I have a a question regarding a constrained optimisation problem where there are three constraints. I was wondering if should include lagrangian multipliers for each of the constraints or whether the solution 1 is correct or whether solution 2 is the only way to solve this problem.
Problem
\begin{equation*} \begin{array}{lllrrrlllll} &&\underset{K_x,K_y,L_x,L_y}{\max} f_y(K_y,L_y)&s.t.& f_x(K_x,L_x)\geq \bar{x}, K_x + K_y = \bar{K}, L_x + L_y = \bar{L}\end{array} \end{equation*}
Solution 1: The Lagrangian is \begin{equation*} L(K_y,L_y,\lambda) = f_y(K_y,L_y) + \lambda[f_x(K_x, L_x)-\bar{x}] \end{equation*}
The First Order Conditions give \begin{equation*}\begin{array}{lclllllll} \frac{\partial L}{\partial K_y} &=& \frac{\partial f_y(K_y,L_y)}{\partial K_y} +\lambda \frac{\partial f_x(K_x,L_x)}{\partial K_x}\cfrac{d K_x}{d K_y} = 0 \\\\ &=& \frac{\partial f_y(K_y,L_y)}{\partial K_y} -\lambda \frac{\partial f_x(K_x,L_x)}{\partial K_x} = 0\\\\ &\implies& \frac{\partial f_y(K_y,L_y)}{\partial K_y} =\lambda \frac{\partial f_x(K_x,L_x)}{\partial K_x}\\\\ &i.e.& MPK_y = \lambda MPK_x\\\\ \frac{\partial L}{\partial L_y} &=& \frac{\partial f_y(K_y,L_y)}{\partial L_y} +\lambda \frac{\partial f_x(K_x,L_x)}{\partial L_x}\cfrac{dL_x}{dL_y} = 0 \\\\ &=& \frac{\partial f_y(K_y,L_y)}{\partial L_y} -\lambda \frac{\partial f_x(K_x,L_x)}{\partial L_x}= 0 \\\\ &\implies& \frac{\partial f_y(K_y,L_y)}{\partial L_y} =\lambda \frac{\partial f_x( K_x,L_x)}{\partial L_x}\\\\ &i.e.& MPL_y = \lambda MPL_x \end{array} \end{equation*}
Solution 2 My lecturer uses the constraints to substitute for $K_x$ and $L_x$ so that the problem becomes
\begin{equation*} \underset{K_y, L_y}{\max} f_y(K_y,L_y)\quad s.t \quad f_x(K-K_y,L-L_y) = \bar{x} \end{equation*}
and then writes the Lagrangian as
\begin{equation*} L = f_y(K_y,L_y) + \lambda[\bar{x} - f_x(K -K_y,L-L_y)] \end{equation*}
The problem with this is that when solving the problem we get $\partial f_x/\partial L_y$ and $\partial f_x/\partial K_y$ instead of $\partial f_x/\partial L_x$ and $\partial f_x/\partial K_x$ (which have an economic interpretation), hence why I try to avoid writing it in this way.
No, we don't get what you write.
$$\frac{\partial f_x}{\partial L_y} = \frac{\partial f_x(\bar K-K_y,\bar L-L_y)}{\partial L_y} = \frac{\partial f_x}{\partial L_x}\cdot \frac{\partial L_x}{\partial L_y}= MPL_x\cdot\frac{\partial (\bar L-L_y)}{\partial L_y} = -MPL_x$$
etc