2023-03: Lack of Compactness of Budget Sets in Infinite Dimensional Commodity Spaces and Its Consequences

This paper demonstrates that the budget sets in the Contingent Markets (CM) and Financial Markets (FM) economies with infinite dimensional commodity spaces might not be norm-bounded and therefore might not be weakly (weakly*) compact. The lack of weak (weak*) compactness of these budget sets has serious implications. In particular, it is no longer guaranteed that weakly (weakly*) continuous utility functions attain their maximum on these budget sets. Thus, the individual and therefore total demand functions need not exist in CM and FM economies with infinite dimensional commodity spaces. The lack of existence of demand functions does not however imply the lack of existence of equilibrium in CM and FM economies with infinite dimensional commodity spaces.