Topology and invertible maps

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Topology and invertible maps

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Chichilnisky, Graciela

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Graciela Chichilnisky

Abstract

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.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 8811.

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Date of creation: 15 Sep 1997
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Handle: RePEc:pra:mprapa:8811

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Paper
- Chichilnisky, G., 1996. "Topology and Invertible Maps," Discussion Papers 1996_15, Columbia University, Department of Economics.

Find related papers by JEL classification:
C68 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Computable General Equilibrium Models
C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
C62 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Existence and Stability Conditions of Equilibrium

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

Chichilnisky, Graciela & Zhou, Yuqing, 1998. "Smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 29(1), pages 27-42, January. [Downloadable!] (restricted)
Other versions:
- Chichilnisky, G & Zhou, Y, 1996. "Smooth Infinite Economiesq," Discussion Papers 1996_30, Columbia University, Department of Economics.
- Chichilnisky, G. & Zhou, Y., 1996. "Smooth Infinite Economies," Discussion Papers 1996_14, Columbia University, Department of Economics.
Dierker, Egbert, 1972. "Two Remarks on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 40(5), pages 951-53, September. [Downloadable!] (restricted)

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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

Elvio Accinelli & Daniel Vaz, 1999. "Inversión bajo incertidumbre," Documentos de Trabajo (working papers) 1299, Department of Economics - dECON. [Downloadable!]

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