Derivative w.r.t exogenous variables

Question

Suppose $f(x_{1}, x_{2})$ defined on a compact set, which guarantee the existence of extrema, $x_1*, x_2*$, and also given the Hessian matrix of $f$ is strictly concave. Suppose there is also a exogenous variable $t$, where $0<t<1$, is in the function $f(x_1,x_2)$, in order to find $\frac{\partial x_1*}{\partial t}$, can I take the derivative with respect to t over the equation of $\frac{\partial f(x_1*, x_2*)}{\partial x_1} = 0$(Assuming I can find it in this way) This equation holds due to the existence of extrema. The only problem I have is can I, actually taking the derivative with respect to exogenous variable? And because in order to find the Hessian matrix, I only consider the two variables $x_1$, $x_2$, so it's a 2 by 2 matrix, because I kind of assumed t to be fixed, therefore it's a constant? Then, what would it mean to take derivative with respect to a constant? Plus, I don't know if t and $x_1$ is dependent or not.

Answer 1

If the solution $x^*(t)=(x_1^*(t),x_2^*(t))$ is interior at the initial value of $t$ (which is not guaranteed by your assumptions), then we can use the implicit function theorem.

Explicitly including $t$ as an argument in $f$, the solutions are defined by the following first-order conditions:

$$\text{$f_1(x^*_1(t),x^*_2(t),t)=0$ and $f_2(x^*_1(t),x^*_2(t),t)$=0}$$

Implicitly differentiating gives (suppressing the function arguments):

$$f_{11}x_1^{*\prime}+f_{12}x_2^{*\prime}+f_t=0$$

and

$$f_{21}x_1^{*\prime}+f_{22}x_2^{*\prime}+f_t=0$$

This is the system

$$\begin{pmatrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{pmatrix}\begin{pmatrix}x_1^*(t)\\ x_2^*(t)\end{pmatrix}=-\begin{pmatrix}f_t\\ f_t\end{pmatrix}$$

The two-by-two matrix on the LHS is the Hessian. If the Hessian is negative definite, then it is invertible.

The solution is

$$\begin{pmatrix}x_1^*(t)\\ x_2^*(t)\end{pmatrix}=-\begin{pmatrix}f_{11} & f_{12} \\ f_{21} & f_{22} \end{pmatrix}^{-1}\begin{pmatrix}f_t\\ f_t\end{pmatrix}=\frac{1}{f_{11}f_{22}-f_{12}^2}\begin{pmatrix}f_t(f_{12}-f_{22})\\ f_t(f_{12}-f_{11})\end{pmatrix}$$