Exact and useful optimization methods for microeconomic theory

This paper points out that the treatment of utility maximization in current textbooks on microeconomic theory is deficient in at least three respects: breadth of coverage, completeness-cumcoherence of solution methods and mathematical correctness. Improvements are suggested in the form of a Kuhn-Tucker type theorem that has been customized for microeconomics. The role of the domain of differentiability of the utility function is emphasized, not only to point out a persistent error in the standard literature on microeconomic theory, but also, more constructively, to call attention to the fact that this domain can be chosen sensibly in order to include the maximization of certain nondifferentiable utility functions, such as Leontiev utility functions. To ensure uniqueness of the optimal solution (S)-quasiconcavity, an apparently new adaptation of the notion of strict quasiconcavity, is introduced. It improves upon an earlier notion formulated by Aliprantis, Brown and Burkinshaw. To underscore the usefulness of the optimality conditions obtained here, five quite different instances of utility maximization are completely solved by a single coherent method.