Getting to a Competitive Equilibrium
提出一个随机价格调整模型，研究分散化交换经济如何以高概率逼近竞争均衡，并证明在价格路径稳定性假设下，市场制造者的库存近似恒定。
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 A ∈ (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 α is the number of agents, then the new price is p + (α -λ )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 tâtonnement 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 tâtonnement 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.