Equilibrium in incomplete markets revisited

Abstract

This paper is devoted to the proof of an equilibrium existence theorem in incomplete markets, concerning the one-period economy with a finite number states of the world. The proof of the main theorem relies on the Brouwer Fixed Point Theorem. The assumption of incomplete markets is crucial for the proof of the main theorem, since the pricing functionals’-space is not one-dimensional. Usually, in General Equilibrium Theory of Incomplete Markets, the utility functions of the consumers -investors are supposed to be smooth-regular. In the present paper, we assume that the utility functions are continuous and the risk-less asset is extremely desirable. This class of utility functions assures that the demand vector is well-defined and unique, as well. The interest of economists about the effect of perturbation of pricing functionals on the existence of equilibrium is satisfied by the reference on the pricing functionals’ dimension. The equilibrium allocation uniqueness is not deduced by the Brouwer Fixed Point Theorem. We also notice that the existence of equilibrium may be deduced in the corresponding multi-period model by using the corresponding event-tree.