Question 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 -

MPL/MPK = PL/PK

Production function is as follows -

q = L0.25K0.75

Calculate MPL -

MPL = dq/dL

MPL = d(L0.25K0.75)/dL

MPL = 0.25K0.75/L0.25

MPK = dq/dK

MPK = d(L0.25K0.75)/dK

MPK = 0.75L0.25/K0.25

Cost minimizing condition is as follows -

MPL/MPK = PL/PK

(0.25K0.75/L0.25)/(0.75L0.25/K0.25) = 10/5

0.25K/0.75L = 2

K/3L = 2

K = 6L

Firm is producing 400 units of output.

So,

q = L0.25K0.75

400 = L0.25K0.75

400 = L0.25(6L)0.75

400 = 60.75 L0.25 L0.75

400 = 3.83 * L

L = 400/3.83 = 104.44

Thus,

The firm will use 104.44 units of labor.

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