The Aggregate Excess Demand Correspondence and the Structure of Economies with Externalities

Abstract

This paper will examine in detail certain properties of static general equilibrium models with externalities. Previous literature has considered the question of existence of equilibrium for such models (Arrow and Hahn [1], McKenzie [3], Osana [4] and Scheffman [5]), and has examined the topological properties of the set of feasible allocations (Osana [4]). The focus of our investigation will be to formulate the question: How are static general equilibrium models with externalities different from such models without in a manner for which we can provide a concise answer. The purpose of our investigation is to increase our understanding of the effects of externalities in static general equilibrium models. We will confine ourselves to static models because of course the dynamics of even models without externalities is not sufficiently well understood. The model we will use is similar to the one presented in Osana [4], except for one important difference: we will not net out intermediate goods by considering only net input-output vectors. It is important to consider intermediate goods explicitly in models with externalities, since intermediate goods are often direct sources of externalities. Stripped to its bare bones, the modern general equilibrium model consists of a compact convex set of feasible net allocations and an aggregate excess demand correspondence which is upper semicontinuous and convex-valued. From a static viewpoint, then, the way to formulate our central question is to ask: How are the net feasible set and aggregate excess demand correspondence different in models with externalities? Osana [4] has verified (under reasonable assumptions) that the net feasible set for models with externalities is compact. We will go further and establish that this set is not generally convex. This result is probably not surprising, and is at least implicit in much of the externality literature. Our basic contribution here is a concise revelation of the mathematical source of the non-convexities. We cannot immediately consider the aggregate excess demand correspondence for a model with externalities, because this concept has hitherto not been defined. The main contribution of this paper is to define what may reasonably be called the aggregate excess demand correspondence for models with externalities, and to establish its properties. In conclusion, we will present some interesting examples of non-convexities.