Oligopoly Pricing and Advertising in Isoelastic Adoption Models

This paper deals with deterministic dynamic pricing and advertising differential games which are stylized models of special durable-good oligopoly markets. We analyze infinite horizon models with constant price and advertising elasticities of demand in the cases of symmetric and asymmetric firms. In particular, we consider general saturation/adoption effects. These effects are modeled as transformations of the sum of the cumulative sales of all competing firms. We specify a necessary and sufficient condition such that a unique Markovian Nash equilibrium for such games exist. For two classes of models we derive solution formulas of the optimal policies and of the value functions, and we show how to compute the evolution of the cumulative sales of each firm. The analysis of these games reveals that the existence of the Nash equilibrium relies on the possibility to separate a component, which is specific for each firm, from a [market] component, which is the same for all firms. The common factor is a function of the decreasing untapped market size. The individual factor of each firm reflects its individual market power and has an impact on equilibrium prices; each such coefficient depends on the price elasticities, unit costs, arrival rates, and discount factors of all competing companies. Formulas for these coefficients reveal how equilibrium prices depend on the number of competing firms, and how the entry or exit of a firm affects the price structure of the oligopoly.