Chapter 20 IS-LM dynamics with forward-looking expectations and lecture notes/Ch20-201… · namic extension of the IS-LM model. ... We assume expectations are rational (model consistent)

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.

663

664

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

C. Groth, Lecture notes in macroeconomics, (mimeo) 2011

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.


666



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


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


668



(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.


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)


670



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.



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


672



= 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.



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


674



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



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.


676



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



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


678



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



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).


680



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.



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.


682



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,



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)


684



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.


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


686



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 )


20.6. Exercises 687

Figure 20.17: An anticipated downward shift in and responses of the long-tern

rate, output, and money supply (regime )


688



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.


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Chapter 20 IS-LM dynamics with forward-looking expectations and lecture notes/Ch20-201… · namic extension of the IS-LM model. ... We assume expectations are rational (model consistent)