Can anyone help answer the below? I know I'm supposed to use the envelope theorem i.e. find the value function then differentiate wrt $p$$x$ and $p$$y$, but I'm struggling to get the right answer. Thank you in advance!
A firm uses input $x$ to produce $y = f(x)$ while facing market prices $(p$$x$, $p$$y$) for $(x, y)$, respectively. The choice of $x$ satisfies the first order condition for profit maximisation, $p$$y$$f′(x) − p_x = 0$.
If the prices change by the small amounts $(p$$x$,$p$$y$), what is the approximate change in the optimal use of $x$?
The first-order condition for profit maximization is $$F(p_x,p_y,x)=p_yf'(x) − p_x = 0$$
The implicit function theorem tells us that as long as $F$ is continuously differentiable, and $F_x\neq 0$ then the partial derivatives of $x$ are given by $$\frac{\partial x}{\partial p_x}=-\frac{F_{p_x}}{F_x}\text{ and } \frac{\partial x}{\partial p_y}=-\frac{F_{p_y}}{F_x}$$
The partial derivatives you get above are the same as you would get from using the chain rule to implicitly differentiate the following with respect to $p_x$ and $p_y$ $$p_yf'[x(p_x,p_y)] − p_x = 0$$
Hopefully, you should be able to take it from there using either method.