Suppose I have functions $f_1$ and $f_2$ that I am maximising with respect to arguments $x_1$ and $x_2$ respectively. First order conditions for optimisation is given by
$$f_1(x_1^*,x_2,a)=1$$
$$f_2(x_1,x_2^*,a)=1$$
where superscript $^*$ indicates that values that maximize the function, and $a$ is some constant. I know that $x_1^*$ is increasing in $a$ if we keep $x_2$ fixed, since
$$\frac{\partial f_i}{\partial a}>0 \hspace{0.2cm} ; \hspace{0.2cm} \frac{\partial f_i}{\partial x_i}<0 $$
For $i=1,2$, which implies that when $a$ increases both $x_1^*$ and $x_2^*$ increase.[1] Since it may be that $\frac{\partial x_i}{\partial x_j}<0$. Does an increase in $a$ (locally) increase $x_1^*$?
My attempt, through the implicit function theorem:
$$ \frac{\partial x_1^*}{\partial a} =- \frac{\frac{\partial f_1}{\partial x_2}\frac{\partial x_2^*}{\partial a} + \frac{\partial f_1}{\partial a}}{\frac{\partial f_1}{\partial x_1}+\frac{\partial f_1}{\partial x_2}\frac{\partial x_2^*}{\partial x_1}} ,$$
whose sign, I think, determines whether the effect is positive or negative.
[1] $f_1$ increases, which needs to be offset by an increase in $x_1$ to satisfy the FOC.