Existence of an Exact Walrasian Equilibrium in Nonconvex Economies

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 economyapproach 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.


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 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.
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 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 p n * and an allocation (x h *) which are a walrasian equilibrium of the convexified version of the original economy (see, e.g., Hildenbrand 1975: 150). It is possible to 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, p n * and (x h *) 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.

Existence of an Exact Walrasian Equilibrium in Nonconvex Economies
Consider the space E n of pure exchange economies E n ((u h ), (ω h )) h∈N with n consumers and k goods satisfying the assumptions of strict positivity of the initial endowment vector ω h and of continuity and strict monotonicity of utility function Given a couple of utility functions ˆ and h h u u , the distance δ between the preferences underlying these functions is defined as follows (see Debreu 1969): We shall use the same metric m used by Postlewaite and Schmeidler (1981): A walrasian equilibrium of economy E n is a non-negative price vector p n * and an allocation (x nh *) h∈N such that: ω (n) ∈ A (n) (p n *, (ω h )) and x nh * ∈A h (p n *,ω h ) for every h∈N.
Remark. It is easy to show that for some economies it is easier to obtain an estimate of the smallest element of the equilibrium vector p n *, say min n p , than the price p n * itself. In this case, an upper bound for the distance between the original economy and the perturbed one is min 1 (( 1) 2 max 1)

Proofs
The next two results are well-known.
www.economics-ejournal.org 7 Lemma 2. (see, e.g., Hildenbrand (1974, p. 150)) For every n, From now on p n * indicates an equilibrium price vector associated to the convexified economy coE n ((u h ), (ω h )) h∈N . It is assumed also that the price vector p n * belongs to the (k-1)-dimensional unit simplex. By Lemma 2 the budget surface

Assertions (ii) and (iii) can immediately be verified by substitution. Fact (iv)
follows from the properties of function γ hε . As for (v), set: Clearly, λ* > 1. By the fact that vector y n is optimal with respect to preferences ˆh u at price vector p n *, while vector x h is optimal with respect to preferences h u at the same price vector.