Chapter 15, Problem 10P

(a)

To determine

Equation that relate planned spending to output and real rate of interest.

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:

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

Thus, the equation for PAE is derived as $\text{PAE}=4,950-3,000\text{r}+0.6\text{Y}$.

(b)

To determine

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

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, then the PAE is calculated as follows:

$\begin{array}{c}\text{PAE}=4,950-3,000\text{r}\\ =4,950-\left(3,000\times 0.02\right)\\ =4,950-60\\ =4,890\end{array}$

Thus, the autonomous expenditure is 4,890.

The equilibrium output in the short run can be calculate by fix PAE equal to the output (Y). When the inflation rate is 0.0 and rate of interest is 0.02, Y is calculated by substituting the equation $\text{PAE}=4,890-0.6\text{Y}$ as follows:

$\begin{array}{c}\text{PAE}=4,890+0.6\text{Y}\\ \text{Y}=4,890+0.6\text{Y}\\ \text{Y}-0.6\text{Y}=4,890\\ \text{0}\text{.4Y=}4,890\\ \text{Y=}\frac{4,890}{0.4}\\ \text{=12,225}\end{array}$

Thus, the equilibrium output in the short run is 12,225.

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

Table 1

Rate of inflation	Real inflation rate	Autonomous expenditure	Equilibrium output
0	0.02	4,890	12,225
0.01	0.03	4,860	12,150
0.02	0.04	4,830	12,075
0.03	0.05	4,800	12,000
0.04	0.06	4,770	11,925

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

In Figure 1, the horizontal axis represents output and the vertical axis represents the rate of inflation. 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.

(c)

To determine

Impact of increasing government purchases and construction of a table 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 is derived as follows:

$\begin{array}{c}\text{PAE}=\text{C}+{\text{I}}^{\text{P}}+\text{G}+\text{NX}\\ \text{PAE}=\left(\text{1,600 }+\text{ 0}\text{.6(Y}-2,000\text{)}-\text{ 2,000r}\right)+\left(\text{2,500}-\text{1,000r}\right)+2,100+50\\ =1,600+0.6\text{Y}-1,200-2,000\text{r}+\text{2,500}-\text{1,000r}+2,100+50\\ =\left(1,600-1,200+2,500+2,100+50\right)\left(-2,000\text{r}-\text{1,000r}\right)+0.6\text{Y}\\ =5,050-3,000\text{r}+0.6\text{Y}\end{array}$

Thus, the equation for PAE is derived as $\text{PAE}=5,050-3,000\text{r}+0.6\text{Y}$. The autonomous expenditure is not related to output (Y). Hence, the equation can be rewritten as $\text{PAE}=5,050-3,000\text{r}$. If the inflation rate is 0.0 and real interest rate is 0.02, the PAE is calculated as follows:

$\begin{array}{c}\text{PAE}=5,050-3,000\text{r}\\ =5,050-\left(3,000\times 0.02\right)\\ =5,050-60\\ =4,990\end{array}$

Thus, the autonomous expenditure is 4,990.

The equilibrium output in the short run is calculated by fix PAE equal to the output (Y). When the inflation rate is 0.0 and rate of interest is 0.02, Y is calculated by substituting equation $\text{PAE}=5,050-3,000\text{r}$ as follows:

$\begin{array}{c}\text{PAE}=4,990+0.6\text{Y}\\ \text{Y}=4,990+0.6\text{Y}\\ \text{Y}-0.6\text{Y}=4,990\\ \text{0}\text{.4Y=}4,990\\ \text{Y=}\frac{4,990}{0.4}\\ \text{=12,475}\end{array}$

Thus, the equilibrium output in the short run is 12,475.

Similarly, substitute each values of the rate of interest in the equation $\text{PAE}=5,050-3,000\text{r}+0.6\text{Y}$. The table below shows the calculated autonomous expenditure and equilibrium output at each level of the rate of interest.

Table 1

Inflation Rate	Real Interest Rate	Autonomous Expenditure	Equilibrium Output
0	0.02	4,990	12,475
0.01	0.03	4,960	12,400
0.02	0.04	4,930	12,325
0.03	0.05	4,900	12,250
0.04	0.06	4,870	12,175

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

In Figure 2, the horizontal axis represents output and the vertical axis represents the rate of inflation. 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.