Consider the following maximization problem: $$ \max_{x,I}G(x,y) $$ subject to the constraints $$ p_x x +p_I I= z \\ y=f(I, q)\\ q=q^0$$
Here is my attempt at writing out the lagrangian $$ \Lambda = G(x,y) - \lambda_1 (p_x x + p_I I - z) - \lambda_2 (y-f(I,q)-\lambda_3 (q-q^0)$$ Is the First order condition with respect to $I$: $$ \frac{\partial G}{\partial y}\frac{\partial y}{\partial I} - \lambda_1 p_I - \lambda_2 (\frac{\partial y}{\partial I} + \frac{\partial f}{\partial I})=0$$ or is it $$- \lambda_1 p_I - \lambda_2 \frac{\partial f}{\partial I}=0$$
In other words, when I take the partial of the lagrange with respect to $I$, do I treat $y$ as a constant, or as a function of $I$?
- (I get confused because we don't choose $y$ directly, so it is more like a parameter than a variable, but we have a constraint that says $y$ depends on $I$