Edgeworth's conjecture with infinitely many commodities: L(1)

Equivalence of the core and the set of Walrasian allocations has long been taken as one of the basic tests of perfect competition. The present paper examines this basic test of perfect competition in economies with an infinite dimensional space of commodities and a large finite number of agents. In this context, we cannot expect equality of the core and the set of Walrasian allocations; rather, as in the finite dimensional context, we look for theorems establishing core convergence (that is, approximate decentralization of core allocations in economies with a large finite number of agents). Previous work in this area has established that core convergence for replica economies and core equivalence for economies with a continuum of agents continue to be valid in the infinite dimensional context under assumptions much the same as those needed in the finite dimensional context. For general large finite economies, however, we present here a sequence of examples of failure of core convergence. These examples point to a serious disconnection between replica economies and continuum economies on the one hand, and general large finite economies on the other hand. We identify the source of this disconnection as the measurability requirements that are implicit in the continuum model, and which correspond to compactness requirements that have especially serious economic content in the infinite dimensional context. We also obtain a positive result. When the commodity space is L(1), the space of integrable functions on a finite measure space, we establish core convergence under the assumptions that marginal utility goes to zero as consumption tends to infinity and the per capita social endowment lies above a consumption bundle which is equidesirable with respect to the preferences. This positive result depends on a version of the Shapley-Folkman theorem for L(1).