Antonio D'Agata, University of Catania

"A Variational Inequality Approach to the Poincaré-Miranda Theorem: Theory and economic applications"

Summary.

Recently, the Poincaré-Miranda (PM) Theorem was proved by using a boundary condition expressed in terms of normal cones (the s.c. avoiding cones (AC) condition). We frame the PM Theorem in terms of variational inequalities and provide two boundary conditions equivalent to the AC condition. The use of the variational inequality approach and of the alternative conditions unveils the common logic behind extant proofs of the PM Theorem, offers generalizations of existing results, and yields economic applications as well. In this context, in particular, we prove the existence of a walrasian equilibrium without the properties and homogeneity and Walras’ Law of the excess demand function. We show also that, whenever the standard simplex is used as price set, the Walras' Law can be substantially weakened but cannot be completely disposed of, as it compensates for the low dimension of the price Space.