Competitive Equilibria on Turnpikes in a Mckenzie Economy, I: A Neighborhood Turnpike Theorem

Abstract

In his seminal paper, Bewley [1982] integrates growth theory and competitive equilibrium theory, considering a model with finitely many infinitely lived consumers who trade commodities in infinitely many competitive markets from the present through the future. In such a model, a competitive equilibrium path is an optimal growth path. Using this property, he proves a turnpike theorem for competitive equilibrium paths. In growth theory, a social welfare function, with respect to which optimal growth paths are considered, has been assumed to be exogenously given. Bewley shows that in the competitive market system the market mechanism endogenously determines the social welfare function with respect to which an equilibrium path is regarded as an optimal path. In Bewley's model, the future is discounted by a factor p. In growth theory, two types of turnpike theorems have been proved when the future is discounted. The first type is called a neighborhood turnpike theorem.' It asserts that given c>0, there is 0< p' <1 such that p'? p < 1 implies that an optimal path at the discount factor p eventually stays within the s-neighborhood of a stationary path. This theorem does not require the social welfare function to be differentiable but only to be strictly concave, and is proved by McKenzie [1979]. The second type is called an asymptotic turnpike theorem. It asserts that there is 0 < p' < 1 such that p' < p < 1 implies that an optimal path at the dicount factor p converges to a stationary path. This theorem has been proved only under differentiability assumptions (Scheinkman [1976]). Bewley proves an asymptotic turnpike theorem, making use of extensive differentiability assumptions. The purpose of this paper is to synthesize these two importnat results of Bewley and McKenzie. Namely, we will prove a neighborhood turnpike theorem of McKenzie type in a model like that of Bewley, making no differentiability assumptions. We will make a few generalizations of Bewley's assumptions. Among them, of importance are relaxing his differentiability assumptions, weakening his interiority assumption, and replacing his assumption of decreasing returns to