A stochastic differential game of duopolistic competition with sticky prices

In this paper, we present a stochastic differential game model of duopolistic competition with sticky prices, which extends the model analyzed in the seminal paper by Fershtman and Kamien (1987), and derive analytically a feedback Nash equilibrium of the game. Uncertainty is modelled by means of a Wiener process affecting the evolution of the price. We show that the expected price converges to a level that can be either higher or lower than the deterministic stationary price, depending on market size. We also show that uncertainty is beneficial to firms in terms of long-run expected profits and may be beneficial to firms in terms of discounted expected profits, depending on market size as well. Furthermore, we show that the long-run stationary probability density of the market price can be computed explicitly.