Optimal Pricing and Advertising Policies for New Product Oligopoly Models

Abstract

In this paper our previous work on monopoly and oligopoly new product models is extended by the addition of pricing as well as advertising control variables. These models contain Bass's demand growth model, and the Vidale-Wolfe and Ozga advertising models, as well as the production learning curve model and an exponential demand function. The problem of characterizing an optimal pricing and advertising policy over time is an important question in the field of marketing as well as in the areas of business policy and competitive economics. These questions are particularly important during the introductory period of a new product, when the effects of the learning curve phenomenon and market saturation are most pronounced. We consider first the monopoly case with linear advertising cost, exponential demand, and three different pricing rules: the optimal variable pricing, the instantaneous marginal pricing, and the optimal constant pricing rules. Several theoretical results are established for these rules including the facts that the instantaneous marginal pricing rule is a myopic version of the optimal pricing rule and the optimal constant pricing rule is a weighted average over time of the instantaneous marginal pricing rule. Another surprising result is that, after the market is at least half saturated, a pulse of advertising must be preceded by a significant drop in price. Numerical solutions of a number of examples are discussed. Oligopolistic models are analyzed as nonzero-sum differential games in the rest of the paper. The state and adjoint equations are easy to write down, but impossible to solve in closed form. Hence we describe how to reformulate these models as discrete differential games, and give a numerical algorithm for finding open loop Nash solutions. The latter was used to solve three triopoly models. In each case it was found that optimal prices and advertising rates start high and steadily decline.