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D'Agata, Antonio

Article Existence of an exact Walrasian equilibrium in nonconvex economies

Economics: The Open-Access, Open-Assessment E-Journal

Provided in Cooperation with: Kiel Institute for the World Economy (IfW)

Suggested Citation: D'Agata, Antonio (2012) : Existence of an exact Walrasian equilibrium in nonconvex economies, Economics: The Open-Access, Open-Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 6, Iss. 2012-12, pp. 1-16, http://dx.doi.org/10.5018/economics-ejournal.ja.2012-12

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Vol. 6, 2012-12 | April 5, 2012 | http://dx.doi.org/10.5018/economics-ejournal.ja.2012-12

Existence of an Exact Walrasian Equilibrium in Nonconvex Economies

Antonio D’Agata University of Catania

Abstract The existence of an exact Walrasian equilibrium in nonconvex economies is still a largely unexplored issue. This paper shows that such an equilibrium exists in nonconvex economies by following the ‘nearby economy’ approach introduced by Postlewaite and Schmeidler (Approximate Walrasian Equilibrium and Nearby Economies, 1981) for convex economies. More precisely, the paper shows that any equilibrium price of the convexified version of a nonconvex economy is an equilibrium price also for a set of ‘perturbed’ economies with the same number of agents. It shows that in this set there are economies that differ from the original economy only as regards preferences or initial endowments. JEL C62, D51 Keywords Exact Walrasian equilibrium; nonconvex economies; perturbed economies

Correspondence Antonio D’Agata, Faculty of Political Science, University of Catania, Via Vittorio Emanuele, 8, 95131 Catania, Italy, e-mail: [email protected]

Citation Antonio D’Agata (2012). Existence of an Exact Walrasian Equilibrium in Nonconvex Economies. Economics: The Open-Access, Open-Assessment E-Journal, Vol. 6, 2012-12. http://dx.doi.org/10.5018/economics-ejournal.ja.2012-12

1 Introduction

The existence of an exact walrasian equilibrium in non-convex economies is still a largely unexplored issue. Mas-Colell (1977) shows that that in the space of differentiable economies there exists an open (in an appropriate topology) and dense set of economies such that if one considers a sequence of finite economies with an increasing number of consumers and with limit in this set then, eventually, an exact walrasian equilibrium exists. Smale (1974) shows the existence of an extended equilibrium in a nonconvex differentiable economy. In addition to the differentiability of the economies, Mas Colell’s work is constrained by the use of sequences of purely competitive economies, while Smale’s work relies upon the use of a nonconventional concept of equilibrium.

Postlewaite and Schmeidler (1981) introduce a “nearby economy” approach to deal with the existence issue in convex economies. They show that if an allocation of any convex economy is “approximately” walrasian at price p, then it is possible to construct an economy “near” (in terms of an “average” metric) the original where that allocation is walrasian at the same price p. Postlewaite and Schmeidler’s result is obtained constructively by perturbing the preferences of agents in the original convex economy in such a way that the indifference surface passing through the bundle of the approximate walrasian equilibrium coincides with the original indifference surfaces outside the budget set while inside the budget set it is flattened onto the budget surface, with continuous extensions also to neighboring surfaces. The motivation of this approach is that “If we don't know

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the characteristics [of the agents in an economy], but rather, we must estimate them, it is clearly too much to hope that the allocation would be walrasian with respect to the estimated characteristics even if it were walrasian with respect to the true characteristics. …… [Thus,] one could not easily pronounce that the procedure generating the allocation was not walrasian by examining the allocations unless one is certain that there have been no errors in determining the agents’ characteristics” (Postlewaite and Schmeidler 1981:105–106). More recent economic applications of the “nearby economy” approach along Postlewaite and

Schmeidler’s interpretation have been provided by Kubler and Schmedders (2005) and Kubler (2007).1

Since large but finite nonconvex economies exhibit approximate equilibria

(see, e.g. Hildenbrand et al. 1971, Anderson et al. 1982), one may wonder whether Postlewaite and Schmeidler’s approach can be used to prove that close to nonconvex economies there exists a nonconvex economy with an exact walrasian equilibrium. As a matter of fact, their approach could immediately be extended in this direction,2 however, their perturbation rule has the disturbing feature that it

______1 Anderson (1986) develops the “nearby economy” argument within a very general framework and, relying on nonstandard analysis and an appropriate formal language, provides an abstract theorem showing that objects “almost” satisfying a property are “near” an object exactly satisfying that property. He emphasizes also that this approach can be used to obtain existence results and applies his abstract result to show the existence of exact decentralization of core allocations (Anderson 1986: 231). 2 Anderson (1986)’s existential result could be used as well although it does not provide any information concerning the economy with an exact equilibrium. www.economics-ejournal.org 2

yields convexity of the individual demand set at the equilibrium price in the nearby economy.

In this paper we introduce a rule for perturbing the original nonconvex economy which allows to retain nonconvexity of preferences in the perturbed economy also at the equilibrium price, and we show that for any nonconvex economy there is a set of perturbed nonconvex economies with the same number of agents as the original which exhibit an exact walrasian equilibrium. We provide also an upper bound on the size of perturbation. More specifically, we show that

(a) the equilibrium price of the convexified version of the original economy is also the equilibrium price of an exact equilibrium of any economy in this set; (b) as the number of agents tends to infinity this set becomes closer and closer (in terms of an appropriate metric) to the original economy; (c) in this set there are economies which differ from the original one in terms of initial endowments or in terms of preferences only. In addition, the walrasian equilibrium of the convexified version of the original nonconvex economy is also an exact walrasian equilibrium of the economy perturbed only in preferences and the economies perturbed only in endowments exhibit also a no-trade equilibrium.

The intuition behind our results is very simple: consider a n consumer, k good pure exchange economy satisfying all standard assumptions except convexity of preferences. Under our hypotheses, there exists a strictly positive price vector pn* and an allocation (xh*) which are a walrasian equilibrium of the convexified version of the original economy (see, e.g., Hildenbrand 1975: 150). It is possible to

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deformate continuously consumers’ indifference curves parallel to the budget surface in such a way that the walrasian equilibrium consumption bundles become optimal with respect to the new preferences. So, pn* and (xh*) are an exact walrasian equilibrium of the economy perturbed in preferences only. In addition,

Shapley and Folkman Theorem ensure that the number of consumers whose preferences have to be perturbed is independent upon the number of consumers (to be precise, is not greater than k+1.) Therefore, as the number of consumers increases the distance between the original economy and the perturbed economy tends to zero. It is shown that this logic can be extended to perturbations in preferences and/or endowments.

2 Existence of an Exact Walrasian Equilibrium in Nonconvex Economies

Consider the space En of pure exchange economies En((uh), (ωh))h∈N with n consumers and k goods satisfying the assumptions of strict positivity of the initial endowment vector ωh and of continuity and strict monotonicity of utility function uh for each consumer h ∈N = {1,2,…, n}. We assume that the consumption set of all consumers is the non-negative orthant of the k-dimensional Euclidean space, and that there exists a compact subset Ω of this space such that ωh ∈Ω for every h

∈N and every n ∈ ∞, where ∞ is the set of natural numbers. Denote by Ah(⋅,ωh), (n) (n) A (⋅,(ωh)) and ω , respectively, the demand correspondence of agent h, the aggregate demand correspondence and the aggregate endowment of economy En.

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Symbols co, d and dH indicate, respectively, the convex hull operator, the

Euclidean distance and the Hausdorff distance. Symbols K and (a)k denote, respectively, the index set of goods (i.e. Kk= {1,2,..., } ) and the k-dimensional vector whose elements are all equal to a. Finally, symbol coEn, denotes the convexified version of economy En; i.e. the economy whose demand correspondence of consumer h is coAh(⋅,ωh). For any utility function uh, set

2k Pu={(, xy ) ∈ℜ+ uhh () x ≥ u () y} . h

Given a couple of utility functions uuhh and ˆ , the distance δ between the preferences underlying these functions is defined as follows (see Debreu 1969):

δε(uu ,ˆ )= d ( PP , ) = inf ∈∞ (0, )P ⊆ NP ( ) and P⊆ NP ( ) where hhHuuhˆh{ uuuuhhhhεεˆˆ }

Nε ()⋅ is the closed ε-ball around a set. We shall use the same metric m used by ˆ Postlewaite and Schmeidler (1981): m(,)EEnn= 1 ωω− ˆ δ (,)uuˆ + hh, where E u , (ω ) and ∑ hN∈ uuhh ()nn () nh(( ) h)hN∈ n ωω+ ˆh Eˆ uˆ , (ωˆ ) are economies in E . A walrasian equilibrium of economy E is nh(( ) h)hN∈ n n (n) a non-negative price vector pn* and an allocation (xnh*)h∈N such that: ω ∈ (n) A (pn*, (ωh)) and xnh* ∈Ah(pn*,ωh) for every h∈N. The set of walrasian equilibria of economy En is indicated by W ()En . A walrasian equilibrium of the convexified economy coEn is defined in an obvious way and the set of these equilibria is indicated by W (coEn ) . In the following main result one should keep in mind that under our assumptions set W (coEn ) is non-empty for every n ∈ ∞ (see Lemma 2 in Section 3).

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Theorem. Let En((uh), (ωh))h∈N ∈En be a pure exchange economy satisfying the stated assumptions and let (pxn *,( nh *) h∈ N )∈W (coE)n . Then, for every ε > 0 there exists a set Xnn(p *) ⊂En such that:

(a) if EXˆ ((upˆ ), (ωˆ )) )∈ ( *) then there exists an allocation ()xˆ such n h h hN∈ n n nh h∈ H ˆ ˆ that (pn*, ()xˆnh h∈ H ) ∈ W ()En . In addition, m(Enn,) E → 0 as n → ∞; in particular, ωε+ ˆ 1 pnh*( ( ) k ) m(Enn, E )≤+ (( kK 1)nε + 1) with Knε = 2 max max ; hN∈∈ iK n pni *

(b) there exist economies E (((u ), (ω )) ) and E ((u ), (ω )) ) in n h h hN∈ n h h hN∈ X (p *) with ωω= and uu = for every h ∈N. In addition, (p *, (x *) ) ∈ nn hh hh n nh h∈N  W ()En and (pn *,(ω h ) hH∈ )∈W (E ) .

Remark. It is easy to show that for some economies it is easier to obtain an

min estimate of the smallest element of the equilibrium vector pn*, say pn , than the price pn* itself. In this case, an upper bound for the distance between the original ωε+ 1 ∑ ∈ hi k economy and the perturbed one is ((k ++ 1) 2 max iK 1) . hN∈ min npn

3 Proofs

The next two results are well-known.

k Lemma 1. (see, e.g., Balasko (1988, p. 77)) Let p ∈ℜ+ be a price vector.

k Then, Aphhhh(,ωω )= Ap (,ˆ ) for everyωωˆhh∈B(, p h ) ={ x ∈ℜ+ px ⋅ = p ⋅ωh}.

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Lemma 2. (see, e.g., Hildenbrand (1974, p. 150)) For every n, W (coEn ) ≠∅.

k Moreover, if (pn*, (xnh*)h∈N) ∈W (coEn ) , then p*∈ℜ++ .

From now on pn* indicates an equilibrium price vector associated to the convexified economy coEn((uh), (ωh))h∈N. It is assumed also that the price vector pn* belongs to the (k-1)-dimensional unit simplex. By Lemma 2 the budget surface

Bh(pn*,ωh) of consumer h is compact. By Urysohn’s Lemma (see, e.g. Willard 1970: 102), for every real number ε > 0 and every h ∈ N there exists a continuous

k function γ hε :,ℜ→+ [01] (which depends also on pn* and ωh) such that γhε(x) = 1 if

k x ∈ Bh(pn*,ωh) and γhε(x) = 0 if x∈ℜ+ \ Shε ( Bp hn ( *,ω h )) where

k Shε ( Bp hn ( *,ω h )) = {y∈ℜ+ pnh** ⋅ωε − < py n ⋅ < pnh*)⋅+ωε}is the open ε -

“slice” containing set Bh(pn*,ωh). The shaded area in Figure 1 illustrates set

Shε ( Bp hn ( *,ω h )) while segment B indicates the budget line Bphn( *,ω h ) .

k Given two vectors xyhh, ∈ℜ+ \0{ } such that pn*(⋅−= xy hh ) 0 , let

th(⋅ ;ε , p n *, xy hh , ) be a mapping defined as follows:

xi txh( ;εγ , p n *, xy hh , )= x+− minhε (xx )( h y h ) , where function γ has been iK∈ hε yhi previously defined. Intuitively, transformation th(⋅ ;ε , p n *, xy hh , ) translates any

l xi point x in ℜ+ by the vector minγ hε (xx )( hh− y ) perpendicular to p *. In iK∈ n yhi Figure 1 the curved arrows describe the effects of transformation th on points on the budget line: for example, point yh is mapped into point xh. Moreover, the action of transformation th decreases as the axes are approached, and vanishes outside the

ε-slice containing the budget line. Given that pn*(⋅−= xy hh ) 0 whenever

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xyhh,∈ Bp h ( n *,ω h ) , mapping th is always defined for every ε > 0 and every couple of vectors lying on the budget surface of consumer h at price pn*.

k Lemma 3. Given ε > 0 and pn *∈ℜ++ , for every xyhh,∈ Bp h ( n *,ω h ) , map

th(⋅ ;ε , p n *, xy hh , ) satisfies the following properties:

k (i) th(⋅; ε, pn*, xh, yh) maps ℜ+ into itself and is continuous;

k (ii) pn*⋅= tx h ( ;ε , p n *, xy hh , ) p n *⋅ x for every x∈ℜ+ ;

(iii) th(yh; ε, pn*, xh, yh) = xh; k (iv) th(x; ε, pn*, xh, yh) = x for every x ∈ ℜ+ \Shε(Bh(pn*,ωh));

(v) for every x ∈Shε(Bh(pn*,ωh)) there exists λ > 1 such that

λx∈Shε(Bh(pn*,ωh)) and th(λx; ε, pn*, xh, yh) > x;

k Proof. (i) Continuity is obvious. Take any x∈ℜ+ , then, for j ∈K,

x j x j thj( x ;ε , p n *, xy h , h ) = xj +minγ hε (xx )( hj−≥ y hj ) xj +−minγ hε (xx ) hj iK∈ iK∈   yhi yhi x j x j yhj = minγ hε (xx ) hj ≥ 0 .  iK∈ yhj yhi

Assertions (ii) and (iii) can immediately be verified by substitution. Fact (iv) follows from the properties of function γhε. As for (v), set:

λλλ*= sup{ ∈∞[1 ,) xS ∈hε ( Bp hn ( *, ω h ))} .

k Clearly, λ* > 1. By the fact that λω*x∈ℜ+ \{ Shε ( Bp hn ( *, h ))} and by fact (iv),

th(λε *;, x pn *,, xy hh )= λ * x > x. By continuity, there exists a real number λ, 1 < λ

< λ*, such that th(λε x ; , p n *, xy hh , ) > x. ♦

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Transformation th(⋅ ;ε , p n *, xy hh , ) is used to “perturbate” preferences by the

following rule uˆh(⋅= ) ut hh ( ( ⋅ ;ε , p n *, xy h , h )) . Figure 1 illustrates the effects of

transformation th on the indifference curve uh(x) = c in case k = 2. The dotted curve

is the part of the indifference curve of uˆh in the ε- slice containing the budget line.

For any other point outside this set, the indifference curve of uˆh coincides with the

indifference curve of uh. Intuitively, under transformation th(⋅ ;ε , p n *, xy hh , )

vector yn is optimal with respect to preferences uˆh at price vector pn*, while vector

xh is optimal with respect to preferences uh at the same price vector.

u (x) = c h

uxˆh ()= c

xh pn* xh-yh

ωh

Figure 1. Transformation th(⋅ ;ε , p n *, xy hh , )

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k Lemma 4. Given ε > 0 and for every pn *∈ℜ++ and every

xyhh,∈ Bp h ( n *,ω h ) , if utility functions uhh and uˆ satisfy

pnh*(ωε+ ( ) k ) uxˆh( )= utx hh ( ( ;ε , p n *, xy h , h )) , thenδ (,)uuhhˆ ≤ 2 max . iK∈ pni *

k Proof. By Lemma 3(iv), uxˆhh()= ux () for x ∈ ℜ+ \Shε(Bh(pn*,ωh)), so, preferences of consumer h perturbed by th differ only inside set Shε(Bh(pn*,ωh)) with respect to the original preferences. Therefore, in order to prove the assertion it is enough to consider couples of vectors only in this set. Take (,xy )∈ Pwith uh

2 xy,∈= Shε Snhεε( Bp h ( n *,ωω h ))× S nh ( Bp h ( n *, h )) and suppose that (,xy )∉ P(otherwise there is nothing to prove), i.e. uxˆˆ()≤ uy () or uˆh hh

utxhh( ( ))≤ uty hh ( ( )) . Since xS∈ hε ( Bp hn ( *,ω h )) , then by Lemma 3(ii),

txh( )∈ S hε ( Bp hn ( *,ω h )) . By Lemma 3(v) there exists λ > 1 such that

λωtyh( )∈ S hε ( Bp hn ( *, h )) and th(λth(y)) > th(y). Set x'= λ tyh () and yy' = . By monotonicity, it follows that uthh( (λ ty h ( )) = uxˆˆh( ')>= uty hh ( ( )) uy h ( ') , that is (xy ', ') ∈ P with (xy ', ') ∈ S2 . Therefore, d(( xy , ),( x ', y '))≤ diam S2 where diam uˆh hε nhε indicates the diameter of a set. Suppose now that (,xy )∈ Pwith (,xy )∈ S2 , uˆh nhε and, again, (,xy )∉ P. Then, u (x) ≤ u (y). Take x′ = λy and y′ = y where λ > 1 is uh h h λω∈ ′ ′ such that ySnhε ( Bp h ( n *, h )) . Hence, uh(x ) ≥ uh(y ), that is, there exists a point (xy ', ') ∈ Pwith (xy ', ') ∈ S2 . Thus, again d(( xy , ),( x ', y '))≤ diam S2 . Therefore, uh hε nhε δ (uu ,ˆ )≤ diam S2 . Notice now that vector ωε+ ()does not belong to h h nhε hk ω ⋅ω + ε − ⋅= ωε Shε(Bh(pn*, h)) because ppn*( h ()) k nh * where the equality follows from the fact that pn* belongs to the (k-1)-dimensional unit simplex. Thus,

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k Snhε ( Bp h ( n *,ω h )) ⊂ {x∈ℜ+ pxpn* ≤ nh *(ωε + ( ) k )} ⊂ ∆nhε , where ∆nhε is the k- dimensional closed simplex with vertices (0, 0,…, 0), (Wnhε , 0, …, 0), …., (0, 0,

pnh*(ωε+ ( ) k ) …, Wnhε ) and Wnhε = max  is well-defined because of the strict iK∈ pni * 2 ⊂ ∆2 =∆ ×∆ positivity of pn* (Lemma 2). Hence, Shε nhε nh εε nh and, consequently,

22 pnh*(ωε+ ( ) k ) diamShε ≤ diam ∆=nh 2 ⋅=Wnhε 2 max . ♦ iK∈ pni *

= ∈ℜkn× Proof of Theorem. (a) Set Np( nh*) {( x)hN∈ + x∈co A()n ( p *,(ωω )), p *⋅= x p * ⋅ , hN ∈} , that is, Np( *) denotes ∑ hN∈ h n hnhn h n ()n the set of allocations which are feasible in terms of vectors in set coAp (nh *,(ω )) and which maintain constant consumers’ income with respect to price pn* and to the initial allocation (ωh)h∈N. By Lemma 1, Aphn( *,ωωˆ h )= Ap hn ( *, h ) for every

()nn() h∈H and, therefore, Ap(nh *,(ωω ))= Ap ( nh *,(ˆ )) whenever (ωˆh ) hN∈ ∈ Np ( n *) .

()nn () Take any ()ωˆh hN∈ ∈ Np( n *) . Hence, ωωˆ ∈=coAp (nh *,( )) ttˆˆ coAp()n ( *,(ωˆ )) . Then ωαˆˆ()nn=xxˆˆ()nn = α ˆ with 0 ≤ αˆ n ≤ 1, nh ∑i=11i i ∑∑i= i hN∈ nhi i tˆ αˆ n = and ≤≤+tkˆ , xxˆˆ()n = ∈ Ap(n) ( *, (ωˆ )) , and ∑i=1 i 1 11 i ∑ hN∈ nhi nh ˆ xˆnhi∈ Ap h( n *, ωˆ h ) for every i = 1,2,…, t and every hH∈ . Therefore, tˆ ωˆ ()n = yˆ where yˆˆ=αωˆ n x ∈co Ap ( *, ) = coAp ( *, ωˆ ) . By ∑ hN∈ nh nh ∑i=1 i nhi h n h hn h Shapley-Folkman Theorem there exists a subset Jˆˆ⊂ N with #1 Jk ≤+ , such nn ωˆ ()n = + ˆ ∈ ωˆ that ∑∑∈∈ˆˆyyˆˆnh' nh'' where ynh'' Ap h( n *, h ' ) and h' NJ\ nnh'' J yˆ ∈co Ap ( *,ωˆ ) . Let {xˆ } be a family of vectors defined as follows: nh'' h '' n h'' nh hN∈

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ˆ xyˆˆnh''= nh for h'∈ NJ\ n and xˆˆnh'' ∈∈{ x Ah'' ( p n *,ωˆ h'' ) dxy ( ,nh'' ) ≤

ˆ dzy ( ,ˆnh'' ), for every z∈ Ah'' ( p n *,ωˆ h'' )} for hJ''∈ n . ˆ Consider now the set Xn(pn*) of perturbed economies En((uˆ h ),(ωˆ h )) hN∈ defined as follows: (ωˆh ) hN∈ ∈ Np ( n *) and uxˆh()= utx hh ( ())where txh ()= x +

x j minγ hε (xx )(ˆˆ nh− y nh ) for h ∈N (in what follows, for the sake of simplicity, iK∈ yˆnhi ε ˆ ˆ ⋅−=ˆˆ we drop parameters , xnh and ynh in th.) By construction, pn*( xy nh nh ) 0 for ˆ h ∈N and, moreover, uuˆhh''= for h'∈ NJ\ n . We show that (py *,(ˆ ) )∈W Eˆ for every Eˆ ((uˆ ),(ωˆ )) in X (p *). n nh h∈ N ( n ) n h h hN∈ n n First, by construction ωωˆˆ()n = = ˆ , so allocation ()yˆ is ∑∑hN∈∈h hNynh nh h∈ N feasible. That yˆnh∈ Bp h( n *,ωˆ h ) for every hH∈ , follows again by construction. Vector yˆ is optimal for agent h'∈ NJ\ ˆ because yˆ ∈ Ap( *,ωˆ ) . We now nh' n nh'' h n h ' ˆ show that uyˆˆh''( nh '' )≥ ux ˆ h '' ()for every xBp∈ hn'' ( *,ωˆ h'' ) and every hJ''∈ n . To this end, notice that, by Lemma 3(ii), transformation th maps the budget surface into itself. So, by monotonicity of preferences, we can focus only on vectors on the latter. Thus, suppose that there exists xBp ∈ hn'' ( *,ωˆ h'' ) with xy ≠ ˆnh'' such that

uxˆh'' () > uy ˆˆh'' ( nh '' ). Hence, by definition, utxhh''( '' ( ))> uty hh'' ( '' (ˆ nh '' )) . By Lemma 3(iii), this implies that utx( ( ))> ux (ˆ ) . Since tx( )∈ Bp ( *,ωˆ ) , this h'' h '' h'' nh '' h'' hn'' h'' contradicts the fact that xˆ ∈ Ap( *,ωˆ ) for hJ''∈ ˆ . nh'' h '' n h'' n We now provide an upper bound for m(,)EEˆ and show that m(,)EEˆ → 0as nn nn

n →∞. First, as already noticed, in the perturbed economy Eˆ , uxˆ ()= ux ()for n hh'' 11ωω− ˆ ∈ ˆ ˆ = δ ˆ + hh h' NJ\ n . Hence, m(,)EEnn ∑∑∈∈ˆ (uuh'' , h '' ) ()nn () . nnh'' Jn hNωω+ ˆ

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k +11 ωω+ ˆ 1 ˆ ≤+ hh= ++ By Lemma 4, mK(,)EEnn nε ∑ ∈ ()nn () ((kK 1)nε 1) nnhNωω+ ˆ n pnh*(ωε+ ( ) k ) where Knε = 2 max max . Under our assumptions, hN∈∈ iK pni * ppni*0→> i (see, e,g,, Hildenbrand and Kirman 1988: 93) and all ˆ vectorsωh belong to the compact set Ω, hence m(EEnn,0) → as n → ∞.

(b) In the particular case ()ωωˆh hN∈∈= () h hN ∈ Np( n *) , notice that, by definition, the walrasian equilibrium (pxn *,( nh *) h∈ N ) of coEn satisfies the n condition: ωω()nn=x* ∈ co Ap() ( *,( )) . Therefore, there are ∑ h=1 nh n h t t()11≤≤tk + real numbers α > 0 such that α = 1 and ni ∑i=1 ni t tn ωα()nn= xx()= α where x()nn∈ Ap ()( *,(ω )) and ∑i=1ni i ∑∑ ih= 11ni= nhi i nh

xnhi∈ Ap h( n *,ω h ) for every i = 1,2., …, t and every h ∈ N. We have also, t x* =αω x ∈co Ap ( *, ) . By Shapley-Folkman Theorem there exists a nh ∑i=1 ni nhi h n h ⊂ ≤+ ω ()n = + subset Jnn N with #1 Jk , such that ∑∑∈∈xxnh' **nh'' h' NJ\ nnh'' J ∈ ω ∈ ω ˆ where xnh''* Ap h ( n *, h ' ) and xnh'' * co Aph'' ( n *, h'' ) . Let {xnh}hN∈ be a family of vectors defined as follows: xxˆ = *for h'∈ NJ\ and xˆ ∈ nh'' nh n nh''

{x∈≤ Ah''( p n *,ω h'' ) dx ( , xnh'' *) dz ( , xnh'' *), for every zAp∈ hn'' ( *,ω h'' )} for

hJ''∈ n . Consider now the perturbed economy E (((uh ),(ω h )) hH∈ ) obtained from the original one by changing only utility functions as follows: uxh()= utx hh ( ())where

xi txh ()= x+− minγ hε ()(xxˆ nh x nh *) for h ∈N. By a similar argument used iK∈ xnhi * before, it is possible to show that uxhnh( *)≥ ux hh ( ) for every xBp∈ hn( *,ω h ) and every h ∈N. Since the allocation (xnh *) h∈ N is feasible, one obtains that

(pxn *,( nh *) h∈ N ) ∈W (En ) .

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Finally, since ωω()nn∈coAp () ( *,( )) , choose (ω )∈ Np ( *) such that nh h hN∈ n ()nn ()  ωω ∈ Ap(nh *,( )) . Consider the perturbed economy En(((u h ),(ω h )) hN∈ ) . By construction, there exist n vectors such that y ∈ Ap( *,ω ) for every h ∈N and nh h n h ω ()n =  . It follows that (py *,( ) )∈ (E ) . It is obviously possible ∑ hN∈ ynh n nh h∈ N W n to choose vectors ω in such a way that ω = y for every h ∈N, which implies h h nh  that (pn *,(ω nh ) h∈ N )∈W (En ) .♦

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