Existence of Exact Walrasian Equilibria in Non Convex Economies

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D'Agata, Antonio

Working Paper Existence of exact Walrasian equilibria in non convex economies

Economics Discussion Papers, No. 2011-47

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

Suggested Citation: D'Agata, Antonio (2011) : Existence of exact Walrasian equilibria in non convex economies, Economics Discussion Papers, No. 2011-47, Kiel Institute for the World Economy (IfW), Kiel

This Version is available at: http://hdl.handle.net/10419/52226

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Existence of Exact Walrasian Equilibria in Non Convex Economies

Antonio D’Agata University of Catania

Abstract The existence of an exact Walrasian equilibrium in non convex economies is still a largely unexplored issue. In this paper an existence result for exact equilibrium in non convex economies is provided by following the “almost-near” approach introduced by Postlewaite and Schmeidler for convex economies. More precisely, we show that for any non convex economy there is a set of “perturbed” economies with the same number of agents exhibiting an exact Walrasian equilibrium; moreover as the number of agents tends to infinity the perturbed economies can be chosen as much close as we like to the original one.

JEL C62, D51 Keywords Exact Walrasian equilibrium; non convex 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] The author would like to thank the Italian Ministry of University and Education (MIUR) and the University of Catania (Fondi PRA) for financial support. The usual caveats apply.

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 an “almost-near” approach to deal with existence issues in convex economies: they show that if an allocation of any convex economy is “almost” 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. The motivation of the approach is that “If we don't know 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

- 2 - 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, pp. 105-106). 1 More recent economic applications of the “almost-near” approach along

Postlewaite and Schmeidler’s interpretation have been provided by Kubler and

Schmedders (2005) and Kubler (2007). 2

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 curves outside the budget set while inside the budget set it is

1 Anderson (1986) develops the “almost-near” 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, p. 231)).

2 Kubler and Schmedders also quote Blum, Cucker, Shub, and Smale (1997, Chapter 8) as an example of application of this approach to computation theory.

- 3 - flattened onto the budget surface, with continuous extensions also to neighboring surfaces.

This method, in principle could be extended to show that close to nonconvex economies with near walrasian equilibria there exists an economy with an exact equilibrium.

However, their perturbation rule requires that at the exact equilibrium price of the nearby economy the demand set of agents is convex, which is a quite disturbing feature.

In this paper we introduce a rule for perturbing the original nonconvex economy which allows to retain nonconvexity of preferences of the perturbed economy also at the equilibrium price, and we show that for any nonconvex economy there is a set of perturbed economies with the same number of agents as the original which exhibit an exact walrasian equilibrium. Moreover, as the number of agents tends to infinity the perturbed economies can be chosen as much close (in terms of an appropriate metric) as we like. The intuition behind our result is very simple: consider a n consumer, k good pure exchange economy satisfying all standard assumptions except convexity of preferences. Since, under our hypotheses, there exists a strictly positive price vector ensuring that the aggregate supply vector belongs the convex hull of the aggregate demand set at that price (see, e.g.,

Hildenbrand (1975, p. 150)). Hence at this price p it is possible to perturb the economy by shrinking and translating the indifference curves and/or changing the initial endowments

- 4 - perpendicularly to the price vector in such a way that the aggregate supply vector of the perturbed economy belongs to the aggregate demand set, that is, the perturbed economy has an exact equilibrium at price p. In addition, by Shapley and Folkman Theorem, the number of consumers whose endowments and/or 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 in terms of Postlewaite and

Schmeidler’s metric.

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}. The consumption set of consumers will always be assumed to be the non-

(n) negative orthant of the k-dimensional Euclidean space. Denote by Ah(⋅,ωh), A (⋅,( ωh)) and

ω(n), respectively, the demand correspondence of agent h, the aggregate demand

correspondence and the aggregate endowment of economy En. Symbols co, d and dH

- 5 - indicate, respectively, the convex hull operator, the Euclidean distance and the Hausdorff

distance. Symbol co En denotes the convexified version of economy En; i.e. the economy

whose demand correspondence of consumer h is co Ah(⋅,ωh). For any utility function uh, set

P=(,) xy ∈ℜ2k uxuy () ≥ () . Given a couple of utility functions u and uˆ , the distance uh { + h h } h h

δ between the preferences underlying these functions is defined as follows (see Debreu

uudPP PNP PNP (1969)): δ(,)hhHuuˆ = ( ,ˆ ) =∈∞⊆ inf ε (0,) uε ( uˆ ) and u ˆ ⊆ ε ( u ) where h h { h h h h }

Nε (⋅ ) is the closed ε-ball around a set. We shall use the same metric mn used by

 ˆ  mˆ 1 u u ωh− ω h Postlewaite and Schmeidler (1981): nnn(,)E E =∑ δ (,) hh ˆ + n n  , where n h∈ N () () ω+ ω ˆh 

ˆ Enu h, (ω h ) and Enuˆ h, (ωˆ h ) are economies in E . A walrasian equilibrium of (( ) )h∈ N (( ) )h∈ N n

(n) economy En is 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 in

economy En is indicated by W(En ) . A walrasian equilibrium of the convexified economy

co En is defined in an obvious way and the set of these equilibria is indicated by W(coEn ) .

In the following result it is worth keeping in mind that under our assumptions, W(coEn ) is non-empty for every n ∈ » (see Lemma 2 in Section 3.)

Theorem. Let En(( uh), ( ωh)) h∈N ∈En be a pure exchange economy satisfying the stated

W E ) X assumptions and let (pn *,( x nh *) h∈ N )∈ (co n . Then, there exists a set n(p n *) ⊂En such - 6 -

Eˆ X W Eˆ (Eˆ E 0 that if n((uˆ h ), (ωˆ hhN ))∈ )∈ nn ( p *) , then (n ) ≠ ∅ ; moreover, mn n, n ) → as n → ∞.

Furthermore , for every n ∈ » there exists in X (p *) economies E (u , (ω )) ) and n n n( h) hhN∈ E* W E* n((u h *), (ω h *)) hN∈ with uh= u h and ωh* = ωh for every h ∈N and (pn*, ( xnh *) h∈N) ∈ (n ) .

The last part of the previous result means that the walrasian equilibrium of co En (which always exists under our assumptions) is the walrasian equilibrium of an appropriately non-

* convex economy En obtained by perturbing only the preferences of the original economy

E E n, while the perturbed non convex economy n exhibiting an exact equilibrium differs from

E* E the original only by the endowments. In addition, economies n and n become as close to

En as we like when the number of consumers is “big enough”. Following Postlewaite and

Schmeidler’s interpretation, a consequence of the previous result is that a walrasian equilibrium of the convexified version of any large nonconvex economy should be interpreted as an exact walrasian equilibrium of a nonconvex economy “close” to the original one obtained by perturbing the preferences.

3. Proofs

The next two results are well-known.

- 7 -

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

A( p ,ωˆ ) for every ωˆ ∈Bp( , ω ) =∈ℜ⋅=⋅ xk pxp ω . h h h h h { + h }

Lemma 2. For every n W E Moreover, if (see, e.g., Hildenbrand (1974, p. 150)) , (co n ) ≠ ∅ .

p x W E , then p k ( n*, ( nh *) h∈N) ∈ (co n ) * ∈ ℜ ++ .

From now on pn* is an equilibrium price vector associated to economy co En(( uh), ( ωh)) h∈N.

By Lemma 2 the budget surface Bh(pn*, ωh) of consumer h is compact. By Urysohn’s

Lemma (see, e.g. Willard (1970, p. 102)), given a real number ε > 0 and for every h ∈ N

k   p there exists a continuous function γ hε :ℜ+ →  0,1  (which depends also on n* and ωh) such

k that γhε(x) = 1 if x ∈ Bh(pn*, ωn) and γhε(x) = 0 if x∈ ℜ + \ SBphε ( hn ( *,ω h )) where

S( B ( p *,ω )) = y∈ℜk p* ⋅−<ωε pyp **) ⋅< ⋅+ ωε is the open ε -“slice” hε hn h { + nh n nh }

containing set Bh(pn*, ωh) (see Figure 1, where segment B is the budget line and the shaded area is the ε-slice containing it.)

x y k p x y t p xy Given two vectors h, h ∈ ℜ + \{ 0 } such that n*(⋅ h − h )0 = , let h(;,⋅ ε n *, hh , ) be a

x  mapping defined as follows: txpxyx(;,*,,)ε= + minj  γ ()( xxy − ) , where hnhhj=1,2,..., k y  hhhε hj 

function γhε has been defined previously. Intuitively, transformation th(;,⋅ ε p n *, xy hh , )

- 8 -

x  l j translates any point x in ℜ+ by the vector min  γ ()(x x− y ) perpendicular to pn*. j=1,2,..., k y  hε h h hj 

In 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.

uh(x)= c

xh pn* xh-yh

ωh B

Figure 1. The continuous segment is the budget line . The shaded area is

the ε-slice Shε( ) cotaining the budget line. Under transformation th point yh is mapped into point xh. .

In the following result, notice that if xyhh,∈ Bp hn ( *,ω h ) , then pn*(⋅ x h − y h )0 = :

k Lemma 3. Given ε > 0 and pn * ∈ ℜ ++ , for every xyhh,∈ Bp hn ( *,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 ;

- 9 -

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

(iii) th(yh; ε, pn*, xh, yh) = xh;

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

k Proof. (i) Continuity is obvious. Take any x ∈ ℜ + , then, for i = 1,2,…, k,

x  x x  txpxyx(;,*,,)ε=+ minj γ ()( xxyx −≥+ ) min j γ () xx −=i  y hi nhhij=1,..., k y hε hihiij= 1,..., k  y hhiε  y  hi hj hj hi  x  minj  γ (x ) x ≥ 0 . Assertions (ii) and (iii) can immediately be verified by j=1,..., k y  hε hi hj  substitution. As for (iv), set: λλλ*= sup ∈∞ 1,x ∈ S ( Bp ( *, ω )) . Clearly, λ* > 1. {  ) hε hn h }

We have: thi(*;,λ xpxy ε nhh *, , )= λ * xx ii > , i = 1,2,…, k. By continuity, the assertion

follows. Fact (v) follows from the stated properties of function γhε.

k Lemma 4. Given ε > 0 and for every pn * ∈ ℜ ++ and xyhh,∈ Bp hn ( *,ω h ) there exists a

positive number K h such that if the utility functions uh and u ˆ h satisfy

uxˆh()= utx hh ((;,ε p nhh *, xy , )) , then δ(uh , uˆ h ) ≤ K 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

- 10 - containing the budget line. For any other point outside this set, the indifference curve of

uˆh coincides with the indifference curve of uh .

k Proof of Lemma 4 . By Lemma 3(v), uxˆh()= ux h () for x ∈ℜ+ \Shε(Bh(pn*, ωh)). So,

preferences of consumer h differ only inside set Shε(Bh(pn*, ωh)). Given the strict positivity

2 of pn*, this set is compact. Hence, also the Cartesian product Shε = Shε(Bh(pn*, ωh)) ×

Shε(Bh(pn*, ωh)) is compact. It follows that there exists a positive real number Kh such that its diameter rS(2 )=∈ℜ= supα α dxyxy (( , ),( ', ')), ( xyxyS , ),( ', ') ∈ 2 is less than K . { + hε } h

Take (x , y ) ∈ P with x, y∈ S 2 and suppose that (x , y ) ∉ P (otherwise there is nothing to uh hε uˆh

prove), i.e. uxˆh()≤ uy ˆ h () or utxhh(())≤ uty hh (()) . Since x∈ Shε ( Bp hn ( *,ω h )) , then by

Lemma 3(ii), txh( )∈ S hhnε ( Bp ( *,ω h )) . By Lemma 3(iv) there exists λ > 1 such that

λtyh( )∈ S hhnε ( Bp ( *, ω h )) and th(λth(y)) > th(y). Set x'= λ th ( y ) and y' = y . By monotonicity, it follows that ut( (λ ty ( )) = uxˆ( ')> uty ( ( )) = uy ˆ ( ') , that is (x ', y ') ∈ P with h h h h hh h uˆh

2 (x ', y ') ∈ S hε . Therefore, dxy(( , ),( xy ', ')) ≤ K h .

2 Suppose now that (x , y ) ∈ P with (x , y ) ∈ S , and, again, (x , y ) ∉ P . Then, uh(x) ≤ uh(y). uˆh hε uh

≥ Take x′ = λy and y′ = y where λ > 1 is such that λy∈ Shε ( Bp hn ( *, ω h )) . Hence, uh(x′)

- 11 -

2 uh(y′), that is, there exists a point (x ', y ') ∈ P with (x ', y ') ∈ S . Therefore, uh hε dxy(( , ),( xy ', ')) ≤ K . It follows that δ(u , uˆ ) ≤ K . h h h h

k ()n () n Proof of Theorem. By definition, pn * ∈ ℜ ++ satisfies the condition: ω∈ coA ( p n *,( ω h )) .

t That is, there are t (1≤t ≤ k + 1 ) real numbers αni > 0 such that ∑ α = 1 and i=1 ni

t t n ω()n= αx () n = α x where x()n∈ A () n ( p *,(ω )) and x∈ A( p *,ω ) for ∑i=1ni i ∑ i = 1 ni ∑ h = 1 nhi i n h nhi h n h

n every i = 1,2., …, t and every h = 1, …, n. Therefore, ω(n ) = y , where ∑h=1 nh

t y=α x ∈ co Ap ( *, ω ) . nh∑i=1 ni nhi h n h

Denote by N( p n *) the set of allocations which are feasible in terms of vectors in set

(n ) coA ( p n *,(ω h )) and which maintains constant consumers’ income with respect to price pn*

N p and to the initial allocation ( ωh)h∈N i.e. ( n *) =

x∈ℜk× n∑ xAp ∈co( n ) (*,()),ω pxp * ⋅=⋅∈ * ω , hN . By Lemma 1, any {( h)h∈ N + h∈ N h nhnhnh }

allocation in N( p n *) maintains unchanged the demand set of agents at price pn*, that is

Ap( *,ωˆ )= Ap ( *, ω ) for every h∈H - and, therefore, Ap()n( *,(ω ))= Ap () n ( *,( ω ˆ )) - hn h hn h nh nh

ˆ whenever ()ωhhN∈ ∈ N (*) p n .

- 12 -

ˆ N p ˆ()nA () n p ˆ()nA () n p ˆ Take any (ωhhN )∈ ∈ ( n * ) . Since ω∈ co (n *,( ω h )) , hence, ω∈ co (n *,( ω h )) .

tˆ tˆ n tˆ Then ωˆ()n= α ˆ nxˆ () n == α ˆ x ˆ with 0 ≤αˆn ≤1, αˆn = 1 and 1≤tˆ ≤ k + 1 , ∑i=1ii ∑ i = 1 ni ∑ h = 1 nhi i ∑i=1 i

(n ) (n) xˆ= x ˆ ∈ A( p *, (ωˆ )) , and xˆ ∈ A( p *, ωˆ ) for every i = 1,2,…, tˆand every i∑h∈ N nhi n h nhi h n h

tˆ h = 1,2,…, n. Therefore, ωˆ(n ) = yˆ where yˆ=αˆn x ˆ ∈ co Ap ( *, ω ) = ∑h∈ N nh nh∑i=1 inhi hn h

coAh ( p n *, ωˆ h ) . By Shapley-Folkman Theorem there exists a subset

JNˆ⊂ with # Jk ˆ ≤ + 1, such that ωˆ(n ) =yˆ + y ˆ where n n ∑hNJ'\∈ˆnh' ∑ hJ '' ∈ ˆ nh '' n n yˆ ∈ A( p *,ωˆ ) and yˆ ∈ co A ( p *,ωˆ ) . Let xˆ be a family of vectors defined as nh' h ' n h ' nh'' hn '' h '' { nh }h∈ N follows: xˆ= y ˆ for h'∈ N \ J ˆ and xˆ∈ xAp ∈(*,ωˆ )(, dxy ˆ ) ≤ dzy (, ˆ ), nh' nh ' n nh''{ hnh '' '' nh '' nh '' z A p ˆ h J ˆ ∈ h''( n *,ω h '' ) } for '' ∈ n .

X Eˆ Consider now the set n(pn*) of perturbed economies n((uˆ h ),(ωˆ hhN )) ∈ defined as follows:

x  ()ωˆ ∈ N (*) p and uxˆ ( )= utx ( ( )) where txx()= + minj  γ ()( xxyˆ − ˆ ) for h ∈N hhN∈ n h h h hy  hε nh nh ˆnhj 

(in what follows, for the sake of simplicity, we drop parameters ε, xˆnh and yˆnh in th.)

ˆ Clearly, pn*(⋅ xˆ nh − y ˆ nh )0 = for h ∈N and, moreover, uˆh'= u h ' for h'∈ N \ J n .

We show that (p *,( yˆ ) ) ∈ W Eˆ for every Eˆ ((uˆ ),(ωˆ )) in X (p *). First, by n nh h∈ N( n ) n h hhN∈ n n construction ωˆ(n ) = ω ˆ = yˆ , so allocation (yˆ ) is feasible. That ∑hN∈h ∑ hN ∈ nh nh h∈ N

- 13 - yˆnh∈ B h( p n *,ωˆ h ) for every h = 1,2, …, n, follows by construction. Vector yˆnh ' is optimal for agent h'∈ N \ J ˆ because yˆ ∈ A( p *,ωˆ ) . We now show that for every h'' ∈ J ˆ n nh' h ' n h ' n

uyˆh''( ˆ nh '' )≥ ux ˆ h '' () for every x∈ Bh''( p n *,ωˆ h '' ) . To this end, notice that, by Lemma 3(ii),

transformation th at price pn* maps the budget hyperplane into itself. So, by monotonicity,

we can focus only on points on the latter. Thus, suppose that there exists x∈ Bh''( p n *,ωˆ h '' )

such that uxˆh''()> uy ˆ hnh '' ( ˆ '' ) . Hence utxh''( h '' ( ))> uty h '' ( h '' (ˆ nh '' )) . By Lemma 3(iii), this implies that utx( ( ))> ux (ˆ ) . Since t( x )∈ B ( p *,ωˆ ) , this contradicts the fact that h'' h '' h '' nh '' h'' hnh '' '' xˆ ∈ A( p *,ωˆ ) for h'' ∈ J ˆ . nh'' h '' n h '' n

ˆ We show that mn(,E n E n )→ 0 as n → ∞ . First, as already noticed, in the perturbed

economy Eˆ , uxˆ ()= ux () for h'∈ N \ J ˆ . Hence, m (E , E ˆ ) = n h' h ' n n n n

1 1  ω− ω ˆ  u u ˆ h h ∑hJˆ δ (h'' , h '' ) + ∑ hN ()n () n  . Therefore, by Lemma 4, n''∈n n ∈ ω+ ω ˆ 

k ω+ ω ˆ mˆ +1 K 1 h h 1 k K m ˆ n(E n , E n ) ≤ +∑ n n = ((+ 1) + 1) . Thus, nE n, E n → 0 as n → ∞. n n h∈ N ω()+ ω ˆ () n ( )

N p As far as the last part of the assertion is concerned, choose (*)ωhhN∈= () ω hhN ∈ ∈ ( n * ) .

By the same argument at the beginning of this proof, there are t (1≤t ≤ k + 1 ) numbers

t ()nt () n t n αni > 0 satisfying the condition ∑ α = 1 such that ω=∑ αx = ∑ α ∑ x i=1 ni i=1ni i i = 1 ni h = 1 nhi

()n () n where xi∈ A( p n *,(ω h )) and xnhi∈ A h( p n *,ω h ) for every i = 1,2., …, t and every h = 1, - 14 -

n t …, n. Therefore, ω(n ) = y , where y=α x ∈ co Ap ( *, ω ) . Thus, ∑h=1 nh nh∑i=1 ni nhi h n h

p y W E (n *,( nhhN )∈ )∈ ( co n ) . By Shapley-Folkman Theorem there exists a subset

JN⊂ with # Jk ≤ + 1, such that ω(n ) =y + y where n n ∑hNJ'\∈nh' ∑ hJ '' ∈ nh '' n n y∈ A( p *,ω ) and y∈ co A ( p *,ω ) . Let x be a family of vectors defined as nh' h ' n h ' nh'' hn '' h '' { nh }h∈ N follows: x= y for h'∈ N \ J and nh' nh ' n x∈∈ xAp(*,ω )(, dxy ) ≤ dzy (, ), zAp ∈ (*, ω ) for h'' ∈ J . Consider now the nh''{ hnh '' '' nh '' nh '' hnh '' '' } n

E* * perturbed economy n((u h ),(ω hhN )) ∈ obtained from the original by changing only utility

x  functions as follows: ux*()= utx (()) where txx()= + minj  γ ()( xxy − ) for h ∈N. h h h hy  hε nh nh nhj 

By a similar argument used before, it is possible to show that

(p *,( y ) )∈ W E* (( u * ),(ω )) . However, we already noticed that n nhhN∈ ( n n hhN∈ )

p y W E ()nA () n p (n *,( hhN )∈ )∈ ( co n ) . Finally, since ω∈ co (n *,( ω h )) , choose

ω ()n∈ ExcoA () n ( p *,( ω )) and ()ω ∈ N (*) p such that ω= ω (n ) , where Ex n h hhN∈ n ∑h∈ N h indicates the set of extreme points. Since, by standard results on convex hulls (see, for example Lay (1992, Chapter 2)) and by Lemma 1,

()n () n () n ExcoAp (nh *,(ω ))= Ex Ap ( nh *,( ω )) = Ex Ap ( nh *,( ω )) , one obtains that

- 15 -

ω()n∈ ExA () n ( p *,( ω )) . Therefore, ω (n ) = y where y ∈ Ex A ( p *,ω ) (see Price n h ∑h∈ N nh nh hn h

W E (1940)). Thus, ynh∈ A hn( p *,ω h ) for every h ∈N. So, (pn *,( y nhhN )∈ )∈ ( n (( u n ),(ω hhN )) ∈ ) .

- 16 -

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Anderson, R.M. (1986), “’Almost’ implies ‘near’”, Transactions of the American

Mathematical Society 296 , 229-237.

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