Regular Infinite Economies

Abstract

The main contribution of this paper is to place smooth infinite economies in the setting of the equilibrium manifold and the natural projection mapà la Balasko. We show that smooth infinite economies have an equilibrium set that has the structure of a Banach manifold and that the natural projection map is smooth. We define regular and critical economies, and regular and critical prices, and we show that the set of regular economies coincides with the set of economies whose excess demand function has only regular prices. Generic determinacy of equilibria follows as a by-product. Keywords: General equilibrium, infinite economies, intertemporal choice, uncertainty. JEL Classification: D5, D50, D51, D80, D90.Resumen La principal contribución de este trabajo es ubicar a las economías infinitas suaves en el marco de la variedad de equilibrio y la proyección naturalà la Balasko. Demostramos que las economías infinitas suaves tienen un conjunto de equilibrio que tiene la estructura de una variedad diferencial de Banach y que la proyección natural es suave. Definimos economías regulares y críticas, y precios regulares y críticos, y demostramos que el conjunto de economías regulares coincide con el conjunto de economías cuya función de exceso de demanda sólo contiene precios regulares. Determinación genérica de equilibrios se obtiene como corolario. Palabras Clave: Equilibrio general, economías infinitas, elección intertemporal, incertidumbre.

The number of equilibria of smooth infinite economies

“…It is shown in Covarrubias (2010) that it satisfies P, Z ω (P ) = 0 for all P ∈ S, which in turn implies that Z ω can be interpreted as a vector field on the set of prices S. When thought of as a vector field, we will write Z ω : S → T S where T S is the tangent bundle of S. Definition 1. We say that P ∈ S is an equilibrium of the economy ω ∈ Ω if Z ω (P ) = 0.…”

“…Theorem 1. (Covarrubias, 2010) The economy ω ∈ Ω is regular if and only if all equilibrium prices of Z ω are regular.…”

“…Theorem 2. (Covarrubias, 2010) Almost all economies are regular. That is, the set of economies ω ∈ Ω that give rise to an excess demand function Z ω with only regular equilibrium prices, are residual in Ω.…”

“…First, subsection 3.1 contains a quick review of the basic definitions of Fredholm theory, which are mathematical tools needed to extend differential topology to infinite dimensions (further mathematical definitions are included in an appendix). In section 3.2 we review the determinacy results obtained previously in Covarrubias (2010) showing that most excess demand functions have isolated zeros; that is, that equilibria are locally unique. This guarantees that it makes sense to actually count the number of equilibria.…”

The equilibrium set of economies with a continuous consumption space

“…Given these trade-offs, in our previous work ( [10], [11]) we set to study models of pure exchange economies with a continuous commodity space. As we will exemplify below, this model is a good framework to study problems in finance, infinite-horizon models and product differentiation.…”