Rutgers University and Centre for Economic Policy Research

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1 Avoiding iquidiy Traps Jess Benhabib New York Universiy Sephanie Schmi-Grohé Rugers Universiy and Cenre for Economic Policy Research Marín Uribe Universiy of Pennsylvania Once he zero bound on nominal ineres raes is aken ino accoun, Taylor-ype ineres rae feedback rules give rise o uninended selffulfilling deceleraing inflaion pahs and aggregae flucuaions driven by arbirary revisions in expecaions. These undesirable equilibria exhibi he essenial feaures of liquidiy raps since moneary policy is ineffecive in bringing abou he governmen s goals regarding he sabiliy of oupu and prices. This paper proposes several fiscal and moneary policies ha preserve he appealing feaures of Taylor rules, such as local uniqueness of equilibrium near he inflaion arge, and a he same ime rule ou he deflaionary expecaions ha can lead an economy ino a liquidiy rap. We wish o hank Mike Woodford for many very helpful commens and suggesions on Benhabib, Schmi-Grohé, and Uribe (2001b) a he 1999 NBER Summer Insiue, which led o he wriing of his paper and on which our analysis is based. We are also graeful o John Cochrane, he edior, and hree anonymous referees for very helpful commens and suggesions. We acknowledge echnical assisance from he C. V. Sarr Cener of Applied Economics a New York Universiy. [Journal of Poliical Economy, 2002, vol. 110, no. 3] 2002 by The Universiy of Chicago. All righs reserved /2002/ $

2 536 journal of poliical economy I. Inroducion In recen years, here has been a revival of empirical and heoreical research aimed a undersanding he macroeconomic consequences of moneary policy regimes ha ake he form of ineres rae feedback rules. One driving force of his renewed ineres can be found in empirical sudies showing ha in he pas wo decades moneary policy in he Unied Saes is well described as following such a rule. In paricular, an influenial paper by Taylor (1993) characerizes he Federal Reserve as following a simple rule whereby he federal funds rae is se as a linear funcion of inflaion and he oupu gap wih coefficiens of 1.5 and 0.5, respecively. Taylor emphasizes he sabilizing role of an inflaion coefficien greaer han uniy, which loosely speaking implies ha he cenral bank raises real ineres raes in response o increases in he rae of inflaion. Afer his seminal paper, ineres rae feedback rules wih his feaure have become known as Taylor rules. Taylor rules have also been shown o represen an adequae descripion of moneary policy in oher indusrialized economies (see, e.g., Clarida, Galí, and Gerler 1998). A he same ime, a growing body of heoreical work has argued ha Taylor rules conribue o macroeconomic sabiliy. Researchers have arrived a his conclusion following differen roues. For example, evin, Wieland, and Williams (1999) use a nonopimizing, raional expecaions model and find ha a Taylor rule is he opimal ineres rae feedback rule in he sense ha i minimizes a quadraic loss funcion of inflaion and oupu deviaions from heir respecive arge levels. Roemberg and Woodford (1999) find a similar resul using a dynamic, opimizing, general equilibrium model and a welfare crierion for policy evaluaion. eeper (1991), Bernanke and Woodford (1997), and Clarida e al. (2000) argue ha Taylor rules conribue o aggregae sabiliy because hey guaranee he uniqueness of he raional expecaions equilibrium, whereas ineres rae feedback rules wih an inflaion coefficien of less han uniy, also referred o as passive rules, are desabilizing because hey render he equilibrium indeerminae, hus allowing for expecaions-driven flucuaions. Two imporan elemens are common across hese mehodologically diverse sudies: firs, hey resric aenion o local dynamics, or small flucuaions, around a arge level of inflaion; and second, hey do no ake ino accoun he fac ha nominal ineres raes are bounded below by zero. These wo simplificaions have serious consequences for aggregae sabiliy. The essence of he problems ha may arise once he zero bound on nominal ineres raes and global equilibrium dynamics are aken ino accoun can be illusraed by considering wo simple relaionships. The

3 avoiding liquidiy raps 537 Fig. 1. The liquidiy rap in a flexible-price model firs one is an ineres rae feedback rule R p R(p) whereby he nominal ineres rae, R, is se as an increasing and nonnegaive funcion of he inflaion rae p. The second relaionship is a seady-sae Fisher equaion, R p r p, sipulaing ha he nominal ineres rae mus equal he sum of he real ineres rae r and inflaion p (see fig. 1). Suppose ha a he arge inflaion rae p he moneary auhoriy follows a Taylor-ype, or acive, ineres rae policy in he sense ha R (p ) 1 1. Then, clearly he presence of a zero bound on nominal ineres raes and he assumpion ha he ineres rae rule is increasing in inflaion necessarily imply he exisence of a second inflaion rae, p, a which he feedback rule and he Fisher equaion inersec. A his second inersecion, inflaion is low and possibly negaive, he nominal ineres rae is low and possibly zero, and moneary policy is passive, R (p )! 1. In Benhabib, Schmi-Grohé, and Uribe (2001b), we show, in he conex of dynamic, opimizing, general equilibrium models, wih and wihou nominal rigidiies, ha he uninended seady sae p is locally indeerminae. For in is neighborhood, equilibria exis in which inflaion, ineres raes, and aggregae aciviy flucuae in response o nonfundamenal revisions in expecaions. More imporan, he second seady sae gives rise o equilibrium rajecories along which inflaion and he nominal ineres rae sar arbirarily close o he inended arges (p, R(p )) and con- 1 verge gradually o (p, R(p )). 1 A branch of he lieraure has focused on he limiaions ha he zero bound on nominal ineres raes imposes on he governmen s abiliy o conduc counercyclical moneary policy. Model-based assessmens of he coss associaed wih hese limiaions

4 538 journal of poliical economy When he economy falls ino his ype of deceleraing inflaion dynamics, i is headed o a siuaion of low and possibly negaive inflaion and low and possibly zero ineres raes in which moneary policy becomes ineffecive in bringing abou he governmen s goals regarding he sabiliy of oupu and prices. This sae of affairs has all he essenial characerisics of a liquidiy rap, a he hear of which here is a cenral bank ha is powerless o reverse he downward slide in prices hrough expansionary moneary policy in he form of lower and lower ineres raes. 2 The cenral focus of his paper is he design of fiscal and moneary policies ha preserve he Taylor rule, and wih i all is desirable local properies, around he arge rae of inflaion and real aciviy, and a he same ime eliminae equilibrium dynamics leading o he liquidiy rap. Two broad approaches o avoiding liquidiy raps are presened. In he firs one, liquidiy raps are ruled ou by means of fiscal policy while mainaining he assumpion ha moneary policy follows everywhere an ineres rae rule. The proposed sabilizaion policy feaures a srong fiscal simulus ha is auomaically acivaed when inflaion begins o decelerae. Specifically, he fiscal rule consiss of an inflaion-sensiive budge surplus schedule ha calls for lowering axes when inflaion subsides. As he economy approaches he liquidiy rap, fiscal deficis become large enough ha he low-inflaion seady sae becomes fiscally unsusainable and ceases o be a raional expecaions equilibrium. The basic insigh ha one can rule ou liquidiy raps by making hem fiscally unsusainable is due o Woodford (1999). Our firs approach o avoiding liquidiy raps hus provides heoreical suppor for he recen policy proposals emanaing mos noably from he U.S. Deparmen of he Treasury suggesing ha economies wih near-zero nominal ineres raes and hus lile room for moneary sabilizaion policy, such as Japan, should spend heir way ou of he liquidiy rap. 3 However, we arrive a his policy recommendaion for very differen reasons. The fiscal simulus eliminaes he liquidiy rap no by using he radiional Keynesian muliplier, as is he convenional are conained in Fuhrer and Madigan (1997), Orphanides and Wieland (1998), Reifschneider and Williams (1999), and Wolman (1999). 2 In moneary economics, he erm liquidiy rap is used o describe a number of differen bu relaed phenomena. For insance, he sandard exbook presenaion of liquidiy raps is concerned wih a siuaion in which he M curve is horizonal, which occurs when he demand for real balances is perfecly elasic. Krugman (1998) refers o liquidiy raps as episodes in which he nominal ineres rae vanishes. Finally, in Woodford (1999), he erm is used o describe equilibria in which real balances are a or above a finie saiaion level for which he marginal uiliy of money is zero. 3 See, e.g., he June 24, 1998, esimony of awrence Summers, hen depuy secreary of he Treasury, before a Senae Foreign Affairs subcommiee and his remarks o he World Economic Developmen Congress held on Ocober 1, 1998, in Washingon.

5 avoiding liquidiy raps 539 wisdom, bu raher by affecing he ineremporal budge consrain of he governmen. The channel hrough which he liquidiy rap is eliminaed here is more akin o Pigou s argumen on he implausibiliy of liquidiy raps. In a closed economy, he ineremporal budge consrain of he governmen is he mirror image of he ineremporal budge consrain of he represenaive household. A decline in axes increases he household s afer-ax wealh, which induces an aggregae excess demand for goods. As a consequence, he price level mus increase in order o reesablish equilibrium in he goods marke. The second approach consiss of swiching from an ineres rae rule o a money growh rae peg should inflaion embark on a self-fulfilling deceleraing pah. This alernaive is a popular one, frequenly menioned in he policy debae: when caugh in a low-inflaion equilibrium, governmens should simply sar prining money o jump-sar he economy. Krugman (1998), for example, argues forcefully for his ype of policy as a way o bring Japan ou of is curren recession. However, his recommendaion is ypically made wihou any reference o he accompanying fiscal policy. As shown in Woodford (1999), a money growh rae rule is a successful ool o avoid or escape a liquidiy rap only if coupled wih he righ fiscal policy. For insance, if he fiscal regime in place a he ime of he swich o a money growh rule is one ha guaranees fiscal susainabiliy under all circumsances, hen prining money may in fac be counerproducive since i is likely o accelerae he deflaionary spiral. In general, wha is needed o make he swich o a money growh rae rule successful is a fiscal policy ha, as nominal ineres raes approach zero, makes he governmen ineremporally insolven. The remainder of he paper is organized in five secions. Secion II presens he model and he baseline moneary-fiscal regime. Secion III characerizes local equilibrium dynamics. Secion IV shows how he economy can fall ino a liquidiy rap when moneary policy akes he form of a Taylor-ype ineres rae feedback rule. Secions V and VI develop, respecively, fiscal and moneary insrumens capable of eliminaing liquidiy raps. Secion VII closes he paper by discussing he robusness of he resuls o imporan model perurbaions such as he inroducion of nominal fricions, he reamen of ime as a discree variable, and he adopion of Gesell axes, which have been suggesed in he relaed lieraure as a possible way o avoid liquidiy raps. II. The Model In his secion we use a simple economic environmen o illusrae how a moneary-fiscal regime frequenly advocaed on he basis of aggregae sabiliy can in fac lead o expecaional raps. There are wo differences

6 540 journal of poliical economy beween our analysis and ha found in he relaed lieraure: firs, we do no resric he analysis o local dynamics around a paricular saionary sae. Second, we ake explici accoun of he zero bound on nominal ineres raes in he specificaion of ineres rae feedback rules. A. Households Consider an endowmen economy populaed by a large number of idenical infiniely lived households wih preferences defined over consumpion and real balances and described by he uiliy funcion M r ( P) 0 e u c, d, (1) where c denoes consumpion, M denoes nominal money balances, and P denoes he price level. The insananeous uiliy index u is assumed o be increasing in boh argumens, o be concave, and o saisfy ucm 1 0, so ha consumpion and real balances are Edgeworh comple- mens. We also assume ha u m(y, m) lim R(p ) Gy 1 0, u (y, m) mr c where R(p ) is a nonnegaive consan o be defined below. 4 In addiion o fia money, he represenaive household has access o nominal governmen bonds, denoed by B, ha pay he nominal ineres rae R. The household is endowed wih a consan sream of perishable goods y and pays real lump-sum axes. Is flow budge consrain is hen given by Pc P M B p RB Py. eing m { M/P denoe real balances and a { (M B)/P denoe real financial wealh, we can wrie he consrain above as c a p (R p)a Rm y, (2) where p { P/P denoes he insan rae of inflaion. The righ-hand side of his budge consrain represens he sources of income: real ineres on he household s asses ne of he opporuniy cos of holding money and he endowmen. The lef-hand side shows he uses of income: 4 These assumpions are made for analyical convenience bu can easily be relaxed. For example, in he Appendix, we consider he case of preferences ha display saiaion in real balances (eq. [A3]). In his case, u m and u cm vanish for finie values of he level of real balances. This implies ha in equilibrium he demand for real balances remains finie as he nominal ineres rae approaches zero.

7 avoiding liquidiy raps 541 consumpion, ax paymens, and savings. Households are also subjec o a borrowing limi of he form { } r 0 lim exp [R(s) p(s)]ds a() 0, (3) which prevens hem from engaging in Ponzi games. This no Ponzi game consrain says ha he household is no permied o implemen consumpion and money-holding plans ha imply ha is real deb posiion ne of money holdings grows a a rae higher han or equal o he real ineres rae. Clearly, because he uiliy funcion is increasing in consumpion and real balances, he household will always find i opimal o saisfy he borrowing limi above wih equaliy. The represenaive household chooses pahs for consumpion, real balances, and wealh so as o maximize (1) subjec o he flow budge consrain (2) and he borrowing limi (3), given is iniial real wealh, a(0), and he pahs of axes, inflaion, and nominal ineres raes. The associaed opimaliy condiions are (2), (3) holding wih equaliy, and u (c, m) p l, (4) c and u (c, m) p lr, (5) m l p l(r p R), (6) where l is a agrange muliplier associaed wih he flow budge consrain. B. Moneary and Fiscal Policy We assume ha he moneary auhoriy follows an ineres rae feedback rule of he form R p R(p). (7) We refer o moneary policy as acive a an inflaion rae p if R (p) 1 1 and as passive if R (p)! 1. oosely speaking, moneary policy is acive if he cenral bank raises he real ineres rae in response o an increase in inflaion and is passive if he cenral bank fails o do so. Three assumpions regarding he form of his feedback rule are of grea consequence for he subsequen resuls: firs, we assume ha he nominal ineres rae is nondecreasing in inflaion, ha is, R (p) 0 for all p. Second, he ineres rae feedback rule saisfies he zero bound on nominal ineres raes in ha, regardless of how low he inflaion

8 542 journal of poliical economy rae may be, he moneary auhoriy will never hreaen o implemen a negaive nominal rae. Thus R(p) 0 for all p. Third, we assume ha he cenral bank arges a rae of inflaion p 1 r and ha, in he spiri of Taylor (1993), i conducs an acive moneary policy around is inflaion arge by responding o increases (decreases) in inflaion wih a more han one-for-one increase (decrease) in he nominal ineres rae. Formally, ap 1 r : R(p ) p r p, R (p ) 1 1. (8) The governmen finances is deficis by prining money, M, and issuing nominal bonds, B, ha pay he nominal ineres rae R. We assume ha public consumpion is zero and ha he governmen levies real lumpsum axes,. Therefore, he flow budge consrain of he governmen is given by B p RB M P, which can be wrien as ȧ p (R p)a Rm. (9) By definiion, he iniial condiion a(0) saisfies A(0) a(0) p, (10) P(0) where A(0) { M(0) B(0) 1 0 denoes he iniial level of oal nominal governmen liabiliies and is predeermined. Following Benhabib e al. (2001a), we classify fiscal policies as eiher Ricardian or non-ricardian. 5 Ricardian fiscal policies are hose ha ensure ha he presen discouned value of oal governmen liabiliies converges o zero, ha is, { } r 0 lim exp [R(s) p(s)]ds a() p 0 (11) is saisfied under all possible equilibrium or off-equilibrium pahs of endogenous variables, such as he price level, he money supply, inflaion, or he nominal ineres rae. In his subsecion, we resric aenion o one paricular Ricardian fiscal policy ha akes he form Rm p aa, (12) where he funcion of ime a is chosen arbirarily by he governmen subjec o he consrain ha i is posiive and bounded below by some a 1 0. This policy saes ha consolidaed governmen revenues, ha is, ax revenues plus ineres savings from he issuance of money, are always higher han a cerain fracion a of oal governmen liabiliies. A special case of his ype of policy is a balanced-budge rule whereby ax revenues 5 Our definiion of Ricardian fiscal policies is slighly differen from ha proposed in Woodford (1995).

9 avoiding liquidiy raps 543 are equal o ineres paymens on he deb. This case emerges when a p R, provided ha R is bounded away from zero (Benhabib e al. 2001a). C. Equilibrium Equilibrium in he goods marke requires ha consumpion be equal o he endowmen c p y. (13) Given he assumpions regarding he form of he insan uiliy funcion, equaions (4) and (5) implicily define a liquidiy preference funcion of he form m p l(c, R) ; lc 1 0, l R! 0. Because in equilibrium consump- ion equals he endowmen, which is assumed o be consan, in equilibrium real balances are only a funcion of he nominal ineres rae. I hen follows direcly from (4) ha he marginal uiliy of consumpion, l, is a decreasing funcion of R: l p (R);! 0. (14) The reason why he marginal uiliy of consumpion falls when he nominal ineres rae increases is ha real balances conrac as he opporuniy cos of holding money rises. In urn, because consumpion and real balances are assumed o be Edgeworh complemens ( ucm 1 0), in equilibrium he marginal uiliy of consumpion increases wih real balances and hus falls wih he nominal ineres rae. Use he feedback rule (7) o eliminae R from (6) and (14). Then ime-differeniae (14) and combine he resuling expression wih (6) o obain he following firs-order differenial equaion in inflaion: (R(p)) R (p)p p [R(p) p r]. (15) (R(p)) The economics behind his differenial equaion are as follows: Suppose ha he real ineres rae exceeds he subjecive rae of discoun, R(p) p 1 r. In his case, households will opimally choose a declining pah for he marginal uiliy of consumpion. Bu in he endowmen economy we are considering, he equilibrium marginal uiliy of consumpion can fall only if real balances are expeced o fall (recall ha real balances and consumpion are Edgeworh complemens). Under he implied liquidiy preference funcion, a decline in real balances requires an increase in he opporuniy cos of holding money, R. Given he ineres rae feedback rule, an increase in he nominal ineres rae will be associaed wih an increase in he inflaion rae, ṗ 1 0.

10 544 journal of poliical economy Combining he governmen budge consrain (9) wih he moneary and fiscal policy rules, equaions (7) and (12), yields ȧ p [R(p) p a]a. (16) Finally, when (7) is used o eliminae R from (11), he ransversaliy condiion becomes { } r 0 lim exp [R(p(s)) p(s)]ds a() p 0. (17) A perfec-foresigh compeiive equilibrium is defined as an iniial price level P(0) 1 0 and funcions of ime p and a saisfying (10) and (15) (17), given he iniial condiion A(0) 1 0. Noe ha because of he assumed Ricardian naure of he fiscal policy regime, given a funcion of ime p, equaions (10) and (16) imply a pah for a ha saisfies he ransversaliy condiion for any iniial price level P(0) 1 0. I follows immediaely ha if an equilibrium exiss, hen he iniial price level is indeerminae. However, his nominal indeerminacy is no he focus of our analysis. We are insead concerned wih real deerminacy, ha is, he deerminacy of he funcion of ime p, which in urn governs he deerminaion of real balances and hus welfare. Thus, in analyzing perfec-foresigh equilibria, we can resric aenion o funcions of ime p ha solve he differenial equaion (15). III. ocal Equilibria Consider firs he seady-sae soluions o equaion (15), ha is, consan inflaion raes saisfying R(p) p r p. By he assumpion given in (8), p represens a seady-sae inflaion rae. Because p is a nonpredeer- mined variable, p is, in fac, a perfec-foresigh equilibrium. Because p represens he inflaion arge of he cenral bank, we refer o his level of inflaion as he inended seady sae. A he inended seady sae, moneary policy is acive, ha is, R (p ) 1 1. Bu p is no he only seady-sae equilibrium. To see his, noe ha he exisence of a seady sae a which moneary policy is acive ogeher wih he assumpions ha he ineres rae feedback rule is nondecreasing and nonnegaive implies he exisence of an inflaion rae p! p saisfying R(p ) p r p. To faciliae he analysis, we assume ha R(p) is smooh and sricly increasing and ha 6 R(p ) 1 0. Clearly, p represens a seady-sae equilibrium. This second seady-sae equilibrium has he properies ha 6 The case is reaed in he Appendix, where we show ha he resuls of his R(p ) p 0 secion also obain in his case.

11 avoiding liquidiy raps 545 moneary policy is passive and ha he inflaion rae is below he arge inflaion rae. 7 We refer o p as he uninended or liquidiy rap seady sae. Consider now he exisence of local equilibria oher han he seady saes. Noe ha, because /R is always posiive, he sign of p in equaion (15) is he same as he sign of R(p) p r. Because R (p ) 1 1, i follows ha near he acive seady sae p, p is increasing in p. Thus, if one limis he analysis o equilibria in which p remains forever in a small neighborhood around p, hen he only perfec-fore- sigh equilibrium is he acive seady sae iself, p() p p for all. This local uniqueness resul has served as an imporan heoreical argumen for advocaing he use of acive, or Taylor-ype, ineres rae feedback rules o ensure aggregae sabiliy (e.g., eeper 1991; Woodford 1996; Clarida e al. 2000). 8 If he economy is expeced o remain forever near he inflaion arge p, hen our model implies a relaionship beween he real ineres rae and inflaion ha is quie sandard. In paricular, our model is consisen wih he mainsream inuiion ha when he Fed raises ineres raes, inflaion comes down. To undersand he local equilibrium dynamics implied by he model, i is necessary o consider he quesion of how in our heoreical framework he real ineres rae and inflaion adjus o exogenous shocks. This ype of exercise is necessary because, as poined ou above, in he absence of fundamenal shocks, he only raional expecaions equilibrium in which he rae of inflaion remains forever near he inended arge is he arge iself; ha is, p() p p and R() p R(p ) for all. Thus, in he absence of fundamenal shocks, no comovemen beween inflaion and ineres raes would ever be observed. In our model economy, he Taylor arge is locally sable in he sense ha exogenous fundamenal shocks ha push he sysem away from he inended seady sae give rise o dynamics ha force he sysem o reurn o he seady sae. As we show in he nex secion, he dynamics of he model can be very differen when one allows for global equilibria, 7 In he conex of he presen flexible-price endowmen economy, he low-inflaion seady-sae equilibrium p is, in fac, preferred o he arge seady-sae equilibrium p, for i is associaed wih higher real balances and hus higher levels of uiliy. However, as shown in Benhabib e al. (2001b), in he presence of nominal rigidiies, he low-inflaion equilibrium may be welfare inferior o he arge seady sae. 8 Noe ha if he assumpion ha real balances and consumpion are Edgeworh complemens ( ucm 1 0) is replaced wih he assumpion of Edgeworh subsiuabiliy ( u cm! 0), hen he perfec-foresigh equilibrium is indeerminae in he sense ha any iniial inflaion rae near he inended arge p gives rise o an inflaion rajecory ha converges o p. For a deailed analysis of his case, see Benhabib e al. (2001a). Tha paper shows ha he local equilibrium dynamics in he case in which u cm is negaive are qualiaively idenical o hose arising in a model in which money eners in he producion funcion, so ha firms coss are increasing in he level of he nominal ineres rae.

12 546 journal of poliical economy ha is, equilibria in which he economy may move permanenly away from he Taylor arge. The local equilibrium dynamics in he neighborhood of he lowinflaion seady sae p are quie differen from hose around he arge seady sae p. Since R (p )! 1, i follows ha p is decreasing in p for p near p. This implies ha inflaion rajecories originaing in he viciniy of p converge o p and hus can be suppored as perfecforesigh equilibria. Therefore, he low-inflaion seady sae is no locally unique. 9 This is one of he reasons why passive moneary policy is considered desabilizing. IV. Falling ino iquidiy Traps A focal poin of his paper is o draw aenion o he fac ha in addiion o he local equilibria described in Secion III, here exiss a large number of equilibrium rajecories originaing arbirarily close o (and o he lef of) he inended arge p ha converge smoohly and mono- onically o he uninended, or liquidiy rap, seady sae p (see fig. 1). 10 Along such equilibrium pahs, he cenral bank, following he prescripion of he Taylor rule, coninuously eases in an aemp o reverse he persisen decline in inflaion. Bu hese effors are in vain, and indeed counerproducive, for hey inroduce furher downward pressure on inflaion. This sae of maers, in which moneary policy fails o sop he deceleraion in prices, is cenral o he noion of a liquidiy rap. Anoher aspec of hese dynamics ha is cenral o he concep of liquidiy raps is heir self-fulfilling naure: all ha is needed o fall ino he liquidiy rap is ha people expec he economy o slide ino a phase of deceleraing inflaion. The inuiion behind he resul ha a Taylor arge may open he door o liquidiy raps is as follows. Suppose ha he public expecs inflaion o fall in he fuure. Then, because moneary policy is public knowledge, agens also expec he nominal ineres rae o fall. In urn, he expeced decline in nominal ineres raes will be associaed wih an expeced increase in he demand for real balances and hence wih an expeced increase in he marginal uiliy of consumpion (recall our mainained assumpion ha ucm 1 0 and ha in equilibrium agens al- ways expec consumpion o be consan). Bu opimizing households will expec he marginal uiliy of consumpion o increase only if he 9 The local indeerminacy of equilibrium around p gives rise o aggregae insabiliy in he form of saionary sunspo equilibria. For a general resul on he exisence of saionary sunspo equilibria in coninuous-ime models displaying local indeerminacy of he perfec-foresigh equilibrium, see Shigoka (1994). 10 As is clear from he figure, here also exis hyperinflaionary rajecories originaing o he righ of p. These equilibrium pahs, however, are no he focus of our sudy.

13 avoiding liquidiy raps 547 curren real ineres rae is below he rae of ime preference. Because he Fed follows a Taylor rule, he real ineres rae is below he rae of ime preference only when curren inflaion is below he Taylor arge. The circle closes because expecaions of fuure declines in inflaion can be suppored as equilibrium oucomes when he curren inflaion rae is below he Taylor arge. I is in his paricular sense ha in his economy expecaions of fuure declines in inflaion and he nominal ineres rae can be self-fulfilling. When hese expecaional dynamics ake hold of he economy, he usual inuiion abou he relaion beween real ineres raes and inflaion breaks down. A decline in he rae of inflaion may now be iniiaed purely by expecaions of fuure declines in inflaion, and he Fed s acive response of lowering he nominal and real ineres raes canno conain he ensuing pah descending ino a liquidiy rap. This is precisely why i is so difficul o undersand he dynamics of liquidiy raps in general and Japan s curren slump in paricular: in spie of he cenral bank s aggressive lowering of ineres raes, inflaion, far from picking up, keeps edging down. V. Avoiding iquidiy Traps hrough Fiscal Policy In his secion, we develop policy schemes designed o eliminae he liquidiy rap. The sraegy is o modify fiscal policy while mainaining he assumed moneary policy. Specifically, we consider fiscal adjusmen mechanisms ha are auomaically acivaed whenever he economy embarks on a self-fulfilling pah of deceleraing inflaion. These mechanisms will rule ou liquidiy raps by making he low-inflaion seady sae fiscally unsusainable. A. An Inflaion-Sensiive Revenue Schedule Consider replacing he Ricardian fiscal policy given by (12) wih one in which he coefficien a, reflecing he sensiiviy of consolidaed governmen revenues wih respec o he level of oal governmen liabiliies, is an increasing funcion of he inflaion rae. Specifically, he fiscal policy now akes he form Rm p a(p)a; a 1 0. (18) We impose he following addiional resricions on he funcion a: and a(p ) 1 0 (19) a(p )! 0. (20)

14 548 journal of poliical economy This policy guaranees ha he inended seady sae p is a perfec- foresigh equilibrium and a he same ime rules ou he liquidiy rap p as an equilibrium oucome. To see his, combine (18) and he Taylor rule (7) wih he insan governmen budge consrain (9) o obain he following equilibrium law of moion for a: Solving his differenial equaion, one obains a() p exp which implies ha ȧ p [R(p) p a(p)]a. (21) { } 0 [R(p(s)) p(s) a(p(s))]ds a(0), { } [ ] 0 r 0 lim exp [R(s) p(s)]ds a() p a(0) lim exp a(p(s))ds. (22) r Consider a consan inflaion rajecory p p p. By condiion (19), he righ-hand side of (22) equals zero. Therefore, he ransversaliy condiion is saisfied and p p p represens a perfec-foresigh equilibrium. On he oher hand, for an inflaion rajecory in which p converges o he liquidiy rap p, condiion (20) implies ha he righ-hand side of (22) does no converge o zero (excep in he special case in which iniial oal governmen liabiliies, a(0), are exacly equal o zero), hus violaing he ransversaliy condiion. As a resul, no inflaion pah leading o he liquidiy rap can be suppored as an equilibrium oucome. Under he proposed inflaion-sensiive revenue schedule, he governmen manages o fend off he uninended low-inflaion equilibrium by hreaening o implemen a fiscal simulus package consising of a severe increase in he consolidaed defici should he inflaion rae become sufficienly low. Ineresingly, his ype of policy prescripion is wha he U.S. Treasury and a large number of academic and professional economiss are advocaing as a way for Japan o lif iself ou of is curren deflaionary rap. However, we arrive a his policy recommendaion for very differen reasons. The fiscal simulus we propose eliminaes he liquidiy rap no by using he radiional Keynesian muliplier, as is he convenional wisdom, bu raher by affecing he ineremporal budge consrain of he governmen. The channel hrough which he liquidiy rap is eliminaed here is more akin o Pigou s argumen on he implausibiliy of liquidiy raps. In a closed economy, he ineremporal budge consrain of he governmen is he mirror image of he ineremporal budge consrain of he represenaive household. A decline in axes increases he household s afer-ax wealh, which induces an aggregae excess demand for goods. Wih aggregae supply fixed, he

15 avoiding liquidiy raps 549 price level mus increase in order o reesablish equilibrium in he goods marke. 1. Targeing he Growh Rae of Nominal Governmen iabiliies The following example o rule ou liquidiy raps under ineres rae feedback rules was firs suggesed by Woodford (1999). In his example, he fiscal policy rule consiss in pegging he growh rae of oal nominal governmen liabiliies. Tha is, where k is assumed o saisfy Ȧ p k, (23) A R(p )! k! R(p ). (24) Rm p [R(p) k]a. Thus his fiscal policy rule is a special case of he one given in equaion (18) when a(p) akes he form R(p) k. In paricular, noe ha, because he Taylor rule is increasing in he rae of inflaion, so is R(p) k. Furhermore, condiion (24) implies ha resricions (19) and (20) are saisfied. I follows immediaely ha argeing he growh rae of nominal governmen liabiliies is a poenial way of eliminaing equilibrium deflaionary spirals. Clearly, his condiion implies ha a he liquidiy rap ( ), oal nom- inal governmen liabiliies grow a a rae larger han he nominal ineres rae. Thus, as he economy approaches he liquidiy rap, he presen discouned value of governmen liabiliies does no converge o zero. Therefore, such pahs for he inflaion rae canno be suppored as equilibrium oucomes. Formally, expressing A/A as (a/a) p and combining he fiscal policy rule above wih he insan governmen budge consrain (9) yields p 2. A Balanced-Budge Requiremen Consider now a fiscal policy rule consising of a zero secondary defici, ha is, P p RB, where B denoes ousanding ineres-bearing public deb. This policy rule requires ha he governmen equaes is primary surplus, P, o ineres paymens on he public deb, RB, so ha he secondary defici, given by RB P, is always equal o zero. Recalling ha a p (B/P) m, we can rewrie he balanced-budge rule as Rm p R(p)a. (25)

16 550 journal of poliical economy I follows ha a balanced-budge requiremen is also a special case of he fiscal policy rule given in (18) in which a(p) p R(p). Clearly, in his case a(p) is increasing because so is R(p). Condiion (19) is saisfied because R(p ) 1 0 by assumpion. On he oher hand, condiion (20) is no saisfied because of our mainained assumpion ha R(p ) 1 0. However, if one relaxes his assumpion by considering he case of a Taylor rule ha sipulaes zero nominal ineres raes a sufficienly low raes of inflaion, i becomes clear ha a balanced-budge rule eliminaes he liquidiy rap as long as nominal ineres rae pahs leading o he 11 liquidiy rap saisfy lim r 0 R(s)ds!. To see why, noe ha under a balanced-budge rule, by (9) and (25), he law of moion of real oal governmen liabiliies is given by ȧ p pa. Using his expression, we can wrie he ransversaliy condiion (11) as [ ] r 0 lim a(0) exp R(s)ds p 0. In he Appendix, we presen wo examples involving a Taylor rule ha ses he nominal ineres rae o zero a low raes of inflaion. In one of he examples, pahs leading o he liquidiy rap feaure nominal ineres raes ha converge o zero bu never acually reach ha floor. In he oher example, he nominal ineres rae reaches he zero bound in finie ime. We show ha in boh cases a balanced-budge rule succeeds in avering he liquidiy rap. These examples show ha i is in principle possible o escape he liquidiy rap wihou he hrea of generaing fiscal deficis. Wha is crucial, however, is a commimen of he cenral bank o lower nominal ineres raes all he way o zero as inflaion becomes sufficienly low. Any ineres rae feedback rule lacking his srong commimen leaves he door open o uninended deflaionary spirals. VI. Avoiding iquidiy Traps hrough a Moneary Regime Swich Thus far, we have sudied he design of fiscal policies capable of eliminaing liquidiy raps when he moneary auhoriy follows an ineres rae feedback rule ha is valid globally (i.e., for all possible values of he inflaion rae). An alernaive roue o avoiding self-fulfilling liquidiy raps is o modify moneary policy when he economy seems o be headed oward a low-inflaion spiral. For example, in he case of Japan, a frequenly advocaed sraegy o lif he economy ou of deflaion is for he Bank of Japan o swich o a money growh rae arge leing 11 This is how Schmi-Grohé and Uribe (2000) rule ou liquidiy raps in a discreeime cash-in-advance economy wih cash and credi goods.

17 avoiding liquidiy raps 551 ineres raes be marke deermined. In his secion, we show ha abandoning an ineres rae feedback rule in favor of a moneary arge when inflaion reaches dangerously low levels can be a successful way o avoid falling ino a deflaionary rap, bu i need no be. The effeciveness of such an alernaive will in general depend on he accompanying fiscal regime. We illusrae his general conclusion by means of wo examples. A. Swiching o a Money Growh Rule Is Ineffecive When Fiscal Policy Is Ricardian We begin by showing ha under he Ricardian fiscal policy rule given by (12), swiching from an ineres rae feedback rule o a money growh rae rule as he nominal ineres rae ges close o zero will no eliminae self-fulfilling deflaions. To esablish his resul, i is enough o show ha under his fiscal regime, self-fulfilling deflaions exis even under a money growh rae rule. 12 Specifically, assume ha moneary policy akes he form Ṁ p m, (26) M where m 1 r is a consan. 13 This moneary policy implies ha ṁ p m p. (27) m A perfec-foresigh equilibrium can hen be defined as funcions of ime c, m, p, l, and R saisfying (4), (5), (6), (13), and (27). As we have shown above, under he assumed fiscal policy he ransversaliy condiion (17) is always saisfied, so we do no include i in our definiion of equilibrium. Combining he equilibrium condiions yields he following differenial equaion in real balances: r m [u m(y, m)/u c(y, m)] ṁ p. (28) (1/m) [u cm(y, m)/u c(y, m)] Any funcion of ime m saisfying his differenial equaion represens a perfec-foresigh equilibrium. Figure 2 depics he corresponding phase diagram. Because m is a jump variable, one possible equilibrium 12 This finding is due o Woodford (1999). We repea he argumen here o make he presenaion self-conained. 13 Noe ha we allow for boh posiive and negaive raes of moneary expansion. This is of ineres for he resuls ha follow because under he fiscal regime ypically assumed in he lieraure on speculaive deflaions (namely, B p 0 a all imes), self-fulfilling deflaions occur only for negaive money growh raes (Woodford 1994).

18 552 journal of poliical economy Fig. 2. Phase diagram of m under a money growh rae arge, eq. (28) is a seady-sae equilibrium given by m p m, ˆ where mˆ is a consan saisfying u (y, m) ˆ m r m p. u (y, m) ˆ Given our mainained assumpions ha consumpion and real balances are Edgeworh complemens ( ucm 1 0) and ha he insan uiliy index is concave (so ha u mm! 0), we have ha here is a unique seady-sae equilibrium and ha m 1 0 for m 1 mˆ and m! 0 for m! m. ˆ I follows immediaely ha here exiss a coninuum of equilibria, originaing o he righ of ˆm, wih he characerisic ha he economy falls ino a deflaionary rap in which real balances grow wihou bound and he nominal ineres rae approaches zero. 14 The policy implicaion of his resul is ha when fiscal policy is Ricardian, swiching from an ineres rae feedback rule o a money growh rae rule as he economy approaches he liquidiy rap only makes hings worse since i pushes he economy o an even more severe case of deflaion. c B. Swiching o a Money Growh Rule May Be Effecive When Fiscal Policy Is No Ricardian I should be clear a his poin ha wheher he adopion of a moneary arge represens a successful ool for escaping liquidiy raps depends 14 Clearly, speculaive hyperinflaions are also possible. Such explosive price-level pahs could be ruled ou by inroducing resricions on individual preferences (as in Brock [1974, 1975]) or by assuming ha he governmen has he abiliy o guaranee a minimum redempion value for money (as in Obsfeld and Rogoff [1983]).

19 avoiding liquidiy raps 553 crucially on he assumed fiscal sance. As he previous discussion demonsraes, when fiscal policy is Ricardian, a swich o a moneary arge is likely o be counerproducive. However, he cenral resul of his subsecion is ha when fiscal policy is no Ricardian, he moneary auhoriy may be able o inflae is way ou of a liquidiy rap by argeing a sufficienly high rae of money growh. We convey his argumen by sudying a fiscal regime under which rajecories leading o a liquidiy rap are possible when moneary policy akes he form of an ineres rae feedback rule like (7) bu are impossible when he cenral bank conrols he rae of expansion of he moneary aggregae. We hen use his insigh o consruc a moneary regime ha keeps he appealing properies of a Taylor rule in he neighborhood of he arge rae of inflaion p and eliminaes he possibiliy of liquidiy raps by swiching o a money growh rae rule when he economy approaches he low-inflaion seady sae p. Consider, for example, a fiscal policy whereby public deb is exogenous, nonnegaive, and bounded by an exponenial funcion of ime. Specifically, g 0 B() Be ; B 0. (29) This expression defines a family of fiscal policies ha includes a number of special cases frequenly considered in moneary economics. Perhaps he mos commonly assumed fiscal regime is one in which public deb equals zero a all imes ( B() p 0). I also includes policies ha limi he growh rae of public deb such as he Maasrich crierion, which ses an upper bound on deb of 60 percen of gross domesic produc ( B p 0.6y and g p 0). To see ha fiscal policies belonging o he class defined in (29) are no Ricardian, consider, for example, a rajecory of nominal ineres raes converging o a consan less han g. Clearly, in his case he presen discouned value of public deb converges o infiniy, violaing he ransversaliy condiion (11). We begin by showing ha if he fiscal policy resricion (29) is combined wih he ineres rae feedback rule (7), hen self-fulfilling liquidiy raps may occur in equilibrium. For he analysis ha follows, i will prove convenien o rewrie he equilibrium condiions in erms of sequences for real balances and nominal deb raher han in erms of inflaion and real wealh, as we did in Secion IIC. Combining (4) (7) and (13) yields u c(y, m) ṁ p [r p(m) R(p(m))], (30) u cm(y, m)

20 554 journal of poliical economy Fig. 3. Phase diagram of m under an ineres rae feedback rule, eq. (30) where p(m) is a sricly decreasing funcion implicily defined by u m(y, m) u c(y, m) p R(p). The ransversaliy condiion (17) and he iniial condiion (10) become, respecively, and B() [ ] P(0) lim exp { } r 0 [R(p(m(s))) p(m(s))]ds m() exp R(p(m(s)))ds p 0 (31) 0 P(0)m(0) B(0) p A(0). (32) A perfec-foresigh equilibrium is defined as a funcion of ime m and an iniial price level P(0) 1 0 saisfying (30) (32), given an exogenous funcion of ime B saisfying (29) and A(0) 1 0. Figure 3 displays he phase diagram associaed wih equaion (30), which, of course, is qualiaively equivalen o ha corresponding o equaion (15) and shown in figure 1. In paricular, here exiss a seady sae m, associaed wih he arge inflaion rae p, and a seady sae m 1 m, associaed wih he low inflaion rae p, ha is, wih he liquidiy rap. Bu here exis oher soluions o he differenial equaion (30). Specifically, here exiss an infinie number of rajecories of real balances ha originae in he viciniy of m and converge o m as well as a coninuum of rajecories saring in a neighborhood o he righ of

21 avoiding liquidiy raps 555 m ha also converge o m. These rajecories will represen perfecforesigh equilibria if hey saisfy he ransversaliy condiion (31). To he exen ha g! R(p(m )) or B p 0, equaion (31) will hold for any rajecory m converging o m. Therefore, if g! R(p(m )) or B p 0, here is a coninuum of perfec-foresigh equilibria saring arbirarily close 15 o he inended seady sae m ha lead o he liquidiy rap m. On he oher hand, when moneary policy akes he form of a money growh rae rule like he one given by equaion (26) wih m 0, he large number of perfec-foresigh equilibria ha arise under he ineres rae feedback rule (7) are reduced o a unique one. To see his, noe ha in his case a perfec-foresigh equilibrium is a funcion of ime m saisfying (28) and he ransversaliy condiion (17), which can be wrien as [ u (y, m) ] u m(y, m) lim exp { [ r 0 u c(y, m) ] } u m(y, m) m ds M(0) exp ds B() p 0, (33) 0 c given an exogenous funcion of ime B saisfying (29) and he iniial condiion M(0) 1 0. We have already characerized he soluions o he differenial equaion (28), which are summarized in figure 2: here exiss a unique seady sae mˆ and a coninuum of rajecories saring o he righ of mˆ and converging o infiniy. However, under he fiscal policy considered here, none of he soluions in which real balances grow wihou bound can be suppored as a compeiive equilibrium. The reason is ha as m converges o infiniy, he nominal ineres rae, u m(y, m)/u c(y, m), con- verges o zero, implying, given he mainained assumpion of a nonnegaive rae of money growh, ha he firs erm of he ransversaliy condiion (33) fails o approach zero as ges large. As a resul, under he fiscal policy resricion (29), a money growh rae peg is a successful ool o fend off self-fulfilling liquidiy raps. 1. A Moneary Policy Regime Swich An ineresing quesion ha emerges from he resuls above is wheher he cenral bank could design a moneary policy ha akes he form of a Taylor rule near he inflaion arge p and swiches o a money growh rae rule when he economy appears o be sliding ino a liquidiy rap. 15 To complee he characerizaion of equilibrium, we noe ha, in conras o he case under he Ricardian fiscal policy (12), he equilibrium displays nominal deerminacy in he sense ha, given funcions B and m, P(0) is uniquely deermined by (32).

22 556 journal of poliical economy One obsacle ha he consrucion of such a policy swich mus ackle is o preven an anicipaed discree jump in he price level a he ime of he regime change. Besides price-level smoohing, cenral bank behavior in developed counries has been described as pursuing a smooh rae of inflaion. This characerizaion is reflecive of he observed remarkable inflaion ineria. In he conex of our model, he equilibrium price level and inflaion rae are coninuous if real balances and heir ime derivaive are coninuous. 16 Accordingly, we show how o design a moneary policy swich from a Taylor rule o a money growh rae rule ha eliminaes he liquidiy rap while guaraneeing coninuiy of m and ṁ. e m be he hreshold value of real balances below which he cenral bank follows he ineres rae feedback rule given by (7) and above which i pegs he growh rae of he money supply as described by equaion (26). The dynamics of real balances are herefore given by (30) for m m and by (28) for m 1 m. Tha is, { u c(y, m) [r p(m) R(p(m))] for m m u cm(y, m) ṁ p 1 (34) 1 u cm(y, m) u m(y, m) r m for m 1 m. m u(y, m) u (y, m) [ ] [ ] c One can choose he money growh rae m and he hreshold m in such a way ha he differenial equaion (34) has a unique seady sae a m, so ha a he seady sae, moneary policy akes he form of a Taylor rule and inflaion coincides wih he arge p. Figure 4 superimposes he phase diagrams corresponding o he ineres rae feedback rule (fig. 3) and he money growh rae rule (fig. 2). I is eviden from figure 4 ha in order for m o be he unique seady sae of (34), m and m mus be such ha m! m ˆ! m! m. In urn, he resricion m! m ˆ! m requires seing m as follows: c p! m! p. (35) To see why, noe ha m, m, ˆ and m are implicily given by, respecively, 16 This follows from he fac ha, regardless of he moneary regime, (4) (6) imply ha in equilibrium l u m(y, m) p p r l u c(y, m) and l u cm(y, m) p. l u c(y, m)m From he fac ha p exiss everywhere, we have ha P exiss and is coninuous.

23 avoiding liquidiy raps 557 Fig. 4. Phase diagram of m under a moneary policy regime swich, eq. (34) u ˆ ˆ m(y, m )/u c(y, m ) p r p, u m(y, m)/u c(y, m) p r m, and u m(y, m )/u c(y, m ) p r p. However, no any value of m in he inerval (m, ˆ m ) guaranees he coninuiy of m. This furher requiremen will be me only if a m he righ-hand sides of (28) and (30) are equal o each oher. I is apparen from figure 4 and from our characerizaion of equaions (28) and (30) ha here exiss a leas one such value of real balances in he inerval (m, ˆ m ). (Figure 4 is drawn under he assumpion ha here exiss a unique such m.) We pick any one of hese values for real balances as he hreshold for he moneary policy swich. The solid line in figure 4 depics he phase diagram of equaion (34) when m and m are chosen so ha m is he only consan soluion o ha differenial equaion and ṁ is coninuous. The arge level of real balances m is no jus a seady-sae soluion o (34) bu indeed represens a perfec-foresigh equilibrium. For, as we showed earlier, i saisfies he ransversaliy condiion (31). In addiion o m, equaion (34) admis a coninuum of soluions ha begin o he righ of m and converge o infiniy. However, none of hese soluions can be suppored as perfec-foresigh equilibria because, as real balances cross he hreshold m, moneary policy swiches from an ineres rae feedback rule o a money growh rae rule and real balances embark on an explosive pah ha, provided ha m is nonnegaive, violaes he ransversaliy condiion (33). 17 We conclude ha he proposed moneary regime swich is successful a ruling ou he liquidiy rap, preserving a Taylor rule around he 17 The requiremen given (35), calls for I follows ha he moneary policy m 0, p 1 0. swich is ineffecive in eliminaing he liquidiy rap if he cenral bank arges a negaive rae of inflaion.

24 558 journal of poliical economy arge rae of inflaion p, and guaraneeing he coninuiy of he price level and inflaion. VII. Discussion and Conclusion The zero bound on nominal ineres raes makes economies in which moneary policy akes he form of an ineres rae feedback rule prone o uninended equilibrium oucomes. When hese undesirable circumsances occur, he moneary auhoriy finds iself powerless o bring abou he policy objecives of he governmen. I is precisely his inabiliy of moneary policy o affec key macroeconomic variables, such as he level of inflaion and he volailiy of oupu and prices, ha is a he hear of he concep of a liquidiy rap. Besides he explici consideraion of he zero bound on nominal ineres raes, perhaps he mos noable difference beween our model and hose ha sress he desirabiliy of Taylor rules is he absence of nominal rigidiies. However, he possibiliy of falling ino a liquidiy rap as a consequence of Taylor-ype rules is no limied o he simple flexibleprice environmen presened in his paper. In Benhabib e al. (2001b), we show ha Taylor rules also engender liquidiy raps in environmens wih sluggish price adjusmen. In his ype of model, he liquidiy rap involves indeerminacy no only of inflaion and real balances, as in he model considered in his paper, bu also of he level of aggregae demand. The policy recommendaions aimed a eradicaing liquidiy raps proposed in Secions V and VI are also effecive in economies wih sicky prices. For hose recommendaions involve he violaion of a ransversaliy condiion in he even ha he economy falls ino a liquidiy rap. The violaion of his long-run resricion depends on he asympoic behavior of he endogenous variables of he model, which is independen of shor-run nominal price rigidiies. A furher difference beween he heoreical environmen considered in his paper and ha sudied in par of he relaed lieraure is our reamen of ime as a coninuous variable. Again, neiher he exisence of a liquidiy rap emerging as a consequence of he adopion of a Taylor rule nor he effeciveness of he proposed remedies is affeced by his assumpion in any imporan way. Schmi-Grohé and Uribe (2000) analyze a discree-ime cash-in-advance model wih cash and credi goods and show ha a Taylor rule in combinaion wih a lower bound on nominal raes gives rise o an uninended liquidiy rap. Because he naure of his undesirable equilibrium is idenical o ha idenified in his paper, he long-run resricions ha are capable of eliminaing liquidiy raps in he coninuous-ime model will also be applicable under discree ime. The policies considered in his paper can be viewed as inended o