Edgeworth's Conjecture with Infinitely Many Commodities

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 under assumptions much the same as those usual in the finite dimensional context. For general large finite economies, however, we present here a sequence of examples of the failure of core convergence. These examples point to a serious disconnection between replica economies and continuum economies on the one hand an 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 positive results. When the commodity space is a Riesz space, we show that familiar assumptions lead to a kind of local core convergence; strong assumptions lead to global core convergence. In the differentiated commodities context, we obtain core convergence results that are quite parallel to known equivalence results for continuum economies. Our positive results depend on infinite dimensional versions of the Shapley-Folkman theorem.