Publisher Summary This chapter summarizes the account of the extension of the classical general equilibrium model to an infinite dimensional setting. The classical finite dimensional theory, the commodity space is the canonical finite dimensional linear space R n . By contrast, there is no canonical infinite dimensional linear space. Different economic applications require models involving different infinite dimensional linear spaces. The mathematical discipline of functional analysis has already been well developed as a tool for the abstract study of linear spaces. The chapter follows the methodology of functional analysis and attacks the existence problem. Advantage of this method is that it yields general results, capable of application in a wide variety of specific models. An important line of research in classical general equilibrium theory has been the relationship of the core to the set of competitive allocations. In the infinite dimensional setting, an extensive body of work has been developed, which centers around the infinite-dimensional version of the Debreu–Scarf core convergence theorem.

Chapter 34 Equilibrium theory in infinite dimensional spaces

Publisher Summary Market demand and market excess demand functions are defined by summing the preference-maximizing actions of consumers. These functions are central to most treatments of value theory. It discusses that the demand theory of the individual consumer has been well developed. Just as consumer demand theory is the study of properties common to demand functions that are generated by the preference-maximization of an individual consumer, market demand theory investigates properties that are shared by market demand functions. As market demand is often the observable and relevant variable for analysis, it is important to see how it is restricted by the utility hypothesis. When preferences are homothetic and the distribution of income is independent of prices, the market demand function has all the properties of a consumer demand function. The chapter discusses conditions under which market demand and market excess demand functions are consumer demand functions. It also provides an overview of Debreu's theorem, which explains that for every continuous function F that satisfies homogeneity and Walra's Law and for every compact set of positive prices K, there exists an economy with market excess demand function F on K.

Chapter 14 Market demand and excess demand functions

Economic equilibria are usually solutions to fixed point problems rather than solutions to convex optimization problems. This leads to two difficulties that are closely related: First, equilibria may be difficult to compute. Second, a model economy may have more than one equilibrium. This paper explores these issues for a number of stylized economies, including static economies that involve both pure exchange and production, economies that have infinite numbers of goods because of time and uncertainty, and economies with distortionary taxes and externalities. There are numerous numerical examples that illustrate the theory and could serve as test problems for algorithms.

Computation and multiplicity of equilibria

We characterize equilibria of general equilibrium models with externalities and taxes as solutions to optimization problems. This characterization is similar to Negishi's characterization of equilibria of economies without externalities or taxes as solutions to social planning problems. It is often useful for computing equilibria or deriving their properties. Frequently, however, finding the optimization problem that a particular equilibrium solves is difficult. This is especially true in economies with multiple equilibria. In a dynamic economy with externalities or taxes there may be a robust continuum of equilibria even if there is a representative consumer. This indeterminacy of equilibria is closely related to that in overlapping generations economies.

/pdf/on-characterizing-equilibria-of-economies-with-externalities-2y4oxahs9z.pdf

William R. Zame

Papers

Determinacy of equilibrium in large-scale economies

A stein manifold topologically but not holomorphically equivalent to a domain in Cn