Notesale: Turn your study into money

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EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics

Lecture 7-8: Fiscal and Monetary Policy in the ISLM Model
Lecture Outline:
- the IS-LM model equilibrium
- how to use the IS-LM model to analyze the effects of fiscal and monetary policy
Essential reading:
Mankiw: Ch
...

With that we mean that the general price level will not suddenly adjust when
economic conditions change
...
If the economic conditions do
not change (meaning: demand and supply do not change) the price will not change
...
Given the
supply an increase in the demand should result in an increase in the price level (and in
the quantity exchanged in the market)
...
How long the price will remain at
the same level after the demand has changed will denote the short-run
...
Here we have the same idea just applied to the general price level instead to
the price of a particular good
...

The curve is given by: Y = C (Y − T ) + I (r ) + G
The LM curve represents money market equilibrium
...


1

In the graph: the values Y1 and r1 represent the values of real income and the interest
rate that ensure that the goods market and the money market are simultaneously in
equilibrium
...
If we know the
mechanism we can try to affect this mechanism through economic policy in order to
affect the equilibrium level of real income
...
In that
lecture we gave an idea about how the real GDP is calculated
...

Now we can ask the following question:
What happens when for some reasons we are not at the equilibrium implied by the ISLM model? For example, suppose we start from a situation where the goods market is
in equilibrium, however, the money market is not
...
In particular, at YH and at rH the goods market is in
equilibrium, however, at that level of income YH, the interest rate is too high for the
equilibrium to take place also in the money market
...
If the interest rate is too high in the money market (rA < rH), we know that the
demand for money will be low
...
A disequilibrium must occur only if that inequality is not
satisfied)
...
This means that rH must decrease
...
As
income increases, money demand increases and this together with the reduction of the
interest rate will contribute to eliminate the excess of money supply in the money
market
...
At point E, no agents in
both markets have an incentive to change their behaviour and therefore we are at a
general equilibrium
...


3

A similar reasoning can be made for other points like point A, where the money
market is in equilibrium but the goods market is not
...
The interest is low and
this means that Investments will be too high (remember that the goods market
equilibrium implies I = S, any disequilibrium implies that equality does not hold)
...
In the money market, an increase in the demand of money, for a given
money supply, will increase the interest rate
...
When this
process will stop? At point E again, where the interest rate will be higher and income
will be higher than at point A
...

So we can say that there are market forces that naturally will establish the equilibrium
in the money market and in the goods market simultaneously
...
If we think that the equilibrium E is not a good equilibrium we may want
to change the market result using policies
...
We can define the concept of Full-employment equilibrium: a
situation where the number of unemployed workers is equal to the number of job
vacancies available
...
It means that unemployment in mainly
based on a voluntary basis
...
A full-employment equilibrium is therefore a
situation where most of the labour resources are used in the production process and so
the real GDP is very close to the Potential real GDP, that is the maximum level of
output that can be produced, given the technology available ( = the production
function), using all the resources available in the economy
...


4

It is clear that an economy would preferred to be close to the potential real GDP most
of the time, since this will imply that the real income is high and there are few wastes
of economic resources
...
Those economists believed that markets work well, and that sooner or later the
economy will converge to the full-employment equilibrium
...
However, if you believe that the description of an economic system made by
Keynes is more correct, your answer to the previous question should be no
...

The main idea is that markets may not work very well, since there are market failures
(externalities), rigidities (prices and wages that are not flexible) and so on
...
Obviously the analysis of Keynes is based
on the short-run (in the long-run we all agree that markets should work well)
...

The analysis of Keynes is highly influenced by the Great Depression that started in
1929 (the General Theory was published in 1936) and affected many countries and
lasted for some years
...

Therefore, since the ideas of Keynes had a great influence on macroeconomic theory,
the IS-LM model has been used by economists to study the effects of fiscal and
monetary policy on the equilibrium of an economic system
...
This is because the IS-LM model is a model that explains
the behaviour of the Aggregate Demand of an economy
...
In particular, we
focus on linear functions
...
Y is
real income and T is the tax level, so (Y-T) is the disposable income
...

Government expenditure is exogenous and equal to G
...

The equilibrium in the goods market is:

Y =C + I +G

Using the functions defined above we have:

Y = C0 + c (Y − T ) + I 0 − br + G
Solving for Y:

Y=

1
[C0 + I 0 + G − cT − br ] 1)
1− c

Equation 1) is our IS curve
...

Therefore, we can rewrite the IS curve as:

r=

1
[C0 + I 0 + G − cT − Y (1 − c)]
b

2)

Equations 1) and 2) are obviously the same thing written in a different way
...
This means that a big change in income will result in a
small change in r
...
We call this
situation as an IS curve that is ELASTIC with respect to Y (if a big change in Y

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results in a small change in r, then reversing the causality, a small change in r will
result in a big change in Y)
...
This means that a small change in Y will result in a big
change in r
...
We say in this case that the IS
curve is INELASTIC with respect to Y
...

The money supply M = M and prices are exogenous
...
This
means that a big change in income will result in a small change in r
...
We call this case as an LM curve ELASTIC with respect
to Y
...
A small change in Y will
result in a big change of r
...
We call this case as an LM INELASTIC with respect to Y
...
Here the two
variables that define a fiscal policy are G and T
...

From equation 5) we have that:

∂r *
k
=
>0
∂G bk + (1 − c)h

8)

∂r *
− ch
=
<0
∂T bk + (1 − c)h

9)

An increase in G (everything else constant) will increase the equilibrium level of r
while an increase in T (everything else constant) will decrease the equilibrium level of
r
...

8

In the Keynesian Cross the multiplier effect of G was given by

1

...
Therefore,
the result of the Keynesian cross model was a Partial Equilibrium Effect, based only
on the goods market framework
...

In particular notice that:

1
h
>
1 − c bk + (1 − c)h

10)

if h, k and b are greater than zero
...
In this case the LM curve is

infinitively elastic (completely flat) and the crowding out is zero:

h
1
=
h →∞ bk + (1 − c ) h
1− c

lim

(see the Appendix of this lecture note)
...
The reason is that an increase in G increases income BUT also the interest
rate (equation 8))
...

Therefore, the increase in income due to the increase in G will be partly compensated
by the decreasing in the Investments
...

The increase in government expenditure crowds out some of private investments
...

The level of crowding out depends on the relative slopes of the IS and the LM curves
...
From equation 2) we
know that an increase in G will increase the vertical intercept of the IS curve and
from equation 1) we know that if G increases, real income must increase by

∆G

...

The reason is that an increase in income will increase money demand for a given
level of money supply, creating an excess of money demand in the money market
...

3) An increase in r will decrease I and so the initial increase in G will partly offset
by this decreasing in I
...


Crowding out and the relative slopes of the IS and LM curves:

a) Suppose the LM is particularly inelastic

10

∆G
,
1− c

LMnew
r

LM0

rnew
r1
r0

IS1
E0
IS0

Y0

Y1

Y2

Y

Ynew

Suppose an increase in G
...
The initial LM is LM0
...
Notice that the LM curve does not move since G does
not enter in the LM equation
...
Now consider an LM curve steeper (meaning less elastic)
than LM0, call it LMnew
...
The equilibrium with this less elastic LM
curve will be rnew and Ynew
...

The reason is that now an increase in G will have a bigger effect on the interest rate
than before and so the reduction in investment will be higher than before
...
The slope is large (curve inelastic) if h is a
really small number
...
If income increases, money demand increases and for a
given level of money supply the interest rate must increase
...


11

b) Suppose the IS curve is particularly elastic

r

LM0

r1
rnew

IS10

r0

IS00

E0
IS1
IS0

Y0

Y1

Y2

Y

Ynew

In this graph we have two IS curves quite inelastic (IS0 and IS1) and two more elastic
(IS00 and IS11)
...
Notice that also the effect on r is lower when the IS curve
is elastic than in the case where the IS curve is less elastic (rnew < r1)
...
The IS curve is elastic if the slope is a small
b

number
...
If b is large, a change in the
interest rate will have a big impact on the level of investment
...
This will increase the money demand and given the
level of money supply will increase the interest rate
...


A similar reasoning can be made for the case where Taxes are reduced, since in this
case the shift in the IS curve is similar to the case where G is increased
...


The effect of an increase in the tax level is less interesting in the IS-LM model since
the result is a reduction in the equilibrium income
...
So we consider a monetary policy as a change in the money
supply
...


An increase in M will shift the LM curve (M does not appear in the IS curve) down,
by an amount given by the change in M
...
Two of them are relatively more elastic
(LM0 and LM1) than the other two (LM00 and LM10)
...
An increase in M by ∆M will shift down
the LM curve (remember that the intercept of the LM curve is negative)
...
As we can see
the effect is an increase in income from Y0 to Y1 and a reduction in r from r0 to r1
...
To eliminate this excess of money, the
theory of liquidity preference says that the interest rate must decrease
...

As Income increases money demand increases helping to restore the equilibrium in
the money market
...

Now consider the same change in money supply, but with a less elastic LM curve like
LM00
...

The same change in M has now a bigger effect on income and on the interest rate
...
In particular, remember that the LM is
inelastic if h is small
...
If money supply increases, in order to increase the money demand and restore
the equilibrium in the money market the interest rate must decrease quite a lot
...


Here we have considered an expansionary monetary policy (M increases)
...

In the case of a restrictive monetary policy the LM will shift to the left and the result
will be a reduction in Income and an increase in the interest rate in equilibrium
...


Obviously the effectiveness of monetary policy depends also on the slope of the IS
curve
...
As we will see an increase in M will tend to have a bigger effect on Y
the more elastic is the IS curve (Ynew > Y1 )
...
A small decrease in the interest rate will have a big impact on I and
therefore on Y
...
Given the slope of the LM curve, monetary policy (expansionary or
restrictive) is more effective in affecting the level of real income the more elastic is
the IS curve

The Liquidity Trap: the Keynesian Case

The liquidity trap is also called the “Keynesian case” of the neoclassical synthesis
since it represents the economic situation that is closer to the original ideas of Keynes
on how an economic system should work
...
At that interest
rate people will hold any amount of money for any given level of income
...
We say that in this case money and less liquid assets are essentially the same
thing
...

In terms of the slope of the LM curve it means that

h→∞,

meaning that the

demand for money is infinitely sensitive to r
...

So the first two basic elements of a liquidity trap are:
1) Interest rate extremely low, possibly zero;
2) The LM curve is completely flat at that low interest rate;
Now we introduce another element that makes the liquidity trap closer to Keynes’
ideas
...
You may think at the
animal spirits as the subjective expectations of the investors about the future
...
You may

16

think at a sort of optimism or pessimism of investors when you talk about animal
spirits
...

The fact that investment are not very related to the interest rate in Keynes’ opinion,
implies that the IS curve is relatively inelastic since b is a small number
...
Otherwise the
economy can be stuck in a lower equilibrium for a long time
...
If the IS curve is
inelastic, it may cut the horizontal axis to the left of Yfull, like at Y1 in the figure
...
In this situation,
independently on where the LM curve will lie, the equilibrium in the market cannot
be the full-employment equilibrium
...

Now we add the LM curve to the previous figure to show the liquidity trap case:

17

IS

IS1

r

IS2

rL

Y0

Y1

Y2

Yfull

Y

Suppose that the interest rate is very low at level rL (notice that this interest rate
should be close to zero, however, for graphical convenience, we plotted it as if it is
much higher than zero)
...
This is the case of
the liquidity trap
...
The idea according to Keynes is: as
income increases, the economic situation improves, after a while investors will start to
gain confidence, they will become optimistic, they will start taking investments again
and the system will start working again as in the usual IS-LM model we have seen
before
...
If we don’t intervene in some
way the system will remain at that equilibrium and we will be stuck in the liquidity
trap
...

There are two ways to do that according the IS-LM model:
1) Expansionary Monetary Policy: in the liquidity trap this policy does not
work
...

2) Expansionary Fiscal Policy: this policy is really effective in a liquidity trap
...
We have seen that when the

18

LM curve is flat, the multiplier of the government expenditure is equal to the
case in the Keynesian Cross case, meaning that this policy will be very
effective in increasing equilibrium income
...
So an expansionary fiscal policy will shift the IS curve to IS1 for
example
...

Are the liquidity traps common in reality?
Not really, however, in the Great Depression of 1929 the nominal interest rate
dropped quite dramatically and its level started to be very close to zero after few
years
...

The recent credit crunch is another example of a situation where nominal interest rate
became very close to zero
...

In particular, monetary policymakers may adjust M in response to changes in fiscal
policy, or vice versa
...

Suppose that the government increases G
...
hold M constant
2
...
hold Y constant
In each case, the effects of the ∆G are different:
Case 1) Holding M constant
If Congress raises G, the IS curve shifts right
...
The result is in the following graph

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r
LM1
r2
r1
IS2
IS1
Y

Y1 Y2

Result: income increases and interest rate increases
...

Case 3) Holding Y constant
In this case the central bank must decrease M to compensate the increase in G
...


Is the general price level really fixed in the short-run?
The analysis done so far is based on the assumption that the general price level is
fixed
...

An empirical experiment of this sort has been done by Bernanke and Gertler (1995)
...
The increase in the
interest rate is interpreted as a change in monetary policy
...
The experiment is as
follow: first we increase the interest rate in the first period, then nothing else changes
and we see over time how long it takes for the general price level to change
...
9, No
...
(Autumn, 1995), pp
...


21

First, the interest rate increases (the Fund Rate), then they look how long it takes for
the general price level to change
...
Output instead starts to decrease quite early, around 4 months after
the increase in the interest rate
...
This is consistent
with the analysis of the IS-LM model
...
This is different from the experiment of Bernanke and Gertler
...
They found that in average a price remains fixed for 5
...
This result seems to suggest that prices are sticky but not as sticky as
we may think
...
Bils and P
...
112, 947-985
...

Suppose we want to find the limit for x that tends to infinity of the ratio of the two
functions:
lim

x→∞

f ( x)
g ( x)

Suppose that the result of that limit is:
lim
x →∞

f ( x) ∞
=
g ( x) ∞

This means that the limit is not defined
...

In that situation we can apply the Hopital theorem that says that we can calculate:
lim
x →∞

where f ' ( x) =

f ' ( x)
g ' ( x)

df ( x)
dg ( x)
and g ' ( x) =
are first derivative
...

g ( x ) x →∞ g ( x )

Suppose that also lim
x→∞

f ' ( x) ∞
= , then we can continue in the process and calculate:
g ' ( x) ∞
f '' ( x )
lim ''
x→∞ g ( x)

where f '' ( x) =

d 2 f ( x)
d 2 g ( x)
and g '' ( x) =

...
Meaning:
convergent or divergent
...
The important thing is that the result of the limit is

∞
0
or
...

Consider the following limit:
x −1
x→∞
x

lim
The result is: lim
x →∞

x −1 ∞
=
...
However, by using the Hopital theorem, and
taking the limit of the ratio of the first derivatives we have that:
lim
x→∞

So we can say that

x −1
1
⇒ lim = 1
x →∞ 1
x

x −1
converges to 1 when x tends to infinity
...
The derivative of the denominator is 3x 2
...
In the denominator we have x to the power
of 3 that grows faster than x to the power of 2
...
That is equal to 1
...

So we can write the limit as:

h
1
1
⇒ lim
=
h →∞ bk + (1 − c ) h
h →∞ (1 − c )
(1 − c)
lim

2) The Algebra of the IS-LM model

The IS-LM model we analysed in the lecture is given by:

r=

1
[C0 + I 0 + G − cT − Y (1 − c)]
b
k
1M
r= Y−
h
h P

We solve this system by equalising the right-hand side of each equation (they are both
equal to the same thing, r, so they must be equal):

1
[C0 + I 0 + G − cT − Y (1 − c)] = k Y − 1 M
b
h
h P
Solving equation 1) for Y:

1
[C0 + I 0 + G − cT ] − (1 − c) Y = k Y − 1 M
b
b
h
h P
1
1M k
(1 − c)
⇒ [C0 + I 0 + G − cT ] +
= Y+
Y
b
h P h
b
1
1M
k 1− c
⇒ [C0 + I 0 + G − cT ] +
=Y +
b
h P
b 
h


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1)

1M
 kb + (1 − c) h  1
Y
= [C0 + I 0 + G − cT ] +
 b
hb
h P


1
hb
1
hb
M
⇒ Y = [C0 + I 0 + G − cT ]
+
b
kb + (1 − c)h h kb + (1 − c)h P
h
b
M
⇒ Y* =
A+
kb + (1 − c)h
kb + (1 − c)h P
where A = C 0 + I 0 + G − cT
...
For example using the LM curve:

k
h
b
M 1 M
−
A+
h  kb + (1 − c)h
kb + (1 − c)h P  h P


k
k
b
M 1M
⇒r=
A+
−
kb + (1 − c)h
h kb + (1 − c)h P h P
r=


k
bk
1M 
A+
− 1
kb + (1 − c)h
h P  kb + (1 − c)h 

k
1 M  bk − bk − (1 − c) h 
⇒r=
A+
kb + (1 − c)h
h P  kb + (1 − c)h 


⇒r=

⇒ r* =

k
(1 − c)
M
A−
kb + (1 − c)h
kb + (1 − c)h P

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