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

A. C. Eberhard; J-P. Crouzeix

doi:10.3934/jimo.2007.3.233

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2007, Volume 3, Issue 2: 233-255. Doi: 10.3934/jimo.2007.3.233

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

A. C. Eberhard^1, and
J-P. Crouzeix^2,

1.
School of Mathematical and Geospatial Sciences, Royal Melbourne Institute of Technology, G.P.O. Box 2476V, Melbourne, Australia 3001
2.
LIMOS, Université Blaise Pascal, Boite Postale 206, F-63174 AUBIERE Cedex

Received: July 31, 2006

Revised: September 30, 2006

Published: March 2007

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Abstract

We investigate a multifunction $x$→Ñ _f $(x)$ derived via normal cones to the level sets Š $(x)$ := { $x^$' | $f(x^$') $ < f(x)$} for an important class of pseudo--convex functions. It is shown that $x$→Ñ _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.

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Mathematics Subject Classification: Primary: 47H05, 90A40, 26A51; Secondary: 90A10, 52A01.

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