Chapter 3, Problem 9NP

a.

To determine

The full employment level of output in each year with the help of given information.

Explanation of Solution

The full employment level of output in each year.

Given table,

Year	Unemployment rate	Output
1	8	950
2	6	1030
3	7	1033.5
4	5	1127.5

The full employment level of output is the output level at which all the available resources are used efficiently.

To calculate the full employment level of output, there is need to use the formula which is as follows:

  $\frac{\overline{Y}-Y}{\overline{Y}}=2\left(u-\overline{u}\right)$

Here, $\overline{Y}$ is the full employment level of output,

$\overline{u}$ is the natural rate of unemployment,

Y is the actual output and u is the actual unemployment rate.

For year 1,

  $\begin{array}{l}\frac{\overline{Y}-Y}{\overline{Y}}=2\left(u-\overline{u}\right)\\ \frac{\overline{Y}-Y}{\overline{Y}}=2\left(0.08-0.06\right)\\ \overline{Y}-Y=0.04\overline{Y}\\ \overline{Y}-0.04\overline{Y}=950\\ 0.96\overline{Y}=950\\ \overline{Y}=989.6\end{array}$

Therefore, in year 1 the full employment level of output is equal to $989.6.

Similarly, the full employment level of output in year 2, year 3, and year 4 can be calculated and the respective result is shown in table.

Year	Unemployment rate u	Natural rate of unemployment(%)	u-   $\overline{u}$	Actual output	Full employment level of output   $\overline{Y}$
1	0.08	0.06	0.02	950	$989.6
2	0.06	0.06	0	1030	$1030
3	0.07	0.06	0.01	1033.5	$1054.6
4	0.05	0.06	-0.01	1127.5	$1105.4

b.

To determine

The growth rate of full employment level of output of year 2, 3, and 4 in two different ways.

Explanation of Solution

The growth rate of full employment level of output is the ratio of change in full employment level of output to the full employment level of output that is, $\frac{\Delta \overline{Y}}{\overline{Y}}$

For year 2, growth rate is:

  $\begin{array}{c}\frac{\Delta \overline{Y}}{\overline{Y}}=\frac{1030-989.6}{989.6}\\ =0.04\end{array}$

The growth rate of full employment level of output with the help of previous solution is given below:

Year	Full employment level of output   $\overline{Y}$	Growth rate   $\frac{\Delta \overline{Y}}{\overline{Y}}$
1	989.6	-
2	1030	0.04
3	1054.6	0.023
4	1105.4	0.048

For the second method, first calculate the change in unemployment rate, then the growth rate of output. And lastly use the Okun’s law, given as:

  $\frac{\Delta Y}{Y}=\frac{\Delta \overline{Y}}{\overline{Y}}-2\Delta u$

To calculate the full employment level of output. The tablegiven below represent the change in unemployment rate and growth rate of output.

Year	Unemployment rate u	Natural rate of unemployment(%)	$\Delta u$	Actual output Y	$\frac{\Delta Y}{Y}$
1	0.08	0.06	-	950	-
2	0.06	0.06	-0.02	1030	0.084
3	0.07	0.06	0.01	1033.5	0.003
4	0.05	0.06	-0.02	1127.5	0.091

Now use the equation:

  $\frac{\Delta Y}{Y}=\frac{\Delta \overline{Y}}{\overline{Y}}-2\Delta u$

To find the full-employment level of output.

Write this equation as:

  $\begin{array}{l}\frac{\Delta Y}{Y}=\frac{\Delta \overline{Y}}{\overline{Y}}-2\Delta u\\ \frac{\Delta \overline{Y}}{\overline{Y}}=\frac{\Delta Y}{Y}+2\Delta u\end{array}$

The following table represent the value of full-employment level of output.

Year	$$   $\Delta u$	$\frac{\Delta Y}{Y}$	$\frac{\Delta \overline{Y}}{\overline{Y}}=\frac{\Delta Y}{Y}+2\Delta u$
1	-	-	-
2	-0.02	0.084	0.044
3	0.01	0.003	0.023
4	-0.02	0.091	0.051

The comparison of full-employment level of output calculated by two different method is given below:

Year	Growth rate   $\frac{\Delta \overline{Y}}{\overline{Y}}$	$\frac{\Delta \overline{Y}}{\overline{Y}}=\frac{\Delta Y}{Y}+2\Delta u$
2	0.04	0.044
3	0.023	0.023
4	0.048	0.051

From the table, it is concluded that both the results are similar.