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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 new`IS` 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,
c- Balance of trade,

 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`

For trade balance,
`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.












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