Suppose the production of a firm is modeled by where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that and the labor force is increasing at the rate of 90 and capital is growing at a rate of $200,000 per year. Determine the rate of change of production.
A) Production is increasing at a rate of 13.27 units per year.
B) Production is increasing at a rate of 2.13 units per year.
C) Production is increasing at a rate of 10.15 units per year.
D) Production is increasing at a rate of 21.29 units per year.

Question

Suppose the production of a firm is modeled by  where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that  and  the labor force is increasing at the rate of 90 and capital is growing at a rate of $200,000 per year. Determine the rate of change of production.
A) Production is increasing at a rate of 13.27 units per year. 
B) Production is increasing at a rate of 2.13 units per year. 
C) Production is increasing at a rate of 10.15 units per year. 
D) Production is increasing at a rate of 21.29 units per year.

Quizplus · Accepted Answer

Using the production function:

P = 11(K^0.3)(L^0.7)

Taking the derivative with respect to time (t):

dP/dt = 11(0.3K^(-0.7))(dK/dt)(L^0.7) + 11(0.7K^0.3)(L^(-0.3))(dL/dt)

Plugging in the given values:

dP/dt = 11(0.3(1 million)^(-0.7))(200,000)(L^0.7) + 11(0.7)(1 Million)^(0.3)((90,000)/1000)^(0.7)

dP/dt ≈ 2.13

Therefore, the production is increasing at a rate of 2.13 units per year.

Suppose the Production of a Firm Is Modeled by