I have a function which looks like: $y = f (x_{1}, g(x_1, y))$. I want to express the cross-partial of $g$ w.r.t $x \;and \; y$ in terms of the function or derivatives of $f$. Here is how I did: First take derivative w.r.t $y$ and I get $1 = f_2g_2$, so $g_2 = 1/f_2$; Secondly, take derivative w.r.t. $x$ and I have $0 = f_1 + f_2g_1$ so $g_1 = - f_1/f_2$. Now take cross partial w.r.t. $x_1 \; and \; y$ and get $g_{21} = (f_1f_{22} - f_{21}f_2)/f_2^3$.
So now I have the question that when I took the derivative w.r.t. $x_1$ (in the second step), is it right that the left hand side is $0$? Isn't $y$ a function of $x_1$? Can it be $0$ then?
Now if I have a function $F = pf(x_1, g(x_1, y))$, and I want to take derivatives of $F$ w.r.t. $x_1 $ and $y$, how could I plug in what I have from previous steps to express $F_{x_1y}$? ($p$ is a constant.)
Actually, the context of this problem is about a one-output, two-input profit-maximization problem, $y$ refers to the output and $x_1$ refers to an input. The original problem is: For the one-output, two-input profit-maximization problem, using supermodularity, under what minimal conditions is $x_1(p, w)$ decreasing in $p$? The construction of the problem is to set a maximization problem like $\underset{x_1}\max \; pf(x_1, g(x_1, y)) - w_1x_1 - w_2x_2$. We need to know how does the optimal choice of $x_1$ change in the price of the output, $p$.
Thanks a lot！