Industrial production
a. Economists often use the expression “rate of growth” in relative rather than absolute terms. For example, let u = f(t) be the number of people in the labor force at time t in a given industry. (We treat this function as though it were differentiable even though it is an integer-valued step function.)
Let v = g(t) be the average production per person in the labor force at time t. The total production is then y = uv. If the labor force is growing at the rate of 4% per year (du/dt = 0.04u) and the production per worker is growing at the rate of 5% per year (dy/dt = 0.05v), find the rate of growth of the total production, y.
b. Suppose that the labor force in part (a) is decreasing at the rate of 2% per year while the production per person is increasing at the rate of 3% per year. Is the total production increasing, or is it decreasing, and at what rate?
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