Walrasian equilibria are almost always finite in number

TL;DR

This paper proves that in typical exchange economies, the number of Walrasian equilibria is finite, using singularity theory, and extends classical results to full open simplices.

Contribution

It introduces the concept of finite singularity type to show generic finiteness of equilibria in exchange economies on the full open simplex.

Findings

Generic economies have finitely many equilibria.

Finite singularity type maps are open and dense among smooth maps.

Extension of classical results to full open price simplices.

Abstract

We show that in the context of exchange economies defined by aggregate excess demand functions on the full open price simplex, the generic economy has a finite number of equilibria. Genericicity is proved also for critical economies and, in both cases, in the strong sense that it holds for an open dense subset of economies in the Whitney topology. We use the concept of finite singularity type from singularity theory. This concept ensures that the number of equilibria of a map appear only in finite number. We then show that maps of finite singularity type make up an open and dense subset of all smooth maps and translate the result to the set of aggregate excess demand functions of an exchange economy. Along the way, we extend the classical results of Sonnenschein-Mantel-Debreu to aggregate excess demand functions defined on the full open price simplex, rather than just compact subsets of the simplex.