A Model of the Evolution of Duopoly: Does the Asymmetry between Firms Tend to Increase or Decrease?

This paper is an attempt to identify some of the factors that affect the evolution of market structure in a model of dynamic competition between two firms. The stochastic evolution of the state of competition depends on the respective effort rates of the firms. The question is whether the current leader works harder than the laggard—does the ‘gap’ between firms tend to increase or decrease? We show that several effects are at work. The state tends to evolve in the direction where joint payoffs are greater. Since joint payoffs are related to joint product-market profits less joint effort costs, there are two classes of effect: the joint-profit effect and various joint-cost effects. The latter result in part from the pattern of profits, and in part from endpoint effects that give relief from efforts. Asymptotic expansions illuminate these influences. Moreover, we show by numerical simulation that there is another kind of joint-cost effect. The pattern of joint effort costs can influence the pattern of evolution of market structure, and the evolution of the pattern of market structure can influence the pattern of efforts, in a mutually self-reinforcing manner. In particular, there may be equilibria in which this last effect means that the laggard works harder than the leader even though all the other effects work in favour of the leader.