Question

Without using a Lagrangian, find the cost minimizing levels of L
and K for the production function **q = L^.6K^.4** if
the price of labor =10, the price of capital = 15, and desired
output = 100. What is the total cost to produce that output?

Answer #1

Cost is minimized when MPL/MPK = w/r = 10/15 = 2/3

MPL =
q/L
= 0.6 x (K/L)^{0.4}

MPK =
q/K
= 0.4 x (L/K)^{0.6}

MPL/MPK = (0.6/0.4) x (K/L) = 3K/2L = 2/3

K = 4L/9

Substituting in production function:

L^{0.6}K^{0.4} = 100

L^{0.6}(4L/9)^{0.4} = 100

L^{0.6}L^{0.4} x (4/9)^{0.4} = 100

L x 0.72 = 100

L = 138.89

K = (4 x 138.89) / 9 = 61.73

Total cost = wL + rK = (10 x 138.89) + (15 x 61.73) = 1,388.9 + 925.95 = 2,314.85

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