Approximate tâtonnement processes

Summary We introduce a probabilistic model for price adjustment in an exchange economy which approximates the classical Walras tâtonnement process while avoiding many of its unrealistic features. The model 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 commodity vectors held by the other agents in the economy. Our results will show that the Walras tâtonnement process can be decentralized without changing its behavior on the macroeconomic scale. Our model has a finite set of commodities, a market maker who adjusts prices, and a large finite set of agents who trade only with the market maker. Each agent has a demand function depending on his commodity vector and the price vector. At each discrete time, one agent is chosen at random and exchanges his current commodity vector for his demand vector. Then the market maker adjusts the price vector by an amount which depends on the selected agent's commodity vector and the current price. Prices are adjusted rapidly enough to avoid prolonged trading at the wrong price, but slowly enough so that a substantial price change will depend on a significant simple of agents. The main result shows that with probability arbitrarily close to one the price will rapidly approach and then remain close to an equilibrium value, following a path which is close to the price path of the corresponding tâtonnement process.