1. ECON 102 Tutorial: Week 20 Shane Murphy

2. Today’s Outline Week 20 worksheet – IS-LM Model:
Please make sure you review all of problems on your own and ask if you have any questions.
If you’re unsure of any solutions here, please see Chapter 24 in your textbook – it provides detailed explanations and examples.
If you didn’t receive your 2nd exam, you may collect it from my office.
Exam 3 will be available some time after Easter holidays.

3. IS-LM Model: Some Important Equations
IS Curve (Derived in Box 24.1 of the textbook):
𝑌= 1 1−𝑐 𝐴 −𝑓𝑖
Or we can write it as:
𝑖= 1 𝑓 𝐴 − 1−𝑐 𝑓 𝑌
LM Curve (Derived in Box 24.2 of the textbook):
𝑘𝑌−ℎ𝑖= 𝑀
Or, we can write this as:
𝑖= 1 ℎ 𝑘𝑌 − 𝑀
Equilibrium in the IS-LM model (Equation (3) of Box 24.3 of the textbook):
𝑌= 1 𝑓𝑘+ℎ(1−𝑐) ℎ 𝐴 +𝑓 𝑀
𝐴 is total autonomous expenditures.
(defined on pg. 580, ch. 22)
𝑐 is the Marginal Propensity to Consume.
(defined on page 576, ch. 22)
𝑓 measures the responsiveness of
consumption and investment
expenditures to changes in the rate of
interest. The larger 𝑓 is, the smaller the
slope of the IS curve will be.
𝑀 is the money supply.
𝑖 is the interest rate (real).
𝑘 models the transactions demand for
money.
ℎ models the idea that interest rate is
the opportunity cost of money.

4. Question 1 Assume that c = 0.75, 𝐴 = 2,000 and f = 500.
Draw a graph plotting the IS curve going through points at which i = 0.01 and i = 0.10.
We know that the IS curve combines the goods & services market and the money market and finds the equilibrium interest rate and income in these markets simultaneously. So we have Y on the X-axis (just like in the Keynesian Cross diagram), and interest rate, i, on the Y-axis.
The equation for the IS curve can be written as:
𝑌= 1 1−𝑐 𝐴 −𝑓𝑖
Please note, these are the graphs that were provided with the solutions – however a proper graph would show both the X- and Y-axes, and any points should be labelled, if not to scale.

5. Question 1
Assume that c = 0.75, 𝐴 = 2,000 and f = 500. Draw a graph plotting the IS curve going through points at which i = 0.01 and i = 0.10.
We know that the IS curve is given by the equation:
𝑌= 1 1−𝑐 𝐴 −𝑓𝑖
To graph our IS Curve, we can plug in the two values of i that we are given, and then plot the values of Y for each i.
If i = 0.01:
𝑌= 1 1− −500∗0.01 =7980
If i = 0.10:
𝑌= 1 1− −500∗0.10 =7800
What if i = 0?
Then
𝑌= 𝐴 1−𝑐 = =8000

6. Question 2
Using the data in Problem 1 draw a graph showing how the IS curve would shift if government expenditure increased by 100.
A rise in government expenditure means a rise in autonomous expenditure 𝐴 from 2,000 to 2,100.
The IS curve would shift to the right by: ∆𝑌= 1 1−𝑐 ∆𝐴, where ΔA is the change in 𝐴 .
So the size of the shift is ∆𝑌= 1 1− =400
Alternatively, recalculate the Y values for i = 0.01 and for i = 0.1 as in Problem 1, but this time using 𝐴 = 2,100.
This illustrates the idea that a change in autonomous expenditures shifts the IS curve:
If 𝐴 ↑, IS Curve → and If 𝐴 ↓, IS Curve ←

7. Question 3
Assume that h = 1,000, k = 0.25 and 𝑀 = 1,240. Draw a graph plotting the LM curve going through points at which i = 0.01 and i = 0.10.
We know the LM curve is given by the equation 𝑘𝑌−ℎ𝑖= 𝑀
In order to graph this, we can rearrange so that Y is on the left hand side: 𝑌= 1 𝑘 ℎ𝑖+ 𝑀 .
Now, we can find values for Y, for each value of i given, just like we did in Q1.
If i = 0.01,
then 𝑌= ∗ =5,000
If i = 0.1,
then 𝑌= ∗ =5,360
What if i = 0?
Then 𝑌= 1 𝑘 𝑀 = ,240 =4,960

8. Question 4
Using the data in Problem 3 draw a graph showing how the LM curve would shift if the money supply is increased by 60.
A rise in the money supply would shift the LM curve to the right by ΔY.
We can write this as: ∆𝑌= 1 𝑘 ∆𝑀
Plugging in values, we can calculate ΔY:
∆𝑌= =240
So, if the money supply is increased by 60, the LM curve would shift to the right by ΔY = 240.
Alternatively, we could recalculate and plot the Y values for i = 0.01 and for i = 0.1, like we did in Problem 3, but this time using 𝑀 = 1,300.
This Problem illustrates the idea that, for a given level of income, the LM curve will shift if the Central Bank changes the Money Supply:
If 𝑀 ↑, then LM curve → And If 𝑀 ↓, then LM Curve ←

9. Question 5 Assume that MD = 0.2Y – 1,000i, Y = 4,000, i = 5%.
By how much would the central bank have to reduce the money supply if it wished to increase the interest rate by 1%?
In equilibrium MS = MD
MS = 0.2Y – 1,000i
If we assume a constant income level Y, then the change in the money supply ΔM is related to a change in interest rate Δi as follows:
ΔM = -1,000 Δi
ΔM = -1,000 * 0.01
ΔM = -10
So the central bank would have to reduce the money supply by 10.
Alternatively, we could get the same result by inserting the value for Y and the two different values for i into the MS = MD equation and calculating the two money supply values, and then fining the change in money supply required.

10. Question 6 Assume that MD = 0.2Y – 1,000i, Y = 4,000, i = 5%.
By how much would the central bank have to let the interest rate change if it cuts the money supply by 100?
In equilibrium MS = MD
MS = 0.2Y – 1,000i
We re-arrange to isolate i: i = (-1/1,000)(0.2Y + M)
So, if we hold income, Y, at a constant, then the change in the interest rate Δi is related to a change in money supply ΔM as follows:
Δi = (-1/1,000) ΔM
Δi = (-1/1,000) * (-100)
Δi = 0.1 = 10%
So the interest rate would rise by 10 percentage points.

11. Question 7
Suppose,in a given economy: 𝐴 = 1,200, 𝑀 = 1,000, c = 0.75, f = 800, h = 1,200 and k = What are the equilibrium values for Y and i?
Looking for equilibrium in the IS-LM model, plug the given values into Equation (3) of Box 24.3 of the textbook to get:
𝑌= 1 𝑓𝑘+ℎ(1−𝑐) ℎ 𝐴 +𝑓 𝑀
𝑌= ∗0.25+1,200(1−0.75) 1,200(1,200)+800(1,000)
𝑌=4,480  
Now that we have Y at the equilibrium, we can substitute Y = 4,480 into the LM equation, to get the equilibrium rate of interest, i:
LM Curve: 𝑖= 1 ℎ 𝑘𝑦 − 𝑀
  𝑖= 1 1, (4,480) −1,000
𝑖=0.1 
So in this economy’s equilibrium, i = 0.1 and Y = 4,480.

12. Question 8
Using the data in problem 7, suppose the economy faced a recessionary gap of 360. By how much would the government have to increase purchases to close the recessionary gap?
If there is a recessionary gap and we want to close it, we need to increase output, Y, by the amount of the gap.
So, in this problem we are trying to find out, how much do we need to increase government expenditure, in order for Y to increase by 360.
In the IS-LM analysis, an increase in government purchases is a change in autonomous expenditures.
So we want to find how much do we need to increase autonomous expenditure, 𝐴 , in order for Y to increase by 360.

13. Question 8
We know: 𝐴 = 1,200, 𝑀 = 1,000, c = 0.75, f = 800, h = 1,200 and k = 0.25.
And we want to find how much do we need to increase autonomous expenditure, 𝐴 , in order for Y to increase by 360.
We start with our Equilibrium output Y in the IS-LM model, equation (3) in Maths Box 24.3:
𝑌= 1 𝑓𝑘+ℎ(1−𝑐) ℎ 𝐴 +𝑓 𝑀
Keeping money supply M constant, a change in equilibrium output ΔY is related to a change in autonomous expenditures ΔA as follows:
∆𝑌= 1 𝑓𝑘+ℎ(1−𝑐) ℎ∆𝐴
Rearranging to solve for ΔA, we get:
∆𝐴= 1 ℎ 𝑓𝑘+ℎ(1−𝑐) ∆𝑌
Inserting values yields ∆𝐴= 1 1, ∗0.25+1,200(1−0.75) 360
∆𝐴=150
So government expenditure would have to be increased by 150 in order to raise equilibrium output Y by 360 and close the recessionary gap.

14. Question 9
From Q7, we know: 𝐴 = 1,200, 𝑀 = 1,000, c = 0.75, f = 800, h = 1,200 and k = 0.25.
Suppose the economy faced an expansionary gap of 480. By how much would the government have to reduce the money supply to close the expansionary gap?
This is just like Q8, but instead of a recessionary gap, we have an expansionary gap. So We want to find out how much do we need to decrease 𝑀 , in order for Y to decrease by 480.

15. Question 9
From Q7, we know: 𝐴 = 1,200, 𝑀 = 1,000, c = 0.75, f = 800, h = 1,200 and k = 0.25.
Suppose the economy faced an expansionary gap of 480. How much do we need to decrease 𝑀 , in order for Y to decrease by 480?
Again, we start with the Equilibrium output Y in the IS-LM model, eq. (3) in Maths Box 24.3:
𝑌= 1 𝑓𝑘+ℎ(1−𝑐) ℎ 𝐴 +𝑓 𝑀
If we assume that autonomous expenditure A is held constant, then a change in equilibrium output ΔY is related to a change in the money supply ΔM as follows:
∆𝑌= 1 𝑓𝑘+ℎ(1−𝑐) 𝑓∆𝑀
Rearranging to solve for ΔM: ∆𝑀= 1 𝑓 𝑓𝑘+ℎ(1−𝑐) ∆𝑌
Now we can plug in values: ∆𝑀= ∗0.25+1,200(1−0.75) (-480)
∆𝑀=−300
So the money supply would have to be reduced by 300 in order to reduce equilibrium output Y by 480 and close the expansionary gap.

16. Next Class Week 21 Worksheet - Fiscal Policy & Monetary Policy
Looks at Policy Applications of the maths that we used in this week’s tutorial.
Chapters 25 and 26 in the Textbook
Have a nice Easter Break!