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Optimal Production, Investment, and Output Price Controls for a Monopoly Firm of the Evans' Type

Russell G. Thompson
Melvin D. George
Perry L. Brown
Michael S. Proctor

Abstract

In this paper, a continuous time model for a monopoly firm of the Evans' type, encompassing operations, investments, and output prices, is formulated as an optimal control problem. In the model the objective of the firm is to maximize, subject to various constraints, the integral of production profits less interest and investment costs over a finite decision-making interval, plus the value of the capacity at the end of the period. The state variables are capacity, debt, and output price; the controls are the scale of operation, rate of purchase of new capacity, and rate of change of the output price. Final capacity, price, and debt are control parameters. There are several inequality constraints. Using results in control theory, the optimal controls are characterized for a model basically linear in structure. It shows that the one case suggested by Evans for further analysis is a trivial problem. These results are interpreted using the properties of the value equation. In addition, the control model is formulated alternatively as a mathematical programming problem. Solutions may then be computed by publishing algorithms.