General Equilibrium Foundation of Partial Equilibrium Analysis 978-3-319-56696-2, 3319566962, 978-3-319-56695-5

Table of contents :
Front Matter ....Pages i-xiv
Introduction (Takashi Hayashi)....Pages 1-6
General Equilibrium Theory (Takashi Hayashi)....Pages 7-35
Income Evaluation of Welfare Change: Equivalent Variation, Compensating Variation, and Consumer Surplus (Takashi Hayashi)....Pages 37-69
The Assumption of No Income Effect and Quasi-linear Preferences (Takashi Hayashi)....Pages 71-92
Is the Approximation Error Large or Small? (Takashi Hayashi)....Pages 93-97
Small Income Effects (Takashi Hayashi)....Pages 99-136
Partial Equilibrium Welfare Analysis Under Uncertainty (Takashi Hayashi)....Pages 137-162
Mechanism Design in Partial Equilibrium (Takashi Hayashi)....Pages 163-180
Back Matter ....Pages 181-185

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GENERAL EQUILIBRIUM FOUNDATION OF PARTIAL EQUILIBRIUM ANALYSIS

Takashi Hayashi

General Equilibrium Foundation of Partial Equilibrium Analysis

Takashi Hayashi

General Equilibrium Foundation of Partial Equilibrium Analysis

Takashi Hayashi Adam Smith Business School University of Glasgow Glasgow UK

ISBN 978-3-319-56695-5 ISBN 978-3-319-56696-2 DOI 10.1007/978-3-319-56696-2

(eBook)

Library of Congress Control Number: 2017940593 © The Editor(s) (if applicable) and The Author(s) 2017 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Image credit: Design Pics/Chris Knorr Printed on acid-free paper This Palgrave Macmillan imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Introductory courses of microeconomics normally start with partial equilibrium analysis, consumer surplus analysis in particular. In order not to torture students, consumer surplus for a given individual is quite often introduced simply as a cardinal utility that is measured in income, which is taken to be interpersonally comparable. Then, we teach efficient allocation of resources as maximization of social surplus, which is the sum of consumer surplus across individuals minus social cost of production. Then, we move on to intermediate or advanced/intermediate courses, in which we introduce the concepts of indifference curves and ordinal utility defined over multiple goods, Edgeworth box, Pareto efficiency, contract curve (or Pareto set), and general equilibrium with simultaneous determination of price and income, as if nothing has been taught before that. In order to teach imperfect competition or incomplete information then, which is difficult to teach in the general equilibrium framework not just because of pedagogical hardness but also because of analytical limitations, we come back to the partial equilibrium framework, calculate

v

vi

Preface

deadweight loss and teach how to extract surplus or how to restore the loss, again as if nothing has been taught before that. Such gap between partial equilibrium analysis and general equilibrium analysis is paid little attention, not only at pedagogical level but also at the academic level, especially, in the era in which academic economists are demanded to provide more “useful applications.” It has not been made clear, however, even to the audiences who should know it, in what sense partial equilibrium analysis is indeed a “part” of general equilibrium analysis. The problem is pervasive. Those who leave the economics subject after introductory courses tend to get an understanding that “money can buy” is the everything of economics, and that maximization of social surplus alone determines an economic outcome uniquely and “scientifically,” whether they decide to be pro or con about it when leaving. According to the “money can buy” understanding, if an item or an action is “really necessary” for an individual she should be able to pay for it. This tautological assertion is coming from a confusion between being able to pay and being willing to pay, while the equivalence between the two requires a certain condition. And the criticism to it, namely “money can’t buy,” would be understood as a reaction to this confusion at least, while I wouldn’t say that it is standing on the same confusion. As economists who have learned general equilibrium theory, we know that such confusion is due to a confusion between income effect and substitution effect, and even when “money can buy” is empirically wrong what should be blamed is a wrong application of the particular condition, or applying the condition without knowing what it is, or not even being aware of relying on the condition, and not economics itself at methodological level. However, in introductory courses, we are in fact teaching that “money can buy.” We also know that maximization of social surplus alone does not determine an economic outcome at a general equilibrium level, as maximization of surplus alone leaves it totally undetermined how we should distribute the maximized social surplus, and as Pareto efficiency alone is totally silent about distributional properties of allocations. However, in introductory courses, very little emphasis is put on this apparent determinacy.

Preface

vii

This book aims at providing a bridge over the gap, or at least illustrating the natures of the gap. It summarizes researches, old and new, which I view are deeply relevant to the gap, as well as the materials which are standard at graduate/upper-division undergraduate level but I guess are not fully shared by the intended audiences. I apologize, if the way how I approach to those is selective and I insist too much on my own researches, though. I also hope that this book serves as a “teacher’s manual,” which helps teachers to have a better understanding of the gap, and provides intuitive stories on the assumptions underlying the partial equilibrium analysis. Glasgow, UK

Takashi Hayashi

Acknowledgements

I thank the reviewers for their helpful comments and suggestions, which improved the manuscript substantially, and my colleague Michele Lombardi for his helpful discussions on some of the materials in the book. I thank Mr. James Safford for his editorial helps, and Ms. Laura Pacey for providing me with the opportunity to write this book. All remaining errors are my own.

ix

Contents

1 Introduction

1

2 General Equilibrium Theory

7

3 Income Evaluation of Welfare Change: Equivalent Variation, Compensating Variation, and Consumer Surplus

37

4 The Assumption of No Income Effect and Quasi-linear Preferences

71

5 Is the Approximation Error Large or Small?

93

6 Small Income Effects

99

7 Partial Equilibrium Welfare Analysis Under Uncertainty

137

8 Mechanism Design in Partial Equilibrium

163 xi

xii

Contents

Bibliography

181

Index

183

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1

Fig. Fig. Fig. Fig. Fig.

3.2 3.3 3.4 4.1 4.2

Fig. Fig. Fig. Fig. Fig.

4.3 4.4 4.5 4.6 4.7

Set of Pareto-efficient allocations Kaldor-improvement Kaldor-improvement Mutual Kaldor-improvements Mutual Kaldor-improvements Hicks improvement Gorman paradox Compensated variation, equivalent variation, and change in consumer surplus Inverse demand function Consumer surplus Change in consumer surplus Consumption space R þ  R Change in income when the background income is sufficiently large Quasi-linear preference Willingness to pay Consumer surplus Substitution and income effect in quasi-linear preference Compensated and equivalent variation in quasi-linear preference

18 19 20 21 22 23 24 40 41 42 42 72 73 74 75 76 80 82 xiii

xiv

Fig. Fig. Fig. Fig. Fig. Fig. Fig.

List of Figures

4.8 5.1 6.1 6.2 6.3 6.4 8.1

Set of efficient allocations under quasi-linear preferences Hicksian deadweight loss and Marshallian deadweight loss Induced preference Asymptotic preference Induced preference Limit preference Two-sector second-price auctions with income effects

88 96 113 114 121 122 166

1 Introduction

1.1

Introduction

Partial equilibrium analysis isolates the objects to allocate, or the decision problem at hand, from the rest of the economy, assuming that other things remain equal. Such simplification allows us to provide not only sharp prediction in positive analysis but also a concrete solution in normative prescription, which we cannot hope for in the general equilibrium approach. What conditions are required in order to justify such practice, however? On what ground can we say that other things remain equal, either precisely or approximately? To think of this question, first, let us go through the partial equilibrium analysis as taught in introductory courses.

1.2

The Econ 101 Presentation

The summary below may look too much caricaturized, but I hope it is to the point. Consider that there is a single output good. Then, for a generic consumer: © The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_1

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• Assume that utility of consumption is measurable in terms of income. Denote the utility of q units of consumption by U (q). • Given the price of a given good p, the consumer’s surplus, which is the net utility obtained by subtracting expenditure pq from the gross utility U (q), is given by U (q) − pq The generic consumer maximizes this. • At the utility-maximizing point, we have equality of marginal utility and price, U  (q) = p. That is, the net utility is maximized when incremental utility from incremental consumption of the given good is equal to the incremental cost of it. • Take q on the horizontal axis and p on the vertical axis, and plot the above maximization point, then we obtain a demand curve p = U  (q). At a social level: • Given p, each individual i is assumed to maximize Ui (qi ) − pqi • At the utility-maximizing point, we have equality of marginal utility and price Ui (qi ) = p. • In a perfectly competitive market, given p, each firm k maximizes pyk − Ck (yk ) where Ck denotes firm k’s cost function exhibiting nondecreasing marginal cost.

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Introduction

3

• At the profit-maximizing point, we have equality of marginal cost and price p = Ck (yk ) • Because marginal utilities and marginal costs are equalized in competitive equilibrium, and marginal cost is assumed to be nondecreasing, it solves the social surplus maximization problem. max q,y

subject to



Ui (qi ) −



i

Ck (yk )

k

 i

qi =



yk .

k

Now, what is the problem? At least we should notice the following points. 1. In the above argument, utility is treated as if it is an income evaluation of consumption. When an individual can receive income transfer as well as she pays for consumption, when such net income change is denoted by t, she is supposed to have utility function over consumption/net income transfer pair (q, t) in the form V (q, t) = U (q) + t which exhibits that marginal utility is income is constant, one. Also, it is presumed that marginal utility of consumption is diminishing. However, utility function is no more than a representation of a given preference ranking and neither utility value or marginal utility value has no quantitative meaning by itself. It might be OK to use it just to describe individual choice, but we indeed take the sum of it across individuals, presuming that utility is comparable across different individuals, and in particular that marginal utility of income is equal across individuals. How can we measure utilities in terms of income in such manner?

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2. Here the object called “income,” which plays the role of the unit of measurement, is give as a primitive. However, from the general equilibrium viewpoint it must be a composite of all the other goods, and the value of such composition and its content must be endogenously determined in market. How can we take it as if it is such a homogeneous material? 3. In the argument the obtained demand is determined only by the relationship between “marginal utility” and price. However, in general equilibrium analysis, which comes after the introductory stuff, we know that demand in general depends on income. Where has the income gone? 4. On the producer side, the concept of cost function looks relatively innocuous. However, it presumes that the firms are price-taking in input markets at least, although they may or may not be pricetaking in the output market. Also, it is left unspecified where the maximized profits go, while it should be explicitly specified in a general equilibrium model. What situation in the general equilibrium setting would justify such assumption?

1.3

It Can Be Restored, but at Some (Maybe Huge?) Cost

The above caricaturization might not be fair, since one may still describe the same content in the framework of ordinal utility, just by using the concept of indifference curves and marginal rate of substitution. In fact, this is what we do in later chapters. However, we should notice that it cannot be done in a costless manner, as such translation relies on a stringent condition, namely the assumption of no income effect on the good under consideration. Under this assumption, we can reduce the dimension of the consumption space by one, and we can describe individuals’ preferences as if they have interpersonally comparable utility functions which are measurable in income, and taking the sum across them has certain economic mean-

1

Introduction

5

ing. Under such reduction of dimension, the law of diminishing marginal utility, which does not make sense under the ordinal utility framework, is restored as one phenomenal form of the law of diminishing marginal rate of substitution.

1.4

Apparent Determinateness

In the above framework, the surplus maximization problem max q,y

subject to



Ui (qi ) −



i

Ck (yk )

k

 i

qi =



yk .

k

has a unique solution as a pair of consumption allocation q = (qi ) and production activity y = (yk ). It sounds as if efficiency alone uniquely determines economic activity and it is equivalent to being in competitive equilibrium. We know, however, that in the general equilibrium framework Pareto efficiency alone implies only that marginal rate of substitution and marginal rate of transformation are equalized across individuals and across firms, and it admits, in general, a continuum of efficient allocations, and the competitive equilibrium solution only selects a point (or a subset in general) from such set. This apparent gap is due to the fact that in the partial equilibrium model it is left totally undetermined who should get the maximized surplus and how much, and the apparent determinateness of allocation is simply due to omitting this dimension. In fact, the above maximization is totally silent about who should receive how much of the maximized social surplus. But how should we present this omission procedure explicitly in a general equilibrium setting?

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1.5

Partial Equilibrium as an Institutional Artifact

Another aspect, which I think is important but covered only in recent studies, is that partial equilibrium is an institutional artifact, in the sense that institutional constraints induce agents to behave as if their course of actions fall in the partial equilibrium analysis. Even when the assumptions made above are in general false, we should note that such a way of seeing things is built into how the bureaucracy of policy-making and administration works. An authority in charge of a particular issue typically isolates the issue at hand from the rest of the society, and collects information such as preferences just about the alternatives at hand, presuming that other things remain equal. This presumption is generally wrong. However, even though it is wrong, the authority can force citizens to behave as if it is in fact true, by forcing them to report “preference” rankings just over alternatives it handles, and refusing to listen to “it depends.” What is the consequence of such practice? In the next chapter, we will briefly review the basic general equilibrium theory, and the come back to the issues raised above.

2 General Equilibrium Theory

Here we briefly review the general equilibrium theory, which is pretty traditional: preference and the concept of ordinal utility, demand and comparative statics, the definition of Arrow–Debreu equilibrium, Pareto efficiency and welfare theorems, welfare comparison and compensation principle, and incomplete asset markets. As they are standard, they are presented without proofs. For a comprehensive treatment of general equilibrium theory, see a standard textbook, such as Mas-Colell et al. [1] as well as reputable books, such as Debreu [2], Mas-Colell [3], Magill and Quinzii [4].

2.1

Preference and Utility Function

Consider that, there are n goods in the economy. The consumption set for each individual is taken to be the nonnegative orthant Rn+ . Let  denote a generic individual’s preference ordering over Rn+ , which satisfy Completeness: for all x, y ∈ Rn+ , it holds either x  y or y  x. Transitivity: for all x, y, z ∈ Rn+ , x  y and y  z imply x  z. © The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_2

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Continuity: for all sequences {x ν } and {y ν } in Rn+ , such that x ν  y ν for all ν and limν→∞ x ν = x and limν→∞ y ν = y it holds x  y. Let  denote the strict preference and ∼ denote indifference, which is defined by x  y ⇐⇒ x  y and not y  x and

x ∼ y ⇐⇒ x  y and y  x

Through the book, we also assume that preference  satisfies Strong Monotonicity:

for all x, y ∈ Rn+ it holds x ≥ y, x = y =⇒ x  y

Strict Convexity:

for all x, y ∈ Rn+ with x = y and λ ∈ (0, 1) it holds x ∼ y =⇒ λx + (1 − λ)y  x

although we may consider a weaker version of convexity at some point: Convexity:

for all x, y ∈ Rn+ with x = y and λ ∈ (0, 1) it holds x ∼ y =⇒ λx + (1 − λ)y  x

A numerical function u : Rn+ → R is said to represent  if it holds x  y ⇐⇒ u(x) ≥ u(y) for all x, y ∈ Rn+ . It is called a utility function. The concept of utility function here is ordinal, in the sense that such a function is no more than a representation of preference ranking, and the assigned numerical values as utilities have no quantitative meaning. To be precise, the following statement holds.

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General Equilibrium Theory

9

Theorem 2.1 Fix a preference ranking . (i) Suppose that a utility function u represents . Take any function f : u(Rn+ ) → R that is strictly increasing. Then a function f ◦ u defined by ( f ◦ u)(x) = f (u(x)) for each x ∈ Rl+ is also a utility function which represents . (ii) Suppose that u and v are utility functions which represent . Then, there is a strictly increasing function f : u(Rn+ ) → R such that v = f ◦u

2.2

Demand and Compensated Demand

2.2.1 Demand Function and Indirect Utility Function Consider a generic consumer, who is supposed to be price-taking throughout. Given, a price vector p ∈ Rn++ and income w > 0, she solves the utility maximization problem max u(x)

x∈Rn+

subject to

p · x ≤ w.

The existence of optimization point is guaranteed by Continuity and compactness of budget set B( p, w) = {x ∈ x ∈ Rn+ : p · x ≤ w}. Under Strong Monotonicity the budget constraint is met with equality, and under Strict Convexity the optimal consumption is unique; hence, it is denoted by x( p, w) and satisfies p · x( p) = w for all ( p, w). The function x : Rn++ × R++ → Rn+ defined as above is called demand function.

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Denote the maximal utility under ( p, w) by v( p, w). The function v : Rn++ × R++ → R defined so is called indirect utility function. Proposition 2.1 Let x( p, w) and v( p, w) denote the demand function and the indirect utility function defined for utility representation u(·), respectively. Let f be any monotone transformation and denote the compensated demand function and the expenditure function defined for representation  u (x) = f (u(x)). Let  x ( p, w) and  v ( p, w) denote the demand function and the indirect utility function defined for utility representation  u (·), respectively. Then, it holds  x ( p, w) = x( p, x)  v ( p, w) = f (v( p, w)).

2.2.2 Compensated Demand Function, Expenditure Function, and Income Compensation Function Given, a price vector p ∈ Rn++ and utility level u, consider the expenditure minimization problem minn p · x x∈R+

subject to

u(x) ≥ u

Denote the solution as a function of ( p, u) by h( p, u), and call it compensated demand function. From Strong Monotonicity and Continuity, the solution for expenditure minimization problem always exists, and from Strict convexity the solution is always unique. Also, denote the minimized expenditure by e( p, u) = p · h( p, u), and call it expenditure function.

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General Equilibrium Theory

11

The expenditure-minimizing point is utility-maximizing given the price when the minimized expenditure is given as the income. Thus, it holds h( p, u) = x( p, e( p, u)) Also, the utility-maximizing point is minimizing the expenditure given the price in order to satisfy the same level of utility as it yields. Thus, it holds x( p, w) = h( p, v( p, w)) We might be uncomfortable with taking “utility level” as an input, as it appears to contradict with the concept of ordinal utility. But it is without loss of generality to formulate compensated demand function and expenditure function with a particular representation, as different formulations obtained from different utility representations are suitably translatable to each other. Proposition 2.2 Let h( p, u) and e( p, u) denote the compensated demand function and the expenditure function defined for utility representation u(·), respectively. Let f be any monotone transformation, and denote the compensated demand function and the expenditure function defined for representation  u (x) = f (u(x)). Let  h( p, u) and  e( p, u) denote the compensated demand function and the expenditure function defined for utility representation  u (·), respectively. Then, it holds  h( p, v) = h( p, f −1 (v))  e( p, v) = e( p, f −1 (v)). The proof follows directly from the definition. Actually, we can get rid of utility representation in order to consider the concept of compensated demand, by introducing income compensation function, which is defined by μ( p | p, w) = e( p , v( p, w))

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Then by definition it holds h( p , v( p, w)) = x( p , μ( p | p, w)) Thus, we can carry out the analysis of compensated demand just by demand function x and income compensation function μ, which turns out to be rather helpful in some cases.

2.3

Comparative Statics

To facilitate comparative statics, we will assume differentiable preferences. Differentiable Preference: On Rn++ ,  allows representation u : Rn++ → R which is twice-continuously differentiable, such that 1. 2.

Du(x)  0 for all x ∈ Rn++ , it has negative definite bordered Hessian at all x ∈ Rn++ .

Under Differentiable Preference, we can define marginal rate of substitution of Good h for Good k at x ∈ Rl++ by M RS

k,h

(x) =

∂u(x) ∂ xk ∂u(x) ∂ xh

We also impose a boundary condition, which rules out corner solutions. Boundary Condition:

For all x ∈ Rn++ it holds

lim M RS k,h (z k , x h , x−{k,h} ) = ∞

z k →0

and

lim M RS k,h (xk , z h , x−{k,h} ) = 0

z h →0

The following claims are standard.1

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General Equilibrium Theory

13

Proposition 2.3 (Interior Solution) Under Differentiable Preference and Boundary Condition, the utility maximization problem has a unique solution x( p, w) in Rn++ , which satisfy the first-order condition Du(x) = λp where λ > 0 is the corresponding Lagrange multiplier. Moreover, the demand function x( p, w) and indirect utility function v( p, w) are differentiable over Rn++ × R++ . Under Differentiable Preference and Boundary Condition, the expenditure minimization problem has a unique solution h( p, u) in Rn++ , which satisfy the first-order condition p = μDu(x) where μ > 0 is the corresponding Lagrange multiplier. Moreover, the compensated demand function h( p, u) and expenditure function e( p, u) are differentiable over Rn++ × u(R++ ). Proposition 2.4 (Shepard’s Lemma) Compensated demand function and expenditure minimization satisfy ∂e( p, u) = h k ( p, u) ∂ pk for all k = 1, . . . , n. Proposition 2.5 (The Slutsky Equation) Demand function and compensated demand function satisfy ∂ xl ( p, w) xl ( p, w) ∂h l ( p, u) = + xk ( p, w) ∂ pk ∂ pk ∂w for all k, l = 1, . . . , n.

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Proposition 2.6 (Roy’s Identity) Demand function and indirect utility function satisfy xk ( p, w) =

∂v( p,w) ∂ pk − ∂v( p,w) ∂w

for all k = 1, . . . , n. From Shepard’s lemma, for income compensation function, we obtain ∂μ( p | p, u) ∂e( p , v( p, w)) = ∂ pk ∂ pk

= h k ( p , v( p, w)) = xk ( p , μ( p | p, w))

for all k = 1, . . . , n.

2.4

General Equilibrium in Exchange Economies

2.4.1 Setting and Definitions Consider that, there are I individuals and n goods. Each i = 1, . . . , I is characterized by 1. Consumption set Rn+ 2. Preference relation i over Rn+ , which is assumed to satisfy the conditions as in the previous section. 3. Initial endowment ωi ∈ Rn++ Here is the definition of competitive equilibrium. Definition 2.1 A competitive equilibrium is a pair of price vector p ∈ Rn+ \ {0} and allocation x such that p · xi ≤ p · ωi and it holds p · xi ≤ p · ωi =⇒ xi  xi

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General Equilibrium Theory

15

for all xi ∈ Rl+ for all i = 1, . . . , I, and I 

xi =

i=1

I 

ωi

i=1

2.4.2 Efficiency Let us briefly review the definition of Pareto efficiency and the two welfare theorems. Definition 2.2 Allocation x = (xi )i=1,...,I ∈ R I n is said to be feasible if I  i=1

xi ≤

I 

ωi .

i=1

Definition 2.3 A feasible allocation x is said to be Pareto-efficient if there is no feasible allocation x such that xi i xi for all i = 1, . . . , m and

xi i xi

for at least one i. Theorem 2.2 (First welfare theorem) Under Strong Monotonicity of preference, any competitive equilibrium allocation is Pareto-efficient.2 Theorem 2.3 (Second welfare theorem) Assume Strong Monotonicity and Convexity of preference. Then, any Pareto-efficient allocation x is obtained as a competitive quasi-equilibrium allocation after suitable redistribution of initial endowments, in the following sense: there is a pair of price vector I such that p ∈ Rn+ \ {0} and income distribution w = (wi )i=1,...,I ∈ R+ it holds p · xi = wi and

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xi  xi =⇒ p · xi ≤ wi for all xi ∈ Rl+ for all i = 1, . . . , I , and I 

xi =

i=1

I 

ωi

i=1

2.4.3 Differential Characterization Under the assumption of differentiable preferences satisfying the boundary condition, an interior equilibrium allocation is characterized by the firstorder condition Du i (xi ) = λi p, where λi is the corresponding Lagrange multiplier for individual i. Hence the marginal rate of substitution is equalized to relative price. The marginal rate of substitution of Good h for Good k for i at xi ∈ Rl++ is given by M RSik,h (xi )

=

∂u i (xi ) ∂ xik ∂u i (xi ) ∂ xi h

Let M RSi (xi ) = (M RSik,h (xi ))k,h=1,...,l . Then, it holds pk M RSik,h (xi ) = ph for all k, h = 1, . . . , n and for all i = 1, . . . , I . Here efficiency of interior allocation is characterized by equalization of the marginal rate of substitution between individuals. In other words, efficiency imposes that subjective relative values between goods are equal for all individuals at margin.

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17

Then the following claim holds. Proposition 2.7 Interior allocation x is Pareto-efficient if and only if M RSi (xi ) = M RS j (x j ) for all i, j = 1, . . . , n. At an interior competitive equilibrium allocation, it follows from the first-order condition that M RSik,h (xi ) =

pk ph

for all k, h. Hence the first welfare theorem follows. At any interior Pareto-efficient allocation it holds M RSi (xi ) = M RS j (x j ) and the second welfare theorem follows by taking the competitive equilibrium price p ∈ Rl++ by pk = M RSik,h (xi ) ph for all k, h, where the definition does not depend on the choice of i because of equalization of MRSs.

The Pareto Set There may be arbitrarily many Pareto-efficient allocations. As illustrated in Fig. 2.1, we can draw arbitrarily many pairs indifference curves which are tangent to each other. We can actually draw a continuous curve by depicting points at which such tangency holds. In the current setting in which the goods are continuously divisible, there is actually a continuum of Pareto-efficient allocations.

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A’s Good 2 IA B’s Good 1

OB

IB

A’s Good 1

OA

B’s Good 2

Fig. 2.1 Set of Pareto-efficient allocations

2.5

The Compensation Principle

Change in economic activity does not always make all individuals better off. Nevertheless, in the partial equilibrium analysis such change is often justified on the ground that it is maximizing social surplus or generating a larger surplus. What normative criteria is it resorting to? Later, we will see that it is resorting to so-called compensation principles. Let me provide the general definition of the principles here. There are several criteria being proposed in the literature. The so-called Kaldor criterion says that a change should be accepted if we can make everybody better off by reallocating the allocation obtained by the change than in the allocation before the change. Formally, it says Definition 2.4 An allocation y = (y1 , . . . , yn ) is a Kaldor-improvement of x = (x1 , . . . , xn ) if there exists an allocation y = (y1 , . . . , yn ) with n  i=1

yi =

n  i=1

yi

2

such that it holds

General Equilibrium Theory

19

yi i xi

for all i and

yi i xi

for at least one i. It is obvious that if y is a Pareto-improvement of x it is a Kaldorimprovement of x. See Fig. 2.2, in which there are two consumers A and B. Then allocation (y A , y B ) is a Kaldor-improvement of (x A , x B ), since we can obtain (y A , y B ) by reallocating (y A , y B ), which is a Pareto-improvement of (x A , x B ). Note that vectors y A − y A and y B − y B are exactly opposite of each other. Or, one can explain this by using utility possibility frontiers. Fix a representation of A’s preference u A and a representation of B’s preference u B . Given, a vector of aggregate resources available to the society e = (e1 , e2 ), let

Good 2 yB

yB xB

yA yA xA Fig. 2.2 Kaldor-improvement

Good 1

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I (e) = {(u A (x A ), u B (x B )) : x A + x B = e1 , x A2 + x B2 = e2 } be the set of pairs of A’s utility and B’s utility which are obtained by allocating e. Of course, this is only for describing trade-offs between A’s gain and B’s gain and utility numbers themselves have no quantitative meanings. See Fig. 2.3, in which two utility possibility frontiers are drawn, I (e) obtained from e and I (e ) obtained from e . Then y on I (e ) makes a Kaldor-improvement of x on I (e) since we can pick y on I (e ) which is in the upper-right of x. Let me state two problems of the Kaldor criterion. One is, It says a change should be accepted if we can reallocate the allocation after the change so as to make everybody better-off. Why not just doing such reallocation? If the reallocation is indeed done it is simply a Paretoimprovement, isn’t it?

The definition of Kaldor-improvement only says “we can reallocate the allocation,” and it does not require that such reallocation is indeed done.

B

I(e)

y

y x

I(e ) A

Fig. 2.3 Kaldor-improvement

2

General Equilibrium Theory

21

Why should one get convinced by such unwarranted story of potential reallocation when he is, in fact, losing because of the change? If the reallocation is left undone such criterion is deceptive, and if the allocation is indeed done we just need the Pareto criterion and it is just redundant. The other problem is that an allocation which Kaldor-improves upon another may be Kaldor-improved upon by the latter. See Fig. 2.4, in which (y A , y B ) is a Kaldor-improvement of (x A , x B ) through the potential reallocation to (y A , y B ), and (x A , x B ) is a Kaldor-improvement of (y A , y B ) through the potential reallocation to (x A , x B ). Hence, the Kaldor-criterion cannot rank properly between allocations in general. One can explain this by using the utility possibility frontiers. See Fig. 2.5, in which two utility possibility frontiers are drawn, I (e) obtained from e and I (e ) obtained from e . Then y on I (e ) makes a Kaldor-improvement of x on I (e) since we can pick y on I (e ) which is in the upper-right of x. However, x makes a Kaldor improvement of y as well, since we can pick x on I (e) which is in the upper-right of y. Let me introduce a criterion which is the “complement” of the Kaldor criterion. The so-called Hicks criterion says that a change should be accepted if we cannot make everybody better off by reallocating the allocation

Good 2 yB

yB xB xB yA yA xA xA

Fig. 2.4 Mutual Kaldor-improvements

Good 1

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T. Hayashi

B

I(e) x y

y x

I(e ) A

Fig. 2.5 Mutual Kaldor-improvements

before the change than in the allocation obtained by the change. In another words, one is a Hicks-improvement of another if the latter is not a Kaldor-improvement of the former. Formally, Definition 2.5 An allocation y = (y1 , . . . , yn ) is a Hicks-improvement of x = (x1 , . . . , xn ) if there exists no allocation x = (y1 , . . . , yn ) with n  i=1

such that it holds

for all i and

for at least one i.

xi

=

n  i=1

xi i yi xi i yi

xi

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General Equilibrium Theory

23

B

I(e) x y I(e ) A Fig. 2.6 Hicks improvement

Let me explain this using utility possibility frontiers. See Fig. 2.6. There, we can never go to the upper-right of y on I (e ) by reallocating x on I (e). Hence y is a Hicks improvement of x. The same comments as above apply to the Hicks criterion. Besides the ethical issue, an allocation which Hicks-improves upon another may be Hicks-improved upon by the latter. See Fig. 2.6 again. There, we can never go to the upper-right of y on I (e ) by reallocating x on I (e). Hence y is a Hicks improvement of x. However, it is also the case that we can never go to the upper-right of x on I (e) by reallocating y on I (e ). Hence x is a Hicks improvement of y. Since the Kaldor criterion and Hicks criterion are the “complement” of each other, if we impose both we can avoid the problem that two allocations dominate each other under the Kaldor criterion alone and under the Hicks criterion alone, respectively. It is called Scitovsky criterion. Definition 2.6 An allocation y = (y1 , . . . , yn ) is a Scitovsky-improvement or of x = (x1 , . . . , xn ) if y is both a Kaldor-improvement and a Hicksimprovement of x.

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T. Hayashi

B

y x

z

w A

Fig. 2.7 Gorman paradox

If one is a Hicks-improvement of another it means the latter is not a Kaldor-improvement of the former. Hence it is always the case that if one is a Scitovsky-improvement of another the latter is not a Scitovskyimprovement of the former. However, the ranking by the Scitovsky-improvement may be intransitive, that is, it may have a cycle. See Fig. 2.7. There y is a Scitovskyimprovement of x, z is a Scitovsky-improvement of y, w is a Scitovskyimprovement of z, but x is a Scitovsky-improvement of w. It is called the Gorman paradox. So the Scitovsky-improvement does not help, unfortunately. Samuelson considered a weakening of the condition, stating that one allocation should be better than another when the entire utility possibility frontier given by the former is above the entire utility possibility frontier by the latter. Let us call this Samuelson criterion. This leads to a cycle again when it is combined with the Pareto criterion, however. The same example works. In Fig. 2.7, y is a Pareto-improvement of x, z is a Samuelsonimprovement of y, w is a Pareto-improvement of z, but x is a Samuelsonimprovement of w. Now, we are pretty much in deadlock.

2

2.6

General Equilibrium Theory

25

Social Welfare Function

2.6.1 Arrovian Social Welfare Function We saw that the Pareto criterion alone is silent about which efficient allocation is socially desirable, and cannot rank between efficient allocation even including indifference. Also, it is orthogonal to any notion of fairness. Consider, for example, between a slightly inefficient allocation and an efficient but extremely unfair (in some sense) allocation. Then, we will be required to give a quantitative judgment over several mutually orthogonal criteria. This motivates us to provide a complete ranking over allocations. Let ω ∈ Rn+ be the social endowment vector, and let  I : X = x ∈ Rn++

I 

 xi = ω

i=1

be the set of feasible allocations in which everybody receives positive consumption. Let R be the set of complete, transitive, continuous convex, and strongly monotone preference relations over Rn+ . Let R0 be the set of complete and transitive orderings over X . An Arrovian social welfare functional is a mapping R : R I → R0 , where R() denotes the social ranking given a profile of individual prefI ∈ R I , and let P() denote the corresponding strict erences = (i )i=1 ranking. The following two axioms are natural to require. Axiom 2.1 Pareto: For all ∈ R I and for all x, y ∈ X , if x i y for all i = 1, . . . , I , then x P()y.

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T. Hayashi

Axiom 2.2 Nondictatorship: There is no i = 1, . . . , I such that for all ∈ R I and for all x, y ∈ X , if x i y then x P()y. Now the well-known independence of irrelevant alternatives axiom basically states that only the ordinal information about preferences should matter. Axiom 2.3 Independence of Irrelevant Alternatives: For all ,  ∈ R I and for all x, y ∈ X , if x i y ⇐⇒ x i y for all i = 1, . . . , I , then x R()y ⇐⇒ x R( )y. Here is a version Arrow theorem stated for exchange economy, which is proven, for example, by Bordes et al. [5]. Theorem 2.4 Let n ≥ 2. Then, there is no social welfare functional R : R I → R0 which satisfies Independence of Irrelevant Alternatives, Pareto, and Nondictatorship.

2.6.2 Bergson–Samuelson Social Welfare Function We saw that it is impossible to aggregate preferences so that the aggregation is independent of the choice of cardinalization of preference representation. We may still accept the idea that evaluation of allocation should depend only on individuals’ utilities, and exclude any “paternalistic” judgment which involves something beyond individual utility functions, while it has to involve a choice of cardinalization.

2

General Equilibrium Theory

27

Let us take utility representation of each individual’s preference as given, which are assumed to be monotone and concave, and consider aggregating them. Then, a Bergson–Samuelson social welfare function is given in the form U (x) = W (u 1 (x1 ), . . . , u I (x I )) I where W : i=1 u i (Rl+ ) → R is an aggregator function. Let us focus on the class of additive Bergson–Samuelson social welfare functions, given the form U (x) =

I 

αi u i (xi )

i=1 I \ {0} be a fixed welfare weight vector. where α ∈ R+ Then the following claims hold. I \ {0}, the maximizer of Theorem 2.5 For any α ∈ R+

U (x) =

I 

αi u i (xi )

i=1

in the set of feasible allocations is Pareto-efficient. Theorem 2.6 Let x be any Pareto-efficient allocation in the set of feasible I \ {0} such that allocations. Then there is a welfare weight vector α ∈ R+ x is maximizing I  U (x) = αi u i (xi ) i=1

in the set of feasible allocations. We should note that unless we have a strong belief or evidence that a particular way of cardinalization is the reasonable one among many, focusing on the additive aggregation like above is no more than for a mathematical convenience. For example, take an exponential transformation of the

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T. Hayashi

original cardinalization, so that vi (xi ) = exp u i (xi ), implying

u i (xi ) = log vi (xi )

Then we obtain U (x) = =

I  i=1 I 

αi u i (xi ) αi log vi (xi )

i=1

= log

I 

vi (xi )αi

i=1

which is ordinally equivalent to another Bergson–Samuelson social welfare function V (x) =

I 

vi (xi )αi

i=1

which is multiplicative. This point is the key point in the so-called Harsanyi-Sen debate on whether establishing an additive aggregation theorem indeed provides a formal foundation of Benthamite utilitarianism. To understand, see the comprehensive treatment by Weymark [6].

2.6.3 Negishi Approach Negishi [7] showed that competitive market maximizes a weighted sum of individual utilities, where the weights are determined endogenously so that each individual’s one is equal to the inverse of his marginal utility of income. Such weight vector is called Negishi weights.

2

General Equilibrium Theory

29

Assume differentiable preference, and go back to the first-order condition for competitive equilibrium, where Du i (xi ) = λi p for each i = 1, . . . , I . Now, let αi = λ1i for each i, and consider the weighted sum of utilities I 

αi u i (xi )

i=1

Then competitive equilibrium allocation maximizes this weighted sum of utilities since it yields an extreme value for the Lagrangean in the form L=

I  i=1

αi u i (xi ) −

n  k=1

μk

 I  i=1

xik −

I 

eik

i=1

as the Lagrange multiplier on Good k is taken to be μk = pk for each k = 1, . . . , n and αi = λ1i for each i. We should be careful, though, because such Negishi weight vector α = (α1 , . . . , α I ) is endogenously determined in equilibrium. This makes a critical difference from Bergson–Samuelson social welfare function in which welfare weights are exogenously chosen by the planner, which reflects his value judgment. For a fixed profile of initial endowment the Negishi “social welfare function” behaves as if it is a Bergson–Samuelson social welfare function, but such welfare weight changes as initial endowment changes, which is not the case in BS.

2.7

General Equilibrium Under Uncertainty

Aggregate expected consumer surplus is a prominent one as an efficiency measure in partial equilibrium welfare analysis under uncertainty. We will give a general equilibrium characterization of when the use of such

30

T. Hayashi

measure is justified. Here, we briefly review the general equilibrium models of resource allocation under uncertainty.

2.7.1 The Environment Focus on the two-period setting, in which there is no consumption or earning taking place in Period 0. There are S states of the world in Period 1. There are n goods at each Sn , where its state in Period 1. Hence, the consumption space is thus R+ element for individual i = 1, . . . , I is denoted by xi = (xi1 , . . . , xi S ). Each individual i has Sn . • Preference i over R+ — Typically, it is assumed to be represented in the expected utility form

u i (xi ) =

S 

πs vi (xis )

s=1

where the function vi is called von-Neumann/Morgenstern index. Sn • Endowment ωi ∈ R++ Note that von-Neumann/Morgenstern index of the utility function, not a utility function, which forms a class of additive representations of preference. As far as we restrict attention to the class of additive representations of a given preference, which is a proper subset of the whole set of representation of the preference, such index has cardinal meaning, as its curvature explains the degree of risk aversion. Note, however, that overall representation is still ordinal. For example, take an exponential transformation of the expected utility form, then we have eu i (xi ) = e

S

s=1 πs vi (x is )

=

S   s=1

evi (xis )

πs

.

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General Equilibrium Theory

31

Now let ui (xi ) = eu i (xi ) and  vi (z) = evi (z) . Then we obtain ui (xi ) =

S 

vi (z))πs . (

s=1

which is a geometric mean rather than arithmetic mean.

2.7.2 Arrow–Debreu Market Let us first describe the case that there is a complete system of markets for all state-contingent consumptions. Definition 2.7 An Arrow–Debreu equilibrium is a pair of price vector Sn \ {0} and an allocation x such that p · x ≤ p · ω and it holds p ∈ R+ i i p · xi ≤ p · ωi =⇒ xi  xi Sn for all i = 1, . . . , I , and for all xi ∈ R+ I  i=1

xi =

I 

ωi

i=1

From the first welfare theorem, allocation in Arrow–Debreu equilibrium is Pareto-efficient according to the individuals’ ex-ante preferences over state-contingent consumptions.

2.7.3 Sequential Trade Now consider that there is not necessary a complete system of markets for state-contingent consumptions. Instead, let us consider a possibly incomplete system of asset markets.

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T. Hayashi

Consider that there are K assets. Let R denote the return matrix, where Rsk denote Asset k’s gross return rate at State s. Asset price vector is denoted by q ∈ R K and spot price vectors are denoted by ps ∈ Rn++ for each s = 1, . . . , S. Then individual i = 1, . . . , I faces a sequence of budget constraints in the form K 

qk z ik ≤ 0

k=1

ps xis ≤

K 

Rsk z ik + ps ωis , s = 1, . . . , S

k=1

The natural restriction on asset price system is that it allows no arbitrage. Definition 2.8 (q, R) admits an arbitrage if there exists z such that qz ≤ 0 and Rs z ≥ 0 for all s and Rs z > 0 for at least one s. Here is the well-known necessary and sufficient condition for the no arbitrage condition. Theorem 2.7 (q, R) admits no arbitrage if and only if there exists μ ∈ S \ {0} such that q = μR. R+ Here is the definition of competitive equilibrium which corresponds to the current setting of incomplete asset markets. Definition 2.9 A Radner equilibrium is a quadruple of asset price vector q, spot price vectors ( ps )s=1,...,S , consumption allocation x and portfolio allocation z such that for each i the consumption-portfolio pair (xi , z i ) is optimal for i under the constraint K 

qk z ik ≤ 0

k=1

ps xis ≤

K  k=1

Rsk z ik + ps ωis , s = 1, . . . , S

2

General Equilibrium Theory

33

The following observations are straightforward. Proposition 2.8 If (q, p, x, z) constitutes a Radner equilibrium given R then (q, R) admits no arbitrage. Proposition 2.9 If (q, p, x, z) constitutes a Radner equilibrium given S \ {0} such that q = μR. R then there exists μ ∈ R+ Let us verify that when the asset markets are in fact complete Radner equilibrium and Arrow–Debreu equilibrium are equivalent. Definition 2.10 The asset markets are complete when rank R = S. Theorem 2.8 Suppose the asset markets are complete. (i) If ( p, x) forms an Arrow–Debreu equilibrium then there is an asset price vector q and portfolio allocation such that (q, p, x, z) forms a Radner equilibrium. (ii) If (q, p, x, z) forms a Radner equilibrium, then there is a vector S \ {0} such that the price vector defined by (μ p , . . . , μ p ), μ ∈ R+ 1 1 S S and x form an Arrow–Debreu equilibrium.

2.7.4 Market Incompleteness and Efficiency Incompleteness of asset markets in general leads to (ex-ante) inefficiency of allocation, we cannot hedge all the uncertainties. What about the secondbest property? Here, assume that n = 1, and Let Ui∗ (z i ) = Ui (R1 z i + ωi1 , . . . , R S z i + ωi S ) for each i = 1, . . . , I Definition 2.11 Asset allocation (z 1 , .

. . , z I ) ∈ R I K is constrained I Pareto-efficient if it is feasible (that is, i=1 z i ≤ 0) and if there is no

other feasible asset allocation (z 1 , . . . , z I ) such that Ui∗ (z i ) ≥ Ui∗ (z i )

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T. Hayashi

for all i and

Ui∗ (z i ) > Ui∗ (z i )

for at least one i. When there is only one consumption good at each state, Radner equilibrium satisfies the constrained Pareto-efficiency. Theorem 2.9 Assume two periods and n = 1. Then asset allocation in Radner equilibrium is constrained Pareto-efficient. This result is not true when there are two or more goods, or there are more than two periods, because you cannot even define the indirect utility function defined with portfolio choices alone. A crude intuition might tell us that when we have more assets our (ex-ante) welfare improves, as we have more devices to hedge uncertainty. This is wrong. That a complete market leads to an efficient allocation and whether our welfare monotonically improves as the market becomes “more complete” are different questions. In fact, Hart [8] shows an example that introducing a new security to trade makes everybody worse-off. We will come back to this problem in the last chapter.

Notes 1. See, for example, Katzner [9] as well as Mas-Colell et al. [1]. 2. Strong Monotonicity can be weakened to Local Nonsatiation, which says that at any point and its open neighborhood (relative to the consumption space) there is always a strictly better point in it.

References 1. Mas-Colell, Andreu, Michael Dennis Whinston, and Jerry R. Green. 1995. Microeconomic theory, vol. 1. New York: Oxford University Press. 2. Debreu, G. 1959.Theory of value: An axiomatic analysis of economic equilibrium. New Haven: Yale University Press. 3. Mas-Colell, A. 1985. The theory of general economic equilibrium: A differentiable approach, No. 9. Cambridge: Econometric Society Monographs.

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4. Magill, Michael J.P., and Martine Quinzii. 2002. Theory of incomplete markets: Vol. 1, Vol. 1. The MIT Press. 5. Bordes, Georges, Donald E. Campbell, and Michel Le Breton. 1995. Arrow’s theorem for economic domains and Edgeworth hyperboxes. International Economic Review 441–454. 6. Weymark, John A. 1991. A reconsideration of the Harsanyi-Sen debate on utilitarianism. Interpersonal Comparisons of Well-Being 255. 7. Negishi, T. 1960. Welfare economics and existence of an equilibrium for a competitive economy. Metroeconomica 12 (2–3): 92–97. 8. Hart, Oliver D. 1975. On the optimality of equilibrium when the market structure is incomplete. Journal of Economic Theory 11 (3): 418–443. 9. Katzner, Donald W. 1968. A note on the differentiability of consumer demand functions. Econometrica: Journal of the Econometric Society 415–418.

3 Income Evaluation of Welfare Change: Equivalent Variation, Compensating Variation, and Consumer Surplus

Consumer surplus is a money metric which measures the desirability of economic outcome. A natural question is whether it is consistent with individual preference maximization and/or maximization of certain social welfare. In the classical general equilibrium demand theory, consumer surplus is introduced as a proxy of less controversial notions of money metric measure, namely equivalent variation and compensating variation. Here, we briefly review whether and how these money metrics can be consistent with individual preference maximization and/or maximization of social welfare.

3.1

Definitions

The original definition of equivalent variation and compensating variation is attributed to Hicks [1]. For the later purpose, however, we follow a general definition due to Chipman and Moore [2].

© The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_3

37

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T. Hayashi

Definition 3.1 Equivalent variation for economic change from ( p 0 , w 0 ) into ( p 1 , w 1 ) is defined by E V ( p 1 , w 1 | p 0 , w 0 ) = e( p 0 , v( p 1 , w 1 )) − e( p 0 , v( p 0 , w 0 )) = e( p 0 , v( p 1 , w 1 )) − w 0 It is the amount of income compensation/lump-sum tax that is required to obtain the welfare level after the economic change under the original price system. Definition 3.2 Compensating variation for economic change from ( p 0 , w 0 ) into ( p 1 , w 1 ) is defined by C V ( p 1 , w 1 | p 0 , w 0 ) = e( p 1 , v( p 1 , w 1 )) − e( p 1 , v( p 0 , w 0 )) = w 1 − e( p 1 , v( p 0 , w 0 )) It is the amount of income compensation/lump-sum tax that is required to obtain the original welfare level under the price system after the economic change.

3.1.1 Single-Good Price Change Let us compare between equivalent variation and compensating variation, for the case that the price of Good k decreases from pk0 to pk1 , with pk1 < pk0 . Note that from Shepard’s lemma, we have 0 , w 0 | p 0 , w 0 ) = e( p 0 , v( p 1 , p 0 , w 0 )) − w 0 E V ( p1 , p−k k −k 0 , w 0 )) − e( p 1 , v( p 1 , p 0 , w 0 )) = e( p 0 , v( pk1 , p−k k −k  p0 k 0 , v( p 1 , p 0 , w 0 ))da h k (a, p−k = k −k pk1

Since it holds 0 0 v( p 0 , w 0 ) < v(a, p−k , w 0 ) < v( pk1 , p−k , w0 )

3

Income Evaluation of Welfare Change …

39

0 , v(a, p 0 , w 0 )) = at all pk1 < a < pk0 and it holds h k (a, p−k −k 0 , w 0 ) because of duality, we have xk (a, p−k

 0 , w0 | p0 , w0 ) = E V ( pk1 , p−k

 ≥  =

pk0 pk1 pk0 pk1 pk0 pk1

0 0 h k (a, p−k , v( pk1 , p−k , w 0 ))da 0 0 h k (a, p−k , v(a, p−k , w 0 ))da 0 xk (a, p−k , w 0 )da

where the last line in the right-hand side is the change in consumer surplus to be defined below (Fig. 3.1). On the other hand, we have 0 0 , w 0 | p 0 , w 0 ) = w 0 − e( pk1 , p−k , v( p 0 , w 0 )) C V ( pk1 , p−k 0 = e( p 0 , v( p 0 , w 0 )) − e( pk1 , p−k , v( p 0 , w 0 ))  p0 k 0 h k (a, p−k , v( p 0 , w 0 ))da = pk1

Again since 0 0 , w 0 ) < v( pk1 , p−k , w0 ) v( p 0 , w 0 ) < v(a, p−k 0 , v(a, p 0 , w 0 )) = at all pk1 < a < pk0 and it holds h k (a, p−k −k 0 0 xk (a, p−k , w ) because of duality, we have

 C V ( pk1 ,

0 p−k , w0 | p0 , w0 )

=  ≤

pk0 pk1 pk0 pk1

0 h k (a, p−k , v( p 0 , w 0 ))da 0 0 h k (a, p−k , v(a, p−k , w 0 ))da

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T. Hayashi

Good k price hk hk pk + Δpk

pk

xk

xk

Good k

Fig. 3.1 Compensated variation, equivalent variation, and change in consumer surplus

 =

pk0 pk1

0 xk (a, p−k , w 0 )da

where the last line the right-hand side is the change in consumer surplus to be defined below. In order to calculate compensated variation and equivalent variation we need to know the compensated demand function, which is not directly observable and we need to know the entire preference in order to know it. We are given only a demand function at best as data in practice, however. Now how can we evaluate welfare change in terms of income given only a demand function. Again, consider the price change of Good k. Given demand function x( p, w) fix income w and p−k , solve the equation xk = xk ( pk , p−k , w)

3

Income Evaluation of Welfare Change …

41

Good k price

Good k Fig. 3.2 Inverse demand function

for pk , and denote the solution by pk = pk (xk ) as w and p−k are fixed. This is called inverse demand function. Plot this on the plane as in Fig. 3.2, where we take a quantity of Good 1 on the horizontal axis and its price on the vertical axis. This inverse demand function describes apparent willingness to pay for the given good, that is, the amount of income the consumer is willing to give up in order to have an extra unit of consumption of it. As the marginal apparent gain from consumption of Good k is pk (z) − pk when its price is pk , the total consumer surplus from consuming xk units of it is given by  xk

pk (z)dz − pk xk

0

This is called consumer surplus, and it describes the apparent amount of net gain from trade measured in terms of income. It is the area surrounded by the inverse demand, horizontal line pk and the vertical axis in Fig. 3.3. Now consider form example that the price of Good k increase from pk to pk +pk . Then, the consumer surplus after the price change is the area

42

T. Hayashi

Good k price

pk

xk

Good k

xk

Good k

Fig. 3.3 Consumer surplus

Good k price pk + Δpk

pk

Fig. 3.4 Change in consumer surplus

surrounded by the inverse demand curve, horizontal line pk + pk and the vertical axis in Fig. 3.4. Hence, the change in consumer surplus due to the price change is the area surrounded by two horizontal lines pk + pk and pk , the inverse demand curve and the vertical axis in Fig. 3.4.

3

Income Evaluation of Welfare Change …

43

By rotating Fig. 3.4 by 90-degrees, we see the integral description of the change in consumer surplus  C S = −

pk +pk pk

xk (a, p−k , w)da,

which is counted as a loss.

3.2

Recovery of Welfare Measure from Marshallian Demand

There is still a way to recover equivalent and compensated variation from observable Marshallian demand function. Here, we follow the argument by Hausman [3]. Consider that we change ( pk , w) so that we maintain the welfare level v( p 0 , w 0 ), that is, consider the equality 0 , w) = v( p 0 , w 0 ) v( pk , p−k 0 , w 0 | p 0 , w 0 ). so that w = w 0 + E V ( pk , p−k Take the total derivative of this, then we get 0 , w) ∂v( pk , p−k

∂ pk

d pk +

0 , w) ∂v( pk , p−k

∂w

which is rewritten into 0 ,w) ∂v( pk , p−k

dw ∂ pk =− 0 ,w) ∂v( pk , p−k d pk ∂w

From Roy’s identity xk ( p, w) =

∂v( p,w) ∂ pk − ∂v( p,w) ∂w

dw = 0

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T. Hayashi

we obtain a differential equation dw( pk ) 0 = xk ( pk , p−k , w) d pk We solve this differential equation with the boundary condition w( pk0 ) = w 0 , and it is solvable for certain simple and well-behaved class of Marshallian demand functions.

3.3

Individual Welfare Judgment: Comparison Between Status Quo and an Alternative

The purpose of measures in income term as introduced above is of course to make a welfare judgment. Hence, the natural question is whether they are consistent with individual utility maximization or maximization of certain social welfare function. Here, we need to be clear about whether we are comparing between some status quo and an alternative, or given the status quo we are comparing between arbitrary alternatives. When the comparison is between status quo ( p 0 , w 0 ) and an alternative 1 ( p , w 1 ), it holds v( p 1 , w 1 ) ≥ v( p 0 , w 0 ) ⇐⇒ e( p 0 , v( p 1 , w 1 )) ≥ e( p 0 , v( p 0 , w 0 )) ⇐⇒ E V ( p 1 , w 1 | p 0 , w 0 ) ≥ 0 Hence equivalent variation makes a judgment which is consistent with the underlying preference. Likewise, since v( p 1 , w 1 ) ≥ v( p 0 , w 0 ) ⇐⇒ e( p 1 , v( p 1 , w 1 )) − e( p 1 , v( p 0 , w 0 )) ⇐⇒ C V ( p 1 , w 1 | p 0 , w 0 ) ≥ 0 compensating variation makes a judgment which is consistent with the underlying preference.

3

3.4

Income Evaluation of Welfare Change …

45

Individual Welfare Judgment: Comparison Between Alternatives

However, do EV and/or CV provide a consistent ranking over all economic variables, not just between the status quo and an alternative but between any pair of alternatives? To answer this question, we need to extend the definition so that it allows comparison between any alternatives, in which both price vectors and income vary. Such in price vectors may be obtained by regulation or taxations, and change in income may be obtained through lump-sum transfers.

3.4.1 Equivalent Variation The equivalent variation for a move from status quo ( p 0 , w 0 ) to an alternative ( p, w) is defined by E V ( p, w| p 0 , w 0 ) = e( p 0 , v( p, w)) − e( p 0 , v( p 0 , w 0 )) Then, the consistency requirement for welfare judgment is formulated by E V ( p 1 , w1 | p0 , w0 ) ≥ E V ( p 2 , w2 | p0 , w0 ) ⇐⇒ v( p 1 , w1 ) ≥ v( p 2 , w2 ).

This is met since E V ( p1 , w1 | p0 , w0 ) ≥ E V ( p2 , w2 | p0 , w0 ) ⇐⇒ (e( p 0 , v( p 1 , w 1 )) − e( p 0 , v( p 0 , w 0 ))) −(e( p 0 , v( p 2 , w 2 )) − e( p 0 , v( p 0 , w 0 ))) ≥ 0 ⇐⇒ e( p 0 , v( p 1 , w 1 )) − e( p 0 , v( p 2 , w 2 )) ≥ 0 ⇐⇒ v( p 1 , w 1 ) ≥ v( p 2 , w 2 ) Thus, we obtain

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T. Hayashi

Theorem 3.1 For any preference, equivalent variation makes consistent welfare ranking between any pair of alternatives, in which both price vector and income may vary.

3.4.2 Compensating Variation The same thing does not hold for compensating variation, however. The consistency condition puts stringent restrictions on the class of preferences, depending on what economic variables are considered to change. Here, we follow the argument by Chipman and Moore [2].

3.4.2.1 General Case What about compensating variation? Here, the consistency requirement is formulated by C V ( p 1 , w1 | p0 , w0 ) ≥ C V ( p 2 , w2 | p0 , w0 ) ⇐⇒ v( p 1 , w2 ) ≥ v( p 2 , w2 ).

Recall the definition of compensating variation for a move from status quo ( p 0 , w 0 ) to an alternative ( p, w), which is given by C V ( p, w| p 0 , w 0 ) = e( p, v( p, w)) − e( p, v( p 0 , w 0 )) = w − e( p, v( p 0 , w 0 )) Since v is homogeneous of degree zero in ( p, w), so so should be C V . However, since C V (λp, λw| p 0 , w 0 ) = λw − e(λp, v( p 0 , w 0 )) = λ(w − e( p, v( p 0 , w 0 ))) = λC V ( p, w| p 0 , w 0 ) it is homogeneous of degree one. It is a contradiction. Thus, we obtain

3

Income Evaluation of Welfare Change …

47

Theorem 3.2 There is no regular preference such that compensating variation makes consistent welfare ranking between any pair of alternatives, when both price vector and income may vary.

3.4.2.2 Preference Condition for Consistency Under Constant Income Compensating variation may lead to consistent individual welfare judgment when we restrict attention limited kinds of economic changes. Here, let us consider that only prices change. Let w be the level of income which is to be maintained constant. Consider the consistency requirement v( p 1 , w) ≥ v( p 2 , w) ⇐⇒ C V ( p 1 , w| p 0 , w 0 ) ≥ C V ( p 2 , w| p 0 , w 0 ) Since V is homogeneous of degree zero for ( p, w), for arbitrary and ( p 2 , w 2 ) we have

( p1 , w1 )

v( p 1 , w 1 ) ≥ v( p 2 , w 2 ) ⇐⇒ v(w p 1 /w 1 , w) ≥ v(w p 2 /w 2 , w) Because of the consistency requirement, we have v(w p 1 /w 1 , w) ≥ v(w p 2 /w 2 , w) ⇐⇒ C V (w p 1 /w 1 , w| p 0 , w 0 ) ≥ C V (w p 2 /w 2 , w| p 0 , w 0 ) Since C V (w p 1 /w 1 , w| p 0 , w 0 ) = w − e(w p 1 /w 1 , v( p 0 , w 0 )) w = w − 1 e( p 1 , v( p 0 , w 0 )) w

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T. Hayashi

and C V (w p 2 /w 2 , w| p 0 , w 0 ) = w − e(w p 2 /w 2 , v( p 0 , w 0 )) w = w − 2 e( p 2 , v( p 0 , w 0 )) w we have C V (w p 1 /w 1 , w| p 0 , w 0 ) ≥ C V (w p 2 /w 2 , w| p 0 , w 0 ) w2 w1 ≥ ⇐⇒ e( p 1 , v( p 0 , w 0 )) e( p 2 , v( p 0 , w 0 )) By combining the above, we obtain the condition v( p 1 , w 1 ) ≥ v( p 2 , w 2 ) ⇐⇒

w1 w2 ≥ e( p 1 , v( p 0 , w 0 )) e( p 2 , v( p 0 , w 0 ))

Thus, we have Theorem 3.3 Assume that income is w at any alternative. Then C V satisfies the consistency condition if and only if v has the form v( p, w) =

w e( p, v( p 0 , w 0 ))

or its monotone transformation. This is equivalent to saying that the preference is homothetic.

3.4.2.3 Preference Condition for Consistency Under Fixed Price of Numeraire Next, consider that the price of some good is fixed, and that we evaluate changes of the other goods and lump-sum income transfer.

3

Income Evaluation of Welfare Change …

49

Fix the price of Good k to be p k . Consider the consistency requirement 1 2 1 , w 1 ) ≥ v( p k , p−k , w 2 ) ⇐⇒ C V ( p k , p−k , w1 | p0 , w0 ) v( p k , p−k 2 ≥ C V ( p k , p−k , w2 | p0 , w0 )

Since V is homogeneous of degree zero for ( p, w), for arbitrary and ( p 2 , w 2 ) we have

( p1 , w1 )

1 2 v( p 1 , w 1 ) ≥ V ( p 2 , w 2 ) ⇐⇒ v( p k , p k p−k / pk1 , p k w 1 / pk1 ) ≥ v( p k , p k p−k / pk2 , p k w 2 / pk2 )

Because of the consistency requirement, we have 1 2 v( p k , p k p−k / pk1 , p k w 1 / pk1 ) ≥ v( p k , p k p−k / pk2 , p k w 2 / pk2 ) 1 2 / pk1 , p k w 1 / pk1 | p 0 , w 0 ) ≥ C V ( p k , p k p−k / pk2 , p k w 2 / pk2 | p 0 , w 0 ) ⇐⇒ C V ( p k , p k p−k

Since 1 C V ( p k , p k p−k / pk1 , p k w 1 / pk1 | p 0 , w 0 ) =

=

pk w1 1 − e( p k , p k p−k / pk1 , v( p 0 , w 0 )) pk1 pk w1 p − k1 e( p 1 , v( p 0 , w 0 )) pk1 pk

and 2 C V ( p k , p k p−k / pk2 , p k w 2 / pk2 | p 0 , w 0 ) =

=

pk w2 2 − e( p k , p k p−k / pk2 , v( p 0 , w 0 )) pk2 pk w1 p − k2 e( p 2 , v( p 0 , w 0 )) pk2 pk

we have 1 2 C V ( p k , p k p−k / pk1 , p k w 1 / pk1 | p 0 , w 0 ) ≥ C V ( p k , p k p−k / pk2 , p k w 2 / pk2 | p 0 , w 0 )

⇐⇒

w1 1 w2 1 − 1 e( p 1 , v( p 0 , w 0 )) ≥ 2 − 2 e( p 2 , v( p 0 , w 0 )) 1 pk pk pk pk

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T. Hayashi

By combining the above, we obtain v( p 1 , w 1 ) ≥ v( p 2 , w 2 ) ⇐⇒

w1 1 w2 1 − 1 e( p 1 , v( p 0 , w 0 )) ≥ 2 − 2 e( p 2 , v( p 0 , w 0 )) 1 pk pk pk pk

Thus, we have Theorem 3.4 Assume that the price of Good k is is fixed to be p k at any alternative. Then C V satisfies the consistency condition if and only if v has the form w e( p, v( p 0 , w 0 )) v( p, w) = − pk pk or its monotone transformation. That is, there is no income effect on every good other than Good k.

3.5

Individual Welfare Judgment Based on Consumer Surplus

Does (change in) consumer surplus induce a ranking over economic variables that is consistent with individual preference maximization? Following Blackorby and Donaldson [4], let us go over the conditions required for consistency.

3.5.1 Path-Independence and Welfare We need to introduce one mathematical concept, called path-independence. a function f : Rm → Rm , and taking the integral of Consider m 0 m 1 m j=1 f j (z), as we move from a point z ∈ R to another point z ∈ R . Then the value of such integral in general depends on choice of a path (rectifiable path, more precisely), which is a function z : [0, 1] → Rm with z(0) = z 0 and z(1) = z 1 .

3

Income Evaluation of Welfare Change …

51

For a particular choice of path, we have the line integral 



m z1  z0

f j (z)dz j ≡

m 1

0

j=1

f j (z(t))dz j (t)

j=1

It is known (see Kaplan [5] for example) that if the value of such line integral is independent of the choice of path then there must be a potential function F such that  F(z ) − F(z ) = 1

0

m z1  z0

f j (z)dz j

j=1

Existence of potential function is the synonym of existence of utility function, as it holds 

m z1  z0

f j (z)dz j ≥ 0 ⇐⇒ F(z 1 ) ≥ F(z 0 )

j=1

Thus, path-independence is essentially equivalent to the existence of preference that justifies a measure using line integral. Since m m   ∂ F(z) dz j = f j (z)dz j d F(z) = ∂z j j=1

for all dz, it holds

for all j = 1, . . . , m. Also, since

j=1

∂ F(z) = f j (z) ∂z j

∂ 2 F(z) ∂ 2 F(z) = ∂z j ∂z h ∂z h ∂z j

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T. Hayashi

it holds

∂ f j (z) ∂ f h (z) = ∂z j ∂z h

for all h, j = 1, . . . , n. Once path-independence holds, such measure provides a consistent ranking not just between status quo and an alternative, but also between any pair of alternatives, since F(z 1 ) − F(z 2 ) = F(z 1 ) − F(z 0 ) − (F(z 2 ) − F(z 0 )) Hence, we restrict attention to the comparison between status quo and an alternative, without loss of generality.

3.5.2 General Case Now, we apply the concept of path-independence in order to see if consumer surplus gives us a consistent ranking. Let us start with the general case. Consider a mapping ( p, w) : [0, 1] → Rn++ × R++ with ( p, w)(0) = ( p 0 , w 0 ) and ( p, w)(1) = ( p 1 , w 1 ). Then define consumer surplus in the line-integral form  C S( p 1 , w 1 | p 0 , w 0 ) = 

( p 1 ,w1 )

( p 0 ,w0 ) 1

≡ 0

 n 

 n 

 xk ( p, w)d pk + dw

k=1



xk ( p, w)d pk (t) + dw(t)

k=1

Then, from the symmetry property obtained above, path-independence of consumer surplus with both price-vector and income varying requires ∂ xk ( p, w) ∂1 = ∂w ∂ pk

3

Income Evaluation of Welfare Change …

for all k = 1, . . . , n. But because

∂1 ∂ pk

53

= 0, it means

∂ xk ( p, w) =0 ∂w for all k, which cannot happen under regular preferences. Thus, we obtain Theorem 3.5 There is no regular preference such that consumer surplus is path-independent when both price vector and income vary.1

3.5.3 When Income Stays Constant Consider a mapping p : [0, 1] → Rn++ such that p(0) = p 0 and p(1) = p 1 . Then we can consider line integral defined by C S( p 1 , w0 | p0 , w0 ) =

 p1  n p 0 k=1

xk ( p, w0 )d pk ≡

 1 n 0 k=1

xk ( p, w0 )d pk (t)

Then, from the symmetry property obtained above, path-independence of consumer surplus requires ∂ xk ( p, w) ∂ xl ( p, w) = ∂ pl ∂ pk for all k and l. It is known that symmetry of cross price derivatives for uncompensated demand function implies homotheticity of preference (see, for example, Blackorby et al. [6]). Thus, we obtain Theorem 3.6 When income is fixed, consumer surplus is path-independent if and only if the preference is homothetic as well as strongly monotone and strictly convex.

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T. Hayashi

3.5.4 When Prices of a Subset of Good and Income Vary Let K be a proper subset of {1, . . . , n}, and consider a mapping ( p K , w) : 0 K ×R 0 [0, 1] → R++ ++ such that ( p K , w)(0) = ( p K , w ) and ( p, w)(1) = ( p 1K , w 1 ). Then, we can consider line integral defined by 0 , w1 | p0 , w0 ) = C S( p 1K , p−K

≡

 ( p1 ,w1 ) K ( p 0K ,w0 )

 1 0

⎛ ⎝



⎛ ⎝

n 

⎞ 0 , w)d p + dw ⎠ xk ( p K , p−K k

k∈K

⎞

0 , w)d p (t) + dw(t)⎠ xk ( p K , p−K k

k∈K

Then, from the symmetry property obtained above, path-independence of consumer surplus with both price-vector and income varying requires ∂ xk ( p, w) ∂1 = ∂w ∂ pk for all k ∈ K . But because

∂1 ∂ pk

= 0, it means

∂ xk ( p, w) =0 ∂w for all k ∈ K , Theorem 3.7 When prices of a subset of goods K and income vary, consumer surplus is path-independent if and only if preference exhibit no income effect on every good in K .

3

3.6

Income Evaluation of Welfare Change …

55

Consistency of Social Welfare Judgment

Now, let us see if the measures provide consistent ranking at a social level.

3.6.1 Equivalent Variation The argument on equivalent variation we discussed for individual welfare judgment extends to the social level. We consider when the welfare judgment on these measures are consistent with individualistic social welfare functions a la Bergson and Samuelson, which evaluates allocation x ∈ Rn I in the form U (x) = W (u 1 (x1 ), . . . , u I (x I )) where u i denotes the individual utility representation for i. Such consistency requirement will put a restriction on the forms of both W and u1, . . . , u I . As we are going to evaluate price vectors or pairs of price vector and income distribution, the social welfare function will take the indirect form v( p, w) = W (v1 ( p, w1 ), . . . , v I ( p, w I )) where each individual i’s indirect utility is defined over price vector p, which is common to all, and own income wi . Then, for all ( p 0 , w 0 ), ( p 1 , w 1 ) and ( p 2 , w 2 ), it holds E Vi ( p 1 , wi1 | p0 , wi0 ) ≥ E Vi ( p 2 , wi2 | p0 , wi0 ) ⇐⇒ vi ( p 1 , wi1 ) ≥ vi ( p 2 , wi2 )

for each i = 1, . . . , I Therefore, for each i there is a monotone transformation φi such that  vi ( p, wi ) = φi E Vi ( p, wi | p 0 , wi0 ) Then, by letting W (a1 , . . . , a I ) = W (φ1−1 (a1 ), . . . , φ −1 I (a I ))

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T. Hayashi

we obtain  W E V1 ( p 1 , w11 | p0 , w10 ), . . . , E V1 ( p 1 , w11 | p0 , w10 )  ≥ W E V1 ( p 2 , w12 | p0 , w10 ), . . . , E V1 ( p 2 , w12 | p0 , w10 ) ⇐⇒ W (v1 ( p 1 , w11 ), . . . , v I ( p 1 , w1I )) ≥ W (v1 ( p 2 , w12 ), . . . , v I ( p 2 , w2I ))

for all ( p 0 , w 0 ), ( p 1 , w 1 ) and ( p 2 , w 2 ). Hence any monotone method of aggregating equivalent variation is consistent with some Bergson–Samuelson social welfare function. We should note, however, that particular choice of status quo ( p 0 , w 0 ) = ( p 0 , w10 , . . . , w 0I ) may be favorable for somebody and opposite for somebody else, which is the case not only for income distribution but also for price vector, since some price vector favor some kind of preferences and vice versa. This problem applies to any kind of money metric of social welfare.

3.6.2 Compensating Variation Now consider compensating variation, which is defined by C Vi ( p, wi | p 0 , wi0 )) = wi − e( p, vi ( p 0 , wi0 )) for each i = 1, . . . , I . The most demanding requirement is that the judgment is consistent with some Bergson–Samuelson social welfare function so that we can compare between any alternatives. That is, C Vi ( p 1 , wi1 | p0 , wi0 ) ≥ C V ( p 2 , wi2 | p0 , wi0 ) ⇐⇒ vi ( p 1 , wi1 ) ≥ vi ( p 2 , wi2 ).

However, we already know that even for the single individual case the above condition cannot be met in general. Here, we follow the argu-

3

Income Evaluation of Welfare Change …

57

ments by Blackorby and Donaldson [7], who looked into the consistency requirement on comparison between status quo and an alternative2 : (C V1 ( p 1 , w11 | p 0 , w10 )), . . . , C VI ( p 1 , w 1I | p 0 , w 0I ))) ≥ 0 ⇐⇒ v( p 1 , w 1 ) ≥ v( p 0 , w 0 ) for all ( p 0 , w 0 ) and ( p 1 , w 1 ). The consistency condition imposes the following restriction on the class of preferences and the form of Bergson–Samuelson social welfare function. Theorem 3.8 (vi )i=1,...,I , v and  satisfy consistency requirement if and only if they have the form vi ( p, wi ) = v i (α( p)wi + βi ( p)), i = 1, . . . , I   I I   ai wi + βi ( p) v( p, w) = W α( p) i=1

(s) =

I 

i=1

ai si

i=1

Lemma 3.1 The consistency condition is met if and only if v can take the form v( p, w) = v( p, φ(w)) where v is increasing in the second argument and φ satisfies φ(λw) = λφ(w) for λ > 0. Proof Let p 1 = p 0 . Then C Vi ( p 1 , wi1 | p 0 , wi0 ) = wi1 − wi0

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T. Hayashi

From the consistency requirement, we have (w 1 − w 0 ) ≥ 0 ⇐⇒ v( p 0 , w 1 ) ≥ v( p 0 , w 0 ) Since the left-hand side is independent of p, v has the form v( p, w) = v( p, φ(w)), where φ is an increasing function and v is increasing in the second argument. Hence, it holds (w 1 − w 0 ) ⇐⇒ φ(w 1 ) ≥ φ(w 0 ) From the consistency again, we have v( p 0 /λ, w 1 ) ≥ v( p 0 /λ, w 0 ) ⇐⇒ (w 1 − w 0 ) ≥ 0 ⇐⇒ v( p 0 , w 1 ) ≥ v( p 0 , w 0 ) Since v is homogeneous of degree zero, it holds v( p 0 /λ, w 1 ) ≥ v( p 0 /λ, w 0 ) ⇐⇒ v( p 0 , λw 1 ) ≥ v( p 0 , λw 0 ) Thus, we obtain v( p 0 , w 1 ) ≥ v( p 0 , w 0 ) ⇐⇒ v( p 0 , λw 1 ) ≥ v( p 0 , λw 0 ) which means v is homothetic in w, and therefore, we obtain φ(w 1 ) ≥ φ(w 0 ) ⇐⇒ φ(λw 1 ) ≥ φ(λw 0 ) which means φ is homothetic. By taking W suitably together we can take φ to be a homogeneous function.  Lemma 3.2 φ has the form φ(w) =

I  i=1

ai wi

3

Income Evaluation of Welfare Change …

59

Proof Given any w 1 ≥ w 0 0, let μ=

φ(w 1 − w 0 ) >0 φ(w 0 )

Then from the definition of μ and homogeneity of φ it holds φ(w 1 − w 0 ) = μφ(w 0 ) = φ(μw 0 ) From the consistency condition we have ((w 1 − w 0 ) − μw 0 ) = 0, which is equivalent to (w 1 − (1 + μ)w 0 ) = 0. From the consistency condition again it holds φ(w 1 ) = φ((1 + μ)w 0 ) and by the homogeneity of φ we obtain φ(w 1 ) = (1 + μ)φ(w 0 ) By the definition of μ, we obtain φ(w 1 ) = φ(w 0 ) + φ(w 1 − w 0 ) Now let z = w 1 − w 0 , then we obtain a Cauchy functional equation φ(w 0 + z) = φ(w 0 ) + φ(z) which holds for arbitrary w 0 and z, and its solution has the form

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T. Hayashi

φ(w) =

I 

ai wi .

i=1

 Proof of the Theorem From the previous lemma, the indirect Bergson– Samuelson social welfare function has the form   I  ai wi v( p, w) = v p, i=1

Hence in terms of utilities as variables it holds   I  W (u 1 , . . . , u I ) = v p, ai ei ( p, u i ) i=1

Since the left-hand side is independent of p, so is the right-hand side, and we may normalize the prices of all goods, say equal to 1. Hence, we may write W as  W (u 1 , . . . , u I ) = W

∗

I 

 ai Fi (u i )

i=1

where W ∗ and F1 , . . . , FI satisfy W∗



 I 

ai Fi (u i ) = v 1,

i=1

Define



I 

 ai ei (1, u i )

i=1

vi∗ ( p, wi ) = Fi (vi ( p, wi ))

for each i = 1, . . . , I .

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Income Evaluation of Welfare Change …

61

Then combining the above we obtain W

∗

 I 

 ai vi∗ ( p, wi )

 =v

p,

I 

i=1

 ai wi

i=1

Now let

z i = ai wi

for each i = 1, . . . , I and vi∗∗ ( p, z i ) = ai vi∗ ( p, wi ). Then, we obtain a Pexider functional equation W

∗

 I 

 vi∗∗ ( p, z i )

 =v

i=1

p,

I 

 zi

i=1

which has the solution vi∗∗ ( p, z i ) = α( p)z i + βi ( p)

3.6.3 Consumer Surplus Continue to follow Blackorby and Donaldson [7], and consider the consistency requirement on the use of consumer surplus, in the form (C S1 ( p 1 , w11 | p 0 , w 0 ), . . . , C S I ( p 1 , w 1I | p 0 , w 0 )) ≥ 0 ⇐⇒ v( p 1 , w 1 ) ≥ v( p 0 , w 0 ) for all ( p 0 , w 0 ) and ( p 1 , w 1 ).

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T. Hayashi

As we already know, when both price vector and income vary consumer surplus cannot be well defined as the definition fails to be pathindependent even for a single individual. Thus, we restrict attention to price changes, where income stays at status quo.

3.6.3.1 When Income Stays Constant: An Impossibility Still, we reach impossibility of using consumer surplus. Consider a mapping p : [0, 1] → Rn++ such that p(0) = p 0 and p(1) = p 1 . Then we can consider line integral defined by C S( p 1 , w0 | p0 , w0 ) =

 p1  n p 0 k=1

xk ( p, w0 )d pk ≡

 1 n 0 k=1

xk ( p, w0 )d pk (t)

It is known that individual preference that yields path-independent consumer surplus have to be homothetic, hence has the form vi ( p, wi ) = v i (αi ( p)wi ) where v i is increasing and αi : Rn++ → R++ satisfies αi (λp) =

1 αi ( p) λ

for λ > 0. From Roy’s identity, we obtain xik ( p, wi ) = for each k = 1, . . . , n.

∂αi ( p) ∂ pk

αi ( p)

wi

3

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Income Evaluation of Welfare Change …

Hence, we obtain  C Si ( p , w | p , w ) = 1

0

0

0

= = =

n 1

xk ( p, w 0 )d pk (t) 0 k=1  1 n ∂αi ( p) ∂ pk wi0 d pk (t) α ( p) 0 k=1 i wi0 (log αi ( p 1 ) − log αi ( p 0 )) wi0 (log αi ( p 1 )wi0 − log αi ( p 0 )wi0 )

= log(αi ( p 1 )wi0 )wi − log(αi ( p 0 )wi0 )wi 0

0

Consider the most natural case of adding consumer surplus across individuals, then from the last expression we obtain I 

C Si ( p 1 , w0 | p0 , w0 ) ≥ 0 ⇐⇒

i=1

I 

log(αi ( p 1 )wi0 )wi ≥ 0

i=1

⇐⇒ log

⇐⇒

i=1

log(αi ( p 0 )wi0 )wi

0

i=1 I

(αi ( p 1 )wi0 )wi ≥ log 0

i=1 I

I 

I

(αi ( p 0 )wi0 )wi

0

i=1

(αi ( p 1 )wi0 )wi ≥ 0

I

(αi ( p 0 )wi0 )wi

0

i=1

Notice that this ranking depends not only on individuals’ utilities given by αi ( p 1 )wi0 for each i, but also on their incomes. This means that even in the simplest case of adding consumer surplus the ranking does not depend solely on individuals’ utilities, and it it cannot be rationalized by a Bergson–Samuelson social welfare function.

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T. Hayashi

3.6.3.2 Consumer Surplus with Normalized Prices The above impossibility argument motivates us to consider consumer surplus with normalized price, which is defined in the line-integral form C Si∗ ( p 1 , wi1 | p 0 , wi0 ) =



n p 1 /wi1  p 0 /wi0 k=1

xk ( p/wi , 1)d( pk /wi )

Note that since the x( p, wi ) is homogeneous of degree zero it holds x( p/wi , 1) = x( p, wi ). In order that this measure is path-independent, it must hold ∂ x j (q, 1) ∂ xk (q, 1) = ∂q j ∂qk for all k, j = 1, . . . , n. It is known that symmetry of cross price derivatives for uncompensated demand function implies homotheticity of preference. Thus, we obtain vi ( p, wi ) = v i (αi ( p)wi ) where v i is increasing and αi ( p) is homogeneous of degree minus one. By Roy’s identity, it holds xik ( p, wi ) =

∂αi ( p) ∂ pk

αi ( p)

wi

Hence  q1  n i qi0 k=1

xk (qi , 1)dqik =

 q1  n ∂αi (qi ) i ∂q ik

qi0 k=1 αi (qi )

dqik = log αi (qi1 ) − log αi (qi0 )

Now taking qi0 = p 0 /wi0 , qi1 = p 1 /wi1 , and from homogeneity of degree minus zero of αi we obtain C Si∗ ( p 1 , wi1 | p 0 , wi0 ) = log αi ( p 1 )wi1 − log αi ( p 0 )wi0

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Income Evaluation of Welfare Change …

65

Thus, for each individual consumer surplus with normalized price is a consistent measure since C Si∗ ( p 1 , wi1 | p 0 , wi0 ) ≥ 0 ⇐⇒ log αi ( p 1 )wi1 ≥ log αi ( p 0 )wi0 ⇐⇒ Vi ( p 1 , wi1 ) ≥ Vi ( p 0 , wi0 )

Theorem 3.9 (vi )i=1,...,I , v and  satisfy consistency requirement if and only if they have the form vi ( p, wi ) = v i (αi ( p)wi ) i = 1, . . . , I   I

bi v( p, w) = W [αi ( p)wi ] i=1

(s) =

I 

bi si

i=1

Lemma 3.3 The consistency condition is met if and only if V can take the form v( p, w) = v( p, φ(w)) where v is increasing in the second argument and φ satisfies φ(λw) = λφ(w) for λ > 0. Proof Let p 1 = p 0 then it holds C Si∗ ( p 1 , wi1 | p 0 , wi0 ) = log wi1 − log wi0 From the consistency requirement, we obtain (log w11 − log w10 , . . . , log w1I − log w0I ) ⇐⇒ v( p 1 , w1 ) ≥ v( p 0 , w0 )

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Since the left-hand side is independent of p, v has the form v( p, w) = v( p, φ(w)), where φ is an increasing function and v is increasing in the second argument. Hence, it holds (log w11 − log w10 , . . . , log w 1I − log w 0I ) ⇐⇒ φ(w 1 ) ≥ φ(w 0 ) From the consistency again, we have v( p0 /λ, w1 ) ≥ v( p 0 /λ, w0 ) ⇐⇒ (log w11 − log w10 , . . . , log w1I − log w0I ) ≥ 0 ⇐⇒ v( p 0 , w1 ) ≥ v( p 0 , w0 )

Since v is homogeneous of degree zero, it holds v( p 0 /λ, w 1 ) ≥ v( p 0 /λ, w 0 ) ⇐⇒ v( p 0 , λw 1 ) ≥ v( p 0 , λw 0 ) Thus, we obtain v( p 0 , w 1 ) ≥ v( p 0 , w 0 ) ⇐⇒ v( p 0 , λw 1 ) ≥ v( p 0 , λw 0 ) which means v is homothetic in w, and therefore, we obtain φ(w 1 ) ≥ φ(w 0 ) ⇐⇒ φ(λw 1 ) ≥ φ(λw 0 ) which means φ is homothetic. By taking W suitably together we can take φ to be a homogeneous function.  Lemma 3.4 φ has the form φ(w) =

I

wibi

i=1

where b1 , . . . , b I > 0 and

I

i=1 bi

= 1.

3

Income Evaluation of Welfare Change …

67

Proof Normalize φ(1) = 1. Define   ψ(z) = log φ e z 1 , . . . , e z I then it holds ψ(0) = 0. Then from the consistency condition, leaving prices unchanged, we obtain (z 1 − z 0 ) ≥ 0 ⇐⇒ ψ(z 1 ) ≥ ψ(z 0 ) Since φ is homogeneous of degree one, we have   ψ(z + λ1) = log φ e z 1 +λ , . . . , e z I +λ   = log φ eλ e z 1 , . . . , eλ e z I   = log eλ φ e z 1 , . . . , e z I   = log φ e z 1 , . . . , e z I + λ = ψ(z) + λ Let

μ = ψ(z 1 ) − ψ(z 0 )

Then it holds ψ(z 0 + μ1) = ψ(z 0 ) + μ = ψ(z 1 ) From the consistency condition we have (z 1 − (z 0 + μ1)) = 0, which is seen as

((z 1 − z 0 ) − μ1) = 0.

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From the consistency condition again we obtain ψ(z 1 − z 0 ) = ψ(μ1) = ψ(1) + μ = μ = ψ(z 1 ) − ψ(z 0 ) Thus, by letting y = z 1 − z 0 we obtain a Cauchy functional equation ψ(y + z 0 ) = ψ(y) + ψ(z 0 ) which has the solution ψ(z) =

I 

bi z i

i=1

By transforming the variables we obtain the result.



Notes 1. Here, we mean regular preference by ordering which satisfies differentiability and the boundary condition as well as strong monotonicity and strict convexity. 2. One might think of the comparison between arbitrary pairs of alternatives for restricted kinds of changes such as keeping constancy of income or keeping the price of some goods unchanged. This delivers similar corresponding preference condition for consistency.

References 1. Hicks, John R. 1939. The foundations of welfare economics. The Economic Journal 49 (196): 696–712. 2. Chipman, John S., and James C. Moore. 1980. Compensating variation, consumer’s surplus, and welfare. The American Economic Review 933–949. 3. Hausman, Jerry A. 1981. Exact consumer’s surplus and deadweight loss. The American Economic Review 71 (4): 662–676. 4. Blackorby, Charles, and David Donaldson. 1999. Market demand curves and Dupuit-Marshall consumers’ surpluses: A general equilibrium analysis. Mathematical Social Sciences 37 (2): 139–163.

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5. Kaplan, Wilfred. 1952. Advanced calculus, vol. 4. Reading, MA: AddisonWesley. 6. Blackorby, Charles, Daniel Primont, and R. Robert Russell. 1978. Duality, separability, and functional structure: Theory and economic applications, Vol. 2. Elsevier Science Ltd. 7. Blackorby, Charles, and David Donaldson. 1985. Consumers’ surpluses and consistent cost-benefit tests. Social Choice and Welfare 1 (4): 251–262.

4 The Assumption of No Income Effect and Quasi-linear Preferences

In the previous chapter, we saw various restrictions on preferences under which income evaluation of economic change provides welfare judgments which are consistent with individual utility maximization and maximization of individualistic social welfare functions. Among them, the case of no income effect on a single good or a limited set of goods is most relevant to partial equilibrium welfare analysis, in which we consider changes of price and allocation of it together with lump-sum transfers. This is indeed the case which Hicks [1] considered as the justification of the use of consumer surplus. I spend one chapter on this case, providing the so-called quasi-linear preference model, explaining how it can be consistent with the concept of ordinal utility and guarantees the equivalence between surplus maximization and Pareto efficiency.

4.1

The Reduced Two-Good Model

Consider the following two-good model, where Good 1 is the object of partial equilibrium analysis, and Good 2 is income transfer to be spent over the other goods. This income transfer is allowed to take negative values © The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_4

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as well. When it is negative it means that the consumer is decreasing income to be spent over the other goods, that is, paying. Therefore, the consumption space is not the negative quadrant, but it is the right half of the plain R+ × R. For example, when x = (x1 , x2 ) is given as in Fig. 4.1 the consumer is losing |x2 | units of income. The substantive assumption behind this is that the market for Good 1 is very small compared to the entire economy so that the consumer’s income is sufficiently large compared to Good 1 and taken to be unlimited in the local sense, and only its relative increase or decrease matters. That is, as in Fig. 4.2 the consumption space R+ × R is obtained by magnifying the portion where the “background income” to be allocated to the other commodities is already sufficiently large.1 In a later chapter, we will see that such limit situation emerges as the number of goods tends to be large the good we are looking at tends to be only a small fraction of the entire set of goods. Then consumer’s preference is supposed to be a quasi-linear in Good 2, the income transfer.

Good 2

x1 x2

Fig. 4.1 Consumption space R+ × R

x

Good 1

4

The Assumption of No Income Effect …

Income

73

Change in income

Good 1

Good 1 Fig. 4.2 Change in income when the background income is sufficiently large

Definition 4.1 The preference is said to be quasi-linear if the indifference curves are parallel along the vertical axis. Indifference curves for a quasi-linear preference are depicted as in Fig. 4.3. Indifference curves being parallel along the Good-2-axis means that marginal rate of substitution is independent of the quantity of Good 2. That is, the amount of income the consumer is willing to give up in order to get one extra unit of Good 1 is independent of how much income he is holding (see the dotted line in Fig. 4.3). It is that there is no income effect on Good 1. When is the no income effect assumption valid? This argument dates back to Marshall [2], who thought that when the commodity under consideration is negligibly small compared to the entire set of commodities the income effect on it is negligible. For example, if you don’t buy a small thing such one can of cola for 2 dollars it is not because you can’t pay 2 dollars or paying 2 dollars seriously affects consumption of the other goods, but because you judge that it does not deserve 2 dollars.

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Good 2

Good 1

Fig. 4.3 Quasi-linear preference

Marshall thought that in such situations in which income effect is negligible consumption is determined only by comparison between willingness to pay and price. Again, in a later chapter, we will see that such limit situation emerges as the number of goods tends to be large the good we are looking at tends to be only a small fraction of the entire set of goods. The no income effect assumption is important in making welfare judgments. If you mistakenly assume that income effect is negligible while it actually matters, when the consumer’s apparent willingness to pay for the good is for example low you cannot distinguish whether it is because he does not care for it or it is because of the actual income effect. Since the indifference curves are parallel along the Good 2 axis, one can pick one of them and let it represent the whole preference. As in Fig. 4.4 pick the difference curve passing through the origin and look at a point on it with Good 1 quantity denoted by x1 . On a given indifference curve if Good 1 quantity is determined it automatically determines the quantity of Good 2, and it is represented as a function of x1 . Let it be denoted by −v(x1 )

4

The Assumption of No Income Effect …

75

Good 2

x1

Good 1

−v(x1 )

Fig. 4.4 Willingness to pay

Here v(x1 ) is the largest possible amount of Good 2 the consumer is willing to give up in order to get x1 units of Good 1, which meets the condition (x1 , −v(x1 )) ∼ (0, 0) Thus v(x1 ) is understood to be the willingness to pay for x1 units of Good 1. Under the assumption that there is no income effect on Good 1 this willingness to pay does not depend on income and depends only on x1 . When the consumer is indifferent between bundle x = (x1 , x2 ) and another bundle consisting only of Good 2, denoted (0, w2 ), that is, when x ∼ (0, w2 ), let us call w2 the consumer surplus generated by x, in the sense that it is the evaluation of net gain from consumption in terms of income. To find the consumer surplus, draw the indifference curve passing through x = (x1 , x2 ) and look at its intercept with the vertical axis,

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Good 2 w2 = v(x1 ) + x2

x2 x1

Good 1

−v(x1 )

Fig. 4.5 Consumer surplus

(0, w2 ), as in Fig. 4.5. Since the indifference curves are parallel along the vertical axis we have w2 = v(x1 ) + x2 , that is, it holds x ∼ (0, v(x1 ) + x2 ) Hence the consumer surplus gained from x is v(x1 ) + x2 . Now, pick any x = (x1 , x2 ) and y = (y1 , y2 ). Since x ∼ (0, v(x1 ) + x2 ), y ∼ (0, v(y1 ) + y2 ), we have x  y if and only if x2 + v(x1 )  y2 + v(y1 ). Thus consumer surplus v(x1 )+x2 is a representation of the quasi-linear preference. Notice, however, that any monotone transformation of a representation of a given preference is again a representation of the same preference, for arbitrary monotone transformation f the function f (v(x1 ) + x2 ) represents the above quasi-linear preference. Thus, we obtain the proposition below.

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The Assumption of No Income Effect …

77

Proposition 4.1 When a preference is quasi-linear with respect to Good 2 it is represented in the form u(x) = f (v(x1 ) + x2 ) where f is arbitrary monotone transformation. Notice that while the whole utility representation u has no quantitative meaning since f is arbitrary, the consumer surplus v(x1 ) + x2 inside f has certain quantitative meaning, in that it is interpreted to be a measure of gains from trade in terms of income. While there can be arbitrarily many utility representations for a given preference, the function describing one’s willingness to pay, that is v, is uniquely determined when the preference is quasi-linear in income. This v is a component of utility representation, not a utility representation by itself. To emphasize the distinction let me call v the willingness to pay function for a given consumer. Willingness to pay has economic content under the assumption of quasi-linearity within the framework of partial equilibrium analysis, while the whole utility representation again has no quantitative meaning. Historically, the willingness to pay function was believed to be “the” utility function and this made people believe that utility function has a quantitative meaning. This is a confusion, however, while nowadays teachers very often make use of this confusion between them in introductory courses for “educational purpose.”

4.2

Marginal Willingness to Pay as Marginal Rate of Substitution

Let us look more into the implication of quasi-linearity. Proposition 4.2 When preference is quasi-linear and represented in the form f (v(x1 ) + x2 ), the marginal rate of substitution of Good 2 (income transfer) for Good 1 (the good under consideration) at x = (x1 , x2 ) is given by MRS(x) = v  (x1 )

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Proof Given representation f (v(x1 ) + x2 ), marginal utility of Good 1 is ∂f (v(x1 ) + x2 ) = f  (v(x1 ) + x2 )v  (x1 ) ∂x1 On the other hand, marginal utility of Good 2, income transfer to be spent over the other goods, is ∂f (v(x1 ) + x2 ) = f  (v(x1 ) + x2 ) ∂x2 Hence, the marginal rate of substitution is MRS(x) =

∂f (v(x1 )+x2 ) ∂x1 ∂f (v(x1 )+x2 ) ∂x2

=

f  (v(x1 ) + x2 )v  (x1 ) = v  (x1 ) f  (v(x1 ) + x2 ) 

The above says two things. 1. Marginal rate of substitution is independent of monotone transformation f . 2. The marginal rate of substitution depends only on x1 and independent of x2 . 1 applies to any (differentiable) preference. Generally, the marginal rate of substitution is independent of which representation to pick for a given preference. 2 is specific to quasi-linearity. Recall that marginal rate of substitution of Good 1 for Good 2 is the amount of Good 2 one is willing to give up in order to have an extra one unit of Good 1, which is exactly the notion of willingness to pay in the current context. In general, the marginal rate of substitution depends on both x1 and x2 , which means willingness to pay is not determined independently of income. Under the assumption of quasi-linearity, however, willingness to pay for Good 1 is independent of the amount of Good 2 held and determined by x1 alone. Thus, v  (x1 ) is understood to be the amount of income the consumer is willing to give

4

The Assumption of No Income Effect …

79

up in order to get an extra one unit of the good under consideration, that is, marginal willingness to pay. “What’s the difference from the argument in the beginning of the chapter?” you might say. There is a difference at a categorical level. There may be arbitrarily many utility representations for a given preference, since one can take the arbitrary monotone transformation of them. Marginal utility has no quantitative meaning either. On the other hand, willingness to pay and marginal willingness to pay have quantitative meanings under the assumption of quasi-linearity or equivalently speaking, the assumption of no income effect.

4.3

No Income Effect and Inverse Demand Function

Let us now look at consumption choice when preference is quasi-linear. Let p be the relative price of Good 1 for Good 2 (that is income transfer), and I be the income (while income would not matter since no income effect is assumed here). Thus, we consider the maximization problem max f (v(x1 ) + x2 ) x

subject to px1 + x2 = w Apply the tangency condition assuming that the willingness to pay function v is “smooth,” meaning MRS(x1 , x2 ) = pp21 . Because MRS (x1 , x2 ) = v  (x1 ) holds under quasi-linearity and we are taking pp21 = p due to the normalization we have v  (x1 ) = p Notice that here the consumption of Good 1 is determined by just one equation independently of income w. Hence by solving this one equation for the one unknown x1 we obtain the demand function for Good 1 x1 (p),

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which is independent of income holding and depends only on the relative price for income. Also, the tangency condition is saying that the price such that the consumer continues to buy Good 1 up to x1 units is v  (x1 ). That is, the marginal willingness to pay is interpreted to be the inverse demand function in which the Good 1 quantity x1 is the independent variable. That is, the inverse function of the demand function x1 (p), which is denoted by p(x1 ), is given by p(x1 ) = v  (x1 ). Now let us reconfirm that the income effect on Good 1 is zero. Assuming that the price of Good 2 is normalized to 1, consider that the price of Good 1 goes up from p to p as in Fig. 4.7, in which the consumption moves x = ( x1 , x2 ) be the compensated from x = (x1 , x2 ) to x = (x1 , x2 ). Let  demand under the price after the change which yields the same welfare level as x does. Because the indifference curves are parallel along the vertical axis,  x is precisely in the above of x , hence the income effect on Good 1, that is x1 − x1 , is zero (Fig. 4.6).

Good 2

x x

x Good 1

Fig. 4.6 Substitution and income effect in quasi-linear preference

4

4.4

The Assumption of No Income Effect …

81

Compensating Variation, Equivalent Variation, and Consumer Surplus

As indicated in the previous chapter, under the assumption of no income effect the three measures of welfare change, change in consumer surplus, compensated variation, and equivalent variation, coincide. Go back to the Slutsky equation ∂x1 ∂x1 ∂h1 = + x1 ∂p1 ∂p1 ∂w When income effect is zero, we have

∂x1 ∂w

= 0 and the equation becomes

∂x1 ∂h1 = , ∂p1 ∂p1 which means that the demand curve and compensated demand curve coincide. In Fig. 3.1, it means that all three curves collapse into one curve. Hence, the three criteria coincide under the assumption of no income effect.. Let me explain this using Fig. 4.7. Assuming that the price of Good 2 is normalized to 1, consider that the price of Good 1 goes up from p to p + p. Let x denote the demand before the price change, x denote the demand after the price change, x denote the compensated demand under p + p which yields the same welfare level as x does, and  x denote the compensated demand under p which yields the same welfare level as x does. Here, the compensated variation is x2 − x2 and the equivalent variation is x2 − x2 , and the change in consumer surplus is the difference between the intercepts of the two indifference curves with the vertical axis. Because, the indifference curves are parallel along the vertical axis, these three coincide.

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Good 2

x x

x Good 1 x

Fig. 4.7 Compensated and equivalent variation in quasi-linear preference

4.5

Partial Equilibrium Reformulated

Again, fix the price of Good 2 equal to one, that is, take the normalization p2 = 1. Then denote the price of Good 1 by p instead of p1 . From the assumption of no income effect we assume that consumers’ preferences are linear in Good 2, that is, each consumer i = 1, . . . , I has preference represented in the form ui (xi1 , xi2 ) = f (vi (xi1 ) + xi2 ) Since consumption Good 1 does not depend on income under the assumption of no income effect, without loss of generality assume that initial income is zero, and consider that consumer i obeys his budget constraint pxi1 + xi2 = 0 That is, the income transfer here is xi2 = −pxi1 , which is the payment for the purchase of Good 1.

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The Assumption of No Income Effect …

83

As seen above, consumption of Good 1 is independent of income and it is determined by the condition of equality between marginal willingness to pay and (relative) price vi (xi1 ) = p From this, we obtain inverse demand function pi (xi1 ) = vi (xi1 ) and demand function xi1 (p) = (vi )−1 (p). Then, the aggregate demand function is given by x1 (p) =

I 

xi1 (p),

i=1

and by taking its inverse we obtain the aggregate inverse demand function p(x1 ). It is helpful later to take an understanding that the buyer who is willing to buy the “x1 -th” unit of the good is willing to pay p(x1 ). Such “x1 -th” consumer is said to be the marginal consumer at x1 . For simplicity, the production side is summarized in the form of a representative firm with a cost function C(y1 ), where y1 denotes its output level of Good 1. In a competitive market, it solves max py1 − C(y1 ) y1

Here, we assume that the shutdown condition does not bind, and the equality between marginal cost and price holds, MC(y1 ) = p From this, we obtain the inverse supply function p(y1 ) = MC(y1 ) and supply function y1 (p) = (MC)−1 (p).

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Then competitive partial equilibrium in which demand matches supply in the market for Good 1 is determined by ∗

x1 (p ) =

I 

xi1 (p∗ ) = y1 (p∗ )

i=1

4.6

Efficiency and Surplus Maximization

In order to talk about efficiency let rewrite down the definition of feasible allocation in the setting of partial equilibrium. Definition 4.2 Consumption allocation x = (x1 , . . . , xI ) is said to be feasible if there exists y1 such that I  i=1 I 

xi1 = y1 xi2 = −C(y1 )

i=1

Note that the second equality says that total payment by the consumers equals to the total cost for production. In the framework of partial equilibrium, Pareto efficiency is equivalent to maximizing social surplus. Sometimes this is confused with the idea of “the greatest happiness of the greatest number.” Since we are maintaining the viewpoint of ordinal utility we cannot simply take the sum of utilities across individuals, since we cannot compare them without bringing in a particular faith. On the other hand, willingness to pay and consumer surplus have quantitative meanings and it has an economic meaning to take the sum of them. Indeed, as is discussed below maximizing social surplus has totally silent about how much surplus each consumer should gain.

4

The Assumption of No Income Effect …

85

Proposition 4.3 Consumption allocation x = (x1 , . . . , xI ) is Paretoefficient if and only if (x11 , . . . , xI 1 ) is given by the solution to   I I   vi (xi1 ) − C xi1 max x,y

i=1

i=1

 , . . . , x  ) such that Proof “Only if ” Part: Suppose there exists (x11 I1 I 

 vi (xi1 )−C

 I 

i=1

  xi1

>

i=1

I 

vi (xi1 ) − C

 I 

i=1

 xi1

i=1

Let S=

 I 

 vi (xi1 )−

i=1

I 

  I    I    −C vi (xi1 ) − C xi1 xi1 

i=1

i=1

i=1

then by assumption we have S > 0. Divide this among all individuals so  that si > 0 for each i and Ii=1 si = S. Now for each i = 1, . . . , I let   xi2 = xi2 + vi (xi1 ) − vi (xi1 ) + si   I Then, from ni=1 xi2 = −C x i=1 i1 and the definition of S it holds I  i=1

 xi2

=

I  i=1

= −C

xi2 +  I 

I 



xi1 +

i=1

= −C

 I 

vi (xi1 ) −

i=1



I 

 vi (xi1 )+

i=1 I  i=1

vi (xi1 ) −

n 

si

i=1 I 

 vi (xi1 )+S

i=1

 xi1

i=1  , x  ), . . . , (x  , x  )) is feasible. which implies x = ((x11 12 I1 I2

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The for all i = 1, . . . , I it holds   ) + xi2 = vi (xi1 ) + xi2 + si > vi (xi1 ) + xi2 vi (xi1

Therefore x is a Pareto improvement of x. Hence x is not Pareto-efficient. “If ” Part: Suppose x = (x1 , . . . , xI ) is not Pareto-efficient. Then, there is  , . . . , x  ) such that a feasible allocation (x11 I1   ) + xi2 ≥ vi (xi1 ) + xi2 vi (xi1

for all i and

  ) + xi2 > vi (xi1 ) + xi2 vi (xi1

for at least one i. Then by summing up the inequalities we obtain I I 

    {vi (xi1 ) + xi2 } vi (xi1 ) + xi2 > i=1

i=1

  I I  By feasibility, we have Ii=1 xi2 = −C x i=1 i1 and i=1 xi2 =  I  x −C i=1 i1 for the corresponding productions. Hence we have I  i=1

 vi (xi1 )−C

 I  i=1

  xi1

>

I  i=1

vi (xi1 ) − C

 I 

 xi1

i=1

 When willingness to pay and cost function are smooth, Pareto efficiency is characterized by equalization of marginal willingness to pay and marginal cost.

4

The Assumption of No Income Effect …

Proposition 4.4 Consumption allocation Pareto-efficient if and only if it holds vi (xi1 ) = MC

 I 

=

x

(x1 , . . . , xI )

87

is

 xi1

i=1

holds for all i = 1, . . . , I .

4.7

Coincidence of Pareto, Kaldor, and Hicks

The problem of inconsistency of Kaldor–Hicks criterion vanishes in the quasi-linear environment in which there is no income effect, while the ethical problem I discussed above still remains. There the Kaldor criterion and the Hicks criterion coincide and the comparisons are determined by the amount of social surplus, that is, “making the pie bigger is better.” Proposition 4.5 Assume the quasi-linear environment. y = (y1 , . . . , yI ) and x = (x1 , . . . , xI ) satisfy I  i=1

{vi (yi1 ) + yi2 } >

I 

Then,

if

{vi (xi1 ) + xi2 }

i=1

y is both a Kaldor-improvement and a Hicks-improvement of x.   Proof Let w = Ii=1 {vi (yi1 ) + yi2 } − Ii=1 {vi (xi1 ) + xi2 } > 0. Then  = y and it is enough to set let’s say yi1 i1  yi2 = vi (xi1 ) + xi2 − vi (yi1 ) +

w n

for all i.  In the framework of partial equilibrium analysis, Pareto efficiency pins down allocation of Good 1 basically uniquely through maximization of social surplus, but we should notice that it is totally silent about how the

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maximized surplus (i.e., Good 2) should be distributed among individuals. Any distribution of maximized social surplus is efficient, indeed. How we should distributed social surplus is a question which is orthogonal to the notion of efficiency. To clarify this point, consider an exchange economy between two individuals. Consumer A has an initial holding of Good 1 denoted by eA1 , and consumer B has eB1 . Without loss of generality let the initial holdings of Good 2 be eA2 = eB2 = 0. Then feasible allocation (xA , xB ) must satisfy xA1 + xB1 = eA1 + eA2 xA2 + xB2 = 0 Then, the set of Pareto-efficient allocations is depicted as in Fig. 4.8, which is a vertical line. This is because when equality of marginal rates of substitution between A and B holds it holds at any point vertically above or below since indifference curves are parallel along the vertical axis due to the assumption of no income effect. Thus, while the allocation of Good 1 is uniquely determined the allocation of Good 2 is totally undetermined.

xA2

xB1

e

xA1

xB2 Fig. 4.8 Set of efficient allocations under quasi-linear preferences

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The Assumption of No Income Effect …

89

In competitive equilibrium marginal willingness to pay for all consumers and marginal cost are equal to the price, that is, vi (xi1 ) = MC (y1 ) = p holds for all i = 1, . . . , n. Thus it is Pareto-efficient. Note, however, that this is choosing only a particular way of distributing maximized social surplus, while efficiency alone is totally silent about how to distribute the surplus.

4.8

Social Welfare Maximization

Efficiency alone, or Kaldor/Hicks criterion alone is totally silent about how we should split the maximized social surplus. If the social planner maximizes social welfare, in which certain inequality concern is taken into account, then such welfare judgment must involve how we should distribute the maximized surplus. Consider that we have fixed utility representation of each individual i’s preference, in the form ui (xi1 , xi2 ) = φi (vi (xi1 ) + xi2 ) where φi is some monotone function. Let us fix a Bergson–Samuelson social welfare function W (u1 (x11 , x21 ), . . . , uI (xI 1 , xI 2 )) Then, it takes the form W (φ1 (v1 (x11 ) + x12 ), . . . , φI (vI (xI 1 ) + xI 2 ))) We maximize this under the feasibility constraint. However, since any such maximizer must be at least Pareto-efficient, and since we already know that efficiency is equivalent to maximization of net social surplus, it has be maximizing net social surplus. The converse is

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of course not true in general, when the Bergson–Samuelson social welfare function exhibits inequality concern. For example, let the Bergson–Samuelson social welfare function be utilitarian including the choice of cardinalization, so that W (φ1 (v1 (x11 ) + x12 ), . . . , φI (vI (xI 1 ) + xI 2 ))) =

I 

{vi (xi1 ) + xi2 }

i=1

then again any allocation maximizing net social surplus is welfaremaximizing, as the utility-possibility frontier is simply a hyperplane with the normal vector being 1 and social indifference curves are also parallel hyperplanes with the normal vector being 1. However, when the Bergson– Samuelson social welfare function is egalitarian including the choice of cardinalization, so that W (φ1 (v1 (x11 ) + x12 ), . . . , φI (vI (xI 1 ) + xI 2 ))) = min {vi (xi1 ) + xi2 } i

then obviously only a subset (typically a single point) of the set of surplusmaximizing allocations is welfare-maximizing. Theorem 4.1 Under the assumption of quasi-linear preferences, maximization of Bergson–Samuelson social welfare function implies maximization of net social surplus, while the converse is not true in general. We should note, however, that from a general equilibrium point of view the quasi-linear utilities are not a primitive, in the sense that the numeraire good is a composite of everything else, and how one is willing to substitute between the good under consideration and such numeraire good implicitly depends on the underlying income distribution. In order that the isolation of partial equilibrium social decision from the rest of the economy is harmless, it is desired that allocation of surplus is independent of such underlying income distribution, hence the social choice should be independent of the choice of origin of quasi-linear utility functions. To be precise, let U ⊂ RI be the (compact) set of possible profiles of individual surpluses before redistribution, and let

4

The Assumption of No Income Effect …

H (U ) =

I 

si = max

i=1

u∈U

I 

91

ui

i=1

be the set of feasible allocations of the maximized social surplus. Then a social choice function f picks f (U ) ∈ H (U ). This is called a quasi-linear social choice function. Now the condition of being independent of the choice of origin will take the form of axiom called Independence of Utility Origin: f (U + {w}) = f (U ) + {w} for all U ⊂ RI and w ∈ RI . It is easy to see that this requirement is incompatible with maximizing a social welfare function except when we are totally indifferent between any allocation of maximized social surplus. It is seen by taking any w such that H (U + {w}) = H (U ), then the maximization in the same set of feasible allocations of maximized social surplus yields f (U ) = f (U + {w}), but this contradicts with f (U + {w}) = f (U ) + {w}. Theorem 4.2 There is no quasi-linear social choice function satisfying Independence of Utility Origin and coming from the maximization of some social welfare function.

Note 1. A similar idea is found in Miyake [3].

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References 1. Hicks, J.R. 1939. Value and capital: An Inquiry into Some Fundamental Principles in Economics. Oxford. 2. Marshall, A. 1920. Principle of Economics. London: Macmillan. 3. Miyake, Mitsunobu. 2006. On the applicability of Marshallian partialequilibrium analysis. Mathematical Social Sciences 52 (2): 176–196.

5 Is the Approximation Error Large or Small?

In order that maximization of consumer surplus is consistent with individual preference maximization, the assumption of no income effect is required. Since this assumption cannot be met in an exact way, it is natural to ask how large is the error in estimating welfare because of that. Here we follow Willig [1], which provides a bound on the approximation error. To illustrate, assume for a moment that income elasticity of demand for Good k ∂xk (p, w) w · ∂w xk (p, w) is constant, say equal to η. For simplicity, let us restrict attention to the case that η = 1. Then, by solving the differential equation dw dxk =η xk w with the initial condition w = w 0 , we obtain  w η xk (p, w) = xk (p, w 0 ) w0

© The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_5

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From Shepard’s lemma we have  η μ(p|p0 , w) ∂μ(p|p0 , w) 0 0 = xk (p, μ(p|p , w)) = xk (p, w ) ∂pk w0 hence we obtain the differential equation μ(p|p0 , w)−η d μ = (w 0 )−η xk (p, w 0 )dpk By integrating both sides from pk0 to pk1 and applying μ(p0 |p0 , w 0 ) = w 0 , we obtain 0 |p0 , w)1−η − (w 0 )1−η μ(pk1 , p−k

1−η

= (w 0 )−η



pk1 pk0

0 xk (pk , p−k , w 0 )dpk ,

which yields  0 |p0 , w) = w 0 μ(pk1 , p−k

Here let

 A=

1−η 1+ w0 pk1

pk0



pk1

pk0

 0 xk (pk , p−k , w 0 )dpk

0 xk (pk , p−k , w 0 )dpk ,

which is the change in consumer surplus. Then from Taylor expansion 1

(1 + t) 1−η ≈ 1 +

t ηt 2 + 1 − η 2(1 − η)2

1 1−η

.

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Is the Approximation Error Large or Small?

95

we obtain 0 |p0 , w 0 ) − μ(p0 |p0 , w 0 ) CV = μ(pk1 , p−k 0 |p0 , w 0 ) − w 0 = μ(pk1 , p−k

1 1 − η 1−η 0 =w 1+ A − w0 0 w

ηA2 A 0 ≈w 1+ 0 + − w0 w 2(w 0 )2 ηA2 =A+ . 2w 0

and 0 0 0 EV = μ(pk1 , p−k |pk1 , p−k , w 0 ) − μ(p0 |pk1 , p−k , w0 ) 0 , w0 ) = w 0 − μ(p0 |pk1 , p−k

1 1 − η 1−η 0 0 =w −w 1− A w0

ηA2 A 0 0 ≈w −w 1− 0 + w 2(w 0 )2 ηA2 =A− . 2w 0

Now drop the assumption of constancy of income elasticity of demand but assume it has upper bound η and lower bound η. Then, we obtain the inequalities η|A| η|A| C−A ≤ ≤ 0 2w |A| 2w 0 and

η|A| 2w 0

≤

A−E η|A| ≤ |A| 2w 0

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hk p0k + t T p0k

B C

x1

xk

Fig. 5.1 Hicksian deadweight loss and Marshallian deadweight loss

This means that the percentage error in approximation of income measure of welfare is small when income elasticity of demand is sufficiently small. However, Hausman [2] points out that percentage error in approximating deadweight loss can be large. Let p0 denote the original price level of the good, and consider taxing t per unit. Then in Fig. 5.1, equivalent variation, or Hicksian consumer surplus (HCS) is T + B, where T stand for the tax revenue, and change in consumer surplus (Marshallian consumer surplus, MCS is T + B + C). Then, the deadweight loss measured from HCS, called Hicksian deadweight loss (HDL), is B, whereas the deadweight loss measure from MCS, called Marshallian deadweight loss (MDL) is B + C. Thus, |HCS − MCS| = |HDL − MDL| = C holds. Although the percentage error of approximation of surplus T C +B tends to be smaller, the percentage error of approximation of deadweight loss CB may be large.

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References 1. Willig, Robert D. 1976. Consumer’s surplus without apology. The American Economic Review 66 (4): 589–597. 2. Hausman, Jerry A. 1981. Exact consumer’s surplus and deadweight loss. The American Economic Review 71 (4): 662–676.

6 Small Income Effects

In the previous chapters, we saw that the assumption of no income effect is necessary for the use of consumer surplus and the approximation error in using it as a proxy for equivalent and compensating variation is small when income effect is small. This leaves a substantive question: when can we assume that income effect is negligibly small? Marshall [1] argues that when the market for a single good under consideration is “small” then the “other things remain equal” assumption is approximately correct. The formal proof of this claim, however, had to wait for the work by Vives [2] in 1987.

6.1

Vives (1987)

Let n denote the number of goods, which tends to be arbitrarily large. Consider a sequence of prices and base incomes {( p n , w n )}, where p n is an n-dimensional price vector, which is uniformly bounded from above and below, and income w n increases at the same rate as n. Then, for each n consumption vector z n ∈ Rn++ must satisfy budget constraint n 

pkn z kn = w n .

k=1 © The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_6

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Then, for an incremental income change w we have n 

  pkn z kn + z kn = w n + w

k=1

which implies so-called the Engel aggregation condition n  k=1

pkn

∂z kn = 1. ∂w

Here, the distinction between base income and incremental change of income should be made clear. As the number of goods n tends to be large, the base income level is also supposed to be large at the same rate. On the other hand, the incremental change of income w stays with a constant order of magnitude. Therefore, as the number of goods tends to be large the impact of incremental change of income tends to be small. When all goods are normal, under suitable assumptions the income ∂z n

derivative of demand ∂wk uniformly converges to zero at rate 1/n. The idea is that when the sequence of prices { p n } is uniformly bounded from above and below, from the Engel aggregation condition the sum of income  ∂z n derivatives nk=1 ∂wk is uniformly bounded from above and below as well, ∂z n

and every ∂wk is shown to have the same degree of magnitude. Let us be more precise about this argument. Let n denote the number of goods, which tends to be arbitrarily large. The consumer is assumed to have a sequence of preference representations indexed by n, denoted by {U n }. For each n, U n (·) : Rn++ → R is defined over the n-dimensional positive consumption space. The sequence {U n } is assumed to satisfy the following properties. Assumption 6.1 1. For every n, U n : Rn++ → R is continuously differentiable and DU n (x) > 0 for all x ∈ Rn++ . 2. For every n, U n : Rn++ → R is twice continuously differentiable and D 2 U n (x) is negative definite for all x ∈ Rn++ .

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3. There exist nonincreasing functions φ, φ : R++ → R++ with φ(z) ≤ φ(z) for all z ∈ R and lim φ(z) = ∞, lim φ(z) = 0, z→∞

z→0

such that φ(xk ) ≤

∂U n (x) ≤ φ(xk ) ∂ xk

for all n, for all x ∈ Rn++ and k = 1, . . . , n. 4. For any z, z > 0 with z ≤ z there exist β, β > 0 with β ≤ β such that for all n, for all x ∈ [z, z]n , all the eigenvalues of D 2 U n (x) lie in [β, β]. The first two assumptions are straightforward. The third assumption states that the Inada-type boundary condition holds uniformly. Let us call it Uniform Inada Property. When the sequence of utility functions consists of additively separable ones, U n (x) =

n 

vkn (xk )

k=1

the condition is equivalent to say that there exist nonincreasing functions φ, φ : R++ → R++ satisfying the above conditions such that φ(xk ) ≤ (vkn ) (xk ) ≤ φ(xk ) for all n, for all x ∈ Rn++ and k = 1, . . . , n. The last condition roughly states that second derivative of the utility function is uniformly bounded from above and away from zero. To see this, consider that sequence consists of additively separable utility functions, then the condition is equivalent to say that there exist β, β > 0 with

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β ≤ β such that for all n, for all x ∈ [z, z]n it holds β ≤ |(vkn ) (xk )| ≤ β. A sequence of price-income pairs {( p n , w n )}, where p n ∈ Rn++ is an n-dimensional vector for each n, is supposed to satisfy the following assumptions Assumption 6.2 1. There exist p, p > 0 with p ≤ p such that for all n it holds p ≤ pkn ≤ p for all k = 1, . . . , n. 2. w n = O(n). Here is the result by Vives [2]. Theorem 6.1 Under the above assumptions, (i) the order of√magnitude of the Euclidian norm of the income derivative of demand is 1/ n. (ii) if preferences are representable by a sequence {U n } of additive separable or homothetic utility functions then the order of magnitude of the income derivative of every good is 1/n. (iii) the associate Slutsky matrices are nondegenerate. Let us get into some details of the proof. For each n, let x n ( p n , w n ) denote the solution to max u n (x) n x∈R++

subject to

pn · x = wn

The first is to establish that the sequence {x n ( p n , w n )} is uniformly bounded away from above and below. Lemma 6.1 There exist x ≥ x > 0 and such that for all n and for all k = 1, . . . , n it holds x ≤ x n ( p n , w n ) ≤ x.

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Proof Suppose that the uniform boundedness from above fails. Then without loss of generality, we can assume maxk=1,...,n xkn → ∞, after taking a subsequence if necessary. Then from Uniform Inada Property and because φ is nonincreasing we have ∂U n (x n ) ≤ min φ(xkn ) = φ min k=1,...,n k=1,...,n ∂ xk



 max

k=1,...,n

xkn

As maxk=1,...,n xkn → ∞, from the property of φ we have φ n n (maxk=1,...,n xkn ) → 0, hence mink=1,...,n ∂U∂ x(xk ) → 0. From the first-order condition, where λn is the Lagrange multiplier corresponding to each n, we have min

k=1,...,n

∂U n (x n ) = λn min pkn ≥ λn p, k=1,...,n ∂ xk

and therefore, we obtain λn → 0. From the first-order condition, we also have max

k=1,...,n

∂U n (x n ) = λn max pkn ≤ λn p, k=1,...,n ∂ xk

we obtain maxk=1,...,n ∂U∂ x(xk On the other hand, since n

max

k=1,...,n

n)

→ 0.

∂U n (x n ) ≥ min φ(xkn ) = φ( min xkn ) k=1,...,n k=1,...,n ∂ xk

it holds φ(mink=1,...,n xkn ) → 0. From the property of φ we have mink=1,...,n xkn → ∞. However, from the budget constraint w = n

n  k=1

pkn xkn

≥p

n  k=1

xkn ≥ n p min xkn , k=1,...,n

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we have min k=1,...,n xkn ≤

wn . np

This contradicts to w n = O(n). In a similar manner, we can prove the uniform boundedness away from zero.  Likewise, the corresponding sequence of Lagrange multipliers is uniformly bounded from above and away from zero. Lemma 6.2 There exist λ ≥ λ > 0 and such that for all n and for all k = 1, . . . , n it holds λ ≤ λn ( p n , w n ) ≤ λ. Proof From the first-order condition we get λn w n = λn

n 

pkn xkn =

k=1

n  ∂U n (x n ) k=1

∂ xk

xkn

which is rewritten into n 1  ∂U n (x n ) n · xk wn n ∂ xk n

λn =

k=1

From the previous lemma, together with w n = O(n) and Uniform Inada property, there exists constants c ≥ c > 0 such that cφ(x)x ≤ λn ≤ cφ(x)x for all n. Proof of the theorem (i) By differentiating the first-order condition DU n = λn p n



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105

by income we obtain D 2 U n Dw x n = Dw λn p n where D 2 U n is evaluated at x n ( p n , w n ), implying Dw x n = Dw λn (D 2 U n )−1 p n Since

it holds

( p n )t D w x n = 1 1 = ( p n )t Dw x n = Dw λn ( p n )t (D 2 U n )−1 p n

which implies Dw λn =

1 ( p n )t (D 2 U n )−1 p n

.

Therefore, we obtain D 2 U n Dw x n =

1 ( p n )t (D 2 U n )−1 p n

pn

Hence, we have Dw x n =

1 (D 2 U n )−1 p n ( p n )t (D 2 U n )−1 p n

and therefore, Dm x n = (D 2 U n )−1 p n | p n (D 2 U n )−1 p n |−1 By the uniform boundedness of the eigenvalues of Hessian, there exist μ ≥ μ > 0 such that p n 2 μ−1 ≤ | p n (D 2 U n )−1 p n | ≤ p n 2 μ−1

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and

p n μ−1 ≤ (D 2 U n )−1 p n ≤ p n μ−1

for all n, which implies μ p n μ

≤ Dm x n ≤

μ p n μ

√ √ Since p n ≤ p n ≤ p n, we get μ μ √ √ ≤ Dm x n ≤ pμ n pμ n (ii) To prove the second assertion, assume that for each n, U n has the form U n (x n ) =

n 

u n,k (xk )

k=1

Then the first-order condition reduces to (u nk ) (xk ) = λn pkn for every n and k = 1, . . . , n. Therefore, we obtain ∂ xkn ∂w ∂ xln ∂w

and hence

μp μp

for all n and k, l = 1, . . . , n.

=

≤

(u ln ) (xln ) pkn (u nk ) (xkn ) pln ∂ xkn ∂w ∂ xln ∂w

≤

μp μp

6

Thus, all

∂ xkn ∂w

Small Income Effects

107

have the same order of magnitude. Because n  k=1

pkn

∂ xkn =1 ∂w

and p ≤ pkn ≤ p hold for all n, such order of magnitude is 1/n. (iii) It suffices to show that absolute values of all eigenvalues of Slutsky matrix restricted to T pn = {z ∈ Rn : ( p n )t z = 0} for each n are uniformly bounded from above and away from zero. For a moment, fix n and forget about it. By taking the derivative of U (h( p, h)) = u it holds

D p h DU = 0.

From the first-order condition DU = λp, we obtain

D p hp = 0.

By taking the first-order condition DU = λp we obtain

D 2 U D p h = D p λp t + λI

By multiplying p to both sides from right, we obtain   0 = D 2 U D p hp = D p λp t + λI p which yields D pλ = −

λ p pt p

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Hence we obtain D2U D p h = −

λ pp t + λI pt p

which yields D p h = λ(D U ) 2

−1

Let A = (D U ) 2

Then it holds

−1





1 I − t pp t p p

1 I − t pp t p p





AD 2 U A = A.

Now let us be explicit about n. Since λn is uniformly bounded from above and away from zero, it suffices to show that absolute values of all eigenvalues of An restricted to T pn are uniformly bounded from above and away from zero. For any z ∈ T pn , it holds z t An z = z t An D 2 U n An z = y t D 2 U y with y = An z. Note that since z ∈ T pn it holds n −1

y = A z = (D U ) n

2



 1 t I − t pp z = (D 2 U n )−1 z p p

implying that y = 0. From the assumption of uniform boundedness of eigenvalues of Hessian, it holds |y t D 2 U y| μ≤ ≤μ yt y which yields μ≤

|z t An z| ≤μ z t (An )t An z

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109

for all z ∈ T pn . Then, it is seen that absolute values of all eigenvalues of

An restricted to T pn are in μ1 , μ1 . Notice that the fourth condition assumed by Vives is a cardinal property of representation. This makes us wonder what its ordinal content is. Hayashi [3] shows that when the slopes of wealth expansion paths are uniformly bounded from above and below on compacta, which presumes that all goods are normal, the order of magnitude of the income derivative of every good is 1/n. To be precise, given an n − 1 positive vector ψ, where its entries are indexed by 2, . . . , n, solve a system of equations n (x) = ψk , k = 1, . . . , n M RS1,k

which obtains a wealth expansion path 

n (x1 ) x2n (x1 ), . . . , xn−1



which is differentiable in x1 under the assumption of differentiable preferences. Now consider the slopes along the given wealth expansion path 

∂xn ∂ x2n , . . . , n−1 ∂ x1 ∂ x1



and assume that all the entries of such vectors are bounded away from above and below on compacta. Then, since ∂ xkn ∂w ∂ x1n ∂w

∂ xkn = ∂ x1

for all k = 2, . . . , n are uniformly bounded from above and below, all ∂ xkn ∂w are positive have the same order of magnitude. Since

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T. Hayashi n 

pkn

k=1

∂ xkn =1 ∂w

and p ≤ pkn ≤ p hold for all n, the order of magnitude of the income derivative of every good is 1/n.

6.2

Deadweight Loss

Let us continue to follow the argument by Vives [2] to see that as n tends to be large the Marshallian deadweight loss tends to approximate the Hicksian deadweight loss. Let E V ( p 1 | p 0 , w) = w − e( p 0 , v( p 1 , w)) denote the equivalent variation for price change from p 0 to p 1 . Then, the Hicksian deadweight loss is given as the difference between the equivalent variation and tax revenue. H DL( p 1 | p 0 , w) = E V ( p 1 | p 0 , w) − ( p 1 − p 0 )x( p 1 , w) Now let us restrict attention to the price change of a single good, Good k. The same argument holds for price changes for any fixed number of goods. Then for each n, the corresponding Marshallian deadweight loss is given by

p1 k n 0 MC S = xk ( pk , p−k , w)d pk pk0

and the Hicksian deadweight loss is given by

HC S = n

pk1 pk0

0 0 h k ( pk , p−k , v( pk1 , p−k , w))d pk .

6

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Small Income Effects

The approximation error is hence  1  pk

 |H C S n − MC S n | = 

 

0 , v( p 1 , p 0 , w)) − x ( p , p , w)d p  (h k ( pk , p−k k k −k k k −k 

 p0 k

p1   k  0 , v( p 1 , p 0 , w)) − x ( p , p , w) d p ≤  k h k ( pk , p−k k k −k k −k pk0

0 , v( p 1 , p 0 , w)) By integrating Slutsky equation, noting h k ( pk1 , p−k k −k 0 , w), we obtain = xk ( pk1 , p−k  

  0 0 0 , v( pk1 , p−k , w)) − xk ( pk , p−k , w) ≤ h k ( pk , p−k

pk pk0

   ∂ x (q , p 0 , w)  k k  −k  dqk  ∂w

 0 xk (qk , p−k , w)  

From the above  result, xk (q k , p−k√, w) ≤ x and there is a constant  , p−k ,w)  c > 0 such that  ∂ xk (qk∂w  ≤ c/ n. Hence   xc   0 0 0 , v( pk1 , p−k , w)) − xk ( pk , p−k , w) = √ | pk − pk0 | h k ( pk , p−k n and therefore xc |H C S n − MC S n | ≤ √ | pk1 − pk0 |2 2 n From (iii) of the theorem above it is seen that all diagonal entries of Slutsky matrix are in absolute values uniformly bounded from above and away from zero. Therefore, the slope of Hicksian compensated demand ∂h n is uniformly bounded away from zero, and we have | ∂ pkk | ≥ b for some

b > 0, which implies |H DL n | ≥ b2 | pk1 − pk0 |2 . Thus, from |H C S n − MC S n | = |H DL n − M DL n | we obtain xc |H DL n − M DL n | ≤ √ . n |H DL | b n

112

6.3

T. Hayashi

Asymptotic Quasi-linearity

Given n commodities, consider that one commodity, say j, is to be the object of partial equilibrium analysis and the remaining n−1 commodities are to be aggregated, and let n, j be the preference induced over pairs of consumption of the commodity under analysis and income transfer to be allocated to the other commodities. For (x, a), (y, b) ∈ R++ × (−w n , ∞), define (x, a) n, j (y, b) by

U n x, z n, j (x, a) ≥ U n y, z n, j (y, b) ,

where z n, j (x, a) ∈ Rn−1 ++ is the solution to   max U n x, z − j

z − j ∈Rn−1 ++

subject to



pkn z k = w n + a

k = j

and similarly for z n, j (y, b). See Fig. 6.1 for how the induced preference typically looks like. We that the sequence of induced 2-good preferences {n, j } converges and the limit preference is quasi-linear, as is illustrated in Fig. 6.2. Hayashi [3] shows that n, j is asymptotically quasi-linear in the sense that the derivative of the marginal rate of substitution with respect to a vanishes as n tends to infinity. Notice again that w n tends to infinity, which means that income transfer a tends to be relatively very small compared to it.

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Small Income Effects

113

a

x

−wn

Fig. 6.1 Induced preference

Fix n and j, where j = 1 without loss of generality, and consider the maximization problem   max U n x, z − j

z − j ∈Rn−1 ++

subject to



pkn z k = w n + a

k = j

From the assumptions made on the finite-dimensional subspaces, the above maximization problem has a unique solution in the interior. Hence, we can resort to the first-order condition: there exists λn, j > 0 such that  ∂ n U x, z − j = λn, j pk ∂z k

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T. Hayashi a

a

x

−→

x

−wn

Fig. 6.2 Asymptotic preference

for all k = j. From the second-order differentiability assumption, the solution, denoted z n, j (x, a), is differentiable in (x, a). Also, we have the corresponding Lagrange multiplier as a differentiable function of (x, a), hence we denote it by λn, j (x, a). By differentiating the budget equation by x, we have  k = j

n, j

∂z (x, a) pk k = 0. ∂x

By differentiating the budget equation by a, we have  k = j

Now let

n, j

pk

∂z k (x, a) = 1. ∂a

V n, j (x, a) = U n x, z n, j (x, a)

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115

be the indirect utility function given by the conditional optimal consumption. Then, we have   ∂ n ∂ V n, j (x, a) = U x, z − j  ∂x ∂x z − j =z n, j (x,a)    ∂    + U n x, z − j  ∂z k  k = j n, j z − j =z

n, j

∂z k (x, a) (x,a)

∂x

  ∂z n, j (x, a)  ∂ n n, j  U x, z − j  + λ (x, a) pk k = ∂x ∂x z − j =z n, j (x,a) k = j   ∂ n U x, z − j  = ∂x z − j =z n, j (x,a)

and ∂ V n, j (x, a) ∂a

   ∂   n U x, z − j  = ∂z k  k = j = λn, j (x, a)



pk

n, j

z − j =z n, j (x,a) n, j ∂z k (x, a)

∂z k (x, a) ∂a

∂a

k = j

= λn, j (x, a) Thus, we obtain the characterization of the induced preference. Proposition 6.1 Given n, the marginal rate of substitution of income transfer for the neighboring good j, at (x, a) ∈ R++ × (−w n ∞), takes the form ∂ V n, j (x, a)  ∂ V n, j (x, a) ∂x ∂a    ∂ n 1 · U x, z − j  = n, j λ (x, a) ∂ x z − j =z n, j (x,a)    ∂ n x, z  − j z − j =z n, j (x,a) ∂x U 1 = n ·   ∂ p1 U n x, z 

S n, j (x, a) =

∂z 1

−j

z − j =z n, j (x,a)

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T. Hayashi

=

  1 n · M RS x, z −j  1, j z − j =z n, j (x,a) p1n

From the above we obtain   n n, j ∂ S n, j (x, a) 1 ∂ M RS1, j x, z (x, a) = n · ∂a p1 ∂a   n, j n n, j 1  ∂ M RS1, j x, z (x, a) z k (x, a) = n · p1 ∂z k ∂a k = j

z

n, j

(x,a)

Since the order of magnitude of k ∂a is 1/n under the assumption that slopes of wealth expansion paths are uniformly bounded from above and below, we obtain the following. Let o(n) denote a sequence such that limn→∞ o(n)/n = 0. Then, the result below says that preference induced over the two goods is asymptotically quasi-linear, as marginal rate of substitution tends to be asymptotically constant in Good 2, income transfer. Theorem 6.2 Assume the condition under which the order of magnitude of the income derivative of every good is 1/n. Also, assume that    n    ∂ M RS1, j x, z − j    = o(n)   ∂z k k = j

for every compact set bounded from below. Then, it holds lim

  n, j  ∂ S (x, a)  =0 sup   ∂a

n→∞ (x,a)∈C

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Note that the additional condition is immediately met under separable preference, as       n   ∂ M RS n x, z − j    ∂ M RS1, j x, z − j  1, j    =      ∂z k ∂z 1 k = j

=−

(v n ) (z j )(v1n ) (z 1 ) ((v n ) (z 1 ))2

which is bounded away from above and below on compacta.

6.4

Smallness of a Commodity

The Vives approach, however, does not allow us to handle eventually (countably) infinitely many commodities, since (i) it assumes roughly that all commodities have the uniform degree of utility weight, which cannot be true when there are indeed countably infinitely many commodities because if so the entire utility function cannot take a finite value; and (ii) it assumes that income increases at the same rate as the number of commodities, which we cannot think of literally in the limit. Let us consider, how we can get an exact relationship in the limit, between preference in the general equilibrium setting and the notion of willingness to pay in the partial equilibrium setting. Here we take a reverse direction, following Hayashi [4]. Present the whole set of commodities in the outset, which is a continuum, and subdivide it into many pieces so that each piece tends to be arbitrarily small. The continuum assumption might look odd, but it applies not only to the case of finely differentiated commodities but also to resource allocation under uncertainty with a continuum of states and intertemporal resource allocation with continuous time. Also, more importantly, it is a reasonable framework for precisely describing what we mean by “negligible.” Consider that the set of commodity characteristics is given as a continuum, say the unit interval [0, 1]. Let μ be the Lebesgue measure over [0, 1]. Then, a price system is given as a density function p : [0, 1] → R++ with

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suitable mathematical properties, and income w > 0 is given as a fixed number. We formulate the process of subdivision in the form a sequence of finite partitions {J n } of the set of commodity characteristics, which tends to be finer and finer for larger n and converges to the finest partition of singletons. For simplicity, we consider that increasing partition is generated by binary expansions,       n   n 1 1 2 2 − 2 2n − 1 2 −1 n J = 0, n , n , n , . . . , , ,1 , 2 2 2 2n 2n 2n for each n. At each step of the subdivision, we consider that the consumer is given a finite number of commodities, namely, |J n | commodities. That is, given n, each K ∈ J n is taken to be one commodity so that the consumption amount is constant over K , denoted by z nK let’s say, and the price of  n subdivision K is given by p K = K p(t)dμ(t). Then, the budget constraint takes the form 

p nK z nK = w

K ∈J n

Let w be the income change then it satisfies 

  p nK z nK + z nK = w + w,

K ∈J n

which implies the corresponding version of Engel aggregation condition  K ∈J n

p nK

∂z nK = 1. ∂w

The budget constraint and the Engel aggregation condition are rewritten into  pn K z nK μ(K ) = w μ(K ) n K ∈J

6

and

 K ∈J n

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119

∂z n p nK · K μ(K ) = 1 μ(K ) ∂w

pn

respectively, where μ(KK ) is the average of prices of the elements of K . Because differentiation of integration a function is the original function itself, as the subdivisions finer and finer the vector of average we make

prices

pK μ(K ) K ∈J n

converges to the density p, and the budget constraint

converges to the integral form

1

p(t) f (t)dμ(t) = w

0

where consumption bundle is given as a function f : [0, 1] → R++ , and the Engel aggregation condition converges to its continuum version

1

p(t) 0

∂ f (t) dμ(t) = 1. ∂w ∂z n

Notice that income effect on each subdivision ∂wK converges to the density f (t) on its element t, a positive term ∂∂w , and does not vanish. However, the ∂z n

income effect adjusted to its smallness ∂wK μ(K ) is negligibly small as its f (t) limit is ∂∂w dμ(t). This is because the income change associated with having one extra unit of subdivision K is w · μ(K ) instead of w, as its limit at commodity element t is w · dμ(t) instead of w. Let U be a representation of preference over the entire consumption space with the continuum of commodity characteristics. Given n, let U n be n the restriction of U on to the finite-dimensional subspace R|J | , defined by ⎞ ⎛  z K 1K ⎠ U n (z) = U ⎝ K ∈J n

120

T. Hayashi |J n |

for z = (z K ) K ∈J n ∈ R++ . Also, given n and J ∈ J n to be fixed, n (x, z −J ) ∈ R|J | denotes the vector such that x is its J -component n z −J ∈ R|J |−1 refers to the rest of the entries. Given n, pick J ∈ J n to be the object of partial equilibrium analysis in the approximate sense. In the limit, the set J is supposed to shrink to a point. Let x be the consumption amount which is constant over J . Let a be the amount of income transfer which is accompanied with the consumption of each element of J . Since the mass of the piece is μ(J ), the total income transfer which is accompanied with the consumption of the commodity piece is aμ(J ). Because income is fixed to be finite here, this adjustment corresponds to the nature that income transfer is relatively very small compared to the pool of income. Thus, given n, J ∈ J n , the preference relation induced over pairs of consumption and income transfer, denoted n,J , is defined through Hicksian aggregation: (x, a) n,J (y, b)

w hold for (x, a), (y, b) ∈ R++ × − μ(J , ∞ if and only if )

U n x, z n,J (x, a) ≥ U n y, z n,J (y, b) , |J n |−1

where z n,J (x, a) = (z n,J K (x, a)) K ∈J n \{J } ∈ R++ maxn

|J |−1 z −J ∈R++

subject to

is the solution to

U n (x, z −J ) 

p K z K = w + aμ(J )

K ∈J n \{J }

and similarly for z n,J (y, b). See Fig. 6.3 for how the induced preference typically looks like. Then, the sequence of induced 2-good preferences {n,J } converges and the limit preference is quasi-linear, as is illustrated in Fig. 6.4. This is a stronger result in the sense that the previous one shows only that

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121

a

x

w − µ(J)

Fig. 6.3 Induced preference

the corresponding sequence exhibits asymptotic constancy of MRS with respect to income transfers but there may not be a limit preference. The rest of the section provides a more precise account of this limit argument. Let T = [0, 1] denote the unit interval,  be the family of Lebesgue measurable sets, and μ be the Lebesgue measure. Let L ∞ (T ) be the space of essentially bounded measurable functions from T to R, which is endowed with the sup norm. Let ∞ L∞ +++ (T ) = { f ∈ L (T ) : ∃l > 0, a.e. t ∈ T, f (t) ≥ l}

be the set of vectors which are bounded away from below, and it is taken to be the consumption space. See appendix for more mathematical details of the consumption space.

122

T. Hayashi a

a

x

−→

x

w − µ(J)

Fig. 6.4 Limit preference

Let U : L ∞ +++ (T ) → R be a representation of the preference. Here, we list the basic assumptions on the representation U . The assumptions below involve some cardinal information about the representation. One may write them down in terms of marginal rate of substitution as is done in Hayashi [3]. Regular Preference: (i) U : L ∞ +++ (T ) → R is norm-continuous and Frechet differentiable. Moreover, DU ( f ) ∈ L 1+++ (T ) for all f ∈ L ∞ +++ (T ), and the 1 (T ) is continuous in the following (T ) → L mapping DU (·) : L ∞ +++ sense: given any compact set C ⊂ Rm , for any sequence of functions ν from C to L ∞ +++ (T ), denoted by { f }, and a function from C to ν L∞ +++ (T ) denoted by f , if { f } weak-∗ converges to f uniformly on ν C then DU ( f ) weakly converges to DU ( f ) uniformly on C. (ii) U : L ∞ +++ (T ) → R is strictly quasi-concave. We make some assumptions about the preference induced on the finitedimensional subspaces generated by {J n }.

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Regular Preference on Finite Dimensions: (i) For all n, the restriction of U onto the finite-dimensional subspace n |J n | R|J | , denoted U n : R++ → R, which is defined by ⎛ U n (z) = U ⎝



⎞ z K 1K ⎠

K ∈J n |J n |

for z = (z K ) K ∈J n ∈ R++ , is twice continuously differentiable. (ii) Denote the first derivative of U n by DU n . Then, |J n |

DU n (z) ∈ R++ |J n |

for all z ∈ R++ . (iii) Denote the second derivative of U n by D 2 U n . Then, for all z ∈ |J n | R++ , the (|J n | + 1) × (|J n | + 1) matrix  H (z) = n

D 2 U n (z) DU n (z)t DU n (z) 0



is invertible. We assume that the Inada-type condition holds ina uniform manner across n, which is parallel to what Vives [2] assumes for increasing numbers of commodities. Uniform Inada Property: There exist nonincreasing functions φ, φ from R++ to R++ such that (i) φ(y) ≤ φ(y) for all y ∈ R++ ; (ii) φ(y) → ∞ as y → 0 and φ(y) → 0 as y → ∞; |J n |

(iii) for all n, z = (z K ) K ∈J n ∈ R++ and K ∈ J n , φ(z K ) ≤

∂U n (z)  μ(K ) ≤ φ(z K ). ∂z K

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The last basic assumption is about prices and the base income level. Wealth and Prices: (i) w > 0; (ii) p ∈ L 1+++ (T ) and there exist p, p with 0 < p < p such that p(t) ∈ [ p, p] for almost all t ∈ T .

6.4.1 Hicksian Aggregation Fix n and J ∈ J n , and consider the maximization problem maxn

|J |−1 z −J ∈R++

U n (x, z −J ) 

subject to

p K z K = w + aμ(J )

K ∈J n \{J }

From the assumptions made on the finite-dimensional subspaces, the above maximization problem has a unique solution in the interior (see Debreu [5], Mas-Colell [6]). Hence, we can resort to the first-order condition: there exists λn,J > 0 such that ∂ U n (x, z −J ) = λn,J p K ∂z K  for all K ∈ J n \ {J }, where p K = K p(t)dμ(t). From the secondorder differentiability assumption, the solution, denoted z n,J (x, a) = (z n,J K (x, a)) K ∈J n \{J } , is differentiable in (x, a). Also, we have the corresponding Lagrange multiplier as a differentiable function of (x, a), hence we denote it by λn,J (x, a). By differentiating the budget equation by x, we have  K ∈J n \{J }

∂z n,J (x, a) pK K = 0. ∂x

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125

By differentiating the budget equation by a, we have 

pK

K ∈J n \{J }

Now let V

n,J

∂z n,J K (x, a) = μ(J ). ∂a

(x, a) = U

n

x, z

n,J

(x, a)

be the indirect utility function defined through Hicksian aggregation. Then, we have   ∂ n ∂ V n,J (x, a) = U x, z −J  ∂x ∂x z −J =z n,J (x,a) +

 K ∈J n \{J }

∂ U ∂z K

n

   x, z −J  

∂z n,J K (x, a) z −J =z n,J (x,a)

∂x

   ∂z n,J (x, a) ∂ n U x, z −J  + λn,J (x, a) pK K ∂x ∂x z −J =z n,J (x,a) K ∈J n \{J }   ∂ n U x, z −J  = ∂x z −J =z n,J (x,a)

=

and ∂ V n,J (x, a) ∂a

   ∂ n U (x, z −J ) = ∂z K  K ∈J n \{J } 

= λn,J (x, a)

 K ∈J n \{J }

pK

z −J =z n,J (x,a) ∂z n,J K (x, a)

∂z n,J K (x, a) ∂a

∂a

= λn,J (x, a)μ(J ) Thus, we obtain the characterization of the induced preference.

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Proposition 6.2 Given n, the marginal rate of substitution of income transfer

w for the neighboring good J ∈ J n , at (x, a) ∈ R++ × − μ(J ) , ∞ , takes the form

∂ V n,J (x, a)  ∂ V n,J (x, a) ∂x ∂a  ∂ n  z U (x, ) −J 1 ∂x z −J =z n,J (x,a) · . = n,J λ (x, a) μ(J )

S n,J (x, a) =

Remark 6.1 Note that when preference is represented in additively separable form

v( f (t), t)dμ(t) U( f ) = T

the MRS formula above reduces to S

n,J

1 (x, a) = n,J · λ (a)



∂ J ∂ x v(x, t)dμ(t)

μ(J )

Next, we derive comparative statics properties of the conditional demand. From the second-order argument, we have 

2 U n (x, z n t D−J −J ) D−J U (x, z −J ) n D−J U (x, z −J ) 0



t dz −J dλ



 =

D J D−J U n (x, z −J )t 0t 0 μ(J )



dx da

 ,

where D J refers to the derivative with regard to z J and D−J refers to the derivative with regard to z −J . Given n and J ∈ J n , let H n,J (z) be the |J n | × |J n | matrix obtained by deleting the J -row and the J -column of H n (z). That is,  H

n,J

(z) =

2 U n (z) D U n (z)t D−J −J D−J U n (z) 0

 .

For each K ∈ J n \ {J }, let HKn,J (z) be the matrix obtained by replacing the K -column of H n,J (z) by (D J D−J U n (z), 0)t . Also, for each K ∈

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127

n,J (z) be the matrix obtained by replacing the K -column J n \ {J }, let H K of H n,J (z) by (0, μ(J ))t . Then, by Cramer’s rule, we have     n,J  HK (x, z −J ) dz K . =  n,J  H (x, z −J ) dx     n,J  HK (x, z −J ) dz K  =  n,J  H (x, z −J ) da

and

for each n, J ∈ J n and K ∈ J n \ {J }. Thus, we can characterize the sensitivity of conditional demand by means of the differential properties of the preference. Here, we assume that the sensitivity terms given above are uniformly bounded as the consumption vectors are uniformly bounded. Uniform Boundedness of Sensitivity: For any fixed z, z > 0, there exist α, α and β, β such that    n,J  (z) H  K  α   n,J   α  H (z)    n,J   HK (z) β   n,J   β  H (z)

and

for all n, z ∈ [z, z]J and J, K ∈ I n . n

   n,J  HK (z)

Remark 6.2 When the preference is additively separable we have | H n,J (z)|

= 0 for all n, z ∈ [z, z]J and J, K ∈ I n , hence the first assertion of the assumption is met in a straightforward manner. n

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T. Hayashi

Here, we are assuming the condition which just makes income effects uniformly bounded, and not assuming that the income effect on each commodity piece vanishes, though it turns out to be true eventually. Also note that the above conditions are stated directly as a property of the preference, the primitive, not as a property of the derived conditional demand function.

6.4.2 The Limit Theorem Now consider making the subdivision finer and finer. We show that the induced 2-good preference converges to a quasi-linear preference (see Fig. 6.4). Hereafter, fix τ ∈ T arbitrarily and let J = J n (τ ) for each n, and fix a compact set C ⊂ R++ × R. Also, for all sufficiently large n’s, let n { f n,J (τ ) } be the sequence of functions from C to [z1, z1] ⊂ L ∞ +++ (T ) given by f n,J

n (τ )

(x, a) = x1 J n (τ ) +



n,J n (τ )

zK

(x, a)1 K

K ∈J n \{J n (τ )}

for each n and (x, a) ∈ C. The proof follows from two lemmata. Lemma 6.3 The sequence { f n,J (τ ) } has a convergent subsequence k(n) { f k(n),J (τ ) } with the limit f ∈ [z1, z1] which is constant over (x, a), in the sense that n

sup | f k(n),J

(x,a)∈C

for all q ∈ L 1 (T ).

k(n) (τ )

(x, a), q −  f, q| → 0

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Moreover, f is the unique solution to the problem (we call it unconditional problem) U( f )

p(t) f (t)dμ(t) = w. subject to max

f ∈L ∞ +++ (T )

T

Lemma 6.4 The corresponding subsequence of {λn,J (τ ) (x, a)} converges to λ > 0 uniformly on C, which is the Lagrange multiplier associated with the solution f given above. We make the following assumption with regard to the limit of shrinking neighborhoods. n

Continuous Marginal Utility Density: For almost every τ ∈ T , for any D ⊂ R++ and f ∈ L ∞ +++ (T ), there exists U (x, τ ; f ) such that   ∂     sup  U x1 J + f 1T \J − U (x, τ ; f )μ(J ) = o(μ(J )), ∂x

x∈D

where J is any interval containing τ with μ(J ) > 0. Moreover, U (x, τ ; f ) is continuous in f in the following sense: Given any compact set C ⊂ Rm , if a sequence of functions from C to L +++ (T ), denoted { f ν }, weak-∗ converges to f uniformly on C, then   sup sup U (x, τ ; f ν (s)) − U (x, τ ; f (s)) → 0. s∈C x∈D

Remark 6.3 In the additive separable case, this is nothing but the result of the Lebesgue differentiation theorem which is applied to 

∂ J ∂ x v(x, t)dμ(t)

μ(J )

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which yields convergence to U (x, τ, f ) =

∂v(x, τ ) . ∂x

Here is the limit theorem. Theorem 6.3 Given almost every τ ∈ T and any compact set C ⊂ R++ × R, there exists a subsequence of {n}, denoted {k(n)}, such that sup |S n,J

k(n) (τ )

(x,a)∈C

where S τ (x) ≡

(x, a) − S τ (x)| → 0

1 · U (x, τ ; f ), λ

f is the unique solution to the problem U( f )

p(t) f (t)dμ(t) = w subject to max

f ∈L ∞ +++ (T )

T

and λ is the corresponding Lagrange multiplier. Let τ be the preference relation over R++ × R which corresponds to the limit. Then by integrating the above marginal rate of substitution formula in the limit we can represent it by (x, a) τ (y, b)

⇐⇒



1 x 1 y U (z, τ ; f )dz + a ≥ U (z, τ ; f )dz + b. λ 0 λ 0

Here are some examples of how to calculate marginal utility density. Example 6.1 Consider the weighted expected utility preference (Chew and MacCrimmon [7]) represented in the form  v( f (t))dν(t) U( f ) = T , T w( f (t))dν(t)

6

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Small Income Effects

where ν is absolutely continuous with respect to μ. By direct calculation, we have  x1 J + f 1T \J = μ(J )



    ν(J ) v (x) w(x)ν(J ) + T \J w( f (t))dμ(t) − v(x)ν(J ) + T \J v( f (t))dμ(t) w (x) . ·

2  μ(J ) w(x)ν(J ) + T \J w( f (t))dμ(t) ∂ ∂x U



Hence, the marginal utility density of commodity τ ∈ T at quantity x is U (x, τ ; f ) =

  dν(τ ) v  (x) w( f (t))dν(t) − w (x) v( f (t))dν(t)  · dμ(τ ) ( w( f (t))dν(t))2

Notice that in the expected utility case with w being constant, say 1, it dν(τ )  reduces to dμ(τ ) v (x). Example 6.2 Let T = [0, T ]. Consider Uzawa preference (Uzawa [8]) represented in the form

T

U( f ) =

u( f (t))e−

t 0

β( f (s))ds

dt.

0

By direct calculation, we have ∂ ∂x U



  

T t  inf J x1 J + f 1T \J − β( f (s))ds u( f (t))e sup J dt . = e− 0 β( f (s))ds u  (x) − β  (x) μ(J ) sup J

Hence the marginal utility density of commodity τ ∈ T at quantity x is U (x, τ ; f ) = e−

τ 0

 β( f (s))ds

u  (x) − β  (x)

T τ

u( f (t))e−

t τ

 β( f (s))ds dt

.

Notice that in the additive case with β being a constant it reduces to

e−βτ u  (x).

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T. Hayashi

Application: General Equilibrium and Partial Equilibrium

Here, we can provide an exact relationship between general equilibrium and partial equilibrium. Note that each individual i’s willingness to pay depends on the price system and income at the general equilibrium level, ( p, wi ). Consumer i’s marginal willingness to pay for an extra one unit of commodity τ ∈ T at consumption amount x is given by Siτ (x; p, wi ) =

1 · U (x, τ ; f i ( p, wi )), λi ( p, wi )

where f i ( p, wi ) is the solution to the problem Ui ( f )

p(t) f (t)dμ(t) = wi subject to max

f ∈L ∞ +++ (T )

T

and λi ( p, wi ) is the corresponding Lagrange multiplier. To illustrate, consider a pure exchange economy in which consumers’ initial endowments are given by ωi ∈ L ∞ +++ (T ), i = 1, . . . , I . Proposition 6.3 Maintain the previous assumptions, and assume that for each i = 1, . . . , I , for almost all τ ∈ T and every f ∈ L ∞ +++ (T ), Ui (x; τ, f ) is decreasing in x. Then, an interior allocation ( f i )i=1,...,I constitutes competitive general equilibrium under price system p and if and only if for almost all τ ∈ T , ( f i (τ ))i=1,...,I satisfies Siτ ( f i (τ ); p,  p, ωi ) = p(τ ) for every i = 1, . . . , I .

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Proof (“=⇒” part) The interior equilibrium condition tells that DUi ( f i )(τ ) = λi p(τ ) for all i = 1, . . . , I and almost all τ ∈ T , where λi is the Lagrange multiplier for the problem max Ui (g) g

subject to g ∈

L∞ +++ (T ),

p(t)g(t)dμ(t) =  p, ωi . T

Pick almost any τ ∈ T , then since Ui ( f i (τ ), τ ; f i ) = DUi ( f i )(τ ) for each i, we have Ui ( f i (τ ), τ ; f i ) = p(τ ) λi for each i. (“⇐=” part) Given p and ( p, ωi )i=1,...,I , suppose fi ) Ui ( f i (τ ), τ ;  = p(τ )  λi for each i and almost all τ ∈ T , where  f i is the interior solution to the problem max Ui (g) g

subject to g ∈

L∞ +++ (T ),

p(t)g(t)dμ(t) =  p, ωi  T

and  λi is the corresponding Lagrange multiplier. The interior optimality condition for  f i is that f i )(τ ) =  λi p(τ ) DUi ( 

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T. Hayashi

for almost all τ ∈ T , hence we have f i ) = DUi (  f i )(τ ) Ui ( f i (τ ), τ ;  for almost all τ ∈ T . From the assumption made above, this implies f i (τ ) =  f i (τ ) for almost all τ ∈ T and f i is optimizing under the budget constraint.  This also implies that competitive equilibrium allocation is viewed as the solution to an unconstrained maximization problem for the integral of consumer surplus across commodities. Proposition 6.4 Maintain the previous assumptions. Then, an interior allocation ( f i )i=1,...,I constitutes competitive general equilibrium under price system p if and only if it is a solution to max

g 1 ,...,g I

6.6



I

 i=1

T

0

gi (t)

 Sit (x; p,  p, ωi )d x − p(t)gi (t) dμ(t).

Appendix: Mathematical Details on L ∞ (T )

Denote the norm dual of L ∞ (T ) by L ∞ (T )∗ . It is known that the norm dual of L ∞ (T ) is the set of finitely additive signed measured over T endowed with the total variation norm, which is denoted by ba(T ). Thus, L ∞ (T )∗ = ba(T ). Let L 1 (T ) be the space of Lebesgue integrable functions from T to R, which is endowed with the integral norm. It is known that L 1 (T ) can be viewed as a subset of L∞ (T )∗ = ba(T ). There the dual operation takes the form  p, f  = T p(t) f (t)dμ(t), where f ∈ L ∞ (T ) and p ∈ L 1 (T ). Given anintegrable function p : T → R and a measurable set K ∈ , let p K = K p(t)dμ(t). 1 ∗ ∞ It is known the dual operation given by  that L (T ) = L (T ), with ∞  f, p = T f (t) p(t)dμ(t) where f ∈ L (T ) and p ∈ L 1 (T ). Hence, one can consider weak convergence in L 1 (T ) and weak-∗ convergence in

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L ∞ (T ). Say that a sequence in L 1 (T ), denoted { p ν }, weakly converges to p if  f, p ν  →  f, p for all f ∈ L ∞ (T ). Say that a sequence in L ∞ (T ), denoted { f ν }, weak-∗ converges to f if  f ν , p →  f, p for all p ∈ L 1 (T ). Let C ⊂ Rm be a compact set and consider a sequence of functions from C to L 1 (T ), denoted by { p ν }. Say that { p ν } weakly converges to p, a function from C to L 1 (T ), uniformly on C if sup | f, p ν (s) − p(s)| → 0 s∈C

for all f ∈ L ∞ (T ). Also, consider a sequence of functions from C to L ∞ (T ), denoted by { f ν }. Then say that { f ν } weak-∗ converges to f , a function from C to L ∞ (T ), uniformly on C if sup | f ν (s) − f (s), p| → 0 s∈C

for all p ∈ L 1 (T ).

References 1. Marshall, A. 1920. Principle of Economics. London: Macmillan. 2. Vives, X. 1987. Small income effects: A Marshallian theory of consumer surplus and downward sloping demand. Review of Economic Studies 54: 87–103. 3. Hayashi, T. 2008. A note on small income effects. Journal of Economic Theory 139: 360–379. 4. Hayashi, T. 2013. Smallness of a commodity and partial equilibrium analysis. Journal of Economic Theory 148: 279–305. 5. Debreu, G. 1972. Smooth preferences. Econometrica 40 (4): 603–615. 6. Mas-Colell, A. 1985. The theory of general economic equilibrium: A differentiable approach, No. 9. Cambridge: Econometric Society Monographs.

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7. Chew, S.H., and K.R. MacCrimmon. 1979. Alpha-nu choice theory; A generalization of expected utility theory. Working Paper No. 686, University of British Columbia, Faculty of Commerce and Business Administration, Vancouver. 8. Uzawa, H. 1968. Time preference, the consumption function and optimum asset holdings, in wolfe, editor, value, capital and growth. Papers in honor of Sir John Hicks, Chigaco, Aline.

7 Partial Equilibrium Welfare Analysis Under Uncertainty

7.1

Expected Consumer Surplus and Its Aggregate

The role of (aggregate) expected consumer surplus as an efficiency measure is prominent in many fields which adopt the partial equilibrium framework under uncertainty. To recall the definition, suppose there are S states of the world, let (x1 , . . . , x S ) denote the vector of an individual’s state-contingent consumptions of the commodity under consideration, where xs denotes his consumption of the good at state s = 1, . . . , S. Let (a1 , . . . , a S ) denote the vector of state-contingent income transfers to him, where as denotes ex-post transfer at state s = 1, . . . , S. Then, the expected consumer surplus for the given individual takes the form S 

(v(xs ) + as ) πs

s=1

where (π1 , . . . , π S ) denotes the probability vector. The use of expected consumer surplus has a strange implication, however, that price instability is good (see, for example, Waugh [1] and Massell © The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_7

137

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T. Hayashi

[2]). Consider that inverse demand curve is linear or that it is locally approximated linearly. Say it is p(x) = 1 − x. Then consumer surplus given price p is (1 − p)2 /2, which is convex in p and implies p being riskier is good. Even when the inverse demand function is non-linear, from the Taylor-expansion argument for inducing a small risk on price increase expected consumer surplus. This makes us wonder what is exactly the preference condition which allows this operation. The usual textbook/classroom remark for this is that it relies on two assumptions: (1) no income effect in the sense that the marginal rate of substitution of income transfer at state s and consumption of the good contingent on state s is independent of as ; (2) risk neutrality in the sense that evaluation of uncertain prospects in the above form depends only on the expectation of consumer surplus S s=1 (v(x s ) + as ) πs without any further adjustment, or in other words the marginal rates of substitution between income transfers at different states are constant.

7.2

The Preference Condition for Expected Consumer Surplus

To understand the above points, first we follow the argument by Rogerson [3], who considers preference over probability distribution of price-income pairs in ex-post spot markets, basically represented in the expected utility form  U (F) = v( p, w)d F, in which V plays the role of both indirect utility function and the vonNeumann/Morgenstern index.1 As a function defined over price-income pairs it is supposed to play the role of the indirect utility function in the ex-post spot markets. At the same time, it describes the consumer’s risk attitude toward price-income uncertainty. We should note also that there are spot markets only.

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On the other hand, consumer surplus in spot market for Good k under price-income pair ( pk , p−k , w) is given by  C S( pk , p−k , w) =

∞ pk

xk (qk , p−k , w)dqk

and its expectation is given by  EC S(F) =

C S( pk , p−k , w)d F( pk , p−k , w)    ∞ xk (qk , p−k , w)dqk d F( pk , p−k , w) = pk

First result is a negative one, saying that expected consumer surplus cannot be consistent with maximizing any underling preference when all probability distributions over prices of all goods and income have to be considered. Theorem 7.1 There is no preference which allows representation by ECS over all lotteries over ( pk , p−k , w) Proof Since EC S represents the consumer’s preference in the expected utility form, from cardinal equivalence of von-Neumann/Morgenstern index there exists a positive constant a and a constant b such that C S( pk , p−k , w) = av( pk , p−k , w) + b holds for all ( pk , p−k , w). Then, from the definition of C S, we obtain  ∞ ∂ ∂C S( pk , p−k , w) = xk (qk , p−k , w)dqk ∂ pk ∂ p k pk = −xk ( pk , p−k , w)

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On the other hand, from Roy’s identity we have ∂v( pk , p−k , w) ∂ (av( pk , p−k , w) + b) = −a xk ( pk , p−k , w) ∂ pk ∂w Therefore, for arbitrary ( pk , p−k , w) it holds −xk ( pk , p−k , w) = −a

∂v( pk , p−k , w) xk ( pk , p−k , w), ∂w

, p−k ,w) which implies that ∂v( pk∂w is constant. ∂ V ( pk , p−k ,w) is a cardinal property, and it Notice that constancy of ∂w requires not only that there is no income effect on all goods but also the consumer is risk-neutral with regard to random variation of income.

∂v( pk , p−k , w) = α = 1/a ∂w By integrating we obtain v( pk , p−k , w) = αw + β( pk , p−k ) where β( pk , p−k ) is a function independent of w. However, from homogeneity of degree zero of v, we have v(λpk , λp−k , λw) = αλw + β(λpk , λp−k ) = v( pk , p−k , w) = αw + β( pk , p−k ) which implies β( pk , p−k ) − β(λpk , λp−k ) = α(λ − 1) w When the left-hand side is zero, it means α = 0, which means marginal utility of income is zero, a contradiction. When the left-hand-side is nonze-

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ro, when we vary w we have a contradiction to the constancy of the righthand-side.  To see some possibility, let us consider that only the price-income pair ( pk , w) is endogenously random and p−k is exogenously random. Let k,w be the set of probability distributions over pairs of Good k price and income. Then, the preference condition consistent with the use of expected consumer surplus is that there is no income effect on Good k and the consumer is risk-neural. Theorem 7.2 The following statements are equivalent. (i) EC represents the consumer’s preference over k,w . (ii) The indirect utility function/von-Neumann Morgenstern index has the form v( pk , p−k , w) = α( p−k )w + β( pk , p−k ) where α is homogeneous of degree minus one, and β is homogeneous of degree zero. Proof Since EC S represents the consumer’s preference in the expected utility form on the domain of k,w , from cardinal equivalence of von-Neumann/Morgenstern index there exists a positive-valued function a( p−k ) and a function b( p−k ) such that C S( pk , p−k , w) = a( p−k )V ( pk , p−k , w) + b( p−k ) holds for all ( pk , p−k , w). Then, from the definition of C S we obtain  ∞ ∂C S( pk , p−k , w) ∂ = xk (qk , p−k , w)dqk ∂ pk ∂ p k pk = −xk ( pk , p−k , w) On the other hand, from Roy’s identity, we have ∂ ∂ V ( pk , p−k , w) xk ( pk , p−k , w) (a( p−k )v( pk , p−k , w) + b( p−k )) = −a( p−k ) ∂ pk ∂w

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Therefore, for arbitrary ( pk , p−k , w) it holds −xk ( pk , p−k , w) = −a( p−k )

∂v( pk , p−k , w) xk ( pk , p−k , w), ∂w

, p−k ,w) which implies that ∂v( pk∂w depends only on p−k , and constant over variables ( pk , w). Hence, it satisfies the condition

∂v( pk , p−k , w) = α( p−k ) ∂w where α( p−k ) depends only on p−k . By integrating we obtain v( pk , p−k , w) = α( p−k )w + β( pk , p−k ) Since v is homogeneous of degree zero, we obtain v(λpk , λp−k , λw) = α(λp−k )λw + β(λpk , λp−k ) = v( pk , p−k , w) = α( p−k )w + β( pk , p−k ) Since w is arbitrary, we obtain α(λp−k ) = and

1 α( p−k ) λ

β(λpk , λp−k ) = β( pk , p−k )

 Practically, the above result says that the use of expected consumer surplus presumes that there is no income effect on the goods under consideration and the consumer is risk-neutral with respect to probabilistic changes in income.

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7.3

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143

Incompleteness of Underlying Asset Markets

We have seen that the use of expected consumer surplus requires the two assumptions, no income effect on the good under consideration and risk neutrality. This leaves it unquestioned when such conditions are met. We already know that when the commodity under consideration is a tiny piece of a large set of commodities income effect on it tends to be negligible. But what about the assumption of risk-neutrality? Also, we would point out one more assumption which is critical in aggregation, which is implicit in the above approach: (3) values of income transfers are equal across states of the world, and equal across individuals; or at least, the way how the values of income transfers differ across states is the same across individuals. To see this explicitly, let us look at the model of incomplete asset markets and go over the Hicksian aggregation operation there, based on Hayashi [4]. Following the argument on the smallness of commodity in the previous chapter, we consider that at each state the deterministic consumption space is defined with a continuum of commodity characteristics. Let T = [0, 1] be the set of commodity characteristics,  be the family of Lebesgue measurable sets, and μ be the Lebesgue measure. Let L ∞ (T ) be the space of essentially bounded measurable functions from T to R, which is endowed with the sup norm. Let ∞ L∞ +++ (T ) = { f ∈ L (T ) : ∃l > 0, a.e. t ∈ T, f (t) ≥ l}.

be the set of essentially bounded measurable functions which are positively bounded away from zero. We take L ∞ +++ (T ) to be the set of deterministic consumptions. There are S states of the world, which is assumed o be finite. We S take L ∞ +++ (T ) to be the set of state-contingent consumptions. Given S f ∈ L∞ +++ (T ) , f s (t) denotes the consumption of commodity t ∈ T to be received at the spot markets to be opened at state s = 1, . . . , S.

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We take L 1+++ (T ) S to be the set of state-contingent price systems in the spot markets. Given p ∈ L 1+++ (T ) S , ps (t) denotes the price of commodity t ∈ T at the spot markets to be opened at state s. S be the vector of state-contingent deliveries of background Let w ∈ R++ income, where ws denotes the income to be received at state s. There are H securities available in the asset markets and let R be an S × H matrix describing the payoff structure, which says that security h pays Rsh units of income at state s which can be used in the spot markets H be the vector of security prices. there. Let q ∈ R++ We make the following assumptions on wealth and prices. Wealth and Prices: S ; (i) w ∈ R++ (ii) p ∈ L 1+++ (T ) S and there exist p, p with 0 < p < p such that ps (t) ∈ [ p, p] for almost all t ∈ T and all s = 1, . . . , S. H . (iii) q ∈ R++ (iv) (q, R) admits no arbitrage. (v) rank R = H .

Condition (v) is imposed without loss of generality, because otherwise there may be many portfolio choices which yield the same state-contingent consumption plans. Given a measurable set of commodity characteristics K ⊂ T and a  state of the world s, let ps K = K ps (t)dμ(t) be the price of K in the spot markets at state s. We assume that the preference over state-contingent consumptions follows the expected utility theory, and it is represented in the form U( f ) =

S 

u( f s )πs ,

s=1

where u : L ∞ +++ (T ) → R is the von-Neumann/Morgenstern index and πs denotes the probability of each state s. First, we impose an assumption on preference regularity.

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Regular Preference: (i) u : L ∞ +++ (T ) → R is norm-continuous and Frechet differentiable. Moreover, Du( f ) ∈ L 1+++ (T ) for all f ∈ L ∞ +++ (T ), and the map∞ 1 ping Du(·) : L +++ (T ) → L (T ) is continuous in the following sense: given any compact set C ⊂ Rm , for any sequence of functions ν from C to L ∞ +++ (T ), denoted by { f }, and a function from C to ∞ ν L +++ (T ) denoted by f , if { f } weak-∗ converges to f uniformly on C then Du( f ν ) weakly converges to Du( f ) uniformly on C. (ii) u : L ∞ +++ (T ) → R is strictly concave. We also make an assumption on the preference induced on the finitedimensional subspaces generated by {J n }. Regular Preference on Finite Dimensions: (i) For all n, the restriction of u onto the finite-dimensional subspace n n RJ , denoted u n : RJ ++ → R, which is defined by ⎛ u n (z) = u ⎝



⎞ z K 1K ⎠

K ∈J n

for z = (z K ) K ∈J n ∈ RJ ++ , is twice continuously differentiable, where 1 K denotes the characteristic function over K ∈ J n . (ii) Denote the first derivative of u n by Du n . Then, n

Du n (z) ∈ RJ ++ n

for all z ∈ RJ ++ . n (iii) Denote the second derivative of u n by D 2 u n . Then, for all z ∈ RJ ++ , the |J n | × |J n | matrix D 2 u n is negative definite. n

We assume that the Inada-type condition holds in a uniform manner across n, which is parallel to what Vives [5] assumes for increasing numbers of commodities.

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Uniform Inada Property: There exist nonincreasing functions φ, φ from R++ to R++ such that (i) φ(y) ≤ φ(y) for all y ∈ R++ ; (ii) φ(y) → ∞ as y → 0 and φ(y) → 0 as y → ∞; n (iii) for all n, z = (z K ) K ∈J n ∈ RJ ++ and K ∈ J , n

φ(z K ) ≤

∂u n (z) μ(K ) ≤ φ(z K ). ∂z K

7.3.1 Hicksian Aggregation Given n, pick J ∈ J n to be the object of partial equilibrium analysis in the approximate sense. Now, consider the preference induced over pairs of state-contingent consumptions of commodity piece J and associated state-contingent income transfers. n Given n and J ∈ J n and for each s = 1, . . . , S, let (xs , z s,−J ) ∈ RJ n denote the vector such that xs refers to its J th entry and z s,−J ∈ RJ \{J } n refers to the rest of the entries. Also, let (x, z −J ) ∈ RJ ×S be given by n (x, z −J ) = (xs , z s,−J )s=1,...,S and z −J ∈ R(J \{J })×S be given by z −J = (z s,−J )s=1,...,S . Consider vectors of state-contingent income transfers associated with the consumption of given commodity piece J ∈ J n , which is to be spent

S w n on the rest of commodity pieces J \ {J }. When a ∈ − μ(J ) , ∞ is given, it means that the consumer receives as μ(J ) units of income transfer at state s = 1, . . . , S, where it is taken to be a payment when it is negative. Note that this income transfer is adjusted to the size of commodity piece J , because as it tends to be small associated income transfers tend to be small as well.

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S Definition 7.1 Given n, J ∈ J n and (x, a), (y, b) ∈ R++ × S

w − μ(J ) , ∞ , the relation

(x, a) n,J (y, b) holds if

V n,J (x, a) ≥ V n,J (y, b),

where V

n,J

(x, a) =

max

(J n \{J })×S

ζ ∈R H ,z −J ∈R++

S 

  u n xs , z s,−J πs

s=1

subject to H 

q h ζh = 0

h=1



ps K z s K = ws +

K ∈J n \{J }

H 

Rsh ζh + as μ(J ) for each s = 1, . . . , S

h=1

Under the current assumption, the finite-dimensional problem above for each fixed n has a unique interior solution, since it falls in the standard demand analysis in the literature of general equilibrium with incomplete markets (GEI), such as Geanakoplos and Polemarch akis [6], Magill and Quinzii [7]. Since ws is positive for all s = 1, . . . , S and the asset markets are assumed to admit no arbitrage, the solution is guaranteed to exist. Because of the regularity of the preference the solution is unique and in the interior. n Let z sn,J K (x, a) denote the demand for commodity piece K ∈ J \ n,J {J } in the spot markets st state s = 1, . . . , S, let z s,−J (x, a) = 



n,J n,J n,J and let z −J (x, a) = z s,−J (x, a) . Let z s K (x, a) n K ∈J \{J }

s=1,...,S

the demand for security h = 1, . . . , H and let ζhn,J (x, a) denote

 n,J n,J ζh (x, a) = ζh (x, a) . Let λn,J 0 (x, a) > 0 be the Lagrange h=1,...,H

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H multiplier on the constraint h=1 qh ζh = 0 and λn,J s (x, a) > 0 be the  H multiplier on the constraint K ∈J n \{J } ps K z s K = ws + h=1 Rsh ζh + as μ(J ) for each s = 1, . . . , S. Thus, we have the first-order condition λn,J 0 (x, a)qh =

S  λn,J s (x, a)Rsh for each h = 1, . . . , H s=1

  ∂ n u n xs , z s,−J = λn,J πs s (x, a) ps K for each K ∈ J \ {J }, s = 1, . . . , S, ∂z s K

where the derivatives are evaluated at optimal consumption and security demand. By direct calculations we obtain  ∂ V n,J (x, a) ∂ n = πs u xs , z s,−J ∂ xs ∂ xs n,J ∂ V (x, a) = λn,J s (x, a)μ(J ) ∂as Let M RSxn,J s ,as (x, a) denote the marginal rate of substitution of ex-post income transfer at state s for the good under consideration to be delivered at state s. Then from the above formula, we obtain   ∂ n πs ∂ xs u x s , z s,−J n,J M RSxs ,as (x, a) = n,J · μ(J ) λs (x, a) Likewise, we obtain (x, a) = M RSan,J s ,as

λn,J s (x, a)

λn,J s (x, a)   ∂ n u , z x π s s s,−J ∂ x s (x, a) = M RSxn,J   s ,x s

πs ∂ x∂ u n xs , z s ,−J s   ∂ n x u , z s s,−J π s ∂x M RSxn,J (x, a) = n,J · s s ,as

μ(J ) λs (x, a)

7

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We derive comparative statics properties of the conditional demand. From the second-order argument, we have ⎞ dζ ⎛ ⎞ ⎜ dz 1,−J ⎟ d x 1 ⎜ . ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎟ ⎜ dz ⎜dx ⎟ ⎟ ⎜ H n,J (x, z n,J (x, a)) ⎜ S,−J ⎟ = G n,J (x, z n,J (x, a)) ⎜ S ⎟ ⎜ da1 ⎟ ⎜ dλ0 ⎟ ⎜ . ⎟ ⎜ dλ ⎟ ⎜ 1 ⎟ ⎝ .. ⎠ ⎜ . ⎟ ⎝ .. ⎠ da S dλ S ⎛

×S is an (H + 2n S + 1) × (H + where H n,J (z) for arbitrary z ∈ RJ ++ 2n S + 1) matrix given by n

H n,J (z) = ⎛ 0J n \{J } q ⎜ p1n,J ⎜ ⎜ .. ⎜ −R . ⎜ ⎜ ⎜ 0J n \{J } ⎜ ⎜ OH 2 O H ×S ⎜ ⎜O n 2 n π D ⎜ (J \{J })×H 1 −J u (z 1 ) ⎜ .. .. ⎜ ⎝ . . O(J n \{J })×H O(J n \{J })2

··· 0J n \{J } 0 ··· 0J n \{J } .. .. . 0tS . n,J ··· pS ··· O H ×S qt t · · · O(J n \{J })2 0J n \{J } ( p1n,J )t .. .. .. .. . . . . t 2 u n (z ) 0t · · · π S D−J S J n \{J } 0J n \{J }

⎞

0S

⎟ ⎟ ⎟ ⎟ OS2 ⎟ ⎟ ⎟ ⎟ t ⎟ −R ⎟ t · · · 0J n \{J } ⎟ ⎟ ⎟ .. ⎟ .. ⎠ . . t · · · ( p n,J S )

and G n,J (z) is an (H + 2n S + 1) × 2S matrix given by G n,J (z) = ⎛

0S

0S ··· .. . ··· O H ×S

⎞

⎜ μ(J ) 0 ⎟ ⎜ ⎟ ⎜ .. ⎟ . ⎜ ⎟ . ⎜ . ⎟ OS2 . ⎜ ⎟ ⎜ 0 μ(J ) ⎟ ⎜ ⎟ ⎜ ⎟ O H ×S ⎜ ⎟ t ⎜ −π1 D J D−J u n (z 1 ) · · · ⎟ 0 ⎜ ⎟ J n \{J } ⎜ ⎟ . . . ⎜ ⎟ . . . . ⎝ ⎠ . . O(J n \{J })×S t n 0J n \{J } · · · −π S D J D−J u (z S )

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and psn,J = ( ps K ) K ∈J n \{J } is a row vector with 2n − 1 entries for each s = 1, . . . , S,  2 D−J u n (z s )

=

∂ 2 u n (z s ) ∂z s L ∂z s K

 K ,L∈J n \{J }

is a (2n − 1) × (2n − 1) matrix for each s = 1, . . . , S, and  D J D−J u (z s ) = n

∂ 2 u n (z s ) ∂z s J ∂z s K

 K ∈J n \{J }

is a column vector with 2n − 1 entries for each s = 1, . . . , S. Also, for given numbers X and Y , let 0 X denote the row zero vector with X entries and let 0 X ×Y denote the X × Y zero matrix. n,J For each h = 1, . . . , H , and s = 1, . . . , S, let Hh,s

(z) be the matrix n,J obtained by replacing the column of H (z) for asset h by the column of G n,J (z) for commodity piece J at s . For each h = 1, . . . , H and s = n,J (z) be the matrix obtained by replacing the column of 1, . . . , S, let H h,s H n,J (z) for asset h by the column of G n,J (z) for income transfer at state s . For each s, s = 1, . . . , S and K , ∈ J n \ {J }, let Hsn,J K ,s (z) be the n,J matrix obtained by replacing the column of H (z) for commodity piece s K by the column of G n,J (z) for commodity piece J at s . For each n,J (z) be the s = 1, . . . , S, s = 1, . . . , S, and K ∈ J n \ {J }, let H s K ,s matrix obtained by replacing the column of H n,J (z) for commodity piece s K by the column of G n,J (z) for income transfer at state s .

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Then, by Cramer’s rule, we obtain    n,J n,J (x, a)) (x, z H   h,s

dζh  , h = 1, . . . , H, s = 1, . . . , S =  n,J  H (x, z n,J (x, a)) d xs

   n,J  n,J  Hh,s (x, z (x, a)) dζh  , h = 1, . . . , H, s = 1, . . . , S =  n,J  H (x, z n,J (x, a)) das

   n,J n,J (x, a)) (x, z H   s K ,s

dz s K  , s, s = 1, . . . , S, K ∈ I n \ {J } =  n,J  H (x, z n,J (x, a)) d xs

   n,J  n,J  Hs K ,s (x, z (x, a)) dz s K  , s, s = 1, . . . , S, K ∈ I n \ {J } =  n,J  H (x, z n,J (x, a)) das

Condition below guarantees that the sensitivity terms given above are uniformly bounded as the consumption vectors are uniformly bounded. Uniform Boundedness of Sensitivity : For any fixed z, z > 0, there exist n α, α and β, β such that for all n, z ∈ [z, z]J , h ∈ H , J, K ∈ J n and s, s = 1, . . . , S, it holds      n,J   n,J   Hh,s (z)  Hs K ,s (z) α   n,J  ,  n,J   α  H (z)  H (z) and

      n,J   n,J  Hh,s (z)  Hs K ,s (z) β   n,J  ,  n,J   β.  H (z)  H (z)

7.3.2 The Limit Theorem Fix any τ ∈ T as the object of partial equilibrium analysis in the limit, S × RS . and let J = J n (τ ) for each n, and fix a compact set C ⊂ R++ n Also, for all sufficiently large n, let { f n,J (τ ) } be the sequence of functions

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S from C to [z1, z1] ⊂ L ∞ +++ (T ) given by

f sn,J

n (τ )



(x, a) = xs 1 J n (τ ) +

n,J n (τ )

zs K

(x, a)1 K

K ∈J n \{J n (τ )}

for each n, s = 1, . . . , S and (x, a) ∈ C. Then, the method as adopted in the previous chapter provides a similar sequence of lemmata. Lemma 7.1 The sequence {ζ n,J (τ ) , f n,J (τ ) } has a convergent subsequence k(n) k(n) {ζ n,J (τ ) , f n,J (τ ) } with the limit ζ ∈ [ζ , ζ ] H and f ∈ [z1, z1], which are constant over (x, a), in the sense that n

sup ζ n,J

k(n) (τ )

(x,a)∈C

n

, −ζ  → 0 as n → ∞

and sup | f n,J

k(n) (τ )

(x,a)∈C

(x, a), q − f, q | → 0 as n → ∞

for all s = 1, . . . , S, and q ∈ L 1 (T ). Moreover, (ζ, f ) is the unique solution to the problem (we call it unconditional problem) S  max u( f s )πs S ζ ∈R H , f ∈L ∞ +++ (T )

s=1

subject to H 

q h ζh = 0

h=1



ps (t) f s (t)dμ(t) = ws + T

H  h=1

Rsh ζh for each s = 1, . . . , S

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Lemma 7.2 The corresponding subsequence of {λn,J (τ ) (x, a)} converges to {0}∪S λ ∈ R++ uniformly on C, which is the vector of Lagrange multipliers associated with the solution (ζ, f ) in the unconditional problem. We impose the following continuity property. n

Continuous Marginal Utility Density : For almost every τ ∈ T , for any compact set D ⊂ R++ and f ∈ L ∞ +++ (T ), there exists a function u(·, τ ; f ) : D → R such that   ∂    sup  u x1 J + f 1T \J − u(x, τ ; f )μ(J ) = o(μ(J )), x∈D ∂ x where J is any interval containing τ with μ(J ) > 0, and o(μ(J )) is a function which vanishes faster than μ(J ). Moreover, u(x, τ ; f ) is continuous in f in the following sense: Given any compact set C ⊂ Rm , if a sequence of functions from C to L +++ (T ), denoted { f ν }, weak-∗ converges to f uniformly on C, then   sup sup u(x, τ ; f ν (c)) − u(x, τ ; f (c)) → 0 as ν → ∞. c∈C x∈D

Note that when preference is separable over deterministic consumptions allowing representation   S  φ v( f s (t), t)dμ(t) πs , U( f ) = s=1

T

the above condition follows from the fundamental theorem of calculus and we have   ∂

v( f s (t), t)dμ(t) v(xs , τ ) u(xs , τ ; f s ) = φ ∂ xs T for each s = 1, . . . , S. Under the continuity condition, we obtain the limit theorem.

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Theorem 7.3 Given almost every τ ∈ T and any compact set C ⊂ R++ × R, there exists a subsequence of {n}, denoted {k(n)}, such that as n → ∞ it holds     u(x , τ, f ) π k(n) (τ ) s s s k(n),J →0 sup  M RSxs ,as (x, a) −  λs (x,a)∈C S    λs  k(n),J k(n) (τ )  (x, a) − sup  M RSas ,as

→0 λs  (x,a)∈C S    πs u(xs , τ, f s )  k(n),J k(n) (τ )  (x, a) − →0 sup  M RSxs ,xs

πs u(xs , τ, f s )  (x,a)∈C S     u(x , τ, f ) π k(n) (τ ) s s s k(n),J →0 sup  M RSxs ,as

(x, a) −  λ

S (x,a)∈C

s

for all s, s = 1, . . . , S, where λ0 , {λs }s=1,...,S are the Lagrange multipliers to the unconditional problem.

7.3.3 Limit Preference Let τ denote the preference over state-contingent allocations of commodity τ ∈ T and associated income transfers, which is defined over a S × R S as in the above theorem, sufficiently large compact set C ⊂ R++ such that the marginal rates of substitutions it induces are the limits of the marginal rate of substitutions: πs u(xs , τ, f s ) λs λ s M RSaτs ,as (x, a) = λs

πs u(xs , τ, f s ) M RSxτs ,xs (x, a) = πs u(xs , τ, f s ) π s u(x s , τ, f s ) M RSxτs ,as (x, a) = λs

M RSxτs ,as (x, a) =

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By integrating back the MRS, we see that the limit preference is represented in the form   xs S   λs 1 u(z s , τ, f s )dz s + a s πs U (x, a) = λ0 0 λ0 πs τ

s=1

or its arbitrary monotone transformation. Here, the limit preference describes a willingness to pay for a “negligible commodity,” which is in our argument given as a “density” notion. In the limit, x describes the state-contingent delivery of a “negligible” commodity, and a describes income transfer to be spent on the other commodities, which is adjusted to the size of the commodity at the general equilibrium level and hence “negligible.” Nevertheless, in the sense of “density” the marginal rate of substitution between them is well defined. The limit preference has the following notable properties. 1. The limit preference is risk neutral, not only that it exhibits no income effect on ex-post allocations. The no income effect property is seen from the marginal rate of substitution of income transfer for consumption being independent of the amount of income at every state. The risk neutrality property is seen from the marginal rates of substitution between income transfers across states being constant. This is because when income effect of a commodity is small because of the smallness of commodity it also means that income transfer associated to it is small as well and we already know from Arrow’s result that smooth expected utility preference exhibits risk neutrality toward small risks (Arrow [8]). Thus, the two conditions are tied together. Note also that the risk attitude as a primitive matters of course at a general equilibrium level, and here it enters the above willingness-topay formula as a fixed constant. 2. The marginal rate of substitution in the limit between extra income transfers at different states of the world equals to the ratio between the Lagrange multipliers associated to those states. Therefore, the values

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of state-contingent income transfers may differ between states and the way how they differ between states may, in general, differ between consumers.

7.4

Is Aggregate Expected Consumer Surplus a Right Efficiency Measure?

By continuing to follow the argument in Hayashi [4], let us see if the aggregate of expected adjusted consumer surplus is consistent with Kaldor– Hicks criterion. I Definition 7.2 Say that (x, a) = (xi , ai )i=1 yields larger aggregate exI if pected adjusted consumer surplus than (y, b) = (yi , bi )i=1

 S    I   1  xis λis u(z is , τ, f s )dz is + ais πis λi0 0 λi0 πis i=1 s=1  S    I   1  yis λis > u(z is , τ, f s )dz is + bis πis . λi0 0 λi0 πis i=1

s=1

First, we can verify that aggregate expected adjusted consumer surplus criterion is consistent with Kaldor criterion when the asset markets are complete. Proposition 7.1 Suppose the asset markets are complete. Then, if (x, a) yields larger aggregate expected adjusted consumer surplus than (y, b), there existI I I s ( ai S )i=1 with i=1  ais = i=1 ais for all s = 1, . . . , S such that τ (xi , ai0 , ai S ) i (yi , bi ) for all i = 1, . . . , I . It says that an option generating larger aggregate expected adjusted consumer surplus can be made Pareto superior by making suitable redistribution of ex-post income transfers. Note that the result holds independently of the distribution of beliefs, this is because in complete markets each individual’s marginal rate of substitution between extra income transfers at different states fully takes

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her subjective belief into account through asset allocation, and the rates are equalized across individuals. Proof Notice that the above condition is equivalent to  S  I   πis

>

λi0 i=1 s=1 S  I   πis i=1 s=1

λi0

xis

0

 0

yis

λis u(z is , τ, f s )dz is + ais λi0



 λis u(z is , τ, f s )dz is + bis . λi0

Without loss of generality, assume that R is an S × S matrix and invertible. Then from the first order condition we have   λi S λi1 ,..., = q R −1 , λi0 λi0 for every i = 1, . . . , I , which is common across all consumers. When the asset markets are incomplete, the aggregate expected adjusted consumer surplus is in general inconsistent with the ex-ante Kaldor criterion, however. Under uncertainty, the value of income may differ across states, and when the asset markets are incomplete the way how the value of income differs across states, in general, differs across individuals. Competitive equilibrium allocation is constrained efficient at partial I equilibrium level in the following sense: if ( f i )i=1 is a competitive equilibrium allocation given equilibrium prices p and q and asset trade ζ , then I solves for almost all τ ∈ T , the partial equilibrium allocation ( f is (τ ))i=1 the maximization problem   xis S  I   λis 1 max u(z is , τ, f s )dz is − ps (τ )xis πis . I λi0 0 λi0 πis (xi )i=1 i=1 s=1

However, just like competitive equilibrium allocation is only constrained efficient at a general equilibrium level it is only constrained efficient at the partial equilibrium level as well and can be improved upon by state-contingent income transfers with ex-post budget balance.

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To illustrate, consider a two-state example with no production, in which the two states are equally likely and there is only one asset which yields one unit of income per one unit regardless of states. Also, let T = [0, 1]. Let ω A = (ω A1 , ω A2 ) and ω B = (ω B1 , ω B2 ) denote state-contingent endowments for A and B, respectively. They are given by ω A1 = (δ + ε)1, ω A2 = ε1 ω B1 = ε1, ω B2 = (δ + ε)1 For simplicity, both A and B have the same preference which additively separable and uniform over commodity characteristics. Thus, it is represented by 1 U( f ) = φ 2



 T

1 v( f i1 (t), t)dμ(t) + φ 2





v( f i2 (t), t)dμ(t) , T

for both i = A, B. Without loss of generality, normalize the spot price system by  ps (t)dμ(t) = 1 T

for all s = 1, 2 and set q = 1 because there is just one asset. Then, in competitive general equilibrium, we have p 1 = p2 = 1 f A1 = (δ + ε)1, f B1 = ε1,

f A2 = ε1

f B2 = (δ + ε)1

1 1 λ A1 = φ (v(δ + ε))v (δ + ε), λ A2 = φ (v(ε))v (ε) 2 2 1 1 λ B1 = φ (v(ε))v (ε), λ B2 = φ (v(δ + ε))v (δ + ε) 2 2

7

λ A0 = λ B0 =

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 1

φ (v(δ + ε))v (δ + ε) + φ (v(ε))v (ε) ≡ λ0 2

According to our result, consumer surplus in the state-contingent allocation problem for a single commodity and associated income transfers is given by 

 φ (v(δ + ε)) φ (v(δ + ε))v (δ + ε) v(x A1 ) + a A1 λ0 λ0   1 φ (v(ε)) φ (v(ε))v (ε) + v(x A2 ) + a A2 + a A0 2 λ0 λ0   1 φ (v(ε)) φ (v(ε))v (ε) U B (x B , a B ) = v(x B1 ) + a B1 2 λ0 λ0   1 φ (v(δ + ε)) φ (v(δ + ε))v (δ + ε) v(x B2 ) + a B2 + a B0 . 2 λ0 λ0 U A (x A , a A ) =

1 2

Then competitive partial equilibrium in the market for a single commodity delivers U A ((δ + ε, ε), (0, −(δ + ε), −ε)) = U B ((ε, δ + ε), (0, −ε, −(δ + ε))) =

u+u , 2

where  φ (v(δ + ε))  v(δ + ε) − v (δ + ε)(δ + ε) λ0  φ (v(ε))  u= v(ε) − v (ε)ε λ0

u=

Now, consider the following reallocation of state-contingent income transfers: move γ1 units of income from A to B at State 1, and move γ2 units of income from B to A at State 2, where γ1 = γ2 =

φ (v(δ

λ0 (u − u) .

+ ε))v (δ + ε) + φ (v(ε))v (ε)

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Then, the reallocation improves both parties’ expected adjusted consumer surplus, since U A ((δ + ε, ε), (0, −(δ + ε) − γ1 , −ε + γ2 )) = U B ((ε, δ + ε), (0, −ε + γ1 , −(δ + ε) − γ2 )) u + u (u − u)(φ (v(ε))v (ε) − φ (v(δ + ε))v (δ + ε)) + = 2 φ (v(δ + ε))v (δ + ε) + φ (v(ε))v (ε) u+u . > 2 How should we interpret the criterion of aggregate expected consumer surplus as used in practice, then? It is interpreted as assuming that how the value of income differ across states is common by assum across individuals,  λi1 λi S ing (πi1 , . . . , πi S ) = (π1 , . . . , π S ) and λi0 , . . . , λi0 = (π1 , . . . , π S ) for all i for some common (π1 , . . . , π S ) and prescribing the maximization of  S    I   1  xis u(z is , τ, f s )dz is + ais πs . λi0 0 i=1

s=1

This is seemingly consistent with the Pareto criterion because the marginal rates of substitution between state-contingent income transfers are assumed to be equalized across individuals. However, such assumption of equality cannot come from underlying preference maximization of the individuals in general when the asset markets are incomplete, even when their beliefs are identical. On the other hand, the criterion based on the aggregate adjusted expected consumer surplus  S    I   1  xis λis u(z is , τ, f s )dz is + ais πis λi0 0 λi0 πis i=1

s=1

is consistent with underlying preference maximization of the individuals whether the asset markets are complete or incomplete. However, when

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the asset markets are incomplete it is not in general consistent with the ex-ante Pareto criterion. What lesson should we draw from this? It is that the value of surplus for a given individual depends on how hard or easy to insure his earning, and it is possible that we hurt potentially valuable resources by means of surplus maximization, because of hardness to insure it. To understand, consider making some partial equilibrium public decision by means of an auction. Such decision may hurt somebody at some contingency. If one wants to avoid such decision he needs to pay, but how does he finance? When the source of his earning is such that it is hard to borrow against, he cannot finance the money to pay for protecting himself, despite that the potential gain from doing this is socially large.

Note 1. For related arguments see also Turnovsky et al. [9] and Schlee [10, 11].

References 1. Waugh, F.V. 1944. Does the consumer benefit from price instability? Quarterly Journal of Economics 58: 602–614. 2. Massell, B.F. 1969. Price stabilization and welfare. Quarterly Journal of Economics 83 (2): 284–298. 3. Rogerson, W.P. 1980. Aggregate expected consumer surplus as a welfare index with an application to price stabilization. Econometrica 48 (2): 423–436. 4. Hayashi, Takashi. 2014. Consumer surplus analysis under uncertainty: A general equilibrium perspective. Journal of Mathematical Economics 55: 154– 164. 5. Vives, X. 1987. Small income effects: A Marshallian theory of consumer surplus and downward sloping demand. Review of Economic Studies 54: 87– 103. 6. Geanakoplos, John, and Heraklis M. Polemarchakis. 1986. Existence, regularity and constrained suboptimality of competitive allocations when the asset market is incomplete. Uncertainty, Information and Communication: Essays in Honor of KJ Arrow 3: 65–96.

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7. Magill, Michael J.P., and Martine Quinzii. 2002.Theory of incomplete markets: Vol. 1, Vol. 1. The MIT Press. 8. Arrow, K.J. 1971. Essays in the theory of risk bearing. Chicago, IL: Markham Publishing Company. 9. Turnovsky, S.J., H. Shalit, and A. Schmitz. 1980. Consumer’s surplus, price instability, and consumer welfare. Econometrica 48 (1): 135–152. 10. Schlee, E. 2008. Expected consumer’s surplus as an approximate welfare measure. Economic Theory 34 (1): 127–155. 11. Schlee, Edward E. 2013. Surplus maximization and optimality. The American Economic Review 103 (6): 2585–2611.

8 Mechanism Design in Partial Equilibrium

8.1

Partial Equilibrium as an Institutional Artifact

Many of the mechanism design practices are grounded on the partial equilibrium framework. An authority handling a particular issue isolates it from the rest of the economy and makes a decision based only on the “preferences” over alternatives at hand. They do not listen to “it depends.” This presumes that preferences are separable, while such assumption is false in general and there is no such thing as “preference” just over alternatives being handled in a single sector. Hence, if one changes policy in one sector it, in general, has general equilibrium effects across sectors. For example, when we change something in a school allocation mechanism it must have a general equilibrium effect on housing markets. However, the institutional and bureaucratic reality is that whether such presumption is met or not the partial equilibrium nature is built in as a part of how a social decision process works. When such misspecification is built in as a given institutional constraint, how does this misspecification constrain the overall economy activities? Based on Hayashi and Lombardi [1], we illustrate the nature of the problem. © The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2_8

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To understand the point, consider auction/public decision with transfers in multiple sectors. There are let’s say two sectors and the outcome space in each sector is given by X1 = Y 1 × H X 2 = Y 2 × H, where Y 1 and Y 2 are the sets of pure public decisions without transfer, in each section, and  H = a ∈ [−b, ∞) I :

I 

 ai ≤ 0

i=1

is the set of closed transfers of the numeraire good. The concept of “income,” if understood properly, already presumes that there is some “market” behind the partial equilibrium problems. Here, we assume that it is given as a numeraire good in a physical sense, as our purpose is to show that nevertheless, the requirement of partial equilibrium analysis is restrictive. Let ei > 2b denote individual i’s initial endowment of the numeraire good. Then his preference over X 1 × X 2 is supposed to be represented in the form Ui (y 1 , y 2 , ei + ti1 + ti2 ) where (y 1 , t 1 ) ∈ X 1 and (y 2 , t 2 ) ∈ X 2 , and Ui is increasing in the third argument, as i’s final amount of the numeraire good is ei + ti1 + ti2 . Convectional partial mechanism design assumes that such preferences are separable across sectors so that one can handle Sector 1 decision without worrying about general equilibrium effects in relation to Sector 2 decisions. However, such separability assumption not only presumes that surplus from (or willingness to pay for) pure public decisions is separable across sectors but also there is no income effect on the pure public decisions. In other words, you cannot allow for the role of income effects while maintaining the assumption of separability.

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Proposition 8.1 Preference is (weakly) separable if and only if it is represented in the form vi1 (y 1 ) + vi2 (y 2 ) + ti1 + ti2 Proof “If ” part is obvious, hence we show the “only if ” part. First, we show that the marginal ordering Ri1 over Sector 1 outcomes induced by Ri exhibits no income effects. Consider any two pure social decisions for issue 1, say y 1 and yˆ 1 , and any four income transfers, say t 1 , t˜1 , tˆ1 and t¯1 . Then the separability requirement applied to outcomes (y 2 , t 2 ) and 2 (y , tˆ2 ) in Sector 2, where tˆi2 = ti2 − q, implies (y 1 , ti1 )Ri1 ( yˆ 1 , tˆi1 ) ⇐⇒ Ui (y 1 , y 2 , ti1 + ti2 + ei ; Ri ) ≥ Ui ( yˆ 1 , y 2 , tˆi1 + ti2 + ei ; Ri )     ⇐⇒ Ui (y 1 , y 2 , ti1 + q + tˆi2 + ei ; Ri ) ≥ Ui ( yˆ 1 , y 2 , tˆi1 + q + tˆi2 + ei ; Ri ) ⇐⇒ (y 1 , ti1 + q)Ri1 ( yˆ 1 , tˆi1 + q).

Thus, the marginal ordering Ri1 exhibits no income effect. Since the arguments for the other marginal ordering Ri2 are entirely symmetric, we conclude that agent i’s separable ordering Ri has a quasi-linear utility representation vi (y 1 , y 2 ) + ti1 + ti2 By the separability property again, the function vi describing wilingness to pay has to be additively separable, hence we obtain the result.  When there is non-separability due to income effect, even if the willingness to pay for an item is independent of that for another the standard strategy-proof mechanism is not strategy-proof when it is run sector-wise. To illustrate, consider that there are two bidders, A and B. There are two sectors, each in which one item is for sale. Consider running the second-price auction in each sector. Suppose that B has a standard quasi-linear preference, in which his willingness to pay for a pair of Item 1 and Item 2 is the sum of willingness to pay for each item. Then, the fact that it is a dominant strategy for B to bid his willingness to pay in each sector does not change.

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Item 1&2

eA − b1B − b2B

eA − b2B

Item 2

Item 1

eA − b1B

No item

eA

eA + t1A + t2A

Fig. 8.1 Two-sector second-price auctions with income effects

On the other hand, A’s preference is as follows: (i) there is no direct complementarity between the two items; (ii) there is no income effect on Item 1; (iii) there is income effect on Item 2, in the sense that as A has more income his willingness to pay for Item 2 increases. Now suppose B’s bids are b1B in Sector 1 and b2B in Sector 2. Then Fig. 8.1 illustrates that A’s best response is to win Item 1 only. However, when B’s bid or valuation in Sector 1 changes let’s say to zero, A’s best response is to win both Item 1 and Item 2. Hence A has no dominant strategy any longer.

8.2

Implementation in Partial Equilibrium

Let us consider, a mechanism design problem with the following constraint: there are multiple sectors, and each individual sends his message to each sector office separately, and the sector office do not communicate with each other. This describes the partial equilibrium nature of our bureaucracy. We submit a message to the school board, we submit messages to housing agency, we submit a messages to transportation agency, etc., separately, and those offices do not communicate each other in order to make a publication as a whole.

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Imagine that there are two kinds of authorities, one is central and the others are sector-specific. The central authority cares about overall social objective, but cannot directly handle how bureaucracy works in each sector. Thus, the central authority has delegate sector-specific objective to each sector authority, who cares about how to implement it just using the information given to him. There are I individuals and n sectors. For each k = 1, . . . , n, let X k be the set of outcomes of that sector. Hence, the set of total outcomes is X=

n 

Xk.

k=1

 Its element x ∈ X is sometimes denoted by x = x k , x −k for a given k. Let Ri denote i’s weak preference ordering over X . For each x −k , we  −k k , on X k by can define the sector-k conditional ordering, Ri x       for all y k , z k ∈ X k : y k Rik x −k z k ⇐⇒ y k , x −k Ri z k , x −k . We say that the ordering Ri is separable if and only if for all x

−k

,y

−k

∈X

−k

:

Rik

 x

−k



=

Rik

 y

−k



≡ Rik .

for all k = 1, . . . , n. Let R (X ) be the set of set of weak orders on X . Let Rsep (X ) be the set of separable orderings on X . Let Ri ⊂ R (X ) be the domain of admissible orderings on X for agent i. Let Dik be the class of sector-k conditional orderings of the outcomes sep of X k induced by elements of Ri , that is,

 Dik = Rik |Ri ∈ Risep .

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Finally

RI =

I 

Ri ,

i=1

A social choice correspondence is a set-valued mapping ϕ : R I  X . Also, later will provide a derivative concept of sector-specific objective, which is given in the form ϕ k : DkI  X k for each  k = 1, . . . , n. Mechanism for Sector k is a game form  k = Mik i∈I , h k , where • Mik is the message space of agent i in Sector k, and • h k : M s → X k : outcome function in Sector k  Game form  k together with R k ∈ DkI defines a game  k , R k in k k Sector Say that a message profile m ∈ M is a Nash equilibrium of  k k. k  , R if for all i = 1, . . . , I , it holds     for all m¯ ik ∈ Mik : h m k Rik h m¯ ik , m k−i .  Let N E( s , R k ) denotes set of Nash equilibrium profiles of  k , R k .  Then h k N E( k , R k ) is the set of Nash equilibrium outcomes in  k k  ,R  I , h , where The whole game form is given by  = (Mi )i=1 • Mi = nk=1 Mik is the message space of agent i • h : M → X is the outcome function, which maps m∈M=

I 

Mi

i=1

into a unique outcome in X such that    h (m) = h k m k

k=1,...,n

.

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 Now the game form  ≡  k k=1,...,n and a profile R ∈ R I induce a game (, R). Say that a message profile in the whole game m ∈ M is a Nash equilibrium of (, R) if for all i = 1, . . . , I , it holds for all m¯ i ∈ Mi : h (m) Ri h (m¯ i , m −i ) , Let N E(, R) denote the set of Nash equilibrium profiles of (, R) and h (N E(, R)) denote the set of Nash equilibrium outcomes of (, R). The following is a basic but helpful observation, saying that preferences are in fact separable the set of Nash equilibrium message profiles is the product of Nash equilibrium message profiles of each component game played with marginal preferences. Lemma 8.1 Let  be a product set of partial equilibrium mechanisms. For sep all R ∈ R I , n  N E(, R) = N E( k , R k ). k=1

Now, the definition of implementability in partial equilibrium mechanisms is provided. Definition 8.1 The social choice correspondence ϕ : R I  X is Nash implementable in partial equilibrium if there exist a product set of par tial equilibrium mechanisms  and a sequence ϕ k k=1,...,n of onedimensional social choice correspondences, where ϕ k : DkI  X k for all k = 1, . . . , n, such that: (i) for all R ∈ R I , it holds ϕ (R) = h(N E(, R)),   (ii) for all k = 1, . . . , n, it holds ϕ k R k = h k N E  k , R k for all R k ∈ DkI , (iii) for ∈ R I : m ∈ N E(, R) for some m ∈ M =⇒ m s ∈  sall R N E  , R¯ s for some R¯ k ∈ DkI , for all k = 1, . . . , n. The first condition states that the overall equilibrium outcomes must coincide with what central authority is trying to achieve. The second condition states that the equilibrium outcomes in each sector must coincide with what the sector authority is trying to achieve. The third condition

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states that central authority cannot notice any inconsistency between the general equilibrium way of generating equilibrium outcomes and the partial equilibrium way of generating equilibrium outcomes since otherwise the central authority will realize that there is something wrong in delegating sector-specific objectives to the sector-authorities.

8.3

Necessary Conditions

Here, we list the conditions which are necessary for implementation in partial equilibrium. First is that when preferences are in fact separable the decision in each sector depends only on a profile of marginal preferences over the alternatives in that sector. Definition 8.2 The social choice correspondence ϕ : R I  X is decomposable provided that there exists a list of correspondence ϕ k : DkI  X k , for k = 1, . . . , n, such that ϕ (R) =

n 

ϕ k (R k )

k=1

for each profile R ∈ R I

Rsep I (X ).

Theorem 8.1 If the social choice correspondence ϕ : R I  X is is Nash implementable in partial equilibrium then it is decomposable. The second condition is the standard Maskin monotonicity defined over the whole social choice correspondence. For any ordering Ri and outcome x, the weak lower contour set of Ri at x is defined by L (x, Ri ) = {x  ∈ X |x Ri x  }. Definition 8.3 The social choice correspondence ϕ : R I  X is Maskin monotonic provided that for all x ∈ X and all R, R  ∈ R I ,if x ∈ ϕ (R) and L(x, Ri ) ⊆ L(x, Ri ) for all i = 1, . . . , I , then x ∈ ϕ R  . Theorem 8.2 If the social choice correspondence ϕ : R I  X is Nash implementable in partial equilibrium then it is monotonic.

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The third condition is that when preferences are in fact separable and hence the social choice correspondence is decomposable the social choice correspondence in each sector is Maskin monotonic. Definition 8.4 The decomposable social choice correspondence ϕ : R I  X is sector-wise Maskin monotonic provided  that for all k = 1, . . ., all x k ∈ X k and all R k , R¯ k ∈ DkI if x k ∈ ϕ k R k and L(x k , Rik ) ⊆  L(x k , R¯ ik ) for all i = 1, . . . , I , then x k ∈ ϕ k R¯ k . Theorem 8.3 If the decomposable social choice correspondence ϕ : R I → X is Nash implementable in partial equilibrium then it is sector-wise monotonic. The last condition connects between the society-wide decision and sector-specific decisions. It says that a socially optimal choice under any given profile of preferences is optimal again when the profile changes into any one of separable preferences such that its sector-wise position in each sector should not go down. For presentation, let L k (x, Ri ) = {z k ∈ X k : x Ri (z k , x −k )} The condition is given as below. Definition 8.5 The social choice correspondence ϕ : R I  X is decomposable Maskin provided that for all x ∈ X , all R ∈ R I

monotonic sep  and all R ∈ R I R I (X ), if x ∈ ϕ (R) and L k (x, Ri ) ⊆ L k (x, Ri ) for all i = 1, . . . , I , for all k = 1, . . . , n, then x ∈ ϕ R  . Theorem 8.4 If the social choice correspondence ϕ : R I  X is Nash implementable in partial equilibrium then it is decomposable Maskin monotonic.

8.4

Impossibility of Efficient Social Choice

Here, let us verify that even when preferences are indeed separable “marginal preferences” do not provide sufficient information for determining what to maximize in which sector.

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To illustrate, consider a profile of separable preferences like below. A B (x 1 , x 2 ) (y 1 , y 2 ) (y 1 , x 2 ) (y 1 , x 2 ) (x 1 , y 2 ) (x 1 , y 2 ) (y 1 , y 2 ) (x 1 , x 2 ) In such abstract setting, application of the standard Gibbard–Satterthwaite argument tells us that we have to predetermine who should be prioritized in which sector independently of preference profiles. Let us say A is prioritized in Sector 1 and B is prioritized in Sector 2. Then in Sector 1 x 1 is chosen and in Sector 2 y 2 is chosen. However, the pair (x 1 , y 2 ) is Pareto-inferior. This may be just a coincidence that such wrong priority is just for this particular preference profile. However, one can provide another preference profile when the priority structure is reversed. As a generalization of the above observation, we obtain the following impossibility result.1 Theorem 8.5 Suppose I ≥ 2 and that n ≥ 2. For all k = 1, . . . , n, X k consists of at least two Sector-k outcomes. Let ϕ be a social choice correspondence satisfying the property of weak Pareto. Then, ϕ can be Nash implemented in partial equilibrium if and only if is dictatorial.

8.5

Sufficiency

Now, let us go in to the sufficiency characterization. Here, we add auxiliary conditions on sector-specific social choice correspondences, which are known to be the weakest ones in the standard sufficiency characterization of Nash-implementable social choice correspondences.2

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Definition 8.6 A decomposable social choice correspondence ϕ : R I  X satisfies sector-wise unanimity if for all k = 1, . . . , n, all x k ∈ X k and  k all  k k k k k k k R ∈ D I , if X ⊆ L x , Ri for all i = 1, . . . , I , then x ∈ ϕ R . Definition 8.7 A decomposable social choice correspondence ϕ : R I  k X satisfies sector-wise weak no veto-power  k ifkfor allk k= 1,k . . . , n,allk x k∈ k k k k k k X and all R , R¯ ∈ D I if x ∈ ϕ R  , y ∈ L x , Ri ⊆ L y , R¯ i for some i = 1, . . . , I and X k ⊆ L y k , R¯ kj for all j ∈ I \ {i}, then  y k ∈ ϕ k R¯ k . The important restriction we rely on is the domain condition named Property 1. P1 if for all R ∈ R I Property 1 (P1, for short): The domain R I satisfies

sep and all x ∈ X there exists a profile R¯ ∈ R I R I (X ) such that for every i = 1, . . . , I it holds that

and that

for all k = 1, . . . , n : L k (x, Ri ) ⊆ L k (x, R¯ i )

(8.1)

 L x, R¯ i ⊆ L (x, Ri ) .

(8.2)

Theorem 8.6 Let n ≥ 3. Suppose that the domain R I satisfies P1. The social choice correspondence ϕ : R I  X is Nash implementable in partial equilibrium if ϕ satisfies decomposability, Maskin monotonicity, sector-wise Maskin monotonicity, decomposable Maskin monotonicity, sector-wise unanimity, and sector-wise weak no veto-power. The proof is relegated to the appendix. The idea of it, which is by construction, is actually simple: just run Maskin’s canonical mechanism sector-wise, where in each sector the authority in charge asks each individual to submit information just about outcomes in that sector, together with a tie-breaking device. To understand, imagine that there are two sectors, one (Sector L) allocates left shoes and the other (Sector R) allocates right shoes. Then Sector L requires you submit just “preferences” about left shoes (together with a tie-breaking device), and Sector R requires to submit just “preferences”

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about right shoes (together with a tie-breaking device). If one receives a message saying that you intend to receive some left shoe, it is normal to understand that you intend to receive its counterpart. But in this mechanism the authority handling Sector L only understands that you intend to receive that particular left shoe just as an individual item, and the authority handling Sector R only understands that you intend to receive a particular right just shoe as an individual item. This sounds stupid, but this is the nature of the institutional constraint we face. We should note that the domain condition P1 is indispensable, not just for our particular construction of the implementing mechanism here. See Hayashi and Lombardi [1] for details, while the point is that if P1 is violated it is possible for one to make a profitable deviation by means of choosing to be punished in every sector.

8.6

Applications

How restrictive is the P1 condition? It is essentially ruling out severe complementarity so that it is possible to gain by means of getting worse off in each sector. Condition 8.1 (Abstract domain) For all Ri ∈ Ri and all x, y ∈ X it holds that x Ri (y k , x −k ) for all k = 1, . . . , n =⇒ x Ri y. In the auction domain, the above condition is equivalent to the following. Condition 8.2 (Auction domain) For all y 1 , z 1 ∈ Y 1 , y 2 , z 2 ∈ Y 2 and t 1 , t 2 ∈ H , if Ui (y 1 , y 2 , ti1 + ti2 + ei ) = Ui (z 1 , y 2 , ti1 + ti1 + ti2 + ei ) = Ui (y 1 , z 2 , ti1 + ti2 + ti2 + ei )

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then Ui (z 1 , z 2 , ti1 + ti1 + ti2 + ti2 + ei ) = Ui (y 1 , y 2 , ti1 + ti2 + ei ). This means that along an indifference set willingness to pay for a pair of items is the sum of willingness to pay for its components. This still allows non-separability due to income effects, though. Thus, we can take a unique equivalent separable/quasi-linear preference which has the same indifference set at a given point. Therefore, we can induce the individual to behave as if they have separable/quasi-linear preference in equilibrium. More precisely, in such a domain, given any Ri ∈ Ri and any y 1 , ∈ Y 1 , 2 y , ∈ Y 2 and t 1 , t 2 ∈ H , there is a unique equivalent separable/quasilinear preference R¯ i represented in the form U i (z 1 , z 2 , ai1 + ai2 + ei ) = v i1 (z 1 ) + v i2 (z 2 ) + ai1 + ai2 + ei such that and

v i1 (z 1 ) − v i1 (y 1 ) = −ti1 v i2 (z 2 ) − v i2 (y 2 ) = −ti2

hold whenever Ui (y 1 , y 2 , ti1 + ti2 + ei ) = Ui (z 1 , y 2 , ti1 + ti1 + ti2 + ei ) = Ui (y 1 , z 2 , ti1 + ti2 + ti2 + ei ) Here, we give two examples of social choice rules which are implementable in partial equilibrium. Provided that P1 is met, one can extend a series of single-sector social choice rules to a multi-sector one in a “monotonic” manner. Example 8.1 The social choice correspondence ϕVs C G : DkI  Y k × H is the sector k VCG solution provided that for all R k ∈ DkI and all  (q k , t k ) ∈ Y k × H , (q k , t k ) ∈ ϕVk C G R k if and only if it is a Vickery– Clarke–Groves outcome under R k .

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The social choice correspondence ϕ : R I  Y is the sector-wise VCG social choice correspondence provided that for all R ∈ R I and all x ∈ Y :  k (q, t) ∈ ϕ SV C G (R) ⇐⇒ (q k , t k ) ∈ ϕVk C G R k

for an arbitrary (R )k=1,...,n ∈ (q, t).

k=1,...,n

DkI which is equivalent to R at

Example 8.2 The social choice correspondence ϕ kM : DkI  Y s × H is the sector k competitive solution provided that for all R k ∈ DkI and all  (q k , t k ) ∈ Y k × H , (q k , t k ) ∈ ϕ kM R k if and only if there exists p k such that tik = − p k qik and qik solves max vik (z ik ) − p k z ik ,

p k z k ∈H

where vik describes willingness-to-pay exhibited by Rik . The social choice correspondence ϕ : R I  Y is the sector-wise competitive social choice correspondence provided that for all R ∈ R I and all x ∈ Y :  k (q, t) ∈ ϕ S M (R) ⇐⇒ (q k , t k ) ∈ ϕ kM R k

for an arbitrary (R )k=1,...,n ∈ (q, t).

8.7

k=1,...,n

DkI which is equivalent to R at

Appendix: Proof of the Sufficiency Theorem

The proof is based on the construction of a product set of partial e k quilibrium mechanisms  =  k=1,...,n , where sector-k mechanism,   k = M k , h k , is a canonical mechanism due to Maskin [4] . Sector kmechanism:

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Agent i’s message space is defined by3 Mik = DkI × X k × Z+ , where Z+ is the set of nonnegative integers. Thus, agent i ’s strategy message of an outcome in X k , a profile of orderings and a nonnegative integer. a typical message sent by agent i is denoted by m ik =   Thus,  i i R k , x k , z i . The message space of agents is the product space Mk =

I 

Mik ,

i=1

with m s as a typical strategy profile. The outcome function h k is defined with the following three rules:     i i Rule 1: If m ik = R k , x k , 0 = R¯ k , x k , 0 for each agent i =   1, . . . , I and x k ∈ ϕ k R¯ k , then h k m k = x k .   Rule 2: If n − 1 agents send m kj = R¯ k , x k , 0 with x k = ϕ k R¯ k , but     i i agent i sends m ik = R k , x k , z i = R¯ k , x k , 0 , then we can have two cases:  i   i 1. If x k R¯ ik x k , then h k m k = x k .  i   2. If x k P R¯ ik x k , then h k m k = x k . Rule 3: Otherwise, an integer game is played: identify the agent who plays the highest integer (if there is a tie at the top, pick the agent with the lowest index among them). This agent is declared the winner of the game and the alternative implemented is the one she selects.  Since ϕ is decomposable, there exists a sequence ϕ k k=1,...,n of onedimensional social choice correspondences, where ϕ k : DkI  X k for each k = 1, . . . , n. Also, note that the proof of part (ii) of Definition 8.1 follows very closely the proof of Repullo (1987; pp. 40–41) given that

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ϕ k satisfies sector-wise Maskin monotonicity, sector-wise weak no vetopower and sector-wise unanimity. To complete the proof, we show that part (i) and part (iii) of Definition 8.1 are satisfied as well. Thus, let us fix any R ∈ R I . Nothing has to be proved orderings because  k ofkseparable if R is a profile it holds N E (, R) = k=1,...,n N E  , R and that part (ii) and part (iii) of Definition 8.1 hold. So let us suppose that R is in general nonseparable. Let us first show that h (N E(, R)) ⊆ ϕ (R). Take any x ∈  E(, R)). Then, there exists m ∈ N E(, R) such that h (m) = h (N h k m k k=1,...,n = x. Then, given that m ∈ N E(, R), for each agent i = 1, . . . , I it holds that      ˆ ˆ for each k = 1, . . . , n : h k m k Rik h k m k

k ˆ =k

  for each mˆ ik ∈ Mik . , h k mˆ ik , m k−i

(8.3) Given that R I satisfies P1, there exists a profile R¯ ∈ R I such that for all i = 1, . . . , I , it holds that

 for all k = 1, . . . , n : L k (h (m) , Ri ) ⊆ L k h (m) , R¯ i and that

 L h (m) , R¯ i ⊆ L (h (m) , Ri ) .

Rsep I (X ) (8.4)

(8.5)

sep Then, from (8.3) and (8.4) and from the fact that R¯ ∈ R I R I (X ), it follows that for all i = 1, . . . , I and all k = 1, . . . , n it holds that     h k m k R¯ ik h k m˜ ik , m k−i for all m˜ ik ∈ Mik , and so

  for all k = 1, . . . , n : m k ∈ N E  k , R¯ k .

 Part (ii) of Definition 8.1 and decomposability imply that h (m) ∈ ϕ R¯ . Since the SCR ϕ is Maskin monotonic and since, moreover, (8.5) holds for every agent, it follows that h (m) ∈ ϕ (R). This completes the proof

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that h (N E(, R)) ⊆ ϕ (R). Note that the preceding arguments also show that part (iii) of Definition 8.1 is met as well. For the converse, suppose that x ∈ ϕ (R). Given that the R I satisfies

sep P1, it follows that there exists a profile R¯ ∈ R I R I (X ) such that for all i = 1, . . . , I , it holds that

and that

 for all k = 1, . . . , n : L k (x, Ri ) ⊆ L k x, R¯ i

(8.6)

 L x, R¯ i ⊆ L (x, Ri ) .

(8.7)

Since x ∈ ϕ (R) and since, moreover, (8.6) holds for each agent i and each  sector s, decomposable Maskin monotonicity

sep implies that x ∈ ϕ R¯ . Furthermore, given that R¯ ∈ R I R I (X ), decom posability implies ϕ R¯ = k=1,...,n ϕ k ( R¯ k ), where R¯ k ∈ DkI is the ¯ profile of sector-k marginal (ii) of Defi R.Part  orderings inducedk by k ¯ ¯ nition 8.1 implies that ϕ R = k=1,...,n h N E  , R k . More   over, given that N E , R¯ = k=1,...,n h k N E  k , R¯ k , we have    ϕ R¯ = h N E , R¯ . Thus, there exists m ∈ M such that h (m) = x and     for all i = 1, . . . , I : h m i , m −i ∈ X |m i ∈ Mi ⊆ L x, R¯ i . Finally, given that (8.7) holds, it follows that    for all i = 1, . . . , I : h m i , m −i ∈ X |m i ∈ Mi ⊆ L (x, Ri ) . We have that h (m) = x ∈ h (N E (, R)). Thus, we have established part (i) of Definition 8.1

Notes 1. See Barberà et al. [2], Le Breton and Sen [3] for related results in the setting of strategy-proof social choice functions over multiple sectors with separable preferences.

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2. See for example Maskin [4], Moore and Repullo [5], Maskin and Sjöström [6], Sjöström [7]. 3. Note that Dik is nonempty for each agent i = 1, . . . , I since P1 holds.

References 1. Hayashi, T., and M. Lombardi. 2017. Implementation in partial equilibrium. Journal of Economic Theory 169: 13–34. 2. Barberà, S., H. Sonnenschein, and L. Zhou. 1991. Voting by committees. Econometrica 59: 595–609. 3. Le Breton, M., and A. Sen. 1999. Separable preferences, strategyproofness and decomposability. Econometrica 67: 605–628. 4. Maskin, E. 1999. Nash equilibrium and welfare optimality. Review of Economic Studies 66: 23–38. 5. Moore, J., and R. Repullo. 1990. Nash implementation: A full characterization. Econometrica 58: 1083–1100. 6. Maskin, E., and T. Sjöström. 2002. Implementation theory. In Handbook of social choice and welfare, ed. K. Arrow, A.K. Sen, and K. Suzumura, 237–288. Amsterdam: Elsevier Science. 7. Sjöström, T. 1991. On the necessary and sufficient conditions for Nash implementation. Social Choice and Welfare 8: 333–340.

Bibliography

1. Arrow, K.J. 1964. Le role des valeurs boursieres pour la repartition la meilleure des risques. In Econometrie, 41–47. Paris: CNRS (translated as The role of securities in the optimal allocation of risk-bearing. Review of Economic Studies 31: 91–96). 2. Bewley, T. 1972. Existence of equilibria in economies with infinitely many commodities. Journal of Economic Theory 4: 514–540. 3. Gibbard, A. 1973. Manipulation of voting schemes: A general result. Econometrica 41: 587–601. 4. Hurwicz, L., and D. Schmeidler. 1978. Outcome function which guarantee the existence and Pareto optimality of Nash equilibria. Econometrica 46: 144– 174. 5. Satterthwaite, M. 1975. Strategyproofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10: 187–217.

© The Editor(s) (if applicable) and The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2

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Index

A Approximation error, 96 Arrovian social welfare function, 25 Arrow-Debreu equilibrium, 31 Asymptotic quasi-linearity, 112, 121, 128 B Bergson–Samuelson social welfare function, 27, 55, 63, 89 C Change in consumer surplus, 42 Compensated demand function, 10, 111 Compensating variation, 38, 46, 56, 81 Compensation principle, 18 Competitive equilibrium, 14, 134

Constrained Pareto-efficiency, 34 Consumer surplus, 41, 50, 61, 75, 81, 134 D Deadweight loss, 96, 110 Demand function, 9 E Engel aggregation condition, 118, 119 Equivalent variation, 38, 45, 55, 81, 110 Exchange economy, 14 Expected consumer surplus, 137, 156 Expected utility, 30, 138, 144 Expenditure function, 10 Expenditure minimization, 10

© The Editor(s) (if applicable) and The Author(s) 2017 T. Hayashi, General Equilibrium Foundation of Partial Equilibrium Analysis, DOI 10.1007/978-3-319-56696-2

183

184

Index

F First welfare theorem, 15 G Game form, 168, 169 Gorman paradox, 24 H Hicks criterion, 22, 87 Hicksian aggregation, 120, 125, 146 I Income compensation function, 11 Income elasticity of demand, 93, 96 Indirect utility function, 10, 138 Induced 2-good preference, 112, 120, 125 Inverse demand function, 41, 80, 83 K Kaldor criterion, 18, 87, 156 M Marginal rate of substitution, 12, 77 Marginal utility density, 129 Market completeness, 156 Mechanism, 168, 169 Mechanism design, 163 Message space, 168, 169 N Nash equilibrium, 168, 169 Negishi weights, 28 No income effect, 73, 82, 138, 142, 155

O Ordinal utility, 8 Outcome function, 168, 169 P Pareto efficiency, 15, 85 Partial equilibrium implementation, 169 Partial equilibrium mechanism, 163 Path-independence, 50, 52, 62 Preference ordering, 7 Q Quasi-linear preference, 72, 82, 121, 128 R Radner equilibrium, 32 Risk neutrality, 138, 142, 155 Roy’s identity, 14 S Samuelson criterion, 24 Scitovsky criterion, 23 Second welfare theorem, 16 Separable preference, 167 Shepard’s lemma, 13 Slutsky equation, 13, 81, 111 Small income effects, 99 Social choice correspondence, 168 Social surplus, 84 U Uniform boundedness of sensitivity, 127, 151

Index

Uniform Inada property, 101, 124, 145

V von-Neumann/Morgenstern index, 30, 138, 144

Utility function, 8 Utility maximization, 9

185

W Willingness to pay, 75