Scenario 7.2:
The production function for earthquake detectors (Q) is given as follows:
Q = 4K 1 /2 L 1/2
where K is the amount of capital employed and L is the amount of labor employed. The price of capital, P K , is $18 and the price of labor, P L , is $2.
-Refer to Scenario 7.2. Suppose that you receive an order for 60 earthquake detectors. How much labor will you use to minimize the cost of 60 earthquake detectors?
A) 1
B) 5
C) 10
D) 45
E) none of the above

Question

Scenario 7.2:
The production function for earthquake detectors (Q) is given as follows:
Q = 4K 1 /2 L 1/2 
where K is the amount of capital employed and L is the amount of labor employed. The price of capital, P K , is $18 and the price of labor, P L , is $2.
-Refer to Scenario 7.2. Suppose that you receive an order for 60 earthquake detectors. How much labor will you use to minimize the cost of 60 earthquake detectors?
A) 1
B) 5
C) 10
D) 45
E) none of the above

Quizplus · Accepted Answer

To minimize the cost of producing 60 earthquake detectors, we need to find the combination of labor and capital that produces 60 units of output at the lowest cost. We can use the cost minimization rule, which states that the ratio of the marginal product of labor to the price of labor should equal the ratio of the marginal product of capital to the price of capital:

MPL/PL = MPK/PK

Taking logarithms and differentiating with respect to labor and capital, we get:

(1/U1)ln(Q)-ln(L)-(1/U1)ln(Ps1u1b1Ls1u1b0) = (1/U1)ln(Q)-ln(K)-(1/U1)ln(Ps1u1b1Ks1u1b0)

Rearranging and solving for L, we get:

L = (PL/PU)(MPK/MPL)K

Substituting the values, we get:

L = (2/18)(4S1U1P1S1S1U1P11S1S1P0)^2 = 45

Therefore, we should use 45 units of labor to minimize the cost of producing 60 earthquake detectors.

Scenario 7.2: the Production Function for Earthquake Detectors (Q) Is Given as Given