The Equilibrium of Duopoly in a Market of Homogeneous Goods

Abstract

In this paper, we consider duopoly equilibria in which firms act rationally on the basis of mutually consistent assumptions about their rivals' behaviour. It turns out that there exist various kinds of equilibria corresponding to various pairs of behaviour patterns. Depending on the relation between the firms' cost functions and the market demand functions there are different outcomes: monopoly, limit-pricing, stable asymmetric duopoly with one firm a price-maker and the other (less efficient) firm a price-taker, and unstable symmetric duopoly which might have price competition (as in Bertrand, 1883) or quasi-agreement (Fellner, 1960). There remain some weak points in the structure of traditional oligopoly theory as developed by Cournot (1838), Bertrand (1883), Edgeworth (1925), Stackelberg (1934) and others. Cournot hypothesizes that an oligopolist assumes rivals' quantity to be fixed, but this assumption typically turns out to be incorrect as the rivals' reactions occur, and Shapley and Shubik (1969) argue that price is the more natural strategic variable. Bertrand and Edgeworth, analysing duopolists' behaviour with price-setting strategies, find that there is price competition only because of a special assumption about cost conditions. Stackelberg considered asymmetric duopoly, with a leader and a follower, and concluded as follows. (a) There cannot be a leader-follower outcome in an oligopolistic market for a homogeneous good with price-setting strategies (see also Fellner, 1960, Chapter 2). The outcome must be price competition as in Bertrand and Edgeworth. (b) Quantity-strategy oligopoly almost always has a leader-leader outcome. In this paper we shall investigate rational behaviour of a duopolist who takes account of price as well as quantity in his strategy, and we shall examine various patterns of duopoly. In Section I we shall analyse rational behaviour of a duopolist who conjectures his rival's reaction with respect to price as well as quantity. In Section II, comparing his profit in various cases, we shall describe optimal behaviour. In Section III we shall show various kinds of duopoly equilibria that correspond to various cost curves of both duopolists. Rigorous mathematical arguments are set out in the Appendix.

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