Definable utility in o-minimal structures

We obtain definable utility representations for both continuous and upper semi-continuous definable preferences in the "tame topology" of o-minimal expansions of real closed ordered fields [Pillay, A., Steinhom, C., 1986. Definable sets in ordered structures. I. Transactions of American Mathematical Society 295, 565-592; Knight, K., Pillay, A., Steinhorn, C., 1986. Definable sets in ordered structures. II. Transactions of American Mathematical Society 295, 593-605; Van den Dries, L., 1998. Tame Topology and O-minimal Structures. Cambridge Univ. Press, Cambridge; etc.]. Such preferences have significant applications, for example, in establishing local determinacy of competitive equilibrium [Blume, L., Zame, W., 1992. The algebraic geometry of competitive equilibrium. In: Neuefeind, W., Riezman, R.G. (Eds.), Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin.], and in modeling bounded rationality [Richter, M.K., Wong, K.-C., 1996. Bounded rationalities and definable economies. Working Paper No. 295, University of Minnesota.]. Our proofs are based on geometric theorems for definable sets, and provide new alternatives to the classical tools of separability [Debreu, G., 1954. Representation of a preference ordering by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. (Eds.), Decision Processes. Wiley, New York (Reprinted in Mathematical Economics by Debreu, G., 1983, Cambridge Univ. Press, Cambridge); Rader, T., 1963. The existence of a utility function to represent preferences. Review of Economic Studies 30, 229-232.] and metric distance [Arrow, K., Hahn, F.H., 1971. General Competitive Analysis. Holden-Day, San Francisco.]. The results extend Theorem 1 of [Blume, L., Zame, W., 1992. The algebraic geometry of competitive equilibrium. In: Neuefeind, W., Riezman, R.G. (Eds.), Economic Theory and International Trade: Essays in Memorium J. Trout Rader. Springer-Verlag, Berlin.] in several directions. (C) 2000 Elsevier Science S.A. All. rights reserved. JEL classification: D11.