Recurrence of a modified random walk and its application to an economic model

A modification of Chung and Fuchs’ (Mem. Amer. Math. Soc., 6 (1951), pp. 1-12) recurrence theorem for random walks leads to an analogous result for a different discrete parameter Markov process. This latter process is applicable to an analysis of price stabilization programs involving purchases and sales from a buffer stock. 1. The stochastic model. By means of the following stochastic model, Salant (1981) analyzes how speculators maximizing expected profits under "rational expectations" would respond to a government price stabilization program involving purchases and sales from a buffer-stock. Let IN denote the set of nonnegative integers, {0, 1, 2,...}. At time IN\{0}, a harvest, Ht, of random size, is produced, drawn independently from an unchanging distribution; part of the harvest is consumed and the remainder is stored by the government or the private sector, with 0t denoting the combined stock at the beginning of period t. Consumer demand for the commodity is assumed to increase as prices decrease. Salant shows that in any period, the market price which induces profit-seeking speculators and the government to sell exactly what consumers demand can be written as a decreasing function of the total stock at the beginning of the period. It is shown that when 0, is in a range of a <= Ot <= b, the government can keep the market price at the official level P. When 0, falls below a, speculators "attack" the government stock, purchasing it in its entirety; the government is no longer able to defend the ceiling, and the price rises, much as it did in the gold market in 1968 when the government’s stockpile was attacked (see, for example, Wolfe (1976)). If Ot climbs above b, buffer-stock managers do not have the funds needed to support the floor and the market price falls. LettingD denote the consumer demand function and P/(. denote the price function, the following stochastic difference equation and associated initial condition, 00 a0, describe the evolution of stocks" (1’) 0t+l Ot "-t’nt+l-D(P+(Ot)). Under the assumption that (2’) ix D(P) E(Ht), we shall show, by means of a theorem proved in 3, that for a buffer-stock manager, speculative attacks and an inability to defend the floor are recurrent events. It is almost certain that there is no way for the manager to avoid speculative attacks altogether, however large the initial stockpile, as long as the official price is set so that demand equals (or exceeds) the expected harvest. Other aspects of the general problem warrant study as well, but are not investigated here; one such is the waiting time until the first attack. Here, we consider the case of a government whose goal is a fixed price (or "peg"), and whose policy is to maintain this price, as was done in the gold market for much of this century (refer to Wolfe (1976)). In another direction, however, we could extend our analysis to the case in which the goal is to keep the price within an interval (or "band"), with corresponding policy of appropriate intervention when the price reaches either endpoint of the * Received by the editors October 2, 1979, and in final revised form April 18, 1980. Department of Mathematical Sciences, Drexel University, Philadelphia, PA 19104. t Federal Trade Commission, Washington, DC 20580.