Walrasian equilibria are almost always finite in number

In the context of exchange economies, defined by aggregate excess demand functions, we extend results on finiteness of equilibria to economies defined on the full open price simplex. Genericity 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 proper maps. We translate the result to the set of aggregate excess demand functions of an exchange economy to show that finiteness of equilibria is a generic property in sets of economies which form an open subset of the space of proper maps. We construct an explicit class of aggregate excess demand for such economies spanned by Cobb-Douglas consumers.
Along the way, we explore the extension of the classical results of Sonnenschein-Mantel-Debreu to functions defined on the full open price simplex, rather than just compact subsets of the simplex. In particular, we identify necessary boundary conditions for such functions to be aggregate excess demand functions.