Sofia B. S. D. Castro

(Centro de Matemática and Faculdade de Economia, Universidade do Porto)

(Centro de Matemática and Faculdade de Economia, Universidade do Porto)

Seminar

Wednesday, 21 April, 2010, at 15:15-16:15, in Aud. D3 (1531-215)

Abstract:

Joint work with: Sami Dakhlia (Department of Economics and Finance, College of Business, University of Southern Mississippi, USA) and Peter Gothen (Centro de Matemática and Faculdade de Ciências, Universidade do Porto, Portugal).

A main concern in economics is the study of the set of equilibria in an economy, that is, the set of prices for which supply equals demand in the economy. Supply and demand are best modelled by the excess demand function defined to be the difference between demand and supply, that is, equilibrium prices are zeros of the aggregate excess demand. In 1970, Debreu ["Economies with a Finite set of equilibria," Econometrica 38, 38-392] showed that regular economies (those repre- sented by a regular aggregate excess demand) have finitely many isolated equilibria. It is however known that critical economies may have an infinite number, even a continuum, of equilibrium prices. By showing that the aggregate excess demand of an economy is generically (that is, for a residual subset of the set of smooth maps) a Boardman map, we are able to perturb any smooth aggregate excess demand function into a function with isolated zeros. Finiteness then follows when we show that, under generic assumptions on the economy, the set of price equilibria is compact. It is by no means obvious that the perturbed aggregate excess demand function represents a perturbation of the initial economy. In order to ensure this, we use a natural topology for the space of preferences and geometric arguments to deal with the perturbation of the indierence level surfaces of agents. After a brief description of the concepts and problems in economics, I shall present the results in Castro and Dakhlia ["Finiteness of Walrasian equilibria", SSRN 1156100] and in Castro, Dakhlia and Gothen ["Direct perturbations of aggregate excess demand", Journal of Mathematical Economics, to appear] that lead to the proof of finiteness of equilibria.

A main concern in economics is the study of the set of equilibria in an economy, that is, the set of prices for which supply equals demand in the economy. Supply and demand are best modelled by the excess demand function defined to be the difference between demand and supply, that is, equilibrium prices are zeros of the aggregate excess demand. In 1970, Debreu ["Economies with a Finite set of equilibria," Econometrica 38, 38-392] showed that regular economies (those repre- sented by a regular aggregate excess demand) have finitely many isolated equilibria. It is however known that critical economies may have an infinite number, even a continuum, of equilibrium prices. By showing that the aggregate excess demand of an economy is generically (that is, for a residual subset of the set of smooth maps) a Boardman map, we are able to perturb any smooth aggregate excess demand function into a function with isolated zeros. Finiteness then follows when we show that, under generic assumptions on the economy, the set of price equilibria is compact. It is by no means obvious that the perturbed aggregate excess demand function represents a perturbation of the initial economy. In order to ensure this, we use a natural topology for the space of preferences and geometric arguments to deal with the perturbation of the indierence level surfaces of agents. After a brief description of the concepts and problems in economics, I shall present the results in Castro and Dakhlia ["Finiteness of Walrasian equilibria", SSRN 1156100] and in Castro, Dakhlia and Gothen ["Direct perturbations of aggregate excess demand", Journal of Mathematical Economics, to appear] that lead to the proof of finiteness of equilibria.

Contact person: Jørgen Ellegaard Andersen