Question

Suppose the production of a firm is modeled by P(k,l)=16k ^2/3 l ^1/3 , where k measures capital (in millions of dollars) and l measures the labor force (in thousands of workers). Suppose that when l=2 and k=3, the labor is increasing at the rate of 90 workers per year and capital is decreasing at a rate of $210,000 per year. Determine the rate of change of production. Round your answer to the fourth decimal place.

Answer 1

The rate of change of production is approximately -1120.6935 units per year.

Step-by-step explanation:

To determine the rate of change of production, we can use the concept of partial derivatives.

The production function P(k,l) = 16k^(2/3) * l^(1/3) represents the output of the firm, where k is the capital and l is the labor force.

To find the rate of change of production, we need to calculate the partial derivatives with respect to k and l.

The partial derivative of P with respect to k is given by: ∂P/∂k = (32/3) k^(-1/3) l^(1/3)

The partial derivative of P with respect to l is given by: ∂P/∂l = (16/3) k^(2/3) l^(-2/3)

Given that when l=2 and k=3, the labor force is increasing at a rate of 90 workers per year (∆l/∆t = 90) and capital is decreasing at a rate of $210,000 per year (∆k/∆t = -210,000), we can use these values to find the rate of change of production.

Using the chain rule for differentiation, we can calculate the rate of change of production as follows:

dP/dt = (∂P/∂k) (dk/dt) + (∂P/∂l) (dl/dt)

Substitute the given values:

dP/dt = (32/3) 3^(-1/3) 2^(1/3) (-210,000) + (16/3) 3^(2/3) 2^(-2/3) 90

Solving this equation yields: dP/dt ≈ -1120.6935

Therefore, the rate of change of production is approximately -1120.6935 units per year.