On Uniqueness of Equilibrium for Complete Markets with Infinitely Many Goods and in Finance Models

Our concern in this work is to obtain conditions for the uniqueness of equilibria, with commodity bundles as consumption patterns which depend on the state of the world. In the first section we consider an economy with complete markets, where consumption spaces are a finite product of measurable function spaces, with separable and proper utility functions and with strictly positive endowments. Using the excess utility function the infinite dimensional problem stated above is reduced to a finite dimensional one. We obtain local uniqueness. The degree theory and specially the Poincar´e-Hopf theorem applied to this excess utility function, allow us to characterize the cardinality of the equilibrium set, and we find conditions for the global uniqueness of this set. On the other hand, we obtain conditions for the uniqueness in economies with incomplete markets and only one good available in each state of the world. When markets are incomplete, equilibrium allocations are typically not Pareto efficient; then the results obtained in section 1, can not be generalized here. Nevertheless we show that for the single consumption good case the first theorem of welfare is satisfied, and then conditions for the uniqueness of equilibrium can be obtained as straightforward extension of our results shown in the first section. This is a particular simple case on incomplete markets but, is a very important one on finance theory.