Determinacy of equilibrium in large-scale economies

Abstract

In this paper we argue that indeterminacy of equilibrium is a possibility inherent in economies with a double infinity of agents and goods, large-square economies. We develop a framework that is quite different from the overlapping generations one and that is amenable to analysis by means of differential calculus in linear spaces. The commodity space is a separable, infinite-dimensional Hilbert space, and each of a continuum of consumers is described by means of an individual excess demand function defined on an open set of prices. In this setting we prove an analog of the Sonnenschein-Mantel-Debreu theorem. Using this result, we show that the set of economies whose equilibrium sets contain manifolds of arbitrary dimension is non-empty and open in the appropriate topology. In contrast, we also show that, if the space of consumers is sufficiently small, then local uniqueness is a generic property of economies. The concept of sufficiently small has a simple mathematical formulation (the derivatives of individual excess demands form a uniformly integrable family) and an equally simple economic interpretation (the variation across consumers is not too great).