Optimal Pricing Strategies for Primary and Contingent Products under Duopoly Environment

Abstract

The purpose of the current paper is to study optimal pricing policies for duopoly firms that introduce two new related products: a primary and a nondurable captive contingent product. The use of the former requires the latter product, and the diffusion and the sales of the latter are contingent upon the diffusion of the former. The competition is modelled as a differential game and the solutions are derived as open-loop Nash equilibria. The results of our analysis show that the prices of the contingency product are constant, determined by constant cost, shadow price and own- and cross-elasticities. On the other hand, the price trajectories of the primary product are characterized by such parameters as the directionality of the diffusion effect of the primary product, the magnitude of the coefficient of the repeat purchase of the contingent product and the convexity or concavity of the cumulative adoption of the primary product with respect to time.

We wish to thank Steffen Jørgensen, George Zaccour, Engelbert Dockner, Gary Erickson and Gila Fruchter who had provided us valuable comments on the previous version of the paper. The first author acknowledges the Research Fund from the Ph.D. Program in Management, Rutgers University. The second author acknowledges the Research Grant from the Faculty of Management, Rutgers University.