This is a question about the envelope theorem. Suppose I have a maximization problem
$$\max_{x,y} f(x,y,\xi)$$ such that $$g(x,y,\xi) \leq c$$ where $x$ and $y$ are control variables and $\xi$ is a given parameter which affects $f$ and/or $g$, but over which we do not maximize.
We write the Lagrangian $$L(x,y,\xi) = f(x,y,\xi) - \lambda (g(x,y,\xi) - c)$$ for which the two first order conditions are given by
$$\frac{\partial L}{\partial x} = \frac{\partial f(x,y,\xi)}{\partial x} -\lambda \frac{\partial g(x,y,\xi)}{\partial x} =0$$ $$\frac{\partial L}{\partial y} = \frac{\partial f(x,y,\xi)}{\partial y} -\lambda \frac{\partial g(x,y,\xi)}{\partial y}=0 $$
The next step in the several sets or notes I have seen say I need to solve this problem say I need to write a function
$$F(\xi) = f(x(\xi),y(\xi),\xi)$$ and take the total derivative.
Let me know if anything isn't clear and I will fix it.
My Question:
Why can I write $x$ and $y$ as functions of $\xi$?