Final answer:
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.