On a Geometric Programming Approach to Profit Maximization: The Case of CES Technology

Abstract

The profit maximization problem takes a central place in the theory of the firm, especially when conditions for perfect competition hold. In this paper, we solve the profit maximization problem of a perfectly competitive firm when the constant elasticity of substitution (CES) production function with n≥2 inputs describes its technology. Commonly, this problem is solved by using multivariable differential calculus. However, to avoid tedious algebraic manipulations and bypass checking nontrivial necessary and sufficient conditions, we employ geometric programming (GP), and the power mean inequality (PMI) as an elegant complementary tool to multivariable calculus. Since the GP and the PMI are simple optimization techniques without derivatives, they can provide new insights into the given problem to managers, students, and other audiences who may be unfamiliar with multivariable differential calculus. Additionally, by using the properties of limits, we show that the solution to the profit maximization problem with Cobb-Douglas technology is a limiting case of our result.