On the equilibrium price set of an exchange economy

Abstract

Consider a pure exchange economy composed of a finite number of traders with strictly positive initial endowments and continuous, monotone, strictly convex preferences. It is well known that the intersection of the unit sphere with the set of price equilibria for such an economy is non-empty and compact. Under smoothness hypotheses on preferences and generic (i.e., non-degeneracy) conditions further (fixed-point index type) strong restrictions on the equilibrium price set can be derived; this was done by Dierker (1972). In this paper we provide converses to the above statements, i.e., we prove that, imprecisely speaking, if the number of commodities is greater than two, then every pattern of equilibria compatible with the above referred to properties can arise from an economy in the class we consider. It is obvious that this characterization of the equilibrium price set problem [already studied by Sonnenschein (1972)] is closely related to the problem of characterizing excess demand functions defined on compact sets of prices. In fact, the solution to the first problem has had to wait for a solution to the second to be found; this has been accomplished only recently; see Sonnenschein (1973), Mantel (1974, 1976), Debreu (1974). Our proof here amounts to a refinement of the one by Debreu (1974) leading to a sharper version of his result, sharp enough for the purposes of this paper.