Existence of Competitive Equilibria without Standard Boundary Behavior

Francesco Ruscitti
DOI: 10.4236/am.2011.211198                

Abstract

We study the existence of competitive equilibria when the excess demand function fails to satisfy the standard boundary behavior. We introduce alternative boundary conditions and we examine their role in proving the existence of strictly positive solutions to a system of non-linear equations (competitive equilibium prices). In addition, we slightly generalize a well-known theorem on the existence of maximal elements, and we unveil the link between the hypothesis of our theorem and one of the boundary conditions introduced in this work.

Keywords

Excess Demand Function, Competitive Equilibrium, Set-Valued Functions, Selections, Fixed Points, Maximal Elements

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. D. Aliprantis, D. J. Brown and O. Burkinshaw, “Existence and Optimality of Competitive Equilibria,” Springer-Verlag, Berlin, 1990. doi:10.1007/978-3-642-61521-4
[2]	K. Arrow and F. Hahn, “General Competitive Analysis,” Holden-Day, San Francisco, 1971.
[3]	A. Mas-Colell, M. D. Whinston and J. R. Green, “Microeconomic Theory,” Oxford University Press, Oxford, 1995.
[4]	J. Geanakoplos, “Nash and Walras Equilibrium via Brouwer,” Economic Theory, Vol. 21, No. 2, 2003, pp. 585-603. doi:10.1007/s001990000076
[5]	G. Q. Tian, “Generalized KKM Theorems, Minimax Inequalities and Their Applications,” Journal of Optimization Theory and Applications, Vol. 83, No. 2, 1994, pp. 375-389. doi:10.1007/BF02190063
[6]	N. C. Yannelis and N. D. Prabhakar, “Existence of Maximal Elements and Equilibria in Linear Topological Spaces,” Journal of Mathematical Economics, Vol. 12, No. 3, 1983, pp. 233-245. doi:10.1016/0304-4068(83)90041-1
[7]	E. Michael, “Continuous Selections, I,” Annals of Mathematics, Vol. 63, No. 2, 1956, pp. 361-382. doi:10.2307/1969615
[8]	C. D. Aliprantis and K. C. Border, “Infinite Dimensional Analysis a Hitchhiker’s Guide,” 3rd Edition, Springer, Berlin, 2006.
[9]	M. Todd, “A Note on Computing Equilibria in Economies with Activity Analysis Models of Production,” Journal of Mathematical Economics, Vol. 6, No. 2, 1979, pp. 135-144. doi:10.1016/0304-4068(79)90002-8
[10]	T. J. Kehoe, “An Index Theorem for General Equilibrium Models with Production,” Econometrica, Vol. 48, No. 5, 1980, pp. 1211-1232. doi:10.2307/1912179
[11]	W. Neuefeind, “Notes on the Existence of Equilibrium Proofs and the Boundary Behavior of Supply,” Econometrica, Vol. 48, No. 7, 1980, pp. 1831-1837. doi:10.2307/1911941
[12]	F. Husseinov, “Boundary Behavior of Excess Demand and Existence of Equilibrium,” Journal of Economic Theory, Vol. 87, No. 2, 1999, pp. 434-449. doi:10.1006/jeth.1999.2547
[13]	G. Impicciatore, L. Panaccione and F. Ruscitti, “Walras’ Theory of Capital Formation: An Intertemporal Equilibrium Reformulation,” Journal of Economics, Forthcoming.
[14]	C. O. Ewald, “Games, Fixed Points and Mathematical Economics,” University of St. Andrews, School of Economics and Finance, SSRN Lecture Notes, St. Andrews, 2007.
[15]	G. Tian, “Fixed Points Theorems for Mappings with Non-Compact and Non-Convex Domains,” Journal of Mathematical Analysis and Applications, Vol. 158, No. 1, 1991, pp. 161-167. doi:10.1016/0022-247X(91)90274-4
[16]	J. Zhou, “On the Existence of Equilibrium for Abstract Economies,” Journal of Mathematical Analysis and Applications, Vol. 193, No. 3, 1995, pp. 839-858. doi:10.1006/jmaa.1995.1271
[17]	F. E. Browder, “The Fixed Point Theory of Multi-Valued Mappings in Topological Vector Spaces,” Mathematische Annalen, Vol. 177, No. 4, 1968, pp. 283-301. doi:10.1007/BF01350721