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Chapter 20

IS-LM dynamics with

forward-looking expectations

A main weakness of the standard IS-LM model as described in Section 19.4of 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 model

allows for a possible deviation of output from demand the adjustmentof output to demand takes time (so that in the meanwhile changes inorder books and inventories occur). In this way the model highlightsthe interaction between fast-moving asset markets and slower-movinggoods markets.

There are three financial assets, money, a short-term bond, and a long-term bond. Accordingly there is a distinction between the short-terminterest rate and the long-term interest rate. Thereby changes in theterm 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 nostochastic elements in the model, in fact perfect foresight is assumed.

These elements lead to a richer model which conveys the central messageof Keynesian theory. The equilibrating role in the output market is takenby output changes generated by discrepancies between aggregate demandand supply. In addition, letting agents have endogenous forward-looking ex-pectations is an important step towards a useful model of macrodynamics.

663

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664CHAPTER 20. IS-LM DYNAMICS WITH FORWARD-

LOOKING EXPECTATIONS

Moreover, the distinction between short-term and long-term interest ratesopens up for a succinct indicator of expectations. Finally, this distinction al-lows a more realistic account of what monetary policy can directly accomplishand what it cannot; while the central bank controls the short-term interestrate (at least under normal circumstances), investment and consumptionprimarily 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 imperfectcompetition and prices set in advance by firms operating under conditions ofexcess capacity.

Let denote the long-term real interest rate at time (the internal rateof return on a consol, cf. below). By replacing the short-term interest rate inthe aggregate demand function from the simple IS-LM model of the previouschapter by the long-term rate, we obtain a better description of effectiveaggregate demand:

= ( (+ ()) ) + ( ) + ( ) + where

0 1 01 0Generally notation is as in Section 19.4. Disposable private income is Twhere T = + ()) 0 0() 1 and is a constant parameterreflecting tightness of discretionary fiscal policy; represents governmentspending on goods and services. In order not to have too many balls in theair at the same time, we ignore the stochastic terms here as well as in themoney demand function below. As the model is in continuous time, includingstochastic 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 seenas reflecting that a substantial fraction of households are credit-constrained.Perceived human wealth (the present value of the expected after-tax laborincome stream), which at the microeconomic level is a major determinant ofconsumption, is also likely to depend positively on current earnings. Simi-larly, capital investment by demand-constrained firms will depend positivelyon current economic activity, to the extent that this activity signals thelevel 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. Gr ot h, Lect ure not es i n macr oeconomi cs, ( mi meo) 2011

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20.1. A dynamic IS-LM model 665

rate. Firms investment in production equipment and structures is normallyan 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 leveldepends on. A rise in induces a negative substitution effect and on av-erage possibly also a negative wealth effect. Changes in households wealth,whether in the form of human wealth, equity shares, or housing estate, aretriggered by changes in the long-term interest rate.

Because of the short-run perspective of the model, explicit reference tothe available capital stock in the investment function, is suppressed.

The continuous-time framework is convenient by making it easy to operatewith different speeds of adjustment for different variables. With respect tothe speed of adjustment to changes in demand, we shall operate with athreepartition as envisaged in Table 20.1 (where output is understood toconsist primarily of goods and services with elastic supply, in contrast toagricultural products and construction).

Table 20.1. Three speeds of adjustment

Item Adjustment to demand shifts

asset prices fastoutput mediumprices on output low or absent

Whereas the model lets asset prices adjust immediately, the adjustment ofoutput to demand takes some time. This is modeled like an error-correction:

=( ) (20.1)

= (( ) + )

where 0 is a constant adjustment speed. During the adjustment processalso demand changes (since the output level and fast-moving asset pricesare among the determinants of demand). The difference between demandand output is made up of changes in order books and inventories behind thescene. In a more elaborate version of the model unintended positive andnegative inventory investment should result in a feedback on future demandand 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 depositsin commercial banks. As financial markets in practice adjust very fast, themodel assumes clearing in the asset markets at any instant. Real moneydemand depends positively on (a proxy for the number of transactionsper time unit for which money is needed) and negatively on the short-termnominal interest rate, the opportunity cost of holding money. For a givenmoney supply, the equilibrium is brought about by immediate adjustmentof the short-term interest rate.

In equation (20.3) appears the new variable which is the real price ofa long-term bond, here identified as an indexed consol (sometimes called aperpetuity) paying to the owner a constant stream of one unit of account pertime unit in the indefinite future. The equation tells us that the long-terminterest rate at time is the reciprocal of the market price of a consol at timeThis is just another way of saying that the long-term rate, , is defined asthe internal rate of return on the consol. Indeed, the internal rate of returnis that number, which satisfies the equation

= Z

1 ()= ()

= 1

(20.7)

Thus the long-term interest rate is that discount rate which transformsthe payment stream on the consol into a present value equal to the marketprice 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 + ) =

1

1+

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 theshort-term nominal interest rate minus the expected inflation rate. Equation(20.5) can be interpreted as a no-arbitrage condition saying that the expectedreal rate of return on the consol (including a possible expected capital gainor capital loss, depending on the sign of ) must in equilibrium equal thereal 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 apositive risk premium should be added on the right-hand side of (20.5). Weshall 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 infl

ation rate is constant at an exogenous level, . The interpre-tation is that price changes mainly reflect changes in costs and that thesechanges are relatively steady.

We assume expectations are rational (model consistent). As there is nouncertainty in the model (no stochastic elements), this assumption amountsto perfect foresight. We thus have

= and

= = Therefore,equation (20.4) reduces to

= = for all.

Whichever monetary regime we are going to consider below, the modelcan be reduced to two coupled first-order differential equations in andThe 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 theconsol equals the fundamental valueof the consol. By fundamental value ismeant the present value of the future dividends from the consol, using the

2 If aconstantrisk premium were added, the dynamics of the model will only be slightlymodified


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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). Inother 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 thehigher isNote that ifis expected to be a constant, (20.10) simplifiesto

= 1R

()

= 1

1 =

Or if for exampleis expected to be increasing, we get

= 1R

1

1=

20.2 Monetary policy regimesAs in the static IS-LM model of Chapter 19, we shall focus on three differentmonetary policy regimes. The two first are regime ,where the real moneysupply is the policy instrument, and regime where the short-term nomi-nal interest rate is the policy instrument. This second regime is by far thesimplest one and in some sense closer to what modern monetary policy isabout. Regime is also of interest, however, both because of its historicalappeal 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 infact 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 shocksmay occur, but only so seldom that agents ignore the possibility that a newsurprise may occur later.3 Note that if a shock occurs, the time derivativesof the variables should be interpreted as right-hand derivatives, e.g., lim0(( +) ())

3In Part VI of this text we consider models where the exogenous variables are stochasticand agents behavior take the uncertainty into ccount.


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20.2. Monetary policy regimes 669

Y

R

E

IS

LM

A0Y

0R

0Y Y

R

Figure 20.1: Phase diagram when is instrument.

In addition to the inflation rate, the fiscal policy variables, andare exogenous. And the initial values0 and0 are historically given sincein 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 followthe path:

= 0= 0

Equation (20.2) then reads ( ) =This equation defines as animplicit 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 toequal a given real money supply, a higher volume of transactions must gohand 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 isillustrated 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 asthe downward sloping curve, IS, in Fig. 20.1. In addition, the figure showsthe 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 mustbe positive, according to the arrows). Also the unstable arm is displayed inthe 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 themodel, output adjustment takes time. Hence, at time = 0 the economymust be somewhere on the vertical line =0. The question is then whetherthere can be rational asset price bubbles. Anasset price bubble, also calleda speculative bubble, is present if the market value of an asset for some time

diff

ers from itsfundamental value

; the fundamental value is the present valueof the expected future dividends from the asset, as defined in (20.10). Ara-tionalasset price bubble is an asset price bubble that is consistent with theno-arbitrage condition (20.5) under rational expectations.

4More formally, the determinant of the Jacobian matrix for the right-hand sidesof 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. Thisis in order to distinguish these pseudo steady states (where underlying slow changesin prices, capital stock, and technology are completely ignored) from genuine long-runsteady states.


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Now, in the present framework apositiverational bubble can not generallybe ruled out. Because consols (like equity shares) have no terminal date, aforever growing asset price bubble generated by self-fulfilling expectations istheoretically possible. At least, within this model there is nothing to ruleit out. In Fig. 20.1 any of the diverging paths with ultimately falling,and therefore the asset priceultimately rising, could reflect such a bubble.Butnegativerational bubbles can be ruled out. Essentially, this is becausethey presuppose that the market price of the consol initially drops below thepresent value of the future dividends (the right-hand side of (20.10)). Butin such a situation everyone would want to buy the consol and enjoy the

dividends. The resulting excess demand immediately drives the asset priceback.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 tosay about bubbles later in this book. Here we simply assume that the marketparticipants never have bubble expectations. Then rational bubbles will notarise and neither the explosive nor the implosive paths of in Fig. 20.1 canmaterialize. 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 thelong-term interest rate is relatively low so that demand exceeds productionand gradually pulls production upward. Hereby demand is stimulated, butless than one-to-one so, both because the marginal propensity to spend isless than one and because also the interest rate rises. Ultimately, say withina year, the economy settles down at the steady state.

Impulse response dynamics

Let us consider the effects of level shifts in and respectively. Supposethat the economy has been in its steady state until time 0 In the steadystate we have = = . Then either fiscal or monetary policy changes. Thequestion is what the effects on , and are. The answer depends very

much on whether we consider an unanticipatedchange in the policy variablein question (or )at time0or ananticipatedchange. As to an anticipatedchange, 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 derivesthe term announcement effects, synonymous with anticipation effects.

To prepare the ground, considerfirst the question: how are the IS and LMcurves 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 = 0locus 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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= 0 locus (the LM curve) depends on and moves downwards, if isincreased, since

| =0 = 0from (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 upwardshift in is shown in Fig. 20.2. Since and are kept unchanged, thehigher must to some extent be debt financed and is thus associated with ahigher amount of outstanding government bonds. When shifts, the long-term interest rate jumps up to , cf. Fig. 20.3, reflecting that the marketvalue of the consol jumps down. The explanation is as follows. The higherimplies higher output demand, by (19.17). This triggers an expectation ofincreasing(see (20.13)), and therefore also an expectation of increasing and in view of (20.11). The implication is, by (20.10), a lower0 and ahigher0, as illustrated in Fig. 20.4. After0outputand the short-termrate gradually increase toward their new steady state values, 0 and 0

respectively, as shown by Fig. 20.4. As time proceeds and the economy getscloser to the expected high future values of, these higher values graduallybecome dominating in the determination of in (20.10). Hence, after0 alsogradually 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 steadystate, we have because anticipates all the future increases in

5The crowding out is only partial, because still increases.


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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 0that is, a capital loss is expected. Tocompensate for this, thelevel of (which always equals1)must be higherthan such that the no-arbitrage condition (20.5) is still satisfied.

Formulas for the steady-state effects of the change in can be found byusing the comparative statics method of Section 19.5 on the two steady-stateequations = ( ) + and = ( ) with the two endogenousvariables and Given the preparatory work already done, a more simplemethod is to substitute = ( ) into the first-mentioned steady-stateequation to get = ( ( ) ) + Taking the total differential onboth sides gives =+ + from which follows, by (20.11),

=

1

1 + 0

From( ) = we get0 = + =( ) + = 0so 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 ratemultipliers w.r.t. are the same.

(b) The effect of an anticipated upward shift in We assume thatthe 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 upwardshift in (regime)

at time 1, cf. the upper panel of Fig. 20.4. The implied expectation thatthe 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 whatlevel? 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 expecteddiscontinuity at time 1, since such ajump would imply an infinite expected capital loss (or capital gain) per timeunit immediately before = 1 by holding long-term bonds. Anticipating forexample a capital loss, the market participants would want to sell long-termbonds in advance. The implied excess supply would generate an adjustmentofdownwards until no longer a jump is expected to occur at time 1. Ifinstead a capital gain is anticipated, an excess demand would arise. Thiswould generate in advance an upward adjustment of thus defeating theexpected capital gain. This is the general principle that arbitrage preventsan expected jump in an asset price.

In the time interval (0 1) the dynamics are determined by the oldphase diagram, based on the no-arbitrage condition which rules up to time1. 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 precisely10units of time to reach the new saddle path. At time 0, therefore, jumps toexactly 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 inFig. 20.5. Then, not only would there be a longer way along the road to the new saddle


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LM

IS

'IS

'E

E

Y

R

Y 'Y

R

'R

AB

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 20.5and 20.6. This is because the potentially counteracting force, the increasein 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 inthe old steady state.

There are two interesting features. First, in regime a credible announce-ment of future expansive fiscal policy can have a temporary contractionaryeffect when the announcement occurs. This is due to financial crowding out.The second feature relates to the term structure of interest rates, also calledtheyield curve. The relationship between the internal rate of return on finan-cial assets and their time to maturity is called the term structure of interestrates. 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 (andthereby a higher ) is expected to show up, is getting nearer. But at thesame time the short-term rate is falling because of the falling transactionneed for money implied by the initially fallingtriggered by the rise in thelong-term interest rate.7

path, but the system would also start from a position closer to the old steady-statepoint, E. This implies an initially lower adjustment speed.

7 A 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 futureaggregate income, so that the hypothetical temporary decrease in demand and outputnever materializes. On the other hand, the model has in fact been seen as an explana-tion that president Ronald Reagans announced tax cut in the USA 1981-83 (combinedwith the strict monetary policy aiming at disinflation) were associated with several yearsrecession.


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Figure 20.6: An anticipated upward shift in and the responses of interest ratesand output (regime )

The theory of the term structure

What we have just seen is the expectations theory of the term structure inaction. Empirically, the term structure of interest rates tends to be upward-sloping, but certainly not always and it may suddenly shift. The theoryof the term structure of interest rates generally focuses on two explanatoryfactors. One is uncertainty and this factor tends to imply a positive slopebecause the greater uncertainty generally associated with long-term bondsgenerates a risk premium on these. About this factor the present model has

nothing to say since it ignores uncertainty. But the model has something tosay about the other factor, namely expectations. Indeed, the model quite wellexemplifies what is called the expectations theory of the term structure. In itssimplest form this theory ignores uncertainty and says that if the short-terminterest rate is expected to rise in the future, the long-term rate today willtend to be higher than the short-term rate today. This is because, absentuncertainty, the long-term rate incorporates (is a kind of average of) theexpected future short-term rates, as indicated by (20.10). Similarly, if theshort-term interest rate is expected to fall in the future, the long-term ratetoday will, everything else equal, tend to be lower than the short-term rate


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IS

E

'EA

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 slopeof the term structure to be positive, changes in expectations are importantin explaining changes in the term structure. In practice, most bonds aredenominated in money. Central to the theory is therefore the link betweenexpected future inflation and the expected future short-term nominal interest

rate. This aspect is not captured by the present model, which ignores changesin the price level.

(c) The effect of an unanticipated downward shift in The shift inis shown in the upper panel of Fig. 20.8. The shift triggers, at time 0anupward jump in the long-term rate to the level where the new saddle pathis (point A in Fig. 20.7). The explanation is that the fall in money supplyimplies an upward jump in the short-term rate at time 0 cf. (20.10). Asindicated by Fig. 20.8, the short-term rate will be expected toremainhigherthan 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. Theresulting decline in the transactions-motivated demand for money leads tothe 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 everypoint in time after 1 lower than the short-term rate.

It is interesting that when the new policy is introduced, both andovershoot in their adjustment to the new long-run levels. This happens,because, after 0, both and have to be decreasing, parallel with thedecreasing 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 interestrates 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( ( ) ) + = toget(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 sameas those for the static IS-LM model.

(d) The effect of an anticipated downward shift in The shift inis announced at time0 to take place at time1cf. Fig. 20.9. At the time 0


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IS

LM

'LMA

B

E

'E

Y

Y'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 time1. This is due to the expectation that the short-term rate will in the longerrun be higher than in the old steady state. The higher implies a loweroutput 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 economyfollows that path (AB in Fig. 20.9) which, starting from a point on thevertical line = takes precisely 1 0 units of time to reach the newsaddle 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 assetprice, exactly at the time1 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 upwardin the short-term rate (this is foreseen by everybody, but it implies nocapital loss because the bond is short-term). Output continues fallingtowards its new low steady state level, cf. Fig. 20.10. The transactions-motivated demand for money decreases and therefore gradually decreasestowards the new long-run level which is above the old because is smallerthan before. The long-term rate discounts this gradual fall in in advanceand is therefore, after 1 always lower than . Nevertheless, in the long runthe long-term rate approaches the short-term rate (in this model where thereis 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 asinstrument

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). Thenthe real money supply, has to be endogenous, which reflects that thecentral 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 themoney market rate, where money market is synonymous with interbankmarket and refers to narrow money, cf. Chapter 16. In the Euro areathe 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 InDenmark what comes closest is Nationalbankens udlnsrente (the weeklylending rate of Danmarks Nationalbank).

8As to the evolution of the Federal Funds Rate in response to economic and politicalevents since 1987 (the Greenspan period), see Appendix C.


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Y

R

E

IS

0Y

0R

0Y Y

iA

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 01 0

Becausedoes not appear in (20.16), this system is simpler. The system de-termines the movement ofandIn the next step the required movementof is determined by = ( ) =0

( ) from (20.2).Using a similar method as before we construct the phase diagram, cf. Fig.

20.11. The = 0locus is now horizontal. The steady state is again a saddlepoint and is saddle-point stable. Notice, that here the saddle path coincideswith the = 0locus.

Dynamic responses to policy changes when the short-term interestrate 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 untiltime0In the steady state we have = = . Then either fiscal policyor monetary policy shifts. We consider the following three shifts in exogenousvariables:

(a) An unanticipated decrease of See figures 20.13 and 20.14.


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E'E

'IS IS

R

YY'Y

i

0Y

0R

Figure 20.12: Phase portrait of an unanticipated downward shift in (regime)

(b) An unanticipated decrease ofSeefigures 20.14 and 20.15 in AppendixB.

(c) An anticipated decrease of See figures 20.16 and 20.17 in AppendixB.

As to the anticipated shift in, we imagine that the central bank at time0 credibly announces the shift into take place at time 1 0

The figures illustrate the responses. The diagrams should, by now, beself-explanatory. The only thing to add is that the reader is free to introduceanother 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 stateof 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 tobe executed. This is what point (c) is about.

20.2.3 Policy regime0: A counter-cyclical interest raterule

Suppose the central bank conducts stabilization policy by using the interestrate rule = 0 +1 where 0 and 1 are constant policy parameters,


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Figure 20.13: An unanticipated downward shift in and responses of interestrates, output, and money supply (regime )

1 0. Then our dynamic system is

= (( ) + ) 0 1 01 0

=

(0 + 1) +

These two differential equations determine the time path of ( ) by aphase diagram similar to that in Fig. 20.1. Responses to unanticipated andanticipated changes in are qualitatively the same as in regime wherethe money stock was the instrument. Qualitatively, the only difference isthat the money stock is no longer an exogenous constant, but has to adjustaccording to

= 0(

0 + 1)


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in order to let the counter-cyclical interest rule work. In Exercise 20.x thereader is asked to show that because 0 , this monetary policyregime 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 andemployment 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 explicitmicroeconomic foundation. Yet the consumption and investment functionscan to some extent be defended on a microeconomic basis.9 The dynamicversion of the IS-LM model incorporates wealth effects through changes inthe long-term interest rate,

The simple process assumed for the adjustment of output to changes indemand is ad hoc. At best it can be seen as a rough approximation to themicroeconomic theory of intended and unintended inventory investment. Of

course, it is also not satisfactory that changes in production and employmenthave no wage and price effects at all. At least after some time there shouldbe wage and price responses. To put it differently: a more satisfactory IS-LMmodel calls for an extension with a Phillips curve.10 Then, might generallyin the medium term tend to its natural level.

20.4 Bibliographic notes

A remark on the relationship between our presentation of the model and theoriginal version in Blanchard (1981) seems appropriate. In fact our presen-tation corresponds to that in Blanchard and Fischer (1989). In the originalBlanchard (1981) paper, however, the forward-looking variable is Tobins rather than the long-term interest rate, . But since the (real) long-terminterest 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.

10 This 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 ofthree coupled differential equations.


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20.5. Appendix 685

to the inverse of Tobins q, there is essentially no difference. Wealth effectscome true whether the source is interpreted as changes in Tobins or thelong-term interest rate.

20.5 Appendix

A. Solving the no-arbitrage equation for in the absence of assetprice bubbles

No text available.

B. More examples of dynamics in policy regime

Thefigures 20.14 and 20.15 illustrate responses to an unanticipated loweringof the short-term interest rate, and figures 20.16 and 20.17 illustrate theresponses to an anticipated lowering.

E

ISR

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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20.6. Exercises 687

Figure 20.17: An anticipated downward shift in and responses of the long-ternrate, output, and money supply (regime )


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25t h of June 200 3 - The US

interest 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 interest rate

togeth er with the European central

ban k.

Sping of 2000 - The IT and

internet boom has driven stock

prices to a very high l evel.

September 1998 - The large

hedge fu nd Long Term Capital

Manag emen t is on the brink of

failure and th reatens to p ull down

with it twelve of the largest

investment banks.

17th of August 1998 - Russ ia

defaults on its debt . The ruble is

de valued creating a huge turmoil

on st ock 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--

vis th e dollar.

Decembe r 1996 - Alan Greensp an

warns about "Irrational

Exubu rance" on th e stock market.

Ma rch 199 4 - Alan Greensp an

observes , as one of the first, that

the United States is en tering a

period o f extra ordinar y high

pro du ctivity g rowth.

1991 - The US economy is in

recession.

19th of October 1987 - "Black

Monday" o n the stoc k exchange

in Wall Street. The Dow Jones

stoc k index falls by 22.6 %.

11th of August 1987 - Alan

Greensp an too k office as chairman

of the Fed, replacing t he inflation

hawk Pau l Volcker.

0%

2%

4%

6%

8%

10%

12%

Q1

1987

Q1

1988

Q1

1989

Q1

1990

Q1

1991

Q1

1992

Q1

1993

Q1

1994

Q1

1995

Q1

1996

Q1

1997

Q1

1998

Q1

1999

Q1

2000

Q1

2001

Q1

2002

Q1

2003

Q1

2004

Q1

2005

Q1

2006

US Federal Funds Rate

Figure 20.18: The movements of the US Federal Funds Rate during the periodAlan Greenspan was chairman of the Fed. Source: International Financial Statis-

tics, IMF.


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