A discrimination model for the dynamic monopoly firm

In this study, the price discrimination problem for the dynamic monopoly firm is formulated as an optimal control problem for the case where the demands are interdependent pricewise. This formulation reflects both the description of the discrimination problem as found in the literature and a dynamic demand law. The objective of the firm is to maximize, subject to constraints, the discounted value of the firm´s capacity at the end of a finite interval plus the integral over a finite interval of the discounted returns from sales in the two markets less the discounted costs of production, product differentiation, and investment. The constraints are four differential equations describing the temporal rates of change in the two output prices and the temporal rates of change in the two types of capacity. There are six sets of inequalities limiting the values of the controls. The state variables are the two output prices and the two capacities; the controls are the temporal rates of change in the two output prices, the two levels of production, and the levels of investment in the two types of capacity. Initial output prices are given; terminal output prices are control parameters. Using results in control theory, decision rules for optimal pricing, production, and investment are derived and interpreted. This interpretation indicates the dynamic interaction between the pricing, producing, and investing decisions in each market.