The Stability of Exchange and Adaptive Expectations

THE STABILITY of dynamic systems has been extensively studied by Arrow and Nerlove [3] and Arrow and Hurwicz [2] under the assumption of gross substitutability and adaptive expectation. In their systems, it is assumed that a current price is changed by current excess demand, and that excess demand is a function of all current and expected future prices, the latter being formed adaptively to the past history of prices. One should note that they explicitly or implicitly assumed that excess demands were independent of transactions among individuals. Two alternative interpretations of this assumption were suggested by Arrow and Hurwicz [2]. One is the recontract interpretation of the Walrasian ta'tonnement, in which no transactions actually take place during the process until equilibrium is achieved. The second interpretation is that transactions can indeed take place, but that they do not affect decisions taken at any subsequent period, because all the commodities are assumed to be instantaneously perishable, and the utility for present and future goods is also assumed to be the sum of utilities of present goods and of future goods, the utility functions being of the same mathematical form. The second interpretation seems more interesting to us, since, once the assumption of perishability is made, the dynamic system can describe the changes of actual prices over time.2 In the following, however, we are concerned with another polar case, dropping the assumption of perishability and substituting another extreme assumption, namely, that all the commodities are permanently durable with negligible depreciation by consumption. We shall keep the hypothesis that the utility function for present goods and that for future goods are of mathematically same form and are independent of the argu