# Econometrica

## Journal Of The Econometric Society

### An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics

Edited by: Guido W. Imbens • Print ISSN: 0012-9682 • Online ISSN: 1468-0262

Econometrica: Jan, 1996, Volume 64, Issue 1

# Getting to a Competitive Equilibrium

H. Jerome Keisler

In the previous paper Keisler (1995) we introduced a random price adjustment model for an exchange economy which is decentralized in that the trades permitted to an agent and the resulting price changes depend only on the commodity vector currently held by that agent, and not on the whole economy. Our model is an exchange economy with a finite set of commodities, a market maker who adjusts prices, a large finite set of agents who trade only with the market maker, and a parameter $\lambda \in (0, 1)$ which determines the rate of price adjustment. Each agent has an initial endowment and a preference relation. At each discrete time, one agent is chosen at random and makes the trade which maximizes his preferences subject to the budget constraint at the current price vector. Then the market maker adjusts the price vector according to the following rule: if $x$ is the change in the commodity vector of the agent who just traded, $p$ is the old price, and $\alpha$ is the number of agents, then the new price is $p + (\alpha^{-\lambda})x$. Thus both the price vector and the commodity allocation change randomly with time. We obtain asymptotic results as the number of agents goes to infinity, subject to stability assumptions on the price paths. It was shown in the earlier paper that with probability arbitrarily close to one the price path in our model will approximate the price path of the corresponding tatonnement process on a rapid time scale, and will then remain close to a limit price. In this paper we show that, with probability arbitrarily close to one, the economy will approach a competitive equilibrium, and the process will be feasible in the sense that the market maker's inventory is approximately constant over time. Our main assumption is that the price adjustment rule is stable throughout the trading process. This is stronger than the classical tatonnement assumption that the price adjustment rule is stable at the initial endowment. We show that many classical examples of exchange economies satisfy our assumptions, and then give an example where stability at the initial endowment holds but the stronger stability assumption needed for our results fails.