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Topology and invertible maps

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  • Chichilnisky, Graciela

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.

Suggested Citation

  • Chichilnisky, Graciela, 1997. "Topology and invertible maps," MPRA Paper 8811, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:8811
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    References listed on IDEAS

    as
    1. Chichilnisky, Graciela & Zhou, Yuqing, 1998. "Smooth infinite economies," Journal of Mathematical Economics, Elsevier, vol. 29(1), pages 27-42, January.
    2. Dierker, Egbert, 1972. "Two Remarks on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 40(5), pages 951-953, September.
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    Cited by:

    1. Elvio Accinelli, 2004. "Inversión Bajo Incertidumbre," Remef - Revista Mexicana de Economía y Finanzas Nueva Época REMEF (The Mexican Journal of Economics and Finance), Instituto Mexicano de Ejecutivos de Finanzas, IMEF, vol. 3(1), pages 21-44, Marzo 200.
    2. Covarrubias, Enrique, 2008. "Necessary and sufficient conditions for global uniqueness of equilibria," MPRA Paper 8833, University Library of Munich, Germany.

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    More about this item

    Keywords

    manifolds; mathematical economics; Jacobian; supply and demand; equilibrium;
    All these keywords.

    JEL classification:

    • C68 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computable General Equilibrium Models
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

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