium more than compensates and thus the expected profit from a policy sold for a red car is 5426 2 025 20000 5 426 versus 3287 2 015 20000 5 287 for a gray car Second best Suppose the insurer does not observe the color of the customers car and knows only that 10 percent of all cars are red and the rest are gray The secondbest menu of insurance policiesconsisting of a premiuminsurance coverage bundle 1 pH xH2 targeted for highrisk red cars and 1 pL xL2 for lowrisk gray carsis indicated by the circled points in Figure 188 Red cars are fully insured xH 5 20000 To solve for the rest of the contractual parameters observe that xL pH and pL can be found as the solution to the maximization of expected insurer profit 01 1 pH 2 025 200002 1 09 1 pL 2 015xL 2 1868 subject to a participation constraint for the low type 085 ln 1100000 2 pL2 1 015 ln 1100000 2 pL 2 20000 1 xL2 114795 1869 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 661 and to an incentive compatibility constraint for the high type ln 1100000 2 pH2 075 ln 1100000 2 pL 2 1 025 ln 1100000 2 pL 2 20000 1 xL 2 1870 Participation and incentive compatibility constraints for the other types can be ignored just as in the nonlinear pricing problem This maximization problem is difficult to solve by hand One way to proceed is to treat Equa tion 1870 as an equality and solve for pH as a function of pL and xL and then to treat Equation 1869 as a equality and solve for xL as a function of pL Substituting these values into Equation 1868 transforms it after carefully rearranging into the following singlevariable objective function 20300 1 07650pL 2 2038 3 107 1100000 2 pL2 23 2 2328 3 1032 1100000 2 pL2 173 1871 This admittedly ugly expression has a wellbehaved graph shown in Figure 189 which reaches a maximum at pL 1985 Substituting this value into Equations 1868 and 1869 yields xL 11638 and pH 4146 QUERY How much profit is earned on the highrisk type in the second best Why doesnt the insurer just refuse to serve that type 1800 1850 1900 1950 2000 2050 2100 2150 2200 Lowrisk policy premium pL Insurer proft 129 130 1295 FIGURE 189 Solving for Second Best in the Numerical Example Graphing the expression for profit in Equation 1871 as a function of pL shows that a maximum is reached at pL 1985 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 662 Part 8 Market Failure 1883 Competitive insurance market Assume now that insurance is provided not by a monopoly but rather by a perfectly com petitive market resulting in fair insurance Figure 1810 depicts the equilibrium in which insurers can observe each individuals risk type Lines EF and EG are drawn with slopes 11 2 πL2πL and 11 2 πH2πH respectively and show the market opportunities for each person to trade W1 for W2 by purchasing fair insurance12 The lowrisk type is sold pol icy F and the highrisk type is sold policy G Each type receives full insurance at a fair premium However the outcome in the figure is unstable if insurers cannot observe risk types The high type would claim to be low risk and take contract F But then insurers that offered F would earn negative expected profit At F insurers break even serving only the lowrisk types so adding individuals with a higher probability of loss would push the company below the breakeven point With perfect information the competitive insurance market results in full insurance at fair premiums for each type The high type is offered policy G the low type policy F G F E 0 W1 W2 Certainty line FIGURE 1810 Competitive Insurance Equilibrium with Perfect Information 12To derive these slopes called odds ratios note that fair insurance requires the premium to satisfy p 5 πx Substituting into W1 and W2 yields W1 5 W0 2 p 5 W0 2 πx W2 5 W0 2 p 2 l 1 x 5 W0 2 l 1 11 2 π2x Hence a 1 increase in the insurance payment x reduces W1 by π and increases W2 by 1 2 π Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 663 The competitive equilibrium with unobservable types is shown in Figure 1811 The equilibrium is similar to the second best for a monopoly insurer A set of policies is offered that separates the types The highrisk type is fully insured at point G the same policy as he was offered in the first best The lowrisk type is offered policy J which fea tures partial insurance The low type would be willing to pay more for fuller insurance preferring a policy such as K Because K is below line EF an insurer would earn positive profit from selling such a policy to lowrisk types only The problem is that K would also attract highrisk types leading to insurer losses Hence insurance is rationed to the lowrisk type With hidden types the competitive equilibrium must involve a set of separating con tracts it cannot involve a single policy that pools both types This can be shown with the aid of Figure 1812 To be accepted by both types and allow the insurer to at least break even the pooling contract would have to be a point such as M within trian gle EFG But M cannot be a final equilibrium because at M there exist further trading opportunities To see this note thatas indicated in the figure and discussed earlier in the chapterthe indifference curve for the high type 1UH2 is flatter than that for the low type 1UL2 Consequently there are insurance policies such as N that are unattractive to highrisk types attractive to lowrisk types and profitable to insurers because such policies lie below EF Assuming that no barriers prevent insurers from offering new contracts pol icies such as N will be offered and will skim the cream of lowrisk individuals from With hidden types the highrisk type continues to be offered firstbest policy G but the lowrisk type is rationed receiving only partial insurance at J in order to keep the highrisk type from pooling G F E J K 0 W1 W2 Certainty line UH UL FIGURE 1811 Competitive Insurance Equilibrium with Hidden Types Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 664 Part 8 Market Failure any pooling equilibrium Insurers that continue to offer M are left with the adversely selected individuals whose risk is so high that insurers cannot expect to earn any profit by serving them13 Pooling contract M cannot be an equilibrium because there exist insurance policies such as N that are profitable to insurers and are attractive to lowrisk types but not to highrisk types G F E M N 0 W1 W2 Certainty line UH UL FIGURE 1812 Impossibility of a Compet itive Pooling Equilibrium 13Demonstrating that a pooling contract is unstable does not ensure the stability of separating contracts In some cases separating contracts such as shown in Figure 1811 can themselves be vulnerable to deviating entry by a pooling contract in which case no purestrategy equilibrium exists One way to generate an equilibrium in this case is to posit duopoly insurers that set policies simultaneously and solve for the mixedstrategy equilibrium Another way would be to ignore entry by deviating contracts taking any set of posted contracts yielding zero profit as a competitive equilibrium This leads to a proliferation of equilibria which can be pared down using an equilibrium refinement such as proposed by E Azevedo and D Gottlieb Perfect Competition in Markets with Adverse Selection Wharton Business School working paper May 2015 EXAMPLE 186 Competitive Insurance for the Little Red Corvette Recall the automobile insurance analysis in Example 185 but now assume that insurance is pro vided by a competitive market rather than a monopolist Under full information the competitive equilibrium involves full insurance for both types at a fair premium of 10252 1200002 5 5000 for highrisk red cars and 10152 1200002 5 3000 for lowrisk gray cars If insurers cannot observe car colors then in equilibrium the coverage for the two types will still be separated into two policies The policy targeted for red cars is the same as under full infor mation The policy targeted for gray cars involves a fair premium pL 5 015xL 1872 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 665 EXAMPLE 187 Certifying Car Color Return to the competitive market for automobile insurance from Example 186 Let R be the most that the owner of a gray car would be willing to pay to have his car color and thus his type cer tified and reported to the market He would then be fully insured at a fair premium of 3000 earning surplus ln 1100000 2 3000 2 R2 In the absence of such a certified report his expected surplus is 085 ln 1100000 2 4532 1 015 ln 1100000 2 453 2 20000 1 30202 5 114803 1874 189 MARKET SIGNALING In all the models studied so far the uninformed principal moved firstmaking a contract offer to the agent who had private information If the information structure is reversed and the informed player moves first then the analysis becomes much more complicated putting us in the world of signaling games studied in Chapter 8 When the signaler is a principal who is offering a contract to an agent the signaling games become complicated because the strategy space of contractual terms is virtually limitless Compare the simpler strategy space of Spences education signaling game in Chapter 8 where the worker chose one of just two actions to obtain an education or not We do not have space to delve too deeply into complex signaling games here nor to repeat Chapter 8s discussion of simpler signaling games We will be content to gain some insights from a few simple applications 1891 Signaling in competitive insurance markets In a competitive insurance market with adverse selection ie hidden risk types we saw that the lowrisk type receives only partial insurance in equilibrium He would benefit from report of his type perhaps hiring an independent auditor to certify that type so the report ing would be credible The lowrisk type would be willing to pay the difference between his equilibrium and his firstbest surplus in order to issue such a credible signal It is important that there be some trustworthy auditor or other way to verify the authen ticity of such reports because a highrisk individual would now have an even greater incentive to make false reports The highrisk type may even be willing to pay a large bribe to the auditor for a false report and an insurance level that does not give redcar owners an incentive to deviate by pooling on the graycar policy 075 ln 1100000 2 pL2 1 025 ln 1100000 2 pL 2 20000 1 xL2 5 ln 1950002 1873 Equations 1872 and 1873 can be solved similarly to how we solved for the second best in the previous example yielding pL c 453 and x L c 3021 QUERY How much more would graycar owners be willing to pay for full insurance Would an insurer profit from selling full insurance at this higher premium if it sold only to owners of gray cars Why then do the companies ration insurance to gray cars by insuring them partially Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 666 Part 8 Market Failure Solving for R in the equation ln 1100000 2 453 2 R2 5 114803 1875 yields R 5 207 Thus the lowrisk type would be willing to pay up to 207 to have a credible report of his type issued to the market The owner of the red car would pay a bribe as high as 2000the difference between his fair premium with full information 5000 and the fair premium charged to an individual known to be of low risk 3000 Therefore the authenticity of the report is a matter of great importance QUERY How would the equilibrium change if reports are not entirely credible ie if there is some chance the highrisk individual can successfully send a false report about his type What incentives would an auditor have to maintain his or her reputation for making honest reports 1892 Market for lemons Markets for used goods raise an interesting possibility for signaling Cars are a leading example Having driven the car over a long period of time the seller has much better infor mation about its reliability and performance than a buyer who can take only a short test drive Yet even the mere act of offering the car for sale can be taken as a signal of car quality by the market The signal is not positive The quality of the good must be below the thresh old that would have induced the seller to keep it As George Akerlof showed in the article for which he won the Nobel Prize in economics the market may unravel in equilibrium so that only the lowestquality goods the lemons are sold14 To gain more insight into this result consider the usedcar market Suppose there is a continuum of qualities from lowquality lemons to highquality gems and that only the owner of a car knows its type Because buyers cannot differentiate between lemons and gems all used cars will sell for the same price which is a function of the average car quality A cars owner will choose to keep it if the car is at the upper end of the quality spectrum since a good car is worth more than the prevailing market price but will sell the car if it is at the low end since these are worth less than the market price This reduction in average quality of cars offered for sale will reduce market price leading wouldbe sellers of the highestquality remaining cars to withdraw from the market The market continues to unravel until only the worstquality lemons are offered for sale The lemons problem leads the market for used cars to be much less efficient than it would be under the standard competitive model in which quality is known Indeed in the standard model the issue of quality does not arise because all goods are typically assumed to be of the same quality Whole segments of the market disappearalong with the gains from trade in these segmentsbecause higherquality items are no longer traded In the extreme the market can simply break down with nothing or perhaps just a few of the worst items being sold The lemons problem can be mitigated by trustworthy usedcar dealers by development of carbuying expertise by the general public by sellers providing proof that their cars are troublefree and by sellers offering moneyback guarantees But anyone who has ever shopped for a used car knows that the problem of potential lemons is a real one 14G A Akerlof The Market for Lemons Quality Uncertainty and the Market Mechanism Quarterly Journal of Economics August 1970 488500 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 667 1810 AUCTIONS The monopolist has difficulty extracting surplus from the agent in the nonlinear pricing problem because highdemand consumers could guarantee themselves a certain surplus by choosing the low demanders bundle A seller can often do better if several consumers compete against each other for her scarce supplies in an auction Competition among con sumers in an auction can help the seller solve the hiddentype problem because highvalue consumers are then pushed to bid high so they dont lose the good to another bidder In the setting of an auction the principals offer is no longer a simple contract or menu of contracts as in the nonlinear pricing problem instead her offer is the format of the auc tion itself Different formats might lead to substantially different outcomes and more or less revenue for the seller so there is good reason for sellers to think carefully about how to design the auction There is also good reason for buyers to think carefully about what bidding strategies to use Auctions have received a great deal of attention in the economics literature ever since William Vickerys seminal work for which he won the Nobel Prize in economics15 Auc tions continue to grow in significance as a market mechanism and are used for selling such goods as airwave spectrum Treasury bills foreclosed houses and collectibles on the Inter net auction site eBay There are a host of different auction formats Auctions can involve sealed bids or open outcries Sealedbid auctions can be first price the highest bidder wins the object and must pay the amount bid or second price the highest bidder still wins but need only pay the nexthighest bid Openoutcry auctions can be either ascending as in the socalled EXAMPLE 188 UsedCar Market Suppose the quality q of used cars is uniformly distributed between 0 and 20000 Sellers value their cars at q Buyers equal in number to the sellers place a higher value on cars q 1 b so there are gains to be made from trade in the usedcar market Under full information about quality all used cars would be sold But this does not occur when sellers have private information about quality and buyers know only the distribution Let p be the market price Sellers offer their cars for sale if and only if q p The quality of a car offered for sale is thus uniformly distributed between 0 and p implying that expected quality is 3 p 0 qa1 pbdq 5 p 2 1876 see Chapter 2 for background on the uniform distribution Hence a buyers expected net sur plus is p 2 1 b 2 p 5 b 2 p 2 1877 There may be multiple equilibria but the one with the most sales involves the highest value of p for which Equation 1877 is nonnegative b 2 p2 5 0 implying that p 5 2b Only a fraction 2b20000 of the cars are sold As b decreases the market for used cars dries up QUERY What would the equilibrium look like in the fullinformation case 15W Vickery Counterspeculation Auctions and Competitive Sealed Tenders Journal of Finance March 1961 837 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 668 Part 8 Market Failure English auction where buyers yell out successively higher bids until no one is willing to top the last or descending as in the socalled Dutch auction where the auctioneer starts with a high price and progressively lowers it until one of the participants stops the auction by accepting the price at that point The seller can decide whether or not to set a reserve clause which requires bids to be over a certain threshold else the object will not be sold Even more exotic auction formats are possible In an allpay auction for example bid ders pay their bids even if they lose A powerful and somewhat surprising result due to Vickery is that in simple settings riskneutral bidders who each know their valuation for the good perfectly no collu sion etc many of the different auction formats listed here and more besides provide the monopolist with the same expected revenue in equilibrium To see why this result is surprising we will analyze two auction formats in turna firstprice and a secondprice sealedbid auctionsupposing that a single object is to be sold In the firstprice sealedbid auction all bidders simultaneously submit secret bids The auctioneer unseals the bids and awards the object to the highest bidder who pays his or her bid In equilibrium it is a weakly dominated strategy to submit a bid b greater than or equal to the buyers valuation v D E F I N I T I O N Weakly dominated strategy A strategy is weakly dominated if there is another strategy that does at least as well against all rivals strategies and strictly better against at least one A buyer receives no surplus if he bids b 5 v no matter what his rivals bid If the buyer loses he gets no surplus if he wins he must pay his entire surplus back to the seller and again gets no surplus By bidding less than his valuation there is a chance that others val uations and consequent bids are low enough that the bidder wins the object and derives a positive surplus Bidding more than his valuation is even worse than just bidding his valuation There is good reason to think that players avoid weakly dominated strategies meaning here that bids will be below buyers valuations In a secondprice sealedbid auction the highest bidder pays the nexthighest bid rather than his own This auction format has a special property in equilibrium All bidding strat egies are weakly dominated by the strategy of bidding exactly ones valuation Vickerys analysis of secondprice auctions and of the property that they induce bidders to reveal their valuations has led them to be called Vickery auctions We will prove that in this kind of auction bidding something other than ones true val uation is weakly dominated by bidding ones valuation Let v be a buyers valuation and b his bid If the two variables are not equal then there are two cases to consider either b v or b v Consider the first case 1b v2 Let b be the highest rival bid If b v then the buyer loses whether his bid is b or v so there is a tie between the strategies If b b then the buyer wins the object whether his bid is b or v and his payment is the same the secondhighest bid b in either case so again we have a tie We no longer have a tie if b lies between b and v If the buyer bids b then he loses the object and obtains no surplus If he bids v then he wins the object and obtains a net surplus of v 2 b 0 so bidding v is strictly better than bidding b v in this case Similar logic shows that bidding v weakly dominates bidding b v The reason that bidding ones valuation is weakly dominant is that the winners bid does not affect the amount he has to pay for that depends on someone elses the second highest bidders bid But bidding ones valuation ensures the buyer wins the object when he should Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 669 With an understanding of equilibrium bidding in secondprice auctions we can com pare first and secondprice sealedbid auctions Each format has plusses and minuses with regard to the revenue the seller earns On the one hand bidders shade their bids below their valuations in the firstprice auction but not in the secondprice auction a plus for secondprice auctions On the other hand the winning bidder pays the highest bid in the firstprice auction but only the secondhighest bid in the secondprice auction a minus for secondprice auctions The surprising result proved by Vickery is that these plusses and minuses balance perfectly so that both auction types provide the seller with the same expected revenue Rather than working through a general proof of this revenue equivalence result we will show in Example 189 that it holds in a particular case EXAMPLE 189 Art Auction Suppose two buyers 1 and 2 bid for a painting in a firstprice sealedbid auction Buyer is val uation vi is a random variable that is uniformly distributed between 0 and 1 and is indepen dent of the other buyers valuation Buyers valuations are private information We will look for a symmetric equilibrium in which buyers bid a constant fraction of their valuations bi 5 kvi The remaining step is to solve for the equilibrium value of k Symmetric equilibrium Given that buyer 1 knows his own type v1 and knows buyer 2s equi librium strategy b2 5 kv2 buyer 1 best responds by choosing the bid b1 maximizing his expected surplus Pr11 wins auction2 1v1 2 b12 1 Pr11 loses auction2 102 5 Pr1b1 b22 1v1 2 b12 5 Pr1b1 kv22 1v1 2 b12 1878 5 Pr1v2 b1k2 1v1 2 b12 5 b1 k 1v1 2 b12 We have ignored the possibility of equal bids because they would only occur in equilibrium if buyers had equal valuations yet the probability is zero that two independent and continuous ran dom variables equal each other The only tricky step in Equation 1878 is the last one The discussion of cumulative distribu tion functions in Chapter 2 shows that the probability Pr1v2 x2 can be written as Pr1v2 x2 5 3 x 2q f1v22dv2 1879 where f is the probability density function But for a random variable uniformly distributed between 0 and 1 we have 3 x 0 f1v22dv2 5 3 x 0 112dv2 5 x 1880 so Pr1v2 b1k2 5 b1k Taking the firstorder condition of Equation 1878 with respect to b1 and rearranging yields b1 5 v12 Hence k 5 12 implying that buyers shade their valuations down by half in forming their bids Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 670 Part 8 Market Failure In more complicated economic environments the many different auction formats do not necessarily yield the same revenue One complication that is frequently considered is sup posing that the good has the same value to all bidders but that they do not know exactly what that value is Each bidder has only an imprecise estimate of what his or her valuation might be For example bidders for oil tracts may have each conducted their own surveys of the likelihood that there is oil below the surface All bidders surveys taken together may give a clear picture of the likelihood of oil but each one separately may give only a rough idea For another example the value of a work of art depends in part on its resale value unless the bidder plans on keeping it in the family forever which in turn depends on others valuations each bidder knows his or her own valuation but perhaps not others An auction conducted in such an economic environment is called a common values auction The most interesting issue that arises in a common values setting is the socalled win ners curse The winning bidder realizes that every other bidder probably thought the good was worth less meaning that he or she probably overestimated the value of the good The winners curse sometimes leads inexperienced bidders to regret having won the auction Sophisticated bidders take account of the winners curse by shading down their bids below their imprecise estimates of the value of the good so they never regret having won the auction in equilibrium Order statistics Before computing the sellers expected revenue from the auction we will intro duce the notion of an order statistic If n independent draws are made from the same distribution and if they are arranged from smallest to largest then the kth lowest draw is called the kthorder statistic denoted X1k2 For example with n random variables the nthorder statistic X1n2 is the largest of the n draws the 1n 2 12thorder statistic X1n212 is the second largest and so on Order statistics are so useful that statisticians have done a lot of work to characterize their properties For instance statisticians have computed that if n draws are taken from a uniform distribution between 0 and 1 then the expected value of the kthorder statistic is E 1X1k22 5 k n 1 1 1881 This formula may be found in many standard statistical references Expected revenue The expected revenue from the firstprice auction equals E 1max1b1 b22 2 5 1 2 E 1max1v1 v22 2 1882 But max1v1 v22 is the largestorder statistic from two draws of a uniform random variable between 0 and 1 the expected value of which is 23 according to Equation 1881 Therefore the expected revenue from the auction equals 1122 1232 5 13 Secondprice auction Suppose that the seller decides to use a secondprice auction to sell the painting In equilibrium buyers bid their true valuations bi 5 vi The sellers expected rev enue is E 1min 1b1 b22 2 because the winning bidder pays an amount equal to the losers bid But min 1b1 b22 5 min 1v1 v22 and the latter is the firstorder statistic for two draws from a random variable uniformly distributed between 0 and 1 whose expected value is 13 according to Equa tion 1881 This is the same expected revenue generated by the firstprice auction QUERY In the firstprice auction could the seller try to boost bids up toward buyers valuations by specifying a reservation price r such that no sale is made if the highest bid does not exceed r What are the tradeoffs involved for the seller from such a reservation price Would a reservation price help boost revenue in a secondprice auction Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 671 Analysis of the common values setting is rather complicated and the different auction formats previously listed no longer yield equivalent revenue Roughly speaking auctions that incorporate other bidders information in the price paid tend to provide the seller with more revenue For example a secondprice auction tends to be better than a firstprice auction because the price paid in a secondprice auction depends on what other bidders think the object is worth If other bidders thought the object was not worth much then the secondhighest bid will be low and the price paid by the winning bidder will be low precluding the winners curse Summary In this chapter we have provided a survey of some issues that arise in modeling markets with asymmetric information Asymmetric information can lead to market inefficiencies relative to the firstbest benchmark which assumes per fect information Cleverly designed contracts can often help recover some of this lost surplus We examined some of the following specific issues Asymmetric information is often studied using a principal agent model in which a principal offers a contract to an agent who has private information The two main vari ants of the principalagent model are the models of hid den actions and of hidden types In a hiddenaction model called a moral hazard model in an insurance context the principal tries to induce the agent to take appropriate actions by tying the agents payments to observable outcomes Doing so exposes the agent to random fluctuations in these outcomes which is costly for a riskaverse agent In a hiddentype model called an adverse selection model in an insurance context the principal cannot extract all the surplus from high types because they can always gain positive surplus by pretending to be a low type In an effort to extract the most surplus possible the principal offers a menu of contracts from which different types of agent can select The principal distorts the quan tity in the contract targeted to low types in order to make this contract less attractive to high types thus extracting more surplus in the contract targeted to the high types Most of the insights gained from the basic form of the principalagent model in which the principal is a monopolist carry over to the case of competing princi pals The main change is that agents obtain more surplus The lemons problem arises when sellers have private information about the quality of their goods Sellers whose goods are higher than average quality may refrain from selling at the market price which reflects the aver age quality of goods sold on the market The market may collapse with goods of only the lowest quality being offered for sale The principal can extract more surplus from agents if several of them are pitted against each other in an auc tion setting In a simple economic environment a variety of common auction formats generate the same revenue for the seller Differences in auction format may generate different levels of revenue in more complicated settings Problems 181 Clare manages a piano store Her utility function is given by Utility 5 w 2 100 where w is the total of all monetary payments to her and 100 represents the monetary equivalent of the disutility of exerting effort to run the store Her next best alternative to managing the store gives her zero utility The stores revenue depends on random factors with an equal chance of being 1000 or 400 a If shareholders offered to share half of the stores reve nue with her what would her expected utility be Would she accept such a contract What if she were only given a quarter share What is the lowest share she would accept to manage the firm b What is the most she would pay to buy out the store if shareholders decided to sell it to her c Suppose instead that shareholders decided to offer her a 100 bonus if the store earns 1000 What fixed salary would she need to be paid in addition to get her to accept the contract d Suppose Clare can still choose to exert effort as above but now can also choose not to exert effort in which case Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 672 Part 8 Market Failure she saves on the disutility cost of effort and the shops revenue is 400 for certain i If shareholders decide to offer her a revenuesharing contract as in part a what is the lowest share that would induce her to exert effort ii If shareholders could design a contract for her involv ing a fixed salary plus bonus what design maximizes their expected profit revenues minus payments to the manager 182 A personalinjury lawyer works as an agent for his injured plaintiff The expected award from the trial taking into account the plaintiffs probability of prevailing and the dam age award if she prevails is l where l is the lawyers effort Effort costs the lawyer l 22 a What is the lawyers effort his surplus and the plaintiffs surplus in equilibrium when the lawyer obtains the cus tomary 13 contingency fee ie the lawyer gets 13 of the award if the plaintiff prevails b Repeat part a for a general contingency fee of c c What is the optimal contingency fee from the plaintiffs perspective Compute the associated surpluses for the lawyer and plaintiff d What would be the optimal contingency fee from the plaintiffs perspective if she could sell the case to her lawyer ie if she could ask him for an upfront pay ment in return for a specified contingency fee possibly higher than in part c Compute the upfront pay ment assuming that the plaintiff makes the offer to the lawyer and the associated surpluses for the lawyer and plaintiff Do they do better in this part than in part c Why do you think selling cases in this way is outlawed in many countries 183 Solve for the optimal linear price per ounce of coffee that the coffee shop would charge in Example 184 How does the shops profit compare to when it uses nonlinear prices Hint Your first step should be to compute each types demand at a linear price p 184 Return to the nonlinear pricing problem facing the monopoly coffee shop in Example 184 but now suppose the proportion of high demanders increases to 23 and the proportion of low demanders decreases to 13 What is the optimal menu in the secondbest situation How does the menu compare to the one in Example 184 185 Suppose there is a 5050 chance that an individual with log arithmic utility from wealth and with a current wealth of 20000 will suffer a loss of 10000 from a car accident Insur ance is competitively provided at actuarially fair rates a Compute the outcome if the individual buys full insurance b Compute the outcome if the individual buys only partial insurance covering half the loss Show that the outcome in part a is preferred c Now suppose that individuals who buy the partial rather than the full insurance policy take more care when driv ing reducing the damage from loss from 10000 to 7000 What would be the actuarially fair price of the partial policy Does the individual now prefer the full or the partial policy 186 Suppose that lefthanded people are more prone to accidents than righthanded Lefties have a certain chance of suffering an injury equivalent to a 500 loss righties only have a 50 percent chance The population contains 10 lefties and 100 righties All individuals have logarithmic utilityofwealth functions and 1000 of initial wealth a Solve for the fullinformation outcome ie suppos ing it can observe the individuals dominant hand for a monopoly insurer How much perconsumer profit does it earn from each contract b Solve for the profitmaximizing outcome for a monopoly insurer when consumers have private information about their types How much perconsumer profit does it earn from each contract Does it make sense to serve both types c Solve for the fullinformation outcome for perfectly competitive insurers d Show if consumers have private information about their types any separating contract for righties that involves a nontrivial level of partial insurance will always attract lefties away from their separating contract if contracts involve fair insurance as required by perfect competition Conclude that the competitive equilibrium must involve righties receiving no insurance Further conclude that the adverse selection problem is so severe here that any competitive equilibrium is equivalent to the complete disappearance of insurance in this market 187 Suppose 100 cars will be offered on the usedcar market Let 50 of them be good cars each worth 10000 to a buyer and let 50 be lemons each worth only 2000 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 673 a Compute a buyers maximum willingness to pay for a car if he or she cannot observe the cars quality b Suppose that there are enough buyers relative to sellers that competition among them leads cars to be sold at their maximum willingness to pay What would the mar ket equilibrium be if sellers value good cars at 8000 At 6000 188 Consider the following simple model of a common values auction Two buyers each obtain a private signal about the value of an object The signal can be either high H or low L with equal probability If both obtain signal H the object is worth 1 otherwise it is worth 0 a What is the expected value of the object to a buyer who sees signal L To a buyer who sees signal H b Suppose buyers bid their expected value computed in part a Show that they earn negative profit conditional on observing signal Han example of the winners curse Analytical Problems 189 Doctorpatient relationship Consider the principalagent relationship between a patient and doctor Suppose that the patients utility function is given by UP 1m x2 where m denotes medical care whose quantity is determined by the doctor and x denotes other consumption goods The patient faces budget constraint Ic 5 pmm 1 x where pm is the relative price of medical care The doctors utility function is given by Ud1Id2 1 Upthat is the doctor derives utility from income but being altruistic also derives utility from the patients wellbeing Moreover the additive specification implies that the doctor is a perfect altruist in the sense that his or her utility increases oneforone with the patients The doctors income comes from the patients med ical expenditures Id 5 pmm Show that in this situation the doctor will generally choose a level of m that is higher than a fully informed patient would choose 1810 Increasing competition in an auction A painting is auctioned to n bidders each with a private value for the painting that is uniformly distributed between 0 and 1 a Compute the equilibrium bidding strategy in a first price sealedbid auction Compute the sellers expected revenue in this auction Hint Use the formula for the expected value of the kthorder statistic for uniform dis tributions in Equation 1881 b Compute the equilibrium bidding strategy in a sec ondprice sealedbid auction Compute the sellers expected revenue in this auction using the hint from part a c Do the two auction formats exhibit revenue equivalence d For each auction format how do bidders strategies and the sellers revenue change with an increase in the num ber of bidders 1811 Team effort Increasing the size of a team that creates a joint product may dull incentives as this problem will illustrate16 Suppose n partners together produce a revenue of R 5 e1 1 c1 en here ei is partner is effort which costs him c 1ei2 5 e2 i 2 to exert a Compute the equilibrium effort and surplus revenue minus effort cost if each partner receives an equal share of the revenue b Compute the equilibrium effort and average surplus if only one partner gets a share Is it better to concentrate the share or to disperse it c Return to part a and take the derivative of surplus per partner with respect to n Is surplus per partner increas ing or decreasing in n What is the limit as n increases d Some commentators say that ESOPs employee stock ownership plans whereby part of the firms shares are distributed among all its workers are beneficial because they provide incentives for employees to work hard What does your answer to part c say about the incen tive properties of ESOPs for modern corporations which may have thousands of workers Behavioral Problem 1812 Nudging consumers into adverse selection The bestselling book Nudge suggests that rather than restrict ing consumer choice nudging them toward wise options lower calorie foods lowerexpense mutual funds either by making those options the default or by providing clearer information about the features of the choices might be a more efficient policy allowing the subset of consumers who would benefit from other options to still choose them17 A recent paper by Benjamin Handel suggests that nudges may not be completely innocuous18 In insurance markets nudging consumers toward policies that are better suited to their risk classes may exacerbate the adverseselection problem on the overall market 16The classic reference on the hiddenaction problem with multiple agents is B Holmström Moral Hazard in Teams Bell Journal of Economics Autumn 1982 32440 17Richard Thaler and Cass Sunstein Nudge Improving Decisions About Health Wealth and Happiness London Penguin Books 2009 18Adverse Selection and Inertia in Health Insurance Markets When Nudging Hurts American Economic Review 103 December 2013 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 674 Part 8 Market Failure To illustrate this point we will return to Example 186 in the text in which competitive firms offer automobile insur ance to drivers with logarithmic utilities and initial wealth 100000 whose types car colors are private information Owners of red cars have a 025 chance that their 20000 car is stolen owners of gray cars have a 015 chance Red cars make up 10 percent of the population and the remaining 90 per cent are gray The example showed that the equilibrium sep arating contracts involved full insurance for the 20000 loss at a 5000 premium which is purchased by redcar owners and partial insurance repaying 3021 rounded off in case of a loss at a 453 premium which is purchased by graycar owners a Following previous studies of insurance to obtain a metric that combines welfare across consumers in the presence of uncertainty Handel suggests computing the certainty equivalent for each consumer the certain wealth that would give the consumer the same expected utility as in equilibriumsee the definition in Chapter 7 and then taking the average weighted by the proportion of each type of these certainty equivalents across consumers in the population Compute the average certainty equivalent in the competitive equilibrium from Example 186 b Suppose that either because of inertia misinforma tion or other behavioral bias some consumers end up choosing the wrong policy for his or her type We will assume a very simple behavioral model here Behavioral consumers are equally likely to choose any one of the available contracts Assume further that all consumers of both types are behavioral Throughout your answer to this part hold the contracts mentioned in the statement of the question fixed Compute the average certainty equivalent across consumers A nudge is proposed that would provide enough information to convert the behav ioral consumers from this part into the rational consum ers of part a Use a comparison of your welfare results from this part to those from part a to argue that the proposed nudge enhances welfare c Instead of fixing the contract terms now suppose con tract terms adjust to account for the actual cost of insur ance provision given consumers behavioral biases Find the competitive equilibrium Compute the average cer tainty equivalent across consumers in this equilibrium Compare your result to that from part a to prove that a nudge to convert behavioral consumers into the rational ones of part a reduces welfare Handels article does more than make this theoretical point According to his estimates of demand for insurance health not auto insurance in his study at a large firm that makes up his sample the average employee has 2000 of inertia that is he or she would not change insurance plans for less than 2000 of savings This inertia keeps some highrisk consum ers from switching into adequate insurance a welfare loss The offsetting welfare gain is that inertia keeps lowrisk types in the pool for the fuller insurance contract reducing the average cost of providing insurance thus reducing competi tive premiums On balance Handel estimates that the bene ficial effects of inertia outweigh the harmful effects implying that a nudge eliminating this inertia would reduce welfare as in part c Suggestions for Further Reading Bolton P and M Dewatripont Contract Theory Cambridge MA MIT Press 2005 Comprehensive graduate textbook treating all topics in this chap ter and many other topics in contract theory Krishna V Auction Theory San Diego Academic Press 2002 Advanced text on auction theory LuckingReiley D Using Field Experiments to Test Equiv alence between Auction Formats Magic on the Internet American Economic Review December 1999 106380 Tests the revenue equivalence theorem by selling Magic playing cards over the Internet using various auction formats Milgrom P Auctions and Bidding A Primer Journal of Eco nomic Perspectives Summer 1989 322 Intuitive discussion of methods used and research questions explored in the field of auction theory Rothschild M and J Stiglitz Equilibrium in Competitive Insurance Markets An Essay on the Economics of Imper fect Information Quarterly Journal of Economics November 1976 62950 Presents a nice graphic treatment of the adverse selection problem Contains ingenious illustrations of various possibilities for sepa rating equilibria Salanié B The Economics of Contracts A Primer Cambridge MA MIT Press 1997 A concise treatment of contract theory at a deeper level than this chapter Shavell S Economic Analysis of Accident Law Cambridge MA Harvard University Press 1987 Classic reference analyzing the effect of different laws on the level of precaution undertaken by victims and injurers Discusses how the availability of insurance affects parties behavior Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 675 EXTENSIONS USING EXPERIMENTS TO MEASURE ASYMMETRICINFORMATION PROBLEMS The chapter made the theoretical case that moral hazard and adverse selection are market imperfections with the potential to reduce welfare relative to a fullinformation setting While a theoretical possibility the empirical question remains whether these market imperfections are substantial in realworld mar kets Perhaps asymmetric information is such a minor prob lem that an insurance or financial firm is safe to ignore it in designing contracts that it offers to consumers Perhaps gov ernment intervention to remedy these market imperfections may cause more harm than good Even if it could somehow be established that asymmetric information is a serious imperfection in realworld markets further study is useful to tease out whichmoral hazard or adverse selection if not bothseems to be the major source of welfare loss As we will discuss the contract features or gov ernment policies that best address moral hazard may exacer bate adverse selection and vice versa Empirical study of asymmetricinformation problems in realworld markets is difficultalmost by definition If the firms expert participants in the market with a profit incentive to obtain as much information as possible about consumers themselves lack the information how can an econometrician as an outside observer hope to measure the information in a study One possibility is for the econometrician to use mar ket outcomes to obtain some indirect estimate of the extent of asymmetricinformation problems In health insurance for example the number of doctor visits can be compared across insured and uninsured consumers and the difference can be attributed to consumer asymmetric information This method would have difficulty distinguishing between moral hazard and adverse selection however Do insured consumers who bear less of the cost out of their own pocket seek more care for minor conditions than uninsured a sort of moral hazard Or do higherrisk consumers disproportionately buy insur ance an adverseselection problem Both would lead to a pos itive correlation between insurance and health expenditures So this research design may not help tease apart the two asym metricinformation problems E181 Natural and field experiments Some of the prominent studies of asymmetricinformation in consumer contracts such as insurance and finance resort to clever experiments to try to measure the importance of the different asymmetricinformation problems The experiments come in two forms natural experiments and field experiments Natural experiments are not personally designed by the researcher but rather arise from significant natural events or governmentpolicy changes that fortuitously produce data much as would a welldesigned experiment An example is the Finkelstein et al 2012 article covered in detail below which examined a program to expand Medicaid free health insurance for poor in Oregon in 2008 The budget was too limited to provide full coverage for all the eligible uninsured so a lottery was used to determine coverage The lottery elim inated adverse selection because selection into the insurance program was random Thus lottery provided experimental conditions to study moral hazard in isolation It is referred to as a natural experiment because Oregon did not design the lottery for research purposes but just as a way to address a budget constraint only later did clever researchers realize the research opportunity the lottery provided Another article we will cover using a natural experiment is Einav Finkelstein and Cullen 2010 The managers of various subdivisions of the large firm providing the study sample had leeway in choosing benefits for their employees including the subsidy provided for various insurance policies As a result the employees in the subdivisions who were otherwise very similar to each other including on the dimension of health risks faced dif ferent premiums for the same insurance The firm did not vary premiums to subdivisions for research purposes this was simply the result of the whims of subdivision managers who set benefits packages Clever researchers realized that this pre mium variation provided an ideal natural experiment to study adverse selection studying whether higher prices resulted in only the higherrisk consumers taking up the policies The two other articles we will study use field experi ments designed by researchers with the explicit intention to identify and measure asymmetricinformation problems The upside of a field experiment is that researchers have full control to make the experiment as clean and informative as possible The downside is that because research budgets tend to be small the scope of the experiments is generally limited involving few participants and small transactions The articles we study are notable in having substantially larger scope than the typical field experiment Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 676 Part 8 Market Failure E182 Extending the theory Before diving into the empirical articles we will extend the the ory from the chapter in two useful ways First we will intro duce a way to measure aggregate welfare in a population facing uncertainty The idea is simple The wellbeing of one consumer facing an uncertain situation can be measured by the certain income that the consumer would accept in lieu of participating in that situation in other words the certainty equivalent The certainty equivalent for each consumer can be summed or aver aged over them to obtain an aggregate welfare measure Con sumer surplus can be added to profits to obtain overall social welfare although in the subsequent analysis we will consider perfectly competitive outcomes in which firms earn zero profit so the average certainty equivalent will be all we need to deter mine social preferences over different outcomes Second we will expand on the point made in the intro duction that policies addressing one form of asymmetricin formation problem can exacerbate the other form We will show that a government mandate that all consumers obtain full insurance can eliminate adverse selection but can reduce welfare if moral hazard is the main information problem So as not to unduly delay our study of the empirical arti cles we will make these theoretical points within the context of a familiar example of adverse selection involving insurance provided to risky drivers of red cars with a 025 chance of theft of the 20000 car and safe drives of gray cars with a 015 chance of accident Consumers have 100000 in initial wealth and logarithmic utility In Example 186 in the text we showed that the equilibrium with competitive insurers who cannot condition the policies on the car type involves two policies full insurance 1xH 5 20000 pH 5 50002 purchased by highrisk drivers and partial insurance 1xL 5 3021 pL 5 4532 purchased by lowrisk drivers The certainty equivalent for highrisk types CEH is easy to com pute because they are fully insured so have certain wealth of CEH 5 100000 2 pH 5 95000 The certainty equivalent for lowrisk drivers CEL satisfies ln 1CEL2 5 015 ln 1100000 2 20000 2 pL 1 xL2 1 085 ln 1100000 2 pL2 which after substituting for xL and pL and exponentiating yields CEL 5 96793 all calculations rounded to the near est digit To obtain overall consumer surplus we can com pute the weighted average across consumers Assuming as in Example 186 that 10 percent of consumers drive red cars and 90 percent gray the weighted average is 1012 1950002 1 1092 1967932 5 96614 This is our measure of welfare in the competitive equilibrium under adverse selection The government could consider various market interven tions to address the adverseselection problem One possi bility is a mandate that all consumers obtain full insurance issuing a large fine to anyone who remains uninsured1 A sufficiently high fine would induce all consumers to buy full insurance whatever the cost With all consumers seek ing full insurance the competitive market ends up offer ing just the one fullinsurance policy The fair premium for this full insurance reflects the pooling of high and lowrisk consumers together implying the average risk of accident is 1012 10252 1 1092 10152 5 016 The equilibrium pre mium under the mandate is p 5 10162 1200002 5 3200 All consumers end up with the same final wealth under this fullinsurance policy 100000 2 p 5 96800 This is the perconsumer certainty equivalent associated with the man date and in fact equals the certainty equivalent in the first best 186 per consumer greater than in the competitive equilibrium without a mandate This exercise illustrates the role of a gov ernmentimposed mandate in addressing adverse selection Now suppose instead of adverse selection from Example 186 insurers confront the moralhazard problem described in Example 183 That example has only one consumer type who has the option of installing a car alarm that costs 1750 that lowers the probability of theft from 025 to 015 The example showed that the competitiveequilibrium contract involved partial insurance with a payout in case of theft of x 5 3374 sold at a premium of p 5 506 The partial insur ance exposes the car owner to just enough risk to induce him or her to install the alarm The certainty equivalent associated with this policy solves ln 1CE2 5 015 ln 1100000 2 20000 2 1750 2 p 1 x2 1 085 ln 1100000 2 1750 2 p2 subtracting the cost of the alarm from wealth in all states Substituting the computed values of x and p and exponentiat ing yields CE 5 95048 If the government mandates full insurance the insured consumer has no incentive to install an alarm The fair pre mium if the competitively supplied fullinsurance policy is thus p 5 102521200002 5 5000 The certainty equivalent associated with this policy is 95000 In this case the mandate results in a 48 per consumer reduction in the certainty equiv alent and thus welfare because the mandate precludes partial insurance and thus any incentives to take precaution This exercise illustrates the potential drawback of a gov ernment mandate and illustrates the value of policymakers knowing whether moral hazard or adverse selection is the major problem The reasoning is not restricted to mandates premium subsidies can have similar pros and cons Armed with this theoretical understanding we now turn to the arti cles using experiments to separately identify the level of each of these asymmetricinformation problems 1The Affordable Care Act popularly known as ObamaCare after the US President who proposed it involves an individual mandate enforced with a substantial fine In 2015 the fine could amount to more than 12000 for a family with three children Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 677 E183 Moral Hazard in health insurance When health economists talk about moral hazard in health care what many have in mind is not that fuller insurance leads individuals to eat less healthy food exercise less or smoke Rather the main issue they have in mind is that individuals will consume more medical services if some or all is covered by insurance and less is paid out of pocket Some health econ omists suggest that the rapidsome would say excessiverise in healthcare expenditures over the decades preceding the Affordable Care Act could be stemmed by offering less than full insurance confronting consumers with copayments and deductibles How much of an effect this policy would have depends on how responsive consumers are to prices or in technical terms familiar to students using this text on the elasticity of healthcare expenditures with respect to price The Rand health insurance experiment conducted by the Rand Corporation funded by the US government was a largescale field experiment run over several years in the 1970s in a handful of cities involving nearly 6000 partici pants Researchers randomized subjects into policies involv ing 0 25 50 and 95 percent copayment rates and examined their expenditures over the sample period Randomly assign ing copayments to subjects removes any possibility of adverse selection in the form of riskier consumers choosing policies with lower copayments The expenditures associated with dif ferent copayments represent those that a random and thus typical consumer would make faced with paying that per centage out of pocket The Manning et al 1987 article reports the results from the study some of the key results are reproduced in Table 182 In essence the entries in Table 182 represent points on the average individuals demand curve for health care taking the various copayment levels to represent prices facing the consumer and taking healthcare expenditures as a money metric for the quantity of health care the consumer utilizes TABLE 182 KEY RESULTS FROM RAND HEALTH INSURANCE EXPERIMENT Assigned Copayment Adjusted Annual Expenditure 1984 Individuals 0 750 1893 25 617 1137 50 573 383 95 540 1120 Source Table 2 from Manning et al 1987 Filled circles are average expenditures for different copayments from the Manning et al 1987 study Individual consumer demand p 1q2 estimated as the regression line fitting the circles weighted by size proportional to number of observations in group Marginal cost curve mc 1q2 is a horizontal line of height 1 5 100 Firstbest social welfare equals the area of triangle ABC A consumer facing no copay ment will overconsume healthcare leading to deadweight loss equal to the area of triangle CDE The deadweight loss from a reduction in copayment from 50 to 25 is shown as the shaded trapezoid 0 50 100 150 200 250 300 0 100 200 300 400 500 600 700 800 Healthcare expenditure 1984 dollars Copayment A B C D E mcq pq FIGURE E181 Welfare Loss from Moral Hazard in Rand Health Insurance Experiment Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 678 Part 8 Market Failure The four points from Table 182 are graphed in Figure E181 as dots2 The size of the dots is proportional to the number of individuals assigned that copayment Assuming the aver age individuals demand is linear it can be estimated as the regression line labeled p1q2 that best fits the points weighted by their size which turns out to have the equation p 5 288 2 039q i The dotted line is the marginal cost curve for health care mc 1q2 Given that we are using dollars expended as the mea sure of the quantity of health care utilized by the consumer it is natural to take mc 1q2 as a horizontal line of height 1 or equivalently 100 percent3 The demand and cost curves can be used to compute consumer surplus and welfare here for health care as we did for other sorts of goods As usual the socially efficient outcome is as usual given by point at which demand intersects marginal cost Hence firstbest surplus is given by the area of shaded triangle ABC Using Equation i one can derive the coordinates of points A B and C and use these to compute the area of triangle ABC 1 2 1AB2 1BC2 5 1 2 1288 2 1002 14832 45400 ii Reducing the copayment below 100 percent leads to over consumption In the extreme with no copayment the con sumer utilizes health care up to point E The deadweight loss from this overconsumption is given by the area of triangle CDE Again using Equation i one can find the coordinates of points D and E and use these one can use to compute area of triangle CDE 1 2 1CD2 1DE2 5 1 2 11002 1739 2 4832 12800 iii Expressed as a percentage of firstbest surplus deadweight loss from moral hazard is 1280045400 5 28 This is a fairly small deadweight loss considering the drastic change involved moving from a situation in which the consumer pays everything out of pocket to one in which he or she pays nothing A more realistic change might to increase drop the copayment from 50 percent to 25 percent increasing dead weight loss by the area of the shaded trapezoid in the figure less than 9 percent of the firstbest surplus given by the area of ABC The reason for the relatively small deadweight loss is that the estimated demand curve is quite inelastic For example at the midway point between points C and E involving a 50 per cent copayment one can show that this elasticity is 02 At a copayment of 25 percent the elasticity is even lower 01 These elasticities are close to the low end for nonexperimen tal studies conducted before the Rand experiment which ranged anywhere from 01 to 21 The larger in absolute value estimates from previous nonexperimental studies likely resulted from the confounding effects of adverse selec tion Once these confounding effects are purged healthcare expenditures show little price sensitivity This is an important result suggesting that confronting consumers with more of the cost of their healthcare decisions will not stem much of the increase in medical expenditures It is worth emphasizing how rare the Rand experiment is in economics It is hard to think of even a handful of cases in which the government or any funder agreed to the tens or hundreds of millions of dollars necessary to fund a largescale field experiment designed by researchers for study purposes The Oregon health insurance experiment studied by Finkelstein et al 2012 is also large scale but is a natural rather than field experiment Wanting to expand its Medicaid program subsidized medical insurance for the poor but with facing a tight budget constraint in 2008 Oregon decided to use a lottery to allocate the insurance to eligible citizens The authors of the study realized that although it was not Oregons intention the lottery provides exactly the random allocation that eliminates adverse selection allowing them to measure the pure effect of moral hazard The scale was enormous with 90000 individuals signing up to be part of the lottery Because this was a natural not a field experiment the design did not allow for clean estimation of a price elasticity of healthcare demand Instead it allows the researchers to measure how much poor people increase their healthcare utilization when they have access to formal insurance rather than having to rely on paying out of pocket borrowing money or skipping bill paying outright The authors found that access to Medicaid increased healthcare expenditures by 778 about a 25 per cent increase The expenditures appeared to have beneficial effects reducing self reports of financial strain by over 30 per cent and increasing self reports of being in good health by 25 percent While health economists have focused on the prob lem that full insurance may cause overconsumption of medi cal services this study hints at the contrasting possibility that a lack of insurance may cause underconsumption E184 Adverse selection in health insurance The experimental studies in the previous section sought to measure pure moral hazard putting aside adverse selection The article by Einav Finkelstein and Cullen 2010 examined a natural experiment that can be used to go in the opposite direction measuring pure adverse selection putting aside 3The assumption that mc 1q2 is a horizontal line of height 1 5 100 while natural is not completely general If health care is supplied by imperfect competitors at some markup over costs then the social cost of health care will be less than the amount expended on it some of the expenditure will flow to the healthcare providers as a rent Alternatively assuming mc 1q2 is less than 100 percent will generally reduce the estimates of deadweight loss from moral hazard 2The idea of estimating demand by fitting a regression line to group means and using this demand curve to calculate welfare and deadweight loss was introduced by Einav Finkelstein and Cullen 2010 studied in the next section They apply the methodology to compute the deadweight loss due to adverse selection We mirror their methods here to compute the deadweight loss due to moral hazard Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 679 moral hazard They studied the administrative data from Alcoa one of the worlds largest lightweight metal manufac turer The conglomerates seven divisions were run by different managers who had the responsibility of setting the terms of the benefits packages including the premiums for two health insurance policies that employees could choose between a basic policy and a gold policy involving fuller insurance and a surcharge The price variation across divisions seemed to be fairly random having little to do with the makeup of the divi sions risk pools which appeared fairly similar to each other based on data on previous years expenditures Table 183 reproduces some key results The different per centages of employees taking up the gold policy at the differ ent surcharges across the divisions in effect trace out points along the demand curve for gold insurance The points are drawn in Figure E182 as filled circles and the solid line fitting the points weighted by number of observations is labeled P 1Q2 This is not an individuals demand curve as in the previ ous figure but market demand where a market in this context is one of Alcoas divisions The average cost of serving con sumers selecting the gold policy in each division are drawn as open circles and the dotted line fitting these points is labeled AC1Q2 The equations for these lines are P 1Q2 5 1081 2 1023Q iv AC1Q2 5 585 2 198Q v Filled circles are percentage of employees taking up gold policy at surcharge in each Alcoa division in Einav Finkelstein and Cullen 2010 study Open circles are average cost of serving consumers selecting gold policy in division Circle size proportional to log of observations in division Solid demand curve P1Q2 fitted to filled circles Dotted average cost curve AC1Q2 fitted to open circles its downward slope indicates presence of adverse selection Firstbest social welfare equals area of triangle ACE Competitive market undersupplies insurance leading to deadweight loss equal to the area of triangle BCD FIGURE E182 Welfare Loss from Adverse Selection in Alcoa Experiment ACQ MCQ PQ 0 200 400 600 800 1000 0 20 40 60 80 100 Q taking up gold policy P gold policy surcharge A B C D E TABLE 183 KEY RESULTS FROM ALCOA EXPERIMENT Premium Difference 2004 Fraction Choosing Gold Contract Average Cost 2004 Individuals 384 067 451 2939 466 066 499 67 495 064 459 526 570 046 493 199 659 049 489 41 Source Table 2 from Einav Finkelstein and Cullen 2010 Division with only seven individual observations omitted to make patterns show up more clearly Weighted regression lines differ slightly from authors because of that omission Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 680 Part 8 Market Failure How does adverse selection show up in this figure The presence of adverse selection boils down to the fact that AC1Q2 is downward sloping The downward slope here has nothing to do with economies of scale in insurance provision By construction the only cost accounted for entering into here comes from payouts for medical expenses The down ward slope here comes from differences in medical risks for consumers who buy at different prices The consumers who choose the gold policy when a high surcharge is imposed have high medical costs on average in the subsequent year The additional consumers who are induced to choose the policy at lower surcharges have lower medical costs pulling down the average In the pareddown model of adverse selection in insurance studied in the chapter there is no other possibility than for AC1Q2 to slope down Consumers only differed in the probability of a claim so highdemand consumers were necessarily highcost consumers This need not be the case in the real world In a richer environment consumers may differ in their health risk aversion and many other dimensions The highest demand consumers may not expect many medical charges but just very risk averse In that case there may be the opposite of adverse selectionadvantageous selectionand AC1Q2 may slope up The estimates from the Alcoa experi ment indicate that in that setting there is adverse selection because AC1Q2 slopes down Assuming insurance is competitively supplied equi librium will be determined by the breakeven point where P 1Q2 5 AC1Q2 Call the associated equilibrium quantity Qc which one can see from the graph is Qc 5 60 This is not the efficient quantity The first best is achieved at the point at which P 1Q2 5 MC1Q2 generating social welfare equal to the area of triangle ACE in the figure Call the socially efficient quantity Qs which one can see from the graph is Qs 5 80 Because MC1Q2 AC1Q2 when AC1Q2 is downward slop ing P 1Q2 intersects MC1Q2 at a greater quantity than where it intersects AC1Q2 ie Qs Qc 4 Welfare is lost when consum ers between Qc and Qs are not served because they value the insurance more than the marginal cost of serving them How ever to induce them to buy would require such a steep price drop that average cost could no longer be covered preventing competitive firms from breaking even The deadweight loss from undersupply of insurance due to adverse selection equals the area of the shaded triangle 1 2 3AC1Qc2 2 MC1Qc2 4 1Qs 2 Qc2 5 1 2 1466 2 3472 180 2 602 12 As we have scaled quantity as percentage of eligible consum ers taking up the gold policy this 12 figure represents the deadweight loss per consumer due to adverse selection in this market a very small number only 6 percent of firstbest wel fare given by the area of triangle ACE Having worked so hard on the calculations it is worth circling back to reconsider what made the Alcoa divisions an ideal natural experiment for these purposes The calcu lations required an estimate of the demand curve for insur ance If managers had chosen the surcharge for the gold plan strategically say to economize on benefit expenses maximize employee wellbeing or achieve some other goal and they shared the same goal then the only reason prices would have differed across the divisions is that the underlying populations were systematically different Instead of having six points on the same demand curve the observations from the different divisions would be individual points on six different demand curves But one point does not determine a line so there would be no way to estimate the required demand curve What made the Alcoa experiment unique is that managers set the surcharges in a seemingly random way resulting in price variation even though the population of employees was fairly similar across divisions as the authors argue E185 Asymmetric Information in Consumer Credit Asymmetric information is not solely a problem in insur ance markets Karlan and Zinman 2009 study the extent of adverse selection and moral hazard in consumer credit mar kets A high interest rate may generate adverse selection if it ends up attracting only those highrisk borrowers unfazed by the interest rate because they are unlikely to pay the loan back in any event High interest rates may also generate moral haz ard Why work hard and live frugally if most of the benefit is siphoned off by burdensome repayments Putting asym metric information aside a higher interest rate can lead to more defaults through a simple liquidity effect Higher inter est rates lead to higher repayments which may be difficult to afford on a given budget Like the Alcoa experiment this one also involves a large firm in this case one of the largest microlenders in South Africa specializing in highinterest loans annual interest rates of around 200 percent to high risk borrowers The authors designed a way to separately measure adverse selection and moral hazard in the same field experiment implemented by the firm One wonders how the researchers were able to convince a forprofit firm to invest the considerable resources needed to run such a largescale field experiment involving over 4000 customers At the same time researchers were uncovering deep insights about asymmetric information the firm likely thought it could learn about its customer base allowing it to 4The expression for AC1Q2 in Equation v can be used to compute total cost which can be differentiated to find MC1Q2 5 585 2 396 Q The demand and cost equations can be used to solve for precise values of Qc and Qs rather than eyeballing them from the figure Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 18 Asymmetric Information 681 offer more profitable loan products in the future The trend toward the use of data analytics to improve a firms perfor mance following the example of the Oakland As baseball team in Moneyball may open up a wealth of opportunities for researchers to run field experiments to answer longstanding economic questions Their ingenious design randomly selected customers to receive one of three different interest rates in a flyer advertis ing a new loan product Upon applying for loan some con sumers were randomly selected to receive a pleasant surprise in the form of a reduction in the actual interest rate Varia tion in the advertised rate tests for pure adverse selection in particular whether only the worst credit risks were attracted by the highest interest rates Surprise variation in the actual rate offered to consumers receiving the same advertised rate holds constant selection customers cannot select whether to apply based on a factor they could not have anticipated isolating the moralhazard and liquidity effects of high inter est rates A final treatment involved a surprise reward for timely repayment in the form of the promise to extend future loans at attractive rates also randomized among customers This treatment allows for a relatively clean test of moral haz ard because the reward only comes in the future so does not relax present liquidity constraints The authors found that variation in the advertised interest rate had little effect on the probability of consumer default suggesting that adverse selection may not be important in this market Surprise variation in the actual interest rate also had little effect The surprise reward for timely repayment had the significant effect eliminating about 15 of total defaults sug gesting that moral hazard is an important force in this market E186 Summary None of these experiments alone provides the final answer on how much welfare is lost to asymmetricinformation prob lems across the economy At best each provides an isolated case study of one type of contract in a small subpopulation one firm or one income group in one state However the accumulation of case studies can start to show general pat terns In addition these prominent studies often provided conceptual and methodological advances aiding followon research The pattern emerging from the handful of experi ments surveyed here is that moral hazard is a more important than adverse selection as a source of asymmetric information and deadweight loss References Einav L A Finkelstein and M R Cullen Estimating Wel fare in Insurance Markets Using Variation in Prices Quar terly Journal of Economics 125 August 2010 877921 Finkelstein A et al The Oregon Health Insurance Exper iment Evidence from the First Year Quarterly Journal of Economics 127 August 2012 10571106 Karlan D and J Zinman Observing Unobservables Identi fying Information Asymmetries with a Consumer Credit Field Experiment Econometrica 77 November 2009 19932008 Lewis M Moneyball The Art of Winning an Unfair Game New York Norton 2003 Manning W G et al Health Insurance and the Demand for Medical Care Evidence from a Randomized Experiment American Economic Review 77 June 1987 25177 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 683 CHAPTER NINETEEN Externalities and Public Goods In Chapter 13 we looked briefly at a few problems that may interfere with the allocational efficiency of perfectly competitive markets Here we will examine two of those problems externalities and public goodsin more detail This examination has two purposes First we wish to show clearly why the existence of externalities and public goods may distort the allocation of resources In so doing it will be possible to illustrate some additional features of the type of information provided by competitive prices and some of the circumstances that may diminish the usefulness of that information Our second reason for looking more closely at externalities and public goods is to suggest ways in which the allocational prob lems they pose might be mitigated We will see that at least in some cases the efficiency of competitive market outcomes may be more robust than might have been anticipated 191 DEFINING EXTERNALITIES Externalities occur because economic actors have effects on third parties that are not reflected in market transactions Chemical makers spewing toxic fumes on their neigh bors jet planes waking up people and motorists littering the highway are from an eco nomic point of view all engaging in the same sort of activity They are having a direct effect on the wellbeing of others that is outside market channels Such activities might be contrasted to the direct effects of markets When I choose to purchase a loaf of bread for example I perhaps imperceptibly increase the price of bread generally and that may affect the wellbeing of other bread buyers But such effects because they are reflected in market prices are not externalities and do not affect the markets ability to allocate resources efficiently1 Rather the increase in the price of bread that results from my increased purchase is an accurate reflection of societal preferences and the price increase helps ensure that the right mix of products is produced That is not the case for toxic chem ical discharges jet noise or litter In these cases market prices of chemicals air travel or disposable containers may not accurately reflect actual social costs because they may take no account of the damage being done to third parties Information being conveyed by market prices is fundamentally inaccurate leading to a misallocation of resources 1Sometimes effects of one economic agent on another that take place through the market system are termed pecuniary externalities to differentiate such effects from the technological externalities we are discussing Here the use of the term externalities will refer only to the latter type because these are the only type with consequences for the efficiency of resource allocation by competitive markets Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 684 Part 8 Market Failure Before analyzing in detail why failing to take externalities into account can lead to a misallocation of resources we will examine a few examples that should clarify the nature of the problem 1911 Externalities in production To illustrate the externality issue in its simplest form we consider two firms one produc ing good x and the other producing good y The production of good x is said to have an external effect on the production of y if the output of y depends not only on the inputs chosen by the yentrepreneur but also on the level at which the production of x is carried on Notationally the production function for good y can be written as y 5 f1k l x2 191 where x appears to the right of the semicolon to show that it is an effect on production over which the yentrepreneur has no control2 As an example suppose the two firms are located on a river with firm y being downstream from x Suppose firm x pollutes the river in its productive process Then the output of firm y may depend not only on the level of inputs it uses itself but also on the amount of pollutants flowing past its factory The level of pollutants in turn is determined by the output of firm x In the production function shown by Equation 191 the output of firm x would have a negative marginal physical pro ductivity yx 0 Increases in x output would cause less y to be produced In the next section we return to analyze this case more fully since it is representative of most simple types of externalities 1912 Beneficial externalities The relationship between two firms may be beneficial Most examples of such positive externalities are rather bucolic in nature Perhaps the most famous proposed by J Meade involves two firms one producing honey raising bees and the other producing apples3 Because the bees feed on apple blossoms an increase in apple production will improve productivity in the honey industry The beneficial effects of having wellfed bees are a pos itive externality to the beekeeper In the notation of Equation 191 yx would now be positive In the usual perfectly competitive case the productive activities of one firm have no direct effect on those of other firms yx 5 0 1913 Externalities in consumption Externalities also can occur if the activities of an economic actor directly affect an indi viduals utility Most common examples of environmental externalities are of this type From an economic perspective it makes little difference whether such effects are created by firms in the form say of toxic chemicals or jet noise or by other individuals litter or 2We will find it necessary to redefine the assumption of no control considerably as the analysis of this chapter proceeds 3J Meade External Economies and Diseconomies in a Competitive Situation Economic Journal 62 March 1952 5467 D E F I N I T I O N Externality An externality occurs whenever the activities of one economic actor affect the activities of another in ways that are not reflected in market transactions As a summary then we have developed the following definition Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 685 perhaps the noise from a loud radio In all such cases the amount of such activities would enter directly into the individuals utility function in much the same way as firm xs output entered into firm ys production function in Equation 191 As in the case of firms such externalities may sometimes be beneficial you may actually like the song being played on your neighbors radio So again a situation of zero externalities can be regarded as the middle ground in which other agents activities have no direct effect on individuals utilities One special type of utility externality that is relevant to the analysis of social choices arises when one individuals utility depends directly on the utility of someone else If for example person A cares about person Bs welfare then we could write his or her utility function 1U A2 as utility 5 UA1x1 xn UB2 192 where x1 xn are the goods that A consumes and U B is Bs utility If A is altruistic and wants B to be well off as might happen if B were a close relative U AU B would be pos itive If on the other hand A were envious of B then it might be the case that U AU B would be negative that is improvements in Bs utility make A worse off The mid dle ground between altruism and envy would occur if A were indifferent to Bs welfare 1U AU B 5 02 and that is what we have usually assumed throughout this book for a brief discussion see the Extensions to Chapter 3 1914 Externalities from public goods Goods that are public or collective in nature will be the focus of our analysis in the second half of this chapter The defining characteristic of these goods is nonexclusion that is once the goods are produced either by the government or by some private entity they provide benefits to an entire groupperhaps to everyone It is technically impossible to restrict these benefits to the specific group of individuals who pay for them so the ben efits are available to all As we mentioned in Chapter 13 national defense provides the traditional example Once a defense system is established all individuals in society are pro tected by it whether they wish to be or not and whether they pay for it or not Choosing the right level of output for such a good can be a tricky process because market signals will be inaccurate 192 EXTERNALITIES AND ALLOCATIVE INEFFICIENCY Externalities lead to inefficient allocations of resources because market prices do not accurately reflect the additional costs imposed on or benefits provided to third parties To illustrate these inefficiencies requires a general equilibrium model because inefficient allocations in one market throw into doubt the efficiency of marketdetermined outcomes everywhere Here we choose a simple general equilibrium model that allows us to make these points in a compact way Specifically we assume there is only one person in our economy whose utility U1x y2 depends on the quantities x and y of two goods consumed the variables will also be used for the goods names The person is endowed with labor l the only input in the economy He or she can allocate lx to the production of good x and ly for good y where lx 1 ly 5 l 193 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 686 Part 8 Market Failure The production function for the good x is straightforward x 5 f1lx2 194 To illustrate externalities we assume that the output of y depends not only on how much labor is used to produce it but also on how much of x is produced This would model a situation say where the firm producing y is downriver from the other firm and must cope with the pollution created when x is produced The production function for y is given by y 5 g 1ly x2 195 Regarding the partial derivatives of this production function we will assume gl 0 more labor input naturally produces more output but will explore various signs for the other partial derivative The case of a negative externality such as pollution flowing downriver can be captured by gx 0 the case of a positive externality such as bees pollinating apple trees can be captured by gx 0 To provide a parallel treatment we will denote the deriv ative of the first production function as fl 5 f r 1lx2 even though it is a function of a single variable and so partial derivative notation would not otherwise be needed 1921 Finding the efficient allocation The economic problem for society then is to maximize utility U1x y2 5 U1 f 1lx2 g 1ly x2 2 5 U1 f 1lx2 g 1ly f 1lx2 2 2 196 subject to the constraint on labor endowment The Lagrangian expression for this maximi zation problem is 5 U1 f 1lx2 g 1ly f 1lx2 2 2 1 λ 1l 2 lx 2 ly2 197 Careful application of the chain rule gives the two firstorder conditions lx 5 Ux fl 1 Uy gx fl 2 λ 5 0 198 ly 5 Uy gl 2 λ 5 0 199 Using Equation 199 to substitute for λ in Equation 198 dividing the resulting equation through by Uy fl and rearranging yields MRS 5 Ux Uy 5 gl fl 2 gx 5 RPT 1910 The ratio of marginal utilities on the lefthand side is the persons MRS in consumption The righthand side although it requires some discussion to see it reflects the tradeoff between the two goods on the production side what we called the RPT in Chapter 13 The first term glfl is the ratio of the marginal products ordinarily showing up in RPT embodying how reallocating labor shifts the production of the two goods in the absence of an externality The second term 2gx represents the externality that the production of x has on y which has to be taken into account to generate an efficient allocation We will show in the next section that Equation 1910 which is required for efficiency does not hold in the competitive allocation proving that the competitive allocation is inef ficient in the presence of externalities Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 687 1922 Inefficiency of the competitive allocation Facing equilibrium prices px and py a utilitymaximizing individual would opt for MRS 5 px py 1911 Facing wage rate w a profitmaximizing producer of x would set choose lx such that the marginal revenue product of labor equals the input price px fl 5 w The producer of y would choose analogously py gl 5 w Combining these conditions for profit maximiza tion px fl 5 py gl or upon rearranging px py 5 gl fl 1912 Combining this with the previous equation gives the equilibrium condition for competitive pricing MRS 5 gl fl 1913 This equation looks like the MRS 5 RPT condition that in Chapter 13 led to efficiency but in this model with an externality the ratio of marginal products glfl is not the true RPT The true RPT on the righthand side of Equation 1910 includes an extra term to account for the externality Its absence from Equation 1913 reflects the fact that the producer of x ignores the effect of its output on the other firms production in the competitive equi librium Whether the competitive equilibrium involves too much or too little x depends on whether the externality is positive or negative If it is a negative externality pollution flowing downriver for example then gx 0 Subtracting a negative term leads to a greater MRS in Equation 1910 than 1913 Recalling that MRS which measures the absolute value of the slope of an indifference curve is decreasing as x increases the greater MRS in Equa tion 1910 means that the socially efficient level of x is lower than in the competitive allo cation In other words the competitive market leads to too much of the good generating a negative externality On the other hand if x generates a positive externality pollinating bees for example repeating the previous arguments reversing the sign of gx shows that the MRS in Equation 1910 is lower than in 1913 meaning that the competitive alloca tion involves too little x compared to the social optimum If gx5 0 there is no difference between Equations 1910 and 1913 allowing us to recover the result that the competitive equilibrium is efficient in the absence of an externality EXAMPLE 191 Production Externalities To illustrate the losses from failure to consider production externalities suppose two newsprint producers are located along a river The upstream firm has a production function of the form x 5 f1lx2 5 2000lx 1914 where lx is the number of workers hired per day and x is newsprint output in feet The down stream firm has a similar production function but its output may be affected by the chemicals the upstream firm dumps into the river as it produces more x y 5 g 1ly x2 5 2000ly 11 1 αx2 1915 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 688 Part 8 Market Failure If α 5 0 then the upstream firms production has no effect on the downstream firm but if α 0 then an increase in x harms the downstream firms production Assuming newsprint sells for P 5 1 per foot and workers earn w 5 100 per day the upstream firm will maximize profits by setting this wage equal to labors marginal revenue product 100 5 P df dlx 5 1000 l212 x 1916 The solution then is lx 5 100 If α 5 0 there are no externalities the downstream firm will also hire 100 workers Each firm will produce 20000 feet of newsprint Effects of a negative externality When the upstream firm generates negative externality 1α 02 its profitmaximizing hiring decision is not affectedit sill hires lx 5 100 and produces x 5 20000 But the downstream firms marginal product of labor is lower because of this exter nality If α 5 2140000 for example then profit maximization requires 100 5 P g ly 5 1000 l212 y 11 1 αx2 5 1000 l212 y 11 2 20000400002 5 500 l212 y 1917 Solving this equation for ly shows that the downstream firm now hires only 25 workers because of this lowered productivity This firms output now is y 5 200025 11 2 20000400002 5 5000 1918 Because of the externality 1α 5 21400002 the downstream firm produces less than without the externality 1α 5 02 Inefficiency We can demonstrate that decentralized profit maximization is inefficient in this situation by imagining that the two firms merge and that the manager must decide how to allo cate the combined workforce If one worker say is transferred from the upstream to the down stream firm then upstream output becomes x 5 200099 5 19900 1919 but downstream output becomes x 5 200026 11 2 19900400002 5 5125 1920 Total output has increased by 25 feet of newsprint with no change in total labor input The mar ketbased allocation was inefficient because the upstream firm did not take into account the neg ative effect of its output on the downstream firm Social marginal cost The inefficiency in the premerger situation can be demonstrated in another way by comparing the upstream firms private marginal cost to the social marginal cost of an increase in x To compute the firms marginal cost we solve for its total cost and differentiate By Equation 1914 to produce x units the firm needs to hire lx 5 x24000000 workers Hence its total cost and marginal costs are TC1x2 5 wlx 5 100 x2 4000000 5 x2 40000 1921 MC1x2 5 TCr 1x2 5 x 20000 1922 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 689 193 PARTIALEQUILIBRIUM MODEL OF EXTERNALITIES Example 191 suggests a way to derive the key insights from a rigorous generalequilib rium analysis quite simply using a supplydemand diagram as illustrated in Figure 191 Dealing with one good allows us to dispense with the cumbersome x and y notation for quantities and just use Q to denote the market quantity of the good that may have an asso ciated externality Private marginal costs of production are denoted MC1Q2 which also represents the competitive supply curve for the good The inverse demand curve is P 1Q2 The competitive equilibrium is given by the intersection between supply and demand equivalently the intersection between P 1Q2 and MC1Q2yielding the quantity Qc Suppose that increases in Q cause harm to third parties a negative externality illustrated in panel a There are several possible ways the negative externality might be generated One is that the production of Q shifts the production function down for goods not shown in the figure much like the production of x did to y in Example 191 Another is that the pro duction of Q harms neighbors besides firms Pollution that a newsprint factory spills into a river may reduce the utility that visitors to a downstream park obtain from swimming and fishing in the river A third possibility is that the act of consuming rather than producing Q causes the harm to other people for example cigarettes may harm the health of those who breathe the secondhand smoke All of these possibilities create a divergence between An increase in x has social costs beyond the upstream firms expenditures on the labor input Here the additional social cost is the harm done to the downstream firm which can be mea sured in monetary terms by the profits it loses when x is increased In principle the downstream firms consumers could also be harmed by the reduction in its output caused by more upstream pollution however because the market price for newsprint is a constant that does not depend on the downstream firms outputpresumably the downstream firm is just a tiny player in that marketthe absence of a price effect means there is no measurable consumer harm Computing lost profits takes a bit of work We can substitute from Equation 1915 to write the downstream firms profits as πy 5 Py 2 wly 5 2000ly 11 1 αx2 2 100ly 1923 Using the envelope theorem dπy dx 5 πy x l y 5 2000αl y 5 20000α 11 1 αx2 1924 The last equality follows by substituting the value of ly that maximizes Equation 1923 namely l y 5 100 11 1 αx2 2 Equation 1924 gives the profit gain from an increase in x the profit loss is the negative of this Combining the profit loss with the upstream firms private marginal cost gives the comprehensive social marginal cost SMC1x2 5 x 20000 2 20000α 11 1 αx2 5 3x 80000 1 1 2 1925 substituting the particular value α 5 2140000 Social marginal cost exceeds private marginal cost leading the upstream firm to produce too much x in the competitive outcome QUERY Suppose α 5 1140000 What would that imply about the relationship between the firms How would such an externality affect the allocation of labor Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 690 Part 8 Market Failure The inverse demand curve for the good is given by PQ and private marginal costs by MCQ which is also the market supply curve Panel a illustrates the case of a negative externality in which the good imposes external costs on third parties Social marginal costs SMCQ exceed MCQ by the extent of these costs market equilibrium quantity Qc exceeds the socially efficient quantity Qs Panel b illus trates the case of a positive externality in which the good benefits third parties Social marginal benefits SMBQ lie above the inverse demand curve implying Qc Qs In both panels deadweight loss from inefficient quantity given by area of shaded region FIGURE 191 PartialEquilibrium Model of Externalities SMCQ MCQ PQ Qc Qs Ps Pc a Negative externality Price costs Output per period SMBQ MCQ PQ Qs Qc Ps Pc b Positive externality Price costs Output per period Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 691 private marginal costs MC1Q2 and overall social marginal costs SMC1Q2 The vertical dis tance between the two curves in panel a represents the harm imposed on third parties Notice that the perunit costs of these externalities need not be constant independent of output Q In the figure for example the size of these external costs given by the gap between the marginal cost curves increases as Q increases At the marketdetermined output level Qc the comprehensive social marginal cost exceeds the market price Pc thereby indicating that output has been pushed too far It is clear from the figure that the socially optimal output level is Qs at which the market price Ps paid for the good now reflects all costs Panel b illustrates the case of a positive externality The positive externality may arise when the goods production enhances the production of other goods as with the example of a honey producers bees pollinating nearby orchards trees The positive externality may arise when the consumption of a good directly benefits other people such as a fresh coat of paint on a house contributing to the overall beauty of a neighborhood enjoyed by all the homeowners in that location However the positive externality is generated the resulting social marginal benefit from output of the good shown as the SMB1Q2 curve will exceed P 1Q2 In this context it is useful to think of P 1Q2 as the benefit of the consumer making the marginal buying decision in other words the marginal private benefit function The competitive price Pc reflecting the marginal private benefit at output Qc lies below the full social marginal benefit at Qc Therefore the competitive output is less than the socially optimal output Qs in the panel b case of a positive externality 194 SOLUTIONS TO NEGATIVE EXTERNALITY PROBLEMS Incentivebased solutions to the harm from negative externalities start from the basic observation that output of the externalityproducing activity is too high under a marketdetermined equilibrium Perhaps the first economist to provide a complete analysis of this distortion was A C Pigou who in the 1920s suggested that the most direct solution would simply be to tax the externalitycreating entity4 All incentivebased solutions to the externality problem stem from this basic insight5 For concreteness this section takes the case of a negative externality Analogous arguments apply to the case of a positive external ity The overproduction problem becomes an underproduction problem the tax solution becomes a subsidy and so forth but the economic logic remains the same 1941 Pigovian tax Figure 192 shows how Pigous taxation solution can be used to eliminate the deadweight loss from the negative externality seen in panel a of Figure 191 As is the case for any tax imposition of a Pigovian tax here creates a vertical wedge between the demand and supply curves for the good In the figure the relevant demand curve is the inverse demand labeled P 1Q2 and the supply curve is determined by private marginal cost MC1Q2 The optimal tax is shown as t Imposition of this tax serves to reduce output from Qc to Qs the social 4A C Pigou The Economics of Welfare London MacMillan 1920 Pigou also recognized the importance of subsidizing goods that yield positive externalities 5We do not discuss purely regulatory solutions here although the study of such solutions forms an important part of most courses in environmental economics See W J Baumol and W E Oates The Theory of Environmental Policy 2nd ed Cambridge Cambridge University Press 2005 and the Extensions to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 692 Part 8 Market Failure optimum The tax is set to reflect the distance between private marginal costs MC1Q2 and social marginal costs SMC1Q2 Notice that the tax must be set to reflect marginal harm at the optimal quantity Qs rather than at the original market equilibrium quantity Qc This point is also made in the next example and more completely in the next section by return ing to our simple general equilibrium model This figure reproduces the panel from the previous figure illustrating a negative externality A tax of amount t that reflects the social marginal costs above and beyond private marginal costs would achieve the socially efficient production level Qs SMCQ MCQ PQ Qc Qs Ps Pc Price costs Output per period a Negative externality t FIGURE 192 Pigovian Tax EXAMPLE 192 A Pigovian Tax on Newsprint The inefficiency in Example 191 arises because the upstream newsprint producer produced output x without taking into account the effect of its production on the downstream firm A suitably chosen tax can cause the upstream firm to reduce output to the efficient level Equilibrium without a tax For comparison we will first review the competitive equilibrium in the absence of a tax An easy way to solve for this uses the marginal cost curve derived at the end of Example 191 for the upstream firm MC1x2 5 x20000 A price taker in the newsprint market the upstream firm maximizes profit by setting the market price P 5 1 equal to marginal cost yielding xc 5 20000 exactly as we showed in the previous example Pigovian tax This output for the upstream firm is too high to be socially efficient The socially effi cient output is not where P 5 MC1x2 but where P 5 SMC1x2 Using the SMC1x2 function derived in the previous example for the particular value of the negative externality α 5 2140000 we have 1 5 P 5 SMC1x2 5 3x 80000 1 1 2 1926 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 693 1942 Taxation in the generalequilibrium model The optimal Pigovian tax in our generalequilibrium model is to set t 5 2py gx That is the perunit tax on good x should reflect the marginal harm that x does in reduc ing y output valued at the market price of good y Notice again that this tax must be based on the value of this externality at the optimal solution because gx will generally be a function of the level of x output a tax based on some other output level would be inappropriate To prove that the proposed Pigovian tax is socially optimal imagine it is levied on consumers implying that total price paid by the consumer is px 1 t and the price received by the firm is px As we know from Chapter 12 the incidence of the tax does not depend on which side consumers or producers it is levied imagining it is levied on consumers here gives us a convention to start from Consumer utility maximization gives the condition MRS 5 px 1 t py 5 px py 2 gx 1928 when the tax is set at the proposed level t 5 2py gx Because the prices received by firms are denoted px and py as before profit maximization by firms continues to give Equation 1912 Combining Equations 1912 and 1928 gives the condition for competitive equilib rium with a tax MRS 5 gl fl 2 gx 1929 identical to the condition for social efficiency in Equation 1910 proving that the proposed Pigovian tax is indeed optimal The Pigovian taxation solution can be generalized in a variety of ways that provide insights about the conduct of policy toward externalities For example in an economy with many xproducers the tax would convey information about the marginal impact that out put from any one of these would have on y output Hence the tax scheme mitigates the need for regulatory attention to the specifics of any particular firm It does require that regulators have enough information to set taxes appropriatelythat is they must know firm ys production function giving the solution for the socially optimal level of upstream output xs 5 13333 The Pigovian tax that arrives at this social optimum can be found by setting t equal to the wedge between mar ket price P and marginal cost MC1xs2 that is t 5 P 2 MC1xs2 5 1 2 13333 20000 5 1 3 1927 A Pigovian tax of t 5 13 confronts the upstream firm with the harm its output and attendant pollution causes to the downstream firm leading to the socially efficient output QUERY The Pigovian tax was set to the wedge between price and private marginal cost at the socially efficient output level xs 5 13333 What happens if the private marginal cost at the com petitive output level xc 5 20000 is used instead to compute the wedge Why does it make a difference what output level is used Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 694 Part 8 Market Failure 1943 Pollution rights An innovative policy that would mitigate the informational requirements involved with Pigovian taxation is the creation of a market for pollution rights Suppose for example that firm x must purchase from firm y rights to pollute the river they share In this case xs decision to purchase these rights is identical to its decision to choose its output level because it cannot produce without them The net revenue x receives per unit is given by px 2 r where r is the payment the firm must make for each unit it produces Firm y must decide how many rights to sell to firm x Because it will be paid r for each right it must choose x output to maximize its profits πy 5 py g 1lx x2 1 rx 1930 the firstorder condition for a maximum is πy x 5 py gx 1 r 5 0 or r 5 2py gx 1931 Equation 1931 makes clear that the equilibrium solution to pricing in the pollution rights market will be identical to the Pigovian tax equilibrium From the point of view of firm x it makes no difference whether a tax of amount t is paid to the government or a royalty r of the same amount is paid to firm y So long as t 5 r a condition ensured by Equation 1931 the same efficient equilibrium will result 1944 The Coase theorem In a famous 1960 paper Ronald Coase showed that the key feature of the pollution rights equilibrium is that these rights be well defined and tradable with zero transaction costs6 The initial assignment of rights is irrelevant because subsequent trading will always yield the same efficient equilibrium In our example we initially assigned the rights to firm y allowing that firm to trade them away to firm x for a perunit fee r If the rights had been assigned to firm x instead that firm still would have to impute some cost to using these rights themselves rather than selling them to firm y This calculation in combination with firm ys decision about how many such rights to buy will again yield an efficient result To illustrate the Coase result assume that firm x is given xT rights to produce and to pollute It can choose to use some of these to support its own production x or it may sell some to firm y an amount given by xT2x Gross profits for x are given by πx 5 px x 1 r1xT 2 x2 5 1 px 2 r2x 1 rxT 5 1 px 2 r2f1lx2 1 rxT 1932 and for y by πy 5 py g 1ly x2 2 r1xT 2 x2 1933 Clearly profit maximization in this situation will lead to precisely the same solution as in the case where firm y was assigned the rights Because the overall total number of rights 1xT 2 is a constant the firstorder conditions for a maximum will be exactly the same in the two cases This independence of initial rights assignment is usually referred to as the Coase theorem Although the results of the Coase theorem may seem counterintuitive how can the level of pollution be independent of who initially owns the rights it is in reality nothing more than the assertion that in the absence of impediments to making bargains all mutually 6R Coase The Problem of Social Cost Journal of Law and Economics 3 October 1960 144 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 695 beneficial transactions will be completed When transaction costs are high or when infor mation is asymmetric initial rights assignments will matter because the sorts of trading implied by the Coase theorem may not occur Therefore it is the limitations of the Coase theorem that provide the most interesting opportunities for further analysis This analysis has been especially far reaching in the field of law and economics7 where the theorem has been applied to such topics as tort liability laws contract law and product safety legislation see Problem 194 195 ATTRIBUTES OF PUBLIC GOODS We now turn our attention to a related set of problems about the relationship between competitive markets and the allocation of resources those raised by the existence of public goods We begin by providing a precise definition of this concept and then examine why such goods pose allocational problems We then briefly discuss theoretical ways in which such problems might be mitigated before turning to examine how actual decisions on pub lic goods are made through voting The most common definitions of public goods stress two attributes of such goods non exclusivity and nonrivalness We now describe these attributes in detail 1951 Nonexclusivity The first property that distinguishes public goods concerns whether individuals may be excluded from the benefits of consuming the good For most private goods such exclusion is indeed possible I can easily be excluded from consuming a hamburger if I dont pay for it In some cases however such exclusion is either very costly or impossible National defense is the standard example Once a defense system is established everyone in a coun try benefits from it whether they pay for it or not Similar comments apply on a more local level to goods such as mosquito control or a program to inoculate against disease In these cases once the programs are implemented no one in the community can be excluded from those benefits whether he or she pays for them or not Hence we can divide goods into two categories according to the following definition 1952 Nonrivalry A second property that characterizes public goods is nonrivalry A nonrival good is one for which additional units can be consumed at zero social marginal cost For most goods of course consumption of additional amounts involves some marginal costs of production Consumption of one more hot dog requires that various resources be devoted to its pro duction However for certain goods this is not the case Consider for example having one more automobile cross a highway bridge during an offpeak period Because the bridge is 7The classic text is R A Posner Economic Analysis of Law 4th ed Boston Little Brown 1992 A more mathematical approach is T J Miceli Economics of the Law New York Oxford University Press 1997 D E F I N I T I O N Exclusive goods A good is exclusive if it is relatively easy to exclude individuals from benefiting from the good once it is produced A good is nonexclusive if it is impossible or costly to exclude individuals from benefiting from the good Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 696 Part 8 Market Failure already in place having one more vehicle cross requires no additional resource use and does not reduce consumption elsewhere Similarly having one more viewer tune in to a television channel involves no additional cost even though this action would result in addi tional consumption taking place Therefore we have developed the following definition 1953 Typology of public goods The concepts of nonexclusion and nonrivalry are in some ways related Many nonexclusive goods are also nonrival National defense and mosquito control are two examples of goods for which exclusion is not possible and additional consumption takes place at zero mar ginal cost Many other instances might be suggested The concepts however are not iden tical Some goods may possess one property but not the other For example it is impossible or at least very costly to exclude some fishing boats from ocean fisheries yet the arrival of another boat clearly imposes social costs in the form of a reduced catch for all con cerned Similarly use of a bridge during offpeak hours may be nonrival but it is possible to exclude potential users by erecting toll booths Table 191 presents a crossclassification of goods by their possibilities for exclusion and their rivalry Several examples of goods that fit into each of the categories are provided Many of the examples other than those in the upper left corner of the table exclusive and rival private goods are often produced by governments That is especially the case for nonexclusive goods because as we shall see it is difficult to develop ways of paying for such goods other than through compulsory taxation Nonrival goods often are privately produced there are after all private bridges swimming pools and highways that consumers must pay to use as long as nonpayers can be excluded from consuming them8 Still we will use the following stringent definition which requires both conditions TABLE 191 EXAMPLES SHOWING THE TYPOLOGY OF PUBLIC AND PRIVATE GOODS Exclusive Yes No Rival Yes Hot dogs automobiles houses Fishing grounds public grazing land clean air No Bridges swimming pools satellite television transmission scrambled National defense mosquito control justice 8Nonrival goods that permit imposition of an exclusion mechanism are sometimes referred to as club goods because provision of such goods might be organized along the lines of private clubs Such clubs might then charge a membership fee and permit unlimited use by members The optimal size of a club is determined by the economies of scale present in the production process for the club good For an analysis see R Cornes and T Sandler The Theory of Externalities Public Goods and Club Goods Cambridge Cambridge University Press 1986 D E F I N I T I O N Nonrival goods A good is nonrival if consumption of additional units of the good involves zero social marginal costs of production D E F I N I T I O N Public good A good is a pure public good if once produced no one can be excluded from benefiting from its availability and if the good is nonrivalthe marginal cost of an additional consumer is zero Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 697 196 PUBLIC GOODS AND RESOURCE ALLOCATION To illustrate the allocational problems created by public goods we again employ a simple general equilibrium model In this model there are only two individualsa single person economy would not experience problems from public goods because he or she would incorporate all of the goods benefits into consumption decisions We denote these two individuals by A and B Their utility functions U A1x yA2 and U B 1x yB2 depend on the amount each consumes of a public good x and a traditional nonpublic good yA for A and yB for B Labor is the only input in the economy Person A is endowed with lA units of labor He or she can allocate lAx to the production of x and lAy to y where lAx 1 lAy 5 lA Person B is endowed with lB units of labor that he or she can similarly allocate The total labor endowment in the economy is l 5 lA 1 lB Production of the public good depends on their combined labor inputs x 5 f 1lAx 1 lBx2 5 f 1 l 2 lAy 2 lBy2 1934 Production of the traditional good depends on their separate labor input yA 5 g 1lAy2 and yB 5 g 1lBy2 1935 Notice how the mathematical notation captures the essential nature of x as a public good characterized by nonexclusivity and nonrivalry Nonexclusivity is reflected by the fact that As labor input increases the amount of x that B consumes A cannot prevent B from enjoying the fruits of As labor and vice versa Nonrivalry is reflected by the fact that the consumption of x by each person is identical to the total amount of x produced As consumption of x does not diminish what B can consume These two characteristics of good x constitute the barriers to efficient production under most decentralized decision schemes including competitive markets To find the socially efficient outcome we will solve the problem of allocating labor to maximize one persons utility say As for any given level of Bs utility The Lagrangian expression for this problem is 5 U A1 f1l 2 lAy 2 lBy2 g 1lAy2 2 1 λ 3U B 1 f1l 2 lAy 2 lBy2 g 1lBy2 2 2 U 4 1936 where U is a constant level of Bs utility and where we have substituted for x yA and yB from Equations 1934 and 1935 The firstorder conditions for a maximum are lAy 5 2U A x f r 1 U A ygr 2 λU B x f r5 0 1937 lBy 5 2U A x f r 2 λU B x f r 1 λU B ygr 5 0 1938 These two equations together imply U A y 5 λU B y and hence that λ 5 UA y UB y Substituting this value of λ into either one of the firstorder conditions lets say Equation 1937 and rearranging yields U A x U A y 1 U B x U B y 5 gr f r 1939 or more succinctly MRSA 1 MRSB 5 RPT 1940 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 698 Part 8 Market Failure The intuition behind this condition which was first articulated by P A Samuelson9 is that it is an adaptation of the efficiency conditions described in Chapter 13 to the case of pub lic goods For such goods the MRS in consumption must reflect the amount of y that all consumers would be willing to give up to get one more x because everyone will obtain the benefits of the extra x output Hence it is the sum of each individuals MRS that should be equated to the rate of product transformation here given by grfr2 1961 Failure of a competitive market Production of goods x and y in competitive markets will fail to achieve this allocational goal With perfectly competitive prices px and py each individual will equate his or her MRS to the price ratio pxpy As we demonstrated earlier in the chapter profit maximiza tion by producers would lead to an equality between the rate of product transformation grfr and the price ratio pxpy This behavior would not achieve the optimality condition expressed in Equation 1940 The price ratio pxpy would be too low in that it would provide too little incentive to produce good x In the private market a consumer takes no account of how his or her spending on the public good benefits others so that consumer will devote too few resources to such production The allocational failure in this situation can be ascribed to the way in which private markets sum individual demands For any given quantity the market demand curve reports the marginal valuation of a good If one more unit were produced it could then be consumed by someone who would value it at this market price For public goods the value of producing one more unit is in fact the sum of each consumers valuation of that extra output because all consumers will benefit from it In this case then individual demand curves should be added vertically as shown in Figure 193 rather than horizontally 9P A Samuelson The Pure Theory of Public Expenditure Review of Economics and Statistics November 1954 38789 D1 D2 D3 D D1 D2 D3 D Price Quantity per period 3 2 1 3 2 FIGURE 193 Derivation of the Demand for a Public Good For a public good the price individuals are willing to pay for one more unit their marginal valuations is equal to the sum of what each individual would pay Hence for public goods the demand curve must be derived by a vertical summation rather than the horizontal summation used in the case of private goods Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 699 as they are in competitive markets The resulting price on such a public good demand curve will then reflect for any level of output how much an extra unit of output would be valued by all consumers But the usual market demand curve will not properly reflect this full marginal valuation 1962 Inefficiency of a Nash equilibrium One might think that the competitive market fails to produce an efficient level of the public good because the market involves a large number of nonstrategic agents Unfortunately the publicgood problem is more general than that Even two agents who behave strategicallyas long as they act independently rather than pursuing one of the policy solutions we will go on to studywill fail to produce enough of the pub lic good It is true that the publicgood problem tends to get worse as the number of agents increases beyond two Each person considers only his or her benefit from investing in the public good taking no account of the benefits spilling over to others With many consum ers the direct benefit may be very small indeed For example how much do one persons taxes contribute to national defense in the United States In the limit as the number of consumers grows into the thousands or millions any one person may opt for providing essentially none of the public good becoming a pure free rider hoping to benefit from the expenditures of others If every person adopts this strategy then no resources will be allocated to public goods Example 193 illustrates the freerider problem in a situation that may be all too familiar starting from two agents and working up to a large number To analyze strategic behavior rigorously we will look for the Nash equilibrium using the tools learned in the chapter on game theory EXAMPLE 193 The Roommates Dilemma To illustrate the nature of the publicgood problem numerically suppose two roommates A and B having identical preferences derive utility from the cleanliness of their room and the knowl edge gained from economics texts read The specific utility function for roommate A is U A 1x yA2 5 x13y23 A 1941 where yA is the number of hours A spends reading and x 5 xA 1 xB is the sum across roommates of the hours spent cleaning Roommate B has the analogous utility function In this problem x is the public good and y is the private good Assume each roommate can spend up to 10 hours on these activities during the week Thus 10 is like income in their budget constraint and the effec tive prices of the activities in terms of time are both 1 one hour Nash equilibrium We first consider the outcome if the roommates make their consumptions decisions independently without coming to a more or less formal agreement about how much time to spend cleaning Roommate As decision depends on how much time B spends and vice versa This is a strategic situation requiring the tools from the chapter on game theory to analyze We will look for the Nash equilibrium in which roommates play mutual best responses To find As best response take as given the number of hours xB that B spends cleaning A max imizes utility 1xA 1 xB2 13y 23 A subject to the time budget constraint 10 5 xA 1 yA leading to the Lagrangian expression 5 1xA 1 xB2 13 y23 A 1 λ110 2 xA 2 yA2 1942 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 700 Part 8 Market Failure The firstorder conditions are xA 5 1 3 1xA 1 xB2 223 y 23 A 2 λ 5 0 1943 xB 5 2 3 1xA 1 xB2 13 y213 A 2 λ 5 0 1944 Solving these equations in the usual way to eliminate λ gives yA 5 2 1xA 1 xB2 Substituting into the budget constraint implies 10 5 xA 1 yA 5 xA 1 2 1xA 1 xB2 5 3xA 1 2xB 1945 Recognizing that the equilibrium will be symmetric and thus x A 5 x B we have 10 5 3x A 1 2xA 5 5xA implying xA 5 2 5 xB Equilibrium utilities are U A 5 U B 5 412 823 63 Efficient allocation There are several ways to compute the efficient allocation One way is to use the result that the sum of each persons MRS equals the price ratio In this example MRSA 5 U A x U A y 5 1132x223y 23 A 1232x13y213 A 5 yA 2x 1946 and similarly for B Hence the condition for efficiency is MRSA 1 MRSB 5 yA 2x 1 yB 2x 5 1 1947 implying yA 1 yB 5 2x Substituting into the combined budget constraint 20 5 x 1 yA 1 yB yields 20 5 x 1 2x implying x 5 203 and thus x A 5 x B 5 103 33 and y A 5 y B 5 203 67 Utilities in the efficient allocation are U A 5 U B 5 12032 13 12032 23 67 Comparison The Nash equilibrium involves too little cleaning 2 hours each compared to the 33 hours each in the efficient allocation It might be possible for them to come to a formal or informal agreement to clean more perhaps deciding on a time during which they both can clean the room at the same time so they can monitor each other continuing to clean as long as the other does for the fully efficient 33 hours In the absence of such an agreement the roommates face a similar dilemma as the players in the Prisoners Dilemma The Nash equilibrium both fink is Pareto dominated by another outcome both remain silent Beyond two roommates Considerable insight can be gained if the situation is generalized to an arbitrary number of roommates n who can all contribute to cleaning Now A maximizes util ity 3xA 1 n 2 1xB4 13y 23 A subject to budget constraint 10 5 xA 1 yA where xB is the time spent cleaning by any one of the other roommates We can compute the Nash equilibrium as above setting up the Lagrangian expression taking firstorder conditions and solving The resulting can be set up as we did earlier in this Example derivatives taken resulting in equation yA 5 2x as before In a symmetric equilibrium with n roommates x 5 nx A Substituting y A 5 2x 5 2nx A into the budget constraint yields 10 5 x A 1 2nx A implying xA 5 10 2n 1 1 1948 As n becomes arbitrarily large each roommate provides essentially no cleaning a clear demon stration of the freerider problem The efficient amount of cleaning satisfies n MRSA 5 n yA 2x 5 1 1949 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 701 197 LINDAHL PRICING OF PUBLIC GOODS An important conceptual solution to the public goods problem was first suggested by the Swedish economist Erik Lindahl10 in the 1920s Lindahls basic insight was that individ uals might voluntarily consent to be taxed for beneficial public goods if they knew that others were also being taxed Specifically Lindahl assumed that each individual would be presented by the government with the proportion of a public goods cost he or she would be expected to pay and then reply honestly with the level of public good output he or she would prefer In the notation of our simple general equilibrium model individual A would be quoted a specific percentage 1αA2 and then asked what level of public goods he or she would want given the knowledge that he or she would have to contribute this fraction of the required labor To answer that question truthfully this person would choose that overall level of public goods output x that maximizes U A1x g 1lA 2 αA f 21x2 2 1950 The firstorder condition for this utilitymaximizing choice of x is given by U A x 2 αAU A ygra 1 f rb 5 0 1951 or MRSA 5 αA gr f r 5 αARPT 1952 Individual B presented with a similar choice would opt for a level of public goods satisfying MRSB 5 αBRPT 1953 An equilibrium would then occur where αA 1 αB 5 1 where the level of public goods expenditure favored by the two individuals precisely generates enough in tax contributions to pay for it For in that case MRSA 1 MRSB 5 1αA 1 αB2RPT 5 RPT 1954 10Excerpts from Lindahls writings are contained in R A Musgrave and A T Peacock Eds Classics in the Theory of Public Finance London Macmillan 1958 because by the symmetry of the problem the sum of the MRS across roommates equals n MRSA Hence ny A 5 2x Substituting into the combined budget constraint 10n 5 x 1 ny A 5 3x implying x 5 10n3 and thus x A 5 103 While the cleaning effort per roommate shrinks to zero in the Nash equilibrium the efficient level remains a constant 103 each no matter how many of them live together The moral of the story is that you shouldnt be surprised if a big group say a fraternity soror ity or even an economics department faculty lives in messy conditions even though a small amount of effort from each would be enough to make the place sparkle The individuals are not necessarily exceptionally lazy they just may be rational players in a Nash equilibrium QUERY How would an increase in the number of roommates affect their ability to enforce an informal or formal agreement to keep the room clean Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 702 Part 8 Market Failure and this equilibrium would be efficient see Equation 1940 Hence at least on a concep tual level the Lindahl approach solves the public good problem Presenting each person with the equilibrium tax share price will lead him or her to opt for the efficient level of public goods production EXAMPLE 194 Lindahl Solution for the Roommates Lindahl pricing provides a conceptual solution to the roommates dilemma of too little cleaning in Example 193 If the government or perhaps social convention suggests that each of the two roommates should contribute half the total cleaning effort then each would face an effective price per hour of total cleaning of a half hour of personal effort The CobbDouglas form of the roommates utility functions imply that 13 of each persons total time budget that is 203 hours should be spent on cleaning Hence the solution will be x 5 203 and y A 5 y B 5 203 This is indeed the efficient solution in Example 193 This solution works if the government knows enough about the roommates preferences that it can set the payment shares in advance and stick to them Knowing that the roommates have symmetric preferences in this example it could set equal payment shares αA 5 αB 5 12 and rest assured that both will honestly report the same demands for the public good x 5 203 If however the government does not know their preferences it would have to tweak the payment shares based on their reports to make sure the reported demands end up being equal as required for the Lindahl solution to be in equilibrium Anticipating the effect of their reports on their payment shares the roommates would have an incentive to underreport demand In fact this underreporting would lead to the same outcome as in the Nash equilibrium from Example 193 QUERY Although the 5050 sharing in this example might arise from social custom in fact the optimality of such a split is a special feature of this problem What is it about this problem that leads to such a Lindahl outcome Under what conditions would Lindahl prices result in other than a 5050 sharing 1971 Shortcomings of the Lindahl solution Unfortunately Lindahls solution is only a conceptual one We have already seen in our examination of the Nash equilibrium for public goods production and in our roommates example that the incentive to be a free rider in the public goods case is very strong This fact makes it difficult to envision how the information necessary to compute equilibrium Lindahl shares might be obtained Because individuals know their tax shares will be based on their reported demands for public goods they have a clear incentive to understate their true preferencesin so doing they hope that the other guy will pay Hence simply asking people about their demands for public goods should not be expected to reveal their true demands We will discuss more sophisticated mechanisms for eliciting honest demand reports at the end of the chapter 1972 Local public goods Some economists believe that demand revelation for public goods may be more tracta ble at the local level11 Because there are many communities in which individuals might reside they can indicate their preferences for public goods ie for their willingness to 11The classic reference is C M Tiebout A Pure Theory of Local Expenditures Journal of Political Economy October 1956 41624 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 703 pay Lindahl tax shares by choosing where to live If a particular tax burden is not utility maximizing then people can in principle vote with their feet and move to a community that does provide optimality Hence with perfect information zero costs of mobility and enough communities the Lindahl solution may be implemented at the local level Sim ilar arguments apply to other types of organizations such as private clubs that provide public goods to their members given a sufficiently wide spectrum of club offerings an efficient equilibrium might result Of course the assumptions that underlie the purported efficiency of such choices by individuals are quite strict Even minor relaxation of these assumptions may yield inefficient results owing to the fragile nature of the way in which the demand for public goods is revealed EXAMPLE 195 The Relationship between Environmental Externalities and Public Goods Production In recent years economists have begun to study the relationship between the two issues we have been discussing in this chapter externalities and public goods The basic insight from this exam ination is that one must take a general equilibrium view of these problems in order to identify solutions that are efficient overall Here we illustrate this point by returning to the computable general equilibrium model firms described in Chapter 13 see Example 134 To simplify matters we will now assume that this economy includes only a single representative person whose utility function is given by utility 5 U1x y l g c2 5 x 05y0 3l 02g 01c 02 1955 where we have added terms for the utility provided by public goods g which are initially financed by a tax on labor and by clean air c Production of the public good requires capital and labor input according to the production function g 5 k05l 05 there is an externality in the production of good y so that the quantity of clean air is given by c 5 10 2 02y The production functions for goods x and y remain as described in Example 134 as do the endowments of k and l Hence our goal is to allocate resources in such a way that utility is maximized Base case Optimal public goods production with no Pigovian tax If no attempt is made to control the externality in this problem then the optimal level of public goods produc tion requires g 5 293 and this is financed by a tax rate of 025 on labor Output of good y in this case is 297 and the quantity of clean air is given by c 5 10 2 594 5 406 Overall utility in this situation is U 5 1934 This is the highest utility that can be obtained in this situation without regulating the externality A Pigovian tax As suggested by Figure 192 a unit tax on the production of good y may improve matters in this situation With a tax rate of 01 for example output of good y is reduced to y 5 274 1c 5 10 2 548 5 4522 and the revenue generated is used to expand public goods production to g 5 377 Utility is increased to U 5 1938 By carefully specifying how the reve nue generated by the Pigovian tax is used a general equilibrium model permits a more complete statement of welfare effects The double dividend of environmental taxes The solution just described is not optimal however Production of public goods is actually too high in this case since the revenues from environmental taxes are also used to pay for public goods In fact simulations show that optimal ity can be achieved by reducing the labor tax to 020 and public goods production to g 5 331 With these changes utility expands even further to U 5 1943 This result is sometimes referred to as the double dividend of environmental taxation Not only do these taxes reduce externali ties relative to the untaxed situation now c 5 10 2 560 5 440 but also the extra governmen tal revenue made available thereby may permit the reduction of other distorting taxes Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 704 Part 8 Market Failure 198 VOTING AND RESOURCE ALLOCATION Voting is used as a social decision process in many institutions In some instances indi viduals vote directly on policy questions That is the case in some New England town meetings many statewide referenda for example Californias Proposition 13 in 1977 and for many of the national policies adopted in Switzerland Direct voting also char acterizes the social decision procedure used for many smaller groups and clubs such as farmers cooperatives university faculties or the local Rotary Club In other cases how ever societies have found it more convenient to use a representative form of government in which individuals vote directly only for political representatives who are then charged with making decisions on policy questions For our study of public choice theory we will begin with an analysis of direct voting This is an important subject not only because such a procedure applies to many cases but also because elected representatives often engage in direct voting in Congress for example and the theory we will illustrate applies to those instances as well 1981 Majority rule Because so many elections are conducted on a majority rule basis we often tend to regard that procedure as a natural and perhaps optimal one for making social choices But even a cursory examination indicates that there is nothing particularly sacred about a rule requir ing that a policy obtain 50 percent of the vote to be adopted In the US Constitution for example two thirds of the states must adopt an amendment before it becomes law And 60 percent of the US Senate must vote to limit debate on controversial issues Indeed in some institutions Quaker meetings for example unanimity may be required for social decisions Our discussion of the Lindahl equilibrium concept suggests there may exist a distribution of tax shares that would obtain unanimous support in voting for public goods But arriving at such unanimous agreements is usually thwarted by emergence of the free rider problem Examining in detail the forces that lead societies to move away from una nimity and to choose some other determining fraction would take us too far afield here We instead will assume throughout our discussion of voting that decisions will be made by majority rule Readers may wish to ponder for themselves what kinds of situations might call for a decisive proportion of other than 50 percent 1982 The paradox of voting In the 1780s the French social theorist M de Condorcet observed an important peculiarity of majority rule voting systemsthey may not arrive at an equilibrium but instead may cycle among alternative options Condorcets paradox is illustrated for a simple case in Table 192 Suppose there are three voters Smith Jones and Fudd choosing among three policy options For our subsequent analysis we will assume the policy options represent three levels of spending A low B medium or C high on a particular public good but Con dorcets paradox would arise even if the options being considered did not have this type of QUERY Why does the quantity of clean air decrease slightly when the labor tax is reduced rela tive to the situation where it is maintained at 025 More generally describe whether environmen tal taxes would be expected always to generate a double dividend Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 705 ordering associated with them Preferences of Smith Jones and Fudd among the three pol icy options are indicated in Table 192 These preferences give rise to Condorcets paradox Consider a vote between options A and B Here option A would win because it is favored by Smith and Fudd and opposed only by Jones In a vote between options A and C option C would win again by 2 votes to 1 But in a vote of C versus B B would win and we would be back where we started Social choices would endlessly cycle among the three alternatives In subsequent votes any choice initially decided upon could be defeated by an alternative and no equilibrium would ever be reached In this situation the option finally chosen will depend on such seemingly nongermane issues as when the balloting stops or how items are ordered on an agendarather than being derived in some rational way from the preferences of voters 1983 Singlepeaked preferences and the median voter theorem Condorcets voting paradox arises because there is a degree of irreconcilability in the pref erences of voters Therefore one might ask whether restrictions on the types of preferences allowed could yield situations where equilibrium voting outcomes are more likely A fun damental result about this probability was discovered by Duncan Black in 194812 Black showed that equilibrium voting outcomes always occur in cases where the issue being voted upon is onedimensional such as how much to spend on a public good and where voters preferences are single peaked To understand what the notion of single peaked means consider again Condorcets paradox In Figure 194 we illustrate the preferences that gave rise to the paradox by assigning hypothetical utility levels to options A B and C that are consistent with the preferences recorded in Table 192 For Smith and Jones pref erences are single peaked As levels of public goods expenditures increase there is only one local utilitymaximizing choice A for Smith B for Jones Fudds preferences on the other hand have two local maxima A and C It is these preferences that produced the cycli cal voting pattern If instead Fudd had the preferences represented by the dashed line in Figure 194 where now C is the only local utility maximum then there would be no par adox In this case option B would be chosen because that option would defeat both A and C by votes of 2 to 1 Here B is the preferred choice of the median voter Jones whose preferences are between the preferences of Smith and the revised preferences of Fudd Blacks result is quite general and applies to any number of voters If choices are unidi mensional13 and if preferences are single peaked then majority rule will result in the selec tion of the project that is most favored by the median voter Hence that voters preferences 12D Black On the Rationale of Group Decision Making Journal of Political Economy February 1948 2334 13The result can be generalized a bit to deal with multidimensional policies if individuals can be characterized in their support for such policies along a single dimension TABLE 192 PREFERENCES THAT PRODUCE THE PARADOX OF VOTING Choices ALow Spending BMedium Spending CHigh Spending Preferences Smith Jones Fudd A B C B C A C A B Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 706 Part 8 Market Failure will determine what public choices are made This result is a key starting point for many models of the political process In such models the median voters preferences dictate pol icy choiceseither because that voter determines which policy gets a majority of votes in a direct election or because the median voter will dictate choices in competitive elections in which candidates must adopt policies that appeal to this voter 199 A SIMPLE POLITICAL MODEL To illustrate how the median voter theorem is applied in political models suppose a com munity is characterized by a large number n of voters each with an income given by yi The utility of each voter depends on his or her consumption of a private good 1ci2 and of a public good g according to the additive utility function utility of person i 5 Ui 5 ci 1 f1 g2 1956 where fg 0 and fgg 0 Each voter must pay income taxes to finance g Taxes are proportional to income and are imposed at a rate t Therefore each persons budget constraint is given by ci 5 11 2 t2yi 1957 The government is also bound by a budget constraint g 5 a n i51 tyi 5 tny A 1958 where yA denotes average income for all voters Fudd Fudd alternate Jones Smith Utility Quantity of public good A B C FIGURE 194 SinglePeaked Preferences and the Median Voter Theorem This figure illustrates the preferences in Table 192 Smiths and Joness preferences are single peaked but Fudds have two local peaks and these yield the voting paradox If Fudds preferences had instead been single peaked the dashed line then option B would have been chosen as the preferred choice of the median voter Jones Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 707 Given these constraints the utility of person i can be written as a function of his or her choice of g only Ui 1 g2 5 ay A 2 g nb yi y A 1 f1 g2 1959 Utility maximization for person i shows that his or her preferred level of expenditures on the public good satisfies dUi dg 52 yi ny A 1 fg 1 g2 5 0 or g 5 f 21 g a yi ny Ab 1960 This shows that desired spending on g is inversely related to income Because in this model the benefits of g are independent of income but taxes increase with income highincome voters can expect to have smaller net gains or even losses from public spending than can lowincome voters 1991 The median voter equilibrium If g is determined here through majority rule its level will be chosen to be that level favored by the median voter In this case voters preferences align exactly with incomes so g will be set at that level preferred by the voter with median income 1ym2 Any other level for g would not get 50 percent of the vote Hence equilibrium g is given by g 5 f 21 g a ym ny Ab 5 f 21 g c a1 nb a ym y Ab d 1961 In general the distribution of income is skewed to the right in practically every political jurisdiction in the world With such an income distribution ym y A and the difference between the two measures becomes larger the more skewed is the income distribution Hence Equation 1961 suggests that ceteris paribus the more unequal is the income dis tribution in a democracy the higher will be tax rates and the greater will be spending on public goods Similarly laws that extend the vote to increasingly poor segments of the pop ulation can also be expected to increase such spending 1992 Optimality of the median voter result Although the median voter theorem permits a number of interesting positive predictions about the outcome of voting the normative significance of these results is more difficult to pinpoint In this example it is clear that the result does not replicate the Lindahl voluntary equilibriumhighincome voters would not voluntarily agree to the taxes imposed14 The result also does not necessarily correspond to any simple criterion for social welfare For example under a utilitarian social welfare criterion g would be chosen so as to maximize the sum of utilities SW 5 a n i51 Ui 5a n i51 c ay A 2 g nb yi y A 1 f1 g2 d 5 ny A 2 g 1 nf1 g2 1962 The optimal choice for g is then found by differentiation dSW dg 52 1 1 nfg 5 0 14Although they might if the benefits of g were also proportional to income Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 708 Part 8 Market Failure or g 5 f 21 g a1 nb 5 f 21 g c a1 nb a y A y Ab d 1963 which shows that a utilitarian choice would opt for the level of g favored by the voter with average income That output of g would be smaller than that favored by the median voter because ym yA In Example 196 we take this analysis a bit further by showing how it might apply to governmental transfer policy EXAMPLE 196 Voting for Redistributive Taxation Suppose voters were considering adoption of a lumpsum transfer to be paid to every person and financed through proportional taxation If we denote the perperson transfer by b then each indi viduals utility is now given by Ui 5 ci 1 b 1964 and the government budget constraint is nb 5 tny A or b 5 ty A 1965 For a voter whose income is greater than average utility would be maximized by choosing b 5 0 because such a voter would pay more in taxes than he or she would receive from the transfer Any voter with less than average income will gain from the transfer no matter what the tax rate is Hence such voters including the decisive median voter will opt for t 5 1 and b 5 y A That is they would vote to fully equalize incomes through the tax system Of course such a tax scheme is unrealisticprimarily because a 100 percent tax rate would undoubtedly create negative work incentives that reduce average income To capture such incentive effects assume15 that each persons income has two components one responsive to tax rates 3 yi 1t2 4 and one not responsive 1zi2 Assume also that the average value of zi is 0 but that its distribution is skewed to the right so zm 0 Now utility is given by Ui 5 11 2 t2 3 yi 1t2 1 zi4 1 b 1966 Assuming that each person first optimizes over those variables such as labor supply that affect yi 1t2 the firstorder condition16 for a maximum in his or her political decisions about t and b then becomes using the government budget constraint in Equation 1965 dUi dt 5 2zi 1 t dy A dt 5 0 1967 Hence for voter i the optimal redistributive tax rate is given by ti 5 zi dy Adt 1968 Assuming political competition under majority rule voting will opt for that policy favored by the median voter the equilibrium rate of taxation will be t 5 zm dy Adt 1969 15What follows represents a much simplified version of a model first developed by T Romer in Individual Welfare Majority Voting and the Properties of a Linear Income Tax Journal of Public Economics December 1978 16368 16Equation 1967 can be derived from 1966 through differentiation and by recognizing that dyidt 5 0 because of the assumption of individual optimization Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 709 1910 VOTING MECHANISMS The problems involved in majority rule voting arise in part because such voting is simply not informative enough to provide accurate appraisals of how people value public goods This situation is in some ways similar to some of the models of asymmetric information examined in the previous chapter Here voters are more informed than is the government about the value they place on various taxspending packages Resource allocation would be improved if mechanisms could be developed that encourage people to be more accurate in what they reveal about these values In this section we examine two such mechanisms Both are based on the basic insight from Vickrey secondprice auctions see Chapter 18 that incorporating information about other bidders valuations into decisionmakers cal culations can yield a greater likelihood of revealing truthful valuations 19101 The Groves mechanism In a 1973 paper T Groves proposed a way to incorporate the Vickrey insight into a method for encouraging people to reveal their demands for a public good17 To illustrate this mechanism suppose that there are n individuals in a group and each has a private and unobservable net valuation vi for a proposed taxationexpenditure project In seek ing information about these valuations the government states that should the project be undertaken each person will receive a transfer given by ti 5 a j2i v j 1970 where v j represents the valuation reported by person j and the summation is taken over all individuals other than person i If the project is not undertaken then no transfers are made Given this setup the problem for voter i is to choose his or her reported net valuation so as to maximize utility which is given by utility 5 vi 1 ti 5 vi 1 a j2i v j 1971 Since the project will be undertaken only if g n i51 v i 0 and since each person will wish the project to be undertaken only if it increases utility ie vi 1 g j2i v i 0 it follows that a utilitymaximizing strategy is to set v i 5 vi Hence the Groves mechanism encourages each person to be truthful in his or her reporting of valuations for the project 17T Groves Incentives in Teams Econometrica July 1973 61731 Because both zm and dyAdt are negative this rate of taxation will be positive The optimal tax will be greater the farther zm is from its average value ie the more unequally income is distributed Similarly the larger are distortionary effects from the tax the smaller the optimal tax This model then poses some rather strong testable hypotheses about redistribution in the real world QUERY Would progressive taxation be more likely to raise or lower t in this model Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 710 Part 8 Market Failure 19102 The Clarke mechanism A similar mechanism was proposed by E Clarke also in the early 1970s18 This mechanism also envisions asking individuals about their net valuations for some public project but it focuses mainly on pivotal votersthose whose reported valuations can change the overall evaluation from negative to positive or vice versa For all other voters there are no special transfers on the presumption that reporting a nonpivotal valuation will not change either the decision or the zero payment so he or she might as well report truthfully For voters reporting pivotal valuations however the Clarke mechanism incorporates a Pigovianlike tax or transfer to encourage truth telling To see how this works suppose that the net valuations reported by all other voters are negative 1 g j2i v j 02 but that a truthful state ment of the valuation by person i would make the project acceptable 1vi 1g j2i v j 02 Here as for the Groves mechanism a transfer of ti 1 g j2i v j which in this case would be negativeie a tax would encourage this pivotal voter to report v i 5 vi Similarly if all other individuals reported valuations favorable to a project 1 g j2i v j 02 but inclu sion of person is evaluation of the project would make it unfavorable then a transfer of ti 5 g j2i v j which in this case is positive would encourage this pivotal voter to choose v i 5 vi also Overall then the Clarke mechanism is also truth revealing Notice that in this case the transfers play much the same role that Pigovian taxes did in our examination of externalities If other voters view a project as unfavorable then voter i must compensate them for accepting it On the other hand if other voters find the project acceptable then voter i must be sufficiently against the project that he or she cannot be bribed by other voters into accepting it 19103 Generalizations The voter mechanisms we have been describing are sometimes called VCG mechanisms after the three pioneering economists in this area of research Vickrey Clarke and Groves These mechanisms can be generalized to include multiple governmental projects alterna tive concepts of voter equilibrium or an infinite number of voters One assumption behind the mechanisms that does not seem amenable to generalization is the quasilinear utility functions that we have been using throughout Whether this assumption provides a good approximation for modeling political decision making remains an open question however 18E Clarke Multipart Pricing for Public Goods Public Choice Fall 1971 1933 Summary In this chapter we have examined market failures that arise from externality or spillover effects involved in the con sumption or production of certain types of goods In some cases it may be possible to design mechanisms to cope with these externalities in a market setting but important limits are involved in such solutions Some specific issues we examined were as follows Externalities may cause a misallocation of resources because of a divergence between private and social marginal cost Traditional solutions to this divergence include mergers among the affected parties and adoption of suitable Pigovian taxes or subsidies If transaction costs are small then private bargaining among the parties affected by an externality may bring social and private costs into line The proof that resources will be efficiently allocated under such circumstances is sometimes called the Coase theorem Public goods provide benefits to individuals on a nonex clusive basisno one can be prevented from consuming such goods Such goods are also usually nonrival in that the marginal cost of serving another user is zero Private markets will tend to underallocate resources to public goods because no single buyer can appropriate all of the benefits that such goods provide Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 711 A Lindahl optimal taxsharing scheme can result in an efficient allocation of resources to the production of pub lic goods However computing these tax shares requires substantial information that individuals have incentives to hide Majority rule voting does not necessarily lead to an effi cient allocation of resources to public goods The median voter theorem provides a useful way of modeling the actual outcomes from majority rule in certain situations Several truthrevealing voting mechanisms have been developed Whether these are robust to the special assumptions made or capable of practical application remain unresolved questions Problems 191 A firm in a perfectly competitive industry has patented a new process for making widgets The new process lowers the firms average cost meaning that this firm alone although still a price taker can earn real economic profits in the long run a If the market price is 20 per widget and the firms mar ginal cost is given by MC 5 04q where q is the daily widget production for the firm how many widgets will the firm produce b Suppose a government study has found that the firms new process is polluting the air and estimates the social marginal cost of widget production by this firm to be SMC 5 05q If the market price is still 20 what is the socially optimal level of production for the firm What should be the rate of a governmentimposed excise tax to bring about this optimal level of production c Graph your results 192 On the island of Pago Pago there are 2 lakes and 20 anglers Each angler can fish on either lake and keep the average catch on his particular lake On Lake x the total number of fish caught is given by F x 5 10lx 2 1 2 l2 x where lx is the number of people fishing on the lake For Lake y the relationship is F y 5 5ly a Under this organization of society what will be the total number of fish caught b The chief of Pago Pago having once read an economics book believes it is possible to increase the total num ber of fish caught by restricting the number of peo ple allowed to fish on Lake x What number should be allowed to fish on Lake x in order to maximize the total catch of fish What is the number of fish caught in this situation c Being opposed to coercion the chief decides to require a fishing license for Lake x If the licensing procedure is to bring about the optimal allocation of labor what should the cost of a license be in terms of fish d Explain how this example sheds light on the connection between property rights and externalities 193 Suppose the oil industry in Utopia is perfectly competitive and that all firms draw oil from a single and practically inex haustible pool Assume that each competitor believes that it can sell all the oil it can produce at a stable world price of 10 per barrel and that the cost of operating a well for 1 year is 1000 Total output per year Q of the oil field is a function of the number of wells n operating in the field In particular Q 5 500n 2 n2 and the amount of oil produced by each well q is given by q 5 Q n 5 500 2 n 1972 a Describe the equilibrium output and the equilibrium number of wells in this perfectly competitive case Is there a divergence between private and social marginal cost in the industry b Suppose now that the government nationalizes the oil field How many oil wells should it operate What will total output be What will the output per well be c As an alternative to nationalization the Utopian govern ment is considering an annual license fee per well to dis courage overdrilling How large should this license fee be if it is to prompt the industry to drill the optimal number of wells 194 There is considerable legal controversy about product safety Two extreme positions might be termed caveat emptor let the buyer beware and caveat vendor let the seller beware Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 712 Part 8 Market Failure Under the former scheme producers would have no responsi bility for the safety of their products Buyers would absorb all losses Under the latter scheme this liability assignment would be reversed Firms would be completely responsible under law for losses incurred from unsafe products Using simple sup ply and demand analysis discuss how the assignment of such liability might affect the allocation of resources Would safer products be produced if firms were strictly liable under law How do possible information asymmetries affect your results 195 Suppose a monopoly produces a harmful externality Use the concept of consumer surplus in a partial equilibrium diagram to analyze whether an optimal tax on the polluter would nec essarily be a welfare improvement 196 Suppose there are only two individuals in society Person As demand curve for mosquito control is given by qa 5 100 2 p for person B the demand curve for mosquito control is given by qb 5 200 2 p a Suppose mosquito control is a pure public good that is once it is produced everyone benefits from it What would be the optimal level of this activity if it could be produced at a constant marginal cost of 120 per unit b If mosquito control were left to the private market how much might be produced Does your answer depend on what each person assumes the other will do c If the government were to produce the optimal amount of mosquito control how much will this cost How should the tax bill for this amount be allocated between the individuals if they are to share it in proportion to benefits received from mosquito control 197 Suppose the production possibility frontier for an economy that produces one public good x and one private good y is given by 100x2 1 y2 5 5000 This economy is populated by 100 identical individuals each with a utility function of the form utility 5 xyi where yi is the individuals share of private good production 15 y1002 Notice that the public good is nonexclusive and that everyone benefits equally from its level of production a If the market for x and y were perfectly competitive what levels of those goods would be produced What would the typical individuals utility be in this situation b What are the optimal production levels for x and y What would the typical individuals utility level be How should consumption of good y be taxed to achieve this result Hint The numbers in this problem do not come out evenly and some approximations should suffice Analytical Problems 198 More on Lindahl equilibrium The analysis of public goods in this chapter exclusively used a model with only two individuals The results are readily generalized to n personsa generalization pursued in this problem a With n persons in an economy what is the condition for efficient production of a public good Explain how the characteristics of the public good are reflected in these conditions b What is the Nash equilibrium in the provision of this public good to n persons Explain why this equilibrium is inefficient Also explain why the underprovision of this public good is more severe than in the twoperson cases studied in the chapter c How is the Lindahl solution generalized to n persons Is the existence of a Lindahl equilibrium guaranteed in this more complex model 199 Taxing pollution Suppose that there are n firms each producing the same good but with differing production functions Output for each of these firms depends only on labor input so the functions take the form qi 5 fi 1li2 In its production activities each firm also produces some pollution the amount of which is determined by a firmspecific function of labor input of the form gi 1li2 a Suppose that the government wishes to place a cap of amount K on total pollution What is the efficient alloca tion of labor among firms b Will a uniform Pigovian tax on the output of each firm achieve the efficient allocation described in part a c Suppose that instead of taxing output the Pigovian tax is applied to each unit of pollution How should this tax be set Will the tax yield the efficient allocation described in part a d What are the implications of the problem for adopting pollution control strategies For more on this topic see the Extensions to this chapter Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Chapter 19 Externalities and Public Goods 713 1910 Vote trading Suppose there are three individuals in society trying to rank three social states A B and C For each of the methods of social choice indicated develop an example to show how the resulting social ranking of A B and C will be intransitive as in the paradox of voting or indeterminate a Majority rule without vote trading b Majority rule with vote trading c Point voting where each voter can give 1 2 or 3 points to each alternative and the alternative with the highest point total is selected 1911 Public choice of unemployment benefits Suppose individuals face a probability of u that they will be unemployed next year If they are unemployed they will receive unemployment benefits of b whereas if they are employed they receive w11 2 t2 where t is the tax used to finance unemployment benefits Unemployment bene fits are constrained by the government budget constraint ub 5 tw112u2 a Suppose the individuals utility function is given by U 5 1yi2 δδ where 1 2 δ is the degree of constant relative risk aversion What would be the utilitymaximizing choices for b and t b How would the utilitymaximizing choices for b and t respond to changes in the probability of unemploy ment u c How would b and t change in response to changes in the risk aversion parameter δ 1912 Probabilistic voting Probabilistic voting is a way of modeling the voting process that introduces continuity into individuals voting decisions In this way calculustype derivations become possible To take an especially simple form of this approach suppose there are n voters and two candidates labeled A and B for elective office Each candidate proposes a platform that promises a net gain or loss to each voter These platforms are denoted by θA i and θB i where i 5 1 c n The probability that a given voter will vote for candidate A is given by π A i 5 f 1Ui 1θA i 2 2 Ui 1θB i 2 2 where f r 0 f s The probability that the voter will vote for candidate B is π B i 5 1 2 π A i a How should each candidate choose his or her platform so as to maximize the probability of winning the election subject to the constraint g i θA i 5 g i θB i 5 0 Do these constraints seem to apply to actual political candidates b Will there exist a Nash equilibrium in platform strategies for the two candidates c Will the platform adopted by the candidates be socially optimal in the sense of maximizing a utilitarian social welfare Social welfare is given by SW 5 g iUi 1θi2 Suggestions for Further Reading Alchian A and H Demsetz Production Information Costs and Economic Organization American Economic Review 62 December 1972 77795 Uses externality arguments to develop a theory of economic organizations Barzel Y Economic Analysis of Property Rights Cambridge Cambridge University Press 1989 Provides a graphical analysis of several economic questions that are illuminated through use of the property rights paradigm Black D On the Rationale of Group Decision Making Jour nal of Political Economy February 1948 2334 Reprinted in K J Arrow and T Scitovsky Eds Readings in Welfare Eco nomics Homewood IL Richard D Irwin 1969 Early development of the median voter theorem Buchanan J M and G Tullock The Calculus of Consent Ann Arbor University of Michigan Press 1962 Classic analysis of the properties of various voting schemes Cheung S N S The Fable of the Bees An Economic Investi gation Journal of Law and Economics 16 April 1973 1133 Empirical study of how the famous beeorchard owner externality is handled by private markets in the state of Washington Coase R H The Market for Goods and the Market for Ideas American Economic Review 64 May 1974 38491 Speculative article about notions of externalities and regulation in the marketplace of ideas The Problem of Social Cost Journal of Law and Economics 3 October 1960 144 Classic article on externalities Many fascinating historical legal cases Cornes R and T Sandler The Theory of Externalities Pub lic Goods and Club Goods Cambridge Cambridge University Press 1986 Good theoretical analysis of many of the issues raised in this chap ter Good discussions of the connections between returns to scale excludability and club goods Demsetz H Toward a Theory of Property Rights Ameri can Economic Review Papers and Proceedings 57 May 1967 34759 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 714 Part 8 Market Failure Brief development of a plausible theory of how societies come to define property rights MasColell A M D Whinston and J R Green Microeco nomic Theory New York Oxford University Press 1995 Chapter 11 covers much of the same ground as this chapter does though at a somewhat more abstract level Olson M The Logic of Collective Action Cambridge MA Harvard University Press 1965 Analyzes the effects of individual incentives on the willingness to undertake collective action Many fascinating examples Persson T and G Tabellini Political Economics Explaining Economic Policy Cambridge MA MIT Press 2000 A complete summary of recent models of political choices Covers voting models and issues of institutional frameworks Posner R A Economic Analysis of Law 5th ed Boston Little Brown 1998 In many respects the bible of the law and economics movement Posners arguments are not always economically correct but are unfailingly interesting and provocative Samuelson P A The Pure Theory of Public Expenditures Review of Economics and Statistics 36 November 1954 38789 Classic statement of the efficiency conditions for pub lic goods production Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 715 Although our discussion of externalities focused on how Pigo vian taxes can make goods markets operate more efficiently similar results also apply to the study of the technology of pol lution abatement In these Extensions we briefly review this alternative approach We assume there are only two firms A and B and that their output levels qA and qB respectively are fixed throughout our discussion It is an inescapable scien tific principle that production of physical goods as opposed to services must obey the conservation of matter Hence production of qA and qB is certain to involve some emission byproducts eA and eB The physical amounts of these emis sions or at least their harmful components can be abated using inputs zA and zB which cost p per unit The resulting levels of emissions are given by f A 1qA zA2 5 eA and f B 1qB zB2 5 eB i where for each firms abatement function f1 0 and f2 0 E191 Optimal abatement If a regulatory agency has decided that e represents the max imum allowable level of emissions from these firms then this level would be achieved at minimal cost by solving the Lagrangian expression 5 pzA 1 pzB 1 λ1 f A 1 f B 2 e2 ii Firstorder conditions for a minimum are p 1 λf A 2 5 0 and p 1 λf B 2 5 0 iii Hence we have λ 5 2pf A 2 5 2pf B 2 iv This equation makes the rather obvious point that costminimizing abatement is achieved when the marginal cost of abatement universally referred to as MAC in the envi ronmental literature is the same for each firm A uniform standard that required equal emissions from each firm would not be likely to achieve that efficient resultconsiderable cost savings might be attainable under equalization of MACs as compared to such uniform regulation E192 Emission taxes The optimal solution described in Equation iv can be achieved by imposing an emission tax t equal to λ on each firm pre sumably this tax would be set at a level that reflects the mar ginal harm that a unit of emissions causes With this tax each firm seeks to minimize pzi 1 tf i 1qi zi2 which does indeed yield the efficient solution t 5 2pf A 2 5 2pf B 2 v Notice that as in the analysis of Chapter 19 one benefit of the taxation solution is that the regulatory authority need not know the details of the firms abatement functions Rather the firms themselves make use of their own private information in determining abatement strategies If these functions dif fer significantly among firms then it would be expected that emissions reductions would also differ Emission taxes in the United Kingdom Hanley Shogren and White 1997 review a variety of emis sion taxation schemes that have been implemented in the United Kingdom They show that marginal costs of pollu tion abatement vary significantly perhaps as much as thir tyfold among firms Hence relative to uniform regulation the cost savings from taxation schemes can be quite large For example the authors review a series of studies of the Tees estuary that report annual cost savings in the range of 10 million 1976 pounds The authors also discuss some of the complications that arise in setting efficient effluent taxes when emission streams do not have a uniform mix of pollutants or when pollutants may accumulate to dangerous levels over time E193 Tradable permits As we illustrated in Chapter 19 many of the results achiev able through Pigovian taxation can also be achieved through a tradable permit system In this case the regulatory agency would set the number of permits 1s2 equal to e and allocate these permits in some way among firms 1sA 1 sB 5 s2 Each EXTENSIONS Pollution AbAtEmEnt Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 716 Part 8 Market Failure firm may then buy or sell any number of permits desired but must also ensure that its emissions are equal to the number of permits it holds If the market price of permits is given by ps then each firms problem is again to minimize pzi 1 ps 1ei 2 si2 vi which yields an identical solution to that derived in Equations iv and v with ps 5 t 5 λ Hence the tradable permit solution would be expected to yield the same sort of cost savings as do taxation schemes SO2 trading The US Clean Air Act of 1990 established the first large scale program of tradable emission permits These focused on sulfur dioxide emissions with the goal of reducing acid rain arising from powerplant burning of coal Schmalensee et al 1998 review early experiences under this program They conclude that it is indeed possible to establish large and wellfunctioning markets in emission permits More than 5 million oneton emission permits changed hands in the most recent year examinedat prices that averaged about 150 per permit The authors also show that firms using the permit system employed a wide variety of compliance strat egies This suggests that the flexibility inherent in the permit system led to considerable cost savings One interesting aspect of this review of SO2 permit trading is the authors specula tions about why the permit prices were only about half what had been expected They attribute a large part of the explana tion to an initial overinvestment in emission cleaning tech nology by power companies in the mistaken belief that permit prices once the system was implemented would be in the 300400 range With such large fixedcost investments the marginal cost of removing a ton of SO2 may have been as low as 65ton thereby exerting a significant downward force on permit prices E194 Innovation Although taxes and tradable permits appear to be mathemat ically equivalent in the models we have been describing this equivalence may vanish once the dynamics of innovation in pollution abatement technology are considered Of course both procedures offer incentives to adopt new technologies If a new process can achieve a given emission reduction at a lower MAC it will be adopted under either scheme Yet in a detailed analysis of dynamics under the two approaches Milli man and Prince 1989 argue that taxation is better Their rea soning is that the taxation approach encourages a more rapid diffusion of new abatement technology because incremental profits attainable from adoption are greater than with permits Such rapid diffusion may also encourage environmental agen cies to adopt more stringent emission targets because these targets will now more readily meet costbenefit tests References Hanley N J F Shogren and B White Environmental Eco nomics in Theory and Practice New York Oxford Uni versity Press 1997 Milliman S R and R Prince Firm Incentive to Promote Technological Change in Pollution Control Journal of Environmental Economics and Management November 1989 24765 Schmalensee R P L Joskow A D Ellerman J P Montero and E M Bailey An Interim Evaluation of the Sulfur Dioxide Trading Program Journal of Economic Perspec tives Summer 1998 5368 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 717 The following brief answers to the queries that accom pany each example in the text may help students test their understanding of the concepts being presented CHAPTER 1 11 If price depends on quantity differentiation of p 1q2 q would be more complicated This would lead to the concept of marginal revenuea topic we encounter in many places in this book 12 The reduced form in Equation 116 shows that dpda 5 1225 So if a increases by 450 p should increase by 2which is what a direct solution shows 13 If all labor is devoted to x production then x 5 200 5 141 with full employment and x 5 180 5 134 with unemployment Hence the efficiency cost of unemployment is 07 units of x Similar calculations show that the efficiency cost in terms of good y is about 15 units of that good With reductions in both goods one would need to know the relative price of x in terms of y in order to aggregate the losses CHAPTER 2 21 The firstorder condition for a maximum is πl 5 50l 2 10 5 0 l 5 25 π 5 250 22 No only the exponential function or a function that approximates it over a range has constant elasticity 23 Putting all the terms over a common denominator gives y 5 165 3p 5 55 p Hence y p 5 255 p2 24 For different constants each production possibility frontier is a successively larger quarter ellipse centered at the origin 25 The arguments are identical The supply curve in ele mentary economics slopes upward because it represents the upward sloping supply marginal cost curves for pricetaking firms The analysis here also relies in increasing marginal costs 26 These would be concentric circles centered at x1 5 1 x2 5 2 For y 5 10 the circle is a single point 27 The total derivative is used because π 1 p2 is a value function depending only on p The partial derivative is used because the general function for profits depends on the exogenous variable p and on the endogenous variable qthat is it has not been optimized The envelope the orem shows that dπ 1p2dp 5 π 1p q2p0 q5q 5 q where the final equation follows because q is treated as a constant in taking the partial derivative and is to be evaluated at its optimal level 28 Assume one of the side y lengths must be doubled Now the perimeter constraint is P 5 2x 1 3y and the first order condition for a maximum imply x 5 3y2 5 P4 y 5 P6 29 The value function from minimizing the perimeter for a given area is P 1A2 5 4A Hence direct compu tation shows dP 1A2dA 5 2A The envelope the orem applied to this constrained minimum problem is dP 1A2dA 5 A 5 λD 5 2x 5 2y 5 2A Hence both approaches yield the same result Brief Answers to Queries Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 718 Brief Answers to Queries 210 This function resembles an inverted cone that has only one highest point 211 A linear constraint would be represented by a plane in these threedimensional figures Such a plane would have a unique tangency to the surfaces in both Figures 24a and 24c For an unconstrained maximum however the plane would be horizontal so only Figure 24a would have a maximum 212 Such a transformation would not preserve homogene ity However it would not affect the tradeoff between the xs for any constant k 2f1f2 5 2x2x1 213 Total variable costs of this expansion would be 3 110 100 02q dq 5 01q2 0 110 100 5 1210 2 1000 5 210 This could also be calculated by subtracting total costs when q 5 100115002 from total costs when q 5 110117102 Fixed costs would cancel out in this subtraction 214 As we show in Chapter 17 a higher value for δ will cause wine to be consumed earlier A lower value for γ will make the consumer less willing to experience con sumption fluctuations 215 If g 1x2 is concave then values of this function will increase less rapidly than does x itself Hence E3 g 1x2 4 g 3E1x2 4 In Chapter 7 this is used to explain why a person with a diminishing marginal utility of wealth will be riskaverse 216 Using the results from Example 215 for the uni form distribution gives μx 5 1b 2 a22 5 6 σ2 x 5 1b 2 a2 212 5 12 and σx 5 1205 5 3464 In this case 577 percent 15 2 3464122 of the distribution is within 1 standard deviation of the mean This is less than the comparable figure for the Normal distribu tion because the uniform distribution is not bunched around the mean However unlike the Normal the entire uniform distribution is within 2 standard devi ations of the mean because that distribution does not have long tails CHAPTER 3 31 The derivation here holds utility constant to create an implicit relationship between y and x Changes in x also implicitly change y because of this relationship Equa tion 311 32 The MRS is not changed by such a doubling in Exam ples 1 and 3 In Example 2 the MRS would be changed because 11 1 x2 11 1 y2 2 11 1 2x2 11 1 2y2 33 For homothetic functions the MRS is the same for every point along a positively sloped ray through the origin 34 The indifference curves here are horizontally parallel That is for any given level of y the MRS is the same no matter what the value of x is One implication of this as we shall see in Chapter 4 is that the effect of additional income on purchases of good y is zeroafter a point all extra income is channeled into the good with constant marginal utility good x CHAPTER 4 41 Constant shares imply xpy 5 0 and ypx 5 0 Notice py does not enter into Equation 423 px does not enter into 424 42 Budget shares are not affected by income but they may be affected by changes in relative prices This is the case for all homothetic functions 43 Since a doubling of all prices and nominal income does not change the budget constraint it will not change util itymaximizing choices Indirect utility is homogeneous of degree zero in all prices and nominal income 44 In the CobbDouglas case with py 5 3 E11322 5 2 1 305 2 5 693 so this person should have his or her income reduced by a lumpsum 107 to compensate for the fall in prices In the fixed proportions case the original consumption bundle now costs 7 so the compensation is 210 Notice that with fixed proportions the consump tion bundle does not change but with the CobbDouglas the new choice is x 5 346 y 5 115 because this person takes advantage of the reduction in the price of y Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Brief Answers to Queries 719 CHAPTER 5 51 The shares equations computed from Equations 55 or 57 show that this individual always spends all of his or her income regardless of px py and I That is the shares sum to one 52 If x 5 05Ipx then I 5 100 and px 5 1 which imply that x 5 50 In Equation 511 x 5 05 110012 5 50 also If px rises to 20 the CobbDouglas predicts x 5 25 The CES implies x 5 1006 5 1667 The CES is more responsive to price 53 Since proportional changes in px and py do not induce substitution effects holding U constant implies that x and y will not change That should be true for all com pensated demand functions 54 A larger exponent for say x in the CobbDouglas function will increase the share of income devoted to that good and increase the relative importance of the income effect in the Slutsky decomposition This is eas iest to see using the Slutsky equation in elasticity form Example 55 55 Consider the CobbDouglas case for which expx 5 21 regardless of budget shares The Slutsky equation in elasticity terms shows that because the income effect here is 2sxexI 5 2sx112 5 2sx the compensated price elasticity is ec xpx 5 expx 1 sx 5 211 2 sx2 More gen erally ec xpx5211 2 sx2σ 52σ 1 sxσ so if the share of income devoted to good x is small the compensated price elasticity is given by the negative of the elasticity of substitution But when the share is larger compen sated demand is less elastic because the individual is still bound by a budget constraint that restricts the overall size of the price response that is possible 56 Typically it is assumed that demand goes to zero at some finite price when calculating total consumer sur plus The specific assumption made does not affect cal culations of changes in consumer surplus CHAPTER 6 61 Since xpy includes both income and substitution effects this derivative could be 0 if the effects offset each other The conclusion that xpy 5 0 implies the goods must be used in fixed proportions would hold only if the income effect of this price change were 0 62 Asymmetry can occur with homothetic preferences since although substitution effects are symmetric income effects may differ in size 63 Since the relationships between py pz and ph never change the maximization problem will always be solved the same way CHAPTER 7 7 1 In case 1 the probability of seven heads is less than 001 Hence the value of the original game is 6 In case 2 the prize for obtaining the first head on the twentieth flip is over 1 million The value of the game in this case is 19 1 1000000219 5 2091 7 2 With linear utility the individual would care only about expected dollar values and would be indifferent about buying actuarially fair insurance When utility U is a convex function of wealth 1Us 02 the individual prefers to gamble and will buy insurance only if it costs less than is actuarially justified 7 3 If A 5 1024 CE112 5 107000 2 05 1024 11042 2 5 102000 CE122 5 102000 2 05 1024 4 106 5 101800 So the riskier allocation is preferred On the other hand if A 5 3 1024 then the less risky allocation is preferred 7 4 Willingness to pay is a declining function of wealth Equa tion 743 With R 5 0 the person will pay 50 to avoid a 1000 bet if W0 5 10000 but only 5 if W0 5 100000 With R 5 2 he or she will pay 149 to avoid a 1000 bet if W0 5 10000 but only 15 if W0 5 100000 7 5 Option value may be low for a riskaverse person if one of the choices is relatively safe Reworking the example with A11x2 5 12 shows that the option value is 0125 for the riskneutral person but only about 011 for the riskaverse one Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 720 Brief Answers to Queries 76 The actuarially fair price for such a policy is 025 19000 5 4750 The maximum amount the individual would pay x solves the equation 1145714 5 075 ln 1100000 2 x2 1 025 ln 199000 2 x2 Solving this yields an approximate value of x 5 5120 This person would be willing to pay up to 370 in administrative costs for the deductible policy CHAPTER 8 81 No dominant strategies Paper scissors isnt a Nash equilibrium because player 1 would deviate to rock 82 If the wife plays mixed strategy 19 89 and the hus band plays 45 15 then his expected payoff is 49 If she plays 1 0 and he plays 45 15 his expected payoff is 45 If he plays 45 15 her best response is to play ballet 83 Players earn 23 in the mixedstrategy Nash equilib rium This is less than the payoff even in the less desir able of the two purestrategy Nash equilibria Symmetry might favor the mixedstrategy Nash equilibrium 84 The Nash equilibrium would involve higher quantities for both if their benefits increased If herder 2s benefit decreased his or her quantity would fall and the others would rise 85 Yes Letting p be the probability that player 1 is type t 5 6 player 2s expected payoff from choosing left is 2p This is at least as high as 2s expected payoff of 4 11 2 p2 from choosing right if p 23 86 Moving from incomplete to full information increases herder 1s output and decreases the rivals if 1 is the high type The opposite is true if 1 is the low type The high type prefers full information and would like to some how signal its type the low type prefers incomplete information and would like to conceal its type 87 Obtaining an education informs the firm about the workers ability and thus may increase the highskill workers salary The separating equilibrium would not exist if the lowskill worker could get an education more cheaply than the highskill one 88 The proposed pooling outcome cannot be an equilibrium if the firms posterior beliefs equal its priors after unex pectedly seeing an uneducated worker Then its beliefs would be the same whether it encountered an educated worker it would have the same best response and work ers would deviate from E If the firm has pessimistic posteriors following NE then the outcome is an equilib rium because the firms best response to NE would be NJ inducing both types of worker to pool on E 89 In equilibrium type H obtains an expected payoff of jw 2 cH 5 cL 2 cH This exceeds the payoff of 0 from deviating to NE Type L pools with type H on E with probability e But ded Pr 1H2 5 1π 2 w2π Since this expression is positive type L must increase its probability of playing E to offset an increase in Pr 1H2 and still keep player 2 indifferent between J and NJ CHAPTER 9 91 Now with k 5 11 q 5 72600l 2 2 1331l 3 MPl 5 145200l 2 3993l 2 APl 5 72600l 2 1331l2 In this case APl reaches its maximal value at l 5 273 rather than at l 5 30 92 Since k and l enter f symmetrically if k 5 l then fk 5 fl and fkk 5 fll Hence the numerator of Equation 919 will be negative if fkl fll Combining Equations 922 and 923 and remembering k 5 l shows this holds for k 5 l 20 93 The q 5 4 isoquant contains the points k 5 4 l 5 0 k 5 1 l 5 1 and k 5 0 l 5 4 It is therefore fairly sharply convex It seems possible that an Lshaped iso quant might be approximated for particular coefficients of the linear and radical terms 94 Because the composite technical change factor is θ 5 αφ 1 11 2 α2ε a value of α 5 03 implies that technical improvements in labor will be weighted more highly in determining the overall result Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Brief Answers to Queries 721 CHAPTER 10 101 If σ 5 2 then ρ 5 05 kl 5 16 l 5 85 k 5 1285 and C 5 96 If σ 5 05 then ρ 5 21 kl 5 2 l 5 60 k 5 120 and C 5 1080 Notice that changes in σ also change the scale of the production function so the total cost figures cannot be compared directly 102 The expression for unit costs is 1v12σ 1 w12σ2 1112σ2 If σ 5 0 then this function is linear in w 1 v For σ 0 the function is increasingly convex showing that large increases in w can be offset by small decreases in v 103 The elasticities are given by the exponents in the cost functions and are unaffected by technical change as modeled here 104 In this case σ 5 With w 5 4v cost minimization could use the inputs in any combination for q con stant without changing costs A rise in w would cause the firm to switch to using only capital and would not affect total costs This shows that the impact on costs of an increase in the price of a single input depends importantly on the degree of substitution 105 Because capital costs are fixed in the short run they do not affect shortrun marginal costs in mathematical terms the derivative of a constant is zero Capital costs do however affect shortrun average costs In Figure 109 an increase in v would shift MC AC and all of the SATC curves upward but would leave the SMC curves unaffected CHAPTER 11 111 If MC 5 5 profit maximization requires q 5 25 Now P 5 750 R 5 18750 C 5 125 and π 5 6250 112 Factors other than p can be incorporated into the con stant term a These would shift D and MR but would not affect the elasticity calculations 113 When w rises to 15 supply shifts inward to q 5 8P5 When k increases to 100 supply shifts outward to q 5 25P6 A change in v would not affect shortrun marginal cost or the shutdown decision 114 A change in v has no effect on SMC but it does affect fixed costs A change in w would affect SMC and short run supply 115 A rise in wages for all firms would shift the market sup ply curve upward raising the product price Because total output must fall given a negatively sloped demand curve each firm must produce less Again both substi tution and output effects would then be negative CHAPTER 12 121 The ability to sum incomes in this linear case would require that each person have the same coefficient for income Because each person faces the same price aggregation requires only adding the price coefficients 122 A value for β other than 05 would mean that the expo nent of price would not be 10 The higher the β is the more price elastic is shortrun supply 123 Following steps similar to those used to derive Equation 1230 yields ePβ 5 2eSβ eSP 2 eDP Here eSβ 5 eSw5205 so eP w 521205231 2 121224 5 0522 5 0227 Multiplication by 020 since wages rose by 20 percent predicts a price rise of 45 percent In the example the price rise is 103819957 5 1043 very close to what is predicted by the equation 124 The shortrun supply curve is given by Qs 5 05P 1 750 and the shortterm equilibrium price is 643 Each firm earns approximately 2960 in profits in the short run 125 Total and average costs for Equation 1255 exceed those for Equation 1242 for q 159 Marginal costs for Equation 1255 always exceed those for Equation 1242 Optimal output is lower with Equation 1255 than with Equation 1242 because marginal costs increase more than average costs Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 722 Brief Answers to Queries 126 Losses from a given restriction in quantity will be greater when supply andor demand is less elastic The actor with the least elastic response will bear the greater share of the loss 127 An increase in t increases deadweight loss but it also decreases quantity Hence the burden of raising a given amount of tax revenue increases at the margin A high enough tax can actually decrease tax revenue If t 1P 1 t2 21eDP then dtQdt 0 CHAPTER 13 131 An increase in labor input will shift the first frontier out uniformly In the second case such an increase will shift the yintercept out farther than the xintercept because good y uses labor intensively 132 In all three scenarios the total value of output is 200w composed half of wages and half of profits With the shift in supply consumers still devote 100w to each good Purchases of x are twice those of y because y costs twice as much With the shift in demand the consumer spends 20w on good x and 180w on good y But good y now costs three times what x costs so consumers buy only three times as much y as they do x 133 All efficient allocations require the ratio of x to y to be relatively high for A and low for B Hence when good x is allocated evenly A must get less than half the amount of y available and B must get more than half Because efficiency requires 2yAxA 5 05yBxB and the symme try of the utility functions requires yBxB 5 xAyA for equal utility we can conclude xA 5 2yA xB 5 05 yB So xA 5 6667 yA 5 3333 xB 5 3333 and yB 5 6667 Utility for both parties is about 496 134 The consumers here also spend some of their total income on leisure For person 1 say total income with the equilibrium prices is 40 0136 1 24 0248 5 114 The CobbDouglas exponents imply that this person will spend half of this on good x Hence total spending on that good will be 57 which is also equal to the quan tity of x bought 157 multiplied by this goods equilib rium price 0363 135 Nosuch redistribution could not make both betteroff owing to the excess burden of the tax CHAPTER 14 141 The increase in fixed costs would not alter the output decisions because it would not affect marginal costs It would however raise average cost by 5 and reduce profits to 12500 With the new C1Q2 function MC would rise to 015Q In this case Qm 5 400 Pm 5 80 C1Qm2 5 22000 and πm 5 10000 142 For the linear case an increase in a would increase price by a2 A shift in the price intercept has an effect similar to an increase in marginal cost in this case In the con stant elasticity case the term a does not enter into the calculation of price For a given elasticity of demand the gap between price and marginal cost is the same no matter what a is 143 With e 5 215 the ratio of monopoly to competi tive consumer surplus is 058 Equation 1422 Profits represent 19 percent of competitive consumer surplus Equation 1424 144 If Q 5 0 P 5 100 Total profits are given by the trian gular area between the demand curve and the MC curve less fixed costs This area is 05 11002 16662 5 33333 So πm 5 33333 210000 5 23333 145 Solving the problem by absent mindedly combining demand functions yields P 5 11 and π 5 75 This profit is lower than the 85 that can be earned by just serving market 2 with a price of 15 The absentminded solution underestimates the profit from prices higher than P 12 because it generates negative quantities and profits on market 1 The correct approach recog nizes that quantity is 0 not negative on market 1 for P 12 in effect market 1 disappears leaving only market 2 146 The monopolist should charge perunit fees equal to marginal cost p 1 5 p 2 5 6 and extract the entire Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Brief Answers to Queries 723 consumer surplus from each type with the fixed fees a 1 5 18 182 5 162 and a 2 5 6 122 5 36 CHAPTER 15 151 Members of a perfect cartel produce less than their best responses so cartels may be unstable 152 A point on firm 1s best response must involve a tan gency between 1s isoprofit and a horizontal line of height q2 This isoprofit reaches a peak at this point Firm 2s isoprofits look something like right parenthe ses that peak on 2s bestresponse curve An increase in demand intercept would shift out both best responses resulting in higher quantities in equilibrium 153 The firstorder condition is the mathematical represen tation of the optimal choice Imposing symmetry before taking a firstorder condition is like allowing firm i to choose the others outputs as well as its own Making this mistake would lead to the monopoly rather than the Cournot outcome in this example 154 An increase in the demand intercepts would shift out both best responses leading to an increase in equilib rium prices 155 Locating in the same spot leads to marginal cost pricing as in the Bertrand model with homogeneous products Locating at opposite ends of the beach results in the softest price competition and the highest prices 156 It is reasonable to suppose that competing gas stations monitor each others prices and could respond to a price change within the day so one day would be a reasonable period length A year would be a reasonable period for producers of small cartons of milk for school lunches because the contracts might be renegotiated each new school year 157 Reverting to the stagegame Nash equilibrium is a less harsh punishment in a Cournot model firms earn pos itive profit than a Bertrand model firms earn zero profit 158 Firms might race to be the first to market investing in research and development and capacity before suffi cient demand has materialized In this way they may compete away all the profits from being first a possible explanation for the puncturing of the dotcom bubble Investors may even have overestimated the advantages of being first in the affected industries 159 In most industries price can be changed quickly perhaps instantlywhereas quantity may be more difficult to adjust requiring the installation of more capacity Thus price is more difficult to commit to Among other ways firms can commit to prices by men tioning price in their national advertising campaigns by offering price guarantees and by maintaining a long run reputation for not discounting list price 1510 Entry reduces market shares and lower prices from tougher competition so one firm may earn enough profit to cover its fixed cost where two firms would not 1511 The social planner would have one firm charge mar ginal cost prices This would eliminate any deadweight loss from pricing and also economize on fixed costs CHAPTER 16 161 Nonlabor income permits the individual to buy lei sure but the amount of such purchases depends on laborleisure substitutability 162 The conclusion does not depend on linearity So long as the demand and supply curves are convention ally shaped the curves will be shifted vertically by the parameters t and k 163 With this sharing Equation 1637 becomes π 5 11 2 α2 v s1s2 2 pg g 2 ps s and profit maximiza tion requires that v ss 5 ps11 2 α2 Hence the firm will invest less in specific human capital In future bar gaining workers might be willing to accept a lower α in exchange for the firms paying some of the costs of general human capital Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 724 Brief Answers to Queries 164 Now MRP 5 30 per hour In this case the monopsony will hire 750 workers and wages will be 15 per hour As before the wages remains at only half the MRP 165 The monopsonist wants to be on its demand for labor curve the union presumably wants to be on the labor supply curve of its members Only the supplydemand equilibrium l 583 w 1167 satisfies both these curves Whether this is indeed a Nash equilibrium depends among other things on whether the union defines its payoffs as being accurately reflected by the labor supply curve 166 If the firm is riskneutral workers riskaverse opti mal contracts might have lower wages in exchange for morestable income CHAPTER 17 171 Using Equation 1717 yields c1c0 5 102 5 11 1 r2 1γ Solving for r gives 1 1 r 5 11022 γ If γ 5 1 r 5 002 If γ 5 3 r 5 0061 Greater fluctuation aversion requires a higher real interest rate to entice the typical person to accept the prevailing consumption growth 172 If g is uncertain the future marginal utility of consump tion will be a random variable If Ur 1c2 is convex its expected value with uncertain growth will be greater than its value when growth is at its expected value The effect is similar to what would occur with a lower growth rate Equation 1730 shows that the riskfree interest rate must fall to accommodate such a lower g 173 With an inflation rate of 10 percent the nominal value of the tree would rise at an additional 10 percent per year But such revenues would have to be discounted by an identical amount to calculate real profits so the opti mal harvesting age would not change 174 For a monopolist an equation similar to Equation 1767 would hold with marginal revenue replacing price With a constant elasticity demand curve price would have the same growth rate under monopoly as under perfect competition CHAPTER 18 181 The manager has an incentive to overstate gross profits unless some discipline is imposed by an audit If audits are costly the efficient arrangement might involve few audits with harsh punishments for false reports If harsh punishments are impossible the power of the managers incentives might have to be reduced 182 The insurer would be willing to pay the difference between its first and secondbest profits 298 2 96 5 202 183 Insurance markets are generally thought to be fairly competitive except where regulation has limited entry It is hard to say which segment is most competitive The fact that the individuals purchase car insurance whereas firms purchase health insurance on behalf of their employees in bulk may affect the nature of competition 184 A linear price would allow the consumer to buy what ever number of ounces desired at the 10 cents per ounce price Here the consumer is restricted to two cup sizes 4 or 16 ounces 185 The profit from serving a high type redcar owner is pH 2 025 20000 5 4146 2 5000 5 2854 There is no avoiding this loss If the insurer tries to drop the option for the high type they will simply buy the option meant for the low type generating even larger losses from serving the high type than 854 186 Graycar owners obtain utility of 114803 in the com petitive equilibrium under asymmetric information They would obtain the same utility under full insurance with a premium of 3210 The difference between this and the equilibrium premium 453 is 2757 Any premium between 3000 and 3210 would allow an insurance company to break even from its sales just for gray cars The problem is that redcar owners would deviate to the policy causing the company to make neg ative profit 187 If the reports are fairly credible then gray cars may still be able to get as full insurance with reporting as Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Brief Answers to Queries 725 without but not as full as with 100 percent credibility Auditors have shortrun incentives to take bribes to issue gray reports In the long run dishonesty will reduce the fees the auditor can charge He or she would like to maintain high fees by establishing a reputation for honest reporting which would be ruined if ever dis covered to be dishonest 188 If there are fewer sellers than buyers then all the cars will sell A car of quality q will sell at a price of q 1 b If there are fewer buyers than sellers then all buyers will purchase a car but some cars will be left unsold a random selection of them The equilibrium price will equal the cars quality q 189 Yes reservation prices can often help The tradeoffs involved in increasing the reservation price are on the one hand that buyers are encouraged to increase their bids but on the other hand that the probability the object goes unsold increases In a secondprice auction buyers bid their valuations without a reservation price and a reservation price would not induce them to bid above their valuations CHAPTER 19 191 The externality is positive if α 0 The downstream firm will now be more productive and thus use more labor than the upstream firm 192 Using the competitive output level the tax would be P 2 MC1xc2 5 1 2 20000 20000 5 0 equivalent to removing the tax returning the firms to the inefficient equilibrium The quantity used to evalu ate marginal cost matters because marginal cost varies with x 193 It is plausible that enforcing an agreement would become increasingly difficult the more roommates there are Monitoring and recording the efforts of many individuals would be a complex job Feeling less per sonal connection one of many roommates might be more inclined to shirk 194 The roommates have identical preferences here and therefore identical marginal rates of substitution If each pays half the price of the public good then the sum of their MRSs will be precisely the ratio of the price of the public good to the price of the private good as required in Equation 1940 With differing MRSs the sharing might depart from 5050 to ensure efficiency 195 Reduction of the labor tax increases aftertax income and the demand for good y With a fixed Pigovian tax pollution rises More generally the likelihood of a dou ble dividend depends on the precise demand relation ship in peoples utility functions between clean air and the other items being taxed here labor 196 Progressive taxation should raise t because the median voter can gain more revenue from highincome tax pay ers without incurring high tax costs Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 727 Only very brief solutions to most of the oddnumbered problems in the text are given here Complete solutions to all of the problems are contained in the Solutions Manual which is available to instructors upon request CHAPTER 2 21 a fx 5 8x fy 5 6y b Constraining f1x y2 5 16 creates an implicit func tion between the variables The slope of this func tion is given by dy dx 5 2 fx fy 5 28x 6y for combinations of x and y that satisfy the constraint c Since f11 22 5 16 dy dx 5 2 8 1 6 2 5 2 2 3 d The f1x y2 5 16 contour line is an ellipse centered at the origin The slope of the line at any point is given by dydx 5 28x6y 23 Both approaches yield x 5 y 5 05 25 a The firstorder condition for a maximum is 2gt 1 40 5 0 so t 5 40g b Substitution yields f1t2 5 205g 140g2 21 40 140g2 5 800g So f1t2g 5 2800g2 c This follows because fg 5 205 1t2 2 d fg 5 205 140g2 2 5 208 so each 01 increase in g reduces maximum height by 008 27 a Firstorder conditions require f1 5 f2 5 1 Hence x2 5 5 With k 5 10 x1 5 5 b With k 5 4 x1 5 21 c x1 5 0 x2 5 4 d With k 5 20 x1 5 15 x2 5 5 Because marginal value of x1 is constant every addition to k beyond 5 adds only to that variable 29 Since fii 0 the condition for concavity implies that the matrix of secondorder partials is negative definite Hence the quadratic form involving 3 f1 f24 will be neg ative as required for quasiconcavity The converse is not true as shown by the CobbDouglas function with α 1 β 1 211 a f s 5 δ1δ 2 12xδ22 0 b Since f11 f22 0 and f12 f21 5 0 Equation 298 obviously holds c This preserves quasiconcavity but not concavity 213 a From Equation 285 a function in one variable is concave if f s 1x 2 0 Using the quadratic Taylor to approximate fx near a point a f1x2 f1a2 1 f r 1a2 1x 2 a2 1 05f s 1a2 1x 2 a2 2 f 1a2 1 f r 1a2 1x 2 a2 because f s 1a2 0 and 1x 2 a2 2 02 b From Equation 298 a function in two vari ables is concave if f11 f22 2 f 2 12 0 and we also know that due to the concavity of the function 05 1 f11dx 2 1 1 2f12dx1dx2 1 f22dx 2 22 0 This is the third term of the quadratic Taylor expansion where dx 5 x 2 a dy 5 y 2 b Thus we have f1x y2 f1a b2 1 f1 1a b2 1x 2 a2 1 f2 1a b2 1y 2 b2 which shows that any concave function must lie on or below its tangent plane Solutions to OddNumbered Problems Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 728 Solutions to OddNumbered Problems 215 a Use Var 1x2 5 E3 1x 2 E1x2 2 24 5 E1x2 2 2xE1x2 1 1E1x2 2 22 b Let y 5 1x 2 μx2 2 and apply Markovs inequality to y c First part is trivial Let 5 Sx i n E1X2 5 nμn 5 μ Var 1X2 5 nσ2n2 5 σ2n d Var 1X2 5 12k2 2 2k 1 12σ2 which is minimized for k 5 05 In this case Var 1X2 5 05σ2 If say k 5 07 Var 1X2 5 058σ2 so it is not changed all that much e If σ2 1 5 rσ2 2 the weighted average is minimized if k 5 r 11 1 r2 CHAPTER 3 31 a No b Yes c Yes d No e Yes 33 The shape of the marginal utility function is not neces sarily an indicator of convexity of indifference curves 35 a U1h b m r2 5 min1h 2b m 05r2 b A fully condimented hot dog c 160 d 210an increase of 31 percent e Price would increase only to 1725an increase of 78 percent f Raise prices so that a fully condimented hot dog rises in price to 260 This would be equivalent to a lumpsum reduction in purchasing power 37 a Indifference curve is linearMRS 5 13 b α 5 2 β 5 1 c Just knowing the MRS at a known point can identify the ratio of the CobbDouglas exponents 39 ac See detailed solutions 311 It follows since MRS 5 MUx MUy MUx doesnt depend on y or vice versa 31b is a counterexample 313 a MRS 5 fx fy 5 y b fxx 5 fxy 5 0 so the condition for quasiconcavity reduces to 21y2 0 c An indifference curve is given by y 5 exp 1k 2 x2 d Marginal utility of x is constant marginal utility of y diminishes As income rises consumers will even tually choose only added x e y could be a particular good whereas x could be everything else 315 a U 5 αβα112β2 5 α Hence b 1U 2 5 U b Because the reference bundle has y 5 0 it is not possible to attain any specified utility level by repli cating this bundle c α is given by the length of a vector in the direction of the reference bundle from the initial endow ment to the target indifference curve See detailed solutions d This follows directly from the convexity of indiffer ence curves See detailed solutions CHAPTER 4 41 a t 5 5 and s 5 2 b t 5 52 and s 5 4 Costs 2 so needs extra 1 43 a c 5 10 b 5 3 and U 5 127 b c 5 4 b 5 1 and U 5 79 45 b g 5 I 1 pg 1 pv 22 v 5 I 12pg 1 pv2 c Utility 5 V1 pg pv I2 5 m 5 v 5 I 12pg 1 pv2 d Expenditures 5 E1 pg pv V2 5 V 12pg 1 pv2 47 a See detailed solutions b Requires expenditure of 12 c Subsidy is 59 per unit Total cost of subsidy is 5 d Expenditures to reach U 5 2 are 971 To reach U 5 3 requires 486 more A subsidy on good x must be 074 per unit and costs 829 e With fixed proportions the lumpsum and single good subsidy would cost the same 49 If px py ab then E 5 pxUa If px py ab then E 5 pyUb If px py 5 ab then E 5 pxUa 5 pyUb 411 a Set MRS 5 px py b Set δ 5 0 c Use px xpy y 5 1 px py2 δ1δ212 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Solutions to OddNumbered Problems 729 413 a See detailed solutions b Multiplying prices and income by 2 does not change V c Obviously VI 0 d Vpx Vpy 0 e Just exchange I and V f Multiplying the prices by 2 doubles E g Just take partials h Show Epx 0 2Ep2 x 0 CHAPTER 5 51 a U1x y2 5 075x 1 2y 3 8 x 1 y b x 5 Ipx if px 3 8 py c x 5 0 if px 3 8 py d Changes in py dont affect demand until they reverse the inequality e Just two points or vertical lines 53 a It is obvious since px py doesnt change b No good is inferior 55 a x 5 I 2 px 2px y 5 I 1 px 2py Hence changes in py do not affect x but changes in px do affect y b V 5 1I 1 px2 2 4px py and so E 5 4px pyV 2 px c The compensated demand function for x depends on py whereas the uncompensated function did not 57 a Use the Slutsky equation in elasticity form Because there are no substitution effects eh ph 5 0 2 sh eh I 5 0 2 05 5 205 b Compensated price elasticity is zero for both goods which are consumed in fixed proportions c Now sh 5 23 so ehph 5 223 d For a ham and cheese sandwich 1sw2 esw psw 5 21 esw ph 5 esw psw epsw ph 5 1212 05 5 205 59 a sx I 5 pxIxI 2 pxx I 2 Multiplication by I sx 5 I 2 pxx gives the result bd All of these proceed as in part a e Use Slutsky equationsee detailed solutions 511 a Just follow the approaches used in the twogood cases in the text see detailed solutions 513 a ln E1px pyU2 5 a0 1 α1 ln px 1 α2 log py 1 1 2 γ11 1ln px2 2 1 1 2 γ22 1ln py2 2 1 γ12 ln px ln py 1 U β0 pβ1 x pβ2 y b Doubling all prices adds ln 2 to the log of the expen diture function thereby doubling it with U held constant c sx 5 α1 1 γ11 ln px 1 γ12 ln py 1 Uβ0 β1 pβ121 x pβ2 y 515 a Decision utility i If px 5 py 5 1 and I 5 10 x 5 8 y 5 2 U2 5 1008 ii x 5 7 y 5 3 U2 5 1030 Hence there is a loss of utility of 022 iii Achieving y 5 3 requires a price of py 5 23 With this price this person chooses x 5 8 y 5 3 U2 5 8 1 3 ln 3 5 1130 so this subsidy would have to be accompanied by an income tax of 1 to arrive at the same bundle as in part ii Arriving at the bundle specified in ii could also be achieved by both taxing good x and subsidizing good y That solution would require a unit tax of 19 on good x and a unit subsidy of 727 on good y iv Utility could also be raised to 1030 from the 1008 calculated in part i with an income grant of 022 all of which would be used to purchase good x So that would not address the problem of the underconsumption of good y b Preference uncertainty i With U1x y2 5 x 1 25 ln y the optimal choices are x 5 75 y 5 25 U1 5 x 1 2 ln y 5 933 U2 5 x 1 3 ln y 5 1025 ii With perfect knowledge U1 5 8 1 2 ln 2 5 939 U2 5 7 1 3 ln 3 5 1030 so in each case there is a utility loss of about 005 iii As a result of part ii this person would pay up to about 005 to learn what his or her preferences will actually be CHAPTER 6 61 a Convert this to a CobbDouglas with α 5 β 5 05 Result follows from prior examples Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 730 Solutions to OddNumbered Problems b Also follows from CobbDouglas c Set mps 5 spm and cancel the symmetric substitution effects d Use the CobbDouglas representation 63 a pbt 5 2pb 1 pt b Since pc and I are constant c 5 I2pc is also constant c Yessince changes in pb or pt affect only pbt 65 a p2x2 1 p3x3 5 p3 1kx2 1 x32 b Relative price 5 1p2 1 t2 1p3 1 t2 Approaches p2p3 1 as t S 0 Approaches 1 as t S q So an increase in t raises the relative price of x2 c Does not strictly apply since changes in t change relative prices d May reduce spending on x2the effect on x3 is uncertain 67 Show xi xj I 5 xj xiI and use symmetry of net substitution effects 69 a CV 5 E1pr1 pr2 p3 c pn U2 2 E1p1 p2 p3 c pnU 2 b See graphs in detailed solutionsnote that change in one price shifts compensated demand curve in the other market c Symmetry of crossprice effects implies that order is irrelevant d Smaller for complements than for substitutes 611 See graphs in detailed solutions or in Samuelson reference 613 a Applying the envelope theorem to both minimiza tion problems yields dE dt 5 dE dp1 dp1 dt 1 dE dp2 dp2 dt 1 dE dp3 dp3 dt 5 0 1 x2 c p2 0 1 x3 c p3 0 5 y 5 dE dt Again applying the envelope theorem to both problems dE dp1 5 x1 c 5 dE dp1 b Because neither the price of x2 or x3 changes the maximum value for the function V depends only on m That is there is a unique correspondence between m and the utility it provides The equality of the Lagrange multipliers is derived by repeated application of the envelope theorem to the various optimization subproblems CHAPTER 7 71 P 5 0525 73 a One trip expected value 5 05 0 1 05 12 5 6 Two trip expected value 5 025 0 1 05 6 1 025 12 5 6 b Twotrip strategy is preferred because of smaller variance c Adding trips reduces variance but at a diminishing rate So desirability depends on the trips cost 75 a E1U2 5 075 ln1100002 1 025 ln190002 5 91840 b E1U2 5 ln197502 5 91850insurance is preferable c 260 77 a E1v22 5 1 b E1h22 5 k2 c r1W2 5 1W d The formula p 5 k22W can be used to compute the six numerical values of the risk premium which is increasing in k and decreasing in W 79 a 1 Her expected utility from investing only in A is EA3U1W2 4 5 1 216 1 1 20 5 2 and from investing equally in the two assets is Eequal split 3U1W2 4 5 1 4125 1 1 4 0 1 1 48 1 1 445 2121 2 One can see from a graph of Ea12a split 3U1W2 4 5 1 416a 1 9 11 2 a2 1 1 4 0 1 1 4 4a 1 1 4 31 2 a that it is maximized by a 5 08 to one decimal b 1 With perfect negative correlation Eequal split 3U1W2 4 5 1 28 1 1 245 2475 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Solutions to OddNumbered Problems 731 2 One can see from a graph of Ea12a split 3U1W2 4 5 1 2 4a 1 1 2 31 2 a that it is maximized by a 5 06 711 a 1r1W2 5 μ 1 wγ b rr1W2 5 γ c lim γSq r1w2 5 1μ d Letting lim γSq r1w2 5 1μ 5 A yields Us 1w2 5 2AUr 1w2 Solving this differential equation yields U1w2 5 2kA21e2Aw the same as the formula in the text for k 5 A e U1w2 5 θ 1μ2 2 2μ w 1 w 22 f Utility is still unbounded for certain values of the parameters 713 a See graph in detailed answers b Mixed portfolios lie on a segment between the risky and riskless assets c Risk aversion is indicated by sharper bend to indif ference curves A person with Lshaped indifference curves infinitely risk averse would hold no risky asset d A CRRA investor has homothetic indifference curves 715 g Riskneutral Stan indifferent among AD h Riskaverse Stan should choose safe option in each scenario B in 1 and D in 2 i Most subjects chose C in Scenario 2 but a risk averse person should choose D j 1 Depends but could make same choices as most experimental subjects 2 See detailed answers for graph Curve has to shift because of kink at anchor point Petes curves are convex below anchor and concave above while Stans are concave everywhere CHAPTER 8 81 a C F b Each player randomizes over the two actions with equal probability c Players each earn 4 in the purestrategy equilib rium Players 1 and 2 earn 6 and 7 respectively in the mixedstrategy equilibrium d The extensive form is similar to Figures 181 and 182 but has three branches from each node rather than two 83 a The extensive form is similar to Figure 89 b Do not veer veer and veer do not veer c Players randomize with equal probabilities over the two actions d Teen 2 has four contingent strategies always veer never veer do the same as Teen 1 and do the oppo site of Teen 1 e The first is do not veer always veer the second is do not veer do the opposite and the third is veer never veer f Do not veer do the opposite is a subgameperfect equilibrium 85 a If all play blond then one would prefer to deviate to brunette to obtain a positive payoff If all play bru nette then one would prefer to deviate to blond for payoff a rather than b b Playing brunette provides a certain payoff of b and blond provides a payoff of a with probabil ity 11 2 p2 n21 the probability no other player approaches the blond Equating the two payoffs yields p 5 1 2 1ba2 11n212 c The probability the blond is approached by at least one male equals 1 minus the probability no males approach her 1 2 11 2 p2 n 5 1 2 1ba2 n1n212 This expression is decreasing in n because n 1n 2 12 is decreasing in n and ba is a fraction 87 a The bestresponse function is lLC 5 35 1 l24 for the lowcost type of player 1 lHC 5 25 1 l24 for the highcost type and l2 5 3 1 l14 for player 2 where l1 is the average for player 1 Solving these equations yields l LC 5 45 l HC 5 35 and l 2 5 4 c The lowcost type of player 1 earns 2025 in the BayesianNash equilibrium and 2055 in the full information game so it would prefer to signal its type if it could Similar calculations show that the highcost player would like to hide its type 89 a The condition for cooperation to be sustainable with one period of punishment is δ 1 so one period of punishment is not enough Two periods of punishment are enough as long as δ2 1 δ 2 1 0 or δ 062 b The required condition is that the present dis counted value of the payoffs from cooperat ing 211 2 δ2 exceed that from deviating 3 1 δ11 2 δ102 11 2 δ2 1 2δ11 11 2 δ2 Simplifying Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 732 Solutions to OddNumbered Problems 2δ 2 δ11 2 1 0 Using numerical or graphi cal methods this condition can be shown to be δ 050025 not much stricter than the condition for cooperation with infinitely many periods of punishment 1δ 122 811 a Responder accepts any r 0 proposer offers r 5 0 b Same as in a c 1 Responder accepts any r a 11 1 2a2 2 Proposer offers exactly r 5 a 11 1 2a2 3 In Dictator Game proposer still offers r 5 0 so less even split than Ultimatum Game CHAPTER 9 91 a k 5 8 and l 5 8 b k 5 10 and l 5 5 c k 5 9 l 5 65 k 5 95 and l 5 575 fractions of hours d The isoquant is linear between solutions a and b 93 a q 5 10 k 5 100 l 5 100 C 5 10000 b q 5 10 k 5 33 l 5 132 C 5 8250 c q 5 1213 k 5 40 l 5 160 C 5 10000 d Carlas ability to influence the decision depends on whether she can impose any costs on the bar if she is unhappy serving the additional tables Such abil ity depends on whether Carla is a draw for Cheers customers 95 Let A 5 1 for simplicity a fk 5 αkα21l β 0 fl 5 βkαl β21 0 fkk 5 α 1α 2 12kα22l β 0 fll 5 β 1β 2 12kαl β22 0 fkl 5 flk 5 αβkα21l β21 0 b eqk 5 fk kq 5 α eql 5 fl lq 5 β c f1tk tl2 5 t α1βf1k l2 f1tk tl2t tf1k l2 5 1α 1 β2t α1β At t 5 1 this is just α 1 β d e Apply the definitions using the derivatives from part a 97 a β0 5 0 b MPk 5 β2 1 1 2 β1lk MPL 5 β3 1 1 2 β1kl c In general σ is not constant If β2 5 β3 5 0 σ 5 1 If β1 5 0 σ 5 q 99 a If f1tk tl2 5 tf1k l2 then eqt 5 f1tk tl2t tf 1tk tl2 If t S 1 then f1k l2f1k l2 5 1 b Apply Eulers theorem and use part a f1k l2 5 fkk 1 fll c eqt 5 2 11 2 q2 Hence q 05 implies eqt 1 and q 05 implies eqt 1 d The production function has an upper bound of q 5 1 911 a Apply Eulers theorem to each fi b With n 5 2 k2fkk 1 2klfkl 1 l 2fll 5 k1k 2 12f1k l2 If k 5 1 this implies fkl 0 If k 1 it is even clearer that fkl must be positive For k 1 the case is not so clear c Implies that fij 0 is more common for k 5 1 d 1Sαi2 2 2 Sαi 5 k1k 2 12 CHAPTER 10 101 By definition C1q1 02 is the cost of producing just good 1 in one firm By assumption C1q1q22 q C1q102 q1 C1q1q22 q C10q22 q2 Multiplying respectively by q1 and q2 and summing gives the economiesofscope condition 103 a C 5 q 1v5 1 w102 AC 5 MC 5 v5 1 w10 b For q 50 SC 5 10v 1 wq10 SAC 5 10vq 1 w10 SMC 5 w10 c AC 5 MC 5 05 For q 50 SAC 5 10q 1 03 SMC 5 03 105 a First show SC 5 125 1 q2 125 1 q2 2100 Set up Lagrangian for cost minimization 5 SC 1 λ1q 2 q1 2 q22 yielding q1 5 025 q2 b SC 5 125 1 q2125 SMC 5 2q125 SAC 5 125q 1 q125 SMC11002 5 160 SMC11252 5 200 SMC12002 5 320 c Distribution across plants irrelevant in long run C 5 2q AC 5 MC 5 2 d Distribute output evenly across plants Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Solutions to OddNumbered Problems 733 107 a Let B 5 v12 1 w12 Then k 5 Cv 5 Bv212q and l 5 Cw 5 Bw212q b q 5 1 k21 1 l21 109 a C 5 q1γ 3 1vα2 12σ 1 1wβ2 12σ4 1112σ2 b C 5 qα2αβ2β vαwβ c wlvk 5 βα d lk 5 3 1vα21wβ2 4 σ so wlvk 5 1vw2 σ21 1βα2 σ Labors relative share is an increasing function of βα If σ 1 labors share moves in the same direction as vw If σ 1 labors relative share moves in the opposite direction to vw This accords with intuition on how substitutability should affect shares 1011 a sij 5 ln Ci ln wj 2 ln Cj ln wj 5 ex c i wj 2 ex c j wj b sij 5 ln Cj ln wi 2 ln Ci ln wi 5 ex c j wi 2 ex c i wi c See detailed solutions CHAPTER 11 111 a q 5 50 b π 5 200 c q 5 5P 2 50 113 a C 5 wq24 b π 1P w2 5 P 2w c q 5 2Pw d l 1P w2 5 P 2w 2 115 a Diminishing returns is needed to ensure that a profitmaximizing output choice exists b C1v w q2 5 1w 1 v2q2100 P 1P v w2 5 25P 2 1w 1 v2 c q 5 PP 5 50P 1w 1 v2 5 20 P 5 6000 d q 5 30 P 5 13500 117 a b q 5 a 1 bP P 5qb 2 ab R 5 Pq 5 1q2 2 aq2b mr 5 2qb 2 ab and the mr curve has double the slope of the demand curve so d 2 mr 5 2qb c mr 5 P 11 1 1e2 5 P 11 1 1b2 d It follows since e 5 qP Pq 119 b Diminishing returns is needed to ensure increasing marginal cost c σ determines how firms adapt to disparate input prices d q 5 PP 5 K 11 2 γ2 P γγ21 1v12σ 1 w12σ2 γ112σ21γ212 The size of σ does not affect the supply elasticity but greater substitutability implies that increases in one input price will shift the supply curve less e See detailed solutions 1111 a Follow the indicated steps By analogy to part c of Problem 1110 qv 5 2kP b As argued in the text lw 0 By similar argu ments kv 0 implying the last term of the dis played equation in part a is positive c First differentiate the definitional relation with respect to w Second differentiate the relation with respect to v and use this expression to sub stitute for l sk Finally substitute the result kw 5 lw d The increase in long versus shortrun costs from a wage increase wr ws can be compared by com bining three facts C1v wr q2 5 SC1v wr q kr2 for kr 5 kc 1v wr q2 C1v ws q2 5 SC1v ws q ks2 for ks 5 kc 1v ws q2 SC1v ws q ks2 SC1v ws q kr2 1113 a See detailed answers for proof b The formula for crossprice elasticity of input demand weighs both terms by the share of the other input The effect of a change in the price of the other input will depend primarily on the importance of this other input c Using Shephards lemma and an implication of Eulers Theorem 1Cww 5 2vCwvw2 shows All 5 2vkCwvC wlCwCv 5 2sk sl Akl 1115 If the assets are separate the equilibrium investments are x s F 5 116 and x s G 5 a216 yielding joint surplus 316 11 1 a22 If GM acquires both assets equilib rium investments are x b F 5 0 and x b G 5 a24 yielding joint surplus a24 The latter joint surplus is higher if a 3 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 734 Solutions to OddNumbered Problems CHAPTER 12 121 a q 5 10P 2 20 b Q 5 1000P 2 2000 c P 5 25 Q 5 3000 123 a P 5 a 2 d g 2 b 1 c g 2 b I Q 5 d 1 g 1a 2 d2 g 2 b 1 cg g 2 b I b dP dI 5 c g 2 b 0 dQ dI 5 cg g 2 b 0 c Differentiation of the demand and supply equations yields b dP dI 1 c 2 dQ dI 5 0 g dP dI 2 dQ dI 5 0 dP dI 5 2c 21 0 21 b 21 g 21 5 c g 2 b dQ dI 5 b 2c g 0 b 21 g 21 5 cg g 2 b d Suppose a 5 10 b 5 21 c 5 01 d 5 210 g 5 1 I 5 100 P 5 10 1 005 100 5 15 Q 5 5 an increase of income of 10 would increase quan tity demanded by 1 if price were held constant This would create an excess demand of 1 that must be closed by a price rise of 05 125 a n 5 50 Q 5 1000 q 5 20 P 5 10 and w 5 200 b n 5 72 Q 5 1728 q 5 24 P 5 14 and w 5 288 c The increase for the makers 5 5368 The linear approximation for the supply curve yields approxi mately the same result 127 a P 5 11 Q 5 500 and r 5 1 b P 5 12 Q 5 1000 and r 5 2 c DPS 5 750 d D rents 5 750 129 a Longrun equilibrium requires P 5 AC 5 MC AC 5 kq 1 a 1 bq 5 MC 5 a 1 2bq Hence q 5 kb P 5 a 1 2kb b Want supply 5 demand nq 5 nkb 5 A 2 BP 5 A 2 B1a 1 2kb2 Hence n 5 A 2 B1a 1 kb2 kb c A has a positive effect on n That makes sense since A reflects the size of the market If a 0 the effect of B on n is clearly negative d Fixed costs k have a negative effect on n Higher marginal costs raise price and therefore reduce the number of firms 1211 a Use the deadweight loss formula from Problem 1210 a n i51 DW1ti2 1 λaT 2 a n i51 ti pi xib λi 5 05 3eD eS 1eS 2 eD2 42ti pi xi 2 λpi xi 5 0 T 5 T 2 a n i51 ti pi xi 5 0 Thus ti 5 2λ 1eS 2 eD2eS eD 5 λ 11eS 2 1eD2 b The above formula suggests that higher taxes should be applied to goods with more inelastic supply and demand c This result was obtained under a set of very restric tive assumptions 1213 More on the comparative statics of supply and demand a dP dβ 5 0 21 2Sβ 21 DP 21 SP 21 5 2Sβ SP 2 DP dQ dβ 5 DP 0 SP 2Sβ DP 21 SP 21 5 2DP Sβ SP 2 DP Hence if Sβ 0 then dP dβ 0 and dQdβ 0 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Solutions to OddNumbered Problems 735 b dP dQ 5 0 21 1 21 DP 21 SP 21 5 1 SP 2 DP 0 dQ DQ 5 DP 0 SP 1 DP 21 SP 21 5 DP SP 2 DP 0 So a positive quantity wedge increases price and reduces the quantity that goes to meet demand c The analysis in the chapter shows that dQdα dP dα 5 SP With sufficient observations on the impact of dif fering values of α one could identify the slope of the supply curve Similarly with sufficient obser vations on the impact of differing values of β one could identify the slope of the demand curve If the same parameter shifts both curves it is not possible to identify the slope of either of them CHAPTER 13 131 b If y 5 2x x2 1 2 12x2 2 5 900 9x2 5 900 x 5 10 y 5 20 c If x 5 9 on the production possibility frontier y 5 8192 5 2024 If x 5 11 on the frontier y 5 7792 5 1974 Hence RPT is approximately 2DyDx 5 2120502 2 5 025 133 Let F 5 Food C 5 Cloth a Labor constraint F 1 C 5 100 b Land constraint 2F 1 C 5 150 c Outer frontier satisfies both constraints d Frontier is concave because it must satisfy both con straints Since the RPT 5 1 for the labor constraint and 2 for the land constraint the production possi bility frontier of part c exhibits an increasing RPT hence it is concave e Constraints intersect at F 5 50 C 5 50 For F 50 dCdF 5 21 so PFPC 5 1 For F 50 dCdF 5 22 so PFPC 5 2 f If for consumers dCdF 5 25 4 then PFPC 5 5 4 g If PFPC 5 19 or PFPC 5 11 consumers will choose F 5 50 C 5 50 since both price lines are tangent to production possibility frontier at its kink h 08F 1 09C 5 100 Capital constraint C 5 0 F 5 125 F 5 0 C 5 1111 This results in the same PPF since capital constraint is nowhere binding 135 a The contract curve is a straight line Only equilib rium price ratio is PHPC 5 43 b Initial equilibrium on the contract curve c Not on the contract curveequilibrium is between 40H 80C and 48H 96C d Smith takes everything Jones starves 137 a px 5 0374 py 5 0238 pk 5 0124 pl 5 0264 x 5 262 y 5 223 b px 5 0284 py 5 0338 pk 5 0162 pl 5 0217 x 5 302 y 5 185 c Raises price of labor and relative price of x 139 Computer simulations show that increasing returns to scale is still compatible with a concave production pos sibility frontier provided the input intensities of the two goods are suitably different 1311 a Doubling prices leaves excess demands unchanged b Since by Walras law p1ED1 5 0 and ED1 5 0 The excess demand in market 1 can be calculated explicitly as ED1 5 13p2 2 2 6p2 p3 1 2p2 3 1 p1 p2 1 2p1 p32p2 1 This is also homogeneous of degree 0 in the prices c p2p1 5 3 p3p1 5 5 1313 a As demand for good x is given by xA 5 2 1 pxA 1 yA2 3p Bs demand for good x is given by xB 5 p 11000 2 xA2 1 1000 2 yA 3p b Setting demand equal to supply for good x yields p 5 yA 1 1000 2000 2 xA c With these initial endowments p 5 1 d Part b shows that increases in the endowment of either good for person A will raise the relative price of good x because that good is favored by this person CHAPTER 14 141 a Q 5 24 P 5 29 and π 5 576 b MC 5 P 5 5 and Q 5 48 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 736 Solutions to OddNumbered Problems c Consumer surplus 5 1152 Under monopoly con sumer surplus 5 288 profits 5 576 deadweight loss 5 288 143 a Q 5 25 P 5 35 and π 5 625 b Q 5 20 P 5 50 and π 5 800 c Q 5 40 P 5 30 and π 5 800 145 a P 5 15 Q 5 5 C 5 65 and π 5 10 b A 5 3 P 5 15 Q 5 605 and π 5 1225 147 a Under competition P 5 10 Q 5 500 CS 5 2500 Under monopoly P 5 16 Q 5 200 CS 5 400 b See graph in detailed solutions c Loss of 2100 of which 800 is transferred to monop oly profits 400 is a loss from increased costs not rel evant in usual analysis and 900 is a deadweight loss 149 Firstorder conditions for a maximum imply X 5 C1X2Cr 1X2that is X is chosen independently of Q 1411 a The solution for monopoly quantity is Qm 5 c 1s 2 12 1a1 2 c12 a0 2 c0 d 1s b Constant average and marginal cost corresponds to c1 5 0 Substituting into the solution from part a gives Qm 5 c a1 1s 2 12 a0 2 c0 d 1s c Letting di 5 ai 2 ci and x 5 Q s the firstorder condition can be turned into a quadratic equation with solution xm 5 d 2 0 1 4d1d2 1s2 2 12 2 d0 2 11 1 s2d2 Monopoly quantity can be recovered from Qm 5 x1s m d See the detailed solutions for a graph illustrating various average cost shapes 1413 a Pm 5 8 Qm 5 2 πm 5 4 CSm 5 2 Wm 5 πm 1 CSm 5 6 b Monopoly profit is π 5 Q 1P 1 s 2 AC2 5 110 2 P2 1P 1 s 2 62 Maximizing yields Pm 5 8 2 s2 implying Qm 5 2 1 s2 and πm 5 12 1 s22 2 c Subtracting consumers expenditure from gross consumer surplus yields CSm 5 1 2a18 2 s 2b a2 1 s 2b 2 a8 1 s 2b a2 1 s 2b 5 1 8 14 2 3s2 14 1 s2 d Welfare equals Wm 5 πm 1 CSm 5 112 2 s2 14 1 s28 maximized at s 5 4 e The solution for the monopoly price is exactly as in part b The difference is that the subsidy expen diture now comes from the government instead of misinformed consumers CHAPTER 15 151 a P m 5 Qm 5 75 Pm 5 5625 b P c 5 qc i 5 50 πc i 5 2500 c P b 5 0 Qb 5 150 πb i 5 0 153 a Equilibrium quantities are qc i 5 11 2 2ci 1 cj23 Further Qc 5 12 2 c1 2 c223 P c 5 11 1 c1 1 c223 πc i 5 11 2 2c1 1 c22 29 Pc 5 πc 1 1 πc 2 CSc 5 12 2 c1 2 c22 218 and W c 5 Pc 1 CSc b The diagram looks like Figure 152 A reduction in firm 1s cost would shift its best response out increasing its equilibrium output and reducing 2s 155 a p i 5 1 12 2 b2 b q i 5 11 2 2b2 12 2 b2 π i 5 1 12 2 b2 2 c The diagram would look like Figure 154 An increase in b would shift out both best responses and result in higher equilibrium prices for both 157 a q 1 5 75 q 2 5 752 b If firm 1 accommodates 2s entry it earns 28125 To deter 2s entry 1 needs to produce q1 5 150 2 2K2 Firm 1s profit from operating alone in the market and producing this output is 1150 2 2K22 12K22 which exceeds 28125 if K2 1206 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Solutions to OddNumbered Problems 737 159 a q i 5 1a 2 c2 1n 1 12b Further Q 5 n1a 2 c2 1n 1 12b P 5 1a 1 nc2 1n 1 12 P 5 nπ i 5 1nb2 3 1a 2 c2 1n 1 12 4 2 CS 5 1n2b2 3 1a 2 c2 1n 1 12 4 2 and W 5 3n 1n 1 12 4 3 1a 2 c2 2b4 Because firms are symmetric si 5 1n implying H 5 n11n2 2 5 1n b We can obtain a rough idea of the effect of merger by seeing how the variables in part a change with a reduction in n Perfirm output price industry profit and the Herfindahl index increase Total out put consumer surplus and welfare decrease c Substituting c1 5 c2 5 14 into the answers for 153 we have q i 5 14 Q 5 12 P 5 12 P 5 18 CS 5 18 and W 5 14 Also H 5 12 d Substituting c1 5 0 and c2 5 14 into the answers for 153 we have q 1 5 512 q 2 5 212 Q 5 712 P 5 512 P 5 29144 CS 5 49288 and W 5 107288 Also H 5 2949 e Comparing part a with b suggests that increases in the Herfindahl index are associated with lower welfare The opposite is evidenced in the compari son of part c to d 1511 a This is the indifference condition for a consumer located distance x from firm i b The profitmaximizing price is p 5 1 p 1 c 1 tn22 c Setting p 5 p and solving for p gives the specified answer d Substituting p 5 p 5 c 1 tn into the profit func tion gives the specified answer e Setting tn2 2 K 5 0 and solving for n yields n 5 tK f Total transportation costs equal the number of halfsegments between firms 2n times the trans portation costs of consumers on the half segment e 12n 0 tx dx 5 t8n2 Total fixed cost equal nF The number of firms minimizing the sum of the two is n 5 1122tK 1513 a The expected margin on each consumer must be zero at the posted prices 11 2 α2 1v 2 c2 1 α 1 pi 2 c2 5 0 Solving p i 5 3c 2 11 2 α2v4α implying s i 5 1v 2 c2α b Color laser printers may earn most of their profits from multiple toner cartridges with shrouded prices c If firm 1 advertises the net surplus that consum ers all sophisticated now obtain from firm 2 is v 2 e 2 p 2 5 1v 2 c2α 2 e Firm 1 may as well dispense with the shrouded price and just charge a posted price optimally leaving consumers with no more surplus than if they buy from firm 2 pd 1 5 v 2 1v 2 c2α 2 e Firm 1s resulting profit margin is negative under the stated condition d Advertising effectively educates myopic consum ers about how to buy at the posted price from the rival firm which is difficult for the advertising firm to undercut and still break even e If the posted price from part a is already non negative the equilibrium is unchanged If it is negative one can show that the firm earns a posi tive margin on each consumer by setting the posted price to 0 CHAPTER 16 161 a Full income 5 40000 l 5 2000 hours b l 5 1400 hours c l 5 1700 hours d Supply is asymptotic to 2000 hours as w rises 163 a Grant 5 6000 2 075 1I2 If I 5 0 grant 5 6000 I 5 2000 grant 5 4500 I 5 4000 grant 5 3000 b Grant 5 0 when 6000 2 075I 5 0 I 5 6000 075 5 8000 c Assume there are 8000 hours in the year Full Income 5 4 3 8000 5 32000 5 c 1 4h d Full income 5 32000 1 grant 5 32000 1 6000 2 075 4 18000 2 h2 5 38000 2 24000 1 3h 5 c 1 4h or 14000 5 c 1 h for I 8000 That is for h 6000 hours welfare grant creates a kink in the budget constraint at 6000 hours of leisure 165 a For MEl 5 MRPl l40 5 10 2 l40 so 2l40 5 10 and l 5 200 Get w from supply curve w 5 l80 5 20080 5 250 b For Carl the marginal expense of labor now equals the minimum wagewm 5 400 Setting this equal to the MRP yields l 5 240 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 738 Solutions to OddNumbered Problems c Under perfect competition a minimum wage means higher wages but fewer workers employed Under monopsony a minimum wage may result in higher wages and more workers employed 167 a Since q 5 240x 2 2x2 total revenue is 5q 5 1200x 2 10x2 MRP 5 TRx 5 1200 2 20x Produc tion of pelts x 5 l Total cost 5 wl 5 10x2 Marginal cost 5 Cx 5 20x Under competition price of pelts 5 MC 5 20x MRP 5 px 5 MC 5 20x x 5 30 px 5 600 b From Dans perspective demand for pelts 5 MRP 5 1200 2 20x R 5 px x 5 1200x 2 20x2 Marginal revenue Rx 5 1200 2 40x set equal to marginal cost 5 20x Yields x 5 20 px 5 800 c From UFs perspective supply of pelts 5 MC 5 20x 5 px total cost px x 5 20x2 and MEx 5 C x 5 40x So MEx 5 40x 5 MRPx 5 1200 2 20x with a solution of x 5 20 px 5 400 169 E3U1yjob12 4 5 100 40 2 05 1600 5 3200 E3U1yjob22 4 5 E3U1wh2 4 5 E3100wh 2 05 1wh2 24 5 800w 2 05 336w2 1 64w24 5 800w 2 50w2 1611 a Vw 5 λ 11 2 h2 5 λl 1w n2 Vn 5 λ l 1w n2 5 1Vw2 1Vn2 b xiw 5 xiw 0 U5Constant l 3xin4 c MEl 5 wll 5 w 1 lwl 5 w 31 1 1 1el w2 4 CHAPTER 17 171 b Income and substitution effects work in opposite directions If c1r 0 then c2 is price elastic c Budget constraint passes through y1 y2 and rotates through this point as r changes Income effect depends on whether y1 c1 or y1 c1 initially 173 25 years 175 a Not at all b Tax would be on opportunity cost of capital c Taxes are paid later so cost of capital is reduced d If tax rates decline the benefit of accelerated depre ciation is reduced 177 Using equation 1766 we get p 1152 5 e075 1p0 2 c02 1 coe203 p 1152 5 e075p0 2 e075c0 1 c0e203 125 5 e075p0 2 7 1e075 1 e2032 p0 5 636 179 a Maximizes expected utility b If marginal utility is convex applying Jensens inequality to that function implies E3Ur 1c12 4 Ur 3E1c12 4 5 Ur 1c02 So must increase next periods consumption to yield equality c Part b shows that this person will save more when next periods consumption is random d Prompting added precautionary savings would require an even higher r exacerbating the paradox 1711 a Use x 11 2 x2 5 x 1 x2 1 c for x 1 b See detailed solutions for derivative c The increased output from a higher t must be bal anced against 1 the delay in getting the first yield and 2 the opportunity cost of a delay in all future rotations d f t is asymptotic to 50 as t S q e t 5 100 f t 5 1041 1713 a The discount factors drop significantly from 1 to 0594 for period t 1 1 and then follow a slow and steady geometric rate of decline of 099 b The significant drop of the discount factors for period t 1 1 means that preferences at time t are inconsistent with preferences at time t 1 1 c In period t the MRS between ct11 and ct12 will be Ur 1ct112δ Ur 1ct122 At time t 1 1 the MRS between ct11 and ct12 will be Ur 1ct112βδUr 1ct122 This means that effectively preferences would change between the two periods d Constraints are necessary so as to avoid changes in the consumption decision from one period to the other e Examples include retirement funds with penalties for early withdrawal of funds real estate saving bonds and certificates of deposit In general illiq uid assets provide a form of commitment against future overconsumption Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Solutions to OddNumbered Problems 739 CHAPTER 18 181 a Both half and quartershare contracts provide her with positive expected utility Ehalf3U1W2 4 5 250 Equarter 3U1W2 4 5 75 The lowest share she would accept solves 1052 11000s2 1 1052 1400s2 2100 5 0 implying s 14 b The most she would pay equals 1052 110002 1 1052 14002 2 100 5 600 c Her fixed salary solves 1052 11002 1 f 2 10 5 0 or f 5 50 d i Her grossprofit share must solve 1052 11000s2 1 1052 1400s2 2 100 400s implying s 13 ii The bonus that would induce her to work hard solves 05b 2 100 0 implying b 200 183 Solving the utilitymaximization problem for each type yields demands qH 5 120p2 2 and qL 5 115p2 2 The monopolists expected profit from a linear price is 1 2 1 p 2 c2 a20 p b 2 1 1 2 1 p 2 c2 a15 p b 2 5 625 1 p 2 c2 2p2 Solving p 5 10 when c 5 5 yielding an expected profit of 15625 185 a The premium satisfies p 5 1052 1100002 5 5000 b The premium satisfies p 5 1052 150002 5 2500 The individuals utility is 96017 less than the 96158 from part a verifying that he or she prefers full to partial insurance c The premium satisfies p 5 1052 1700022 5 1750 The individuals utility from partial insurance is now 97055 more than from part a 187 a 1122 1100002 1 1122 120002 5 6000 b If sellers value cars at 8000 only lemons will be sold at a market price of 2000 If sellers value cars at 6000 all cars will be sold at a market price of 6000 189 The optimum of the fully informed patient satisfies 1Upm2 1Upx2 5 pm or MRS 5 pm The doctors optimum satisfies pmUrd 1 Upm 2 pm Upx 5 0 Rearranging MRS pm implying that the doctor chooses more medical care 1811 a Equilibrium effort is e i 5 1n yielding surplus for one partner 2n 2 1 2n2 b The worker receiving the 100 percent share exerts effort e i 5 1 and obtains surplus 12 c Differentiating the surplus computed in a d dna2n 2 1 2n2 b 5 1 2 n n3 which is negative for n 1 The limit of the sur plus as n S q equals 0 d The analysis suggests it is unlikely that the stock plan provides incentives in a rational model but there may be unmodeled behavioral or bargaining effects CHAPTER 19 191 a P 5 20 and q 5 50 b P 5 20 q 5 40 MC 5 16 and tax 5 4 193 a n 5 400 The externality arises because one wells drilling affects all wells output b n 5 200 c Fee 5 2000well 195 The tax will improve matters only if the output restric tion required by the externality exceeds the output restriction brought about by the monopoly 197 a Roughly speaking individuals would freeride on each other under perfect competition producing x 0 and obtaining utility 0 More rigorously in the Nash equilibrium each sets RPT 5 MRSi yielding x 5 0704 y i 5 0704 y 5 704 and utility 5 0704 b x 5 5 y 5 50 y i 5 05 and utility 5 158 199 a Want gri to be the same for all firms b A uniform tax will not achieve the result in part a c In g e n e r a l opt i m a l p o l lut i on t a x i s t 5 1 p 2 wf r2 1gr which will vary from firm to firm However if firms have simple linear produc tion functions given by qi 5 ali then a uniform tax Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 740 Solutions to OddNumbered Problems can achieve efficiency even if gi differs among firms In this case the optimal tax is t 5 λ 1a 2 w2a where λ is the value of the Lagrange multiplier in the social optimum described in part a d It is more efficient to tax pollution than to tax output 1911 a Choose b and t so that y is the same in each state Requires t 5 U b b always equals 11 2 t2w and t 5 U c No Because this person is risk averse he or she will always opt for equal income in each state Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 741 Some of the terms that are used frequently in this book are defined below The reader may wish to use the index to find those sections of the text that give more complete descriptions of these concepts A Absolute Risk Aversion See Risk Aversion Adverse Selection The problem facing insurers that risky types are both more likely to accept an insurance policy and more expensive to serve Asymmetric Information A situation in which an agent on one side of a transaction has information that the agent on the other side does not have Average Cost Total cost per unit of output AC1q2 5 C1q2q Average Product Output per unit of a given input For example the average product of labor is denoted APl 5 ql 5 f1k l2l B Barriers to Entry Factors limiting otherwise profitable entry into a market preventing the emergence of per fect competition Bayes Rule Formula used in an environment of uncer tainty for updating beliefs based on new information BayesianNash Equilibrium A strategy profile in a two player simultaneousmove game in which player 1 has private information This generalizes the Nash equilib rium concept to allow for player 2s beliefs about player 1s type Bertrand Paradox The Nash equilibrium in a simulta neousmove pricing game is competitive pricing even when there are only two firms Best Response A strategy for player i that leads to at least as high a payoff as any other strategy i could play given rivals specified strategies C Ceteris Paribus Assumption The assumption that all other relevant factors are held constant when examin ing the influence of one particular variable in an eco nomic model Reflected in mathematical terms by the use of partial differentiation Coase Theorem Result attributable to Ronald Coase if bargaining costs are zero an efficient allocation of resources can be attained in the presence of externali ties through reliance on bargaining among the parties involved CobbDouglas Function A tractable functional form used in consumer and producer theory An example of a CobbDouglas utility function is U1x y2 5 αxβy Compensated Demand Function Function showing rela tionship between the price of a good and the quantity consumed while holding real income or utility con stant Denoted by xc 1 px py U2 Compensated Price Elasticity The price elasticity of the compensated demand function xc 1 px py U2 That is ex c px 5 xcpx pxxc Compensating Variation CV The compensation required to restore a persons original utility level when prices change Compensating Wage Differentials Differences in real wages that arise when the characteristics of occupations cause workers in their supply decisions to prefer one job over another Glossary of Frequently Used Terms Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Complements Gross Two goods such that if the price of one rises the quantity consumed of the other will fall Goods x and y are gross complements if xpy 0 See also Substitutes Gross Complements Net Two goods such that if the price of one rises the quantity consumed of the other will fall holding real income utility constant Goods x and y are net complements if xpy 0U5U 0 Such compensated crossprice effects are symmetric that is xpy0U5U 5 ypx 0U5U See also Substitutes Net Also called Hicksian substi tutes and complements Composite Commodity A group of goods whose prices all move togetherthe relative prices of goods in the group do not change Such goods can be treated as a single commodity in many applications Concave Function A function that lies everywhere below its tangent plane Condorcet Paradox Outcome of voting over pairs of policies can cycle endlessly if voter preferences are not singlepeaked Constant Cost Industry An industry in which expansion of output and entry by new firms has no effect on the cost curves of individual firms Constant Returns to Scale See Returns to Scale Consumer Surplus The area below the Marshallian demand curve and above market price Shows what an individual would pay for the right to make voluntary transactions at this price Changes in consumer sur plus can be used to measure the welfare effects of price changes Contingent Input Demand See Input Demand Functions Contour Line The set of points along which a function has a constant value Useful for graphing threedimen sional functions in two dimensions Individuals indif ference curve maps and firms production isoquant maps are examples Contract Curve The set of all the efficient allocations of goods among those individuals in an exchange econ omy Each of these allocations has the property that no one individual can be made better off without making someone else worse off Cost Function See Total Cost Function Cournot Equilibrium Equilibrium in a quantitysetting game involving two or more firms Crossprice Elasticity of Demand For the demand func tion x 1px py I2 ex py 5 xpy pyx D Deadweight Loss A loss of mutually beneficial trans actions Losses in consumer and producer surplus that are not transferred to another economic agent Decreasing Cost Industry An industry in which expan sion of output generates costreducing externalities that cause the cost curves of those firms in the industry to shift downward Decreasing Returns to Scale See Returns to Scale Demand Curve A graph showing the ceteris paribus relationship between the price of a good and the quan tity of that good purchased A twodimensional repre sentation of the demand function x 5 x 1px py I2 This is referred to as Marshallian demand to differ entiate it from the compensated Hicksian demand concept Diminishing Marginal Productivity See Marginal Physical Product Diminishing Marginal Rate of Substitution See Marginal Rate of Substitution Discount Factor The degree to which a payoff next period is discounted in making this periods decisions denoted by δ in the text If r is the singleperiod interest rate then usually δ 5 1 11 1 r2 Dominant Strategy A strategy s i for player i that is a best response to the allstrategy profile of other players Duality The relationship between any constrained maximization problem and its related dual con strained minimization problem E Economic Cost The relevant cost for making economic decisions Includes explicit payments for inputs as well as opportunity costs that may only be implicit Economic Efficiency Exists when resources are allo cated so that no activity can be increased without cutting back on some other activity See also Pareto Efficient Allocation Edgeworth Box Diagram A graphic device used to demonstrate economic efficiency Most frequently used 742 Glossary of Frequently Used Terms Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 to illustrate the contract curve in an exchange economy but also useful in the theory of production Elasticity A unitfree measure of the proportional effect of one variable on another If y 5 f1x2 then ey x 5 yx xy Elasticity of Substitution A measure of how substitut able inputs are in production related to the curvature of the isoquant Entry Conditions Characteristics of an industry that determine the ease with which a new firm may begin production Under perfect competition entry is assumed to be costless whereas in a monopolistic industry there are significant barriers to entry Envelope Theorem A mathematical result the deriva tive of a value function with respect to an exogenous variable is equal to the partial derivative of the original optimization problem with respect to that variable when all endogenous variables take on their optimal values Equilibrium A situation in which no actors have an incentive to change their behavior At an equilibrium price the quantity demanded by individuals is exactly equal to that which is supplied by all firms Equivalent Variation The added cost of attaining the new utility level when prices change Eulers Theorem A mathematical theorem if f1x1 c xn2 is homogeneous of degree k then f1x1 1 f2x2 1 c1 fnxn 5 kf1x1 c xn2 Exchange Economy An economy in which the supply of goods is fixed ie no production takes place The available goods however may be reallocated among individuals in the economy Expansion Path The locus of those costminimizing input combinations that a firm will choose to produce various levels of output when the prices of inputs are held constant Expected Utility The average utility expected from a risky situation If there are n outcomes x1 c xn with probabilities p1 c pn then the expected utility is given by E1U2 5 p1U1x12 1 p2U1x22 1 c1 pnU1xn2 Expenditure Function A value function derived from the individuals expenditure minimization problem Shows the minimum expenditure necessary to achieve a given utility level expenditures 5 E1 px py U2 Extensive Form Diagram of the game tree showing sequence of players moves Externality An effect of one economic agent on another that is not taken into account by normal market behavior F Financial Option Contract A contract offering the right but not the obligation to buy or sell an asset during some future period at a certain price First Best A theoretical benchmark given by the socially efficient outcome attainable by a social planner in the absence of relevant constraints FirstMover Advantage The advantage that may be gained by the player who moves first in a game FirstOrder Conditions Mathematical conditions that must necessarily hold if a function is to take on its maximum or minimum value Usually show that any activity should be increased to the point at which mar ginal benefits equal marginal costs First Theorem of Welfare Economics Every Walrasian equilibrium is Pareto optimal Fixed Costs Costs that do not change as the level of output changes Examples include expenditures on an input that cannot be varied in the short run or expen ditures involved in a products invention See also Vari able Costs Folk Theorem General understanding that a wide range of equilibria can arise in a repeated game with infinitely patient players G Game An abstract representation of a strategic situa tion constituted by players strategies and payoffs General Equilibrium Model A model of an economy that portrays the operation of many markets simultaneously Giffens Paradox A situation in which the increase in a goods price leads individuals to consume more of the good Arises because the good in question is inferior and because the income effect induced by the price change is stronger than the substitution effect H Hidden Action An action taken by one party to a con tract that cannot be directly observed by the other party Glossary of Frequently Used Terms 743 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 744 Glossary of Frequently Used Terms Hidden Type A characteristic of one party to a contract that cannot be observed by the other party prior to agreeing to the contract Homogeneous Function A function f 1x1 x2 c xn2 is homogeneous of degree k if f1mx1 mx2 c mxn2 5 mkf1x1 x2 c xn2 Homothetic Function A function that can be repre sented as a monotonic transformation of a homoge neous function The slopes of the contour lines for such a function depend only on the ratios of the variables that enter the function not on their absolute levels and so are constant along a ray from the origin Hotelling Model A model of price competition among sellers differentiated by their location along a line I Income and Substitution Effects Two different effects that come into play when an individual is faced with a changed price for some good Income effects arise because a change in the price of a good will affect an individuals purchasing power Even if purchasing power is held constant however substitution effects will cause individuals to reallocate their expectations Substitution effects are reflected in movements along an indifference curve whereas income effects entail a movement to a different indifference curve See also Slutsky Equation Income Elasticity of Demand For the demand function x 1 px py I2 exI 5 xI Ix Increasing Cost Industry An industry in which the expansion of output creates costincreasing externali ties which cause the cost curves of those firms in the industry to shift upward Increasing Returns to Scale See Returns to Scale Indifference Curve Map A contour map of an individ uals utility function showing those alternative bundles of goods from which the individual derives equal levels of welfare Indirect Utility Function A value function representing utility as a function of all prices and income Individual Demand Curve The ceteris paribus relation ship between the quantity of a good an individual chooses to consume and the goods price A two dimensional representation of x 5 x 1 px py I2 for one person Inferior Good A good that is bought in smaller quanti ties as an individuals income rises Inferior Input A factor of production that is used in smaller amounts as a firms output expands Input Demand Functions These functions show how input demand for a profitmaximizing firm is based on input prices and on the demand for output The input demand function for labor for example can be written as l 5 l 1P v w2 where P is the market price of the firms output Contingent input demand functions 3l c 1v w q2 4 are derived from cost minimization and do not necessarily reflect profitmaximizing output choices Isoquant Map A contour map of the firms production function The contours show the alternative combina tions of productive inputs that can be used to produce a given level of output K KuhnTucker Conditions Firstorder conditions for an optimization problem in which inequality constraints are present These are generalizations of the firstorder conditions for optimization with equality constraints L Lerner Index A measure of market power given by the percent markup over marginal cost L 5 P 2 MC P Limit Pricing Choice of lowprice strategies to deter entry Lindahl Equilibrium A hypothetical solution to the pub lic goods problem the tax share that each individual pays plays the same role as an equilibrium market price in a competitive allocation Long Run See Short RunLong Run Distinction Lump Sum Principle The demonstration that general purchasing power taxes or transfers are more efficient than taxes or subsidies on individual goods M Marginal Cost MC The additional cost incurred by producing one more unit of output MC 5 Cq Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Glossary of Frequently Used Terms 745 Marginal Input Expense The increase in total costs that results from hiring one more unit of an input Marginal Physical Product MP The additional output that can be produced by one more unit of a particu lar input while holding all other inputs constant If q 5 f 1k l2 MPl 5 ql It is usually assumed that an inputs marginal productivity diminishes as additional units of the input are put into use while holding other inputs fixed Marginal Rate of Substitution MRS The rate at which an individual is willing to trade one good for another while remaining equally well off The MRS is the abso lute value of the slope of an indifference curve MRS 5 2 dy dx U5U It is widely assumed MRS is diminishing ie MRS falls the more x is added and y taken away Marginal Revenue MR The additional revenue obtained by a firm when it is able to sell one more unit of output MR 5 dRdq Marginal Revenue Product MRP The extra revenue that accrues to a firm when it sells the output that is pro duced by one more unit of some input In the case of labor for example MRPl 5 MR MPl Marginal Utility MU The extra utility that an individ ual receives by consuming one more unit of a particular good Market Demand The sum of the quantities of a good demanded by all individuals in a market Will depend on the price of the good prices of other goods each consumers preferences and on each consumers income Mixed Strategy A strategy in which a player chooses which action to play probabilistically Monopoly One supplier itself serving the whole market Monopsony An industry in which there is only a single buyer of the good Moral Hazard The effect of insurance coverage on individuals decisions to undertake activities that may change the likelihood or sizes of losses N Nash Equilibrium Strategies for each player that are mutual best responses Fixing others equilibrium strat egies no player has a strict incentive to deviate Normal Good A good for which quantity demanded increases or stays constant as an individuals income increases Normative Analysis Economic analysis that takes a position on how economic actors or markets should operate O Oligopoly A market served by few firms but more than one Opportunity Cost Doctrine The simple though farreach ing observation that the true cost of any action can be measured by the value of the best alternative that must be forgone when the action is taken Output and Substitution Effects Come into play when a change in the price of an input that a firm uses causes the firm to change the quantities of inputs it will demand The substitution effect would occur even if output were held constant and it is reflected by move ments along an isoquant Output effects on the other hand occur when output levels change and the firm moves to a new isoquant P Paradox of Voting Illustrates the possibility that major ity rule voting may not yield a determinate outcome but may instead cycle among alternatives Pareto Efficient Allocation An allocation of resources in which no one individual can be made better off without making someone else worse off Partial Equilibrium Model A model of a single market that ignores repercussions in other markets Perfect Competition The most widely used economic model there are assumed to be a large number of buy ers and sellers for any good and each agent is a price taker See also Price Taker Pigouvian Tax A tax instituted to correct the prob lem of overconsumption in the presence of a negative externality Positive Analysis Economic analysis that seeks to explain and predict actual economic events Present Discounted Value PDV The current value of a sum of money that is payable sometime in the future Takes into account the effect of interest payments Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Price Discrimination Selling identical goods at different prices There are three types first degreeselling each unit at a different price to the individual willing to pay the most for it perfect price discrimination second degreeadopting price schedules that give buyers an incentive to separate themselves into differing price categories third degreecharging different prices in separated markets Price Elasticity of Demand For the demand function x 1 px py I2 expx 5 xpx pxx Price Taker An economic agent that makes decisions on the assumption that these decisions will have no effect on prevailing market prices PrincipalAgent Relationship The hiring of one person the agent by another person the principal to make economic decisions Prisoners Dilemma Originally studied in the theory of games but has widespread applicability The crux of the dilemma is that each individual faced with the uncer tainty of how others will behave may be led to adopt a course of action that proves to be detrimental for all those individuals making the same decision Producer Surplus The extra return that producers make by making transactions at the market price over and above what they would earn if nothing were pro duced It is illustrated by the size of the area below the market price and above the supply curve Production Function A conceptual mathematical func tion that records the relationship between a firms inputs and its outputs If output is a function of capital and labor only this would be denoted by q 5 f1k l2 Production Possibility Frontier The locus of all the alternative quantities of several outputs that can be pro duced with fixed amounts of productive inputs Profit Function A value function showing the relation ship between a firms maximum profits 1P2 and the output and input prices it faces profit 5 P 1P v w2 Profits The difference between the total revenue a firm receives and its total economic costs of production Economic profits equal zero under perfect competition in the long run Monopoly profits may be positive however Property Rights Legal specification of ownership and the rights of owners Public Good A good that once produced is available to all on a nonexclusive basis Many public goods are also nonrivaladditional individuals may benefit from the good at zero marginal costs Pure Strategy A single choice involving no randomization Q Quasiconcave Function A function for which the set of all points for which f1X2 k is convex R Rate of Product Transformation RPT The rate at which one output can be traded for another in the productive process while holding the total quantities of inputs con stant The RPT is the absolute value of the slope of the production possibility frontier Rate of Return The rate at which present goods can be transformed into future goods For example a one period rate of return of 10 percent implies that forgoing 1 unit of output this period will yield 110 units of output next period Rate of Technical Substitution RTS The rate at which one input may be traded off against another in the productive process while holding output constant The RTS is the absolute value of the slope of an isoquant RTS 5 2dk dl q5q0 Real Option An option arising in a setting outside of financial markets Relative Risk Aversion See Risk Aversion Rent Payments to a factor of production that are in excess of that amount necessary to keep it in its current employment Rental Rate The cost of hiring one machine for 1 hour Denoted by v in the text RentSeeking Activities Economic agents engage in rentseeking activities when they utilize the political process to generate economic rents that would not ordinarily occur in market transactions Returns to Scale A way of classifying production functions that records how output responds to propor tional increases in all inputs If a proportional increase in all inputs causes output to increase by a smaller 746 Glossary of Frequently Used Terms Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 proportion the production function is said to exhibit decreasing returns to scale If output increases by a greater proportion than the inputs the production function exhibits increasing returns Constant returns to scale is the middle ground where both inputs and outputs increase by the same proportions Mathemati cally if f1mk ml2 5 mkf 1k l2 k 1 implies increasing returns k 5 1 constant returns and k 1 decreasing returns Risk Aversion Unwillingness to accept fair bets Arises when an individuals utility of wealth function is con cave ie when Us 1W2 0 Absolute risk aversion is measured by r1W2 5 2Us 1W2Ur 1W2 Relative risk aversion is measured by rr1W2 5 2WUs 1W2 Ur 1W2 S Second Best The best that a decision maker can do under relevant constraints which falls short of the unconstrained first best Third best fourth best and so on are yet less efficient outcomes as further constraints are added to the problem SecondOrder Conditions Mathematical conditions required to ensure that points for which firstorder conditions are satisfied are indeed true maximum or true minimum points These conditions are satisfied by functions that obey certain convexity assumptions Second Theorem of Welfare Economics Any Pareto opti mal allocation can be attained as a Walrasian equilib rium by suitable transfers of initial endowments Shephards Lemma Application of the envelope the orem which shows that a consumers compensated demand functions and a firms constant output input demand functions can be derived from partial differen tiation of expenditure functions or total cost functions respectively Shifting of a Tax Market response to the imposition of a tax that causes the incidence of the tax to be on some economic agent other than the one who actually pays the tax Short Run Long Run Distinction A conceptual distinction made in the theory of production that differentiates between a period of time over which some inputs are regarded as being fixed and a longer period in which all inputs can be varied by the producer Signaling Actions taken by individuals in markets characterized by hidden types in an effort to identify their true type Slutsky Equation A mathematical representation of the substitution and income effects of a price change on utilitymaximizing choices xpx 5 xpx0U5U 2 x 1xI2 Social Welfare Function A hypothetical device that records societal views about equity among individuals SubgamePerfect Equilibrium A strategy profile 1s 1 s 2 c s n2 that constitutes a Nash equilibrium for every proper subgame Substitutes Gross Two goods such that if the price of one increases more of the other good will be demanded That is x and y are gross substitutes if xpy 0 See also Complements Slutsky Equation Substitutes Net Two goods such that if the price of one increases more of the other good will be demanded if utility is held constant That is x and y are net substitutes if xpy0U5U 0 Net substitutability is symmetric in that xpy0U5U 5 xpx0U5U See also Complements Slutsky Equation Substitution Effects See Income and Substitution Effects Output and Substitution Effects Slutsky Equation Sunk Cost An expenditure on an investment that can not be reversed and has no resale value Supply Function For a profitmaximizing firm a func tion q 1P v w2 that shows quantity supplied q as a function of output price P and input prices v w Supply Response Increases in production prompted by changing demand conditions and market prices Usu ally a distinction is made between shortrun and long run supply responses T Tacit Collusion Choice of cooperative monopoly strategies without an explicit agreement to form a cartel Total Cost Function A function C1v w q2 showing the minimum cost 1C2 of producing q units of output when input prices are v and w Glossary of Frequently Used Terms 747 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Tragedy of the Commons Environmental problem of overconsumption arising when scare resources are treated as common property U Utility Function A mathematical conceptualization of the way in which individuals rank alternative bundles of commodities If there are only two goods x and y utility is denoted by U1x y2 V Value Function Result of an optimization problem showing the optimized value as a function only of exogenous variables Variable Costs Costs that change in response to changes in the level of output being produced by a firm This is in contrast to fixed costs which do not change VickeryClarkeGroves Mechanism Citizens announce their values for a public good receiving positive or neg ative payments calibrated to induce truthful announce ments May be able to eliminate the inefficiencies associated with simple voting von NeumannMorgenstern Utility A ranking of out comes in uncertain situations such that individuals choose among these outcomes on the basis of their expected utility values W Wage The cost of hiring one worker for 1 hour Denoted by w in the text Walrasian Equilibrium An allocation of resources and an associated price vector such that quantity demanded equals quantity supplied in all markets at these prices assuming all parties act as pricetakers Walrasian Price Adjustment The assumption that mar kets are cleared through price adjustments in response to excess demand or supply Z ZeroSum Game A game in which winnings for one player are losses for the other player 748 Glossary of Frequently Used Terms Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 749 Author names are in italics glossary terms are in boldface A AC See Average cost function AC Accounting cost 326 Addiction 112 Adverse selection 221 658665 competitive insurance market and 662665 firstbest contract 658 secondbest contract 658661 Agents asymmetric information and 635 defined 635 principalagent model 635637 Aggregation Cournot 163 Engel 162163 of goods 190191 202203 AIDS almost ideal demand system 140 180 Aizcorbe Ana M 179 Alcoa entry deterrence by 551552 Aleskerov Fuad 111 Allocation of time 575578 graphical analysis 577578 income and substitution effects of change in real wage rate 577 twogood model 575576 utility maximization 576577 Almost ideal demand system AIDS 140 180 Altruism 112 115116 685 Anderson E 396 Annuities 626 Antiderivatives calculating 59 defined 58 Antitrust laws Alcoa 552 explicit cartels and 541 Standard Oil Company 555 Appropriability effect 557 Assumptions of nonsatiation 95 118 testing 4 See also Ceteris paribus assumption Asymmetric information 237 633671 adverse selection in insurance 658665 auctions 667671 complex contracts as response to 633634 gross definitions 187188 hidden actions 637638 hidden types 647 market signaling 665667 moral hazard in insurance 642647 nonlinear pricing 648657 680682 ownermanager relationship 638642 principalagent model 635637 Atkeson Andrew 323 Attributes model 194195 Attributes of goods See Household production models Auctions 667671 Automobiles flexibility in fuel usage 223224 227229 tied sales 524 usedcar market signaling in 667 Average cost AC 334 defined 334 graphical analysis of 336337 properties of 341 Average physical productivity 299300 Average revenue curve 371 Axioms of rational choice 8990 B Backward induction 268269 Bairam E 323 Barriers to entry 491493 creation of 492493 legal 492 oligopolies and 556 technical 491492 Battle of the Sexes backward induction in 268269 expected payoffs in 257258 extensive form for 265 formal definitions 255 mixed strategies in 258259 Nash equilibrium in 253255 266267 Sequential 264 subgameperfect equilibrium 267268 Bayesian games 273278 BayesianNash equilibrium 274278 285288 defined 276 games of incomplete information 276 Tragedy of the Commons 276278 Bayes rule 273 280281 Becker Gary 112 290 Behrman Jere R 138 Beliefs of players 279281 posterior 279281 prior 279281 Benefitcost ratio 4243 Benefits mandated 584 Bentham Jeremy 90 Bernat G A 488 Bernoulli Daniel 208210 Bertrand J 527 Bertrand game 261 521528 534 Cournot game versus 534 differentiated products 535541 568 feedback effect 559 Nash equilibrium of 527528 naturalspring duopoly in 530531 tacit collusion in 542543 Bertrand paradox 528 Best response Cournot model 531532 defined 250 imperfect competition 567 payoffs in 251253 Tragedy of the Commons 262263 Beta coefficients 245 Binomial distribution 69 expected values of 7172 variances and standard deviations for 73 Index Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 750 Index Black Duncan 705 Blackorby Charles 202 Bolton P 655 Bonds 621622 Borjas G J 324 Brander J A 570 Brouwers theorem 292 468 Brown D K 487 Buckley P A 361 Budget constraints attributes model 195 mathematical model of exchange 465 476 in twogood case 117 Budget shares 125126 138140 almost ideal demand system 140 CES utility 139 linear expenditure system 139 variability of 138 Burniaux J M 488 Businessstealing effect 557 C Calculus fundamental theorem of 6061 Capacity constraints 534535 Capital 599620 accumulation of 599600 capitalization of rents 430 costs 325326 demand for 610612 energy substitutability and 323 natural resource pricing 617620 present discounted value approach 613616 pricing of risky assets 608610 rate of return 599608 time and 625630 Capital asset pricing model CAPM 245 CARA constant absolute risk aversion function 218 237 Cardinal properties 5758 Cartels 525526 antitrust laws and 541 naturalspring duopoly 530531 CDF cumulative distribution function 70 Central limit theorem 69 CEOs chief executive officers 364 Certainty equivalent 215 CES utility 104105 313314 323 budget shares and 125126 139 cost functions 339 demand elasticities and 164165 labor supply 581582 Ceteris paribus assumption 56 partial derivatives and 27 in utilitymaximizing choices 9091 CGE models See Computable general equilibrium CGE models Chain rule 25 3032 Chance nodes 274 Change in demand 403 Change of variable 59 Changes in income 143144 Chief executive officers CEOs 364 China changing demands for food in 181 Choice 111112 individual portfolio problem 243245 rational axioms of 8990 special preferences 111112 See also Statepreference model Utility Clarke E 710 Clarke mechanism 710 Classification of longrun supply curves 423424 Closed shops 592 CO2 reduction strategies 488 Coase Ronald 393 512513 694 Coase conjecture 512513 Coase theorem 694695 CobbDouglas production function 312313 cost functions 345 envelope relations and 353354 shifting 344345 Solow growth model 322323 technical progress in 317318 CobbDouglas utility 103 183 corner solutions 123126 labor supply and 580582 Commitment versus flexibility 545 Comparative statistics analysis 414415 changes in input costs 426428 Cramers rule 85 in general equilibrium model 460462 industry structure 425 of monopoly 501502 shifts in demand 425426 Compensated crossprice elasticity of demand 161 Compensated demand curves 151155 compensating variation and 166168 defined 153 relationship between compensated uncompensated curves 154155 relationship to uncompensated curves 156159 Shephards lemma 153154 Compensated demand functions 151152 155 Compensated ownprice elasticity of demand 161 Compensating variation CV 166 Compensating wage differentials 585589 Competition allocative inefficiency and 687689 failure of competitive market 698699 for innovation 560561 perfect 407 419 See also Competitive insurance market Imperfect competition Competitive insurance market adverse selection and 658665 equilibrium with hidden types 663 equilibrium with perfect information 662 moral hazard and 646647 signaling in 665 See also Insurance Competitive price system 449450 behavioral assumptions 450 law of one price 449450 Complements 186189 asymmetry of gross definitions 187188 gross 186 imperfect competition 567570 net 188189 perfect 103104 Completeness and preferences 89 Composite commodities 190193 generalizations and limitations 191193 housing costs as 191193 theorem 190191 twostage budgeting and 202203 Compound interest mathematics of 625630 Computable general equilibrium CGE models 478482 487 economic insights from 480482 solving 479 structure of 479 Computers and empirical analysis 19 Concave functions 51 5355 8283 Concavity of production possibility frontier 457458 quasiconcave functions 5355 Condorcet M de 704 Consols perpetuities 626 Constant absolute risk aversion CARA function 218 242243 Constant cost industry 419421 defined 424 infinitely elastic supply 420421 initial equilibrium 419420 responses to increase in demand 420 Constant elasticity 372 Constant elasticity of substitution CES function See CES utility Production Function Constant relative risk aversion 220 Constant relative risk aversion CRRA function 220 Constant returns to scale 305 Constant risk aversion 218219 Constrained maximization 4045 8485 duality 4345 envelope theorem in 4546 firstorder conditions and 41 formal problem 40 Lagrange multiplier method 40 4143 optimal fences and 4345 secondorder conditions and 5152 Consumer price index CPI 178181 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Index 751 Consumer search 540541 Consumer surplus 166170 consumer welfare and expenditure function 166 defined 173 overview 168 using compensated demand curve to show CV 166168 welfare changes and Marshallian demand curve 168170 Consumer theory relationship of firm to 364365 Consumption convexity and balance in 9699 of goods utility from 91 See also Indifference curves Contingent commodities fair markets for 233 prices of 232 states of world and 232 Contingent input demand 346348 costminimizing input choices 330 Shephards lemma and 345346 Continuity partial equilibrium competitive model 445 preferences and 8990 Continuous actions games with 293 Continuous random variables 6768 Continuous time 627630 continuous growth 628629 duration 629630 payment streams 629 Continuum of actions 261263 Contour lines 35 111 Contract curves 470471 473 Contracts 633634 asymmetric information 633634 firstbest 635636 643644 652 660 secondbest 635636 644646 660661 value of 634 Controlled experiments 6 Convex functions 8283 Convex indifference curves 9596 97 100101 Convexity 9699 Corn Laws debate 462463 Correspondences functions versus 291292 Costbenefit analysis 230 Cost curves perunit 354355 shifts in 337339 See also Cost functions Cost functions 325356 average and marginal 334 336337 costminimizing input choices 328333 definitions of costs 325327 graphical analysis of total costs 334335 homogeneity 339 input prices and 340 profit maximization and cost minimization relationship 327328 Shephards lemma and elasticity of substitution 348 shifts in cost curves and 337348 shortrun longrun distinction 348356 translog 360361 Cost minimization illustration of process 332333 principle of 330 relationship between profit maximization and 327328 Costs accounting 326 economic 326327 sunk 546 Cournot Antoine 163 528 Cournot aggregation 163 Cournot equilibrium 547548 Cournot game 261 528534 feedback effect 559 imperfect competition 568 longrun equilibrium and 557558 Nash equilibrium of Cournot game 529532 naturalspring duopoly 530531 prices versus quantities 534 tacit collusion in 543544 varying number of firms and 533534 Covariance 7476 CPI consumer price index 178181 Cramers rule 85 Crosspartial derivatives 50 Crossprice effects asymmetry in 187188 net substitutes and complements 188 profit maximization and input demand 385 Slutsky decomposition 185 Crossprice elasticity of demand 160 Crossproductivity effects 303304 CRRA constant relative risk aversion function 220 Cumulative distribution function CDF 70 CV compensating variation 166 D Deadweight loss 437438 Deaton Angus 140 Decrease in price graphical analysis of 145 Decreasing cost industry 423 Decreasing returns to scale 306307 Definite integrals defined 60 differentiating 6263 Delay option value of 229 Demand See Supply and demand Demand aggregation and estimation 445447 Demand curves compensated 151155 defined 148 demand functions and 150151 importance of shape of supply curve 413414 importance to supply curves 412413 individual 148151 shifts in 150 412 uncompensated 154155 See also Compensated demand curves Demand elasticities 159165 compensated price elasticities 161162 Marshallian 159160 price elasticity and total spending 160161 price elasticity of demand 160 relationships among 162165 Demand functions 141143 demand curves and 150151 indirect utility function 126127 mathematical model of exchange 465466 Demand relationships among goods 183196 attributes of goods 193196 composite commodities 190193 home production 193196 implicit prices 193196 net substitutes and complements 188189 overview 183 simplifying demand and twostage budgeting 202203 substitutability with many goods 189190 substitutes and complements 186188 twogood case 183185 Derivatives crosspartial 50 defined 22 homogeneity and 56 partial 2630 rules for finding 2425 second 2324 value of at point 2223 Deterring entry See Entry deterrence accommodation Dewatripont M 655 Diamond Peter 540 Dictator game 285286 Diewert W Erwin 203 Differentiated products See Product differentiation Diminishing marginal productivity See Marginal physical product MP Diminishing marginal rate of substitution See Marginal rate of substitution MRS Diminishing returns 455 Diminishing RTS See Rate of technical substitution RTS Direct approach 4 156 Discount factor 271272 541544 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 752 Index Discrete random variables 6768 Discrimination price See Price discrimination Disequilibrium behavior 435 Dissipation effect 560 Diversification 222223 Dominant strategies defined 253 Nash equilibrium 253 260 Doucouliagos H 361 Dual expenditureminimization problem 130 131 Duality 4345 Duffield James A 111 Durability of goods 504 Dutch MIMIC model 487 Dynamic optimization 6367 maximum principle 6467 optimal control problem 6364 Dynamic views of monopoly 516517 E Economic costs 326327 defined 327 Economic efficiency concept of 17 welfare analysis and 431434 Economic goods in utility functions 92 Economic models 319 ceteris paribus assumption 56 economic theory of value 918 modern developments in 1819 optimization assumptions 78 positivenormative distinction 89 structure of economic models 69 theoretical models 3 verification of 45 Economic profits 366 Edgeworth Francis Y17 Edgeworth box diagram 451452 470 Efficiency allocative inefficiency 685689 concept of 17 efficient allocations 452453 686 Pareto efficient allocation 469 welfare analysis and 431434 Elasticity general definition of 2829 interpretation in mathematical model of market equilibrium 416417 marginal revenue and 369370 of substitution 307309 348 of supply 424 Elasticity of demand compensated crossprice 161 compensated ownprice 161 crossprice 160 monopolies and 500501 price 159 160 Elasticity of substitution 105 307309 342 defined 308 graphic description of 308309 See also CES utility Empirical analysis computers and 19 importance of 5 Empirical estimates 424 Endogenous variables 67 Energy capital and 323 homothetic functions and 203 Engel Ernst 138 Engel aggregation 162163 Engels law 138 Entrepreneurial service costs 326 Entry conditions See Entry deterrence accommodation Entry deterrenceaccommodation barriers to entry 491493 entrydeterrence model 553554 imperfect competition 555559 570 in sequential game 568569 strategic entry deterrence 550552 Envelope theorem 3640 CobbDouglas cost functions and 353354 in constrained maximization problems 4546 manyvariable case 3840 profit function 377378 Shephards lemma and 345346 specific example of 3637 Environmental externalities 703704 Equilibrium BayesianNash 274278 281284 computable 478482 487 existence of 260 median voter 707 separating 282283 554 subgameperfect 267268 Walrasian 466 476477 See also General equilibrium Nash equilibrium Partial equilibrium model Equilibrium path 266267 Equilibrium point 12 Equilibrium price defined 410 determination of 410411 458460 of future goods 605 606 supplydemand equilibrium 1213 Equilibrium rate of return 606 Equity premium paradox 623 Eulers theorem 56 189 Evolutionary games and learning 286 Exact price indices 180 Exchange mathematical model of 464475 demand functions and homogeneity 465466 equilibrium and Walras law 466 existence of equilibrium in exchange model 466468 first theorem of welfare economics 468471 second theorem of welfare economics 471473 social welfare functions 474475 utility initial endowments and budget constraints 465 vector notation 464465 Exchange economy 472473 Exchange value labor theory of 10 Exclusive goods 695 Exogenous variables 67 Expansion path 330333 Expected utility 208212 Expected value 7072 207 Expenditure functions 166 defined 131132 properties of 132134 substitution bias and 179 Expenditure minimization 129132 Experimental games 284286 Dictator game 285286 Prisoners Dilemma 285 Ultimatum game 285 Exponential distribution 69 expected values of random variables 72 variances and standard deviations 73 Extensive form games of incomplete information 275 of sequential games 265 Externalities 683711 allocative inefficiency and 685689 defining 683685 graphic analysis of 691 partialequilibrium model of 689691 in production 684 solutions to externality problem 691695 F Factor intensities 455456 Factor prices 462463 Fair bets 213217 Fair gambles 208209 213214 Fair markets for contingent goods 233 Fama E F 245 Farmland reserve pricing 524 Feedback effect 559 Feenstra Robert C 180 Field experiment 675 Financial option contracts 224 Finitely repeated games 270 541 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Index 753 Firms 363365 complicating factors 363364 expansion path costminimizing input choices 330333 in oligopoly setting 555559 profit maximization 393396 relationship to consumer theory 364365 simple model of 363 Firstbest contracts 641 652 adverse selection and 654655 monopoly insurers 660 moral hazard and 643644 ownermanager relationship 639 principalagent model 635637 Firstbest nonlinear pricing 649651 Firstdegree price discrimination 505506 Firstmover advantage 546548 Firstorder conditions 121122 Lagrange multiplier method 41 for maximum 23 3435 118119 First theorem of welfare economics 468470 defined 469 Edgeworth box diagram 471 Fisher Body 364 393396 Fixed costs practical examples of 355356 shortrun 349 sunk costs versus 546 Fixed point 292 Fixedproportions production function 310312 338339 Fixed supply allocating 6567 Flexibility 223230 commitment versus 551552 computing option value 227229 implications for costbenefit analysis 230 model of real options 224226 number of options 226 option value of delay 229 types of options 223224 Folk theorem for infinitely repeated games 271272 541542 Foundations of Economic Analysis Samuelson 18 Friedman Milton 4 Fudenberg D 291 569 Fullinformation case 639 Functional form and elasticity 2829 Fundamental theorem of calculus 6061 Fuss M 360 Future goods 601602 605606 G Game theory 247286 basic concepts 247248 continuum of actions 261263 evolutionary games and learning 286 existence of equilibrium 260 291293 experimental games 284286 incomplete information 273 mixed strategies 256260 Nash equilibrium 250255 payoffs 248 players 248 Prisoners Dilemma 248250 repeated games 270272 sequential games 263269 signaling games 278284 simultaneous Bayesian games 273278 strategies 248 Garcia S 361 Gaussian Normal distribution 69 7374 Gelauff G M M 487 General equilibrium 449482 comparative statistics analysis 460463 mathematical model of exchange 475478 mathematical model of production and exchange 464475 modeling and factor prices 462464 perfectly competitive price system 449450 with two goods 452460 General equilibrium model 14 462464 693 computable 478482 simple 480481 welfare and 487488 General Motors GM 364 391 Giffen Robert 147148 Giffens paradox 147148 Glicksberg I L 293 GM General Motors 364 391 Goods changes in price of 145148 demand relationships among 183196 202203 durability of 504 exclusive 695 fair markets for contingent 233 future 601602 606 inferior 143144 146147 information as 230231 nonrival 695 normal 143144 substitutability with many 189190 See also Demand relationships among goods Public goods Gorman W M 445 Gould Brain W 181 Government procurement 488 Graaflund J J 487 Grim strategy 272 Gross complements 184 186 Gross definitions asymmetry of 187188 Grossman Michael 112 Grossman Sanford 393 Gross substitutes 184 186187 Groves T 709 Groves mechanism 709 Growth accounting 316318 Gruber Jonathan 112 H Habits and addiction 112 Hanley N 715 Hanson K 488 Harsanyi John 273 Hart Oliver 393 Hausman Jerry 179180 Hayashi Fumio 138 Hessian matrix 8283 Hicks John 188189 Hicksian demand curves 151155 relationship between compensated and uncompensated 154155 Shephards lemma 153154 Hicksian demand functions See Demand functions Hicksian substitutes and complements 188189 Hicks second law of demand 189 Hidden actions 635 637638 639642 Hidden types 647 663 Hoffmann S 488 Holdup problem 395 Homogeneity cost functions 339 of demand 142143 162 and demand functions 465466 and derivatives 56 expenditure functions 132 income aggregation and 445 mathematical model of exchange 465466 profit functions 377 Homogeneous functions 5558 derivatives and 56 Eulers theorem 56 homothetic functions 5658 Homothetic functions 5658 203 306307 Homothetic preferences 105 Hone P 361 Hotelling Harold 378 538 Hotellings beach model 538540 Hotellings lemma 381 Household production models 193196 corner solutions 195196 illustrating budget constraints 195 linear attributes model 194195 overview 193194 Housing costs as composite commodity 191192 Human capital 585587 Hybrid equilibria 282 284 Hyperbolic discounting 623 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 754 Index I Immigration 324 Imperfect competition 525562 Bertrand model 527528 capacity constraints 534535 Cournot model 528534 entry of firms 555559 innovation 559561 longerrun decisions 545550 pricing and output 525527 product differentiation 535541 signaling 552555 strategic entry deterrence 550552 strategic substitutes and complements 567570 tacit collusion 541544 Implicit shadow prices 193196 Implicit functions 3233 Income aggregation 445 Income effects 141173 consumer surplus 166170 demand concepts and evaluation of price indices 178181 demand curves and functions 141143 148155 demand elasticities 159165 income changes 143144 preference and substitution effect 171172 price changes 145148 156159 real wage rate changes 577 twogood case 183185 See also Substitution effects Income elasticity of demand 159 Incompleteinformation games 273276 Increasing cost industry 422 Increasing returns to scale 305 307 312 313 Independent variables 32 Indifference curve maps 9495 102 Indifference curves convexity of 9596 100101 defined 93 maps 9495 102 mathematics of 99101 and transitivity 95 twogood case 183 utility maximization in attributes model 195196 Indirect approach 4 Indirect utility function 126127 Individual demand curves 148151 Industry structure 425 Inequality constraints 4648 complementary slackness 4748 slack variables 4647 solution using Lagrange multipliers 47 twovariable example 46 Inferior goods 144 146147 Inferior inputs 330 331 Infinitely elastic longrun supply 420421 Infinitely repeated games 271272 541544 Information 230231 in economic models 18 as good 230231 quantifying value of 231 See also Asymmetric information Initial endowments 465 Innovation 559561 competition for 560561 monopoly on 560 Input costs changes in 426428 industry structure and 427428 Input demand decomposing into substitution and output components 387388 profit maximization and 381388 Input demand functions 383 Inputs contingent demand for and Shephards lemma 345348 substitution 341343 supply and longrun producer surplus 430431 See also Cost minimization Labor markets Insurance adverse selection 221 658665 asymmetric information 634 competitive theft 646647 moral hazard 642647 precaution against car theft 645646 premiums 216217 risk aversion and 215216 in statepreference model 234235 willingness to pay for 215216 See also Competitive insurance market Integration 5863 antiderivatives 5860 definite integrals 60 differentiating definite integral 6263 fundamental theorem of calculus 6061 by parts 59 Interest rates 607608 Interfirm externalities 684 Inverse elasticity rule 494495 Investments 546 613616 diversification 222223 portfolio problem 242245 theory of 612 Isoquant maps 300304 constant returnstoscale production function 306 elasticity of substitution 309 importance of crossproductivity effects 303304 input inferiority 331 rate of technical substitution 301303 simple production functions 311 technical progress 315 Isoquants defined 300 See also Isoquant maps Rate of technical substitution RTS J Jackman Patrick C 179 Jensen M 245 Jensens inequality 214 225 Jobmarket signaling 279280 hybrid equilibrium in 284 pooling equilibrium in 283284 separating equilibrium in 282283 Jorgenson Dale W 203 K Kakutanis fixed point theorem 292 Kehoe Patrick J 139 323 Kehoe Timothy J 139 Koszegi Botond 112 KuhnTucker conditions 48 Kwoka J E 524 L Labor costs 325 mandated benefits 584 productivity 298299 Labor markets 575595 allocation of time 575578 equilibrium in 583584 labor unions 592595 market supply curve for labor 582583 mathematical analysis of labor supply 578582 monopsony in labor market 589591 wage variation 585589 Labor supply 578582 dual statement of problem 579 Slutsky equation of labor supply 579582 Labor theory of exchange value 10 Labor unions 592595 bargaining model 594595 goals 592595 modeling 593594 Lagrangian multiplier as benefitcost ratio 4243 interpreting 4142 method for 40 in ngood utility maximization 122 solution using 47 Lancaster KJ 194 Latzko D 361 Law of one price 449450 Leading principal minors 82 Learning games 286 Legal barriers to entry 492 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Index 755 Lemons market for 666667 Leontief Wassily 314 Leontief production functions 313314 323 Lerner Abba 370 Lerner index 370 494 LES linear expenditure system 139 Lewbel Arthur 203 Lightning calculations 115 Limitations and composite commodities 191193 Lindahl Erik 701 Lindahl equilibrium 701704 local public goods 702704 shortcomings of 702 Linear attributes model 194195 Linear expenditure system LES 139 Linear pricing 648 Linear production function 310 Local public goods 702704 Locay L 524 Long run See Shortrun longrun distinction Longrun analysis elasticity of supply 424 longrun equilibrium 419421 424428 overview 418 producer surplus in 428431 shape of supply curve 421424 Longrun competitive equilibrium 418 Longrun cost curves 351354 Longrun elasticity of supply 424 Longrun equilibrium comparative statistics analysis of 424428 conditions for 418 constant cost case 419421 Cournot model 557558 in oligopoly 556557 Longrun producer surplus 428431 Longrun supply curves 423424 Lump sum principle 127129 M MacBeth J 245 Majority rule 704 Malthus Thomas 298 617 Marginal benefit 42 Marginal costs MC 334 336337 351 defined 334 graphical analysis of 336337 pricing 513 Marginal expense ME 590 Marginalism 11 365366 Marginal physical product MP 298 Marginal productivity 297300 average physical productivity 299300 diminishing 298 marginal physical product 298 rate of technical substitution 302 Marginal rate of substitution MRS defined 93 indifference curves 99100 with many goods 106107 Marginal revenue MR 367372 curves 371372 defined 366 and elasticity 369370 from linear demand functions 369 pricemarginal cost markup 370371 Marginal revenue product MRP 382 Marginal utility MU 99100 122 213214 242 Market basket index 178179 Market demand 401405 defined 404 elasticity of market demand 405 generalizations 404 market demand curve 402 shifts in 403 shifts in market demand curve 402403 simplified notation 404405 Market period 405 Markets meaning of 535536 reaction to shift in demand 411412 rental rates 611 separation thirddegree price discrimination through 507509 tools for studying 18 Market supply curve 407408 582583 Marshall Alfred 11 17 391 401 Marshallian demand 159160 168170 179180 Marshallian substitutes and complements 186 Marshallian supplydemand synthesis 1114 Masten S E 396 Mathematical statistics 6776 Matrix algebra 8285 Cramers rule 85 constrained maxima 8384 quasiconcavity 84 Maximal punishment principle for crime 272 Maximization 83 of one variable 2125 constrained 4045 5152 of several variables 3436 Maximum principle 6467 MC See Marginal costs MC McFadden D 360 ME marginal expense 590 Meade J 684 Median voter theorem 705708 median voter equilibrium 707 optimality of median voter result 707708 overview 706707 singlepeaked preferences and 705706 MES minimum efficient scale 337 Mexico NAFTA and 139 487 Microsoft 560 Milliman S R 716 Minimization of costs 327328 329 332333 of expenditures 129132 Minimum efficient scale MES 337 Mixed strategies 256260 computing mixedstrategy equilibria 258260 formal definitions 256257 Modern economics founding of 10 Monjardet Bernard 111 Monopolies 491522 allocational effects of 499 barriers to entry 491493 coffee shop example 654655 comparative statics analysis of 501502 defined 491 distributional effects of 499 dynamic views of 516517 on innovation 560 linear twopart tariffs 523524 misallocated resources under 498501 natural 513 price determination for 494 price discrimination 504513 product quality and durability 502504 profit maximization and output choice 493498 regulation of 513516 resource allocation and 498501 simple demand curves 497498 welfare losses and elasticity 500501 Monopoly output 496497 502 Monopoly rents 495 Monopsonies 589591 Monotonic transformations 5658 Monteverde K 396 Moore John 393 Moral hazard 221 642647 competitive insurance market 646647 defined 643 firstbest insurance contract 643644 mathematical model 643 secondbest insurance contract 644646 Morgenstern Oscar 210 Morishima M 342 Morishima elasticities 342 Mostfavored customer program 570 MP marginal physical product 298 MR See Marginal revenue MR MRP marginal revenue product 382 MRS See Marginal rate of substitution MRS MU marginal utility 99100 122 213214 242 Muellbauer John 140 Multiself model 176 Murphy Kevin M 112 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 756 Index Multivariable Calculus 2635 calculating partial derivatives 2627 chain rule with many variables 3032 elasticity 2829 firstorder conditions for maximum 3435 implicit functions 3233 partial derivatives 2630 secondorder conditions 3536 Youngs theorem 30 Mutual funds 245 N NAFTA North American Free Trade Agreement 139 487 Nash John 250 260 288 291 Nash bargaining 394 Nash equilibrium 250255 in Battle of the Sexes 253255 258259 of Bertrand game 527528 of Cournot game 529532 defined 251 dominant strategies 253 existence of 291293 formal definition 250251 imperfect competition 567 inefficiency of 699701 in Prisoners Dilemma 251252 in sequential games 266267 underlining bestresponse payoffs 252 Natural experiment 675 Natural monopolies 491 513 Natural resource pricing 617620 decrease in prices 618619 profitmaximizing pricing and output 617619 renewable resources 619620 social optimality 619 substitution 619 Natural spring example 530534 547548 551552 Negative definite 8284 Negative externalities 263 Nested production functions 323 Net complements 188189 Net substitutes 188189 New goods bias 179180 ngood case 120126 corner solutions 122126 firstorder conditions 121 implications of firstorder conditions 121122 interpreting Lagrange multiplier 122 Nicoletti G 488 ninput case elasticity of substitution 308309 returns to scale 307 Nominal interest rates 607608 Nondepreciating machines 611 Nonexclusive goods 231 695 Nonhomothetic preferences 106 Nonlinear pricing 648657 with continuum of types 655657 firstbest case 649651 mathematical model 649 secondbest case 651655 Nonoptimality of shortrun costs 350 Nonrival goods 695696 Nonuniqueness of utility measures 90 Normal Gaussian distribution 69 7374 Normal form for Battle of the Sexes 253 264 for Prisoners Dilemma 249 Normal goods 144 Normative analysis 89 North American Free Trade Agreement NAFTA 139 487 Nudge 673674 O Oczkowski E 139 Oi Walter 510 Oligopolies 525566 Bertrand model 527528 capacity constraints 534535 Cournot model 528534 defined 525 entry of firms 555559 innovation 559561 longerrun decisions 545550 naturalSpring 533534 pricing and output 525527 product differentiation 535541 signaling 552555 strategic entry deterrence 550552 strategic substitutes and complements 567570 tacit collusion 541544 See also Cournot game OlivieraMartins J 488 Opportunity cost doctrine 15 457458 Optimal control problem 6364 Optimization assumptions 78 dynamic 6367 Ordinal properties 5758 Output choice 366367 Output effects principle of 385 profit maximization and input demand 384385 Outputs imperfect competition 525527 monopolies and 493494 496497 profitmaximizing for natural resources 617619 Ownermanager relationship 638642 comparison to standard model of firm 642 fullinformation case 639 hiddenaction case 639642 Ownership of machines 611612 P Paradox of voting 704705 Pareto Vilfredo 17 469 Pareto efficient allocation 469 Pareto superiority 523524 Partial derivatives calculating 2627 ceteris paribus assumption and 27 defined 26 secondorder 29 units of measurement and 2728 Partial equilibrium model 14 401444 comparative statistics analysis 424428 demand aggregation and estimation 445447 economic efficiency and welfare analysis 431434 of externalities 689691 longrun analysis 418419 longrun elasticity of supply 424 longrun equilibrium 419421 market demand 401405 mathematical model of market equilibrium 414417 price controls and shortages 434435 pricing in very short run 405407 producer surplus in long run 428431 shape of longrun supply curve 421424 shifts in supply and demand curves 412414 shortrun price determination 407412 tax incidence analysis 435440 timing of supply response 405 Payoffs 248 in Battle of the Sexes 257258 in best response 252 in Rock Paper Scissors game 255 PDF See Probability density function PDF PDV See Present discounted value PDV Perfect Bayesian equilibrium 281282 Perfect competition 449450 behavioral assumptions 450 defined 407 law of one price 449450 longrun equilibrium 419 Perfect complements 103104 Perfect price discrimination 505506 Perfect substitutes 103 Perpetual rate of return 600 Perpetuities consols 626 Philip N E 139 Pigou A C 691 Pigovian taxes 691693 710 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Index 757 Players 248 273274 279281 Pointslope formula 34 Political support for trade policies 464 Pollution abatement of 715716 CO2 reduction strategies 488 emission taxes in the United Kingdom 715 pollution rights 694 Pontryagin L S 64 Pooling equilibrium 282 554555 in competitive insurance market 664 in jobmarket signaling game 283284 Portfolio problem 242245 CARA utility 242243 individual choices 243245 many risky assets 243 mutual funds 245 one risky asset 242 optimal portfolios 243 studies of CAPM 245 Positive analysis 9 Positive definite 8284 Positivenormative distinction 89 Posterior beliefs 279282 Pratt J W 216217 219 Pratts risk aversion measure 217219 Predatory pricing 555 Predictions testing 45 Preferences 89107 axioms of rational choice 8990 manygood case 106107 mathematics of indifference curves 99101 overview 89 trades and substitution 9299 utility 9092 utility functions for specific 102105 Present discounted value PDV 625627 annuities and perpetuities 626 bonds 626627 investment decisions 613616 Price controls and shortages 434435 disequilibrium behavior 435 welfare evaluation 435 Price discrimination 504510 across segmented markets 507510 defined 504 dynamic 512513 firstdegree 505 perfect 505506 seconddegree 510513 thirddegree 507 through nonuniform schedules 510513 Price dispersion 540541 Price elasticity 159 160 Pricemarginal cost markup 370371 Prices of contingent commodities 232 of future goods 601 606 imperfect competition 525527 implicit 193196 law of one 449450 perfectly competitive 449450 predatory 555 response to changes in 156159 in shortrun analysis 407412 shrouded 522 versus value 910 in very short run 405407 welfare effects of 168170 See also Bertrand game Consumer surplus Equilibrium price Expenditure functions Natural resource pricing Nonlinear pricing Price discrimination Price schedules 510513 Price takers 369 372376 Primont Daniel 202 Prince R 716 Principalagent relationship 635637 Principles of Economics Marshall 11 Prior beliefs 279280 Prisoners Dilemma 248250 experiments with 285 finitely repeated games 270 infinitely repeated games 271272 Nash equilibrium in 251252 normal form 249 thinking strategically about 249250 underlining procedure in 252 Private information See Asymmetric information Probability density function PDF defined 68 207 examples of 6869 random variables and 67 Producer surplus defined 380 428 in long run 428431 in short run 378381 Product differentiation 535541 Bertrand competition with 536540 Bertrand model 568 consumer search and price dispersion 540541 Hotellings beach model 538540 meaning of market 535536 toothpaste as a differentiated product 537538 Production and exchange mathematical model of 475478 budget constraints and Walras law 476 Walrasian equilibrium 476477 welfare economics in Walrasian model with production 477478 Production externalities 687689 Production functions 297321 322324 CES 313314 323 CobbDouglas 312313 317318 322323 defined 297 elasticity of substitution 307309 fixed proportions 310312 generalized Leontief 323 homothetic 306307 isoquant maps and rate of technical substitution 300304 linear 310 marginal productivity 297300 nested 323 returns to scale 304307 technical progress 314318 translog 324 twoinput 299300 Production possibility frontier 1517 453454 concavity of 457458 defined 454 and economic inefficiency 1617 implicit functions and 3233 Profit functions 376381 defined 376 envelope results 377378 properties of 376377 shortrun 380381 Profit maximization 363391 boundaries of firm 393396 cost minimization and 327328 decisions 372373 defined 365 finding derivatives and 25 functions of variable 49 graphical analysis 367 input demand and 381388 marginalism and 365366 marginal revenue and 367372 by monopolies 493498 nature and behavior of firms 363365 optimization assumptions and 78 output choice and 366367 493498 principle of 367 profit functions 376381 secondorder conditions and 367 shortrun supply by pricetaking firm 372376 testing assumptions of 4 testing predictions of 45 Profits 366 monopolies 495 See also Profit functions Profit maximization Proper subgames 267268 Properties of expenditure functions 132134 Property rights 394395 Public goods attributes of 695696 defined 696 derivation of the demand for 698 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 758 Index environmental externalities and production of 703704 externalities 685 Lindahl pricing of 701704 resource allocation and 697701 Roommates dilemma 699701 simple political model 706709 voting and resource allocation 704706 voting mechanisms 709710 Puppy dog strategy 548550 552 567 Pure inflation 142 Pure strategies 259260 Q Quality choice models and 111112 monopoly product 502503 Quantifying value of information 231 Quantitative size of shifts in cost curves 343 Quasiconcave function 5355 concave functions and 5355 convex indifference curves 100 Quasiconcavity 84 R Random variables continuous 6768 defined 207 discrete 6768 expected value of 7172 and probability density functions 67 variance and standard deviation of 7374 Rate of product transformation RPT 454 Rate of return demand for future goods 602 determining 601608 effects of changes in 604605 equilibrium 606 interest rates 607608 overview 599601 perpetual 600 price of future goods and 601 regulation of 514515 riskfree 609 singleperiod 600 supply of future goods 605606 utility maximization 603604 Rate of technical substitution RTS defined 301 diminishing 303304 importance of crossproductivity effects 303304 marginal productivities and 302 reasons for diminishing 302303 Rational choice axioms of 8990 Real interest rates 607608 Real option theory 224226 Reinsdorf Marshall B 180 Relative risk aversion 219220 Renewable resources 619620 Rent capitalization of 430 monopoly 495 Ricardian 429430 Rental rates 611 Repeated games 270272 finitely 270 541 infinitely 271272 541544 Replacement effect 560 Resource allocation monopoly and 498501 public goods and 697701 voting and 704706 Returns to scale 304307 constant 305 defined 304 homothetic production functions 306307 ninput case 307 Revealed preference theory 171172 graphical approach 172 negativity of substitution effect 172 Ricardian rent 429430 Ricardo David 1011 429 430 Risk aversion 212216 constant 218219 constant relative 220 defined 215 fair gambles and 213214 fair bets and 213214 insurance and 215217 measuring 216220 relative 219220 risk premiums and 236237 statepreference approach to choice 233 wealth and 217218 See also Uncertainty Risk premiums 236237 Robinson S 488 Rock Paper Scissors game 255 Rockefeller John D 498 525 Rodriguez A 524 Roys identity 179180 RPT rate of product transformation 453 454455 686687 RTS See Rate of technical substitution RTS Russell R Robert 202 S SAC shortrun average total cost function 351 354355 St Petersburg paradox 208209 Samuelson Paul 18 171 Scarf Herbert 479 Scharfstein D S 245 Schmalensee R 716 Schmittlein D C 396 Schumpeter J A 516517 Secondbest contracts 636 adverse selection 658665 defined 658 monopoly insurer 660 moral hazard 644646 nonlinear pricing 648657 principalagent model 635 Secondbest nonlinear pricing 651657 Seconddegree price discrimination 510513 Second derivatives 2324 Secondorder conditions 23 367 382383 concave and convex functions 51 8285 constrained maxima 8384 curvature and 4855 matrix algebra and 8285 for maximum 83 119120 quasiconcavity 84 several variables 3536 Secondorder partial derivatives 29 Secondparty preferences 112 Second theorem of welfare economics 471473 Selfishness 115116 Selten Reinhard 270 271 Separating equilibrium 282283 554 Sequential Battle of the Sexes game 264 Sequential games 263269 backward induction 268269 Battle of the Sexes 264 extensive form 265 Nash equilibria 266267 subgameperfect equilibrium 267268 Shadow implicit prices 193196 Sharpe W F 243 Shephard R W 153 Shephards lemma 153154 contingent demand for inputs and 345348 defined 153 elasticity of substitution and 348 net substitutes and complements 188189 Shogren J F 715 Short run long run distinction 348356 fixed and variable costs 349 graphs of perunit cost curves 354355 nonoptimality of 350 relationship between longrun cost curves and 351354 shortrun marginal and average costs 351 total costs 349 Shortrun analysis 348356 fixed and variable costs 349 graphs of perunit cost curves 354355 nonoptimality of 350 price determination 407412 producer surplus in 378381 relationship between longrun cost curves and 351354 shortrun marginal and average costs 351 total costs 349 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Index 759 Shortrun average total cost function SAC 351 354355 Shortrun fixed costs 349 Shortrun marginal cost function SMC 351 354355 Shortrun market supply function 408 Shortrun supply curve 373374 413 Shortrun supply elasticity 408410 Shortrun variable costs 349 Shrouded prices 522 Shutdown decision 374376 Signaling 273 552555 665667 in competitive insurance markets 665666 entrydeterrence model 553554 jobmarket 279280 282284 market for lemons 666 pooling equilibrium 554555 predatory pricing 555 separating equilibrium 554 Signaling games 273 278284 Bayes rule 280281 jobmarket signaling 279280 perfect Bayesian equilibrium 281284 Simplexes 292 Simultaneous games 273278 BayesianNash equilibrium 274278 player types and beliefs 273274 sequential games versus 263269 Singleinput case 383 Singlepeaked preferences 705706 Singleperiod rate of return 600 Single variable calculus 2125 derivatives 22 firstorder condition for maximum 23 rules for finding derivatives 2425 second derivatives 2324 secondorder conditions and curvature 23 4849 value of derivative at point 2223 Slesnick Daniel T 203 Slutsky Eugen 157 Slutsky equation 157159 for crossprice effects 184 of labor supply 579582 twogood case 183185 SMC shortrun marginal cost function 351 354355 Smith Adam 10 11 18 116 304 468 470 477 Smith John Maynard 286 Smith R B W 524 Smith Vernon 284 Social optimality 619 Social welfare function 474475 Solow R M 316317 322 Solow growth model 322323 Special preferences 111112 habits and addiction 112 quality 111112 secondparty preferences 112 threshold effects 111 Spence Michael 278 Spencer B J 570 Spurious product differentiation 200 Stackelberg H von 546 Stackelberg model 546548 Stage games 270272 Standard deviation 7274 Statepreference model 231237 contingent commodities 232 fair markets for contingent goods 233 graphic analysis of 233235 insurance in 236237 prices of contingent commodities 232 risk aversion in 233 236237 risk premiums 236237 states of world and contingent commodities 232 utility analysis 232 States of the world 232 Stein J 245 Stochastic discount factor 609 Stigler George J 112 Stock options 223224 Stocks 61 Stoker Thomas M 203 StoneGeary utility function 136 Strategic entry deterrence 553554 Strategies 248 dominant 253 260 grim 272 mixed 256260 portfolio problem 242245 in Prisoners Dilemma 249250 puppy dog and top dog 548550 552 567 pure 256257 trigger 270272 Strictly mixed strategies 257 Subgameperfect equilibrium 267268 Subramanian S 488 Substitutes 186188 asymmetry of gross definitions 187188 elasticity of Shephards lemma and 348 gross 184 186 imperfect competition 567570 with many goods 189190 of natural resources 617620 net 188189 perfect 103 strategic 567570 See also Trades and substitution Substitution bias expenditure functions and 179 market basket index 179 Substitution effects 145147 157 384 consumer surplus 166170 demand concepts and evaluation of price indices 178181 demand curves and functions 148151 demand elasticities 163165 demand functions 141143 impact on demand elasticities 163165 importance of 163165 negativity of 172 net substitutes and complements 188 price changes 145148 156159 principle of 385 profit maximization and input demand 384 386 real wage rate changes 577 revealed preference and 171172 twogood case 117120 See also Income effects Sunk costs 546 Sun Tzu 226 Supply and demand 111112 equilibrium 1214 450451 shifts in 425426 special preferences 111112 synthesis 1114 Supply curve importance of shape of 413414 importance of shape of demand curve 412413 importance to demand curves 413414 longrun 421424 monopoly 495498 reasons for shifts in 412 shifts in 412413 shortrun 373376 Supply elasticity 408410 elasticity of 424 Supply function 374376 380381 387388 409410 Supply response 405 Swan Peter 504 Swans independence assumption 504 T Tacit collusion 541544 in Bertrand model 542543 in Cournot model 542543 in finitely repeated games 541 in infinitely repeated games 541544 Tariffs twopart 510512 523524 Taxation environmental 703 excess burden of 438439 481482 in general equilibrium model 487 693 lump sum principle of 128 Pigovian 691693 703 voting for redistributive 708709 Tax incidence analysis 435439 deadweight loss and elasticity 437438 effects on attributes of transactions 439 mathematical model of tax incidence 435437 transaction costs 438439 welfare analysis 437 Taylors series 79 216 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 760 Index Technical barriers to entry 491492 Technical progress 314318 in CobbDouglas production function 317318 effects on production 460 growth accounting 316318 measuring 315 Teece D J 396 Testing assumptions 4 predictions 45 Theil H 446 Theoretical models 3 Theory of Games and Economic Behavior The von Neumann and Morgenstern 210 Thirdbest outcome 635637 Thirddegree price discrimination 507 Thomas A 361 Threshold effects 111 Tied sales 524 Time allocation of 575578 capital and 625630 continuous 627630 Timing of supply response 405 Tirole J 291 568569 Tobin J 243 Top dog strategy 548550 552 567 Total cost function 333 Trade general equilibrium models 487 imperfect competition 570 political support for 464 prices 463464 Trades and substitution 9299 convexity 9599 indifference curve map 9495 marginal rate of substitution 9294 transitivity 95 Tragedy of the Commons 261263 276278 Transaction costs 395396 438439 Transitivity indifference curves and 95 preferences and 89 Translog cost function applications of 361 manyinput 361 with two inputs 360361 Translog production function 324 Trigger strategies 270272 Tucker A W 248 Twogood model allocation of time 575576 demand relationships among goods 183185 Twogood utility maximization 117120 budget constraint 117 corner solutions 120 firstorder conditions for maximum 118119 secondorder conditions for maximum 119120 Twoinput case 383384 Twopart pricing 648 Twopart tariffs 510512 523524 Twostage budgeting homothetic functions and energy demand 203 relation to composition commodity 202203 theorem 202203 theory of 202 Twotier pricing systems 513514 Typology of public goods 696 U Ultimatum game 285 289290 Uncertainty 207237 asymmetry of information 237 diversification 222223 in economic models 18 expected utility hypothesis 208210 fair gambles 208209 flexibility 223230 information as a good 230231 insurance 221 mathematical statistics 207 measuring risk aversion 216220 methods for reducing risk and 221 portfolio problem 242245 risk aversion 212216 statepreference approach to choice under 231237 von NeumannMorgenstern theorem 210212 Uncompensated demand curves 154155 Uniform distribution 6970 7273 Usedcar market signaling in 667 Utility 9092 arguments of functions 9192 ceteris paribus assumption 9091 from consumption of goods 91 defined 92 economic goods 92 externalities in 684685 functions for specific preferences 102106 mathematical model of exchange 465 maximization 576577 603604 nonuniqueness of measures 90 See also CES utility CobbDouglas utility Indifference curves Preferences Utility maximization 115134 altruism and selfishness 115116 in attributes model 194195 budget shares and 138140 expenditure minimization 129132 graphical analysis of twogood case 117120 indirect utility function 126127 individuals intertemporal 602 initial survey 116 labor supply 576577 and lightning calculations 115 lump sum principle 127129 ngood case 120126 properties of expenditure functions 132134 See also Demand relationships among goods Income effects Substitution effects V Value early economic thoughts on 910 economic theory of 918 labor theory of exchange 10 of options 227229 Value and Capital Hicks 189 191 Value function 39 127 Value in exchange concept 10 Value in use concept 10 Variable costs 349 Variables chain rule with many 3032 change of variable 59 endogenous 67 exogenous 67 functions of one 4849 functions of several 2636 functions of two 49 independent 32 independent implicit functions and 32 random 6768 7374 207 slack 4647 Variance 7274 207 Vector notation 464465 Vedenov Dmitry V 111 Verification of economic models 45 importance of empirical analysis 5 profitmaximization model 4 testing assumptions 4 testing predictions 45 Vickery William 667669 Villarreal Hector J 181 von Neumann John 210211 von NeumannMorgenstern theorem 210 212 expected utility maximization 211 212 utility index 210211 von NeumannMorgenstern utility 210211 215 242 Voting 704706 Clarke mechanism 710 Groves mechanism 709 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Index 761 mechanisms generalizations of 710 median voter theorem 705706 paradox of 704705 708709 resource allocation and 704706 W Wages compensating differentials 585589 variation in 585589 Wales Terrence J 203 Walras Leon 14 466468 Walrasian price adjustment 466 476477 Walras law equilibrium and 466 mathematical model of production and exchange 475478 Waterdiamond paradox 10 14 Weakly dominated strategy 668 Wealth and risk aversion measurement 217219 Wealth of Nations The Smith 10 Welfare analysis 437 applied analysis 432434 changes and the Marshallian demand curve 168170 consumer and the expenditure function 166 economic efficiency and 431434 economics 1718 economics in the Walrasian model with production 477478 effects of price changes 168170 evaluation price controls and shortages 435 first theorem of welfare economics 468470 general equilibrium and 478482 general equilibrium models and 487488 loss computations 433434 loss from a price increase 170 monopolies and 500501 second theorem of welfare economics 471473 Westbrook M D 361 Wetzstein Michael E 111 White B 715 Williamson Oliver 393 Y Yatchew A 361 Youngs theorem 30 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203 Copyright 2017 Cengage Learning All Rights Reserved May not be copied scanned or duplicated in whole or in part WCN 02200203