The Zero Bound on Interest Rates and Optimal Monetary Policy

GAUTI B. EGGERTSSON InternationalMonetary Fund MICHAEL WOODFORD Princeton University

The Zero Bound on InterestRates and OptimalMonetary Policy

THE CONSEQUENCES FOR THE PROPER conductof monetarypolicy of the existenceof a lower boundof zero for overnightnominal 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 within50 basis points of zero since October1995, and it has been essentially equal to zero for most of the past four years (figure 1). Thusthe Bankof Japanhas hadlittle roomto furtherreduce short- termnominal interest rates in all thattime. MeanwhileJapan's growth has remainedanemic, and prices have continuedto fall, suggestinga need for monetary stimulus. Yet the usual remedy-lower short-termnominal interestrates-is plainly unavailable.Vigorous expansionof the monetarybase has also seemedto do little to stimulatedemand under these circumstances:as figure 1 also shows, the monetarybase is now more than twice as large,relative to GDP, as it was in the early 1990s. In the United States, meanwhile,the federalfunds rate has now been reducedto only 1 percent,and signs of recoveryremain exceedingly frag- ile. This has led many to wonderif this countrymight not also soon find itself in a situationwhere interestrate policy is no longer availableas a

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. 139 140 BrookingsPapers on EconomicActivity, 1:2003

Figure 1. Japan: Call Rate on OvernightLoans and Ratio of MonetaryBase to GDP, 1990-2002

Call rate Percenta year

2 1

Monetarybase-to-GDP ratio Index,1992= 1.0

2.2-

2.0-

1.8-

1.6-

1.4 -

1.2-

1.0

1992 1994 1996 1998 2000 2002

Source: Nomuradatabase, Bank of Japan. Gauti B. Eggertsson and Michael Woodford 141 tool for macroeconomicstabilization. A numberof other countriesface similarquestions. John Maynard Keynes first raisedthe questionof what can be done to stabilize the economy when it has fallen into a liquidity trap-when interestrates have fallen to a level below which they cannot be drivenby furthermonetary expansion-and whethermonetary policy can be effective at all undersuch circumstances.Long treatedas a mere theoreticalcuriosity, Keynes's questionnow appearsto be one of urgent practical importance, but one with which theorists have become unfamiliar. The questionof how policy shouldbe conductedwhen the zero bound is reached-or when the possibility of reaching it can no longer be ignored-raises many fundamentalissues for the theoryof monetarypolicy. Some would arguethat awareness of the possibilityof hittingthe zero boundcalls for fundamentalchanges in the way policy is conductedeven before the boundhas been reached.For example,Paul Krugmanrefers to deflationas a "blackhole,"1 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 monetarypolicy in a liquiditytrap is that it is vital to steerfar clearof circumstancesin which deflationaryexpecta- tions could ever begin to develop-for example, by targetinga sufficiently high positive rate of inflationeven undernormal circumstances. Othersare more sanguineabout the continuingeffectiveness of monetary policy even when the zero bound is reached.For example, it is often arguedthat deflationneed not be a black hole, because monetarypolicy can affect aggregatespending, and hence inflation, through channels other thancentral bank control of short-termnominal interest rates. Thus there has been much recent discussion, with respect to both Japan and the United States-of the advantagesof vigorousexpansion of the monetary base even withoutany furtherreduction in interestrates, of the desirabil- ity of attemptsto shift longer-terminterest rates through central bank purchases of longer-maturitygovernment securities, and even of the desirabilityof centralbank purchases of otherkinds of assets. Yet if these views are correct,they challengemuch of the recentcon- ventionalwisdom regardingthe conductof monetarypolicy, both within centralbanks and among academicmonetary economists. That wisdom has stresseda conceptionof the problemof monetarypolicy in termsof

1. Paul Krugman,"Crisis in Prices?"New YorkTimes, December 31, 2002, p. A19. 142 BrookingsPapers on EconomicActivity, 1:2003 the appropriateadjustment of an operatingtarget for overnightinterest rates, and the prescriptionsformulated for monetarypolicy, such as the celebratedTaylor rule,2are typically cast in these terms. Indeed, some have arguedthat the inabilityof such a policy to preventthe economy fromfalling into a deflationaryspiral is a criticalflaw of the Taylorrule as a guide to policy.3 Similarly,concern over the possibility of enteringa liquiditytrap is sometimespresented as a serious objectionto anothercurrently popular monetarypolicy prescription,namely, inflationtargeting. The definition of a policy prescriptionin termsof an inflationtarget presumes that there is in fact some level of the nominalinterest rate that can allow the target to be hit (or at least projectedto be hit, on average).But, some argue,if the zero interestrate bound is reachedunder circumstances of deflation,it will not be possible to hit any higher inflationtarget, because further interestrate decreasesare not possible. Is there, in such circumstances, any point in having an inflationtarget? The Bank of Japanhas frequently offered this argumentas a reason for resisting inflationtargeting. For example, Kunio Okina, directorof the Institutefor Monetaryand Eco- nomic Studiesat the Bank of Japan,was quotedas arguingthat "because short-terminterest rates are alreadyat zero, settingan inflationtarget of, say, 2 percentwouldn't carry much credibility."4 We seek to shed light on these issues by consideringthe consequences of the zero lower boundon nominalinterest rates for the optimalconduct of monetarypolicy, in the context of an explicitly intertemporalequilibrium model of the monetary transmissionmechanism. Although our modelis extremelysimple, we believe it can help clarifysome of the basic issues just raised.We are able to considerthe extent to which the zero boundrepresents a genuineconstraint on attainableequilibrium paths for inflationand real activity,and the extentto which open-marketpurchases of variouskinds of assets by the centralbank can mitigatethat constraint. We are also able to show how the existenceof the zero boundchanges the characterof optimal monetarypolicy, relative to the policy rules that would be judged optimalin its absenceor in the case of real disturbances small enoughfor the boundnever to matterunder an optimalpolicy.

2. Taylor(1993). 3. Benhabib,Schmitt-Grohe, and Uribe (2001). 4. "JapanBOJ Official:Hard to Set InflationTargets," Dow Jones News, August 11, 1999. Gauti B. Eggertsson and Michael Woodford 143 To preview our results,we find that the zero bounddoes representan importantconstraint on whatmonetary stabilization policy can achieve,at least when certainkinds of real disturbancesare encounteredin an envi- ronmentof low inflation.We arguethat the possibilityof expandingthe monetarybase throughcentral bank purchasesof a variety of types of assets does little if anythingto expandthe set of feasible paths for inflation and real activity that are consistent with equilibriumunder some (fully credible)policy commitment. Hence the relevanttrade-offs can correctlybe studiedby simply con- sideringwhat alternativeanticipated state-contingent paths of the short- termnominal interest rate can achieve, takinginto accountthe constraint that this rate must be nonnegativeat all times. Doing so, we find that the zero interestrate bound can indeedbe temporarilybinding, and when it is, it inevitably results in lower welfare than could be achieved in the absenceof such a constraint.5 Nonetheless,we arguethat the zero boundrestricts possible stabilization outcomes under sound policy to a much more modest degree than the deflationpessimists presume. Even thoughthe set of feasible equilibrium outcomes correspondsto those that can be achieved throughalter- native interest rate policies, monetarypolicy is far from powerless to mitigatethe contractionaryeffects of the kind of disturbancesthat would make the zero bound a binding constraint.The key to dealing with this sort of situationin the least damagingway is to create the right kind of expectationsregarding how monetarypolicy will be used after the constraint is no longer binding, and the central bank again has room to maneuver.We use our intertemporalequilibrium model to characterize

5. We do not explorehere the possibilityof relaxingthe constraintby taxing money balances,as originallyproposed by Gesell (1929) andKeynes (1936), andmore recently by Buiterand Panigirtzoglou(2001) and Goodfriend(2000). Althoughthis representsa solution to the problemin theory,it presentssubstantial practical difficulties, not the least of which is the politicalopposition that such an institutionalchange would be likely to gener- ate. Ourconsideration of the problemof optimalpolicy also abstractsfrom the availability of fiscal instruments,such as the time-varyingtax policy recommendedby Feldstein (2002). We agree with Feldsteinthat thereis a particularlygood case for state-contingent fiscal policy to deal with a liquiditytrap, even if fiscal policy is not a very useful tool for stabilizationpolicy moregenerally. Nonetheless, we considerhere only the problemof the properconduct of monetarypolicy, takingas given the structureof tax distortions.As long as one does not thinkthat state-contingentfiscal policy can (or will) be used to eliminate even temporarydeclines in the naturalrate of interestbelow zero, the problemfor monetary policy thatwe considerhere remains relevant. 144 BrookingsPapers on EconomicActivity, 1:2003 the kind of expectationsregarding future policy that it would be desirable to create,and we discuss a form of price-leveltargeting rule thatif credibly committedto-should bring about the constrained-optimal equilibrium.We also discuss, moreinformally, how othertypes of policy actions could help increase the credibility of the central bank's announcedcommitment to this kind of futurepolicy. Our analysis will be recognized as a developmentof several key themes in Paul Krugman'streatment of the same topic in these pages a few years ago.6Like Krugman,we give particularemphasis to the role of expectationsregarding future policy in determiningthe severity of the distortionsthat result from hitting the zero bound.Our primary contribu- tion, relativeto Krugman's earliertreatment, will be the presentationof a more fully dynamicanalysis. For example,our assumptionof staggered pricing,rather than Krugman's simple hypothesisof prices that are fixed for one period, allows for richer (and at least somewhatmore realistic) dynamicresponses to disturbances.In our model, unlikein Krugman's,a real disturbancethat lowers the naturalrate of interestcan cause outputto remainbelow potentialfor years (as shown in figure2 laterin the paper), ratherthan only for a single "period,"even whenthe averagefrequency of price adjustmentsis more than once a year. These richerdynamics are also importantfor a realisticdiscussion of the kind of policy commitment that can help to reduce economic contractionduring a liquiditytrap. In our model a commitmentto create subsequentinflation involves a commitmentto keep interestrates low for some time in the future,whereas in Krugman'smodel a commitmentto a higherfuture price level does not involve any reductionin futurenominal interest rates. We are also better able to discuss such questionsas how the creationof inflationaryexpec- tationswhile the zero boundis bindingcan be reconciledwith maintain- ing the credibilityof the centralbank's commitmentto long-runprice stability. Ourdynamic analysis also allows us to furtherclarify the severalways in which the centralbank's managementof private sector expectations can be expected to mitigate the effects of the zero bound. Krugman emphasizesthe fact thatincreased expectations of inflationcan lower the real interestrate impliedby a zero nominalinterest rate. This might suggest, however,that the centralbank can affect the economy only insofar

6. Krugman (1998). Gauti B. Eggertsson and Michael Woodford 145 as it affects expectationsregarding a variable that it cannot influence except quite indirectly;it might also suggest that the only expectations that should matterare those regardinginflation over the relatively short horizon correspondingto the term of the nominal interestrate that has fallen to zero. Such interpretationseasily lead to skepticismabout the practicaleffectiveness of the expectationschannel, especially if inflation is regardedas being relatively "sticky"in the short run. Our model is insteadone in which expectationsaffect aggregatedemand through several channels. Firstof all, it is not merelyshort-term real interestrates that matter for currentaggregate demand;our model of intertemporalsubstitution in spendingimplies that the entire expected futurepath of short-termreal rates shouldmatter, or alternativelythat very long termreal rates should matter.7This means that the creationof inflationexpectations, even with regardto inflationthat should not occuruntil at least a yearinto the future, should also be highly relevantto aggregatedemand, as long as it is not accompaniedby correspondinglyhigher expected future nominal interest rates. Furthermore,the expected future path of nominal interest rates matters,and not just their currentlevel, so that a commitmentto keep nominal interestrates low for a longer period of time should stimulate aggregatedemand, even when currentinterest rates cannot be lowered further,and even underthe hypothesisthat inflationexpectations would remain unaffected. Because the central bank can clearly control the futurepath of short-termnominal interest rates if it has the will to do so, any failureof such a commitmentto be crediblewill not be due to skep- ticism about whetherthe central bank is able to follow throughon its commitment. The richerdynamics of our model are also importantfor the analysisof optimalpolicy. Krugmanmainly addresses the questionof whethermonetary policy is completelyimpotent when the zero bound binds, and he arguesfor the possibilityof increasingreal activityin the liquiditytrap by

7. In the simple model presentedhere, this occurs solely as a result of intertemporal substitutionin privateexpenditure. But thereare a numberof reasonsto expect long-term rates,rather than short-termrates, to be the criticaldeterminant of aggregatedemand. For example,in an open-economymodel, the real exchangerate becomes an importantdeter- minantof aggregatedemand. But the real exchangerate shouldbe closely linkedto a very long domesticreal rateof return(or alternativelyto the expectedfuture path of short-term rates)as a resultof interestrate parity, together with an anchorfor the expectedlong-term real exchangerate (deriving, for example,from long-run purchasing power parity). 146 BrookingsPapers on EconomicActivity, 1:2003 creatingexpectations of inflation.Although we agree with this conclusion, it does not answer the question of whether,or to what extent, it would be desirableto create such expectations,given the well-founded reasonsthat the centralbank shouldhave to not preferinflation at a later time. Nor is Krugman's model well suited to addresssuch a question, insofaras it omits any reasonfor even an extremelyhigh subsequentinflation to be deemedharmful. Our staggered-pricingmodel insteadimplies that inflation(whether anticipated or not) does createdistortions, justify- ing an objectivefunction for stabilizationpolicy that tradesoff inflation stabilizationand outputgap stabilizationin termsthat are often assumed to representactual central bank concerns. We characterizeoptimal policy in such a setting and show that it does indeed involve a commitmentto history-dependentpolicy of a sort that should result in higher inflation expectationsin responseto a binding zero bound.We can also show to what extent it should be optimalto create such expectations,assuming thatthis is possible.We find,for example,that it is not optimalto commit to so muchfuture inflation that the zerobound ceases to bind,even though this is one possible type of equilibrium;this is why the zero bounddoes remaina relevantconstraint, even underan optimalpolicy commitment.

Is Quantitative Easing a Separate Policy Instrument?

A firstquestion we wish to consideris whetherexpansion of the monetary base representsa policy instrumentthat should be effective in preventingdeflation and an associatedoutput decline, even undercircum- stances where overnightinterest rates have fallen to zero. Accordingto Keynes's famous analysis,8monetary policy ceases to be an effective instrumentto head off economic contractionin a "liquiditytrap," which can arise if interestrates fall so low that furtherexpansion of the money supplycannot drive them lower. Othershave arguedthat monetary expansion shouldincrease nominal aggregate demand even undersuch circumstances,and the suppositionthat this is correctlies behindJapan's explicit adoption,since March2001, of a policy of "quantitativeeasing" in addi- tion to the zero interestrate policy thatcontinues to be maintained.9

8. Keynes(1936). 9. See Kimuraand others (2002) for a discussionof this policy as well as an expression of doubtsabout its effectiveness. Gauti B. Eggertsson and Michael Woodford 147 Here we considerthis questionin the contextof an explicitlyintertem- poral equilibriummodel, which models both the demandfor money and the role of financialassets (includingthe monetarybase) in privatesector budgetconstraints. The model we use for this purposeis moredetailed in several senses than that used in subsequentsections to characterizeoptimal policy. We do this to make it clear that we have not excludeda role for quantitativeeasing simplyby failingto model the role of moneyin the economy.10 Ourkey resultis an irrelevanceproposition for open-marketoperations in a varietyof types of assets that the centralbank might acquire,under the assumption that the open-marketoperations do not change the expected futureconduct of monetaryor fiscal policy (in senses that we specify below). It is perhapsworth noting at the outset that our intention in statingsuch a result is not to vindicatethe view that a centralbank is powerlessto halt a deflationaryslump, and hence to absolve the Bank of Japan,for example,of any responsibilityfor the continuingstagnation in thatcountry. Although our propositionestablishes that there is a sense in which a liquiditytrap is possible, this does not meanthat the centralbank is powerlessunder the circumstanceswe describe.Rather, our intentis to show thatthe key to effective centralbank action to combata deflationary slumpis the managementof expectations.Open-market operations should be largely ineffective to the extent that they fail to change expectations regardingfuture policy; the conclusionwe draw is not that such actions are futile, but ratherthat the centralbank's actions should be chosen with a view to signalingthe natureof its policy commitments,and not for the purposeof creatingsome sort of "direct"effects.

A Neutrality Proposition for Open-Market Operations Our model abstractsfrom endogenousvariations in the capital stock and assumesperfectly flexible wages (or some othermechanism for effi- cient laborcontracting), but it assumesmonopolistic competition in goods marketsand stickyprices that are adjustedat randomintervals in the man- ner assumedby GuillermoCalvo, so that deflationhas real effects.1"We

10. Woodford(forthcoming, chapter 4) discussesthe model in moredetail and consid- ers the consequencesof various interest rate rules and money growth rules under the assumptionthat disturbances are not largeenough for the zero boundto bind. 11. Calvo (1983). 148 BrookingsPapers on EconomicActivity, 1:2003 assumethat the representativehousehold seeks to maximizea utilityfunc- tion of the form

EtyET-t {u(Ct, Mt/t; 9t) - J [H, (j); g ]dj} where Ctis a Dixit-Stiglitzaggregate of consumptionof each of a contin- uum of differentiatedgoods,

C, - [lo ct(i) with an elasticityof substitution0 > 1; Mtmeasures end-of-period household money balances,12Pt is the Dixit-Stiglitzprice index,

(1) P, -- [J|o Pt(i)j-e di]1-e and Ht(j) is the quantitysupplied of labor of type j. Real balances are includedin the utility function,"3as a proxy for the services that money balancesprovide in facilitatingtransactions.14 Each industry j employsan industry-specifictype of labor,with its own wage. For each value of the disturbances tt, u(, *; tt) is a concave function, increasingin the firstargument and increasingin the secondfor all levels of real balancesup to a satiationlevel Fn(Ct;tt). The existenceof a satiation level is necessaryin orderfor it to be possible for the zero interest ratebound ever to be reached;we regardJapan's experience over the past severalyears as having settledthe theoreticaldebate over whethersuch a level of real balancesexists. Unlike many papersin the literature,we do not assume additiveseparability of the functionu between the first two

12. We do not introducefractional-reserve banking into our model. Technically,M, refers to the monetarybase, and we representhouseholds as obtainingliquidity services fromholding this base, eitherdirectly or throughintermediaries (not modeled). 13. FollowingSidrauski (1967) andBrock (1974, 1975). 14. We use this approachto modelingthe transactionsdemand for money becauseof its familiarity.As shown in Woodford(forthcoming, appendix section A.16), a cash-in- advancemodel leads to equilibriumconditions of essentiallythe same generalform, and the neutralityresult that we presentbelow wouldhold in essentiallyidentical form were we to modelthe transactionsdemand for moneyafter the fashionof Lucasand Stokey (1987). Gauti B. Eggertsson and Michael Woodford 149 arguments;this (realistic)complication allows a furtherchannel through which money can affect aggregatedemand, namely, by an effect of real moneybalances on the currentmarginal utility of consumption.Similarly, for each value of gt, i(.; g,) is an increasing convex function. The vector of exogenous disturbancest, may contain several elements, so that no assumptionis made aboutcorrelation of the exogenousshifts in the functions u and v. For simplicitywe assume complete financialmarkets and no limit on borrowingagainst future income. As a consequence,a householdfaces an intertemporalbudget constraint of the form

El XQ,,T [PTCT +8TMT] T=t

< W+ Et EQt,T fonT(i)di + foWT ()HT( )dj -TTh

lookingforward from any periodt. HereQtT iS the stochasticdiscount factor thatthe financialmarkets use to value randomnominal income at date T in monetaryunits at date t; 8t is the opportunitycost of holdingmoney and is equal to itl(l + it), where it is the riskless nominal interest rate on one-periodobligations purchased in periodt, in the case thatno interestis paid on the monetarybase; Wtis the nominal value of the household's financialwealth (includingmoney holdings)at the beginningof periodt; fHt(i)represents the nominalprofits (revenue in excess of the wage bill) in period t of the supplierof good i; wt(j) is the nominal wage earnedby laborof typej in periodt, and Tthrepresents the net nominaltax liabilities of each household in period t. Optimizing household behavior then implies the following necessary conditions for a rationalexpectations equilibrium. Optimal timing of household expenditurerequires that aggregatedemand Yt for the compositegood satisfy an Eulerequation of the form'5

(2) uC(Y,,M,/PF;,,) = E{uj(Yt+,AMt+i/eP+t;t,+,)(l +it

15. For simplicity,we abstractfrom governmentpurchases of goods. Ourequilibrium conditionsdirectly extend to the case of exogenous governmentpurchases, as shown in Woodford(forthcoming, chapter 4). 150 BrookingsPapers on EconomicActivity, 1:2003 Optimal substitutionbetween real money balances and expenditure leads to a staticfirst-order condition of the form

Um(YIMt/PI;t) _ it

u1(Y1,M,lP,;9, l+i, underthe assumptionthat zero interestis paid on the monetarybase, and thatpreferences are such as to excludethe possibilityof a cornersolution with zero money balances.If both consumptionand liquidityservices are normalgoods, this equilibriumcondition can be solved uniquelyfor the level of real balancesL(Y,, it; a) that satisfy it in the case of any positive nominalinterest rate. The equilibriumrelation can then equivalentlybe writtenas a pairof inequalities:

M (3) >L(Y, ,i,;,

(4) it 2 09 togetherwith the "complementaryslackness" condition that at least one musthold with equalityat any time. Here we defineL(Y, 0; t) = m-(Y; t), the minimumlevel of real balancesfor which um= 0, so thatthe function L is continuousat i = 0. Householdoptimization similarly requires that the paths of aggregatereal expenditureand the price index satisfy the bounds

(5) ,j3TE,[u(YTcMT/PT; TA)YT+ Um(YTaMT/PT; eT)(MT/PT)] < T=t

E,[u (YTaMT/PT; =0 (6) limDT- 9T)DT/PT] looking forwardfrom any periodt, whereDt measuresthe total nominal value of governmentliabilities (monetarybase plus governmentdebt) at the end of periodt underthe monetary-fiscalpolicy regime. (The condition in expression5 is requiredfor the existence of a well-definedinter- temporal budget constraint, under the assumption that there are no limitations on households' ability to borrow against future income, whereasthe transversalitycondition in equation6 musthold if the house- Gauti B. Eggertsson and Michael Woodford 151

hold hits its intertemporalbudget constraint.)The conditionsin expressions 2 through6 also suffice to imply that the representativehousehold chooses optimalconsumption and portfolioplans (includingits planned holdings of money balances) given its income expectations and the prices (including financial asset prices) that it faces, while making choices that are consistentwith financialmarket clearing. Each differentiatedgood i is supplied by a single, monopolistically competitiveproducer. There are assumedto be manygoods in each of an infinitenumber of industries;each industryj uses a type of laborthat is specificto thatindustry, and all goods in an industrychange their prices at the same time. Eachgood is producedin accordancewith a commonpro- ductionfunction

y,(i) = A,f [h,(i)],

where At is an exogenous productivityfactor common to all industries; f(.) is an increasing,concave function;and ht(i) is the industry-specific labor hired by firm i. The representativehousehold supplies all types of laborand consumesall types of goods.16 The supplierof 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 responseto firms' pricing decisions, on the one hand, and the terms on which optimizinghouseholds are willing to supplyeach type of labor,on the other, we can show that nominalprofits (sales revenue in excess of laborcosts) in periodt of the supplierof good i are given by the function

1 Pi,pJ, P; 1t,MtI /Jt, = P t[P p

hf {f' [yt(P;/e t)-O/A];tt Jr~ p,(i), p,i,P,;Y,M,lP,,Ptf]Ytp(i)Y,M[p,(it)lP, u,(Yt, MtIP lp;t ) p I{,[p il,]0A

16. We mightalternatively assume specialization across households in the type of labor supplied;in the presenceof perfectsharing of laborincome risk acrosshouseholds, household decisionsregarding consumption and laborsupply would all be as assumedhere. 152 BrookingsPapers on EconomicActivity, 1:2003

wherepi is the commonprice chargedby the other firmsin industryj.17 (We introducethe notationgt for the completevector of exogenous disturbances,including variations in technologyas well as in preferences.)If prices were fully flexible, pt(i) would be chosen each period to maximize this function. Insteadwe suppose that prices remainfixed in monetaryterms for a randomperiod of time. Following Calvo, we supposethat each industry has an equalprobability of reconsideringits priceseach period,and we let O< x < 1 be the fractionof industrieswhose prices remainunchanged each period. In any industrythat revises its prices in period t, the new pricep* will be the same.This priceis implicitlydefined by the first-order condition

Tt (7) Et{X IQIt1.(Pt*Pg* PT;YT,MT/PT,1T)} =0?

We note furthermorethat the stochastic discount factor used to price futureprofit streams will be given by

(8) QT= T-tU (CT,MT/PT;r 9T)

Finally,the definitionin equation1 implies a law of motionfor the aggregate price index of the form

(9) t= [(1 -x)p*6- aoU++ !i-e Equations7 and 9, which jointly determinethe path of prices given demandconditions, represent the aggregatesupply block of our model. It remainsto specify the monetaryand fiscal policies of the government.18

17. In equilibrium,all firmsin an industrycharge the sameprice at any given time. But we must defineprofits for an individualsupplier i in the case of contemplateddeviations fromthe equilibriumprice. 18. The particularspecification of monetaryand fiscal policy proposedhere is not intendedto suggestthat either monetary or fiscal policy mustbe expectedto be conducted accordingto rulesof the sortassumed here. Indeed, in latersections we recommendpolicy commitmentson the partof boththe monetaryand the fiscalauthorities that do not conform to the assumptionsmade here. The pointis to definewhat we meanby the qualificationthat open-marketoperations are irrelevantif they do not change expectedfuture monetary or fiscal policy. To make sense of such a statement,we must define what it would mean for Gauti B. Eggertsson and Michael Woodford 153 To addressthe questionof whetherquantitative easing represents an addi- tional tool of policy, we supposethat the centralbank's operatingtarget for the short-termnominal interest rate is determinedby a feedbackrule in the spiritof the Taylorrule,'9

(10) i, = O(PI/Pl- I YI;u ), wherenow gt may also includeexogenous disturbances in additionto the ones listed above, to which the central bank happens to respond. We assume that the function0 is nonnegativefor all values of its arguments (otherwisethe policy would not be feasible, given the zero lower bound), but thatthere are conditionsunder which the rule prescribesa zero interest ratepolicy. Such a rule impliesthat the centralbank supplies the quan- tity of base money thathappens to be demandedat the interestrate given by this formula;hence equation10 implies a path for the monetarybase, so long as the value of 4 is positive. However,under those conditionsin which the value of 4 is zero, the policy commitmentin equation 10 impliesonly a lower boundon the monetarybase thatmust be supplied.In these circumstanceswe may ask whetherit matterswhether a greateror a smallerquantity of base money is supplied.We assume that the central bank'spolicy in this regardis specifiedby a base-supplyrule of the form

(11) Mt = PtL Yt,O(PtIP, Y; t); t]v(t/t-l Y; t), wherethe multiplicativefactor Nv satisfies the following two conditions:

u(P,/P, I, Yt,)=t 1 if O(/P II,t; ) >0 , otherwise x(P,/P,- 9,t; ,) 21I. for all values of its arguments.(The second conditionimplies that Ni= 1 wheneverit > 0.) Note that a base-supplyrule of this form is consistent with boththe interestrate operating target specified in equation10 andthe equilibriumrelations in expressions 3 and 4. The use of quantitative these policies to be specifiedin a way thatprevents them from being affectedby pastopen- marketoperations. The specific classes of policy rules discussedhere show not only that ourconcept of "unchangedpolicy" is logicallypossible, but indeed that it could correspond to a policy commitmentof a fairlyfamiliar sort, one thatwould represent a commitmentto "soundpolicy" in the views of some. 19. Taylor(1993). 154 BrookingsPapers on EconomicActivity, 1:2003 easing as a policy tool can thenbe representedby a choice of a functioniy thatis greaterthan 1 undersome circumstances. It remainsto specify which sort of assets should be acquired(or dis- posed of) by the centralbank when it variesthe size of the monetarybase. We allow the asset side of the centralbank balance sheet to includeany of k different types of securities, distinguishedfrom each other by their state-contingentreturns. At the end of periodt, the vectorof nominalval- ues of centralbank holdings of the various securitiesis given by Mtew", wherecom is a vectorof centralbank portfolio shares. These sharesare in turndetermined by a policy rule of the form

m (12) (W)m= om(tJ/t-I 4t; t ) where the vector-valuedfunction cWm-(.) has the propertythat its compo- nents sum to 1 for all possible values of its arguments.The fact thatCO(.) dependson the same argumentsas 4(*)means that we allow for the possibility thatthe centralbank changes its policy whenthe zero boundis binding (for example,buying assets thatit would not hold at any othertime). The fact thatit dependson the same argumentsas x( ) allows us to specify changesin the compositionof the centralbank portfolio as a function of the particularkinds of purchasesassociated with quantitativeeasing. The payoffs on these securitiesin each state of the worldare specified by exogenouslygiven (state-contingent)vectors at andbt andmatrix Ft. A vectorof asset holdingszt-I at the end of periodt - 1 resultsin delivery,to the ownerof a quantitya'zt-I of money, a quantityb'zt of the consumption good and a vector Ftzt-I 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 specificationallows us to treat a wide rangeof types of assets that may differ as to maturity,degree of indexa- tion, and so on.20 The gross nominalreturn Rt(j) on the jth asset between periodst - 1 and t is then given by

20. For example,security j in periodt - 1 is a one-periodriskless nominal bond if b,(j) andF,( ;j) arezero in all states,while a,(j) > 0 is the samein all states.Security j is instead a one-periodreal (or indexed)bond if a,(j) andF,( ;j) are zero, while b,(j) > 0 is the same in all states.It is a two-periodriskless nominal pure discount bond if insteada,(j) andb,(j) arezero, F,(i,j) = 0 for all i ? k; F,(k,j) > 0 is the samein all states,and security k in period t is a one-periodriskless nominal bond. Gauti B. Eggertsson and Michael Woodford 155

(13) R (j) = ~~~~a,(j) + P, b, (j) + q,'F,(,,j) q,3 (J) where qt is the vector of nominal asset prices in (ex-dividend)period-t trading.The absenceof arbitrageopportunities implies as usualthat equilibriumasset pricesmust satisfy

T-l , (14) q= E + Pb"]HFs T2t+1 EQt,T[a s=t+1 where the stochasticdiscount factor is again given by equation8. Under the assumptionthat no interestis paid on the monetarybase, the nominal transferby the centralbank to the publictreasury each periodis equalto

(15) = Tcb Rt'wmjMt-Mt whereR, is the vectorof returnsdefined by equation13. We specify fiscal policy in termsof a rulethat determines the evolution of total governmentliabilities Dt, here definedto be inclusiveof the monetarybase, as well as a rule that specifiesthe compositionof outstanding nonmonetaryliabilities (debt) among different types of securitiesthat the governmentmight issue. We assumethat the pathof totalgovernment liabilities accordswith a rule of the form

(16) d('' IP t. ) which specifies the acceptablelevel of real governmentliabilities as a function of the preexistinglevel and various aspects of currentmacro- economic conditions. This notation allows for such possibilities as an exogenouslyspecified state-contingent target for real governmentliabilities as a proportionof GDP, or for the governmentbudget deficit (inclu- sive of intereston the publicdebt) as a proportionof GDP, amongothers. The partof total liabilitiesthat consists of base money is specifiedby the base rule in equation11. We suppose,however, that the rest may be allo- cated among any of a set of differenttypes of securitiesthat the govern- mentmay issue; for convenience,we assumethat this is a subsetof the set of k securities that the central bank may purchase.If ojftindicates the shareof governmentdebt (nonmonetaryliabilities) at the end of periodt 156 BrookingsPapers on EconomicActivity, 1:2003 that is of type j, then the flow governmentbudget constrainttakes the form

D, = R',*(of- R'wI B,B l_Tb_-Tcb _T whereB, - Dt - Mt is the total nominalvalue of end-of-periodnonmone- taryliabilities, and Tt is the nominalvalue of the primarybudget surplus (taxes net of transfers,if we abstractfrom governmentpurchases). This identitycan then be invertedto obtainthe net tax collections Thimplied by a given rule (equation16) for aggregatepublic liabilities; this depends in general on the composition of the public debt as well as on total borrowing. Finally,we assumethat debt managementpolicy (the determinationof the compositionof the government'snonmonetary liabilities at each point in time) is specifiedby the function

(17) c[ = x(e/e- t; gt)9 which specifies the shares as a functionof aggregateconditions, where the vector-valuedfunction cf also has componentsthat sum to 1 for all possible values of its arguments.Together the two relationsin equations 16 and 17 complete our specificationof fiscal policy and close our model.21 We may now definea rationalexpectations equilibrium as a collection of stochasticprocesses {p*, Pt, Y,, it, qt, Mt, wom,Dt, w{f , with each endogenousvariable specified as a functionof the historyof exogenous disturbancesto thatdate, that satisfy each of the conditionsin expressions 2 through6 of the aggregatedemand block of the model,the conditionsin equations7 and 9 of the aggregatesupply block, the asset-pricingrelations equation14, the conditionsin equations10 through12 specifying monetarypolicy, and the conditionsin equations16 and 17, specifying fiscal policy in each period. We then obtain the following irrelevance resultfor the specificationof certainaspects of policy:

21. We might, of course, allow for other types of fiscal decisions from which we abstracthere-government purchases,tax incentives,and so on-some of which may be quiterelevant to dealingwith a liquiditytrap. But ourconcern here is solely with the question of what monetarypolicy can achieve;we introducea minimalspecification of fiscal policy only for the sake of closing our general-equilibriummodel, and to allow discussion of the fiscal implicationsof possibleactions by the centralbank. Gauti B. Eggertsson and Michael Woodford 157

PROPOSITION.The set of pathsfor the variables{p*, Pt, Yt,it, qt, Dt1 that are consistentwith the existence of a rationalexpectations equilibriumis independentof the specificationof the functionsvf (equation11), (Om(equation 12), and cf (equation 17). The reason for this is fairly simple. The set of restrictionson the processes {p*, Pt, Y,, i,, qt, Dt} impliedby our model can be writtenin a form that does not involve the variables {Mt, com,c{f}, and hence that does not involve the functions com,i, or cf. To show this, we first note that,for all m ? m-(C;g),

u(C,m; ) = u[C,m-(C; t); t],

because additionalmoney balancesbeyond the satiationlevel provideno furtherliquidity services. By differentiatingthis relation,we see further thatuc(C, m; t) does not dependon the exact value of m either,as long as m exceeds the satiationlevel. It follows that,in our equilibriumrelations, we can replacethe expressionuc(Yt, MtlPt; tt) with

X(Y,,9P,/P,I; 9') =_-ucY,[,,(P ,Y.;9,); 9,]; 9.,1 using the fact thatexpression 3 holds with equalityat all levels of realbal- ances at which uc depends on the level of real balances.Hence we can writeuc as a functionof variablesother than Mt/Pt, withoutusing the relation in equation11, and so in a way thatis independentof the functionW. We can similarly replace the expression um(Yt,MtlPt; gt)(MtIPt)in expression5 with

U. {YL[Yt,, (Pt1PtI, Yt; gt ); gt]; gt }L[Yt,, (Pt1Pt_I,Y.;9t.); ,t],

since MtlPtmust equal L[Y,,4(Pt/Pt, Yt; t); t] when real balancesdo not exceed the satiation level, whereas um= 0 when they do. Finally, we can expressnominal profits in periodt as a function:

after substitutingX(Y, PtlPt-1;a) for the marginalutility of real income in the wage demandfunction that is used in derivingthe profitfunction 158 BrookingsPapers on EconomicActivity, 1:2003 f1.22 Using these substitutions,we can write each of the equilibriumrelations in expressions2, 5, 6, 7, and 14 in a way thatno longermakes refer- ence to the money supply. It then follows thatin a rationalexpectations equilibrium the variables {p,*,Pt, Y,, i,, qt, Dt} must satisfyin each periodthe following relations:

(18) X(1",f'/J; g,) = f3Et[X(Y,+iP,+,lP,;9t+,)(1 +i,)

(19) X3EjX(YT,PT/PT_I; 9T)YT + 9(YT, PT/PT-1; ET)] < ?? T=t

(20) lim, TE[X(YT,PT/PT; 9T)DT/PT] 0

PTT--

(21) q, =t AT_; ;T)[P77iaT + bT] F X(Yt 9 Pt/P-s gtt) T?t+1 s=t+l

(22) Et{(ap) )T-tX(YT, PT'PT-i; 9T)Pf'1l (P7*iPt*7 PT;YT9 PT/PT-, 9T)}

=0, along with equations9, 10, and 16 as before. None of these involve the variables{Mt, t7, ot}, nor do they involve the functions1f, cIm, or Wf Furthermore,this is the completeset of restrictionson these variables that are requiredin orderfor them to be consistentwith a rationalexpec- tationsequilibrium. For any given processes {p*, Pt, Y,, i,, qt, Dt} that satisfy the equationsjust listed in each period, the implied path of the money supplyis given by equation11, which clearly has a solution,and this path for the money supplynecessarily satisfies expression 3 and the complementaryslackness condition, as a resultof our assumptionsabout the form of the function i. Similarly, the implied compositions of the centralbank portfolio and of the public debt at each point in time are given by equations12 and 17. We thenhave a set of processesthat satisfy

22. See Woodford(forthcoming, chapter 3). Gauti B. Eggertsson and Michael Woodford 159

all of the requirementsfor a rationalexpectations equilibrium, and the resultis established.

Discussion The above propositionimplies thatneither the extent to which quantitativeeasing is employedwhen the zero boundbinds, nor the natureof the assets that the central bank may purchasethrough open-market operations, has any effect on whethera deflationaryprice-level path represents a rationalexpectations equilibrium. Hence ourgeneral-equilibrium analysis of inflationand outputdetermination does not supportthe notion that expansionsof the monetarybase representan additionaltool of policy, independentof the specificationof the rule for adjustingshort-term nominal interestrates. If the commitmentsof policymakersregarding the rule by which interestrates will be set, on the one hand,and the rule by which total privatesector claims on the governmentwill be allowedto grow, on the other, are fully credible,then it is only the choice of those commit- mentsthat matters. Other aspects of policy shouldmatter in practiceonly insofaras they help to signal the natureof these policy commitments. Of course, the validity of our result dependson the reasonablenessof our assumptions,and these deservefurther discussion. Like any economic model, ours abstractsfrom the complexityof actualeconomies in many respects.Have we abstractedfrom featuresof actualeconomies that are crucialfor a correctunderstanding of the issues underdiscussion? It mightbe suspectedthat an importantomission is our neglect of portfolio-balanceeffects, which play an importantrole in muchrecent discussion of the policy options that would remain available to the Federal Reserveshould the federalfunds rate reach zero.23 The idea is thata central bank should be able to lower longer-terminterest rates even when overnightrates are alreadyat zero, throughpurchases of longer-maturity governmentbonds. This would shift the compositionof the publicdebt in the handsof the public in a way that affects the termstructure of interest rates. (Because it is generally admittedin such discussions that base money and very shortterm Treasury securities have become near-perfect substitutesonce short-terminterest rates have fallen to zero, the desired effect shouldbe achievedequally well by a shift in the maturitystructure

23. See, for example,Clouse andothers (2003) andOrphanides (2003). 160 BrookingsPapers on EconomicActivity, 1:2003 of Treasurysecurities held by the centralbank, without any changein the monetarybase, as by an open-marketpurchase of long-termbonds with newly createdbase money.) No such effects arise in our model, whetherfrom centralbank securities purchasesor debt managementby the publictreasury. But this is not, as some mightexpect, because we have simplyassumed that bonds of dif- ferentmaturities (or for that matter,other kinds of assets that the central bank might choose to purchaseinstead of the shortest-maturityTreasury bills) are perfectsubstitutes. Our framework allows for centralbank purchases of differentassets having differentrisk characteristics(different state-contingentreturns), and our model of asset market equilibrium incorporatesthose termpremiums and risk premiumsthat are consistent with the absenceof arbitrageopportunities. Ourconclusion differs from that of the literatureon portfoliobalance effects for a differentreason. The classic theoreticalanalysis of portfolio balance effects assumes a representativeinvestor with mean-variance preferences.This has the implicationthat if the supplyof assets thatpay off disproportionatelyin certainstates of the world is increased(so that the extent to which the representativeinvestor's portfolio pays off in those statesmust also increase),the relativemarginal valuation of income in those particularstates is reduced,resulting in a lower relativeprice for the assets thatpay off in those states.But in our general-equilibriumasset pricingmodel, there is no such effect. The marginalutility to the representativehousehold of additionalincome in a given state of the world dependson the household'sconsumption in that state, not on the aggregate payoff of its asset portfolioin that state.And changesin the composition of the securities in the hands of the public do not change the state-contingentconsumption of the representativehousehold-this dependson equilibriumoutput, and althoughoutput is endogenous,we have shownthat the equilibriumrelations that determine it do not involve the functions A, com,or C0.24 Our assumptionof complete financialmarkets and no limits on borrowing against future income may also appearextreme. However, the

24. Ourgeneral-equilibrium analysis is in the spiritof the irrelevanceproposition for open-marketoperations of Wallace(1981). Wallace's analysisis often supposedto be of little practicalrelevance for actual monetarypolicy because his model is one in which money servesonly as a storeof value, so thatan equilibriumin which short-termTreasury securities dominatemoney in terms of rate of returnis not possible, althoughthis is Gauti B. Eggertsson and Michael Woodford 161 assumptionof completefinancial markets is only a convenience,allowing us to write the budgetconstraint of the representativehousehold in a simple way. Even in the case of incompletemarkets, each of the assets thatis tradedwill be priced accordingto equation14, where the stochasticdiscount factoris given by equation8, and once again therewill be a set of relationsto determineoutput, goods prices, and asset prices that do not involve f, cIo, or cf The absenceof borrowinglimits is also innocuous, at least in the case of a representative-householdmodel, becausein equilibriumthe representativehousehold must hold the entire net supply of financial claims on the government.As long as the fiscal rule (equation 16) implies positive governmentliabilities at each date, any borrowing limits that might be assumed can never bind in equilibrium. Borrowinglimits can mattermore in the case of a model with heteroge- neous households.But in this case the effects of open-marketoperations should depend not merely on which sorts of assets are purchasedand which sortsof liabilitiesare issued to financethose purchases,but also on how the centralbank's trading profits are eventually rebated to the private sector (thatis, with what delay and how distributedacross the heteroge- neous households),as a result of the specificationof fiscal policy. The effects will not be mechanicalconsequences of the changein composition of assets in the handsof the public,but insteadwill resultfrom the fiscal transfersto which the transactiongives rise; it is unclearhow quantita- tively significantsuch effects shouldbe. Indeed,leaving aside the questionof whethera clear theoreticalfoun- dationexists for the existence of portfoliobalance effects, there is not a great deal of empiricalsupport for quantitativelysignificant effects. The attemptto separatelytarget short-term and long-terminterest rates under OperationTwist in the early 1960s is generallyregarded as havinghad a modesteffect at best on the term structure.25The empiricalliterature that has sought to estimate the effects of changes in the compositionof the publicdebt on relativeyields has also, on the whole, foundeffects thatare

routinelyobserved. However, in the case of open-marketoperations conducted at the zero bound,the liquidityservices provided by moneybalances at the marginhave fallen to zero, so thatan analysisof the kindproposed by Wallaceis correct. 25. Okun(1963) andModigliani and Sutch(1966) are importantearly discussions that reachedthis conclusion.Meulendyke (1998) summarizesthe literatureand finds that the predominantview is thatthe effect was minimal. 162 BrookingsPapers on EconomicActivity, 1:2003 not large when presentat all.26For example,Jonas Agell and Mats Pers- son summarizetheir findingsas follows: "Itturned out that these effects were rathersmall in magnitude,and that their numericalvalues were highly volatile. Thus the policy conclusionto be drawnseems to be that thereis not muchscope for a debt managementpolicy aimedat systematically affectingasset yields."27Moreover, even if one supposesthat large enoughchanges in the compositionof the portfolioof securitiesleft in the hands of the privatesector can substantiallyaffect yields, it is not clear how relevantsuch an effect shouldbe for realactivity and the evolutionof goods prices. For example, James Clouse and others argue that a sufficiently largereduction in the numberof long-termTreasuries in the hands of the public shouldlower the marketyield on those securitiesrelative to short-termrates, because certain institutions will find it importantto hold long-term Treasury securities even when they offer an unfavorable yield.28But even if this is true, the fact that these institutionshave idio- syncraticreasons to hold long-termTreasuries-and that,in equilibrium, no one else holds any or plays any role in pricingthem-means that the lower observedyield on long-termTreasuries may not correspondto any reductionin the perceivedcost of long-termborrowing for otherinstitu- tions. If one is able to reducethe long-termbond rate only by decoupling it from the rest of the structureof interestrates, and from the cost of financinglong-term investment projects, it is unclearthat such a reduction should do much to stimulateeconomic activity or to halt deflationary pressures. Hence we are not inclinedto supposethat our irrelevanceproposition representsso poor an approximationto realityas to depriveit of practical relevance.Even if the effects of open-marketoperations under the conditions the propositiondescribes are not exactly zero, it seems unlikelythat they should be large. In our view it is more importantto note that our

26. Examplesof studies findingeither no effects or only quantitativelyunimportant ones includeModigliani and Sutch (1967), Frankel(1985), Agell andPersson (1992), Wal- lace and Warner(1996), and Hess (1999). Roley (1982) and Friedman(1992) find some- whatlarger effects. 27. Agell andPersson (1992, p. 78). 28. Clouse and others (2003). StephenG. Cecchetti("Central Banks Have Plenty of Ammunition,"Financial Times, March17, 2003, p. 13) similarlyargues that it shouldbe possiblefor the FederalReserve to independentlyaffect long-term bond yields if it is deter- mined to do so, given that it can printmoney withoutlimit to buy additionallong-term Treasuriesif necessary. Gauti B. Eggertsson and Michael Woodford 163 irrelevanceproposition depends on an assumptionthat interest rate policy is specifiedin a way that implies that these open-marketoperations have no consequencesfor interest rate policy, either immediately(which is trivial,because it would not be possible for them to lower currentinterest rates,which is the only effect thatwould be desired),or at any subsequent date. We have also specifiedfiscal policy in a way that implies that the contemplatedopen-market operations have no effect on the path of total governmentliabilities {D,} either, whetherimmediately or at any later date. Althoughwe thinkthese definitionsmake sense, as a way of isolat- ing the pure effects of open-marketpurchases of assets by the central bank from either interestrate policy on the one hand or fiscal policy on the other,those who recommendmonetary expansion by the centralbank may intend for this to have consequencesof one or both of these other sorts. For example,when it is arguedthat a "helicopterdrop" of money into the economy would surely stimulate nominal aggregate demand, the thoughtexperiment that is usuallycontemplated is not simply a changein the functionv in our policy rule equation 11. First of all, it is typically supposedthat the expansionof the money supply will be permanent.If this is the case, then the function0 thatdefines interest rate policy is also being changed,in a way that will become relevantat some futuredate, when the money supply no longer exceeds the satiationlevel.29 Second, the assumptionthat the money supply is increasedthrough a helicopter droprather than an open-marketoperation implies a changein fiscal policy as well. Such an operationwould increasethe value of nominalgov- emnmentliabilities, and it is generallyat least tacitly assumedthat this is a permanentincrease as well. Hence the experimentthat is imaginedis not one that our irrelevanceproposition implies shouldhave no effect on the equilibriumpath of prices.

29. This explainsthe apparentdifference between our result and that obtained by Auer- bach and Obstfeld (2003) in a similar model. These authorsassume explicitly that an increasein the money supplyat a time when the zero boundbinds carries with it the impli- cationof a permanentlylarger money supply, and that there exists a futuredate at whichthe zero boundceases to bind, so thatthe largermoney supplywill imply a differentinterest ratepolicy at that laterdate. Clouse and others(2003) also stressthat maintenanceof the largermoney supplyuntil a date at which the zero boundwould not otherwisebind represents one straightforwardchannel through which open-marketoperations while the zero boundis bindingcould have a stimulativeeffect, althoughthey discussother possible channels as well. 164 BrookingsPapers on EconomicActivity, 1:2003 Even more important,our irrelevanceresult applies only given a correct private sector understandingof the central bank's commitments regardingfuture policy. Suchunderstanding may be lacking.We havejust arguedthat the key to lowering long-terminterest rates, in a way that actuallyprovides an incentivefor increasedspending, is to changeexpectations regardingthe likely future path of short-termrates, ratherthan throughintervention in the marketfor long-termTreasuries. As a matter of logic, this need not requireany open-marketpurchases of long-term Treasuriesat all. Nonetheless,the privatesector may be uncertainabout the natureof the centralbank's policy commitment,and so it may scruti- nize the bank'scurrent actions for furtherclues. In practice,the manage- ment of privatesector expectations is an artof considerablesubtlety, and shifts in the portfolioof the centralbank could be of some value in making credibleto the privatesector the centralbank's own commitmentto a particularkind of futurepolicy, as we discuss furtherin the penultimate section of the paper.Signaling effects of this kind are often arguedto be an importantreason for the effectiveness of interventionsin foreign- exchangemarkets, and they might well providea justificationfor open- marketoperations when the zero boundbinds.30 We do not wish, then,to arguethat asset purchasesby the centralbank are necessarilypointless under the circumstancesof a bindingzero lower bound on short-termnominal interest rates. However, we do think it importantto observethat, insofar as such actionscan have any effect, it is not because of any necessaryor mechanicalconsequence of the shift in the portfolioof assets in the handsof the privatesector itself. Instead,any effect of such actionsmust be due to the way in whichthey changeexpec- tationsregarding future interest rate policy or, perhaps,the futurepath of total nominalgovernment liabilities. Later we discuss reasonswhy open- marketpurchases by the centralbank might plausibly have consequences for expectationsof these types. But because it is only througheffects on expectationsregarding future policy that these actions can matter,we focus our attentionon the questionof whatkind of commitmentsregard- ing future policy are in fact to be desired. And this question can be addressedwithout explicit considerationof the role of centralbank open- market operationsof any kind. Hence we will simplify our model-

30. Clouse and others (2003) arguethat this is one importantchannel through which open-marketoperations can be effective. Gauti B. Eggertsson and Michael Woodford 165

abstractingfrom monetaryfrictions and the structureof governmentliabilities altogether-and insteadconsider what is the desirableconduct of interestrate policy, and what kind of commitmentsabout this policy are desirableto makein advance.

How Severe a Constraint Is the Zero Bound?

We turnnow to the questionof how the existence of the zero bound restrictsthe degreeto which a centralbank's stabilization objectives, with regardto both inflationand real activity, can be achieved, even under ideal policy. The discussionin the previous section establishedthat the zero bounddoes representa genuineconstraint. It is not truethat equilib- ria that cannot be achieved througha suitable interest rate policy can somehowbe achievedthrough other means, and the zero bounddoes limit the set of possible equilibriumpaths for prices and output,although the quantitativeimportance of this constraintremains to be seen. Nonetheless,we will see thatit is not at all the case thata centralbank can do nothingto mitigatethe severityof the destabilizingimpact of the zero bound.The reasonis that inflationand outputdo not dependsolely on the currentlevel of short-termnominal interest rates, or even solely on the historyof such rates up until the present(so that the currentlevel of interest rates would be the only thing that could possibly change in response to an unanticipateddisturbance). The expected characterof future interestrate policy is also a critical determinantof the degree to which the central bank achieves its stabilizationobjectives, and this allows importantscope for policy to be improvedupon, even when there is little choice aboutthe currentlevel of short-terminterest rates. In fact, the managementof expectationsis the key to successfulmone- tarypolicy at all times, not just in those relativelyunusual circumstances when the zero boundis reached.The effectivenessof monetarypolicy has little to do with the directeffect of changingthe level of overnightinterest rates, since the currentcost of maintainingcash balancesovernight is of fairly trivialsignificance for most businessdecisions. What actually matters is the private sector's anticipationof the future path of short-term rates,because this determinesequilibrium long-term interest rates as well as equilibriumexchange rates and other asset prices-all of which are quiterelevant for manycurrent spending decisions, and hence for optimal 166 BrookingsPapers on EconomicActivity, 1:2003 pricing behavior as well. How short-termrates are managed matters becauseof the signals that such managementgives abouthow the private sectorcan expectthem to be managedin the future.But thereis no reason to supposethat expectations regarding future monetary policy, and hence regardingthe futurepaths of nominal variablesmore generally, should change only insofar as the current level of overnight interest rates changes. A situationin which there is no decision to be made aboutthe currentlevel of overnightrates (as in Japanat present)is one that gives urgencyto the questionof what expectationsregarding future policy one should wish to create, but this is in fact the correctway to think about soundmonetary policy at all times. Of course,the questionof whatfuture policy one shouldwish peopleto expect does not ariseif currentconstraints leave no possibilityof commit- ting oneself to a differentsort of policy in the futurethan one would otherwise have pursued. This means that the private sector must be convincedthat the centralbank will not conductpolicy in a way that is purelyforward looking,that is, takingaccount 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 undesirablefor the central bankto pursuea given inflationtarget, once the zerobound is expectedno longerto preventthat target from being achieved,even in the case thatthe pursuitof this targetwould be optimalif the zero bounddid not exist (or wouldnever bind under an optimalpolicy). The reasonis thatan expectation that the centralbank will pursuethe fixed inflationtarget after the zero bound ceases to bind gives people no reason to hold the kind of expectations,while the boundis binding,that would mitigatethe distortions createdby it. A history-dependentinflation target3"-if the central bank'scommitment to it can be madecredible-can insteadyield a supe- rioroutcome. But this, too, is an importantfeature of optimalpolicy rules more generally.32Hence the analyticalframework and institutionalarrangements used in makingmonetary policy need not be changedin any fundamental way in orderto deal with the specialproblems created by a liquiditytrap. As we explainlater in the paper,the optimalpolicy in the case of a binding zero bound can be implementedthrough a targetingprocedure that

31. As we will show, it is easierto explainthe natureof the optimalcommitment if it is describedas a history-dependentprice-level target. 32. See, for example,Woodford (forthcoming, chapter 7). Gauti B. Eggertsson and Michael Woodford 167 representsa straightforwardgeneralization of a policy thatwould be optimal even if the zero boundwere expectednever to bind.

Feasible Responses to Fluctuations in the Natural Rate of Interest In orderto characterizehow stabilizationpolicy is constrainedby the zero bound, we make use of a log-linearapproximation to the structural equations presented in the previous section, of a kind that is often employed in the literatureon optimal monetary stabilizationpolicy.33 Specifically, we log-linearize the structuralequations of our model (except for the zero boundin expression4) aroundthe pathsof inflation, output,and interestrates associatedwith a zero-inflationsteady state, in the absence of disturbances(g, = 0). We choose to expandaround these particularpaths because the zero-inflationsteady state represents optimal policy in the absenceof disturbances.34In the event of small enoughdis- turbances,optimal policy will still involve paths in which inflation,output, and interestrates are at all times close to those of the zero-inflation steadystate. Hence an approximationto ourequilibrium conditions that is accuratein the case of inflation,output, and interestrates near those values will allow an accurateapproximation to the optimalresponses to disturbancesin the case thatthe disturbancesare small enough. In the zero-inflationsteady state, it is easily seen that the real rate of interestis equal to r-_ -1 - 1 > 0; this is also the steady-statenominal interestrate. Hence, in the case of small enough disturbances,optimal policy will involve a nominalinterest rate that is always positive, and the zero boundwill not be a bindingconstraint. (Optimal policy in this case is characterizedin the referencescited in the previousparagraph.) However, we are interestedin the case in which disturbancesare at least occasion- ally largeenough for the zero boundto bind, thatis, to preventattainment of the outcomethat would be optimalin the absenceof such a bound.It is

33. See, for example,Clarida, Galf, and Gertler(1999); Woodford(forthcoming). 34. See Woodford(forthcoming, chapter 7) for moredetailed discussion of this point. The fact thatzero inflation,rather than mild deflation,is optimaldepends on ourabstracting fromtransactions frictions, as discussedfurther in footnote40 below. As Woodfordshows, a long-runinflation target of zero is optimalin this model, even when the steady-stateout- put level associatedwith zero inflationis suboptimal,owing to marketpower. The reasonis thata commitmentto inflationin some periodt resultsboth in increasedoutput in periodt andin reducedoutput in periodt - 1 (owing to the effect of expectedinflation on the aggregate supplyrelation, equation 25 below);because of discounting,the secondeffect on welfare fully offsets the benefitof the firsteffect. 168 BrookingsPapers on EconomicActivity, 1:2003 possible to consider this problem rigorously using only a log-linear approximationto the structuralequations in the case where the lower bound on nominalinterest is assumedto be not much below r. We can arrangefor this gap to be as small as we may wish, without changing other crucial parametersof the model such as the assumedrate of time preference,by supposingthat interestis paid on the monetarybase at a rateitm > 0 thatcannot (for some institutionalreason) be reduced.Then the lower boundon interestrates actually becomes

(23) it 2i?.

We will characterizeoptimal policy subjectto a constraintof the form of expression23, in the case that both a bound on the amplitudeof disturbances Il1l I and the steady-stateopportunity cost of holding money 6 = (r - im)/(1 + r) > 0 are small enough. Specifically, both our structural equationsand our characterizationof the optimalresponses of inflation, output,and interestrates to disturbanceswill be requiredto be exact only up to a residual of order 0(1 It; 6 112): We then hope (without here seeking to verify) that our characterizationof optimal policy in the case of a small opportunitycost of holding money and small disturbancesis not too inaccuratein the case of an opportunitycost of several percentage points (the case in which im= 0) and disturbanceslarge enough to cause the naturalrate of interestto varyby severalpercentage points (as will be requiredin orderfor the zero boundto bind). As Woodfordhas shown elsewhere,35the log-linearapproximate equilibriumrelations may be summarizedby two equationseach period: a forward-looking"IS relation"

(24) xt = Etxt+, - (i - Ett+, - rtn), and a forward-looking"AS relation"(or New KeynesianPhillips curve)

(25) TEt= Kxt + PEtTct+l+ ut.

Here it - log(P,/P,1) is the inflationrate, x, is a welfare-relevantoutput gap, and it is now the continuouslycompounded nominal interestrate, correspondingto log (1 + it) in the notationused in the previoussection.

35. Woodford(forthcoming). Gauti B. Eggertsson and Michael Woodford 169

The termsu, and rn are compositeexogenous disturbance terms that shift the two equations;the formeris commonlyreferred to as a cost-pushdis- turbance,whereas the latterindicates exogenous variation in the Wicksel- lian naturalrate of interest,that is, the equilibriumreal rate of interestin the case that outputgrowth is at all times equal to its naturalrate. The coefficientsco and K areboth positive, and 0 < ,3< 1 is againthe utilitydis- countfactor of the representativehousehold. Equation 24 is a log-linear approximationto equation 2, whereas equation25 is derivedby log-linearizingequations 7 through9 and then eliminating log (p*/PJ). We omit the log-linear version of the money demandrelation in expression3, becausehere we are interestedsolely in characterizingthe possible equilibriumpaths of inflation, output, and interestrates, and we may abstractfrom the question of what might be the requiredpath for the monetarybase that is associatedwith any such equilibrium.(It suffices that thereexist a monetarybase that will satisfy the money demandrelation in each case, and this will be true as long as the interestrate bound is satisfied.) The other equilibriumrequirements of the earlierdiscussion can be ignoredin the case thatwe are interested only in possible equilibriathat remain forever near the zero-inflation steady state, because they are automaticallysatisfied in that case. Equa- tions 24 and 25 representa pair of equationseach period to determine inflationand the outputgap, given the centralbank's interestrate policy. We will seek to comparealternative possible pathsfor inflation,the output gap, and the nominal interest rate that satisfy these two log-linear equationstogether with expression23. Note that our conclusionswill be identical(up to a scale factor)in the event thatwe multiplythe amplitude of the disturbancesand the steady-stateopportunity cost 6 by any common factor;alternatively, if we measurethe amplitudeof disturbancesin units of 6, our resultswill be independentof the value of 6 (to the extent that our log-linear approximationremains valid). Hence we choose the normalization6 = 1 - f, correspondingto im= 0, to simplify the presen- tation.In that case the lower boundfor the nominalinterest rate is again given by expression4.

Deflation under Forward-Looking Policy We begin by consideringthe degreeto which the zero boundimpedes the achievementof the centralbank's stabilizationobjectives in the case 170 BrookingsPapers on EconomicActivity, 1:2003 that the bankpursues a strictinflation target. We interpretthis as a commitmentto adjustthe nominalinterest rate so that

(26) t each period,insofar as it is possible to achieve this with some nonnega- tive interestrate. It is easy to verify, by the IS and AS equationsabove, thata necessarycondition for this targetto be satisfiedis

(27)(27) it=rn+t*i, ~~~~~=rtn +X*.

Wheninflation is on target,the real interestrate is equalto the naturalreal rate at all times, and the output gap is at its long-runlevel. The zero bound,however, prevents equation 27 from holdingif rn < -t*. Thus, if the naturalrate of interestis low, the zero bound frustratesthe central bank's ability to implementan inflationtarget. Suppose the inflationtarget is zero, so that n* = 0. Then the zero boundis bindingif the natural rate of interestis negative, and the centralbank is unableto achieve its inflationtarget. To illustratethis, considerthe following experiment.Suppose the naturalrate of interestis unexpectedlynegative in period0 and revertsback to its steady-statevalue r > 0 with a fixed probabilityin every period.Fig- ure 2 shows the state-contingentpaths of the output gap and inflation under these circumstancesfor each of three differentpossible inflation targetsn*. We assumein period0 thatthe naturalrate of interestbecomes -2 percenta yearand then reverts back to the steady-statevalue of +4 percent a year with a probabilityof 0.1 each quarter.Thus the naturalrate of interestis expectedto be negativefor ten quarterson averageat the time the shock occurs. The dashedlines in figure2 show the state-contingentpaths of the output gap and inflationif the centralbank targetszero inflation.36Starting

36. In our numericalanalysis, we interpretperiods as quarters,and we assumecoeffi- cient values of cx= 0.5, K = 0.02, and D = 0.99. The assumedvalue of the discountfactor impliesa long-runreal rate of interestr equalto 4 percenta year.The assumedvalue of K is consistentwith the empiricalestimate of Rotembergand Woodford(1997). The assumed value of a representsa relativelylow degree of interestsensitivity of aggregateexpendi- ture.We preferto bias our assumptionsin the directionof only a modesteffect of interest rateson the timingof expenditure,so as not to exaggeratethe size of the outputcontraction that is predictedto result from an inabilityto lower interestrates when the zero bound Gauti B. Eggertsson and Michael Woodford 171

Figure 2. State-ContingentResponses of Inflationand the Output Gap to a Shock to the Natural Rate of Interest under Strict InflationTargetinga

Inflation Percenta year

R*=2%

?0-

- -5 I I I I I I I I I I I I I I I I I Z*=1% II I I I I I I IIIIIIIIIIII

-10 7z*= LLLLLLLLLLLLLLLLLLLI

Output gap Percentof GDP

-5-

-10 I I I I I I I I I I I I I I I I I I I l I I I I I I I I I I I I I I I I I L I llI I I I I I I I I I I I I I I I I I I zI I I I I I I I I I I I I I I I I I I I LLLLLLLLLLLLLLLLLLLI 0 5 10 15 20 Quartersafter shock to naturalrate of interest

Source: Authors' calculations. a. The targetedrate of inflationis designatedby ir*. Each upward-slopingline representsthe response of inflationor the output gap if the naturalrate of interestreturns to its steady-statevalue in that period. 172 BrookingsPapers on EconomicActivity, 1:2003 from the left, the first dashedline shows the equilibriumthat prevailsif the naturalrate of interestreturns to the steadystate in period 1, the next line if it returnsin period2, and so on. The inabilityof the centralbank to set a negativenominal interest rate results in a 14 percentoutput gap and 10 percentannual deflation. The fact thatin each quarterthere is a 90 percent chanceof the naturalrate of interestremaining negative for the next quartercreates the expectationof futuredeflation and a continuednega- tive outputgap, which createseven furtherdeflation. Even if the central bank lowers the short-termnominal interest rate to zero, the real rate of returnis positive,because the privatesector expects deflation. The shadedlines in figure2 show the equilibriumthat prevailsif the centralbank insteadsets a 1 percentannual inflation target. In this case the privatesector expects 1 percentinflation once the economy is out of the liquiditytrap. This, however, is not enough to offset the -2 percent naturalrate of interest,so that in equilibriumthe privatesector expects deflationinstead of inflation.The resultof this anda negativenatural rate of interestis 4 percentannual deflation (when the naturalrate of interestis negative)and an outputgap of 7 percent. Finally, the solid horizontalline shows the evolution of output and inflationin the case wherethe centralbank targets 2 percentannual inflation. In this case the centralbank can satisfyequation 4 even whenthe natural rate of interest is negative. When the naturalrate of interest is -2 percent,the centralbank lowers the nominalinterest rate to zero. Since the inflationtarget is 2 percent,the realrate is -2 percent,which is enough to close the outputgap andkeep inflationon target.If the inflationtarget is high enough,therefore, the centralbank is able to accommodatea negative naturalrate of interest.This is the argumentgiven by EdmundPhelps, LawrenceSummers, and StanleyFischer for a positive inflationtarget.37 Krugmanmakes a similarargument and suggestsmore concretely that, in 1998, Japanneeded a positive inflationtarget of 4 percentunder its then- currentcircumstances to achievenegative real rates and curb deflation.38 Althoughit is clear that commitmentto a higher inflationtarget will indeedguard against the need for an outputgap in periodswhen the nat- binds.As figure2 shows, even for this valueof a, the outputcontraction that results from a slightlynegative value of the naturalrate of interestis quite substantial. 37. Phelps(1972); Summers(1991); Fischer(1996). 38. Krugman(1998). Gauti B. Eggertsson and Michael Woodford 173 uralrate of interestfalls, the priceof this solutionis the distortionscreated by the inflation,both when the naturalrate of interestis negative and undermore normal circumstances as well. Hencethe optimalinflation target (fromamong the strictinflation targeting policies just considered)will be some value thatis at least slightlypositive, in orderto mitigatethe distortionscreated by the zero boundwhen the naturalrate of interestis negative, but not so high as to keep the zero boundfrom ever binding(see the tablein the next section).An intermediateinflation target, in contrast(like the 1 percenttarget considered in the figure),leads to a substantialreces- sion when the naturalrate of interestbecomes negative, and chronic inflation at all othertimes. Hence no suchpolicy allows a completesolution of the problemposed by the zero boundin the case that the naturalrate of interestis sometimesnegative. Nor can one do betterthrough commitment to any policy rule that is purelyforward looking in the sense discussedelsewhere by Woodford.39 A purely forward-lookingpolicy is one underwhich the centralbank's actionat any time dependsonly on an evaluationof the possible pathsfor the centralbank's target variables (here, inflation and the outputgap) that are possible from the currentdate forward,neglecting past conditions exceptinsofar as they constrainthe economy's possiblefuture path. In the log-linearmodel presentedabove, the possible pathsfor inflationand the outputgap from period t onwarddepend only on the expectedevolution of the naturalrate of interestfrom period t onward.If one assumesa Mar- kovian process for the naturalrate, as in the numericalanalysis above, thenany purelyforward-looking policy will resultin an inflationrate, output gap, and nominalinterest rate in periodt thatdepend only on the natural rate in period t-in our numericalexample, on whetherthe natural rate is still negative or has alreadyreturned to its long-runsteady-state value. It is easily shown in the case of our two-state example that the optimal state-contingentpath for inflationand outputfrom among those with this propertywill be one in which the zero boundbinds if and only if the naturalrate is in the low state;hence it will correspondto a strict inflationtarget of the kindjust considered,for some r * betweenzero and 2 percent. But one can actuallydo considerablybetter, through commitment to a history-dependentpolicy, in which the centralbank's actionswill depend

39. Woodford (2000). 174 BrookingsPapers on EconomicActivity, 1:2003 on past conditionseven thoughthese areirrelevant to the degreeto which its stabilizationgoals could in principlebe achievedfrom then on. In the next sectionwe characterizethe optimalform of history-dependentpolicy and determinethe degree to which it improvesupon the stabilizationof both outputand inflation.

The Optimal Policy Commitment We now characterizeoptimal monetary policy, by optimizingover the set of all possible state-contingentpaths for inflation, output, and the short-termnominal interest rate consistentwith the log-linearizedstruc- tural relationsin equations24 and 25. It is assumed(for now) that the expectationsregarding future state-contingent policy thatare requiredfor such an equilibriumcan be madecredible to the privatesector. In consideringthe centralbank's optimization problem under the assumptionthat a crediblecommitment is possible regardingfuture policy, we do not mean to minimize the subtlety of the task of actually communicatingsuch a commitmentto the public and making it credible.However, we do not believe it makes sense to recommenda policy that would systematically seek to achieve an outcome other than a rationalexpectations equilibrium.That is, we areinterested in policies thatwill have the desiredeffect even when correctlyunderstood by the public. Optimizationunder the assumptionof crediblecommitment is simply a way of findingthe best possible rationalexpectations equilibrium. Once the equilibriumthat one would like to bringabout has been identified,along with the interestrate policy that it requires,one can turnto the questionof how best to signal these intentionsto the public (an issue that we briefly address in the paper'spenultimate section). We assumethat the governmentminimizes

(28) minEo{Pt(x,2 + C t=o

This loss functioncan be derivedby a second-orderTaylor expansion of the utility of the representativehousehold.40 The optimalprogram can be

40. See Woodford(forthcoming, chapter 6) for details.This approximationapplies in the case thatwe abstractfrom monetary frictions as assumedin this section.If transactions frictions are instead nonnegligible,the loss function should include an additionalterm proportionalto (it- i_)2. This would indicatewelfare gains from keepingnominal interest Gauti B. Eggertsson and Michael Woodford 175 found by a Lagrangianmethod, extending the methodsused by Richard Clarida,Jordi Gali, and Mark Gertlerand by Woodfordto the case in which the zero bound can sometimesbind, as shown by TaehunJung, Yuki Teranishi,and TsutomuWatanabe.4" We combine the zero bound and the IS equationto yield the following inequality:

X, ?E,x,+ + (rtn +ER,i+1).

The Lagrangianfor this problemis then

"co= Eo P3t t=O

2 [IC + tX + (pit [X,-,+-+ tn ] + 92t [Tt-X +]}

Jung,Teranishi, and Watanabeshow thatthe first-orderconditions for an optimalpolicy commitmentare

(29) z. + (P2t- (P2t-1 - -'(P1t-1 = 0

(30) Xx, + (plt -1 Plt- -Kp2t = 0

(31) (Plt 0, it 20, (pit =0.

One cannot solve this system by applyingstandard solution methods for rationalexpectations models, becauseof the complicationsof the nonlin- ear constraintin equation 31. The appendix describes the numerical rates as close as possible to the zero bound (or, more generally, the lower bound i-). Nonetheless,because of the stickinessof prices, it would not be optimalfor interestrates to be at zero at all times, as impliedby the flexible-pricemodel discussedby Uhlig (2000). The optimalinflation rate in the absenceof shocks would be slightly negative,rather than zero as in the "cashless"model consideredin this section;but it would not be so low that the zero boundwould be reached,except in the event of temporarydeclines in the natural rateof interest,as in the analysishere. Note also thatequation 28 implies thatthe optimal outputgap is zero. Moregenerally, there should be an outputgap stabilizationobjective of the form (x, - x*)2;the utility-basedloss functioninvolves x* = 0 only if one assumesthe existence of an outputor employmentsubsidy that offsets the distortiondue to the market powerof firms.However, the value of x* affects neitherthe optimalstate-contingent paths derivedin this section, and shown in figures 3 and4, nor the formulasgiven in the earlier section for the optimaltargeting rule. 41. Clarida,Gali, and Gertler(1999); Woodford(1999; forthcoming,chapter 7); Jung, Teranishi,and Watanabe (2001). 176 BrookingsPapers on EconomicActivity, 1:2003 method we use to solve these equationsinstead.42 Here we discuss the resultsthat we obtainfor the particularnumerical experiment considered in the previoussection. Whatis apparentfrom the first-orderconditions is that optimalpolicy 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 endogenousvariables. This can be seen by the appearanceof a lagged value of the Lagrangemultipliers in the first-orderconditions. To get a sense of how this history dependencematters, it is useful to consider again the numericalexample shown in figure 2. Supposethe naturalrate of interestbecomes negative in period 0 and then revertsto the steady state with a fixed probabilityin each period.Figure 3 shows the optimal output gap, the inflation rate, and the price level from period 0 to period25. As in figure 2, the separatelines in each panel show the evolution of the variablesin the case that the disturbanceslast for different lengthsof time rangingfrom one quarterto twentyquarters. One observesthat the optimalpolicy involves committingto the creationof an outputboom once the naturalrate again becomes positive, and hence to the creationof futureinflation. Such a commitmentstimulates aggregatedemand and reducesdeflationary pressure while the economy remainsin the liquiditytrap, through each of severalchannels. As Krugmanpoints out,43creating the expectationof futureinflation can lower real interestrates, even when nominalinterest rates cannotbe reduced. In the context of Krugman'smodel, it might seem that this requiresthat inflation be promisedquite quickly (that is, by the following "period").Our fully intertemporalmodel shows how even the expectation of laterinflation-which nominalinterest rates are not expectedto rise to offset-can stimulate current demand, because in our model current spending decisions depend on real interestrate expectationsfar in the future.For the same reason, the expectationthat nominal interestrates will be kept low later,when the centralbank might otherwise have raised them, will also stimulatespending while the zero bound still binds. And

42. Jung,Teranishi, and Watanabe(2001) discussthe solutionof these equationsonly for the case in which the numberof periodsfor which the naturalrate of interestwill be negativeis knownwith certaintyat the time thatthe disturbanceoccurs. Here we show how the systemcan be solved in the case of a stochasticprocess for the naturalrate of a particularkind. 43. Krugman(1998). Figure 3. Inflation,the Output Gap, and Prices under the Optimal Policy Commitmenta

Inflation Percent a year

0.3

0.2-

0.1 I

0.0

-0.1I

Output gap Percent of GDP

Price level Index, quarter -1 =100

100.2j

100.0

0 5 10 15 20 Quarters after shock to natural rate of interest

Source: Authors' calculations. a. See figure 2 for explanationof diagram. 178 BrookingsPapers on EconomicActivity, 1:2003 finally, the expectationof higherfuture income should stimulatecurrent spending,in accordancewith the permanentincome hypothesis.In addi- tion, pricesare less likely to fall, even given the currentlevel of realactivity, to the extent that future inflation is expected. This reduces the distortionscreated by deflationitself. On the otherhand, these gains from the change in expectationswhile the economyis in the liquiditytrap can be achieved(given rationalexpectations on the part of the private sector) only if the central bank is expected to actuallypursue the inflationarypolicy after the naturalrate returnsto its normallevel. This will in turncreate distortions at thatlater time, and this limits the extentto which this tool is used underan optimal policy. Hence some contractionof outputand some deflationoccur during the time that the naturalrate is negative, even underthe optimalpolicy commitment. Also, andthis is a key point,although the optimalpolicy involves commitmentto a higherprice level in the future,the pricelevel will ultimately be stabilized.This is in sharpcontrast to a constantpositive-inflation target, whichwould imply an ever-increasingprice level. Figure4 shows the correspondingstate-contingent nominal interest rate under the optimal commitmentand contrastsit with the evolution of the nominal interest rateunder a zero-inflationtarget. To increaseinflation expectations in the trap,the centralbank commits to keepingthe nominalinterest rate at zero afterthe naturalrate of interestbecomes positive again.In contrast,if the centralbank targetszero inflation,it raises the nominal interestrate as soon as the naturalrate of interestbecomes positive again. The optimal commitmentis an exampleof history-dependentpolicy, in which the central bank commitsitself to raise interestrates slowly at the time the natural rate becomes positive in orderto affect expectationswhen the zero boundis binding. The natureof the additionalhistory dependence of the optimalpolicy may perhapsbe moreeasily seen if we considerthe pathsof inflation,output, and interestrates undera single possible realizationof the random fundamentals.Figure 5 comparesthe equilibriumpaths of all three variables,both under the zero-inflationtarget and under optimal policy, in the case wherethe naturalrate of interestis negativefor fifteenquarters (t = 0 through14), but whereit is not knownuntil quarter 15 thatthe naturalrate will returnto its normallevel in that quarter.Under the optimalpolicy the nominal interestrate is kept at zero for five more quarters(t = 15 Gauti B. Eggertsson and Michael Woodford 179

Figure 4. Response of the Nominal Interest Rate under a Zero-InflationTarget and under the Optimal Policy Commitment

Percenta year

n=O Optimal 4- r=?r

2 -

0 5 10 15 20 Quartersafter shock to naturalrate of interest

Source: Authors' calculations.

through19), whereasit immediatelyreturns to its long-runsteady-state level in quarter15 underthe forward-lookingpolicy. The consequenceof the public anticipatingpolicy of this kind is that both the contractionof real activity and the deflationthat occur underthe strictinflation target arelargely avoided,as shownin the secondand third panels of the figure.

Implementing Optimal Policy

We turn now to the question of how policy should be conductedin orderto bringabout the optimalequilibrium characterized in the previous section. The question of the implementationof optimal policy remains nontrivial,even after the optimal state-contingentpaths of all variables have been identified,because in generalthe solutionobtained for the optimal state-contingentpath of the policy instrument(the short-termnominal interestrate) does not in itself representa useful descriptionof a policy Figure 5. Response of the Nominal Interest Rate, Inflation,and the Output Gap to a Shock of SpecificDurationa

Interest rate Percenta year *= 0 Optimal 5 4- 3- 2-

Inflation Percenta year

0 - -2

-4

-6 -8

-10

Output gap Percentof GDP

5 -

0 5 10 15 20 Source: Authors' calculations. a. Response to a fall in the naturalrate of interestbelow zero for a period of fifteen quarters. Gauti B. Eggertsson and Michael Woodford 181 rule.' For example,in the contextof the presentmodel, a commitmentto a state-contingentnominal interestrate path, even when fully credible, does not imply determinaterational expectations equilibrium paths for inflationand output; it is insteadnecessary for the centralbank to be committed (and be understoodto be committed) to a particularway of respondingto deviationsof inflationand the outputgap fromtheir desired paths.Another problem is thata completedescription of the optimalstate- contingentinterest rate path is unlikelyto be feasible.In the previoussection we showed that one can characterize(at least numerically)the optimalstate-contingent interest rate path in the case of one very particular kindof stochasticprocess for the naturalrate of interest.But a solution of this kind, allowing for all possible states of belief aboutthe probabili- ties of various futurepaths of the naturalrate (and disturbancesto the aggregatesupply relationas well), would be difficultto write down, let alone explainto the public. Here we show that optimal policy can nonethelessbe implemented throughcommitment to a policy rule that specifies the central bank's short-runtargets at each pointin time as a (fairlysimple) function of what has occurredbefore that date. How can this be done?One may be tempted to believe thatour suggestedpolicy is not entirelyrealistic or operational. Figures3 and 4, for example,indicated that the optimalpolicy involves a complicatedstate-contingent plan for the nominal interest rate, which would be hardto communicateto the public. Furthermore,it may appear thatit dependson knowledgeof a special statisticalprocess for the natural rateof interest,which is in practicehard to estimate.Our discussion of the fixed inflationtarget suggests that the effectivenessof increasinginflation expectationsto close the outputgap dependson the differencebetween the announcedinflation target and the naturalrate of interest. It may thereforeseem crucial to estimatethe naturalrate of interestin orderto implementthe optimalpolicy. Below, however, we presentthe striking result that the optimalpolicy rule can be implementedwithout any esti- mateor knowledgeof the statisticalprocess for the naturalrate of interest. This is an exampleof a robustlyoptimal direct policy rule of the kind discussed by MarcGiannoni and Woodford for the case of a generalclass of linear-quadraticpolicy problems.45An interestingfeature of the present

44. For furtherdiscussion in a more general context, see Woodford(forthcoming, chapter7). 45. Giannoniand Woodford (2003). 182 BrookingsPapers on EconomicActivity, 1:2003 exampleis that we show how to constructa robustlyoptimal rule in the same spirit,in a case where not all of the relevantconstraints 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 observethe price level and the outputgap. The rule suggestedreplicates exactly the historydependence discussed in the last section. The rule is implementedas follows. First,in each and every periodthere is a predeterminedprice-level target p*. The centralbank chooses the interestrate it to achieve the target relation

(32) P,=P*, if this is possible.If it is not possible,even by loweringthe nominalinterest rateto zero, then it= 0. Herept is an outputgap-adjusted price index,46 definedby

Pt Pt +-xt. K The targetfor the next periodis then determinedas

(33) P* - p, + -'(1+ Ka)A - , whereAt is the targetshortfall in periodt:

(34) A =-P,t.

It can be verifiedthat this rule does indeed achieve the optimalcommitment solution.If the price-leveltarget is not reached,because of the zero

46. On the desirabilityof a targetfor this indexin the case thatthe zero bounddoes not bind,see Woodford(forthcoming, chapter 7). This wouldcorrespond to a nominalGDP target in the case that X = iKand thatthe naturalrate of outputfollows a deterministictrend. However, the utility-basedloss functionderived in Woodford(forthcoming, chapter 6) involvesX = K/0, where0 >1 is the elasticityof demandfaced by the suppliersof differenti- ated goods, so thatthe optimalweight on outputis considerablyless thanunder a nominal GDP target.Furthermore, the welfare-relevantoutput gap is unlikely to correspondtoo closely to deviationsof realGDP from a deterministictrend. Gauti B. Eggertsson and Michael Woodford 183

Figure 6. Responses of the Price-LevelTarget and the Gap-AdjustedPrice Level to a Shock to the Natural Rate of Interest

Gap-adjustedindex, quarter-1= 100 101.4- 101.2 Target(pt*)

101.0

100.8-

100.6

100.4-

100.2

100.0 Actual path (pt) 0 5 10 15 20 Quartersafter shock to naturalrate of interest

Source: Authors' calculations. bound, the centralbank increasesits targetfor the next period. This, in turn,increases inflation expectations further in the trap,which is exactly what is neededto reducethe real interestrate. Figure6 shows how the price-level targetp* would evolve over time, dependingon the numberof periodsin which the naturalrate of interest remainsnegative, in the same numericalexperiment as in figure3. (Here the darklines show the evolution of the gap-adjustedprice level Pt, and the shadedlines show the evolution of p*.) One observes that the target price level is ratchetedsteadily higher during the periodin which the naturalrate remains negative, as the actualprice level continuesto fall below the target by an increasing amount. Once the naturalrate of interest becomes positive again, the degree to which the gap-adjustedprice level undershootsthe targetbegins to shrink,although the targetoften continues 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 naturalrate has been negative.) As the degree of undershoot- ing begins to shrink,the price-level targetbegins to fall again, as a result 184 BrookingsPapers on EconomicActivity, 1:2003 of the dynamicsspecified by equation33. This hastensthe date at which the targetcan actuallybe hit with a nonnegativeinterest rate. Once the targetceases to be undershot,it no longer changes, and the centralbank targetsand achievesa new constantvalue for the gap-adjustedprice level Pt, one slightly higher than the target in place before the disturbance occurred. Note that this approachto implementingoptimal policy answersthe questionof whetherthere is any pointin announcingan inflationtarget (or price-level target) if one knows that it is extremely unlikely to be achievedin the shortrun, because the zero boundis likely to continueto bind.The answerhere is yes. The centralbank wishes to makethe private sector aware of its commitmentto the time-varyingprice-level target describedby equations32 through34, because eventuallyit will be able to hit the target.The anticipationof that fact (that is, of the level that prices will eventuallyreach, as a resultof the policies that the bank will follow afterthe naturalrate of interestagain becomes positive) while the naturalrate is still negative is importantin mitigatingthe distortions caused by the zero bound.The fact that the targetis not hit immediately should not create doubts about whether central bank announcements regardingits targethave any meaning,if it is explainedthat the bank is committedto hittingthe targetif this is possible at a nonnegativeinterest rate, so that, at each point in time, eitherthe targetwill be attainedor a zero interestrate policy will be followed. The existence of the targetis relevanteven when it is not being attained,because it allows the private sector to judge how close the centralbank is to a situationin which it would feel justifiedin abandoningthe zero interestrate policy; hence the currentgap betweenthe actualand the targetprice level shouldshape private sector expectationsof the time when interest rates are likely to remain low.47 Would the private sector have any reason to believe that the central bankwas seriousabout the price-leveltarget, if in each periodall thatis

47. An interesting feature of the optimal rule is that it involves a form of history dependence 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 lowered interest rates in the past had the zero bound not prevented it. Gauti B. Eggertsson and Michael Woodford 185 observedis a zero nominalinterest rate and yet anothertarget shortfall? The best way of makinga rule credibleis for the centralbank to conduct policy over time in a way that demonstratesits commitment.Ideally, the centralbank's commitmentto the price-leveltargeting framework would be demonstratedbefore the zero boundcame to bind (at which time the centralbank would have frequentopportunities to show thatthe targetdid determineits behavior).The rule proposedabove is one that would be equally optimalboth undernormal circumstances and in the case of the relativelyunusual kind of disturbancethat causes the naturalrate of interest to be substantiallynegative. To understandhow the rule works outside of the trap,it is useful to note that, when the nominalinterest rate is positive, A, = 0 at all times. The centralbank should thereforedemonstrate a commitmentto subse- quentlyundo any over- or undershootingof the price-leveltarget. In this case any deflationthat occurs when the economyfinds itself in a liquidity trapshould create expectations of futureinflation, as mandatedby optimal policy. The additionalterm A, implies that,when the zero boundis binding, the centralbank should raise its long-runprice-level target even further,thus increasinginflation expectations even more. It may be wonderedwhy we discuss our proposalin terms of a (gap- adjusted)price-level target rather than an inflationtarget. In fact, we could equivalentlydescribe the policy in termsof a time-varyingtarget for the gap-adjustedinflation rate f t - A-1 The reasonwe preferto describe the rule as a price-leveltargeting rule is thatthe essence of the rule is easily describedin those terms.As we show below, afixed targetfor the gap- adjustedprice level would actuallyrepresent quite a good approximation to optimalpolicy, whereasa fixed inflationtarget would not come close, becauseit would fail to allow for any of the historydependence of policy necessaryto mitigatethe distortionsresulting from the zero bound.

A Simpler Proposal One may arguethat an unappealingaspect of the rule suggestedabove is that it involves the termAt, which determinesthe change in the price- level target,and is nonzeroonly when the zero boundis binding.Suppose that the centralbank's commitmentto a policy rule can become credible over time only throughrepeated demonstrations of its commitmentto act in accordancewith it. In that case the part of the rule that involves the adjustmentof the target in response to target shortfallswhen the zero 186 BrookingsPapers on EconomicActivity, 1:2003 boundbinds might not come to be well understoodby the privatesector for a very long time, because the occasions when the zero boundbinds will presumablybe relativelyinfrequent. Fortunately,most of the benefits that can be achieved in principle through a credible commitmentto the optimal targeting rule can be achievedthrough commitment to a much simplerrule, which would not involve any special provisos that are invoked only in the event of a liquiditytrap. Consider the following simplerrule:

(35) Pt +-xx = * K wherenow the targetfor the gap-adjustedprice level is fixed at all times. The advantageof this rule, althoughit is not fully optimalwhen the zero boundis binding,is thatit may be moreeasily communicatedto the public. Note that the simple rule is fully optimalin the absence of the zero bound.In fact, even if the zero boundoccasionally binds, this rule results in distortionsonly a bit more severe thanthose associatedwith the fully optimalpolicy. Figures 7 and 8 comparethe results for these two rules. The shaded lines show the equilibriumunder the constant-price-leveltarget rule in equation35, whereasthe darklines show the fully optimalrule in equations 32 through34. As the figuresshow, the constant-price-leveltargeting rule resultsin state-contingentresponses of outputand inflationthat are very close to those underthe optimalcommitment, even if underthis rule the price level falls furtherduring the period when the zero bound binds, and only asymptoticallyrebounds to its level before the disturbance.The table below shows thatthe simple rule alreadyachieves most of the welfaregain thatthe optimalpolicy achieves;the table reportsthe value of expected discountedlosses, as a percentageof what could be achievedby a strictzero-inflation target (equation 28), conditionalon the occurrenceof the disturbancein period0, underthe variouspolicies discussed above:

Policy Loss (percent) Strict inflation target, n* = 0 100 Strict inflation target, n* = 1 24.1 Strict inflation target, n* = 2 32 Constant price-level target 0.0725 Optimal rule 0.036 Gauti B. Eggertsson and Michael Woodford 187

Figure 7. Responsesof Inflationand the Output Gap under the Optimal Targeting Rule and under the Simple Rulea

Inflation Percenta year

0.4-

0.2

0.0

-0.2- 4).2 ~~~~'Sim iple rule/ Optimalrule

Output gap Percentof GDP

-1 - W-

0 5 10 15 20 Quartersafter shock to naturalrate of interest

Source: Authors' calculations. a. The simple rule is describedin equation35. 188 Brookings Papers on EconomicActivity, 1:2003

Figure 8. Responses of the Nominal Interest Rate and Prices under the Optimal Targeting Rule and under the Simple Rule Nominal interest rate Percenta year

Simple Optimal 4-

3 -

Price level Index,quarter-I = 100

100.4

100.2

100.0

99.8-

0 5 10 15 20 Quartersafter shock to naturalrate of interest

Source: Authors' calculations. Gauti B. Eggertsson and Michael Woodford 189 Both of the lattertwo, history-dependentpolicies are vastly superiorto any of the strict inflationtargets. Although it is true that losses remain twice as large underthe simple rule as underthe optimalrule, they are nonethelessfairly small. As with the fully optimalrule, no estimateof the naturalrate of interest is needed to implementthe constant-price-leveltargeting rule. It may seem puzzlingat firstthat a constant-price-leveltargeting rule does well, because no accountis taken of the size of the disturbanceto the natural rateof interest.This comes aboutbecause a price-leveltarget commits the governmentto undo any deflationwith subsequentinflation; a largerdis- turbance,which creates a larger initial deflation,automatically creates greaterinflation expectations in response.Thus an automaticstabilizer is built into the price-level target,which is lacking undera strict inflation targetingregime.48 A proper strategyfor the central bank to use in communicatingits objectivesand targetswhen outsidethe liquiditytrap is of crucialimpor- tance for this policy rule to be successful.To see this, considera rule that is equivalentto equation35 when the zero boundis not binding.Taking the differenceof equation35, we obtain

(36) it +-(xt - xt-1)= . K

Although this rule results in an equilibriumidentical to that under the constant-price-leveltargeting rule when the zero boundis not binding,the result is dramaticallydifferent when the zero bound is binding,because this ruleimplies that the inflationrate is proportionalto the negativeof the growthrate of output.Thus it mandatesdeflation when thereis growthin the outputgap. This implies that the centralbank will deflate once the economyis out of a liquiditytrap, because the economy will then be in a periodof outputgrowth. This is exactly the oppositeof whatis optimal,as we have observedabove. Thus the outcomeunder this rule is even worse than under a strict zero-inflationtarget, even if this rule replicatesthe price-leveltargeting rule when out of the trap.What this underlinesis that it is not enoughto replicatethe equilibriumbehavior that correspondsto

48. Wolman (forthcoming)also stresses this advantageof rules that incorporatea price-leveltarget over rules thatonly respondto the inflationrate, such as a conventional Taylorrule. 190 BrookingsPapers on EconomicActivity, 1:2003 equation35 in normaltimes to inducethe correctset of expectationswhen the zero boundis binding.It is crucialto communicateto the public that the governmentis committedto a long-runprice-level target. This commitmentis exactly what createsthe desired inflationexpectations when the zero boundis binding.

Should the Central Bank Keep Some Powder Dry? Thus far we have consideredonly alternativepolicies that might be followed after the naturalrate of interesthas unexpectedlyfallen to a negative value, causing the zero boundto bind. A questionof consider- able currentinterest in countrieslike the United States,however, is how policy should be affected by the anticipationthat the zero bound might well bindbefore long, even if it is not yet binding.Some have arguedthat, in such circumstances,the FederalReserve should be cautiousabout low- eringinterest rates all the way to zero too soon, in orderto save its ammunition for futureemergencies. This suggests thatthe anticipationthat the zero boundcould bindin the nearfuture should lead to tighterpolicy than would otherwisebe justifiedgiven currentconditions. Others argue, however, that policy shouldinstead be more inflationarythan one might otherwise prefer,to reducethe probabilitythat a furthernegative shock will resultin a bindingzero bound. Ourcharacterization of the optimaltargeting rule can shed light on this debate.Recall that the rule laid out in equations32 through34 describes optimalpolicy regardlessof the assumedstochastic process for the natural rate of interest,and not only in the case of the particulartwo-state Markovprocess assumedin figure3. In particular,the same rule is optimal in the case that informationis received indicatingthe likelihoodof the naturalrate of interestbecoming negative before this actuallyoccurs. How shouldthat news affectthe conductof policy?Under the optimaltargeting rule, the optimaltarget for PA is unaffectedby such expectations,as long as the zero boundis not yet binding,because only targetshortfalls that have alreadyoccurred can justify a change in the target value pt*. Thus an increasedassessment of the likelihoodof a bindingzero bound over the coming year or two would not be a reason for increasingthe price-leveltarget (or the impliedtarget rate of inflation).49

49. This conclusion,however, is likely to dependon a relativelyspecial feature of our model, namely, the fact that our targetvariables (inflation and the outputgap) are both Gauti B. Eggertsson and Michael Woodford 191

On the otherhand, this news will affect the pathsof inflation,output, and interestrates, even in the absence of any immediatechange in the centralbank's price-leveltarget, owing to the effect on forward-looking privatesector spendingand pricingdecisions. The anticipationof a coming state in which the naturalrate of interestwill be negative,and actual interestrates will not be able to fall as much, owing to the zero bound, will reduceboth desiredreal expenditure(at unchangedshort-term interest rates)and desiredprice increases,because of the anticipationof negative output gaps and price declines in the future. This change in the behaviorof the privatesector's outlook will requirea changein the central bank's conductof policy in orderto hit its unchangedtarget for the modified price level, likely in the directionof a preemptiveloosening. This is illustratedby the numericalexperiment shown in figure 9. Here we supposethat in quarter0 both the centralbank and the privatesector learnthat the naturalrate of interestwill fall to -2 percenta year only in period4. It is known that the naturalrate will remainat its normallevel of +4 percenta year until then;after the drop,it will returnto the normal level with a probabilityof 0.1 each quarter,as in the case consideredear- lier. We now consider the characterof optimal policy from period 0 onward,given this information.Figure 9 again shows the optimal state- contingentpaths of inflationand output in the case thatthe disturbanceto the naturalrate, when it arrives,lasts for one quarter,two quarters,and so on. We observe that, underthe optimalpolicy commitment,prices begin to decline mildly as soon as the news of the coming disturbanceis received. The centralbank is nonethelessable to avoid undershootingits targetfor A,at first, by stimulatingan increasein real activity sufficient to justify the mild deflation. (Given the private sector's shift to pes- simism, this is the policy dictatedby the targetingrule, given thateven a mild immediateincrease in real activity is insufficientto preventa price decline, owing to the anticipateddecline in real demandwhen the disturbance hits.) By quarter3, however, this is no longer possible, and the

purelyforward-looking variables: their equilibrium values at any point in time depend(in oursimple model) only on the economy'sexogenous state and the expectedconduct of policy from the currentperiod onward.There are a varietyof reasonswhy a more realistic model may well imply thatthese variablesare functions of laggedendogenous variables as well, and hence of past policy. In such a case, the optimaltarget criterion will be at least somewhatforward looking, as discussedin Giannoniand Woodford(2003). Figure 9. Responsesof Inflation,the Output Gap, and Prices to an AnticipatedShock under Optimal Policy"

Inflation Percent a year

0.3

0.2-

0.1

0.0

-0.1I

Output gap Percentof GDP

2.0- 1.5- 1.0 0.5 0.0 -I -0.5-

Price level Index, quarter -1 =100

100.2 -

100.0

0 5 10 15 20 Quarters after shock to natural rate of interest

Source: Authors' calculations. a. Response in a scenario where the shock to the naturalrate of interestis anticipatedfour quartersin advance of its occurrence. Gauti B. Eggertsson and Michael Woodford 193 central bank undershootsits target for Pt (as both prices and output decline), even thoughthe nominalinterest rate is at zero. Thus, optimal policy involves drivingthe nominalinterest rate to zero even before the naturalrate of interesthas turnednegative, when that developmentcan alreadybe anticipatedfor the near future.The fact that the zero bound binds even before the naturalrate of interest becomes negative means that the price-level target is higher than it otherwise would have been when the disturbanceto the naturalrate arrives. As a result,deflation and the outputgap duringthe periodwhen the naturalrate is negativeare less severe than in the case where the disturbanceis unanticipated.Optimal policy in this scenario is somewhat more inflationaryafter the disturbance occurs thanin the case consideredin figure 3, becausein this case the optimal policy commitmenttakes into account the contractionary effects, in periodsbefore the disturbancetakes effect, of the anticipation that the disturbancewill result in price-level and output declines. The fact that optimal policy after the disturbanceoccurs is differentin this case, despite the fact that the disturbancehas exactly the same effects as before fromquarter 4 onward,is anotherillustration of the historydependence of optimalpolicy.

Preventing a Self-FulfillingDeflationary Trap

Ouranalysis thus far has assumedthat the real disturbancethat results in a negative naturalrate of interestdoes so only temporarily.We have thereforesupposed that price-level stabilization will eventuallybe consistent with positive nominalinterest rates and, accordingly,that a time will foreseeably be reached when the central bank can create inflation by keeping short-termnominal rates at a low but nonnegativelevel. But is it possible for the zero boundto bindforever in equilibrium,not becauseof a permanentlynegative natural rate, but simply becausedeflation continues to be (correctly)expected indefinitely? If so, the centralbank's commitmentto a nondecreasingprice-level target might seem irrelevant;the pricelevel would fall furtherand further short of the target,but becauseof the bindingzero bound,the centralbank could neverdo anythingabout it. In the model presentedin the first part of the paper,a self-fulfilling, permanentdeflation is indeed consistent with both the Euler equation (equation 2) for aggregate expenditure, the money-demandrelation 194 BrookingsPapers on EconomicActivity, 1:2003 (expression3), andthe pricingrelations (equations 7 through9). Suppose that, from some date t onward,all disturbancestt = 0 with certainty, so that the naturalrate of interestis expected to take the constantvalue r = - 1 > 0, as in the scenarios considered earlier in the paper. Then the possible pathsfor inflation,output, and interestrates consistent with each of the relationsjust listed in all periodst ? t are given by

it =0 =/- J3<1 pr P <1- <

Yt=Y, where Y< Yis implicitlydefined by the relation

Hli[*,p*,1;Y,i (Y;0),0]= 0.

Note thatthis deflationarypath is consistentwith monetarypolicy as long as real balancessatisfy M,lP, > miI(Y;0) each period;faster growth of the money supply does nothingto preventconsistency of this path with the requirementthat money supplyequal money demandin each period. Thereremains, however, one furtherrequirement for equilibriumin the earliermodel, namely,the transversalitycondition (equation 6) or, equiv- alently, the requirementthat householdshit their intertemporalbudget constraint.Whether the deflationarypath is consistentwith this condition as well depends,properly speaking, on the specificationof fiscal policy: it is a matterof whetherthe governmentbudget results in contractionof the nominalvalue of totalgovernment liabilities Dt at a sufficientrate asymp- totically. Under some assumptionsabout the characterof fiscal policy, such as the Ricardianfiscal policy rule assumed by Jess Benhabib, StephanieSchmitt-Grohe, and MartinUribe,50 the nominalvalue of governmentliabilities will necessarilycontract as the price level falls, so that equation6 is also satisfied,and the processesdescribed above will indeed representa rationalexpectations equilibrium. In such a case, then, a commitmentto the price-leveltargeting rule proposedin the previoussection will be equally consistent with more than one equilibrium:if people

50. Benhabib,Schmitt-Grohe, and Uribe (2001). Gauti B. Eggertsson and Michael Woodford 195 expect the optimal price-level process characterizedearlier, that will indeedbe an equilibrium,but if they expect perpetualdeflation, that will be an equilibriumas well. However, this outcome can be excluded througha suitable commitment with regardto the asymptoticevolution of total governmentliabilities. Essentially,there needs to be a commitmentto policies that ensure thatthe nominalvalue of governmentliabilities cannot contract at the rate requiredfor satisfactionof the transversalitycondition, despite perpetual deflation.One exampleof a commitmentthat would suffice is a commitment to a balanced-budgetpolicy of the kind analyzedby Schmitt-Grohe and Uribe.51These authorsshow thatself-fulfilling deflations are not possible when monetarypolicy is committedto a Taylorrule andthe government to a balancedbudget. The key to their result is that the fiscal rule includes a commitmentthat is as bindingagainst running large surpluses as it is againstrunning large deficits;then the nominalvalue of government liabilitiescannot contract, even when the price level falls exponen- tially forever. The credibilityof this sortof fiscal commitmentmight be doubted,and so anotherway of maintaininga floorunder the asymptoticnominal value of total governmentliabilities is througha commitmentnot to contractthe monetarybase, togetherwith a commitmentof the governmentto maintain a nonnegativeasymptotic present value of the publicdebt. In particular, supposethat the centralbank commitsitself to follow a base-supply rule of the form

(37) M, = P:*ii(Y;;,, )

in each periodwhen the zero boundbinds (thatis, when it is not possible to hit the price-leveltarget with a positive nominalinterest rate), where

P*I exp{P, x,}

is the currentprice-level target implied by the adjustedprice-level target p*. When the zero bound does not bind, the monetarybase is whatever level is demandedat the nominal interestrate requiredto hit the price- level target.This is a rule in the same spiritas equation11, specifying a

51. Schmitt-Groh6and Uribe(2000). 196 BrookingsPapers on EconomicActivity, 1:2003 particularlevel of excess supply of base money when the zero bound binds, but letting the monetarybase be endogenouslydetermined by the centralbank's other targets at othertimes. Equation37 is a more compli- cated formulathan is necessaryto make our point, but it has the advan- tage of making the monetary base a continuous function of other aggregatestate variablesat the point wherethe zero boundjust ceases to bind. This particularform of commitmenthas the advantagethat it may be consideredless problematicfor the centralbank to commititself to maintain a particularnominal value for its liabilitiesthan for the public trea- suryto do so. It can also be justifiedas entirelyconsistent with the central bank'scommitment to the price-leveltargeting rule; even when the target cannotbe hit, the centralbank supplies the quantityof money thatwould be demandedif the price level were at the target.Doing so-refusing to contractthe monetarybase even in a deflation-is a way of signalingto the public that the centralbank is serious about its intentionto see the price level restoredto the targetlevel. If one then assumesa fiscal commitmentthat guarantees that

(38) lim E,Q,TBT =0, that is, that the governmentwill asymptoticallybe neithercreditor nor debtor,the transversalitycondition reduces to

(39) limT8E,[uC( YT,MT/ PT; 9T)MT/PT] =?0

In the case of the base-supplyrule in equation37, this conditionis violated in the candidateequilibrium described above, since the price-level andoutput paths specified would imply that

= 3TEt[uc(YT, MT/PT; 9T)MT/PT] PWuc[Yfii(Y;0);0]fii(Y;O)PT/PT

> ,BTUC Y9fi(Y;O);O fi(Y;O)P, /PT9 wherethe last inequalitymakes use of the fact that,under the price-level targetingrule, {p*} is a nondecreasingseries. Note that the final expression on the right-handside is independentof T, for all dates T ? T. Hence the series is boundedaway from zero, andthe conditionin equation39 is violated. Gauti B. Eggertsson and Michael Woodford 197 Thus a commitmentof this kind can exclude the possibilityof a self- fulfilling deflationof the sort describedabove as a rationalexpectations equilibrium.It follows that there is a possible role for quantitative easing-understood to mean the supplyof base money beyondthe mini- mum quantityrequired for consistency with the zero nominal interest rate-as an elementof an optimalpolicy commitment.A commitmentto supply base money in proportionto the target price level, and not the actualcurrent price level, when the zero boundprevents the centralbank fromhitting its price-leveltarget, can be desirableboth as a way of ruling out self-fulfillingdeflations and as a way of signalingthe centralbank's continuingcommitment to the price-level target,even thoughit is tem- porarilyunable to hit it. Note that this result does not contradictour irrelevanceproposition, becausehere we have madea differentassumption about the natureof the fiscal commitment.Equation 38 implies that the evolutionof total nominal governmentliabilities will not be independentof the centralbank's targetfor the monetarybase. As a consequence,the neutralityproposition no longer holds. The importof that propositionis that expansionof the monetarybase when the economy is in a liquidity trap is necessarily pointless;rather, any effect of such action must dependeither on changing expectations regardingfuture interest rate policy or on changing expectationsregarding the futurepath of total nominalgovernment liabilities. The present discussion has illustrated circumstancesin which expansionof the monetarybase-or, at any rate, a commitmentnot to contractit-could serve both these ends. Nonetheless,the presentdiscussion does not supportthe view thatthe centralbank should be able to hit its price-leveltarget at all times, simply by floodingthe economy with as much base money as is requiredto prevent the price level from falling below the targetat any time. Ourearlier analysisstill describesall possiblepaths for the pricelevel consistentwith rationalexpectations equilibrium, and we have seen that even if the central bank were able to choose the expectationsthat the private sector should have (as long as it were willing to act in accordancewith them), the zero boundwould prevent it frombeing able to fully stabilizeinflation and the outputgap. Furthermore,the degree of monetarybase expansion duringa liquiditytrap called for by the rule in equation37 is quite modest. The monetarybase will be raisedgradually, if the zero boundcontin- ues to bind, as the price-level target is ratchetedup to steadily higher 198 BrookingsPapers on EconomicActivity, 1:2003 levels. But our calibratedexample above indicatesthat this would typically involve only a very modest increasein the monetarybase, even if the liquiditytrap lasts for severalyears. There would be no obviousbene- fit to the kind of rapidexpansion of the monetarybase actuallytried in Japanover the pasttwo years.Such an expansionis evidentlynot justified by any intentionregarding the futureprice level, and hence regardingthe size of the monetarybase once Japanexits fromthe trap.But an injection of base money thatis expectedto be removedonce the zero boundceases to bind should have little effect on spendingor pricingbehavior, as we showedin the firstsection of the paper.

Further Aspects of the Managementof Expectations

In the first section we arguedthat neitherexpansion of the monetary base as such nor open-marketpurchases of particulartypes of assets should have any effect on either inflationor real activity, except to the extentthat these actionsmight change expectations regarding future interest ratepolicy (or possibly expectationsregarding the asymptoticbehav- ior of total nominal governmentliabilities, and hence the question of whether the transversalitycondition should be satisfied). This then allowed us to characterizethe optimalpolicy commitmentwithout any referenceto the use of such instrumentsof policy; a considerationof the differentpossible joint paths of interestrates, inflation,and outputthat would be consistent with rationalexpectations equilibrium sufficed to allow us to determinethe best possibleequilibrium that one could hope to arrange,and to characterizeit in termsof the interestrate policy thatone shouldwish the privatesector to expect. However,this does not mean that other aspects of policy-beyond a mere announcementof the rule to which the centralbank wishes to be understoodto be committedin settingfuture interest rate policy-cannot matter.They may matterinsofar as certainkinds of presentactions may help to signal the bank's intentionsregarding future policy, or make it more crediblethat the centralbank will indeedcarry out those intentions. A full analysis of the ways in which policy actions may be justified as helpingto steerexpectations is beyondthe scope of this paper,and in any event the question is one that has as much to do with psychology and effective communicationas with economic analysis. Nonetheless, we Gauti B. Eggertsson and Michael Woodford 199 offer a few remarkshere aboutthe kinds of policies thatmight contribute to the creationof desirableexpectations.

Demonstrating Resolve One way in which currentactions may help to createdesirable expectationsregarding future policy is by being seen to be consistentwith the principlesthat the centralbank wishes the private sector to understand will guide that policy. We have alreadymentioned one example of this: one way to convince the privatesector that the centralbank will follow the optimal price-level targetingrule after a period in which the zero boundhas been hit is by following this rule beforesuch a situationarises. Our discussion in the previous section provides a furtherexample. Adjustmentof the supplyof base money while the zero boundis binding, so as to keep the monetarybase proportionalto the target price level ratherthan the actualcurrent price level, can be helpful,even thoughirrel- evant to interestrate control, as a way of communicatingto the private sectorthe centralbank's belief aboutwhere the pricelevel oughtproperly to be (and hence the quantityof base money that the economy ought to need). By puttingthe existence of the price-leveltarget in greaterrelief, such an action can help createthe expectationsregarding future interest rate policy necessaryto mitigate the distortionscreated by the binding zero bound. As a furtherexample, Clouse andothers argue that open-market operations may be stimulative,even when the zero bound has been reached, because they "demonstrateresolve" to keep the nominalinterest rate at zero for a longertime thanwould otherwisebe expected.52But an expansion of the monetarybase when the zero bound is binding need not be interpretedin this way. Consider,for example,a centralbank with a con- stantzero inflationtarget, as discussedpreviously. When the zero bound binds, such a bank is unableto hit its inflationtarget and should exhibit frustrationwith this state of affairs. If some within the bank believe it should always be possible to hit the target with sufficiently vigorous monetaryexpansion, one might well observe substantialgrowth in the monetarybase at a time when the inflation target is being undershot. Nonetheless, this would not imply any commitmentto looser policy

52. Clouse and others (2003) 200 BrookingsPapers on EconomicActivity, 1:2003 subsequently;such a centralbank would never intentionallyallow the monetarybase to be higherthan required to hit the inflationtarget, if the targetcan be hit. The resultshould be the equilibriumpath shown in fig- ure2, andthere should be no effect fromthe quantitativeeasing that occurs while the zerobound binds. This showsthat it matterswhat the privatesector understandsto be the principlethat motivates quantitative easing; it is not simplya questionof how largeis the increasein the monetarybase. Similarly,open-market purchases of long-termtreasury bonds when short-termrates are at zero, as advocatedby Ben Bernankeand Stephen Cecchetti,53among others, may well have a stimulativeeffect even if portfolio balance effects are quantitativelyunimportant. We argued previously that under such circumstancesit is desirablefor the central bankto commititself to maintainlow short-termrates even afterthe naturalrate of interestrises again. The level of long-termrates can indicate the extent to which the marketsactually believe such a commitment.If a centralbank's judgment is that long-termrates remainhigher than they shouldbe underthe optimalequilibrium, owing to privatesector skepti- cism aboutwhether the history-dependentinterest rate policy will actually be followed, then a willingness to buy long-termbonds from the privatesector at a price it regardsas more appropriateis one way for the centralbank to demonstratepublicly thatit expects to carryout its com- mitmentregarding future interest rate policy. Given that the privatesector is likely to be uncertain about the nature of the central bank's commitment(in the case of imperfectcredibility), and thatit can reason- ably assume that the centralbank knows more about its own degree of resolve thanothers do, actionby the centralbank that is consistentwith a belief on its own partthat it will keep short-termrates low in the futureis likely to shift private beliefs in the same direction.If so, open-market purchasesof long-termbonds could lower long-terminterest rates, stimulate the economy immediately,and bring the economy closer to the optimalrational expectations equilibrium. However, that effect follows not from the purchasesthemselves, but from how they are interpreted. For them to be interpretedas indicatinga particularkind of commitment with regardto future policy, it is importantthat the central bank have itself formulatedsuch an intention,and that it so inform the public, so thatits open-marketpurchases will be seen in this light.

53. Bernanke(2002); StephenG. Cecchetti,"Central Banks Have Plentyof Ammuni- tion,"Financial Times, March17, 2003, p. 13. Gauti B. Eggertsson and Michael Woodford 201 Similarremarks apply to the proposalsby BennettMcCallum and Lars Svensson that purchasesof foreign exchange be used to stimulatethe economythrough devaluation of the currency.54Under the optimalpolicy commitmentdescribed in an earliersection, a declinein the naturalrate of interest should be accompaniedby depreciationof the currency,both because nominal interestrates fall (and are expected to remainlow for some time) and becausethe expectedlong-run price level (andhence the expectedlong-run nominal exchange rate) should increase. It follows that the extent to which the currencydepreciates can providean indicatorof the extentto which the marketsbelieve thatthe centralbank is committed to such an optimalpolicy; andif the depreciationis insufficient,purchases of foreignexchange by the centralbank provide one way for it to demon- strateits own confidencein its policy intentions.Again, the effect in question is not a mechanicalconsequence of the bank'spurchases, but instead dependson theirinterpretation.55

Providing Incentives to Improve Credibility A relatedbut somewhatdistinct argumentis that actions at the zero boundmay help renderthe centralbank's commitment to an optimalpolicy more credible,by providingthe bank with a motive to behave in the futurein the way thatit would currentlywish thatpeople 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 centralbank is in need of a "commitmenttechnology" because it will be unableto resist the tempta- tion to breakits commitmentslater in the absence of such a constraint. Rather,it is thatthe centralbank may well need a way of makingits com-

54. McCallum(2000); Svensson(2001). Svensson'sproposal includes a targetpath for the price level, which the exchangerate policy is used to (eventually)achieve, and in this respectit is similarto the policy advocatedhere. However,Svensson's discussionof the usefulnessof interventionin the marketfor foreignexchange does not emphasizethe role of such interventionsas a signalregarding future policy. 55. The numerical analysis by Coenen and Wieland (forthcoming)finds that an exchangerate policy can be quite effective in creatingstimulus when the zero bound is binding.But what is actually shown is that a rationalexpectations equilibrium exists in which the currencydepreciates and deflationis halted;these effects could be viewed as resultingfrom a crediblecommitment to a targetpath for the price level, similarto the one discussedin this paper,and not requiringany interventionin the foreignexchange market at all. 202 BrookingsPapers on EconomicActivity, 1:2003 mitmentvisible to the privatesector. Taking actions now that imply that the centralbank will be disadvantagedlater if it were to deviatefrom the policy to which it wishes to commititself can serve this purpose. To considerwhat kind of currentactions provide useful incentives,it is helpfulto analyze(Markov) equilibrium under the assumptionthat policy is conductedby a discretionaryoptimizer, unable to commit to specific futureactions at all.56Consider first what a Markovequilibrium under discretionaryoptimization would be like in the case that the only policy instrumentavailable is a short-termnominal interest rate, whose value is chosen each period,and the objectiveof the centralbank is minimization of the loss functionin equation28. As shown above, if the centralbank can crediblycommit itself, this problemhas a solutionin which the zero bounddoes not resultin too seriousa distortion,although it does bind. Under the assumptionof central bank discretion,however, the outcome will be much inferior.Note that discretionarypolicy (underthe assumptionof Markovequilibrium in the dynamic policy game) is an exampleof a purelyforward-looking policy. It then follows fromour earlier argumentthat the equilibriumoutcome will correspondto the kind of equilibriumdiscussed there in the case of a strict inflationtarget. More specifically,it is obviousthat the equilibriumis the same as undera strict inflationtarget 7t* = 0, since this is the inflationrate that the discretionary optimizer will choose once the naturalrate of interest is again at its steady-statelevel. (From that point onward, a policy of zero inflation clearlyminimizes the remainingterms in the discountedloss function.) As shownin figure2, if the privatesector expects thatthe centralbank will behavein this fashion,and the naturalrate of interestremains negative for severalquarters, the resultwill be a deep and prolongedcontrac- tion of economicactivity and a sustaineddeflation. We have also seen that these effects could largelybe avoided,even in the absenceof otherpolicy instruments,if the centralbank were able to crediblycommit itself to a history-dependentmonetary policy in later periods.Thus, in the kind of situationconsidered here, there is a deflationary bias to discretionary monetarypolicy, although,at its root, the problemis againthe one identified in the classic analysis of Finn Kydlandand EdwardPrescott.57 We now considerinstead the extentto which the outcomecould be improved,

56. As in Eggertsson (2003a, 2003b). 57. Kydland and Prescott (1977). Gauti B. Eggertsson and Michael Woodford 203

even in a Markovequilibrium with discretionaryoptimization, by changing the natureof the policy game. One exampleof a currentpolicy action, availableeven when the zero boundbinds, thatcan help shift expectationsregarding future policy in a desirableway is for the governmentto cut taxes andissue additionalnom- inal debt.58Alternatively, the tax cut can be financedby money creation, because when the zero bound binds, there is no difference between expandingthe monetarybase and issuing additionalshort-term Treasury debt at zero interest.This is essentiallythe kind of policy imaginedwhen people speak of a "helicopterdrop" of additionalmoney into the economy, but here it is the fiscal consequencesof such an action with which we are concerned. Of course,if the objectiveof the centralbank in settingmonetary policy remainsas assumedabove, this will makeno differenceto the discre- tionaryequilibrium: the optimal policy once the naturalrate of interest becomespositive againwill once moreappear to be the immediatepursuit of a strict zero-inflationtarget. However, if the centralbank also cares aboutreducing the social costs of increasedtaxation-whether becauseof collection costs or because of other distortions-as it ought if it really takessocial welfareinto account,the resultis different.As Eggertssonhas shown elsewhere,59the tax cut will then increaseinflation expectations, even if the governmentcannot commit to futurepolicy. It may be asked why, if such an incentive exists, Japancontinues to sufferdeflation, given the growthin Japanesegovernment debt during the 1990s. One possible answer is that althoughthe gross nationaldebt is 140 percentof GDP in Japantoday, this does not reflectthe trueinflation incentives of the government.The ratio of gross nationaldebt to GDP overestimatesthe government'sinflation incentives, because a substantial portionof governmentdebt is held by othergovernment institutions.60 Net governmentdebt is only 67 percentof GDP, and, as a result, inflation

58. As discussedin Eggertsson(2003a). 59. Eggertsson(2003a). 60. Governmentinstitutions such as the social securitysystem, the postal savings system, postallife insurance,and the TrustFund Bureau hold muchof this nominaldebt. If the partof the publicdebt held by these institutionsis subtractedfrom total gross government debt, the remainderis only 67 percentof GDP. Most of the governmentinstitutions that hold the government'snominal debt have real liabilities.For example,the social security system(which holds roughly25 percentof the nominaldebt held by the government)pays Japanesepensions and medical expenses. Those pensions are indexed to the consumer 204 BrookingsPapers on EconomicActivity, 1:2003 incentivesmay not be much greaterin Japanthan in a numberof other countries. An even morelikely reasonfor continuedlow inflationexpectations in Japan,despite the size of the nominal public debt, is skepticismabout whether the central bank can be expected to care about reducing the burdenof the public debt when determiningfuture monetary policy. The public may believe that the Bank of Japanlacks such an objective;the expressedresistance of the Bank of Japanto suggestionsthat it increase its purchasesof Japanesegovernment bonds, on the groundthat this could encouragea lack of fiscal discipline,6'certainly suggests that reducing the burden of governmentfinance is not among its highest priorities.As Eggertssonhas stressedelsewhere,62 in orderfor fiscal policy to be effective as a means of increasinginflationary expectations, fiscal and monetary policy must be coordinatedso as to maximize social welfare. The consequencesof a narrowconcern with inflationstabilization on the part of the centralbank, together with an inabilityto crediblycommit future monetarypolicy, can be dire, even from the point of view of the bank's own stabilizationobjectives. Anotherinstrument that may be used to changeexpectations regarding futuremonetary policy is open-marketpurchases of real assets or foreign exchange.Purchases of real assets (say, real estate) can be thoughtof as anotherway of increasingnominal government liabilities, which should affect inflationincentives in much the same way as deficit spending.63 Purchasesof real assets have the advantageof not worseningthe overall fiscal position of the government-a currentconcern in Japan,given its existing gross debt-while still increasingthe fiscal incentive for inflation. A furtheradvantage of this approachis thatit need not dependon a perceivedcentral bank interest in reducingthe burdenof the public debt. Since the (nominal)capital gains frominflation accrue to the centralbank itself under this policy, the centralbank may be perceived to have an price index. If inflationincreases, the real value of social securityassets will fall, but the realvalue of mostits liabilitieswill remainunchanged. Thus the Ministryof Financewould eventuallyhave to step in to makeup for any loss in the valueof social securityassets if the governmentis to keep its pensionprogram unchanged. Therefore the gains from reducing the real value of outstandingdebt are partlyoffset by a decreasein the real value of the assets of governmentinstitutions such as social security. 61. Asahi Shimbun,"Bank of JapanAdvised to 'PrintMoney' to Escape Deflation," Dow JonesNews, February10, 1999. 62. Eggertsson(2003a). 63. Eggertsson(2003a). Gauti B. Eggertsson and Michael Woodford 205 incentive to inflate simply on the groundthat it cares aboutits own balance sheet, for example because doing so will help ensure its indepen- dence. (One can easily arguethat, under a rationalscheme of cooperation betweenthe centralbank and the government,the centralbank should not choose policy on the basis of concernsabout its balancesheet. But under such an ideal regime, it should choose monetarypolicy with a view to reducingthe burdenof the publicdebt, amongother goals.) The incentive effects of open-marketoperations in foreign exchange are even simpler.64Open-market purchases of foreignassets give the central bank an incentive to inflate in the futurein order to realize capital gains at the expense of foreigners.These will be valuableif the central bankcares eitherabout its own balancesheet or aboutreducing the burden of the public debt, as in the case of real asset purchases.However, capital gains on foreign exchange that result from depreciationof the domesticcurrency will be valuableeven if the centralbank cares neither aboutits balancesheet (for example,because it cooperatesperfectly with the public treasury)nor about the burden of the debt (for example, becausenondistorting sources of revenueare availableto the public treasury). Capitalgains at the expense of foreignerswould allow an increase in domesticspending, by eitherthe governmentor the privatesector, and a centralbank must value this if it has the nationalinterest at heart. Underrational expectations, of course, no such capitalgains are real- ized on average.Still, the purchaseof foreign assets can work as a commitment device, because if the central bank reneged on its inflation commitment,it would cause capital losses if the governmentholds foreign assets. Purchasesof foreign assets are thus a way of committingthe governmentto looser monetarypolicy in the future.This createsa reason for purchasingforeign exchange in orderto cause a devaluation(which will also stimulatecurrent demand), even without any assumptionof a deviationfrom interestrate parity of the kind reliedupon by authorssuch as McCallum in recommendingdevaluation for Japan.65Clouse and others arguethat open-marketpurchases of long-termTreasuries by the FederalReserve should also change expectationsin a way that resultsin immediatestimulus.66 The argumentis thatif the centralbank were not to follow throughon its commitmentto keep short-termrates low, it would

64. As shown by Eggertsson (2003b). 65. McCallum (2000). 66. Clouse and others (2003). 206 BrookingsPapers on EconomicActivity, 1:2003 suffera capitalloss on the long-termbonds that it purchasedat a pricethat made sense only on the assumptionthat it wouldkeep interestrates low. Similarly,Peter Tinsley has proposeda policy thatwould createthis kind of incentiveeven more directly,namely, the sale by the FederalReserve of optionsto obtainfederal funds at a futuredate at a certainprice.67 The FederalReserve would then stand to lose money if it did not keep the fundsrate at the level to which it had previouslycommitted itself. Althoughthese proposalsshould also help reinforcethe credibilityof the kind of policy commitmentassociated with the optimalequilibrium (as characterizedin the first section of the paper),they have at least one importantdisadvantage relative to purchasesof real assets or of foreign exchange. They only provide the centralbank an incentive to maintain low nominalinterest rates for a certainperiod; they do not provideit with an incentive to ensure that the price level eventuallyrises to a higher level. Thus they may do little to counterprivate sector expectationsthat nominal interest rates will remain low for years-but because goods prices are going to continueto fall, not becausethe centralbank is committed to eventual reflation,as in the self-fulfilling deflationtrap discussed above. This is arguablythe kind of expectationthat has now taken root in Japan,where even ten-yearbond yields are alreadywell below 1 percent, even though prices continue to fall and economic activity remains anemic. Creatingthe perceptionthat the central bank has an incentiveto continuetrying to raise the price level, and thatit will not be contentas long as nominalinterest rates remain low, may be a more successful way of generatingthe sort of expectationsassociated with the optimalequilibrium.

Conclusion

We have arguedthat the key to dealingwith a situationin which monetarypolicy is constrainedby the zero lowerbound on short-termnominal interest rates is the skillful managementof expectationsregarding the futureconduct of policy. By "managementof expectations"we do not mean thatthe centralbank should imagine that, if it uses sufficientguile, it can lead the privatesector to believe whateverthe centralbank wishes it to believe, no matterwhat it actuallydoes. Insteadwe have assumedthat

67. Tinsley (1999). Gauti B. Eggertsson and Michael Woodford 207 there is no point in the centralbank trying to get the private sector to expect something that the central bank does not itself intend to bring about.But we do contendthat it is highly desirablefor a centralbank to be able to commit itself in advanceto a course of action that is desirable because of the benefitsthat flow from its being anticipated,and then to workto makethat commitment credible to the privatesector. In the contextof a simple optimizingmodel of the monetarytransmis- sion mechanism,we have shown that a purelyforward-looking approach to policy-which allows for no possibilityof committingfuture policy to respondto past conditions-can lead to quite bad outcomesin the event of a temporarydecline in the naturalrate of interest,regardless of the kind of policy pursuedat the time of the disturbance.We have also characterized optimal policy, underthe assumptionthat credible commitmentis possible, and shownthat it involves a commitmentto eventuallybring the generalprice level back up to a level even higher than would have pre- vailed had the disturbancenever occurred.Finally, we have describeda type of history-dependentprice-level targetingrule with the following properties:that a commitmentto base interest rate policy on this rule determinesthe optimalequilibrium, and that the same form of targeting rule continuesto describeoptimal policy regardlessof which of a large numberof types of disturbancesmay affect the economy. Given the role of privatesector anticipationof history-dependentpolicy in realizinga desirableoutcome, it is importantfor centralbanks to develop effective methodsof signaling their policy commitmentsto the privatesector. An essentialprecondition for this, certainly,is for the centralbank itself to clearlyunderstand the kind of history-dependentbehav- ior to which it shouldbe seen to be committed.It can then communicate its thinkingon the matterand act consistentlywith the principlesthat it wishes the private sector to understand.Simply conductingpolicy in accordancewith a rule may not suffice to bring about an optimal, or nearlyoptimal, equilibrium, but it is the place to start.

APPENDIX A The Numerical Solution Method

HERE WE ILLUSTRATEa solution method for the optimal commitment solution discussed in the first section of the text. This same methodcan 208 BrookingsPapers on EconomicActivity, 1:2003 also be applied,with appropriatemodification of each of the steps,to find- ing the solutionin the case wherethe centralbank commits to a constant price-leveltarget rule or to a constantinflation target. We assumethat the naturalrate of interestbecomes unexpectedlynegative in period 0 and then reverts back to normalwith probabilityaxt in every period t. Our numericalwork assumesthat there is a final date S at which the natural rate becomes positive with a probabilityof 1, althoughthis date may be arbitrarilyfar in the future. The solutiontakes the form

it=O V t if OO V t if t>-r+k.

It follows that

Etxt+l -xt+ G(Etn,j + rn)=0 if t?-+k,.

HereX is the stochasticdate at which the naturalrate of interestreturns to the steady state. We assumethat X can take any value between 1 and the terminaldate S. The numberX + k, is the periodin which the zero bound ceases to bindcontingent on the naturalrate of interestbecoming positive in period . Note thatthe value of k, can dependon the value of . We will firstshow the solutionfor the problemas if we knew the sequence{k,)s=. We then describea numericalmethod to findthe sequence {k,)s=.

The Solution for t > T + kT

The system can be writtenin the following form:

(Al) [EtZt+P M 1t where

Z- and Pt Gauti B. Eggertsson and Michael Woodford 209 If there are two eigenvaluesof the matrixM outside the unit circle, this system has a uniquebounded solution of the form

(A2) P. = floP-

(A3) Z, = AoPt-1.

The Solution for T < t < T + k

Again this is a perfect-foresightsolution, but with the zero boundbind- ing. The solutionsatisfies the following equations:

,tt = KXt +fit+1 = (A4) xt o(Y,t 1t,+)+ x,+1 7t + (P2t - (P2t-1 - [lPPt-p = 0

Xxxt + (Pit - I3Npli - K(p2, = 0-

The system can be writtenas

[Z,] [C D] [Zt+] [V] This system has a solutionof the form

(A5) P = flk-jp + 4Dkr-j

(A6) Z+j = Ak,-jP j-I + Ok,-j where j = 0, 1, 2, ..., k. Here flktj iS the coefficient in the solution when there are k, - j periodsuntil the zero boundstops being binding(that is, when k,- j = 0, the zero boundis no longer bindingand the solutionis equivalentto thatin equationsA2 and A3). We can find the numbersAi, f), 0i, v forj = 1, 2, 3, ..., k by solving the equations below using the initialconditions (D = 00 = 0 for] = 0 andthe initialconditions for A' and fli given in equationsA2 and A3:

tf = [I - BA-11]-'A Ai =C+DAA-f1 (Di = (I - BAj-1)-1[B0i-1 + M] 'i = DAi-I (Di + D0i-I + V. 210 BrookingsPapers on EconomicActivity, 1:2003

The Solution for t < T

The solutionsatisfies the following equations:

t =Kt { at+l)t+l+ at+l II It + Alt'2t + e+l)}

it = + (1- at+l)7t+l + t,+l(A I It + 1'2 2t + E) )} +

{(1 -t+o )it+l + t+l (2 t 2 + 2 )}

1t + P2t - P2t-i - 5-i(yp = 0

XXXt + PIt - P-It-l K(2, = 0.

Here a tilde on a variabledenotes the value of thatvariable contingent on the naturalrate of interestbeing negative.Akijt+1 is the ijth elementof the matrixAkt+1. The value kt+1depends on the numberof additionalperiods thatthe zero boundis binding(recall that here we are solving for the equilibrium on the assumptionthat we know the value of the sequence {k s 1). We can writethe system as

Lz,I LCt DtJ Lit+,I [t J, We can solve this backwardfrom the date S on which the naturalrate returnsto normalwith a probabilityof 1. We can then calculatethe path for each variableto date 0. Note that

BS-1 =DS- = 0.

By recursivesubstitution we can find a solutionof the form

(A7) Pt = Q.P. l + 4X,

(A8) it = tP, l + ot, where the coefficients are time dependent.To find the numbersAi, Qn, Ot, and Dt, considerthe solutionof the system in periodS - 1 when Bs-, = Ds-, = 0. We have Gauti B. Eggertsson and Michael Woodford 211

QS_1= As-1 (DS-l= MS-, As-, = Cs-, OS-1 =VS-1.

We can findthe numbersAt, Qt, Ot,and (Ft for periods0 to S - 2 by solving the systembelow (using the initialconditions shown above for S - 1):

nt = [I - BtAt+l]-'At At = Ct + DtAt+flIt (Ft = (I - BAt+1)-l[Bt,t+l + Mt] et = DtAt+l4Dt + DtOt+l + Vt.

Using the initial conditionP 1 = 0, we can solve for each of the endogenous variablesunder the contingencythat the liquidity trap lasts until periodS, using equationsA7 andA8. We thenuse the solutionfrom equations A2 to A6 to solve for each of the variableswhen the naturalrate revertsback to the steadystate.

Solving for {k,}ltl

A simple way to findthe value for {kIs I is to firstassume that k, is the same for all t and find the lowest k, = k so that the zero boundis never violated. Using this initial guess for IkJs one then finds the lowest value of ks so that the zero boundis never violated.Using this value for ks, and k for all otherkT, one then findsthe lowest value of ks-1so thatthe zero boundis neverviolated, and so on untilthe lowest possible value for k1is found.The value thusfound for the sequence{kIs I can be used as a new initial guess for {k}s=-1, and the procedurejust describedcan be repeateduntil the solutionconverges. For this paperwe wrotea routinein MATLABthat applies this method.The solutionconverged, and we veri- fied thatthe resultsatisfied all the necessaryconditions.