Question

Use the production​ function: Q​ = 4L​1/2K​1/2. Refer to Scenario 3. Suppose that the price of labor is​ $5 and the price of capital is​ $20. Your firm desires to produce 200 units of output. How much labor will be hired to minimize the costs of producing 200 units of​ output?

Answer 1

Answer:

100 labor will be hired to minimize the costs of producing 200 units of​ output.

Explanation:

Maximising profit

Profit (Pr) = P*Q - Cost

                = P*Q - 5L - 20K

                 = P*4(LK)^0.5 - 4L - 20K

       dPr/dL= P*2(K/L)^0.5 - 4 = 0

       dPr/dK = P*2(L/K)^0.5 - 20 =0

We know equilibrium condition

MPL/MPK = w/ r

                 = 5/20

                  = 0.25  

MPL = 2(K/L)^0.5  MPK = 2(L/K)^0.5

K/L = 0.25

K = 0.25L

200 = 4(0.25L^2)^0.5

       =4*0.5*L

L = 100

Therefore, 100 labor will be hired to minimize the costs of producing 200 units of​ output.