# Calculus/Optimization - Implicit fuction theorem with equality constraints

I have the following constrained maximization problem, written as a Lagrangian: $$L(x,y,\lambda) = f(x,y) - \lambda(g(x,y))$$ I can derive a set of implicit equations that characterize the solution, however, no closed form solution exists. I was thinking that I could apply the implicit function theorem to at least sign a set of partial derivatives (for example, $\partial x^* / \partial \alpha$, where $\alpha$ is some exogenous parameter).

Say I have, $$\frac{\partial L(.)}{\partial x} = f_x - \lambda g_x = 0$$ $$\frac{\partial L(.)}{\partial y} = f_y - \lambda g_y = 0$$ $$\frac{\partial L(.)}{\partial \lambda} = g(x,y) = 0$$ I can eliminate $\lambda$ and derive an implicit equation, $$\frac{f_x}{g_x} - \frac{f_y}{g_y} = 0$$ Which I combine with the budget constraint to get: $$H(x,y) = \frac{f_x}{g_x} - \frac{f_y}{g_y} + g(x,y) = 0$$ Can I then apply the implicit function theorem to $H(x,y)$ to evaluate the sign of $\partial x / \partial \alpha$ (where $\alpha$ is an exogenous parameter) evaluated at the optimal $x^*$?

I'm aware that I could go the long route and set up a system of first order conditions to be solved using Cramer's rule. This would look something like, let $H_1 = \frac{\partial L(.)}{\partial x}$ , $H_2 = \frac{\partial L(.)}{\partial y}$ and $H_3 = \frac{\partial L(.)}{\partial \lambda}$. At the optimal $x^*$, $y^*$, $\lambda^*$ each of these equations are equal to zero. Given an exogenous parameter $\alpha$ (in both $f(x,y$ and $g(x,y)$), what I'm after is (an element in) the solution to the following equation: $$\left( \begin{matrix} \frac{\partial H_1}{\partial x} & \frac{\partial H_1}{\partial y} & \frac{\partial H_1}{\partial \lambda} \\ \frac{\partial H_2}{\partial x} & \frac{\partial H_2}{\partial y} & \frac{\partial H_1}{\partial \lambda} \\ \frac{\partial H_3}{\partial x} & \frac{\partial H_3}{\partial y} & \frac{\partial H_3}{\partial \lambda} \end{matrix} \right) \left( \begin{matrix} \frac{\partial x^*}{\partial \alpha} \\ \frac{\partial y^*}{\partial \alpha} \\ \frac{\partial \lambda^*}{\partial \alpha} \end{matrix} \right) = - \left( \begin{matrix} \frac{\partial H_1}{\partial \alpha} \\ \frac{\partial H_2}{\partial \alpha} \\ \frac{\partial H_3}{\partial \alpha} \end{matrix} \right)$$ I was just hoping there would be an easier way to extract the information since $f(x,y)$ and $g(x,y)$ are very complicated functions. Thanks again.