Exam 1

  1. Question

    A firm has the following production function: F(K,L)=KL2. F(K,L)= K L^{2}. The price for one unit of capital is pK=13p_K = 13 and the price for one unit of labor is pL=13p_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. minK,LC(K,L)=pKK+pLL=13K+13Lsubject to:F(K,L)=QKL2=430 \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. ℒ(K,L,λ)=C(K,L)−λ(F(K,L)−Q)=13K+13L−λ(KL2−430) \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. ∂ℒ∂K=13−λL2=0∂ℒ∂L=13−2λKL2−1=0∂ℒ∂λ=−(KL2−430)=0 \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 KK, LL, and λ\lambda.

    Solving the first two equations for λ\lambda and equating them gives: 13L2=132KL2−1K=132⋅13⋅L2−(2−1)K=1326⋅L \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: KL2=430(1326⋅L)L2=4301326L3=430L=(2613⋅430)13=9.5096854≈9.51K=1326⋅L=4.7548427≈4.75 \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: C(K,L)=13K+13L=61.812955+123.62591=185.438865≈185.44 \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 185.44185.44.