Market with Many More Agents than Commodities

This paper shows that, indeed , Aumann's classical existence and equivalence theorems depend on there being many more agents than commodities. We show that for an arbitrary atomless measure space of agents three is a fixed non-separable infinite dimensional commodity space in which one can construct an economy that satisfies all the standard assumptions but which has no equilibrium, a core allocation that is not Walrasian, and a Pareto efficient allocation that is not a valuation equilibrium. We identify the source of the failure as the technical requirement that allocations to be strongly measurable. Our main example is set in a commodity-measure space pair that displays an "acute scarcity" of strongly measurable allocations - and where strong measurability necessitates that consumer choices be closely correlated no matter the prevailing prices. This makes the core large since there may not be any strongly measurable improvements even though there are many weakly measurable strict improvements. Moreover, at some prices that aggreate demand correspondence is empty since disaggregated demand has no strongly measurable selections, though it does have weakly measurable selections. We also prove a positive core equivalence result for economies in non-separable commodity spaces.