Globally and universally convergent price adjustment processes

Abstract

We discuss three processes of price adjustment in an exchange economy, proposed by Smale (1976), van der Laan and Talman (1987), and Kamiya (1990), respectively. Under a regularity condition on the economy, the first process is guaranteed to converge to a competitive equilibrium for almost every initial price system such that one of the prices is equal to zero. The process of Kamiya (1990) is guaranteed to converge to the set of competitive equilibria for almost every initial price system, under a condition on the boundary behavior of the excess demand function of the economy. The (van der Laan and Talman 1987) process was shown by Herings (1997) to converge to a competitive equilibrium for a generic set of exchange economies for any initial price system. The simplest way to describe these processes is by characterizing the path of prices that they generate. Convergence proofs then rely on results from differential topology and establish that these paths have a manifold structure. The required tools, involving regular constraint sets and manifolds with generalized boundary, are explained in detail and can be fruitfully applied in other domains as well. The paper concludes with an overview of globally and universally convergent processes in other environments like production economies, economies with price rigidities, and game theory.