A Note on Concavity Conditions of Cobb–Douglas and CES Production Function with at Least Two Inputs

Abstract

In microeconomics, the strict concavity is a very important property of a production function. In 2010, Avvakumov et al. gave a necessary and sufficient condition for the strict concavity of the Cobb–Douglas and constant elasticity of substitution (CES) production function with at least two inputs. To derive these conditions, the negative definiteness of the Hessian for both production functions was examined using certain recurrences for the principal corner minors. The purpose of this note is to complement the proof of Avvakumov, Kiselev, Orlov & Taras’ev ( 2010 , Computational Mathematics and Modeling, 21(3), 336–378) by showing that the use of recurrences and mathematical induction is not necessary, and that a necessary and sufficient condition for the strict concavity can be obtained by considering a particular square matrix, whose determinant can be calculated directly using the rule for the determinant of a lower or upper triangular matrix. JEL Classifications: C60, C65, D21, D24