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

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.

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On characterizing equilibria of economies with externalities and taxes as solutions to optimization problems

https://hal.archives-ouvertes.fr/hal-00630200/document

Demographic structure and capital accumulation

Stochastic olg models, market structure and optimality

On the existence of price equilibria in dynamic economies

http://seaver-faculty.pepperdine.edu/jburke2/articles/Overlapping.pdf

Equilibrium for Overlapping Generations in Continuous Time

Inactive transfer policies and efficiency in general overlapping-generations economies

A benchmark for comparative dynamics and determinacy in overlapping-generations economies

We combine and strengthen optimality and robustness theorems for the overlapping-generations model of money. Roughly, we find a Pareto-optimal monetary equilibrium of a generic stationary economy that is near an optimal monetary equilibrium of each nearby non-stationary economy. Since the nearby equilibria are monetary, the general problem of macroeconomic stabilization reduces to maintaining the money supply. And since the nearby equilibria are optimal, stabilization is socially desirable.

Jonathan L. Burke

Papers

On the existence of price equilibria in dynamic economies

Equilibrium for Overlapping Generations in Continuous Time

Inactive transfer policies and efficiency in general overlapping-generations economies

A benchmark for comparative dynamics and determinacy in overlapping-generations economies

The robustness of optimal equilibrium among overlapping generations