ECON 102 Tutorial: Week 20
Ayesha Ali
www.lancaster.ac.uk/postgrad/alia10/econ102.html
a.ali11@lancaster.ac.uk
office hours: 8:00AM – 8:50AM tuesdays LUMS C85
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.
IS-LM Model: Some Important Equations
𝐴 is total autonomous expenditures.
(defined on pg. 580, ch. 22)
IS Curve (Derived in Box 24.1 of the textbook):
1
1−𝑐
𝑌=
𝐴 − 𝑓𝑖
Or we can write it as:
1
𝑓
𝑖=
𝐴 −
1−𝑐
𝑓
LM Curve (Derived in Box 24.2 of the textbook):
𝑘𝑌 − ℎ𝑖 = 𝑀
Or, we can write this as:
𝑖=
1
ℎ
𝑌
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.
𝑘𝑌 − 𝑀
Equilibrium in the IS-LM model (Equation (3) of Box 24.3 of the textbook):
1
𝑌 = 𝑓𝑘+ℎ(1−𝑐) ℎ𝐴 + 𝑓𝑀
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.
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:
1
𝑌=
1−𝑐
𝐴 − 𝑓𝑖
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:
1
𝑌=
𝐴 − 𝑓𝑖
1−𝑐
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 − 0.75
2000 − 500 ∗ 0.01 = 7980
If i = 0.10:
𝑌=
1
1 − 0.75
2000 − 500 ∗ 0.10 = 7800
What if i = 0?
Then
𝑌=
𝐴
2000
=
= 8000
1−𝑐
0.25
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 =
So the size of the shift is
∆𝑌 =
1
∆𝐴, where ΔA
1−𝑐
1
100 = 400
1−0.75
is the change in 𝐴.
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 ←
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
1
In order to graph this, we can rearrange so that Y is on the left hand side: 𝑌 = 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,
1
then 𝑌 = 0.25 1000 ∗ 0.01 + 1240 = 5,000
If i = 0.1,
1
then 𝑌 = 0.25 1000 ∗ 0.10 + 1240 = 5,360
What if i = 0?
1
Then 𝑌 = 𝑘 𝑀 =
1
0.25
1,240 = 4,960
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.
1
We can write this as:
∆𝑌 = 𝑘 ∆𝑀
Plugging in values, we can calculate ΔY:
1
∆𝑌 =
60 = 240
0.25
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 ←
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.
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
We re-arrange to isolate i:
MS = MD
MS = 0.2Y – 1,000i
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.
Question 7
Suppose,in a given economy: 𝐴 = 1,200, 𝑀= 1,000, c = 0.75, f = 800, h = 1,200 and k = 0.25.
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:
1
𝑌 = 𝑓𝑘+ℎ(1−𝑐) ℎ𝐴 + 𝑓𝑀
𝑌=
1
800 ∗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:
1
LM Curve:
𝑖 = ℎ 𝑘𝑦 − 𝑀
𝑖=
1
1,200
0.25(4,480) − 1,000
𝑖 = 0.1
So in this economy’s equilibrium, i = 0.1 and Y = 4,480.
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.
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:
1
𝑌=
ℎ𝐴 + 𝑓 𝑀
𝑓𝑘 + ℎ(1 − 𝑐)
Keeping money supply M constant, a change in equilibrium output ΔY is related to a change in
autonomous expenditures ΔA as follows:
1
∆𝑌 =
ℎ∆𝐴
𝑓𝑘 + ℎ(1 − 𝑐)
Rearranging to solve for ΔA, we get:
Inserting values yields
1
∆𝐴 =
𝑓𝑘 + ℎ(1 − 𝑐) ∆𝑌
ℎ
1
∆𝐴 = 1,200 800 ∗ 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.
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.
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:
1
𝑌=
ℎ𝐴 + 𝑓 𝑀
𝑓𝑘 + ℎ(1 − 𝑐)
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:
1
∆𝑌 =
𝑓∆𝑀
𝑓𝑘 + ℎ(1 − 𝑐)
1
𝑓𝑘 + ℎ(1 − 𝑐) ∆𝑌
𝑓
1
800 ∗ 0.25 + 1,200(1 −
800
Rearranging to solve for ΔM:
∆𝑀 =
Now we can plug in values:
∆𝑀 =
∆𝑀 = −300
0.75) (-480)
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.
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!