Exam 1

  1. Question

    A firm has the following production function:
    F(K,L)=K L2 .

    The price for one unit of capital is pK =13 and the price for one unit of labor is pL =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,L C(K,L) = pK K+ pL L = 13K+13L subject to: F(K,L)=Q K L2 =430

    Step 2: Lagrange function.
    L(K,L,λ) = C(K,L)-λ(F(K,L)-Q) = 13K+13L-λ(K L2 -430)

    Step 3: First order conditions.
    L K = 13-λ L2 =0       (1) L L = 13-2λK L2-1 =0       (2) L λ = -(K L2 -430)=0       (3)

    Step 4: Solve the system of equations for K, L, and λ.
    Equating Equations (1) and (2) after solving for λ gives:
    13 L2 = 13 2K L2-1 K = 13 2·13 · L2-(2-1) K = 13 26 ·L

    Substituting this in the optimization constraint gives:
    K L2 = 430 ( 13 26 ·L) L2 = 430 13 26 L3 = 430 L = ( 26 13 ·430) 1 3 =9.50968541  9.51 K = 13 26 ·L=4.7548427  4.75

    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 = 185.438865185.44

    Given the target output, the minimal costs are 185.44.
    lagrange-003.png