Adv. Math. Econ. 12, 97-127 (2009)

Smooth feasible solutions to a dual Monge—Kantorovich problem with applications to best approximation and utility theory in mathematical economics

Vladimir L. Levin

Central Economics and Mathematics Institute of the Russian Academy of Sciences, Nakhimovskii Prospect 47, 117418 Moscow, Russia

Received: January 21, 2008 Revised: October 9, 2008

JEL classification: C65

Mathematical Subject Classification (2000): 46N10, 41A50, 91B16

Abstract. Given a (closed or open) subset X in IR^{n}, which is stable with respect to shifts in positive directions, we consider inequalities u(x) − u(y) ≤ c(x, y), x, y ∈ X, and for a wide class of functions c on X × X, derive a smooth solution to these inequalities from a Lebesgue measurable one. Applications are given to a best approximation problem and to several problems of mathematical economics relating to preferences that admit smooth (or Lipschitz continuous) utility functions, smooth-utility-rational choice, and smooth representations of interval orders.

Key words: axioms of revealed preference, dual Monge—Kantorovich problem, exact solutions to a best approximation problem, Lipschitz continuous and smooth utility functions, preference, representations of interval orders, smoothutility-rational choice, strengthened acyclicity assumption orders