Chapter 20 IS-LM dynamics with forward-looking expectations and lecture notes/Ch20-201… · namic...
Transcript of Chapter 20 IS-LM dynamics with forward-looking expectations and lecture notes/Ch20-201… · namic...
Chapter 20
IS-LM dynamics with
forward-looking expectations
A main weakness of the standard IS-LM model as described in Section 19.4
of the previous chapter is the absence of dynamics and endogenous forward-
looking expectations. This motivated (Blanchard, 1981) to develop a dy-
namic extension of the IS-LM model. The key elements are:
• The focus is manifestly on short-run mechanisms. Therefore the modelallows for a possible deviation of output from demand− the adjustmentof output to demand takes time (so that in the meanwhile changes in
order books and inventories occur). In this way the model highlights
the interaction between fast-moving asset markets and slower-moving
goods markets.
• There are three financial assets, money, a short-term bond, and a long-term bond. Accordingly there is a distinction between the short-term
interest rate and the long-term interest rate. Thereby changes in the
term structure of interest rates, known as the yield curve, can be stud-
ied.
• Agents have endogenous forward-looking expectations. These expecta-tions are assumed to be rational (model consistent). Since there are no
stochastic elements in the model, in fact perfect foresight is assumed.
These elements lead to a richer model which conveys the central message
of Keynesian theory. The equilibrating role in the output market is taken
by output changes generated by discrepancies between aggregate demand
and supply. In addition, letting agents have endogenous forward-looking ex-
pectations is an important step towards a useful model of macrodynamics.
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CHAPTER 20. IS-LM DYNAMICS WITH FORWARD-
LOOKING EXPECTATIONS
Moreover, the distinction between short-term and long-term interest rates
opens up for a succinct indicator of expectations. Finally, this distinction al-
lows a more realistic account of what monetary policy can directly accomplish
and what it cannot; while the central bank controls the short-term interest
rate (at least under “normal circumstances”), investment and consumption
primarily depend on the long-term rate.
20.1 A dynamic IS-LM model
As in the previous chapter we consider a closed industrialized economy where
manufacturing goods and services are supplied in markets with imperfect
competition and prices set in advance by firms operating under conditions of
excess capacity.
Let denote the long-term real interest rate at time (the internal rate
of return on a consol, cf. below). By replacing the short-term interest rate in
the aggregate demand function from the simple IS-LM model of the previous
chapter by the long-term rate, we obtain a better description of effective
aggregate demand:
= ( − ( + ()) ) + ( ) + ≡ ( ) + where
0 1 0−1 0
Generally notation is as in Section 19.4. Disposable private income is −Twhere T = + ( )) 0 ≤ 0( ) 1 and is a constant parameter
reflecting “tightness” of discretionary fiscal policy; represents government
spending on goods and services. In order not to have too many balls in the
air at the same time, we ignore the stochastic terms here as well as in the
money demand function below. As the model is in continuous time, including
stochastic terms would require the use of stochastic differential calculus.
The positive dependency of aggregate demand on current aggregate in-
come, reflects primarily that private consumption depends positively on
disposable income. That current disposable income has this role can be seen
as reflecting that a substantial fraction of households are credit-constrained.
Perceived human wealth (the present value of the expected after-tax labor
income stream), which at the microeconomic level is a major determinant of
consumption, is also likely to depend positively on current earnings. Simi-
larly, capital investment by demand-constrained firms will depend positively
on current economic activity, to the extent that this activity signals the
level of demand in the future.
The negative dependency of aggregate demand on reflects first and fore-
most that the capital investment depends negatively on the long-term interest
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20.1. A dynamic IS-LM model 665
rate. Firm’s investment in production equipment and structures is normally
an endeavour with a lengthy time horizon. Similarly, the households’ invest-
ment in durable consumption goods (including housing) is based on medium-
or long-term considerations. Indeed, households’ general consumption level
depends on . A rise in induces a negative substitution effect and on av-
erage possibly also a negative wealth effect. Changes in household’s wealth,
whether in the form of human wealth, equity shares, or housing estate, are
triggered by changes in the long-term interest rate.
Because of the short-run perspective of the model, explicit reference to
the available capital stock in the investment function, is suppressed.
The continuous-time framework is convenient by making it easy to operate
with different speeds of adjustment for different variables. With respect to
the speed of adjustment to changes in demand, we shall operate with a
threepartition as envisaged in Table 20.1 (where “output” is understood to
consist primarily of goods and services with elastic supply, in contrast to
agricultural products and construction).
Table 20.1. Three speeds of adjustment
Item Adjustment to demand shifts
asset prices fast
output medium
prices on output low or absent
Whereas the model lets asset prices adjust immediately, the adjustment of
output to demand takes some time. This is modeled like an error-correction:
≡
= (
− ) (20.1)
= (( ) +− )
where 0 is a constant adjustment speed. During the adjustment process
also demand changes (since the output level and fast-moving asset prices
are among the determinants of demand). The difference between demand
and output is made up of changes in order books and inventories behind the
scene. In a more elaborate version of the model unintended positive and
negative inventory investment should result in a feedback on future demand
and supply; in this first approach we ignore this.
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The rest of the model is more standard:
= ( ) 0 0 (20.2)
=1
(20.3)
≡ − (20.4)
1 +
= (20.5)
≡
= (20.6)
Equation (20.2) is the usual equilibrium condition for the money market,
“money” being interpreted as currency in circulation and checkable deposits
in commercial banks. As financial markets in practice adjust very fast, the
model assumes clearing in the asset markets at any instant. Real money
demand depends positively on (a proxy for the number of transactions
per time unit for which money is needed) and negatively on the short-term
nominal interest rate, the opportunity cost of holding money. For a given
money supply, the equilibrium is brought about by immediate adjustment
of the short-term interest rate.
In equation (20.3) appears the new variable which is the real price of
a long-term bond, here identified as an indexed consol (sometimes called a
perpetuity) paying to the owner a constant stream of one unit of account per
time unit in the indefinite future. The equation tells us that the long-term
interest rate at time is the reciprocal of the market price of a consol at time
This is just another way of saying that the long-term rate, , is defined as
the internal rate of return on the consol. Indeed, the internal rate of return
is that number, which satisfies the equation
=
Z ∞
1 · −(−) =
∙−(−)
−
¸∞
=1
(20.7)
Thus the long-term interest rate is that discount rate which transforms
the payment stream on the consol into a present value equal to the market
price of the consol at time . Inverting (20.7) gives (20.3).1
1Similarly, in discrete time, with payments at the end of each period, we would have
=
∞X=+1
1
(1 +)−=
11+
1− 11+
=1
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20.1. A dynamic IS-LM model 667
Equation (20.4) defines the ex ante short-term real interest rate as the
short-term nominal interest rate minus the expected inflation rate. Equation
(20.5) can be interpreted as a no-arbitrage condition saying that the expected
real rate of return on the consol (including a possible expected capital gain
or capital loss, depending on the sign of ) must in equilibrium equal the
real rate of return on the alternative asset, the short-term bond. In general,
in view of the higher risk associated with long-term claims, presumably a
positive risk premium should be added on the right-hand side of (20.5). We
shall ignore uncertainty, however, so that there is no risk premium.2 Finally,
equation (20.6) says that within the relatively short time perspective of the
model, the inflation rate is constant at an exogenous level, . The interpre-
tation is that price changes mainly reflect changes in costs and that these
changes are relatively steady.
We assume expectations are rational (model consistent). As there is no
uncertainty in the model (no stochastic elements), this assumption amounts
to perfect foresight. We thus have = and = = Therefore,
equation (20.4) reduces to = = − for all .
Whichever monetary regime we are going to consider below, the model
can be reduced to two coupled first-order differential equations in and
The first differential equation is (20.1) above. As to the second, note that
from (20.3) we have = − Substituting into (20.5), where = ,
and using again (20.3) gives
1
+
= −
= = − (20.8)
in view of (20.4) with = = − . By reordering,
= ( − + ) (20.9)
where the determination of depends on the monetary regime.
Before considering alternative policy regimes, we shall emphasize an equa-
tion which is very useful for the economic interpretation of the ensuing dy-
namics. Assuming no speculative bubbles (see below), the no-arbitrage for-
mula (20.5) is equivalent to a statement saying that the market value of the
consol equals the fundamental value of the consol. By fundamental value is
meant the present value of the future dividends from the consol, using the
2If a constant risk premium were added, the dynamics of the model will only be slightly
modified
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(expected) future short-term interest rates as discount rate:
=
Z ∞
1 · − so that (20.10)
=1
=
1R∞1 · −
This equivalence result can be obtained by solving the differential equa-
tion (20.8), given that there are no speculative bubbles (see Appendix A). In
other words: the long-term rate, is a kind of average of the (expected)
future short-term rates, The higher are these, the lower is and the
higher is Note that if is expected to be a constant, (20.10) simplifies
to
=1R∞
−(−)
=1
1=
Or if for example is expected to be increasing, we get
=1R∞
−
1
1=
20.2 Monetary policy regimes
As in the static IS-LM model of Chapter 19, we shall focus on three different
monetary policy regimes. The two first are regime , where the real money
supply is the policy instrument, and regime where the short-term nomi-
nal interest rate is the policy instrument. This second regime is by far the
simplest one and in some sense closer to what modern monetary policy is
about. Regime is also of interest, however, both because of its historical
appeal and because it yields impressive dynamics. In addition, regime
is of significance because of its partial affinity with what happens under a
counter-cyclical interest rate rule. Our third monetary policy regime is in
fact an example of such a rule, which we name regime 0The assumption of perfect foresight means that the agents’ expectations
coincide with the prediction of our deterministic model. Once-for-all shocks
may occur, but only so seldom that agents ignore the possibility that a new
surprise may occur later.3 Note that if a shock occurs, the time derivatives
of the variables should be interpreted as right-hand derivatives, e.g., ≡lim∆→0( (+∆)− ())∆
3In Part VI of this text we consider models where the exogenous variables are stochastic
and agents’ behavior take the uncertainty into ccount.
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20.2. Monetary policy regimes 669
Y
R
E
IS
LM
A
0Y
0R
0Y Y
R
Figure 20.1: Phase diagram when is instrument.
In addition to the inflation rate, the fiscal policy variables, and
are exogenous. And the initial values 0 and 0 are historically given since
in this model not only the price level but also the output level is sluggish.
20.2.1 Policy regime : Money stock as instrument
Here we assume that the central bank maintains the real money supply,
at a constant level, by letting the nominal money supply follow
the path:
= 0 =0
Equation (20.2) then reads ( ) = This equation defines as an
implicit function of and , i.e.,
= () with = − 0 = 1 0 (20.11)
Inserting this into (20.9), we have
= [ − () + ] (20.12)
which together with (20.1) constitutes our dynamic system in the two en-
dogenous variables, and . For convenience, we repeat (20.1) here:
= (( ) +− ) 0 0 1 0 ∈ (−1 0)(20.13)
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Phase diagram
Given 0, (20.12) implies
T 0 for T ()− respectively. (20.14)
We have |=0 = = − 0 that is, for real money demand to
equal a given real money supply, a higher volume of transactions must go
hand in hand with a higher nominal short-term interest rate which in turn,
for given inflation, requires a higher real interest rate. The = 0 locus is
illustrated as the upward sloping curve, LM, in Fig. 20.1.
From (20.13) we have
T 0 for ( ) + T respectively. (20.15)
We have |=0 = (1 − ) 0 that is, higher aggregate demand in
equilibrium requires a lower interest rate. The = 0 locus is illustrated as
the downward sloping curve, IS, in Fig. 20.1. In addition, the figure shows
the direction of movement in the different regions, as described by (20.14)
and (20.15). We see that the steady state point, E, with coordinates ( )
is a saddle point.4 This implies that two and only two solution paths −one from each side − converges towards E. These two saddle paths, whichtogether make up the stable arm, are shown in the figure (their slope must
be positive, according to the arrows). Also the unstable arm is displayed in
the figure (the negatively sloped stippled line).
The initial value of output, 0 is in this model predetermined, i.e., de-
termined by ’s previous history; relative to the short time horizon of the
model, output adjustment takes time. Hence, at time = 0 the economy
must be somewhere on the vertical line = 0. The question is then whether
there can be rational asset price bubbles. An asset price bubble, also called
a speculative bubble, is present if the market value of an asset for some time
differs from its fundamental value; the fundamental value is the present value
of the expected future dividends from the asset, as defined in (20.10). A ra-
tional asset price bubble is an asset price bubble that is consistent with the
no-arbitrage condition (20.5) under rational expectations.
4More formally, the determinant of the Jacobian matrix for the right-hand sides
of the two differential equations, evaluated at the steady-state point, ( , ), is
£( − 1) +
¤ 0. Hence, the two eigenvalues are of opposite sign. In this
and the next chapter steady-state values are marked by a bar rather than an asterix. This
is in order to distinguish these “pseudo steady states” (where underlying slow changes
in prices, capital stock, and technology are completely ignored) from “genuine” long-run
steady states.
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20.2. Monetary policy regimes 671
Now, in the present framework a positive rational bubble can not generally
be ruled out. Because consols (like equity shares) have no terminal date, a
forever growing asset price bubble generated by self-fulfilling expectations is
theoretically possible. At least, within this model there is nothing to rule
it out. In Fig. 20.1 any of the diverging paths with ultimately falling,
and therefore the asset price ultimately rising, could reflect such a bubble.
But negative rational bubbles can be ruled out. Essentially, this is because
they presuppose that the market price of the consol initially drops below the
present value of the future dividends (the right-hand side of (20.10)). But
in such a situation everyone would want to buy the consol and enjoy the
dividends. The resulting excess demand immediately drives the asset price
back.
In any case, in view of the simplistic character of the model, it does not
provide an appropriate framework for bubble analysis. We will have more to
say about bubbles later in this book. Here we simply assume that the market
participants never have bubble expectations. Then rational bubbles will not
arise and neither the explosive nor the implosive paths of in Fig. 20.1 can
materialize. We are then left with the saddle path, the path AE in the figure,
as the unique solution to the model. As the figure is drawn, 0 And the
long-term interest rate is relatively low so that demand exceeds production
and gradually pulls production upward. Hereby demand is stimulated, but
less than one-to-one so, both because the marginal propensity to spend is
less than one and because also the interest rate rises. Ultimately, say within
a year, the economy settles down at the steady state.
Impulse response dynamics
Let us consider the effects of level shifts in and respectively. Suppose
that the economy has been in its steady state until time 0 In the steady
state we have = = . Then either fiscal or monetary policy changes. The
question is what the effects on , and are. The answer depends very
much on whether we consider an unanticipated change in the policy variable
in question ( or) at time 0 or an anticipated change. As to an anticipated
change, we can imagine that the government or the central bank at time 0credibly announces a shift to take place at time 1 0 From this derives
the term “announcement effects”, synonymous with “anticipation effects”.
To prepare the ground, consider first the question: how are the IS and LM
curves affected by shifts in and respectively? We have, from (20.15),|=0 = −1 0 that is, a shift to a higher moves the = 0 locus
(the IS curve) upwards. But the = 0 locus is not affected by a shift in .
On the other hand, the = 0 locus is not affected by a shift in . But the
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LOOKING EXPECTATIONS
= 0 locus (the LM curve) depends on and moves downwards, if is
increased, since |=0 = 0 from (20.14).
We now consider a series of policy changes, some of which are unantici-
pated, whereas others are anticipated.
t 0t
G
'G
G
Figure 20.2: Unanticipated upward shift in
(a) The effect of an unanticipated upward shift in The upward
shift in is shown in Fig. 20.2. Since and are kept unchanged, the
higher must to some extent be debt financed and is thus associated with a
higher amount of outstanding government bonds. When shifts, the long-
term interest rate jumps up to , cf. Fig. 20.3, reflecting that the market
value of the consol jumps down. The explanation is as follows. The higher
implies higher output demand, by (19.17). This triggers an expectation of
increasing (see (20.13)), and therefore also an expectation of increasing
and in view of (20.11). The implication is, by (20.10), a lower 0 and a
higher 0 , as illustrated in Fig. 20.4. After 0 output and the short-term
rate gradually increase toward their new steady state values, 0 and 0respectively, as shown by Fig. 20.4. As time proceeds and the economy gets
closer to the expected high future values of , these higher values gradually
become dominating in the determination of in (20.10). Hence, after 0 also
gradually increases toward its new steady state value, the same as that for
By dampening output demand the higher implies a financial crowding-
out effect on production.5 After 0 during the transition to the new steady
state, we have because “anticipates” all the future increases in
5The crowding out is only partial, because still increases.
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20.2. Monetary policy regimes 673
A
E
'E
IS
'IS
LM
R
Y Y 'Y
R
'R
Figure 20.3: Phase portrait of an unanticipated upward shift in (regime )
and incorporates them, cf. (20.10). Note also that (20.8) implies
= + T for T 0 respectively
For example, 0 reflects that 0 that is, a capital loss is expected. To
compensate for this, the level of (which always equals 1) must be higher
than such that the no-arbitrage condition (20.5) is still satisfied.
Formulas for the steady-state effects of the change in can be found by
using the comparative statics method of Section 19.5 on the two steady-state
equations = ( ) + and = ( ) with the two endogenous
variables and Given the preparatory work already done, a more simple
method is to substitute = ( ) into the first-mentioned steady-state
equation to get = ( ( ) ) + Taking the total differential on
both sides gives = + + from which follows, by (20.11),
=
1
1− +
0
From ( ) = we get 0 = + = ( )+ = 0
so that
= −
1− +
0
Since our steady-state equations corresponds exactly to the IS and LM equa-
tions for the static IS-LM model of Chapter 19, the output and interest rate
multipliers w.r.t. are the same.
(b) The effect of an anticipated upward shift in We assume that
the private sector at time 0 becomes aware that will shift to a higher level
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Figure 20.4: Responses of interest rates and output to an unanticipated upward
shift in (regime )
at time 1, cf. the upper panel of Fig. 20.4. The implied expectation that
the short-term interest rate will in the future rise towards a higher level, 0,immediately triggers an upward jump in the long-term rate, . To what
level? In order to find out, note that the market participants understand
that from time 1 the economy will move along the new saddle path cor-
responding to the new steady state, E’, in Fig. 20.5. The market price,
of the consol cannot have an expected discontinuity at time 1, since such a
jump would imply an infinite expected capital loss (or capital gain) per time
unit immediately before = 1 by holding long-term bonds. Anticipating for
example a capital loss, the market participants would want to sell long-term
bonds in advance. The implied excess supply would generate an adjustment
of downwards until no longer a jump is expected to occur at time 1. If
instead a capital gain is anticipated, an excess demand would arise. This
would generate in advance an upward adjustment of thus defeating the
expected capital gain. This is the general principle that arbitrage prevents
an expected jump in an asset price.
In the time interval (0 1) the dynamics are determined by the “old”
phase diagram, based on the no-arbitrage condition which rules up to time
1. In this time interval the economy must follow that path (AB in Fig. 20.5),
which, starting from a point on the vertical line = , takes precisely 1−0units of time to reach the new saddle path. At time 0, therefore, jumps to
exactly the level in Fig. 20.5.6 This upward jump has a contractionary
6Note that is unique. Indeed, imagine that the jump, − was smaller than in
Fig. 20.5. Then, not only would there be a longer way along the road to the new saddle
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20.2. Monetary policy regimes 675
LM
IS
'IS
'E
E
Y
R
Y 'Y
R
'R
A
B
0Y
0R
Figure 20.5: Phase portrait of an anticipated upward shift in (regime )
effect on output demand. So output starts falling as shown by figures 20.5
and 20.6. This is because the potentially counteracting force, the increase
in has not yet taken place. Not until time 1 when shifts to 0, doesoutput begin to rise. In the long run both , and are higher than in
the old steady state.
There are two interesting features. First, in regime a credible announce-
ment of future expansive fiscal policy can have a temporary contractionary
effect when the announcement occurs. This is due to financial crowding out.
The second feature relates to the term structure of interest rates, also called
the yield curve. The relationship between the internal rate of return on finan-
cial assets and their time to maturity is called the term structure of interest
rates. Fig. 20.6 shows that the term structure “twists” in the time interval
(0, 1). The long-term rate rises, because the time where a higher (and
thereby a higher ) is expected to show up, is getting nearer. But at the
same time the short-term rate is falling because of the falling transaction
need for money implied by the initially falling triggered by the rise in the
long-term interest rate.7
path, but the system would also start from a position closer to the “old” steady-state
point, E. This implies an initially lower adjustment speed.7A conceivable objection to the model in this context is that it does not take into ac-
count that consumption and investment are likely to depend positively on expected future
aggregate income, so that the hypothetical temporary decrease in demand and output
never materializes. On the other hand, the model has in fact been seen as an explana-
tion that president Ronald Reagan’s announced tax cut in the USA 1981-83 (combined
with the strict monetary policy aiming at disinflation) were associated with several years’
recession.
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Figure 20.6: An anticipated upward shift in and the responses of interest rates
and output (regime )
The theory of the term structure
What we have just seen is the expectations theory of the term structure in
action. Empirically, the term structure of interest rates tends to be upward-
sloping, but certainly not always and it may suddenly shift. The theory
of the term structure of interest rates generally focuses on two explanatory
factors. One is uncertainty and this factor tends to imply a positive slope
because the greater uncertainty generally associated with long-term bonds
generates a risk premium on these. About this factor the present model has
nothing to say since it ignores uncertainty. But the model has something to
say about the other factor, namely expectations. Indeed, the model quite well
exemplifies what is called the expectations theory of the term structure. In its
simplest form this theory ignores uncertainty and says that if the short-term
interest rate is expected to rise in the future, the long-term rate today will
tend to be higher than the short-term rate today. This is because, absent
uncertainty, the long-term rate incorporates (is a kind of average of) the
expected future short-term rates, as indicated by (20.10). Similarly, if the
short-term interest rate is expected to fall in the future, the long-term rate
today will, everything else equal, tend to be lower than the short-term rate
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20.2. Monetary policy regimes 677
IS
E
'E
A
LM
'LM
Y
R
Y'Y
R
'R
0Y
0R
Figure 20.7: Phase portrait of an unanticipated downward shift in (regime)
today. Thus, rather than explaining the statistical tendency for the slope
of the term structure to be positive, changes in expectations are important
in explaining changes in the term structure. In practice, most bonds are
denominated in money. Central to the theory is therefore the link between
expected future inflation and the expected future short-term nominal interest
rate. This aspect is not captured by the present model, which ignores changes
in the price level.
(c) The effect of an unanticipated downward shift in The shift in
is shown in the upper panel of Fig. 20.8. The shift triggers, at time 0 an
upward jump in the long-term rate to the level where the new saddle path
is (point A in Fig. 20.7). The explanation is that the fall in money supply
implies an upward jump in the short-term rate at time 0 cf. (20.10). As
indicated by Fig. 20.8, the short-term rate will be expected to remain higher
than before the decline in . The rise in triggers a fall in output demand
and so output gradually adjusts downward as depicted in Fig. 20.8. The
resulting decline in the transactions-motivated demand for money leads to
the gradual fall in the short-term rate towards the new steady state level.
This fall is anticipated by the long-term rate, which is, therefore, at every
point in time after 1 lower than the short-term rate.
It is interesting that when the new policy is introduced, both and
“overshoot” in their adjustment to the new long-run levels. This happens,
because, after 0, both and have to be decreasing, parallel with the
decreasing which implies lower money demand.
Not surprisingly, there is not money neutrality. This is due, of course, to
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Figure 20.8: An unanticipated downward shift in and the responses of interest
rates and output (regime )
the price level being rigid in this model.
To find expressions for the steady-state effects of the change in we
first take the total differential on both sides of ( ( ) ) + = to
get (1− − ) = By (20.11), this gives
=
1− +
0
Hence, = ( ) 0 for 0 From = ( ) we get
= + = ( )+ so that
=
(1− )
1− +
0
Hence, = () 0 for 0 These multipliers are the same
as those for the static IS-LM model.
(d) The effect of an anticipated downward shift in The shift in
is announced at time 0 to take place at time 1 cf. Fig. 20.9. At the time 0
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20.2. Monetary policy regimes 679
IS
LM
'LM
A
B
E
'E
YY'Y
R
R
'R
0Y
0R
Figure 20.9: Phase portrait of an anticipated downward shift in (regime )
of “announcement” jumps to and then gradually increases until time
1. This is due to the expectation that the short-term rate will in the longer
run be higher than in the old steady state. The higher implies a lower
output demand and so output gradually adjusts downward. Then also the
short-term rate moves downward until time 1. In the time interval (0 1)
the dynamics are determined by the “old” phase diagram and the economy
follows that path (AB in Fig. 20.9) which, starting from a point on the
vertical line = takes precisely 1 − 0 units of time to reach the new
saddle path. Since in the time interval (0, 1) increases, while decreases,
we again witness a “twist” in the term structure of interest rates, cf. Fig.
20.10.
Due to the principle that arbitrage prevents an expected jump in an asset
price, exactly at the time 1 of implementation of the tight monetary policy,
the economy reaches the new saddle path generated by the lower money
supply (cf. the point in Fig. 20.9). The fall in triggers a jump upward
in the short-term rate (this is foreseen by everybody, but it implies no
capital loss because the bond is short-term). Output continues falling
towards its new low steady state level, cf. Fig. 20.10. The transactions-
motivated demand for money decreases and therefore gradually decreases
towards the new long-run level which is above the old because is smaller
than before. The long-term rate discounts this gradual fall in in advance
and is therefore, after 1 always lower than . Nevertheless, in the long run
the long-term rate approaches the short-term rate (in this model where there
is no risk premium).
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Figure 20.10: An anticipated downward shift in and responses of interest rates
and output (regime )
20.2.2 Policy regime : The short-term interest rate as
instrument
Here we shall analyze a monetary policy regime where the short-term inter-
est rate is the instrument. The model now takes as an exogenous constant
(a policy parameter, together with the fiscal instruments, and ). Then
the real money supply, has to be endogenous, which reflects that the
central bank through open market operations adjusts the monetary base so
that the actual short-term rate equals the one desired (and usually explic-
itly announced) by the central bank. A common name for this rate is the
money market rate, where “money market” is synonymous with “interbank
market” and refers to “narrow money”, cf. Chapter 16. In the Euro area
the more technical term for the targeted money market rate is the EONIA
(euro overnight index average) and in the US the Federal Funds Rate.8 In
Denmark what comes closest is “Nationalbankens udlånsrente” (the weekly
lending rate of Danmarks Nationalbank).
8As to the evolution of the Federal Funds Rate in response to economic and political
events since 1987 (the Greenspan period), see Appendix C.
C. Groth, Lecture notes in macroeconomics, (mimeo) 2011
20.2. Monetary policy regimes 681
Y
R
E
IS
0Y
0R
0Y Y
i
A
Figure 20.11: Phase diagram when is instrument.
The dynamic system
With exogenous and endogenous, the dynamic system consists of (20.9)
and (20.1), which we repeat here for convenience:
= ( − + ) (20.16)
= (( ) +− ) where (20.17)
0 1 0−1 0
Because does not appear in (20.16), this system is simpler. The system de-
termines the movement of and In the next step the required movement
of is determined by = ( ) = 0( ) from (20.2).
Using a similar method as before we construct the phase diagram, cf. Fig.
20.11. The = 0 locus is now horizontal. The steady state is again a saddle
point and is saddle-point stable. Notice, that here the saddle path coincides
with the = 0 locus.
Dynamic responses to policy changes when the short-term interest
rate is the instrument
Let us again consider effects of permanent level shifts in exogenous variables,
here and Suppose that the economy has been in its steady state until
time 0 In the steady state we have = = − . Then either fiscal policy
or monetary policy shifts. We consider the following three shifts in exogenous
variables:
(a) An unanticipated decrease of See figures 20.13 and 20.14.
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CHAPTER 20. IS-LM DYNAMICS WITH FORWARD-
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E 'E
'IS IS
R
Y Y'Y
i
0Y
0R
Figure 20.12: Phase portrait of an unanticipated downward shift in (regime )
(b) An unanticipated decrease of See figures 20.14 and 20.15 in Appendix
B.
(c) An anticipated decrease of See figures 20.16 and 20.17 in Appendix
B.
As to the anticipated shift in , we imagine that the central bank at time
0 credibly announces the shift in to take place at time 1 0
The figures illustrate the responses. The diagrams should, by now, be
self-explanatory. The only thing to add is that the reader is free to introduce
another interpretation of, say, the exogenous variable . For example,
could be interpreted as measuring consumers’ and investors’ “degree of opti-
mism”. The shift (a) could then be seen as reflecting the change in the “state
of confidence” associated with the worldwide recession in 2001 or in 2008.
The shift (b) could be interpreted as the immediate reaction of the Fed in
the USA. As the public becomes aware of the general recessionary situation,
further decreases of the federal funds rate, are expected and tends also to
be executed. This is what point (c) is about.
20.2.3 Policy regime 0: A counter-cyclical interest raterule
Suppose the central bank conducts stabilization policy by using the interest
rate rule = 0 + 1 where 0 and 1 are constant policy parameters,
C. Groth, Lecture notes in macroeconomics, (mimeo) 2011
20.2. Monetary policy regimes 683
Figure 20.13: An unanticipated downward shift in and responses of interest
rates, output, and money supply (regime )
1 − 0. Then our dynamic system is
= (( ) +− ) 0 1 0−1 0
=£ − (0 + 1) +
¤
These two differential equations determine the time path of ( ) by a
phase diagram similar to that in Fig. 20.1. Responses to unanticipated and
anticipated changes in are qualitatively the same as in regime where
the money stock was the instrument. Qualitatively, the only difference is
that the money stock is no longer an exogenous constant, but has to adjust
according to
= 0(
0 + 1)
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CHAPTER 20. IS-LM DYNAMICS WITH FORWARD-
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in order to let the counter-cyclical interest rule work. In Exercise 20.x the
reader is asked to show that because 0 − , this monetary policy
regime is more stabilizing w.r.t. output than regime
20.3 Discussion
In this chapter we have analyzed a dynamic version of the IS-LM model.
The framework captures the empirically motivated principle that output and
employment in the short run tend to be demand-driven − with quantitiesas the equilibrating factor, while prices only respond little to changes in
aggregate demand (in the model they do not respond at all).
It is a weakness of the simple IS-LM model considered here that the
aggregate behavior of the agents is postulated and not based on an explicit
microeconomic foundation. Yet the consumption and investment functions
can to some extent be defended on a microeconomic basis.9 The dynamic
version of the IS-LM model incorporates wealth effects through changes in
the long-term interest rate,
The simple process assumed for the adjustment of output to changes in
demand is ad hoc. At best it can be seen as a rough approximation to the
microeconomic theory of intended and unintended inventory investment. Of
course, it is also not satisfactory that changes in production and employment
have no wage and price effects at all. At least after some time there should
be wage and price responses. To put it differently: a more satisfactory IS-LM
model calls for an extension with a Phillips curve.10 Then, might generally
in the medium term tend to its natural level.
20.4 Bibliographic notes
A remark on the relationship between our presentation of the model and the
original version in Blanchard (1981) seems appropriate. In fact our presen-
tation corresponds to that in Blanchard and Fischer (1989). In the original
Blanchard (1981) paper, however, the forward-looking variable is Tobin’s
rather than the long-term interest rate, . But since the (real) long-term
interest rate can, in this context, be considered as in essence proportionate
9The proviso “to some extent” refers primarily to the investment function. In a reason-
able investment function, expected future output demand should appear as an argument
(with a positive partial derivative), given we are in a world of imperfect competition.10This makes the model substantially more complicated,. But in fact Blanchard (1981,
last section) does take a first step towards such an extension, ending up with a system of
three coupled differential equations.
C. Groth, Lecture notes in macroeconomics, (mimeo) 2011
20.5. Appendix 685
to the inverse of Tobin’s q, there is essentially no difference. Wealth effects
come true whether the source is interpreted as changes in Tobin’s or the
long-term interest rate.
20.5 Appendix
A. Solving the no-arbitrage equation for in the absence of asset
price bubbles
No text available.
B. More examples of dynamics in policy regime
The figures 20.14 and 20.15 illustrate responses to an unanticipated lowering
of the short-term interest rate, and figures 20.16 and 20.17 illustrate the
responses to an anticipated lowering.
E
IS
R
Y'YY
i
'i 'EA
0Y
0R
Figure 20.14: Phase portrait of an unanticipated downward shift in (regime )
C. The movements of the US federal funds rate 1987-2007
See Fig. 20.18
20.6 Exercises
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CHAPTER 20. IS-LM DYNAMICS WITH FORWARD-
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Figure 20.15: An unanticipated downward shift in and the responses of the
long-term rate, output, and money supply (regime )
E
IS
R
Y 'YY
i
'i 'E
A
0Y
0R B
Figure 20.16: Phase portrait of an anticipated downward shift in (regime )
C. Groth, Lecture notes in macroeconomics, (mimeo) 2011
20.6. Exercises 687
Figure 20.17: An anticipated downward shift in and responses of the long-tern
rate, output, and money supply (regime )
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CHAPTER 20. IS-LM DYNAMICS WITH FORWARD-
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25th of June 2003 - The US
inte rest rate is reduced to a record low of one percent.
11th of September - The terrorist
attacks hit parts of the financial infrastructure. Shortly after the
Fed lowers the inte rest rate toge ther with the European central
bank.
Sping of 2000 - The IT and
internet boom has driven stock prices to a very high level.
September 1998 - The large
hedge fund Long Term Capital Management is on the brink of
failure and threatens to pull down with it twelve of the largest
investment banks.
17th of August 1998 - Russ ia
defaults on its debt. The ruble is devalued creating a huge turmoil
on stock markets.
1st of July 1997 - The financial
crisis in Asia sets off when Thailand announces it is no longer able to fix its exchange rate vis-à-
vis the dolla r.
December 1996 - Alan Greens pan
warns about "Irrational Exuburance" on the stock marke t.
March 1994 - Alan Greenspan
observes , as one of the first, that the United States is entering a
period of extraordinary high productivity growth.
1991 - The US economy is in
recession.
19th of October 1987 - "Black
Monday" on the stock exchange in Wall Street. The Dow Jones
s tock index falls by 22.6 %.
11th of August 1987 - Alan
Greenspan took office as chairman of the Fed, replacing the infla tion
hawk Paul Volcke r.
0%
2%
4%
6%
8%
10%
12%
Q11987
Q11988
Q11989
Q11990
Q11991
Q11992
Q11993
Q11994
Q11995
Q11996
Q11997
Q11998
Q11999
Q12000
Q12001
Q12002
Q12003
Q12004
Q12005
Q12006
US Federal Funds Rate
Figure 20.18: The movements of the US Federal Funds Rate during the period
Alan Greenspan was chairman of the Fed. Source: International Financial Statis-
tics, IMF.
C. Groth, Lecture notes in macroeconomics, (mimeo) 2011