The number of labor that will be hired to minimize the costs of producing 200 units of output can be determined as follows:
Q = 4L^(1/2)K^(1/2) = 4L^0.5K^0.5 = 4(LK)^0.5
MPL = dQ/dL = 2L^-0.5K^0.5 = 2(K/L)^0.5
MPK = dQ/dK = 2L^0.5K^-0.5 = 2(L/K)^0.5
Therefore, we have:
MPL / MPK = 2(K/L)^0.5 / 2(L/K)^0.5
MPL / MPK = K / L
From the question, we have:
r = Price of capital =$20 or 20
w = Price of labor = $5 or 5
Therefore, we have:
w/r = 5 / 20 = 0.25
The equilibrium condition states that:
MPL/MPK = w/ r
Substituting, we have:
K / L = 0.25
K = 0.25L
Since Q = 200 ans K = 0.25L, we substitute both in the production function and solve for L as follows:
200 = 4(0.25L)^0.5
200 = 4(L0.25L)^0.5 =
200 = 4(0.25L^2)^0.5
200 = 4(0.25^0.5 * L^(2 * 0.5))
200 = 4 * 0.5 * L
200 = 2L
L = 200 / 2
L = 100
Therefore, the number of labor that will be hired to minimize the costs of producing 200 units of output is a. 100.
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