Macroeconomics
Macroeconomics
10th Edition
ISBN: 9780134896441
Author: ABEL, Andrew B., BERNANKE, Ben, CROUSHORE, Dean Darrell
Publisher: PEARSON
Question
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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.

a)

Expert Solution
Check Mark

Explanation of Solution

An economy with the following equations is presented:

Desired consumption Cd=250+0.5Y(Y-T)-500r

Desired investment Id=250-500r

Real money demand L=0.5Y-500i

Full employment output Y¯=1000

Expected inflation πe=0

Using the equation Y=Cd+Id+G , to obtain the IS equation

Substituting the desired consumption and investment and solving for Y

  Y=Cd+Id+GY=(250+0.5( Y-T)-500r)+(250-500r)+GY=500+0.5Y-0.5T+1000r+G0.5Y=500-0.5T-1000r+GY=1000-T-2000r+2G

Setting the real money demand to MP and let i=r+πe where πe=0

  MP=LMP=0.5Y-500iMP=0.5Y-500(r+πe)MP=0.5Y-500r

The aggregate demand curve is Y=400+10200P

The equilibrium interest rate is r=0.1

The equilibrium value of the price level is P=17

The equilibrium value of consumption is C=600

The equilibrium value of investment is I=200

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

  Y=1000-T-2000r+2GY=1000-200-2000r+2(200)Y=1200-2000r

Equation is Y=1200-2000r

Substituting the value of M into the Lm equation to obtain the equation for LM curve,MP=0.5Y-500r7650P=0.5Y-500r

Thus, the equation for LM curve is 7650P=0.5Y-500r

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

  4( 7650P)=4(0.5Y-500r)30600P=2Y-2000r2000r=2Y-30600P

Solving the IS equation for 2000r,Y=1200-2000r2000r=1200-Y

Equating the IS and LM equations, following is derived:

  1200-Y=2Y-30600P1200+30600P=3Y12003+306003P=3Y3400+10200P=Y

Thus, the aggregate demand curve is 400+10200P .

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

  Y=1200-2000r1000=1200-2000r2000r=200r=0.1

Therefore, the equilibrium interest rate is 0.1.

Using the aggregate demand curve to obtain the price level,

Substitute the full employment output value.

  1000=400+10200P600=10200PP=10200600P=17

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

  Cd=250+(Y-T)-500rCd=250+0.5(1000-200)-500(0.1)Cd=250+400-50C=600

The equilibrium consumption is 600

Substituting the value of real interest rate into the desired investment equation to obtain the equilibrium investment valueId=250-500rId=250-500(0.1)Id=250-50Id=200

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.

b)

Expert Solution
Check Mark

Explanation of Solution

An economy with the following equations is presented:

Desired consumption Cd=250+0.5Y(Y-T)-500r

Desired investment Id=250-500r

Real money demand L=0.5Y-500i

Full employment output Y¯=1000

Expected inflation πe=0

Using the equation Y=Cd+Id+G , to obtain the IS equation

Substituting the desired consumption and investment and solving for Y

  Y=Cd+Id+GY=(250+0.5( Y-T)-500r)+(250-500r)+GY=500+0.5Y-0.5T+1000r+G0.5Y=500-0.5T-1000r+GY=1000-T-2000r+2G

Setting the real money demand to MP and let i=r+πe where πe=0

  MP=LMP=0.5Y-500iMP=0.5Y-500(r+πe)MP=0.5Y-500r

The new aggregate demand curve is Y=400+12000P

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:

  9000P=0.5Y-500r

Multiplying the LM equation by 4 and solving it for 2000r4( 9000P)=4(0.5Y-500r)36000P=2Y-2000r2000r=2Y-36000P

Solving the IS equation for 2000rY=1200-2000r2000r=1200-Y

Equating the IS and LM equations, following is derived:

  1200-Y=2Y-36000P1200+36000P=3Y12003+360003P=3Y3400+12000P=Y

Thus, the new aggregate demand curve is Y=400+12000P

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.

  1000=400+12000P600=12000PP=12000600P=20

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.

c)

Expert Solution
Check Mark

Explanation of Solution

An economy with the following equations is presented:

Desired consumption Cd=250+0.5Y(Y-T)-500r

Desired investment Id=250-500r

Real money demand L=0.5Y-500i

Full employment output Y¯=1000

Expected inflation πe=0

Using the equation Y=Cd+Id+G , to obtain the IS equation

Substituting the desired consumption and investment and solving for Y

  Y=Cd+Id+GY=(250+0.5( Y-T)-500r)+(250-500r)+GY=500+0.5Y-0.5T+1000r+G0.5Y=500-0.5T-1000r+GY=1000-T-2000r+2G

Setting the real money demand to MP and let i=r+πe where πe=0

  MP=LMP=0.5Y-500iMP=0.5Y-500(r+πe)MP=0.5Y-500r

The aggregate demand equation is Y=13003+10200P

The equilibrium value of real interest rate is

The equilibrium value of price level is P=18

The equilibrium value of consumption is C=525

The equilibrium value of investment is I= 175

The values for taxes have increased to T=300 which is equal to the government purchases G=300

Recalculating the IS equation

  Y=1000-T-2000r+2GY=1000-300-2000r+2(300)Y=1300-2000r

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.

  4( 7650P)=4(0.5Y-500r)30600P=2Y-2000r2000r=2Y-30600P

Solving the IS equation for 200rY=1300-2000r2000r=1300-Y

Equating the IS and LM equations1300-Y=2Y-30600P1300+30600P=3Y13003+306003P=3Y313003+10200P=Y

The aggregate demand equation is Y=13003+10200P

Substituting the given full employment output into the IS equation to obtain the real interest rateY=1300-2000r1000=1300-2000r2000r=300r=3002000

The equilibrium value of real interest rate is therefore 0.15.

Using the aggregate demand curve for obtaining the value of price level,1000=13003+10200P17003=10200PP=102003.31700P=18

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,

  Cd=250+0.5(Y-T)-500rCd=250+0.5(1000-300)-500(0.15)Cd=250+350-75Cd=525

The equilibrium value of consumption is therefore 525

Using the desired investment equation to find the equilibrium value of interestId=250-500r=250-500(0.15)=250-75=175

The equilibrium value of investment is 175.

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