A Non-Calculus Approach to Solving the Cost Minimization Problem with the Modified Spillman Production Function

Abstract

Cost minimization is a fundamental problem for all firms. In this study, we consider the cost minimization problem in the context of a farm business, where output is described by the modified Spillman production function. This problem holds significant importance in agricultural economics. Mathematically, it represents an application of unconstrained optimization to economic and business problems. The conventional approach to solving this problem involves the use of calculus, specifically the Lagrange multiplier method. However, the application of differential calculus requires nontrivial steps that may not be easily understood without a solid background in mathematical analysis. The aim of this paper is to introduce a novel non-calculus approach to solving the given problem. Our approach is new and relies solely on the weighted arithmetic mean–geometric mean inequality and basic algebra, making it more accessible and intuitive compared to calculusbased techniques. Furthermore, we provide an elementary derivation of the interpretation of the Lagrange multiplier as a tool for sensitivity analysis.