EDGEWORTH 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 prese nt paper examines this basic test of perfect competition in economies with an infinite dimensional space of commodities and a large finite n umber of agents. In this context, we cannot expect equality of the cor e and the set of Walrasian allocations; rather, as in the finite dimen sional context, we look for theorems establishing core convergence (th at is, approximate decentralization of core allocations in economies w ith a large finite number of agents). Previous work in this area has e stablished that core convergence for replica economies and core equiva lence for economies with a continuum of agents continue to be valid in the infinite dimensional context under assumptions much the same as t hose needed in the finite dimensional context. For general large finit e economies, however, we present here a sequence of examples of failur e of core convergence. These examples point to a serious disconnection between replica economies and continuum economies on the one hand, an d general large finite economies on the other hand. We identify the so urce of this disconnection as the measurability requirements that are implicit in the continuum model, and which correspond to compactness r equirements that have especially serious economic content in the infin ite dimensional context. We also obtain a positive result. When the co mmodity space is L(1), the space of integrable functions on a finite m easure space, we establish core convergence under the assumptions that marginal utility goes to zero as consumption tends to infinity and th e per capita social endowment lies above a consumption bundle which is equidesirable with respect to the preferences. This positive result d epends on a version of the Shapley-Folkman theorem for L(1).