A Complete Method of Comparative Statics for Optimization Problems (Unabbreviated Version)

Abstract

A new method of deriving comparative statics information using generalized compensated derivatives is presented which yields constraint-free semidefiniteness results for any differentiable, constrained optimization problem. More generally, it applies to any differentiable system governed by an extremum principle, be it a physical system subject to the minimum action principle, the equilibrium point of a game theoretical problem expressible as an extremum, or a problem of decision theory with incomplete information treated by the maximum entropy principle. The method of generalized compensated derivatives is natural and powerful, and its underlying structure has a simple and intuitively appealing geometric interpretation. Several extensions of the main theorem such as envelope relations, symmetry properties and invariance conditions, transformations of decision variables and parameters, degrees of arbitrariness in the choice of comparative statics results, and rank relations and inequalities are developed. The relationship of the new method to existing formulations is established, thereby providing a unification of the main differential comparative statics methods currently in use. A second theorem is also established which yields exhaustive, constraint-free comparative statics results for a general, constrained optimization problem. This theorem subsumes all other comparative statics formulations. The method is illustrated with a variety of models, some well known, such as profit and utility maximization, where several novel extensions and results are derived, and some new, such as the principal-agent problem, the efficient portfolio problem, a model of a consumer with market power, and a cost-constrained profit maximization model.