Topology and invertible maps

Graciela Chichilnisky

MPRA Paper from University Library of Munich, Germany

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.

Keywords: manifolds; mathematical economics; Jacobian; supply and demand; equilibrium (search for similar items in EconPapers)
JEL-codes: C02 C62 C68 (search for similar items in EconPapers)
Date: 1997-09-15
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Citations: View citations in EconPapers (2)

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