Editing Comparative statics (section)

==Without constraints==
Suppose <math>p(x;q)</math> 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 <math>x^*(q)= \arg \max p(x;q) </math>.
Let <math>f(x;q)=D_xp(x;q)</math>, the ''n'' by ''n'' matrix of first partial derivatives of <math>p(x;q)</math> with respect to its first ''n'' arguments ''x''<sub>1</sub>,...,''x''<sub>''n''</sub>.
The maximizer <math>x^*(q)</math> is defined by the ''n''×1 first order condition <math>f(x^*(q);q)=0</math>.

Comparative statics asks how this maximizer changes in response to changes in the ''m'' parameters. The aim is to find <math>\partial x^*_i/ \partial q_j, i=1,...,n, j=1,...,m</math>.

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, <math>x^*(q)</math> may be viewed locally as a continuously differentiable function, and the local response of <math>x^*(q)</math> to small changes in ''q'' is given by 
:<math>D_qx^*(q)=-[D_xf(x^*(q);q)]^{-1}D_qf(x^*(q);q).</math>
Applying the chain rule and first order condition,
:<math>D_qp(x^*(q),q)=D_qp(x;q)|_{x=x^*(q)}.</math>
(See [[Envelope theorem]]).

===Application for profit maximization===
Suppose a firm produces ''n'' goods in quantities <math>x_1,...,x_n</math>. The firm's profit is a function ''p'' of <math>x_1,...,x_n</math> and of ''m'' exogenous parameters <math>q_1,...,q_m</math> which may represent, for instance, various tax rates. Provided the profit function satisfies the smoothness and concavity requirements, the comparative statics method above describes the changes in the firm's profit due to small changes in the tax rates.