A Dynamic Oligopoly Game with Lags in Demand: More on the Monotonicity of Price in the Number of Sellers

Abstract

This paper explores a game-theoretic model of oligopoly. The game is played by sellers of a single product. In each (discrete) time period, each seller sets a single price for his output and satisfies all demand at that price. His costs are assumed to be zero. Each seller's demand is the sum of two parts: a monopolistic part that is unaffected by other sellers' prices, and a competitive part that is positive only for the low-price setter. So far the model is just a repeated version of the game in Rosenthal [1980]. The added feature of this paper is that competitive demand at any time depends not only on this period's low price but also on last period's low price. This feature introduces a dynamic aspect to the situation. Monopolistic demand, on the other hand, is assumed to be the same function at every time for every seller. Although the buyers are not modeled as players in the game, the kind of competitive demand behavior mentioned above might result, for example, from a stationary overlapping-generations model in which consumers live for two periods and can purchase their second-period consumption in either period. If on the basis of currently available information a lower price could be forecast for the next period, some delay in the purchase decisions of the young and hence a functional form of the type specified might be expected. It turns out that if competitive demand is monotone nondecreasing in previous-period price then the process of low prices produced by the equilibrium of primary interest in this paper is a stationary ergodic Markov process that has the property that the conditional expectation of low price at time t given low price at (t 1) is a nonincreasing function of low price at (t1). Hence a decision to delay increasing quantities of purchases the higher the price in the current period can be justified on average by correspondingly lower prices in the succeeding period. The economic environment is assumed to be deterministic and known with certainty by the sellers. The only sources of randomness are then the decisions of the sellers. Notice that if there were no monopoly demand, competition (Bertrand) among the sellers would drive the price in every period to zero. The presence of monopoly demand, as in the static model in Rosenthal [1980], leads to randomized decisions (at least at the principal equilibrium here) and therefore,