Chapter 12, Problem 4NP

a.

To determine

To describe: The change in the unemployment rate and the percentage by which the output fall short of full employment output along with the sacrifice ratio.

Answer to Problem 4NP

In the first year, the unemployment rate is $\text{8\%}$ , cyclical unemployment is $\text{2\%}$ Therefore, a $\text{2\%}$ increase in cyclical unemployment would make the output fall short of the full employment output by $\text{4\%}$ . And in the second year, the unemployment rate is $\text{6\%}$ , cyclical unemployment is zero percent. Since, there is no increase in cyclical unemployment, therefore the output does not change and remains at full employment.

Explanation of Solution

Given data,

Inflation rate $\text{(π)}$ is $\text{12\%}$

Natural unemployment rate $\text{(ū)}$ is $\text{6\%}$

In the first year $\text{π=0}\text{.04}$ and ${\pi }^e=0.08$

In the second year, $\text{π=0}\text{.04}$ and ${\text{π}}^e\text{=0}\text{.04}$

And, The equation for expectations-augmented Philips curve is as follows:

${\text{π = π}}^{\text{e}}\text{-2(u-}\overline{\text{u}}\text{)}$

Substitute the values from above given data in the equation, we get:

$\begin{array}{l}0.04=0.08-2(u-0.06)\\ 2(u-0.06)=0.08-0.04\\ 2(u-0.06)=0.04\\ 2u-0.12=0.04\\ 2u=0.16\\ u=0.08\end{array}$

Hence, the unemployment rate in the first year is $8\%$ .

Therefore, cyclical unemployment is $\left(0.08-0.06\right)=0.02$ i.e. $2\%$ .

Now, calculate the unemployment rate for the second year, as calculated in first case:

$\begin{array}{l}0.04=0.04-2(u-0.06)\\ 2(u-0.06)=0.04-0.04\\ 2(u-0.06)=0\\ u=0.06\end{array}$

Thus, the unemployment rate in the first year is $6\%$ .

Therefore, cyclical unemployment is $\left(0.06-0.06\right)=0.00$ i.e. $0\%$ .

The formula used to calculate sacrifice ratio is given below, whereas sacrifice ratio is the amount of output lost when the inflation rate reduced by $1\%$ .

$\text{SR=}\frac{\text{ΔY}}{\text{Δπ}}$

Where,

$\text{ΔY}$ is the change in output and,

$\text{Δπ}$ is the change in inflation.

Since the long run equilibrium of the economy is at inflation rate of $12\%$ and the short run inflation rate for the two year is given to be $4\%$ , then the change in inflation rate (disinflation) equals $12-4=8\%$ i.e. $\text{Δπ=8\%}$ , during the two year disinflation the output had reduced by 4%. Therefore, the sacrifice ratio is $\frac{4}{8}=0.5\%$

Economics Concept Introduction

Introduction: Okun’s Law graph representation helps in defining relationship between unemployment and losses in the GDP of an economy. It explains the link between unemployment and output levels. Output is directly dependent on the amount of labor used in the production. This law intended to tell us how much of a country’s GDP may be lost when the unemployment is above its natural rate. Therefore, okun’s law explains the negative correlation between GDP growth and unemployment.

b.

To determine

To describe: The change in the unemployment rate and the percentage by which output falls short of full employment output along with the sacrifice ratio.

Answer to Problem 4NP

The unemployment rate for each of the four years are $0.07,0.08,0.07$ and $0.06$ respectively, the output fall short of full employment output each of the four years are $2\%,4\%,2\%$ and zero percent and the sacrifice ratio for this disinflation is 1%.

Explanation of Solution

Given data,

Year	1	2	3	4
$\pi$	$0.08$	$0.04$	$0.04$	$0.04$
${\pi }^e$	$0.10$	$0.08$	$0.06$	$0.04$
$\overline{u}$	$0.06$	$0.06$	$0.06$	$0.06$

The formula used to calculate unemployment is as under:

$(u=\frac{{\pi }^e-\pi }{2}+\overline{u})$

Substitute the values from given data in the equation, we get:

Year	1	2	3	4
$\pi$	$0.08$	$0.04$	$0.04$	$0.04$
${\pi }^e$	$0.10$	$0.08$	$0.06$	$0.04$
$\overline{u}$	$0.06$	$0.06$	$0.06$	$0.06$
$(u=\frac{{\pi }^e-\pi }{2}+\overline{u})$	$0.07$	$0.08$	$0.07$	$0.06$
Increase in Cyclical Unemployment	$0.01$	$0.02$	$0.01$	$0.00$
Fall in Output each year	$2\%$	$4\%$	$2\%$	$0\%$

And,

The formula used to calculate sacrifice ratio is given below,

$\text{SR=}\frac{\text{ΔY}}{\text{Δπ}}$

The total loss of the output in the 4 years is $8\%\left(2\%+4\%+2\%+0\%\right)$ and the disinflation accounted for $8\%\left(12-4\right)$ . Therefore, the sacrifice ratio equals to $1\%$ .

Economics Concept Introduction

Introduction: Okun’s Law graph representation helps in defining relationship between unemployment and losses in the GDP of an economy. It explains the link between unemployment and output levels. Output is directly dependent on the amount of labor used in the production. This law intended to tell us how much of a country’s GDP may be lost when the unemployment is above its natural rate. Therefore, okun’s law explains the negative correlation between GDP growth and unemployment.