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

<p>$$\frac{\partial ^2 f(x^*,t)}{\partial x\partial t} \cdot \frac {\partial x^*}{\partial t} &gt; 0$$</p>

<p>The matter was initially treated in <strong>P. Samuelson's "Foundations of Economic Analysis".</strong></p>

<p><a href="https://en.wikipedia.org/wiki/Topkis%27s_theorem" rel="nofollow noreferrer">Topkis's Theorem</a> is a more abstract mathematical treatment that states the condition required in order to have $\partial x^*/\partial t \geq0$.</p>

CC BY-SA 3.0
2017-04-15T11:50:56.650Z
2017-04-15T12:44:43.270Z
2017-04-15T12:44:43.270Z
Alecos Papadopoulos
<p>Suppose my optimization problem is stated as follows </p>

<p>$\max\limits_x f(x,t)$</p>

<p>$s.t.$   $g(x,t) \leq 0$  </p>

<p>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 $t$?</p>

<optimization><comparative-statics>
Topkis' Theorem
esperanto
# Topkis' Theorem

(Asked by: username_0 on 2017-04-15)

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 $t$?

**Tags:** optimization, comparative-statics

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## 1 answers

### Answer by username_1 on 2017-04-15. Score: 2

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](https://en.wikipedia.org/wiki/Topkis%27s_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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