Chapter 10, Problem 3NP

a)

To determine

The equilibrium values of output, real interest rate, price level, consumption and investment and the equation of IS curve, LM curve and the aggregate demand curve.

Explanation of Solution

An economy with the following equations is presented:

Desired consumption ${\text{C}}^{\text{d}}\text{=250+0}\text{.5Y}\left(\text{Y-T}\right)\text{-500r}$

Desired investment ${\text{I}}^{\text{d}}\text{=250-500r}$

Real money demand $\text{L=0}\text{.5Y-500i}$

Full employment output $\overline{\text{Y}}\text{=1000}$

Expected inflation ${\text{π}}^{\text{e}}\text{=0}$

Using the equation ${\text{Y=C}}^{\text{d}}{\text{+I}}^{\text{d}}\text{+G}$ , to obtain the IS equation

Substituting the desired consumption and investment and solving for Y

  $\begin{array}{l}{\text{Y=C}}^{\text{d}}{\text{+I}}^{\text{d}}\text{+G}\\ \text{Y=}\left(\text{250+0}\text{.5}\left(\text{Y-T}\right)\text{-500r}\right)\text{+}\left(\text{250-500r}\right)\text{+G}\\ \text{Y=500+0}\text{.5Y-0}\text{.5T+1000r+G}\\ \text{0}\text{.5Y=500-0}\text{.5T-1000r+G}\\ \text{Y=1000-T-2000r+2G}\end{array}$

Setting the real money demand to $\frac{\text{M}}{\text{P}}$ and let ${\text{i=r+π}}^{\text{e}}$ where ${\text{π}}^{\text{e}}\text{=0}$

  $\begin{array}{l}\frac{\text{M}}{\text{P}}\text{=L}\\ \frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500i}\\ \frac{\text{M}}{\text{P}}\text{=0}{\text{.5Y-500(r+π}}^{\text{e}}\text{)}\\ \frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500r}\end{array}$

The aggregate demand curve is $\text{Y=400+}\frac{\text{10200}}{\text{P}}$

The equilibrium interest rate is $\text{r=0}\text{.1}$

The equilibrium value of the price level is $\text{P=17}$

The equilibrium value of consumption is $\text{C=600}$

The equilibrium value of investment is $\text{I=200}$

Substituting the given values of T and G into the general IS equation to obtain the IS curve equation

  $\begin{array}{l}\text{Y=1000-T-2000r+2G}\\ \text{Y=1000-200-2000r+2}\left(\text{200}\right)\\ \text{Y=1200-2000r}\end{array}$

Equation is $\text{Y=1200-2000r}$

Substituting the value of M into the Lm equation to obtain the equation for LM curve,$\begin{array}{l}\frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500r}\\ \frac{\text{7650}}{\text{P}}\text{=0}\text{.5Y-500r}\end{array}$

Thus, the equation for LM curve is $\frac{\text{7650}}{\text{P}}\text{=0}\text{.5Y-500r}$

Using the IS and LM equations to find out the aggregate demand equation

  $\begin{array}{l}\text{4}\left(\frac{\text{7650}}{\text{P}}\right)\text{=4}\left(\text{0}\text{.5Y-500r}\right)\\ \frac{\text{30600}}{\text{P}}\text{=2Y-2000r}\\ \text{2000r=2Y-}\frac{\text{30600}}{\text{P}}\end{array}$

Solving the IS equation for 2000r,$\begin{array}{l}\text{Y=1200-2000r}\\ \text{2000r=1200-Y}\end{array}$

Equating the IS and LM equations, following is derived:

  $\begin{array}{l}\text{1200-Y=2Y-}\frac{\text{30600}}{\text{P}}\\ \text{1200+}\frac{\text{30600}}{\text{P}}\text{=3Y}\\ \frac{\text{1200}}{\text{3}}\text{+}\frac{\text{30600}}{\text{3P}}\text{=}\frac{\text{3Y}}{\text{3}}\\ \text{400+}\frac{\text{10200}}{\text{P}}\text{=Y}\end{array}$

Thus, the aggregate demand curve is $\text{400+}\frac{\text{10200}}{\text{P}}$ .

Substituting the full employment output into the IS equation to obtain the real interest rate.

  $\begin{array}{l}\text{Y=1200-2000r}\\ \text{1000=1200-2000r}\\ \text{2000r=200}\\ \text{r=0}\text{.1}\end{array}$

Therefore, the equilibrium interest rate is 0.1.

Using the aggregate demand curve to obtain the price level,

Substitute the full employment output value.

  $\begin{array}{l}\text{1000=400+}\frac{\text{10200}}{\text{P}}\\ \text{600=}\frac{\text{10200}}{\text{P}}\\ \text{P=}\frac{\text{10200}}{\text{600}}\\ \text{P=17}\end{array}$

The equilibrium price level is 17

To obtain the value of consumption, substitute the given values of output and tax and the calculated value of interest rate into the desired consumption equation

  $\begin{array}{l}{\text{C}}^{\text{d}}\text{=250+}\left(\text{Y-T}\right)\text{-500r}\\ {\text{C}}^{\text{d}}\text{=250+0}\text{.5(1000-200)-500}\left(\text{0}\text{.1}\right)\\ {\text{C}}^{\text{d}}\text{=250+400-50}\\ {\text{C}}^{}\text{=600}\end{array}$

The equilibrium consumption is 600

Substituting the value of real interest rate into the desired investment equation to obtain the equilibrium investment value$\begin{array}{l}{\text{I}}^{\text{d}}\text{=250-500r}\\ {\text{I}}^{\text{d}}\text{=250-500}\left(\text{0}\text{.1}\right)\\ {\text{I}}^{\text{d}}\text{=250-50}\\ {\text{I}}^{\text{d}}\text{=200}\end{array}$

Thus, the equilibrium investment is 200

b)

To determine

The equilibrium values of output, real interest rate, price level, consumption and investment and the equation of the aggregate demand curve.

Explanation of Solution

An economy with the following equations is presented:

Desired consumption ${\text{C}}^{\text{d}}\text{=250+0}\text{.5Y}\left(\text{Y-T}\right)\text{-500r}$

Desired investment ${\text{I}}^{\text{d}}\text{=250-500r}$

Real money demand $\text{L=0}\text{.5Y-500i}$

Full employment output $\overline{\text{Y}}\text{=1000}$

Expected inflation ${\text{π}}^{\text{e}}\text{=0}$

Using the equation ${\text{Y=C}}^{\text{d}}{\text{+I}}^{\text{d}}\text{+G}$ , to obtain the IS equation

Substituting the desired consumption and investment and solving for Y

  $\begin{array}{l}{\text{Y=C}}^{\text{d}}{\text{+I}}^{\text{d}}\text{+G}\\ \text{Y=}\left(\text{250+0}\text{.5}\left(\text{Y-T}\right)\text{-500r}\right)\text{+}\left(\text{250-500r}\right)\text{+G}\\ \text{Y=500+0}\text{.5Y-0}\text{.5T+1000r+G}\\ \text{0}\text{.5Y=500-0}\text{.5T-1000r+G}\\ \text{Y=1000-T-2000r+2G}\end{array}$

Setting the real money demand to $\frac{\text{M}}{\text{P}}$ and let ${\text{i=r+π}}^{\text{e}}$ where ${\text{π}}^{\text{e}}\text{=0}$

  $\begin{array}{l}\frac{\text{M}}{\text{P}}\text{=L}\\ \frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500i}\\ \frac{\text{M}}{\text{P}}\text{=0}{\text{.5Y-500(r+π}}^{\text{e}}\text{)}\\ \frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500r}\end{array}$

The new aggregate demand curve is $\text{Y=400+}\frac{\text{12000}}{\text{P}}$

The equilibrium price level is P=20

Now, the money supply has increased to M=9000

Use the IS and LM equations to find the new aggregate demand equation

By substituting, following is derived:

  $\frac{\text{9000}}{\text{P}}\text{=0}\text{.5Y-500r}$

Multiplying the LM equation by 4 and solving it for 2000r$\begin{array}{l}\text{4}\left(\frac{\text{9000}}{\text{P}}\right)\text{=4}\left(\text{0}\text{.5Y-500r}\right)\\ \frac{\text{36000}}{\text{P}}\text{=2Y-2000r}\\ \text{2000r=2Y-}\frac{\text{36000}}{\text{P}}\end{array}$

Solving the IS equation for 2000r$\begin{array}{l}\text{Y=1200-2000r}\\ \text{2000r=1200-Y}\end{array}$

Equating the IS and LM equations, following is derived:

  $\begin{array}{l}\text{1200-Y=2Y-}\frac{\text{36000}}{\text{P}}\\ \text{1200+}\frac{\text{36000}}{\text{P}}\text{=3Y}\\ \frac{\text{1200}}{\text{3}}\text{+}\frac{\text{36000}}{\text{3P}}\text{=}\frac{\text{3Y}}{\text{3}}\\ \text{400+}\frac{\text{12000}}{\text{P}}\text{=Y}\end{array}$

Thus, the new aggregate demand curve is $\text{Y=400+}\frac{\text{12000}}{\text{P}}$

The change in money supply does not affect the equilibrium values of real interest rate consumption or investment

Therefore, using the aggregate demand curve to obtain the value for price level.

  $\begin{array}{l}\text{1000=400+}\frac{\text{12000}}{\text{P}}\\ \text{600=}\frac{\text{12000}}{\text{P}}\\ \text{P=}\frac{\text{12000}}{\text{600}}\\ \text{P=20}\end{array}$

The equilibrium price level is 20.

c)

To determine

The equilibrium values of output, real interest rate, price level, consumption and investment and the equation of the aggregate demand curve.

Explanation of Solution

An economy with the following equations is presented:

Desired consumption ${\text{C}}^{\text{d}}\text{=250+0}\text{.5Y}\left(\text{Y-T}\right)\text{-500r}$

Desired investment ${\text{I}}^{\text{d}}\text{=250-500r}$

Real money demand $\text{L=0}\text{.5Y-500i}$

Full employment output $\overline{\text{Y}}\text{=1000}$

Expected inflation ${\text{π}}^{\text{e}}\text{=0}$

Using the equation ${\text{Y=C}}^{\text{d}}{\text{+I}}^{\text{d}}\text{+G}$ , to obtain the IS equation

Substituting the desired consumption and investment and solving for Y

  $\begin{array}{l}{\text{Y=C}}^{\text{d}}{\text{+I}}^{\text{d}}\text{+G}\\ \text{Y=}\left(\text{250+0}\text{.5}\left(\text{Y-T}\right)\text{-500r}\right)\text{+}\left(\text{250-500r}\right)\text{+G}\\ \text{Y=500+0}\text{.5Y-0}\text{.5T+1000r+G}\\ \text{0}\text{.5Y=500-0}\text{.5T-1000r+G}\\ \text{Y=1000-T-2000r+2G}\end{array}$

Setting the real money demand to $\frac{\text{M}}{\text{P}}$ and let ${\text{i=r+π}}^{\text{e}}$ where ${\pi }^e=0$

  $\begin{array}{l}\frac{\text{M}}{\text{P}}\text{=L}\\ \frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500i}\\ \frac{\text{M}}{\text{P}}\text{=0}{\text{.5Y-500(r+π}}^{\text{e}}\text{)}\\ \frac{\text{M}}{\text{P}}\text{=0}\text{.5Y-500r}\end{array}$

The aggregate demand equation is $\text{Y=}\frac{\text{1300}}{\text{3}}\text{+}\frac{\text{10200}}{\text{P}}$

The equilibrium value of real interest rate is

The equilibrium value of price level is $\text{P=18}$

The equilibrium value of consumption is $\text{C=525}$

The equilibrium value of investment is $\text{I= 175}$

The values for taxes have increased to $\text{T=300}$ which is equal to the government purchases $\text{G=300}$

Recalculating the IS equation

  $\begin{array}{l}\text{Y=1000-T-2000r+2G}\\ \text{Y=1000-300-2000r+2(300)}\\ \text{Y=1300-2000r}\end{array}$

The equation for LM curve remains the same.

Using the M and IS equations to find the aggregate demand equation,

Multiplying the Lm equation by 4 sand solving for 2000r.

  $\begin{array}{l}\text{4}\left(\frac{\text{7650}}{\text{P}}\right)\text{=4}\left(\text{0}\text{.5Y-500r}\right)\\ \frac{\text{30600}}{\text{P}}\text{=2Y-2000r}\\ \text{2000r=2Y-}\frac{\text{30600}}{\text{P}}\end{array}$

Solving the IS equation for 200r$\begin{array}{l}\text{Y=1300-2000r}\\ \text{2000r=1300-Y}\end{array}$

Equating the IS and LM equations$\begin{array}{l}\text{1300-Y=2Y-}\frac{\text{30600}}{\text{P}}\\ \text{1300+}\frac{\text{30600}}{\text{P}}\text{=3Y}\\ \frac{\text{1300}}{\text{3}}\text{+}\frac{\text{30600}}{\text{3P}}\text{=}\frac{\text{3Y}}{\text{3}}\\ \frac{\text{1300}}{\text{3}}\text{+}\frac{\text{10200}}{\text{P}}\text{=Y}\end{array}$

The aggregate demand equation is $\text{Y=}\frac{\text{1300}}{\text{3}}\text{+}\frac{\text{10200}}{\text{P}}$

Substituting the given full employment output into the IS equation to obtain the real interest rate$\begin{array}{l}\text{Y=1300-2000r}\\ \text{1000=1300-2000r}\\ \text{2000r=300}\\ \text{r=}\frac{\text{300}}{\text{2000}}\end{array}$

The equilibrium value of real interest rate is therefore 0.15.

Using the aggregate demand curve for obtaining the value of price level,$\begin{array}{l}\text{1000=}\frac{\text{1300}}{\text{3}}\text{+}\frac{\text{10200}}{\text{P}}\\ \frac{\text{1700}}{\text{3}}\text{=}\frac{\text{10200}}{\text{P}}\\ \text{P=}\frac{\text{10200}}{\text{3}}\text{.}\frac{\text{3}}{\text{1700}}\\ \text{P=18}\end{array}$

The equilibrium value of price level is 18.

Substituting the values of output and tax and calculated value of real interest rate into the desired consumption to obtain the value of consumption,

  $\begin{array}{l}{\text{C}}^{\text{d}}\text{=250+0}\text{.5(Y-T)-500r}\\ {\text{C}}^{\text{d}}\text{=250+0}\text{.5}\left(\text{1000-300}\right)\text{-500}\left(\text{0}\text{.15}\right)\\ {\text{C}}^{\text{d}}\text{=250+350-75}\\ {\text{C}}^{\text{d}}\text{=525}\end{array}$

The equilibrium value of consumption is therefore 525

Using the desired investment equation to find the equilibrium value of interest$\begin{array}{l}{\text{I}}^{\text{d}}\text{=250-500r}\\ \text{=250-500}\left(\text{0}\text{.15}\right)\\ \text{=250-75}\\ \text{=175}\end{array}$

The equilibrium value of investment is 175.