Lagrangian Multiplier Question

Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.

Answer 1

Here is an idea.

Using the FOCs from the Lagrangian, i.e., $\nabla f_1 = \lambda \cdot \nabla f_2$, you should have obtained $\frac{\partial f_1}{\partial L_1} = \lambda \frac{\partial f_2}{L_1}$ and $\frac{\partial f_1}{\partial K_1} = \lambda \frac{\partial f_2}{\partial K_2}$. Then we have \begin{align} \frac{dy_1}{dy_2} &= \frac{\partial f_1}{\partial L_1} \frac{\partial L_1}{\partial y_2} + \frac{\partial f_1}{\partial K_1} \frac{\partial K_1}{\partial y_2} \\ &= \lambda \left( \frac{\partial f_2}{\partial L_1} \frac{\partial L_1}{\partial y_2} + \frac{\partial f_2}{\partial K_1} \frac{\partial K_1}{\partial y_2} \right) \\ &= \lambda \frac{dy_2}{dy_2} = \lambda. \end{align}