Chapter 15, Problem 11P

(a)

To determine

Construction of a table that shows the relation of equilibrium output in the short run and inflation rate.

Explanation of Solution

The planned aggregate expenditure is the sum of consumption, planned investment, government spending, and net export. The equation that connects the planned spending to output and real rate interest can be derived as follows:

$\begin{array}{c}\text{PAE}=\text{C}+{\text{I}}^{\text{P}}+\text{G}+\text{NX}\\ \text{PAE}=\left(\text{1,600 }+\text{ 0}\text{.6(Y}-2,000\text{)}-\text{ 2,000r}\right)+\left(\text{2,500}-\text{1,000r}\right)+2,000+50\\ =1,600+0.6\text{Y}-1,200-2,000\text{r}+\text{2,500}-\text{1,000r}+2,000+50\\ =\left(1,600-1,200+2,500+2,000+50\right)+0.6\text{Y}-\left(2,000\text{r}+\text{1,000r}\right)\\ \text{PAE}=4,950+0.6\text{Y}-3,000\text{r}\end{array}$

Thus, the equation for PAE is derived as $\text{PAE}=4,950-3,000\text{r}+0.6\text{Y}$. The autonomous expenditure is not related to the output (Y). Hence, the equation can be rewritten as $\text{PAE}=4,950-3,000\text{r}$. If the inflation rate is 0.0 and real interest rate is 0.04, the PAE can be calculated as follows:

$\begin{array}{c}\text{PAE}=4,950-3,000\text{r}\\ =4,950-\left(3,000\times 0.04\right)\\ =4,950-120\\ =4,830\end{array}$

Thus, the autonomous expenditure is 4,830.

The equilibrium output in the short run can be calculated by fixing PAE, which is equal to the output (Y). The equilibrium output (Y) can be calculated by substituting the equation $\text{PAE}=4,830+0.6\text{Y}$ as follows:

$\begin{array}{c}\text{PAE}=4,830+0.6\text{Y}\\ \text{Y}=4,830+0.6\text{Y}\\ \text{Y}-0.6\text{Y}=4,830\\ \text{0}\text{.4Y=}4,830\\ \text{Y=}\frac{4,830}{0.4}\\ \text{=12,075}\end{array}$

Thus, the equilibrium output in the short run is 12,075.

Similarly, substitute each value of the rate of interest in the equation $\text{PAE}=4,950-3,000\text{r}+0.6\text{Y}$. Table-1 below shows the calculated autonomous expenditure and equilibrium output at each level of rate of interest.

Table 1

Rate of inflation	Real inflation rate	Autonomous expenditure	Equilibrium output
0	0.04	4,830	12,075
0.01	0.045	4,815	12,038
0.02	0.05	4,800	12,000
0.03	0.055	4,785	11,963
0.04	0.06	4,770	11,925

With the equilibrium output, the aggregate demand curve is graphically represented below:

In Figure 1, the horizontal axis represents the output, and the vertical axis represents the rate of inflation. The curve AD is the aggregate demand curve, which refers to the total value of the goods and services that are demanded at each price or inflation rate in a given period of time.

(b)

To determine

Construction of a table that shows the relation of equilibrium output in the short run and inflation rate when the Fed reduces the real rate of interest.

Explanation of Solution

The equation for PAE is derived as $\text{PAE}=4,950-3,000\text{r}+0.6\text{Y}$. The autonomous expenditure is not related to the output (Y). Hence, the equation can be rewritten as $\text{PAE}=4,950-3,000\text{r}$. If the inflation rate is 0.0 and real interest rate is 0.02, the PAE can be calculated as follows:

$\begin{array}{c}\text{PAE}=4,950-3,000\text{r}\\ =4,950-\left(3,000\times 0.035\right)\\ =4,950-105\\ =4,845\end{array}$

Thus, the autonomous expenditure is 4,845.

The equilibrium output in the short run can be calculated by fixing PAE, which is equal to the output (Y). The equilibrium output (Y) can be calculated by substituting the equation $\text{PAE}=4,830+0.6\text{Y}$ as follows:

$\begin{array}{c}\text{PAE}=4,845+0.6\text{Y}\\ \text{Y}=4,845+0.6\text{Y}\\ \text{Y}-0.6\text{Y}=4,845\\ \text{0}\text{.4Y=}4,845\\ \text{Y=}\frac{4,845}{0.4}\\ \text{=12,112}\text{.5}\end{array}$

Thus, the equilibrium output in the short run is around 12,113.

Similarly, substitute each value of rate of interest in the equation $\text{PAE}=4,950-3,000\text{r}+0.6\text{Y}$. Table-2 below shows the calculated autonomous expenditure and equilibrium output at each level of rate of interest.

Table 2

Rate of inflation	Real inflation rate	Autonomous expenditure	Equilibrium output
0	0.035	4,845	12,113
0.01	0.04	4,830	12,075
0.02	0.045	4,815	12,038
0.03	0.05	4,800	12,000
0.04	0.055	4,785	11,963

Table-2 shows the autonomous expenditure and equilibrium output at each level of inflation rate with a new real rate of interest. While comparing Tables 1 and 2, it is clear that the autonomous expenditure increases about 15 at each level of inflation rate. In the case of equilibrium output, it is increased about 38.  This increase in aggregate demand will lead to a shift in the AD curve to the right because the Fed adopts easing of monetary policy, which means, the Fed reduced the real rate of interest and thereby cause a downward shift of reaction curve. As a result, AD curve shifts to the right.