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Working paper

Remarks on the Afriat's theorem and the Monge-Kantorovich problem

Kolesnikov A., Kudryavtseva O., Nagapetyan T.
The classical concept of the revealed preferences was introduced by P. Samuelson and studied by H.S. Houthakker,  M. Richter, S. Afriat, H. Varian and many others. It was shown by Afriat  that the so called SARP (or cyclically consistence) axiom  is a necessary and sufficient condition for existence of an appropriate concave utility function for a finite set of choices and prices observed (this is called the rationalization of the preferences relations by a concave utility function). Later Varian suggested some effective tests for SARP in the homogeneous case. The result on existence of a homogeneous concave utility function can be considered as a particular fact of the Monge-Kantorovich mass transportation theory, which has found numerous applications in many fields of mathematics during the last decade. In this paper we explain this viewpoint and discuss some related questions. We give an instructive and short description of the relation between these concepts, which seems somehow missing in the literature. Applying some recent results on the Monge–Kantorovich problem, we give a complete characterization of the homogeneous rationalizable data sets in the continuous case from the ”optimal mass transportation” viewpoint.