Optimal Operations & Investments of the Firm

In this paper, a dynamic continuous time model of the firm, encompassing operations and investments, 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 operating profits less the costs of new capacity and the interest on borrowed funds (or plus the interest on savings) over a fixed decision-making interval plus the discounted value of capacity at the end of the period. The state variables are capacity and debt, and the controls are scale of operation and rate of purchase of new capacity; the final capacity and debt are control parameters. There are several inequality constraints. Using results in control theory, the optimal controls are determined for the case the rate of production is linear in the scale variable, the growth of capacity is linear in the rate of purchase of new capacity and the attrition of capacity is linear in the amount of capacity. Extensions of the model are made to include multiple processes.