Numerical Applications of Macroeconomics

This document provides numerical applications of macroeconomics including goods market equilibrium, money market equilibrium, IS-LM model, Phillips curve, AD-AS model, and more.

Running head: NUMERICAL APPLICATIONS OF MACROECONOMICS
Numerical Applications of Macroeconomics
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1NUMERICAL APPLICATIONS OF MACROECONOMICS
Table of Contents
Answer to Question 1.................................................................................................................3
Answer to Question 1.A.............................................................................................................3
Answer to Question 1.B.............................................................................................................4
Answer to Question 1.C.............................................................................................................6
Answer to Question 1.D.............................................................................................................7
Answer to Question 2.A.............................................................................................................7
Answer to Question 2.B.............................................................................................................8
Answer to Question 2.C.............................................................................................................9
Answer to Question 2.D.............................................................................................................9
Answer to Question 2.E...........................................................................................................10
Answer to Question 3.A...........................................................................................................10
Answer to Question 3.B...........................................................................................................10
Answer to Question 3.C...........................................................................................................11
Answer to Question 3.D...........................................................................................................11
Answer to Question 3.E...........................................................................................................11
Answer to Question 3.F............................................................................................................12
Answer to Question 4.A...........................................................................................................12
Answer to Question 4.B...........................................................................................................12
Answer to Question 4.C...........................................................................................................13
Answer to Question 4.D...........................................................................................................14
Answer to Question 5.A...........................................................................................................14

2NUMERICAL APPLICATIONS OF MACROECONOMICS
Answer to Question 5.B...........................................................................................................16
Answer to Question 6...............................................................................................................17
Answer to Question 6.A...........................................................................................................18
Answer to Question 6.B...........................................................................................................19
Answer to Question 6.C...........................................................................................................19
Reference..................................................................................................................................21

3NUMERICAL APPLICATIONS OF MACROECONOMICS
Answer to Question 1
The model of a closed economy:
C=250+0.75 (Y −T )
I =600−50 r
G=325
T =100
PE=C + I +G
M d
P =1.5 Y −60 r
M s=3000
P=1
Answer to Question 1.A
Goods market equilibrium: Y =PE
⟹ Y =C + I+G
⟹ Y =250+0.75(Y −T )+600−50 r +325
⟹ Y −0.75 Y =250−0.75 (100)+600−50 r +325
⟹ 0.25 Y +50 r=1100
1. IS equation: Y +200 r=44 00
Money market equilibrium: Md
P = M s
P

4NUMERICAL APPLICATIONS OF MACROECONOMICS
⟹ 1.5Y −60 r= 3000
1
2. LM equation: Y −40 r=20 00
Solving the IS and LM curve, the equilibrium output and rate of interest is obtained.
Replacing the value of Y from LM equation in to the IS equation,
40 r + 2000+ 200 r=44 00
⟹ r=1 0
Replacing the value of r in IS model:
Y +200∗10=44 00
⟹ Y =44 00−2000
⟹ Y =24 00
The equilibrium output, Y is 2400 and the rate of interest, r is 10.
Answer to Question 1.B
Goods market equilibrium: Y =PE
⟹ Y =C + I+G
⟹ Y =250+0.75(Y −T )+600−50 r +325
⟹ Y −0.75 Y =250−0.75 (250)+600−50 r +325
⟹ 0.25 Y +50 r=987.5
3. IS equation: Y +200 r=3950
Money market equilibrium: Md
P = M s
P

5NUMERICAL APPLICATIONS OF MACROECONOMICS
⟹ 1.5Y −60 r= 3000
1
4. LM equation: Y −40 r=20 00
Solving the IS and LM curve, the equilibrium output and rate of interest is obtained.
Replacing the value of Y from LM equation in to the IS equation,
40 r + 2000+200 r=3950
⟹ r=8.125
Replacing the value of r in IS model:
Y +200∗8.125=3950
⟹ Y =3950−1625
⟹ Y =2325
The equilibrium output, Y is 2325 and the rate of interest, r is 8.125.
Increase in tax reduces the disposable income of individuals which leads to a
reduction in aggregate demand and thus the IS curve shifts inward (Fontana and Setterfield
2016). The below figure shows the shift of the IS curve.

6NUMERICAL APPLICATIONS OF MACROECONOMICS
Answer to Question 1.C
The initial output, Y is 2400. Now, the rate of interest at this level of output can be
determined from the IS curve,
2400+200 r =3950
⟹ r= 1550
200 =7.75
The rate of interest will fall below than the initial rate of interest (r=10) and also the
below the rate of interest (r=8.125) before imposition of the monetary policy.
The monetary policy is taken to raise the money supply with the constant demand and
liquidity preference, the rate of interest rate will fall and the LM curve will shift rightward.
The increase in money supply, state of liquidity preference or demand for money
remaining unchanged, will lead to the fall in rate of interest.

7NUMERICAL APPLICATIONS OF MACROECONOMICS
Answer to Question 1.D
The new output level, Y is 2400 and rate of interest, r is 7.75. Now, the demand for
money is calculated which is equal to the money supply in the equilibrium.
M d
P =1.5∗2400−60∗7.75=3135
⟹ M s
P =M s= M d
P =3135
Hence, the new money supply is 3135 and the previous money supply is 3000. Hence, the
increased money supply is 135.
Answer to Question 2.A
5. The planned expenditure can be explained by the following equation:
EP=C (Y −T )+ I P +G+NX
⟹ EP =a+ b ( Y −T )+ IP +G+ NX
⟹ EP =a+bY −bT + IP +G+ NX
Now, MPC=b=0.75 and the present tax is reduced by 100 units:
EP=a+bY −0.75(T −100)+I P +G+ NX
The planned expenditure will rise by ( 0.75∗100 )=75 units.
The disposable income will rise by 100 units.
The change in planned expenditure is shown by the shift in planned expenditure curve
from EP1 to EP2 in the below diagram with the help of Keynesian Cross model:

8NUMERICAL APPLICATIONS OF MACROECONOMICS
Output
Expenditure
EP2
450
EP1
Y1 Y2
The change in planned expenditure due to fall in tax.
Output
Rate of interest
IS2
LM
IS1
Shift in IS curve due to change in tax
Answer to Question 2.B
Decrease in tax increases the disposable income of individuals which leads to a rise in
aggregate demand and thus the IS curve shifts outward. The below figure shows the shift of
the IS curve.

9NUMERICAL APPLICATIONS OF MACROECONOMICS
Output
Rate of interest
IS2
LM
IS1
Y
1 Y2
Shift in equilibrium due to change in tax
r1
r2
E1
E2
Output
Rate of interest
IS2
LM
IS1
Y
1 Y2
Crowding out effect due to change in tax
r1
r2
E1
E2
E’
Y’
Answer to Question 2.C
The below figure shows that the equilibrium point shifts inward from E1 to E2 due to
an inward shift of IS curve because of a rise in tax.
Answer to Question 2.D

10NUMERICAL APPLICATIONS OF MACROECONOMICS
The crowding out effect can be measured by (Y’-Y2). The shift in IS curve should
have raised the output level at Y’ and the equilibrium should have been at E’. However, the
equilibrium is at E2 which shows the crowding out effect.
Answer to Question 2.E
Decrease in tax raises the disposable income of individuals. This enables the
consumers to purchase more or in simple words it raises the consumption demand. This leads
to a rise in aggregate demand.
Answer to Question 3.A
Goods market equilibrium: Y =C +I+ G
Consumption as a linear function of disposable income: C ( Y −T ) =a+b (Y −T )
Investment as a linear function of the interest rate: I ( r )=c−dr
6. The IS equation: Y =a+b (Y −T )+ c−dr +G
⟹(1−b)Y =(a−bT +c−dr +G)
⟹ Y = 1
1−b (a−bT +c−dr +G)
Answer to Question 3.B
From the equation (6), the sign of “d” explains the relation between rate of interest
and output level. The relation is negative which implies a rise in r will reduce the level of
output. Rise in interest rate sensitivity of investment will raise the effect of fall or rise in rate
of interest. For example, if the rate of interest rate goes up by 1%, then the return from the
investment will fall by d% where “d” is the interest rate sensitivity of investment. Hence the
slope of the IS curve is downward sloping and will be steeper if the value of “d” or the
interest rate sensitivity of investment rises (Friedman 2017).

11NUMERICAL APPLICATIONS OF MACROECONOMICS
Answer to Question 3.C
The bigger horizontal shift will occur due to a rise of $100 in government spending
than a cut in tax by $100. As, a tax cut will raise the disposable income which will raise the
consumption demand as well as savings. A part of the raised disposable income will be the
saving which will not affect the IS curve. However, $100 of government expenditure will
raise the output level by $100 as there is no absorption off government expenditure.
Answer to Question 3.D
Demand for real money balance: L(r , Y )=eY −fr
Let the nominal money supply is M and the price level is P then the money market
equilibrium: L(r , Y )= M
P
7. The LM curve: eY −fr= M
p
⟹ r= 1
f (eY − M
P )
Answer to Question 3.E
The equation (6) and (7) will be used to derive the expression for AD curve. The
value of r from equation (7) is replaced in equation (6),
8. Equation for AD curve: Y = 1
1−b {a−bT + c−d . 1
f (eY − M
P )+G }
⟹ Y + 1
1−b d . 1
f . eY = 1
1−b (a−bT +c +d . 1
f
M
P +G )
⟹ Y { ( 1−b ) f +ed
1−b . 1
f }= 1
1−b (a−bT +c +d . 1
f
M
P +G )

12NUMERICAL APPLICATIONS OF MACROECONOMICS
⟹ Y = f
( 1−b ) f +ed ( a−bT + c+ d . 1
f
M
P +G )
Answer to Question 3.F
The final expression of equation (8) can be presented as below:
9. Y = A
P + B
Where, A= dM
( 1−b ) f +ed >0 ;∧B= f
( 1−b ) f +ed ( a−bT +c +G ) >0.
The equation (9) shows that a rise in P reduces the value of Y which indicates that
there exist a negative relationship between Y and P that indicates the AD is downward
sloping.
Answer to Question 4.A
10. Phillips curve: π=π−1 −0.5 ( u−0.06 )
According to the model, at the natural rate of unemployment, inflation rate is equal to the
expected natural rate of unemployment, that is, π=π−1. With this condition unemployment is
0.06 that is 6% which is equal to the natural rate of employment.
Answer to Question 4.B
In the short run, the expected inflation rate is nothing but the inflation rate of the
previous period. Hence, the Phillips curve presents the short-run relationship between
inflation and unemployment.
11. Phillips curve in short-run: π−π−1 =−0.5(u−0.06)
In the long run, expected inflation is equal to the actual inflation, that is, π=π−1.
12. The long-run Phillips curve: −0.5 ( u−0.06 )=0;⟹ u=6 %

13NUMERICAL APPLICATIONS OF MACROECONOMICS
0.5
Unemployment
Inflation rate
LRPC
SRPC
Short-run and long-run Phillips
curve
Π-
1
0.06
Answer to Question 4.C
Reduction of 5% of inflation means: π−1−π =5 %=0.05
The Philips curve: π−π−1 =−0.5(u−0.06)
⟹−0.05=−0.5 (u−0.06)
⟹ 0.5 u=0.03+ 0.05
⟹ u=0.16
Hence, u−un=0.16−0.6=0.1=10 % of higher cyclical unemployment than natural
unemployment is needed.
Okun’s law: 1 percentage point change in unemployment changes 2 percentage points of
GDP. Hence, 10 percentage of rise in unemployment reduces output by 20 percentage points.
Sacrificeratio= Change∈GDP
Change∈inflation =20
5 =4

14NUMERICAL APPLICATIONS OF MACROECONOMICS
Output
Rate of interest
IS
LM1= LM3
Y
1
Y2
IS-LM model in the Long-run
r1
r2
E1
E2
LM2
Answer to Question 4.D
To bring the inflation rate down from10 percent to 5 percent, the unemployment
should be set to be very high for a short period of time and the other way is to set the cyclical
unemployment at very small amount of spread out over a long period of time. These two
scenario can reduce the inflation rate at different speed (Wun 2016).
Answer to Question 5.A
The Central Bank raises the money supply, this will lead the economy to the
expansionary phase. In the IS-LM model, a rise of money supply will shift the LM curve to
the outward which in will reduce interest rate and raise the level of output at equilibrium E2.
In the long run, this will increase price level for which real-money balances will fall. As a
result, an inward shift of LM curve to the initial position occurs. There will be no change in
the real interest rate and level of output in long-run.
A rise of the money supply shifts the aggregate demand to the outward direction in
AD-AS model. This increases level of output and price level at E2 for which, expected
inflation rises and shifts the SRAS curve in the upward direction in the long run. The final

15NUMERICAL APPLICATIONS OF MACROECONOMICS
Output
Price Level
AD2
SRAS2
Y
1
Y2
AD-AS model in the Long-run
P2
P1 E1
E2
SRAS1
AD1
E3
LRAS
Unemployment
Inflation rate
SRPC2
U
1
UC
Phillips Curve
Π3
Π1 E1
E2
SRPC1
E3
LRPC
Π2
equilibrium is at E3 where the output is at initial level and the price level is at higher position
than the initial level (Weintraub 2018).
A rise in money supply raises output higher than the initial level. This results in a fall
in unemployment rate for which the short-run Phillips curve reaches at equilibrium E2. At
this point, it raises the inflation and reduces the unemployment. Expected inflation rises and

16NUMERICAL APPLICATIONS OF MACROECONOMICS
Output
Rate of interest
LM
IS
IS-LM Model in Long-
run
r
Y
shifts the Phillips curve upward in the long run. Finally, the economy reaches at E3 where
unemployment rate is unchanged and inflation is greater than before.
Answer to Question 5.B
The central bank raises money supply for which people expected that inflation will
rise. This only changes the inflation rate and price level. For the IS-LM model, rise in the
money supply raises the price level in the same rate. This means real balances has not been
changed. So, no change in equilibrium is observed.
A rise in the money supply shifts the aggregate demand curve in the upward direction
and the rise in expected inflation shifts the SRAS curve upward. The output level remains
same and the price level will rise as shown in the below figure.

17NUMERICAL APPLICATIONS OF MACROECONOMICS
Output
Price Level
AD2
SRAS2
Y
1
AD-AS model in the Long-run
P2
P1 E1
SRAS1
AD1
E2
LRAS
Unemployment
Inflation rate
SRPC2
UC
Phillips Curve
Π2
Π1 E1
SRPC1
E2
LRPC
A rise in expected inflation leads to shift short-run Phillips curve in the upward
direction and does not change the unemployment rate. The rise in money supply and the
public expectation of higher inflation does not move the economy to the expansionary cycle.
Answer to Question 6
13. The equation for reserve deposit ratio is
rr =0.4−2 r

18NUMERICAL APPLICATIONS OF MACROECONOMICS
Currency deposit ratio is given as cr = 0.4. The price level is fixed at P = 1. The monetary
base is B = 60. The equation for real money demand is
14. L ( Y , r ) =0.5 Y −10 i
Answer to Question 6.A
It is assumed that the reserves would be taken at $100 million
If r = i = 0.05, the obtained reserve deposit ratio is
rr =0.4− ( 2 ×0.05 )
¿ 0.4−0.1
¿ 0.3
Money Multiplier= 1
Reserve Requirement
¿ 1
0.3
¿ 3.33
Money Supply=Money multiplier × Monetary Base
¿ 3.33 ×60
¿ 200
Money market clears at the point where real money supply equals real money demand. Given
price level is fixed at P = 1, real money supply can be determined as
M S = M
P
¿ 200
1

19NUMERICAL APPLICATIONS OF MACROECONOMICS
¿ 200
M S =M D
⟹ 200=0.5 Y −10i
⟹ 200=0.5 Y − ( 10× 0.05 )
⟹ 200=0.5 Y −0.5
⟹ 0.5 Y =200.5
⟹ Y =401
The money market clears corresponding to the real output level of 401.
Answer to Question 6.B
15. The equation of LM curve is obtained as
Money supply=Money demand
⟹ 200=0.5 Y −10 r
⟹ 10 r=0.5 Y −200
⟹ r=0.05Y −20
Answer to Question 6.C
If reserve ratio does not depend on the real interest rate, then there is a fixed reserve deposit
given by
rr=0.4
Money Multiplier= 1
Reserve Requirement

20NUMERICAL APPLICATIONS OF MACROECONOMICS
¿ 1
0.4
¿ 2.5
Money Supply=Money multiplier × Monetary Base
¿ 2.5 ×60
¿ 150
16. The equation of LM curve then will be
Money supply=Money demand
⟹ 150=0.5 Y −10 r
⟹ 10 r=0.5 Y −150
⟹ r=0.05Y −15

21NUMERICAL APPLICATIONS OF MACROECONOMICS
Reference
Fontana, G. and Setterfield, M. eds., 2016. Macroeconomic theory and macroeconomic
pedagogy. Springer.
Friedman, M., 2017. Quantity theory of money (pp. 1-31). Palgrave Macmillan UK.
Mankiw, N.G., 2016. Brief principles of macroeconomics. Cengage Learning.
Michl, T.R., 2015. Macroeconomic Theory: A Short Course: A Short Course. Routledge.
Weintraub, S., 2018. Classical Keynesianism, monetary theory, and the price level. Pickle
Partners Publishing.
Wun, C., 2016. Analysis On Money Supply, Inflation Rate and Economic Growth Through
Empirical Study.