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

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April 2007, 3(2): 233-255. doi: 10.3934/jimo.2007.3.233

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, Australia
2.	LIMOS, Université Blaise Pascal, Boite Postale 206, F-63174 AUBIERE Cedex, France

Received August 2006 Revised October 2006 Published April 2007

Abstract
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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.

Keywords: revealed preferences., cyclic quasi/pseudo-monotonicity, utility functions, normal cone mappings.

Mathematics Subject Classification: Primary: 47H05, 90A40, 26A51; Secondary: 90A10, 52A0.

Citation: A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial & Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233

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