Solving Profit Maximization Problem in Case of the Cobb-Douglas Production Function via Weighted AG Inequality and Geometric Programming

Abstract

The long-run profit maximization is a standard and important problem having significant implications on a firm’s competitiveness. The common approach is to consider the profit maximization problem for production function with two inputs and use calculus to solve it. However, checking the necessary and sufficient conditions in case of more than two inputs can be difficult. Geometric programming provides a way to solve that problem for any number of inputs without the use of derivatives. Hereby the results are obtained much faster and the solution procedure is more elegant then when using calculus. Liu used the technique of signomial geometric programming to solve the problem in case of the Cobb-Douglas production function with two inputs. However, he was unable to prove that the result obtained is indeed the global maximum. Therefore, in this paper we solve the problem in question by using the weighted arithmetic-geometric inequality (WAG) in case of one input and some transformations of geometric programming in case of two or more inputs and prove that the result obtained is indeed the global optimum.