The Equality of the Core and the Set of Equilibria in Economies with Infinitely Many Commodities and a Continuum of Agents

Abstract

IT IS BY NO MEANS CLEAR that Robert Aumann's Theorem on the equality of the core and the set of equilibria in atomless markets [1] applies to economies with infinitely many commodities. The existence of infinitely many commodities makes it possible for there to be an enormous variation in the characteristics of the agents in an economy. If these characteristics were sufficiently dispersed, it could be very hard to find a non-null set of agents with interests sufficiently similar or compatible to make it possible to form a blocking coalition. Thus, blocking coalitions could be so rare that almost all feasible allocations would be in the core. We show that suitable restrictions remove this difficulty and that Aumann's Theorem can be made to apply to economies whose commodity space is . (M, . p), the set of essentially bounded measurable functions on the measure space (M, ,//, pe). Our key restrictions are measurability assumptions about the maps defining agents' characteristics. Another way to restrict an economy in order to avoid the difficulty we have mentioned is to restrict the size of the commodity space. Jean Gabsewicz has shown that under very general assumptions, Aumann's Theorem applies to economies whose commodity space is C(S), the set of continuous functions on a compact metric space [6, (Chapter 5)]. The key property possessed by this space, from the point of view of size, is topological separability. A justification and explanation of the use of 9cO as a commodity space and of many of our assumptions may be found in [3]. In particular, we defend there the use of the Mackey topology for the duality (i2.c, 56). In Section 4 of this paper, we check that our model is internally consistent by proving that under certain acceptable assumptions, equilibria are measurable in the sense that we require of core allocations. Though consistent, our theory may be nearly empty, for only in a very special case have atomless markets with infinitely many commodities been shown to have equilibria [2, (Chapter 5)]. Jean-Francois Mertens has independently obtained results similar to our own.