Chapter 12, Problem 4PA

(a)

To determine

The IS and the LM equilibrium of the economy.

Explanation of Solution

The investment function of the economy is given by $\text{I}=1,000-50\text{r}$ , whereas the consumption function of the economy is given by $\text{C}=500+0.75\left(\text{Y}-\text{T}\right)$. The government purchases and the taxes of the economy are given to be 1,000 each. Thus, the IS curve satisfies that $\text{Y}=\text{C}\left(\text{Y}-\text{T}\right)+\text{I}\left(\text{r}\right)+\text{G}$.

The values of the C, T, I and G can be plugged into the IS equation as follows:

$\begin{array}{c}\text{Y}=\text{C}\left(\text{Y}-\text{T}\right)+\text{I}\left(\text{r}\right)+\text{G}\\ =500+0.75\left(\text{Y}-1,000\right)+1,000-50\text{r}+1,000\\ =500+0.75\text{Y}-750+1,000-50\text{r}+1,000\\ \text{Y}-0.75\text{Y}=500-750+1,000+1,000-50\text{r}\\ \text{0}\text{.25Y}=1,750-50\text{r}\\ \text{Y}=\frac{1,750-50\text{r}}{0.25}\\ =7,000-200\text{r}\end{array}$

The money demand function of the economy is given as ${\left(\frac{\text{M}}{\text{P}}\right)}^{\text{d}}=\text{Y}-\text{200r}$,  the money supply M is given as6,000, and the price level is given as2. The LM curve can be obtained by equating the supply of money and the demand for the real money balances in the economy with each other.

The supply of real money balances can be calculated by dividing the money supply with the price level in the economy. Since the values of the two are given, the value of the supply of the real money balances can be calculated as follows:

$\begin{array}{c}\text{Supply of real money balances}=\frac{\text{Money supply}}{\text{Price level}}\\ =\frac{6,000}{2}\\ =3,000\end{array}$

Thus, the supply of real money balance is 3,000. The LM curve can be calculated by setting the demand equation equal to the supply equation as follows:

$\begin{array}{c}\text{3,000}=\text{Y}-\text{200r}\\ \text{Y}=3,000+200\text{r}\end{array}$

The IS - LM equilibrium can be calculated by equating the IS equation equal to the LM equation. Thus, the IS-LM equilibrium can be calculated as follows:

$\begin{array}{c}\text{IS}=\text{LM}\\ \text{7,000}-200\text{r}=3,000+200\text{r}\\ \text{7,000}-3,000=200\text{r}+200\text{r}\\ 4,000=400\text{r}\\ \text{r}=\frac{4,000}{400}\\ =10\end{array}$

By substituting the value of r in any equation, it can provide the value of Y as follows:

$\begin{array}{c}\text{Y}=\text{3,000}+\text{200r}\\ =3,000+\left(200\times 10\right)\\ =3,000+2,000\\ =5,000\end{array}$

Thus, the rate of interest and the Y is 10 and 5,000, respectively. These values can be illustrated at point A through the graph as follows:

In Figure 1, horizontal axis measures the income or output and vertical axis measures the interest rate.

Economics Concept Introduction

Monetary policy: The monetary policy is the policy of the central bank that directs the money supply as well as the interest rate of the economy.

Fiscal policy: The fiscal policy is the policy of the government regarding the government expenditures and taxes of the economy.

(b)

To determine

The impact of tax cuts of 20 percent on the IS-LM equilibrium and the tax multiplier.

Explanation of Solution

When the tax rate decreases by 20 percent, the tax will be 800 in the economy. The IS curve will be then as follows:

$\begin{array}{c}\text{Y}=\text{C}\left(\text{Y}-\text{T}\right)+\text{I}\left(\text{r}\right)+\text{G}\\ =500+0.75\left(\text{Y}-800\right)+1,000-50\text{r}+1,000\\ =500+0.75\text{Y}-600+1,000-50\text{r}+1,000\\ \text{Y}-0.75\text{Y}=500-600+1,000+1,000-50\text{r}\\ \text{0}\text{.25Y}=1,900-50\text{r}\\ \text{Y}=\frac{1,900-50\text{r}}{0.25}\\ =7,600-200\text{r}\end{array}$

The IS - LM equilibrium can be calculated by equating the IS equation equal to the LM equation. Thus, the IS-LM equilibrium can be calculated as follows:

$\begin{array}{c}\text{IS}=\text{LM}\\ \text{7,600}-200\text{r}=3,000+200\text{r}\\ \text{7,600}-3,000=200\text{r}+200\text{r}\\ 4,600=400\text{r}\\ \text{r}=\frac{4,600}{400}\\ =11.5\end{array}$

By substituting the value of r in any equation, it can provide the value of Y as follows:

$\begin{array}{c}\text{Y}=\text{3,000}+\text{200r}\\ =3,000+\left(200\times 11.5\right)\\ =3,000+2,300\\ =5,300\end{array}$

Thus, the rate of interest and the Y is 11.5 and 5,300, respectively. These values can be illustrated through the graph as follows:

In Figure 2, horizontal axis measures the income or output and vertical axis measures the interest rate. Thus, the decrease in the tax rate by 20 percent leads to a shift in the IS curve toward the right by 300 forming a new equilibrium point B. The tax multiplier can be calculated by subtracting the change in the total output by the negative change in the taxes as follows:

$\begin{array}{c}\text{Tax multiplier}=\frac{\text{Change in output}}{\text{Change in tax}}\\ =\frac{\left(5,300-5,000\right)}{\left(800-1,000\right)}\\ =\frac{300}{-200}\\ =-1.5\end{array}$

Thus, the tax multiplier is -1.5.

(c)

To determine

The impact of holding the interest rate constant by Fed on Equilibrium.

Explanation of Solution

The supply of real money balances can be calculated by dividing the money supply with the price level in the economy. Since the values of the two are given, the value of the supply of the real money balances can be calculated as follows:

$\begin{array}{c}\text{Supply of real money balances}=\frac{\text{Money supply}}{\text{Price level}}\\ =\frac{\text{M}}{2}\end{array}$

The LM curve can be calculated by setting the demand equation equal to the supply equation as follows:

$\begin{array}{c}\frac{\text{M}}{2}=\text{Y}-\text{200r}\\ \text{Y}=\frac{\text{M}}{2}+200\text{r}\end{array}$

The IS - LM equilibrium can be calculated by equating the IS equation equal to the LM equation. Thus, the rate of interest is held constant which is equal to 10 and this value can be substituted in the equilibrium equation in order to calculate the money supply as follows:

$\begin{array}{c}\text{IS}=\text{LM}\\ \text{7,600}-200\text{r}=\frac{\text{M}}{2}+200\text{r}\\ \text{7,600}-\left(200\times 10\right)=\frac{\text{M}}{2}+\left(200\times 10\right)\\ 7,600-2,000=\frac{\text{M}}{2}+2,000\\ \text{5,600}-2,000=\frac{\text{M}}{2}\\ \text{M}=3,600\times 2\\ =7,200\end{array}$

By substituting the value of r in any equation, it can provide the value of Y as follows:

$\begin{array}{c}\text{Y}=\frac{\text{M}}{2}+\text{200r}\\ =\frac{7,200}{2}+\left(200\times 10\right)\\ =3,600+2,000\\ =5,600\end{array}$

Thus, the rate of interest is held constant by the fed and adjusted the money supply to 7,600. Then, the Y is calculated as 5,600. These values can be illustrated through the graph as follows:

In Figure 3, horizontal axis measures the income or output and vertical axis measures the interest rate. Thus, the change leads to a shift in the LM curve toward the right by 600 by forming a new equilibrium point C. The tax multiplier can be calculated by subtracting the change in the total output by the negative change in the taxes as follows:

$\begin{array}{c}\text{Tax multiplier}=\frac{\text{Change in output}}{\text{Change in tax}}\\ =\frac{\left(5,600-5,000\right)}{\left(800-1,000\right)}\\ =\frac{600}{-200}\\ =-3\end{array}$

Thus, the tax multiplier is -3.

(d)

To determine

The impact keeping income constant through money supply on equilibrium and the tax multiplier.

Explanation of Solution

When the Fed gives importance to keep the income of the economy constant at 5,000 after the tax deduction, the value of the rate of interest can be calculated by plugging the value of the Y into the equation as follows:

$\begin{array}{c}\text{Y}=7,600-200\text{r}\\ \text{5,000}=7,600-200\text{r}\\ \text{200r}=7,600-5,000\\ \text{r}=\frac{2,600}{200}\\ =13\end{array}$

Thus, the rate of interest in the economy will be then 13, which can be substituted in the LM curve to calculate the value of M as follows:

$\begin{array}{c}\frac{\text{M}}{2}=\text{Y}-200\text{r}\\ \frac{\text{M}}{2}=5,000-\left(200\times 13\right)\\ \frac{\text{M}}{2}=5,000-2,600\\ \text{M}=2,400\times 2\\ =4,800\end{array}$

Thus, the value of M is 4,800. Thus, M decreases to 4,800 from 7,600 which means the LM curve will shift toward the left reducing the equilibrium to point D. Therefore, it can be illustrated as follows:

In Figure 4, horizontal axis measures the income or output and vertical axis measures the interest rate. Since the output is not changing and maintaining the same level of 5,000, there will be zero tax multiplier in the economy.

(d)

To determine

The illustration of all the equilibrium points of all the changes.

Explanation of Solution

The various IS-LM equilibrium points can be illustrated on the single graph as A, B, C, and D as follows:

In Figure 5, horizontal axis measures the income or output and vertical axis measures the interest rate.