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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-revenue`T`, 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,
b- Foreign trade multiplier,
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`

b- Foreign trade multiplier,
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.

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