Chapter 4.6, Problem 47E

a.

To determine

Frequently, economists use relative terminology as opposed to absolute ones.

Answer to Problem 47E

The rate of production growth is $9\%$ per year.

Explanation of Solution

Given: The total number of workers in a specific industry at time $t.$

  $y^{\prime }=u^{\prime }v+u{v}^{\prime }=0.04uv+0.05uv=0.09y$

Utilizing the derivatives of $u$ and $v,$ differentiate the total production function.

  $\text{t=8}$

Conclusion:

The rate of production growth is $9\%$ per year.

b.

To determine

A portion of the labour force is ageing at a pace of $2\%$ per year, while production per person is growing at a rate of $3\%$ per year.

Answer to Problem 47E

The overall production is growing by $\text{1\%}$ annually.

Explanation of Solution

Given: Increasing or declining in terms of overall

  $\begin{array}{c}y=uv\frac{dy}{dt}\\ =u\frac{dv}{dt}+v\frac{du}{dt}\end{array}$

Using the product rule, determine the derivative of $y.$

  $\begin{array}{c}\frac{dy}{dt}=u(0.03v)+v(-0.02u)\\ =0.03uv-0.02uv\\ =0.01uv\end{array}$

If the labour force is vanishing at a rate of $2\%$ annually, then $\frac{du}{dt}=-0.02u$ . It follows that $\frac{dv}{dt}=0.03v$ if the production per person increases by $3\%$ each year. After substituting these, simplify $\frac{dy}{dt}$ .

  $\frac{dy}{dt}=0.01y$

Replace with $\text{y=u v}\text{.}$ The entire production then grows by $\text{1\%}$ year.

Conclusion:

The overall production is growing by $\text{1\%}$ annually.