Lecture Notes Nov 26 2008 Econ610 Fall2008

Lecture Notes on Microeconomic Theory ∗

Takashi Kunimoto†

Department of EconomicsMcGill University

First Version: December 2005This Version: March 25, 2009

Abstract. The theme of this note is economics of resource allocations in perfectly com-petitive markets. The topics which will be covered include the theories of consumer and thefirm, competitive equilibrium of an economy and its properties (existence, local and globaluniqueness, the Sonnenschein-Mantel-Debreu Theorem, the first and second welfare theo-rems, the core), decision making under uncertainty , and consumer and producer surplus.Finally, I extend all the analyses to economies under uncertainty in which the Arrow-Debreucontingent markets, sequential markets, and their (non-)equivalence are discussed.

∗I am thankful to the students for their comments, questions, and suggestions. Yet, I believethat there are still many errors in this manuscript. Of course, all remaining ones are my own.

†Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec,H3A2T7, CANADA, [email protected]: http://people.mcgill.ca/takashi.kunimoto/

Syllabus

Econ 610: Microeconomic Theory I

Fall 2008, McGill UniversityTuesdays and Thursdays, 1:05pm - 2:25pm; at Leacock 15

Instructor: Takashi KunimotoEmail: [email protected] 3

Class Web: the WebCTOffice: Leacock 438

READING:

1. “Advanced Microeconomic Theory Second Edition,” by Geoffrey A. Jehle andPhilip J. Reny, Addison Wesley, 2000 (This is the main textbook. I abbre-viate “AMT” for this book.)

2. “Microeconomic Analysis Third Edition,” by Hal R. Varian, W.W. Norton andCompany, 1992 (This is a supplementary text book. I abbreviate “Varian” forthis book.)

3. “Microeconomic Theory,” by Andreu Mas-Colell, Michael D. Whinston, andJerry R. Green, Oxford University Press, 1995. (This is also a supplementarytextbook. I abbreviate “MWG” for this book.)

4. “Lecture Notes on Microeconomic Theory,” by Takashi Kunimoto, 2007. (Thisis the main content of the course. This note is based largely on AMT andpartly on MWG. You can find it on the WebCT. Note also that this note willhave been continuously updated in the course until the end of the course.)

5. “Lecture Notes for “Econ633 Mathematics for Economists,” by Takashi Kuni-moto, 2007. (This is the lecture note on mathematics which was taught in thissummer. You can find it on the WebCT).

All the above three books, AMT, Varian, and MWG should be available in thereserve desk at the library. I also assume that you have knowledge on mathematicscovered by my lecture notes on mathematics. If you don’t, please study them duringthe semester. In any case, it is your responsibility to understand what is requiredfor this course by checking the lecture notes on mathematics at the WebCT.

TA: Maryam Esmaeilpour

3In the semester, I might send emails to all students through the WebCT. But, do not email (orreply to) me through the WebCT. You should directly use [email protected] to contactme.

1

COURSE DESCRIPTION:

The theme of Econ 610 is economics of resource allocations in perfectly compet-itive markets. The topics which will be covered include the theories of consumer(Chapters 1 and 2 of AMT, Chapters 7, 8, and 9 of Varian, and Chapters 2 and 3of MWG) and the firm (Chapter 3 of AMT, Chapters 1, 2, 3, 4, and 5 of Varian,and Chapter 5 of MWG), competitive equilibrium of an economy and its properties,— existence, uniqueness, the Sonnenschein-Mantel-Debreu Theorem, the first andsecond welfare theorems, the core of an economy — (Chapter 5 of AMT, Chapters17 and 21 of Varian, and Chapters 15, 16, 17, 18.B of MWG), preferences underuncertainty (Chapter 2.4 of AMT, Chapter 11 of Varian, and Chapter 6 of MWG),and consumer and producer surplus (Chapters 4.1 and 4.3 of AMT, Chapter 10 ofVarian, and Chapter 10 of MWG). Finally, we extend all the analyses to economiesunder uncertainty in which the Arrow-Debreu contingent markets, sequential mar-kets, and their equivalence are discussed (Chapter 20 of Varian and Chapter 19 ofMWG).

I will not answer how to use the WebCT. If you don’t officially register this coursebut want to access to the WebCT, please give your name, affiliation, and ID numbervia email by September 12 (Fri), 2007 to [email protected] I will notgive the authorization to access to the WebCT to any person who did notemail me by that date.

OFFICE HOURS: Wednesdays and Thursdays, 11:30am - 12:50pm

PROBLEM SETS: There will be approximately 7 problem sets. Problem setsare essential to help you understand the course and to develop your skill to analyzeeconomic problems. Besides, it should be expected that these problems sets arevery good proxies for the exams. You have to hand in your work on each problemset to our TA. Do not hand in your homework to me. Please take the copy of youranswer for the problem set before the submission. Because there might be somedelay for the TA to return your homework. All problem sets are given to you onlythrough the WebCT. Again, I will not provide any copy of them with you.

ASSESSMENT: Problem sets 10%, midterm exam 30%, and final exam(comprehensive) 60%. 4 Only if he/she has a serious reason why he/she cannottake the midterm exam, the grade of that person will be solely based on the finalexam and the problem sets. In other words, there is no makeup midterm exam.However, this treatment is very exceptional. You must take both exams.

4McGill University values academic integrity. Therefore all students must understand the mean-ing and consequences of cheating, plagiarism and other academic offences under the code of studentconduct and disciplinary procedures (See www.mcgill.ca/integrity for more information.

2

Contents

1 Introduction 7

2 Consumer Theory 102.1 Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Preference Relations . . . . . . . . . . . . . . . . . . . . . . . 102.2 The Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The Consumer’s Optimization Problem . . . . . . . . . . . . . . . . 152.4 The Indirect Utility Function . . . . . . . . . . . . . . . . . . . . . . 192.5 The Expenditure Function . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Relations between the Indirect Utility Function and the Expenditure

Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 The Relation between x(p, y) and xh(p, u) . . . . . . . . . . . . . . . 242.8 Income and Substitution Effects . . . . . . . . . . . . . . . . . . . . . 242.9 More Restrictions on the Ordinary Demand . . . . . . . . . . . . . . 282.10 Weak Axiom of Revealed Preference (WARP) . . . . . . . . . . . . . 302.11 Appendix: Integrability . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.11.1 Recovering Preferences from the Expenditure Function . . . . 342.11.2 Recovering the Expenditure Function from Demand . . . . . 35

3 Production 363.1 Properties of Production Sets . . . . . . . . . . . . . . . . . . . . . . 363.2 Production Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 More about Production Functions . . . . . . . . . . . . . . . 403.2.2 Returns to Scale of the Production Function . . . . . . . . . 41

3.3 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Properties of the Cost Functions . . . . . . . . . . . . . . . . 43

3.4 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.1 More about Profit Maximization . . . . . . . . . . . . . . . . 48

3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5.1 More on Homothetic Functions . . . . . . . . . . . . . . . . . 50

3

CONTENTS

4 Partial Equilibrium 524.1 Price and Individual Welfare . . . . . . . . . . . . . . . . . . . . . . 524.2 Quasi-Linear Preference . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Pareto Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4 The Profit Maximization Problem Revisited . . . . . . . . . . . . . . 554.5 The Market Supply Function . . . . . . . . . . . . . . . . . . . . . . 564.6 The Market Demand Function . . . . . . . . . . . . . . . . . . . . . 564.7 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 General Equilibrium 585.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Exchange Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3 Equilibrium in Competitive Market Systems . . . . . . . . . . . . . . 605.4 Existence of Walrasian Equilibrium . . . . . . . . . . . . . . . . . . . 615.5 Regular Economies and Local Uniqueness . . . . . . . . . . . . . . . 655.6 Anything Goes: The Sonnenschein-Mantel-Debreu Theorem . . . . . 675.7 Properties of the Set of Walrasian Allocations . . . . . . . . . . . . . 68

5.7.1 The Edgeworth Box Diagram . . . . . . . . . . . . . . . . . . 685.7.2 Core of an Economy and the First Welfare Theorem . . . . . 695.7.3 The Second Welfare Theorem . . . . . . . . . . . . . . . . . . 70

5.8 Walrasian Equilibrium with Production . . . . . . . . . . . . . . . . 745.8.1 Producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.8.2 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.8.3 Feasibility and Efficiency in Production Economies . . . . . . 765.8.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.9 Walrasian Allocations in Economies with Production . . . . . . . . . 785.10 Uniqueness of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.10.1 The Weak Axiom for Aggregate Excess Demand . . . . . . . 815.10.2 Gross Substitution . . . . . . . . . . . . . . . . . . . . . . . . 83

5.11 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.12 Appendix: “Generalization” of General Equilibrium Theory . . . . . 84

6 Choice under Uncertainty 866.1 Preferences over Gambles . . . . . . . . . . . . . . . . . . . . . . . . 866.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Von Neumann-Morgenstern (VNM) Utility . . . . . . . . . . . . . . 896.4 Existence of VNM Utility Function . . . . . . . . . . . . . . . . . . . 896.5 Uniqueness up to Positive Affine Transformations . . . . . . . . . . . 916.6 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.7 Measures of Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . 95

6.7.1 Comparisons across Individuals . . . . . . . . . . . . . . . . . 966.7.2 Comparisons across wealth levels . . . . . . . . . . . . . . . . 98

6.8 State-Dependent Utility . . . . . . . . . . . . . . . . . . . . . . . . . 99

4

CONTENTS

6.8.1 State-Dependent Preferences and the Extended VNM UtilityRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.8.2 Existence of an Extended VNM Utility Representation . . . . 99

7 General Equilibrium under Uncertainty 1017.1 A Market Economy with Contingent Commodities . . . . . . . . . . 1017.2 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 1037.3 The Working of the Arrow-Debreu Economy . . . . . . . . . . . . . . 104

7.3.1 Ex Ante V.S. Ex Post . . . . . . . . . . . . . . . . . . . . . . 1057.3.2 No market at date 1 and No real transaction at date 0 . . . . 1057.3.3 No need to open spot markets at date 1 . . . . . . . . . . . . 1057.3.4 Each consumer has a single budget . . . . . . . . . . . . . . . 106

7.4 Sequential Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067.4.2 Arrow Security . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.4.3 Implementing the Arrow-Debreu equilibrium allocations . . . 109

7.5 Asset Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.6 Multi-Period Exchange Economies . . . . . . . . . . . . . . . . . . . 118

7.6.1 Implementing the A-D equilibria by Trading Long-lived Secu-rities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.6.2 Genericity of the Case M = X with K or More Securities . . 1247.7 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5

List of Figures

5.1 The Edgeworth Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1 The Relationship between u and v through h . . . . . . . . . . . . . 96

7.1 Construction of the No-arbitrage Weights . . . . . . . . . . . . . . . 1137.2 Existence of an inadmissible z if qT is not proportional to μ

′ · R. . . 1147.3 An information tree: gradual release of information . . . . . . . . . . 1197.4 Dividend Structure of a Radner Equilibrium (a) . . . . . . . . . . . . 1227.5 Dividend Structure of a Radner Equilibrium (b) . . . . . . . . . . . . 123

6

Chapter 1

Introduction

I start my lecture with Rakesh Vohra’s message about what economic theory is. Heis a professor at Northwestern university. 1

All of economic theorizing reduces, in the end to the solution of oneof three problems.

Given a function f and a set S:

1. Find an x such that f(x) is in S. This is the feasibility question.

2. Find an x in S that optimizes f(x). This is the problem of optimal-ity.

3. Find an x in S such that f(x) = x, this is the fixed point problem.

These three problems are, in general, quite difficult. However, if oneis prepared to make assumptions about the nature of the underlyingfunction (say it is linear, convex or continuous) and the nature of the setS (convex, compact etc.) it is possible to provide answers and very niceones at that.

I think this is the biggest picture of microeconomic theory you could have as yougo along this course. Whenever you are at a loss, please come back to this message.

Since we want to study economic theory from “micro” perspective, we build ourtheory on individuals. Assume that all commodities are traded in the centralizedmarkets. Throughout this course, we assume that each individual (consumer andfirm) takes prices as given. We call this the price taking behavior assumption. Youmight ask why individuals are price takers. My answer would be “why not?” Letus go as far as we can with this behavioral assumption and thereafter try to see thelimitation of the assumption. However, you have to wait for Microeconomic theory II

1See http://www.kellogg.northwestern.edu/faculty/vohra/htm/vohra.htm for more infor-mation.

7

CHAPTER 1. INTRODUCTION

for how to relax this assumption. So, stick with this assumption. For each consumer,we want to know

1. What is the set of “physically feasible bundles? Is there any such a bundle atall (feasibility)? We call this set the consumption set.

2. What is the set of “financially feasible bundles? Is there any such a bundle atall (feasibility)? We call this set the budget set.

3. What is the best bundle to the consumer among all feasible bundles (optimal-ity)? We call this bundle the consumer’s demand.

We can make the exact parallel argument for the firm. What is the set of techni-cally feasible inputs (feasibility)? We call this the production set of the firm. Whatis the best combination of inputs to maximize its profit (optimality)? We call thisthe firm’s supply. Once we figure out what are feasible and best choices to each con-sumer and each firm under any possible circumstance, we want to know if there is anycoherent state of affairs where everybody makes her best choice. In particular, allmarkets must clear. We call this coherent state Walrasian (competitive) equilibrium.(a fixed point). The next important question is whether such an equilibrium existsor under what circumstances it exists. This is a problem of existence of competitiveequilibrium of an economy.

Suppose now that there is at least one Walrasian equilibrium of a given economy.Then we want to know what properties this equilibrium possesses: Is the equilib-rium allocation efficient? Under what circumstances is it efficient? Under whatcircumstances, is any efficient allocation can be regarded as a result of a competitiveequilibrium with income transfer (the second welfare theorem)? Is the equilibriumunique? Under what circumstances is it unique? Are there many equilibria? Howmany?

Under some “reasonable” assumptions on the economy, what we know is

1. equilibrium exists,

2. equilibrium allocation is efficient (the First Welfare theorem),

3. equilibrium allocation is in the core 2 and the set of competitive equilibriumallocations coincides with the core allocations when the economy is large (Thecore limit theorem).

4. the equilibrium is not unique, in general, but we know some of conditions whichguarantee the uniqueness of equilibria.

2Given an allocation x, a coalition S makes a justifiable objection to x if there is a redistributionof goods feasible only within S such that every member of S is better off from the redistribution.An allocation x is said to be in the core if no coalition has any justifiable objection to x.

8

CHAPTER 1. INTRODUCTION

5. the number of equilibria is “typically” finite and odd number (local uniquenessof equilibria) 3,

6. in general, equilibrium is not stable in the sense of dynamical systems, butunder some known conditions which guarantee the uniqueness, the equilibriumis stable, as well. Very loosely speaking, the equilibrium is unique, it is stable,as well.

7. there is a precise sense in which we (economists) cannot know anything aboutthe economy beyond the properties mentioned above unless we make more as-sumptions on the economy (The Sonnenschein-Mantel-Debreu theorem: Any-thing goes).

So far, I have assumed that each individual acts in a world of absolute certainty.The consumer knows the qualities of all commodities, for example. Clearly, this isnot true in the real world. Then, many economic decisions contain some elementof uncertainty. Let A = {a1, . . . , an} denote a finite set of outcomes. We referto u(ai) as simply the utility of the outcome ai. Let g = (p1 ◦ a1, . . . , pn ◦ an)denote a gamble that outcome ai occurs with probability pi for each i = 1, . . . , n.An individual evaluates this gamble g according to the von Neumann-Morgensternutility V if

V (g) =n∑

i=1

piu(ai).

One of the most important questions is to ask under what conditions the vonNeumann-Morgenstern utility representation is valid. Furthermore, what we wantto know is

1. when we can unambiguously say that one gamble is better than another,

2. when we can unambiguously say that one individual is more risk averse thananother.

This note takes the Definition-Theorem-Proof style. You have to provide anystatement in terms of mathematics. If you think the statement you made is true,you must prove it. If you think the statement is wrong, you must provide a coun-terexample.

3Technically, it is very important to articulate what I mean by “typically.” But I will not discussthis here.

9

Chapter 2

Consumer Theory

2.1 Environment

• The number of commodities is finite and equal to n (indexed by i = 1, . . . , n).

• each commodity is measured in some infinitely divisible units.

• x = (x1, . . . , xn) ∈ Rn+ is a consumption bundle.

• X is a consumption set that is the set of bundles the consumer can conceive.Usually, we take X = Rn

+.

The minimal requirements on the consumption set are

• ∅ �= X ⊂ Rn+; there is a choice

• X is closed; a technical condition needed for divisible commodities

• X is convex; a technical condition needed for divisible commodities

• 0 ∈ X; no consumption is always possible

2.1.1 Preference Relations

We represent the consumer’s preferences by a binary relation, �, defined on theconsumption set, X. If x � x

′, we say that “x is at least as good as x

′,” for this

consumer. Consumer preferences are characterized axiomatically. These axioms ofconsumer choice formalize the view that the consumer can choose and that choicesare consistent in a particular way.

Definition 2.1.1 (Axiom 1: Completeness) The preference relation � on X iscomplete if, for any x, x

′ ∈ X, we have either x � x′or x

′ � x.

Whenever the consumer is asked which bundle is better for him, he never say “Idon’t know,” if his preference is complete.

10

CHAPTER 2. CONSUMER THEORY

Definition 2.1.2 (Axiom 2: Transitivity) The preference relation � on X istransitive if, for all x, y, z ∈ X, if x � y and y � z, then x � z.

Transitivity is a particular form of consistency of the consumer’s choice. We saythat the consumer is rational if her preference relation on X is complete and transi-tive. Throughout this course, these two axioms continue to be satisfied. In particular,completeness and transitivity together imply that the consumer can completely rankany finite number of elements in the consumption set, X, from best to worst, possiblywith some ties.

Definition 2.1.3 The binary relation � on X is said to be strict preference re-lation if, x � x

′if and only if x � x

′but x

′� x.

Definition 2.1.4 The binary relation ∼ on X is said to be indifference relationif, x ∼ x

′if and only if x � x

′and x

′ � x.

Let x0 be any point in the consumption set X. Relative to any such point, wecan define the following subsets of X:

1. � (x0) ≡ {x ∈ X|x � x0}; the “at least as good as” set or the upper contourset.

2. � (x0) ≡ {x ∈ X|x0 � x}; the “no better than” set or the lower contour set.

3. ≺ (x0) ≡ {x ∈ X|x0 � x}; the “worse than set, or the strict lower contour set.

4. � (x0) ≡ {x ∈ X|x � x0}; the “preferred to” set, or the strict upper contourset.

5. ∼ (x0) ≡ {x ∈ X|x ∼ x0}; the “indifference” set.

Definition 2.1.5 (Axiom 3: Continuity (Cont)) Let X = Rn+. The preference

relation � on Rn+ is continuous if, for all x ∈ Rn

+, the “at least as good as” set� (x) and the “no better than” set � (x) are closed in Rn

+. That is, for any sequenceof pairs {(xn, yn)}∞n=1 with xn � yn for all n, x = limn→∞ xn, and y = limn→∞ yn,we have x � y.

The continuity axiom guarantees that sudden preference reversals do not occur.

Example: The lexicographic preference relation:

For simplicity, assume that X = R2+. Define x � y if either “x1 > y1” or

“x1 = y1 and x2 ≥ y2.” This is known as the lexicographic preference relation. Iclaim that lexicographic preferences are not continuous. To see this, consider thesequence of bundles xn = (1/n, 0) and yn = (0, 1). We have that xn � yn for each n.However, limn→∞ yn = (0, 1) � (0, 0) = limn→∞ xn. This contradicts the continuityrequirement.

11


Exercise 2.1.1 Verify the lexicographic preferences are complete, transitive, stronglymonotone, and strictly convex.

Let Bε(x0) denote the open ball of radius ε centered at x0. That is,

Bε(x0) ={x ∈ Rn

+

∣∣ ‖x − x0‖ < ε}

.

Definition 2.1.6 (Axiom 4′: Local Nonsatiation (LNS)) The preference rela-

tion � on Rn+ is locally nonsatiated if, for all x0 ∈ Rn

+, and for all ε > 0, thereexists some x ∈ Bε(x0) ∩ Rn

+ such that x � x0.

For example, a thick indifference curve violates local nonsatiation. The preferencerelation � is globally nonsatiated if there is no bundle x0 ∈ Rn

+ such that x0 � y forany y ∈ Rn

+. So, a thick indifference curve is compatible with global nonsatiation.

If the bundle x contains at least as much of every good as x′, i.e., xi ≥ x

′i for

i = 1, . . . , n, we write x ≥ x′, while if x contains strictly more of every good than

x′, i.e., xi > x

′i for all i = 1, . . . , n, we write x � x

′′.

Definition 2.1.7 (Axiom 4: Monotonicity (M)) The preference relation � onRn

+ is monotone if for all x, x′ ∈ Rn

+, if x ≥ x′then x � x

′, while if x � x

′, then

x � x′.

The preference relation � on Rn+ is weakly monotone if x � y whenever x ≥

y. If the consumer is completely indifferent among all bundles, his preference iscompatible with weak monotonicity. Therefore, in some cases, weak monotonicity isnot compatible with local nonsatiation. The preference relation � on Rn

+ is stronglymonotone if y ≥ x and y �= x implies y � x.

Exercise 2.1.2 Show the following:

1. If � is strongly monotone, then it is monotonic.

2. If � is monotone, then it is locally non-satiated.

Definition 2.1.8 (Axiom 5′: Convexity (Conv)) The preference relation � on

Rn+ is convex if, whenever x � x

′, tx + (1 − t)x

′ � x′for all t ∈ [0, 1].

The convexity of preferences respects for preferences for diversified bundles. Itcan be interpreted in term of diminishing marginal rates of substitution.

Exercise 2.1.3 Verify the preference relation � on X is convex if and only if forevery x ∈ X, the at least as good as set � (x) is convex.

Exercise 2.1.4 Let u(x) = x21 + x2

2. Check that u represents non-convex preferencerelation.

12


Definition 2.1.9 (Axiom 5: Strict Convexity) The preference relation � on Rn+

is strictly convex if, whenever x �= x′and x � x

′, then tx + (1 − t)x

′ � x′for all

t ∈ (0, 1)

Exercise 2.1.5 Let u(x) = min{x1, x2} represent the Leontief preferences. Checkthe Leontief utility function represents convex preferences but not strictly convexpreferences.

Exercise 2.1.6 Let u(x) = xα1 x

(1−α)2 be the Cobb-Douglas utility function for some

α ∈ (0, 1). Check the Cobb-Douglas utility function exhibits strict convexity.

2.2 The Utility Function

In modern theory, the preference relation is taken to be the primitive, most funda-mental characterization of preferences. The utility function merely “represents,” orsummarizes, the information conveyed by the preference relation.

Definition 2.2.1 A real-valued function u : Rn+ → R is called a utility function

representing the preference relation � on Rn+, if for all x, x

′ ∈ Rn+,

u(x) ≥ u(x′) ⇔ x � x

′

.

Exercise 2.2.1 Show that if a preference relation � on Rn+ can be represented by a

utility function, then it is rational, i.e., complete and transitive.

An important question is one of existence of a “continuous” utility functionrepresenting preference relation.

Theorem 2.2.1 (Existence of a utility function, Debreu (59)) Suppose that thepreference relation � on Rn

+ is complete, transitive (so, rational), and continuous.Then there is a continuous utility function u : Rn

+ → R that represents �.

Proof of Theorem 2.3.1: Only for the sake of simplification of the proof, weassume that preferences are monotone as well as continuous. 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:

u(x)e ∼ x. for any x ∈ Rn+ (∗)

In order to prove the theorem, there are four steps to be checked:

1. There always exists a number u(x) satisfying property (∗).2. The number u(x) is uniquely determined, i.e., x ∼ y ⇔ u(x) = u(y).

13


3. u(·) represents �, i.e., x � y ⇔ u(x) ≥ u(y).

4. The constructed u(·) is indeed continuous.

Step 1: Note that for every x ∈ Rn+, monotonicity implies that x � 0 ∈ Rn. Fix

x ∈ Rn+ and consider the following two subsets of real numbers:

A ≡ {t ≥ 0|te � x} and B ≡ {t ≥ 0|x � te}.

If t∗ ∈ A ∩B, then t∗e ∼ x, so that setting u(x) = t∗ would satisfy property (∗).Thus, we must show that A ∩ B is guaranteed to be nonempty.

The continuity of � implies that both A and B are closed in R+. By monotonicity,t ∈ A implies that t

′ ∈ A if t′ ≥ t. Consequently, A must be a closed interval of the

form [t,∞]. Similarly, monotonicity and continuity imply that B must be a closedinterval of the form [0, t]. Completeness implies that t ∈ A ∪ B for any t ≥ 0. Butthis means that R+ = [0,∞) ⊂ A ∪ B = [0, t] ∪ [t,∞]. So, we conclude that t ≤ t sothat A ∩ B �= ∅.

Step 2: We must show that there is only one number t ≥ 0 such that te ∼ x. Ift1e ∼ x and t2e ∼ x, by transitivity, t1e ∼ t2e. So, by monotonicity, it must be thecase that t1 = t2.

Step 3: Consider two bundles x and y, and their associated utility numbers u(x)and u(y), which by definition satisfy u(x)e ∼ x and u(y)e ∼ y. Then,

x � y ⇔ u(x)e ∼ x � y ∼ u(y)e⇔ u(x)e � u(y)e (∵ transitivity)⇔ u(x) ≥ u(y) (∵ monotonicity)

Step 4: Consider a sequence of bundles {xk} for which xk ∈ Rn+ for each k and

xk → x as k → ∞. Suppose, by way of contradiction, that the constructed u(·) isnot continuous. Then, we have limk→∞ u(xk) �= u(x). Without loss of generality,we can assume that limk→∞ u(xk) > u(x). By Step 1, for each xk, there is a scalartk ≥ 0 such that tke ∼ xk. Let te ∼ x. Let t∗ ≡ limk→∞ tk. Then, the hypothesisthat limk→∞ u(xk) > u(x) means that t∗ > t by monotonicity of �. Since tk → t∗

as k → ∞, we must have tk > t for k large enough. By monotonicity of �, we havetke � te for k large enough. By Step 1, this means that xk � x for k large enough.By continuity of �, we have x � x in the limit. Hence, we have u(x) > u(x) by Step3. This is a contradiction to the uniqueness of u(·) in Step 2. �

Exercise 2.2.2 Show that if u(·) is a continuous utility function representing �,then � is continuous.

f : R → R is said to be strictly increasing if f(x) > f(y) whenever x > y.

14


Theorem 2.2.2 (Uniqueness up to positive monotone transformation) Let� be a preference relation on Rn

+ and suppose that u(·) is a utility function that rep-resents it. Then v(·) also represents � if and only if v(x) = f(u(x)) for every x,where f : R → R is strictly increasing on the set of values taken on by u(·).

Proof of Theorem 2.3.2: (⇐=). This direction is easier. Let v(x) = f(u(x))where f : R → R is strictly increasing. Then,

x � y ⇐⇒ u(x) ≥ u(y) ⇐⇒ v(x) ≥ v(y).

(=⇒) Suppose u(·) and v(·) represent the same preference relation �. Then, theremust exist a function f : R → R such that v(x) = f(u(x)) for any x ∈ Rn

+. Whatwe want to show is that f(·) must be strictly increasing. Suppose not, that is, thereare x, y ∈ Rn

+ with u(x) > u(y) such that v(x) = f(u(x)) ≤ f(u(y)) = v(y). Sinceu(·) represents �, we have x � y. At the same time, v(·) also represents �, we havey � x. This is a contradiction. �

Exercise 2.2.3 Provide several examples of positive monotonic transformation.

u(·) is said to be increasing if u(x) ≥ u(y) whenever x ≥ y, while if u(x) > u(y)whenever x � y. u(·) is said to be strongly increasing if u(x) > u(y) whenever x ≥ yand x �= y. u(·) is said to be quasi-concave if for any x, x

′ ∈ Rn+, u(tx+ (1− t)x

′) ≥

min{u(x), u(x′)} for any t ∈ [0, 1]. u(·) is said to be strictly quasi-concave if for any

x, x′

with x �= x′, u(tx + (1 − t)x

′) > min{u(x), u(x

′)} for any t ∈ (0, 1). See my

lecture notes on mathematics for the details.

Theorem 2.2.3 Let � be represented by u : Rn+ → R. Then:

1. u(·) is (strongly) increasing if and only if � is (strongly) monotone.

2. u(·) is quasi-concave if and only if � is convex.

3. u(·) is strictly quasi-concave if and only if � is strictly convex.

Exercise 2.2.4 Prove Theorem 2.3.3.

2.3 The Consumer’s Optimization Problem

We collect all the consumer’s circumstances into a feasible set, B ⊂ Rn+. The con-

sumer seeks

x∗ ∈ B such that x∗ � x for all x ∈ B.

The consumer’s preference relation � is complete, transitive, continuous, mono-tone, and strictly convex on Rn

+. Therefore, it can be represented by a real-valuedutility function, u(·), that is continuous, increasing, and strictly quasi-concave onRn

+.

15


We are concerned with an individual consumer operating within a market econ-omy. By a market economy, we mean an economic system in which transactionsbetween agents are mediated by markets. There is a market for each commodity,and in these markets, a price pi prevails for each commodity i. We suppose thatprices are strictly positive, so pi > 0, i = 1, . . . , n. We take the vector of marketprices, p � 0, as fixed from the consumer’s point of view. We call this the price tak-ing behavior assumption. The consumer is endowed with a fixed amount of moneyincome y ≥ 0.

We summarize these assumptions on the economic environment of the consumerby specifying the following structure on the feasible set, B(p, y), called the budgetset :

B(p, y) = {x ∈ Rn+ | px ≤ y}.

Lemma 2.3.1 Suppose that p � 0 and y ≥ 0. Then, the budget set B(p, y) isnonempty, convex, and compact.

Recall that a set S in Rn+ is said to be compact if it is closed and bounded.

Exercise 2.3.1 Prove Lemma 2.4.1.

Theorem 2.3.1 (Weierstrass’ theorem) Any continuous real-valued function whosedomain is a compact nonempty set has both the maximum and minimum of it.

Proof of Weierstrass’ Theorem: See my Lecture Note on Mathematics forEconomists. �

Now, the consumer’s utility maximization problem (UMP) is written

maxx∈ n

+

u(x) subject to x ∈ B(p, y).

If x∗ solves the UMP, then u(x∗) ≥ u(x) for all x ∈ B(p, y), which means thatx∗ � x for all x ∈ B(p, y).

Theorem 2.3.2 Suppose that p � 0, y ≥ 0 and u : Rn+ → R is continuous. Then,

the utility maximization problem (UMP) has a solution. Furthermore, if u(·) isstrictly quasi-concave, the solution is unique.

Proof of Theorem 2.4.2: The first part directly comes from Weierstrass’ the-orem. Suppose, for the second part, that there are two distinct solutions x, x

′to the

UMP even if u(·) is strictly quasi-concave. Let xt = tx+(1− t)x′for some t ∈ (0, 1).

Since x and x′

are within the budget set and the budget set is convex, xt is alsowithin the budget set. By strict quasi-concavity, we have that u(xt) > u(x) = u(x

′),

which contradicts our hypothesis that both x and x′are solutions to the UMP. �

16


If the solution x∗ to the UMP is unique for given values of p and y, we canproperly view the solution to the UMP as a function from the set of prices andincome to the set of quantities, X = Rn

+. When viewed as functions of p and y, thesolution to the UMP are known as demand functions, x∗ = x(p, y).

Proposition 2.3.1 (Walras’ law) If u(·) is increasing, the solution x∗ to the util-ity maximization problem has the property that px∗ = y for each p ∈ Rn

++ andy ∈ R+.

Proof of Proposition 2.4.1: Suppose px∗ < y. Let e = (1, . . . , 1) ∈ Rn+.

Choose ε > 0 small enough so that p(x∗ + εe) < y. This new bundle (x∗ + εe) isstill within the budget set and, by monotonicity, gives the consumer strictly higherutility than u(x∗). Then, this contradicts our hypothesis that x∗ is the solution tothe UMP. �

Exercise 2.3.2 Prove that if the underlying preference relation � is locally non-satiated, the solution x∗ to the UMP has the property that px∗ = y, i.e., Walras’law.

Proposition 2.3.2 (Homogeneity of Degree 0 of the Demand Function) Assumethat p � 0 and y ≥ 0. Suppose that u(·) is a continuous utility function representingthe preference relation � on Rn

+. Then, the demand function x(p, y) is homoge-neous of degree of 0 in (p, y). Namely, x(αp, αy) = x(p, y) for any α > 0.

Proof of Proposition 2.10.1: For any scalar α > 0, we have

{x ∈ Rn+ | αpx ≤ αy} = {x ∈ Rn

+ | px ≤ y} = B(p, y).

That is, the set of feasible bundles in the UMP does not change when all prices andwealth are multiplied by a constant α > 0. Therefore, the solution to the UMP mustbe the same. �

The consumer’s utility maximization problem is

maxx∈ n

+

u(x) subject to x ∈ B(p, y) = {x ∈ Rn+| px ≤ y}.

If we impose differentiability on u(·), we can characterize the solution x∗ via thefirst order conditions (FOCs). Assume that x∗ � 0. Then, if x∗ ∈ x(p, y) is asolution to the UMP, then there exists a Lagrange multiplier λ ≥ 0 such that for alli = 1, . . . , n:

∂u(x∗)∂xi

= λpi.

In particular, these FOCs are reduced to

MRSjk ≡ ∂u(x∗)/∂xj

∂u(x∗)/∂xk=

pj

pkfor any j and k,

where MRSjk stands for the marginal rate of substitution of j for k at x∗.

17


Theorem 2.3.3 (Sufficiency of the FOCs) Suppose that u(·) is continuous andquasi-concave on Rn

+, and that (p, y) � 0. If u(·) is differentiable at x∗, and(x∗, λ∗) � 0 solves the FOCs, then x∗ solves the UMP at prices p and incomey.

Proof of Theorem 2.4.3: Our proof relies on the following theorem. You canfind this theorem and its proof in my lecture note on mathematics.

Theorem 5.10 in Lecture Note on Mathematics for Economists: Letu : S → R be a C1 function defined on an open convex set S in Rn. Then, u(·) isquasiconcave on S if and only if for all x, x0 ∈ S,

u(x) ≥ u(x0) =⇒ ∇u(x0) · (x − x0) ≥ 0

Now, suppose that ∇u(x∗) �= 0 and (x∗, λ∗) � 0 solves the first order conditionof the Lagrangian function. Then,

∇u(x∗) = λ∗pp · x∗ = y

If x∗ is not a utility maximizing plan, there must be some x0 ∈ Rn+ such that

u(x0) > u(x∗)p · x0 ≤ y

Because u(·) is continuous and y > 0, there exists α ∈ (0, 1) sufficiently close to1 such that

u(αx0) > u(x∗)p · αx0 < y

Let x = αx0. We execute a series of computations:

∇u(x∗) · (x − x∗) = (λ∗p) · (x − x∗) (∵ ∇u(x∗) = λ∗p)= λ∗ (p · x − p · x∗)< λ∗ (y − y) (∵ p · x < y)= 0

Hence, we have ∇u(x∗) · (x − x∗) < 0 which contradicts the above theorem on thecharacterization of quasiconcavity. �

In order to ensure the existence of competitive equilibrium, we want the de-mand function to be continuous. It turns out that we have already made enoughassumptions that x(p, y) is continuous on Rn

++.

18


Lemma 2.3.2 (Continuity of the demand function) Suppose that u(·) is a con-tinuous, increasing, and strictly quasi-concave utility function. Then, the deriveddemand function x(p, y) is continuous at all p � 0 and y > 0.

If you are interested in the proof of this lemma, please consult Appendix A inChapter 3 of MWG. However, I do not expect you to understand this proof. Justaccept the result.

In many cases, we shall want the demand function to be more than continuous,namely, differentiable.

Theorem 2.3.4 (Differentiable Demand) Let x∗ � 0 solve the UMP at pricesp0 � 0 and y0 > 0. Suppose that the following three conditions are satisfied.

• u(·) is twice continuously differentiable on Rn++.

• ∇u(x∗) �= 0 ⇐⇒ ∂u(x∗)/∂xi > 0 for some i = 1, . . . , n.

• the Hessian of u(·) has a nonzero determinant at x∗ on the subspace M = {z ∈Rn|∇u(x∗)z = 0}.

Then, x(p, y) is differentiable at (p0, y0).

The last condition of the above theorem says that the indifference set throughx∗ is not flat. The same remark applies here as well. Just understand that we needmore assumptions to guarantee differentiable demand than Lemma 2.4.2 above.

2.4 The Indirect Utility Function

Given prices p and income y, the consumer chooses a utility-maximizing bundlex(p, y). The level of utility achieved when x(p, y) is chosen thus will be the highestlevel permitted by the consumer’s budget constraint facing prices p and income y.The relationship among prices, income, and the maximized value of utility can besummarized by a real-valued function v : Rn

+ × R+ → R defined as follows:

v(p, y) = maxx∈ n

+

u(x) subject to px ≤ y.

The function v(p, y) is called the indirect utility function. When u(·) is continuous,v(p, y) is well-defined for all p � 0 and y ≥ 0 because a solution to the UMP isguaranteed to exist. If, in addition, u(·) is strictly quasi-concave, then the solutionis unique. Then,

v(p, y) = u(x(p, y)).

Theorem 2.4.1 (Properties of the Indirect Utility Function) Suppose that u(·)is continuous and increasing. Then, the indirect utility function v(p, y) is

19


1. Continuous on Rn++ × R+,

2. Homogeneous of degree zero in (p, y), i.e., v(αp,αy) = v(p, y) for any α > 0

3. Increasing in y, i.e., v(p, y′) ≥ v(p, y) whenever y

′ ≥ y and v(p, y′) > v(p, y)

whenever y′ � y

4. Decreasing in p, i.e., v(p′, y) ≤ v(p, y) whenever p ≤ p

′and v(p

′, y) < v(p, y)

whenever p � p′

5. Quasi-convex in (p, y),

6. Roy’s identity: If v(p, y) is differentiable at (p0, y0) and ∂v(p0, y0)/∂y �= 0,then

xi(p0, y0) = −∂v(p0, y0)/∂pi

∂v(p0, y0)/∂y, i = 1, . . . , n.

v(p, y) is said to be quasi-convex if for any (p, y), (p′, y

′) ∈ Rn

++×R+, v(pα, yα) ≤max{v(p, y), v(p

′, y

′)} for any α ∈ [0, 1], where pα = αp + (1 − α)p

′and yα =

αy + (1 − α)y′. Property 1 is proved by the theorem of maximum. So, just accept

property 1 without proof. Here I relegate the proof of Roy’s identity to the sectionof expenditure function.

Exercise 2.4.1 Show properties 2,3,4, and 5 in the previous theorem.

2.5 The Expenditure Function

To construct the expenditure function, we ask: “What is the minimum level of moneyexpenditure the consumer must make facing a given set of prices to achieve a givenlevel of utility?” This is called the expenditure minimization problem (EMP). Wedefine the expenditure function as the minimum-value function,

e(p, u) ≡ minx∈ n

+

px subject to u(x) ≥ u,

for all p � 0 and all attainable utility levels u. Let U = {u(x)| x ∈ Rn+} denote the

set of attainable utility levels. If u(·) is continuous and strictly quasi-concave, thesolution will be unique, so we can denote the solution as the function xh(p, u) ≥ 0.Thus, if xh(p, u) is the solution to the EMP, we have

e(p, u) = pxh(p, u).

xh(p, u) is called the compensated or Hicksian demand function if it is single-valued.

20


Theorem 2.5.1 (Properties of the Expenditure Function) Suppose that u(·)is continuous and increasing. Then e(p, u) is

1. Zero when u(·) takes on the lowest level of utility in U .

2. Continuous on its domain Rn++ × U .

3. For all p � 0, increasing and unbounded above in u, i.e., e(p, u′) ≥ e(p, u)

whenever u′ ≥ u and e(p, u

′) > e(p, u

′) whenever u

′> u; and for any M ∈ R+,

there exists u ∈ U such that e(p, u) > M .

4. Increasing in p, i.e., e(p′, u) ≥ e(p, u) whenever p

′ ≥ p and e(p′, u) > e(p, u)

whenever p′ � p.

5. Homogeneous of degree 1 in p, i.e, e(αp, u) = αe(p, u) for any α > 0.

6. Concave in p, i.e., e(αp + (1 − α)p′, u) ≥ αe(p, u) + (1 − α)e(p

′, u) for any

p, p′ ∈ Rn

++ and any α ∈ [0, 1]. Let e(·, u) : Rn++ → R be a twice differentiable

function. If e(·, u) is concave, then, for any p ∈ Rn++, we have

∂2e(p, u)∂p2

i

≤ 0 for any i = 1, . . . , n.

7. Shephard’s lemma: If, in addition, u(·) is strictly quasi-concave and e(p, u) isdifferentiable in p at (p0, u0) with p0 � 0, then we have

∂e(p0, u0)∂pi

= xhi (p0, u0), i = 1, . . . , n.

Proof of Shephard’s lemma: We omit the proof of property 2 which comesfrom the theorem of maximum. We focus only on property 7 and leave the rest asexercise. For simplicity, we focus on the case where xh(p, u) � 0 and we assume thatxh(p, u) is differentiable at (p0, u0). Using the chain rule, the change in expenditurecan be written as

∇pe(p0, u0) = ∇p[pxh(p0, u0)]= xh(p0, u0) + [pDpx

h(p0, u0)]T .

Or, for each commodity i, we have

∂e(p0, u0)∂pi

= xhi (p0, u0) +

n∑j=1

p0j

∂xhj (p0, u0)∂pi

where e(p0, u0) =∑n

j=1 p0jx

hj (p0, u0). Substituting from the FOCs for an interior

solution to the EMP, p = λ∇xu(xh(p0, u0)) (or p0j = λ∂u/∂xj), yields

∇pe(p0, u0) = xh(p0, u0) + λ[∇xu(xh(p0, u0)) · Dpxh(p0, u0)]T .

21


Or, for each i, we have

∂e(p0, u0)∂pi

= xhi (p0, u0) + λ

n∑j=1

∂u

∂xj


But since the constraint u(xh(p, u0)) = u0 holds for all p in the EMP, we know that∇xu(xh(p0, u0)) · Dpx

h(p0, u0) = 0, and so we have the result. Or, for each i, weknow that

n∑j=1

∂u

∂xj


Again, we obtain the result. �

Exercise 2.5.1 Prove properties 1, 3, 4, 5, and 6 in the previous theorem.

With the concept of the expenditure functions, we can now prove Roy’s identity.

Proof of Roy’s Identity: Let u0 = v(p0, y0). Because the identity v(p, e(p, u0)) =u0 holds for all p, differentiating v(p, e(p, u0)) = u0 with respect to p and evaluatingit at p = p0 yields

∇pv(p0, e(p0, u0)) +∂v(p0, e(p0, u0))

∂y∇pe(p0, u0) = 0.

But ∇pe(p0, u0) = xh(p0, u0) by Shephard’s lemma, and so we can substitute andget

∇pv(p0, e(p0, u0)) +∂v(p0, e(p0, u0))

∂yxh(p0, u0) = 0.

Finally, since y0 = e(p0, u0), we can write

∇pv(p0, y0) +∂v(p0, y0)

∂yx(p0, y0) = 0.

Rearranging, this yields the result. �

The idea behind Shephard’s lemma is as follows: If we are at an optimum in theEMP, the changes in demand caused by price changes have no first-order effect onthe consumer’s expenditure. The proof uses the chain rule to break the total effectof the price change into two effects: a direct effect on expenditure from the changein prices holding demand fixed (the first term) and an indirect effect on expenditurecaused by the induced change in demand holding prices fixed (the second term).However, because we are at an expenditure minimizing bundle, the FOCs for theEMP imply that this latter effect is zero. 1

1More general argument can be made by the envelope theorem. Those who are interested in thistheorem are referred to either pp. 504-509, Chapter A2 of AMT or my lecture notes on mathematics.

22


2.6 Relations between the Indirect Utility Function andthe Expenditure Function

Though the indirect utility function and the expenditure function are conceptuallydistinct, there is a close relationship between them. Fix (p, y) and let u = v(p, y).Recall now that e(p, u) is the smallest expenditure needed to attain a level of utilityat least u. Hence, we must have e(p, u) ≤ y. Consequently,

e(p, v(p, y)) ≤ y, ∀(p, y) � 0.

Example 2.6.1 Consider the consumer’s UMP whose preference admits a thick in-difference curve. Then, we have e(p, v(p, y)) < y.

Next, fix (p, u) and let y = e(p, u). Because v(p, y) is the largest utility levelattainable at prices p and with income y, this implies that v(p, y) ≥ u. Consequently,

v(p, e(p, u)) ≥ u, ∀(p, u) ∈ Rn++ × U .

Example 2.6.2 If u(·) is not continuous, it is possible to have v(p, e(p, u)) > u.

The next theorem demonstrates that under certain familiar conditions on pref-erences, both of these inequalities, in fact, must be equalities.

Theorem 2.6.1 (When UMP=EMP) Let v(p, y) and e(p, u) be the indirect util-ity function and expenditure function for some consumer whose utility function iscontinuous and increasing. Then, for all p � 0, y ≥ 0, and u ∈ U :

1. e(p, v(p, y)) = y.

2. v(p, e(p, u)) = u.

Proof of Theorem 2.6.1: (1) UMP ⇒ EMP: Let x∗ be the solution to theUMP given p and y. Suppose, to the contrary, that x∗ is not the solution to theEMP to achieve the required level of utility u(x∗). Then, there exists x

′such that

u(x′) ≥ u(x∗) and px

′< px∗ ≤ y. Let e = (1, . . . , 1) as usual. We can choose ε > 0

small enough so that px′

< p(x′+ εe) < y. This implies that x

′+ εe ∈ B(p, y),

i.e., a feasible bundle. Since u(·) is increasing, we have u(x′+ εe) > u(x∗), which

contradicts the hypothesis that x∗ is the solution to the UMP. (2) EMP ⇒ UMP:Let x∗ be the solution to the EMP given p and u. Assume, for simplicity, thatu > u(0). Then, x∗ �= 0, which implies that px∗ > 0 because p � 0. Suppose, onthe contrary, that x∗ is not the solution to the UMP. Then, there exists x

′such that

u(x′) > u(x∗) and px

′ ≤ y. Due to the fact that x∗ is the solution to the EMP, wealso have px∗ ≤ px

′. However, if u(·) is increasing, we must have px∗ = y. As a

result, we must assume that px∗ = px′. Consider a bundle xα = αx

′for α ∈ (0, 1).

By continuity of u(·), if α is close enough to 1, we must have u(xα) ≥ u(x∗) andpxα < px∗. Because p � 0. However, this contradicts the hypothesis that x∗ is thesolution to the EMP. �

23


2.7 The Relation between x(p, y) and xh(p, u)

Theorem 2.7.1 (When x = xh) Assume that u(·) is continuous, increasing, andstrictly quasi-concave on Rn

+. For p � 0, y ≥ 0, and u ∈ U :

1. x(p, y) = xh(p, v(p, y)).

2. xh(p, u) = x(p, e(p, u)).

The first relation says that the ordinary demand at prices p and income y is equalto the compensated demand at prices p and the utility level that is the maximumthat can be achieved at prices p and income y. The second says that the compensateddemand at any prices p and utility level u is the same as the ordinary demand atthose prices and an income level equal to the minimum expenditure necessarily atthose prices to achieve that utility level.

Proof of Theorem 2.8.1: (UMP ⇒ EMP): Let x0 = x(p0, y0) and let u0 =u(x0). Then, we have v(p0, y0) = u0 by definition of the indirect utility functionv(·). Since u(·) is increasing, px0 = y0 (Walras’ law). We also know the relation thate(p0, v(p0, y0)) = e(p0, u0) = y0 (Remember Theorem 2.7.1 (UMP=EMP)). Now, wehave that u(x0) = u0 and p0x0 = y0, which imply that x0 solves the EMP as wellwhen (p, u) = (p0, u0). Hence, x0 = xh(p0, u0), that is, x(p0, y0) = xh(p0, v(p0, y0)).(EMP ⇒ UMP): Let x0 = xh(p0, u0) and let y0 = p0x0. Then, we have e(p0, u0) = y0

by definition of the expenditure function e(·). Since u(·) is continuous, u(x0) = u0.We also know the relation that v(p0, e(p0, u0) = v(p0, y0) = u0. Then, the fact thatxh(p0, u0) = x(p0, v(p0, v0) = x(p0, y0) and v(p0, u0) = u0 implies that x0 also solvesthe UMP. �

2.8 Income and Substitution Effects

When the price of a good declines, there are at least two conceptually separatereasons why we expect some change in the quantity demanded. First, that goodbecomes relatively cheaper compared to other goods. Because all goods are desirable,even if the consumer’s purchasing power over goods were unchanged, we would expecther to substitute the relatively cheaper good for the now relatively more expensiveones. This is the substitution effect. At the same time, however, whenever a pricechanges, the consumers purchasing power is effectively increased, allowing her tochange her purchases of all goods in any way she sees fit. The effect on quantitydemanded of this generalized increase in purchasing power is called the income effect.

John Hicks proposed a way of decomposing the total effect of a price change. Itis based on the observation that the consumer achieves some level of utility at theoriginal prices before any change has occurred. The formalization of this is givenas follows: The substitution effect is that hypothetical change in consumption that

24


would occur if relative prices were to change to their new levels but the maximumutility the consumer can achieve were kept the same as before the price change.The income effect is then defined as whatever is left of the total effect after thesubstitution effect.

The relationships between total effect, substitution effect, and income effect aresummarized in the Slutsky equation. For the most of time in this course, I assumethat u(·) is continuous, increasing, strictly quasi-concave and assume furthermorethat we will freely differentiate whenever necessary.

Theorem 2.8.1 (The Slutsky Equation) Let x(p, y) be the consumer’s ordinaldemand function. Let u∗ be the level of utility the consumer achieves at prices p andincome y. Then, for any i, j = 1, . . . , n, we have

∂xi(p, y)∂pj︸︷︷︸

Total

=∂xh

i (p, u∗)∂pj︸︷︷︸

Substitution

−xj(p, y)∂xi(p, y)

∂y︸︷︷︸Income

.

Proof of Theorem 2.9.1: Because of the dual relation between the ordinarydemand and the compensated demand, that is, x(p, y) = xh(p, u), we have xh(p, u) =x(p, e(p, u)). Differentiating this with respect to p, we have

∂xhi (p, u)∂pj

=∂xi(p, y)

∂pj+

∂xi(p, y)∂y

· ∂e(p, u)∂pj

, ∀ i, j = 1, . . . , n.

By Shephard’s lemma, we can rewrite the above equation as follows:

∂xhi (p, u)∂pj

=∂xi(p, y)

∂pj+

∂xi(p, y)∂y

· xhj (p, u), ∀ i, j = 1, . . . , n.

Rearranging this and taking into account xh(p, u) = x(p, y), we complete the proof.�

Thanks to the Slutsky equation, whatever we learn about substitution terms thencan be translated into knowledge about “observable” ordinary demands.

Classical statements of the Law of Demand were rather strong: “If price goesdown, quantity demanded goes up.” This seemed generally to conform to observa-tions of how people behave, but there were some troubling exceptions. The famousGiffen’s paradox was the most outstanding of these. What is really true is as follows:“prices and demanded quantities move in opposite directions for price changes thatleave the achieved level of utility unchanged.” This is called the Compensated Lawof Demand.

25


Theorem 2.8.2 (The Compensated Law of Demand) Let xhi (p, u) be the com-

pensated demand for good i. Then

∂xhi (p, u)∂pi

≤ 0, i = 1, . . . , n.

Proof of Theorem 2.9.3: By Shephard’s lemma, we know that

∂xhi (p, u)∂pi

=∂2e(p, u)

∂p2i

, ∀ i = 1, . . . , n.

Since e(p, u) is concave in p, ∂2e(p, u)/∂p2i ≤ 0 for all i = 1, . . . , n. We are done. �

Consider a 2 × 2 matrix A given below.

A =(

a11 a12

a21 a22

)

Then, the matrix A is negative semidefinite if and only if a11 ≤ 0, a22 ≤ 0, anda11a22 − a12a21 ≥ 0. See my lecture notes on mathematics for more details.

Theorem 2.8.3 (Symmetric Substitution Terms) Let xh(p, u) be the consumer’scompensated demand and suppose that e(·, u) is twice continuously differentiable.Then, for any i, j = 1, . . . , n, we have

∂xhi (p, u)∂pj

=∂xh

j (p, u)∂pi

.

Proof of Theorem 2.8.3: Taking into account Shephard’s lemma, what wewant is

∂2e(p, u)∂pi∂pj

=∂2e(p, u)∂pj∂pi

, for any i, j = 1, . . . , n.

This is proved by the following Young’s theorem.

Theorem 2.8.4 (Young’s Theorem) For any twice continuously differentiable func-tion f(·),

∂2f(x)∂xi∂xj

=∂2f(x)∂xj∂xi

, for all i, j = 1, . . . , n.

We accept Young’s theorem without proof. �

Before stating other results, we quickly review some of mathematical notionsused for the results.

26


Definition 2.8.1 The n × n matrix M is negative semidefinite if

z · Mz ≤ 0, for all z ∈ Rn.

If the inequality is strict for all z �= 0, then the matrix M is negative definite.

The next theorem is a characterization of quasi-concavity of a function in termsof negative semidefiniteness of the matrix. See also my lecture notes on mathematicsfor more details.

Theorem 2.8.5 Let u∗ be the level of utility. The twice continuously differentiablefunction e(·, u∗) : Rn

++ → R is concave if and only if D2pe(p, u∗) is negative semidef-

inite for every p ∈ Rn++. If D2

pe(p, u∗) is negative definite for every p ∈ Rn++, then

the function is strictly concave.

Theorem 2.8.6 (Negative Semidefinite Substitution Matrix) Let xh(p, u) bethe consumer’s compensated demand, and let

σ(p, u) ≡

⎛⎜⎜⎝

∂xh1 (p,u)∂p1

· · · ∂xh1 (p,u)∂pn

.... . .

...∂xh

n(p,u)∂p1

· · · ∂xhn(p,u)∂pn

⎞⎟⎟⎠ ,

called the substitution matrix, contain all the compensated substitution terms.Then the matrix σ(p, u) is negative semidefinite.

Proof of Theorem 2.8.6: By Shephard’s lemma, we have⎛⎜⎜⎝

∂xh1 (p,u)∂p1

· · · ∂xh1 (p,u)∂pn

.... . .

...∂xh

n(p,u)∂p1

· · · ∂xhn(p,u)∂pn

⎞⎟⎟⎠ =

⎛⎜⎜⎝

∂2e(p,u)∂p2

1· · · ∂e(p,u)

∂pn∂p1

.... . .

...∂e(p,u)∂p1∂pn

· · · ∂e(p,u)∂p2

n

⎞⎟⎟⎠ .

By theorem 2.8.3, this matrix is symmetric. Using theorem 2.8.5 and taking intoaccount the fact that e(·, u) is concave in p, this matrix is also negative semidefinite.�

Note that a square matrix A is said to be symmetric if A = A′, where A

′is the

transpose of A. See my lecture note on mathematics for these definitions.

Theorem 2.8.7 Let x(p, y) be the consumer’s ordinal demand function. Define the(i, j)-th Slutsky term as

∂xi(p, y)∂pj

+ xj(p, y)∂xi(p, y)

∂y,

27


and form the entire n×n Slutsky matrix of prices and income responses as follows:

S(p, y) =

⎛⎜⎜⎝

∂x1(p,y)∂p1

+ x1(p, y)∂x1(p,y)∂y · · · ∂x1(p,y)

∂pn+ xn(p, y)∂x1(p,y)

∂y...

. . ....

∂xn(p,y)∂p1

+ x1(p, y)∂xn(p,y)∂y · · · ∂xn(p,y)

∂pn+ xn(p, y)∂xn(p,y)

∂y

⎞⎟⎟⎠ .

Then S(p, y) is symmetric and negative semidefinite.

Proof of Theorem 2.8.7: Let u∗ be the maximum utility the consumer achievesat prices p and income y, so u∗ = v(p, y). By Theorem 2.9.1, we know that

∂xhi (p, u∗)∂pj

=∂xi(p, y)

∂pj+ xj(p, y)

∂xi(p, y)∂y

.

By Shephard’s lemma, we have

∂xhi (p, u∗)∂pj

=∂2e(p, u∗)∂pj∂pi

.

By Theorem 2.8.3, S(p, y) is symmetric. Since e(p, u) is concave in p, S(p, y) is alsonegative semidefinite. �

The Slutsky matrix is given as follows:

S(p, y) =

⎛⎜⎜⎜⎝

s11(p, y) s12(p, y) · · · s1n(p, y)s21(p, y) s22(p, y) · · · s2n(p, y)

......

. . ....

sn1(p, y) sn2(p, y) · · · snn(p, y)

⎞⎟⎟⎟⎠ ,

where the (i, j)-th entry is

sij(p, y) =∂xi(p, y)

∂pj+ xj(p, y)

∂xi(p, y)∂y

.

2.9 More Restrictions on the Ordinary Demand

The requirements that consumer demand satisfy homogeneity of degree 0 in (p, y)and Walras’ law (i.e., px∗ = y whenever x∗ ∈ x(p, y)), and that the associated Slutskymatrix be symmetric and negative semidefinite, provide a set of restrictions on allow-able values for the parameters in any empirically estimated ordinary demand system- if that system is to be viewed as belonging to a price-taking, utility-maximizingconsumer. There are other testable restrictions implied by the theory

28


Definition 2.9.1 (Demand Elasticities and Income Shares) Let xi(p, y) be theconsumer’s ordinary demand for good i. Then let

ηi ≡ ∂xi(p, y)∂y

y

xi(p, y)

εij ≡ ∂xi(p, y)∂pj

pj

xi(p, y)

and let

si ≡ pixi(p, y)y

so that si ≥ 0 andn∑

i=1

si = 1.

The symbol ηi denotes the income elasticity of demand for good i, and measuresthe percentage change in the quantity of i demanded per 1 percent change in income.The symbol εij denotes the price elasticity of demand for good i, and measures thepercentage change in the quantity of i demanded per 1 percent change in the pricepj. If j = i, εii is called the own-price elasticity of demand for good i. If j �= i, εij iscalled the cross-price elasticity of demand for good i with respect to pj . The symbolsi denotes the income share, spent on purchases of good i.

Theorem 2.9.1 (Aggregation in Consumer Demand) Let x(p, y) be the con-sumer’s demand. Then, the following relations must hold among income shares,price, and income elasticities of demand:

1. Engel aggregation:∑n

i=1 siηi = 1.

2. Cournot aggregation:∑n

i=1 siεij = −sj, j = 1, . . . , n.

Proof of Theorem 2.9.1: (Engel aggregation): From Walras’ law, we havey = p · x(p, y) for all p and y. Differentiating this with respect to y yields

1 =n∑

i=1

pi∂xi

∂y.

Multiplying and dividing each element in the summation by xi, y, we obtain

1 =n∑

i=1

pixi

y

∂xi

∂y

y

xi.

Using the notations introduced above, we have

1 =n∑

i=1

siηi.

29


(Cournot aggregation): Differentiating y =∑n

i=1 pixi(p, y) with respect to pj

yields

0 =

⎡⎣∑

i�=j

pi∂xi

∂pj

⎤⎦ + xj + pj

∂xj

∂pj.

Note that y = pjxj(p, y) +∑

i�=j pixi(p, y). We can rearrange the above expressionas follows:

−xj =n∑

i=1

pi∂xi

∂pj.

Multiplying pj/y on the both hand sides, we obtain

−pjxj

y=

n∑i=1

pi

y

∂xi

∂pjpj.

Rearranging the above expression, we get

−pjxj

y=

n∑i=1

pixi

y

∂xi

∂pj

pj

xi.

Using the notations introduced above, we have

−sj =n∑

i=1

siεij, j = 1, . . . , n. �

2.10 Weak Axiom of Revealed Preference (WARP)

So far, we have approached demand theory by assuming that the consumer has pref-erences satisfying certain properties (completeness, transitivity, and monotonicity);then we have tried to deduce all of the observable properties of market demand thatfollow as a consequence (homogeneity of degree 0 in (p, y), Walras’ law, symmetryand negative semidefiniteness of the Slutsky matrix). Thus, we have begun by as-suming something we cannot observe - preferences - to ultimately make predictionsabout something we can observe - consumer demand behavior.

In his remarkable “Foundations of Economic Analysis,” Paul Samuelson (1947)suggested an alternative approach. Why not start and finish with observable behav-ior? He showed how virtually every prediction ordinary consumer theory makes canalso be derived from a few simple and sensible assumptions about the consumer’sobservable choices themselves, rather than about his unobservable preferences.

30


Definition 2.10.1 (WARP) A consumer’s choice behavior satisfies WARP if forevery distinct pair of bundles x, x

′for which x is chosen at p and x

′is chosen at p

′,

px′ ≤ px =⇒ p

′x > p

′x

′.

In other words, WARP holds if whenever x is revealed preferred to x′, x

′is never

revealed preferred to x.

Suppose that a consumer’s choice behavior satisfies WARP. Let x(p, y) denotethe choice made by this consumer when faced with prices p and income y. Inaddition to WARP, we will requires the consumer’s choice to satisfy Walras’ law,i.e., p · x(p, y) = y.

Proposition 2.10.1 If x(p, y) satisfies WARP and Walras’ law, then it is homoge-neous of degree 0 in (p, y).

Proof of Proposition 2.10.1: Suppose that x(p, y) is chosen when prices arep and income y, and suppose that x(p

′, y

′) is chosen when prices are p

′= αp and

income is y′= αy for α > 0. What we want is x(p, y) = x(p

′, y

′). From Walras’ law,

we have

p′ · x(p

′, y

′) = y

′

= αy

= αp · x(p, y)

Since p′= αp, we have

αp · x(p′, y

′) = αp · x(p, y). (∗)

If x(p, y) = x(p′, y

′), the above equality (∗) is satisfied and we obtain what we

want. Thus, assume, on the contrary, that x(p, y) �= x(p′, y

′). Note that x(p, y) is

affordable under (p′, y

′), i.e., p

′ · x(p, y) ≤ p′ · x(p

′, y

′). Then, WARP implies that

p · x(p′, y

′) > p · x(p, y). However, this contradicts the equality (∗). �

Proposition 2.10.2 (The Compensated Law of Demand) Suppose that the choicefunction x(p, y) is homogeneous of degree zero and satisfies Walras’ law. Then x(p, y)satisfies the WARP if and only if the following property holds:

For any compensated price change from an initial situation (p, y) to a new price-income pair (p

′, y

′) = (p

′, p

′ · x(p, y)), we have

(p′ − p) ·

[x(p

′, y

′) − x(p, y)

]≤ 0,

with strict inequality whenever x(p, y) �= x(p′, y

′).

31


Proof of Proposition 2.10.2: (WARP ⇒ the compensated law of demand):If x(p

′, y

′) = x(p, y), it follows that (p − p

′) · [x(p

′, y

′) − x(p, y)] = 0. Then we are

done. Thus, assume that x(p, y) �= x(p′, y

′). Consider the following.

(p′ − p) ·

[x(p

′, y

′) − x(p, y)

]= p

′ · [x(p′, y

′) − x(p, y)] − p · [x(p

′, y

′) − x(p, y)]

= −p · [x(p′, y

′) − x(p, y)] (∵ p

′ · x(p′, y

′) = y

′from Walras’ law and y

′= p

′x(p, y))

= −[p · x(p′, y

′) − p · x(p, y)]

= −[p · x(p′, y

′) − y] (by Walras’ law)

Since p′ · x(p, y) = y

′, x(p, y) is affordable under (p

′, y

′). The WARP implies that

x(p′, y

′) must not be affordable under (p, y). Hence, we have p · x(p

′, y

′) > y. Since

p · x(p, y) = y from Walras’ law, we obtain the desired inequality.

(The compensated law of demand ⇒ WARP): Suppose, on the contrary, that theWARP is not satisfied. Then, the following properties are given:

1. p · x(p, y) = y (Walras’ law)

2. p′ · x(p, y) = y

′(The Slutsky compensation)

3. x(p, y) �= x(p′, y

′) (WARP)

4. p · x(p′, y

′) ≤ y (our hypothesis for contradiction)

Then, using the properties used in the first part of the proof, we have

(p′ − p) ·

[x(p

′, y

′) − x(p, y)

]= −p · [x(p

′, y

′) − x(p, y)]

≥ 0 (∵ p · x(p, y) = y and p · x(p′, y

′) ≤ y.)

< 0 (∵ the compensated law of demand if x(p, y) �= x(p′, y

′))

⇒⇐ Contradiction!

Thus, we complete the proof. �

Proposition 2.10.3 If a differentiable choice function x(p, y) satisfies Walras’ law,homogeneity of degree zero, and the WARP, then at any (p, y), the Slutsky matrixS(p, y) is negative semi-definite, i.e., z · S(p, y)z ≤ 0 for any z ∈ Rn.

Proof of Proposition 2.11.3: If the choice function x(p, y) is differentiable,the compensated law of demand is summarized as “dp · dx ≤ 0.” Proposition 2.10.2tells us that

dp · dx ≤ 0.

32


Using the chain rule, the differential change in demand induced by this compensatedprice change can be written as

dx = Dpx(p, y)dp + Dyx(p, y)dy

=[Dpx(p, y) + Dyx(p, y) (x(p, y))

′]dp (because dy = (x(p, y))

′dp)

Then, we obtain

dp · dx ≤ 0 ⇐⇒ dp[Dpx(p, y) + Dyx(p, y) (x(p, y))

′]dp ≤ 0

⇐⇒ dp · S(p, y)dp ≤ 0 �

You should understand this differential version of the compensated law of demandis equivalent to the negative semidefiniteness of the Slutsky matrix. However, thecompensation we have to take care is not the Hicksian compensation but the Slutskycompensation. The Slutsky compensation means that we consider any price changeunder which the consumer’s original bundle (not utility!) is just affordable.

So far, we have seen that if a choice function satisfies WARP and Walras’ law,then homogeneity of degree zero and negative semidefiniteness of the Slutsky matrixare implied by utility maximization. Then, a natural question is whether symmetryof the Slutsky matrix are also implied by WARP and Walras’ law. The answer is“yes” if there are only two goods in the economy and “no” in general.

Exercise 2.10.1 (Hicks (1957)) In a three-commodity world, consider the threebudget sets determined by the price vectors p1 = (2, 1, 2), p2 = (2, 2, 1), and p3 =(1, 2, 2) and income y = 8 (the same for the three budgets). Suppose that the respec-tive unique choices are x1 = (1, 2, 2), x2 = (2, 1, 2), and x3 = (2, 2, 1). Verify thatany two pairs of choices satisfy the WARP but that x3 is reveled preferred to x2, x2

is revealed preferred to x1, and x1 is revealed preferred to x3. This implies that therevealed preference is not transitive.

The next question is as follows: How must we strengthen WARP to obtain atheory of revealed preference that is equivalent to the theory of utility maximization?The answer lies in the “Strong Axiom of Revealed Preference.”

Definition 2.10.2 The choice function x(p, y) satisfies the strong axiom of re-vealed preference (the SARP) if for any list, (p1, y1), . . . , (pK , yK) with x(pk+1, yk+1) �=x(pk, yk) for all k ≤ K−1, we have pK ·x(p1, y1) > yK whenever pk ·x(pk+1, yk+1) ≤yk for all k ≤ K − 1.

Then, we have the following result. Just accept the result.

Proposition 2.10.4 (Houthakker (1950) and Richter (1966)) If the choice func-tion x(p, y) satisfies the SARP, then there is a complete and transitive preference �such that for all (p, y), x(p, y) � z for every z �= x(p, y) with z ∈ B(p, y).

33


Proof of Proposition 2.10.4: Define a relation �1 on commodity vectors byletting x �1 x

′whenever x �= x

′and we have x = x(p, y) and p · x

′ ≤ y for some(p, y). The relation �1 can be read as “directly revealed preferred to.” From �1

define a new relation �2, to be read as “directly or indirectly revealed preferred to,”by letting x �2 x

′whenever there is a chain x1 �1 x2 �1 · · · �1 xN with x1 = x

and xN = x′. Observe that, by construction, �2 is transitive. According to the

SA, �2 is also irreflexive (i.e., x �2 x is impossible). A certain axiom of set theory(known as Zorn’s lemma) tells us the following: Every relation �2 that is transitiveand irreflexive (called a partial order) has a total extension �3, an irreflexive andtransitive relation such that, first, x �2 x

′implies x �3 x

′and, second, whenever

x �= x′, we have either x �3 x

′or x

′ �3 x. Finally, we can define � by letting x � x′

whenever x = x′or x �3 x

′. It is not difficult now to verify that � is complete and

transitive and that x(p, y) � x′whenever p · x′ ≤ y and x

′ �= x(p, y). �

2.11 Appendix: Integrability

If a continuously differentiable demand function x(p, y) is generated by rational pref-erences, then we have seen that it must be homogeneous of degree zero, satisfyWalras’ law, and have a substitution matrix S(p, y) that is symmetric and negativesemidefinite (NSD) at all (p, y). We now pose the reverse question: If we observea demand function x(p, y) that has these properties, can we find preferences thatrationalize x(·). As we show in this section, the answer is yes; these conditions aresufficient for the existence of rational generating preferences. This problem, knownas the integrability problem, has a long tradition in economic theory.

This result tells us that not only are the properties of homogeneity of degree zero,satisfaction of Walras’ law, and a symmetric and negative semidefinite substitutionmatrix necessary consequences of the preference-based demand theory, but these arealso all of its consequences. As long as consumer demand satisfies these properties,there is some rational preference relation that could have generated this demand.The problem of recovering preferences � from x(p, y) can be subdivided into twoparts: (i) recovering an expenditure function e(p, u) from x(p, y), and (ii) recoveringpreferences from the expenditure function e(p, u).

2.11.1 Recovering Preferences from the Expenditure Function

Proposition 2.11.1 Suppose that e(p, u) is strictly increasing in u and is continu-ous, increasing, homogeneous of degree one, concave, and differentiable in p. Then,for every utility level u, e(p, u) is the expenditure function associated with the at-least-as-good-as set

Vu ={x ∈ Rn

+ | p · x ≥ e(p, u) ∀p � 0}

.

That is, e(p, u) = min{p · x | x ∈ Vu} for all p � 0.

34


2.11.2 Recovering the Expenditure Function from Demand

35

Chapter 3

Production

Many aspects enter a full description of a firm: Who owns it? Who manages it?How is it managed? How is it organized? How is it financed? What can it do? Ofall these questions, we concentrate on the last one, “what can the firm do?” Ourjustification is not that the other questions are not interesting (indeed, they are),but that we want to arrive as quickly as possible at a minimal conceptual apparatusthat allows us to analyze market behavior. Then, the firm is viewed merely as a“black box,” able to transform inputs into outputs.

The most general way is to think of the firm as having a production possibilityset, Y ⊂ Rm, where each vector y = (y1, . . . , ym) ∈ Y is a production plan whosecomponents indicate the amounts of the various inputs and outputs. We write ele-ments of y ∈ Y so that yi < 0 if resource i is used up in the production plan, andyi > 0 if resource i is produced in the production plan.

3.1 Properties of Production Sets

1. Y is nonempty. Otherwise, there is no production problem!

2. Y is closed. Consider a sequence {yk} converging to y for which yk ∈ Y foreach k. If Y is closed, y ∈ Y .

3. Y satisfies no free lunch if, whenever y ∈ Y and y ≥ 0, then y = 0. It is notpossible to produce something from nothing. Geometrically, Y ∩ Rn

+ ⊂ {0}.4. Y has the possibility of inaction if 0 ∈ Y . In particular, Y is then nonempty.

Because the firm always has the option of producing nothing.

5. Y satisfies free disposal if y ∈ Y and y′ ≤ y, then y

′ ∈ Y . Namely, Y − Rn+.

The extra amount of inputs can be disposed of or eliminated at no cost. Thisassumption is important for the competitive equilibrium price to be nonnega-tive.

36

CHAPTER 3. PRODUCTION

6. Y exhibits nonincreasing returns to scale if for any y ∈ Y , we have αy ∈ Y forany α ∈ [0, 1].

7. Y exhibits nondecreasing returns to scale if for any y ∈ Y , we have αy ∈ Y forany α ≥ 1.

8. Y exhibits constant returns to scale if y ∈ Y implies that αy ∈ Y for anyα ≥ 0.

9. Y is convex if for any y, y′ ∈ Y , we have αy + (1− α)y

′ ∈ Y for any α ∈ [0, 1].

10. Y is strictly convex if for any y, y′ ∈ Y with y �= y

′, we have αy + (1 − α)y

′ ∈Int(Y ) for any α ∈ (0, 1). Here Int(Y ) denotes the interior of Y .

11. Y is additive if for any y, y′ ∈ Y , we have y + y

′ ∈ Y . Additivity is relatedto the idea of entry. If y ∈ Y is being produced by a firm and another firmenters and produces y

′ ∈ Y , then the net result is the vector y +y′. Hence, the

aggregate production set must satisfy additivity whenever free entry is possible.Then, the number of firms in the market will be determined at the point whereall firms make zero profit.

12. Y is a convex cone if for any y, y′ ∈ Y and any α ≥ 0, and any β ≥ 0, we have

αy + βy′ ∈ Y .

Proposition 3.1.1 The production set Y is additive and exhibits the nonincreasingreturns to scale if and only if it is a convex cone.

Proof of Proposition 3.1.1: (⇐=) Fix y, y′ ∈ Y . Let α ∈ [0, 1] and β = 0.

Since Y is convex cone, we have αy + βy′

= αy ∈ Y . This implies that Y isnonincreasing returns to scale. Fix y, y

′ ∈ Y . Let α = 1 and β = 1. Since Y isconvex cone, we have αy + βy

′= y + y

′ ∈ Y , which shows the additivity of Y .(=⇒) Fix y, y

′ ∈ Y and α > 0 and β > 0. Let k > max{α,β}. By additivity of Y ,ky ∈ Y and ky

′ ∈ Y . Since (α/k) < 1 by construction and thus αy = (α/k)ky, thenonincreasing returns to scale of Y implies that αy ∈ Y . Similarly, βy

′ ∈ Y . Finally,again by additivity of Y , αy + βy

′ ∈ Y as desired. �

Here is a more important thing than the proof per se: If Y is convex cone, Yis, in particular, convex. Thus, Proposition 3.1.1 is a justification for the convexityassumption in production

It is sometimes convenient to describe the production set Y using a functionF (·), called the transformation function. The transformation function F (·) has theproperty that Y = {y ∈ Rm | F (y) ≤ 0} and F (y) = 0 if and only if y is an elementof the boundary of Y . The set of boundary points of Y, {y ∈ Rm | F (y) = 0}, isknown as the transformation frontier.

37


If F (·) is differentiable, and if the production vector y satisfies F (y) = 0, thenfor any commodities i and j, the ratio

MRTSij(y) =∂F (y)/∂yi

∂F (y)/∂yj

is called the marginal rate of technical substitution (MRTS) of good i for good j aty.

Exercise 3.1.1 Let F (y) = 3y1 + y2. Define Y = {y ∈ R2| F (y) ≤ 0}. Draw thegraph of Y . What is the marginal rate of technical substitution?

3.2 Production Functions

The production set is by far the most general way to characterize the firm’s tech-nology because it allows for multiple inputs and multiple outputs. However, for thecase in which the firm produces only a single output from many inputs, it is moreconvenient to describe the firm’s technology in terms of production function.

Then, we shall denote the amount of output by y, and the amount of input i byxi, so that with n inputs, the entire vector of inputs is denoted by x = (x1, . . . , xn).Of course, the input vector as well as the amount of output must be nonnegative,so we require x ≥ 0 and y ≥ 0. The production function, f , is therefore a mappingfrom Rn

+ → R+ with a generic element y = f(x). We shall maintain the followingassumption on the production function f(·).

Assumption 3.2.1 (Properties of the Production Function) The productionfunction, f : Rn

+ → R+, is

• continuous (cf. closedness of Y ),

• increasing: x′ ≥ x ⇒ f(x

′) ≥ f(x) and x

′ � x ⇒ f(x′) > f(x),

• strictly quasi-concave (cf. Strict convexity of Y ), and

• f(0) = 0 (No free lunch and the possibility of inaction).

Exercise 3.2.1 Show that for a single-output technology, the production set Y isconvex if and only if the production function f(·) is concave.

When the production function is differentiable, its partial derivative, ∂f(x)/∂xi,is called the marginal product of input i and gives the rate at which output changesper additional unit of input i employed. For any fixed level of output, y, the set ofinput vectors producing y units of output is called the y-level isoquant. An isoquantis then just a level set of f , which is denoted as Q(y) as follows:

Q(y) ≡ {x ∈ Rn+ | f(x) = y}.

38


For the case of production function, the marginal rate of technical substitutionof good i for good j at x is given as follows:

MRTSij(x) =∂f(x)/∂xi

∂f(x)/∂xj.

The MRTS is one local measure of substitutability between inputs in producing agiven unit of output. Since economists favor unit-free elasticities as such things, themost common one is the elasticity of substitution, σ. Then, σ measures curvature ofan isoquant.

Definition 3.2.1 (The Elasticity of Substitution) For a production function f(x),the elasticity of substitution between inputs i and j at the point x is defined as

σij ≡ MRTSij

xj/xi

d(xj/xi)d(MRTSij)

=d(xj/xi)xj/xi

fi(x)/fj(x)d(fi(x)/fj(x))

=d ln(xj/xi)

d ln (fi(x)/fj(x))

Let y = xj/xi. Note that

d ln(y) =∂ ln y

∂y· dy

=dy

y= d(xj/xi)/(xj/xi).

Exercise 3.2.2 Consider the following CES (constant elasticity of substitution) pro-duction f : R2

+ → R+ with the following form:

f(x) = [α1xρ1 + α2x

ρ2]

1/ρ,

where 0 �= ρ = 1 and α1, α2 > 0. Answer the following questions:

1. Show that when ρ = 1, isoquant curves are linear. What is the elasticity ofsubstitution of this technology?

2. Show that as ρ → 0, this production function f(·) comes to represent f(x) =xα1

1 xα22 . Here assume that α1 + α2 = 1 (Hint: Use L’Hopital’s rule). What is

the elasticity of substitution of this technology?

3. Show that as ρ → −∞, the production function becomes f(x) = min{x1, x2}.What is the elasticity of substitution of this technology?

39


3.2.1 More about Production Functions

A function f : Rn+ → R+ is said to be homogeneous of degree one if, for any x ∈ Rn

+

and any α > 0, f(αx) = αf(x). In particular, the CES production function whichis widely used in empirical researches, is homogeneous of degree one. Do you seewhy? The next theorem says that under the assumption that f(·) is continuous,increasing, and has the property that f(0) = 0, homogeneity of degree one as well asquasiconcavity implies concavity. Note that concavity always implies quasiconcavity.To see this, you are referred to my lecture note on mathematics.

Theorem 3.2.1 Let f(·) be a production function that is continuous, increasing,strictly quasi-concave with the property that f(0) = 0. If f(·) is homogeneous ofdegree one, then f(·) is concave.

Proof of Theorem 3.2.1: Take any x � 0 and x′ � 0 and let y = f(x) and

y′

= f(x′). Then y, y

′> 0 because f(0) = 0 and f(·) is increasing. Since f(·) is

homogeneous of degree one, we have

f(αx) = αy ∀α > 0 and f(βx′) = βy

′ ∀β > 0.

Plugging α = 1/y and β = 1/y′into the above equations, respectively, we obtain

f

(x

y

)= f

(x

′

y

)= 1.

Since f(·) is quasi-concave, we have

f

(γx

y+

(1 − γ)x′

y′

)≥ 1 for all γ ∈ [0, 1].

Choosing γ = y/(y + y′), we rearrange the above equation as follows:

f

(x

y + y′ +x

′

y + y′

)≥ 1.

Again, appealing to homogeneity of degree one, we have

f(x + x′) = f

((y + y

′)

{x

y + y′ +x

′

y + y′

})

= (y + y′)f

(x

y + y′ +x

′

y + y′

)

≥ y + y′

(∵ f

(x

y + y′ +x

′

y + y′

)≥ 1

)

= f(x) + f(x′) (∗).

40


Thus, the above equation (∗) holds for all x, x′ � 0. But the continuity of f guar-

antees that this (∗) also holds for all x, x′ ≥ 0. Consider any two vectors x, x

′ ≥ 0and any γ ∈ [0, 1]. Homogeneity of degree one of f ensures that

f(γx) = γf(x),

f((1 − γ)x

′)= (1 − γ)f(x

′).

Taking into account the equation (∗) which is satisfied for any x, x′ ≥ 0, we have

f(γx + (1 − γ)x

′) ≥ γf(x) + (1 − γ)f(x′),

as desired. �

Exercise 3.2.3 Suppose that f(·) is the production function associated with a single-output technology, and let Y be the production set of this technology. Show that Ysatisfies constant returns to scale if and only if f(·) is homogeneous of degree one.

3.2.2 Returns to Scale of the Production Function

Definition 3.2.2 A production function f(·) is

1. constant returns to scale if f(αx) = αf(x) for all α > 0 and all x ∈ Rn+,

2. increasing returns to scale if f(αx) > αf(x) for all α > 1 and all x ∈ Rn+,

3. decreasing returns to scale if f(αx) < αf(x) for all α > 1 and all x ∈ Rn+.

3.3 Cost Minimization

If the objective of the firm is to maximize profits, it will necessarily choose the leastcostly, or cost-minimizing, production plan for every level of output. We will assumethroughout that firms are perfectly competitive on their input markets and thattherefore they face fixed input prices and take them as given. This is indeed theprice-taking behavior. Let w = (w1, . . . , wn) ≥ 0 be a vector of prevailing marketprices at which the firm can buy inputs x = (x1, . . . , xn).

Definition 3.3.1 The cost function, defined for all input prices w � 0 and alloutput levels y ∈ f(Rn

+) is the minimum-value function,

c(w, y) ≡ minx∈ n

+

w · x subject to f(x) ≥ y.

If x(w, y) solves the cost minimization problem, then

c(w, y) = w · x(w, y).

41


If f(·) is increasing, the constraint will be always binding at a solution. Conse-quently, the cost minimization problem (CMP) is equivalent to

minx∈ n

+

w · x subject to f(x) = y.

Let x∗ be the solution to the CMP. To keep things simple, we will assume x∗ � 0, andthat f(·) is differentiable at x∗ with ∇f (x∗) � 0. Thus, we are able to characterizethe following FOCs: There is a λ∗ ∈ R such that

wi = λ∗∂f(x∗)∂xi

, i = 1, . . . , n

Since w � 0, we have

∂f(x∗)/∂xi

∂f(x∗)/∂xj=

wi

wj.

Thus, cost minimization implies that the marginal rate of substitution between anytwo inputs is equal to the ratio of their prices. The solution x(w, y) to the CMP isreferred to as the firm’s conditional input demand, because it is conditional on thelevel of output y, which at this point is arbitrary and so may or may not be profitmaximizing.

With two inputs, an interior solution corresponds to a point of tangency betweenthe y-level isoquant and an isocost line of the form w · x = α for some α > 0. Ifx1(w, y) and x2(w, y) are solutions, then c(w, y) = w1x1(w, y) + w2x2(w, y).

Theorem 3.3.1 (Properties of the Cost Function) If f is continuous and in-creasing, then c(w, y) is

1. Zero when y = 0.

2. Continuous on its domain.

3. For all w � 0, increasing and unbounded above in y.

4. Increasing in w.

5. Homogeneous of degree one in w.

6. Concave in w.

7. Shephard’s lemma: Assume further that f(·) is strictly quasi-concave. Thenc(w, y) is differentiable in w at (w0, y0) whenever w0 � 0, and

∂c(w0, y0)∂wi

= xi(w0, y0), i = 1, . . . , n.

42


Exercise 3.3.1 Prove Theorem 3.3.1. The proof must be similar to that of Theorem2.6.1 (Properties of the Expenditure Function).

Theorem 3.3.2 (Properties of Conditional Input Demands) Suppose the pro-duction function is continuous, increasing, and strictly quasi-concave and satisfiesthe property that f(0) = 0, and that the associated cost function is twice continuouslydifferentiable. Then

1. x(w, y) is homogeneous of degree zero in w.

2. The substitution matrix, defined and denoted

σ∗(w, y) ≡

⎛⎜⎜⎝

∂x1(w,y)∂w1

· · · ∂x1(w,y)∂wn

.... . .

...∂xn(w,y)

∂w1· · · ∂xn(w,y)

∂wn

⎞⎟⎟⎠ ,

is symmetric and negative semi-definite. In particular, the negative semidefi-niteness property implies that ∂xi(w, y)/∂wi ≤ 0 for all i = 1, . . . , n.

Exercise 3.3.2 Prove Theorem 3.3.2. The latter part of the proof is similar to thatof Theorem 2.9.6.

3.3.1 Properties of the Cost Functions

Definition 3.3.2 An increasing function F : Rn+ → R is said to be homothetic if,

there exist an increasing function f : R → R and a function g : Rn+ → R which is

homogeneous of degree one such that F (x) = f(g(x)) for any x ∈ Rn+.

Proposition 3.3.1 (A Characterization of Homothetic Functions) An increas-ing function F : Rn

+ → R is homothetic only if, whenever F (x) = F (y), thenF (αx) = F (αy) for any α ≥ 0. 1

Proof of Proposition 3.3.1: Assume that F (x) = F (y). I execute the followingseries of deductions.

F (x) = F (y) =⇒ f(g(x)) = f(g(y)) (by definition)=⇒ g(x) = g(y) (because f(·) is increasing.)=⇒ αg(x) = αg(y) ∀ α ≥ 0=⇒ g(αx) = g(αy) (because g(·) is homogeneous of degree one.)=⇒ f(g(αx)) = f(g(αy))=⇒ F (αx) = F (αy) (by definition) �

Homotheticity of the production function means that all isoquant sets (in partic-ular, curves when n = 2) are related by proportional expansion along rays. Consider,

1In the Appendix, I introduce an extra condition under which if part can also be proved.

43


for example, that F (x) = xα1 x

(1−α)2 , where α ∈ (0, 1). You can check this F (·) is

homothetic.

Theorem 3.3.3 When the production function F (·) ≡ f(g(·)) is continuous, in-creasing, strictly quasi-concave, and homothetic and satisfies the property that f(0) =0.

• the cost function c(·) is multiplicatively separable in input prices and outputand can be written c(w, y) = h(y)c(w,1), where h(·) is increasing and c(w, 1)is the unit cost function.

• the conditional input demands x(·) are multiplicatively separable in input pricesand output and can be written x(w, y) = h(y)x(w, 1), where h

′(y) > 0 and

x(w, 1) is the conditional input demand for one unit of output.

Proof of Theorem 3.3.3: Let F (·) denote the production function which isincreasing. Because it is homothetic, it can be written as F (x) = f(g(x)), wheref(·) is strictly increasing, and g(·) is homogeneous of degree one. First, I claim thefollowing:

Claim 3.3.1 f−1(y) > 0 for all y > 0. For all y > 0, there exists x ∈ Rn+ such that

y = F (x).

This is straightforward from the properties that f(·) is increasing and f(0) = 0.For simplicity, we shall assume that the image of F is all of R+. 2 Fix such y > 0.Let α = f−1(1)/f−1(y) > 0. Consider the following chain of relations:

F (x) ≥ y

⇔ f(g(x)) ≥ y (∵ F (x) = f(g(x)))⇔ g(x) ≥ f−1(y)⇔ g(αx) ≥ αf−1(y) (∵ g(·) is homogeneous of degree one)⇔ g(αx) ≥ f−1(1) (∵ α = f−1(1)/f−1(y))⇔ f(g(αx)) ≥ 1

Accordingly, we may express the cost function as follows:

c(w, y) = minx∈ n

+

w · x subject to f(g(x)) ≥ y

= minx∈ n

+

w · x subject to f(g(αx)) ≥ 1

=1α

minx∈ n

+

w · αx subject to f(g(αx)) ≥ 1

=1α

minz∈ n

+

w · z subject to f(g(z)) ≥ 1

=f−1(y)f−1(1)

c(w, 1),

2The image of F is all of + if for any y ∈ +, there exists x ∈ n+ such that f(x) = y.

44


where we let z ≡ αx. When defining h(y) = f−1(y)/f−1(1), we complete the prooffor any y > 0. Since F (0) = 0 and g(·) is homogeneous of degree one, we know thatc(w, 0) = 0 and g(0) = 0. This implies that the result holds for any y ≥ 0. �

When the firm is “stuck” with fixed amounts of certain inputs in the short run,rather than being free to choose those inputs optimally as it can in the long run, weshould expect its costs in the short run to differ from its costs in the long run.

Definition 3.3.3 (The Short-Run Cost Function) Let the production functionbe f(z), where z ≡ (x, x). Suppose that x is a subvector of variable inputs and x is asubvector of fixed inputs. Let w and w be the associated input prices for the variableand fixed inputs, respectively. The short-run total cost function is defined as

SC(w, w, y; x) ≡ minx

w · x + w · x subject to f(x, x) ≥ y.

If x(w, w, y; x) solves this minimization problem, then

SC(w, w, y; x) = w · x(w, w, y; x) + w · x.

The optimized cost of the variable inputs, w · x(w, w, y; x), is called total variablecost. The cost of the fixed inputs, w · x, is called total fixed cost.

Let x(y) denote the optimal choice of the fixed inputs to minimize short-run costof output y at the given input prices. The following must be true:

c(w, w, y) ≡ SC(w, w, y; x(y)) ∀y > 0.

Furthermore, because we have chosen the fixed inputs to minimize short-runcosts, the optimal amounts x(y) must satisfy the FOCs for a minimum:

∂SC(w, w, y; x(y))∂xi

≡ 0,

for all fixed inputs i. Now differentiate the above identity with respect to y, we have

dc(w, w, y)dy

=

direct effect︷︸︸︷∂SC(w, w, y; x(y))

∂y+

indirect effect︷︸︸︷∑i

∂SC(w, w, y; x(y))∂xi︸︷︷︸

=0 ∀i

∂xi(y)∂y

︸︷︷︸=0 as a result

=∂SC(w, w, y; x(y)

∂y.

We summarize: First, the short-run cost minimization problem involves moreconstraints on the firm than the long-run problem, so we know that SC(w, w, y; x) ≥c(w, w, y) for all levels of output and levels of the fixed inputs. Second, for every

45


level of output, the short-run and long-run costs will coincide for some short-runcost function associated with some level of the fixed inputs. Finally, the slope ofthis short-run cost function will be equal to the slope of the long-run cost functionin the cost-output plane. In other words, the long-run total cost curve is the lowerenvelope of the entire family of short-run total cost curves!

3.4 Profit Maximization

Profit is the difference between revenue from selling output and the cost of acquiringthe inputs necessary to produce it. The competitive (i.e., price-taking) firm cansell each unit of output at the market price, p. Its revenues are therefore a simplefunction of output, R(y) = py. Here we assume that the objective of the firm is tomaximize its profit. Then, the firm’s profit maximization problem (PMP) is givenas follows:

max(x,y)≥0

py − w · x subject to f(x) ≥ y.

where f(·) is a production function which is continuous, increasing, and strictlyquasi-concave and satisfies the property that f(0) = 0. Since f(·) is increasing, theconstraint is binding, that is, f(x) = y. Then, the PMP is rewritten as follows.

maxx∈ n

+

pf (x) − w · x.

Let x∗ be the solution to the PMP under (p,w). Assume that x∗ � 0. Then, Ican characterize the solution to the PMP by way of the first order conditions (FOCs).Namely,

p∂f(x∗)

∂xi= wi, ∀i = 1, . . . , n.

Assuming further that w � 0, we derive the following implication from the FOCsfor the profit maximization:

MRTSij(x∗) =∂f(x∗)/∂xi

∂f(x∗)/∂xj=

wi

wj, ∀i, j.

This means that

Profit Maximization ⇒ Cost Minimization.

This fact allows us to simply the PMP further.

maxy≥0

py − c(w, y)

Let y∗ = f(x∗). The FOC for the profit maximization is reduced to

p − dc(w, y∗)dy

= 0.

46


The profit maximizing choice of output, y∗ ≡ y(p,w), is called the firm’s outputsupply function, and the profit maximizing choice of inputs, x∗ ≡ x(p,w), gives thevector of firm input demand functions.

Definition 3.4.1 The firm’s profit function depends only on input and outputprices and is defined as the maximum value function,

π(p,w) ≡ max(x,y)≥0

py − w · x subject to f(x) ≥ y.

Note that the price system may be such that there is no bound on how highprofits may be. Suppose, for example, that a firm with constant returns to scaletechnology produces one unit of a single output (the price is p) for every unit of asingle input (the price is w). Then, π(p,w) = 0 whenever p ≤ w. But π(p,w) = +∞if p > w.

Theorem 3.4.1 (Properties of the Profit Function) Suppose that the produc-tion function f(·) is continuous, increasing, and strictly quasi-concave and satisfiesthe property that f(0) = 0. Then, for p ≥ 0 and w ≥ 0, the profit function π(p,w),where well-defined, is continuous and

1. increasing in p.

2. decreasing in w.

3. homogeneous of degree one in (p,w).

4. convex in (p,w).

5. If differentiable in (p,w) � 0, Hotelling’s lemma follows:

∂π(p,w)∂p

= y(p,w) and∂π(p,w)

∂wi= −xi(p,w), ∀i = 1, . . . , n.

Proof of Theorem 3.4.1: I prove only convexity of π and Hotelling’s lemma.The proofs of the rest of the properties are left as exercise.

Convexity of π(p,w): Let (x, y) and (x′, y

′) be the solutions to the PMP under

(p,w) and under (p′, w

′), respectively. Thus, we have

π(p,w) = py − w · x (3.1)π(p

′, w

′) = p

′y′ − w

′ · x′. (3.2)

47


Let pα = αp + (1− α)p′and wα = αw + (1− α)w

′for α ∈ [0, 1]. Let (xα, yα) be

the solution to the PMP under (pα, wα). Then, we do the following computation:

π(pα, wα) = pαyα − wα · xα

=[αp + (1 − α)p

′]yα −

[αw + (1 − α)w

′] · xα

= α [pyα − w · xα] + (1 − α)[p′yα − w

′ · xα]

≤ α [py − w · x] + (1 − α)[p′y′ − w

′ · x′]= απ(p,w) + (1 − α)π(p

′, w

′).

This completes the proof. �

Hotelling’s Lemma: Suppose that y∗ is a profit maximizing output vector andx∗ is a profit maximizing input vector at prices (p∗, w∗). Define the function

g(p,w) = π(p,w) −(

py∗ −n∑

i=1

wix∗i

).

Since the profit maximizing production plan at prices (p,w) will always be atleast as profitable as the production plan (x∗, y∗). However, the plan (x∗, y∗) will bea profit maximizing plan at prices (p∗, w∗), so the function g(·) reaches a minimumvalue of 0 at (p∗, w∗). Namely, g(p,w) ≥ 0 for all p,w and g(p∗, w∗) = 0. The FOCsfor a minimum then imply that

∂g(p∗, w∗)∂p

=∂π(p∗, w∗)

∂p− y∗ = 0 and

∂g(p∗, w∗)∂wi

=∂π(p∗, w∗)

∂wi+ x∗

i = 0 ∀i = 1, . . . , n.

We obtain the results. �

Exercise 3.4.1 Verify properties 1, 2, and 3 in Theorem 3.4.1.

3.4.1 More about Profit Maximization

Theorem 3.4.2 Let π(p,w) be a twice continuously differentiable profit function forsome competitive firm. Then, for all p > 0 and w � 0 where it is well defined:

1. Homogeneity of degree zero:

y(αp,αw) = y(p,w) ∀α > 0,

xi(αp,αw) = xi(p,w) ∀α > 0 and i = 1, . . . , n.

48


2. (Law of Supply) Own-price effects:

∂y(p,w)∂p

≥ 0,

∂xi(p,w)∂wi

≤ 0 ∀i = 1, . . . , n.

3. The (n + 1) × (n + 1) substitution matrix⎛⎜⎜⎜⎜⎜⎝

∂π(p,w)∂p2

∂π(p,w)∂w1∂p · · · ∂π(p,w)

∂wn∂p∂π(p,w)∂p∂w1

∂π(p,w)∂w2

1· · · ∂π(p,w)

∂wn∂w1

......

. . ....

∂π(p,w)∂p∂wn

∂π(p,w)∂w1∂wn

· · · ∂π(p,w)∂w2

n

⎞⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝

∂y(p,w)∂p

∂y(p,w)∂w1

· · · ∂y(p,w)∂wn

−∂x1(p,w)∂p −∂x1(p,w)

∂w1· · · −∂x1(p,w)

∂wn

......

. . ....

−∂xn(p,w)∂p −∂xn(p,w)

∂w1· · · −∂xn(p,w)

∂wn

⎞⎟⎟⎟⎟⎠

is symmetric and positive semidefinite.

Theorem 3.4.3 Let the production function be f(x, x), where x is a subvector ofvariable inputs and x is a subvector of fixed inputs. Let w and w be the associatedinput prices for variable and fixed inputs, respectively. The short-run profit functionis defined as

π(p,w, w, x) ≡ maxy,x

py − w · x − w · x subject to f(x, x) ≥ y.

The solutions y(p,w, w, x) and x(p,w, w, x) are called the short-run output supplyand variable input demand functions, respectively.

Exercise 3.4.2 Consider the following short-run profit maximization problem:

maxy,x1

py − w1x1 − w2x2 subject to xα1 x1−α

2 ≥ y,

where 0 < α < 1. Assume an interior solution to the short-run PMP. Answer thefollowing questions:

1. Derive the short-run output supply and variable input demand.

2. Derive the short-run profit function.

3. Confirm Hotelling’s lemma.

4. Show that the short-run profit function is convex in (p,w1).

49


3.5 Appendix

3.5.1 More on Homothetic Functions

Here, I shall follow “Homothetic functions revisited,” by P.O. Lindberg, E. AndersEriksson, and Lars-Goran Mattsson (henceforth, LEM), in Economic Theory, 2002,vol. 19, 417-427. This paper examines the proposition that homotheticity is equiva-lent to the property that the marginal rate of substitution is constant along any rayfrom the origin. This claim is made in many places, but hitherto the prerequisiteshave not been stated explicitly. Thus, they show that an additional condition (called“nowhere ray constancy”) is required for the claim to hold. It turns out that thiscondition is implied by assumptions often made in production theory.

The standard definition of homotheticity is that the real-valued function f ishomothetic if

f(x) = h(g(x)) (∗)

where g is homogeneous of degree one and h is strictly increasing. The economic ra-tionale behind homotheticity is that it is sufficient for marginal rates of substitutionto be constant along rays, viz.

fi(λx)fj(λx)

=fi(x)fj(x)

, ∀i �= j and ∀λ > 0. (2∗)

Many authors seem to believe that homotheticity is also necessary for Equation(2∗). It is easy to see, however, that (2∗) also holds for f homogeneous of degreezero, e.g., f(x) = x1/x2 for x ∈ R2

++. This function is not representable in the formof Equation (∗).

The ray through x �= 0 is defined by {x ∈ Rn|x = λx, λ > 0}. A union of a set ofrays (possibly with addition of the origin) is termed a cone. Assume that (2∗) to bevalid for some function f on a cone C in Rn. If fi(x) = 0, we must have fi(λx) = 0for all λ > 0. For all i with fi(x) �= 0, on the other hand, (∗∗) can be rewritten inthe form

fi(λx)fi(x)

=fj(λx)fj(x)

= k(λ,x).

Combining these two cases, we have

∇f (λx) = k(λ,x)∇f(x), (3∗)

i.e., the gradient at λx is parallel to that at x. We will say that a function f satisfying(3∗) for all nonzero x ∈ C and λ > 0, has ray parallel gradients. Note that usingcondition (3∗) rather than (2∗) we can drop the requirement that ∇f (x) �= 0. Ifequation (3∗) holds, then ∇f (x) = 0 implies ∇f (λx) = 0 for all λ > 0. We alsoneed some topological concepts. A set C is termed weakly solid if its interior, IntC ,

50


is connected and contains C in its closure. Connected open cones such as Rn\0, andconvex cones with nonempty interior, such as Rn

+ are weakly solid.We say that a function f is ray constant at x if f(λx) = f(x) for all λ > 0.

Further, we say that f is nowhere ray constant on a cone C in Rn if for all x ∈ Cwith x �= 0, there is λ > 0 such that f(λx) �= f(x). The set C\0 is denoted by C0.

Theorem 3.5.1 (LEM (2002)) Let the continuous function f be defined on aweakly solid cone C in Rn, and continuously differentiable on IntC0. Suppose thatf is nowhere ray constant. Then f is homothetic if and only if it has ray parallelgradients on IntC0.

I can restate the condition of nowhere ray constancy into a more technical onewhich is more effective in the proofs, viz, the following nonorthogonality condition:

∇f (x) · x �= 0 ∀x �= 0 (4∗)

The following lemma shows that this condition in our setting is equivalent to thecondition of nowhere ray constancy.

Lemma 3.5.1 (LEM (2002)) Let the continuously differentiable function f be de-fined on a cone K in Rn. If f fulfills the nonorthogonality condition (4∗), then it isnowhere ray constant. If moreover f has ray parallel gradients on K, then also thereverse implication holds.

A set D in Rn is called locally connected if for each x ∈ D (where D stands forthe closure of D) and every ε > 0, there is an open neighborhood Oε of x + εB(where B is the open unit ball) such that Oε ∩ D is connected.

In our situation with a function f defined on a cone C, let CE = {x ∈ C|f(λx) =f(x) ∀λ > 0} and CNE = C\CE. Note that CE and hence CNE are cones. Moreover,0 ∈ CE and hence CNE ⊆ C0. It turns out that CE is closed in C. When f isdifferentiable at x, then x ∈ CE implies ∇f (x) · x = 0.

Theorem 3.5.2 (LEM (2002)) Let the continuous function f be defined on aweakly solid cone C in Rn, and continuously differentiable on IntC0. Suppose thatIntC0 is locally connected and that f has ray parallel gradients on IntC0. Then

1. on CE, f is ray constant, and

2. CNE decomposes into a denumerable disjoint union CNE =⋃

i Ci of cones Ci,closed in CNE, such that f has a homothetic representation f(x) = hi(gi(x))on each Ci.

51

Chapter 4

Partial Equilibrium

4.1 Price and Individual Welfare

In this note, I consider an entire economy in which consumers and firms interactthrough perfectly competitive markets. The approach I take here is somewhat re-strictive. It is called the partial equilibrium approach, which envisions the marketfor a single good for which each consumer’s expenditure constitutes only a smallfraction of his overall budget. When this is the case, it is reasonable to assume thatchanges in the market for this good will leave the prices of all other commodities ap-proximately unaffected and that there will be, in addition, negligible income effectsin the market under investigation.

If the price of the good q is p, and the vector of all other prices is p, then Ishall simply write the consumer’s indirect utility function as v(p, y). Then, it will beconvenient to introduce a composite commodity, m, as the amount of income spenton all goods other than the current good q. The consumer’s optimization problemis given as follows:

maxq,m

u(q,m) subject to pq + m ≤ y,

and the maximized value of u is v(p, y).

It is often the case that the effect of a new policy essentially reduces to a changein prices that consumers face. Taxes and subsidies are obvious examples. Considera particular consumer whose income is y0. Suppose that the initial price of the goodis p0 and that it will fall to p1. Let v(p0, y0) the consumer’s utility before the pricefall and v(p1, y0) his utility after the price falls. Letting CV denote this change inthe consumer’s income that would leave him as well off after the price falls as he wasbefore, we have

v(p1, y0 + CV ) = v(p0, y0).

52

CHAPTER 4. PARTIAL EQUILIBRIUM

This change in income, CV , required to keep a consumer’s utility constant as a resultof a price change, is called the compensating variation.

Using the dual relation between indirect utility and expenditure function, I have

e(p1, v(p0, y0)) = e(p1, v(p1, y0 + CV )

)= y0 + CV.

Because I know that y0 = e(p0, v(p0, y0)), I have

CV = e(p1, v0) − e(p0, v0),

where I let v0 ≡ v(p0, y0).

By Shephards’ lemma, we obtain the following:

CV = e(p1, v0) − e(p0, v0)

=∫ p1

p0

∂e(p, v0)∂p

dp

=∫ p1

p0

qh(p, v0)dp.

The compensating variation makes good sense as a dollar-denominated measureof the welfare impact a price change will have. Unfortunately, CV will always bethe area to the left of some Hicksian demand curve, which is unobservable fromeconomists’ point of view. Despite this, I can still take advantage of the relationbetween Hicksian and Marshallian demands expressed by the Slutsky equation toobtain an estimate of CV . The two demands generally diverge, and diverge preciselybecause of the income effect of a price change. Recall that at the price-income pair(p0, y0), consumer surplus, CS(p0, y0), is simply the area under the Marshalliandemand curve (given y0) and above the price, p0. The gain in consumer surplus dueto the price fall from p0 to p1 is

ΔCS ≡ CS(p1, y0) − CS(p0, y0) =∫ p1

p0

q(p, y0)dp.

We now turn to producer surplus, which is simply the firm’s revenue over andabove its variable costs. Define p(·) as the inverse demand. We express the sum of

53


consumer surplus and producer surplus as

TS︸︷︷︸Total Surplus

= CS + PS =[∫ q

0p(ξ)dξ − p(q)q

]+ [p(q)q − TV C(q)]︸︷︷︸

(Revenue − Total Variable Cost)

=∫ q

0p(ξ)dξ − TV C(q)

=∫ q

0

⎡⎢⎣p(ξ) − MC(ξ)︸︷︷︸

Marginal Cost

⎤⎥⎦ dξ.

4.2 Quasi-Linear Preference

To fill the gap between compensating variation (CV ) and consumer surplus (CS),we focus on the following special form of preferences:

Definition 4.2.1 The preference relation � on (−∞,∞) × Rn−1+ is quasi-linear

with respect to commodity 1 if

1. If x ∼ y, then (x + αe1) ∼ (y + αe1) for e1 = (1, 0, . . . , 0) and any α ∈ R.

2. x + αe1 � x for all x and all α > 0.

The quasi-linear preference of the consumer is represented by

u(q,m) = φ(q) + m.

where φ(·) is twice continuously differentiable, with φ′(q) > 0 and φ

′′(q) < 0 at all

q ≥ 0. The form of quasi-linearity implies that there is no income effects for thecommodity we are concerned with. Therefore, there is no difference between theMarshallian demand curve and Hicksian demand curve for the commodity.

The quasi-linear preference of consumer i is represented by

ui(q,m) = φi(q) + m.

Consider the marginal rate of substitution for the good q over m.

MRSqm =∂u(q,m)/∂q

∂u(q,m)/∂m= φ

′i(q)

This implies that the demand for the good q does not depend upon m. Hence, aslong as we are concerned only with good q, we can focus only on this good q market.

54


4.3 Pareto Efficiency

When it is possible to make someone better off and no one worse off, we say thata Pareto improvement can be made. If there is no way at all to make a Paretoimprovement, then we say that the situation is Pareto efficient (optimal). If mon-etary transfer is possible across individuals (i.e., each individual has a quasi-linerutility function), the situation is Pareto efficient if and only if the total surplus ismaximized.

4.4 The Profit Maximization Problem Revisited

Consider the competitive firm who must choose output y so as to solve

maxy

py − c(y) = y [p − c(y)/y] = y [p − AC(y)] ,

where AC(y) stands for the average cost when the firm produces output y. Now,we have a familiar formula: p = AC if and only if the firm makes zero profit. TheFOCs and second-order conditions (SOCs) for an interior solutions are

p = c′(y∗) and c

′′(y∗) ≥ 0.

The supply function gives the profit-maximizing output at each price. Therefore, thesupply function y(p) satisfies the FOCs:

p ≡ c′(y(p)),

and the SOC

c′′(y(p)) ≥ 0.

Let us write the cost function as c(y) = cv(y) + F , so that total cost (TC) areexpressed as the sum of variable costs (V C) and fixed costs (FC). We assume thatthe fixed cost must be paid even if output is zero. Then, the firm will find it profitableto produce a positive level of output when the profits from doing so exceed the profitsfrom producing zero:

py(p) − cv(y(p)) − F ≥ −F

py(p) − cv(y(p)) ≥ 0y(p) [p − cv(y(p))/y] ≥ 0

y(p) [p − AV C(y(p))] ≥ 0,

where AV C(y) stands for the average variable cost when the firm produces outputy.

Example: Let c(y) = ay2 + by + c be the cost function, where a, b, and c > 0.We calculate the following:

55


• MC(y) = c′(y) = 2ay + b,

• AC(y) = c(y)/y = ay + b + c/y

• V C(y) = ay2 + by

• AV C(y) = ay + b

We can derive the supply function as the expression that p = MC(y) = 2ay + b:

y =p − b

2aif p ≥ b,

y = 0 if p < b.

Note that b = AV C(0). Consider the following relation between MC and V C.

V C(y) = ay2 + by =[ay2 + by

]y

0=

∫ y

0(2ay + b) dy =

∫ y

0MC(y)dy.

4.5 The Market Supply Function

If qj(p) is the supply function for firm j in a market for the commodity q with Jfirms, the market supply function is given by

qs(p) ≡J∑

j=1

qj(p,w)

When we do partial equilibrium analysis for the good q, we ignore the dependenceof the supply upon input prices, or we consider the case in which the input pricesare fixed throughout our exercise.

4.6 The Market Demand Function

If qi(p, p, yi) is consumer i’s demand for the good q as a function of its own price, p,and prices, p, for all other goods, the market demand function is given by

qd(p) ≡I∑

i=1

qi(p, p, yi).

When we do partial equilibrium analysis for good q, we ignore the dependence ofthe demand upon all other prices, or we consider the case in which the effects of allother prices are negligible for the good q. For example, this is indeed the case if eachconsumer has a quasi-linear preference over good q and the composite good m.

56


4.7 Market Equilibrium

An equilibrium price is a price where the amount demanded equals the amountsupplied. If we let qi(p) be the demand function of consumer i and yj(p) be thesupply function of firm j, then an equilibrium price is simply a solution to theequation

I∑i=1

qi(p) =J∑

j=1

qj(p).

57

Chapter 5

General Equilibrium

5.1 Introduction

It is time to bring building blocks all together and to ask if there is any coherentstate of affairs (namely, general equilibrium) in which each individual optimizes hisobjective subject to his constraint and all markets clear. We start our discussionwith the following quotation from “The Wealth of Nations,” by Adam Smith.

He (each individual) generally, indeed, neither intends to promote thepublic interest, nor knows how much he is promoting it. By preferringthe support of domestic to that foreign industry, he intends only his ownsecurity; and by directing that industry in such a manner as its producemay be of the greatest value, he intends only his own gain, and he isin this, as in many other cases, led by “an invisible hand” (emphasissupplied) to promote an end which was no part of his intention. Nor is italways the worse for the society that it was no part of it. By pursuing hisown interest he frequently promotes that of the society more effectuallythan when he really intends to promote it.

Economists came up with the two concepts, the competitive market and Paretoefficiency in order to articulate what Adam Smith meant and to what extent he wasright. 1 The strategy we adopt in the section of general equilibrium is as follows:First, we consider an exchange economy (i.e., no production at all) and derive allthe conclusions. Second, we will argue how we can introduce production into theeconomy and argue how to extend all our conclusions to production economies. Thissecond part is going to be as brief as our discussion of production was quick comparedto the consumer theory.

1Surprisingly, at least to me, Adam Smith mentioned the “invisible hand” only once in his“gigantic” book.

58

CHAPTER 5. GENERAL EQUILIBRIUM

5.2 Exchange Economy

Consider now the case of many consumers and many goods. Let

I = {1, . . . , I}index the set of consumers, and suppose there are n goods. Each consumer i ∈ I hasa preference relation, �i, and is endowed with a nonnegative vector of the n goods,ei = (ei

1, . . . , ein). Altogether, the collection E =

(�i, ei)i∈I defines an exchange

economy.

Let

e ≡ (e1, . . . , eI)

denote the economy’s endowment vector, and define an allocation as a vector

x ≡ (x1, . . . , xI),

where xi ≡ (xi1, . . . , xi

n) denotes consumer i’s bundle according to the allocation.The set of feasible allocations in this economy is given by

F (e) ≡{

x ∈ RnI+

∣∣∣∣∣ ∑i∈I

xi ≤∑i∈I

ei

}.

Definition 5.2.1 A feasible allocation, x ∈ F (e), is Pareto efficient if there is noother feasible allocation, y ∈ F (e), such that yi � xi for each consumer i ∈ I, withat least one preference strict.

So, an allocation is Pareto efficient if it is not possible to make someone strictlybetter off without making someone else strictly worse off.

Definition 5.2.2 Let S ⊂ I denote a coalition of consumers. We say that S blocksx ∈ F (e) if there is an allocation y such that:

1.∑

i∈S yi ≤ ∑i∈S ei.

2. yi �i xi for all i ∈ S, with at least one preference strict.

Consider the following scenario: All members of the economy agreed to hire onemediator. The jobs of the mediator are threefold. (1) the mediator must proposean allocation to the consumers until all consumers unanimously agree to it; (2)The mediator must give up his original proposal and makes a new proposal to theconsumer as long as a group of consumers makes a “justifiable” objection to theoriginal proposal; and (3) the mediator implements a given allocation if the allocationis unanimously agreed. We say that a group of consumers S makes a justifiableobjection to a given allocation x if S blocks x in the sense of the above definition.We say that a given allocation is unanimously agreed if there is no group of consumerswhich makes a justifiable objection to it.

59


Definition 5.2.3 The core of an exchange economy with endowment e, denoted“Core(e),” is the set of all unblocked feasible allocations.

5.3 Equilibrium in Competitive Market Systems

In a perfectly competitive market system, all transactions between individuals aremediated by impersonal markets. By “impersonal” I mean that the market treatsevery individual anonymously. The market does not care about who you are. Itcares only about whether you are willing to buy or sell the good at the prevailingprice.

Assumption 5.3.1 For each i ∈ I, utility function ui is continuous, strongly in-creasing, and strictly quasi-concave on Rn

+. 2

On competitive markets, every consumer takes prices as given (remember theprice taking behavior assumption), whether acting as a buyer or a seller. If p ≡(p1, . . . , pn) � 0 is the vector of market prices, then each consumer solves

maxxi∈ n

+

ui(xi) subject to p · xi ≤ p · ei.

Theorem 5.3.1 If ui(·) is continuous, strongly increasing, and strictly quasi-concave,then for each p � 0, the consumer’s optimization problem has a unique solutionxi(p, p · ei). In addition, xi(p, p · ei) is continuous in p ∈ Rn

++.

The proof of Theorem 5.3.1 is obtained by appealing to Theorem 2.4.2 andLemma 2.4.2.

Definition 5.3.1 The aggregate excess demand function for good k is the realvalued function,

zk(p) ≡∑i∈I

xik(p, p · ei) −

∑i∈I

eik.

The aggregate excess demand function is the vector-valued function

z(p) ≡ (z1(p), . . . , zn(p)) .

When zk(p) > 0, there is excess demand for good k. When zk(p) < 0, there is excesssupply of good k.

Theorem 5.3.2 Suppose that for each consumer i ∈ I, ui(·) is continuous, stronglyincreasing, and strictly quasi-concave on Rn

+. Then, for all p � 0, the aggregatedemand function z : Rn → Rn has the following three properties:

2ui is strongly increasing if u(x′) > u(x) whenever x

′ ≥ x and x′ �= x.

60


1. Continuity: z(·) is continuous at p.

2. Homogeneity: z(λp) = z(p) for all λ > 0.

3. Walras’ law: p · z(p) = 0.

Proof of Theorem 5.3.2: (Continuity): This comes from the fact that the sumof continuous functions is continuous. (Homogeneity): It follows that each individualdemand function is homogeneous of degree zero in (p, y). Here, note that for eachconsumer i ∈ I, y = p · ei means that λy = (λp) · ei. (Walras’ law): Since ui(·) isstrongly increasing, each individual demand satisfies Walras’ law, i.e., pxi(p, yi) = yi,where yi = pei. We do the following computation:

p · z(p) =∑i∈I

n∑k=1

pk

(xi

k(p, pei) − eik

)

=∑i∈I

n∑k=1

[pkx

ik(p, pei) − pke

ik

]=

∑i∈I

[p · xi(p, pei) − p · ei

]=

∑i∈I

[p · xi(p, yi) − yi

](∵ yi = pei)

= 0 (∵ pxi(p, yi) = yi ∀i ∈ I) �

Consider a market system described by some excess demand function z(·). If, atsome prices p, we had z(p) = 0, or demand equal to supply in every market, thenwe would say that the system of markets is in general equilibrium.

Definition 5.3.2 A vector p∗ ∈ Rn++ is said to be a Walrasian equilibrium price

vector if z(p∗) = 0.

Definition 5.3.3 Let p ∈ Rn++ be a Walrasian equilibrium price vector. Define

x = (xi)i∈I ≡ (ei + zi(p)

)i∈I, where zi(·) is the excess demand function for consumer

i ∈ I. Then, x is said to be a Walrasain allocation.

5.4 Existence of Walrasian Equilibrium

Consider the following scenario. This is sometimes called the verification scenario.There is an auctioneer in this economy. The auctioneer is an omniscient oracle.So, he knows everything. By “everything” I mean everything. What the auctioneerwants is to implement a Walrasian allocation. Even if the auctioneer is an oracleand therefore, he knows what is a Walrasian allocation, he has to “verify” it to allthe members of the economy who are ignorant about the economy but sophisticatedenough to solve the constrained optimization problem. Then, the auctioneer proposes

61


the following communication protocol to find a Walrasian equilibrium price vector pso that all the members of the economy will be convinced that the final allocationgenerated by the protocol is indeed Walrasian.

Define

Δ ≡{

p ∈ Rn+

∣∣∣∣∣n∑

i=1

pi = 1

}.

Define also a mapping f : Δ → Δ with the property that for any p ∈ Δ,

f(p) = arg maxp∈Δ

pz(p).

The communication protocol I describe here is a game between an auctioneerand all the members of the economy. The game is played alternatively betweenthe auctioneer and the consumers as follows: First, the auctioneer announces somep. Given this p, the consumers announce z(p). 3 Given this z(p), the auctioneerannounces p

′such that p

′= arg maxp pz(p). Given this p

′, the consumers announce

z(p′). Given this z(p

′), the auctioneer announces p

′′such that p

′′= arg maxp pz(p

′).

And so on so forth. Therefore, the communication between them is described as asequence of prices {pk}∞k=1 with the property that pk+1 = f(pk) for each k ≥ 1. Thiscommunication stops if there is a p such that p = f(p). Here, p is called a fixed pointof f(·).

The above communication protocol is the one proposed by the auctioneer to verifyto the consumers that it yields a Walrasian allocation. To complete this verificationprocess, the auctioneer must show the two more things: (1) There exists a fixedpoint of f(·); and (2) the fixed point p is a Walrasian equilibrium price.

The first part is easy to establish. Since Δ is a compact set and f(·) is contin-uous (because z(·) is continuous), we know that f(Δ) is also compact. That is, forany sequence {pk} generated by f(·), there exists a subsequence {pkm}∞m=1 whichconverges to p ∈ Δ. This p is a fixed point of f(·). The second part entails moresubtle issues. Define

Δε ≡ {p ∈ Δ| pi ≥ ε > 0 ∀i = 1, . . . , n} .

By construction Δε is a compact set. 4 Note also that Δε → Δ as ε → 0. Supposethat there exists ε > 0 small enough so that the fixed point p ∈ Δε. Then, we musthave that p ∈ Rn

++. I claim the following.3This simply means that each consumer announces his demand after solving the UMP under p.

So, this is a sense in which each consumer cares only about himself. Remember the comments byAdam Smith.

4Those who are concerned with this compactness should pay attention to the following mathe-matical result: Any closed subset of a compact set is compact. Why is it closed? You should beable to prove it yourself.

62


Claim 5.4.1 Suppose that p ∈ Rn++ is a fixed point of f(·). Then, z(p) = 0.

Proof of Claim 5.4.1: Suppose not, that is, z(p) �= 0. By Walras’ law, pz(p) =0. Then, there must exist two commodities i and j such that zi(p) > 0 and zj(p) < 0.Let p∗ �= p with the property that p∗i = pi + pj, p∗j = 0, and p∗k = pk for any k �= i, j.Thus, we have

p∗ · z(p) > p · z(p) = 0,

which contradicts the hypothesis that p is a fixed point of f(·). �

This claim shows that if a fixed point belongs to the interior of Δ, i.e., p ∈ Δε,it corresponds to a Walrasian equilibrium price vector, as well. However, there isone more problem. We really do not know a priori whether or not the fixed pointbelongs to the interior of Δ. All we know is that p ∈ Δ. The final question we areconcerned with is to ask under what conditions the fixed point cannot be on theboundary of Δ, i.e., we want to avoid the case in which p /∈ Δε for any ε > 0. Let∂Δ be the boundary of Δ. I claim the following:

Claim 5.4.2 Let p be a fixed point of f(·). Assume that if pm → p ∈ ∂Δ as m → ∞,then there exists a commodity k such that zk(pm) → ∞ as m → ∞. Then, thereexists ε > 0 such that p ∈ Δε.

Proof of Claim 5.4.2: Assume that p ∈ ∂Δ. Let

p∗ = (0, . . . , 0︸︷︷︸1,... ,k−1

,

k︷︸︸︷1 , 0, . . . , 0︸︷︷︸

k+1,... ,n

).

Since zk(p) = ∞ by our assumption, we have

p∗ · z(p) = ∞ = arg maxp∈Δ

pz(p).

Since p is a fixed point, we must have pz(p) = ∞, which contradicts Walras’ law.Thus, p /∈ ∂Δ. There is no fixed points on the boundary of Δ. By taking ε > 0small enough, we can always make sure that p ∈ Δε. �

Claim 5.4.2 shows that under the assumption described above, a fixed point p off(·) belongs to the interior of Δ. Combining Claim 1 and 2 together, we have thefollowing corollary.

Corollary 5.4.1 Let p be a fixed point of f(·). Assume that if pm → p ∈ ∂Δ asm → ∞, then there exists a commodity k such that zk(pm) → ∞ as m → ∞. Then,z(p) = 0.

63


This corollary says that any fixed point p generated by the communication pro-tocol corresponds to a Walrasian equilibrium price vector. So, the auctioneer indeedverified to the consumers that the communication protocol guarantees a Walrasianallocation. We should, however, be worried about how to justify the assumptionto rule out the possibility that p ∈ ∂Δ. For this, we need strong increasingness ofutility function.

Theorem 5.4.1 Assume that∑

i∈I ei � 0. 5Suppose that each consumer i’s utilityfunction is continuous, strictly quasi-concave, and strongly increasing. If pm → p ∈∂Δ as m → ∞, then there exists a commodity k such that zk(pm) → ∞ as m → ∞.

Proof of Theorem 5.4.1: Consider a sequence of price vectors {pm}∞m=1 con-verging to p �= 0 such that pm � 0 for each m and pk = 0 for some commodity k.Because

∑i∈I ei � 0, we must have p

∑i∈I ei > 0. This implies that

∑i∈I pei > 0.

Therefore, there must exist at least one consumer i ∈ I for whom pei > 0. Thismeans that consumer i’s demand behaves continuously when pm → p as m → ∞.6 Let zi(pm) be the excess demand of consumer i at pm. Since ui(·) is stronglyincreasing, Walras’ law follows, and therefore we have for each m,

pm · zi(pm) = 0.

Define xi(pm) = ei + zi(pm) as consumer i’s demand at pm. Define also

ωk = (0, . . . , 0︸︷︷︸1,... ,k−1

,

k︷︸︸︷∞ , 0, . . . , 0︸︷︷︸k+1,... ,n

).

Let xi(pm) = xi(pm) + ωk. By construction, we have xi(p) ≥ xi(p) for any p. Ifxi(p) �= xi(p), the strong increasingness of ui(·) concludes that

ui(xi(p)) > ui(xi(p)).

However, this contradicts the fact that xi(p) is the solution to the UMP. The onlyway to avoid this contradiction is to have xi

k(p) = ∞. Since xi(·) is continuousprovided that p � 0 and yi > 0 (his income is positive). Thus, we must havexi

k(pm) → ∞ as m → ∞. Since 0 < pei < ∞ (i.e., consumer i’s income is bounded),

this implies zik(p

m) → ∞. As a result, we have zk(pm) → ∞. �

Combining Corollary 1 and Theorem 1 together, I have the following corollarywhich guarantees the existence of Walrasian equilibrium.

Corollary 5.4.2 (Existence of Walrasian Equilibrium) Assume that∑

i∈I ei �0. Suppose that each consumer i’s utility function is continuous, strictly quasi-concave, and strongly increasing. Then, there exists at least one Walrasian equilib-rium.

5This assumption is innocuous. If it is not satisfied for some commodity i, then we should excludethe commodity i from the beginning of analysis.

6See Lemma 2.4.2 for this fact.

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5.5 Regular Economies and Local Uniqueness

Because we can only hope to determine relative prices, we normalize pn = 1 anddenote by

z(p) = (z1(p), . . . , zn−1(p))

the vector of excess demands for the first n − 1 goods. A normalized price vectorp = (p1, . . . , pn−1, 1) constitutes a Walrasian equilibrium if and ony if it solves thesystem of n − 1 equations in n − 1 unknowns:

z(p) = 0.

Definition 5.5.1 An equilibrium price vector p = (p1, . . . , pn−1) is regular if the(n−1)× (n−1) matrix of price effects Dz(p) is nonsingular, that is, has rank n−1.If every normalized equilibrium price vector is regular, we say that the economy isregular.

Proposition 5.5.1 Any regular (normalized) equilibrium price vector (p1, . . . , pn−1, 1)is locally unique (locally isolated). That is, there is an ε > 0 such that if p

′ �= p,p′n = pn = 1, and ‖p′ − p‖ < ε, then z(p

′) �= 0. Moreover, if the economy is regular,

then, the number of normalized equilibrium price vector is finite.

Definition 5.5.2 Suppose that p = (p1, . . . , pn−1, 1) is a regular equilibrium of theeconomy. Then, we denote

index p = (−1)n−1sign |Dz(p)|,

where |Dz(p)| is the determinant of the (n − 1) × (n − 1) matrix Dz(p).

If n = 2, then |Dz(p)| is merely the slope of z1(·) at p. Hence, we see that for thiscase, the index is +1 or −1 according to whether the slope is negative or positive. Aregular economy has a finite number of equilibria. Therefore, for a regular economy,the expression ∑

{p| z(p)=0, pn=1}index p

makes sense. The next proposition (the index theorem) says that the value of thisexpression is always equal to +1.

Proposition 5.5.2 (The Index Theorem) For any regular economy, we have∑{p| z(p)=0, pn=1}

index p = +1

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Note first that the number of equilibria of a regular economy is odd. In particular,this number cannot be zero; so the existence of at least one equilibrium in a regulareconomy is a particular case of the above proposition. Second, the index conceptprovides a classification of equilibria into two types: the type with positive index ismore fundamental because the presence of at least one equilibrium of positive typeis unavoidable.

I next proceed to argue that typically (or, in the usual jargon, generically),economies are regular. Hence, generically, the solutions to the excess demand equa-tions are locally isolated and finite in number, and the index formula holds.

The essence of genericity analysis rests on counting equations and unknowns.Suppose we have a system of M equations in N unknowns:

f1(v1, . . . , vN ) = 0,

...fM (v1, . . . , vN ) = 0,

or, more compactly, f(v) = 0. The normal situation should be one in which, withN unknowns and M equations, we have N − M degrees of freedom available forthe description of the solution set. In particular, if M > N , the system shouldbe over-determined and have no solution; if M = N , the system should be exactlydetermined with the solutions locally isolated; and if M < N , the system should beunder-determined and the solutions not locally isolated.

Definition 5.5.3 The system of M equations in N unknowns f(v) = 0 is regularif rank Df(v) = M whenever f(v) = 0.

For a regular system, the implicit function theorem yields the existence of theright number of degrees of freedom. If M < N , we can choose M variables corre-sponding to M linearly independent columns of Df(v) and we can express the valuesof these M variables that solve the M equations f(v) = 0 as a function of the N −Mremaining variables. 7 If M = N , equilibria must be locally isolated for the samereasons. And if M > N , then rank Df(v) ≤ N < M for all v; in this case, the abovedefinition simply says that the equation system f(v) = 0 is regular if and only if thesystem admits no solution.

It remains to be argued that the regular case is the “normal” one. Suppose thereare some parameters q = (q1, . . . , qS) such that, for every q, I have a system ofequations f(v; q) = 0 as above. The set of possible parameter values is RS (or anopen subset of RS). I can then justifiably say that f(·; q′

) is a perturbation of f(·; q) ifq′is close to q. Hence, the notion that the regularity of a system f(·; q) = 0 is typical,7See my lecture note on Mathematics for the detail.

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or generic, could be captured by demanding that for almost every q, f(·; q) = 0 beregular. In other words, nonregular systems have probability zero of occurring.

Proposition 5.5.3 (The Transversality Theorem) If the M × (N + S) matrixDf(v; q) has rank M whenever f(v; q) = 0, then, for almost every q, the M × Nmatrix Dvf(v; q) has rank M whenever f(v; q) = 0.

If Df(v; q) has rank M whenever f(v; q) = 0, then from any solution, it isalways possible to (differentially) alter the values of the function f in any prescribeddirection by adjusting the v and q variables. The conclusion of the theorem is that,if this can always be done, then whenever we are initially at a nonregular situation,an arbitrary random displacement in q breaks us away from nonregularity.

Let me now specialize our discussion to the case of a system of n − 1 excessdemand equations in n − 1 unknowns, z(p) = 0. A natural set of parameters is theinitial endowments:

e = (e11, . . . , e1

n, . . . , eI1, . . . , eI

n) ∈ RnI++.

I can write the dependence of the economy’s excess demand function on endowmentsexplicitly as z(p; e). I then have the following proposition.

Proposition 5.5.4 For any p and e, rank Dez(p; e) = n − 1.

I skip the proof. I am now ready to state the main result of this section due toDebreu (1970).

Proposition 5.5.5 (Debreu (1970)) For almost every vector of initial endow-ments (e1, . . . , eI) ∈ RnI

++, the economy defined by (�i, ei)i∈I is regular.

Corollary 5.5.1 The set of Walrasian equilibrium prices is locally unique for almostevery economy E = (ui, ei)i∈N . Moreover, the number of Walrasian equilibriumprices is finite, “most of the time.”

5.6 Anything Goes: The Sonnenschein-Mantel-Debreu

Theorem

Theorem 5.6.1 (Sonnenschein (73), Mantel (74), Debreu (74), and Mantel (76))Suppose that z(·) is a continuous function defined on

Pε ={p ∈ Rn

+| pi/pj ≥ ε ∀i, j}

and with values in Rn. Assume that, in addition, z(·) is homogeneous of degreezero and satisfies Walras’s law. Then, there is an economy of n consumers whoseaggregate excess demand function coincides with z(p) in the domain Pε.

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I skip the proof of Theorem 5.6.1 because it is extremely tedious. Interestedreaders should refer to Proposition 17.E.3 in MWG in pp. 602-603 for the proof whenthe number of commodities is 2 (n = 2). The question was posed by Sonnenschein(1973). He conjectured that the answer was that, indeed, on the domain wherepi ≥ ε for all i, the three properties were not only necessary but also sufficient; thatis, we could always such an economy. He also proved that this is so for the two-commodity case. The problem was then solved by Mantel (1974) for any numberof commodities. Mantel made use of 2n consumers. Shortly afterwards, Debreu(1974) gave a different and very simple proof requiring the indispensable minimumof n consumers. This was topped by Mantel (1976), who refined his earlier proof toshow that n homothetic consumers (with no restrictions in their initial endowments)would do.

The Sonnenschein-Mantel-Debreu theorem implied that the aggregate excess de-mand function as a function of prices only has no structure. The situation is quitedifferent when one explicitly assumes the distribution of endowments in the economy.

Proposition 5.6.1 (Brown and Matzkin (1996)) There are prices and individ-ual endowments (p, (ei)i∈I) and (q, (f i)i∈I) such that it is impossible that p is aWalrasian equilibrium price vector for the economy (ui, ei)i∈I and q is a Walrasianequilibrium price vector for the economy (ui, f i)i∈I .

This theorem is remarkable because it provides a very simple way to show thatnot “anything goes” in general equilibrium theory.

5.7 Properties of the Set of Walrasian Allocations

5.7.1 The Edgeworth Box Diagram

Suppose that there are only two consumers in this society, consumer 1 and consumer2, and only two goods, x1 and x2. Let e1 ≡ (e1

1, e12) denote the nonnegative endow-

ment of the two goods owned by consumer 1, and e2 ≡ (e21, e

22) the endowment of

consumer 2. The total amount of each good available in this society then can besummarized by the vector e1 + e2 = (e1

1 + e21, e

12 + e2

2).

In the figure, units of x1 are measured along each horizontal side and units ofx2 along each vertical side. The southwest corner is consumer 1’s origin and thenortheast corner consumer 2’s origin. Increasing amounts of x1 for consumer 1 aremeasured rightward from O1 along the bottom side, and increasing amounts of x1

for consumer 2 are measured leftward from O2 along the top side. Similarly, x2

for consumer 1 is measured vertically up from O1 on the left, and for consumer 2,vertically down on the right. The Edgeworth box is constructed so that its widthmeasured the total endowment of x1 and its height the total endowment of x2.

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O1

O2

e12

x12

(e1, e2)

(x1, x2)

x11 e1

1

e22

x22

x21

e21

e11 + e2

1

e12 + e2

2

Figure 5.1: The Edgeworth Box

Notice that each point in the Edgeworth box has four coordinates - two indicatingsome amount of each good for consumer 1 and two indicating some amount of eachgood for consumer 2. Because the dimensions of the box are fixed by the totalendowments, each set of four coordinates represents some division of the total amountof each good between the two consumers. Every possible allocation of the totalsbetween the consumers is represented by some point in the box. The Edgeworthbox therefore provides a complete picture of every feasible distribution of existingcommodities between consumers.

5.7.2 Core of an Economy and the First Welfare Theorem

Lemma 5.7.1 Suppose that ui(·) is increasing on Rn+. Let xi ∈ Rn

+ be consumer i’sdemand (i.e., the solution to the UMP) at p ≥ 0. Then, for any xi ∈ Rn

+, we havethe following:

1. ui(xi) > ui(xi) ⇒ pxi > pxi: any strictly better bundle cannot be affordable.

2. ui(xi) ≥ ui(xi) ⇒ pxi ≥ pxi: any weakly better bundle cannot be cheaper.

I leave the proof of this lemma as an exercise.

Exercise 5.7.1 Show Lemma 5.6.1.

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Theorem 5.7.1 Consider an exchange economy (ui, ei)i∈I . If each consumer i’sutility function ui(·) is increasing on Rn

+, then every Walrasian allocation is in thecore.

Proof of Theorem 5.6.2: Let x(p∗) be a Walrasain allocation associated withthe Walrasian equilibrium price vector p∗. Suppose, on the contrary, that x(p∗) isnot in the core. Then, we can find a coalition S and another allocation y such that∑

i∈S

yi =∑i∈S

ei

ui(yi) ≥ ui(xi(p∗, p∗ · ei)) ∀i ∈ S,

with at least one strict inequality. The feasibility of (y)i∈S within S described aboveimplies

p∗ ·∑i∈S

yi = p∗ ·∑i∈S

ei (∗)

By Lemma 5.6.1, we conclude that for each i ∈ S,

p∗ · yi ≥ p∗ · xi(p∗, p∗ · ei) = p∗ · ei,

with at least one inequality strict. Summing over all consumers in S, we obtain

p∗ ·∑i∈S

yi > p∗ ·∑i∈S

ei,

which contradicts (∗). �

Note that all core allocations are indeed Pareto efficient. Hence, we have theimmediate corollary below. This is called the First Welfare Theorem (FWT). Forcompetitive market economies, it provides a formal and very general confirmation ofAdam Smith’s asserted “invisible hand” property of the market.

Corollary 5.7.1 (The First Welfare Theorem) Consider an exchange economy(ui, ei)i∈I . If each consumer i’s utility function ui(·) is increasing on Rn

+, then everyWalrasian allocation is Pareto efficient.

Proof of Corollary 5.6.1: It directly follows from the previous theorem whenwe take S as the set of all consumers, I. �

5.7.3 The Second Welfare Theorem

The second welfare theorem gives conditions under which any Pareto efficient al-location can be supported as a Walrasian equilibrium with the appropriate incometransfers. It is a converse of the first welfare theorem in some sense. To establish

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the second welfare theorem, we rely on the following mathematical result with noproof. 8

Theorem 5.7.2 (Separating Hyperplane Theorem) Suppose that the convex setsA,B ⊂ Rn are disjoint (i.e., A ∩ B = ∅). Then, there is p ∈ Rn with p �= 0 and avalue r ∈ R, such that p · x ≥ r for every x ∈ A and p · y ≤ r for every y ∈ B. Thatis, there is a hyperplane that separates A and B, leaving A and B on different sidesof it.

Now, we are ready to state the second welfare theorem.

Theorem 5.7.3 (The Second Welfare Theorem) Consider an exchange econ-omy (ui, ei)i∈I with e =

∑i∈I ei � 0. Assume that each ui(·) is continuous,

strictly quasi-concave, and increasing. Then, for any Pareto efficient allocationx = (x1, . . . , xI), there are a price vector p �= 0 and an assignment of income levels(y1, . . . , yI) with

∑i∈I yi = p·e such that (x1, . . . , xI) is equivalent to

(x1(p, y1), . . . , xI(p, yI)

)which constitutes a Walrasian allocation associated with p, where xi(p, yi) is con-sumer i’s ordinary demand at (p, yi).

Note that continuity and quasi-concavity of utility functions are only needed toguarantee the existence of equilibrium. This point is enlarged in the Appendix.

Proof of Theorem 5.7.3: The proof consists of 9 steps. I must admit that itis indeed a long proof. But hang on the chain of the argument. For every consumeri ∈ I, we define the set of consumptions strictly preferred to xi, that is

Bi(xi) = {zi ∈ Rn+ | ui(zi) > ui(xi)}}.

Define also,

B(x) =∑i∈I

Bi(xi) =

{z ∈ Rn

∣∣∣∣∣ z =∑i∈I

zi and zi ∈ Bi(xi) ∀i ∈ I}

.

Step 1 through 3 are preliminary lemmas for Step 4. Step 4 is an applicationof the separating hyperplane theorem. Hence, in Step 4, we have a candidate fora Walrasian equilibrium price vector. Step 5 through 7 are preliminary lemmasfor Step 8, where we are able to make sure that the Pareto efficient allocation towhich our attention is paid indeed corresponds to each consumer’s utility maximizingbundle subject to his budget constraint associated with the supporting price vectorwe found in Step 4.

Step 1: Every Bi(xi) is convex.

It follows from the fact that ui(·) is quasi-concave. Do you see why?8See my lecture note on mathematics for the detail.

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Step 2: B(x) and {e} are convex.

It follows from the fact that the sum of any two convex sets are convex. I leavethis as an exercise.

Step 3: B(x) ∩ {e} = ∅Suppose not, that is, there is z ∈ B(x) ∩ {e}. Then, there is (z1, . . . , zI) with

z =∑

i∈I zi such that ui(zi) > ui(xi) for every i ∈ I. Because z ∈ {e}, this(z1, . . . , zI) is a feasible allocation as well. This contradicts the hypothesis that x isPareto efficient.

Step 4: There is p = (p1, . . . , pn) �= 0 and a number r such that p · z ≥ r for everyz ∈ B(x) and p · z ≤ r for every z = e.

This is a direct consequence of the separating hyperplane theorem. Check if wehave made enough assumptions to apply the separating hyperplane theorem.

Step 5: ui(zi) ≥ ui(xi) ∀i ∈ I =⇒ p · (∑i∈I zi) ≥ r.

Let 1 = (1, . . . , 1) ∈ Rn. Consider a sequence {εk}∞k=1 for which εk > 0 for eachk and εk → 0 as k → ∞. Since ui(·) is increasing, we have, for each i and for each k,

ui(zi + εk1) > ui(zi) ≥ ui(xi).

Together with Step 4 this implies that for each i and each k, we have

p ·[(∑

i∈Izi

)+ Iεk1

]≥ r.

When k → ∞, then Iεk1 → 0, because of the continuity of the inner productoperation (i.e., p · x is continuous with respect to x), we have

p ·(∑

i∈Izi

)≥ r,

as desired.

Step 6: p · (∑i∈I xi)

= p · e = r.

By step 5, we know that p · (∑i∈I xi) ≥ r. 9 Since (x1, . . . , xI) is a Pareto

efficient allocation and in particular, a feasible allocation, we have∑

i∈I xi ≤ e.Because each ui(·) is increasing, we have

∑i∈I xi = e. 10 This concludes that

p · (∑i∈I xi)

= p · e. From Step 4, this implies that p · x ≤ r. Therefore, we havep · (∑i∈I xi

)= r.

9This is because when you replace zi with xi, the identical argument goes through in Step 5.10Namely, no resources are wasted.

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Step 7: ui(zi) > ui(xi) =⇒ p · zi ≥ p · xi for every i ∈ I (any strictly better bundlecannot be cheapter)

Consider any zi for which ui(zi) > ui(xi). From Step 5 and 6, we have

p ·⎛⎝zi +

∑k �=i

xk

⎞⎠ ≥ r = p ·

⎛⎝xi +

∑k �=i

xk

⎞⎠ .

Hence, p · xi ≥ p · xi.

Step 8: Assume p · xi > 0 for each i 11. Then, ui(zi) > ui(xi) ⇒ p · zi > p · xi. (anystrictly better bundle should be more expensive)

From Step 7, we know

ui(zi) > ui(xi) ⇒ p · zi ≥ p · xi.

Now, what we want to show is that the inequality is strict. Suppose not, that is,there is a bundle zi such that ui(zi) > ui(xi) and p ·zi = p ·xi > 0. Since p ·xi > 0 wecan find a cheaper bundle xi such that p · xi < p · xi. 12 Define xi

α = αzi + (1−α)xi

for α ∈ (0, 1). Since 0 < α < 1, we know that p ·xiα < p ·xi. If we take α close enough

to 1, because of the continuity of the utility function, we have ui(xiα) > ui(xi) and

p · xiα < p · xi. However, this contradicts our hypothesis supported by Step 7.

Step 9: xi = xi(p, yi) for each i ∈ I, where yi = p · xi.

If we take the contraposition of Step 8, we have

p · xi ≤ yi =⇒ ui(xi) ≥ ui(xi).

This means that xi is the solution to the UMP under (p, yi). Thus, the income levelsyi = p · xi for i = 1, . . . , I support (x1, . . . , xn) as a Walrasian allocation associatedwith the Walrasian equilibrium price p.

We complete the proof. �11This assumption is not innocuous. But we ignore this point in this course. One sufficient

condition for this is that xi � 0 for each i ∈ I. That is, each consumer’s demand lies on the interiorof Rn

+.12If p · xi = 0, you have no way of finding a cheaper bundle so that the rest of the argument does

not go through. This is a reason why I need the assumption that p ·xi > 0 for each i ∈ I. However,there is a generalized version of the second welfare theorem which does not require p · xi > 0 foreach i ∈ I. Interested readers should be referred to Chapter 16.D of MWG.

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5.8 Walrasian Equilibrium with Production

5.8.1 Producers

We suppose there is a fixed number J of firms that we index by the set

J = {1, . . . , J}.

We now let yj ∈ Rn be a production plan for firm j, and observe the convention ofwriting yj

k < 0 if commodity k is an input used in the production plan and yjk > 0 if

it is an output produced from the production plan. To summarize the technologicalpossibilities in production, I suppose that firm j possesses a production possibilityset, Y j . We make the following assumption throughout this section

Assumption [The Individual Firm]

1. 0 ∈ Y j ⊂ Rn. (Possibility of inaction)

2. Y j ∩ Rn+ ⊂ {0} (No free lunch)

3. Y j is closed and bounded, i.e., compact.

4. Y j is strongly convex. That is, for all distinct y, y′ ∈ Y j and all α ∈ (0, 1),

yα = αy + (1 − α)y′ ∈ Int(Y ). 13

The first of these guarantees firm profits are bounded from below by zero. Thesecond guarantees that production of output always requires some inputs. Theclosedness assumption imposes continuity. It says that the limits of possible produc-tion plans are themselves possible production plans. The boundedness assumptionis particulary strong and more than necessary. But it is going to be very useful formaking simple the argument for existence of equilibrium. So, I keep assuming this.14 Strong convexity rules out constant and increasing returns to scale in productionand ensures that the firm’s profit-maximizing production plan is unique.

Each firm faces fixed commodity prices p ≥ 0 and chooses a production plan tomaximize its profit. Thus, each firm solves the problem

maxyj∈Y j

p · yj.

13Int(Y ) stands for the interior of Y .14Those who are concerned with the restrictiveness of the boundedness assumption should be

referred to Chapter 17BB of MWG. The basic idea is as follows. First, we put the bound on boththe consumption set and the production set. Accordingly, we can define the truncated economy.Then, we can establish an existence of equilibrium of the truncated economy. Next, we make thebound unlimited. After this operation, we can make sure that the equilibrium of the truncatedeconomy continues to be an equilibrium of the untruncated economy.

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Because the objective function is continuous and the constraint set is compact,a maximum of firm profit will exist. So, for all p ≥ 0, let

Πj(p) ≡ maxyj∈Y j

p · yj

denote firm j’s profit function. By the theorem of maximum, Πj(·) is continuous onRn

+.

Next I consider aggregate production possibilities economy-wide. I suppose thereare no externalities in production between firms, and define the aggregate productionpossibility set,

Y ≡⎧⎨⎩y ∈ Rn

∣∣∣∣∣ y =∑j∈J

yj and yj ∈ Y j ∀j ∈ J⎫⎬⎭

The set Y will inherit all the properties of the individual production sets. Thefollowing theorem says that y ∈ Y maximizes aggregate profit if and only if it canbe decomposed into individual firm profit-maximizing production plans.

Theorem 5.8.1 (Aggregate Profit Maximization) For any prices p ≥ 0, wehave

p · y ≥ p · y ∀y ∈ Y

if and only if there is (y1, . . . , yJ) with the property that y =∑

j∈J yj and yj ∈ Y j

for each j ∈ J such that

p · yj ≥ p · yj ∀yj ∈ Y j and ∀j ∈ J .

We leave the proof of this theorem as an exercise. Check p. 208 in AMT.

5.8.2 Consumers

In a private ownership economy, which we shall consider here, consumers own sharesin firms and firms profits are distributed to shareholders. Consumer i’s shares infirm j entitle her to some proportion 0 ≤ θi

j ≤ 1 of the profits of firm j. Of course,these shares, summed over all consumers in the economy, must sum to 1. Thus,

0 ≤ θij ≤ 1 for all i ∈ I and for all j ∈ J ,

where ∑i∈I

θij = 1 ∀j ∈ J .

In our economy with production and private ownership of firms, a consumer’sincome can arise from two sources - from selling an endowment of commodities

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already owned, and from shares in the profits of any number of firms. If p ≥ 0 is thevector of market prices, one for each commodity, the consumer’s budget constraintis

p · xi ≤ p · ei +∑j∈J

θijΠ

j(p).

By letting mi(p) denote the right hand side of the above expression, the consumer’sproblem is summarized as follows:

maxxi∈ n

+

ui(xi) subject to p · xi ≤ mi(p).

Now, under the assumptions we have made, each firm will earn nonnegative profitsbecause each can always choose the zero production vector. Consequently, mi(p) ≥ 0because p ≥ 0 and ei ∈ Rn

+. We denote consumer i’s demand by xi(p,mi(p)), wheremi(p) is just the consumer i’s income.

5.8.3 Feasibility and Efficiency in Production Economies

An allocation (x, y) =((x1, . . . , xI), (y1, . . . , yJ )

), of bundles to consumers and pro-

duction plans to firms is feasible if

1. xi ∈ Rn+ for all i ∈ I;

2. yj ∈ Y j for all j ∈ J ; and

3.∑

i∈I xi =∑

i∈I ei +∑

j∈J yj.

Definition 5.8.1 The feasible allocation (x, y) is Pareto efficient if there is noother feasible allocation (x, y) such that ui(xi) ≥ ui(xi) for all i ∈ I with at least onestrict inequality.

5.8.4 Equilibrium

Aggregate excess demand for commodity k is

zk(p) ≡∑i∈I

xik(p,mi(p)) −

∑j∈J

yjk(p) −

∑i∈I

eik,

and the aggregate excess demand vector is

z(p) ≡ (z1(p), . . . , zn(p)) .

As before, a vector p ∈ Rn++ is said to be a Walrasian equilibrium price vector if

z(p) = 0.

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Definition 5.8.2 Let p be a Walrasian equilibrium price vector. Define x = (xi)i∈I ≡(ei + zi(p))i∈I and y = (yj)j∈J ≡ yj(p), where zi(p) is the excess demand at p forconsumer i and yj(p) is the profit maximizing production plan at p for firm j. Then,a profile (p, x, y) is said to be a Walrasian equilibrium.

Theorem 5.8.2 (Existence of Walrasian Equilibrium with Production) Considerthe economy (ui, ei, θi

j , Yj)i∈I, j∈J . Suppose that each ui(·) is continuous, strongly

increasing, and strictly quasi-concave. Assume that Y j is compact, strictly convex,and satisfies possibility of inaction and no free lunch. Assume further that there isan aggregate production plan y ∈ Y for which y +

∑i∈I ei � 0. Then, there exists

at least one Walrasian equilibrium.

Proof of Theorem 5.7.2: Our proof is reduced to checking the following fourproperties of the aggregate excess demand function. Remember the existence proofin exchange economies.

1. z(·) is continuous.

2. z(λp) = z(p) for any λ > 0. (Homogeneity of degree zero)

3. p · z(p) = 0. (Walras’ law)

4. If pm → p ∈ ∂Δ as m → ∞, then there is a commodity k such that zk(pm) → ∞as m → ∞. 15

I focus only on property 4 here, while I leave the proof for the rest of the propertiesas an exercise. Consider a sequence of price vectors {pm}∞m=1 with the propertiesthat pm � 0 for each m, pm → p �= 0 as m → ∞, and pk = 0.

Because y +∑

i∈I ei � 0 for some aggregate production vector y, and p ≥ 0, wemust have

p ·(

y +∑i∈I

ei

)> 0.

Recall also that both mi(p) and Πj(p) are well defined for all p ≥ 0. We do thefollowing computation:

15∂Δ stands for the boundary of Δ.

77


∑i∈I

mi(p) =∑i∈I

⎡⎣p · ei +

∑j∈J

θijΠ

j(p)

⎤⎦

=∑i∈I

p · ei +∑j∈J

Πj(p)

(∵

∑i∈I

θij = 1 ∀j ∈ J

)

≥∑i∈I

p · ei + p · y⎛⎝∵

∑j∈J

Πj(p) ≥ p · y from Theorem 1

⎞⎠

= p ·(

y +∑i∈I

ei

)

> 0 (∵ p ≥ 0 and p �= 0).

Therefore, there must exist at least one consumer whose income at price vector pis positive, i.e., mi(p) > 0. Since ui is strongly increasing, we are able to mimic theargument we made in the existence theorem without production. In effect, consumeri’s demand for commodity k goes to infinity, i.e., xi

k(pm,mi(pm)) → ∞ as m → ∞.

Because of the compactness of the aggregate set of feasible allocations, we must havezk(pm) → ∞ as m → ∞. �

5.9 Walrasian Allocations in Economies with Produc-tion

I consider an economy with I consumers. There is also publicly available constantreturns convex technology, Y ⊂ Rn. The set of feasible allocations in the productioneconomy is given by

F (e, Y ) =

{x ∈ RnI

+

∣∣∣∣ ∑i∈I

xi ≤ y +∑i∈I

ei for some y ∈ Y

}

Definition 5.9.1 Let S ⊂ I denote a coalition of consumers. We say that S blocksx ∈ F (e, Y ) if there is an allocation x such that

1.∑

i∈S xi = y +∑

i∈S ei for some y ∈ Y

2. xi �i xi for all i ∈ S, with at least one preference strict.

The above definition says that a coalition S can improve upon a feasible allocationx if there is some way that, by using only their endowments

∑i∈S ei and the publicly

available technology Y , the coalition can produce an aggregate commodity bundlethat can then be distributed to the members of S so as to make each of them betteroff. An allocation x ∈ F (e, Y ) is said to be in the core if there is no coalition Swhich blocks x in the above sense.

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Theorem 5.9.1 (The Core Property of Production Economies) Consider aproduction economy (ui, e

i, Y )i∈I, j∈J where Y is the publicly available productiontechnology to which any consumer can access. If ui(·) is increasing and Y is aconstant returns to scale convex, then every Walrasian allocation is in the core.

Theorem 5.9.2 (The First Welfare Theorem with Production) Assume thateach ui(·) is increasing. Then, every Walrasian allocation is Pareto efficient.

We omit the proof. The proof is almost the same as the counterpart in exchangeeconomies.

Theorem 5.9.3 (The Second Welfare Theorem with Production) Considera production economy (ui, ei, θi

j , Yj)i∈I,j∈J with y+e � 0 for some aggregate produc-

tion plan y ∈ Y , where e =∑

i∈I ei. Assume that each ui(·) is continuous, strictlyquasi-concave, and increasing. Assume further that each Y j is closed, convex, andsatisfies the possibility of inaction and no free lunch. Then, for any Pareto efficientallocation (x, y), there are a price vector p �= 0 and an assignment of income levels(τ1, . . . , τ I) with

∑i∈I τ i = p·e+p·

(∑j∈J yj

)such that

((x1, . . . , xI), (y1, . . . , yJ)

)is equivalent to

((x1(p, τ1), . . . , xI(p, τ I)

),((y1(p), . . . , yJ(p)

))which constitutes a

Walrasian allocation associated with p, where xi(p, τ i) is consumer i’s ordinary de-mand at (p, τ i), yj(p) is firm j’s profit-maximizing production plan under p.

Proof of Theorem 5.8.2: The proof consists of 9+1 steps. This extra stepis needed for establishing each firm j’s profit maximization plan is indeed yj underp. Try to remember what we did in proving the same existence theorem withoutproduction. In other words, if you fully understand the proof for exchange economies,it is not difficult at all to go through this proof. For every consumer i ∈ I, we definethe set of consumptions strictly preferred to xi, that is

Bi(zi) = {zi ∈ Rn+ | ui(zi) > ui(xi)}}.

Define also,

B(x) =∑i∈I

Bi(xi) =

{z ∈ Rn

∣∣∣∣∣ z =∑i∈I

zi and zi ∈ Bi(xi) ∀i ∈ I}

.

Step 1 through 3 are preliminary lemmas for Step 4. Step 4 is an applicationof the separating hyperplane theorem. Then, from Step 4, we have a candidatefor a Walrasian equilibrium price vector. Step 5 through 7 are preliminary lemmasfor Step 8, where we are able to make sure that the Pareto efficient allocation towhich our attention is paid indeed corresponds to each consumer’s utility maximizingbundle subject to his budget constraint associated with the supporting price vectorwe found in Step 4. Finally, we establish each firm’s profit maximizing behavior inStep 9.

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Step 1: Every Bi(xi) is convex.

This is the same as exchange economies.

Step 2: B(x) and Y + {e} are convex.

It follows from the fact that the sum of any two convex sets are convex. I leavethis as an exercise.

Step 3: B(x) ∩ (Y + {e}) = ∅

Suppose not, that is, there is z ∈ B(x)∩ (Y + {e}). Then, there are (z1, . . . , zI)with z =

∑i∈I zi and (y1, . . . , yJ ) with z =

∑j∈J yj ∈ Y + {e} such that ui(zi) >

ui(xi) for every i ∈ I and yj ∈ Y j for each j ∈ J . Thus,((z1, . . . , zI), (y1, . . . , yJ)

)is a feasible allocation as well. This contradicts the hypothesis that (x, y) is Paretoefficient.

Step 4: There is p = (p1, . . . , pn) �= 0 and a number r such that p · z ≥ r for everyz ∈ B(x) and p · z ≤ r for every z ∈ Y + {e}.

This is a direct consequence of the separating hyperplane theorem. Check if wehave made enough assumptions to apply the separating hyperplane theorem.

Step 5: ui(zi) ≥ ui(xi) ∀i ∈ I ⇒ p · (∑i∈I zi) ≥ r.


Step 6: p · (∑i∈I xi)

= p ·(e +

∑j∈J yj

)= r.

By step 5, we know that p·(∑i∈I xi) ≥ r. 16 Since (x, y) = (x1, . . . , xI , y1, . . . , yJ )

is a Pareto efficient allocation and in particular, a feasible allocation, we have∑i∈I xi ≤ e +

∑j∈J yj. Because each ui(·) is increasing, we have

∑i∈I xi =

e +∑

j∈J yj. 17 From Step 4, this implies that p · (∑i∈I xi) ≤ r. Therefore,

we have p · (∑i∈I xi)

= p ·(e +

∑j∈J yj

)= r.

Step 7: ui(zi) > ui(xi) ⇒ p · zi ≥ p · xi for every i ∈ I


Step 8: Assume p · xi > 0 for each i 18. Then, ui(zi) > ui(xi) ⇒ pzi > pxi.16When you replace zi with xi, the identical argument goes through in Step 5.17Namely, no resources are wasted.18This assumption is not innocuous. One sufficient condition for this is that xi � 0 for each i ∈ I.

That is, each consumer’s demand lies on the interior of Rn+. You should be referred to footnotes in

the second welfare theorem in exchange economies.

80



Step 9: p · yj ≤ p · yj for any yj ∈ Y j and j ∈ J (Profit Maximization)

For any yj ∈ Y j , we have

yj +∑h �=j

yh ∈ Y.

From Steps 4 (the separating hyperplane theorem) and 6, we have

p ·⎛⎝e + yj +

∑h �=j

yh

⎞⎠ ≤ r = p ·

⎛⎝e + yj +

∑h �=j

yh

⎞⎠ ,

which implies that p · yj ≤ p · yj.

Step 10: xi = xi(p, τ i) for each i ∈ I, where τ i = p · xi.

If we take the contraposition of Step 8, we have

p · zi ≤ τ i ⇒ ui(xi) ≥ ui(zi).

This means that xi is the solution to the UMP under (p, τ i). Thus, the income levelsτ i = p · xi for i = 1, . . . , I support (x, y) as a Walrasian allocation associated withthe Walrasian equilibrium price p.

We complete the proof. �

5.10 Uniqueness of Equilibria

Up to this point, I have concentrated on the determination of the general propertiesof the competitive market. In this section, I focus on a special class of environmentsin which the uniqueness of equilibria obtains.

5.10.1 The Weak Axiom for Aggregate Excess Demand

To begin, suppose that z(p) =∑

i∈I(xi(p, p · ei) − ei

)is the aggregate excess de-

mand function of the consumers.

Proposition 5.10.1 Given an economy specified by the constant returns technologyY and the aggregate excess demand function of the consumers z(·), a price vector pis a Walrasian equilibrium price vector if and only if

1. p · y ≤ 0 for every y ∈ Y , and

2. z(p) is a feasible production; that is, z(p) ∈ Y .

81


Proof of Proposition 5.9.1: (=⇒) If p is a Walrasian equilibrium price vector,then (2) follows from market clearing and (1) is a necessary condition for profitmaximization with a constant returns technology. (⇐=) Let the consumptions bex∗

i = xi(p, p · ei) for i = 1, . . . , I and the production vector be y∗ = z(p) ∈ Y forsome price vector p. To verify p is a Walrasian equilibrium price vector, the onlycondition that is not immediate is profit maximization. Because p · y ≤ 0 for ally ∈ Y from (2) and p · y∗ = p · z(p) = 0 from Walras’ law, y∗ = z(p) ∈ Y is profitmaximizing. �

Definition 5.10.1 (The Weak Axiom for Excess Demand Functions) The ex-cess demand function z(·) satisfies the weak axiom of revealed preference (WA) if forany pair of price vectors p and p

′, we have

z(p) �= z(p′) and p · z(p

′) ≤ 0 =⇒ p

′ · z(p) > 0.

Lemma 5.10.1 (Uniqueness implies WA) Suppose that the excess demand func-tion z(·) is such that, for any constant returns convex technology Y , the economyformed by z(·) and Y has a unique (normalized) equilibrium price vector. Then z(·)satisfies the weak axiom.

Proof of Lemma 5.10.1: We argue by contradiction. Suppose that the WA wasviolated; that is, suppose that for some p and p

′we have z(p) �= z(p

′), p · z(p

′) ≤ 0,

and p′ · z(p) ≤ 0. Then we claim that both p and p

′are equilibrium prices for the

convex constant returns production set given by

Y ∗ ={

y ∈ Rn| p · y ≤ 0 and p′ · y ≤ 0

}Thus, by the above proposition, p is an equilibrium price vector. The same is truefor p

′. Since z(p) �= z(p

′), we conclude that the equilibrium is not unique for the

economy formed by z(·) and the production set Y ∗. This is a contradiction. �

Proposition 5.10.2 (WA generally implies Uniqueness) If z(·) satisfies the weakaxiom, then, for any constant returns convex technology Y , the set of equilibriumprice vectors is convex. Furthermore, if the set of normalized price equilibria isfinite, there can be at most one normalized price equilibrium.

Proof of Proposition 5.10.2: Suppose that p and p′

are equilibrium pricevectors for the constant returns convex technology Y ; that is, z(p) ∈ Y, z(p

′) ∈ Y ,

and, for any y ∈ Y, p · y ≤ 0 and p′ · y ≤ 0. Let p

′′= αp + (1 − α)p

′for α ∈ [0, 1].

Note that

p′′ · y = αp · y + (1 − α)p

′ · y ≤ 0.

Hence (1) follows in Proposition 5.9.1. To show that p′′

is an equilibrium pricevector, we must show that z(p

′′) ∈ Y by Proposition 5.9.1. Because p

′′ · z(p′′) = 0

(by Walras’ law) and p′′ · z(p

′′) = αp · z(p

′′) + (1 − α)p

′ · z(p′′), we must have either

82


p · z(p′′) ≤ 0 or p

′ · z(p′′) ≤ 0. Suppose that the first possibility holds, so that

p · z(p′′) ≤ 0 (a parallel argument applies if, instead, p

′z(p

′′) ≤ 0). Since z(p) ∈ Y ,

we have p′′ · z(p) ≤ 0. This is because

p′′ · z(p) = αp · z(p) + (1 − α)p

′ · z(p) ≤ 0 (∵ p · y ≤ 0 and p′ · y ≤ 0 for any y ∈ Y )

But with p′′ · z(p) ≤ 0 and p · z(p

′′) ≤ 0, a contradiction to the WA can be avoided

only if z(p′′) = z(p). Hence z(p

′′) ∈ Y . 19 �

5.10.2 Gross Substitution

Definition 5.10.2 (Gross Substitution) The function z(·) has the gross sub-stitute (GS) property if whenever p and p

′are such that for some i, p

′i > pi and

p′k = pk for k �= i, we have zk(p

′) > zk(p) for k �= i.

Proposition 5.10.3 (GS implies Uniqueness) An aggregate excess demand func-tion z(·) that satisfies the gross substitute property has at most one exchange equi-librium; that is, z(p) = 0 has at most one (normalized) solution.

Proof of Proposition 5.10.3: It suffices to show that z(p) = z(p′) cannot occur

whenever p and p′

are two price vectors that are not collinear. By homogeneity ofdegree zero, we can assume that p ≥ p

′and pi = p

′i for some i (This can be done by

choosing αp ≥ p′

by choosing α > 0). Now consider altering the price vector p toobtain the price vector p

′in n− 1 steps, lowering (or keeping unaltered) the price of

every commodity k �= i one at a time. By gross substitution, the excess demand ofgood i cannot decrease in any step, and because p �= p

′, it will actually increase in

at least one step. Hence, zi(p′) > zi(p). This implies that z(p) �= z(p

′). �

5.11 Stability

I consider an exchange economy formalized by means of an excess demand functionz(·). Suppose that I have an initial p that is not an equilibrium price vector, so thatz(p) �= 0. For example, the economy may have undergone a shock and p may be thepreshock equilibrium price vector. Then the demand-and-supply principle suggeststhat prices will adjust upward for goods in excess demand and downward for thosein excess supply. This is what was proposed by Walras; in a differential equationversion put forward by Samuelson (1947), it takes the specific form

dpk

dt= ckzk(p) ∀k = 1, . . . , n (∗)

where dpk/dt is the rate of change of the price for the k-th good and ck > 0 is aconstant affecting the speed of adjustment.

19We now establish that either z(p′′) = z(p) or z(p

′′) = z(p

′). Since this is true for any α ∈ [0, 1],

and since the function z(·) is continuous, this implies that z(p) = z(p′) for any two equilibrium price

vectors p and p′.

83


Proposition 5.11.1 Suppose that z(p∗) = 0 and p∗ · z(p) > 0 for every p not pro-portional to p∗. Then, the relative prices of any solution trajectory of the differentialequation (∗) converge to the relative prices of p∗.

5.12 Appendix: “Generalization” of General Equilib-

rium Theory

Here, we follow “On the fundamental theorems of general equilibrium,” by EricMaskin and Kevin Roberts in Economic Theory, 2008, vol. 35, 233-240. The mainobjective of this section is to provide a very simple proof of the second welfare theo-rem. This can be done by taking a roundabout of establishing a simple generalizationof the existence theorem to economies where Walras’ law need not be satisfied outof equilibrium. Therefore, it is demonstrated that the only difficulty for the proof ofthe standard second welfare theorem lies in establishing existence part.

A generalized competitive mechanism (GCM) is a rule that, for each vector ofprices p (in the unit simplex) and each specification {yj}j∈J of production plansby firms (where yj ∈ Y j for all j ∈ J), assigns to each consumer i an incomeW i(p, {yj}j). One example of a GCM is, of course, the ordinary competitive mecha-nism, in which consumer i is assigned income p · ei +

∑j θi

j(p · yj). Another exampleis the mechanism that, given some fixed allocation ({xi}i, {yj}j) and prices p, givesconsumer i income p · xi.

Definition 5.12.1 An equilibrium of a GCM is a price vector p and an allocation({xi}i∈I , {yj}j∈J) such that

1. for each i ∈ I, xi is preference-maximizing in Rn+, subject to the constraint

p · xi ≤ W i(p, {yj}j);

2. for each j ∈ J , yj is profit-maximizing in Y j given prices p; and

3.∑

i xi =∑

j yj +∑

i ei.

Given a GCM, let Z(·) be the corresponding aggregate excess demand correspon-dence. That is, for any prices p,

Z(p) =

⎧⎨⎩z ∈ Rn

∣∣∣∣∣z =∑i∈N

(xi − ei) −∑j∈J

yj such that requirement 1 and 2 below hold

⎫⎬⎭

1. xi maximizes �i subject to p · xi ≤ W i(p, {yj}j)

2. For all j ∈ J , yj maximizes firm j’s profit in Y j given prices p.

Lemma 5.12.1 (Maskin and Roberts (2008)) Given a GCM, suppose that Z(·)is well-defined, upper-hemicontinuous, and convex- and compact-valued. Supposethat if p is such that pk = 0 for some commodity k, then for all z ∈ Z(p), zk > 0.

84


Finally, suppose that, for all p and all z ∈ Z(p), either (a) z=0 or (b) there existk and � such that zk > 0 and z� ≤ 0. (Note that this last hypothesis constitutes aweakening of Walras’ law.) Then, there exists an equilibrium.

Theorem 5.12.1 (Maskin and Roberts (2008)) (Existence of equilibrium ata Pareto efficient allocation): Let the allocation ({xi}i, {yj}j) be Pareto efficient,and suppose that, for all i, all components of xi are strictly positive. Suppose that,for all i ∈ N and p, W i(p, {yj}j) = p · xi. Assume that preferences are convex,continuous, and strongly monotone, and that production sets are convex, closed, andbounded. Then, an equilibrium exists.

The next theorem is nothing but the first welfare theorem.

Theorem 5.12.2 (Maskin and Roberts (2008), First Welfare Theorem) If pref-erences are strongly monotone, then any equilibrium of a GCM is Pareto efficient.

Theorem 5.12.3 (Maskin and Roberts (2008)) Assume that preferences are stronglymonotone. Suppose that allocation ({xi}i, {yj}j) is Pareto efficient. Consider theGCM in which, given prices p, consumer i receives income W i = p · xi. Then, ifan equilibrium of this GCM exists, ({xi}i, {yj}j) is an equilibrium allocation of theGCM.

It is worth emphasizing that the above theorem requires no convexity assump-tions. The theorem illustrates that convexity in the second welfare theorem is neededonly to show that equilibrium exists; it is not required to show that the equilibriumoccurs at the Pareto efficient allocation. Indeed, it follows directly that if a Paretoefficient allocation cannot be supported as an equilibrium, then starting at this al-location, no equilibrium can exist.

Theorem 5.12.4 (Maskin and Roberts (2008), Second Welfare Theorem)Suppose that preferences and production sets satisfy the hypotheses of the previoustheorem. Then, if ({xi}i, {yj}j) is Pareto efficient and, for all i ∈ I, all componentsof xi are strictly positive, there exists a price p and balanced transfers {T i} (i.e., sum-ming to zero) such that ({xi}i, {yj}j) is an equilibrium allocation with respect to theGCM mechanism that, for each p, gives consumer i the income p·ei+

∑j θi

jp· yj +T i.

85

Chapter 6

Choice under Uncertainty

6.1 Preferences over Gambles

Until now, we have assumed that decision makers act in a world of absolute certainty.As soon as you apply this assumption to the real world, you immediately realize thatthis assumption is simply a lie, but, I hope you think, a useful one. Many economicdecisions contain a significant amount of uncertainty. Here, I would like to discusshow to incorporate the aspects of uncertainty into the decision making problem.

We will maintain the notion of preference relation but, instead of consumptionbundles, the individual will be assumed to have a preference relation over gambles.Let A = {a1, . . . , an} denote a finite set of outcomes (consequences). The ai’s mightwell be consumption bundles, amounts of money (positive or negative), or anythingat all. The main point is that the ai’s themselves involve no uncertainty. g issaid to be a simple gamble if it assigns a probability pi, to each of the outcomesai, in A. Note that pi ≥ 0 and

∑ni=1 pi = 1. We denote this simple gamble g by

(p1 ◦ a1, . . . , pn ◦ an).

Definition 6.1.1 Let A = {a1, . . . , an} be the set of outcomes. Then GS, the set ofsimple gambles (on A), is given by

GS ≡{

(p1 ◦ a1, . . . , pn ◦ an)

∣∣∣∣∣n∑

i=1

pi = 1 and pi ≥ 0 for each i = 1, . . . , n

}.

The simple gamble (α ◦ a1, 0 ◦ a2, . . . , 0 ◦ an−1, (1− α) ◦ an) would be written as(α◦a1, (1−α)◦an). Note that GS contains A because for each i, (1◦ai), the gambleyielding ai with probability one, is in GS . Namely, we consistently expand the set ofchoices, A, for the individual to a bigger one, GS .

Gambles whose prizes are themselves gambles are called compound gambles. Forexample, let A = {a1, a2, a3}. Consider the following gamble g∗ = (1/3 ◦ g1, 1/3 ◦

86

CHAPTER 6. CHOICE UNDER UNCERTAINTY

g2, 1/3 ◦ g3), where gk ∈ GS for k = 1, 2, 3 with the properties that

g1 = (1 ◦ a1);g2 = (1/4 ◦ a1, 3/8 ◦ a2, 3/8 ◦ a3); andg3 = (1/4 ◦ a1, 3/8 ◦ a2, 3/8 ◦ a3).

Let gs be the simple lottery induced by g∗. Then,

gs = (1/2 ◦ a1, 1/4 ◦ a2, 1/4 ◦ a3)

When I discuss the preference relations over set of gambles, the decision maker issupposed to be indifferent between g∗ and gs.

Let G denote the set of all gambles, both simple and compound. G is definedinductively as follows: Let G0 = A. For each j ≥ 1, let

Gj =

{(p1 ◦ g1, . . . , pk ◦ gk

) ∣∣∣∣∣ k ≥ 1; pi ≥ 0 and gi ∈ Gj−1 ∀i = 1, . . . , k; andk∑

i=1

pi = 1

}.

Then, G =⋃∞

j=0 Gj.

6.2 Axioms

Analogous to the case of consumer theory, we shall suppose that the decision makerhas preferences, �, over the set of gambles, G. We proceed by positing a number ofaxioms for the decision maker’s preference relation, �.

Axiom G1: (Completeness) ∀g, g′ ∈ G, we have either g � g

′or g

′ � g.

Axiom G2: (Transitivity) ∀g, g′, g

′′ ∈ G, if g � g′and g

′ � g′′, then g � g

′′.

Let us assume without loss of generality that the elements of A have been indexedso that a1 � a2 � · · · � an.

Axiom G3: (Continuity) ∀g ∈ G, there is some probability α ∈ [0, 1] such thatg ∼ (α ◦ a1, (1 − α) ◦ an).

The continuity axiom is, typically, the one that causes most people to expressdoubts. Consider the following example (adapted from “Notes on the Theory ofChoice,” by David Kreps (1988)): f is a gamble in which you get $1000 for sure; gis a gamble in which you get $10 for sure; and h is a gamble in which you are killedfor sure. Most people would express the preference f � g � h. And so, the axiomholds, there must exist a probability α ∈ (0, 1), presumably close to 1, such thatαf + (1 − α)h � g. That is, you are willing to risk a small (but nonzero) chance ofyour death, to trade up from $10 to $1000. This, many people say, is rather dubious.

87


But consider: Suppose I told you that you could either have $10 right now, or ifyou were willing to drive five miles (pick some location five miles away from whereyou are), an envelope with $1000 was waiting for you. Most people would get outtheir car keys at such a prospect, even though driving the five miles increases everso slightly the chances of a fatal accident. So, perhaps the axiom isn’t so bad.

Axiom G4: (Monotonicity) ∀α, β ∈ [0, 1],

(α ◦ a1, (1 − α) ◦ an) ∼ (β ◦ a1, (1 − β) ◦ an) ⇐⇒ α = β.

Moreover, ∀α, β ∈ [0, 1],

(α ◦ a1, (1 − α) ◦ an) � (β ◦ a1, (1 − β) ◦ an) ⇐⇒ α > β.

Note that monotonicity implies that a1 � an, and so the case in which thedecision maker is indifferent among all the outcomes in A is ruled out.

Axiom G5: (Substitution) If g =(p1 ◦ g1, . . . , pk ◦ gk

), and

h =(p1 ◦ h1, . . . , pk ◦ hk

)are in G, and if hi ∼ gi for every i = 1, . . . , k, then h ∼ g.

Together with the completeness axiom, the substitution axiom implies that whenthe decision maker is indifferent between two gambles, he must be indifferent betweenall convex combinations of them. 1 Most people, viewing the substitution axiom forthe first time, think it looks pretty convincing on first principles for choice underuncertainty. There are, however, a number of well-known cases in which the sub-stitution axiom, as a description of how people do choose, is falsified empirically.The next question you might ask is how often the axiom is falsified empirically. Theanswer is “not often,” though.

Axiom G6: (Reduction to Simple Gambles) ∀g ∈ G, if (p1 ◦ a1, . . . , pn ◦ an) is thesimple gamble induced by g, then (p1 ◦ a1, . . . , pn ◦ an) ∼ g.

The reduction to simple gambles axiom rests on a basic consequentialist premise:We assume that for any risky alternatives, only the reduced simple gamble is ofrelevance to the decision maker. In other words, the decision maker should careabout what outcome he “finally” receives (i.e., consequences) but should not careabout how the gamble results in the final outcome (i.e., procedure or process ofresolution of the uncertainty).

1Formally, I have the following: for any f, g ∈ G, if f ∼ g, then αf + (1 − α)g ∼ f ∼ g for anyα ∈ [0, 1]. This is also known as the betweeness axiom in Dekel (1986). In some other times, theexpected (VNM) utility function is axiomatized using the independence axiom. We say that � over Gsatisfies the independence axiom if, for any f, g, h ∈ G, f � g ⇐⇒ αf+(1−α)h � αg+(1−α)h for anyα ∈ [0, 1]. In particular, the independence axiom implies that f ∼ g ⇐⇒ αf+(1−α)h ∼ αg+(1−α)hfor any α ∈ [0, 1]. Since h can be any gamble, I take h to be either f or g. Hence, with thecompleteness axiom, the independence axiom implies the substitution axiom, but the converse isnot true.

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6.3 Von Neumann-Morgenstern (VNM) Utility

In this section, we would like to know not only whether there is a representation forpreferences over gambles which satisfy all the axioms provided above (indeed, thereis!) but also how “nice” (yet to be defined) the structure of the representation is.Suppose that V : G → R is a utility function representing � on G. So, for everyg ∈ G, V (g) denotes the utility number assigned to the gamble g. Let u(ai) denotethe utility generated from a simple gamble that ai occurs with probability one.

Definition 6.3.1 The function V : G → R has the expected utility property ifthere is an assignment of numbers (u(a1), . . . , u(an)) to the n outcomes such thatfor every g ∈ G,

V (g) =n∑

i=1

piu(ai),

where (p1 ◦ a1, . . . , pn ◦ an) is the simple gamble induced by g.

If an individual’s preferences are represented by a utility function with the ex-pected utility property, and if that person always chooses his most preferred alterna-tive available, then that individual will choose one gamble over another if and onlyif the expected utility of the one exceeds that of the other. Consequently, such anindividual is an expected utility maximizer.

6.4 Existence of VNM Utility Function

Theorem 6.4.1 (Existence of VNM Utility Function on G) Let preferences �over gambles in G satisfy axioms G1 through G6. Then there is a function V : G → Rrepresenting � on G such that V has the expected utility property.

Proof of Theorem 6.4.1:

Step 1: For any g ∈ G, there is a unique V (g) ∈ [0, 1] such thatg ∼ (V (g) ◦ a1, (1 − V (g)) ◦ an).

Consider an arbitrary gamble g from G. Define V (g) to be the number satisfying

g ∼ (V (g) ◦ a1, (1 − V (g)) ◦ an) .

By the continuity axiom (G3), such a number exists. The uniqueness follows fromthe monotonicity axiom (G4). Do you see why?

Step 2: g � g′ ⇐⇒ V (g) ≥ V (g

′).

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Let g, g′ ∈ G be arbitrary gambles. Assume without loss of generality that g � g

′.

By step 1, we have

(V (g) ◦ a1, (1 − V (g)) ◦ an) ∼ g � g′ ∼

(V (g

′) ◦ a1, (1 − V (g

′)) ◦ an

).

By the transitivity axiom (G2), we obtain

(V (g) ◦ a1, (1 − V (g)) ◦ an) �(V (g

′) ◦ a1, (1 − V (g)) ◦ an

).

By the monotonicity axiom (G4), we conclude that V (g) ≥ V (g′).

Step 3: For any g ∈ G, V (g) has the expected utility property.

Let g ∈ G be an arbitrary gamble and let gs ≡ (p1 ◦ a1, . . . , pn ◦ an) ∈ GS be thesimple gamble induced by g. By Step 1, for each outcome ai for i = 1, . . . , n, thereis a unique assignment of number u(ai) ∈ [0, 1] such that

(1 ◦ ai) ∼ (u(ai) ◦ a1, (1 − u(ai)) ◦ an) .

By the reduction to simple gambles axiom (G6), we have g ∼ gs. By Step 2, weknow that V represents � and therefore, we must have V (g) = V (gs). Hence, whatwe want to show is

V (gs) =n∑

i=1

piu(ai).

For every i = 1, . . . , n, let hi denote the simple gamble with the property that hi =(u(ai) ◦ a1, (1 − u(ai)) ◦ an). Define g

′ ≡ (p1 ◦ h1, · · · , pn ◦ hn

). By construction,

hi ∼ ai for every i = 1, . . . , n. By the substitution axiom (G5), we have

g′=

(p1 ◦ h1, · · · , pn ◦ hn

) ∼ (p1 ◦ a1, · · · , pn ◦ an) = gs =⇒ g′ ∼ gs

We now wish to derive the simple gamble g′s induced by the compound gamble g

′.

For each i = 1, . . . , n, there is a probability of piu(ai) that a1 will result. Becausethe occurrences of the hi’s are mutually exclusive, the effective probability thata1 results is the sum

∑ni=1 piu(ai). Similarly, for each i = 1, . . . , n, there is a

probability of pi[1 − u(ai)] that an will result. The effective probability that an

results is∑n

i=1 pi [1 − u(ai)], which is equal to 1 − ∑ni=1 piu(ai). Therefore, the

simple gamble g′s induced by g

′is

g′s ≡

([n∑

i=1

piu(ai)

]◦ a1,

[1 −

n∑i=1

piu(ai)

]◦ an

).

By the reduction to simple gambles axiom (G6), we have g′ ∼ g

′s. By the transitivity

axiom (G2), we obtain

gs ∼([

n∑i=1

piu(ai)

]◦ a1,

[1 −

n∑i=1

piu(ai)

]◦ an

)(∵ g

′ ∼ g′s and g

′ ∼ gs) (∗).

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By Step 1, there is a unique number V (gs) satisfying

gs ∼ (V (gs) ◦ a1, [1 − V (gs)] ◦ an) (∗∗).Therefore, comparing (∗) with (∗∗), we conclude that

V (gs) =n∑

i=1

piu(ai),

as desired. �

6.5 Uniqueness up to Positive Affine Transformations

In the consumer theory case, the utility numbers themselves have only ordinal mean-ing. Any strictly monotonic transformation of one utility representation yields an-other one. On the other hand, the utility numbers associated with a VNM utilityrepresentation of preferences over gambles have content beyond ordinality.

Suppose that A = {a, b, c}, where a � b � c, and that � satisfies all the six axiomsI provided in the previous section. By the continuity axiom and the monotonicityaxiom, there is a unique α ∈ (0, 1) satisfying

b ∼ (α ◦ a, (1 − α) ◦ c) .

Let V be some VNM utility representation of �. Then, the preceding indifferencerelation implies that

V (b) = V ((α ◦ a, (1 − α) ◦ c)= αV (a) + (1 − α)V (c). (because V has the expected utility property).

This equality can be rearranged to yield

V (a) − V (b)V (b) − V (c)

=1 − α

α.

Consequently, the ratios of the differences between the preceding utility numbersare uniquely determined by α. But because the number α was uniquely determinedby the decision maker’s preferences, so is the preceding ratio of utility differences.Then, a strictly increasing transformation of a VNM utility representation mightnot yield another VNM utility representation. Here, we ask the following question:What is the class of VNM utility representations of a given preference ordering?

Theorem 6.5.1 (Uniqueness up to Positive Affine Transformation) Supposethat the VNM utility function V (·) represents �. Then the VNM utility function,V (·) represents the same preferences if and only if there exists α > 0 and β ∈ R suchthat

V (g) = αV (g) + β,

for any gamble g ∈ G.

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Proof of Theorem 6.5.1: It is very easy to show that V and V represents thesame preference relation � on G when V (g) = αV (g) + β, where α > 0 and β ∈ R.Do you agree? Hence, we focus on the opposite direction. Suppose that both V andV represents the same preference over G with the expected utility property. By themonotonicity axiom (G4), we know that V (a1) > V (an). By the continuity axiom(G3), for any g ∈ G, there exists λg ∈ [0, 1] such that

V (g) = λgV (a1) + (1 − λg)V (an).

This implies

λg =V (g) − V (an)V (a1) − V (an)

.

By the reduction to simple gambles axiom (G6), we have

λgV (a1) + (1 − λg)V (an) = V ((λg ◦ a1, (1 − λg) ◦ an)) .

Because V represents �, it follows that g ∼ (λg ◦ a1, (1 − λg) ◦ an). Since V repre-sents the same preference, we have

V (g) = V ((λg ◦ a1, (1 − λg) ◦ an))= λgV (a1) + (1 − λg)V (an)

= λg

(V (a1) − V (an)

)+ V (an) (∵ V is a VNM utility function.)

=V (g) − V (an)V (a1) − V (an)

(V (a1) − V (an)

)+ V (an)

(∵ λg =

V (g) − V (an)V (a1) − V (an)

)

=V (a1) − V (an)V (a1) − V (an)

(V (g) − V (an)) + V (an)

=

[V (a1) − V (an)V (a1) − V (an)

]︸︷︷︸

=α

V (g) +

[V (an) − V (a1) − V (an)

V (a1) − V (an)V (an)

]︸︷︷︸

=β

= αV (g) + β. �

6.6 Risk Aversion

In many economic settings, individuals seem to display aversion to risk. In thissection, we formalize the notion of risk aversion and study some of its property.For simplicity, we shall confine our attention to gambles whose outcomes consist ofdifferent amounts of wealth (monetary income). Thus, let A = R+. Even thoughthe set of outcomes now contain infinitely many elements, we continue to considergambles giving only finitely many outcomes a strictly positive effective probability. Asimple gamble takes the form (p1 ◦w1, . . . , pn ◦wn), where n is some positive integer,the wi’s are nonnegative wealth levels, and the nonnegative probabilities, p1, . . . , pn,

92


sum to 1. Let u(wi) be the level of utility from wealth wi for each i = 1, . . . , n. Wecall this u : R+ → R the Bernoulli utility function. Finally, we shall assume that theindividual’s Bernoulli function, u(·) is differentiable with u

′(w) > 0 for any w ≥ 0.

The expected value of the simple gamble g offering wi with probability pi forany i = 1, . . . , n is given by E(g) =

∑ni=1 piwi. Now suppose the agent is given a

choice between accepting the gamble g on the one hand or receiving with certainty(i.e., with probability one) the expected value of g on the other. If V : G → R is theagent’s VNM utility function, we can evaluate these two alternatives as follows:

V (g) =n∑

i=1

piu(wi),

V (E(g)) = u

(n∑

i=1

piwi

).

When someone would rather receive the expected value of a gamble with certaintythan face the risk inherent in the gamble itself, we say that he is risk averse.

Definition 6.6.1 Let V : G → R be an individual’s VNM utility function for gamblesover nonnegative levels of wealth. Then, for the simple gamble g = (p1 ◦w1, . . . , pn ◦wn), the individual is said to be

1. risk averse at g if V (E(g)) > V (g);

2. risk neutral at g if V (E(g)) = V (g);

3. risk loving at g if V (E(g)) < V (g).

If the individual is risk averse at every nondegenerate simple gamble, g, then theindividual is said simply to be risk averse. 2

Consider a simple gamble involving two outcomes:

g ≡ (p ◦ w1, (1 − p) ◦ ww).

Now suppose the individual is offered a choice between receiving wealth equal toE(g) = pw1 +(1−p)w2 with certainty (,i.e., probability one) or receiving the gambleg itself. We can assess the alternatives as follows:

V (g) = pu(w1) + (1 − p)u(w2),V (E(g)) = u (pw1 + (1 − p)w2) .

2A simple gamble is nondegenerate if it assigns strictly positive probability to at least two distinctwealth levels.

93


If the Bernoulli utility function u(·) is strictly concave, we have V (E(g)) > V (g),so the individual is risk averse. 3 In the case of the risk averse individual, therewill be some amount of money we could offer with certainty that would make himindifferent between accepting that wealth with certainty and facing the gamble g.We call this amount of money the certainty equivalent of the gamble g.

Definition 6.6.2 The certainty equivalent of any simple gamble g over wealthlevels is an amount of wealth, CE(g), offered with certainty, such that V (g) ≡u(CE(g)). The risk premium is an amount of wealth, P (g), such that V (g) ≡u(E(g) − P (g)). Clearly, P (g) ≡ E(g) − CE(g).

Exercise 6.6.1 Consider an individual whose preferences over gambles is repre-sented by the VNM utility function. Verify that the individual is risk averse overgambles involving nonnegative wealth levels if and only if her Bernoulli utility func-tion is strictly concave on R+.

I want to attach meaning to the expression: “A gamble g yields unambiguouslyhigher returns than a gamble g

′.” The following result does not depend upon how

rich and how risk averse the decision maker is.

Proposition 6.6.1 (The First-Order Stochastic Dominance (FOSD)) Assumewithout loss of generality that w1 ≥ w2 ≥ · · · ≥ wn ≥ 0. Assume further that u(0) =0 and u(·) is increasing. Let g = (p1◦w1, . . . , pn◦wn) and g

′= (p

′1◦w1, . . . , p

′n◦wn).

Then, V (g) ≥ V (g′) if

∑ki=1 pi ≥

∑ki=1 p

′i for each k = 1, . . . , n 4

Proof of Proposition 6.6.1 Let u(·) be the Bernoulli utility function of the3This comes from Jensen’s inequality. Those who don’t know what Jensen’s inequality is should

consult my lecture notes on mathematics. Or, you can talk to people who do either statistics oreconometrics, or both.

4Suppose, instead that the domain of monetary incomes is given as +, i.e., a continuum. Then,any gamble is represented by a probability distribution function. Let F (·) and G(·) be two distribu-tion functions defined on +. Then, F (·) first-order stochastically dominates G(·) if F (x) ≤ G(x)for any x ∈ +. Then, Proposition 6.6.1 is re-formalized as follows: V (F ) =

�u(x)dF (x) ≥�

u(x)dG(x) = V (G) if F (·) first-order stochastically dominates G(·).

94


decision maker. What we want to show is V (g) − V (g′) ≥ 0.

V (g) − V (g′)

=n∑

i=1

piu(wi) −∑i=1

p′iu(wi)

=n∑

i=1

(pi − p′i)u(wi)

= (p1 − p′1)u(w1) +

n∑i=2

(pi − p′i)u(wi)

≥ (p1 − p′1)u(w2) +

n∑i=2

(pi − p′i)u(wi) (∵ p1 ≥ p

′1 and u(w1) ≥ u(w2) ≥ 0)

= u(w2)2∑

i=1

(pi − p′i) +

n∑i=3

(pi − p′i)

≥ u(w3)2∑

i=1

(pi − p′i) +

n∑i=3

(pi − p′i)

(∵

2∑i=1

pi ≥2∑

i=1

p′i and u(w2) ≥ u(w3) ≥ 0

)

= u(w3)3∑

i=1

(pi − p′i) +

n∑i=4

(pi − p′i)u(wi)

......

......

...

≥ u(wn)n∑

i=1

(pi − p′i)

≥ 0

(∵

n∑i=1

pi ≥n∑

i=1

p′i and u(wn) ≥ 0

).�

6.7 Measures of Risk Aversion

Many times, we not only want to know whether someone is risk averse, but also howrisk averse they are. Ideally, we’d like a summary measure that allows us both tocompare the degree of risk aversion across individuals and to gauge how the degree ofrisk aversion for a single individual might vary with the level of their wealth. Arrow(1970) and Pratt (1964) have proposed the following measure of risk aversion.

Definition 6.7.1 Given a (twice-differentiable) Bernoulli utility function u(·) formoney, the Arrow-Pratt measure of absolute risk aversion is given by

Ra(w) ≡ −u′′(w)

u′(w)

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R+R+

R

v

hu

R+R+

R

v−1

uh

Figure 6.1: The Relationship between u and v through h

The sign of this measure immediately tells us the basic attitude toward risk:Ra(w) is positive, negative, or zero as the agent is risk averse, risk loving, or riskneutral, respectively. In addition, any positive affine transformation of VNM utilitywill leave the measure unchanged.

Exercise 6.7.1 The Bernoulli utility function of the decision maker is given as:

u(x) = − exp(−αx)

where α > 0. Show that the Arrow-Pratt measure of absolute risk aversion of thedecision maker is constant for any x.

6.7.1 Comparisons across Individuals

Suppose that there are two individuals, 1 and 2, and that individual 1 has a Bernoulliutility function u(·), and individual 2 has a Bernoulli utility function v(·). Let’ssuppose that at every wealth level, w, individual 1’s Arrow-Pratt measure of riskaversion is larger than individual 2’s. That is,

R1a(w) = −u

′′(w)

u′(w)

> −v′′(w)

v′(w)

= R2a(w) ∀w ≥ 0 (∗),

where both u′and v

′are always strictly positive. Assume that v(·) takes on all values

in [0,∞). Then, I can always define a mapping h : R → R such that u(x) = h(v(x))for any x ∈ R+. Consequently, we may describe h : R → R as follows:

h(x) = u(v−1(x)) ∀x ≥ 0.

Therefore, h inherits twice differentiability from u and v with

h′(x) =

u′(v−1(x)

v′(v−1(x))> 0, and

h′′(x) =

u′(v−1(x))

[u

′′(v−1(x))/u

′(v−1(x)) − v

′′(v−1(x))/v

′(v−1(x))

][v′(v−1(x))]2

< 0

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for all x > 0, where the first inequality follows because u′, v

′> 0, and the second

inequality follows from the relation (∗). 5 Therefore, h is a strictly increasing, strictlyconcave function.

Consider a gamble g = (p1 ◦ w1, . . . , pn ◦ wn) over wealth levels. Let CEi(g)denote consumer i’s certainty equivalent for the gamble g.

n∑i=1

piu(wi) = u(CE1(g)),

n∑i=1

piv(wi) = v(CE2(g))

Claim 6.7.1 CE1(g) < CE2(g).

Proof of Claim 6.7.1: Recall

u(CE1(g)) =n∑

i=1

piu(wi) (6.1)

h(x) = u(v−1(x)). (6.2)

Substituting v−1(x) for x in equation (2), we have

h(v(x)) = u(x). (∗)Plugging (∗) into the above equation (1), we do the following computation.

u(CE1(g)) =n∑

i=1

pih(v(wi))

< h

(n∑

i=1

piv(wi)

)(because h(·) is strictly concave.)

= h(v(CE2(g))

)= u(CE2(g)).

Since u′is increasing, we have CE1(g) < CE2(g). �

In particular, Jensen’s inequality is employed in the second line of the abovederivation. 6

5

h′(x) = u

′(v−1(x)) · d

dx

�v−1(x)

�

Let f(x) = v−1(x). Then, we have x = v(f(x)). Differentiating this equation with respect to

x yields 1 = v′(f(x))f

′(x). Taking into account the relation that f

′(x) = d

dx(v−1(x)), we have

ddx

(v−1(x)) = 1/v′(v−1(x)).

6You should consult my lecture notes on mathematics for Jensen’s inequality.

97


Corollary 6.7.1 (Comparisons across individuals) Suppose that two Bernoulliutility functions u(·) for individual 1 and v(·) for individual 2 are given. We sayunambiguously say that individual 1 is more risk averse than individual 2 when

1. R1a(x) > R2

a(x) for every x;

2. There exists an increasing, strictly concave function h(·) such that u(x) =h(v(x)) at all x;

3. CE1(g) < CE2(g) for any gamble g ∈ G4. P 1(g) > P 2(g) for any gamble g ∈ GIn fact, these four definitions are equivalent.

6.7.2 Comparisons across wealth levels

It is a common contention that wealthier people are willing to bear more risk thanpoorer people. Although this might be due to differences in utility functions acrosspeople, it is more likely that the source of the difference lies in the possibility thatricher people “can afford to take a chance.” Hence, I shall consider the followingcondition.

Definition 6.7.2 The Bernoulli utility function u(·) for money exhibits decreasingabsolute risk aversion if Ra(x) is a decreasing function of x.

Definition 6.7.3 Given a Bernoulli utility function u(·), the coefficient of rela-tive risk aversion (CRRA) at x is Rr(x) = −xu

′′(x)/u

′(x).

Exercise 6.7.2 The Bernoulli utility function of the decision maker is given as

u(x) =x1−σ

1 − σ

where −∞ < σ �= 1. Show that the coefficient of relative risk aversion (CRRA) isconstant at any x.

Definition 6.7.4 The Bernoulli utility function u(·) for money exhibits nonin-creasing relative risk aversion if Rr(x) is nonincreasing function of x.

The assumption of decreasing absolute risk aversion yields many economicallyreasonable results concerning risk-bearing behavior. However, in applications, it isoften complemented by a stronger assumption: nonincreasing relative risk aversion.As an exercise, you are encouraged to show that if Rr(x) is nonincreasing functionof x, then Ra(x) is decreasing function of x.

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6.8 State-Dependent Utility

In the previous section, we assumed that the decision maker cares solely about thedistribution of monetary payoffs he receives. Here we consider the possibility thatthe decision maker may care not only about his monetary returns but also aboutthe underlying events, or states of nature, that cause them. We often know that therandom outcome is generated by some underlying causes. For example, the monetarypayoff of an insurance policy might depend on whether or not a certain accident hashappened, the payoff on a corporate stock on whether the economy is in a recession,and the payoff of a casino gamble on the number selected by the roulette wheel.

I call these underlying causes states or state of nature. I denote the set of statesby S and an individual state by s ∈ S. For simplicity, we assume that the set ofstates S is finite and that each state s has a well defined, objective probability ps > 0that it occurs.

Definition 6.8.1 A random variable is a function g : S → R that maps statesinto monetary outcomes.

Every random variable g(·) gives rise to a monetary lottery describable by thedistribution function F (·) with

F (x) =∑

{s∈S|g(s)≤x}ps ∀x

Note that there is a loss in information in going from the random variable repre-sentation of uncertainty to the lottery representation; I do not keep track of whichstates give rise to a given monetary outcome, and only the aggregate probability ofevery monetary outcome is retained. Because I take S to be finite, we can representa random variable with monetary payoffs by the vector (x1, . . . , xS), where ss is themonetary payoff in state s ∈ S. The set of all random variables is then RS .

6.8.1 State-Dependent Preferences and the Extended VNM UtilityRepresentation

Definition 6.8.2 The preference relation � has an extended VNM utility rep-resentation if for every s ∈ S, there is a function us : R → R such that for any(x1, . . . , xS) ∈ RS and (x

′1, . . . , x

′S) ∈ RS,

(x1, . . . , xS) � (x′1, . . . , x

′S) ⇐⇒

∑s∈S

psus(xs) ≥∑s∈S

psus(x′s).

6.8.2 Existence of an Extended VNM Utility Representation

Observe first that since ps > 0 for every s ∈ S, we can formally include ps in thedefinition of the utility function at state s. That is, to find an extended VNM utility

99


representation, it suffices that there be functions us(·) such that

(x1, . . . , xS) � (x′1, . . . , x

′S) ⇐⇒

∑s∈S

us(xs) ≥∑s∈S

us(x′s).

This is because if such functions us exist, then we can define us(·) = (1/ps)us(·) foreach s ∈ S, and we will have∑

s∈S

us(xs) ≥∑s∈S

us(x′s) ⇐⇒

∑s∈S

psus(xs) ≥∑s∈S

psus(x′s).

Thus, from now on, I focus on the existence of an additively separable form∑

s∈S us(·),and the ps’s cease to play any role in the analysis.

I now allow for the possibility that within each state s, the monetary payoff isnot a certain amount of money xs but a random amount with distribution functionFs(·). We denote these uncertain alternatives by g = (F1, . . . , FS). Thus, g is a kindof compound lottery that assigns well-defined monetary gambles as prizes contingenton the realization of the state of the world s. We denote G the set of all such possiblelotteries.

My starting point is now a a rational preference relation � on G. Note that

αg + (1 − α)g′=

(αF1 + (1 − α)F

′1, . . . , αFS + (1 − α)F

′S

)has the usual interpretation as the reduced gamble arising from a randomizationbetween g and g

′, although here I deal with a reduced lottery within each state s.

Definition 6.8.3 (The Extended Expected Utility Property) The function V :G → R has the extended expected utility property if for each s ∈ S, there is anassignment of numbers (us(a1), . . . , us(an)) to the n outcomes such that for everyxs ∈ G,

V (g) =∑s∈S

n∑i=1

pi,sus(ai),

where xs ≡ (p1,s ◦ a1, . . . , pn,s ◦ an) is the simple gamble induced by xs ∈ G.

If I extend the domain of our Axioms G1 through G6 from G to G, I can easilyestablish the existence of extended VNM utility function below.

Theorem 6.8.1 (Existence of the Extended VNM Utility Function on G)Let preferences � over gambles in G satisfy axioms G1 through G6. Then there isa function V : G → R representing � on G such that V has the extended expectedutility property.

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Chapter 7

General Equilibrium underUncertainty

7.1 A Market Economy with Contingent Commodities

I consider an environment with n physical commodities, I consumers, and J firms.The new element I have to take into account is the fact that technologies, endow-ments, and preferences are now uncertain. Throughout the argument, I representuncertainty by assuming that technologies, endowments, and preferences depend onthe state of the world. A state of the world is to be understood as a complete de-scription of a possible outcome of uncertainty, the description being sufficiently finefor any two distinct states of the world to be mutually exclusive. I assume that anexhaustive set S of states of the world is given to us. For simplicity I take S to befinite set with (abusing notation slightly) S elements. A typical element is denoteds = 1, . . . , S.

An approach I will take here is, I think, the most cheapest but the most boringone. Namely, I will change the concept of commodity in such a way that all theprevious general equilibrium analyses in economies without uncertainty will carryover straightforwardly. Of course, whether or not this new concept of commoditymakes sense to you is another issue, though. Indeed, you probably won’t find ituseful concept to pay a serious consideration.

Definition 7.1.1 For every physical commodity � = 1, . . . , n and state s = 1, . . . , S,a unit of state-contingent commodity � is a title (right) to receive a unit of thephysical good � if and only if s occurs. Accordingly, a state-contingent commodityvector is specified by

x =

⎛⎜⎝x11, . . . , xn1︸︷︷︸

state 1

, . . . , x1S , . . . , xnS︸︷︷︸state S

⎞⎟⎠ ∈ RnS,

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CHAPTER 7. GENERAL EQUILIBRIUM UNDER UNCERTAINTY

and is understood as an entitlement to receive the commodity vector (x1s, . . . , xns) ifstate s occurs. Moreover, a negative entry is understood as an obligation to deliver.

With the help of the concept of contingent commodity vectors, we can nowdescribe how the characteristics of economic agents depend on the state of the world.We let the endowments of consumer i = 1, . . . , I be a contingent commodity vector:

ei =

⎛⎜⎝ei

11, . . . , ein1︸︷︷︸

state 1

, . . . , ei1S , . . . , ei

nS︸︷︷︸state S

⎞⎟⎠ ∈ RnS

The meaning of this is that if state s occurs then consumer i has endowment vector(ei

1s, . . . , eins) ∈ Rn.

The preferences of consumer i may also depend on the state of the world (e.g.,the consumer’s enjoyment of wine may well depend on the state of his health). 1 Irepresent this dependence formally by defining the consumer’s preferences over con-tingent commodity vectors. That is, I let the preferences of consumer i be specifiedby a rational (complete and transitive) relation �i defined on a consumption setXi ⊂ RnS.

Definition 7.1.2 Consumer i evaluates contingent commodity vectors by first as-signing to state s a probability πi

s (which could be different across consumers), thenevaluating the physical commodity vectors at state s according to a Bernoulli state-dependent utility function ui

s(xi1s, . . . , xi

ns), and finally computing the expected util-ity. That is, the preferences of consumer i over two contingent commodity vectorsxi, xi ∈ Xi ⊂ RnS satisfy

xi �i xi ⇐⇒S∑

s=1

πisu

is(x

i1s, . . . , xi

ns) ≥S∑

s=1

πisu

is(x

i1s, . . . , xi

ns).

Similarly, the technological possibilities of firm j are represented by a productionset Y j ⊂ RnS. The interpretation is that a state-contingent production plan yj ∈ RnS

is a member of Y j if for every s, the input-output vector (yj1s, . . . , yj

ns) of physicalcommodities is feasible for firm j when state s occurs.

Example 7.1.1 Suppose that there are two states s1 and s2, representing the econ-omy being in the boom and in the recession, respectively. There are two physicalcommodities: input (commodity 1) and output (commodity 2). In this case, the el-ements of Y j are four-dimensional vectors. Assume that inputs must be investedbefore the resolution of the uncertainty about the economic condition and that a unitof inputs produces a unit of output if and only if the economy is in the boom. Then,

yj = (yj11, y

j21, y

j12, y

j22) = (−1, 1,−1, 0)

1In the future, you might get married or not, you might get a baby or not, you might get a newjob or lose it. These things potentially affects the way you evaluate the commodities.

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is a feasible plan.

To complete the description of an economy in the same manner as we describeeconomies without uncertainty, it only remains to specify ownership shares for everyconsumer i and firm j. In principle, these shares could also be state-contingent. Itwill be simple, however, to let θi

j ≥ 0 be the share of firm j owned by consumer i

whatever the state is. Note that∑

i θij = 1 for every j.

7.2 Arrow-Debreu Equilibrium

We have seen in the previous section how an economy where uncertainty matters canbe extended to an artificial economy which is described by means of a set of statesof the world S, a consumption set Xi ⊂ RnS (n stands for the number of physicalcommodities), an endowment vector ei ∈ Xi ⊂ RnS, and a preference relation �i onXi for every consumer i, together with a production set Y j ⊂ RnS and profit shares(θ1

j , . . . , θIj ) for every firm j.

Here I have a big assumption. I postulate the existence of a market for everycontingent commodity (�, s). 2 These markets open at date 0 before the resolution ofuncertainty. The price of the commodity is denoted p(�,s). What is being purchased(or sold) in the market for the contingent commodity (�, s) is commitments to receive(or to deliver) amounts of the physical good � if, and when, state of the world soccurs. It is important to note that although deliveries are contingent, the paymentsare not. Furthermore, I should recognize that for this market to be well defined,it is indispensable that all economic agents are able to recognize the occurrence ofs. That is, information should be symmetric across economic agents. I can, then,apply the concept of Walrasian equilibrium to our contingent claim market economy.When dealing with contingent commodities it is customary to call the Walrasianequilibrium an Arrow-Debreu equilibrium.

Definition 7.2.1 An allocation

(x1, . . . , xI , y1, . . . , yJ) ∈ X1 × · · · × XI × Y 1 × · · · × Y J ⊂ RnS(I+J)

and a system of prices for contingent commodities p = (p(1,1), . . . , p(n,S)) ∈ RnS

constitute an Arrow-Debreu equilibrium if:

1. (Profit Maximization): For every firm j, yj satisfies p · yj ≥ p · yj for allyj ∈ Y j.

2. (Utility Maximization): For every consumer i, xi is maximal for �i in thebudget set ⎧⎨

⎩xi ∈ Xi

∣∣∣∣∣ p · xi ≤ p · ei +∑j∈J

θijp · yj

⎫⎬⎭ .

2Contingent commodity (�, s) means that a physical commodity � when state s occurs.

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3. (Market Clearing):∑

i∈I xi =∑

j∈J yj +∑

i∈I ei.

Standard assumptions we have made for the economy with certainty can bestraightforwardly extended to the Arrow-Debreu contingent commodity market econ-omy. As a result, we are able to ensure that the Arrow-Debreu equilibrium exists,equilibrium allocation is Pareto efficient (the first welfare theorem), and the extendedversion of the second welfare theorem is established. The efficiency implication ofArrow-Debreu equilibrium says, effectively, that the possibility of trading in con-tingent commodities leads, at equilibrium, to an efficient allocation of risk. It isimportant to realize that at any production plan, the profit of a firm, p · yj, is anon-random amount of dollars. Productions and deliveries of goods do, of course,depend on the state of the world, but the firm is active in all the contingent marketsand manages to insure completely.

7.3 The Working of the Arrow-Debreu Economy

Imagine the following story which aims at accounting for the Arrow-Debreu economy.There is only one bank in the world. Don’t even distinguish the central bank fromcommercial banks. In any case, there is only one bank. There is only one centralizedmarket place in which all the commitments are made at date 0 but all the realtransactions are carried out at date 1 according to the commitments made at date0. Every consumer has his personal credit card account provided by the bank.Consumer i’s credit limit is given as p · ei +

∑j θi

jΠj(p), i.e., his budget set at date

0. Each firm also has its business credit card account, but with no credit limit. Ifeach consumer or firm buys one unit of contingent commodity (�, s) in the marketat date 0, the amount p(�,s) will be debited from his account. In turn, he will receivea receipt from the bank. His receipt says that “The bank gives consumer i (firm j)the right to receive one unit of physical commodity � at date 1 if and only when states occurs.” When he wakes up tomorrow, then at date 1, he is able to see if state soccurs. It is also common knowledge that all people see what state occurs at date1. If and only when state s occurs, he brings his receipt with him to the marketand receives one unit of good �. If each consumer or firm sells one unit of contingentcommodity (�, s) in the market at date 0, the amount p�,s will be credited in hisaccount. At the same time, the bank has the receipt which says that “The bank hasthe right to obtain one unit of physical commodity � at date 1 from consumer i (firmj) if and only when state s occurs.” When he wakes up tomorrow, at date 1, he seesstate s occurs. Then, the bank comes to consumer i’s home (firm j’s office) with itsreceipt and demands one unit of good �. Here it is assumed that no one can walkaway. After obtaining the commodity, the bank brings it to the market. If states does not occurs, there is no right whatsoever. The Arrow-Debreu equilibrium ofthis economy requires that there be a system of state contingent prices {p�,s} suchthat (1) each consumer maximizes his utility subject to his credit limit; (2) each firmmaximizes its profit; (3) all markets clear at date 0; and (4) no one goes bankrupt.

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7.3.1 Ex Ante V.S. Ex Post

For simplicity, we ignore production. We set up the consumer’s optimization problemas follows:

maxx∈ nS

S∑s=1

πsus(xs)

s.t.

{x ∈ X

∣∣∣∣∣ p · x ≤ p · e}

(1∗)

Let x∗ be the solution to the optimization problem (1∗). For each state s ∈ S,we set up the following optimization problem.

maxxs∈ n

us(xs)

s.t.

{(xs, x

∗−s) ∈ X

∣∣∣∣∣ p · (xs, x∗−s) ≤ p · e

}(1∗∗)

Note that x∗−s = (x∗1, . . . , x∗

s−1, x∗s+1, . . . , x∗

S). The optimization problem (1∗∗)tries to answer what the optimal consumption plan in state s is if the decision makermade a commitment to x∗−s, which is a set of promises (or contracts) he made (at date0!) if any other states occur. I claim that one of the solutions to the optimizationproblem (1∗∗) is indeed x∗

s. Namely, when he contemplates at date 0 what he can doat date 1, he knows at date 0 that he will not change his mind at date 1. Supposenot, that is, there is xs such that us(xs) > u(x∗

s) and (xs, x∗−s) is in the budget set.

Since πs > 0 for each s = 1, . . . , S, we must have

πsus(xs) +∑s′ �=s

πs′us

′ (x∗s′ ) >

S∑s=1

πsus(x∗s).

This contradicts the fact that x∗ is the solution to the optimization problem (1∗).Therefore, the optimization (1∗) implies that the optimization (1∗∗) for each s. Thisis what I mean by “Ex ante optimality implies ex post optimality.”

7.3.2 No market at date 1 and No real transaction at date 0

This means that each individual carries out all transactions according to the promisesall market participants made in the date 0 market. Each individual keeps thepromise.

7.3.3 No need to open spot markets at date 1

Consider the Arrow-Debreu economy with no production. Let (x1, . . . , xI) ∈ RnSI

be an Arrow-Debreu equilibrium allocation with prices (p(1,1), . . . , p(n,S)) ∈ RnS .

105


When date 1 arrives, a state of the world s is revealed, contracts (promises) areexecuted, and every consumer i receives xi

s = (xi(1,s), . . . , xi

(n,s)) ∈ Rn. Imagine nowthat, after this but before the actual consumption of xi

s, markets for the n physicalgoods were to open at date 1. I ask if there would be any incentive to trade in thesemarkets. I claim that the answer is “no.”

I argue by contradiction. Suppose that there were potential gains from tradeamong the consumers. That is, there were xi

s = (xi(1,s), . . . , xi

(n,s)) for i = 1, . . . , I,such that

∑i x

is ≤ ∑

i eis and (xi

1, . . . , xis, . . . , xi

S) �i (xi1, . . . , xi

s, . . . , xiS) for all i,

with at least one preference strict. This contradicts the conclusion of the first welfaretheorem of the Arrow-Debreu economy. 3 Hence, there is no reason for further tradeto take place.

7.3.4 Each consumer has a single budget

As I explained above, in the context of the Arrow-Debreu economy, there is no marketat date 1 and there is no need to reopen markets at date 1. Since uncertainty willbe resolved at date 1, each consumer must contemplate all possible consequencessimultaneously when making his contingent plan in the market at date 0. Thisimplies that each consumer has only a single budget in the Arrow-Debreu economy.

7.4 Sequential Trade

7.4.1 Preliminaries

The Arrow-Debreu framework provides a remarkable illustration of the power ofgeneral equilibrium theory. Yet, it hardly be realistic. Indeed, at an Arrow-Debreuequilibrium all trade takes place simultaneously and before the uncertainty is re-solved. Trade, so to speak, is a one-shot affair. In reality, however, trade takes placeto a large extent sequentially over time, and frequently as a consequence of informa-tion disclosures. The aim of this section is to introduce a model of sequential tradeand show that Arrow-Debreu equilibria can be reinterpreted by means of tradingprocesses that actually unfold through time.

For the analysis to be as simple as possible I consider only exchange economies.I take Xi = RnS

+ for every consumer i. To begin with, I assume that there are twodates, t = 0 and t = 1, that there is no information whatsoever at t = 0, and thatthe uncertainty has resolved completely at t = 1. Again for simplicity, I assume thatthere is no consumption at t = 0.

As I discussed in the previous section, the Arrow-Debreu equilibrium allocation3You can generalize this argument into the subset of consumers so that you appeal to the core

property of the Arrow-Debreu equilibrium allocation.

106


is Pareto efficient and there is no need to reopen markets at t = 1. Matters, however,are different if not all the n×S contingent commodity markets are available at t = 0.Then the initial trade needed for a Pareto efficient allocation may not be feasible andit is quite possible that ex post (i.e., after the revelation of the state s) the resultingconsumption allocation is not Pareto optimal. There would then be an incentive toreopen the markets and retrade.

A most interesting possibility, first observed by Arrow (1953, 1964), is that, evenif not all contingent commodities are available at t = 0, it may still be the caseunder some conditions that the retrading possibilities at t = 1 guarantee that Paretoefficiency is reached, nevertheless. That is, the possibility of ex post trade can makeup for an absence of some ex ante markets. In what follows, I shall verify that thisis the case whenever at least one physical commodity can be traded contingently att = 0 if, in addition, spot markets open at t = 1 and the spot equilibrium prices arecorrectly anticipated at t = 0.

At t = 0 consumers have expectations regarding the spot market prices prevailingat t = 1 for each possible state s ∈ S. Denote the price vector expected to prevailin state s spot market by ps ∈ Rn, and the overall expectation vector by p =(p1, . . . , pS) ∈ RnS . Suppose that, in addition, at date t = 0 there is trade in the Scontingent commodities denoted by (1, 1) to (1, S); that is, there is contingent tradeonly in the physical good with the label “1.” We denote the vector of prices for thesecontingent commodities traded at t = 0 by (q1, . . . , qS) ∈ RS .

Faced with security prices q ∈ RS at t = 0 and correctly expected spot marketprices (p1, . . . , pS) ∈ RnS at t = 1, every consumer i formulates a portfolio plan(zi

1, . . . , ziS) ∈ RS for contingent commodities at t = 0, as well as a set of spot

market consumption plans (xi1, . . . , xi

S) ∈ RnS for the different states that mayoccur at t = 1. Let U i(·) be a utility function for �i. Then the problem of consumeri can be expressed formally as

max(xi

1, . . . , xiS) ∈ RnS

+

(zi1, . . . , zi

S) ∈ RS

U i(xi1, . . . , xi

S)

s.t. (1)∑s∈S

qszis ≤ 0; (∗)

(2) ps · xis ≤ ps · ei

s + p(1,s)zis ∀s ∈ S.

Restriction (1) is the budget constraint corresponding to security trade at t = 0.The family of restrictions (2) are the budget constraints for the different spot mar-kets. Observe that we are not imposing any restriction on the sign or the magnitudeof zi

s. If zis < −ei

s then one says that at t = 0, consumer i is selling good 1 short.This is because he is selling at t = 0, contingent on state s occurring, more thanhe has at t = 1 if s occurs. Hence, if s occurs he will actually have to buy in the

107


spot market the extra amount of the first good required for the fulfillment of hiscommitments. The possibility of selling short is, however, indirectly limited by thefact that consumption, and therefore ex post wealth, must be nonnegative for everys. Observe also that we have taken the wealth at t = 0 to be zero. This is simply aconvention.

To define an appropriate notion of sequential trade I shall impose a key condition:Consumers’ expectations must be self-fulfilled, or rational; that is, I require thatconsumers’ expectations of the prices that will clear the spot markets for the differentstates s do actually clear them once date t = 1 has arrived and state s is revealed.In other words, all agents’s expectations about prices are not only common but alsocorrect.

Definition 7.4.1 A collection formed by a price vector q = (q1, . . . , qS) ∈ RS forcontingent first good commodities at t = 0, a spot price vector

ps = (p(1,s), . . . , p(n,s)) ∈ Rn

for every s ∈ S, and for every consumer i, consumption plans zi = (zi1, . . . , zi

S) ∈ RS

at t = 0 and xi = (xi1, . . . , xi

S) ∈ RnS at t = 1 constitutes a Radner equilibriumif:

1. For every i, the portfolio and consumption plans zi, xi solve the constrainedmaximization problem (∗).

2.∑

i∈I zis ≤ 0 and

∑i∈I xi

s ≤∑

i∈I eis for every s ∈ S (Market Clearing).

7.4.2 Arrow Security

Let us set p(1,s) = 1 for each state s as a normalization. This implies that at eachstate we use commodity 1 as a unit of account. You could imagine one unit ofcommodity 1 as one dollar. 4 Note that this still leaves one degree of freedom, thatcorresponding to the forward trades at date 0 (so I could put q1 = 1, or perhaps∑

s qs = 1). Then, I can rewrite the consumer’s optimization problem as follows:

max(xi

1, . . . , xiS) ∈ RnS

+

(zi1, . . . , zi

S) ∈ RS

U i(xi1, . . . , xi

S)

s.t. (1)∑s∈S

qszis ≤ 0; (∗)

(2) ps · xis ≤ ps · ei

s + zis ∀s ∈ S.

Definition 7.4.2 One unit of Arrow Security s (s = 1, . . . , S) is a promise todeliver one unit of physical commodity 1 when state s occurs. Its price qs, payableat date 0, is measured in the unit of account at date 0.

4No matter what state occurs, one dollar is one dollar (the nominal value is always the same).However, the purchasing power of one dollar can be different across states.

108


7.4.3 Implementing the Arrow-Debreu equilibrium allocations

The following two propositions effectively say that the set of Radner equilibriumallocations is the same as the set of Arrow-Debreu equilibrium allocations. Then,the number of asset markets needed at date 0 can be significantly reduced from nSto S.

Proposition 7.4.1 (Arrow-Debreu implies Radner) If the allocation x ∈ RnSI

and the contingent commodities price vector (p1, . . . , pS) ∈ RnS++ constitutes an

Arrow-Debreu equilibrium, then there are prices q ∈ RS++ for contingent first good

commodities and consumption plans for these commodities z = (z1, . . . , zI) ∈ RSI

such that the consumption plans x, z, the prices q, and the spot prices (p1, . . . , pS)constitutes a Radner equilibrium.

Proof of Proposition 7.4.1: Consumer i’s budget set of the Arrow-Debreuproblem is

BiAD =

{(xi

1, . . . , xiS) ∈ RnS

+

∣∣∣∣∣ ∑s

ps · (xis − ei

s) ≤ 0

}.

Consumer i’s budget set of the Radner problem is

BiR =

{(xi

1, . . . , xiS) ∈ RnS

+

∣∣∣∣∣ ∃(zi1, . . . , zi

S) s.t.∑

s

qszis ≤ 0 and ps · (xi

s − eis) ≤ zi

s ∀s

}.

What we want to show are the following three: (1) BiAD ⊂ Bi

R for each i ∈ I; (2)∑i z

i ≤ 0; and (3)∑

i(xis − ei

s) ≤ 0 for each s. Let xi ∈ BiAD for each i ∈ I. Set

zis = ps · (xi

s − eis) for each s and each i ∈ I. Because xi ∈ Bi

AD, we have∑s

zis =

∑s

ps · (xis − ei

s) ≤ 0 (∗).

Setting qs = 1 for each s, we have

qszis = zi

s = ps · (xis − ei

s) ∀s (∗∗).

Summing the above term over s, we have∑s

qszis =

∑s

ps · (xis − ei

s) ≤︸︷︷︸xi∈Bi

AD

0.

Setting ps = ps for each s, we have ps · (xis − ei

s) ≤ zis for each s. Thus, xi ∈ Bi

R.Hence, we are done about (1). Since (x, p) constitutes an A-D equilibrium, for eachs, we have ∑

i

(xis − ei

s) ≤ 0

109


This implies that all spot markets clear, which confirms (3). Summing over i, foreach s, we obtain∑

i∈Izis = ps ·

∑i∈I

(xis − ei

s) ≤︸︷︷︸(x,p) is an A-D equilibrium

0

Thus, the security market clear, which completes (2). �

Proposition 7.4.2 (Radner implies Arrow-Debreu) If the consumption plansx ∈ RnSI , z ∈ RSI and prices q ∈ RS

++, (p1, . . . , pS) ∈ RnS++ constitute a Radner

equilibrium, then there is a vector (p1, . . . , pS) ∈ RnS++ such that the allocation x and

the contingent commodities price vector (p1, . . . , pS) ∈ RnS++ constitute an Arrow-

Debreu equilibrium.

Proof of Proposition 7.4.2: What we want to show are the following two: (1)Bi

R ⊂ BiAD for each i and (2)

∑i(x

is − ei) ≤ 0 for each s. Note first that, since

(x, z, q, p) is a Rander equilibrium, we have∑

i(xis − ei

s) ≤ 0 for each s. Thus, (2) isverified. Let xi ∈ Bi

R. Then,

ps · (xis − ei

s) ≤ zis ∀s =⇒ qsps · (xi

s − eis) ≤ qsz

is ∀s.

Set ps = qsps for each s. Summing the above expression over s, we have∑s

ps(xis − ei

s) ≤∑

s

qszis ≤ 0.

This implies that xi ∈ BiAD associated with the contingent commodity price vector

p = (p1, . . . , pS) = (q1p1, . . . , qSpS). Hence (1) is also verified. �

7.5 Asset Markets

I begin again with the simplest situation, in which we have two dates, t = 0 andt = 1, and all the information is revealed at t = 1. Further, for notational simplicityI assume that consumption takes place only at t = 1.

I view an asset, or more precisely, a unit of an asset, as a title to receive eitherphysical goods or dollars at t = 1 in amounts that may depend on which state occurs.The payoffs of an asset are known as its returns. If the returns are in physical goods,the asset is called real. If they are in paper money, they are called financial. Here Ideal only with the real case and, moreover, I assume that the returns of assets areonly in amounts of physical good 1. It is then convenient to normalize the spot priceof that good to be 1 in every state, so that, in effect, I am using it as numeraire.

Definition 7.5.1 A unit of an asset, or security, is a title to receive an amountrs of good 1 at t = 1 if state s occurs. An asset is therefore characterized by itsreturn vector r = (r1, . . . , rS) ∈ RS.

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Examples of assets include the following:

1. r = (1, . . . , 1). This asset promises the future noncontingent delivery of oneunit of good 1. Its real-world counterparts are the markets for commodityfutures. In the special case where there is a single consumption good, I callthis asset the safe (or riskless asset. It is important to realize that with morethan one physical good, a futures contract is not riskless: its return in termsof purchasing power depends on the spot prices of all the goods.

2. r = (0, . . . , 0, 1, 0, . . . , 0). This asset pays one unit of good 1 if and only if acertain state occurs. In the current theoretical setting, they are often calledArrow securities.

Example 7.5.1 (Options) This is an example of a so-called derivative asset,that is, of an asset whose returns are somehow derived from the returns of anotherasset. Suppose there is a primary asset with return vector r ∈ RS. Then a (Eu-ropean) call option on the primary asset at the strike price c ∈ R is itself anasset. A unit of this asset gives the option to buy, after the state is revealed (butbefore the returns are pair), a unit of the primary asset at price c (the price c is inunits of the “numeraire,” that is, of good 1).

What is the return vector r(c) of the option? In a given state s, the option willbe exercised if and only if rs > c (I neglect the case rs = c). Hence

r(c) = (max{0, r1 − c}, . . . ,max{0, rS − c}) .

For a primary asset with returns r = (4, 3, 2, 1) specific examples are

r(3.5) = (0.5,0, 0, 0);r(2.5) = (1.5,0.5,0, 0);r(1.5) = (2.5,1.5,0.5,0).

I proceed to extend the analysis in the previous section by assuming that there isa given set of assets, known as an asset structure, and that these assets can be freelytraded at date t = 0. Each asset k is characterized by a vector of returns rk ∈ RS .The number of assets is K. As before, I assume that there are no initial endowmentsof assets and that short sales are possible. The price vector for the assets traded att = 0 is denoted q = (q1, . . . , qK). A vector of trades in these assets, denoted byz = (z1, . . . , zK) ∈ RK , is called a portfolio.

Definition 7.5.2 A collection formed by a price vector q = (q1, . . . , qK) ∈ RK forassets traded at t = 0, a spot price vector ps = (p1s, . . . , pns) ∈ Rn for every s ∈ S,and, for every consumer i ∈ I, portfolio plan zi = (zi

1, . . . , ziK) ∈ RK at t = 0

and consumption plans xi = (xi1, . . . , xi

S) ∈ RnS at t = 1 constitutes a Radnerequilibrium if:

111


1. For every i ∈ I, the portfolio and consumption plans (zi, xi) solves the problem

max(xi

1, . . . , xiS) ∈ RnS

(zi1, . . . , zi

K) ∈ RK

Ui(xi1, . . . , xi

S)

s.t. (a)∑

k

qkzik ≤ 0

(b) ps · xis ≤ ps · ei

s +∑

k

p1szikrsk ∀ s ∈ S

2.∑

i∈I zik ≤ 0 and

∑i∈I xi

s ≤∑

i∈I eis for every k and s

In the budget set of the above definition, the wealth of consumer i at state s isthe sum of the spot value of his initial endowment and the spot value of the returnof his portfolio. Note that, without loss of generality, I can put p1s = 1 for all s ∈ S.It is convenient to introduce the concept the return matrix R. This is an S × Kmatrix whose kth column is the return vector of the kth asset. Hence, its generic skentry is rsk:

R =

⎛⎜⎝ r11 · · · r1K

.... . .

...rS1 · · · rSK

⎞⎟⎠

With this notation, the budget constraint of consumer i becomes

Bi(p, q, R) ={x ∈ RnS

+ | ∃zi ∈ RK with q · zi ≤ 0 s.t. (p1 · (x1 − ei1), . . . , pS · (xS − ei

S))T ≤ Rzi}

Call the system q ∈ RK of asset prices arbitrage-free if there is no portfolioz = (z1, . . . , zK) such that q · z ≤ 0, Rz ≥ 0, and Rz �= 0. In words, there is noportfolio that is budgetary feasible and that yields a nonnegative return in every stateand a strictly positive return in some state. Note that whether an asset price vectoris arbitrage free or not depends on the returns of the assets and not on preferences.If I assume that preferences are strongly monotone, then an equilibrium asset pricevector q ∈ RK must be arbitrage free: if it were not, it would be possible to increaseutility merely by adding to any current portfolio a portfolio yielding an arbitrageopportunity. Because there are not restrictions on short sales, this addition is alwayspossible.

I now present a very important implication of the assumption that unlimitedshort sales are possible. Namely, I will establish that knowledge of the return matrixR suffices to place significant restrictions on the asset price vector q = (q1, . . . , qK)that could conceivably arise at equilibrium.

Proposition 7.5.1 Assume that rk ≥ 0 and rk �= 0 for all k. Then, for every(column) vector q ∈ RK of asset prices arising in a Radner equilibrium, we canfind multiplies μ = (μ1, . . . , μS) ≥ 0, such that qk =

∑s μsrsk for all k (in matrix

notation, qT = μ · R).

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μ′

RS+

0

V

Figure 7.1: Construction of the No-arbitrage Weights

Proof : The proof is based on the following lemma.

Lemma 7.5.1 If the asset price vector q ∈ RK is arbitrage free, then there is avector of multipliers μ = (μ1, . . . , μS) ≥ 0 satisfying qT = μ · R.

Proof : Since we deal only with assets having nonnegative, nonzero returns, anarbitrage-free price vector q must have qk > 0 for every k. Also, without loss ofgenerality, we assume that no row of the return matrix R has all of its entries equalto zero.

Given an arbitrage-free asset price vector q ∈ RK , consider the convex set

V = {v ∈ RS | v = Rz for some z ∈ Rk with q · z = 0}.

The arbitrage freeness of q implies that V ∩{RS+\{0}} = ∅. Since both V and RS

+\{0}are convex sets and the origin ({0}) belongs to V , we can apply the separatinghyperplane theorem to obtain a nonzero vector μ

′= (μ

′1, . . . , μ

′S) such that μ

′ · v ≤ 0for any v ∈ V and μ

′ · v ≥ 0 for any v ∈ RS+. Note that it must be that μ

′ ≥ 0.Moreover, because v ∈ V implies −v ∈ V (this is because q · z = 0 ⇔ q · (−z) =−q · z = 0), it follows that μ

′ · v = 0 for any v ∈ V .

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μ′ · R

{z ∈ RK | μ′ · Rz = 0}

0T

qT

z

RK

Figure 7.2: Existence of an inadmissible z if qT is not proportional to μ′ · R.

We now argue that the row vector qT must be proportional to the row vectorμ

′ ·R ∈ RK . The entries of μ′and of R are all nonnegative and no row of R is null.

Therefore μ′ · R ≥ 0T and μ

′ · R �= 0T . If qT is not proportional to μ′ · R, then we

can find z ∈ RK such that q · z = 0 and μ′ · Rz > 0. But letting v = Rz, we would

then have v ∈ V and μ′ · v �= 0, which we have just seen cannot happen. Hence qT

must be proportional to μ′ · R; that is, qT = αμ

′ · R for some real number α > 0.Letting μ = αμ

′, we have the conclusion of the lemma. �

This completes the proof of the proposition. �

As we have already argued, if short sales of assets are possible and preferencesare strongly monotone (e.g., if preferences admit an expected utility representationwith strictly positive subjective probabilities for the states), then equilibrium assetprices must be arbitrage-free.

Definition 7.5.3 An asset structure with an S × K return matrix R is completeif rank R = S, that is, if there is some subset of S assets with linearly independentreturns.

Example 7.5.2 (Asset Structure for Arrow Securities and More) In the caseof S Arrow securities, the return matrix is the S × S identity matrix. This is the

114


canonical example of complete markets. But there are many other ways for a matrixto be nonsingular. Thus, with three states and three assets, we could have the returnmatrix

R =

⎡⎣ 1 0 0

1 1 01 1 1

⎤⎦ ,

which has rank equal to 3, the number of states.

Example 7.5.3 (Spanning through Options) Suppose that S = 4 and there isa primary asset with returns r = (4, 3, 2, 1). We have seen that, for every strike pricec, the option defined by c constitutes an asset with return vector

r(c) = (max{0, r1 − c},max{0, r2 − c},max{0, r3 − c},max{0, r4 − c}) .

Using options I can create a complete asset structure supported entirely on the pri-mary asset r. For example, the return vectors r(3.5), r(2.5), r(1.5), and r are linearlyindependent. Thus, the asset structure consisting of the primary asset plus three op-tions with strike prices 3.5,2.5, and 1.5 is complete. More generally, whenever theprimary asset is such that rs �= rs′ for all s �= s

′, it is possible to generate a complete

asset structure by means of options. If the primary asset does not distinguish betweentwo states, no derived asset can do so either.

Proposition 7.5.2 (A-D ⇔ Radner) Suppose that the asset structure is complete.Then:

1. (A-D ⇒ Radner): If the consumption plans x = (x1, . . . , xI) ∈ RnSI and theprice vector p = (p1, . . . , pS) ∈ RnS

++ constitutes an Arrow-Debreu equilibrium,then there are asset prices q ∈ RK

++ and portfolio plans z = (z1, . . . , zI) ∈ RKI

such that the consumption plans x, portfolio plans z, asset prices q, and spotprices p = (p1, . . . , pS) constitutes a Radner equilibrium.

2. (Radner ⇒ A-D): If the consumption plans x ∈ RnSI , portfolio plans z ∈RKI , and prices q ∈ RK

++, p = (p1, . . . , pS) ∈ RnS++ constitutes a Radner equilib-

rium, then there are multipliers (μ1, . . . , μS) ∈ RS++ such that the consumption

plans x and the contingent commodities price vector (μ1p1, . . . , μSpS) ∈ RnS

constitutes an Arrow-Debreu equilibrium. (The multiplier μs is interpreted asthe value, at t = 0, of a dollar at t = 1 and state s; recall that p1s = 1 for eachs.)

Proof : (A-D ⇒ Radner): Define qk =∑

s p1srsk for every k. Denote by Λ theS × S diagonal matrix whose s diagonal entry is p1s:

Λ =

⎛⎜⎜⎜⎝

p11 0 · · · 00 p12 · · · 0...

.... . .

...0 0 · · · p1S

⎞⎟⎟⎟⎠

115


Then qT = 1·ΛR, where 1 ∈ RS is a column vector with all its entries equal to 1. Forevery i, the (column) vector of wealth transfers across states (at the Arrow-Debreuequilibrium) is

mi =(p1 · (xi

1 − ei1), . . . , pS · (xi

S − eiS)

)T

By the budget balance, we have 1 · mi = 0 for every i and we have∑

i∈I mi = 0 bymarket clearing. By completeness, rank ΛR = S and therefore, we can find vectorszi ∈ RK such that mi = ΛRzi for i = 1, . . . , I − 1. Letting

zI = −(z1 + · · · zI−1),

we also have mI = −(m1+· · ·+mI−1) = ΛRzI . Therefore, for each i, the portfolio zi

allows consumer i to reach the Arrow-Debreu consumption in the different states atthe spot prices (p1, . . . , pS). To verify the budget balance note that q ·zi = 1·ΛRzi =1 · mi = 0 for each i.

(Radner ⇔ A-D): Assume, without loss of generality, that p1s = 1 for eachs. By the previous proposition, we have qT = μ · R for some arbitrage weightsμ = (μ1, . . . , μS). We show that x is an Arrow-Debreu equilibrium with respect to(μ1p1, . . . , μSpS). Let xi ∈ Bi

R. By the completeness assumption, there is zi ∈ RK

such that (p1 · (xi1−ei

1), . . . , pS · (xiS −ei

S))T = Rzi and, therefore, q ·zi = μ ·Rzi ≤ 0.Since xi satisfies the budget constraint of the Radner economy, we must satisfy(

(p1 · (xi1 − ei

1), . . . , pS · (xiS − ei

S))T ≤ Rzi.

Moreover,∑s

μsps · (xis − ei

s) ≤ μ · Rzi = q · zi (∵ qT = μ · R)

≤ 0 (∵ the budget constraint of the Radner economy)

�

In discussing Radner equilibria, what matters is not so much the particular assetstructure but the linear space,

Range R ={v ∈ RS| v = Rz for some portfolio z ∈ R

} ⊂ RS,

the set of wealth vectors that can be spanned by means of the existing assets. It isquite possible for two different asset structures to give rise to the same linear space.The next result tells us that, whenever this is so, the set of Radner equilibriumallocations for the two asset structures is the same.

Proposition 7.5.3 Suppose that the asset price vector q ∈ RK , the spot pricesp = (p1, . . . , pS) ∈ RnS, the consumption plans x = (x1, . . . , xI) ∈ RnSI

+ , and the

116


portfolio plans z = (z1, . . . , zI) ∈ RKI constitutes a Radner equilibrium for an assetstructure with S×K return matrix R. Let R

′be the S×K

′return matrix of a second

asset structure. If Range R′= Range R, then x is still the consumption allocation

of a Radner equilibrium in the economy with the second asset structure.

Proof : By the previous proposition, the asset prices satisfy the no-arbitragecondition qT = μ · R for some μ ∈ RS

+. Denote q′

= [μ · R′]T . We claim that if

Range R = Range R′, then

Bi(p, q′, R

′) = Bi(p, q, R) ∀i.

We show that if xi ∈ Bi(p, q, R) then xi ∈ Bi(p, q′, R

′). To see this, let(

p1 · (xi1 − ei

1), . . . , pS · (xiS − ei

S))T ≤ Rz

and q · z ≤ 0. Since Range R = Range R′, we can find z

′ ∈ RK′

such that Rz =R

′z′ ∈ Range R

′. But then

q′ · z′

= μ · R′z′

(∵ q′= [μ · R′

]T )= = μ · Rz (∵ Rz = R

′z′)

= q · z (∵ qT = μ · R)≤ 0.

To argue that the asset prices q′, the spot prices p = (p1, . . . , pS), and the

consumption allocation x ∈ RnSI are part of a Radner equilibrium in the economywith an asset structure having return matrix R, it suffices to find portfolios z

′=

(z′1, . . . , z

′I) ∈ RKI with the following two properties:

1.∑

i∈I z′i = 0; and

2. for every consumer i,

mi =(p1 · (xi

1 − ei1), . . . , pS · (xi

S − eiS)

)T = R′z′i.

By strong monotonicity of preferences, we have mi = Rzi for every consumeri. Hence, mi ∈ Range R and therefore mi ∈ Range R

′for every i. Choose then

z′1, . . . , z

′I−1 such that mi = R′z′i for every i = 1, . . . , I − 1. Finally, let z

′I =−(z

′1 + · · · + z′I−1). Then

∑i z

′i = 0 and also

mI = −(m1 + · · ·mI−1) = −R′(z

′1 + · · · + z′I−1) = R

′z′I . �

Example 7.5.4 (Pricing an Option) Suppose that, with S = 2, there is an assetwith non-contingent returns, say r1 = (1, 1) and a second asset r2 = (3 + α, 1 − α),with α > 0. The asset prices are q1 = 1 and q2. I now consider an (call) option onthe second asset that has strike price c ∈ (1, 3). Then, there are a, b ∈ R such that

r2(c) = (3 + α − c, 0) = ar2 + br1 = (a(3 + α) + b, a(1 − α) + b).

117


If I solve the above equation with respect to a and b, I obtain

a =3 + α − c

2 + 2αand b = −(1 − α)(3 + α − c)

2 + 2α

Thus, the asset structure R is

R =(

1 3 + α 3 + α − c1 1 − α 0

).

The no-arbitrage condition requires that there exists μ = (μ1, μ2) ∈ R2 with qT = μR.That is,

qT = (1, q2, q2(c)) = (μ1, μ2)(

1 3 + α 3 + α − c1 1 − α 0

)= (μ1 + μ2, μ1(3 + α) + μ2(1 − α), μ1(3 + α − c))

Thus, I obtain

q2(c) =3 + α − c

2 + 2α[q2 − (1 − α)] .

7.6 Multi-Period Exchange Economies

The same methodology developed in the previous section can be used to considermore general multi-period exchange economies. Suppose I have T + 1 dates t =0, 1, . . . , T and, as before, S states, but assume that the states emerge graduallythrough a tree. Here final nodes stand for the possible states realized by time t = T ,that is, for complete histories of the uncertain environment. When the path throughthe tree coincides for two states, s and s

′, up to time t, this means that in all

periods up to and including period t, s and s′

cannot be distinguished. Subsets ofS are called events. A collection of events F is an information structure if it isa partition, that is, if for every state s, there is E ∈ F with s ∈ E and for anytwo E,E

′ ∈ F , E �= E′, we have E ∩ E

′= ∅. The interpretation is that if s

and s′belong to the same event in F then s and s

′cannot be distinguished in the

information structure F . To capture formally a situation with sequential revelationof information, I look at a family of information structures: (F0, . . . ,Ft, . . . ,FT ).The process of information revelation makes the Ft increasingly fine: Ft+1 is atleast as fine as Ft. It is assumed that F0 is trivial, i.e., F0 = {S} and FT is thediscrete partition, i.e., FT = {1, . . . , S} = S. Once one has information sufficientto distinguish between two states, the information is not forgotten. The partitionscould in principle be different across individuals. However, I shall assume that theinformation structure is the same for all consumers. A pair (t, E) where t is a dateand E ∈ Ft is called a date-event. Date-events are associated with the nodes of thetree. Each date-event except the first has a unique predecessor, and each date-eventnot at the end of the tree has one or more successors.

118


1

2

3

4

5

6

t = 1t = 0 t = 2

s

Figure 7.3: An information tree: gradual release of information

There is a single consumption good which cannot be stored. This good is con-sumed at each date and the amount consumed at date t can vary across cells ofFt. Thus, I then say that a vector z is measurable with respect to the familyof information partitions (F0, . . . ,FT ) if, for every (t, s) and (t, s

′), I have that

z(t, s) = z(t, s′) whenever s, s

′belong to the same element of the partition Ft.

That is, whenever s and s′

cannot be distinguished at time t, the consumptiongoods assigned to the two states cannot be different. Finally, I impose on en-dowments ei = {ei(t, s) ∈ R+| t = 0, . . . , T and s ∈ S} and consumption setsXi = {xi(t, s) ∈ R+| t = 0, . . . , T and s ∈ S} with the restriction that all theirelements be measurable with respect to the family of information partitions. 5 Withthis, I have reduced the multi-period structure to our original formulation.

There are N assets or securities in this economy. There are claims to (statecontingent) consumption at date T . They are indexed by n = 1, . . . ,N . Securityn entitles the bearer (on date T ) to rks units of the consumption good at date Tif the state is s. Denote rs ≡ (r1s, . . . , rNs). The net supply of these securities iszero. It is assumed that for every state, there is one of these securities that pays offa nonnegative amount in every state and a strictly positive amount in that state.

At each date t ≤ T and in every state s ∈ S, markets open in which these N

5For any t, we must have the following property: ei(t, s) = ei(t, s′) and xi(t, s) = xi(t, s

′)

whenever s and s′

belong to the same cell of Ft.

119


securities can be traded for one another and for the consumption good. The price(in units of the consumption good) of security n at date t in state s will be denotedby qn(t, s). These markets have no transaction costs and no restrictions on shortsales. A price system is a vector stochastic process q = {qn(t, s)| n = 1, . . . ,N, t =0, . . . , T, s ∈ S and for each t, qn(t, s) = qn(t, s

′) whenever s and s

′belong to the

same cell of Ft}. The consumer’s problem in this economy is to manage a portfolioof these N securities in order to obtain for himself the best possible vector of statecontingent consumption. A portfolio strategy is a N dimensional vector stochasticprocess z = {zk(t, s)| for any t, zn(t, s) = zn(t, s

′) whenever s and s

′belong to the

same cell of Ft}. The interpretation is that zn(t, s) is the number of shares of securityn held from date t until t + 1 in state s. (For t = T, zn(T, s) is the number of sharesfrom which the dividend is received.) For any state s ∈ S, define the following:

xi(z, q, s) =(xi

0(z, q, s), . . . , xiT (z, q, s)

)xi

0(z, q, s) = ei(0, s) − zi(0, s) · q(0, s)xi

t(z, q, s) = ei(t, s) +(zi(t − 1, s) − zi(t, s)

) · q(t, s) ∀ t = 1, . . . , T

xiT (z, q, s) = ei(T, s) +

(zi(T − 1, s) − zi(T, s)

) · q(T, s) + zi(T, s) · rs

Denote xit(z

i, q) ≡ {xit(s, z

i, q)| s ∈ S} and xi(zi, q) ≡ {xit(z

i, q)| t = 0, . . . , T}. Aconsumption bundle xi is feasible for consumer i at prices q if xi ∈ Xi and thereexists a portfolio strategy zi such that xi ≤ xi(zi, q). The set of feasible consumptionbundles for consumer i at prices q is denoted Xi(q).

Definition 7.6.1 A collection formed by a price system q and for every consumeri ∈ I, consumption plan xi and portfolio strategy zi constitutes a Radner equilib-rium of a multi-period exchange economy if:

1. xi ≤ xi(zi, q) and xi ∈ Xi for all i ∈ I;

2. xi is �i maximal among all x ∈ Xi(q) for each i ∈ I; and

3.∑

i∈I zi = 0.

The first condition says that xi is a feasible consumption plan for i and thatxi is feasible if consumer i adopts the portfolio strategy zi. The second conditionsays that taking prices q as given, consumer i can do no better than xi. The thirdcondition says that securities markets clear exactly. It is relatively easy to see thatmarkets for the consumption good clear as well at a Radner equilibrium.

The following alternate characterization of a Radner equilibrium will be useful.Fix a price system q. Define

M i = {x ∈ Xi| x = xi(z, q) for some portfolio strategy z}Then, define

M i = {x ∈ Xi| x = m + (τ, 0, . . . , 0) for some m ∈ M i and τ ∈ R}; (7.1)π(m + (τ, 0, . . . , 0)) = π(ei) + τ for m ∈ M i and τ ∈ R. (7.2)

120


Clearly M i is a subspace of Xi.

Proposition 7.6.1 If (q, (xi, zi)i∈I) is a Radner equilibrium of a multi-period ex-change economy, then there is a strictly positive linear functional π : M → R suchthat:

1. M i ∩ {x ∈ Xi| x ≥ ei, x �= ei} = ∅ for all i ∈ I (No Arbitrage); and

2. For each i ∈ N, xi ∈ M i ∩ Xi, π(xi) ≤ π(ei), and xi is �i maximal in{x ∈ M i ∩ Xi| π(x) ≤ π(ei)}.

Sketch of Proof : Suppose xi and q are given as part of a Radner equilibriumof a multi-period exchange economy. By strong monotonicity of �i, there exist zi

such that xi = xi(zi, q). Let zi = zi for i �= 1 and z1 = −∑i�=1 zi.

7.6.1 Implementing the A-D equilibria by Trading Long-lived Se-curities

Suppose that for a Radner equilibrium price system q, the corresponding space M isX. Then, the equilibrium allocation (xi)i∈I is an equilibrium allocation for an Arrow-Debreu economy with a complete set of contingent claims markets and therefore isPareto efficient. Thus, it is natural to seek conditions that yield M = X. Define fort < T and E ∈ Ft,

K(t, E) = cardinality{E′ ∈ Ft+1| E′ ⊂ E}

K = max{K(t, E)| t < T, E ∈ Ft}

In words, K(t, E) is the number of “subcells” of E in Ft+1. This is a measure ofthe amount of information that might be received by date t+1 if at date t, the eventE is known to prevail: If K(t, E) = 1, then no new information will be received. IfK(t, E) = 2, then new information of an either type will be received, and so on.

Proposition 7.6.2 (Kreps (1982)) Let q be a Radner equilibrium price system,and let M be defined from q. A necessary and sufficient condition for M = X is thatfor each t < T and E ∈ Ft,

Rank {span{q(t + 1, s)| s ∈ E} = K(t, E) (7.3)

A paraphrase of this condition is that the conditional support of qt+1 give thats ∈ E consists of K(t, E) linearly independent vectors. There are at most K(t, E)vector in this conditional support (because qt+1 is Ft+1 measurable). Thus, K(t, E)is an upper bound on Rank { span {q(t + 1, s)| s ∈ E}. The condition is thatthis upper bound is hit in every instance. Rather than work through the details,a full example will be given which should make both the proposition and its prooftransparent.

121


0 1 2 3Date:

s1

s2

s3

s4

s5

s6

State

0.811.26

0.7292.4

0.813.75

0.91.4

0.94.2

0.9094.2

111213161415

Figure 7.4: Dividend Structure of a Radner Equilibrium (a)

Suppose that there are six states S = {s1, s2, s3, s4, s5, s6} and four dates t =0, 1, 2, 3. The exogenous information structure is given by the partitions:

F0 = {S};F1 = {{s1, s2}, {s3, s4, s5, s6}};F2 = {{s1, s2}, {s3, s4}, {s5, s6}};F3 = S = {{s1}, {s2}, {s3}, {s4}, {s5}, {s6}}

Thus, K(1, {s1, s2}) = 1 while K(2, {s1, s2}) = 2. Suppose that there are twosecurities whose dividends at date 3 are as the following table:

state s1 s2 s3 s4 s5 s6

payoff of security #1: r1(·) 1 1 1 1 1 1payoff of security #2: r2(·) 1 2 3 6 4 5

Consider two possible equilibrium price systems arising from these data, as de-picted in Figure 4 and 5.

The column vector in these event trees give the prices of the two securities as afunction of the date and state. For example, the column vector(

0.94.2

)which is interpreted to mean q1(2, s4) = 0.9 and q2(2, s4) = 4.2. Note that the treestructure corresponds to the information structure.

122


0 1 2 3Date:

s1

s2

s3

s4

s5

s6

State

111213161415

0.7292.4

0.811.26

0.813.78

0.91.4

0.94.2

0.9094.242

Figure 7.5: Dividend Structure of a Radner Equilibrium (b)

Does M = X in either or both cases? The answer is yes if and only if for everyt > 0 and E ∈ Ft, the vector x = (x(0), . . . , x(T )) that is given by x(t) = 0 for t �= tand x(t) = 1E is in M . That is, there must exist a portfolio strategy that producesone unit of consumption in event E at date t and nothing at any other date-eventpair. Let me begin by asking if this is true for t = 1 and for every E ∈ F1. In eachcase, the answer is yes – the two possible values of q(1) are linearly independent,thus there exist (z1, z2) and (z

′1, z

′2) such that

q1(1)z1 + q2(1)z2 = 1{s1,s2}q1(1)z

′1 + q2(1)z

′2 = 1{s3,s4,s5,s6}

This clearly suffices. Now proceed to ask the question for t = 2. For E = {s1, s2},there is no problem in either case. But matters are not so simple for E = {s3, s4}.In case (a), it can be done: First solve

0.9z1 + 4.2z2 = 10.909z1 + 4.1z2 = 0.

This can done because the two column vectors are linearly independent. Letz∗1 , z∗2) be the solution. Next solve

0.81z1 + 1.26z2 = 00.81z1 + 3.75z2 = 0.81z∗1 + 3.75z∗2 .

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This can be done by the first step: The solution, denote it (z∗∗1 , z∗∗2 ), just a scalarmultiple of (z∗1 , z∗2). Then, the strategy of starting with (z∗∗1 , z∗∗2 ) at date 0, changingto (0, 0) at date 1 if {s1, s2} occurs and to (z∗1 , z∗2) if {s3, s4, s5, s6} occurs, and theconsuming everything at date 2 yields one unit of consumption at date 2 if and onlyif {s3, s4} occurs.

But consider case (b). One cannot solve

0.9z1 + 4.2z2 = 10.909z1 + 4.242z2 = 0,

because the tow column vectors are linear dependent. Thus, if one consumes oneunit at date 2 and nothing at date 3 when {s3, s4} occurs, one must either consumesomething at date 2 or at date 3 when {s5, s6} occurs. By inductively applying thissort of logic, one can see that M = X in case (a), but that M �= X in case (b).

A Radner equilibrium (a) makes the basic idea clear. In this economy there aresix states of the world and only two securities, yet markets are complete. This isbecause the process of learning which of the six states is the true state takes placenot all at once but in three steps. Agents can revise their portfolios after each stepin the learning process. At each step, at most two “signals” are possible. And theequilibrium prices of the two securities are “well behaved” – they are “linearly inde-pendent” in a fashion that enables agents to take full advantage of new informationas it is received.

7.6.2 Genericity of the Case M = X with K or More Securities

Fix the economic setting; that is, fix the state space (S), information structure (F )and the agents (I). Suppose N securities are selected at “random.” By a selectionof N securities is meant a selection of a point r from the set R which is defined asfollows:

R ≡ {rn(s) ∈ R| n = 1, . . . ,N, s ∈ S}

A subset of R will be called sparse if its closure has Lebesgue measure zero. If theselection of r is done “randomly enough,” there is zero probability that the outcomewill land in a given sparse set. Following the terminology of Radner (1979), a resultthat holds off of a sparse set is called generic.

Proposition 7.6.3 (Kreps (1982)) Fix the economic setting (S,F ,I). Supposethat, in this setting, there is an equilibrium with equilibrium allocation (xi)i∈I in anArrow-Debreu economy. Then, if N ≥ K, there is a sparse set R ⊂ R such that forall r ∈ R\R, the economy with K long-lived securities paying r at date T admits aRadner equilibrium with M = X and with equilibrium allocation (xi)i∈I .

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Proof : In the Arrow-Debreu economy, there is a linear functional φ : X → Rthat is strictly positive and that satisfies

xi ∈ Xi, φ(xi) ≤ φ(ei), and xi is �i maximal in {x ∈ Xi| φ(x) ≤ φ(ei)}. (7.4)

That is, φ gives the equilibrium prices. Normalize φ so that φ((1,0, . . . , 0)) = 1. Forevery t = 0, . . . , T and E ∈ Ft, define χt,E by

χt,E(t) = 0 for t �= t and χt,E(t) = 1E .

That is, χt,E is the claim that pays one unit of consumption at date t in the eventE. For any r ∈ R, define q from r and φ as follows: For t ≤ T and s ∈ S, let E ∈ Ft

be such that s ∈ E. Then, let

qk(t, s) =

∑s∈E rk(s)φ

(χT,{s}

)φ(χt,E)

. (7.5)

Two things, once demonstrated, give the result. First, except for r from a sparsesubset of R, q so defined satisfies (7.3), and thus M = X. Second, for all such r, thelinear functional π defined in (7.2) is φ.

For the first result, it is necessary to show that except for r from a sparse set, theset {q(t + 1, s)| s ∈ E} contains K(t, E) linearly independent vectors for every t andE ∈ Ft. Since there are finitely many such pairs (t, E) and since the union of a finitenumber of sparse sets is sparse, it suffices to show that for every t and E, the setof r ∈ R for which the corresponding {q(t + 1, s)| s ∈ E} does not contain K(t, E)linearly independent vectors is sparse. Using (7.5), the set {qn(t + 1, s)| s ∈ E} canbe written {∑

s∈E′ rn(s)φ

(χT,{s}

)φ(χt,E

′ )

∣∣∣∣∣E′ ∈ Ft+1, E′ ⊂ E

}

which, letting α(t, s) denote the strictly positive scalar φ(χT,{s}

)/φ (χt,E), is⎧⎨

⎩∑s∈E′

r(s)α(t + 1, s)

∣∣∣∣∣ E′ ∈ Ft+1, E

′ ⊂ E

⎫⎬⎭

The set of r for which this set of K(t, E) vector is linearly dependent is clearlyclosed. That it has Lebesgue measure zero is also apparent as follows: Let T : R →(RN

)K(t,E) be the map

T (r) =

⎡⎣∑

s∈E′r(s)α(t + 1)

⎤⎦

E′∈Ft+1, E

′⊂E

,

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and let λ denote Lebesgue measure on R. Then, the measure λ ◦T−1 on(RN

)K(t,E)

is absolutely continuous with respect to Leabesgue measure because α(t + 1, s) arestrictly positive. And the Lebesgue measure in

(RN

)K(t,E) of vector [(r)Nn=1]K(t,E)k=1

such that (r)n=1,... ,N are linearly dependent is zero, if N ≥ K.

For the second result, it suffices to show that for all portfolio strategies z,φ(x(z, q)) = φ(ei). There is nothing to do but grind this out:

7.7 Incomplete Markets

Proposition 7.7.1 (Sunspot Free Equilibria) Suppose that the following fourassumptions are satisfied.

1. U i(x1, . . . , xS) =∑S

s=1 πisus(xs) for any i

2. πis = πs for any i and any s ∈ S

3. uis(·) = ui(·) and ei

s = ei for any i and any s ∈ S

4. ui(·) is strictly concave.

Then, any Pareto efficient allocation must be uniform across states.

Proof of Proposition 7.5.3: Let x = (xi1, . . . , xi

S)i=1,... ,n be a Pareto efficientallocations. For each i and each s, define xi =

∑s πsx

is. The new allocation x is

state independent, and it is also feasible because

n∑i=1

xi =n∑

i=1

∑s

πsxis =

∑s

πs

(n∑

i=1

xis

)≤

∑s

πs

(n∑

i=1

ei

)=

n∑i=1

ei

Since ui(·) is concave,

∑s

πsui(xi) = ui(xi) = ui

(∑s

πsxis

)≥

∑s

πsui(xi

s) ∀ i

Because of the Pareto efficiency of x, the above inequalities must in fact be equality.But, if so, then the strict concavity of ui(·) yields xi

s = xi for any s. This impliesthat any Pareto efficient allocation is state uniform. �

From the state independence of Pareto efficient allocations and the first welfaretheorem, I reach the important conclusion that if a system of complete marketsover the states S can be organized, then the equilibria are sunspot-free, that is,consumption is uniform across states. It turns out, however, that if there is not acomplete set of insurance opportunities, then the above conclusion does not holdtrue. Sunspot-free Pareto efficient equilibria always exist (just make the market

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“not pay attention” to the sunspot). But it is now possible for the consumptionallocation of some Radner equilibria to depend on the state, and consequently to failthe Pareto efficiency test. In such an equilibrium, consumers expect different pricesin different states, and their expectations end up being self-fulfilling. The simplest,and most trivial example is when there are not assets whatsoever. Then a systemof spot prices (p1, . . . , pS) ∈ RnS is a Radner equilibrium if and only if every ps isa Walrasian equilibrium price vector for the spot economy defined by {ui(·), ei}i∈I .If, as is perfectly possible, this economy admits several distinct Walrasain equilibria,then by selecting different equilibrium price vectors for different states, I obtain ssunspot equilibrium, and hence a Pareto inefficient Radner equilbirium.

I have confirmed that Radner equilibrium allocations need not be Pareto efficientand so, in principle, there may exist reallocations of consumption that make allconsumers at least as well off, and at least one consumer strictly better off. It isimportant to recognize, however, that this need not imply that a welfare authoritywho is “as constrained in interstate transfers as the market is” can achieve a Paretoefficiency. An allocation that cannot be Pareto improved by such an authority iscalled a constrained Pareto efficiency. A more significant and reasonable welfarequestion to ask is, therefore, whether Radner equilibrium allocations are constrainedPareto efficient.

For the sake of simplicity, it is assumed here that there is a single commodityper state. The important implication of this assumption is that then the amountof consumption good that any consumer i gets in the different states is entirelydetermined by the portfolio zi. Indeed, xi

s =∑

n zinrsn + ei

s. Hence, I can let

U i∗(z

i) = U i∗(z

i1, . . . , zi

N ) = U i

(∑n

zinr1n + ei

1, . . . ,∑n

zinrSn + ei

S

)

denote the utility induced by the portfolio zi.

Definition 7.7.1 The asset allocation (z1, . . . , zI) ∈ RNI is constrained Paretoefficient if it is feasible (i.e.,

∑i z

i ≤ 0) and if there is no other feasible assetallocation (z1, . . . , zI) ∈ RNI such that

U i∗(z

i) ≥ U i∗(z

i) ∀ i ∈ Iwith at least one inequality strict.

In this context, the utility maximization problem of consumer i becomes

maxzi∈ N U i∗(z

i1, . . . , zi

N )s.t. q · zi ≤ 0.

Suppose that zi∗ ∈ RN for i ∈ I, is a family of solutions to these individualproblems, for the asset price vector q ∈ RN . Then, q ∈ RN is a Radner equilibrium

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price if and only if∑

i zi∗ ≤ 0. Note that this has become now a perfectly conventional

equilibrium problem with N commodities. To it I can apply the first welfare theoremand reach the following conclusion.

Proposition 7.7.2 Suppose that there are only two periods and only one consump-tion good in the second period. Then, any Radner equilibrium is constrained Paretoefficient in the sense that there is no possible redistribution of securities in the firstperiod that leaves every consumer as well off and at least one consumer strictly betteroff.

The situation here is very particular in that once the initial asset portfolio ofa consumer is determined, his overall consumption is fully determined: with onlyone consumption good, there are no possibilities for trade once the state occurs. Inparticular, second-period relative prices do not matter, simply because there are nosuch prices. Things change if there is more than one consumption good in the secondperiod, or if there are more than two periods (See the previous section). Considerthe two-period case with two consumption goods: Then, I cannot summarize theindividual decision problem by means of an indirect utility of the asset portfolio.The relative prices expected in the second period also matter. 6 This substantiallycomplicates the formulation of a notion of constrained Pareto efficiency. In it I havean economy with several Radner equilibria where two of them are Pareto ordered.That is, I have a Rander equilibrium that is Pareto dominated by another Radnerequilibrium. To the extent that it seems natural to allow a welfare authority, at thevery least, to select equilibria, it follows that the first equilibrium is not constrainedPareto efficient.

6Or the relative prices of goods between the second and third period, if I am considering morethan two dates, instead.

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Lecture Notes Nov 26 2008 Econ610 Fall2008