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Unformatted text preview: Lagrange Multiplier Problems in Economics John V. Baxley and John C. Moorhouse, Wake Forest University, Winston-Salem, NC 27109 American Mathematical Monthly, August–September 1984, Volume 91, Number 7, pp. 404–412. 1. Introduction. Several surprises are in store for the mathematics student who looks for the first time at nontrivial constrained optimization problems in economics. The usual constrained problem in a mathematics course has only one or two critical points and the selection of the absolute maximum is clear from the geometric nature of the problem. Mathematics texts often ignore sufficient conditions (involving bordered Hessian determinants) for relative extrema and provide no interpretation of the Lagrange multiplier leaving the student with the impression that has no significance beyond providing an extra variable which magically transforms the constrained problem into an unconstrained higher dimensional problem. Our purpose here is to examine carefully one example in order to highlight the following features that are typical in the formulation of problems in economics: (i) Functions are not usually explicitly given, but are assumed to have characteristic qualitative properties. Thus the problems have an air of “theory” rather than “computation.” Yet the problems are meaningful and often generate important insights into economic behavior. (ii) Normally the economist is not interested in solving for a constrained optimum; rather his starting assumption is that an optimum is achieved and seeks to base predictions of behavioral responses on the assumption that optimizing will continue. For example, assuming a firm minimizes the cost of producing a given output, one wishes to know how changes in input prices will affect the firm’s behavior. Thus the problem is not “find the minimum,” but “assuming the minimum is attained, what consequences can be deduced?” An interesting result is that the economist is primarily interested in necessary conditions for an optimum and wishes profoundly that sufficient conditions were really necessary. (iii) The Lagrange multiplier generally has significance in economic problems. It can usually be interpreted as the rate of change of the optimal value relative to some parameter. (iv) The implicit function theorem plays a pivotal role in the problem analysis, both theoretically and computationally, since the analysis requires solving a system of nonlinear equations for the endogenous (dependent) variables and computing partial derivatives of these variables with respect to the exogenous (independent) variables. 2. An Illustrative Example. We analyze in detail an example in utility maximization....
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