Suppose my optimization problem is stated as follows

$\max\limits_x f(x,t)$

$s.t.$ $g(x,t) \leq 0$

I am interested in finding the direction $x^*$ changes with the parameter $t$. Can someone provide me a reference that describes conditions that if satisfied by $f$ and $g$ will let me answer the question of monotonicity of $x^*$ with respect to $t$?


1 Answer 1


I believe the analysis in E. Silberberg's "The Structure of Economics" (2n ed. 1990), is illuminating, chapter 7. The fundamental comparative-statics result (for constrained and unconstrained problems) is (eq. 7-10 of the book)

$$\frac{\partial ^2 f(x^*,t)}{\partial x\partial t} \cdot \frac {\partial x^*}{\partial t} > 0$$

The matter was initially treated in P. Samuelson's "Foundations of Economic Analysis".

Topkis's Theorem is a more abstract mathematical treatment that states the condition required in order to have $\partial x^*/\partial t \geq0$.


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