Download Lecture 9: Extensions to the IS-LM Model

EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics
Lecture 9: Extensions to the IS-LM Model
Lecture Outline:
- shocks and fluctuations in the IS-LM model;
- the credit crunch and the IS-LM model
- the IS-LM model with a Taylor Rule
Essential reading:
Mankiw: Ch. 12 and this lecture note
Shocks in the IS-LM model: Short-run Fluctuations1
We have seen how the IS-LM model can be used to explain how economic policy
may achieve the full-employment equilibrium. However, we can use the IS-LM
model to explain why income fluctuates over time. The reason, according the IS-LM
model, is that the IS curve and the LM curve can move over time, changing the
equilibrium value of real income over time. We have seen that economic policies can
shift the IS and the LM curve up and down. However, we do not think that movement
in economic policies can explain short-run fluctuations. For example, it is unlikely
that a government wants to create a recession by decreasing government expenditure
or by restricting money supply. Furthermore, short-run fluctuations have the property
that they are not fully predictable. This means that short-run fluctuations, booms and
recessions, have some random component that makes them not completely
predictable. To study those facts we need to introduce the concepts of exogenous
shocks in our IS-LM model. Exogenous shocks are economic events that cannot be
completely anticipated by the people. For example, a change in tastes of consumers
may change the aggregate consumption and this will affect the equilibrium level of
income. However, this change is exogenous because it cannot be explained by our ISLM model in the way we have developed it. Another example of an exogenous shock
that affect our model is a crash in the stock market that will reduce the consumption
and investment and so on. Those shocks have the effect to shift the IS and the LM
curve over time and those shifts cause the real income to fluctuate. Policy makers
started to use the IS-LM model to understand how to stabilise the economy that is
subjected to those fluctuations. The reason is that being in a recession can have
1
This section is adapted from William Poole (1970) “Optimal Choice of Monetary Policy Instruments
in a Simple Stochastic Macro Model” Quarterly Journal of Economics vol. 84, n.2 pp. 197-216.
1
important economic costs that we would like to avoid. But also a boom can have some
costs. For example if the boom lasts for a long period, it means that demand is very
high. Higher demand after a while may result in high prices and so high inflation. In
particular we will see how monetary policy can be used to stabilise the economic
fluctuations. To study how to stabilise the real economy when there are shocks using
the IS-LM model we define two different types of shocks:
1) Real shocks: shocks that affect the IS curve (examples: a change in consumer
tastes, a crash in the stock market);
2) Monetary shocks: shocks that affect the demand for money (examples: a wave
of financial fraud, an increase of internet banking);
The issue is the following: the policy maker (in this case the central bank) knows that
the economy will face some shocks over time. The policy maker only know that they
can be real or monetary but it does not know with certainty which one will occur and
when. Moreover the policy maker knows that shocks can be positive or negative but
again it does not know which one will occur with certainty.
We assume that before any shock is realised the central bank HAS TO COMMIT to
a particular policy. By committing we mean that the central bank has to choose a
policy and when that is decided it has to stick to it and cannot change it over time.
The objective of the central bank is to minimise the effects of the shock on the level
of output.
Question: what policy should the central bank commit to?
Answer: it depends on the type of shock that will hit the economy.
Problem: the central bank when it has to decide which policy to commit to does not
know the type of shocks that will hit the economy.
In particular we are going to look at two possible policies that the central bank can
commit to:
a) Keeping fixed the money supply (a special case of Money Targeting);
b) Keeping fixed the interest rate in the market (a special case of Interest Rate
Targeting)
So the timing of our model is the following:
First the central bank decides a policy, a) or b). Once the decision is made the central
bank has to commit to that rule forever. Then a shock is realised and we can see what
the effects on output are depending on the rule decided by the central bank.
2
Notice that BY ASSUMPTION the central bank cannot just wait, see which shocks
hit the economy and then act accordingly. This will the case where the central bank
can use discretion in deciding which policy to use. Here we assume that the central
bank has to commit to a rule in advance and before any shock is realised. Even
when the shock is realised the central bank has to stick to the rule.
1) Suppose a real shock
The IS curve may shift to the left or to the right, depending if the shock is positive or
negative. However we do not know with certainty where the IS will end up.
ISU
LM2
r
LM
IS
LM1
ISL
r0
YL
Y1
Y0
Y2
YU
Y
Suppose that the IS curve can shift to ISU or to ISL because of a shock with equal
probability. We know that the IS will shift but we do not know where it will end up.
What policy does minimise the variation of the output created by a possible real
shock?
The initial equilibrium is r0, Y0.
Suppose that before the real shock occurs the central bank commits in keeping fixed
the interest rate in the market (interest rate targeting). To keep constant the interest
rate we must move the LM curve to LM1 if the shock moves the IS to ISU. Or we
move the LM to LM2 if the IS moves to ISL. As we can notice by doing that, the real
income in equilibrium can be YL if the shock moves the IS to ISL, or YU if the shock
moves the IS to ISU. The expected variability of income due to the real shock is very
large: from Y0 to YL or from Y0 to YU. This is expected because we do not know if
the shock is going to be positive or negative. Since there is an equal probability that
3
the real shock is positive or negative there is an equal probability to end up in YL or in
YU. Suppose the central bank keeps fixed the money supply (money targeting). In this
case the LM does not move, and now the expected variability of real income is much
less than before: from Y0 to Y1 or from Y0 to Y2.
Conclusion: if the central bank believes that the economy is more likely to face real
shocks in the future and it has to commit in advance to a policy rule then it is better to
commit to money targeting rule where money supply is kept fixed while the market
interest rate is free to change. By doing that the central bank reduces the expected
variability of output due to a real shock.
2) Suppose a monetary shock:
LML
r
LM
IS
LMU
r0
YL
Y0
YU
Y
Here the LM can shift to LMU or to LML according to a given shock with equal
probability. Suppose now that the central bank commits to an interest rate rule where
the interest rate is kept constant. If the LM shifts to LMU, we can decrease the money
supply to bring back the system to the original LM. If the LM shifts to LML, we must
increase the money supply. By doing this, the interest rate will be stabilised and the
expected income fluctuation will be reduced to zero (notice: we don’t know if the
shock is positive is positive or negative but it does not matter here. BY fixing the
interest rate output will be stabilised no matter what).
Conclusion: if the central bank believes that the economy is more likely to face
monetary shocks then it is better to commit to an interest rate rule where the interest
4
rate is kept constant. Committing to such a rule ensures that the level of output is
stabilised in the event of a monetary shock (positive or negative).
Nowadays many central banks (especially in developed countries) conduct monetary
by setting first the interest rate in the market and then they supply money according to
what is the demand of money in the market at that particular interest rate. This is
pretty much the same as an interest rate targeting. Nowadays such policy rule (of
setting the interest rate first) is known as Inflation Targeting in the sense that the
interest rate is fixed (i.e. targeted) in a way that is consistent with a particular target of
inflation that the central bank may have but the idea is very similar to the interest rate
targeting rule that we used in our analysis.
A question could be: why central banks prefer to set the interest rate instead of
targeting money supply?
Our analysis provides a simple answer to that question: because central banks may
believe that monetary shocks are more likely to hit the economy than real shocks. If
that is the case, then committing to a policy rule that sets the interest rate and let
money supply to be decided by the market is a better policy because it reduces the
possible effects of the monetary shocks on output.
Notice that we haven’t explained why the central bank must commit in advance to a
rule. This will become clearer when we will discuss the issue of Credibility and Time
Inconsistency of monetary policy.
The IS-LM model and the Credit Crunch
In the Autumn 2008 the world economy entered the deepest recession since the Great
Depression. The origin of the crisis was the financial crisis in US in 2007, in the so
called “sub-prime” market. The “sub-prime” market is a relatively small part of the
mortgage market intended for borrowers with a relatively high probability of not
being able to repay their loan. How can it be possible that problems in such a small
market can create such a big recession? Furthermore, can we study the current
recession using the IS-LM model? To study the credit crunch using the IS-LM model,
we need to introduce in the model the role of banks. Implicitly we have the banks in
the IS-LM since the interest rate is also the price of credit. However, we need to
introduce the role of banks more explicitly. Banks are financial intermediaries, they
collect money from savers and lend it to borrowers. Here we consider a commercial
5
bank and not an investment bank. The balance sheet of a commercial bank is given
by:
Assets
Loans
Other assets
Liabilities
Capital
Deposits
The value of the bank’s assets is mostly the value of the loans it has made to
households and firms (although the bank may own other assets, such as the buildings
it uses or financial assets such as bonds.) Capital, which appears on the right side, is
the difference between the value of the assets and the value of the liabilities since
Assets = Capital + Deposits, this the accounting identity for a commercial bank.
Leverage: when you buy an asset (financial or real like a house) the leverage is the
ratio between how much you borrow and how much your own capital you put up to
buy the asset. The capital you put up is called “equity”. As the expression below
shows, the smaller the amount of equity for any given value of the asset you buy, the
higher the leverage.
Debt
Debt
Leverage = ____________ = _________________________________________
Capital
Cost of the investment – Amount borrowed (Debt)
For a commercial bank, the leverage is:
Assets
Capital
Consider the leverage for a commercial bank: lower is the level of capital compared to
the debt and riskier the bank becomes.
This implies that if there is a loss for the bank (for example some customers decide to
withdraw the money from the bank, like for the case of Northern Rock), then the bank
is in trouble because it may not have enough capital to absorb the loss and it can
become insolvent. Loans are the biggest part of commercial bank assets as we can see
from the data below: Total bank assets of commercial bank in US December 2008:
9,972 billion $ of which:
Loans
Government securities
Other securities
7,195 (72,2 %)
1,262 (12,6 %)
1,516 (15.2 %)
In order to introduce the banking sector in the IS-LM model we need to see the
following: when firms want to make investments they will do so by borrowing some
6
money from commercial banks. However, when a firm borrows money from a bank it
must pay an interest rate that is normally different (higher) from the interest rate that
savers receive from deposits. The rate borrowers have to pay, i.e. the cost of a loan
from the bank, ρ, is usually equal to the rate savers receive (i) plus a spread, x:
ρ=i+x
Here we use i for the interest rate but you should remember that i = r in the IS-LM
model. Therefore, when a firm has to decide whether to buy a machine, ρ is the
interest rate she has to look at. Investment demand therefore depends on the cost of
bank loans (and not simply on the interest rate ) and can be expressed as:
I = I (ρ)
What determines the spread x between the lending rate ( ρ ) and the deposit rate ( i )?
1) The Banks’ capital: K B . Suppose that the capital of the banks decrease (for
example to absorb a loss), then the leverage increases and the banks become
riskier. The first reaction of the banks is therefore to reduce their assets,
meaning they stop to lend. If banks stop lending, then the spread x increases.
2) The firms’ capital: K F . If a firm wants to make an investment A, it has to
borrow A from a bank. Suppose the firm has a level of capital K F . The level
of capital K F is used as to guarantee the loan. If the firm fails to repay the
loan then bank can get K F . It is clear that if A > K F , then the capital is not
enough to guarantee the loan and so the investment for the bank becomes
riskier and it has to charge a higher lending interest rate, therefore the spread x
increases.
Therefore anything that makes the Banks’ capital and/or the Firms’ capital to
decrease will increase the lending rate and therefore it will reduce the Firms’
investments.
Given those information we can write the IS equation as:
Y = C (Y − T ) + I (i + x( K B , K F )) + G
where x( K B , K F ) means that the spread x depends on the banks’ capital and firms’
capital as outlined above. The LM equation is the usual one:
M
= L(Y , i )
P
Now suppose that for any reason, the capital of the banks decreases. For example,
suppose that some borrowers do not repay their loans and so the bank needs to use its
capital to cover for that. The leverage of the bank increases and so the bank becomes
7
riskier. The bank then can look for new capital, but this takes time and it may be
difficult, or can reduce their assets for example by stopping to lend money. We know
that when the capital of the bank is reduced the spread x increases and so the cost of
borrowing for the firms increases and investment decreases. Now suppose that for any
reason, also the capital of firms decreases, for example because the stock market is
going down. Also this effect will increase the spread x and this will reduce investment
even further. Graphically, a reduction in investment shifts the IS curve to the left:
The new equilibrium is now E’ where real output is now lower. Notice that the
nominal interest rate is reduced after the shift of the IS curve, however this is not very
helpful since for firms is now relevant the lending rate that is the nominal interest rate
plus the spread x. Notice that the reduction in the interest rate is consistent with what
happened in reality, where the interest rate is indeed decreased over the crisis. This
shows how a change in the level of capital of commercial banks and firms can have
indeed an effect in the equilibrium level of real output. From a policy point of view
there are two things that we can do (possible at the same time):
1) We can try to increases government expenditure and try to shift the IS to the
right. This is what many governments have done during the crisis by putting
money into the system to nationalise some of the troubled institutions;
2) The central bank can increase money supply and try to shift the LM to the
right. This is indeed what happened during the crisis.
8
Graphically we can have the following situation:
The IS shifts to the left because of the decrease in the capital of Banks and firms. The
new IS curve is IS’ and the equilibrium is E’. By increasing government expenditure,
the IS can shift to the right, for example to IS’’. By increasing money supply the LM
shifts to the right. However, as we can see from the graph, we may have the case in
which there is not a positive interest rate consistent with the previous equilibrium of Y
(Y*), and we may be stuck to Y’’ that is much lower. This is exactly a situation
consistent with the liquidity trap studied previously. How long are we going to be
stuck to the low equilibrium? This depends on how effective are the policies used.
After a while the low interest rate associated with more liquidity in the system may
start to restore investment spending and so the IS will start shifting to the right and we
can escape the liquidity trap.
A Final Note on the Monetary Policy instrument
In the IS-LM model by definition monetary policy meant a change in money supply
which will then result in a change in the interest rate as a consequence of the initial
9
change in the money supply. However, we know that most of the central banks, move
(i.e. change or control) the interest rate first and then let the money supply level be
decided by the demand for money in the market at that particular interest rate. In
practice central banks tend to target the overnight interest rate – the interest rate
banks charge one another on overnight loans. By setting first the interest rate, money
supply becomes endogenous since it depends on the level of the interest rate that the
central bank wants to achieve (this is the opposite of our assumption on the IS-LM
model where money supply was assumed to be exogenous). In the US that interest
rate is called the federal funds rate. In UK that interest rate use to be called the repo
but it is now called the Bank Rate. Until the 1980s most central banks used a
monetary policy known as Monetary Targeting. The central bank had a target about
the level (or growth) of the money supply and so it changes money supply to achieve
that target. In this case the LM curve that is derived for an exogenous money supply
makes sense in describing such a policy. However, nowadays monetary targeting is
out of fashion. Inflation targeting is now becoming more popular. Inflation targeting
means that the central bank has a target over the inflation rate (say 2%) and it changes
the interest rate accordingly to reach that target. In this case money supply is
endogenous since it depends on the interest rate chosen by the central bank. The LM
curve makes less sense for such a policy.
Why do central banks set interest rates instead of the money supply?
1)
They are easier to measure than the money supply.
2)
The central bank might believe that monetary shocks are more prevalent than
IS shocks. We have seen this in the first section of this lecture note.
How does it work an inflation targeting policy? The graph below explains the
monetary transmission mechanism according to the Bank of England.
10
The monetary transmission mechanism explains how monetary policy, in this case
changing the short-run interest rate affects economic activity and so inflation. In the
IS-LM model we do not have inflation; however we can still model it in such a way
that the central bank changes the interest rate instead of money supply as monetary
policy. As you may guess there are many interest rates in the economy that are
relevant (interest rates that banks charge borrowing firms, interest rate on mortgages,
etc. etc.). The central bank controls directly only one interest rate (the interest rate at
which it lends to other banks overnight). However by controlling that single interest
rate it is like controlling all others interest rates (especially short-run rates) in the
economy. This is because interest rates tend to move together over time. This is
shown in the following graph taken by the Central Bank of Canada.
The central bank of Canada controls directly only the Bank of Canada Rate, but every
time that rate is changed (increased or decreased) also all other interest rates in the
economy will also change and in the same direction.
The IS-LM model with the Taylor Rule
As we noted above, monetary policy is done by setting the interest rate and so money
supply becomes endogenous and not exogenous as we have assumed in deriving the
LM curve. How do we change the IS-LM to take into account this issue?
11
First we need to know how monetary policy is conducted. Central banks perform
monetary policy to achieve low and stable inflation and to avoid large fluctuations in
output and employment due to shocks. A rule for monetary policy that has been
proposed is the Taylor Rule. This is a rule proposed by John Taylor (economist at
Stanford) in 1993 that appeared to be consistent with the actual behaviour of the
Federal Reserve (and also of other central bank institutions). The idea is that the
Central Bank is targeting a particular interest rate to respond to changes in the
inflation and in the real economic activity (changes in Real GDP). Suppose that the
Central Bank wants to achieve an inflation rate of 2%. This is the inflation target of
the central bank. Then the typical Taylor Rule is:
Y −Y 

i = π + 2 + 0.5(π − 2) − 0.5
 Y 
where 100 x
Y −Y
= GDP gap = = percent by which real GDP is below its natural
Y
rate (denoted by Y ). If π = 2 (actual inflation is equal to the target) and output is at
its natural rate, then interest rate is targeted at 4 percent (meaning that the central bank
sets the interest rate at 4%). For each one-point increase in π, monetary policy is
automatically tightened to raise the interest rate by 1.5. For each one percentage point
that GDP falls below its natural rate, monetary policy automatically eases to reduce
the interest rate by 0.5. In the graph below we see how the interest rate implied by the
Taylor Rule matches the interest rate set by the Federal Reserve in US (the federal
funds rate).
12
It appears indeed that the Taylor Rule was consistent with the behaviour of the FED
since 1987. Therefore, it is like the FED was implicitly behaving according to the
Taylor Rule even if the FED has never officially communicated to the public that it
followed that rule. If this is the way monetary policy is conducted we can use this
Taylor Rule in our IS-LM model instead of the LM curve.
Consider the following version of the Taylor Rule:
i = i + ϕ π (π − π ) + ϕ Y (Y − Y )
1)
Equation 1) looks slightly different than the empirical version of the Taylor Rule
described above but it is exactly the same thing.
In equation 1) i is the natural nominal interest rate that is the interest rate that the
central bank would set if both inflation and real output were at the target (for example,
the 4% described above), π is the actual inflation rate, π is the target inflation rate,
Y is real output (real GDP) and Y is the target output that is equal to the natural level
(full employment). The coefficients ϕ π and ϕ Y are positive and denotes the
responsiveness of monetary policy to deviations of inflation from the target and of
output from the target respectively. Since we are in the IS-LM world prices are fixed
in the short-run (meaning inflation is zero) and therefore our Taylor Rule can be
written as:
i = i + ϕ Y (Y − Y )
2)
We call equation 2) the TR schedule when prices are fixed and so inflation is zero.
Notice that in this case the nominal interest rate and the real interest rate are the same
13
as in the usual IS-LM model. This tells us that the central bank raises the interest rate
whenever output is above the natural level and it decreases the interest rate whenever
output is below the target. In this case monetary policy is particularly passive. The TR
schedule is positively sloped in the (i, Y)-space since ϕ Y > 0 , like the LM curve. Here
we use the nominal and not real interest rate, but with fixed price it does not make any
difference which one you use. However, the central bank in reality does control
directly the nominal interest rate (not the real). Graphically the IS-TR model looks
like:
i
TR
i
IS
Y
Y
This is pretty much similar to the IS-LM model. Assume for simplicity that the IS and
the TR schedules intersect at the natural level of output. Now we can see the impact
of a change in fiscal policy (i.e. a shift in the IS curve) on output and the interest rate,
taking the endogenous change in money supply into account. Suppose an increase in
public expenditure so that the IS curve shifts to the right to IS’.
14
i
TR
i’
i
IS’
IS
Y
Y’
YA
Y
The result is an increase in output above the target and so the response of the central
bank is to increase the nominal interest rate as described by the Taylor Rule. Notice
the slight difference with the IS-LM model. In the IS-LM model the increase in the
interest rate after an increase in public expenditure (crowding out) is due to the
changes in the money market (higher Y means higher money demand, for a given
money supply, interest rate must increase). Here, the increase in the interest rate is
due to the fact that the central bank is following a Taylor Rule. Notice that the
fluctuations in output is reduced because if the interest rate was kept constant the new
level of output would have been YA > Y’. What happens here in the money market?
Still an increase in public expenditure increases the money demand. However, now
the central bank changes the money supply in such a way to achieve the interest rate
prescribed by the Taylor Rule. Then we can ask the following: will the result we had
be equal to one we would obtain with an LM curve? Or another way to say: is the TR
schedule the same as the LM curve? The answer is generally no.
To see this, we bring together the IS-TR model with the money market equilibrium:
15
i
(M / P)
( M / P)1
i
TR
ia
C
B
D
La
i’
IS’
A
i
L
IS
M/P
Y
Y’
Suppose that after the increase in output money demand increases from L to La. The
new interest rate in the money market, if money supply remains fixed, should be ia,
that is higher than the interest rate set by according to the Taylor Rule. In particular,
we should be at point C in the old IS-LM model, and the LM curve should be steeper
than the Taylor Rule (the LM curve will be the curve that connects point A with point
C. So depending on how the money demand reacts to changes in output we may have
a different response in terms of the interest rate depending if we use the IS-LM or the
IS-TR model (obviously we can have also the case where the TR schedule is steeper
than the LM curve). In this case the central bank will increase the money supply in
order to reach point D on the money demand schedule La, since that will guarantee the
interest rate i’ to arise in equilibrium. How do we model a change in monetary policy
in the IS-TR model? For example the central bank may change the natural interest
rate. This will shift the TR schedule. Another possibility is that central bank may
change the way it responds to deviation of output from the natural level. In this case it
will be the parameter ϕ Y to change. This will rotate the TR schedule.
However, as you can see the qualitative results of the IS-LM model are still the same
even if we consider a more realistic form of monetary policy. What is different is the
mechanism behind those results (for example, we still have the crowding out effect,
but here the way it arises is different from the way it arises in the classical IS-LM
model). Another main difference with the IS-LM model, is that here the Central Bank
does not decide to be expansionary or contractionary. In the IS-LM model, by setting
the level of money supply, the central bank can decide to increase money supply in
16
Y
order to increase output (expansionary monetary policy). In the IS-TR model, the
Central Bank merely reacts to economic conditions according to the rule it follows (in
this case a Taylor Rule). This seems to be consistent with what central banks tend to
do in normal times: caring about price stability without trying to affect too much
economic output. However, you should notice that during the last credit crunch some
central banks (Bank of England, Federal Reserve) have adopted the quantitative
easing approach that is like the increasing money supply in the original IS-LM model.
17