Markets with many more agents than commodities: Aumann's "hidden" assumption

We address a question posed by J.-F. Mertens and show that, indeed, R. J. A umann'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 there is a fixed non-separable infinite dimensiona l commodity space in which one can construct an economy that satisfies all the standard assumptions but which has no equilibrium, a core allocation th at is not Walrasian, and a Pareto efficient allocation that is not a valuat ion equilibrium. We identify the source of the failure as the requirement t hat allocations be strongly measurable. Our main example is set in a commod ity-measure space pair that displays an "acute scarcity" of strongly measur able allocations - where strong measurability necessitates that consumer ch oices be closely correlated no matter the prevailing prices, This makes the core large since there may not be any strongly measurable improvements eve n though there are many weakly measurable strict improvements. Moreover, at some prices the aggregate demand correspondence is empty since disaggregat ed demand has no strongly measurable selections, though it does have weakly measurable selections. We note that our example can be constructed in any vector space whose dimension is greater than the cardinality of the continu um-that is, whenever there are at least as many Commodities as agents. We a lso prove a positive core equivalence result for economics in non-separable commodity spaces. (C) 2001 Academic Press.