## Exam 1

1. #### Question

A firm has the following production function: $F(K,L)= K L^{2}.$ The price for one unit of capital is $p_K = 13$ and the price for one unit of labor is $p_L = 13$. Minimize the costs of the firm considering its production function and given a target production output of 430 units.

How high are in this case the minimal costs?

#### Solution

Step 1: Formulating the minimization problem. \begin{aligned} \min_{K,L} C(K,L) & = p_K K + p_L L\\ & = 13 K + 13 L\\ \mbox{subject~to:} & F(K,L) = Q \\ & K L^{2} = 430 \end{aligned} Step 2: Lagrange function. \begin{aligned} \mathcal{L}(K, L, \lambda) & = C(K, L) - \lambda (F(K, L) - Q) \\ & = 13 K + 13 L - \lambda (K L^{2} -430) \end{aligned} Step 3: First order conditions. \begin{aligned} \frac{\partial {\mathcal {L}}}{\partial K} & = 13 - \lambda L^{2} = 0\\ \frac{\partial {\mathcal {L}}}{\partial L} & = 13 - {2} \lambda K L^{2 - 1} = 0 \\ \frac{\partial {\mathcal {L}}}{\partial \lambda} & = -(K L^{2}-430) = 0 \end{aligned} Step 4: Solve the system of equations for $K$, $L$, and $\lambda$.

Solving the first two equations for $\lambda$ and equating them gives: \begin{aligned} \frac{13}{L^{2}} & = \frac{13}{{2} K L^{2 - 1}}\\ K & = \frac{13}{2 \cdot 13} \cdot L^{2 - (2 - 1)}\\ K & = \frac{13}{26} \cdot L \end{aligned} Substituting this in the optimization constraint gives: \begin{aligned} K L^{2} & = 430\\ \left(\frac{13}{26}\cdot L \right) L^{2} & = 430\\ \frac{13}{26} L^{3} & = 430\\ L & = \left(\frac{26}{13} \cdot 430\right)^{\frac{1}{3}} = 9.5096854 \approx 9.51\\ K & = \frac{13}{26} \cdot L = 4.7548427 \approx 4.75 \end{aligned}

The minimal costs can be obtained by substituting the optimal factor combination in the objective function: \begin{aligned} C(K, L) & = 13 K + 13 L\\ & = 61.812955 + 123.62591 \\ & = 185.438865 \approx 185.44 \end{aligned}

Given the target output, the minimal costs are $185.44$.