Advanced Microeconomic Theory (3rd Ed) PDF

GEOFFREY A. JEHLE GEOFFREY A. JEHLE...

GEOFFREY A. JEHLE GEOFFREY A. JEHLE THIRD EDITION ADVANCED MICROECONOMIC THEORY PHILIP J. RENY ADVANCED PHILIP J. RENY MICROECONOMIC THEORY THIRD EDITION The classic text in advanced microeconomic theory, revised and expanded. ADVANCED Advanced Microeconomic Theory remains a rigorous, up-to-date standard in microeconomics, giving all the core mathematics and modern theory the advanced student must master. Long known for careful development of complex theory, together with clear, patient explanation, this MICROECONOMIC student-friendly text, with its efficient theorem-proof organisation, and many examples and exercises, is uniquely effective in advanced courses. THEORY New in this edition General equilibrium with contingent commodities Expanded treatment of social choice, with a simplified proof of Arrow’s theorem and complete, step-by-step development of the Gibbard – Satterthwaite theorem THIRD EDITION PHILIP J. RENY GEOFFREY A. JEHLE Extensive development of Bayesian games New section on efficient mechanism design in the quasi-linear utility, private values environment. The most complete and easy-to-follow presentation of any text. Over fifty new exercises Essential reading for students at Masters level, those beginning a Ph.D and advanced undergraduates. A book every professional economist wants in their collection. Cover photograph © Getty Images www.pearson-books.com CVR_JEHL1917_03_SE_CVR.indd 1 10/11/2010 16:08 Advanced Microeconomic Theory We work with leading authors to develop the strongest educational materials in economics, bringing cutting-edge thinking and best learning practice to a global market. Under a range of well-known imprints, including Financial Times Prentice Hall, we craft high quality print and electronic publications that help readers to understand and apply their content, whether studying or at work. To find out more about the complete range of our publishing, please visit us on the World Wide Web at: www.pearsoned.co.uk. Advanced Microeconomic Theory THIRD EDITION G E O F F R E Y A. J E H L E Vassar College P H I L I P J. R E N Y University of Chicago Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk First published 2011  c Geoffrey A. Jehle and Philip J. Reny 2011 The rights of Geoffrey A. Jehle and Philip J. Reny to be identified as author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. ISBN: 978-0-273-73191-7 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress 10 9 8 7 6 5 4 3 2 1 14 13 12 11 Typeset in 10/12 pt and Times-Roman by 75 Printed and bound in Great Britain by Ashford Colour Press Ltd, Gosport, Hampshire  To Rana and Kamran G.A.J. To Dianne, Lisa, and Elizabeth P.J.R.  CO N T E N T S PREFACE xv PART I ECONOMIC AGENTS 1 CHAPTER 1 CONSUMER THEORY 3 1.1 Primitive Notions 3 1.2 Preferences and Utility 4 1.2.1 Preference Relations 5 1.2.2 The Utility Function 13 1.3 The Consumer’s Problem 19 1.4 Indirect Utility and Expenditure 28 1.4.1 The Indirect Utility Function 28 1.4.2 The Expenditure Function 33 1.4.3 Relations Between the Two 41 1.5 Properties of Consumer Demand 48 1.5.1 Relative Prices and Real Income 48 1.5.2 Income and Substitution Effects 50 1.5.3 Some Elasticity Relations 59 1.6 Exercises 63 viii CONTENTS CHAPTER 2 TOPICS IN CONSUMER THEORY 73 2.1 Duality: A Closer Look 73 2.1.1 Expenditure and Consumer Preferences 73 2.1.2 Convexity and Monotonicity 78 2.1.3 Indirect Utility and Consumer Preferences 81 2.2 Integrability 85 2.3 Revealed Preference 91 2.4 Uncertainty 97 2.4.1 Preferences 98 2.4.2 Von Neumann-Morgenstern Utility 102 2.4.3 Risk Aversion 110 2.5 Exercises 118 CHAPTER 3 THEORY OF THE FIRM 125 3.1 Primitive Notions 125 3.2 Production 126 3.2.1 Returns to Scale and Varying Proportions 132 3.3 Cost 135 3.4 Duality in Production 143 3.5 The Competitive Firm 145 3.5.1 Profit Maximisation 145 3.5.2 The Profit Function 147 3.6 Exercises 154 PART II MARKETS AND WELFARE 163 CHAPTER 4 PARTIAL EQUILIBRIUM 165 4.1 Perfect Competition 165 4.2 Imperfect Competition 170 4.2.1 Cournot Oligopoly 174 CONTENTS ix 4.2.2 Bertrand Oligopoly 175 4.2.3 Monopolistic Competition 177 4.3 Equilibrium and Welfare 179 4.3.1 Price and Individual Welfare 179 4.3.2 Efficiency of the Competitive Outcome 183 4.3.3 Efficiency and Total Surplus Maximisation 186 4.4 Exercises 188 CHAPTER 5 GENERAL EQUILIBRIUM 195 5.1 Equilibrium in Exchange 196 5.2 Equilibrium in Competitive Market Systems 201 5.2.1 Existence of Equilibrium 203 5.2.2 Efficiency 212 5.3 Equilibrium in Production 220 5.3.1 Producers 220 5.3.2 Consumers 223 5.3.3 Equilibrium 225 5.3.4 Welfare 232 5.4 Contingent Plans 236 5.4.1 Time 236 5.4.2 Uncertainty 236 5.4.3 Walrasian Equilibrium with Contingent Commodities 237 5.5 Core and Equilibria 239 5.5.1 Replica Economies 240 5.6 Exercises 251 CHAPTER 6 SOCIAL CHOICE AND WELFARE 267 6.1 The Nature of the Problem 267 6.2 Social Choice and Arrow’s Theorem 269 6.2.1 A Diagrammatic Proof 274 x CONTENTS 6.3 Measurability, Comparability, and Some Possibilities 279 6.3.1 The Rawlsian Form 282 6.3.2 The Utilitarian Form 284 6.3.3 Flexible Forms 285 6.4 Justice 288 6.5 Social Choice and the Gibbard-Satterthwaite Theorem 290 6.6 Exercises 296 PART III STRATEGIC BEHAVIOUR 303 CHAPTER 7 GAME THEORY 305 7.1 Strategic Decision Making 305 7.2 Strategic Form Games 307 7.2.1 Dominant Strategies 308 7.2.2 Nash Equilibrium 311 7.2.3 Incomplete Information 319 7.3 Extensive Form Games 325 7.3.1 Game Trees: A Diagrammatic Representation 328 7.3.2 An Informal Analysis of Take-Away 330 7.3.3 Extensive Form Game Strategies 331 7.3.4 Strategies and Payoffs 332 7.3.5 Games of Perfect Information and Backward Induction Strategies 333 7.3.6 Games of Imperfect Information and Subgame Perfect Equilibrium 337 7.3.7 Sequential Equilibrium 347 7.4 Exercises 364 CHAPTER 8 INFORMATION ECONOMICS 379 8.1 Adverse Selection 380 8.1.1 Information and the Efficiency of Market Outcomes 380 CONTENTS xi 8.1.2 Signalling 385 8.1.3 Screening 404 8.2 Moral Hazard and the Principal–Agent Problem 413 8.2.1 Symmetric Information 414 8.2.2 Asymmetric Information 416 8.3 Information and Market Performance 420 8.4 Exercises 421 CHAPTER 9 AUCTIONS AND MECHANISM DESIGN 427 9.1 The Four Standard Auctions 427 9.2 The Independent Private Values Model 428 9.2.1 Bidding Behaviour in a First-Price, Sealed-Bid Auction 429 9.2.2 Bidding Behaviour in a Dutch Auction 432 9.2.3 Bidding Behaviour in a Second-Price, Sealed-Bid Auction 433 9.2.4 Bidding Behaviour in an English Auction 434 9.2.5 Revenue Comparisons 435 9.3 The Revenue Equivalence Theorem 437 9.3.1 Incentive-Compatible Direct Selling Mechanisms: A Characterisation 441 9.3.2 Efficiency 444 9.4 Designing a Revenue Maximising Mechanism 444 9.4.1 The Revelation Principle 444 9.4.2 Individual Rationality 445 9.4.3 An Optimal Selling Mechanism 446 9.4.4 A Closer Look at the Optimal Selling Mechanism 451 9.4.5 Efficiency, Symmetry, and Comparison to the Four Standard Auctions 453 9.5 Designing Allocatively Efficient Mechanisms 455 9.5.1 Quasi-Linear Utility and Private Values 456 9.5.2 Ex Post Pareto Efficiency 458 xii CONTENTS 9.5.3 Direct Mechanisms, Incentive Compatibility and the Revelation Principle 458 9.5.4 The Vickrey-Clarke-Groves Mechanism 461 9.5.5 Achieving a Balanced Budget: Expected Externality Mechanisms 466 9.5.6 Property Rights, Outside Options, and Individual Rationality Constraints 469 9.5.7 The IR-VCG Mechanism: Sufficiency of Expected Surplus 472 9.5.8 The Necessity of IR-VCG Expected Surplus 478 9.6 Exercises 484 MATHEMATICAL APPENDICES 493 CHAPTER A1 SETS AND MAPPINGS 495 A1.1 Elements of Logic 495 A1.1.1 Necessity and Sufficiency 495 A1.1.2 Theorems and Proofs 496 A1.2 Elements of Set Theory 497 A1.2.1 Notation and Basic Concepts 497 A1.2.2 Convex Sets 499 A1.2.3 Relations and Functions 503 A1.3 A Little Topology 505 A1.3.1 Continuity 515 A1.3.2 Some Existence Theorems 521 A1.4 Real-Valued Functions 529 A1.4.1 Related Sets 530 A1.4.2 Concave Functions 533 A1.4.3 Quasiconcave Functions 538 A1.4.4 Convex and Quasiconvex Functions 542 A1.5 Exercises 546 CHAPTER A2 CALCULUS AND OPTIMISATION 551 A2.1 Calculus 551 CONTENTS xiii A2.1.1 Functions of a Single Variable 551 A2.1.2 Functions of Several Variables 553 A2.1.3 Homogeneous Functions 561 A2.2 Optimisation 566 A2.2.1 Real-Valued Functions of Several Variables 567 A2.2.2 Second-Order Conditions 570 A2.3 Constrained Optimisation 577 A2.3.1 Equality Constraints 577 A2.3.2 Lagrange’s Method 579 A2.3.3 Geometric Interpretation 584 A2.3.4 Second-Order Conditions 588 A2.3.5 Inequality Constraints 591 A2.3.6 Kuhn-Tucker Conditions 595 A2.4 Optimality Theorems 601 A2.5 Separation Theorems 607 A2.6 Exercises 611 LIST OF THEOREMS 619 LIST OF DEFINITIONS 625 HINTS AND ANSWERS 631 REFERENCES 641 INDEX 645  PR E FA C E In preparing this third edition of our text, we wanted to provide long-time readers with new and updated material in a familiar format, while offering first-time readers an accessible, self-contained treatment of the essential core of modern microeconomic theory. To those ends, every chapter has been revised and updated. The more significant changes include a new introduction to general equilib- rium with contingent commodities in Chapter 5, along with a simplified proof of Arrow’s theorem and a new, careful development of the Gibbard- Satterthwaite theorem in Chapter 6. Chapter 7 includes many refinements and extensions, especially in our presentation on Bayesian games. The biggest change – one we hope readers find interesting and useful – is an extensive, integrated presentation in Chapter 9 of many of the cen- tral results of mechanism design in the quasi-linear utility, private-values environment. We continue to believe that working through exercises is the surest way to master the material in this text. New exercises have been added to virtually every chapter, and others have been updated and revised. Many of the new exercises guide readers in developing for themselves exten- sions, refinements or alternative approaches to important material covered in the text. Hints and answers for selected exercises are provided at the end of the book, along with lists of theorems and definitions appearing in the text. We will continue to maintain a readers’ forum on the web, where readers can exchange solutions to exercises in the text. It can be reached at http://alfred.vassar.edu. The two full chapters of the Mathematical Appendix still provide students with a lengthy and largely self-contained development of the set theory, real analysis, topology, calculus, and modern optimisation theory xvi PREFACE which are indispensable in modern microeconomics. Readers of this edi- tion will now find a fuller, self-contained development of Lagrangian and Kuhn-Tucker methods, along with new material on the Theorem of the Maximum and two separation theorems. The exposition is formal but pre- sumes nothing more than a good grounding in single-variable calculus and simple linear algebra as a starting point. We suggest that even stu- dents who are very well-prepared in mathematics browse both chapters of the appendix early on. That way, if and when some review or reference is needed, the reader will have a sense of how that material is organised. Before we begin to develop the theory itself, we ought to say a word to new readers about the role mathematics will play in this text. Often, you will notice we make certain assumptions purely for the sake of mathemat- ical expediency. The justification for proceeding this way is simple, and it is the same in every other branch of science. These abstractions from ‘reality’ allow us to bring to bear powerful mathematical methods that, by the rigour of the logical discipline they impose, help extend our insights into areas beyond the reach of our intuition and experience. In the physical world, there is ‘no such thing’ as a frictionless plane or a perfect vacuum. In economics, as in physics, allowing ourselves to accept assumptions like these frees us to focus on more important aspects of the problem and thereby helps to establish benchmarks in theory against which to gauge experience and observation in the real world. This does not mean that you must wholeheartedly embrace every ‘unrealistic’ or purely formal aspect of the theory. Far from it. It is always worthwhile to cast a critical eye on these matters as they arise and to ask yourself what is gained, and what is sacrificed, by the abstraction at hand. Thought and insight on these points are the stuff of which advances in theory and knowledge are made. From here on, however, we will take the theory as it is and seek to understand it on its own terms, leaving much of its critical appraisal to your moments away from this book. Finally, we wish to acknowledge the many readers and colleagues who have provided helpful comments and pointed out errors in previous editions. Your keen eyes and good judgements have helped us make this third edition better and more complete than it otherwise would be. While we cannot thank all of you personally, we must thank Eddie Dekel, Roger Myerson, Derek Neal, Motty Perry, Arthur Robson, Steve Williams, and Jörgen Weibull for their thoughtful comments. PART I ECONOMIC AGENTS  CHAPTER 1 CONSUMER THEORY In the first two chapters of this volume, we will explore the essential features of modern consumer theory – a bedrock foundation on which so many theoretical structures in eco- nomics are built. Some time later in your study of economics, you will begin to notice just how central this theory is to the economist’s way of thinking. Time and time again you will hear the echoes of consumer theory in virtually every branch of the discipline – how it is conceived, how it is constructed, and how it is applied. 1.1 PRIMITIVE NOTIONS There are four building blocks in any model of consumer choice. They are the consump- tion set, the feasible set, the preference relation, and the behavioural assumption. Each is conceptually distinct from the others, though it is quite common sometimes to lose sight of that fact. This basic structure is extremely general, and so, very flexible. By specifying the form each of these takes in a given problem, many different situations involving choice can be formally described and analysed. Although we will tend to concentrate here on specific formalisations that have come to dominate economists’ view of an individual consumer’s behaviour, it is well to keep in mind that ‘consumer theory’ per se is in fact a very rich and flexible theory of choice. The notion of a consumption set is straightforward. We let the consumption set, X, represent the set of all alternatives, or complete consumption plans, that the consumer can conceive – whether some of them will be achievable in practice or not. What we intend to capture here is the universe of alternative choices over which the consumer’s mind is capable of wandering, unfettered by consideration of the realities of his present situation. The consumption set is sometimes also called the choice set. Let each commodity be measured in some infinitely divisible units. Let xi ∈ R repre- sent the number of units of good i. We assume that only non-negative units of each good are meaningful and that it is always possible to conceive of having no units of any particular commodity. Further, we assume there is a finite, fixed, but arbitrary number n of different goods. We let x = (x1 ,... , xn ) be a vector containing different quantities of each of the n commodities and call x a consumption bundle or a consumption plan. A consumption 4 CHAPTER 1 bundle x ∈ X is thus represented by a point x ∈ Rn+. Usually, we’ll simplify things and just think of the consumption set as the entire non-negative orthant, X = Rn+. In this case, it is easy to see that each of the following basic requirements is satisfied. ASSUMPTION 1.1 Properties of the Consumption Set, X The minimal requirements on the consumption set are 1. X ⊆ Rn+. 2. X is closed. 3. X is convex. 4. 0 ∈ X. The notion of a feasible set is likewise very straightforward. We let B represent all those alternative consumption plans that are both conceivable and, more important, realis- tically obtainable given the consumer’s circumstances. What we intend to capture here are precisely those alternatives that are achievable given the economic realities the consumer faces. The feasible set B then is that subset of the consumption set X that remains after we have accounted for any constraints on the consumer’s access to commodities due to the practical, institutional, or economic realities of the world. How we specify those realities in a given situation will determine the precise configuration and additional properties that B must have. For now, we will simply say that B ⊂ X. A preference relation typically specifies the limits, if any, on the consumer’s ability to perceive in situations involving choice the form of consistency or inconsistency in the consumer’s choices, and information about the consumer’s tastes for the different objects of choice. The preference relation plays a crucial role in any theory of choice. Its spe- cial form in the theory of consumer behaviour is sufficiently subtle to warrant special examination in the next section. Finally, the model is ‘closed’ by specifying some behavioural assumption. This expresses the guiding principle the consumer uses to make final choices and so identifies the ultimate objectives in choice. It is supposed that the consumer seeks to identify and select an available alternative that is most preferred in the light of his personal tastes. 1.2 PREFERENCES AND UTILITY In this section, we examine the consumer’s preference relation and explore its connec- tion to modern usage of the term ‘utility’. Before we begin, however, a brief word on the evolution of economists’ thinking will help to place what follows in its proper context. In earlier periods, the so-called ‘Law of Demand’ was built on some extremely strong assumptions. In the classical theory of Edgeworth, Mill, and other proponents of the utilitarian school of philosophy, ‘utility’ was thought to be something of substance. ‘Pleasure’ and ‘pain’ were held to be well-defined entities that could be measured and com- pared between individuals. In addition, the ‘Principle of Diminishing Marginal Utility’ was CONSUMER THEORY 5 accepted as a psychological ‘law’, and early statements of the Law of Demand depended on it. These are awfully strong assumptions about the inner workings of human beings. The more recent history of consumer theory has been marked by a drive to render its foundations as general as possible. Economists have sought to pare away as many of the traditional assumptions, explicit or implicit, as they could and still retain a coherent theory with predictive power. Pareto (1896) can be credited with suspecting that the idea of a measurable ‘utility’ was inessential to the theory of demand. Slutsky (1915) undertook the first systematic examination of demand theory without the concept of a measurable sub- stance called utility. Hicks (1939) demonstrated that the Principle of Diminishing Marginal Utility was neither necessary, nor sufficient, for the Law of Demand to hold. Finally, Debreu (1959) completed the reduction of standard consumer theory to those bare essen- tials we will consider here. Today’s theory bears close and important relations to its earlier ancestors, but it is leaner, more precise, and more general. 1.2.1 PREFERENCE RELATIONS Consumer preferences are characterised axiomatically. In this method of modelling as few meaningful and distinct assumptions as possible are set forth to characterise the struc- ture and properties of preferences. The rest of the theory then builds logically from these axioms, and predictions of behaviour are developed through the process of deduction. These axioms of consumer choice are intended to give formal mathematical expres- sion to fundamental aspects of consumer behaviour and attitudes towards the objects of choice. Together, they formalise the view that the consumer can choose and that choices are consistent in a particular way. Formally, we represent the consumer’s preferences by a binary relation,  , defined on the consumption set, X. If x1  x2 , we say that ‘x1 is at least as good as x2 ’, for this consumer. That we use a binary relation to characterise preferences is significant and worth a moment’s reflection. It conveys the important point that, from the beginning, our theory requires relatively little of the consumer it describes. We require only that consumers make binary comparisons, that is, that they only examine two consumption plans at a time and make a decision regarding those two. The following axioms set forth basic criteria with which those binary comparisons must conform. AXIOM 1: Completeness. For all x1 and x2 in X, either x1  x2 or x2  x1. Axiom 1 formalises the notion that the consumer can make comparisons, that is, that he has the ability to discriminate and the necessary knowledge to evaluate alternatives. It says the consumer can examine any two distinct consumption plans x1 and x2 and decide whether x1 is at least as good as x2 or x2 is at least as good as x1. AXIOM 2: Transitivity. For any three elements x1 , x2 , and x3 in X, if x1  x2 and x2  x3 , then x1  x3. Axiom 2 gives a very particular form to the requirement that the consumer’s choices be consistent. Although we require only that the consumer be capable of comparing two 6 CHAPTER 1 alternatives at a time, the assumption of transitivity requires that those pairwise compar- isons be linked together in a consistent way. At first brush, requiring that the evaluation of alternatives be transitive seems simple and only natural. Indeed, were they not transitive, our instincts would tell us that there was something peculiar about them. Nonetheless, this is a controversial axiom. Experiments have shown that in various situations, the choices of real human beings are not always transitive. Nonetheless, we will retain it in our description of the consumer, though not without some slight trepidation. These two axioms together imply that the consumer can completely rank any finite number of elements in the consumption set, X, from best to worst, possibly with some ties. (Try to prove this.) We summarise the view that preferences enable the consumer to con- struct such a ranking by saying that those preferences can be represented by a preference relation. DEFINITION 1.1 Preference Relation The binary relation  on the consumption set X is called a preference relation if it satisfies Axioms 1 and 2. There are two additional relations that we will use in our discussion of consumer preferences. Each is determined by the preference relation,  , and they formalise the notions of strict preference and indifference. DEFINITION 1.2 Strict Preference Relation The binary relation  on the consumption set X is defined as follows: x1  x 2 if and only if x1  x 2 and x2  x1. The relation  is called the strict preference relation induced by  , or simply the strict preference relation when  is clear. The phrase x1  x2 is read, ‘x1 is strictly preferred to x2 ’. DEFINITION 1.3 Indifference Relation The binary relation ∼ on the consumption set X is defined as follows: x1 ∼ x 2 if and only if x1  x 2 and x2  x1. The relation ∼ is called the indifference relation induced by  , or simply the indifference relation when  is clear. The phrase x1 ∼ x2 is read, ‘x1 is indifferent to x2 ’. Building on the underlying definition of the preference relation, both the strict prefer- ence relation and the indifference relation capture the usual sense in which the terms ‘strict preference’ and ‘indifference’ are used in ordinary language. Because each is derived from CONSUMER THEORY 7 the preference relation, each can be expected to share some of its properties. Some, yes, but not all. In general, both are transitive and neither is complete. Using these two supplementary relations, we can establish something very concrete about the consumer’s ranking of any two alternatives. For any pair x1 and x2 , exactly one of three mutually exclusive possibilities holds: x1  x2 , or x2  x1 , or x1 ∼ x2. To this point, we have simply managed to formalise the requirement that prefer- ences reflect an ability to make choices and display a certain kind of consistency. Let us consider how we might describe graphically a set of preferences satisfying just those first few axioms. To that end, and also because of their usefulness later on, we will use the preference relation to define some related sets. These sets focus on a single alternative in the consumption set and examine the ranking of all other alternatives relative to it. DEFINITION 1.4 Sets in X Derived from the Preference Relation Let x0 be any point in the consumption set, X. Relative to any such point, we can define the following subsets of X: 1.  (x0 ) ≡ {x | x ∈ X, x  x0 }, called the ‘at least as good as’ set. 2.  (x0 ) ≡ {x | x ∈ X, x0  x}, called the ‘no better than’ set. 3. ≺ (x0 ) ≡ {x | x ∈ X, x0  x}, called the ‘worse than’ set. 4.  (x0 ) ≡ {x | x ∈ X, x  x0 }, called the ‘preferred to’ set. 5. ∼ (x0 ) ≡ {x | x ∈ X, x ∼ x0 }, called the ‘indifference’ set. A hypothetical set of preferences satisfying Axioms 1 and 2 has been sketched in Fig. 1.1 for X = R2+. Any point in the consumption set, such as x0 = (x10 , x20 ), represents a consumption plan consisting of a certain amount x10 of commodity 1, together with a certain amount x20 of commodity 2. Under Axiom 1, the consumer is able to compare x0 with any and every other plan in X and decide whether the other is at least as good as x0 or whether x0 is at least as good as the other. Given our definitions of the various sets relative to x0 , Axioms 1 and 2 tell us that the consumer must place every point in X into Figure 1.1. Hypothetical preferences satisfying Axioms 1 and 2. 8 CHAPTER 1 one of three mutually exclusive categories relative to x0 ; every other point is worse than x0 , indifferent to x0 , or preferred to x0. Thus, for any bundle x0 the three sets ≺ (x0 ), ∼ (x0 ), and  (x0 ) partition the consumption set. The preferences in Fig. 1.1 may seem rather odd. They possess only the most limited structure, yet they are entirely consistent with and allowed for by the first two axioms alone. Nothing assumed so far prohibits any of the ‘irregularities’ depicted there, such as the ‘thick’ indifference zones, or the ‘gaps’ and ‘curves’ within the indifference set ∼ (x0 ). Such things can be ruled out only by imposing additional requirements on preferences. We shall consider several new assumptions on preferences. One has very little behavioural significance and speaks almost exclusively to the purely mathematical aspects of representing preferences; the others speak directly to the issue of consumer tastes over objects in the consumption set. The first is an axiom whose only effect is to impose a kind of topological regularity on preferences, and whose primary contribution will become clear a bit later. From now on we explicitly set X = Rn+. AXIOM 3: Continuity. For all x ∈ Rn+ , the ‘at least as good as’ set,  (x), and the ‘no better than’ set,  (x), are closed in Rn+. Recall that a set is closed in a particular domain if its complement is open in that domain. Thus, to say that  (x) is closed in Rn+ is to say that its complement, ≺ (x), is open in Rn+. The continuity axiom guarantees that sudden preference reversals do not occur. Indeed, the continuity axiom can be equivalently expressed by saying that if each element yn of a sequence of bundles is at least as good as (no better than) x, and yn converges to y, then y is at least as good as (no better than) x. Note that because  (x) and  (x) are closed, so, too, is ∼ (x) because the latter is the intersection of the former two. Consequently, Axiom 3 rules out the open area in the indifference set depicted in the north-west of Fig. 1.1. Additional assumptions on tastes lend the greater structure and regularity to prefer- ences that you are probably familiar with from earlier economics classes. Assumptions of this sort must be selected for their appropriateness to the particular choice problem being analysed. We will consider in turn a few key assumptions on tastes that are ordinarily imposed in ‘standard’ consumer theory, and seek to understand the individual and collec- tive contributions they make to the structure of preferences. Within each class of these assumptions, we will proceed from the less restrictive to the more restrictive. We will generally employ the more restrictive versions considered. Consequently, we let axioms with primed numbers indicate alternatives to the norm, which are conceptually similar but slightly less restrictive than their unprimed partners. When representing preferences over ordinary consumption goods, we will want to express the fundamental view that ‘wants’ are essentially unlimited. In a very weak sense, we can express this by saying that there will always exist some adjustment in the compo- sition of the consumer’s consumption plan that he can imagine making to give himself a consumption plan he prefers. This adjustment may involve acquiring more of some com- modities and less of others, or more of all commodities, or even less of all commodities. CONSUMER THEORY 9 By this assumption, we preclude the possibility that the consumer can even imagine hav- ing all his wants and whims for commodities completely satisfied. Formally, we state this assumption as follows, where Bε (x0 ) denotes the open ball of radius ε centred at x0 :1 AXIOM 4’: Local Non-satiation. For all x0 ∈ Rn+ , and for all ε > 0, there exists some x ∈ Bε (x0 ) ∩ Rn+ such that x  x0. Axiom 4 says that within any vicinity of a given point x0 , no matter how small that vicinity is, there will always be at least one other point x that the consumer prefers to x0. Its effect on the structure of indifference sets is significant. It rules out the possibility of having ‘zones of indifference’, such as that surrounding x1 in Fig. 1.2. To see this, note that we can always find some ε > 0, and some Bε (x1 ), containing nothing but points indifferent to x1. This of course violates Axiom 4 , because it requires there always be at least one point strictly preferred to x1 , regardless of the ε > 0 we choose. The preferences depicted in Fig. 1.3 do satisfy Axiom 4 as well as Axioms 1 to 3. A different and more demanding view of needs and wants is very common. Accor- ding to this view, more is always better than less. Whereas local non-satiation requires Figure 1.2. Hypothetical preferences satisfying Axioms 1, 2, and 3. Figure 1.3. Hypothetical preferences satisfying Axioms 1, 2, 3, and 4. 1 See Definition A1.4 in the Mathematical Appendix. 10 CHAPTER 1 that a preferred alternative nearby always exist, it does not rule out the possibility that the preferred alternative may involve less of some or even all commodities. Specifically, it does not imply that giving the consumer more of everything necessarily makes that consumer better off. The alternative view takes the position that the consumer will always prefer a consumption plan involving more to one involving less. This is captured by the axiom of strict monotonicity. As a matter of notation, if the bundle x0 contains at least as much of every good as does x1 we write x0 ≥ x1 , while if x0 contains strictly more of every good than x1 we write x0 x1. AXIOM 4: Strict Monotonicity. For all x0 , x1 ∈ Rn+ , if x0 ≥ x1 then x0  x1 , while if x0 x1 , then x0  x1. Axiom 4 says that if one bundle contains at least as much of every commodity as another bundle, then the one is at least as good as the other. Moreover, it is strictly better if it contains strictly more of every good. The impact on the structure of indifference and related sets is again significant. First, it should be clear that Axiom 4 implies Axiom 4 , so if preferences satisfy Axiom 4, they automatically satisfy Axiom 4. Thus, to require Axiom 4 will have the same effects on the structure of indifference and related sets as Axiom 4 does, plus some additional ones. In particular, Axiom 4 eliminates the possibility that the indifference sets in R2+ ‘bend upward’, or contain positively sloped segments. It also requires that the ‘preferred to’ sets be ‘above’ the indifference sets and that the ‘worse than’ sets be ‘below’ them. To help see this, consider Fig. 1.4. Under Axiom 4, no points north-east of x0 or south-west of x0 may lie in the same indifference set as x0. Any point north-east, such as x1 , involves more of both goods than does x0. All such points in the north-east quadrant must therefore be strictly preferred to x0. Similarly, any point in the south-west quadrant, such as x2 , involves less of both goods. Under Axiom 4, x0 must be strictly preferred to x2 and to all other points in the south-west quadrant, so none of these can lie in the same indifference set as x0. For any x0 , points north-east of the indifference set will be contained in  (x0 ), and all those south-west of the indifference set will be contained in the set ≺ (x0 ). A set of preferences satisfying Axioms 1, 2, 3, and 4 is given in Fig. 1.5. Figure 1.4. Hypothetical preferences x2 satisfying Axioms 1, 2, 3, and 4. x1 x0 x2 x1 CONSUMER THEORY 11 Figure 1.5. Hypothetical preferences satisfying Axioms 1, 2, 3, and 4. The preferences in Fig. 1.5 are the closest we have seen to the kind undoubtedly familiar to you from your previous economics classes. They still differ, however, in one very important respect: typically, the kind of non-convex region in the north-west part of ∼ (x0 ) is explicitly ruled out. This is achieved by invoking one final assumption on tastes. We will state two different versions of the axiom and then consider their meaning and purpose. AXIOM 5’: Convexity. If x1  x0 , then tx1 + (1 − t)x0  x0 for all t ∈ [0, 1]. A slightly stronger version of this is the following: AXIOM 5: Strict Convexity. If x1  =x0 and x1  x0 , then tx1 + (1 − t)x0  x0 for all t ∈ (0, 1). Notice first that either Axiom 5 or Axiom 5 – in conjunction with Axioms 1, 2, 3, and 4 – will rule out concave-to-the-origin segments in the indifference sets, such as those in the north-west part of Fig. 1.5. To see this, choose two distinct points in the indifference set depicted there. Because x1 and x2 are both indifferent to x0 , we clearly have x1  x2. Convex combinations of those two points, such as xt , will lie within ≺ (x0 ), violating the requirements of both Axiom 5 and Axiom 5. For the purposes of the consumer theory we shall develop, it turns out that Axiom 5 can be imposed without any loss of generality. The predictive content of the theory would be the same with or without it. Although the same statement does not quite hold for the slightly stronger Axiom 5, it does greatly simplify the analysis. There are at least two ways we can intuitively understand the implications of con- vexity for consumer tastes. The preferences depicted in Fig. 1.6 are consistent with both Axiom 5 and Axiom 5. Again, suppose we choose x1 ∼ x2. Point x1 represents a bun- dle containing a proportion of the good x2 which is relatively ‘extreme’, compared to the proportion of x2 in the other bundle x2. The bundle x2 , by contrast, contains a propor- tion of the other good, x1 , which is relatively extreme compared to that contained in x1. Although each contains a relatively high proportion of one good compared to the other, the consumer is indifferent between the two bundles. Now, any convex combination of x1 and x2 , such as xt , will be a bundle containing a more ‘balanced’ combination of x1 12 CHAPTER 1 Figure 1.6. Hypothetical preferences satisfying x2 Axioms 1, 2, 3, 4, and 5 or 5. x1 xt x0 x2 x1 and x2 than does either ‘extreme’ bundle x1 or x2. The thrust of Axiom 5 or Axiom 5 is to forbid the consumer from preferring such extremes in consumption. Axiom 5 requires that any such relatively balanced bundle as xt be no worse than either of the two extremes between which the consumer is indifferent. Axiom 5 goes a bit further and requires that the consumer strictly prefer any such relatively balanced consumption bundle to both of the extremes between which he is indifferent. In either case, some degree of ‘bias’ in favour of balance in consumption is required of the consumer’s tastes. Another way to describe the implications of convexity for consumers’ tastes focuses attention on the ‘curvature’ of the indifference sets themselves. When X = R2+ , the (abso- lute value of the) slope of an indifference curve is called the marginal rate of substitution of good two for good one. This slope measures, at any point, the rate at which the con- sumer is just willing to give up good two per unit of good one received. Thus, the consumer is indifferent after the exchange. If preferences are strictly monotonic, any form of convexity requires the indifference curves to be at least weakly convex-shaped relative to the origin. This is equivalent to requiring that the marginal rate of substitution not increase as we move from bundles such as x1 towards bundles such as x2. Loosely, this means that the consumer is no more willing to give up x2 in exchange for x1 when he has relatively little x2 and much x1 than he is when he has relatively much x2 and little x1. Axiom 5 requires the rate at which the consumer would trade x2 for x1 and remain indifferent to be either constant or decreasing as we move from north-west to south-east along an indifference curve. Axiom 5 goes a bit further and requires that the rate be strictly diminishing. The preferences in Fig. 1.6 display this property, sometimes called the principle of diminishing marginal rate of substitution in consumption. We have taken some care to consider a number of axioms describing consumer pref- erences. Our goal has been to gain some appreciation of their individual and collective implications for the structure and representation of consumer preferences. We can sum- marise this discussion rather briefly. The axioms on consumer preferences may be roughly classified in the following way. The axioms of completeness and transitivity describe a consumer who can make consistent comparisons among alternatives. The axiom of conti- nuity is intended to guarantee the existence of topologically nice ‘at least as good as’ and CONSUMER THEORY 13 ‘no better than’ sets, and its purpose is primarily a mathematical one. All other axioms serve to characterise consumers’ tastes over the objects of choice. Typically, we require that tastes display some form of non-satiation, either weak or strong, and some bias in favour of balance in consumption, either weak or strong. 1.2.2 THE UTILITY FUNCTION In modern theory, a utility function is simply a convenient device for summarising the information contained in the consumer’s preference relation – no more and no less. Sometimes it is easier to work directly with the preference relation and its associated sets. Other times, especially when one would like to employ calculus methods, it is easier to work with a utility function. In modern theory, the preference relation is taken to be the primitive, most fundamental characterisation of preferences. The utility function merely ‘represents’, or summarises, the information conveyed by the preference relation. A utility function is defined formally as follows. DEFINITION 1.5 A Utility Function Representing the Preference Relation  A real-valued function u : Rn+ → R is called a utility function representing the preference relation  , if for all x0 , x1 ∈ Rn+ , u(x0 ) ≥ u(x1 )⇐⇒x0  x1. Thus a utility function represents a consumer’s preference relation if it assigns higher numbers to preferred bundles. A question that earlier attracted a great deal of attention from theorists concerned properties that a preference relation must possess to guarantee that it can be represented by a continuous real-valued function. The question is important because the analysis of many problems in consumer theory is enormously simplified if we can work with a utility function, rather than with the preference relation itself. Mathematically, the question is one of existence of a continuous utility function rep- resenting a preference relation. It turns out that a subset of the axioms we have considered so far is precisely that required to guarantee existence. It can be shown that any binary relation that is complete, transitive, and continuous can be represented by a continuous real-valued utility function.2 (In the exercises, you are asked to show that these three axioms are necessary for such a representation as well.) These are simply the axioms that, together, require that the consumer be able to make basically consistent binary choices and that the preference relation possess a certain amount of topological ‘regularity’. In partic- ular, representability does not depend on any assumptions about consumer tastes, such as convexity or even monotonicity. We can therefore summarise preferences by a continuous utility function in an extremely broad range of problems. Here we will take a detailed look at a slightly less general result. In addition to the three most basic axioms mentioned before, we will impose the extra requirement that preferences be strictly monotonic. Although this is not essential for representability, to 2 See, for example, Barten and Böhm (1982). The classic reference is Debreu (1954). 14 CHAPTER 1 require it simultaneously simplifies the purely mathematical aspects of the problem and increases the intuitive content of the proof. Notice, however, that we will not require any form of convexity. THEOREM 1.1 Existence of a Real-Valued Function Representing the Preference Relation  If the binary relation  is complete, transitive, continuous, and strictly monotonic, there exists a continuous real-valued function, u : Rn+ →R, which represents . Notice carefully that this is only an existence theorem. It simply claims that under the conditions stated, at least one continuous real-valued function representing the preference relation is guaranteed to exist. There may be, and in fact there always will be, more than one such function. The theorem itself, however, makes no statement on how many more there are, nor does it indicate in any way what form any of them must take. Therefore, if we can dream up just one function that is continuous and that represents the given preferences, we will have proved the theorem. This is the strategy we will adopt in the following proof. Proof: Let the relation  be complete, transitive, continuous, and strictly monotonic. Let e ≡ (1,... , 1) ∈ Rn+ be a vector of ones, and consider the mapping u : Rn+ →R defined so that the following condition is satisfied:3 u(x)e ∼ x. (P.1) Let us first make sure we understand what this says and how it works. In words, (P.1) says, ‘take any x in the domain Rn+ and assign to it the number u(x) such that the bundle, u(x)e, with u(x) units of every commodity is ranked indifferent to x’. Two questions immediately arise. First, does there always exist a number u(x) satisfying (P.1)? Second, is it uniquely determined, so that u(x) is a well-defined function? To settle the first question, fix x ∈ Rn+ and consider the following two subsets of real numbers: A ≡ {t ≥ 0 | te  x} B ≡ {t ≥ 0 | te  x}. Note that if t∗ ∈ A ∩ B, then t∗ e ∼ x, so that setting u(x) = t∗ would satisfy (P.1). Thus, the first question would be answered in the affirmative if we show that A ∩ B is guaranteed to be non-empty. This is precisely what we shall show. 3 For t ≥ 0, the vector te will be some point in Rn+ each of whose coordinates is equal to the number t, because te = t(1,... , 1) = (t,... , t). If t = 0, then te = (0,... , 0) coincides with the origin. If t = 1, then te = (1,... , 1) coincides with e. If t > 1, the point te lies farther out from the origin than e. For 0 0, for some i = 1,... , n. Because pi > 0 for all i, it is clear from (1.7) that the Lagrangian multiplier will be strictly positive at the solution, because λ∗ = ui (x∗ )/pi > 0. Consequently, for all j, ∂u(x∗ )/∂xj = λ∗ pj > 0, so marginal utility is proportional to price for all goods at the optimum. Alternatively, for any two goods j and k, we can combine the conditions to conclude that ∂u(x∗ )/∂xj pj =. (1.11) ∂u(x∗ )/∂xk pk This says that at the optimum, the marginal rate of substitution between any two goods must be equal to the ratio of the goods’ prices. In the two-good case, conditions (1.10) therefore require that the slope of the indifference curve through x∗ be equal to the slope of the budget constraint, and that x∗ lie on, rather than inside, the budget line, as in Fig. 1.10 and Fig. 1.11(a). In general, conditions (1.10) are merely necessary conditions for a local optimum (see the end of Section A2.3). However, for the particular problem at hand, these necessary first-order conditions are in fact sufficient for a global optimum. This is worthwhile stating formally. THEOREM 1.4 Sufficiency of Consumer’s First-Order Conditions Suppose that u(x) is continuous and quasiconcave on Rn+ , and that (p, y) 0. If u is differentiable at x∗ , and (x∗ , λ∗ ) 0 solves (1.10), then x∗ solves the consumer’s maximisation problem at prices p and income y. Proof: We shall employ the following fact that you are asked to prove in Exercise 1.28: For all x, x1 ≥ 0, because u is quasiconcave, ∇u(x)(x1 − x) ≥ 0 whenever u(x1 ) ≥ u(x) and u is differentiable at x. Now, suppose that ∇u(x∗ ) exists and (x∗ , λ∗ ) 0 solves (1.10). Then ∇u(x∗ ) = λ∗ p, (P.1) ∗ p · x = y. (P.2) If x∗ is not utility-maximising, then there must be some x0 ≥ 0 such that u(x0 ) > u(x∗ ), p · x0 ≤ y. CONSUMER THEORY 25 Because u is continuous and y > 0, the preceding inequalities imply that u(tx0 ) > u(x∗ ), (P.3) p · tx < y. 0 (P.4) for some t ∈ [0, 1] close enough to one. Letting x1 = tx0 , we then have ∇u(x∗ )(x1 − x∗ ) = (λ∗ p) · (x1 − x∗ ) = λ∗ (p · x1 − p · x∗ ) < λ∗ (y − y) = 0, where the first equality follows from (P.1), and the second inequality follows from (P.2) and (P.4). However, because by (P.3) u(x1 ) > u(x∗ ), (P.5) contradicts the fact set forth at the beginning of the proof. With this sufficiency result in hand, it is enough to find a solution (x∗ , λ∗ ) 0 to (1.10). Note that (1.10) is a system of n + 1 equations in the n + 1 unknowns x1∗ ,... , xn∗ , λ∗. These equations can typically be used to solve for the demand functions xi (p, y), i = 1,... , n, as we show in the following example. ρ ρ EXAMPLE 1.1 The function, u(x1 , x2 ) = (x1 + x2 )1/ρ , where 0 =ρ 0. If 0 u is twice continuously differentiable on Rn++ , ∂u(x∗ )/∂xi > 0 for some i = 1,... , n, and the bordered Hessian of u has a non-zero determinant at x∗ , then x(p, y) is differentiable at (p0 , y0 ). 28 CHAPTER 1 1.4 INDIRECT UTILITY AND EXPENDITURE 1.4.1 THE INDIRECT UTILITY FUNCTION The ordinary utility function, u(x), is defined over the consumption set X and represents the consumer’s preferences directly, as we have seen. It is therefore referred to as the direct utility function. Given prices p and income y, the consumer chooses a utility- maximising bundle x(p, y). The level of utility achieved when x(p, y) is chosen thus will be the highest level permitted by the consumer’s budget constraint facing prices p and income y. Different prices or incomes, giving different budget constraints, will generally give rise to different choices by the consumer and so to different levels of maximised utility. The relationship among prices, income, and the maximised value of utility can be summarised by a real-valued function v: Rn+ × R+ → R defined as follows: v(p, y) = maxn u(x) s.t. p · x ≤ y. (1.12) x∈R+ The function v(p, y) is called the indirect utility function. It is the maximum-value function corresponding to the consumer’s utility maximisation problem. When u(x) is continuous, v(p, y) is well-defined for all p 0 and y≥0 because a solution to the maximi- sation problem (1.12) is guaranteed to exist. If, in addition, u(x) is strictly quasiconcave, then the solution is unique and we write it as x(p, y), the consumer’s demand function. The maximum level of utility that can be achieved when facing prices p and income y therefore will be that which is realised when x(p, y) is chosen. Hence, v(p, y) = u(x(p, y)). (1.13) Geometrically, we can think of v(p, y) as giving the utility level of the highest indifference curve the consumer can reach, given prices p and income y, as illustrated in Fig. 1.13. x2 y/p 2 x (p, y) p 1/p 2 u  v (p,y) x1 y/p 1 Figure 1.13. Indirect utility at prices p and income y. CONSUMER THEORY 29 There are several properties that the indirect utility function will possess. Continuity of the constraint function in p and y is sufficient to guarantee that v(p, y) will be contin- uous in p and y on Rn++ ×R+. (See Section A2.4.) Effectively, the continuity of v(p, y) follows because at positive prices, ‘small changes’ in any of the parameters (p, y) fixing the location of the budget constraint will only lead to ‘small changes’ in the maximum level of utility the consumer can achieve. In the following theorem, we collect together a number of additional properties of v(p, y). THEOREM 1.6 Properties of the Indirect Utility Function If u(x) is continuous and strictly increasing on Rn+ , then v(p, y) defined in (1.12) is 1. Continuous on Rn++ ×R+ , 2. Homogeneous of degree zero in (p, y), 3. Strictly increasing in y, 4. Decreasing in p, 5. Quasiconvex in (p, y). Moreover, it satisfies 6. Roy’s identity: If v(p, y) is differentiable at (p0 , y0 ) and ∂v(p0 , y0 )/∂y =0, then ∂v(p0 , y0 )/∂pi xi (p0 , y0 ) = − , i = 1,... , n. ∂v(p0 , y0 )/∂y Proof: Property 1 follows from Theorem A2.21 (the theorem of the maximum). We shall not pursue the details. The second property is easy to prove. We must show that v(p, y) = v(tp, ty) for all t > 0. But v(tp, ty) = [max u(x) s.t. tp · x ≤ ty], which is clearly equivalent to [max u(x) s.t. p · x ≤ y] because we may divide both sides of the constraint by t > 0 without affecting the set of bundles satisfying it. (See Fig. 1.14.) Consequently, v(tp, ty) = [max u(x) s.t. p · x ≤ y] = v(p, y). Intuitively, properties 3 and 4 simply say that any relaxation of the consumer’s bud- get constraint can never cause the maximum level of achievable utility to decrease, whereas any tightening of the budget constraint can never cause that level to increase. To prove 3 (and to practise Lagrangian methods), we shall make some additional assumptions although property 3 can be shown to hold without them. To keep things simple, we’ll assume for the moment that the solution to (1.12) is strictly positive and differentiable, where (p, y) 0 and that u(·) is differentiable with ∂u(x)/∂xi > 0, for all x 0. As we have remarked before, because u(·) is strictly increasing, the constraint in (1.12) must bind at the optimum. Consequently, (1.12) is equivalent to v(p, y) = maxn u(x) s.t. p · x = y. (P.1) x∈R+ 30 CHAPTER 1 x2 ty/tp2 = y/p 2 tp1/tp 2  p 1/p 2 v(tp, ty)  v(p, y) x1 ty/tp1 = y/p 1 Figure 1.14. Homogeneity of the indirect utility function in prices and income. The Lagrangian for (P.1) is L(x, λ) = u(x) − λ(p · x − y). (P.2) Now, for (p, y) 0, let x∗ = x(p, y) solve (P.1). By our additional assumption, x∗ 0, so we may apply Lagrange’s theorem to conclude that there is a λ∗ ∈ R such that ∂ L(x∗ , λ∗ ) ∂u(x∗ ) = − λ∗ pi = 0, i = 1,... , n. (P.3) ∂xi ∂xi Note that because both pi and ∂u(x∗ )/∂xi are positive, so, too, is λ∗. Our additional differentiability assumptions allow us to now apply Theorem A2.22, the Envelope theorem, to establish that v(p, y) is strictly increasing in y. According to the Envelope theorem, the partial derivative of the maximum value function v(p, y) with respect to y is equal to the partial derivative of the Lagrangian with respect to y evaluated at (x∗ , λ∗ ), ∂v(p, y) ∂ L(x∗ , λ∗ ) = = λ∗ > 0. (P.4) ∂y ∂y Thus, v(p, y) is strictly increasing in y > 0. So, because v is continuous, it is then strictly increasing on y ≥ 0. For property 4, one can also employ the Envelope theorem. However, we shall give a more elementary proof that does not rely on any additional hypotheses. So con- sider p0 ≥ p1 and let x0 solve (1.12) when p = p0. Because x0 ≥ 0, (p0 − p1 ) · x0 ≥ 0. Hence, p1 ·x0 ≤ p0 ·x0 ≤ y, so that x0 is feasible for (1.12) when p = p1. We conclude that v(p1 , y) ≥ u(x0 ) = v(p0 , y), as desired. Property 5 says that a consumer would prefer one of any two extreme budget sets to any average of the two. Our concern is to show that v(p, y) is quasiconvex in the vector of prices and income (p, y). The key to the proof is to concentrate on the budget sets. CONSUMER THEORY 31 Let B1 , B2 , and Bt be the budget sets available when prices and income are (p1 , y1 ), (p2 , y2 ),and (pt , yt ), respectively, where pt ≡ tp1 + (1 − t)p2 and yt ≡ y1 + (1 − t)y2. Then, B1 = {x | p1 · x ≤ y1 }, B2 = {x | p2 · x ≤ y2 }, Bt = {x | pt · x ≤ yt }. Suppose we could show that every choice the consumer can possibly make when he faces budget Bt is a choice that could have been made when he faced either budget B1 or budget B2. It then would be the case that every level of utility he can achieve facing Bt is a level he could have achieved either when facing B1 or when facing B2. Then, of course, the maximum level of utility that he can achieve over Bt could be no larger than at least one of the following: the maximum level of utility he can achieve over B1 , or the maximum level of utility he can achieve over B2. But if this is the case, then the maximum level of utility achieved over Bt can be no greater than the largest of these two. If our supposition is correct, therefore, we would know that v(pt , yt ) ≤ max[v(p1 , y1 ), v(p2 , y2 )] ∀ t ∈ [0, 1]. This is equivalent to the statement that v(p, y) is quasiconvex in (p, y). It will suffice, then, to show that our supposition on the budget sets is correct. We want to show that if x ∈ Bt , then x ∈ B1 or x ∈ B2 for all t ∈ [0, 1]. If we choose either extreme value for t, Bt coincides with either B1 or B2 , so the relations hold trivially. It remains to show that they hold for all t ∈ (0, 1). Suppose it were not true. Then we could find some t ∈ (0, 1) and some x ∈ Bt such that x∈B / 1 and x∈B/ 2. If x∈B / 1 and x∈B / 2 , then p1 ·x > y1 and p2 ·x > y2 , respectively. Because t ∈ (0, 1), we can multiply the first of these by t, the second by (1 − t), and preserve the inequalities to obtain tp1 ·x > ty1 and (1 − t)p2 · x > (1 − t)y2. Adding, we obtain (tp1 + (1 − t)p2 ) · x > ty1 + (1 − t)y2 32 CHAPTER 1 or pt ·x > yt. But this final line says that x∈B / t , contradicting our original assumption. We must conclude, therefore, that if x ∈ B , then x ∈ B1 or x ∈ B2 for all t ∈ [0, 1]. By our previous argument, t we can conclude that v(p, y) is quasiconvex in (p, y). Finally, we turn to property 6, Roy’s identity. This says that the consumer’s Marshallian demand for good i is simply the ratio of the partial derivatives of indirect utility with respect to pi and y after a sign change. (Note the minus sign in 6.) We shall again invoke the additional assumptions introduced earlier in the proof because we shall again employ the Envelope theorem. (See Exercise 1.35 for a proof that does not require these additional assumptions.) Letting x∗ = x(p, y) be the strictly positive solution to (1.12), as argued earlier, there must exist λ∗ satisfying (P.3). Applying the Envelope theorem to evaluate ∂v(p, y)/∂pi gives ∂v(p, y) ∂ L(x∗ , λ∗ ) = = −λ∗ xi∗. (P.5) ∂pi ∂pi However, according to (P.4), λ∗ = ∂v(p, y)/∂y > 0. Hence, (P.5) becomes ∂v(p, y)/∂pi − = xi∗ = xi (p, y), ∂v(p, y)/∂y as desired. EXAMPLE 1.2 In Example 1.1, the direct utility function is the CES form, u(x1 , x2 ) = ρ ρ (x1 + x2 )1/ρ , where 0 =ρ 0, v(tp, ty) = ty((tp1 )r + (tp2 )r )−1/r  −1/r = ty tr pr1 + tr pr2  −1/r = tyt−1 pr1 + pr2  −1/r = y pr1 + pr2 = v(p, y). To see that it is increasing in y and decreasing in p, differentiate (E.2) with respect to income and any price to obtain ∂v(p, y)  r −1/r = p1 + pr2 > 0, (E.3) ∂y ∂v(p, y)  (−1/r)−1 r−1 = − pr1 + pr2 ypi < 0, i = 1, 2. (E.4) ∂pi To verify Roy’s identity, form the required ratio of (E.4) to (E.3) and recall (E.1) to obtain    (−1/r)−1 r−1 ∂v(p, y)/∂pi − pr1 + pr2 ypi (−1) = (−1)  r  −1/r ∂v(p, y)/∂y p1 + pr2 ypr−1 i = = xi (p, y), i = 1, 2. pr1 + pr2 We leave as an exercise the task of verifying that (E.2) is a quasiconvex function of (p, y). 1.4.2 THE EXPENDITURE FUNCTION The indirect utility function is a neat and powerful way to summarise a great deal about the consumer’s market behaviour. A companion measure, called the expenditure function, is equally useful. To construct the indirect utility function, we fixed market prices and 34 CHAPTER 1 x2 u e*/p2 e3/p2 x2h(p, u) xh p 1/p 2 u x1 x1h(p, u) e3/p 1 e*/p 1 e1/p 1 e2/p 1 Figure 1.15. Finding the lowest level of expenditure to achieve utility level u. income, and sought the maximum level of utility the consumer could achieve. To construct the expenditure function, we again fix prices, but we ask a different sort of question about the level of utility the consumer achieves. Specifically, we ask: what is the minimum level of money expenditure the consumer must make facing a given set of prices to achieve a given level of utility? In this construction, we ignore any limitations imposed by the consumer’s income and simply ask what the consumer would have to spend to achieve some particular level of utility. To better understand the type of problem we are studying, consider Fig. 1.15 and contrast it with Fig. 1.13. Each of the parallel straight lines in Fig. 1.15 depicts all bundles x that require the same level of total expenditure to acquire when facing prices p = (p1 , p2 ). Each of these isoexpenditure curves is defined implicity by e = p1 x1 + p2 x2 , for a dif- ferent level of total expenditure e > 0. Each therefore will have the same slope, −p1 /p2 , but different horizontal and vertical intercepts, e/p1 and e/p2 , respectively. Isoexpenditure curves farther out contain bundles costing more; those farther in give bundles costing less. If we fix the level of utility at u, then the indifference curve u(x) = u gives all bundles yielding the consumer that same level of utility. There is no point in common with the isoexpenditure curve e3 and the indiffer- ence curve u, indicating that e3 dollars is insufficient at these prices to achieve utility u. However, each of the curves e1 , e2 , and e∗ has at least one point in common with u, indi- cating that any of these levels of total expenditure is sufficient for the consumer to achieve utility u. In constructing the expenditure function, however, we seek the minimum expen- diture the consumer requires to achieve utility u, or the lowest possible isoexpenditure curve that still has at least one point in common with indifference curve u. Clearly, that will be level e∗ , and the least cost bundle that achieves utility u at prices p will be the bun- dle xh = (x1h (p, u), x2h (p, u)). If we denote the minimum expenditure necessary to achieve utility u at prices p by e(p, u), that level of expenditure will simply be equal to the cost of bundle xh , or e(p, u) = p1 x1h (p, u) + p2 x2h (p, u) = e∗. CONSUMER THEORY 35 More generally, we define the expenditure function as the minimum-value function, e(p, u) ≡ minn p · x s.t. u(x) ≥ u (1.14) x∈R+ for all p 0 and all attainable utility levels u. For future reference, let U = {u(x) | x ∈ Rn+ } denote the set of attainable utility levels. Thus, the domain of e(·) is Rn++ ×U. Note that e(p, u) is well-defined because for p ∈ Rn++ , x ∈ Rn+ , p · x ≥ 0. Hence, the set of numbers {e|e = p · x for some x with u(x) ≥ u} is bounded below by zero. Moreover because p 0, this set can be shown to be closed. Hence, it contains a smallest number. The value e(p, u) is precisely this smallest number. Note that any solution vector for this minimisation problem will be non-negative and will depend on the parameters p and u. Notice also that if u(x) is continuous and strictly quasiconcave, the solution will be unique, so we can denote the solution as the function xh (p, u) ≥ 0. As we have seen, if xh (p, u) solves this problem, the lowest expenditure necessary to achieve utility u at prices p will be exactly equal to the cost of the bundle xh (p, u), or e(p, u) = p · xh (p, u). (1.15) We have seen how the consumer’s utility maximisation problem is intimately related to his observable market demand behaviour. Indeed, the very solutions to that problem – the Marshallian demand functions – tell us just how much of every good we should observe the consumer buying when he faces different prices and income. We shall now interpret the solution, xh (p, u), of the expenditure-minimisation problem as another kind of ‘demand function’ – but one that is not directly observable. Consider the following mental experiment. If we fix the level of utility the consumer is permitted to achieve at some arbitrary level u, how will his purchases of each good behave as we change the prices he faces? The kind of ‘demand functions’ we are imagin- ing here are thus utility-constant ones. We completely ignore the level of the consumer’s money income and the utility levels he actually can achieve. In fact, we know that when a consumer has some level of income and we change the prices he faces, there will ordinarily be some change in his purchases and some corresponding change in the level of utility he achieves. To imagine how we might then construct our hypothetical demand functions, we must imagine a process by which whenever we lower some price, and so confer a utility gain on the consumer, we compensate by reducing the consumer’s income, thus conferring a corresponding utility loss sufficient to bring the consumer back to the original level of utility. Similarly, whenever we increase some price, causing a utility loss, we must imag- ine compensating for this by increasing the consumer’s income sufficiently to give a utility gain equal to the loss. Because they reflect the net effect of this process by which we match any utility change due to a change in prices by a compensating utility change from a hypothetical adjustment in income, the hypothetical demand functions we are describing are often called compensated demand functions. However, because John Hicks (1939) was the first to write about them in quite this way, these hypothetical demand functions are most commonly known as Hicksian demand functions. As we illustrate below, the 36 CHAPTER 1 x2 x 2h(p 10, p 20, u) p 10/p 20 1 0 h 1 0 p 1/p 2 x 2 (p 1 , p 2 , u) u x1 h 0 x 1 (p 1, p 20, u) x 1h(p 11, p 20, u) (a) p1 p 10 p 11 h 0 x 1 (p 1, p 2 , u) x1 h 0 0 h 1 0 x 1 (p 1, p 2 , u) x 1 (p 1, p 2, u) (b) Figure 1.16. The Hicksian demand for good 1. solution, xh (p, u), to the expenditure-minimisation problem is precisely the consumer’s vector of Hicksian demands. To get a clearer idea of what we have in mind, consider Fig. 1.16. If we wish to fix the level of utility the consumer can achieve at u in Fig. 1.16(a) and then confront him with prices p01 and p02 , he must face the depicted budget constraint with slope −p01 /p02. Note that his utility-maximising choices then coincide with the expenditure-minimising quantities x1h (p01 , p02 , u) and x2h (p01 , p02 , u). If we reduce the price of good 1 to p11 u(0), and that u(·) is differentiable with ∂u(x)/∂xi > 0, ∀ i on Rn++. Now, because u(·) is continuous and strictly increasing, and p 0, the constraint in (1.14) must be binding. For if u(x1 ) > u, there is a t ∈ (0, 1) close enough to 1 such that u(tx1 ) > u. Moreover, u ≥ u(0) implies u(x1 ) > u(0), so that x1  =0. Therefore, p · (tx1 )< p · x1 , because p · x1 > 0. Consequently, when the constraint is not binding, there is a strictly cheaper bundle that also satisfies the constraint. Hence, at the optimum, the constraint must bind. Consequently, we may write (1.14) instead as e(p, u) ≡ minn p · x s.t. u(x) = u. (P.1) x∈R+ 38 CHAPTER 1 The Lagrangian for this problem is L(x, λ) = p · x − λ[u(x) − u]. (P.2) Now for p 0 and u > u(0), we have that x∗ = xh (p, u) 0 solves (P.1). So, by Lagrange’s theorem, there is a λ∗ such that ∂ L(x∗ , λ∗ ) ∂u(x∗ ) = pi − λ∗ = 0, i = 1,... , n. (P.3) ∂xi ∂xi Note then that because pi and ∂u(x∗ )/∂xi are positive, so, too, is λ∗. Under our addi- tional hypotheses, we can now use the Envelope theorem to show that e(p, u) is strictly increasing in u. By the Envelope theorem, the partial derivative of the minimum-value function e(p, u) with respect to u is equal to the partial derivative of the Lagrangian with respect to u, evaluated at (x∗ , λ∗ ). Hence, ∂e(p, u) ∂ L(x∗ , λ∗ ) = = λ∗ > 0. ∂u ∂u Because this holds for all u > u(0), and because e(·) is continuous, we may conclude that for all p 0, e(p, u) is strictly increasing in u on U (which includes u(0)). That e is unbounded in u can be shown to follow from the fact that u(x) is continuous and strictly increasing. You are asked to do so in Exercise 1.34. Because property 4 follows from property 7, we shall defer it for the moment. Property 5 will be left as an exercise. For property 6, we must prove that e(p, u) is a concave function of prices. We begin by recalling the definition of concavity. Let p1 and p2 be any two positive price vectors, let t ∈ [0, 1], and let pt = tp1 + (1 − t)p2 be any convex combination of p1 and p2. Then the expenditure function will be concave in prices if te(p1 , u) + (1 − t)e(p2 , u) ≤ e(pt , u). (P.4) To see that this is indeed the case, simply focus on what it means for expenditure to be minimised at given prices. Suppose in particular that x1 minimises expenditure to achieve u when prices are p1 , that x2 minimises expenditure to achieve u when prices are p2 , and that x∗ minimises expenditure to achieve u when prices are pt. Then the cost of x1 at prices p1 must be no more than the cost at prices p1 of any other bundle x that achieves utility u. Similarly, the cost of x2 at prices p2 must be no more than the cost at p2 of any other bundle x that achie