Abstract
We prove the existence of smooth utility functions for a class of preferences (closed preorders) on a subset X in \({\rm I}\!{\rm R}^n\) which satisfies \(X=X+{\rm I}\!{\rm R}^n_+\). This class of preferences is given by the condition that adding one and the same positive vector to each of two comparable alternatives cannot affect the preference relation between them. Moreover, some its subclass consisting of total preferences admits linear utility functions. Also, we prove the existence of universal smooth utilities for preferences depending on a parameter. Our approach relies on our earlier results on continuous utilities for closed (non-total) preorders on metrizable spaces along with a particular device that enable to pass from a continuous utility to a smooth one.
Vladimir L. Levin: Supported in part by Russian Foundation for Basic Research (grant 07-01-00048) and by the Russian Leading Scientific School Support Programme (grant NSh-6417.2006.6).
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References
Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. LN in Economics and Mathematical Systems, vol. 422. Springer, Heidelberg (1995)
Debreu G. Representation of a preference ordering by a numerical function. In: Thrall, R., et al. (eds.) Decision Processes, pp. 159–165. Wiley (1954)
Debreu, G.: Continuity properties of Paretian utility. Intern. Econ. Rev. 5, 285–293 (1964)
Debreu, G.: Smooth preferences. Econometrica 40, 603–615 (1972)
Debreu, G.: Smooth preferences: a corrigendum. Econometrica 44, 831–832 (1976)
Gevers, L.: On interpersonal comparability and social welfare orderings. Econometrica 47, 75–89 (1979)
Kuratowski, K.: Topology, vol.1. Academic Press, Warsaw (1966)
Levin, V.L.: Some applications of duality for the problem of translocation of masses with a lower semicontinuous cost function. Closed preferences and Choquet theory. Soviet Math. Dokl. 24(2), 262–267 (1981)
Levin, V.L.: A continuous utility theorem for closed preorders on a σ-compact metrizable space. Soviet Math. Dokl. 28(3), 715–718 (1983)
Levin, V.L.: General Monge–Kantorovich problem and its applications in measure theory and mathematical economics. In: Leifman, L.J. (ed.) Functional Analysis, Optimization, and Mathematical Economics. A Collection of Papers Dedicated to Memory of L.V.Kantorovich. pp. 141–176. Oxford University Press, N.Y., Oxford (1990)
Levin, V.L.: The Monge–Kantorovich problems and stochastic preference relations. Adv. Math. Econ. 3, 97–124 (2001)
Levin, V.L.: Best approximation problems relating to Monge–Kantorovich duality. Sbornik Math. 197(9), 1353–1364 (2006)
Mas-Colell, A.: The Theory of General Economic Equilibrium: A Differentiable Approach. Cambridge University Press, London (1985)
Neuefeind, W., Trockel, W.: Continuous linear representations for binary relations. Econ. Theory 6, 351–356 (1995)
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Levin, V.L. (2008). On preference relations that admit smooth utility functions. In: Kusuoka, S., Yamazaki, A. (eds) Advances in Mathematical Economics Volume 11. Advances in Mathematical Economics, vol 11. Springer, Tokyo. https://doi.org/10.1007/978-4-431-77784-7_5
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DOI: https://doi.org/10.1007/978-4-431-77784-7_5
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