The Zero Bound on Interest Rates and Optimal Monetary Policy

The Zero Bound on Interest Rates and

Optimal Monetary Policy

The consequences for the proper conduct of monetary policy of the

existence of a lower bound of zero for overnight nominal interest rates has

recently become a topic of lively interest. In Japan the call rate (the

overnight cash rate analogous to the federal funds rate in the United

States) has been within 50 basis points of zero since October 1995, and it

has been essentially equal to zero for most of the past four years (ﬁg-

ure 1). Thus the Bank of Japan has had little room to further reduce short-

term nominal interest rates in all that time. Meanwhile Japan’s growth has

remained anemic, and prices have continued to fall, suggesting a need for

monetary stimulus. Yet the usual remedy—lower short-term nominal

interest rates—is plainly unavailable. Vigorous expansion of the mone-

tary base has also seemed to do little to stimulate demand under these cir-

cumstances: as ﬁgure 1 also shows, the monetary base is now more than

twice as large, relative to GDP, as it was in the early 1990s.

In the United States, meanwhile, the federal funds rate has now been

reduced to only 1 percent, and signs of recovery remain exceedingly frag-

ile. This has led many to wonder if this country might not also soon ﬁnd

itself in a situation where interest rate policy is no longer available as a

139

GAUTI B. EGGERTSSON

International Monetary Fund

MICHAEL WOODFORD

Princeton University

We would like to thank Tamim Bayoumi, Ben Bernanke, Robin Brooks, Michael

Dotsey, Benjamin Friedman, Stefan Gerlach, Mark Gertler, Marvin Goodfriend, Kenneth

Kuttner, Maurice Obstfeld, Athanasios Orphanides, Kenneth Rogoff, David Small, Lars

Svensson, Harald Uhlig, Tsutomu Watanabe, and Alex Wolman for helpful comments, and

the National Science Foundation for research support through a grant to the National

Bureau of Economic Research. The views expressed in this paper are those of the authors

and do not necessarily represent those of the International Monetary Fund or IMF policy.

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 139

140 Brookings Papers on Economic Activity, 1:2003

Figure 1. Japan: Call Rate on Overnight Loans and Ratio of Monetary Base to GDP,

1990–2002

Source: Nomura database, Bank of Japan.

Call rate

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1992 1994 1996 1998 2000 2002

Monetary base-to-GDP ratio

Index, 1992 = 1.0

Percent a year

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 140

tool for macroeconomic stabilization. A number of other countries face

similar questions. John Maynard Keynes first raised the question of what

can be done to stabilize the economy when it has fallen into a liquidity

trap—when interest rates have fallen to a level below which they cannot

be driven by further monetary expansion—and whether monetary policy

can be effective at all under such circumstances. Long treated as a mere

theoretical curiosity, Keynes’s question now appears to be one of urgent

practical importance, but one with which theorists have become

unfamiliar.

The question of how policy should be conducted when the zero bound

is reached—or when the possibility of reaching it can no longer be

ignored—raises many fundamental issues for the theory of monetary pol-

icy. Some would argue that awareness of the possibility of hitting the zero

bound calls for fundamental changes in the way policy is conducted even

before the bound has been reached. For example, Paul Krugman refers to

deﬂation as a “black hole,”

from which an economy cannot expect to

escape once it has entered. A conclusion often drawn from this pes-

simistic view of the efﬁcacy of monetary policy in a liquidity trap is that

it is vital to steer far clear of circumstances in which deﬂationary expecta-

tions could ever begin to develop—for example, by targeting a sufﬁ-

ciently high positive rate of inﬂation even under normal circumstances.

Others are more sanguine about the continuing effectiveness of monetary

policy even when the zero bound is reached. For example, it is often

argued that deﬂation need not be a black hole, because monetary policy

can affect aggregate spending, and hence inﬂation, through channels other

than central bank control of short-term nominal interest rates. Thus there

has been much recent discussion, with respect to both Japan and the

United States—of the advantages of vigorous expansion of the monetary

base even without any further reduction in interest rates, of the desirabil-

ity of attempts to shift longer-term interest rates through central bank pur-

chases of longer-maturity government securities, and even of the

desirability of central bank purchases of other kinds of assets.

Yet if these views are correct, they challenge much of the recent con-

ventional wisdom regarding the conduct of monetary policy, both within

central banks and among academic monetary economists. That wisdom

has stressed a conception of the problem of monetary policy in terms of

Gauti B. Eggertsson and Michael Woodford 141

1. Paul Krugman, “Crisis in Prices?” New York Times, December 31, 2002, p. A19.

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 141

the appropriate adjustment of an operating target for overnight interest

rates, and the prescriptions formulated for monetary policy, such as the

celebrated Taylor rule,

are typically cast in these terms. Indeed, some

have argued that the inability of such a policy to prevent the economy

from falling into a deﬂationary spiral is a critical ﬂaw of the Taylor rule as

a guide to policy.

Similarly, concern over the possibility of entering a liquidity trap is

sometimes presented as a serious objection to another currently popular

monetary policy prescription, namely, inﬂation targeting. The deﬁnition

of a policy prescription in terms of an inﬂation target presumes that there

is in fact some level of the nominal interest rate that can allow the target

to be hit (or at least projected to be hit, on average). But, some argue, if

the zero interest rate bound is reached under circumstances of deﬂation, it

will not be possible to hit any higher inﬂation target, because further

interest rate decreases are not possible. Is there, in such circumstances,

any point in having an inﬂation target? The Bank of Japan has frequently

offered this argument as a reason for resisting inﬂation targeting. For

example, Kunio Okina, director of the Institute for Monetary and Eco-

nomic Studies at the Bank of Japan, was quoted as arguing that “because

short-term interest rates are already at zero, setting an inﬂation target of,

say, 2 percent wouldn’t carry much credibility.”

We seek to shed light on these issues by considering the consequences

of the zero lower bound on nominal interest rates for the optimal conduct

of monetary policy, in the context of an explicitly intertemporal equilib-

rium model of the monetary transmission mechanism. Although our

model is extremely simple, we believe it can help clarify some of the basic

issues just raised. We are able to consider the extent to which the zero

bound represents a genuine constraint on attainable equilibrium paths for

inﬂation and real activity, and the extent to which open-market purchases

of various kinds of assets by the central bank can mitigate that constraint.

We are also able to show how the existence of the zero bound changes the

character of optimal monetary policy, relative to the policy rules that

would be judged optimal in its absence or in the case of real disturbances

small enough for the bound never to matter under an optimal policy.

142 Brookings Papers on Economic Activity, 1:2003

2. Taylor (1993).

3. Benhabib, Schmitt-Grohé, and Uribe (2001).

4. “Japan BOJ Ofﬁcial: Hard to Set Inﬂation Targets,” Dow Jones News, August 11,

1999.

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 142

To preview our results, we ﬁnd that the zero bound does represent an

important constraint on what monetary stabilization policy can achieve, at

least when certain kinds of real disturbances are encountered in an envi-

ronment of low inﬂation. We argue that the possibility of expanding the

monetary base through central bank purchases of a variety of types of

assets does little if anything to expand the set of feasible paths for inﬂa-

tion and real activity that are consistent with equilibrium under some

(fully credible) policy commitment.

Hence the relevant trade-offs can correctly be studied by simply con-

sidering what alternative anticipated state-contingent paths of the short-

term nominal interest rate can achieve, taking into account the constraint

that this rate must be nonnegative at all times. Doing so, we ﬁnd that the

zero interest rate bound can indeed be temporarily binding, and when it is,

it inevitably results in lower welfare than could be achieved in the

absence of such a constraint.

Nonetheless, we argue that the zero bound restricts possible stabiliza-

tion outcomes under sound policy to a much more modest degree than

the deflation pessimists presume. Even though the set of feasible equilib-

rium outcomes corresponds to those that can be achieved through alter-

native interest rate policies, monetary policy is far from powerless to

mitigate the contractionary effects of the kind of disturbances that would

make the zero bound a binding constraint. The key to dealing with this

sort of situation in the least damaging way is to create the right kind of

expectations regarding how monetary policy will be used after the con-

straint is no longer binding, and the central bank again has room to

maneuver. We use our intertemporal equilibrium model to characterize

Gauti B. Eggertsson and Michael Woodford 143

5. We do not explore here the possibility of relaxing the constraint by taxing money

balances, as originally proposed by Gesell (1929) and Keynes (1936), and more recently by

Buiter and Panigirtzoglou (2001) and Goodfriend (2000). Although this represents a solu-

tion to the problem in theory, it presents substantial practical difﬁculties, not the least of

which is the political opposition that such an institutional change would be likely to gener-

ate. Our consideration of the problem of optimal policy also abstracts from the availability

of ﬁscal instruments, such as the time-varying tax policy recommended by Feldstein

(2002). We agree with Feldstein that there is a particularly good case for state-contingent

ﬁscal policy to deal with a liquidity trap, even if ﬁscal policy is not a very useful tool for

stabilization policy more generally. Nonetheless, we consider here only the problem of the

proper conduct of monetary policy, taking as given the structure of tax distortions. As long

as one does not think that state-contingent ﬁscal policy can (or will) be used to eliminate

even temporary declines in the natural rate of interest below zero, the problem for monetary

policy that we consider here remains relevant.

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 143

the kind of expectations regarding future policy that it would be desir-

able to create, and we discuss a form of price-level targeting rule that—

if credibly committed to—should bring about the constrained-optimal

equilibrium. We also discuss, more informally, how other types of policy

actions could help increase the credibility of the central bank’s

announced commitment to this kind of future policy.

Our analysis will be recognized as a development of several key

themes in Paul Krugman’s treatment of the same topic in these pages a

few years ago.

Like Krugman, we give particular emphasis to the role of

expectations regarding future policy in determining the severity of the

distortions that result from hitting the zero bound. Our primary contribu-

tion, relative to Krugman’s earlier treatment, will be the presentation of a

more fully dynamic analysis. For example, our assumption of staggered

pricing, rather than Krugman’s simple hypothesis of prices that are ﬁxed

for one period, allows for richer (and at least somewhat more realistic)

dynamic responses to disturbances. In our model, unlike in Krugman’s, a

real disturbance that lowers the natural rate of interest can cause output to

remain below potential for years (as shown in ﬁgure 2 later in the paper),

rather than only for a single “period,” even when the average frequency of

price adjustments is more than once a year. These richer dynamics are

also important for a realistic discussion of the kind of policy commitment

that can help to reduce economic contraction during a liquidity trap. In

our model a commitment to create subsequent inflation involves a com-

mitment to keep interest rates low for some time in the future, whereas in

Krugman’s model a commitment to a higher future price level does not

involve any reduction in future nominal interest rates. We are also better

able to discuss such questions as how the creation of inflationary expec-

tations while the zero bound is binding can be reconciled with maintain-

ing the credibility of the central bank’s commitment to long-run price

stability.

Our dynamic analysis also allows us to further clarify the several ways

in which the central bank’s management of private sector expectations

can be expected to mitigate the effects of the zero bound. Krugman

emphasizes the fact that increased expectations of inﬂation can lower the

real interest rate implied by a zero nominal interest rate. This might sug-

gest, however, that the central bank can affect the economy only insofar

144 Brookings Papers on Economic Activity, 1:2003

6. Krugman (1998).

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 144

as it affects expectations regarding a variable that it cannot inﬂuence

except quite indirectly; it might also suggest that the only expectations

that should matter are those regarding inﬂation over the relatively short

horizon corresponding to the term of the nominal interest rate that has

fallen to zero. Such interpretations easily lead to skepticism about the

practical effectiveness of the expectations channel, especially if inﬂation

is regarded as being relatively “sticky” in the short run. Our model is

instead one in which expectations affect aggregate demand through sev-

eral channels.

First of all, it is not merely short-term real interest rates that matter for

current aggregate demand; our model of intertemporal substitution in

spending implies that the entire expected future path of short-term real

rates should matter, or alternatively that very long term real rates should

matter.

This means that the creation of inﬂation expectations, even with

regard to inﬂation that should not occur until at least a year into the future,

should also be highly relevant to aggregate demand, as long as it is not

accompanied by correspondingly higher expected future nominal interest

rates. Furthermore, the expected future path of nominal interest rates

matters, and not just their current level, so that a commitment to keep

nominal interest rates low for a longer period of time should stimulate

aggregate demand, even when current interest rates cannot be lowered

further, and even under the hypothesis that inflation expectations would

remain unaffected. Because the central bank can clearly control the

future path of short-term nominal interest rates if it has the will to do so,

any failure of such a commitment to be credible will not be due to skep-

ticism about whether the central bank is able to follow through on its

commitment.

The richer dynamics of our model are also important for the analysis of

optimal policy. Krugman mainly addresses the question of whether mon-

etary policy is completely impotent when the zero bound binds, and he

argues for the possibility of increasing real activity in the liquidity trap by

Gauti B. Eggertsson and Michael Woodford 145

7. In the simple model presented here, this occurs solely as a result of intertemporal

substitution in private expenditure. But there are a number of reasons to expect long-term

rates, rather than short-term rates, to be the critical determinant of aggregate demand. For

example, in an open-economy model, the real exchange rate becomes an important deter-

minant of aggregate demand. But the real exchange rate should be closely linked to a very

long domestic real rate of return (or alternatively to the expected future path of short-term

rates) as a result of interest rate parity, together with an anchor for the expected long-term

real exchange rate (deriving, for example, from long-run purchasing power parity).

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 145

creating expectations of inﬂation. Although we agree with this conclu-

sion, it does not answer the question of whether, or to what extent, it

would be desirable to create such expectations, given the well-founded

reasons that the central bank should have to not prefer inﬂation at a later

time. Nor is Krugman’s model well suited to address such a question,

insofar as it omits any reason for even an extremely high subsequent inﬂa-

tion to be deemed harmful. Our staggered-pricing model instead implies

that inﬂation (whether anticipated or not) does create distortions, justify-

ing an objective function for stabilization policy that trades off inﬂation

stabilization and output gap stabilization in terms that are often assumed

to represent actual central bank concerns. We characterize optimal policy

in such a setting and show that it does indeed involve a commitment to

history-dependent policy of a sort that should result in higher inﬂation

expectations in response to a binding zero bound. We can also show to

what extent it should be optimal to create such expectations, assuming

that this is possible. We ﬁnd, for example, that it is not optimal to commit

to so much future inﬂation that the zero bound ceases to bind, even though

this is one possible type of equilibrium; this is why the zero bound does

remain a relevant constraint, even under an optimal policy commitment.

Is Quantitative Easing a Separate Policy Instrument?

A ﬁrst question we wish to consider is whether expansion of the mone-

tary base represents a policy instrument that should be effective in

preventing deﬂation and an associated output decline, even under circum-

stances where overnight interest rates have fallen to zero. According to

Keynes’s famous analysis,

monetary policy ceases to be an effective

instrument to head off economic contraction in a “liquidity trap,” which

can arise if interest rates fall so low that further expansion of the money

supply cannot drive them lower. Others have argued that monetary expan-

sion should increase nominal aggregate demand even under such circum-

stances, and the supposition that this is correct lies behind Japan’s explicit

adoption, since March 2001, of a policy of “quantitative easing” in addi-

tion to the zero interest rate policy that continues to be maintained.

146 Brookings Papers on Economic Activity, 1:2003

8. Keynes (1936).

9. See Kimura and others (2002) for a discussion of this policy as well as an expression

of doubts about its effectiveness.

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 146

Here we consider this question in the context of an explicitly intertem-

poral equilibrium model, which models both the demand for money and

the role of ﬁnancial assets (including the monetary base) in private sector

budget constraints. The model we use for this purpose is more detailed in

several senses than that used in subsequent sections to characterize opti-

mal policy. We do this to make it clear that we have not excluded a role

for quantitative easing simply by failing to model the role of money in the

economy.

Our key result is an irrelevance proposition for open-market operations

in a variety of types of assets that the central bank might acquire, under

the assumption that the open-market operations do not change the

expected future conduct of monetary or ﬁscal policy (in senses that we

specify below). It is perhaps worth noting at the outset that our intention

in stating such a result is not to vindicate the view that a central bank is

powerless to halt a deﬂationary slump, and hence to absolve the Bank of

Japan, for example, of any responsibility for the continuing stagnation in

that country. Although our proposition establishes that there is a sense in

which a liquidity trap is possible, this does not mean that the central bank

is powerless under the circumstances we describe. Rather, our intent is to

show that the key to effective central bank action to combat a deﬂationary

slump is the management of expectations. Open-market operations should

be largely ineffective to the extent that they fail to change expectations

regarding future policy; the conclusion we draw is not that such actions

are futile, but rather that the central bank’s actions should be chosen with

a view to signaling the nature of its policy commitments, and not for the

purpose of creating some sort of “direct” effects.

A Neutrality Proposition for Open-Market Operations

Our model abstracts from endogenous variations in the capital stock

and assumes perfectly ﬂexible wages (or some other mechanism for efﬁ-

cient labor contracting), but it assumes monopolistic competition in goods

markets and sticky prices that are adjusted at random intervals in the man-

ner assumed by Guillermo Calvo, so that deﬂation has real effects.

Gauti B. Eggertsson and Michael Woodford 147

10. Woodford (forthcoming, chapter 4) discusses the model in more detail and consid-

ers the consequences of various interest rate rules and money growth rules under the

assumption that disturbances are not large enough for the zero bound to bind.

11. Calvo (1983).

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 147

assume that the representative household seeks to maximize a utility func-

tion of the form

where C

is a Dixit-Stiglitz aggregate of consumption of each of a contin-

uum of differentiated goods,

with an elasticity of substitution θ>1; M

measures end-of-period house-

hold money balances,

is the Dixit-Stiglitz price index,

and H

( j) is the quantity supplied of labor of type j. Real balances are

included in the utility function,

as a proxy for the services that money

balances provide in facilitating transactions.

Each industry j employs an

industry-speciﬁc type of labor, with its own wage.

For each value of the disturbances 

, u(⋅, ⋅; 

) is a concave function,

increasing in the ﬁrst argument and increasing in the second for all levels

of real balances up to a satiation level m

–

; 

). The existence of a satia-

tion level is necessary in order for it to be possible for the zero interest

rate bound ever to be reached; we regard Japan’s experience over the past

several years as having settled the theoretical debate over whether such a

level of real balances exists. Unlike many papers in the literature, we do

not assume additive separability of the function u between the ﬁrst two

() () ,

–

Ppidi

≡

[]

∫

Ccidi

≡













∫

() ,

–

EuCMP Hjdj

tttt t t

βυ

–

(, /;)– (); ,

[]

{}

∫

∑

∞

148 Brookings Papers on Economic Activity, 1:2003

12. We do not introduce fractional-reserve banking into our model. Technically, M

refers to the monetary base, and we represent households as obtaining liquidity services

from holding this base, either directly or through intermediaries (not modeled).

13. Following Sidrauski (1967) and Brock (1974, 1975).

14. We use this approach to modeling the transactions demand for money because of

its familiarity. As shown in Woodford (forthcoming, appendix section A.16), a cash-in-

advance model leads to equilibrium conditions of essentially the same general form, and

the neutrality result that we present below would hold in essentially identical form were we

to model the transactions demand for money after the fashion of Lucas and Stokey (1987).

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 148

arguments; this (realistic) complication allows a further channel through

which money can affect aggregate demand, namely, by an effect of real

money balances on the current marginal utility of consumption. Similarly,

for each value of 

, υ(⋅; 

) is an increasing convex function. The vector

of exogenous disturbances 

may contain several elements, so that no

assumption is made about correlation of the exogenous shifts in the func-

tions u and υ.

For simplicity we assume complete ﬁnancial markets and no limit on

borrowing against future income. As a consequence, a household faces an

intertemporal budget constraint of the form

looking forward from any period t. Here Q

t,T

is the stochastic discount fac-

tor that the ﬁnancial markets use to value random nominal income at date

T in monetary units at date t; δ

is the opportunity cost of holding money

and is equal to i

/(1 + i

), where i

is the riskless nominal interest rate on

one-period obligations purchased in period t, in the case that no interest is

paid on the monetary base; W

is the nominal value of the household’s

ﬁnancial wealth (including money holdings) at the beginning of period t;

(i) represents the nominal proﬁts (revenue in excess of the wage bill) in

period t of the supplier of good i; w

( j) is the nominal wage earned by

labor of type j in period t, and T

represents the net nominal tax liabilities

of each household in period t. Optimizing household behavior then

implies the following necessary conditions for a rational expectations

equilibrium. Optimal timing of household expenditure requires that

aggregate demand Y

for the composite good satisfy an Euler equation of

the form

() ( , / ; ) ( , / ; )( ) ,21

1111

uYMP EuY M P i

ct tt t t ct t t t t

=+













++++

EQPC M

W E Q i di w j H j dj T

ttTTTTT

tt tT T T T T

[]

() () () – ,

≤+ +

[]

∞

∑

∫∫

∑

Gauti B. Eggertsson and Michael Woodford 149

15. For simplicity, we abstract from government purchases of goods. Our equilibrium

conditions directly extend to the case of exogenous government purchases, as shown in

Woodford (forthcoming, chapter 4).

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 149

Optimal substitution between real money balances and expenditure

leads to a static ﬁrst-order condition of the form

under the assumption that zero interest is paid on the monetary base, and

that preferences are such as to exclude the possibility of a corner solution

with zero money balances. If both consumption and liquidity services are

normal goods, this equilibrium condition can be solved uniquely for the

level of real balances L(Y

, i

; 

) that satisfy it in the case of any positive

nominal interest rate. The equilibrium relation can then equivalently be

written as a pair of inequalities:

together with the “complementary slackness” condition that at least one

must hold with equality at any time. Here we deﬁne L(Y, 0; ) = m

–

(Y; ),

the minimum level of real balances for which u

= 0, so that the function

L is continuous at i = 0. Household optimization similarly requires that

the paths of aggregate real expenditure and the price index satisfy the

bounds

looking forward from any period t, where D

measures the total nominal

value of government liabilities (monetary base plus government debt) at

the end of period t under the monetary-fiscal policy regime. (The condi-

tion in expression 5 is required for the existence of a well-defined inter-

temporal budget constraint, under the assumption that there are no

limitations on households’ ability to borrow against future income,

whereas the transversality condition in equation 6 must hold if the house-

( ) lim ( , / ; ) /60

tcT TT T TT

EuY M P DP

→∞

[]

=β 

() ( , / ; ) ( , / ; )( / )5 β

tcT TT TT mT TT T TT

EuYMP Y uYMP MP+

[]

<∞

∞

∑

() ,40i

≥

() ( ,; )3

LY i

tt t

≥ 

uYMP

mt tt t

ct tt t

(, /; )



150 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 150

hold hits its intertemporal budget constraint.) The conditions in expres-

sions 2 through 6 also suffice to imply that the representative household

chooses optimal consumption and portfolio plans (including its planned

holdings of money balances) given its income expectations and the

prices (including financial asset prices) that it faces, while making

choices that are consistent with financial market clearing.

Each differentiated good i is supplied by a single, monopolistically

competitive producer. There are assumed to be many goods in each of an

inﬁnite number of industries; each industry j uses a type of labor that is

speciﬁc to that industry, and all goods in an industry change their prices at

the same time. Each good is produced in accordance with a common pro-

duction function

where A

is an exogenous productivity factor common to all industries;

f(⋅) is an increasing, concave function; and h

(i) is the industry-speciﬁc

labor hired by ﬁrm i. The representative household supplies all types of

labor and consumes all types of goods.

The supplier of good i sets a price for that good at which it satisﬁes

demand in each period, hiring the labor inputs necessary to meet that

demand. Given households’ allocation of demand across goods in

response to ﬁrms’ pricing decisions, on the one hand, and the terms on

which optimizing households are willing to supply each type of labor, on

the other, we can show that nominal proﬁts (sales revenue in excess of

labor costs) in period t of the supplier of good i are given by the function

Π pi p PYMP piYpi P

fYpP A

uYMP

Pf Y p i P A

tt tt t t t t t

htt

ttt

ct tt t

ttttt

(), , ; , / ,

() ()/

–

(/)/ ;

(, /; )

()/ / ,

–

––

–



[]

≡

[]

{}

[]

{}

yi Afhi

ttt

() (),=

[]

Gauti B. Eggertsson and Michael Woodford 151

16. We might alternatively assume specialization across households in the type of labor

supplied; in the presence of perfect sharing of labor income risk across households, house-

hold decisions regarding consumption and labor supply would all be as assumed here.

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 151

where p

is the common price charged by the other ﬁrms in industry j.

(We introduce the notation 

for the complete vector of exogenous dis-

turbances, including variations in technology as well as in preferences.) If

prices were fully ﬂexible, p

(i) would be chosen each period to maximize

this function.

Instead we suppose that prices remain ﬁxed in monetary terms for a

random period of time. Following Calvo, we suppose that each industry

has an equal probability of reconsidering its prices each period, and we let

0 <α<1 be the fraction of industries whose prices remain unchanged

each period. In any industry that revises its prices in period t, the new

price p

* will be the same. This price is implicitly deﬁned by the ﬁrst-order

condition

We note furthermore that the stochastic discount factor used to price

future proﬁt streams will be given by

Finally, the deﬁnition in equation 1 implies a law of motion for the aggre-

gate price index of the form

Equations 7 and 9, which jointly determine the path of prices given

demand conditions, represent the aggregate supply block of our model. It

remains to specify the monetary and ﬁscal policies of the government.

() (– ).

–

PpP

ttt

[]

αα

θθ

()

(, /;)

–

8 Q

uCMP

cT TTT

ct ttt

=β



() ( , , ; , / ,

–

EQppPYMP

tT t t T T T T T

αΠ 

∞

∑













152 Brookings Papers on Economic Activity, 1:2003

17. In equilibrium, all ﬁrms in an industry charge the same price at any given time. But

we must deﬁne proﬁts for an individual supplier i in the case of contemplated deviations

from the equilibrium price.

18. The particular speciﬁcation of monetary and ﬁscal policy proposed here is not

intended to suggest that either monetary or ﬁscal policy must be expected to be conducted

according to rules of the sort assumed here. Indeed, in later sections we recommend policy

commitments on the part of both the monetary and the ﬁscal authorities that do not conform

to the assumptions made here. The point is to deﬁne what we mean by the qualiﬁcation that

open-market operations are irrelevant if they do not change expected future monetary or

ﬁscal policy. To make sense of such a statement, we must deﬁne what it would mean for

1440-03 BPEA/Eggertsson 07/17/03 08:11 Page 152

To address the question of whether quantitative easing represents an addi-

tional tool of policy, we suppose that the central bank’s operating target

for the short-term nominal interest rate is determined by a feedback rule

in the spirit of the Taylor rule,

where now 

may also include exogenous disturbances in addition to the

ones listed above, to which the central bank happens to respond. We

assume that the function φ is nonnegative for all values of its arguments

(otherwise the policy would not be feasible, given the zero lower bound),

but that there are conditions under which the rule prescribes a zero inter-

est rate policy. Such a rule implies that the central bank supplies the quan-

tity of base money that happens to be demanded at the interest rate given

by this formula; hence equation 10 implies a path for the monetary base,

so long as the value of φ is positive. However, under those conditions in

which the value of φ is zero, the policy commitment in equation 10

implies only a lower bound on the monetary base that must be supplied. In

these circumstances we may ask whether it matters whether a greater or a

smaller quantity of base money is supplied. We assume that the central

bank’s policy in this regard is speciﬁed by a base-supply rule of the form

where the multiplicative factor ψ satisﬁes the following two conditions:

for all values of its arguments. (The second condition implies that ψ=1

whenever i

> 0.) Note that a base-supply rule of this form is consistent

with both the interest rate operating target speciﬁed in equation 10 and the

equilibrium relations in expressions 3 and 4. The use of quantitative

ψφ

(/ ,;

)(/,;

(/ ,;

––

–

PP Y PP Y

PP Y

tt t t tt t t

tt t t





≥

if otherwise

() ,(/ ,;

); ( / , ;

––

MPLYPPY PPY

ttt ttttt tttt

[]

φψ 

() (/ ,;

–

iPPY

ttttt

=φ 

Gauti B. Eggertsson and Michael Woodford 153

these policies to be speciﬁed in a way that prevents them from being affected by past open-

market operations. The speciﬁc classes of policy rules discussed here show not only that

our concept of “unchanged policy” is logically possible, but indeed that it could correspond

to a policy commitment of a fairly familiar sort, one that would represent a commitment to

“sound policy” in the views of some.

19. Taylor (1993).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 153

easing as a policy tool can then be represented by a choice of a function ψ

that is greater than 1 under some circumstances.

It remains to specify which sort of assets should be acquired (or dis-

posed of) by the central bank when it varies the size of the monetary base.

We allow the asset side of the central bank balance sheet to include any of

k different types of securities, distinguished from each other by their

state-contingent returns. At the end of period t, the vector of nominal val-

ues of central bank holdings of the various securities is given by M



where 

is a vector of central bank portfolio shares. These shares are in

turn determined by a policy rule of the form

where the vector-valued function 

(⋅) has the property that its compo-

nents sum to 1 for all possible values of its arguments. The fact that 

(⋅)

depends on the same arguments as φ(⋅) means that we allow for the possi-

bility that the central bank changes its policy when the zero bound is bind-

ing (for example, buying assets that it would not hold at any other time).

The fact that it depends on the same arguments as ψ(⋅) allows us to spec-

ify changes in the composition of the central bank portfolio as a function

of the particular kinds of purchases associated with quantitative easing.

The payoffs on these securities in each state of the world are speciﬁed

by exogenously given (state-contingent) vectors a

and b

and matrix F

. A

vector of asset holdings z

t–1

at the end of period t – 1 results in delivery, to

the owner of a quantity a′

t–1

of money, a quantity b′

t–1

of the consump-

tion good and a vector F

t–1

of securities that may be traded in the

period-t asset markets, each of which may depend on the state of the

world in period t. This ﬂexible speciﬁcation allows us to treat a wide

range of types of assets that may differ as to maturity, degree of indexa-

tion, and so on.

The gross nominal return R

( j) on the jth asset between periods t – 1

and t is then given by

() (/ ,;

–

 

tt t t

PP Y=

154 Brookings Papers on Economic Activity, 1:2003

20. For example, security j in period t – 1 is a one-period riskless nominal bond if b

( j)

and F

(⋅ ; j) are zero in all states, while a

( j) > 0 is the same in all states. Security j is instead

a one-period real (or indexed) bond if a

( j) and F

(⋅ ; j) are zero, while b

( j) > 0 is the same

in all states. It is a two-period riskless nominal pure discount bond if instead a

( j) and b

( j)

are zero, F

(i, j) = 0 for all i ≠ k; F

(k, j) > 0 is the same in all states, and security k in period

t is a one-period riskless nominal bond.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 154

where q

is the vector of nominal asset prices in (ex-dividend) period-t

trading. The absence of arbitrage opportunities implies as usual that equi-

librium asset prices must satisfy

where the stochastic discount factor is again given by equation 8. Under

the assumption that no interest is paid on the monetary base, the nominal

transfer by the central bank to the public treasury each period is equal to

where R

is the vector of returns deﬁned by equation 13.

We specify ﬁscal policy in terms of a rule that determines the evolution

of total government liabilities D

, here deﬁned to be inclusive of the mon-

etary base, as well as a rule that speciﬁes the composition of outstanding

nonmonetary liabilities (debt) among different types of securities that the

government might issue. We assume that the path of total government lia-

bilities accords with a rule of the form

which speciﬁes the acceptable level of real government liabilities as a

function of the preexisting level and various aspects of current macro-

economic conditions. This notation allows for such possibilities as an

exogenously speciﬁed state-contingent target for real government liabili-

ties as a proportion of GDP, or for the government budget deﬁcit (inclu-

sive of interest on the public debt) as a proportion of GDP, among others.

The part of total liabilities that consists of base money is speciﬁed by the

base rule in equation 11. We suppose, however, that the rest may be allo-

cated among any of a set of different types of securities that the govern-

ment may issue; for convenience, we assume that this is a subset of the set

of k securities that the central bank may purchase. If 

indicates the

share of government debt (nonmonetary liabilities) at the end of period t

() , ,;

–

––















() – ,

–– –

11 1

TMM

′

R 

() ,

–

′

[]

≥+

∏

∑

qabF

tttTTtTs

EQ P

() ()

() () (,)

()

–

aj Pbj j

ttt tt

′

⋅qF

Gauti B. Eggertsson and Michael Woodford 155

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 155

that is of type j, then the ﬂow government budget constraint takes the

form

where B

≡ D

– M

is the total nominal value of end-of-period nonmone-

tary liabilities, and T

is the nominal value of the primary budget surplus

(taxes net of transfers, if we abstract from government purchases). This

identity can then be inverted to obtain the net tax collections T

implied

by a given rule (equation 16) for aggregate public liabilities; this depends

in general on the composition of the public debt as well as on total

borrowing.

Finally, we assume that debt management policy (the determination of

the composition of the government’s nonmonetary liabilities at each point

in time) is speciﬁed by the function

which speciﬁes the shares as a function of aggregate conditions, where

the vector-valued function 

also has components that sum to 1 for all

possible values of its arguments. Together the two relations in equations

16 and 17 complete our speciﬁcation of ﬁscal policy and close our

model.

We may now deﬁne a rational expectations equilibrium as a collection

of stochastic processes {p

*, P

, Y

, i

, q

, M

, 

, D

, 

}, with each

endogenous variable speciﬁed as a function of the history of exogenous

disturbances to that date, that satisfy each of the conditions in expressions

2 through 6 of the aggregate demand block of the model, the conditions in

equations 7 and 9 of the aggregate supply block, the asset-pricing rela-

tions equation 14, the conditions in equations 10 through 12 specifying

monetary policy, and the conditions in equations 16 and 17, specifying

ﬁscal policy in each period. We then obtain the following irrelevance

result for the speciﬁcation of certain aspects of policy:

() (/ ,;

–

 

tt t t

PP Y=

DBTT

ttt

′

R 

––

––,

156 Brookings Papers on Economic Activity, 1:2003

21. We might, of course, allow for other types of ﬁscal decisions from which we

abstract here—government purchases, tax incentives, and so on—some of which may be

quite relevant to dealing with a liquidity trap. But our concern here is solely with the ques-

tion of what monetary policy can achieve; we introduce a minimal speciﬁcation of ﬁscal

policy only for the sake of closing our general-equilibrium model, and to allow discussion

of the ﬁscal implications of possible actions by the central bank.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 156

PROPOSITION

. The set of paths for the variables {p

*, P

, Y

, i

, q

, D

}

that are consistent with the existence of a rational expectations equilib-

rium is independent of the speciﬁcation of the functions ψ (equation 11),



(equation 12), and 

(equation 17).

The reason for this is fairly simple. The set of restrictions on the

processes {p

*, P

, Y

, i

, q

, D

} implied by our model can be written in a

form that does not involve the variables {M

, 

}, and hence that

does not involve the functions ψ, 

, or 

. To show this, we ﬁrst note

that, for all m ≥ m

–

(C; ),

because additional money balances beyond the satiation level provide no

further liquidity services. By differentiating this relation, we see further

that u

(C, m; ) does not depend on the exact value of m either, as long as

m exceeds the satiation level. It follows that, in our equilibrium relations,

we can replace the expression u

, M

; 

) with

using the fact that expression 3 holds with equality at all levels of real bal-

ances at which u

depends on the level of real balances. Hence we can

write u

as a function of variables other than M

, without using the rela-

tion in equation 11, and so in a way that is independent of the function ψ.

We can similarly replace the expression u

, M

; 

)(M

) in

expression 5 with

since M

must equal L[Y

, φ(P

t–1

, Y

; 

); 

] when real balances do

not exceed the satiation level, whereas u

= 0 when they do. Finally, we

can express nominal proﬁts in period t as a function:

after substituting λ(Y

, P

t–1

; 

) for the marginal utility of real income

in the wage demand function that is used in deriving the proﬁt function

(), , ; , / ,

–

Π pi p PYPP

tttt t1



[]

φφ

(,/ ; )

,( / , ; ); ; ,( / , ; ); ,

–

––

YPP

uYLY PPY LY PPY

ttt t

mt t tt t t t t t tt t t t



 

≡

[]

{}

[]

λφ(,/;) , ,(/,;);; ,

––

YPP u YLY PP Y

ttt t ct t tt t t t t11

≡

[]

{}

uCm uCmC(, ;) , (;); ,=

[]

Gauti B. Eggertsson and Michael Woodford 157

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 157

Π.

Using these substitutions, we can write each of the equilibrium rela-

tions in expressions 2, 5, 6, 7, and 14 in a way that no longer makes refer-

ence to the money supply.

It then follows that in a rational expectations equilibrium the variables

*, P

, Y

, i

, q

, D

} must satisfy in each period the following relations:

along with equations 9, 10, and 16 as before. None of these involve the

variables {M

, ω

}, nor do they involve the functions ψ, 

, or 

Furthermore, this is the complete set of restrictions on these variables

that are required in order for them to be consistent with a rational expec-

tations equilibrium. For any given processes {p

*, P

, Y

, i

, q

, D

} that

satisfy the equations just listed in each period, the implied path of the

money supply is given by equation 11, which clearly has a solution, and

this path for the money supply necessarily satisﬁes expression 3 and the

complementary slackness condition, as a result of our assumptions about

the form of the function ψ. Similarly, the implied compositions of the

central bank portfolio and of the public debt at each point in time are

given by equations 12 and 17. We then have a set of processes that satisfy

() ( ) (,/ ; )

(,,;,/ ,

)

–

EYPPPppPYPP

TTT TT t

tTTTT T

αβ λ Π

∞

∑













()

(,/ ; )

( , / ; )[ ]

–

′

≥+

∑

∏

qabF

ttt t

tTTT TTT T

YPP

EYPP P

βλ



( ) lim ( , / ; ) /

–

20 0

tTTTTTT

EYPP DP

→∞

[]

=βλ 

() (,/ ; ) (,/ ; )

––

βλ µ

tTTTTT TTTT

E Y PP Y Y PP+

[]

<∞

∞

∑

() (,/ ;) ( , /; )( )

–

18 1

1111

λβλYPP E Y P P i

ttt t t t t t t t

=+













++ +

158 Brookings Papers on Economic Activity, 1:2003

22. See Woodford (forthcoming, chapter 3).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 158

all of the requirements for a rational expectations equilibrium, and the

result is established.

Discussion

The above proposition implies that neither the extent to which quanti-

tative easing is employed when the zero bound binds, nor the nature of the

assets that the central bank may purchase through open-market opera-

tions, has any effect on whether a deﬂationary price-level path represents

a rational expectations equilibrium. Hence our general-equilibrium analy-

sis of inﬂation and output determination does not support the notion that

expansions of the monetary base represent an additional tool of policy,

independent of the speciﬁcation of the rule for adjusting short-term nomi-

nal interest rates. If the commitments of policymakers regarding the rule

by which interest rates will be set, on the one hand, and the rule by which

total private sector claims on the government will be allowed to grow, on

the other, are fully credible, then it is only the choice of those commit-

ments that matters. Other aspects of policy should matter in practice only

insofar as they help to signal the nature of these policy commitments.

Of course, the validity of our result depends on the reasonableness of

our assumptions, and these deserve further discussion. Like any economic

model, ours abstracts from the complexity of actual economies in many

respects. Have we abstracted from features of actual economies that are

crucial for a correct understanding of the issues under discussion?

It might be suspected that an important omission is our neglect of port-

folio-balance effects, which play an important role in much recent discus-

sion of the policy options that would remain available to the Federal

Reserve should the federal funds rate reach zero.

The idea is that a cen-

tral bank should be able to lower longer-term interest rates even when

overnight rates are already at zero, through purchases of longer-maturity

government bonds. This would shift the composition of the public debt in

the hands of the public in a way that affects the term structure of interest

rates. (Because it is generally admitted in such discussions that base

money and very short term Treasury securities have become near-perfect

substitutes once short-term interest rates have fallen to zero, the desired

effect should be achieved equally well by a shift in the maturity structure

Gauti B. Eggertsson and Michael Woodford 159

23. See, for example, Clouse and others (2003) and Orphanides (2003).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 159

of Treasury securities held by the central bank, without any change in the

monetary base, as by an open-market purchase of long-term bonds with

newly created base money.)

No such effects arise in our model, whether from central bank securi-

ties purchases or debt management by the public treasury. But this is not,

as some might expect, because we have simply assumed that bonds of dif-

ferent maturities (or for that matter, other kinds of assets that the central

bank might choose to purchase instead of the shortest-maturity Treasury

bills) are perfect substitutes. Our framework allows for central bank pur-

chases of different assets having different risk characteristics (different

state-contingent returns), and our model of asset market equilibrium

incorporates those term premiums and risk premiums that are consistent

with the absence of arbitrage opportunities.

Our conclusion differs from that of the literature on portfolio balance

effects for a different reason. The classic theoretical analysis of portfolio

balance effects assumes a representative investor with mean-variance

preferences. This has the implication that if the supply of assets that pay

off disproportionately in certain states of the world is increased (so that

the extent to which the representative investor’s portfolio pays off in

those states must also increase), the relative marginal valuation of income

in those particular states is reduced, resulting in a lower relative price for

the assets that pay off in those states. But in our general-equilibrium asset

pricing model, there is no such effect. The marginal utility to the repre-

sentative household of additional income in a given state of the world

depends on the household’s consumption in that state, not on the aggre-

gate payoff of its asset portfolio in that state. And changes in the compo-

sition of the securities in the hands of the public do not change the

state-contingent consumption of the representative household—this

depends on equilibrium output, and although output is endogenous, we

have shown that the equilibrium relations that determine it do not involve

the functions ψ, 

, or 

Our assumption of complete ﬁnancial markets and no limits on bor-

rowing against future income may also appear extreme. However, the

160 Brookings Papers on Economic Activity, 1:2003

24. Our general-equilibrium analysis is in the spirit of the irrelevance proposition for

open-market operations of Wallace (1981). Wallace’s analysis is often supposed to be of

little practical relevance for actual monetary policy because his model is one in which

money serves only as a store of value, so that an equilibrium in which short-term Treasury

securities dominate money in terms of rate of return is not possible, although this is

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 160

assumption of complete ﬁnancial markets is only a convenience, allowing

us to write the budget constraint of the representative household in a sim-

ple way. Even in the case of incomplete markets, each of the assets that is

traded will be priced according to equation 14, where the stochastic dis-

count factor is given by equation 8, and once again there will be a set of

relations to determine output, goods prices, and asset prices that do not

involve ψ, 

, or 

. The absence of borrowing limits is also innocuous,

at least in the case of a representative-household model, because in equi-

librium the representative household must hold the entire net supply of

ﬁnancial claims on the government. As long as the ﬁscal rule (equa-

tion 16) implies positive government liabilities at each date, any borrow-

ing limits that might be assumed can never bind in equilibrium.

Borrowing limits can matter more in the case of a model with heteroge-

neous households. But in this case the effects of open-market operations

should depend not merely on which sorts of assets are purchased and

which sorts of liabilities are issued to ﬁnance those purchases, but also on

how the central bank’s trading proﬁts are eventually rebated to the private

sector (that is, with what delay and how distributed across the heteroge-

neous households), as a result of the speciﬁcation of ﬁscal policy. The

effects will not be mechanical consequences of the change in composition

of assets in the hands of the public, but instead will result from the ﬁscal

transfers to which the transaction gives rise; it is unclear how quantita-

tively signiﬁcant such effects should be.

Indeed, leaving aside the question of whether a clear theoretical foun-

dation exists for the existence of portfolio balance effects, there is not a

great deal of empirical support for quantitatively signiﬁcant effects. The

attempt to separately target short-term and long-term interest rates under

Operation Twist in the early 1960s is generally regarded as having had a

modest effect at best on the term structure.

The empirical literature that

has sought to estimate the effects of changes in the composition of the

public debt on relative yields has also, on the whole, found effects that are

Gauti B. Eggertsson and Michael Woodford 161

routinely observed. However, in the case of open-market operations conducted at the zero

bound, the liquidity services provided by money balances at the margin have fallen to zero,

so that an analysis of the kind proposed by Wallace is correct.

25. Okun (1963) and Modigliani and Sutch (1966) are important early discussions that

reached this conclusion. Meulendyke (1998) summarizes the literature and ﬁnds that the

predominant view is that the effect was minimal.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 161

not large when present at all.

For example, Jonas Agell and Mats Pers-

son summarize their ﬁndings as follows: “It turned out that these effects

were rather small in magnitude, and that their numerical values were

highly volatile. Thus the policy conclusion to be drawn seems to be that

there is not much scope for a debt management policy aimed at systemat-

ically affecting asset yields.”

Moreover, even if one supposes that large

enough changes in the composition of the portfolio of securities left in the

hands of the private sector can substantially affect yields, it is not clear

how relevant such an effect should be for real activity and the evolution of

goods prices. For example, James Clouse and others argue that a sufﬁ-

ciently large reduction in the number of long-term Treasuries in the hands

of the public should lower the market yield on those securities relative to

short-term rates, because certain institutions will ﬁnd it important to hold

long-term Treasury securities even when they offer an unfavorable

yield.

But even if this is true, the fact that these institutions have idio-

syncratic reasons to hold long-term Treasuries—and that, in equilibrium,

no one else holds any or plays any role in pricing them—means that the

lower observed yield on long-term Treasuries may not correspond to any

reduction in the perceived cost of long-term borrowing for other institu-

tions. If one is able to reduce the long-term bond rate only by decoupling

it from the rest of the structure of interest rates, and from the cost of

ﬁnancing long-term investment projects, it is unclear that such a reduction

should do much to stimulate economic activity or to halt deﬂationary

pressures.

Hence we are not inclined to suppose that our irrelevance proposition

represents so poor an approximation to reality as to deprive it of practical

relevance. Even if the effects of open-market operations under the condi-

tions the proposition describes are not exactly zero, it seems unlikely that

they should be large. In our view it is more important to note that our

162 Brookings Papers on Economic Activity, 1:2003

26. Examples of studies ﬁnding either no effects or only quantitatively unimportant

ones include Modigliani and Sutch (1967), Frankel (1985), Agell and Persson (1992), Wal-

lace and Warner (1996), and Hess (1999). Roley (1982) and Friedman (1992) ﬁnd some-

what larger effects.

27. Agell and Persson (1992, p. 78).

28. Clouse and others (2003). Stephen G. Cecchetti (“Central Banks Have Plenty of

Ammunition,” Financial Times, March 17, 2003, p. 13) similarly argues that it should be

possible for the Federal Reserve to independently affect long-term bond yields if it is deter-

mined to do so, given that it can print money without limit to buy additional long-term

Treasuries if necessary.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 162

irrelevance proposition depends on an assumption that interest rate policy

is speciﬁed in a way that implies that these open-market operations have

no consequences for interest rate policy, either immediately (which is

trivial, because it would not be possible for them to lower current interest

rates, which is the only effect that would be desired), or at any subsequent

date. We have also speciﬁed ﬁscal policy in a way that implies that the

contemplated open-market operations have no effect on the path of total

government liabilities {D

} either, whether immediately or at any later

date. Although we think these deﬁnitions make sense, as a way of isolat-

ing the pure effects of open-market purchases of assets by the central

bank from either interest rate policy on the one hand or ﬁscal policy on

the other, those who recommend monetary expansion by the central bank

may intend for this to have consequences of one or both of these other

sorts.

For example, when it is argued that a “helicopter drop” of money into

the economy would surely stimulate nominal aggregate demand, the

thought experiment that is usually contemplated is not simply a change in

the function ψ in our policy rule equation 11. First of all, it is typically

supposed that the expansion of the money supply will be permanent. If

this is the case, then the function φ that deﬁnes interest rate policy is also

being changed, in a way that will become relevant at some future date,

when the money supply no longer exceeds the satiation level.

Second,

the assumption that the money supply is increased through a helicopter

drop rather than an open-market operation implies a change in ﬁscal pol-

icy as well. Such an operation would increase the value of nominal gov-

ernment liabilities, and it is generally at least tacitly assumed that this is a

permanent increase as well. Hence the experiment that is imagined is not

one that our irrelevance proposition implies should have no effect on the

equilibrium path of prices.

Gauti B. Eggertsson and Michael Woodford 163

29. This explains the apparent difference between our result and that obtained by Auer-

bach and Obstfeld (2003) in a similar model. These authors assume explicitly that an

increase in the money supply at a time when the zero bound binds carries with it the impli-

cation of a permanently larger money supply, and that there exists a future date at which the

zero bound ceases to bind, so that the larger money supply will imply a different interest

rate policy at that later date. Clouse and others (2003) also stress that maintenance of the

larger money supply until a date at which the zero bound would not otherwise bind repre-

sents one straightforward channel through which open-market operations while the zero

bound is binding could have a stimulative effect, although they discuss other possible chan-

nels as well.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 163

Even more important, our irrelevance result applies only given a cor-

rect private sector understanding of the central bank’s commitments

regarding future policy. Such understanding may be lacking. We have just

argued that the key to lowering long-term interest rates, in a way that

actually provides an incentive for increased spending, is to change expec-

tations regarding the likely future path of short-term rates, rather than

through intervention in the market for long-term Treasuries. As a matter

of logic, this need not require any open-market purchases of long-term

Treasuries at all. Nonetheless, the private sector may be uncertain about

the nature of the central bank’s policy commitment, and so it may scruti-

nize the bank’s current actions for further clues. In practice, the manage-

ment of private sector expectations is an art of considerable subtlety, and

shifts in the portfolio of the central bank could be of some value in mak-

ing credible to the private sector the central bank’s own commitment to a

particular kind of future policy, as we discuss further in the penultimate

section of the paper. Signaling effects of this kind are often argued to be

an important reason for the effectiveness of interventions in foreign-

exchange markets, and they might well provide a justiﬁcation for open-

market operations when the zero bound binds.

We do not wish, then, to argue that asset purchases by the central bank

are necessarily pointless under the circumstances of a binding zero lower

bound on short-term nominal interest rates. However, we do think it

important to observe that, insofar as such actions can have any effect, it is

not because of any necessary or mechanical consequence of the shift in

the portfolio of assets in the hands of the private sector itself. Instead, any

effect of such actions must be due to the way in which they change expec-

tations regarding future interest rate policy or, perhaps, the future path of

total nominal government liabilities. Later we discuss reasons why open-

market purchases by the central bank might plausibly have consequences

for expectations of these types. But because it is only through effects on

expectations regarding future policy that these actions can matter, we

focus our attention on the question of what kind of commitments regard-

ing future policy are in fact to be desired. And this question can be

addressed without explicit consideration of the role of central bank open-

market operations of any kind. Hence we will simplify our model—

164 Brookings Papers on Economic Activity, 1:2003

30. Clouse and others (2003) argue that this is one important channel through which

open-market operations can be effective.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 164

abstracting from monetary frictions and the structure of government lia-

bilities altogether—and instead consider what is the desirable conduct of

interest rate policy, and what kind of commitments about this policy are

desirable to make in advance.

How Severe a Constraint Is the Zero Bound?

We turn now to the question of how the existence of the zero bound

restricts the degree to which a central bank’s stabilization objectives, with

regard to both inﬂation and real activity, can be achieved, even under

ideal policy. The discussion in the previous section established that the

zero bound does represent a genuine constraint. It is not true that equilib-

ria that cannot be achieved through a suitable interest rate policy can

somehow be achieved through other means, and the zero bound does limit

the set of possible equilibrium paths for prices and output, although the

quantitative importance of this constraint remains to be seen.

Nonetheless, we will see that it is not at all the case that a central bank

can do nothing to mitigate the severity of the destabilizing impact of the

zero bound. The reason is that inﬂation and output do not depend solely

on the current level of short-term nominal interest rates, or even solely on

the history of such rates up until the present (so that the current level of

interest rates would be the only thing that could possibly change in

response to an unanticipated disturbance). The expected character of

future interest rate policy is also a critical determinant of the degree to

which the central bank achieves its stabilization objectives, and this

allows important scope for policy to be improved upon, even when there

is little choice about the current level of short-term interest rates.

In fact, the management of expectations is the key to successful mone-

tary policy at all times, not just in those relatively unusual circumstances

when the zero bound is reached. The effectiveness of monetary policy has

little to do with the direct effect of changing the level of overnight interest

rates, since the current cost of maintaining cash balances overnight is of

fairly trivial signiﬁcance for most business decisions. What actually mat-

ters is the private sector’s anticipation of the future path of short-term

rates, because this determines equilibrium long-term interest rates as well

as equilibrium exchange rates and other asset prices—all of which are

quite relevant for many current spending decisions, and hence for optimal

Gauti B. Eggertsson and Michael Woodford 165

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 165

pricing behavior as well. How short-term rates are managed matters

because of the signals that such management gives about how the private

sector can expect them to be managed in the future. But there is no reason

to suppose that expectations regarding future monetary policy, and hence

regarding the future paths of nominal variables more generally, should

change only insofar as the current level of overnight interest rates

changes. A situation in which there is no decision to be made about the

current level of overnight rates (as in Japan at present) is one that gives

urgency to the question of what expectations regarding future policy one

should wish to create, but this is in fact the correct way to think about

sound monetary policy at all times.

Of course, the question of what future policy one should wish people to

expect does not arise if current constraints leave no possibility of commit-

ting oneself to a different sort of policy in the future than one would oth-

erwise have pursued. This means that the private sector must be

convinced that the central bank will not conduct policy in a way that is

purely forward looking, that is, taking account at each point in time only

of the possible paths that the economy could follow from that date

onward. For example, we will show that it is undesirable for the central

bank to pursue a given inﬂation target, once the zero bound is expected no

longer to prevent that target from being achieved, even in the case that the

pursuit of this target would be optimal if the zero bound did not exist (or

would never bind under an optimal policy). The reason is that an expecta-

tion that the central bank will pursue the ﬁxed inﬂation target after the

zero bound ceases to bind gives people no reason to hold the kind of

expectations, while the bound is binding, that would mitigate the distor-

tions created by it. A history-dependent inﬂation target

—if the central

bank’s commitment to it can be made credible—can instead yield a supe-

rior outcome.

But this, too, is an important feature of optimal policy rules more gen-

erally.

Hence the analytical framework and institutional arrangements

used in making monetary policy need not be changed in any fundamental

way in order to deal with the special problems created by a liquidity trap.

As we explain later in the paper, the optimal policy in the case of a bind-

ing zero bound can be implemented through a targeting procedure that

166 Brookings Papers on Economic Activity, 1:2003

31. As we will show, it is easier to explain the nature of the optimal commitment if it is

described as a history-dependent price-level target.

32. See, for example, Woodford (forthcoming, chapter 7).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 166

represents a straightforward generalization of a policy that would be opti-

mal even if the zero bound were expected never to bind.

Feasible Responses to Fluctuations in the Natural Rate of Interest

In order to characterize how stabilization policy is constrained by the

zero bound, we make use of a log-linear approximation to the structural

equations presented in the previous section, of a kind that is often

employed in the literature on optimal monetary stabilization policy.

Speciﬁcally, we log-linearize the structural equations of our model

(except for the zero bound in expression 4) around the paths of inﬂation,

output, and interest rates associated with a zero-inﬂation steady state, in

the absence of disturbances (

= 0). We choose to expand around these

particular paths because the zero-inﬂation steady state represents optimal

policy in the absence of disturbances.

In the event of small enough dis-

turbances, optimal policy will still involve paths in which inﬂation, out-

put, and interest rates are at all times close to those of the zero-inﬂation

steady state. Hence an approximation to our equilibrium conditions that is

accurate in the case of inﬂation, output, and interest rates near those val-

ues will allow an accurate approximation to the optimal responses to dis-

turbances in the case that the disturbances are small enough.

In the zero-inﬂation steady state, it is easily seen that the real rate of

interest is equal to r

–

≡ β

–1

– 1 > 0; this is also the steady-state nominal

interest rate. Hence, in the case of small enough disturbances, optimal

policy will involve a nominal interest rate that is always positive, and the

zero bound will not be a binding constraint. (Optimal policy in this case is

characterized in the references cited in the previous paragraph.) However,

we are interested in the case in which disturbances are at least occasion-

ally large enough for the zero bound to bind, that is, to prevent attainment

of the outcome that would be optimal in the absence of such a bound. It is

Gauti B. Eggertsson and Michael Woodford 167

33. See, for example, Clarida, Galí, and Gertler (1999); Woodford (forthcoming).

34. See Woodford (forthcoming, chapter 7) for more detailed discussion of this point.

The fact that zero inﬂation, rather than mild deﬂation, is optimal depends on our abstracting

from transactions frictions, as discussed further in footnote 40 below. As Woodford shows,

a long-run inﬂation target of zero is optimal in this model, even when the steady-state out-

put level associated with zero inﬂation is suboptimal, owing to market power. The reason is

that a commitment to inﬂation in some period t results both in increased output in period t

and in reduced output in period t – 1 (owing to the effect of expected inﬂation on the aggre-

gate supply relation, equation 25 below); because of discounting, the second effect on wel-

fare fully offsets the beneﬁt of the ﬁrst effect.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 167

possible to consider this problem rigorously using only a log-linear

approximation to the structural equations in the case where the lower

bound on nominal interest is assumed to be not much below r

–

. We can

arrange for this gap to be as small as we may wish, without changing

other crucial parameters of the model such as the assumed rate of time

preference, by supposing that interest is paid on the monetary base at a

rate i

≥ 0 that cannot (for some institutional reason) be reduced. Then the

lower bound on interest rates actually becomes

We will characterize optimal policy subject to a constraint of the form of

expression 23, in the case that both a bound on the amplitude of distur-

bances |||| and the steady-state opportunity cost of holding money δ

–

≡

–

– i

)/(1 + r

–

) > 0 are small enough. Specifically, both our structural

equations and our characterization of the optimal responses of inflation,

output, and interest rates to disturbances will be required to be exact only

up to a residual of order O(||; δ

–

): We then hope (without here seeking

to verify) that our characterization of optimal policy in the case of a

small opportunity cost of holding money and small disturbances is not

too inaccurate in the case of an opportunity cost of several percentage

points (the case in which i

= 0) and disturbances large enough to cause

the natural rate of interest to vary by several percentage points (as will be

required in order for the zero bound to bind).

As Woodford has shown elsewhere,

the log-linear approximate equi-

librium relations may be summarized by two equations each period: a

forward-looking “IS relation”

and a forward-looking “AS relation” (or New Keynesian Phillips curve)

Here π

≡ log(P

/ P

t–1

) is the inﬂation rate, x

is a welfare-relevant output

gap, and i

is now the continuously compounded nominal interest rate,

corresponding to log (1 + i

) in the notation used in the previous section.

() .25

πκ βπ

ttttt

xE u=+ +

() – ( ––),24

xEx iE r

ttt tttt

σπ

() .23 ii

≥

168 Brookings Papers on Economic Activity, 1:2003

35. Woodford (forthcoming).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 168

The terms u

and r

are composite exogenous disturbance terms that shift

the two equations; the former is commonly referred to as a cost-push dis-

turbance, whereas the latter indicates exogenous variation in the Wicksel-

lian natural rate of interest, that is, the equilibrium real rate of interest in

the case that output growth is at all times equal to its natural rate. The

coefﬁcients σ and κ are both positive, and 0 < β < 1 is again the utility dis-

count factor of the representative household.

Equation 24 is a log-linear approximation to equation 2, whereas

equation 25 is derived by log-linearizing equations 7 through 9 and then

eliminating log (p

*/P

). We omit the log-linear version of the money

demand relation in expression 3, because here we are interested solely in

characterizing the possible equilibrium paths of inflation, output, and

interest rates, and we may abstract from the question of what might be

the required path for the monetary base that is associated with any such

equilibrium. (It suffices that there exist a monetary base that will satisfy

the money demand relation in each case, and this will be true as long as

the interest rate bound is satisfied.) The other equilibrium requirements

of the earlier discussion can be ignored in the case that we are interested

only in possible equilibria that remain forever near the zero-inflation

steady state, because they are automatically satisfied in that case. Equa-

tions 24 and 25 represent a pair of equations each period to determine

inflation and the output gap, given the central bank’s interest rate policy.

We will seek to compare alternative possible paths for inflation, the out-

put gap, and the nominal interest rate that satisfy these two log-linear

equations together with expression 23. Note that our conclusions will be

identical (up to a scale factor) in the event that we multiply the amplitude

of the disturbances and the steady-state opportunity cost δ

–

by any com-

mon factor; alternatively, if we measure the amplitude of disturbances in

units of δ

–

, our results will be independent of the value of δ

–

(to the extent

that our log-linear approximation remains valid). Hence we choose the

normalization δ

–

= 1 – β, corresponding to i

= 0, to simplify the presen-

tation. In that case the lower bound for the nominal interest rate is again

given by expression 4.

Deﬂation under Forward-Looking Policy

We begin by considering the degree to which the zero bound impedes

the achievement of the central bank’s stabilization objectives in the case

Gauti B. Eggertsson and Michael Woodford 169

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 169

that the bank pursues a strict inﬂation target. We interpret this as a com-

mitment to adjust the nominal interest rate so that

each period, insofar as it is possible to achieve this with some nonnega-

tive interest rate. It is easy to verify, by the IS and AS equations above,

that a necessary condition for this target to be satisﬁed is

When inﬂation is on target, the real interest rate is equal to the natural real

rate at all times, and the output gap is at its long-run level. The zero

bound, however, prevents equation 27 from holding if r

< –π*. Thus, if

the natural rate of interest is low, the zero bound frustrates the central

bank’s ability to implement an inﬂation target. Suppose the inﬂation tar-

get is zero, so that π* = 0. Then the zero bound is binding if the natural

rate of interest is negative, and the central bank is unable to achieve its

inﬂation target.

To illustrate this, consider the following experiment. Suppose the nat-

ural rate of interest is unexpectedly negative in period 0 and reverts back

to its steady-state value r

–

> 0 with a ﬁxed probability in every period. Fig-

ure 2 shows the state-contingent paths of the output gap and inﬂation

under these circumstances for each of three different possible inﬂation

targets π*. We assume in period 0 that the natural rate of interest becomes

–2 percent a year and then reverts back to the steady-state value of +4 per-

cent a year with a probability of 0.1 each quarter. Thus the natural rate of

interest is expected to be negative for ten quarters on average at the time

the shock occurs.

The dashed lines in ﬁgure 2 show the state-contingent paths of the out-

put gap and inﬂation if the central bank targets zero inﬂation.

Starting

() *.27 ir

=+π

() *26 ππ

170 Brookings Papers on Economic Activity, 1:2003

36. In our numerical analysis, we interpret periods as quarters, and we assume coefﬁ-

cient values of σ=0.5, κ = 0.02, and β = 0.99. The assumed value of the discount factor

implies a long-run real rate of interest r

equal to 4 percent a year. The assumed value of κ is

consistent with the empirical estimate of Rotemberg and Woodford (1997). The assumed

value of σ represents a relatively low degree of interest sensitivity of aggregate expendi-

ture. We prefer to bias our assumptions in the direction of only a modest effect of interest

rates on the timing of expenditure, so as not to exaggerate the size of the output contraction

that is predicted to result from an inability to lower interest rates when the zero bound

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 170

Gauti B. Eggertsson and Michael Woodford 171

Figure 2. State-Contingent Responses of Inﬂation and the Output Gap to a Shock to

the Natural Rate of Interest under Strict Inﬂation Targeting

Source: Authors’ calculations.

a. The targeted rate of inﬂation is designated by π*. Each upward-sloping line represents the response of inﬂation or the output

gap if the natural rate of interest returns to its steady-state value in that period.

–10

–5

π* = 2%

π* = 1%

π* = 0

Inflation

Percent a year

–10

–5

0 5 10 15

Quarters after shock to natural rate of interest

Output gap

Percent of GDP

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 171

from the left, the ﬁrst dashed line shows the equilibrium that prevails if

the natural rate of interest returns to the steady state in period 1, the next

line if it returns in period 2, and so on. The inability of the central bank to

set a negative nominal interest rate results in a 14 percent output gap and

10 percent annual deﬂation. The fact that in each quarter there is a 90 per-

cent chance of the natural rate of interest remaining negative for the next

quarter creates the expectation of future deﬂation and a continued nega-

tive output gap, which creates even further deﬂation. Even if the central

bank lowers the short-term nominal interest rate to zero, the real rate of

return is positive, because the private sector expects deﬂation.

The shaded lines in ﬁgure 2 show the equilibrium that prevails if the

central bank instead sets a 1 percent annual inﬂation target. In this case

the private sector expects 1 percent inﬂation once the economy is out of

the liquidity trap. This, however, is not enough to offset the –2 percent

natural rate of interest, so that in equilibrium the private sector expects

deﬂation instead of inﬂation. The result of this and a negative natural rate

of interest is 4 percent annual deﬂation (when the natural rate of interest is

negative) and an output gap of 7 percent.

Finally, the solid horizontal line shows the evolution of output and

inﬂation in the case where the central bank targets 2 percent annual inﬂa-

tion. In this case the central bank can satisfy equation 4 even when the nat-

ural rate of interest is negative. When the natural rate of interest is

–2 percent, the central bank lowers the nominal interest rate to zero. Since

the inﬂation target is 2 percent, the real rate is –2 percent, which is enough

to close the output gap and keep inﬂation on target. If the inﬂation target is

high enough, therefore, the central bank is able to accommodate a negative

natural rate of interest. This is the argument given by Edmund Phelps,

Lawrence Summers, and Stanley Fischer for a positive inﬂation target.

Krugman makes a similar argument and suggests more concretely that, in

1998, Japan needed a positive inﬂation target of 4 percent under its then-

current circumstances to achieve negative real rates and curb deﬂation.

Although it is clear that commitment to a higher inﬂation target will

indeed guard against the need for an output gap in periods when the nat-

172 Brookings Papers on Economic Activity, 1:2003

binds. As ﬁgure 2 shows, even for this value of σ, the output contraction that results from a

slightly negative value of the natural rate of interest is quite substantial.

37. Phelps (1972); Summers (1991); Fischer (1996).

38. Krugman (1998).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 172

ural rate of interest falls, the price of this solution is the distortions created

by the inﬂation, both when the natural rate of interest is negative and

under more normal circumstances as well. Hence the optimal inﬂation tar-

get (from among the strict inﬂation targeting policies just considered) will

be some value that is at least slightly positive, in order to mitigate the dis-

tortions created by the zero bound when the natural rate of interest is neg-

ative, but not so high as to keep the zero bound from ever binding (see the

table in the next section). An intermediate inﬂation target, in contrast (like

the 1 percent target considered in the ﬁgure), leads to a substantial reces-

sion when the natural rate of interest becomes negative, and chronic inﬂa-

tion at all other times. Hence no such policy allows a complete solution of

the problem posed by the zero bound in the case that the natural rate of

interest is sometimes negative.

Nor can one do better through commitment to any policy rule that is

purely forward looking in the sense discussed elsewhere by Woodford.

A purely forward-looking policy is one under which the central bank’s

action at any time depends only on an evaluation of the possible paths for

the central bank’s target variables (here, inﬂation and the output gap) that

are possible from the current date forward, neglecting past conditions

except insofar as they constrain the economy’s possible future path. In the

log-linear model presented above, the possible paths for inﬂation and the

output gap from period t onward depend only on the expected evolution

of the natural rate of interest from period t onward. If one assumes a Mar-

kovian process for the natural rate, as in the numerical analysis above,

then any purely forward-looking policy will result in an inﬂation rate, out-

put gap, and nominal interest rate in period t that depend only on the nat-

ural rate in period t—in our numerical example, on whether the natural

rate is still negative or has already returned to its long-run steady-state

value. It is easily shown in the case of our two-state example that the

optimal state-contingent path for inflation and output from among those

with this property will be one in which the zero bound binds if and only

if the natural rate is in the low state; hence it will correspond to a strict

inflation target of the kind just considered, for some π* between zero and

2 percent.

But one can actually do considerably better, through commitment to a

history-dependent policy, in which the central bank’s actions will depend

Gauti B. Eggertsson and Michael Woodford 173

39. Woodford (2000).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 173

on past conditions even though these are irrelevant to the degree to which

its stabilization goals could in principle be achieved from then on. In the

next section we characterize the optimal form of history-dependent policy

and determine the degree to which it improves upon the stabilization of

both output and inﬂation.

The Optimal Policy Commitment

We now characterize optimal monetary policy, by optimizing over the

set of all possible state-contingent paths for inﬂation, output, and the

short-term nominal interest rate consistent with the log-linearized struc-

tural relations in equations 24 and 25. It is assumed (for now) that the

expectations regarding future state-contingent policy that are required for

such an equilibrium can be made credible to the private sector. In consid-

ering the central bank’s optimization problem under the assumption that a

credible commitment is possible regarding future policy, we do not mean

to minimize the subtlety of the task of actually communicating such a

commitment to the public and making it credible. However, we do not

believe it makes sense to recommend a policy that would systematically

seek to achieve an outcome other than a rational expectations equilib-

rium. That is, we are interested in policies that will have the desired effect

even when correctly understood by the public. Optimization under the

assumption of credible commitment is simply a way of ﬁnding the best

possible rational expectations equilibrium. Once the equilibrium that one

would like to bring about has been identiﬁed, along with the interest rate

policy that it requires, one can turn to the question of how best to signal

these intentions to the public (an issue that we brieﬂy address in the

paper’s penultimate section).

We assume that the government minimizes

This loss function can be derived by a second-order Taylor expansion of

the utility of the representative household.

The optimal program can be

( ) min ( ) .28

βπ λ+













∞

∑

174 Brookings Papers on Economic Activity, 1:2003

40. See Woodford (forthcoming, chapter 6) for details. This approximation applies in

the case that we abstract from monetary frictions as assumed in this section. If transactions

frictions are instead nonnegligible, the loss function should include an additional term

proportional to (i

– i

)

. This would indicate welfare gains from keeping nominal interest

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 174

found by a Lagrangian method, extending the methods used by Richard

Clarida, Jordi Galí, and Mark Gertler and by Woodford to the case in

which the zero bound can sometimes bind, as shown by Taehun Jung,

Yuki Teranishi, and Tsutomu Watanabe.

We combine the zero bound

and the IS equation to yield the following inequality:

The Lagrangian for this problem is then

Jung, Teranishi, and Watanabe show that the ﬁrst-order conditions for an

optimal policy commitment are

One cannot solve this system by applying standard solution methods for

rational expectations models, because of the complications of the nonlin-

ear constraint in equation 31. The appendix describes the numerical

() , , .31 0 0 0

ϕϕ

tt tt

ii≥≥ =

() ––

–

30 0

11 2

λϕβϕ κϕx

tt t t

() ––

–

29 0

221

πϕ ϕ βσϕ

ttt t

111 2 1

[]













∞

++ +

∑

xxx r x

tt ttt t t

tt t t

πλ ϕ σπ σ ϕπκ βπ–– – –– .

xEx rE

ttt t

≤++

++11

σπ().

Gauti B. Eggertsson and Michael Woodford 175

rates as close as possible to the zero bound (or, more generally, the lower bound i

Nonetheless, because of the stickiness of prices, it would not be optimal for interest rates

to be at zero at all times, as implied by the flexible-price model discussed by Uhlig (2000).

The optimal inflation rate in the absence of shocks would be slightly negative, rather than

zero as in the “cashless” model considered in this section; but it would not be so low that

the zero bound would be reached, except in the event of temporary declines in the natural

rate of interest, as in the analysis here. Note also that equation 28 implies that the optimal

output gap is zero. More generally, there should be an output gap stabilization objective of

the form (x

– x*)

; the utility-based loss function involves x* = 0 only if one assumes the

existence of an output or employment subsidy that offsets the distortion due to the market

power of firms. However, the value of x* affects neither the optimal state-contingent paths

derived in this section, and shown in figures 3 and 4, nor the formulas given in the earlier

section for the optimal targeting rule.

41. Clarida, Galí, and Gertler (1999); Woodford (1999; forthcoming, chapter 7); Jung,

Teranishi, and Watanabe (2001).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 175

method we use to solve these equations instead.

Here we discuss the

results that we obtain for the particular numerical experiment considered

in the previous section.

What is apparent from the ﬁrst-order conditions is that optimal policy

is history dependent, so that the optimal choice of inﬂation, the output

gap, and the nominal interest rate depends on the past values of the

endogenous variables. This can be seen by the appearance of a lagged

value of the Lagrange multipliers in the ﬁrst-order conditions. To get a

sense of how this history dependence matters, it is useful to consider

again the numerical example shown in figure 2. Suppose the natural rate

of interest becomes negative in period 0 and then reverts to the steady

state with a fixed probability in each period. Figure 3 shows the optimal

output gap, the inflation rate, and the price level from period 0 to

period 25. As in figure 2, the separate lines in each panel show the evo-

lution of the variables in the case that the disturbances last for different

lengths of time ranging from one quarter to twenty quarters.

One observes that the optimal policy involves committing to the cre-

ation of an output boom once the natural rate again becomes positive, and

hence to the creation of future inﬂation. Such a commitment stimulates

aggregate demand and reduces deﬂationary pressure while the economy

remains in the liquidity trap, through each of several channels.

As Krugman points out,

creating the expectation of future inﬂation

can lower real interest rates, even when nominal interest rates cannot be

reduced. In the context of Krugman’s model, it might seem that this

requires that inﬂation be promised quite quickly (that is, by the following

“period”). Our fully intertemporal model shows how even the expectation

of later inﬂation—which nominal interest rates are not expected to rise to

offset—can stimulate current demand, because in our model current

spending decisions depend on real interest rate expectations far in the

future. For the same reason, the expectation that nominal interest rates

will be kept low later, when the central bank might otherwise have raised

them, will also stimulate spending while the zero bound still binds. And

176 Brookings Papers on Economic Activity, 1:2003

42. Jung, Teranishi, and Watanabe (2001) discuss the solution of these equations only

for the case in which the number of periods for which the natural rate of interest will be

negative is known with certainty at the time that the disturbance occurs. Here we show how

the system can be solved in the case of a stochastic process for the natural rate of a particu-

lar kind.

43. Krugman (1998).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 176

Figure 3. Inﬂation, the Output Gap, and Prices under the Optimal Policy

Commitment

Source: Authors’ calculations.

a. See ﬁgure 2 for explanation of diagram.

–0.1

0.0

0.1

0.2

0.3

Inflation

Output gap

100.0

100.2

0 5 10 15 20

Price level

Index, quarter -1 =100

Percent of GDP

Percent a year

Quarters after shock to natural rate of interest

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 177

ﬁnally, the expectation of higher future income should stimulate current

spending, in accordance with the permanent income hypothesis. In addi-

tion, prices are less likely to fall, even given the current level of real activ-

ity, to the extent that future inﬂation is expected. This reduces the

distortions created by deﬂation itself.

On the other hand, these gains from the change in expectations while

the economy is in the liquidity trap can be achieved (given rational expec-

tations on the part of the private sector) only if the central bank is

expected to actually pursue the inﬂationary policy after the natural rate

returns to its normal level. This will in turn create distortions at that later

time, and this limits the extent to which this tool is used under an optimal

policy. Hence some contraction of output and some deﬂation occur during

the time that the natural rate is negative, even under the optimal policy

commitment.

Also, and this is a key point, although the optimal policy involves com-

mitment to a higher price level in the future, the price level will ultimately

be stabilized. This is in sharp contrast to a constant positive-inﬂation tar-

get, which would imply an ever-increasing price level. Figure 4 shows the

corresponding state-contingent nominal interest rate under the optimal

commitment and contrasts it with the evolution of the nominal interest

rate under a zero-inﬂation target. To increase inﬂation expectations in the

trap, the central bank commits to keeping the nominal interest rate at zero

after the natural rate of interest becomes positive again. In contrast, if the

central bank targets zero inﬂation, it raises the nominal interest rate as

soon as the natural rate of interest becomes positive again. The optimal

commitment is an example of history-dependent policy, in which the cen-

tral bank commits itself to raise interest rates slowly at the time the nat-

ural rate becomes positive in order to affect expectations when the zero

bound is binding.

The nature of the additional history dependence of the optimal policy

may perhaps be more easily seen if we consider the paths of inﬂation, out-

put, and interest rates under a single possible realization of the random

fundamentals. Figure 5 compares the equilibrium paths of all three vari-

ables, both under the zero-inﬂation target and under optimal policy, in the

case where the natural rate of interest is negative for ﬁfteen quarters (t = 0

through 14), but where it is not known until quarter 15 that the natural rate

will return to its normal level in that quarter. Under the optimal policy

the nominal interest rate is kept at zero for five more quarters (t = 15

178 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 178

through 19), whereas it immediately returns to its long-run steady-state

level in quarter 15 under the forward-looking policy. The consequence of

the public anticipating policy of this kind is that both the contraction of

real activity and the deflation that occur under the strict inflation target

are largely avoided, as shown in the second and third panels of the figure.

Implementing Optimal Policy

We turn now to the question of how policy should be conducted in

order to bring about the optimal equilibrium characterized in the previous

section. The question of the implementation of optimal policy remains

nontrivial, even after the optimal state-contingent paths of all variables

have been identiﬁed, because in general the solution obtained for the opti-

mal state-contingent path of the policy instrument (the short-term nominal

interest rate) does not in itself represent a useful description of a policy

Gauti B. Eggertsson and Michael Woodford 179

Figure 4. Response of the Nominal Interest Rate under a Zero-Inﬂation Target and

under the Optimal Policy Commitment

Source: Authors’ calculations.

15 20

Quarters after shock to natural rate of interest

Percent a year

Optimal

π* = 0

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 179

Interest rate

Optimalπ* = 0

Percent a year

–10

–8

–6

–4

–2

1050

15 20

Output gap

Inflation

Percent a year

Percent of GDP

Figure 5. Response of the Nominal Interest Rate, Inﬂation, and the Output Gap to a

Shock of Speciﬁc Duration

Source: Authors’ calculations.

a. Response to a fall in the natural rate of interest below zero for a period of ﬁfteen quarters.

1440-03 BPEA/Eggertsson 07/18/03 16:19 Page 180

rule.

For example, in the context of the present model, a commitment to

a state-contingent nominal interest rate path, even when fully credible,

does not imply determinate rational expectations equilibrium paths for

inﬂation and output; it is instead necessary for the central bank to be com-

mitted (and be understood to be committed) to a particular way of

responding to deviations of inﬂation and the output gap from their desired

paths. Another problem is that a complete description of the optimal state-

contingent interest rate path is unlikely to be feasible. In the previous sec-

tion we showed that one can characterize (at least numerically) the

optimal state-contingent interest rate path in the case of one very particu-

lar kind of stochastic process for the natural rate of interest. But a solution

of this kind, allowing for all possible states of belief about the probabili-

ties of various future paths of the natural rate (and disturbances to the

aggregate supply relation as well), would be difﬁcult to write down, let

alone explain to the public.

Here we show that optimal policy can nonetheless be implemented

through commitment to a policy rule that speciﬁes the central bank’s

short-run targets at each point in time as a (fairly simple) function of what

has occurred before that date. How can this be done? One may be tempted

to believe that our suggested policy is not entirely realistic or operational.

Figures 3 and 4, for example, indicated that the optimal policy involves a

complicated state-contingent plan for the nominal interest rate, which

would be hard to communicate to the public. Furthermore, it may appear

that it depends on knowledge of a special statistical process for the natural

rate of interest, which is in practice hard to estimate. Our discussion of the

ﬁxed inﬂation target suggests that the effectiveness of increasing inﬂation

expectations to close the output gap depends on the difference between

the announced inﬂation target and the natural rate of interest. It may

therefore seem crucial to estimate the natural rate of interest in order to

implement the optimal policy. Below, however, we present the striking

result that the optimal policy rule can be implemented without any esti-

mate or knowledge of the statistical process for the natural rate of interest.

This is an example of a robustly optimal direct policy rule of the kind dis-

cussed by Marc Giannoni and Woodford for the case of a general class of

linear-quadratic policy problems.

An interesting feature of the present

Gauti B. Eggertsson and Michael Woodford 181

44. For further discussion in a more general context, see Woodford (forthcoming,

chapter 7).

45. Giannoni and Woodford (2003).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 181

example is that we show how to construct a robustly optimal rule in the

same spirit, in a case where not all of the relevant constraints are linear

(owing to the fact that the zero bound binds at some times and not at

others).

An Optimal Targeting Rule

To implement the rule proposed here, the central bank need only

observe the price level and the output gap. The rule suggested replicates

exactly the history dependence discussed in the last section. The rule is

implemented as follows.

First, in each and every period there is a predetermined price-level tar-

get p

*. The central bank chooses the interest rate i

to achieve the target

relation

if this is possible. If it is not possible, even by lowering the nominal inter-

est rate to zero, then i

= 0. Here p˜

is an output gap-adjusted price index,

deﬁned by

The target for the next period is then determined as

where ∆

is the target shortfall in period t:

It can be veriﬁed that this rule does indeed achieve the optimal commit-

ment solution. If the price-level target is not reached, because of the zero

() –

34 ∆

tt t

pp≡

() ( ) – ,

––

–

33 1

tt t t+

=+ +βκσβ∆∆

.pp x

tt t

≡+

()

32 pp

182 Brookings Papers on Economic Activity, 1:2003

46. On the desirability of a target for this index in the case that the zero bound does not

bind, see Woodford (forthcoming, chapter 7). This would correspond to a nominal GDP tar-

get in the case that λ = κ and that the natural rate of output follows a deterministic trend.

However, the utility-based loss function derived in Woodford (forthcoming, chapter 6)

involves λ = κ/θ, where θ >1 is the elasticity of demand faced by the suppliers of differenti-

ated goods, so that the optimal weight on output is considerably less than under a nominal

GDP target. Furthermore, the welfare-relevant output gap is unlikely to correspond too

closely to deviations of real GDP from a deterministic trend.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 182

bound, the central bank increases its target for the next period. This, in

turn, increases inﬂation expectations further in the trap, which is exactly

what is needed to reduce the real interest rate.

Figure 6 shows how the price-level target p

* would evolve over time,

depending on the number of periods in which the natural rate of interest

remains negative, in the same numerical experiment as in ﬁgure 3. (Here

the dark lines show the evolution of the gap-adjusted price level p˜

, and

the shaded lines show the evolution of p

*.) One observes that the target

price level is ratcheted steadily higher during the period in which the nat-

ural rate remains negative, as the actual price level continues to fall below

the target by an increasing amount. Once the natural rate of interest

becomes positive again, the degree to which the gap-adjusted price level

undershoots the target begins to shrink, although the target often contin-

ues to be undershot (as the zero bound continues to bind) for several

more quarters. (How long this is true depends on how high the target

price level has risen relative to the actual index; it will be higher the

longer the natural rate has been negative.) As the degree of undershoot-

ing begins to shrink, the price-level target begins to fall again, as a result

Gauti B. Eggertsson and Michael Woodford 183

Figure 6. Responses of the Price-Level Target and the Gap-Adjusted Price Level to a

Shock to the Natural Rate of Interest

Source: Authors’ calculations.

101.4

101.2

101.0

100.8

100.6

100.4

100.2

100.0

0 5 10 15

Gap-adjusted index, quarter –1 = 100

Target (p

Actual path (p

)

Quarters after shock to natural rate of interest

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 183

of the dynamics speciﬁed by equation 33. This hastens the date at which

the target can actually be hit with a nonnegative interest rate. Once the

target ceases to be undershot, it no longer changes, and the central bank

targets and achieves a new constant value for the gap-adjusted price level

p˜

, one slightly higher than the target in place before the disturbance

occurred.

Note that this approach to implementing optimal policy answers the

question of whether there is any point in announcing an inﬂation target (or

price-level target) if one knows that it is extremely unlikely to be

achieved in the short run, because the zero bound is likely to continue to

bind. The answer here is yes. The central bank wishes to make the private

sector aware of its commitment to the time-varying price-level target

described by equations 32 through 34, because eventually it will be able

to hit the target. The anticipation of that fact (that is, of the level that

prices will eventually reach, as a result of the policies that the bank will

follow after the natural rate of interest again becomes positive) while the

natural rate is still negative is important in mitigating the distortions

caused by the zero bound. The fact that the target is not hit immediately

should not create doubts about whether central bank announcements

regarding its target have any meaning, if it is explained that the bank is

committed to hitting the target if this is possible at a nonnegative interest

rate, so that, at each point in time, either the target will be attained or a

zero interest rate policy will be followed. The existence of the target is

relevant even when it is not being attained, because it allows the private

sector to judge how close the central bank is to a situation in which it

would feel justiﬁed in abandoning the zero interest rate policy; hence the

current gap between the actual and the target price level should shape pri-

vate sector expectations of the time when interest rates are likely to

remain low.

Would the private sector have any reason to believe that the central

bank was serious about the price-level target, if in each period all that is

184 Brookings Papers on Economic Activity, 1:2003

47. An interesting feature of the optimal rule is that it involves a form of history depen-

dence that cannot be summarized solely by the past history of short-term nominal interest

rates; if the nominal interest rate has fallen to zero in the recent past, it matters to what

extent the zero bound has prevented the central bank from pursuing as stimulative a policy

as it otherwise would have. In this respect the optimal policy rule derived here is similar to

the rules advocated by Reifschneider and Williams (1999), under which the interest rate

operating at each point in time should depend on how low the central bank would have low-

ered interest rates in the past had the zero bound not prevented it.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 184

observed is a zero nominal interest rate and yet another target shortfall?

The best way of making a rule credible is for the central bank to conduct

policy over time in a way that demonstrates its commitment. Ideally, the

central bank’s commitment to the price-level targeting framework would

be demonstrated before the zero bound came to bind (at which time the

central bank would have frequent opportunities to show that the target did

determine its behavior). The rule proposed above is one that would be

equally optimal both under normal circumstances and in the case of the

relatively unusual kind of disturbance that causes the natural rate of inter-

est to be substantially negative.

To understand how the rule works outside of the trap, it is useful to

note that, when the nominal interest rate is positive, ∆

= 0 at all times.

The central bank should therefore demonstrate a commitment to subse-

quently undo any over- or undershooting of the price-level target. In this

case any deﬂation that occurs when the economy ﬁnds itself in a liquidity

trap should create expectations of future inﬂation, as mandated by optimal

policy. The additional term ∆

implies that, when the zero bound is bind-

ing, the central bank should raise its long-run price-level target even fur-

ther, thus increasing inﬂation expectations even more.

It may be wondered why we discuss our proposal in terms of a (gap-

adjusted) price-level target rather than an inﬂation target. In fact, we could

equivalently describe the policy in terms of a time-varying target for the

gap-adjusted inﬂation rate π

≡ p˜

– p˜

t –1

. The reason we prefer to describe

the rule as a price-level targeting rule is that the essence of the rule is eas-

ily described in those terms. As we show below, a ﬁxed target for the gap-

adjusted price level would actually represent quite a good approximation

to optimal policy, whereas a ﬁxed inﬂation target would not come close,

because it would fail to allow for any of the history dependence of policy

necessary to mitigate the distortions resulting from the zero bound.

A Simpler Proposal

One may argue that an unappealing aspect of the rule suggested above

is that it involves the term ∆

, which determines the change in the price-

level target, and is nonzero only when the zero bound is binding. Suppose

that the central bank’s commitment to a policy rule can become credible

over time only through repeated demonstrations of its commitment to act

in accordance with it. In that case the part of the rule that involves the

adjustment of the target in response to target shortfalls when the zero

Gauti B. Eggertsson and Michael Woodford 185

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 185

bound binds might not come to be well understood by the private sector

for a very long time, because the occasions when the zero bound binds

will presumably be relatively infrequent.

Fortunately, most of the beneﬁts that can be achieved in principle

through a credible commitment to the optimal targeting rule can be

achieved through commitment to a much simpler rule, which would not

involve any special provisos that are invoked only in the event of a liq-

uidity trap. Consider the following simpler rule:

where now the target for the gap-adjusted price level is ﬁxed at all times.

The advantage of this rule, although it is not fully optimal when the zero

bound is binding, is that it may be more easily communicated to the pub-

lic. Note that the simple rule is fully optimal in the absence of the zero

bound. In fact, even if the zero bound occasionally binds, this rule results

in distortions only a bit more severe than those associated with the fully

optimal policy.

Figures 7 and 8 compare the results for these two rules. The shaded

lines show the equilibrium under the constant-price-level target rule in

equation 35, whereas the dark lines show the fully optimal rule in equa-

tions 32 through 34. As the ﬁgures show, the constant-price-level target-

ing rule results in state-contingent responses of output and inﬂation that

are very close to those under the optimal commitment, even if under this

rule the price level falls further during the period when the zero bound

binds, and only asymptotically rebounds to its level before the distur-

bance. The table below shows that the simple rule already achieves most

of the welfare gain that the optimal policy achieves; the table reports the

value of expected discounted losses, as a percentage of what could be

achieved by a strict zero-inﬂation target (equation 28), conditional on the

occurrence of the disturbance in period 0, under the various policies dis-

cussed above:

Policy Loss (percent)

Strict inﬂation target, π* = 0 100

Strict inﬂation target, π* = 1 24.1

Strict inﬂation target, π* = 2 32

Constant price-level target 0.0725

Optimal rule 0.036

() *,35 pxp

186 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 186

Figure 7. Responses of Inﬂation and the Output Gap under the Optimal Targeting

Rule and under the Simple Rule

Source: Authors’ calculations.

a. The simple rule is described in equation 35.

–0.2

0.2

0.4

0.0

Optimal rule

Simple rule

–1

0 5 10 15

Quarters after shock to natural rate of interest

Inflation

Output gap

Percent of GDP

Percent a year

Gauti B. Eggertsson and Michael Woodford 187

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 187

188 Brookings Papers on Economic Activity, 1:2003

Figure 8. Responses of the Nominal Interest Rate and Prices under the Optimal

Targeting Rule and under the Simple Rule

Source: Authors’ calculations.

Simple Optimal

Nominal interest rate

Percent a year

99.8

100.0

100.2

100.4

105015

Quarters after shock to natural rate of interest

Price level

Index, quarter -1 = 100

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 188

Both of the latter two, history-dependent policies are vastly superior to

any of the strict inﬂation targets. Although it is true that losses remain

twice as large under the simple rule as under the optimal rule, they are

nonetheless fairly small.

As with the fully optimal rule, no estimate of the natural rate of interest

is needed to implement the constant-price-level targeting rule. It may

seem puzzling at ﬁrst that a constant-price-level targeting rule does well,

because no account is taken of the size of the disturbance to the natural

rate of interest. This comes about because a price-level target commits the

government to undo any deﬂation with subsequent inﬂation; a larger dis-

turbance, which creates a larger initial deﬂation, automatically creates

greater inﬂation expectations in response. Thus an automatic stabilizer is

built into the price-level target, which is lacking under a strict inﬂation

targeting regime.

A proper strategy for the central bank to use in communicating its

objectives and targets when outside the liquidity trap is of crucial impor-

tance for this policy rule to be successful. To see this, consider a rule that

is equivalent to equation 35 when the zero bound is not binding. Taking

the difference of equation 35, we obtain

Although this rule results in an equilibrium identical to that under the

constant-price-level targeting rule when the zero bound is not binding, the

result is dramatically different when the zero bound is binding, because

this rule implies that the inﬂation rate is proportional to the negative of the

growth rate of output. Thus it mandates deﬂation when there is growth in

the output gap. This implies that the central bank will deﬂate once the

economy is out of a liquidity trap, because the economy will then be in a

period of output growth. This is exactly the opposite of what is optimal, as

we have observed above. Thus the outcome under this rule is even worse

than under a strict zero-inﬂation target, even if this rule replicates the

price-level targeting rule when out of the trap. What this underlines is that

it is not enough to replicate the equilibrium behavior that corresponds to

() (– ).

–

36 0

ttt

xx+=

Gauti B. Eggertsson and Michael Woodford 189

48. Wolman (forthcoming) also stresses this advantage of rules that incorporate a

price-level target over rules that only respond to the inﬂation rate, such as a conventional

Taylor rule.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 189

equation 35 in normal times to induce the correct set of expectations when

the zero bound is binding. It is crucial to communicate to the public that

the government is committed to a long-run price-level target. This com-

mitment is exactly what creates the desired inﬂation expectations when

the zero bound is binding.

Should the Central Bank Keep Some Powder Dry?

Thus far we have considered only alternative policies that might be

followed after the natural rate of interest has unexpectedly fallen to a

negative value, causing the zero bound to bind. A question of consider-

able current interest in countries like the United States, however, is how

policy should be affected by the anticipation that the zero bound might

well bind before long, even if it is not yet binding. Some have argued that,

in such circumstances, the Federal Reserve should be cautious about low-

ering interest rates all the way to zero too soon, in order to save its ammu-

nition for future emergencies. This suggests that the anticipation that the

zero bound could bind in the near future should lead to tighter policy than

would otherwise be justiﬁed given current conditions. Others argue, how-

ever, that policy should instead be more inﬂationary than one might oth-

erwise prefer, to reduce the probability that a further negative shock will

result in a binding zero bound.

Our characterization of the optimal targeting rule can shed light on this

debate. Recall that the rule laid out in equations 32 through 34 describes

optimal policy regardless of the assumed stochastic process for the nat-

ural rate of interest, and not only in the case of the particular two-state

Markov process assumed in ﬁgure 3. In particular, the same rule is opti-

mal in the case that information is received indicating the likelihood of

the natural rate of interest becoming negative before this actually occurs.

How should that news affect the conduct of policy? Under the optimal tar-

geting rule, the optimal target for p˜

is unaffected by such expectations, as

long as the zero bound is not yet binding, because only target shortfalls

that have already occurred can justify a change in the target value p

Thus an increased assessment of the likelihood of a binding zero bound

over the coming year or two would not be a reason for increasing the

price-level target (or the implied target rate of inﬂation).

190 Brookings Papers on Economic Activity, 1:2003

49. This conclusion, however, is likely to depend on a relatively special feature of our

model, namely, the fact that our target variables (inﬂation and the output gap) are both

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 190

On the other hand, this news will affect the paths of inﬂation, output,

and interest rates, even in the absence of any immediate change in the

central bank’s price-level target, owing to the effect on forward-looking

private sector spending and pricing decisions. The anticipation of a com-

ing state in which the natural rate of interest will be negative, and actual

interest rates will not be able to fall as much, owing to the zero bound,

will reduce both desired real expenditure (at unchanged short-term inter-

est rates) and desired price increases, because of the anticipation of nega-

tive output gaps and price declines in the future. This change in the

behavior of the private sector’s outlook will require a change in the cen-

tral bank’s conduct of policy in order to hit its unchanged target for the

modiﬁed price level, likely in the direction of a preemptive loosening.

This is illustrated by the numerical experiment shown in figure 9. Here

we suppose that in quarter 0 both the central bank and the private sector

learn that the natural rate of interest will fall to –2 percent a year only in

period 4. It is known that the natural rate will remain at its normal level

of +4 percent a year until then; after the drop, it will return to the normal

level with a probability of 0.1 each quarter, as in the case considered ear-

lier. We now consider the character of optimal policy from period 0

onward, given this information. Figure 9 again shows the optimal state-

contingent paths of inflation and output in the case that the disturbance to

the natural rate, when it arrives, lasts for one quarter, two quarters, and

so on.

We observe that, under the optimal policy commitment, prices begin

to decline mildly as soon as the news of the coming disturbance is

received. The central bank is nonetheless able to avoid undershooting its

target for p˜

at first, by stimulating an increase in real activity sufficient

to justify the mild deflation. (Given the private sector’s shift to pes-

simism, this is the policy dictated by the targeting rule, given that even a

mild immediate increase in real activity is insufficient to prevent a price

decline, owing to the anticipated decline in real demand when the distur-

bance hits.) By quarter 3, however, this is no longer possible, and the

Gauti B. Eggertsson and Michael Woodford 191

purely forward-looking variables: their equilibrium values at any point in time depend (in

our simple model) only on the economy’s exogenous state and the expected conduct of pol-

icy from the current period onward. There are a variety of reasons why a more realistic

model may well imply that these variables are functions of lagged endogenous variables as

well, and hence of past policy. In such a case, the optimal target criterion will be at least

somewhat forward looking, as discussed in Giannoni and Woodford (2003).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 191

Figure 9. Responses of Inﬂation, the Output Gap, and Prices to an Anticipated Shock

under Optimal Policy

Source: Authors’ calculations.

a. Response in a scenario where the shock to the natural rate of interest is anticipated four quarters in advance of its occurrence.

–0.1

0.0

0.1

0.2

0.3

Inflation

–0.5

0.0

0.5

1.0

1.5

2.0

100.0

100.2

0101520

Index, quarter -1 =100

Percent of GDP

Percent a year

Price level

Output gap

Quarters after shock to natural rate of interest

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 192

central bank undershoots its target for p˜

(as both prices and output

decline), even though the nominal interest rate is at zero. Thus, optimal

policy involves driving the nominal interest rate to zero even before the

natural rate of interest has turned negative, when that development can

already be anticipated for the near future. The fact that the zero bound

binds even before the natural rate of interest becomes negative means

that the price-level target is higher than it otherwise would have been

when the disturbance to the natural rate arrives. As a result, deflation and

the output gap during the period when the natural rate is negative are less

severe than in the case where the disturbance is unanticipated. Optimal

policy in this scenario is somewhat more inflationary after the distur-

bance occurs than in the case considered in figure 3, because in this case

the optimal policy commitment takes into account the contractionary

effects, in periods before the disturbance takes effect, of the anticipation

that the disturbance will result in price-level and output declines. The

fact that optimal policy after the disturbance occurs is different in this

case, despite the fact that the disturbance has exactly the same effects as

before from quarter 4 onward, is another illustration of the history depen-

dence of optimal policy.

Preventing a Self-Fulﬁlling Deﬂationary Trap

Our analysis thus far has assumed that the real disturbance that results

in a negative natural rate of interest does so only temporarily. We have

therefore supposed that price-level stabilization will eventually be consis-

tent with positive nominal interest rates and, accordingly, that a time will

foreseeably be reached when the central bank can create inﬂation by

keeping short-term nominal rates at a low but nonnegative level. But is it

possible for the zero bound to bind forever in equilibrium, not because of

a permanently negative natural rate, but simply because deﬂation contin-

ues to be (correctly) expected indeﬁnitely? If so, the central bank’s com-

mitment to a nondecreasing price-level target might seem irrelevant; the

price level would fall further and further short of the target, but because of

the binding zero bound, the central bank could never do anything about it.

In the model presented in the ﬁrst part of the paper, a self-fulﬁlling,

permanent deﬂation is indeed consistent with both the Euler equation

(equation 2) for aggregate expenditure, the money-demand relation

Gauti B. Eggertsson and Michael Woodford 193

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 193

(expression 3), and the pricing relations (equations 7 through 9). Suppose

that, from some date τ onward, all disturbances 

= 0 with certainty,

so that the natural rate of interest is expected to take the constant value

–

=β

–1

– 1 > 0, as in the scenarios considered earlier in the paper. Then the

possible paths for inﬂation, output, and interest rates consistent with each

of the relations just listed in all periods t ≥τare given by

where Y

< Y

–

is implicitly deﬁned by the relation

Note that this deﬂationary path is consistent with monetary policy as long

as real balances satisfy M

≥ m

–

; 0) each period; faster growth of the

money supply does nothing to prevent consistency of this path with the

requirement that money supply equal money demand in each period.

There remains, however, one further requirement for equilibrium in the

earlier model, namely, the transversality condition (equation 6) or, equiv-

alently, the requirement that households hit their intertemporal budget

constraint. Whether the deﬂationary path is consistent with this condition

as well depends, properly speaking, on the speciﬁcation of ﬁscal policy: it

is a matter of whether the government budget results in contraction of the

nominal value of total government liabilities D

at a sufﬁcient rate asymp-

totically. Under some assumptions about the character of ﬁscal policy,

such as the Ricardian ﬁscal policy rule assumed by Jess Benhabib,

Stephanie Schmitt-Grohé, and Martin Uribe,

the nominal value of gov-

ernment liabilities will necessarily contract as the price level falls, so that

equation 6 is also satisﬁed, and the processes described above will indeed

represent a rational expectations equilibrium. In such a case, then, a com-

mitment to the price-level targeting rule proposed in the previous section

will be equally consistent with more than one equilibrium: if people

1000

*, ;

;), .pp YmY

[]

pP p

=≡













–

αβ

194 Brookings Papers on Economic Activity, 1:2003

50. Benhabib, Schmitt-Grohé, and Uribe (2001).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 194

expect the optimal price-level process characterized earlier, that will

indeed be an equilibrium, but if they expect perpetual deﬂation, that will

be an equilibrium as well.

However, this outcome can be excluded through a suitable commit-

ment with regard to the asymptotic evolution of total government liabili-

ties. Essentially, there needs to be a commitment to policies that ensure

that the nominal value of government liabilities cannot contract at the rate

required for satisfaction of the transversality condition, despite perpetual

deﬂation. One example of a commitment that would sufﬁce is a commit-

ment to a balanced-budget policy of the kind analyzed by Schmitt-Grohé

and Uribe.

These authors show that self-fulﬁlling deﬂations are not pos-

sible when monetary policy is committed to a Taylor rule and the govern-

ment to a balanced budget. The key to their result is that the ﬁscal rule

includes a commitment that is as binding against running large surpluses

as it is against running large deﬁcits; then the nominal value of govern-

ment liabilities cannot contract, even when the price level falls exponen-

tially forever.

The credibility of this sort of ﬁscal commitment might be doubted, and

so another way of maintaining a ﬂoor under the asymptotic nominal value

of total government liabilities is through a commitment not to contract the

monetary base, together with a commitment of the government to main-

tain a nonnegative asymptotic present value of the public debt. In particu-

lar, suppose that the central bank commits itself to follow a base-supply

rule of the form

in each period when the zero bound binds (that is, when it is not possible

to hit the price-level target with a positive nominal interest rate), where

is the current price-level target implied by the adjusted price-level target

*. When the zero bound does not bind, the monetary base is whatever

level is demanded at the nominal interest rate required to hit the price-

level target. This is a rule in the same spirit as equation 11, specifying a

Ppx

ttt

exp –≡













() (;)

37 MPmY

tt tt

= 

Gauti B. Eggertsson and Michael Woodford 195

51. Schmitt-Grohé and Uribe (2000).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 195

particular level of excess supply of base money when the zero bound

binds, but letting the monetary base be endogenously determined by the

central bank’s other targets at other times. Equation 37 is a more compli-

cated formula than is necessary to make our point, but it has the advan-

tage of making the monetary base a continuous function of other

aggregate state variables at the point where the zero bound just ceases to

bind.

This particular form of commitment has the advantage that it may be

considered less problematic for the central bank to commit itself to main-

tain a particular nominal value for its liabilities than for the public trea-

sury to do so. It can also be justiﬁed as entirely consistent with the central

bank’s commitment to the price-level targeting rule; even when the target

cannot be hit, the central bank supplies the quantity of money that would

be demanded if the price level were at the target. Doing so—refusing to

contract the monetary base even in a deﬂation—is a way of signaling to

the public that the central bank is serious about its intention to see the

price level restored to the target level.

If one then assumes a ﬁscal commitment that guarantees that

that is, that the government will asymptotically be neither creditor nor

debtor, the transversality condition reduces to

In the case of the base-supply rule in equation 37, this condition is vio-

lated in the candidate equilibrium described above, since the price-level

and output paths speciﬁed would imply that

where the last inequality makes use of the fact that, under the price-level

targeting rule, {p

*} is a nondecreasing series. Note that the ﬁnal expres-

sion on the right-hand side is independent of T, for all dates T ≥τ. Hence

the series is bounded away from zero, and the condition in equation 39 is

violated.

ββ

ττ

tcT TT T TT c T

EuY M P M P uYmY mY P P

u YmY mY P P

(, /; ) /

;); (

;) /

;); (

;) / ,



[]

≥

[]

00 0

( ) lim ( , / ; ) / .39 0

tcT TT T TT

EuY M P M P

→∞

[]

=β 

( ) lim ,

38 0

ttTT

EQ B

→∞

196 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 196

Thus a commitment of this kind can exclude the possibility of a self-

fulﬁlling deﬂation of the sort described above as a rational expectations

equilibrium. It follows that there is a possible role for quantitative

easing—understood to mean the supply of base money beyond the mini-

mum quantity required for consistency with the zero nominal interest

rate—as an element of an optimal policy commitment. A commitment to

supply base money in proportion to the target price level, and not the

actual current price level, when the zero bound prevents the central bank

from hitting its price-level target, can be desirable both as a way of ruling

out self-fulﬁlling deﬂations and as a way of signaling the central bank’s

continuing commitment to the price-level target, even though it is tem-

porarily unable to hit it.

Note that this result does not contradict our irrelevance proposition,

because here we have made a different assumption about the nature of the

ﬁscal commitment. Equation 38 implies that the evolution of total nomi-

nal government liabilities will not be independent of the central bank’s

target for the monetary base. As a consequence, the neutrality proposition

no longer holds. The import of that proposition is that expansion of the

monetary base when the economy is in a liquidity trap is necessarily

pointless; rather, any effect of such action must depend either on chang-

ing expectations regarding future interest rate policy or on changing

expectations regarding the future path of total nominal government liabil-

ities. The present discussion has illustrated circumstances in which

expansion of the monetary base—or, at any rate, a commitment not to

contract it—could serve both these ends.

Nonetheless, the present discussion does not support the view that the

central bank should be able to hit its price-level target at all times, simply

by ﬂooding the economy with as much base money as is required to pre-

vent the price level from falling below the target at any time. Our earlier

analysis still describes all possible paths for the price level consistent with

rational expectations equilibrium, and we have seen that even if the cen-

tral bank were able to choose the expectations that the private sector

should have (as long as it were willing to act in accordance with them),

the zero bound would prevent it from being able to fully stabilize inﬂation

and the output gap. Furthermore, the degree of monetary base expansion

during a liquidity trap called for by the rule in equation 37 is quite mod-

est. The monetary base will be raised gradually, if the zero bound contin-

ues to bind, as the price-level target is ratcheted up to steadily higher

Gauti B. Eggertsson and Michael Woodford 197

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 197

levels. But our calibrated example above indicates that this would typi-

cally involve only a very modest increase in the monetary base, even if

the liquidity trap lasts for several years. There would be no obvious bene-

ﬁt to the kind of rapid expansion of the monetary base actually tried in

Japan over the past two years. Such an expansion is evidently not justiﬁed

by any intention regarding the future price level, and hence regarding the

size of the monetary base once Japan exits from the trap. But an injection

of base money that is expected to be removed once the zero bound ceases

to bind should have little effect on spending or pricing behavior, as we

showed in the ﬁrst section of the paper.

Further Aspects of the Management of Expectations

In the ﬁrst section we argued that neither expansion of the monetary

base as such nor open-market purchases of particular types of assets

should have any effect on either inﬂation or real activity, except to the

extent that these actions might change expectations regarding future inter-

est rate policy (or possibly expectations regarding the asymptotic behav-

ior of total nominal government liabilities, and hence the question of

whether the transversality condition should be satisﬁed). This then

allowed us to characterize the optimal policy commitment without any

reference to the use of such instruments of policy; a consideration of the

different possible joint paths of interest rates, inﬂation, and output that

would be consistent with rational expectations equilibrium sufﬁced to

allow us to determine the best possible equilibrium that one could hope to

arrange, and to characterize it in terms of the interest rate policy that one

should wish the private sector to expect.

However, this does not mean that other aspects of policy—beyond a

mere announcement of the rule to which the central bank wishes to be

understood to be committed in setting future interest rate policy—cannot

matter. They may matter insofar as certain kinds of present actions may

help to signal the bank’s intentions regarding future policy, or make it

more credible that the central bank will indeed carry out those intentions.

A full analysis of the ways in which policy actions may be justiﬁed as

helping to steer expectations is beyond the scope of this paper, and in any

event the question is one that has as much to do with psychology and

effective communication as with economic analysis. Nonetheless, we

198 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 198

offer a few remarks here about the kinds of policies that might contribute

to the creation of desirable expectations.

Demonstrating Resolve

One way in which current actions may help to create desirable expec-

tations regarding future policy is by being seen to be consistent with the

principles that the central bank wishes the private sector to understand

will guide that policy. We have already mentioned one example of this:

one way to convince the private sector that the central bank will follow

the optimal price-level targeting rule after a period in which the zero

bound has been hit is by following this rule before such a situation arises.

Our discussion in the previous section provides a further example.

Adjustment of the supply of base money while the zero bound is binding,

so as to keep the monetary base proportional to the target price level

rather than the actual current price level, can be helpful, even though irrel-

evant to interest rate control, as a way of communicating to the private

sector the central bank’s belief about where the price level ought properly

to be (and hence the quantity of base money that the economy ought to

need). By putting the existence of the price-level target in greater relief,

such an action can help create the expectations regarding future interest

rate policy necessary to mitigate the distortions created by the binding

zero bound.

As a further example, Clouse and others argue that open-market opera-

tions may be stimulative, even when the zero bound has been reached,

because they “demonstrate resolve” to keep the nominal interest rate at

zero for a longer time than would otherwise be expected.

But an expan-

sion of the monetary base when the zero bound is binding need not be

interpreted in this way. Consider, for example, a central bank with a con-

stant zero inﬂation target, as discussed previously. When the zero bound

binds, such a bank is unable to hit its inﬂation target and should exhibit

frustration with this state of affairs. If some within the bank believe it

should always be possible to hit the target with sufﬁciently vigorous

monetary expansion, one might well observe substantial growth in the

monetary base at a time when the inﬂation target is being undershot.

Nonetheless, this would not imply any commitment to looser policy

Gauti B. Eggertsson and Michael Woodford 199

52. Clouse and others (2003)

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 199

subsequently; such a central bank would never intentionally allow the

monetary base to be higher than required to hit the inﬂation target, if the

target can be hit. The result should be the equilibrium path shown in ﬁg-

ure 2, and there should be no effect from the quantitative easing that occurs

while the zero bound binds. This shows that it matters what the private sec-

tor understands to be the principle that motivates quantitative easing; it is

not simply a question of how large is the increase in the monetary base.

Similarly, open-market purchases of long-term treasury bonds when

short-term rates are at zero, as advocated by Ben Bernanke and Stephen

Cecchetti,

among others, may well have a stimulative effect even if

portfolio balance effects are quantitatively unimportant. We argued

previously that under such circumstances it is desirable for the central

bank to commit itself to maintain low short-term rates even after the nat-

ural rate of interest rises again. The level of long-term rates can indicate

the extent to which the markets actually believe such a commitment. If a

central bank’s judgment is that long-term rates remain higher than they

should be under the optimal equilibrium, owing to private sector skepti-

cism about whether the history-dependent interest rate policy will actu-

ally be followed, then a willingness to buy long-term bonds from the

private sector at a price it regards as more appropriate is one way for the

central bank to demonstrate publicly that it expects to carry out its com-

mitment regarding future interest rate policy. Given that the private sec-

tor is likely to be uncertain about the nature of the central bank’s

commitment (in the case of imperfect credibility), and that it can reason-

ably assume that the central bank knows more about its own degree of

resolve than others do, action by the central bank that is consistent with a

belief on its own part that it will keep short-term rates low in the future is

likely to shift private beliefs in the same direction. If so, open-market

purchases of long-term bonds could lower long-term interest rates, stim-

ulate the economy immediately, and bring the economy closer to the

optimal rational expectations equilibrium. However, that effect follows

not from the purchases themselves, but from how they are interpreted.

For them to be interpreted as indicating a particular kind of commitment

with regard to future policy, it is important that the central bank have

itself formulated such an intention, and that it so inform the public, so

that its open-market purchases will be seen in this light.

200 Brookings Papers on Economic Activity, 1:2003

53. Bernanke (2002); Stephen G. Cecchetti, “Central Banks Have Plenty of Ammuni-

tion,” Financial Times, March 17, 2003, p. 13.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 200

Similar remarks apply to the proposals by Bennett McCallum and Lars

Svensson that purchases of foreign exchange be used to stimulate the

economy through devaluation of the currency.

Under the optimal policy

commitment described in an earlier section, a decline in the natural rate of

interest should be accompanied by depreciation of the currency, both

because nominal interest rates fall (and are expected to remain low for

some time) and because the expected long-run price level (and hence the

expected long-run nominal exchange rate) should increase. It follows that

the extent to which the currency depreciates can provide an indicator of

the extent to which the markets believe that the central bank is committed

to such an optimal policy; and if the depreciation is insufﬁcient, purchases

of foreign exchange by the central bank provide one way for it to demon-

strate its own conﬁdence in its policy intentions. Again, the effect in ques-

tion is not a mechanical consequence of the bank’s purchases, but instead

depends on their interpretation.

Providing Incentives to Improve Credibility

A related but somewhat distinct argument is that actions at the zero

bound may help render the central bank’s commitment to an optimal pol-

icy more credible, by providing the bank with a motive to behave in the

future in the way that it would currently wish that people would expect it

to behave. Here we brieﬂy discuss how policy actions that are possible

while the economy remains in a liquidity trap may be helpful in this

regard. Our point is not so much that the central bank is in need of a

“commitment technology” because it will be unable to resist the tempta-

tion to break its commitments later in the absence of such a constraint.

Rather, it is that the central bank may well need a way of making its com-

Gauti B. Eggertsson and Michael Woodford 201

54. McCallum (2000); Svensson (2001). Svensson’s proposal includes a target path for

the price level, which the exchange rate policy is used to (eventually) achieve, and in this

respect it is similar to the policy advocated here. However, Svensson’s discussion of the

usefulness of intervention in the market for foreign exchange does not emphasize the role

of such interventions as a signal regarding future policy.

55. The numerical analysis by Coenen and Wieland (forthcoming) ﬁnds that an

exchange rate policy can be quite effective in creating stimulus when the zero bound is

binding. But what is actually shown is that a rational expectations equilibrium exists in

which the currency depreciates and deﬂation is halted; these effects could be viewed as

resulting from a credible commitment to a target path for the price level, similar to the one

discussed in this paper, and not requiring any intervention in the foreign exchange market

at all.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 201

mitment visible to the private sector. Taking actions now that imply that

the central bank will be disadvantaged later if it were to deviate from the

policy to which it wishes to commit itself can serve this purpose.

To consider what kind of current actions provide useful incentives, it is

helpful to analyze (Markov) equilibrium under the assumption that policy

is conducted by a discretionary optimizer, unable to commit to speciﬁc

future actions at all.

Consider ﬁrst what a Markov equilibrium under dis-

cretionary optimization would be like in the case that the only policy

instrument available is a short-term nominal interest rate, whose value is

chosen each period, and the objective of the central bank is minimization

of the loss function in equation 28. As shown above, if the central bank

can credibly commit itself, this problem has a solution in which the zero

bound does not result in too serious a distortion, although it does bind.

Under the assumption of central bank discretion, however, the out-

come will be much inferior. Note that discretionary policy (under the

assumption of Markov equilibrium in the dynamic policy game) is an

example of a purely forward-looking policy. It then follows from our ear-

lier argument that the equilibrium outcome will correspond to the kind of

equilibrium discussed there in the case of a strict inﬂation target. More

speciﬁcally, it is obvious that the equilibrium is the same as under a strict

inﬂation target π* = 0, since this is the inﬂation rate that the discretionary

optimizer will choose once the natural rate of interest is again at its

steady-state level. (From that point onward, a policy of zero inﬂation

clearly minimizes the remaining terms in the discounted loss function.)

As shown in ﬁgure 2, if the private sector expects that the central bank

will behave in this fashion, and the natural rate of interest remains nega-

tive for several quarters, the result will be a deep and prolonged contrac-

tion of economic activity and a sustained deﬂation. We have also seen that

these effects could largely be avoided, even in the absence of other policy

instruments, if the central bank were able to credibly commit itself to a

history-dependent monetary policy in later periods. Thus, in the kind of

situation considered here, there is a deﬂationary bias to discretionary

monetary policy, although, at its root, the problem is again the one identi-

ﬁed in the classic analysis of Finn Kydland and Edward Prescott.

now consider instead the extent to which the outcome could be improved,

202 Brookings Papers on Economic Activity, 1:2003

56. As in Eggertsson (2003a, 2003b).

57. Kydland and Prescott (1977).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 202

even in a Markov equilibrium with discretionary optimization, by chang-

ing the nature of the policy game.

One example of a current policy action, available even when the zero

bound binds, that can help shift expectations regarding future policy in a

desirable way is for the government to cut taxes and issue additional nom-

inal debt.

Alternatively, the tax cut can be ﬁnanced by money creation,

because when the zero bound binds, there is no difference between

expanding the monetary base and issuing additional short-term Treasury

debt at zero interest. This is essentially the kind of policy imagined when

people speak of a “helicopter drop” of additional money into the econ-

omy, but here it is the ﬁscal consequences of such an action with which

we are concerned.

Of course, if the objective of the central bank in setting monetary pol-

icy remains as assumed above, this will make no difference to the discre-

tionary equilibrium: the optimal policy once the natural rate of interest

becomes positive again will once more appear to be the immediate pursuit

of a strict zero-inﬂation target. However, if the central bank also cares

about reducing the social costs of increased taxation—whether because of

collection costs or because of other distortions—as it ought if it really

takes social welfare into account, the result is different. As Eggertsson has

shown elsewhere,

the tax cut will then increase inﬂation expectations,

even if the government cannot commit to future policy.

It may be asked why, if such an incentive exists, Japan continues to

suffer deﬂation, given the growth in Japanese government debt during the

1990s. One possible answer is that although the gross national debt is

140 percent of GDP in Japan today, this does not reﬂect the true inﬂation

incentives of the government. The ratio of gross national debt to GDP

overestimates the government’s inﬂation incentives, because a substantial

portion of government debt is held by other government institutions.

Net

government debt is only 67 percent of GDP, and, as a result, inﬂation

Gauti B. Eggertsson and Michael Woodford 203

58. As discussed in Eggertsson (2003a).

59. Eggertsson (2003a).

60. Government institutions such as the social security system, the postal savings sys-

tem, postal life insurance, and the Trust Fund Bureau hold much of this nominal debt. If the

part of the public debt held by these institutions is subtracted from total gross government

debt, the remainder is only 67 percent of GDP. Most of the government institutions that

hold the government’s nominal debt have real liabilities. For example, the social security

system (which holds roughly 25 percent of the nominal debt held by the government) pays

Japanese pensions and medical expenses. Those pensions are indexed to the consumer

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 203

incentives may not be much greater in Japan than in a number of other

countries.

An even more likely reason for continued low inﬂation expectations in

Japan, despite the size of the nominal public debt, is skepticism about

whether the central bank can be expected to care about reducing the

burden of the public debt when determining future monetary policy. The

public may believe that the Bank of Japan lacks such an objective; the

expressed resistance of the Bank of Japan to suggestions that it increase

its purchases of Japanese government bonds, on the ground that this could

encourage a lack of ﬁscal discipline,

certainly suggests that reducing the

burden of government ﬁnance is not among its highest priorities. As

Eggertsson has stressed elsewhere,

in order for ﬁscal policy to be effec-

tive as a means of increasing inﬂationary expectations, ﬁscal and mone-

tary policy must be coordinated so as to maximize social welfare. The

consequences of a narrow concern with inﬂation stabilization on the part

of the central bank, together with an inability to credibly commit future

monetary policy, can be dire, even from the point of view of the bank’s

own stabilization objectives.

Another instrument that may be used to change expectations regarding

future monetary policy is open-market purchases of real assets or foreign

exchange. Purchases of real assets (say, real estate) can be thought of as

another way of increasing nominal government liabilities, which should

affect inﬂation incentives in much the same way as deﬁcit spending.

Purchases of real assets have the advantage of not worsening the overall

ﬁscal position of the government—a current concern in Japan, given its

existing gross debt—while still increasing the ﬁscal incentive for inﬂa-

tion. A further advantage of this approach is that it need not depend on a

perceived central bank interest in reducing the burden of the public debt.

Since the (nominal) capital gains from inﬂation accrue to the central bank

itself under this policy, the central bank may be perceived to have an

204 Brookings Papers on Economic Activity, 1:2003

price index. If inﬂation increases, the real value of social security assets will fall, but the

real value of most its liabilities will remain unchanged. Thus the Ministry of Finance would

eventually have to step in to make up for any loss in the value of social security assets if the

government is to keep its pension program unchanged. Therefore the gains from reducing

the real value of outstanding debt are partly offset by a decrease in the real value of the

assets of government institutions such as social security.

61. Asahi Shimbun, “Bank of Japan Advised to ‘Print Money’ to Escape Deﬂation,”

Dow Jones News, February 10, 1999.

62. Eggertsson (2003a).

63. Eggertsson (2003a).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 204

incentive to inﬂate simply on the ground that it cares about its own bal-

ance sheet, for example because doing so will help ensure its indepen-

dence. (One can easily argue that, under a rational scheme of cooperation

between the central bank and the government, the central bank should not

choose policy on the basis of concerns about its balance sheet. But under

such an ideal regime, it should choose monetary policy with a view to

reducing the burden of the public debt, among other goals.)

The incentive effects of open-market operations in foreign exchange

are even simpler.

Open-market purchases of foreign assets give the cen-

tral bank an incentive to inﬂate in the future in order to realize capital

gains at the expense of foreigners. These will be valuable if the central

bank cares either about its own balance sheet or about reducing the bur-

den of the public debt, as in the case of real asset purchases. However,

capital gains on foreign exchange that result from depreciation of the

domestic currency will be valuable even if the central bank cares neither

about its balance sheet (for example, because it cooperates perfectly with

the public treasury) nor about the burden of the debt (for example,

because nondistorting sources of revenue are available to the public trea-

sury). Capital gains at the expense of foreigners would allow an increase

in domestic spending, by either the government or the private sector, and

a central bank must value this if it has the national interest at heart.

Under rational expectations, of course, no such capital gains are real-

ized on average. Still, the purchase of foreign assets can work as a com-

mitment device, because if the central bank reneged on its inﬂation

commitment, it would cause capital losses if the government holds for-

eign assets. Purchases of foreign assets are thus a way of committing the

government to looser monetary policy in the future. This creates a reason

for purchasing foreign exchange in order to cause a devaluation (which

will also stimulate current demand), even without any assumption of a

deviation from interest rate parity of the kind relied upon by authors such

as McCallum in recommending devaluation for Japan.

Clouse and

others argue that open-market purchases of long-term Treasuries by the

Federal Reserve should also change expectations in a way that results in

immediate stimulus.

The argument is that if the central bank were not to

follow through on its commitment to keep short-term rates low, it would

Gauti B. Eggertsson and Michael Woodford 205

64. As shown by Eggertsson (2003b).

65. McCallum (2000).

66. Clouse and others (2003).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 205

suffer a capital loss on the long-term bonds that it purchased at a price that

made sense only on the assumption that it would keep interest rates low.

Similarly, Peter Tinsley has proposed a policy that would create this kind

of incentive even more directly, namely, the sale by the Federal Reserve

of options to obtain federal funds at a future date at a certain price.

The

Federal Reserve would then stand to lose money if it did not keep the

funds rate at the level to which it had previously committed itself.

Although these proposals should also help reinforce the credibility of

the kind of policy commitment associated with the optimal equilibrium

(as characterized in the ﬁrst section of the paper), they have at least one

important disadvantage relative to purchases of real assets or of foreign

exchange. They only provide the central bank an incentive to maintain

low nominal interest rates for a certain period; they do not provide it with

an incentive to ensure that the price level eventually rises to a higher

level. Thus they may do little to counter private sector expectations that

nominal interest rates will remain low for years—but because goods

prices are going to continue to fall, not because the central bank is com-

mitted to eventual reﬂation, as in the self-fulﬁlling deﬂation trap dis-

cussed above. This is arguably the kind of expectation that has now taken

root in Japan, where even ten-year bond yields are already well below 1

percent, even though prices continue to fall and economic activity

remains anemic. Creating the perception that the central bank has an

incentive to continue trying to raise the price level, and that it will not be

content as long as nominal interest rates remain low, may be a more suc-

cessful way of generating the sort of expectations associated with the

optimal equilibrium.

Conclusion

We have argued that the key to dealing with a situation in which mon-

etary policy is constrained by the zero lower bound on short-term nominal

interest rates is the skillful management of expectations regarding the

future conduct of policy. By “management of expectations” we do not

mean that the central bank should imagine that, if it uses sufﬁcient guile,

it can lead the private sector to believe whatever the central bank wishes it

to believe, no matter what it actually does. Instead we have assumed that

206 Brookings Papers on Economic Activity, 1:2003

67. Tinsley (1999).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 206

there is no point in the central bank trying to get the private sector to

expect something that the central bank does not itself intend to bring

about. But we do contend that it is highly desirable for a central bank to be

able to commit itself in advance to a course of action that is desirable

because of the beneﬁts that ﬂow from its being anticipated, and then to

work to make that commitment credible to the private sector.

In the context of a simple optimizing model of the monetary transmis-

sion mechanism, we have shown that a purely forward-looking approach

to policy—which allows for no possibility of committing future policy to

respond to past conditions—can lead to quite bad outcomes in the event

of a temporary decline in the natural rate of interest, regardless of the kind

of policy pursued at the time of the disturbance. We have also character-

ized optimal policy, under the assumption that credible commitment is

possible, and shown that it involves a commitment to eventually bring the

general price level back up to a level even higher than would have pre-

vailed had the disturbance never occurred. Finally, we have described a

type of history-dependent price-level targeting rule with the following

properties: that a commitment to base interest rate policy on this rule

determines the optimal equilibrium, and that the same form of targeting

rule continues to describe optimal policy regardless of which of a large

number of types of disturbances may affect the economy.

Given the role of private sector anticipation of history-dependent pol-

icy in realizing a desirable outcome, it is important for central banks to

develop effective methods of signaling their policy commitments to the

private sector. An essential precondition for this, certainly, is for the cen-

tral bank itself to clearly understand the kind of history-dependent behav-

ior to which it should be seen to be committed. It can then communicate

its thinking on the matter and act consistently with the principles that it

wishes the private sector to understand. Simply conducting policy in

accordance with a rule may not sufﬁce to bring about an optimal, or

nearly optimal, equilibrium, but it is the place to start.

APPENDIX A

The Numerical Solution Method

Here we illustrate a solution method for the optimal commitment

solution discussed in the ﬁrst section of the text. This same method can

Gauti B. Eggertsson and Michael Woodford 207

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 207

also be applied, with appropriate modiﬁcation of each of the steps, to ﬁnd-

ing the solution in the case where the central bank commits to a constant

price-level target rule or to a constant inﬂation target. We assume that the

natural rate of interest becomes unexpectedly negative in period 0 and

then reverts back to normal with probability α

in every period t. Our

numerical work assumes that there is a ﬁnal date S at which the natural

rate becomes positive with a probability of 1, although this date may be

arbitrarily far in the future.

The solution takes the form

= 0 ∀ t if 0 ≤ t < τ + k

> 0 ∀ t if t ≥ τ + k

It follows that

t+1

– x

+σ(E

t+1

+ r

) = 0ift < τ + k

= 0ift ≥ τ + k

Here τ is the stochastic date at which the natural rate of interest returns to

the steady state. We assume that τ can take any value between 1 and the

terminal date S. The number τ+k

is the period in which the zero bound

ceases to bind contingent on the natural rate of interest becoming positive

in period τ. Note that the value of k

can depend on the value of τ. We will

ﬁrst show the solution for the problem as if we knew the sequence {k

}

τ=1

We then describe a numerical method to ﬁnd the sequence {k

}

τ=1

The Solution for t ≥ k



The system can be written in the following form:

where

≡













≡













and

() ,

–

























208 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 208

If there are two eigenvalues of the matrix M outside the unit circle, this

system has a unique bounded solution of the form

The Solution for  ≤ t ≤   k



Again this is a perfect-foresight solution, but with the zero bound bind-

ing. The solution satisﬁes the following equations:

The system can be written as

This system has a solution of the form

where j = 0, 1, 2, …, k. Here 

–j

is the coefﬁcient in the solution when

there are k

– j periods until the zero bound stops being binding (that is,

when k

– j = 0, the zero bound is no longer binding and the solution is

equivalent to that in equations A2 and A3). We can ﬁnd the numbers 



, 

, 

for j = 1, 2, 3, …, k by solving the equations below using the

initial conditions 

= 

= 0 for j = 0 and the initial conditions for 

and



given in equations A2 and A3:



= [I – B

j–1

]

–1



= C + D

j–1





= (I – B

j–1

)

–1

[B

j–1

+ M ]



= D

j–1



+ D

j–1

+ V.

() ,

–

ττ



()

–

ττ



















































–

()

––

–– .

–

221

11 2

πκ βπ

σπ

πϕ ϕ βσϕ

λϕβϕκϕ

ttt

ttt t

xt t t t

xr x

=++

() .

–

= 

()

–

= 

Gauti B. Eggertsson and Michael Woodford 209

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 209

The Solution for t < 

The solution satisﬁes the following equations:

Here a tilde on a variable denotes the value of that variable contingent on

the natural rate of interest being negative. Λ

t+1

is the ijth element of the

matrix 

t+1

. The value k

t+1

depends on the number of additional periods

that the zero bound is binding (recall that here we are solving for the equi-

librium on the assumption that we know the value of the sequence

}

τ=1

). We can write the system as

We can solve this backward from the date S on which the natural rate

returns to normal with a probability of 1. We can then calculate the path

for each variable to date 0. Note that

S–1

= D

S–1

= 0.

By recursive substitution we can ﬁnd a solution of the form

where the coefﬁcients are time dependent. To ﬁnd the numbers 

, 



, and 

, consider the solution of the system in period S – 1 when B

S–1

= D

S–1

= 0. We have

()

˜˜

–

ttt t

=+

()

˜˜

–

ttt t

=+

–

















































–

˜˜

–

˜˜

πκ β απ α ϕ ϕ

σαπαϕϕ

tt tt t

tt t

()

+++

()

{}

()

+++

()

++ +

11 1111 122 1

11 1

ΛΛΘ

{{}

()

+++

()

{}

++ +

11 1211 222 2

221

11 2

11 1

–

˜˜

–

ααϕϕ

πϕ ϕ βσϕ

λϕβϕκϕ

tt t

ttt t

xt t t t

tt t

ΛΛΘ

210 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 210



S–1

= A

S–1



S–1

= M

S–1



S–1

= C

S–1



S–1

= V

S–1.

We can ﬁnd the numbers 

, 

, 

, and 

for periods 0 to S – 2 by solv-

ing the system below (using the initial conditions shown above for S – 1):



= [I – B



t+1

]

–1



= C

+ D



t+1





= (I – B



t+1

)

–1



t+1

+ M

]



= D



t+1



+ D



t+1

+ V

Using the initial condition P

–1

= 0, we can solve for each of the endoge-

nous variables under the contingency that the liquidity trap lasts until

period S, using equations A7 and A8. We then use the solution from equa-

tions A2 to A6 to solve for each of the variables when the natural rate

reverts back to the steady state.

Solving for {k



}



t1

A simple way to ﬁnd the value for {k

}

τ=1

is to ﬁrst assume that k

is the

same for all τ and ﬁnd the lowest k

= k

–

so that the zero bound is never

violated. Using this initial guess for {k

}

τ=1

, one then ﬁnds the lowest

value of k

so that the zero bound is never violated. Using this value for

, and k

–

for all other k

, one then ﬁnds the lowest value of k

S–1

so that the

zero bound is never violated, and so on until the lowest possible value for

is found. The value thus found for the sequence {k

}

τ=1

can be used as a

new initial guess for {k

}

S–1

τ=1

, and the procedure just described can be

repeated until the solution converges. For this paper we wrote a routine in

MATLAB that applies this method. The solution converged, and we veri-

ﬁed that the result satisﬁed all the necessary conditions.

Gauti B. Eggertsson and Michael Woodford 211

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 211

Comments and

Discussion

Benjamin M. Friedman: The pathetic ﬂoundering of the Japanese econ-

omy, with 1 percent real growth or less in seven of the last eleven years,

and falling prices in eight of the last nine, has called new attention to a

variety of economic issues that last assumed practical prominence during

and in the aftermath of the depression of the 1930s. These issues include

debt deﬂation, an insolvent banking system, bankrupt corporations, and

the potential impotence of monetary policy. The resulting discussion of

monetary policy has been particularly lively. Just as there is a difference

between a policy that is right and a policy that is wrong under any given

set of circumstances, so there is also a difference between a policy that is

merely wrong and a policy that is also wrongheaded. For years, teachers

of courses on such matters had one real-world example of wrongheaded

monetary policy to which to point, namely, the conduct of the U.S. Fed-

eral Reserve System during the depression. Now the Bank of Japan has

provided a second example for study.

More recently, the weak performance of the American economy has

begun to raise some of the same questions about U.S. monetary policy.

With the federal funds rate now only 1 percent, the fact that nominal inter-

est rates cannot fall below zero has suddenly appeared relevant. The idea

of a “liquidity trap”—whatever that may mean—is likewise attracting

widespread interest. In the same month that this conference was held,

even the Federal Reserve Bank of St. Louis (yes, St. Louis!) published a

short article titled “Pushing on a String”—and the point of the article was

not to dismiss the idea but to give credence to it.

212

1. Piger (2003).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 212

The neoclassical synthesis that emerged from the debates of the 1930s

was, as the name implies, a compromise. At the theoretical level, the eco-

nomics profession’s collective judgment awarded victory to the classical

school. Yes, an underemployed economy, left to its own devices, would

return to full employment. Excess supply of goods and services would

depress prices, and as long as the nominal value of outstanding money

balances remained constant—in other words, as long as the central bank

did not allow the money stock to shrink, as the Federal Reserve did in the

1930s—the consequent increase in the real value of monetary wealth

would stimulate demand and thus restore full employment. The alleged

underemployment equilibrium was not, in fact, a true equilibrium.

By contrast, at the level of empirical relevance the prize went to the

Keynesians. Yes, the Pigou effect (or, in more general terms that also

allowed for nominal government debt other than money, the real balance

effect) would eventually restore full employment. But the emphasis in

that conclusion was decidedly on the adverb. For purposes of practical

policymaking, the presumption was that this process would take far too

long for responsible authorities simply to wait it out. Hence the focus on

monetary policy that came to dominate so much of the last four decades

was relevant after all.

As James Tobin often pointed out, however, the victory claimed at the

theoretical level by the classicals was a thinner one than met the unsus-

pecting eye. On further thought, the Pigou effect was not a general propo-

sition, applicable always and everywhere. Instead, the positive effect of

falling prices on aggregate demand depended on the assumption of non-

extrapolative expectations. It was true, irrespective of expectations, that

falling prices raised the real value of a given nominal money stock. But,

as Tobin emphasized in numerous important papers, falling prices also

meant a positive return on money balances held, and the more rapidly

prices fell, the greater that return was. When prices fell, people found that

their money holdings had greater purchasing power. But, if they expected

prices to continue falling, they would also expect a further positive return

on their money balances, and therefore, under conventional assumptions

about portfolio behavior, would want to hold more of them. For the Pigou

effect to work, the increase in the supply of real money balances due to

the past fall in prices has to outweigh the increase in the demand for real

money balances due to any expected future fall in prices. Or, as Tobin put

it, inﬂationary expectations have to be nonextrapolative.

Gauti B. Eggertsson and Michael Woodford 213

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 213

Like an unwanted ghost out of some well-forgotten past, the issue of

nonextrapolative inﬂation expectations has now returned to haunt the

modern discussion of monetary policy. This was, in part, the point under-

lying Paul Krugman’s widely discussed 1998 Brookings Paper,

which

argued that the Bank of Japan should commit itself to a target of (positive)

4 percent annual inﬂation. It has lurked not far beneath the surface of

many arguments since then about the problems of monetary policy in

Japan, or potential similar problems in the United States. It is also the

point behind this paper by Gauti Eggertsson and Michael Woodford.

Eggertsson and Woodford’s main conclusion is that the central bank

should commit itself not to an inﬂation target but to a speciﬁc form of

price-level target. The reason, as they forthrightly acknowledge, is “to

create the right kind of expectations.” To be sure, a fully credible inﬂation

target, with a nonnegative target value, also commits monetary policy to

stop any deﬂation. But, as Eggertsson and Woodford explain, the price-

level target goes one better: it “commits the government to undo any

deﬂation by subsequent inﬂation.” Or, to use their descriptive phrase

(which I ﬁnd highly apt), the price-level target is automatically equivalent

to a “history-dependent” inﬂation target. If the economy has suffered a

deﬂation, the central bank should now aim for positive inﬂation. In a

further reﬁnement of this idea, Eggertsson and Woodford build some

additional history dependence into the price-level target itself. Not sur-

prisingly, their model simulations indicate that there is a further gain in

policy performance from doing so. (This aspect of their results is analo-

gous to A. W. Phillips’ demonstration, nearly a half century ago, that a

combination of what he called “proportional,” “integral,” and “deriva-

tive” policy components typically achieved superior stabilization results,

compared with the use of any one or even two of these components

alone.) But the central point remains: what matters in their analysis is “the

management of expectations,” speciﬁcally, expectations about inﬂation.

All this strikes me as interesting, correct, and even important. That

said, I have two substantive reservations about what the authors have

done in this paper, and ﬁve concerns about what they have not done. I will

begin with the lacunae.

214 Brookings Papers on Economic Activity, 1:2003

2. Krugman (1998).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 214

First, a true price-level target means that the central bank is committed

not only to undo deﬂations with subsequent inﬂations, but also to undo

inﬂations with subsequent deﬂations. Most discussion of proposals for a

price-level target for monetary policy emphasizes precisely this point.

Many historical episodes suggest that deﬂation is not a desirable outcome

for an economy arranged as ours is, and much economic analysis has

explained why. (Familiar names in this line of work include Irving Fisher,

Albert Hart, and Ben Bernanke.) Having the central bank deliberately cre-

ate a deﬂation therefore usually seems like a bad idea. Eggertsson and

Woodford’s model includes none of the mechanisms (debt defaults, for

example) that make deﬂation harmful. Hence their recommendation of a

price-level target, based on simulations of their model, is weaker than

their forceful, unqualiﬁed prose lets on.

Second, the fact that their model excludes a foreign currency asset sim-

ilarly qualiﬁes their discussion of what effects, if any, follow from mone-

tary expansion once the interest rate has hit the zero bound. The nominal

interest rate is the margin of substitution between some currency today

and the same currency in the future. In a model with only one currency

(and no equity capital), that is the only ﬁnancial margin to discuss. But

when holders of that one currency can exchange it for another, an addi-

tional margin is put in play. The fact that one margin is at a corner solu-

tion does not necessarily mean that there can be no movement at the other.

Hence the “irrelevance result” that the authors present, based on the fact

that certain variables do not enter “the complete set of restrictions . . . to

be consistent with a rational expectations equilibrium” is persuasive

about the model but not necessarily about the world.

It usually goes without saying that analysis of this kind is contingent on

the speciﬁc model used as the engine of that analysis. But in light of the

importance of the issues being addressed here for matters of economic

policy that are currently under discussion in several countries, the point

seems to deserve particular emphasis with respect to this paper. I would

have been more comfortable if the authors had explicitly added the phrase

“in this model” to the statement of some of their key conclusions, includ-

ing especially the sentence that reads, “The above proposition [that is, the

irrelevance proposition] implies that neither the extent to which quantita-

tive easing is employed when the zero bound binds, nor the nature of the

assets that the central bank may purchase through open-market operations,

Gauti B. Eggertsson and Michael Woodford 215

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 215

has any effect on whether a deﬂationary price-level path will represent a

rational expectations equilibrium.”

Third, the same point applies to the authors’ discussion of ﬁscal policy.

The authors’ model excludes any potential direct effects of ﬁscal policy

such as arise, for example, when government transfer payments are not

perfect substitutes for reduced taxes, or when private agents are liquidity

constrained and cannot borrow at the government bond rate, or when gov-

ernment spending is not a perfect substitute for private consumption.

Fourth, yet the same point also applies to debt management policy. The

authors reject the idea that some form of Operation Twist, in which either

the central bank or the ﬁscal authority buys long-term securities and sells

short-term securities, would stimulate spending. The reason is that

“changes in the composition of the securities in the hands of the public do

not change the state-contingent consumption of the representative house-

hold.” As they go on to explain, this result does not follow from assuming

that bondholders do not care about risk, or that short- and long-term secu-

rities are perfect substitutes. What matters is instead the more fundamen-

tal assumption of a representative-agent model (which requires that each

investor hold the full market portfolio of outstanding securities) together

with the further assumption that what matters to each of these representa-

tive agents is only the expected future stream of consumption. Hence the

familiar story by which changing the mix of securities outstanding affects

long- relative to short-term interest rates, which in turn matters for output

and employment because ﬁrms prefer to ﬁnance their capital spending by

issuing long-term liabilities, cannot play out in this model.

The authors acknowledge (in their discussion, not in their model) the

possibility of such effects but dismiss them on two grounds: that in prin-

ciple they further depend on whether and how the relevant government

authorities remit any reduced debt management costs back to the public,

and that empirical evidence from the Operation Twist experiment in the

early 1960s mostly shows only minor results, if any. The theoretical claim

is certainly correct, although it further highlights the extent of the ratio-

nality ascribed to economic agents throughout the analysis: people under-

stand that when they earn less in interest on their holdings of Treasury

debt (because the Federal Reserve has bought up their higher-yielding

long-term bonds), the Federal Reserve’s earnings go back to the Treasury,

which then lowers their taxes, so that they (the investors) end up unaf-

fected. This is not the place to renew the empirical debate over the efﬁ-

216 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 216

cacy of debt management policy, but two points are worth noting. One is

that, as is well known, our ability to judge the effectiveness of Operation

Twist is clouded by the fact that the Treasury and the Federal Reserve

were working at cross purposes, one acting to shorten the average matu-

rity of publicly held government debt and the other to lengthen it. The

other is that the tiny scale of that historical attempt at debt management is

far from what people have in mind today when they suggest that, in the

event of a potential deﬂation problem in the United States, the Federal

Reserve could affect markets by buying up long-term bonds.

Finally, an entirely different point arises with respect to the rule that

determines the price-level target at which monetary policy aims. In the

authors’ model, the optimal path of the price level represents a trade-off

between, on the one hand, the advantages of an upward-sloping path (that

is, positive inﬂation), which reduces the likelihood that the sum of inﬂa-

tion and the natural rate of interest will be negative and therefore the zero

bound on nominal interest rates becomes binding, and, on the other, “the

distortions created by . . . inﬂation.” But what if, up to some point, the net

of those distortions due to inﬂation is a not a negative but a positive for

the economy? For example, what if George Akerlof, William Dickens,

and George Perry are right that, because of nominal wage rigidities, the

economy operates at a higher level of resource utilization, and reallocates

resources more effectively as circumstances change over time, with 2 to

3 percent inﬂation rather than zero?

Then, on the authors’ logic, it would

be a win-win choice to aim for an upward-sloping price trajectory. Doing

so would reduce the likelihood of hitting the zero bound on nominal inter-

est rates, and it would improve the functioning of the real economy away

from that bound.

I turn in closing to two issues of a more fundamental character. First, to

return to the historical context in which I believe this paper ﬁts, the impor-

tant issue is to avoid extrapolative expectations of price declines. Put in

terms of the current debate over Japan, or perhaps even the United States,

it is important that people believe prices will rise, not fall, in the future.

But what if they do not? If the central bank simply announces a 4 percent

inﬂation target, as Krugman recommended for Japan, but monetary policy

is impotent as long as prices are falling and the natural rate of interest (to

use the term favored in this paper) is negative, why should people attach

Gauti B. Eggertsson and Michael Woodford 217

3. Akerlof, Dickens, and Perry (2000).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 217

credence to the central bank’s announced target in the ﬁrst place? And if

they don’t, then how will that announcement restore the potency of pol-

icy? It is the conundrum of Tinker Bell’s dust: If I believe in it, I can ﬂy. If

not, I can’t. And if I don’t, there’s nothing that will prove me wrong.

Most of the authors’ analysis is set in a hypothetical context in which

the central bank has already been operating according to the kind of mon-

etary policy rule that they recommend, and has done so for however long

it takes the public to understand that this is what the central bank does and

to have conﬁdence that it will keep on doing so. The point of their analy-

sis is then to show—in their model—the advantages of having in place

this form of rule rather than some other. But the whole point of the current

discussion is precisely that neither the Bank of Japan, nor the Federal

Reserve System, nor any other central bank for that matter, currently fol-

lows such a rule. (If the authors’ object were to conﬁrm the optimality of

the monetary policy rule that most of these central banks already fol-

lowed, the paper would presumably read quite differently.) Even taking at

face value the paper’s claims for the optimality of this kind of rule, the

question is how to achieve a transition from a time when the central bank

has not been following such a rule, so that there is no reason for the pub-

lic to think it is doing so, to a new regime in which the central bank is fol-

lowing this kind of rule and is fully understood to be doing so.

The authors are well aware of this issue. They discuss ways, beyond

mere announcements, for the central bank to “demonstrate resolve,” and

they appeal to what is now a fairly rich literature (but only scant experi-

ence) of giving central bankers incentives to carry out policy in particular

ways. It is no criticism to say that they have not solved this problem. But

the fact that they have not solved it does not mean that it has gone away or

that it is unimportant.

Finally, without speciﬁcally criticizing the paper on this count, I want to

just here but in today’s monetary policy literature more generally, with

what the authors call “the management of expectations.” Over the past few

decades the literature on monetary policy has traveled a path from ignoring

expectations altogether, to taking expectations into account, to putting

expectations at the center of the analysis, and now, to making expectations

virtually the entirety of the analysis. Eggertsson and Woodford are clear: if

the central bank takes actions that do not affect expectations, those actions

simply do not matter. (Again, I would add “in their model.”)

218 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 218

The reverse of this proposition (which, of course, does not necessarily

follow from the proposition itself) is that the central bank need not ever

do anything. All that matters is that it affect expectations. The operating

arm of monetary policy is then not the trading desk but the press ofﬁce.

Or, to use a different metaphor, from a paper I wrote a few years ago, all

this army needs is a signal corps. On inspection, such views cannot stand

up. The situation they describe is not self-sustaining, at least not for long.

The problem to which this line of thinking nonetheless gives rise is the

increasingly exclusive focus on speciﬁcally managing expectations. The

product the ﬁrm sells may be terrible—if it’s food, it tastes awful; if it’s

wallpaper, it looks ugly; if it’s a machine, it doesn’t work; if it’s medi-

cine, it doesn’t cure anything—but the solution is to be found not in the

design laboratory but at the advertising agency. No matter what the cen-

tral bank is doing, always write the press release to say that the intended

purpose is to keep inﬂation on the straight and narrow, because that is

what the public needs to believe for the central bank to enjoy the fruits of

“credibility.” Even if inﬂation is already somewhat higher than desired,

but the central bank is cutting interest rates anyway in order to spur the

real economy out of a recession (a situation that observers of the U.S.

economy will easily recognize from the very recent past), claim nonethe-

less that the sole purpose of these actions is to preserve price stability.

Ridiculous as it sounds when put in plain language, this is precisely the

ﬂavor of some of the advice the Federal Reserve was receiving not so

long ago.

Eggertsson and Woodford do not fall into this trap. To repeat, their

analysis is all about the advantages they claim for a central bank’s actu-

ally following a particular kind of policy rule and being understood to do

so. But for many readers, I fear, their emphasis on “managing expecta-

tions” may convey the wrong message. Fortunately, the ofﬁcials actually

in charge of U.S. monetary policy have also, most of the time, been more

sensible than to fall into this trap. But the repeated and central emphasis

on the “management of expectations,” not just in this carefully crafted

paper but in so much of both the research literature and the public discus-

sion of monetary policy today as well, is worrisome nonetheless.

Mark Gertler: The topic of this paper is important and highly relevant to

current events. As one would expect from these authors, it also contains

some interesting and innovative theoretical contributions. The paper picks

Gauti B. Eggertsson and Michael Woodford 219

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 219

up two themes from Paul Krugman’s earlier work on this subject. The ﬁrst

is that a central bank may be able to lift an economy out of a liquidity trap

if it can create expectations that its policy will be expansionary in the

future. The second is that creating these expectations is a nontrivial mat-

ter. It involves making a credible commitment to stick to an inﬂationary

policy in the future, after the economy has emerged from the liquidity

trap. The paper goes beyond Krugman’s analysis, however, in several

significant ways. First, it presents a theoretical model that adds some

empirical richness. Second, it characterizes the optimal policy. Finally, it

translates that optimal policy into an operational rule that has a distinct

real-world interpretation.

A major theme of the paper is that an economy slips into a liquidity

trap when the natural real interest rate (that is, the equilibrium interest

rate under flexible prices) becomes negative, and it emerges from the

trap when this rate becomes positive again. The authors treat the natural

real interest rate as an exogenous process, and they then study the behav-

ior of monetary policy conditional on this process. As I discuss below,

however, policymakers facing an economy on the verge of a liquidity

trap, or already enmeshed in one, may also want to consider policies that

directly influence the natural real rate. A natural candidate is a transitory

fiscal policy. In this regard, for an economy truly threatened by a liquid-

ity trap, the coordinated exercise of fiscal and monetary policy may be

desirable. This point, of course, goes back to Keynes, but it can also be

illustrated clearly within a modest variation of the authors’ contemporary

framework.

I begin by summarizing the key aspects of the authors’ analysis and

then offer a few comments on it. I then describe how a slight extension of

the framework suggests a role for coordinated monetary and ﬁscal policy.

For countries with malfunctioning credit systems, such as Japan today,

ﬁnancial restructuring should also be factored into the policy mix.

THE MODEL

. The model is a simple general-equilibrium framework

with money and nominal rigidities in the form of staggered multiperiod

price setting. Let x

be the percentage deviation of output from its natural

(ﬂexible-price equilibrium) level; let π

be inﬂation, i

the nominal rate of

interest, and r

the natural real rate of interest. After log-linearizing

around the deterministic steady state, it is possible to collapse the model

into the following simple system, consisting of an IS curve and a Phillips

curve, speciﬁed by

220 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 220

The IS curve relates the output gap positively to the gap between the

natural real interest rate and the current real market rate and to the

expected future output gap. The Phillips curve, in turn, relates inﬂation to

the output gap and to expected future inﬂation. The nominal interest rate i

is the instrument of monetary policy. The model thus describes the behav-

ior of x

and π

, conditional on the exogenous path of r

and the central

bank’s choice of i

The central bank would like to maintain price stability (that is, keep π

close to zero) and stabilize the output gap (keep x

close to zero). It manip-

ulates i

in order to accomplish these goals. It cannot, however, reduce i

below zero; that is, it faces the following lower-bound constraint:

As the authors’ analysis makes quite clear, the lower bound binds

when the natural real rate is negative. In this situation the economy slips

into a liquidity trap, assuming there is no expectation of excess demand in

the future. When r

> 0 and E

t+1

≤ 0, a negative real market rate is

required to keep output from slipping below the natural level, as equation

1 makes clear. Given the lower bound, however, a negative real rate can

arise only if inﬂation is expected over the next period, and that cannot

happen if the private sector does not expect excess demand to arise in the

future (that is, if E

t+1+i

≤ 0 for all i). In this situation, accordingly, excess

supply (x

< 0) and deﬂation (π

< 0) emerge.

As Krugman originally emphasized, even if the lower bound is bind-

ing, a central bank can still provide stimulus if it can inﬂuence beliefs

about the future course of monetary policy. The authors’ framework is

very useful for illustrating this point: here expectations of the future path

of the nominal rate affect current economic behavior. To see this directly,

iterate equations 1 and 2 forward to obtain

() .5 πβκ

Ex=

{}

∑

() – ( – )4

xE r i E

tt ti

ti ti t i

[]

{}

+++++

∑

σπ

() .30i

≥

() .2

πκ βπ

tttt

xE=+

() – ( – )1

xriE Ex

ttt tt

[]

σπ

Gauti B. Eggertsson and Michael Woodford 221

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 221

Beliefs about future policy translate into expectations about the future

path of the real interest rate gap. In turn, these expectations affect the cur-

rent output gap. They also affect current inﬂation by affecting both the

current and the expected future path of the output gap.

MONETARY POLICY IN THE LIQUIDITY TRAP

. It follows that even if the

central bank is currently powerless to reduce the nominal rate, it can still

stimulate current economic activity by credibly committing to adopt an

expansionary policy once the economy is free of the liquidity trap. Con-

sider the following simple example. Suppose that the natural real interest

rate is expected to be negative for T periods before turning positive indef-

initely; that is, r

t+i

< 0 for i ∈ [0, T – 1], and r

t+i

> 0 for i > T. Given that the

lower bound is binding for the ﬁrst T periods, one can express the output

gap as

Observe that the central bank can raise x

by committing to reduce i

by a

sufﬁcient amount in periods t + T and after. This transmission mechanism

involves two channels. First, the expected path of i

in the post-liquidity

trap period affects current spending. As equation 6 indicates, holding

expected inﬂation constant, a decline in expected future nominal rates

will directly raise x

. Second, the resulting increase in current and

expected future values of x

stimulates inﬂation, which in turn reduces real

interest rates, further stimulating current demand. Krugman has empha-

sized this latter channel. The analysis of the former, however, is new.

Equation 6 also reveals some intuition for the authors’ proposition that

open-market purchases are ineffective when the economy is stuck in a liq-

uidity trap, unless these operations inﬂuence beliefs about the behavior of

interest rates once the economy is out of the trap. When the zero bound is

binding and ﬁscal policy is held constant, an open-market purchase affects

neither nominal interest rates nor private sector wealth.

It thus cannot

() ( )

– ( – ).

xE r x

Er ri

tt ti

ti tT

tti

ti ti

ti t i

[]

{}

()

[]

{}

+++ +

+++ ++++

∑

∑∑

σπ

σπ σ π

222 Brookings Papers on Economic Activity, 1:2003

1. The authors derive their irrelevance proposition for a more general case where util-

ity is nonseparable in real money balances and consumption. However, in the liquidity trap,

real money balances are beyond the satiation point and do not affect the marginal utility of

consumption. Open-market purchases, accordingly, have no effect in this more general

case, either.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 222

affect current spending unless it somehow inﬂuences beliefs about the

path of nominal rates in the post-liquidity trap era. This proposition is

highly relevant to the discussion of what policies to pursue in the event a

liquidity trap threatens, particularly the proposal to buy long-term bonds.

The authors’ proposition shows that such a policy will be ineffective

unless it alters beliefs about future short-term rates.

Equation 6 also makes transparent the nature of the time-consistency

problem. To stimulate the economy, the central bank clearly would like to

create the expectation that it will pursue an expansionary policy even after

it regains its ability to manipulate short-term rates directly. This requires

committing to keep the interest rate gap, r

t+i

– (i

– E

t+i

t+1+i

), positive for a

number of periods after T – 1. Once the economy is out of the liquidity

trap, however, the central bank will prefer to concentrate on maintaining

price and output gap stability. The best way to achieve these goals is to

adjust the nominal rate to ﬁx the interest rate gap at zero. Doing so, how-

ever, would involve reneging on the earlier pledge to keep this gap posi-

tive for a period of time.

OPTIMAL MONETARY POLICY IN A LIQUIDITY TRAP

. A signiﬁcant con-

tribution of the paper is its characterization of the optimal policy in the

liquidity trap. Here the central bank trades off the current gain from creat-

ing expectations that future policy will be expansionary against the cost of

having to stick to this expansionary policy once the economy is free of the

liquidity trap. As the authors show, when the central bank is again free to

manage nominal rates, it should do this so as to adjust demand to stabilize

the price level around the target p*

. This strategy, in turn, results in x

responding in a “lean against the wind” fashion (when i

> 0) as follows:

However, if the lower bound is binding, the central bank should set i

= 0

and ratchet up the target price level for the next period, using the follow-

ing updating rule:

() –(/ )(

– ).7 xpp

txtt

=κλ

Gauti B. Eggertsson and Michael Woodford 223

2. Note also that the authors’ proposition still applies even if the government buys risky

assets. In this instance the private sector will still bear the risk through ﬂuctuating taxes.

However, the authors’ irrelevance theorem (like all Miller-Modigliani theorems) is based

on the assumption of perfect capital markets. This suggests that it may be necessary to

appeal to capital market frictions to justify intervention in the long-term bond market (or to

the notion that this would send a signal about future short-term rates, in line with the

authors’ arguments).

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 223

Intuitively, by committing to the price-level targeting rule when the

economy is free of the liquidity trap, the central bank creates the expecta-

tion that it will offset any deﬂationary pressure in the present with expan-

sionary policy in the future. This threat to adopt an expansionary policy,

in turn, mitigates the pain of the liquidity trap. By ratcheting the price-

level target up if the liquidity trap materializes, the central bank meets

strong deﬂationary pressure with a commitment to intensify its future

expansionary policy. The authors show, however, that a simpler policy

that keeps the path of the price-level target invariant to current conditions

closely approximates the optimal policy, under reasonable assumptions

about parameter values.

The beneﬁt of a price-level target over an inﬂation target to ﬁght deﬂa-

tion is reasonably straightforward. It meets enhanced deﬂationary pres-

sure with an intensiﬁed commitment to pursue expansionary policy in the

future (even if the target price level is unchanged). An inﬂation target, on

the other hand, lets bygones be bygones. That is, an unusual drop in prices

today does not affect the course of policy in the future, since under inﬂa-

tion targeting a central bank is focused only on the current rate of change

in prices. Thus inﬂation targeting does not induce the same kind of stabi-

lizing adjustment of expectations about the future course of policy as does

price-level targeting.

Of course, the authors’ result that price-level targeting is optimal

requires several qualiﬁcations. First, the simple form of the price-level tar-

geting rule is in large part a product of the purely forward-looking Phillips

curve given by equation 2. Although this form of the Phillips curve is use-

ful for gaining insight into how central banks should factor private sector

expectations into policy management, the baseline version used by the

authors does not capture the high persistence of inﬂation observed in the

actual data. However, as Jordi Galí and I have shown, a hybrid variant of

equation 2 that allows for a mix of forward- and backward-looking behav-

ior does a reasonably good job.

Because forward-looking behavior

remains important under this speciﬁcation, the authors’ qualitative con-

clusions regarding the importance of managing future expectations will

survive. However, because inﬂation depends on lagged inﬂation as well

()

–

(

– )

– .

–

pp app

tt tttt+

[]

βππ

224 Brookings Papers on Economic Activity, 1:2003

3. Galí and Gertler (1999). The hybrid variation is given by π

=κx

+γ

t+1

+ γ

t–1

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 224

as on expected future inﬂation, it will no longer be optimal to simply tar-

get the price level and ignore past inﬂation.

Second, the issue of time consistency remains. That is, although the

price-level target helps minimize the damage to the economy from being

in a liquidity trap, once the economy is out of the trap, the central bank

would like to abandon that target. The authors clearly recognize this issue

and propose a number of ways to properly align the central bank’s incen-

tives. Whether these strategies would work in practice, especially for an

economy like the United States, remains an open question. To date, price-

level targeting has not had much appeal. One reason may be that, as I sug-

gested earlier, price-level targeting is most appealing when price setting is

purely forward looking. The belief that at least a component of inﬂation is

backward looking naturally raises concerns about adopting a simple

price-level target.

FISCAL POLICY AND FINANCIAL RESTRUCTURING

. The point that the

liquidity trap is ultimately a product of having a negative natural rate of

interest, although highly transparent in the authors’ analysis, is also inher-

ent in the traditional IS/LM description of this phenomenon. Within this

traditional apparatus, a liquidity trap emerges when the IS curve intersects

the long-run aggregate supply curve at a negative interest rate. Expecta-

tions of future policy play no role in this description, however, in contrast

to the authors’ analysis.

The traditional prescription for a liquidity trap, of course, is expansion-

ary ﬁscal policy. Fiscal stimulus shifts the IS curve outward to the point

where it intersects the long-run aggregate supply curve at a positive inter-

est rate. A suitably accommodative monetary policy, of course, should

also be part of the overall package.

Expansionary ﬁscal policy (along with monetary accommodation) is

also a natural path to take within the authors’ framework. As in the tradi-

tional analysis, this policy, if used effectively, attacks the heart of the

problem by pushing the natural rate of interest into positive territory.

Because the authors’ framework is more highly structured than the IS/LM

model, some subtleties emerge about the nature of the desired interven-

tion that are not present in the traditional analysis.

For example, suppose we modify the authors’ framework to allow for

government consumption as well as private consumption. Then let g

the logarithm of government consumption and let a

be the logarithm of

technology. Then it is straightforward to show that the natural rate of

Gauti B. Eggertsson and Michael Woodford 225

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 225

interest is (approximately) the following implicit function of the expected

growth rate of technology and the expected growth rate of government

expenditure:

The equilibrium real interest rate depends positively on expected produc-

tivity growth and negatively on the expected growth rate of government

expenditure. Intuitively, the latter raises expected consumption growth

(thus pushing up the real interest rate) whereas the former reduces it.

Suppose now that, holding ﬁscal policy constant, r

becomes negative

for a period of time because of a transitory period of negative productiv-

ity growth. In the absence of any policy response, the economy enters a

liquidity trap. However, by pursuing a sufﬁciently aggressive transitory

increase in government expenditure, the ﬁscal authority can push the nat-

ural rate into the positive region, thus avoiding the trap. An important dif-

ference from the traditional analysis is that the government commits to

making the expansion transitory: if the private sector perceives the expan-

sion as permanent, it will not affect the natural rate.

I am not suggesting ﬁscal policy as a substitute for the authors’ mone-

tary prescription but rather as a complementary policy initiative. One

virtue of this approach is that it involves offering direct stimulus to the

economy as opposed to resting one’s hopes entirely on private sector

expectations of future (monetary) stimulus. Of course, before any ﬁrm

conclusions may be drawn, a formal analysis of ﬁscal policy along the

lines of the authors’ analysis of monetary policy would be desirable.

Along these lines, modifying the authors’ framework to allow for ﬁscal

tools would seem to provide a good starting point.

Finally, it is important to recognize that malperformance of credit mar-

kets is a key feature of economies truly enmeshed in a liquidity trap, such

as the U.S. economy during the Great Depression and the Japanese econ-

omy today. At a conceptual level, credit market frictions raise the likeli-

hood that the zero bound will bind. They do so by reducing the real market

interest rate required to produce zero excess demand (that is, x

= 0). To see

this, consider the following very stylized example. Suppose that χ

is the

premium for external ﬁnance that borrowers must pay, owing to the pres-

() ( – , – ); , .900

1112

rEaaEgg

tt t tt t

=><

ρρρ

226 Brookings Papers on Economic Activity, 1:2003

4. In a model with capital, a permanent shift in government expenditure can affect the

natural rate of interest, although transitory movements will have a larger effect.

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 226

ence of capital market frictions. This premium will depend on such fac-

tors as borrowers’ collateral and the overall conditions of financial insti-

tutions. In this environment the opportunity cost of investing is given by

+ i

– E

t+1

, implying that the interest rate gap is now given by r

– [χ

+ i

– E

t+1

], where r

now has the interpretation of being the natural real

rate in the absence of credit market frictions. In this instance the zero

bound will bind if r

– χ

< 0.

Since χ

> 0, ﬁnancial market frictions raise the likelihood that the

economy will slip into a liquidity trap. Intuitively, the rise in the cost of

credit owing to these frictions requires lower risk-free market rates than

otherwise to keep overall borrowing costs from stiﬂing demand and edg-

ing the economy into a deﬂation. To the extent ﬁnancial reforms and

ﬁnancial market restructuring reduce χ

, they help ease the economy out

of the liquidity trap. As with ﬁscal policy, credit market improvements

potentially provide direct stimulus for an economy in the midst of a liq-

uidity trap.

General discussion: James Duesenberry was skeptical of the authors’

assumption that expectations and credibility are the crucial elements in

policymaking. He noted that policy credibility depends on economic

agents believing that policy is effective—that when the Federal Reserve

says it is going to do something to stimulate the economy, it can actually

make it happen. He observed that even the Federal Reserve’s ability to

control the term structure of interest rates was uncertain. Certainly, expec-

tations about future federal funds rates are an important determinant of

the term structure, but other expectations, for example about the level of

capital utilization or the demand for housing, also play an important role,

and these are not exclusively influenced by monetary policy. Even more

problematic is whether the Federal Reserve has the ability (except when

in a liquidity trap) to steer the economy exactly where the Federal

Reserve wants it to go. Many believe that the efforts of monetary policy

are sometimes like “pushing on a string,” and even monetary economists

are uncertain about its effectiveness. Moreover, monetary policy is sup-

posed to work though a variety of channels, affecting investment through

changes in the cost of capital, consumption through wealth effects in the

stock market, and the balance of payments through changes in the

exchange rate, and other examples could be cited. Duesenberry noted

that a recent study by the Federal Reserve Bank of New York arrived at a

Gauti B. Eggertsson and Michael Woodford 227

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 227

wide range of estimates of the magnitude of these effects, and that the

authors were themselves pessimistic about the precision of monetary

policy interventions.

Christopher Sims thought it unfortunate that the paper followed a

recent practice in the literature of making very strong and artiﬁcial

assumptions about ﬁscal policy. In the authors’ model, the ﬁscal authority

pegs the real value of total government liabilities without regard to the

proportions of high-powered money and interest-bearing debt. Sims gave

two reasons for ﬁnding this objectionable. First, this policy is not optimal

in a simple Lucas-Stokey or Barro model. In both models real debt should

respond endogenously to shocks. Second, the policy implies that a ﬁscal

authority, confronted with a large amount of liabilities in the form of

high-powered money, would feel just as committed to raise taxes to retire

that money stock as it would to raise taxes to reduce the same amount of

interest-bearing debt. This is implausible: the division of liabilities

between interest-bearing debt and money should affect the amount of

pressure—or lack of pressure—on the legislature to raise taxes.

William Brainard likewise emphasized the composition of government

debt. In the authors’ model, short-term interest rates link consumption

across successive periods, and therefore current and expected short-term

rates are what matter. To achieve a desired effect on the economy, it then

sufﬁces for the monetary authority to announce the rule it will follow in

setting future short-term rates. The long-term bond rate reﬂects this rule

but plays no separate role in affecting private actions. In reality, however,

announcing a rule for future short-term rates may be quite different from

intervening today in order to inﬂuence the long-term rate. Investments in

capital equipment are irreversible in the short run; as a result, the risk of

borrowing long term for such investments is different from undertaking a

sequence of short-term borrowings. Setting today’s long-term rate elimi-

nates all manner of uncertainties, including uncertainty about the mone-

tary authority’s credibility.

Christopher Sims agreed with Benjamin Friedman that a price-level

target is undesirable in times of high inﬂation if disinﬂation has real costs.

In a rational expectations model, the commitment to a price-level target,

and hence to deﬂation following an inﬂation, makes inﬂation less likely,

but it might be quite difﬁcult to make this costly commitment credible.

Sims suggested that an asymmetric commitment might be more appropri-

228 Brookings Papers on Economic Activity, 1:2003

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 228

ate. Although, historically, liquidity traps have involved deﬂation, the

rates of deﬂation experienced were very low. As a consequence, the

amount of inﬂation tomorrow to which one would have to commit today

in order to reach the price-level target is not high. On the other hand, we

have seen episodes of very rapid and large inﬂation, and Sims thought it

would not be possible to credibly commit to returning prices to their ini-

tial level after such an episode.

Gauti B. Eggertsson and Michael Woodford 229

1440-03 BPEA/Eggertsson 07/17/03 08:12 Page 229

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