A dynamic game of differentiated duopoly with demand enhancing investment

We study a dynamic model of a differentiated product duopoly in which one firm's investments have positive effects on demand for its product as well as demand for its rival firm's product. These investments can be thought of as research and development (R&D) for new products or as advertisement. One salient feature of the model is that we consider each firm’s cumulative investment by demand enhancing investments as a state variable in its dynamic problem. We assume that demand enhancing investments by one firm have positive spillover effects on demand for the rival firm's product. Specifically we examine a differential game between the duopolists with an infinite horizon and analyze their open-loop strategies as well as Markov feedback strategies. We consider two settings regarding the firms' investment decisions, one in which the firms undertake investments noncooperatively and the other in which they choose investment levels cooperatively while they undertake Bertrand competition in the product market. For results, first we show that under each of the two settings, there exists a unique symmetric open-loop Nash equilibrium, and that each firm's investment level under the setting of cooperation is twice as large as that under noncooperation at open-loop Nash equilibrium. We also present dynamic stability conditions for the open-loop Nash equilibrium under noncooperative investment as well as for that under cooperative investment. Next, we derive linear Markov perfect equilibrium under cooperation and that under competition, and examine stability conditions for the linear Markov perfect equilibria. Furthermore, we analyze non-linear Markov feedback strategies and characterize Markov perfect equilibrium under cooperation and that under competition.