Intuitions About Lagrangian Optimization

Abstract

We expose a weakness in an intuitive description popularly associated with the method of Lagrange multipliers and propose an alternative intuition. According to the deficient intuition, the Lagrange technique transforms a constrained optimization problem into an unconstrained optimization problem. This is both mathematically incorrect, and in some contexts contrary to a basic understanding of economic principles. In fact, as is probably understood by most instructors, solutions to the Lagrange conditions for a constrained optimization problem are generally saddle points, an observation typically included in advanced treatments of mathematical economics. At the introductory level, however, instructors often ‘cut corners’ and emphasize that the first-order conditions associated with the method of Lagrange multipliers are essentially the same as for an unconstrained optimization problem, hence leading to an incorrect intuition. We propose an alternative intuition that introduces the Lagrangian function as a perturbation of the original objective function. We characterize a constrained optimum as a point at which all the derivatives of a suitable perturbation must vanish. The paper is both useful for instructors of introductory courses in mathematical methods for economics and also can be used to provide enrichment to students for this very important mathematical technique.