Manifolds

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Manifolds

BIBLIOGRAPHY

In mathematics, a manifold is a space that looks like a Euclidean space, locally around every point. Since a set is defined by its elements, two spaces can be seen as equivalent if there is a bijection (one-to-one correspondence) between them. There are different versions of this equivalence, depending on the structure of the bijection. When the two spaces are topological, the bijection is a homeomorphism if it is continuous and has a continuous inverse. If both spaces are Euclidean, the bijection is a CK diffeormorphism if both it and its inverse are (or can be extended to be) of class CK. A topological space is a J-dimensional topological manifold if for every point there exists a homeomorphism from some open subset of the J-dimensional Euclidean space to an open neighborhood of the point. If, in the Euclidean case, what one has is a CK diffeormorphism, then the space is a CK differentiable manifold.

A manifold has the property that, at every point, one can find a neighborhood that is just a transformation of a Euclidean space, preserving topological and/or differential properties. An example of this type of transformation is from a circle to an ellipse, whereas going to a number eight would not be bijective, to a line would violate continuity, and to a triangle would violate differentiability.

Differentiable manifolds are the natural objects for the application of calculus, and the study of that application, differential topology, has proved useful in economic theory. These techniques were introduced to economics in 1970 by Gerard Debreus proof that, generically in the space of exchange economies (that is, in almost all of them), competitive equilibria are isolated from one another. The argument is an application of the Transversal Density Theorem of differential topology.

Later, Darrel Duffie and Wayne Shafer used Grassmanian manifolds (the sets of linear subspaces of a given Euclidean space) to show that in exchange economies subject to uncertainty and endowed with real assets (those whose payoffs depend on the prices of commodities), a competitive equilibrium generically exists. This application was necessary because the dependence of payoffs on prices makes the space of revenue transfers across states of nature variable. The idea of Duffie and Shafer was to break that dependence, effectively weakening the concept of equilibrium but allowing the use of the manifold structure of the Grassmanian to guarantee its existence, and then to show that, generically, the weaker equilibrium is indeed equilibrium.

Another remarkable application of techniques of differential topology in economics is the proof by John Geanakoplos and Herakles Polemarchakis that in a generic economy with uninsurable risks, just by changing their trade in assets, and then competitively trading in commodity markets, all individuals could be better off than at the competitive equilibrium allocation. This result was also obtained from the Transversal Density Theorem.

But perhaps the most visible application of the concept is the result, claimed by Yves Balasko, that in economies described by the preferences of the individuals, the set of pairs of profiles of endowments and associated competitive equilibrium prices is a differentiable manifold. After this result, the equilibrium set of an economy is usually referred to as equilibrium manifold.

SEE ALSO Topology

BIBLIOGRAPHY

Balasko, Yves. 1975. The Graph of the Walras Correspondence. Econometrica 43 (5-6): 907-912.

Debreu, Gerard. 1970. Economies with a finite set of equilibria. Econometrica 38 (3): 387-392.

Duffie, Darrell and Wayne Shafer. 1985. Equilibrium in Incomplete Markets I: A Basic Model of Generic Existence. Journal of Mathematical Economics 14 (3): 285-300.

Geanakoplos, John D., and Herakles M. Polemarchakis. 1986. Existence, Regularity and Constrained Suboptimality of Competitive Allocations when the Asset Market is Incomplete. In Uncertainty, Information and Communication. Essays in Honor of Kenneth J. Arrow, Vol. III, edited by Walter P. Heller, Ross M. Starr, and David A. Starret. New York: Cambridge University Press.

Andrés Carvajal

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