Managing a Liquidity Trap: Monetary and Fiscal Policy

Transcription

1 Managing a Liquidiy Trap: Moneary and Fiscal Policy Iván Werning, MIT This Version: March 212 Absrac I sudy moneary and fiscal policy in liquidiy rap scenarios, where he zero bound on he nominal ineres rae is binding. I work wih a coninuous-ime version of he sandard New Keynesian model. Wihou commimen he economy suffers from deflaion and depressed oupu. I show ha, surprisingly, boh are exacerbaed wih greaer price flexibiliy. I find ha he opimal ineres rae is se o zero pas he liquidiy rap and jumps discreely up upon exi. Inflaion may be posiive hroughou, so he absence of deflaion is no evidence agains a liquidiy rap. Oupu, on he oher hand, always sars below is efficien level and rises above i. Thus, moneary policy promoes inflaion and an oupu boom. I show ha he opimal prolongaion of zero ineres raes is relaed o he laer, no he former. I hen sudy fiscal policy and show ha, regardless of parameers ha govern he value of fiscal mulipliers during normal or liquidiy rap imes, a he sar of a liquidiy rap opimal spending is above is naural level. However, i declines over ime and goes below is naural level. I propose a decomposiion of spending according o opporunisic and simulus moives. The former is defined as he level of governmen purchases ha is opimal from a saic, cos-benefi sandpoin, aking ino accoun ha, due o slack resources, shadow coss may be lower during a slump; he laer measures deviaions from he former. I show ha simulus spending may be zero hroughou, or swich signs, depending on parameers. Finally, I consider he hybrid where moneary policy is discreionary, bu fiscal policy has commimen. In his case, simulus spending is posiive and iniially increasing hroughou he rap. For useful discussions I hank Manuel Amador, George-Marios Angeleos, Emmanuel Farhi, Jordi Galí, Karel Merens, Ricardo Reis, Pedro Teles as well as seminar paricipans. Special hanks o Ameya Muley and Mahew Rognlie for deailed commens and suggesions. All remaining errors are mine. 1

2 1 Inroducion The 27-8 crisis in he U.S. led o a seep recession, followed by aggressive policy responses. Moneary policy wen full il, cuing ineres raes rapidly o zero, where hey have remained since he end of 28. Wih convenional moneary policy seemingly exhaused, fiscal simulus worh $787 billion was enaced by early 29 as par of he American Recovery and Reinvesmen Ac. Unconvenional moneary policies were also pursued, saring wih quaniaive easing, purchases of long-erm bonds and oher asses. In Augus 211, he Federal Reserve s FOMC saemen signaled he inen o keep ineres raes a zero unil a leas mid 213. Similar policies have been followed, a leas during he peak of he crisis, by many advanced economies. Forunaely, he kind of crises ha resul in such exreme policy measures have been relaively few and far beween. Perhaps as a consequence, he debae over wheher such policies are appropriae remains largely unseled. The purpose of his paper is o make progress on hese issues. To his end, I reexamine moneary and fiscal policy in a liquidiy rap, where he zero bound on nominal ineres rae binds. I work wih a sandard New Keynesian model ha builds on Eggersson and Woodford (23). 1 In hese models a liquidiy rap is defined as a siuaion where negaive real ineres raes are needed o obain he firs-bes allocaion. I adop a deerminisic coninuous ime formulaion ha urns ou o have several advanages. I is well suied o focus on he dynamic quesions of policy, such as he opimal exi sraegy, wheher spending should be fron- or back-loaded, ec. I also allows for a simple graphical analysis and delivers several new resuls. The alernaive mos employed in he lieraure is a discree-ime Poisson model, where he economy sars in a rap and exis from i wih a consan exogenous probabiliy each period. This specificaion is especially convenien o sudy he effecs of subopimal and simple Markov policies because he equilibrium calculaions hen reduce o finding a few numbers bu does no afford any comparable advanages for he opimal policy problem. I consider he policy problem under commimen, under discreion and for some inermediae cases. I am ineresed in moneary policy, fiscal policy, as well as heir inerplay. Wha does opimal moneary policy look like? How does he commimen soluion compare o he discreionary one? How does i depend on he degree of price sickiness? How can fiscal policy complemen opimal moneary policy? Can fiscal policy miigae he problem creaed by discreionary moneary policy? To wha exen is spending gov- 1 Eggersson (21, 26) sudy governmen spending during a liquidiy rap in a New Keynesian model, wih he main focus is on he case wihou commimen and implici commimen o inflae afforded by rising deb. Chrisiano e al. (211), Woodford (211) and Eggersson (211) consider he effecs of spending on oupu, compuing fiscal mulipliers, bu do no focus on opimal policy. 2

3 erned by a concern o influence he privae economy as capured by "fiscal mulipliers", or by simple cos-benefi public finance consideraions? I firs sudy moneary policy in he absence of fiscal policy. When moneary policy lacks commimen, deflaion and depression ensue. Boh are commonly associaed wih liquidiy raps. Less familiar is ha boh oucomes are exacerbaed by price flexibiliy. Thus, one does no need o argue for a large degree of price sickiness o worry abou he problems creaed by a liquidiy rap. In fac, quie he conrary. I show ha he depression becomes unbounded as we converge o fully flexible prices. The inuiion for his resul is ha he main problem in a liquidiy rap is an elevaed real ineres rae. This leads o depressed oupu, which creaes deflaionary pressures. Price flexibiliy acceleraes deflaion, raising he real ineres rae furher and only making maers worse. As firs argued by Krugman (1998), opimal moneary policy can improve on his dire oucome by commiing o fuure policy in a way ha affecs curren expecaions favorably. In paricular, I show ha, i is opimal o promoe fuure inflaion and simulae a boom in oupu. I esablish ha opimal inflaion may be posiive hroughou he episode, so ha deflaion is compleely avoided. Thus, he absence of deflaion, far from being a odds wih a liquidiy rap, acually may be evidence of an opimal response o such a siuaion. I show ha oupu sars below is efficien level, bu rises above i owards he end of he rap. Indeed, he boom in oupu is larger han ha simulaed by he inflaionary promise. There are a number of ways moneary policy can promoe inflaion and simulae oupu. Moneary easing does no necessarily imply a low equilibrium ineres rae pah. Indeed, as in mos moneary models, he nominal ineres rae pah does no uniquely deermine an equilibrium. Indeed, an ineres rae of zero during he rap ha becomes posiive immediaely afer he rap is consisen wih posiive inflaion and oupu afer he rap. 2 I show, however, ha he opimal policy wih commimen involves keeping he ineres rae down a zero longer. The coninuous ime formulaion helps here because i avoids ime aggregaion issues ha may oherwise obscure he resul. Some of my resuls echo findings from prior work based on simulaions for a Poisson specificaion of he naural rae of ineres. Chrisiano e al. (211) repors ha, when he cenral bank follows a Taylor rule, price sickiness increases he decline in oupu during a liquidiy rap. Eggersson and Woodford (23), Jung e al. (25) and Adam and Billi 2 For example, a zero ineres during he rap and an ineres equal o he naural rae ouside he rap. This is he same pah for he ineres rae ha resuls wih discreionary moneary policy. However, in ha case, he oucome for inflaion and oupu is pinned down by he requiremen ha hey reach zero upon exiing he rap. Wih commimen, he same pah for ineres raes is consisen wih higher inflaion and oupu upon exi. 3

4 (26) find ha he opimal ineres rae pah may keep i a zero afer he naural rae of ineres becomes posiive. To he bes of my knowledge his paper provides he firs formal resuls explaining hese findings for inflaion, oupu and ineres raes. An implicaion of my resul is ha he ineres rae should jump discreely upon exiing he zero bound a propery ha can only be appreciaed in coninuous ime. Thus, even when fundamenals vary coninuously, opimal policy calls for a disconinuous ineres rae pah. Turning o fiscal policy, I show ha, here is a role for governmen spending during a liquidiy rap. Spending should be fron-loaded. A he sar of he liquidiy rap, governmen spending should be higher han is naural level. However, during he rap spending should fall and reach a level below is naural level. Inuiively, opimal governmen spending is counercyclical, i leans agains he wind. Privae consumpion sars ou below is efficien level, bu reaches levels above is efficien level near he end of he liquidiy rap. The paern for governmen spending is jus he opposie. The opimal paern for oal governmen spending masks wo poenial moives. Perhaps he mos obvious, especially wihin he conex of a New Keynesian model, is he macroeconomic, counercyclical one. Governmen spending affecs privae consumpion and inflaion hrough dynamic general equilibrium effecs. In a liquidiy rap his may be paricularly useful, o miigae he depression and deflaion associaed wih hese evens. However, a second, ofen ignored, moive is based on he idea ha governmen spending should reac o he cycle even based on saic, cos-benefi calculaions. In a slump, he wage, or shadow wage, of labor is low. This makes i is an opporune ime o produce governmen goods. During he debaes for he 29 ARRA simulus bill, varians of his argumen were pu forh. Based on hese noions, I propose a decomposiion of spending ino "simulus" and "opporunisic" componens. The laer is defined as he opimal saic level of governmen spending, aking privae consumpion as given. The former is jus he difference beween acual spending and opporunisic spending. I show ha he opimum calls for zero simulus a he beginning of a liquidiy rap. Thus, my previous resul, showing ha spending sars ou posiive, can be aribued enirely o he opporunisic componen of spending. More surprisingly, I hen show ha for some parameer values simulus spending is everywhere exacly zero, so ha, in hese cases, opporunisic spending accouns for all of governmen spending policy during a liquidiy rap. Of course, opporunisic spending does, incidenally, influence consumpion and inflaion. Bu he poin is ha hese consideraions need no figure ino he calculaion. In his sense, public finance rumps macroeconomic policy. 4

5 Anoher implicaion is ha, in such cases, commimen o a pah for governmen spending is superfluous. A naive, fiscal auhoriy ha acs wih full discreion and performs he saic cos-benefi calculaion chooses he opimal pah for spending. These resuls assume ha moneary policy is opimal. Things can be quie differen when moneary policy is subopimal due o lack of commimen. To address his I sudy a mixed case, where moneary policy is discreionary bu fiscal policy has he power o commi o a governmen spending pah. Posiive simulus spending emerges as a way o figh deflaion. Indeed, he opimal inervenion is o provide posiive simulus spending ha rises over ime during he liquidiy rap. Back-loading simulus spending provides a bigger bang for he buck, boh in erms of inflaion and oupu. Since price seing is forward looking, spending near he end promoes inflaion boh near he end and earlier. In addiion, any improvemen in he real rae of reurn near he end of he liquidiy rap improves he oupu oucome level for earlier daes. Boh reasons poin owards increasing simulus spending. If he fiscal auhoriy can commi pas he rap, hen i is opimal o promise lower spending immediaely afer he rap, and converge owards he naural rae of spending afer ha. Spending feaures a discree downward jump upon exiing he rap. Inuiively, afer he rap, once he flexible price equilibrium is aainable, lower governmen spending leads o a consumpion boom. This is beneficial, for he same reasons ha moneary policy wih commimen promoes a boom, because i raises he consumpion level during he rap. Thus, he commimen o lower spending afer he rap aemps o mimic he expansionary effecs ha he missing moneary commimens would have provided. The model is cas in coninuous ime and his is one of he disinguishing feaures of my analysis. Why is coninuous ime simpler and more powerful here? One answer is ha coninuous ime is useful whenever one needs o solve for endogenous swiching imes, as in he balance of paymen crisis model in Krugman (1979), he Baumol-Tobin model of invenory money demand, or in oher ss menu-cos models. This is also he siuaion here because he soluion has a bang-bang propery, wih he ineres rae being kep a zero up o some endogenous exi ime. In oher words, he advanage has lile o do wih echnical ools, such as he use of Ponryagin s maximum principle, and more o do wih he fac ha ime is of he essence, ha is, we are solving for an exi ime and i is simpler and naural o allow ha key choice variable o be coninuous. The res of he paper is organized as follows. Secion 2 inroduces he model. Secion 3 sudies he equilibrium wihou fiscal policy when moneary policy is conduced wih discreion. Secion 4 sudies opimal moneary policy wih commimen. Secion 5 adds fiscal policy and sudies he opimal pah for governmen spending alongside opimal 5

6 moneary policy. Secion 6 considers mixed cases where moneary policy is discreionary, bu fiscal policy enjoys commimen. 2 A Liquidiy Trap Scenario The model is a coninuous-ime version of he sandard New Keynesian model. The environmen feaures a represenaive agen, monopolisic compeiion and Calvo-syle sicky prices; i absracs from capial invesmen. I spare he reader anoher rendering of he deails of his sandard seing (see e.g. Woodford, 23, or Galí, 28) and skip direcly o he well-known log-linear approximaion of he equilibrium condiions which I use in he remainder of he paper. Euler Equaion and Phillips Curve. zero inflaion, are The equilibrium condiions, log linearized around ẋ() = σ 1 (i() r() π()) π() = ρπ() κx() i() (1a) (1b) (1c) where ρ, σ and κ are posiive consans and he pah {r()} is exogenous and given. We also require a soluion (π(), x()) o remain bounded. The variable x() represens he oupu gap: he log difference beween acual oupu and he hypoheical oupu ha would prevail a he efficien, flexible price, oucome. Inflaion is denoed by π() and he nominal ineres rae by i(). Finally, r() sands for he naural rae of ineres, i.e. he real ineres rae ha would prevail in an efficien, flexible price, oucome wih x() = hroughou. Equaion (1a) represens he consumer s Euler equaion. Oupu growh, equal o consumpion growh, is an increasing funcion of he real rae of ineres, i() π(). The naural rae of ineres eners his condiion because oupu has been replaced wih he oupu gap. Equaion (1b) is he New-Keynesian, forward-looking Phillips curve. I can be resaed as saying ha inflaion is proporional, wih facor κ >, o he presen value of fuure oupu gaps, ˆ π() = κ e ρs x( + s)ds. Thus, posiive oupu gaps simulae inflaion, while negaive oupu gaps produce deflaion. Finally, inequaliy (1c) is he zero-lower bound on nominal ineres raes (hereafer, 6

7 ZLB). As for he consans, ρ is he discoun rae, σ 1 is he ineremporal elasiciy of subsiuion and κ conrols he degree of price sickiness. Lower values of κ imply greaer price sickiness. As κ we approach he benchmark wih perfecly flexible prices, where high levels of inflaion or deflaion are compaible wih minuscule oupu gaps. A number of caveas are in order. The model I use is he very basic New Keynesian seing, wihou any bells and whisles. Basing my analysis on his simple model is convenien because i lies a he cener of many richer models, so we may learn more general lessons. I also faciliaes he normaive analysis, which could quickly become inracable oherwise. On he oher hand, he analysis absracs from unemploymen, and omis disorionary axes, financial consrains and oher fricions which may be relevan in hese siuaions. Quadraic Welfare Loss. I will evaluae oucomes using he quadraic loss funcion L 1 2 ˆ e ρ ( x() 2 + λπ() 2) d. (2) According o his loss funcion i is desirable o minimize deviaions from zero for boh inflaion and he oupu gap. The consan λ conrols he relaive weigh placed on he inflaionary objecive. The quadraic naure of he objecive is convenien and can be derived as a second order approximaion o welfare around zero inflaion when he flexible price equilibrium is efficien. 3 Such an approximaion also suggess ha λ = λ/κ for some consan λ, so ha λ as κ, as prices become more flexible, price insabiliy becomes less harmful. The Naural Rae of Ineres. The pah for he naural rae {r()} plays a crucial role in he analysis. Indeed, if he naural rae were always posiive, so ha r() for all, hen he flexible price oucome wih zero inflaion and oupu gap, π() = x() = for all, would be feasible and obained by leing i() = r() for all. This oucome is also opimal, since i is ideal according o he loss funcion (2). The siuaion described in he previous paragraph amouns o he case where he ZLB consrain (1c) is always slack. The focus of his paper is on siuaions where he ZLB consrain binds. Thus, I am ineresed in cases where r() < for some range of ime. 3 In order o be efficien, he equilibrium requires a consan subsidy o producion o undo he monopolisic markup. An alernaive quadraic objecive ha does no assume he flexible price equilibrium is efficien is 1 ( 2 e ρ (x() x) 2 + λπ() 2) d for x >. Mos of he analysis would carry hrough o his case. 7

8 For a few resuls i is useful o furher assume ha he he economy sars in a liquidiy rap ha i will evenually and permanenly exi a some dae T > : r() < < T r() T. I call such a case a liquidiy rap scenario. A simple example is he sep funcion r [, T) r() = r [T, ) where r > > r. I use he sep funcion case in some figures and simulaions, bu i is no required for any of he resuls in he paper. Finally, I also make a echnical assumpion: ha r(s) is bounded and ha he inegral r(s)ds be well defined and finie for any. 3 Moneary Policy wihou Commimen Before sudying opimal policy wih commimen, i is useful o consider he siuaion wihou commimen, where he cenral bank is benevolen bu canno credibly announce plans for he fuure. Insead, i acs opporunisically a each poin in ime, wih absolue discreion. This provides a useful benchmark ha illusraes some feaures commonly associaed wih liquidiy raps, such as deflaionary price dynamics and depressed oupu. I will also derive some less expeced implicaions on he role of price sickiness. The oucome wihou commimen is laer conrased o he opimal soluion wih commimen. 3.1 Deflaion and Depression To isolae he problems creaed by a complee lack of commimen, I rule ou explici rules as well as repuaional mechanisms ha bind or affec he cenral bank s acions direcly or indirecly. I consruc he unique equilibrium as follows. 4 For T he naural rae is posiive, r() = r >, so ha, as menioned above, he ideal oucome (π(), x()) = (, ) is aainable. I assume ha he cenral bank can guaranee his ou- 4 In his secion, I proceed informally. Wih coninuous ime, a formal sudy of he no-commimen case requires a dynamic game wih commimen over vanishingly small inervals. 8

9 x ẋ = = T r π = π Figure 1: The equilibrium wihou commimen, feauring i() = for T and reaching (, ) a = T. come so ha (π(), x()) = (, ) for T. 5 Taking his as given, a all earlier daes < T he cenral bank will find i opimal o se he nominal ineres rae o zero. The resuling no-commimen oucome is hen uniquely deermined by he ODEs (1a) (1b) wih i() = for T and he boundary condiion (π(t), x(t)) = (, ). This siuaion is depiced in Figure 1 which shows he dynamical sysem (1a) (1b) wih i() = and depics a pah leading o (, ) precisely a = T. Oupu and inflaion are boh negaive for < T as hey approach (, ). Noe ha he loci on which (π(), x()) mus ravel owards (, ) is independen of T, bu a larger T requires a saring poin furher away from he origin. Thus, iniial inflaion and oupu are boh decreasing in T. Indeed, as T we have ha π(), x(). Proposiion 1. Consider a liquidiy rap scenario, wih r() < for < T and r() for T. Le π nc () and x nc () denoe he equilibrium oucome wihou commimen. Then 5 Alhough his seems like a naural assumpion, i presumes ha he cenral bank somehow overcomes he indeerminacy of equilibria ha plagues hese models. A few ideas have been advanced o accomplish his, such as adhering o a Taylor rule wih appropriae coefficiens, or he fiscal heory of he price level. However, boh assume commimen on and off he equilibrium pah. Alhough his issue is ineresing, i seem compleely separae from he zero lower bound. Thus, he assumpion ha (π(), x()) = (, ) can be guaraneed for T allows us o focus on he ineracion beween no commimen and a liquidiy rap scenario. 9

10 inflaion and oupu are zero afer = T and sricly negaive before ha: π nc () = x nc () = T π nc () < x nc () < < T. Moreover, π() and x() are sricly increasing in for < T. In he limi as T, if he naural rae saisfies T r(; T)ds, hen π nc (, T), x nc (, T). The equilibrium feaures deflaion and depression. The severiy of boh depend, among oher hings, on he duraion T of he liquidiy rap. Boh becomes unbounded as T. In his sense, discreionary policy making may have very adverse welfare implicaions. This oucome coincides wih he opimal soluion wih commimen if one consrains he problem by imposing (π(t), x(t)) = (, ). In oher words, he abiliy o commi o oucomes wihin he inerval [, T) is irrelevan; also, he abiliy o commi once = T is reached is also irrelevan. Wha is crucial is he abiliy o commi ex ane a < T o oucomes for = T. How can he oucome be so dire? The main disorion is ha he real ineres rae is se oo high during he liquidiy rap. This depresses consumpion. Imporanly, his effec accumulaes over ime. Even wih zero inflaion consumpion becomes depressed by σ 1 T r()ds. For example, wih log uiliy σ = 1 if he naural rae is -4% and he rap lass wo years he loss in oupu is a leas 8%. Moreover, maers are jus made worse by deflaion, which raises he real ineres rae even more, furher depressing oupu, leading o even more deflaion, in a vicious cycle. Noe ha i is he lack of commimen during he liquidiy rap < T o policy acions and oucomes afer he liquidiy rap T ha is problemaic. Policy commimen during he liquidiy rap < T is no useful. Neiher is he abiliy o announce a credible plan a = T for he enire fuure T. Indeed, if we add (π(t), x(t)) = (, ) as a consrain, hen he no commimen oucome is opimal, even when he cenral bank enjoys full commimen o any choice over (π(), x(), i()) =T saisfying (1a) (1b) for < T and > T. Wha is valuable is he abiliy o commi during he liquidiy rap o policy acions and oucomes afer he liquidiy rap. In paricular, o somehing oher han (π(t), x(t)) = (, ). 1

11 3.2 Elbow Room wih a Higher Inflaion Targe: The Value of Commimen Before sudying opimal policy i is useful o consider he effecs of commimen o simple non-opimal policies ha avoid he depression and deflaion oucomes obained wih full discreion. Consider a plan ha keeps inflaion and oupu gap consan a π() = r > x() = 1 r > for all. κ I follows ha i() = r() + π(), so ha i() = for < T while i() = r + π > r > for T. Alhough his policy is no opimal, i behaves well in he limi as prices become fully flexible. Indeed, in his limi as κ he oupu gap converges uniformly o zero while inflaion remains consan. Thus, if we adop he naural case where λ = λ/κ, he loss funcion converges o is ideal value of zero, L(κ). Compare his o he dire oucome wihou commimen in Proposiion 2, where he oupu gap and losses converge o. Jus as in he case wihou commimen, his simple policy ses he nominal ineres rae o zero during he liquidiy rap, for < T. Noe ha afer he rap, for > T, he nominal ineres rae is acually se o a higher level han he case wihou commimen. Thus, he advanages of his simple policy do no hinge on lower nominal ineres raes. Quie he conrary, higher inflaion here coincides wih higher nominal ineres raes, due o he Fischer effec. One may sill describe he oucome as resuling from looser moneary policy, bu he poin is ha he kind of moneary easing needed o avoid he deflaion and depression does no require lower equilibrium nominal ineres raes. Obviously, hese observaions ranslaes ino long erm ineres raes a = : a commimen o looser fuure moneary policy does no necessarily ranslae ino lower yields on long erm bonds. As we shall see in he nex secion, he opimal policy wih commimen does feaure lower, indeed zero, nominal ineres raes. This idea is more general. For any pah for he naural ineres rae {r()}, se a consan inflaion rae given by π() = π = min r() and an oupu gap of x() = x = κ π. This plan is feasible wih a non-negaive nominal ineres raes i(). These simple policy capure he main idea behind calls o olerae higher inflaion arges ha leave more elbow room for moneary policy during 11

12 liquidiy raps (e.g. Summers, 1991; Blanchard e al., 21). However, given he forward looking naure of inflaion in his model, wha is crucial is he commimen o higher inflaion afer he liquidiy rap. This conrass wih he convenional argumen, where a higher inflaion rae before he rap serves as a precauionary sacrifice for fuure liquidiy raps. I is perhaps surprising ha commimen o a simple policy can avoid deflaion and depressed oupu alogeher. Of course, hey do so a he expense of inflaion and oversimulaed oupu. If he required inflaion arge π or oupu gap x are large, or if he duraion of he rap T is small, hese plans may be quie far from opimal, since hey require a permanen sacrifice for he loss funcion. 6 This moivaes he sudy of opimal moneary policy which I ake up in he nex secion. 3.3 Harmful Effecs from Price Flexibiliy I now reurn o he case wihou wihou commimen. How is his bleak oucome affeced by he degree of price sickiness? One migh expec hings o improve when prices are more flexible. Afer all, he main fricion in New Keynesian models is price rigidiies, suggesing ha oucomes should improve as prices become more flexible. The nex proposiion shows, perhaps counerinuiively, ha he reverse is acually he case. Proposiion 2. Wihou commimen higher price flexibiliy leads o more deflaion and lower oupu: if κ < κ hen π nc (, κ ) < π nc (, κ) < and x nc (, κ ) < x nc (, κ) < for all < T. Indeed, for given T > and < T in he limi as κ hen π(, κ), x(, κ) and L(κ). Sicky prices are beneficial because hey dampen deflaion, his in urn miigaes he depression. In fac, he mos favorable oucome is obained when prices are compleely rigid, κ =. A he oher end of he specrum, in he limi of perfecly flexible prices, as κ, he depression and deflaion become unbounded. 6 The reason he oupu gap x is sricly posiive is he New Keynesian model s non-verical long-run Phillips curve. Some papers have explored modificaions of he New Keynesian model ha inroduce indexaion o pas inflaion. Some forms of full indexaion imply ha a consan level of inflaion affecs neiher oupu nor welfare. Thus, wih he righ form of indexaion very simple policies may be opimal or close o opimal. Of course, his is no he case in he presen model wihou indexaion. 12

13 To see his more clearly, noe ha he Phillips curve equaion (1b) implies ha, for a given negaive oupu gap, a higher κ creaes more deflaion. More deflaion, in urn, increases he real ineres rae i π. By he Euler equaion (1a), his requires higher growh in he oupu gap, bu since x(t) = his ranslaes ino a lower level of x() for earlier daes < T. In words, flexible prices lead o more vigorous deflaion, raising he real ineres rae, increasing he desire for saving, lowering demand and depressing oupu. Lower oupu reinforces he deflaionary pressures, creaing a vicious cycle. The proof in he appendix echoes his inuiion closely. A similar resul is repored in he analysis of fiscal mulipliers by Chrisiano e al. (211). They compue he equilibrium when moneary policy follows a Taylor rule and he naural rae of ineres is a Poisson process, aking wo values. They show numerically ha oupu may be more depressed when prices are more flexible. They do no pursue a limiing resul owards full flexibiliy. 7 My resul is somewha disinc, since i applies o a siuaion wih opimal discreionary moneary policy, insead of a given Taylor rule. Also, i holds for any deerminisic pah for he naural rae. Anoher difference is ha wih Poisson uncerainy an equilibrium fails o exis when prices are sufficienly flexible. Despie hese differences, he logic for he effec is he same in boh cases. 8 The zero lower bound and he lack of commimen are no criical for his resul. The same conclusions follow for any pah of he naural rae {r()} if we assume he cenral bank ses he nominal ineres rae above he naural rae i() = r() + wih > for some period of ime [, T] and hen reurns o implemening he firs-bes oucome x() = π() = and i() = r() for > T. 9 The zero lower bound and he lack of commimen serve o moivae such a scenario. However, anoher jusificaion may be policy misakes of his paricular form, where ineres raes are se oo high (or oo low) for a fixed amoun of ime. As I discuss laer, when he cenral bank commis o an opimal policy, price flexibiliy can be beneficial. Surprisingly, i may sill be harmful, bu i depends on parameers. 7 Basically he same Poisson calculaions in Chrisiano e al. (211) appear also in Woodford (211) and Eggersson (211), alhough he effecs of price flexibiliy are no heir focus and so hey do no discuss is effecs. 8 De Long and Summers (1986) make he poin ha, for given moneary policy rules, price flexibiliy may be desabilizing, even away from a liquidiy rap, in he sense of increasing he variance of oupu. 9 Of course a symmeric resul holds for <. There is a boom in oupu alongside inflaion. The undesirable boom and inflaion are amplified when prices are more flexible, in he sense of a higher κ. 13

14 Is here a Disconinuiy a Full Flexibiliy? No really. The resul ha price flexibiliy makes maers worse may seem puzzling, especially in he limi, since i seems o conradic he noion ha perfecly flexible prices lead o zero oupu gaps. Tha is, a κ = we expec x() = for all, bu, paradoxically, as κ we obain x() insead. Does his reveal an inheren disconinuiy in he New Keynesian model? No. The resul is bes seen as arising from a disconinuiy in moneary policy a κ =, no a disconinuiy in he model iself. The equilibria obained for finie κ described in Proposiion 2 saisfy π() and π(t) =. If one akes hese wo feaures as a requiremen hen here is no equilibrium wih κ =. Economically, his suggess a form of coninuiy: an explosive oucome converges o a siuaion where an equilibrium seizes o exis. In any case, when κ =, moneary policy mus allow sricly posiive inflaion for an equilibrium o exis. In his sense, he disconinuiy in oucomes is produced by a disconinuiy in moneary policy regarding inflaion a κ =. To see his more clearly, consider a liquidiy rap scenario wih r() = r < for < T. Consider indexing moneary policy by a single parameer, π, a arge rae of inflaion. Specifically, suppose π() = π for all T and i() = for < T. For any finie κ his pins down an equilibrium uniquely. Noe ha if π( T) = hen he equilibrium coincides wih he discreionary case. Indeed, Proposiion 2 sill describes he oucome for T in he limi (1/κ n, π n ) (, ). However, suppose ha as 1/κ n we le {π n } o be an increasing sequence converging o r >. Provided his convergence is fas enough, he oucome converges o he one wih flexible price so ha lim n x n () for all. Of course, he paricular limi (1/κ n, π n ) (, ) is moivaed by he lack of commimen, bu he poin is ha his can be seen as moivaing a jump in moneary policy a κ =, which creaes he disconinuiy in oucomes. A more pracical lesson from his discussion is ha he benefis of flexible prices only accrue wih moneary policies ha accep higher inflaion. In oher words, price flexibiliy only does is magic if we use i. 4 Opimal Moneary Policy wih Commimen I now urn o opimal moneary policy wih commimen. The cenral bank s problem is o minimize he objecive (2) subjec o (1a) (1c) wih boh iniial values of he saes, π() and x(), free. The problem seeks he mos preferable oucome, across all hose compaible wih an equilibrium. In wha follows I focus on characerizing he opimal 14

15 pah for inflaion, oupu and he nominal ineres rae Opimal Ineres Raes, Inflaion and Oupu The problem can be analyzed as an opimal conrol problem wih sae (π(), x()) and conrol i(). The associaed Hamilonian is H 1 2 x λπ2 + µ x σ 1 (i r π) + µ π (ρπ κx). The maximum principle implies ha he co-sae for x mus be non-negaive hroughou and zero whenever he nominal ineres rae is sricly posiive µ x (), (3a) i()µ x () =. (3b) The law of moion for he co-saes are Finally, because boh iniial saes are free, we have µ x () = x() + κµ π () + ρµ x (), (3c) µ π () = λπ() + σ 1 µ x (). (3d) µ x () =, (3e) µ π () =. (3f) Taken ogeher, equaions (1a) (1c) and (3a) (3f) consiue a sysem for {π(), x(), i(), µ π (), µ x ()} [, ). Since he opimizaion problem is sricly convex, hese condiions, ogeher wih appropriae ransversaliy condiions, are boh necessary and sufficien for an opimum. Indeed, he opimum coincides wih he unique bounded soluion o his sysem. Suppose he zero-bound consrain is no binding over some inerval [ 1, 2 ]. Then i mus be he case ha µ x () = µ x () = for [ 1, 2 ], so ha condiion (3c) implies 1 I do no dedicae much discussion o he quesion of implemenaion, in erms of a choice of (possibly ime varying) policy funcions ha would make he opimum a unique equilibrium. I is well undersood ha, once he opimum is compued, a ime varying ineres rae rule of he form i() = i () + ψ π (π() π ()) + ψ x (x() x ()) ensures ha his opimum is he unique local equilibrium for appropriaely chosen coefficiens ψ π and ψ x. Eggersson and Woodford (23) propose a differen policy, described in erms of an adjusing arge for a weighed average of oupu and he price level, ha also implemens he equilibrium uniquely. 15

16 x() = κµ π (), while condiion (3d) implies µ π () = λπ(). As a resul, ẋ() = κ µ π () = κλπ() = σ 1 (i() r() π()). Solving for i() gives where i() = I(r(), π()), I(r, π) r() + (1 κσλ)π, is a funcion ha gives he opimal nominal rae whenever he zero-bound is no binding. This is he ineres rae condiion derived in he radiional analysis ha assumes he ZLB never binds (see e.g. Clarida, Gali and Gerler, 1999, pg. 1683). Noe ha his rae equals he naural rae when inflaion is zero, I(r, ) = r. Thus, i encompasses he well-known price sabiliy resul from basic New-Keynesian models. Away from zero inflaion, he ineres rae generally depars from he naural rae, unless σλκ = 1. Given his resul, i follows ha I (r(), π ()) is a necessary condiion for he zero-bound no o bind. The converse, however, is no rue. Proposiion 3. Suppose {π (), x (), i ()} is opimal. Then a any poin in ime eiher i () = I(r(), π ()) or i () =. Moreover if I(r(), π ()) < [, 1 ) hen i () = [, ˆ 1 ]. for 1 < ˆ 1. Likewise, if > hen i () = for [ˆ, ] for ˆ <. According o his resul, he nominal ineres rae should be held down a zero longer han wha curren inflaion warrans. Tha is, he opimal pah for he nominal ineres rae is no he upper envelope i () = max{, I(r(), π ())}. Insead, he nominal ineres rae should be se below his envelope for some ime, a zero. The noion ha commiing o fuure moneary easing is beneficial in a liquidiy rap was firs pu forh by Krugman (1998). His analysis capures he benefis from fuure inflaion only. I is based on a cash-in-advance model where prices canno adjus wihin a period, bu are fully flexible beween periods. The firs bes is obained by commiing 16

17 o money growh and inducing higher fuure inflaion. Thus, inflaion is easily obained and cosless in he model. Eggersson and Woodford (23) work wih he same New Keynesian model as I do here. They repor numerical simulaions where a prolonged period of zero ineres raes are opimal. My resul provides he firs formal explanaion for hese paerns. I also clarifies ha he relevan comparison for he nominal ineres rae i () is he unconsrained opimum I(r(), π ()), no he naural rae r(); he wo are no equivalen, unless κσλ = 1. The coninuous ime framework employed here helps capure he bang-bang naure of he soluion. A discree-ime seing can obscure hings due o ime aggregaion. One ineresing implicaion of my resul is ha he opimal exi sraegy feaures a discree jump in he nominal ineres rae. Whenever he zero-bound sops binding he nominal ineres mus equal I(r(), π ()), which given Proposiion 3, will generally be sricly posiive. Thus, opimal policy requires a discree upward jump, from zero, in he nominal ineres rae. Even when economic fundamenals vary smoohly, so ha I(r(), π ()) is coninuous, he bes exi sraegy calls for a disconinuous hike in he nominal ineres rae. Does commimen o he opimal policy, wih prolonged zero ineres raes, imply lower long erm ineres raes? Relaive o he discreionary equilibrium are yields on long erm bonds available a = lower? No necessarily. Consider he liquidiy rap scenario. Zero ineres raes are generally prolongued pas T, his lowers he yield rae on medium-erm bonds a =. However, upon exi from he zero lower bound a ˆT > T, he shor-erm ineres rae is se o I(r( ˆT), π( ˆT)) r( ˆT) wih sric inequaliy as long as long as κσλ = 1. Once again, as in Secion 3.2, looser moneary policy, in his case opimal moneary policy, is no necessarily unambiguously associaed wih lower longerm ineres raes. By implicaion, his may imply higher yield on very long erm bonds. The previous resul characerizes nominal ineres raes, bu wha can be said abou he pahs for inflaion and oupu? This quesion is imporan for a number of reasons. Firs, oupu and inflaion are of direc concern, since hey deermine welfare. In conras, he nominal ineres rae is merely an insrumen o influence oupu and inflaion. Second, as in mos moneary models, he equilibrium oucome is no uniquely deermined by he equilibrium pah for he nominal ineres rae. A cenral bank wishing o implemen he opimum needs o know more han he pah for he nominal ineres rae. For example, he cenral bank may employ a Taylor rule cenered around he arge pah for inflaion i() = i () + ψ(π() π ()) wih ψ > 1. Finally, undersanding he oucome for inflaion and oupu sheds ligh on he kind of policy commimen required. The nex proposiion characerizes opimal inflaion and oupu. 17

18 Proposiion 4. Suppose he firs-bes oucome is no aainable and ha {π (), x (), i ()} is opimal: 1. Inflaion mus be sricly posiive a some poin in ime: π() > for some. 2. Oupu is iniially negaive x(), bu becomes sricly posiive a some poin, x() > for some >. 3. Furhermore, if (a) κσλ = 1 hen inflaion is iniially zero and is nonnegaive hroughou, π () = and π () for all ; (b) κσλ < 1 hen π () > for all ; (c) κσλ > 1 hen π () wih sric inequaliy if x() <. Inflaion mus be posiive a some poin. Depending on parameers, iniial inflaion may posiive or negaive. In some cases, inflaion may be posiive hroughou. Oupu, on he oher hand, mus swich signs. The iniial recession is never compleely avoided. According o his proposiion, here are wo hings opimal moneary policy accomplishes. Firs, i promoes inflaion o miigae or reverse he deflaionary spiral during he liquidiy rap. This lowers he real rae of ineres, which lessens he recession. Second, i simulaes fuure oupu o creae a boom afer he rap. This percolaes back in ime, making consumers, who anicipae a boom, lower heir desired savings. In oher words, he roo problem during a liquidiy rap is ha desired savings and he real ineres rae are oo high. Opimal policy addresses boh. In his model he wo goals are relaed: inflaion requires a boom in oupu. Thus, pursuing he firs goal already leads, incidenally, o he second, and vice versa. However, he nominal ineres rae pah implied by Proposiion 3 simulaes a larger boom han wha is required by he inflaion promise alone. To see his, suppose ha along he opimal plan I(r(), π ()) for 1, and I(r(), π ()) < oherwise. The opimal plan hen calls for i () = over some inerval [ 1, 2 ]. However, consider an alernaive plan ha has he same inflaion a 1, so ha π( 1 ) = π ( 1 ), bu, in conradicion wih Proposiion 3, feaures i() = I(r(), π()) for all Suppose also ha, for boh plans, he long-run oupu gap is zero: lim x() = lim x () =. I hen follows ha x( 1 ) < x ( 1 ). In his sense, holding down he ineres rae o zero simulaes a boom ha is greaer han he one implied by he inflaion promise. Proposiion 4 singles ou a case wih κσλ = 1 where inflaion sars and ends a zero and is posiive hroughou. This case occurs when he cosae µ π () on he Phillips curve 11 Noe ha, depending on he value of κσλ, he ineres rae may even be greaer han he naural rae r(). The fac ha his policy is consisen wih posiive inflaion and oupu afer he rap even hough i may have higher ineres raes han he discreionary soluion underscores, once again, ha moneary easing does no necessarily manifes iself in lower equilibrium ineres raes. 18

19 Figure 2: A numerical example showing he full discreion case (black) and opimal commimen case (blue). is zero for all. This case urns ou o be an ineresing benchmark wih oher ineresing implicaions for governmen spending. Figure 2 plos he equilibrium pahs for a numerical example. The parameers are se o T = 2, σ = 1, κ =.5 and λ = 1/κ. These choices are made for illusraive purposes and o ensure ha κσλ = 1. They do no represen a calibraion. The choices are iled owards a flexible price siuaion. Relaive o he New Keynesian lieraure, he degree of price sickiness is low (high κ) and he planner is quie oleran of inflaion (low λ). I is also common o se a lower value for σ, on he grounds ha invesmen, which may be quie sensiive o he ineres rae, has been omied from he analysis. The black line represens he equilibrium wih discreion; he blue line, he opimum wih commimen. Wih discreion oupu is iniial depressed by abou 11%, a he opimum his is reduced o jus under 4%. The opimum feaures a boom which peaks a abou 3% a = T. The discreionary case feaures significan deflaion. In conras, because κσλ = 1 opimal inflaion sars a zero and is always posiive. Boh pahs end a origin, which represens he ideal firs-bes oucome. However, alhough he opimum reaches i laer a ˆT = 2.7, i circles around i, managing o say closer o i on average. This improves welfare. One implicaion of Proposiion 4 is ha, whenever he firs bes is unaainable, opimal moneary policy requires commimen. Oupu is iniially negaive x (), bu 19

20 mus urn sricly posiive x ( ) > a some fuure dae for >. This implies ha, if he planner can reopimize and make a new credible plan a ime, hen his new plan would involve iniially negaive oupu x ( ). Hence, i canno coincide wih he original plan which called for posiive oupu. Noe ha he kind of commimen needed in his model involves more han a promise for fuure inflaion, a ime T, as in Krugman (1998). Indeed, my discussion here emphasizes commimen o an oupu boom. More generally, he planning problem feaures boh π and x as sae variables, so commimen o deliver promises for boh inflaion and oupu are generally required. Liquidiy raps are commonly associaed wih deflaion, bu hese resuls sugges ha he opimum compleely avoids deflaion in some cases. This is more likely o be he case if prices are less flexible (low κ), if he ineremporal elasiciy of subsiuion is high (low σ), or if he cenral bank is no oo concerned abou inflaion (low λ). Noe ha if we se λ = λ/κ, hen κσλ = λσ, so he degree of price flexibiliy κ drops ou of he condiion deermining he sign of iniial inflaion. In he oher case, when κσλ < 1, he opimum does feaure deflaion iniially, bu ransiions hrough a period of posiive inflaion as shown by Proposiion 4. Numerical simulaions reurn o deflaion and a negaive oupu gap. I is worh noing ha prolonged zero nominal ineres raes are no needed o promoe posiive inflaion and simulae oupu afer he rap. Indeed, here are equilibria wih boh feaures and a nominal ineres rae pah given by i() = max{, I(r(), π())}. In he liquidiy rap scenario, he same is rue for he ineres rae pah considered under pure discreion, i() = for < T and i() = r() for T. Wihou commimen, a unique equilibrium was obained by adding he condiion ha he firs bes oucome π() = x() = was implemened for T. However, posiive inflaion and oupu, π(t), x(t) are also compaible wih his very same ineres rae pah. This is possible because equilibrium oucomes are no uniquely deermined by equilibrium nominal ineres raes. Policy may sill be described as one of moneary easing, even if his is no necessarily refleced in equilibrium nominal ineres raes To be specific, suppose policy is deermined endogenously according o a simple Taylor rule, wih a ime varying inercep, i() = ī() + φ π π() wih φ π > 1. In he unique bounded equilibrium, a emporarily low value for ī() ypically leads o higher inflaion π(), bu no necessarily a lower equilibrium ineres rae i(). The oucome for he nominal ineres rae i() depends on various parameers. Eiher way, he siuaion wih emporarily low ī() may be described as one of moneary easing. 2

21 4.2 A Simple Case: Fully Rigid Prices To gain inuiion i helps o consider he exreme case wih fully rigid prices, where κ = and π() = for all. 13 Consider he liquidiy rap scenario, where r() < for < T and r() > for > T, and suppose we keep he nominal ineres rae a zero unil some ime ˆT T, and implemen x() = π() = afer ˆT. Oupu is hen ˆ ˆT x(; ˆT) σ 1 r(s)ds. When ˆT = T, he inegrand is always negaive, so ha oupu is negaive: x(, T) < for < T. In fac, he equilibrium coincides wih he full discreion case isolaed by Proposiion 1. When ˆT > T he inegral includes sricly posiive values for r() for (T, ˆT]. This increases he pah for x(; ˆT). For any dae T oupu increases by he consan σ 1 ˆT T r(s)ds >. Saring a =, oupu rises and peaks a T, hen falls unil i reaches zero a ˆT. The boom induced a T percolaes o earlier daes, increasing oupu in a parallel fashion. Larger values of ˆT shrink he iniially negaive oupu gaps, bu lead o larger posiive gaps laer. Saring from ˆT = T an increase in ˆT improves welfare because he loss from posiive oupu gaps are second order, while he gain from reducing exising negaive oupu gaps is firs order. Formally, minimizing he objecive V( ˆT) 2 1 e ρ x(; ˆT) 2 d yields ˆ ˆT V ( ˆT ) = r( ˆT )σ 1 e ρ x(; ˆT )d =, and i follows ha T < ˆT < T where x(, T) =. According o his opimaliy condiion, he presen value of oupu should be zero e ρ x()d =, implying ha he curren recession and subsequen boom should average ou, in presen value. When prices are fully rigid inflaion is zero regardless of moneary policy. Hence, creaing inflaion canno be he purpose of moneary easing. Insead, commiing o zero nominal ineres raes is useful here because i creaes an oupu boom afer he rap. This boom helps miigae he earlier recession. The logic here is compleely differen from he one in Krugman (1998), which isolaed he inflaionary moive for moneary easing. Nex I urn o a graphical analysis of inermediae cases, where boh moives are presen. 13 The same condiions we will obain for κ = here can be obained if we consider he limi of he general opimaliy condiions derived above as κ. However, i is more revealing o derive he opimaliy condiion from a separae perurbaion argumen. 21

22 x ˆT > T T ˆT = T Figure 3: Fully rigid prices. The pah for oupu wih ˆT = T and ˆT > T. 4.3 Siching a Soluion Togeher: A Graphical Represenaion To see he soluion graphically, consider he paricular liquidiy rap scenario wih he sep funcion pah for he naural rae of ineres: r() = r < for < T bu r() = r for T. I is useful o break up he soluion ino hree separae phases, from back o fron. I firs consider he soluion afer some ime ˆT > T when he ZLB consrain is no longer binding (Phase III). I hen consider he soluion beween ime T and ˆT wih he ZLB consrain (Phase II). Finally, I consider he soluion during he rap [, T] (Phase I). Afer he Sorm: Slack ZLB Consrain (Phase III). Consider he problem where he ZLB consrain is ignored, or no longer binding. If his were rue for all ime hen he soluion would be he firs bes π() = x() =. However, here I am concerned wih a siuaion where he ZLB consrain is slack only afer some dae ˆT > T >, a which poin he sae (π( ˆT), x( ˆT)) is given and no longer free, so he firs bes is generally no feasible. The planning problem now ignores he ZLB consrain bu akes he iniial sae (π, x ) as given. Because he ZLB consrain is absen, he consrain represening he Euler equaion is no binding. Thus, i is appropriae o ignore his consrain and drop he oupu gap x() as a sae variable, reaing i as a conrol variable insead. The only remaining sae is inflaion π(). 14 Also noe ha he pah of he naural ineres rae {r()} is irrelevan when he ZLB consrain is ignored. 14 One can pick any absoluely coninuous pah for x() and solve for he required nominal ineres rae as a residual: i() = σẋ() + π() + r(). Disconinuous pahs for x() can be approximaed arbirarily well by coninuous ones. Inuiively, i is as if disconinuous pahs for {x()} are possible, since upward or downward jumps in x() can be engineered by seing he ineres rae o or for an infiniesimal momen in ime. Formally, he supremum for he problem ha ignores he ZLB consrain, bu carries boh π() and x() as saes, is independen of he curren value of x(). Since he curren value of x() does no meaningfully consrain he planning problem, i can be ignored as a sae variable. 22

23 x x = φπ π Figure 4: The soluion wihou he ZLB consrain. I seek a soluion for oupu x as a funcion of inflaion π. Using he opimaliy condiions wih µ x () = one can show ha i() = I(π(), r()) as discussed earlier, wih oupu saisfying x() = φπ() and cosae µ π () = φ κ π(), where φ ρ+ ρ 2 +4λκ 2 2κ so ha φ > ρ/κ. The las inequaliy implies ha he ray x = φπ is seeper han ha for π =. Thus, saring wih any iniial value of π he soluion converges over ime along he locus x = φπ o he origin (π(), x()) (, ). These dynamics are illusraed in Figure 4. Jus ou of he Trap (Phase II). Consider nex he problem for T incorporaing he ZLB consrain for any arbirary saring poin (π(t), x(t)). The problem is saionary since r() = r > for T. If he iniial sae lies on he locus x = φπ, hen he soluion coincides wih he one above. This is essenially also he case when he iniial sae saisfies x < φπ, since one can engineer an upward jump in x o reach he locus x = φπ. 15 Afer his jump, one proceeds wih he soluion ha ignores he ZLB consrain. In conras, he opimum feaures an 15 For example, se i() = /ε > for a shor period of ime [, ε) and choose so ha x(ε) = φπ(ε). As ε his approximaes an upward jump up o he x = φπ locus a =. 23