A Characterization of Egalitarian EquivalenceAuthor(s): Bhaskar Dutta and Rajiv VohraReviewed work(s):Source: Economic Theory, Vol. 3, No. 3 (Jul., 1993), pp. 465-479Published by: SpringerStable URL: http://www.jstor.org/stable/25054715 .Accessed: 27/02/2012 15:33

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Econ. Theory 3,465-479 (1993) "i^

Economic Theory

? Springer-Verlag 1993

A characterization of egalitarian equivalence*

Bhaskar Dutta1 and Rajiv Vohra2 1

Indian Statistical Institute, 7 S.J.S. Sansanwal Marg. New Delhi 110016, INDIA 2

Department of Economics, Brown University, Providence, R.I. 02912, USA

Received: April 21,1992; revised version September 14,1992

Summary. Consider a solution (an allocation rule) for an economy which satisfies

the following criteria: (1) Pareto efficiency, (2) monotonicity, in the sense that if the

set of attainable allocations of the economy becomes larger then the solution makes no consumer worse-off, (3) a weak and primitive notion of fairness with respect to some commodity, say commodity h, in the sense that in an exchange economy in

which the aggregate endowment consists only of commodity h, the solution is equal division. We show that in the class of economies which includes non-convex

technologies the only such solution is egalitarian equivalence with respect to

commodity h. It is also shown that this characterization of egalitarian equivalence holds in convex exchange economies if we add a weak version of a positive association requirement. [JEL Classification Nos.: 021, 022, 024, 025.]

1 Introduction

Consider a solution which allocates resources efficiently and equitably. While it is

natural to consider Pareto optimality as the efficiency criterion, there does not seem

to be a universally accepted notion of fairness. In particular, the no-envy concept of Foley (1967) and the notion of egalitarian equivalence due to Pazner and

Schmeidler (1978) have quite different properties. For instance, they differ with

respect to the monotonicity property, which imposes the requirement that if the set

of attainable allocations becomes larger, then the solution should make no con

sumer worse-off. This requirement has been termed resource monotonicity in

the case where endowments increase and technological monotonicity in the case

when the technology improves; see Chun and Thomson (1988), Roemer (1986) and

Moulin (1987a, 1987b). It has been shown by Moulin and Thomson (1988) that

there is no solution which satisfies resource monotonicity, Pareto efficiency and the

* We are grateful to William Thomson and three anonymous referees for extensive comments on an

earlier version. We also acknowledge helpful comments of the participants of the Social Choice and

Welfare Conference held in Caen, June 1992.

466 B. Dutta and R. Vohra

no-envy property. On the other hand, the notion of egalitarian equivalence is

compatible with monotonicity; see, for example, Thomson (1987, Theorem 8).1 The purpose of this paper is to characterize monotonie solutions which are

efficient and equitable. In view of the negative result of Moulin and Thomson (1988), we adopt a weak and natural notion of fairness; one which is satisfied by the no-envy solution as well as by certain kinds of egalitarian solutions. Consider an exchange economy in which the endowment consists only of one commodity. We take the

position that any notion of fairness of equity must prescribe equal division as the

solution in this case.2 Notice that if preferences are monotonie, then equal division

is also efficient. We show that in the class of economies which includes non-convex

technologies, the only efficient solution which satisfies this equity axiom with respect to commodity h and monotonicity is egalitarian equivalence with commodity h as

the reference bundle. This characterization also holds in convex exchange economies if we add a weak version of a positive association requirement.

Of course, in most situations, one would like to impose the equity axiom on all

commodities since there does not seem to be any compelling argument to

distinguish between commodities in this context. However, a corollary of our

characterization results is that this is incompatible with monotonicity. Thus, these

results, once again, demonstrate the power of the monotonicity axiom in a setting different from those of Kalai (1977) and Moulin (1987a, 1987b).

2 Definitions and main results

We shall consider an economy with n consumers, / commodities and, for the sake

of notational simplicity, a single firm. We shall denote by Rz the / dimensional

Euclidean space and by R+ its positive orthant. We shall use the convention ?,

>, > to order vectors. For a vector zeR', we denote by zh the /i-th coordinate of z. For he {1,...,/}, let Xh denote the vector in R' such that xh. = 1 if/

= h and xhj = 0

if y 7e h. The set of consumers is denoted N = {1,..., n}. We shall assume that each

consumer i has a consumption set Xt which is R+ and preferences which can be

represented by a continuous utility function w^R+h+R. We have chosen to

describe preferences by utility functions only for notational convenience. We shall

be concerned only with ordinal preferences; all our definitions and results can be

presented for economies with preference relations which are reflexive, transitive and

complete. The production set is denoted Y s R' and the aggregate endowment

coeR+. We shall use x to denote a consumption plan, i.e., x = (xi)6f[jA'?. For any

i and k, xik denotes consumer f s consumption of commodity k. For a consumption

plan x, we shall use u(x) to denote the corresponding vector of utilities, i.e.,

u(x) =

(Ui(x?))ni=1. Similarly, we shall use u to denote the collection of utility functions

(Ui)ni= j. An economy can now be defined as e = (u, , Y).

1 It is interesting to note that in some models of indivisible commodities there do exist solutions which

are monotonie, efficient, egalitarian equivalent as well as envy-free; see Alkan, D?mange and Gale

(1991) and Tadenuma and Thomson (1989). 2

Recognizing that in contrast to this position, it may also be legitimate to consider equity notions based

on inter personal utility comparisons that do not necessarily prescribe equal consumption in this

case.

Egalitarian equivalence 467

The set o? allocations or the attainable set of an economy e = (u, 9 Y) is defined as

5>| y+ {?}!.

An allocation x is said to be Pareto optimal if there does not exist another allocation x such that u(x) > u(x), i.e., if there does not exist another allocation x such that

Ui(Xi) > Ui(Xi) for all ieN and for some consumer ?, uv(xv) > uv(xv). We shall use @>(e) to denote the set of Pareto optimal allocations of e.

Let S be the class of economies satisfying the following conditions:

(A) for all isN, ut is quasi-concave, monotonie (in the sense that x?, z?eR + and

z? > Xi implies that ut(z^ > ??(xj); u, is normalized such that m?(0) = 0; for all

x^R^ and all fce{l,..., /}, there exists / > 0 such that u^X;) =

uf(?x*)

(B) Y is closed, contains 0, satisfies free disposal (in the sense that Y ? R+ ^ Y)

andrnRf+ =

{0}. (C) ?/(e) is bounded.

Notice that S is a class of economies in which the production set is not

necessarily convex. This will play an important role in our first result. The last part of assumption (A) requires that no indifference curve is asymptotic to any axis. While

this is not a weak assumption3 it is essential for both our characterization results.

Assumptions (A) and (B) imply that s?(?) is non-empty and closed. An exchange

economy is an economy for which Y = ? R+ and we shall use Se to refer to the

class of such economies.

A solution for a class of economies S is a correspondence <j) which assigns to

every economy in ? a non-empty subset of its allocations, i.e., <\>:?\-+s?(?) is a

mapping such that for all ee?9 </>(e) is_a non-empty subset of s/(e). A solution for a class of economies S is said to be Pareto optimal if for all esS,

<f>(e) ? 0>(e). A solution for a class of economies S is said to be essentially single-valued if

for all eeS9 x,x'e</>(e), u(x) =

u(x'). Thus, for an essentially single-valued solution <j>9 we can denote by u((?>(e)) the

unique utility profile corresponding to the set of solutions for economy e.

Suppose in an exchange economy, the only endowment consists of a single

commodity, say commodity h. Given the monotonicity of preferences, all allocations

in this economy such that aggregate consumption equals the aggregate endowment

are Pareto optimal. Moreover, in this special setting, equal division of the

endowment is a very weak and natural requirement of equity. We shall take this to

be our primitive equity axiom.

Axiom Qh: a solution (?> is said to satisfy the axiom of equal division with respect to

commodity h if for all e = (u9 9 ?JRJ^e?'6 such that k = 0 for all k^h, (?)(e)

=

\

^(e) = Lsl\Xi

3 It does not follow, for instance, from the monotonicity of preferences; see footnote 4.

468 B. Dutta and R. Vohra

Let the utility possibility set of an economy e = (u,co, Y) be denoted

U(e) =

{weRw|there exists xes?(?) such that u = u(x)}.

Let

dU(e) =

{meU(e)\there does not exist u'eU(e) such that u' > u}.

Given the assumption that the utility functions are monotonie, for all ee?, the set

of Pareto optimal allocations is simply the set of allocations corresponding to dU(e), i.e.,

P(e) = {xes/(e)\u(x)edU(e)}.

Notice also that, by the monotonicity of the utility functions, if xe3?(e) and e is an

exchange economy, then x must satisfy the feasibility condition ?. xx = co, despite the fact that we have allowed for free disposal by defining an exchange economy to

have the production set -Rz+ Given our normalization of the utility functions, it is easy to see that for all ee?,

the sets U(e) and dU(e) are non-empty compact subsets of R"+. Moreover, if

dU(e) # {0}, dU(e) is homeomorphic to the unit simplex in R".

We shall formulate our monotonicity axiom to encompass both resource

monotonicity and technological monotonicity.

Axiom M (Monotonicity): a solution <f>for a class of economies ? is said to satisfy the monotonicity axiom if

for all e = (u, co, Y), e' =

(u, co', Y')eS such that U(e) ? U(e'),

x'e(j)(e') implies that u(x') > u(x) for some xe<j)(e)

and

xe(f>(e) implies that u(x') > u(x) for some x'e(e').

An essentially single-valued solution concept (?> for a class of economies ? satisfies

axiom M if for all e = (u, co, Y), e' = (u, co', Y')e? such that U(e)

<= U(e'), u(<?>(e')) >

u((j>(e)). Notice that this axiom can be easily restated in ordinal terms by replacing

U(e) ? U(e') with ??(e) ? s?(e') and by making obvious changes to the statements

made in terms of utilities.

Pazner and Schmeidler (1978) introduced the following solution.

An allocation x is said to be egalitarian equivalent if there exists a reference

commodity bundle zeR+ such that u{(x^ =

ut(z) for all i.

Notice that the reference consumption plan corresponding to z, i.e., (z,..., z) may

not be a feasible allocation. Suppose ee?. Let zeR'+ be such that z?0. For all

non-negative real numbers X, the consumption plan (Xz,...,Xz) traces out a

monotonie path in the utility space. Given monotonicity of preferences and the

compactness of the attainable set, it is easy to see that if z ? 0 then there exists a

unique number X* such that u* = u(X*z,..., X*z)edU(e). This implies that all xe s/(e)

such that u(x) = u* are Pareto optimal and egalitarian equivalent allocations. This

is essentially the argument used by Pazner and Schmeidler to prove the existence

of such allocations. Notice also that this construction guarantees that such solutions

Egalitarian equivalence 469

satisfy axiom M. Corresponding to every zeR' such that z ? 0, we can define Pareto

optimal, egalitarian, equivalent allocations, and we shall denote the corresponding solution ?*, where

E*(e) = {xe&(e)\ there exists ? > 0 such that ??(xj = ut(Xz) for all f}.

It is easy to check that the above argument for the existence of such solutions can

be used for any non-zero vector zeR+ provided we assume that all indifference curves cross every axis,4 are is indeed assumed in condition (A) above. Given a

commodity h we shall consider the egalitarian equivalent solution based on taking as a reference commodity bundle one which consists only of commodity h. Clearly, this solution will satisfy axiom Qh.

The egalitarian equivalent solution with respect to h is defined as

E*(e) =

{x 0>(e)\ there exists ? > 0 such that u^x,) =

u^?Xh) for all ?}.

While E* satisfies axiom M for any z > 0, unless z = Xxh it may not satisfy axiom

Qh. This can be illustrated by a simple example of a two-commodity, two-consumer

exchange economy. The aggregate endowment is (3,0) and the utility functions are

w1(x1) =

0.5x11 +x12,

^2(^2)= x2X + x22.

Clearly, E*(e), the egalitarian equivalent allocation with respect to commodity 1 is

the allocation ((1.5,0), (1.5,0)). However, E*(e) is the allocation ((2,0), (1,0)). Thus,

E\ violates axiom Qx. As is well known, the set of Pareto optimal and envy-free allocations do satisfy

axiom Qh for all h but not axiom M; Moulin and Thomson (1988). Vohra (1992) shows that while the set of Pareto optimal and envy-free allocations may be empty for some eeS, it is possible to weaken the no-envy concept and define the set of

essentially envy-free allocations which exists generally and has many of the

properties of envy-free allocations. In particular, the set of Pareto optimal and

essentially envy-free allocations is non-empty for all ee<? and satisfies axiom Qh for

all h but does not satisfy axiom M.

The last two paragraphs establish that axioms Qh and M are independent. We

have also argued that ?jf is a solution which satisfies both axiom Qh and M. Our

first result shows that every solution defined over S and satisfying Qh and M must

be contained in ?J.

Theorem 1. Suppose <?> is a Pareto optimal solution defined over S and satisfies axioms

QhandM. Then<?><^E*.

Remark 1. Theorem 1 shows that ?? must contain any solution satisfying the

hypotheses of Theorem 1. In order to obtain a "complete" characterization of ?J,

4 Monotonicity of preferences is not enough. For example, if utility functions have the form

u(x) =1-1- x2 and z = (1,0) it is possible that the utility path traced by (Az,..., Xz) does not cross

1 +Xj

dU(e).

470 B. Dutta and R. Vohra

we need to specify that <?> satisfies the additional property of "Pareto indifference",5

namely that if x is in 0(e) for some e and if all individuals are indifferent between x

and y, where y is also a feasible allocation, then y should also be in <f)(e). Of course, this characterization of E% depends on the choice of commodity h to

which the axiom of equal division is applied. It would be natural to explore the

possibilities of imposing the axiom of equal division on every commodity. However, no such solution can satisfy axiom M. This is an immediate corollary of

Theorem 1 based on the observation that for two commodities h,k,h^ k, E% / Ef (as shown by the simple example above).

Corollary 1. There does not exist a Pareto optimal solution defined over ? which

satisfies axioms M, Qh and Qk for k^h.

A weakness of Theorem 1 is that it concerns a characterization for a class of

economies in which the production sets are not necessarily convex. It would be

interesting to see if this characterization holds even if the monotonicity axiom is

imposed only on convex economies. Our next result shows that in convex exchange

economies, such a characterization is indeed possible with an added axiom of

positive association.

Axiom WPAh: a solution (?> defined over a class of economies ? is said to satisfy weak

positive association with respect to h if for all (u,co, Y), (?,co, Y)e?, xe4>(u, , Y)

implies that xe(p(u, , Y) whenever

(i) for all i, and y.-eR^, ufa) > ufa),

(ii) for all i, if ut(yi) # fif(y?) for some ^/eRz+ then there exists Xt > 0 such that

ufa) =

ufaXhX and yih > Xt.

This axiom relates the solution across economies in which attainable sets remain

the same but preferences change. It can be stated purely in ordinal terms. It states

that if x is a solution under some preferences then it must remain a solution if

preferences are changed as follows. For every i define /? such that x? is indifferent

for i to XiXh- By assumption (A) there does exist such a Xt. (If indifference curves do

not cross the h axis, axiom WPAh is vacuously satisfied.) Preferences remain

unchanged over all commodity bundles which involve no more than Xt units of

commodity h and a commodity bundle involving more than /f units of commodity h is made less desirable under the new preferences. See figure 1, where the dashed

curves represent the change in the indifference map. It is worthwhile to observe that axiom WPAh is weaker than Maskin's (1977)

monotonicity assumption. There are two differences. In conditions (i) and (ii) of

WPAh the commodity vectors (yt) are not required to constitute a feasible allocation.

Moreover, w, and m? are required to agree over a subset of the consumption set. In

particular, (this is the crucial difference) the indifference curve through x? remains

unchanged. Axiom WPAh is, therefore, considerably weaker than Maskin's

monotonicity axiom. ?* need not satisfy Maskin's monotonicity axiom but it does

satisfy WPAh for any z > 0. The same is true of the Walrasian correspondence.

5 It bears mentioning that Pareto indifference may not be an innocuous requirement. For instance,

it is not satisfied by the no-envy solution.

Egalitarian equivalence

Figure 1

Theorem 2. Suppose (?> is a Pareto optimal solution defined over Se and satisfies axioms Qh,M and WPAh. Then (?)^E*.

Remark 2. It is natural to ask if assumption WPAh can be dropped from the

statement of Theorem 2. We can provide a partial answer to this question. We shall

show that in a certain class of economies, axiom WPAh cannot be dropped from

the statement of Theorem 2. Consider the class of two-consumer exchange economies in which both consumers have monotonie, convex preferences and no

good is inferior for any agent. Theorem 2 applies to this class of economies, as will

be clear from the proof of Theorem 2. However, it is possible to use results in Moulin

(1989) to prove the following Proposition, the proof of which is provided in the

following section.

Proposition 1. Consider the class of two-consumer exchange economies in which both

consumers have monotonie, convex preferences and no good is inferior for any agent. In this class of economies, there exists a Pareto optimal solution which satisfies M and

Qkforallk =

{l,..:,l}.

Since El does not satisfy Qk for all fe = {l,...,/}, this establishes that axiom

WPAh cannot be dropped from the statement of Theorem 2. This also establishes

that WPAh is independent of Qh and M. Indeed all the axioms of Theorem 2 are

independent; the set of envy-free and Pareto optimal allocations satisfies WPAh and

472 B. Dutta and R. Vohra

Qh for ail h but not necessarily M, and ?* (for z # #*) satisfies WPAh and M but not

necessarily Qh. Remark 1 applies to this result as well. The same argument that allowed us to

derive Corollary 1 from Theorem 1 can be used to derive from Theorem 2:

Corollary 2. There does not exist a Pareto optimal solution defined over ?e which

satisfies axioms M, WPAh, Qh and Qk for k + h.

Since El does not satisfy Maskin's monotonicity axiom, which is stronger than

WPAh, we can appeal to Maskin's (1977) result to state:

Corollary 3. There does not exist a Pareto optimal solution defined over ?e which

satisfies axioms M and Qhfor some h and can be implemented in Nash equilibrium.

Remark 3.6 A conclusion similar to that of Corollary 3 can also be obtained from results of Hurwicz (1979) and Thomson (1979). Hurwicz (1979) showed that under certain conditions a solution which is efficient and individually rational and can be

implemented must contain the Walrasian correspondence. Thomson (1979) showed that this result remains valid if we replace the individual rationality requirement by an equity axiom7 and the conclusion concerning 'Walrasian correspondence' to 'Walrasian correspondence from equal division'. Since the Walrasian corre

spondence from equal division does not satisfy M (Moulin and Thomson (1988)), this establishes the tension between monotonicity, implementation and certain kinds of equity notions. Corollary 3 is concerned with Qh as the equity notion. We do not know whether Corollary 3 itself can be derived from the arguments of Hurwicz (1979) and Thomson (1979).

3 Proofs

Proof of Theorem 1: Let 0 be a solution defined over ? satisfying the hypotheses of Theorem 1. Suppose <^?jf. This implies that there exists an economy e =

(u9co9 Y)e? and xe(?>(e) such that x??jf(e). As pointed out earlier, ?jf defined over ? is a well-defined, Pareto optimal, essentially single-valued solution. Thus, there exists x*eEh?(e) and >l>0 such that ut(xf)

= Ui(Xxh) for all i. Let v* =

(ux(Xxt)9..., un(Xxk)) =

u(El(e)) and let v = w(x). Since Ejf satisfies Pareto indifference

and x$E%, this implies that v # v*. Of course, v, v*edU(e) which, given monotonicity of the utility functions, yields

v? > vf for some i and Vj< vj for some j. (1)

Define an exchange economy ? = (u9a>'9

? Rz+) where d =

nXxh. Notice that the

utility functions are identical in the economies e and ? and that ?e?. By axiom

Qh, <j>(?) =

(Xxh9..., Xxh). Thus, u(x') = v* for all x'e<j)(?y, see8 Figure 2.

6 We owe this observation to William Thomson.

7 For example, one weak equity axiom which can be used here is the following. In an exchange economy

in which all indifference curves are linear and equal division is Pareto optimal, the solution

contains all allocations Pareto indifferent to equal division. 8

The convexity of the utility sets in figure 2 is not important in the proof. Note also that v could be to

the right of v* and in the interior of U(e').

Egalitarian equivalence

Figure 2

Now define an economy in which the attainable set is the union of the attainable

sets of e and ?. In particular, let e" = (u, co', Y') where

r = {/eRV < y for some ye(Y + {o

- co'})u {0}}.

It is easy to see that Y' is closed and satisfies free disposal.9 Moreover,

Y'nR+ =

{0}. In fact, y'eY' implies that y'h < 0. Otherwise there would exist ye Y

such that yh + (oh ?

coh>0, implying that co'eY + {w} and U(?)^ U(e). Axiom M

then implies that v>v*, contradicting (1). Thus, Y' satisfies assumption (B) and

e"eS. Notice that y'eY' if and only if /<0 or there exists ye Y such that

y < y + (o ? ', i.e., either y' < 0 or y' +

' < for some ye Y. Given free disposal,

this establishes that s/(e") =

s?(e)Kjst(e'\ Thus, U(e")=U(e)KjU(e'). Since U(e") contains U(e') and w(x')

= v* for all x'e<t>(e')9 by axiom M, it now follows that

u(x")>v* forallx"E#?").

Since v*edU(e')and v*edU(e)9 it must be the case that v*edU(e")9 which implies that

u(x") = v* for all x"etf>(e"). (2)

On the other hand, since U(e") also contains U(e)9 axiom M implies that there exists

9 Note that Y' may not be convex even if Y is convex.

474 B. Dutta and R. Vohra

x"e(j)(e") such that u(x") > v. But this along with (2) must mean that v* > v9 which

contradicts (1) and completes the proof.

Proof of Theorem 2: Let 0 be a solution defined over Se satisfying the hypotheses of Theorem 2. As before, ?jf defined over Se is a well defined, Pareto optimal,

essentially single-valued solution. Suppose 0 ? ?*. Then there exists an economy e =

(u9co9 ?

1R\)eSe and xe<t>(e) such that x$E^(e). Let x*eE^(e) and A>0 such

that ut(xf) =

Ui(Xxh) for all i. Let v = u(x) and v* =

(m?(Axa)). Of course, v, v*edU(e). Since x$El(e) and ?j* satisfies Pareto indifference, this implies that

v ̂ v*. (3)

We can now define an economy e' = (u, ', ?R+), where a/ =

nXxh and, by axiom

Qh, u(<j)(e')) = v*. The utility possibility sets of these economies are sketched in

figure 2.

Let Di ? max (vf, vt) for all i. Certainly, since v # v*, we know that v > v, v > v*.

Since v*edU(e'), vedll(e) and the utility functions are monotonie this implies that

ti$U(e') and ?$U(e). Given assumption (A), we can now define an economy eb =

(u,(o + ?xk, -R+), where fc / /i, <5 > 0 and ve?U(e?). Commodity k^h will be

kept fixed in the rest of this proof. Of course, v* and v belong to the interior of U(e?). Notice that if U(e') is also contained in the interior of U(e?), then we could construct

an economy eb with b < S, such that v$JJ(e?) and both U(e) and U(e') are contained

in the interior of U(es). The result would then follow from axiom M. Of course, it is

possible that U(e') is not contained in U(e?). What we shall prove, however, is that

it is possible to change the utility functions of the consumers is accordance with the

requirements of axiom WPAh and construct three sequences of economies in which

this property holds. More precisely, we will construct eq, e'q and eq,s, where

s?(eq) =

st(e), s?(?q) =

rf(e'), s?(eq*) =

s?(e?) and for some q,

U(eq)ciInt[JJ(eq^)\ ve<j>(eq) U(e,q)aIntlU(eq>?)l v*e(j)(e,q) and ?edU(eq>?).

(4) (See figure 3.) The conclusion of the proof will then follow from axiom M.

Before we change the preferences, however, we shall identify, in claim 1, some

useful properties of U(e').

Claim 1. Suppose xes?(e') and u{(x^ > vt for some i. Then there exists y > 0 and (xt) such that:

(ii) u? Xj + ^Xk

) > Uj(xj) for some;,

(iii) Ui(Xi) = ?i for all i ?j, . ? ? . ( ^ y \ .<* , . *

IS *V

* n '""/

such that ut(?HXh) =

tV

Proof: For every / define A? such that ufa) =

u^Xh). By assumption (A), At is^well defined and unique. Let \

= max(A, A?). Certainly, xih < A, for all i and xfh < X for

Vm; *i\*i) ?

vx lui an t -r-j,

( y \ (iv) there exists z > 0 such that for all i, uA xf + -

xk > u&iXh + zlk\ where A?

Egalitarian equivalence

Figure 3

all i. Notice that ufaxn) =

#/ for all i. Since v$ U(?), this must mean that ?JL t %> nX.

Let ? =

?"= j X ? nX>0. Suppose xe U(?) and ufa) > v{ for some i, i.e., xi? > 1?. We

now claim:

there exists j such that xJ? < X?

? ?

n-\ (5)

If this were false, we would have ?"= t xih > ?"= \\

? ?

= nK which contradicts the

feasibility of x. Pick; satisfying (5) and let

Vi = uA X? ?

n-\ Xk)<Vy

By (5), Vj > Uj(Xj). Since veU(ea), it follows that

(vj9(?i)ii:j) belongs to the interior of

U(e?). This implies that we can choose 0 < a' < 5 such that (vj9 (?i)i^j)e?U(e?')9 where

e?' = (u9 + <5'xk,

? R'+). Of course, there exists xes/(e?) such that w(x)

= (vj9 (vt)ii:j).

Let y = 5 - ?'. Certainly, I x{r + -& I 6j/(e*) and satisfies conditions (i), (ii) and (iii)

of the claim. We shall now prove (iv). Let J((0?) be the indifference curve of consumer

i passing through 1^, i.e.,

Wi) = {yi?*\\uAyd

= ti}.

476 B. Dutta and R. Vohra

Now for every y?E/?(tSj), define

ij/fa) = <z?eR+ u? yt + -xk J = ufaXh + ZiXk)

Since preferences are monotonie, for every y^I^v^, \?tfa) is a unique positive number. Since the utility functions are continuous and I fa) is closed, it follows that

the set {\?ii(yi)\yi li(?d} is closed. Moreover, this set does not contain 0. Thus, for

every / there exists zi > 0 such that

if ufa) > vt then u? xt + J~xk J

> ufaxh + ?iXk)- (6)

For z = min???, it now follows that (iv) is satisfied and this completes the proof of

Claim 1.

We shall now define a transformation of the utility functions in accordance with the hypotheses of axiom WPAh. Let N be the set of positive integers. For every

qeN and for every i, let 0?:RV h->R'+ be the function defined as

g?(y)k =

yk if k * h

mjnf?>?i + *ziA iffc = fc

Rewriting % + ?- as yh + ?-?- it follows that gq(y) < y for all y and

gi(y) = y if ̂ < X' Now define, for every <je1K and every i, uq(-)

= Ui(gq(-)). Since ut

is monotonie,

for all i, q and for all y eRz+ , ufa > uq(y). (1)

Since gq(y) = y if yh < Xh we can also assert that

for all i and q, yeHl+, yh < Xh implies uq(y) =

ufa. (8)

In particular,

ufa) = wf(xf) for all i and q and u?(xf)

= uq(xf ) for all i and q. (9)

Claim 2: wf satisfies assumption (A) for all q and i.

Proof: From (7) and (8) it follows that uq maps from Rz+ to R+ and u?(0) = 0. Since

gq maps from R'+ to Rz+, it follows from assumption (A) that for every x^R^ and

k there exists X such that Mf(?f (xf)) =

Ui(X'xk)> Given the construction of g\, it is easy to see that there exists X such that u^X'xk)

= Ui{gqG>Xk))' We can now assert that for

every x^R^ and k, there exists I such that uq(xt) =

uq(XXk)- Since gq(y) is continuous

and monotonie, it follows that so is uqr It remains only to be shown that u\ is

quasi-concave. Let xjeR'+ and x = ax + (1 ?

oc)y for some <xe[0, 1]. We need to

show that u?(x) > min[wf(x), u^y)']. Notice that gqih as a function from R to R is

concave, i.e., gq(x)h > agq(x)h + (1 -

ot)gq(y)h. Thus,

^(x)>a^(x) + (l-a)^(>;).

Egalitarian equivalence 477

Since uq(x) =

Ui(gq(x)) and w? is monotonie, this implies that

W?(x)>Wf(a^(x) + (l-a)^(>;)) > min MgKx), u^yU

= min [uq (x), uq(y)\

where the second inequality follows from the quasi-concavity of t/f. And this

completes the proof of the claim.

Let eq and e,q be economies defined as:

eq = (uq9co9-Kl+)9 etq =

(uqMxh,-Rl+), eq>? = (uq9 + ?xk, -R'+).

Notice that for q = 1, eq = e, e'9 = e' and e*,d = e*. As q is increased, l/(e4) shrinks

but, given (9), always contains v and v*. Similarly, U(e'q) shrinks but always contains

v* and l)(eq,?) shrinks but always contains ?; see figures 2 and 3.

From claim 2 we know that eq9e,qeSe for all q. From (7) and (8), using axiom

WPAh we can assert that xe(/>(eq) for all q. Using (9), this yields,

xe(?>(eq)anduq(x) = v9 forall<?. (10)

By axiom Qh we can assert that (Xxh,..., Ax*) =

<?(e'*) for all q. Since m^Ax/,) =

ut(xf) =

!?f for all i, (9) now yields,

uq((f)(e,q)) = v* foralU. (11)

Certainly,

?/(^)c=/w?[1/(^)] for all 4. (12)

In order to complete the proof of (4), we will now show that q can be chosen large

enough so that U(e,q) is also contained in the interior of U(eq,?)9 i.e.,

xest(e'q) implies uq(x)elnt[U(eq?)l (13)

Suppose xes?(e') =

st(e'q). \ixih < X for all i, then uq(x) =

u(x) ? v. Since ?eU(eq,?)9 this completes the proof of (13). Suppose, therefore, that for some i, xih > 2?. The

hypotheses of Claim 1 are now satisfied. Let x, y and z be as in conditions (i)-(iv)

of Claim 1. Since ??(x,) < vi9 for all i, it follows from (8) that for all i, uA xf + ~xk ) =

( y \ v ? / u1[ x? + ~Xk ) Moreover, from the proof of claim 1, we also know that Uj(xj) < vj9

and, therefore, uqj(Xj) =

Uj(Xj). Thus, from (ii) of claim 1,

M?(^' +

nZfc)>M?(Xy)' (14)

Since Ui(Xi) =

ut for all i #7, we can appeal to (iv) of claim 1 and assert that

u'hi + y-x^Ui?ah

+ zXk) foralhV;. (15)

Notice that for all i, xf < nkxh. Thus

uq(Xi) < uq(nXXh) < u^X

+ n-^)x)j-

(16)

478 B. Dutta and R. Vohra

We can now find q large enough so that for all i,

Uif^X +

'^k) < ufaxn + zxk\ (17)

Now, (15), (16) and (17) yield,

mxi + -Xk)> u?(xi) f?r aU i *j for some <? ( 18)

Since I x? + -/k j est(eq\ (18) and (14) imply (13) and this completes the proof.

Proof of Proposition l:10 Throughout this proof we shall be considering exchange economies with the same preferences. It will, therefore, be notationally convenient to denote an economy by co and its utility possibility set by U(co). Moreover, for

simplicity, we assume that there are only two commodities, and leave it to the reader to check that the argument applies to any number of commodities. Moulin (1989) considers a very general framework for analyzing monotonie solutions. In the

present context, it reduces to the following. Suppose a solution <j> is such that for

every exchange economy satisfying the assumptions of Proposition 2, there is function d(co) that prescribes the lower bound on the consumers' utilities, i.e., for all

xecj>(co)9 u(x) > d(d). Of course, d( ) ? U(co). Suppose further, that if d > co, then

d(d) > d(co). Let u^(co) denote the maximum utility of consumer 1 subject to this

restriction, i.e., consumer l's utility when this consumer gets the entire surplus and consumer 2 gets utility d2(a>)). Thus (u*(co)9d2(co))edU( ). For co and d9 let

a d = (min (coh9 dh))lh=x. From Example 2, Lemma 2 and Theorem 3 of Moulin,

it follows that there exists a solution satisfying Pareto optimality and monotonicity if and only if:

(u*(co')9u^(co))iIntlU(u9co a co')], for all 9d. (19)

Suppose we define d(d) as follows

di(d)= max Ui(0.5 hxh). * = i,...,z

It is easy to see that this restriction on a solution implies that it satisfies axiom Qh for all h = 1,...,/. Moreover, d>co implies that d(d)>d(d). To complete the

proof, we will show that (19) holds with this specification of d(d). Recall that ?1(a>)=max[u1(0.5c>1,0), ?j1(0,0.5a>2)] and d2(d)

= max[u2(0.5d290)9

w2(0,0.5cy'2)]. This implies that either u\(d)>u^.Sd^d^ or u\(d)>Ui(d190.5d2). Ifd > co, then it would follow that u$(d) > d^co), implying that u^(co')9 u*(co)$ U(co). Since co = co a co', this establishes (19). Without loss of generality, therefore, we can assume that w\ > cox and d2 < co2. Thus co a

' = (c?x,d2). Suppose (19) does not

hold. Then

(u*(d), u*(co))e U(co a d). (20)

10 We are grateful to Herve Moulin for suggesting this argument.

Egalitarian equivalence 479

Recall that either dx( ) =

ux(0.5 x,0) or dx((o) =

u1(0,0.5co2). Suppose dx(co) =

ux(0.5cox,0). Since u^(of) > ux(0.5(o') and \ > x, this implies that

u*( ')>dx( ). (21)

On the other hand, (20) implies that (uj((?'),i4(co))e[/(co). And this contradicts

(dx( ),u*( ))edU((D). Now consider the case where dx(a>)

= ux(0,0.5co2). There are two possibilities:

(a) u*(co')>ux(0.5a)'x,a)'2) and (b)9 u'x(c?')>ux(co'x,0.5(o2). Consider (a). Clearly,

co2>0.5co2, would yield (21) and we can then follow the previous argument to

complete the proof. Suppose, therefore, that d2 < 0.5co2. From (20), we know that

there exists (xx,x2)est( /\ ') such that u(xx,x2) =

(ux*(a)'),u2K( )). Of course,

(xx + (0, co2 ?

co'2), x2)es/( ). Since co2 > 2co2, it follows that ux(xx + (0, co2 ?

co2)) >

dx( '). But this, contradicts dx( '),u^(o)))edU( ). Now consider (b). In this case

d2(co') =

u2(0,0.5co'2). Since co2>co2, this implies that u^( )> d2( '). Since (20)

implies that (u^((o% ul( ))eU( '), this contradicts (uf(co'),dx( '))edU( ') and

completes the proof.

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