A firm produces output in the cost minimizing way when it used that
quantity of two inputs (Labor and capital) at which following
condition is fulfilled -
MP_{L}/MP_{K} = P_{L}/P_{K}
Production function is as follows -
q = L^{0.25}K^{0.75}
Calculate MPL -
MPL = dq/dL
MPL = d(L^{0.25}K^{0.75})/dL
MPL = 0.25K^{0.75}/L^{0.25}
MPK = dq/dK
MPK = d(L^{0.25}K^{0.75})/dK
MPK = 0.75L^{0.25}/K^{0.25}
Cost minimizing condition is as follows -
MP_{L}/MP_{K} = P_{L}/P_{K}
(0.25K^{0.75}/L^{0.25})/(0.75L^{0.25}/K^{0.25}) = 10/5
0.25K/0.75L = 2
K/3L = 2
K = 6L
Firm is producing 400 units of output.
So,
q = L^{0.25}K^{0.75}
400 = L^{0.25}K^{0.75}
400 = L^{0.25}(6L)^{0.75}
400 = 6^{0.75} L^{0.25} L^{0.75}
400 = 3.83 * L
L = 400/3.83 = 104.44
Thus,
The firm will use 104.44 units of labor.
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