Leapfrogging in a vertical product differentiation

The model considers a two-period duopoly game where in the first period the leader produces a good with a given quality and the other firm can only imitate it. It is the Stackelberg case where, in addition, the leader has the choice of the quality of the good and the imitation is costly, but not prohibitively so. Under this assumption quantities and profits in terms of the quality are derived as subgame perfect equilibrium. In the second period there exists the possibility for the leader and/or the follower to make an investment. The outcome of this is uncertain: it could either be the case that a good of better quality can be introduced, or that a cost-reduction in producing the existing good is attained. The former case is a product innovation, whereas the latter case is a process innovation. By solving the game backwards as a function of the quality of the first period, there exists the possibility of an equilibrium where the follower chooses to invest and the leader does not invest.