Chichilnisky, Graciela (1997): Topology and invertible maps. Published in: Advances in Applied Mathematics , Vol. 21, (1998): pp. 113-123.
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I study connected manifolds and prove that a proper map f: M -> M is globally invertible when it has a nonvanishing Jacobian and the fundamental group pi (M) is finite. This includes finite and infinite dimensional manifolds. Reciprocally, if pi (M) is infinite, there exist locally invertible maps that are not globally invertible. The results provide simple conditions for unique solutions to systems of simultaneous equations and for unique market equilibrium. Under standard desirability conditions, it is shown that a competitive market has a unique equilibrium if its reduced excess demand has a nonvanishing Jacobian. The applications are sharpest in markets with limited arbitrage and strictly convex preferences: a nonvanishing Jacobian ensures the existence of a unique equilibrium in finite or infinite dimensions, even when the excess demand is not defined for some prices, and with or without short sales.
|Item Type:||MPRA Paper|
|Original Title:||Topology and invertible maps|
|Keywords:||manifolds; mathematical economics; Jacobian; supply and demand; equilibrium|
|Subjects:||C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C68 - Computable General Equilibrium Models
C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods
C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62 - Existence and Stability Conditions of Equilibrium
|Depositing User:||Graciela Chichilnisky|
|Date Deposited:||22. May 2008 04:40|
|Last Modified:||14. Feb 2013 18:02|
K. Arrow and F. Hahn, "General Competitive Analysis," North-Holland, Amsterdam, New York, and Tokyo, 1986.
R. Abraham and J. Robbin, "Transversal Mappings and Flows," W. A. Benjamin, New York and Amsterdam, 1967.
G. Chichilnisky, "Manifolds of Preferences and Equilibria," Ph.D. dissertation, University of California, Berkeley, 1976; Project on Efficient Decision Making in Economic Systems, Harvard University, 1976, in "Essays in Honor of G. Debreu" (Hildenbrand and MasColell, Eds.), pp. 131-142, North Holland, 1986.
G. Chichilnisky, Market, arbitrage, social choice and the core, in "Columbia Celebrates Arrow's Contributions," Columbia University, New York, October 27, 1991; CORE Discussion Paper No. 9342, CORE, Universite Catolique de Louvain, Louvain la Neuve, Belgium, 1993, Social Choice and Welfare 14 (1997), 161-198.
G. Chichilnisky, Limited arbitrage is necessary and sufficient for the existence of a competitive equilibrium and the core and limits voting cycles, Economic Lett. December (1994); The paper reappeared in Economic Lett. 52 (1996), 177-180.
G. Chichilnisky, "Limited Arbitrage is Necessary and Sufficient for the Existence of a Competitive Equilibrium with or Without Short Sales," Working Paper No. 650, Columbia University, December 1992; published in Econom. Theory 5 (1995), 79-108.
G. Chichilnisky, "A Unified Perspective on Resource Allocation: Limited Arbitrage Is Necessary and Sufficient for the Existence of a Competitive Equilibrium, the Core and Social Choice," CORE Discussion Paper No. 9527, 1995 (to appear in "Social Choice Reexamined" (K. Arrow, A. Sen, and T. Suzumura, Eds), International Economic Association, also in Metroeconomica 47, No. 3 (1996), 266-280.
G. Chichilnisky, "A Topological Invariant for Competitive Markets," Working Paper, Columbia University, May 1996, J. Math. Econom. 28 (1997), 445-469.
G. Chichilnisky and Y. Zhou, Smooth infinite economies, J. Math. Econom. 29, No. 1 (1998), 27-41.
E. Dierker, Two remarks on the number of equilibria of an economy, Econometrica 40 (1972), 951-955.
C. Ehresman, Sur les espaces fibres differentiables, C. R. Hebd. Seanc. Acad. Paris 224 (1947), 1611-1612.
M. J. Greenberg, "Lectures on Algebraic Topology," Mathematics Lecture Note Series, W. A. Benjamin, Reading, MA, 1967.
J. Hadamard, Sur les transformations ponetuelles, Bull. Soc. Math. de France, 34, (1906).
E. Spanier, "Algebraic Topology," McGraw-Hill, New York, 1966.