# Income Determination Solved Problems

This page consists of various solved problems related to income determination that were asked in different university examinations in different years.

1-Given, C=60+0.8Y_d, I=100,  T=80 & G=80,
Find; a- Equilibrium income of the economy,
b- Value of multiplier,

Solution;
a- Calculating equilibrium income,
The equilibrium identity is given as;
Y=C+I+G.......(i)

substituting the value of C, I & G, we get;
Y=60+0.8Y_d+100+80
Y=0.8(Y-T)+240             \becauseY_d=Y-T

Substituting the value of T, we get;
Y=0.8(Y-80)+240
Y=0.8Y-64+240
Y-0.8Y=176
Y(1-0.8)=176
Y=\frac{176}{0.2}
Y=880  Equilibrium income,

b- Calculating multiplier,
Let us consider consumption function,
Where, a=60
b=0.8
Multiplier =\frac{1}{1-b}

Substituting the value of b we get,
=\frac{1}{1-0.8}
=\frac{1}{0.2}
=5
2- Given the following data for an economy,
C=40+0.9Y_d
I=60, G=30 and T=20
Find;
a- Equilibrium income,
b- Equilibrium income if taxes increases by 10, and T-multiplier,
c- Equilibrium income if government expenditure falls by 20, and G-multiplier,

Solution;
a- Calulating equilibrium income,
The equilibrium identity is given as;
Y=C+I+G

substituting the value of C, I & G, we get;
Y=40+0.9Y_d+60+30
Y=0.9(Y-T)+130             \becauseY_d=Y-T

Substituting the value of T, we get;
Y=0.9(Y-20)+130
Y=0.9Y-18+130
Y-0.9Y=112
Y(1-0.9)=112
Y=\frac{112}{0.1}
Y=1120  Equilibrium income,

b-T-multiplier and Equilibrium income when taxes increases by 10;
Change in income after an increase in tax by 10;
△Y=Y+(△T×\frac{-b}{1-b})                            \becauseT_m=\frac{-b}{1-b}
△Y=1120+(10×\frac{-0.9}{1-0.9})                 Given\because△T=10, and b=0.9
△Y=1120+(-90)
△Y=1030

T-Multiplier,
T_m=\frac{-b}{1-b}
T_m=\frac{-0.9}{1-0.9}                                    Given \because\b=0.9
T_m=\frac{-0.9}{0.1}
T_m=-9

c- G-multiplier and Equilibrium when government expenditure falls by 20,
Change in income after a fall in government expenditure by 20;
△Y=Y+(-△G×\frac{1}{1-b})                            \becauseG_m=\frac{1}{1-b}
△Y=1120-(20×\frac{1}{1-0.9})
△Y=1120-(\frac{20}{0.1})
△Y=1120-200
△Y=920

G-Multiplier
G_m=\frac{1}{1-b}
G_m=\frac{1}{1-0.9}                                       Given \because\b=0.9
G_m=\frac{1}{0.1}
G_m=10

3-  The information about the economy of a country are as follows.
C=100+0.6Y_d
I=90,      G=60,     and    T=20+0.2Y
On the basis of the information find;
a- Equilibrium income and consumption expenditure,
b- The amount of tax that the government collects,
c- What budget policy the government adopts, surplus or deficit,

Solution;
a- Equlibrium income & consumption expenditure,
The equilibrium identity is given as;
Y=C+I+G

substituting the value of C, I & G, we get;
Y=100+0.6Y_d+90+60
Y=0.6(Y-T)+250             \becauseY_d=Y-T

Substituting the value of T, we get;
Y=0.6{Y-(20+0.2Y)}+250
Y=0.6(Y-20-0.2Y)+250
Y=0.6Y-112-0.12Y+250
Y=0.48Y+138
Y-0.48Y=138
Y(1-0.48)=138
Y=\frac{138}{0.52}
Y=265.38  Equilibrium income,

Consumption expenditure,
Given the consumption function as;
C=100+0.6Y_d
C=100+0.6(Y-T)                      \becauseY_d=Y-T

Substituting the value of T, we get;
C=100+0.6{Y-(20+0.2Y)}     \because\T=20+0.2Y

Subtituting the value of Y we get;
C=100+0.6{265.38-(20+0.2×265.38)}
C=100+159.23-20-53.07
C=186.16    Consumption expenditure,

b- The total amount of taxes that the government collects;
T=20+0.2Y

Substituting the value of Y we get;
T=20+0.2×265.38                \becauseY=265.38
T=20+53.07
T=73.07 Amount of taxes,

c- The budgetary policy that the government adopts;
If the amount of government expenditure G equals its tax-revenueT, the government is adopting balanced buget policy. If G>T, it is adopting a deficit budget policy and if G<T, it is adopting a surplus budget policy.

In this problem G=60 and T=73.07,
Here, G<T by 13.07
It implies that the government is adopting surplus budget policy.

4-Given C=150+b(Y-40-tY),    I=50,    G=40,    X=15,    and    M=10+0.12Y
The marginal propensity to consume is equal to 0.9 and proportional income tax rate is equal to 0.2;
Find;
a- Equilibrium national income,
c- Equilibrium value of imports,
d- If equilibrium NI falls short of full employment income by 60, how much government should increase its expenditure to attain full-employment?

Solution;
a- Equilibrium national income,
The equilibrium identity is as follows,
Y=C+I+G+(X-M)

Substituting the value of C, I, G, X and M we get;
Y=150+b(Y-40-tY)+50+40+{15-(10+0.12Y)}

Substituting the value of b and t we get;
Y=150+0.9(Y-40-0.2Y)+50+40+{15-(10+0.12Y)}
Y=150+0.9Y-36-0.18Y+50+40+15-10-0.12Y
Y=209+0.6Y
Y-0.6Y=209
Y(1-0.6)=209
Y=\frac{209}{0.4}
Y=522.5

Foreign trade multiplier is calculated as;
F_m=\frac{1}{1-b(1-t)+tm}

Substituting the values of b=0.9, t=0.2 and m=0.12 we get;
F_m=\frac{1}{1-0.9(1-0.2)+0.2×0.12}
F_m=\frac{1}{1-0.72+0.024}
F_m=\frac{1}{0.304}
F_m=3.29

c- Equilibrium value of imports,
Equilibrium value of imports can be calculated as;
M=10+0.12Y            Given
Substituting the value of Y we get;
M=10+0.12×522.5
M=10+62.7
M=72.7

d-Government expenditure to attain full-employment is calculated as;
△Y=F_m×△G
60=3.29×△G                \because\F_m=3.29 and \because\△Y=60
\frac{60}{3.29}=△G
△G=18.23

To attain full-employment, government expenditure should be increased by 18.23.

5- Given, C=1200+0.8Y_d,    I=1500,    T=2500,    and    G=5300
Based on the information, find;
a- Equilibrium income level,
b- Government expenditure multiplier,
c- By how much the level of income will change, if tax changes to T=2500+0.2Y?

Solution;
a- Equilibrium income level,
The equilibrium identity is given as;
Y=C+I+G

Substituting the value of C, I and G we get;
Y=1200+0.8Y_d+1500+5000
Y=1200+0.8(Y-T)+1500+5300            \because\Y_d=(Y-T)
Y=0.8(Y-2500)+8000                            \becauseT=2500
Y=0.8Y-2000+8000
Y-0.8Y=6000
Y(1-0.8)=6000
Y(0.2)=6000
Y=\frac{6000}{0.2}
Y=30000

b- Government expenditure multiplier,
Goverment expenditure multiplier is calculated as;
G_m=\frac{1}{1-b}
G_m=\frac{1}{1-0.8}                        \because\b=0.8
G_m=\frac{1}{0.2}
G_m=5

c- Equilibrium income level after a change in tax function;
The equilibrium identity is given as;
Y=C+I+G

Substituting the value of C, I and G we get;
Y=1200+0.8Y_d+1500+5000
Y=1200+0.8(Y-T)+1500+5300            \because\Y_d=(Y-T)
Y=0.8(Y-T)+8000

Substituting 2500+0.2Y for T we get;
Y=0.8{Y-(2500+0.2Y)}+8000
Y=0.8Y-2000-0.16Y+8000
Y=0.64Y+6000
Y-0.64Y=6000
Y(1-0.64)=6000
Y(0.36)=6000
Y=\frac{6000}{0.36}
Y=1666.67

When the government imposes tax by 20%, equilibrium income will fall by (30000-16666.67)=13333.33.

6- Given, C=2000+0.8(Y-T),    I=1000,    G=1500    and    T=500+0.2Y,
Find;
a- Equilibrium level of income,
b- Tax multiplier,
c- What happens to the equilibrium level of income if the government expenditure decreases to 1000 and investment increases to 1500?

Solution;
a- Equilibrium level of income,
The equilibrium identity is as follows.
Y=C+I+G

Substituting the values of C, I, and G we get;
Y=2000+0.8(Y-T)+1000+1500

Substituting the value of T we get;
Y=2000+0.8{Y-(500+0.2Y)}+1000+1500
Y=0.8Y-400-0.16Y+4500
Y=0.64Y+4100
Y-0.64Y=4100
Y(1-0.64)=4100
Y(0.36)=4100
Y=\frac{4100}{0.36}
Y=11388.89

b- Tax multiplier,
Tax multiplier T_m is calcilated as;
T_m=\frac{-b}{1-b}
T_m=\frac{-0.64}{1-0.64}      \because\b=0.64 Taken from the above equation.
T_m=\frac{-0.64}{0.36}
T_m=-1.78

c- If the government expenditure decreases to 1000 and investment increases to 1500 the equlibrium income level changes as;
The equilibrium identity is as follows.
Y=C+I+G

Substituting the values of C, I, and G we get;
Y=2000+0.8(Y-T)+1500+1000

Substituting the value of T we get;
Y=2000+0.8{Y-(500+0.2Y)}+1500+1000
Y=0.8Y-400-0.16Y+4500
Y=0.64Y+4100
Y-0.64Y=4100
Y(1-0.64)=4100
Y(0.36)=4100
Y=\frac{4100}{0.36}
Y=11388.89

Due to a decrease in government expenditure by 500 and an increase in investment, there is no change in equilibrium income level.