Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma

Le, Thanh and Le Van, Cuong and Pham, Ngoc-Sang and Sağlam, Çağrı (2020): Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma.

Abstract

In this paper, we use Sperner’s lemma to prove the existence of general equilibrium for a competitive economy with production or with uncertainty and financial assets. We then show that the direct use of Sperner’s lemma together with Carathéodory’s convexity theorem and basic properties of topology such as partition of unit, finite covering of a compact set allow us to bypass the Kakutani fixed point theorem even in establishing the Gale-Nikaido-Debreu Lemma. We also provide a new proof of the Kakutani fixed point theorem based on Sperner’s lemma.

• Item Type: MPRA Paper
• Language: English
• Keywords: Sperner lemma, Simplex, Subdivision, Fixed Point Theorem, Gale-Nikaido-Debreu Lemma, General Equilibrium.
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium; D - Microeconomics > D5 - General Equilibrium and Disequilibrium

References

Aliprantis, C. D., Border, K. C. (2006), Infinite dimensional analysis: a Hitchhiker’s guide, third Edition, Springer-Verlag Berlin Heidelberg.

Arrow,K.J. and Debreu, G. (1954), Existence of an equilibrium for a competitive economy, Econometrica 22, pp. 265-290.

Brouwer, L.E.J. (1911), Uber Abbildung von Mannigfaltigkeiten, Mathematische Annalen 71, pp. 97–115.

Berge, C. (1959), Espaces topologiques et fonctions multivoques, Dunod, Paris. ̈

Border, K. C. (1985), Fixed point theorems with applications to economics and game theory, Cambridge University Press.

Carath ́eodory, C. (1907), Uber den Variabilita ̈tsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Mathematische Annalen (in German). 64 (1): 95-115.

Cass, D., (2006), Competitive equilibrium with incomplete financial markets, Journal of Mathematical Economics 42, pp. 384- 405

Cellina, A. (1969), A theorem on the approximation of compact multivalued map- pings, Atti. Mat. Naz. Lincei 8, 149–153.

Cohen, D.I.A. (1967), On Sperner lemma, Journal of Combinatorial Theory 2, 585- 587.

Debreu, G. (1952), A social equilibrium existence theorem, Proceedings of the National Academy of Sciences 38 (10) 886-893.

Debreu, G. (1956), Market equilibrium, Proceedings of the National Academy of Sci- ences 42 (11) 876-878.

Debreu, G. (1959), Theory of Value - An Axiomatic Analysis of Economic Equilib- rium, Wiley, New York.

Gale, D., Mas-Colell, A. (1975), An equilibrium existence theorem for a general model without ordered preferences, Journal of Mathematical Economics 2, 9-15.

Gale, D., Mas-Colell, A. (1975), Corrections to an equilibrium existence theorem for a general model without ordered preferences, Journal of Mathematical Economics 6, 297–298.

Florenzano, M. (1981), L’Equilibre ́economique g ́en ́eral transitif et intransitif: problemes d’existence, Monographies du s ́eminaire d’ ́econom ́etrie, Editions du CNRS.

Florenzano, M. (1982), The Gale-Nikaido-Debreu lemma and the existence of transi- tive equilibrium with or without the free-disposal assumption, Journal of Mathematical Economics 9, 113-134.

Florenzano, M., (1999), General equilibrium of financial markets: An introduction. CES working paper series.

Florenzano, M., (2009), Two lemmas that changed general equilibrium theory. CES working paper series.

Florenzano, M. and C. Le Van (1986), A note on the Gale-Nikaido-Debreu Lemma and the Existence of General Equilibrium, Economics Letters, 22, 107-110.

Florenzano, M., and Le Van, C. (2001), Finite Dimensional Convexity and Optimization, Springer-Verlag Berlin-Heidelberg.

Gale, D. (1955), The law of supply and demand, Mathematica Scandinavica, 3, 155- 169.

Hadamard, J. (1910), Note sur quelques applications de l’indice de Kronecker. In Introduction `a la th ́eorie des fonctions d’une variable, (volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477.

Kakutani, S. (1941), A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal, Volume 8, Number 3, 457-459.

Knaster, B., Kuratowski K., Mazurkiewicz, S., (1929), A Ein Beweis des Fixpunkt- satzes fu ̈r n- Dimensionale Simplexe, Fund. Math., 14, 132–137.

Kuhn, H. W. (1968), Simplicial aproximations of fixed points, Proc. Nat. Acad: Sci. U.S.A. 61, 1238-1242.

Le Van, C. (1982), Topological degree and the Sperner lemma, Journal of Optimiza- tion Theory and Applications, 37, 371-377.

McKenzie, L.W. (1959), On the existence of general equilibrium for a competitive market, Econometrica, 27, 54-71.

Michael, E. (1956), Continuous selections. I, Annals of Mathematics, Second Series, 63 (2), 361–382.

Nikaido, H. (1956), On the classical multilateral exchange problem. Metroeconomica 8, 135–145.

Park, S. (1999), Ninety years of the Brouwer fixed point theorem. Vietnam Journal of Mathematics 27:3, 187-222.

Park, S. and K.S. Jeong (2003), The proof of the Sperner lemma from the Brouwer fixed point theorem, https://www.researchgate.net/publication/264969230

Scarf, H. (1967), The Approximation of Fixed Points of a Continuous Mapping, SIAM Journal on Applied Mathematics, 15, 5, 1328-1343.

Scarf, H. and T. Hansen (1973), The Computation of Economic Equilibria, Yale University Press, New Haven and London.

Scarf, H. (1982), The Computation of Equilibrium Prices: An Exposition. In Handbook of Mathematical Economics, Volume II, eds. K. Arrow and Alan Kirman, chapter 4.

Sperner, E. (1928), Neuer Beweis fur die Invarianz der Dimensionszahl und des Ge- bietes, Abh. Math. Seminar Univ. Hambourg, 6, 265-272.

Sondjaja, M. (2008), Sperner lemma Implies Kakutani’s Fixed Point Theorem, HMC Senior Theses. 214. https://scholarship.claremont.edu/hmc theses/214

Su, F. E. (1999), Rental Harmony: Sperner lemma in Fair Division, The American Mathematical Monthly 106 (10), 930-942.

Tanaka, Y. (2012), Proof of constructive version of Kakutani’s fixed point theorem directly by Sperner’s lemma and approximate mini-max theorem: A constructive analysis, Advances in Applied Mathematical Analysis, Volume 7 Number 1, pp. 11–27.

Uzawa, H. (1962), Walras existence theorem and Brouwer’s fixed point theorem, Economic Studies Quarterly, 13, pp. 59-62

von Neumann, J. (1937), Uber ein Okonomisches Gleichungs-System und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. In K. Menger (ed.) Ergebnisse eines Mathematischen Kolloquiums. Wien, 1937, pp. 73–83.

Walras, L. (1877), Eléments d’économie politique pure, Lausanne, Corbaz, 1874-1877.

Yoseloff, M. (1974), Topological proofs of some combinatorial theorems, Journal of Combinatorial Theory Series A, 17, 95–111.