The Existence of Nash Equilibria in Reaction Functions for Dynamic Models of Oligopoly

Abstract

Consider Cournot's stability analysis of his mineral spring duopoly [Cournot, 1838]. Each of the two firms believes that its rival's output in the next period will simply equal that observed currently. Profit maximization yields reaction which are explicitly dynamic and are assumed to govern the actual evolution of the system. Cournot proved that the system is asymptotically stable in this simplest case. Cournot's result was shown by Theocharis [1960], not to generalize to three or more firms. Fisher [1961] discusses more general cost and demand functions than those used by Cournot. Even in the case considered by Cournot, however, there is an obvious conceptual objection. Each firm's output will generally vary from one period to the next, flagrantly violating its rival's conjecture. Some of James Friedman's work [especially 1968, 1973, 1976 and 1977] might best be viewed as an attempt to eliminate this delusional quality from dynamic models of oligopoly. Consider a typical duopoly scenario with two firms making repeated simultaneous quantity determinations. Friedman more commonly discusses price setting models, but these are straightforwardly analogous. Each firm believes that the other firm's behavior is determined only by the quantity pair selected in the previous period. In Friedman's work, as distinct from Cournot's, these beliefs are ultimately required to be consistent with the other firm's actual behavior. For example, firm 2 believes that firm l's output in the current period is given as