Determinacy of equilibria in nonsmooth economies

Abstract

Concavifiable preferences are representable by a function which is twice differentiable almost everywhere, by theorem of Alexandroff [Alexandroff, A.D., 1939. Almost Everywhere Existence of Second Differentials of Convex Functions and Some Related Properties of Convex Surfaces (Russian), Leningrad State University Annals, 3, Math. Series 6, 3–35]. We show that if the bordered hessian determinant of a concave utility representation vanishes only on a null set of bundles, then demand is approximately pointwise Lipschitzian outside of the preimage of a null set of bundles. Given this property, equilibrium prices will be locally unique, for almost every endowment. We give an example of an economy satisfying these conditions but not the smoothness conditions of Katzner [Katzner, D., 1968. A note on the differentiability of consumer demand functions. Econometrica 36, 415–418], Debreu [Debreu, G., 1970. Economies with a finite set of equilibria. Econometrica 44, 387–392], and Debreu [Debreu, G., 1972. Smooth preferences. Econometrica 40, 603–615].