Comparative statics of a maximum Suppose that $f(x,a)=\max[g(x,a),h(x,a)]$ and $x^{∗}=\arg\max_{x}f(x,a)$.
Also, let $x_1(a)=\arg\max_{x}g(x,a)$ and $x_2(a)=\arg\max_{x}h(x,a)$.
If both $x'_1(a)$ and $x'_2(a)$ are nonnegative, under what conditions is $\frac{dx^∗}{da}\ge 0$?
As Jesper Hypel has pointed out in a comment, the derivative need not exist. Actually, the function $a\mapsto x^*(a)$ need not even be nondecreasing. Here is an example: Let $g$ and $h$ be given by $g(x,a)=-(a-x)^2$ and $h(x,a)=a-(a-1-x)^2$, respectively. Then $x_1(a)=a$ and $x_2(a)=a-1$. Moreover, $f\big(x^*(a),a\big)=g\big(x_1(a),a\big)$ for $a&lt;0$ and $f\big(x^*(a),a\big)=h\big(x_2(a),a\big)$ for $a&gt;0$. As a consequence, the function $a\mapsto x^*(a)$ jumps down after $0$.