Pricing strategies in a dynamic duopoly: A differential game model

We formulate a differential game model for dynamic pricing in a duopolistic market. Firms' demand functions are derived from utility maximizing behavior of consumers with the demand for a brand given by the logit model. Preferences for brands are assumed to evolve over time in the market in a manner akin to learning models postulated in the marketing literature. We derive the differential equations governing the equilibrium open-loop price paths over time and show that in steady state, the brand with the higher preference level charges the higher price. The formulation is extended to include the effects of consumer heterogeneity, and equilibrium steady-state prices are compared with those obtained when heterogeneity is ignored. A comparison of steady-state dynamic prices with myopic prices is provided. An empirical example is discussed to show how steady-state model predictions may be obtained from actual longitudinal purchase data.