Problem

# Industrial production a. Economists often use the expression “rate of growth” in relati...

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?

#### Step-by-Step Solution

Solution 1

a) $$u=f(t)$$ be the number of people in the labor force at time $$t$$.

$$v=g(t)$$ be the average production per person in the labor force at time $$t$$. $$y=u v$$ is the total production.

Given $$\frac{d u}{d t}=0.04 u$$

\begin{aligned} \frac{d v}{d t} &=0.05 v \\ \frac{d y}{d t} &=u \frac{d v}{d t}+v \frac{d u}{d t} \\ &=0.05(u v)+0.04(u v) \\ &=0.09 u v \\ &=0.09 y \end{aligned}

b) $$\frac{d u}{d t}=-0.02 u$$

\begin{aligned} \frac{d v}{d t} &=0.03 v \\ \frac{d y}{d t} &=u \frac{d v}{d t}+v \frac{d u}{d t} \\ &=0.03(u v)-0.02(u v) \\ &=0.01 u v \end{aligned}