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

To minimize the costs of producing 200 units of output, the company should use Method 1 with 50 units of labor and 10 units of capital. If the cost of labor rises to $200/unit, the company should reassess the costs and select the method with the lowest cost.

Step-by-step explanation:

To minimize the costs of producing 200 units of output, we need to determine the optimal combination of labor and capital. Using the production function Q = 4L^1/2K^1/2, where Q is the quantity of output, L is the quantity of labor, and K is the quantity of capital, we can set up the cost minimization problem.

1. Calculate the cost of Method 1: 50 units of labor and 10 units of capital. Cost = (50 * $5) + (10 * $20) = $500
2. Calculate the cost of Method 2: 20 units of labor and 40 units of capital. Cost = (20 * $5) + (40 * $20) = $900
3. Calculate the cost of Method 3: 10 units of labor and 70 units of capital. Cost = (10 * $5) + (70 * $20) = $1,500

Based on the costs, the best production method to minimize costs for producing 200 units of output is Method 1 with 50 units of labor and 10 units of capital. If the cost of labor rises to $200/unit, the company should reassess the costs using the new labor cost and select the method with the lowest cost.

Answer 2

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

Step-by-step 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.