Comparative Statics Without Constraints
Suppose is a smooth and strictly concave objective function where x is a vector of n endogenous variables and q is a vector of m exogenous parameters. Consider the unconstrained optimization problem . Let, the n by n matrix of first partial derivatives of with respect to its first n arguments x1,...,xn. The maximizer is defined by the n×1 first order condition .
Comparative statics asks how this maximizer changes in response to changes in the m parameters. The aim is to find .
The strict concavity of the objective function implies that the Jacobian of f, which is exactly the matrix of second partial derivatives of p with respect to the endogenous variables, is nonsingular (has an inverse). By the implicit function theorem, then, may be viewed locally as a continuously differentiable function, and the local response of to small changes in q is given by
Applying the chain rule and first order condition,
(See Envelope theorem).
Read more about this topic: Comparative Statics
Famous quotes containing the words comparative and/or constraints:
“If you believe that a nation is really better off which achieves for a comparative few, those who are capable of attaining it, high culture, ease, opportunity, and that these few from their enlightenment should give what they consider best to those less favored, then you naturally belong to the Republican Party. But if you believe that people must struggle slowly to the light for themselves, then it seems to me that you are a Democrat.”
—Eleanor Roosevelt (1884–1962)
“The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.”
—Gerald M. Edelman (b. 1928)