Optimal pricing in a duopoly: A noncooperative differential games solution

Abstract

This paper deals with a differential games model of an oligopoly ofn profit-maximizing firms competing for the same stock of customers. For the sale dynamics, it is assumed that the customers of each firm are driven away gradually by increasing product prices. Since the state variable is absent from the Hamiltonian maximizing conditions as well as from the adjoint equations, open-loop Nash solutions can be obtained. By using phase diagram analysis, for two players the behavior of the optimal pricing strategies can be characterized qualitatively. The main importance of the paper lies in the solution technique, rather than in the economic significance of the proposed model. Under the proposed assumptions, the two-point boundary-value problem resulting from the maximum principle is reduced to a terminal-value problem. It turns out that, for special salvage values of the market shares and if the planning horizon is not too short, nonmonotonic Nash-optimal price trajectories occur.