Edgeworth’s conjecture and the number of agents and commodities

We answer the question asked by Robert Aumann as to whether core equivalence depends on there being “many more agents than commodities.” We show that for a large class of commodity spaces, which might be infinite-dimensional and even non-separable, core equivalence is indeed equivalent to the presence of “many more agents than commodities” when allocations are Bochner integrable. By contrast, we show that in a classical model of an atomless economy with an infinite-dimensional commodity space, the model where the commodity space is $$L_\infty (\mu )$$ L ∞ ( μ ) with the Mackey topology and allocations are Gelfand integrable, core equivalence holds in full generality, even when there are “many more commodities than agents.” The assumptions we make on economies are much weaker than what is commonly used in core equivalence results for infinite-dimensional commodity spaces and reduce to Aumann’s original assumptions when there are finitely many commodities.