Generalized Comparative Statics with Applications to Consumer Theory and Producer Theory

Abstract

THIS PAPER analyzes parametric variation for a general class of constrained static optimization problems. The problem treated is that of maximizing a real-valued objective function F(x, a) which depends on an n-dimensional vector of choice variables x and a q-dimensional vector of parameters a, subject to a constraint which requires that a real-valued constraint function g(x, a), dependent on the n variables and q parameters, equal a given value b. Section 1 establishes the existence of a generalized Slutsky equation everywhere on a subset of n + q dimensional Euclidean space except on a set of Lebesgue measure zero that is, almost everywhere (denoted a.e.) under concavity assumptions on the objective and constraint functions and nonsingularity of the relevant bordered Hessian. This is a new result; previous papers have treated only special cases. In Section 2 the restriction that the generalized matrix of substitution effects be symmetric is imposed, and the most general form of the objective and constraint functions satisfying this requirement is exhibited (linearity of the constraint function is not needed). It is further shown that the generalized matrix of substitution effects exhibits negative semidefiniteness in this case. Section 3 applies these results to the case of a consumer for whom prices enter the utility function. Section 4 applies these results to the Baumol producer maximizing sales subject to a profit constraint. The existence and basic properties (symmetry and negative semidefiniteness) of the generalized Slutsky equation a.e., even when a vector of parameters enters the objective function and the applications of these results to consumer theory and producer theory are the major results of this paper. The classic treatment of comparative statics and its applications in economics is that of Samuelson.2 Samuelson however did not develop the full properties of the generalized Slutsky system developed below. The problem treated in Section 3 has been treated by Kalman.3 His earlier results are extended in this paper. Finally we conjecture that the above properities of the generalized Slutsky system can be derived even with k nonlinear