Optimal production, investment, advertising, and price controls for the dynamic monopoly firm

In this paper, a dynamic continuous time model, encompassing operations, investments, output prices, informative advertising, and brand advertising, is formulated as an optimal control problem in an activity analysis context. In the model, the objective of the firm is to maximize, subject to various constraints, the discounted value of accumulated savings over the finite decision-making interval plus the discounted value of capacity at the end of the interval. The state variables are capacity, goodwill of type 1 reflecting informative advertising, goodwill of type 2 reflecting brand advertising, net debt, and the output price. The controls are the scale of operation, the purchase of new capacity, informative advertising, brand advertising, and the rate of change of the output price. Final capacity, goodwill of types 1 and 2, net debt, and price are control parameters. There are several inequality constraints. Using results in control theory, the optimal controls are completely characterized. These results are interpreted using the properties of the Value Equation. In addition, the control problem is transformed into an approximate programming problem.