## On the equilibrium prices of a regular locally Lipschitz exchange economy

## Main Article Content

### Abstract

We extend classical results by Debreu and Dierker about equilibrium prices of a regular economy with continuously differentiable demand functions/excess demand function to a regular exchange economy with these functions being locally Lipschitz. Our concept of a regular economy is based on Clarke’s concept of regular value and we show that such a regular economy has a finite, odd number of equilibrium prices, the set of economies with infinite number of equilibrium prices has Lebesgue measure zero and there exist locally Lipschitz selections of equilibrium prices around a regular economy.

## Article Details

### Keywords

Brower’s degree, Sard’s theorem, essentially smooth Lipschitz map, nonsmooth equations, regular value, regular exchange economies, equilibrium price

### References

[2] Blume, L.E., Zame, W.R. The algebraic geometry of competitive equilibrium. In: Neuefeind, W. (ed.) General Equilibrium and International Trade. In Memoraum Trout Rader, pp. 53–66. Springer, New York (1993)

[3] Borwein, J. M. Minimal cuscos and subgradients of Lipschitz functions. In: Baillion J.-B., Thera, M. (Eds.) Fixed Point Theory and Its Applications, Pitman Lecture Notes in Math., pp. 57-82. Longman, Essex, UK (1991)

[4] Clarke, F. Optimization and Nonsmooth Analysis. Wiley, New York (1983)

[5] Debreu, G. Economies with a Finite Set of Equilibria. Econometrica, Vol. 38, No. 3, 387-392 (May, 1970)

[6] Deimling, K.Nonlinear functional analysis, Springer-Verlag, New York(1985)

[7] Dierker, E. Two remarks on the number of equilibria of an economy. Econometrica 40, 951-953 (1972)

[8] Dierker, E. Regular economies, Chapter 17, Handbook of Mathematical Economics, edited by K.J. Arrow and M.D. Intriligator. North-Holland Publishing Company (1982)

[9] Ha, T.X.D. The Sard’s theorem for a class of locally Lipschitz mappings. S´eminaire d’analyse convexe, Vol. 17, Exp. No. 9, 13 pp., Univ. Sci. Tech. Languedoc, Montpellier (1989)

[10] Ha, T.X.D. Versions of the Sard theorem for essentially smooth Lipschitz maps and applications in optimization and nonsmooth equations, Journal of Convex Analysis 28, No.1, 157-178 (2021)

[11] Ioffe, Alexander D. Variational analysis of regular mappings. Theory and applications. Springer Monographs in Mathematics. Springer, Cham, 2017.

[12] Lebourg, G. Generic differentiability of Lipschitz functions. Trans. AMS. 256, 125-144 (1979)

[13] Pourciau, B. Analysis and Optimization of Lipschitz continuous mappings. J. Optim. Theory Appl. 22, 311-351 (1977)

[14] Pourciau, B. Univalence and degree for Lipschitz continuous functions, Archive for Rational Mechanics and Analysis 81, no. 3, 289-299 (1983)

[15] Rader, T. Nice demand functions, Econometrica 41, 913-935 (1973)

[16] Sard, A. The measure of critical values of differentiable maps. Bull. AMS. 48, 883-890 (1942)

[17] Shannon, C. Regular nonsmooth equations. Jour. Math. Economics 23,147-165 (1994)

[18] Shannon, C.A prevalent transversality theorem for Lipschitz functions, Proceedings AMS 134, Num. 9, 2755-2765 (2006)