# IS-LM Model: Solved Numerical Problems

1- Given the following functions;
C=100+0.8Y,  I=120-5i,    M_s=120    and    M_d=0.2Y-5i
Find;
a- IS function,
b- LM function,
c- Equilibrium income and interest rate,

Solution;

#### a-Derivation of IS function,

We know that Y=C+I......(i)
Substituting 100+0.8Y for C and 120-5i for I in the equation (i) we get;
Y=100+0.8Y+120-5i
Y-0.8Y=220-5i
Y(1-0.8)=220-5i
Y=\frac{1}{0.2}\(220-5i)
Y=1100-25i

#### b-Derivation of LM function,

We know that M_d=M_s.......(ii)
Substituting 0.2Y for M_d and 120 for M_s in equation (ii) we get;
0.2Y-5i=120
0.2Y=120+5i
Y=\frac{1}{0.2}\(120+5i)
Y=600+25i

#### c-Finding interest rate and income level,

As we know that the simultaneous equilibrium attains where IS equals LM. Hence, the equilibrium condition is expressed as follows.
IS=LM.......(iii)
Plotting IS and LM functions in equation (iii) we get;

1100-25i=600+25i
1100-600=25i+25i
50i=500
i=10
Or,\i=10%

Substituting 10 for i in the LM function, we get;
Y=600+25×10
Y=600+250
Y=850
\therefore\i=10
\therefore\Y=850

2- Given the product and money market variables as;
C=100+0.8(Y-T)
i=100-100i
G=200
T=0.25Y
M_t=0.5Y
M_sp=100-37.5i and
M_s=200
Find;
a-IS and LM functions,
b- Equilibrium level of incom and interest,
c-Disposable income,

Solution;

#### a- IS and LM functions,

We know that the equilibrium identity of the product market is as follows.
Y=C+I+G.........(i)
Plotting the values in the equation (i) we get,
Y=100+0.8(Y-T)+100-100i+200
Y=400+0.8(Y-0.25Y)-100i
Y=400+0.8Y-0.2Y-100i
Y-0.6Y=400-100i
Y(1-0.6)=400-100i
Y=\frac{1}{0.4}\(400-100i)
Y=1000-25i     IS function,

The equilibrium identity of the money market is as follows.
M_d=M_s........(ii)
Plotting the values in the equation (ii), we get;
0.5Y+100-37.5i=200        \because\M_d=M_t+M_{sp}
0.5Y=200-100+37.5i
0.5Y=100+37.5i
Y=\frac{1}{0.5}\(100+37.5i)
Y=200+75i        LM function,

#### b- For equilibrium IS=LM........(iii)

Plotting the IS and LM functions in the equation (iii), we get;
1000-25i=200+75i
1000-200=75i+25i
800=100i
\frac{800}{100}=i
\therefore\i=8

Substitute 8 for i in LM function, we get;
Y=200+75×8
Y=200+600
\therefore\Y=800

#### c- Disposable income

We know that the identity equation for disposable income is expressed as follows.
Y_d=Y-T..........(iv)
Plotting the values in the equation (iv), we get;
Y_d=800-0.25Y
Y_d=800-0.25(800)
Y_d=800-200
\therefore\Y_d=600

3- Given the product and money market variables as;
Consumption function    C=40+0.75Y_d
Investment function        I=140-10i
Govt. expenditure            G=100
Lump-sum tax                  T=80
Money demand                M_d=0.2Y-5i
Money supply                   M_s=85
Find out:
a- Equilibrium income and interest rate,
b- What happens to the equilibrium income and interest rate when the federal government increases its expenditure by 65?

Solution:

#### a- Equilibrium income and interest rate,

Calculating IS function,
The equilibrium equation is Y=C+I+G......(i)
Plotting the values in the equation (i) we get,
Y=40+0.75Y_d+140-10i+100
Or,\Y=280+0.75(Y-T)-10i            \because\Y_d=Y-T
Plotting the value of T we get,
Y=280+0.75(Y-80)-10i
Or,\Y=280+0.75Y-60-10i
Or,\Y-0.75Y=220-10i
Or,\Y(1-0.75)=220-10i
Or,\Y=\frac{1}{0.25}\(220-10i)
\thereforeY=880-40i..........(ii)

Calculating LM function,
Money market equilibrium is expressed as,
M_d=M_s..........(iii)
Plotting the given values in the equation (iii) we get,
0.2Y-5i=85
Or,\0.2Y=85+5i
\therefore\Y=425+25i.........(iv)

Calculating equilibrium income and interest rate,
The economy is in equilibrium at which IS equals LM, hence, it is expressed as follows,
IS=LM...........(v)
plotting the IS and LM functions in the equation (v) we get,
880-40i=425+25i
Or,\880-425=25i+40i
Or,\455=65i
i=\frac{455}{65}
\therefore\i=7

We can substitute 7 for i either in the IS or LM function to calculate income level. Let us substitute in the LM function.
Y=425+25×7
Or,\Y=425+175
\thereforeY=600

#### b- When the federal government increases its expenditure by 65, the government total expenditure will alter as follows.

G=G+△G
Or,\G=100+65
G=165
Now the IS function will also change as a result of a change in the government expenditure.

Calculating the newIS function,
The equilibrium equation is Y=C+I+G......(vi)
Plotting the values in the equation (vi) we get,
Y=40+0.75Y_d+140-10i+165
Or,\Y=345+0.75(Y-T)-10i            \because\Y_d=Y-T
Plotting the value of T we get,
Y=345+0.75(Y-80)-10i
Or,\Y=345+0.75Y-60-10i
Or,\Y-0.75Y=285-10i
Or,\Y(1-0.75)=285-10i
Or,\Y=\frac{1}{0.25}\(285-10i)
\thereforeY=1140-40i..........(vii)

Calculating new new equilibrium income and interest rate,
IS=LM........(viii)
plotting new IS function and with existing LM function in equation (viii) we get,
1140-40i=425+25i
Or,\1140-425=25i+40i
Or,\715=65i
\therefore\i=11
Plotting the value of  i in the IS function, we get,
Y=1140-40×11
Y=1140-440
\thereforeY=700

When the federal government increases its expenditure by 65, the equilibrium level of income will increase by 100, and the rate of interest will also rise by 4.

4- An economy shows the following functions,
C=200+0.75(Y-T)
T=80+0.2Y
I=200-2000i
G=$300 million M_t=0.5Y M_{sp}=200-250i M_s=$400  million

a- Compute equilibrium output and rate of interest.
b- It is realized that the economy is trapped in an economic recession. In order to remove it, the central bank has followed an expansionary monetary policy and increased the money supply by $200 million. At the same time, the federal government has also implemented expansionary fiscal policy and increased its expenditure by$200 million. What will be the effect on the equilibrium level of income and rate of interest in the economy?

Solution;

#### a- Calculating the equilibrium output and interest,

IS function;
We know that the equilibrium identity in a three-sector economy is as follows.
Y=C+I+G.......(i)
Plotting the given functions in the equation (i) we get;
Y=200+0.75(Y-T)+200-2000i+300
Or,\200+0.75[Y-(80+0.2Y)]+200-2000i+300
Or,\Y=200+0.75Y-60-0.15Y+200-2000i+300
Or,\Y-0.60Y=640-2000i
Or,\Y(1-0.60)=640-2000i
Or,\Y=\frac{1}{0.40}\(640-2000i)
Or,\Y=1600-5000i

LM function;
Money market equilibrium is expressed as follows;
M_d=M_s.........(ii)
Plotting the values in the equation (ii) we get;
\0.5Y+200-250i=400        \because\M_d=M_t+M_{sp}
Or,\0.5Y=400-200+250i
Or,\0.5Y=200+250i
Or,\Y=\frac{1}{0.5}\(200+250i)
Or,\Y=400+500i

For simultaneous equilibrium,
IS=LM.......(iii)
plotting the IS and LM functions,
1600-5000i=400+500i
Or,\1600-400=500i+5000i
Or,\1200=5500i
Or,\i=\frac{1200}{5500}
\therefore\0.218

Plotting the value in the LM function,
Y=400+500×0.218
Or,\Y=400+109
\therefore\Y=509 million

#### b- When the central bank increases the money supply by $200 million and the government also increases its expenditure by$200 million at the same time, the money supply function and the government expenditure both get changed as follows.

M_s=M_s+△M_s
Or,\M_s=400+200
\therefore\M_s=600

G=G+△G
Or,\G=300+200
\therefore\G=500

New IS function is as follows,
Y=200+0.75(Y-T)+200-2000i+500
Or,\Y=200+0.75[Y-(80+0.2Y)]+200-2000i+500
Or,\Y=200+0.75Y-60-0.15Y+200-2000i+500
Or,\Y-0.60Y=840-2000i
Or,\Y(1-0.60)=840-2000i
Or,\Y=\frac{1}{0.40}\(840-2000i)
Or,\Y=2100-5000i

New LM function is as follows,
0.5Y+200-250i=600        \because\M_d=M_t+M_{sp}
Or,\0.5Y=600-200+250i
Or,\0.5Y=400+250i
Or,\Y=\frac{1}{0.5}\(400+250i)
Or,\Y=800+500i

The new rate of interest and income level is as follows,
2100-5000i=800+500i
Or,\2100-800=5000i+500i
Or,\1300=5500i
Or,i=\frac{1300}{5500}
therefore\i=0.236

Plotting the value of i in the LM function, we get;
Y=800+500×0.236
Or,\Y=800+118
\therefore\Y=918
Hence, due to the increase in money supply and government expenditure, there is an increase in the level of equilibrium income by 409, and the rate of interest increases by 0.018.

5- Given the following functions of an economy as follows,
C=50+0.9(Y-T)
T=100
I=150-5i
G=100
M_d=0.2Y-10i
M_s=100
X=20
M=10+0.1Y
Find:
a- IS and LM functions,
b- Equilibrium income and interest rate,

Solution:
a- Calculating IS and LM functions,
We know that the product market equilibrium identity of a four-sector economy as follows,
Y=C+I+G+(X-M)...............(i)
Plotting the given values in the equation (i) we get,
Y=50+0.9(Y-T)+150-5i+100+[20-(10+0.1Y)]
Or,\Y=50+0.9Y-0.9(100)+150-5i+100+20-10-0.1Y
Or,\Y=50+0.8Y-90+150-5i+100+20-10
Or,\Y-0.8Y=220-5i
Or,\Y(1-0.8)=220-5i
Or,\Y=\frac{1}{0.2}\(220-5i)
Or,\Y=1100-25i

The money attains its equilibrium when M_d equals M_s. Hence, it is expressed as follows.
M_d=M_s.................(ii)
Plotting the given values in the equation (ii) we get,
0.2Y-10i=100
Or,\0.2Y=100+10i
Or,\Y=500+50i

Hence, the IS and LM functions are as follows,
IS function, Y=1100-25i
LM function Y=500+50i

b- Calculating the equilibrium income and rate of interest,
For equilibrium the IS must be equal to the LM. Hence, it is expressed as follows.
IS=LM...............(iii)
Plotting the IS and LM functions in the equation (iii), we get,
1100-25i=500+50i
Or,\1100-500=50i+25i
Or,\600=75i
Or,\i=\frac{600}{75}
\therefore\i=8,\Or,\8%

Ploting the value  of i in the LM function we get the income level as follows,
Y=500+50i
Or,\Y=500+50×8
Or,\Y=500+400
\therefor\Y=900

c- For balance of trade X must be equal to M.
Here,
X=20
M=10+0.1Y
Or,\M=10+0.1×900
Or,\M=10+90
Or,\M=100
It shows that export is less than import by (20-100)=-80. Hence, the economy is facing a deficit trade balance of 80.

6- Consider the following equations of a sector economy,
C=200+0.8Y_d            Y_d=Y-T
T=60+0.2Y                    I=$300 G=$350                          X=$100 M=20+0.15Y Now find: a- Compute equilibrium output and trade balance, b- It is realized that the economy is trapped in BOP disequilibrium (Balance of payment deficit). It is mainly caused by the mounting trade deficit. The government has implemented an expansionary fiscal policy to promote export-oriented and import-substituting industries and then increased its planned expenditure by$100 million and reduces tax rate by 5%. As a result, export increases by \$40 million and import function decreases for M=10+0.08Y. Compute new equilibrium output and trade balance. Does this fiscal policy be effective to correct BOP disequilibrium? Give reason.

Solution:

#### a- Computing equilibrium output,

As we know that the equilibrium identity of a product market in a four-sector economy is expressed as follows.
Y=C+I+G+(X-M)...........(i)
Substituting the given values for the variables in equation (i), we get;
Y=200+0.8Y_d+300+350+[100-(20+0.15Y)]
Y=200+0.8(Y-T)+300+350+100-20-0.15Y
Y=200+0.8[Y-(60+0.2Y)]+730-0.15Y
Y=200+0.8Y-48-0.16Y+73o-0.15Y
Y=882+0.49Y
Y-0.49Y=882
Y(1-0.49)=882
Y=\frac{882}{0.51}
Y=1729.41

#### For trade balance, let us compare X and M,

Here, X=100
M=20+0.15Y
Or,\M=20+0.15×1729.41            \because\Y=1729.41
Or,\M=20+259.41
Or,\279.41
The amount of export is less than import by 179.41. Hence, the economy is facing a deficit trade of -179.41.

b- Equilibrium output trade balance after the implementation of expansionary fiscal policy,
G=G+△G            X=X+△X
G=350+100        X=100+40
G=450    and       X=140
New importa function is M=10+0.08Y
New tax function is T=60+[0.2Y-(0.2Y)×0.05]
T=60+0.2Y-0.01Y
T=60+0.19Y

For equilibrium output,
Y=C+I+G+(X-M)
Or,\Y=200+0.8(Y-T)+300+400+[140-(10+0.08Y)
Or,\Y=200+0.8[Y-(60+0.19Y)]+300+450+140-10-0.08Y
Or,\Y=200+0.8Y-48-0.152Y+300+450+140-10-0.08Y
Or,\Y=1032+0.568Y
Or,\Y-0.568Y=1032
Or,\Y(1-0.568)=1032
Or,\Y=\frac{1032}{0.432}
Or,\Y=2388.88

X=140
M=10+0.08Y
Or,\M=10+0.08×2388.88
Or,\M=10+191.11
Or,\M=201.11

This shows that the policy adopted by the government is not as effective as it should be. It is because the amount of export is still less than import by 61.11. The economy is still facing deficit trading by -61.11.