Mathematical Model for a Duopolistic Market

A mathematical model is developed for a marketing situation with two competitors when each competitor has two control variables, viz. price and promotional effort. The model discussed in this paper is an extension of Mills' model [Bass, Frank M., Buzzell, Robert D., Greene, Mark R., Lazer, William, Pessemier, Edgar A., Shawver, Donald L., Shuchman, Abraham, Theodore, Chris A., Wilson, George W. Mathematical Models and Methods in Marketing, article by Harland D. Mills, “A Study of Promotional Competition,” pp. 271–301, Richard D. Irwin, Inc., Homewood, Illinois, 1961.]. Mills has assumed that each competitor has only one control variable, viz. promotional effort and the profit margins of the two competitors are known. In this paper the profit margins are also control variables. The paper derives conditions under which non-boundary equilibrium solutions exist and the sensitivity of the model for small deviations in the decision variables from their equilibrium values is tested. Mills' results are found to be valid only under certain conditions.