Topology and invertible maps
AbstractI 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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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 8811.
Date of creation: 15 Sep 1997
Date of revision:
manifolds; mathematical economics; Jacobian; supply and demand; equilibrium;
Other versions of this item:
- 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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- Dierker, Egbert, 1972. "Two Remarks on the Number of Equilibria of an Economy," Econometrica, Econometric Society, vol. 40(5), pages 951-53, September.
- Chichilnisky, G. & Zhou, Y., 1996.
"Smooth Infinite Economies,"
1996_14, Columbia University, Department of Economics.
- Elvio Accinelli & Daniel Vaz, 1999. "Inversión bajo incertidumbre," Documentos de Trabajo (working papers) 1299, Department of Economics - dECON.
- Covarrubias, Enrique, 2008. "Necessary and sufficient conditions for global uniqueness of equilibria," MPRA Paper 8833, University Library of Munich, Germany.
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