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. 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 there 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 requirement that allocations be strongly measurable. Our main example is set in a commodity measure space pair that displays an ``acute scarcity'' of strongly measurable allocations 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 the aggregate demand correspondence is empty since disaggregated 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 continuum that is, whenever there are at least as many commodities as agents. We also prove a positive core equivalence result for economies in non-separable commodity spaces. Journal of Economic Literature Classification Numbers: C62, C71, D41, D50. 2001 Elsevier Science doi:10.1006 jeth.2000.2705, available online at http: www.idealibrary.com on