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).
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