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Faculty Working Paper 91-0139-2.

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Commodity Pair Desirability and the Core Equivalence Theorem

Aldo Rustichini

Department of Economics Northwestern University

Nicholas C Yannelis

Department of Economics University of Illinois

Bureau of Economic and Business Research

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 91-0139-2. College of Commerce and Business Administration University of Illinois at Urbana-Champaign May 1991

Commodity Pair Desirability and the Core Equivalence Theorem

Aldo Rustichini*

Nicholas C. Yannelis**

Department of Economics

* Department of Economics, Northwestern University, Evanston, IL 60208.

**Department of Economics, University of Illinois at Urbana-Champaign, Champaign, IL 61820.

We would like to thank two anonymous referees for several useful comments. Tom Armstrong made several useful suggestions and special thanks are due to him. Of course, we are responsible for any remaining shortcomings.

ABSTRACT

In infinite dimensional commodity spaces whose positive cone has an empty norm interior the core equivalence theorem fails. To this end we introduce a new condition called commodity pair desirability which enables us to preserve the core equivalence theorem. This condition is related to the assumption of an extremely desirable commodity or uniform properness and if the norm interior of the positive cone of commodity space is nonempty and preferences are monotone then it is automatically satisfied.

1. INTRODUCTION

The purpose of this paper is to study the core-Walras equivalence in economies with a continuum of agents and with an infinite dimensional commodity space.

It may be useful to discuss briefly the general importance of infinite dimensional commodity spaces in economics. As others have noted (e.g., Court (1941), Debreu (1954), Gabszewicz (1991), Bewley (1970) and Peleg-Yaari (1970)), infinite dimensional commodity spaces arise quite naturally in economics. In particular, an infinite dimensional commodity space may be desirable in problems involving an infinite time horizon, uncertainty about the possibly infinite number of states of nature of the world, or infinite varieties of commodity characteristics. For instance, the Lebesgue space L,,, of bounded measurable functions on a measure space considered by Bewley (1970), Gabszewicz (1991) and Mertens (1991) is useful in modeling uncertainty or an infinite time horizon. The space Ly of square-integrable functions on a measure space is useful in modeling the trading of long-lived securities over time. Finally, the space M(fi) of measures on a compact metric space considered by Mas-Colell (1975), is useful in modeling differentiated commodities.

In this paper, we study core-Walras equivalence results for perfectly competitive economies with an infinite dimensional commodity space which is general enough to include all of the spaces that have been found most useful in equilibrium analysis. In particular, we cover all the Lebesgue spaces L , (1 < p < oo), the space of measures, M(fi) and the space of continuous functions on a compact metric space C(X).

2

It turns out that in infinite dimensional commodity space whose positive cone has a non-empty (norm) interior one can obtain core-Walras equivalence results under quite mild assumptions. However, in infinite dimensional commodity spaces whose positive cone has an empty (norm) interior, as Rustichini-Yannelis (1991) showed, even under quite strong assumptions on preferences and endowments, core-Walras equivalence fails. In particular, the above authors showed, that even when preferences are strictly convex, monotone, and weak* continuous and initial endowments are strictly positive, core-Walras equivalence fails to hold. It is important to note that this failure results despite the fact that these assumptions are much stronger than the standard assumptions which guarantee equivalence in the Aumann (1964) finite dimensional commodity space setting.

Our main objective is to obtain core-Walras equivalence for infinite dimensional commodity spaces (in particular, Banach lattices) whose positive cone may have an empty (norm) interior and are general enough to cover the space L (1 < p < ») and M(fl). In view of the Rustichini-Yannelis counterexample to the core-Walras equivalence in spaces whose positive cone has an empty interior, we introduce a new condition on preferences called commodity pair desirability. In essence, this assumption is a strengthening of the assumption of an extremely desirable commodity used in Yannelis-Zame (1986), which in turn is related to the condition of uniform properness in Mas-Colell (1986) (see also Chichilnisky-Kalman (1980)). All of these assumptions are essentially bounds on the marginal rate of substitution, and in practice turn out to be quite weak. For example, all three of these

3 assumptions are automatically satisfied whenever preferences are monotone and the positive cone of the commodity space has a non-empty (norm) interior. Hence, this assumption is implicit in the infinite dimensional work of Gabszewicz (1991), Mertens (1991), and Bewley (1973), and is automatically satisfied in the finite dimensional work of Aumann (1964) and Schmeidler-Hildenbrand (Hildenbrand (1972), (1974)).1 We also wish to note that in addition to the commodity pair desirability assumption, the lattice structure of the commodity space will play a crucial role in our analysis.

The remainder of the paper is organized as follows: Section 2 contains notation, definitions and some results on Banach lattices and the integration of correspondences. The economic model is outlined in Section 3 where we also prove a core-Walras equivalence theorem for an ordered separable Banach space of commodities, whose positive cone has a non-empty (norm) interior. The central assumption of the paper, commodity pair desirability, is introduced in Section 4. In Section 5 we prove a core-Walras equivalence result for a commodity space which can be any arbitrary separable Banach lattice, provided that the assumption of commodity pair desirability holds. Finally, some concluding remarks are given in Section 6.

2. PRELIMINARIES 2 . 1 Notation

Rc denotes the £-fold Cartesian product of the set of real numbers R.

intA denotes the interior of the set A.

2 denotes the set of all nonempty subsets of the set A.

4

0 denotes the empty set.

/ denotes the set theoretic subtraction.

dist denotes distance.

If A c x where X is a Banach space, c£h denotes the norm closure of A.

If X is a Banach space its dual is the space X* of all continuous linear functionals on X.

If q e X* and y e X the value of q at y is denoted by q • y.

2.2 Definitions

We begin by collecting some useful notions on Banach lattices (a more detailed exposition may be found in Aliprantis-Burkinshaw (1978, 1985)), which will be needed in the sequel.

A normed vector square is a real vector space E equipped with a norm |*|: E -» [0,o°) satisfying:

(i) | x| > 0 for all x in E, and JxJ = 0 if and only if x = 0; (ii) |ax] = |a| | x ] for all x in E and all a in R; (iii) ||x+y] < JxJ + |y| for all x,y in E. A Banach space is a normed vector space for which the metric induced by the norm is complete.

If E is a Banach space, then its dual space E* is the set of continuous linear functionals on E. The dual space E* is itself a Banach space, when equipped with the norm

||4>i = sup{ |<(>U) | '. x e E, Jxj <; 1} .

A Banach lattice is a Banach space L endowed with a partial order < (i.e., < is a reflexive, antisymmetric, transitive relation) satisfying:

(1) x < y implies x + z < y + z (for all x,y,z e L) ;

(2) x < y implies tx < ty (for all x,y e L, all real numbers t > 0);

(3) every pair of elements x,y e L has a supremum (least upper bound) x V y and an infimum (greatest lower bound) x A y;

(4) |x| < |y| implies Jx] < Iy| (for all x,y e L) .

Here we have written, as |x| = x+ + x" where x+ = x V 0, x" = (-x) V 0; we call x+, x" the positive and negative parts of x respectively and |x| the absolute value of x. Note that x = x+ - x" , and that x+ A x" = 0. We say that x e L is positive if x > 0; we write L+ (or L+) for the set of all positive elements of L and refer to L+ (or L+) as the positive cone of L.

We will say that an element x of L is strictly positive (and write

* x » 0) if <p(x) > 0 whenever <p is a positive non-zero element of L .

Strictly positive elements are usually called quasi- interior to L+.

Note that if the positive cone L+ of L has a non-empty (norm) interior,

then the set of strictly positive elements coincides with the interior

of L+. However, many Banach lattices contain strictly positive elements

even though the positive cone L+ has an empty interior (see

Aliprantis-Burkinshaw (1985, p. 259)). We will now give basic examples

of separable Banach lattices.

(i) the Euclidean space RN;

(ii) the space £ (1 < p < °°) of real sequences (an) for which the norm l(an)| = (2 |ajp)1/p is finite; (iii) the space L (Q,R,/j) (1 < p < °°) of equivalence classes of

measurable function f on the separable measure space (Q,R,/j) for which the norm |f| = ( JQ| f | pdp<) 1/p is finite; (iv) the space C(K) of continuous, real-valued functions on the

compact Hausdorf f space K (with supremum norm) ; A basic property of Banach lattices which will play a crucial role in the sequel, is the Riesz Decomposition Property.

Riesz Decomposition Property; Let L be a Banach lattice and let

n x, y-,.../y be positive elements of L such that 0 < x < S y ■ • Then

n i=l there are positive elements x. , ...,,x in L such that 2 x- = x and

i=l 0 < x- < y. for each i.

We now define some measure theoretic notions as well as the concepts of a Bochner integrable function and the integral of a correspondence .

Let X, Y be sets. The graph of the correspondence <p : X -» 2Y is

denoted by G = {(x,y) e X x Y : y e <p(x)}. Let (T,t,/v) be a finite

measure space (i.e., \i is a real-valued, non-negative countably additive

measure defined on a a-field t of subsets of T such that jj(T) < °o) , and

X a Banach space. The correspondence <p : T -» 2X is said to have a

measurable graph if G. e t » 3(x)/ where 3(x) denotes the Borel

o-algebra on X and ® denotes the product o-algebra. A function

f : T -» X is called simple if there exist x. , x2, ..., xn in X and

n a., a,, ..., a„ in t such that f = 2 xy where y (t) = 1 if t e a-

i=l L 1

and y (t) = 0 if t € a-. A function f : T -* X is said to be

o-measurable if there exists a sequence of simple functions fn : T -* X

such that lim |fn(t) - f(t)| = 0 fj - a.e. A /j-measurable function

n-xr> f : T -» X is said to be Bochner inteqrable if there exists a sequence of

simple functions {fn : n = 1,2,...} such that

lim f \fn[t) - f(t)|dn(t) = 0

T n—oo

In this case we define for each E e t the integral to be

Lf(t)d/j(t) = lim JEfn(t )d/j(t ) . It can be easily shown (see Diestel-Uhl

n-*» (1977, p. 45)) that if f : T -» X is a /j-measurable function then f is

Bochner integrable if and only if J-| f(t) |d/i(t) < °°. we denote by

Lj(fu,X) the space of equivalence classes of X-valued Bochner integrable

functions x : T -» X normed by ||x|| = J_|x(t) | d/i(t ) . Note that one can

easily show that if (T,T,/i) is atomless, the subset of simple functions

1 given by B = {x : T -» X, x = 23 x-xT , A*(T,-) = — } is norm-dense in

i=l i m

L^(/i,X). Moreover, we denote by S^ the set of all X-valued Bochner

inteqrable selections from the correspondence <p : T -» 2X, i.e.

S^ = lx6L,(|l,D : X(t) £ <|>(t) \i - a.e. } .

The integral of the correspondence <p : T -» 2X is defined as: f$(t)d\i(t) = {fxU)d\i(t) : x e 5^}

In the sequel we will denote the above integral by

/4>or /*.

2 . 3 Lemmata

t If (T,t,/j) is atomless and X = R , it follows from Lyapunov's

Theorem that the integral of the correspondence <p : T -» 2X, i.e., j<p, is

convex. However, this result is false in infinite dimensional spaces

(see for instance Yannelis (1991)). Nevertheless, it can be easily

deduced (see for instance Hiai-Umegaki (1977), Khan (1985) or Yannelis

(1991)) from the approximate version of Lyapunov's Theorem in infinite

dimensional spaces that the norm closure of j<p, i.e., c£j<p, is convex.

More formally the following Lemma is true.

Lemma 2.1; Let (T,T,/i) be a finite atomless measure space, X be a Banach space and <p : T -» 2X be a correspondence. Then c£j<p is convex.

We will also need the following result whose proof follows from the measurable selection theorem and can be found in Hiai-Umegaki (1977, Theorem 2.2, p. 156).

Lemma 2.2: Let (T,r,/i) be a finite measure space, X be a separable Banach space, and <p : T -* 2X be a correspondence having a measurable graph. Suppose that j<p * 0. Then for every p e X* we have that

inf p • z = f inf p • y. ze/<J> ye<|> ( •)

It should be noted that Lemma 2.2 has been proved in Hildenbrand (1974, Proposition 6, p. 63) for X = R . However, by recalling that the Aumann measurable selection theorem holds in separable metric spaces, one can easily see that Hildenbrand ' s argument remains true in separable

9 Banach spaces. In fact, it is even true in arbitrary Hausdorff separable and metrizable linear topological spaces.

With all these preliminaries out of the way, we can now turn to our model.

3. THE MODEL AND A PRELIMINARY THEOREM Denote by E the commodity space. Throughout this section the commodity space E will be an ordered Banach space.

An economy % is a quadruple [ {T,t ,/u) , X, >t, e] where

(1) (T,t,/j) is a measure space of agents,

(2) X : T -» 2E is the consumption correspondence,

(3) >- c X(t) xx(t) is the preference relation of agent t, and

(4) e : T -» E is the initial endowment, where e is Bochner integrable and e(t) e X(t) for all t e T.

An allocation for the economy lis a Bochner integrable function x : T -* E+. An allocation x is said to be feasible if

JTx(t)d/j(t) = JTe(t )d/j( t ) . A coalition S is an element of r such that p/(S) > 0. The coalition S can improve upon the allocation x if there exists an allocation g such that

(i) g(t)>t x(t) fj - a.e. in S, and (ii) /sg(t)d/u(t) = Jse(t)d/i(t).

The set of all feasible allocations for the economy £* that no coalition can improve upon is called the core of the economy & and it is denoted by c(£) .

An allocation x and a price p e E /{0} are said to be a competitive equilibrium (or a Walras equilibrium) for the economy %, if

10 (i) x(t) is a maximal element for >t in the budget set {y e X(t) : p • y < p • e(t)} n - a.e., and (ii) JTx(t)d/i(t) = JTe(t)d^(t). We denote by W(^) the set of all competitive equilibria for the economy

r.

We begin by stating some assumptions needed for the proof of our core-Walras equivalence result.

(a.O) E is an ordered separable Banach space whose positive cone

E+ has a non-empty norm interior, i.e., intE+ * 0. (a.l) (Perfect Competition); (T,r,ji) is a finite atomless measure

space, (a. 2) X(t) = E+ for all t e T. (a. 3) (Resource availability) ; The aggregate initial endowment

JTe(t)d^(t) is strictly positive, i.e., Je » 0. (a. 4) (Continuity) : For each x e E+ the set {y e E+ : y >■ x} is

norm open in E+ for all t e T, (a. 5) >t is irreflexive and transitive for all t e T. (a. 6) (Measurability) : The set {(t,y) e T x E+ : y > x} belongs

to t ® 3(E+) . (a. 7) (Monotonicity) ; If x e E+ and v e E+/{0}, then x + v > x for all t e T. Theorem 3.1 below is taken from Rustichini-Yannelis (1991). Since the second part of the proof of Theorem 3.1 is the same as that of Theorem 5.1, we will provide the argument for the sake of completeness. It should be noted, that in view of Remark 3.1 (see below), Theorem 3.1 is

11

more general than Theorem 4.1 of Rustichini-Yannelis (1991). In particular, E need not be separable.

Theorem 3.1: Under assumption (a.O) - (a. 7), C(&) = W(#) .

Proof; The fact that W(^) c c(£) is well known, and therefore its proof is not repeated here. We begin the proof by assuming that the allocation x is an element of the core of &. We wish to show that for

some price p, the pair (x,p) is a competitive equilibrium for &.

E To this end, define the correspondence <p : T -» 2 + by

(3.0) 4>(t) = {Z 6 E^ : z>c x(t)} U {e(t)} .

We claim that:

(3.1) c<t(( <J> - f e) fl int E. = 0,

J T J T

or equivalently ,

(3-2) (f $ _ [ e) f| int E = 0.

J T J T

Suppose otherwise, i.e.,

(f«J) - f e) fl int E. * 0

J T J T

then there exists v e int E+ such that <3-3) f e - v 6 f<|> .

It follows from (3.3) that there exists a function y : T -* E+ such that

(3.4)

J T J 7

e - v,

T J T

12 and y(t) e (p{t) /j - a.e.

Let

S = {t : y(t)>c x{t)} , and

S' = {C : y(t) = e(t)} .

Since J y * J e we have that /j(S) > 0. Define y : S -» E+ by

v

y(t) = y(t) + for all t e S. By monotonicity (assumption (a. 7))

A* (S )

Yft)^ y(t). Since y(t)>t x(t) for all t e S, by transitivity (assumption (a. 5)) y(t)>t x(t) for all t e S. Moreover, it can be easily seen that y(') is feasible for the coalition S, i.e.

[?=(y+v=[y-[e+v

J S J S J T J S

= f e - f e = [ e, {recall {3 .4) ) .

J T J S' J 3

Therefore, we have found an allocation y(') which is feasible for the coalition S and is also preferred to the allocation x, which in turn was assumed to be in the core of %, a blatant contradiction. The above contradiction establishes the validity of (3.1).

We are now in a position to separate the set c£(j<p - Je) = c£j<p - Je from int E. . Clearly the set int E. is convex and non-empty as well. Observe first that by the definition of $(•), 0 is an element of \<p - je and this shows that c£j<p - Je is non-empty. Since, (T,r,ji) is atomless (assumption (a.l)) by Lemma 2.1 c£j<p is convex. Thus, by Theorem 9.10 in Aliprantis-Burkinshaw (1985, p. 136) there exists a continuous linear functional p e E*/{0}, p > 0 such that

13

(3.5) p • y 2 p • f e for all y e (<$> .

Since by assumption (a. 6), > has a measurable graph, so does <p('), i.e., G . e t ® 3(E+)« Therefore, it follows from Lemma 2.2 that

(3.6) inf p ■ y = f inf p - z X J p • e.

ye/<J> ze<$> ( •)

It follows from (3.6) that

(3.7) \i-a.e. p-zlp-e(t) foi all z>t x{t) .

To see this, suppose that for zs<£(*),p«z<p* e(t) for all t e S, Ai(S) > 0.

Define the function z : T -* E+ by

z(t) if t € S

z(t) =

e(t) if t $ S.

Obviously, z e <£(')• Moreover,

f p • z = f p ■ z + f p-e

J: J s J t/s

< f p ■ e + f p ■ e = f p ■ e,

J S J T/S J

a contradiction of (3.6).

We now show that ^ - a.e. p • x(t) = p • e(t). First note that it follows directly from (3.7) that p • x(t) > p • e(t) \jl - a.e. If now p • x(t) > p • e(t) for all t e S, /j ( S ) > 0 then,

p • f x = p • f x + p • f x

J T J T/S J S

> p • f e + p ■ [ e = p ■ f e,

J T/S iS J T

14 contradicting JTx = JTe, since p > 0, p * 0.

To complete the proof we must show that x(t) is maximal in the budget set {z e E+ : p • e(t)} ^ - a.e. The argument is now routine. Since JTe is strictly positive (assumption (a. 3)) it follows that fj({t : p • e(t)}) > 0, for if p • e(t) = 0 \jl - a.e. then p • Je = 0 contradicting the fact that Je is strictly positive since p > 0, p * 0.

Thus, we can safely pick an agent t with positive income, i.e., p • e(t) > 0. Since p • e(t) > 0 there exists an allocation x' such that p • x' < p • e(t). Let y be such that p • y < p • e(t) and let y(X) = \x' + (l-X)y for X e (0,1). Then for any X e (0,1), p • y(X) < p • e(t) and by (3.7) y(X) *t x(t). It follows from the norm continuity of > (assumption (a. 4)) that y } x(t). This proves that x(t) is maximal in the budget set of agent t, i.e., {w : p • w < p • e(t)}. This, together with the monotonicity of preferences (assumption (a. 7)) implies that prices are strictly positive, i.e., p >> 0. Indeed, if there exists v e E+/{0} such that p • v = 0 then p • (x(t) + v) = p • e(t) and by monotonicity x(t) + v > x(t) contradicting the maximality of x(t) in the budget set.

Thus p » 0 and x(t) is maximal in the budget set whenever p • e(t) > 0. Consider now an agent t with zero income, i.e., p • e(t) = 0. Since p » 0 his/her budget set {z : p • z = 0} consists of zero only, and moreover, p • x(t) = p • e(t) = 0. Hence, x(t) = 0 for almost all t e T, with p • e(t) = 0; i.e., zero in this case is the maximal element in the budget set. Consequently, (p,x) is a competitive equilibrium for &, and this completes the proof of Theorem 3.1.

15 Remark 3.1; It is possible to relax the assumption that E is separable. The argument (which is given to us by one of the referees) proceeds as follows:

As before we can obtain (3.5), i.e., there exists p e E*/{0}, p > 0 such that

p • y 2. P • \e for all y e f<J> ,

We now show that p • x(t) = p • e(t) /j-a.e. Let ScT, p(S) >0, e > 0 and v e E++. Define x' : T -» E by

,,, Qx ( x(t) + tv if c e S

<3-8> x'U) =

e(t) if t <? S.

Then x' e S. for all S c T. Hence, 9

p ■ [f x + ev \i(S) + / e] > p • e

J S J T/3

and rearranging we have that Lp • x > Lp • e for any S c T since e > 0

is arbitrary. Thus, it follows (since S is arbitrary) that

p • x(t) > p • e(t) p-a.e. Since x is feasible, i.e., Jx = Je we must

have that Jp • x = Jp • e and therefore p • x(t) = p • e(t) jj-a.e. Now

(3.7) follows by replacing in (3.8) x + ev by z e </>(•)• We then have

that Jsp • z + Jp • e > Jp • e. Rearranging we obtain J_p • z > Jp • e,

teT/S z e S. and we can conclude (since S is arbitrary) that ^/-a.e.,

p • z > p • e(t) for all z > x(t). The rest of the proof is now

identical with the one outlined above in the proof of Theorem 3.1.

Note that in the above step we avoided Lemma 2.2 which requires

the use of the measurable selection theorem (recall that for Lemma 2.2,

E needs to be separable) and as a consequence we do not need to assume

16 that E is separable. In fact, one does not even need to assume that >t has a measurable graph (recall (a.6)).

It turns out that if one drops the assumption that the positive cone of the space E has a non-empty interior, then the above Theorem fails. A counterexample to that effect can be found in Rustichini-Yannelis (1991). In order to remedy this difficulty we introduce the assumption of commodity pair desirability.

4. EXTREMELY DESIRABLE COMMODITIES AND COMMODITY PAIR DESIRABILITY

For notational convenience, below we drop the subscript t on the

preference relation >. We begin by defining the notion of an extremely

desirable commodity. Let E be a Banach lattice and denote its positive

cone (which may have an empty norm interior) by E+. Let v e E+, v * 0

and U be an open neighborhood of zero. We say that v e E+ is an

extremely desirable commodity if there exists U such that for each

x e E+ we have that x + av - z > x whenever a > 0, z < av and z e aU.

In other words, v is extremely desirable if an agent would prefer to

trade any commodity bundle z for an additional increment of the

commodity bundle v, provided that the size of z is sufficiently small

compared to the increment of v. The above notion has a natural

geometric interpretation. In particular, let v e E+, v * 0, U be an

open neighborhood and define the open cone C as follows:

C = {av - z : a > 0, z e E, z e aU } .

The bundle v is said to be an extremely desirable commodity, if for each x e E+, we have y > x whenever y is an element of (C + x) n E+. Note this implies that v is an extremely desirable commodity if for each

17 x e E+ we have that ((-C + x) n E+) n {y : y > x} = 0, or equivalently -cn{y-xeE+ : y > x} = 0. The latter property is precisely the assumption we need for the core-Walras equivalence if we consider L as a commodity space 1 < p < °° [see Rustichini-Yannelis (1991)].

Recall that if the preference relation > is monotone and intE+ * 0, then the assumption of an extremely desirable commodity is automatically satisfied [see for instance Yannelis-Zame (1984)].

We now turn to a strengthening of the above assumption.

A pair (x,y) e E+ x e+ is said to be a desirable commodity pair if

for every z e E+ we have z + x - y > z whenever y < x + z for each

t e T. The pair (x,y) e E x E is said to have the splitting property if

m for any m-tuple (s.,...,s ) e E x ... x e such that E s- = (x-y)" there

i=l m exists an m-tuple (d.,...,d ) e E x ... x e such that E d- = (x-y)+

i=l and the pair (d-,s) is a desirable commodity pair.

We are now ready to define our key notion.

Definition 4.1; Commodity pair desirability obtains if there exists a v e E+, v * 0 and a neighborhood U of zero such that any commodity pair (w,u) of the form w = av, a > 0 and u e aU has the splitting property.

Let us discuss briefly the intuitive meaning of the commodity pair desirability condition. It may be considered as an extension to a multiperson setting of the idea of desirable commodity pair for the case of a single player decision. In this last case a pair (x,y) is desirable if the player is always willing to trade the bundle y in exchange for a bundle x; that is to accept the gain (x-y)+ in exchange

18 for the loss (x-y)". In the case of the splitting property we can imagine that the offer of exchanging (x-y)" for (x-y)+ is presented to a group of m players. The splitting property that we introduce asks that no matter how the reduction in the consumption bundle (given by (x-y)") is allocated over the members of the group, they can always find a way of allocating the surplus ( (x-y)+) and make everyone better off.

Obviously the above concept is a substitutability condition which roughly speaking says that an agent would accept a sufficiently small amount of the commodity bundle s- if he/she would be compensated by consuming more of the desirable commodity bundle dj .

A couple of comments are in order. First notice that for m = 1 in the above definition we have that for any z e E+,

z + (w-u)+ - (w-u)" = z - w-u > z whenever u < z + w, w = av, a > 0, and u e all, i.e., for m = 1 we are reduced to the assumption of an extremely desirable commodity.

Moreover, it is easy to show that if intE+ * 0 and the preference relation >■ is monotone then the condition of commodity pair desirability is automatically satisfied. Specifically, let v e intE+ and U be such that v + U c E+, then for any pair (w,u) with w = av, a > 0, u e aU we have (w-u) e E+ so (w-u)" = 0 and therefore by monotonicity for any z e E+, z + w-u > z.

5. CORE-WALRAS EQUIVALENCE IN BANACH LATTICES WHOSE POSITIVE CONE HAS AN EMPTY INTERIOR

In this section we state and prove our main result, i.e., a

core-Walras equivalence theorem for a commodity space which can be any

19 arbitrary separable Banach lattice whose positive cone may have an empty norm interior. We begin by stating the following assumptions: (a.O') E is any separable Banach lattice.

(a. 8) (Commodity pair desirability): There exists v e E+/{0} and an open neighborhood U of zero such that any commodity pair (w,u) of the form w = av, a > 0 and u e aU, has the splitting property. We are now ready to state and prove the following result:

Theorem 5.1: Under assumption (a.O'), (a.l) - (a. 8), c(£) = W(^) .

Proof: It can be easily shown that W(^) c c(&). Hence, we will

show that if x e c(^) then for some price p, the pair (x,p) is a

E competitive eguilibrium for ¥. Define the correspondence <p : T -» 2 + by

(5.1) (J>(t) = ize E, : z >cx(t)} U {e(t)}.

Let C = u a(v - U) where v e E+/{0} and U given as in (a. 8). We a>0 claim that:

<5-2) ctifo - fe)H - C = 0

or equivalently

<5-3> (/$ - [e)0 - C =

Since -C is open it suffices to show that for any ye . there exists a sequence {(yk, ek) : k=l,2,...} such that yk converges in the L^/jfE) norm to y, J ek -» J e, and

20

(5.4) fy* - [ek <£

J T J T

Let S = {t : y(t)>t x(t)}, S' = T/S. Without loss of generality we may assume that /j(S) > 0 (for if ^(S) = 0 then y(t) = e(t) \i - a.e. which implies that j y - j e = { - C. Consequently (5.3) holds.). In the argument below y and e are restricted to S. Moreover denote by /j_ the restriction of fi to S.

Since y : S -» E+ is Bochner integrable and > is norm continuous

k k k k k

(assumption (a. 4)) there exist y„,...,y in E. and T., T„ ,...,T in r

1 m, T 12 m, k k

such that y converges in the L.|(/js,E) norm to y, and

(5-5) yk - £ vitri

i-1

(5-6) y* >t x{t) for all t e T* and all i, i = l, . . . ,mk. and

<5*7) V-s(Ti) = i, i-1 m*.

mk Let ek = 2 (J ke(t)d/i(t)) x fe.

i=l T. T.

i i

In order to establish (5.4) we first show that (5-8) f yk _ ( ek $ c_

J S J 5

Suppose that (5.8) is false, then

5T y& ~ 52 e^ € " a(-v + & ' and therefore

21

(5.9)

E/i + "-« = E •* .

a a

where x = -=- v, ue-r-U*

mk k Since 2 e. > 0, it follows from (5.9) that

i=l (5.10) lw_ u)- < V vt.

(w-u)-ij; y*

2-1

Applying the Riesz Decomposition Property to (5.10) we can find

(s1,...,s ) e E+ x ... x E+ such that k

i5-11) V s± = (w - u)- 0 i Sj i y* for all i.

i-l

It follows from the assumption of commodity pair desirability that there

exists an mk~tuple (&.,..., d ) e E+ x ... x e+, such that

m, k

2 d- = (w - u)+ and i=l

(5.12) $k m yk + di _ Si >t yk for all t e Tk and for all ±

k k

Note that since y. > s- it follows that y. e E+. Moreover, since

k k k k k

y. >-„ x(t) for all t e T. and all i, and y. >„ y. for all t e T. and all

k k

i, by transitivity of >t we have that y. >t x(t) for all t e T. and all

i. Also,

a m*

i-l i-l J

v mk k l

Define yK = 2 y . X ■• Notice that Jsyk = Js e. Therefore, we

i=l T.

i

have found an allocation yk(-) feasible for the coalition S and

22

preferred to x(*) which in turn was assumed to be in the core of %, a contradiction. Hence, (5.8) holds.

We are now ready to construct the sequence {(yk,ek) : k=l,2,...}. In particular, define yk : T -* E+ by

. yk(t) if t e S yk{t) =

y(c) if t € S.

Similarly define ek : T -* E+ by

, e*(t) if c € s ek(c) =

e(C) if c $ S.

Note that JTyk - JTek $ - C and therefore (5.4) holds. We can now separate the convex nonempty set c£f<p - J e from the convex nonempty set -C. Proceeding as in the proof of Theorem 3.1 one can now complete the proof.

6. CONCLUDING REMARKS Remark 6.1: Since in Banach lattices whose positive cone has a nonempty (norm) interior and preferences are monotone, assumption (a. 8) is automatically satisfied, it follows that in such spaces Theorem 3.1 becomes a corollary of Theorem 5.1. However, since in Theorem 3.1 the commodity space is any arbitrary ordered Banach space (i.e., no lattice structure is required) we cannot derive Theorem 3.1 as a corollary of Theorem 5.1.

Remark 6.2: Notice that in Theorem 5.1 the commodity space was assumed to be a Banach lattice. However, in Theorem 3.1 we only needed the commodity space to be an ordered Banach space, i.e., no lattice

23

structure was required. The reason we needed the lattice structure in Theorem 5.1 was to apply the Riesz Decomposition Property which in turn has a natural economic meaning as was indicated in Yannelis-Zame (1986, p. 89). However, we do not know whether or not one can dispose with the lattice structure in Theorem 5.1. Aliprantis-Burkinshaw (1991) have shown that the lattice structure is a necessary and sufficient condition for the existence of Edgeworth equilibria in economies with finitely many agents.

Remark 6.3: Using the notion of commodity pair desirability one can easily prove the second welfare economics theorem for economies with an atomless measure space of agents and with a commodity space which can be any arbitrary separable Banach lattice.

Remark 6.4: It is of interest to know whether or not under the assumption of Theorem 3 . 1 we have core-Walras existence as well. It is known, for instance, that this is not true if X = Rn (see Aumann (1966, Section 8, p. 17)); this is also the case here. For instance, under (a.O) - (1.7) a competitive equilibrium may not exist. However, by replacing (a. 2) by (a. 2') X(t) = K for all t e T, where K is a weakly compact convex

non-empty subset of E+, and adding the assumption: (a. 11) for all t e T, and for all x(t) e X(t) the set

<Y : Y yt *(t)} is convex and the set {y : x(t) >t y} is

norm open,

24 one can conclude, by virtue of the main theorem in Khan-Yannelis (1986) that both sets, i.e., c{&) and W(&) are non-empty. Notice that in Khan-Yannelis it is assumed that preferences are convex (see also Bewley (1991)), an assumption which is dispensible in the finite dimensional case. In fact as Aumann (1964) showed, the Lyapunov convexity theorem convexifies the aggregate demand set. However, in infinite dimensional spaces, the Lyapunov convexity result fails. It is also worth mentioning that in addition to the failure of Lyapunov 's theorem, Fatou's Lemma fails in infinite dimensional spaces as well (see Yannelis (1988) or Rustichini (1989)). Hence, there is no exact analogue to Schmeidler's version of Fatou's Lemma in infinite dimensions. However, approximate versions of Fatou's Lemma have been obtained by Khan-Majumdar (1986) and Yannelis (1988). Moreover, with additional assumptions exact versions of Fatou's Lemma in infinite dimensional spaces can be obtained as well (see Rustichini (1989) and Yannelis (1988)). Nevertheless, the latter versions of the Fatou Lemma are not sufficient to prove the existence of Walrasian equilibrium. For more information on this problem see Rustichini-Yannelis (1991a), who have showed that in economies with "many more" agents than commodities, there is a nice convexifying effect on aggregation and the Fatou Lemma still holds. As a consequence of this, the theorem on the existence of a Walrasian equilibrium is still true.

NY. 1-4

25

FOOTNOTES

1Hildenbrand (1972, p. 85), attributes the proof to Schmeidler.

^This is so since intE_ is an open set. In particular if A and B are subsets of any topological space an B is open, then it can be easily seen that A n B = (8 if and only if c£A n B = p.

Note that since we have allowed splitting for (w - u)" and (w - u)+, we may think of i ( i=l, 2 , . . . ,m) as agents and the splitting property as a kind of redistribution. Hence, the notion of commodity pair desirability is a "coalitional-type" of uniform properness.

26

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