On the closed-form solution of the monopolist long-run profit maximization problem with linear demand and Cobb-Douglas technology

Abstract

This paper provides an economic and mathematical analysis of the long-run profit maximization problem of a monopolist with linear demand and a two-input Cobb-Douglas technology to derive the conditions that guarantee the solution to this problem. In addition, the conditions under which the closed-form solution can be derived and the conditions under which the monopolist's profit is positive are discussed. Whenever the problem has a unique solution, the closed-form defines the profit function well for the given demand function. The closed-form solution to the problem depends on the returns to scale. The problem has a unique solution for decreasing returns to scale, for which in general no closed-form can be found. On the other hand, the closed-form of the unique solution can easily be found for constant returns to scale. However, three sub-cases are identified for the case of increasing returns to scale. The analysis is supported by economic interpretations and numerical illustrations.