Oligopoly Models for Optimal Advertising When Production Costs Obey a Learning Curve

Business policy questions frequently involve competitive encounters among several different firms. Oligopoly theory in economics was devised to answer similar questions, but its results so far are largely confined to cases of monopoly (one firm), duopoly (two firms), and many firms (wheat farmers). The cases with n firms, where 3 \le n \le 10, are of great interest to business policy, but are rarely treated in the economics literature because of their extreme difficulty. The natural mathematical model for studying such questions is the theory of differential games, originally devised by R. Isaacs (Isaacs, R. 1975. Differential Games. Robert E. Krieger Publishing Co., Huntington, New York. Originally published by John Wiley & Sons, New York, 1965.) for pursuer-evader games; later it was extended by J. Case (Case, J. H. 1979. Economics and the Competitive Process. New York University Press, New York.) to nonzero-sum differential games for application to questions of economic competition. The problem of characterizing an optimal advertising policy over time is an important question in the field of marketing. It is especially important during the period of the introduction of a new product; and it is also during this time that the production learning curve phenomenon is most pronounced. In this paper we study oligopoly models which combine elements of Bass's demand growth model (Bass, F. M. 1969. A new product growth model for consumer durables. Management Sci. 15 (January) 215--227.), the Vidale-Wolfe (Vidale, M. L., H. B. Wolfe. 1957. An operations research study of sales response to advertising. Oper. Res. 5 370--381.) and Ozga (Ozga, S. 1960. Imperfect markets through lack of knowledge. Quart. J. Econom. 74 29--52.) advertising models with linear or quadratic costs, and the production learning curve (Hirschmann, W. P. 1964. Profit from the learning curve. Harvard Bus. Rev. 42 (January--February) 125--139; The Boston Consulting Group. 1972. Perspectives on Experience. The Boston Consulting Group, Boston.). In the monopoly case (n = 1), the optimal advertising function is derived by applying Green's theorem. In the duopoly (n = 2) and triopoly (n = 3) cases we use discrete differential game models, and find Nash equilibrium solutions. In each of these models the form of the optimal policy is easy to find, but because of the complicated nature of the resulting differential equations, closed form solutions for the optimal state and control trajectories are impossible to find. Hence we use a computer to compute these optimal trajectories and plot them for various parameter settings. Because 13n parameters must be chosen to define a model with n players, a complete search of the parameter space is clearly impossible. We content ourselves in this paper with presenting computer plots of four different interesting encounters of the triopoly model. Our methods can be extended to work just as well for competitive encounters among n players, where 1 \le n \le 10. We believe that in this paper we have opened up the way to studying oligopoly theory with a small number of participants and under widely varying conditions. Although the lack of closed form solutions may disappoint some readers, we feel that the easy availability of graphical solutions is a more than adequate substitute.