Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory

Abstract

We investigate a multifunction $x$→N f $(x)$ derived via normal cones to the level sets S $(x)$ := { $x^$' | $f(x^$') $ < f(x)$} for an important class of pseudo--convex functions. It is shown that $x$→N f $(x)$ is simultaneously both a maximally cyclically pseudo--monotone and a maximally pseudo-monotone relation within neighbourhoods on which $f$ is nonconstant. The relevance of this work to the problem of construction of a utility function from observations of revealed preferences of a consumer is discussed.