Monetary Policy Frameworks and the Effective Lower Bound on Interest Rates

Transcription

1 FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES Moneary Policy Frameworks and he Effecive Lower Bound on Ineres Raes Thomas M. Merens and John C. Williams Federal Reserve Bank of San Francisco January 2019 Working Paper hps:// Suggesed ciaion: Merens, Thomas M., John C. Williams Moneary Policy Frameworks and he Effecive Lower Bound on Ineres Raes, Federal Reserve Bank of San Francisco Working Paper hps://doi.org/ /wp The views in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem.

2 Moneary Policy Frameworks and he Effecive Lower Bound on Ineres Raes Thomas M. Merens and John C. Williams January 11, 2019 Absrac This paper applies a sandard New Keynesian model o analyze he effecs of moneary policy in he presence of a low naural rae of ineres and a lower bound on ineres raes. Under a sandard inflaionargeing approach, inflaion expecaions will be anchored a a level below he inflaion arge, which in urn exacerbaes he deleerious effecs of he lower bound on he economy. Two key hemes emerge from our analysis. Firs, he cenral bank can miigae his problem of a downward bias in inflaion expecaions by following an average-inflaion argeing framework ha aims for above-arge inflaion during periods when policy is unconsrained. Second, a dynamic sraegy such as price-level argeing ha raises inflaion expecaions when inflaion is low can boh anchor expecaions a he arge level and poenially furher reduce he effecs of he lower bound on he economy. JEL Classificaion Sysem: E52 Merens: Federal Reserve Bank of San Francisco, 101 Marke Sree, Mailsop 1130, San Francisco, CA 94105; Thomas.Merens@sf.frb.org. Williams: Federal Reserve Bank of New York, 33 Libery Sree, New York, NY 10045; John.C.Williams@ny.frb.org. We hank Parick Adams for ousanding research assisance and Ben Bernanke and audiences a he ASSA meeings, a he Bank for Inernaional Selemens, and a he Federal Reserve Banks of New York and San Francisco for helpful discussions. The views expressed in his paper are hose of he auhors and do no necessarily reflec he posiions of he Federal Reserve Bank of San Francisco, he Federal Reserve Bank of New York, or he Federal Reserve Sysem. 1

3 The adopion of inflaion argeing by many cenral banks succeeded in bringing high and variable inflaion raes of he 1970s and early 1980s under conrol and hereby anchoring inflaion expecaions a he argeed rae. The significan decline in he naural rae of ineres observed in many counries over he pas quarer-cenury implies ha cenral banks are now likely o be consrained by he lower bound on nominal ineres raes relaively frequenly, inerfering wih heir abiliy o offse negaive shocks o he economy (see, for example, Laubach and Williams (2016) and Holson, Laubach and Williams (2017)). The experiences of many advanced economies over he pas decade are a esamen o his new realiy. As a resul, cenral banks now face he challenge of inflaion expecaions poenially being anchored a oo low a level, raher han oo high. In his paper, we invesigae alernaive moneary policy frameworks designed o anchor inflaion expecaions a he desired level even if he lower bound frequenly consrains moneary policy acions. We use a simple New Keynesian model as a laboraory for our analysis. The economy is governed by a Phillips curve ha links inflaion o a supply shock, he oupu gap, and expeced fuure inflaion and an IS-curve ha links he oupu gap o a demand shock, he ex ane real ineres rae, and expecaions of he fuure oupu gap. The cenral bank ses he nominal ineres rae o minimize he variabiliy of he inflaion rae and he oupu gap around heir arge values. We assume ha he ineres rae is he cenral bank s sole policy ool and absrac from unconvenional policies such as asse purchases and quaniaive easing. We sar by assuming ha he cenral bank follows opimal policy under discreion and hen explore alernaive policies ha incorporae feaures designed o miigae he deleerious effecs of he lower bound. Absen a lower bound on ineres raes, he opimal moneary policy under discreion fully offses demand shocks, parially offses supply shocks, and anchors inflaion expecaions a he arge level. This policy behaves like a sandard exbook inflaion-argeing policy. However, when a lower bound on ineres raes consrains policy, ineres raes will no be able o respond opimally o negaive shocks o he economy and he oupu gap and inflaion will be lower han would oherwise occur. Owing o he inheren asymmery of he lower bound, he average inflaion rae will be below he arge rae, and inflaion expecaions will likewise be anchored a oo low a level. This reducion in expeced inflaion furher exacerbaes he effecs of he lower bound on he economy. We conras oucomes from he opimal policy under discreion wih hree main alernaive approaches ha seek o raise inflaion expecaions. Each of hese requires some degree of commimen o fuure acions ha a discreionary policymaker would no ake. The firs dovish policy alernaive reduces he moneary policy responses o shocks in order o limi he asymmery implied by he lower bound. The second average- 2

4 inflaion argeing policy implicily aims for above-arge inflaion when policy is unconsrained, hereby offseing he effecs of he lower bound on expeced inflaion. The hird price-level argeing sraegy, along wih is offshoos, arges he price level raher han he inflaion rae. The main conclusion of his analysis is ha all hree of hese approaches work hrough he same mechanism of raising he inflaion rae above he arge rae when policy is no consrained. Bu, some do so a a greaer cos in erms of sabilizing inflaion and he economy. In paricular, we find ha average-inflaion argeing dominaes he dovish policy sraegies, which allow excessive pass-hrough of shocks o he economy. In addiion, we find ha dynamic sraegies like price-level argeing dominae average-inflaion argeing because he former creae expecaions of relaively high inflaion and oupu gaps following periods when he lower bound is binding. Imporanly, he success of all of hese hree approaches depends crucially on affecing privae-secor expecaions and herefore on boh he credibiliy and he public s clear undersanding of he policy. 1 Model economy and moneary policy frameworks We augmen he sandard New Keynesian model as described, for example, in Clarida, Gali and Gerler (1999) o include a lower bound on ineres raes. The model consiss of hree equaions describing he evoluion of hree endogenous variables: he rae of inflaion, π, he oupu gap, x, and he shor-erm nominal ineres rae, i. Inflaion is deermined by a forward-looking Phillips curve π µ + κx + βe π +1, (1) where E denoes mahemaical expecaions based on informaion a ime, µ is a supply shock, β (0, 1) is he discoun facor, κ > 0, and µ iid U( ˆµ, ˆµ). We assume ha all shocks are uniformly disribued i.i.d over ime and independen from each oher. An IS-curve relaionship describes he deerminaion of he oupu gap x ϵ α(i E π +1 r ) + E x +1, (2) where α > 0, r is he long-run naural real rae of ineres, ϵ is a demand shock, and ϵ iid U( ˆϵ, ˆϵ). Calculaions for he model are provided in he appendix. The cenral bank s objecive is o minimize he expeced weighed sum of he squared values of he oupu gap and inflaion rae. We assume a long-run inflaion arge of zero, bu i is sraighforward o exend he 3

5 analysis o alernaive values of he inflaion arge. Specifically, he cenral bank ses he nominal ineres rae, i, o minimize he expeced quadraic loss: [ ] L E 0 β (π 2 + λx2 ), (3) 0 where λ 0 is he relaive weigh he cenral bank places on he sabilizaion of he oupu gap. The cenral bank decision for i is assumed o occur afer he realizaions of he shocks in he curren period. 1.1 Opimal policy under discreion Assuming ha he lower bound does no consrain policy, opimal policy under discreion can be implemened by seing he nominal ineres rae according o he following policy rule: i u θ 0 + θ µ µ + θ ϵ ϵ + θ E E π +1. (4) The coefficien values describing he opimal policy under discreion are given by: θ 0 r, θ ϵ 1 α, θ µ κ α(κ 2 +λ), and θ E 1+ 1 ακ λβ. This policy fully offses demand shocks and parially offses supply shocks ακ(κ 2 +λ) depending on he degree of concern for oupu sabilizaion in he cenral bank objecive. In his model, he lower bound alers he opimal policy under discreion only in ha he ineres rae is se o he lower bound when he unconsrained ineres rae is below he lower bound. The opimal values of he coefficiens of he policy rule are unaffeced. Tha is, he realized ineres rae is given by: i max{i LB, i u }. Under he model assumpions, we can combine he wo equaions o make he expression for inflaion independen of he oupu gap. Plugging in he rule for ineres raes resuls in wo equaions, one for when moneary policy is consrained π µ + κϵ ακ(i LB r ) + (1 + ακ)e π +1 and one when i is unconsrained π ακ(r θ 0 ) + (1 + ακ(1 θ E ))E π +1 + (1 ακθ µ )µ + κ(1 αθ ϵ )ϵ. The consrain binds when he realizaion of he wo shocks saisfies θ ϵ ϵ + θ µ µ i LB θ 0 θ E E π +1. Wih 4

6 a policy rule of he form (4), we can solve he model analyically. Under hese assumpions, expeced inflaion in all fuure periods is consan and is below arge if he lower bound ever consrains policy.1 This downward bias in inflaion expecaions relaive o he arge sems from expecaions aking ino accoun fuure inflaion raes under boh consrained and unconsrained policy. The resuling reducion in inflaion expecaions in urn implies ha he lower bound consrains policy more ofen and ha moneary policy provides less simulus when policy is consrained due o he higher resuling real ineres rae when a he lower bound. We now consider hree alernaive policy approaches ha assume some form of commimen. 1.2 Dovish policy As shown above, he lower bound on ineres raes leads o below-arge inflaion expecaions ha, in urn, pu downward pressure on inflaion in he curren period hrough he forward-looking Phillips curve. One way o miigae his problem is for he cenral bank o follow a more dovish policy wih smaller policy responses o shocks. In his way, he cenral bank can limi he asymmery implied by he lower bound and he resuling reducion in inflaion expecaions. A simple way o reduce he variance of ineres raes would be for he cenral bank o impose an upper bound on ineres raes. The cenral bank ses he nominal ineres rae as i would under discreion bu imposes an addiional consrain prevening he ineres rae from exceeding he upper bound.2 For example, an upper bound symmeric o he lower bound around r fully eliminaes he downward bias o inflaion expecaions. However, i achieves his by responding subopimally o large posiive shocks, which increases heir pass-hrough o he economy. A similar, more nuanced approach is o reduce he overall response o shocks (see, e.g., Nakaa and Schmid (2016)). This works hrough he same mechanism as an upper bound on ineres raes and increases expeced inflaion, bu also a he cos of greaer pass-hrough of shocks o he economy. Eiher of hese dovish policy approaches he imposiion of an upper bound or more mued overall responses o shocks may change he cenral bank loss relaive o he opimal discreionary policy in his model by reducing he downward bias o expeced inflaion relaive o arge. However, here are more direc 1There are wo seady-sae equilibria, a arge equilibrium and a liquidiy rap equilibrium, as in Benhabib, Schmi-Grohé and Uribe (2001) and Merens and Williams (2018). As is sandard in he lieraure and suppored by he empirical analysis of Merens and Williams (2018), we focus on he arge equilibrium. 2In addiion o he wo seady-sae equilibria under discreion, a hird equilibrium associaed wih he ineres rae a he upper bound emerges. We again resric our analysis here o he arge equilibrium. 5

7 : r Discreion (no lower bound) Discreion AIT PLT Discreion (no lower bound) Discreion AIT PLT Figure 1: The above figures show he pah of inflaion (lef panel) and real ineres rae (righ panel) in response o a negaive supply shock µ 0 ˆµ for various moneary policy frameworks. Discreion refers o opimal moneary policy under discreion, AIT o average-inflaion argeing such ha inflaion expecaions are a arge, and PLT o price-level argeing. ways o achieve inflaion expecaions anchored a he arge level, o which we now urn. 1.3 Average-inflaion argeing The second approach is o aim for above-arge inflaion whenever policy is unconsrained o offse he below-arge inflaion oucomes when policy is consrained. This average-inflaion argeing framework can achieve he desired level of inflaion expecaions hrough an adjusmen of he inercep of he policy rule, θ 0. In paricular, a downward adjusmen of he inercep raises inflaion expecaions, wih he size of he adjusmen needed o achieve zero expeced inflaion depending on he probabiliy of a binding lower bound and he poenial magniude of negaive shocks o he economy. In fac, in his model i can be opimal o se he inercep below his level, hus pushing expeced inflaion above zero. This higher-han-arge mean inflaion rae furher helps reduce he effecs of he lower bound on he economy. Given his value of θ 0, i is opimal o respond o he shocks exacly as one would under he opimal discreionary policy. In his sense, he decisions on level and variance are separable. Taken ogeher, he opimal average-inflaion argeing policy dominaes he dovish policy sraegies. Noe ha he calibraion of he opimal value of θ 0 requires deailed knowledge of he probabiliies and associaed coss of hiing he lower bound implied by he model. 1.4 Price-level argeing The wo alernaive policy approaches described above illusrae how policies ha raise inflaion when he cenral bank is unconsrained can lif inflaion expecaions, wih he beneficial side-effec of miigaing he effec of adverse shocks when he lower bound binds. The same logic suggess ha a more refined approach 6

8 can yield even beer oucomes by aiming o raise inflaion expecaions when inflaion is running below he arge rae, say due o he effecs of he lower bound. These dynamic sraegies are he focus of price-level argeing policies and heir varians such as emporary price-level argeing and nominal GDP argeing. An example of such a dynamic sraegy is a policy rule ha responds o he price level, i upl θ 0 + θ p p + θ µ µ + θ ϵ ϵ + θ E E p π +1, where θ p > 0 and he log of he price level p evolves according o p p 1 + π. As a resul, inflaion expecaions become a funcion of he price level E p π +1 E [π +1 p ]. To solve he model wih his policy rule, we subsiue he ineres rae rule ino he wo equaions for he economy. We hen ake condiional expecaions of hese wo model equaions (condiional on he price level). We approximae expecaions for fuure inflaion and oupu gaps as a funcion of he price level and ierae backwards unil he process converges and he fixed poins for condiional expecaions funcions emerge. Unlike he opimal discreionary policy, he price-level argeing policy delivers mean inflaion equal o he arge rae. Inflaion expecaions are sae-dependen wih heir mean a he arge rae, reflecing he fac ha he policy delivers above-arge inflaion when policy is unconsrained by he lower bound. This resul does no require deailed knowledge of model parameers, as was he case for average-inflaion argeing, bu relies on he naure of a price-level arge ha does no rea bygones as bygones in erms of pas misses of he inflaion arge. In addiion, because his policy raises inflaion expecaions during periods when he lower bound consrains policy, i lowers he real ineres rae during hose periods, hereby reducing he effecs of he lower bound. One varian of price-level argeing, called emporary price-level argeing, imposes a price-level arge only following an episode when he lower bound consrains policy (see Bernanke (2017)). The main benefi of his policy is ha i aims o raise inflaion and oupu gap expecaions when he lower bound consrains policy in he same way ha sandard price-level argeing does. Unlike sandard price-level argeing, i does no auomaically deliver mean inflaion a he arge rae. This is because of he asymmeric naure of he policy rule, which inroduces a second source of asymmery ino he model. To achieve a mean inflaion rae a he arge rae, he inercep of he rule, θ 0, mus be calibraed o ake ino accoun he effecs of he lower bound and he asymmeric naure of he emporary price-level arge. 7

9 1.5 Comparison of policies The various aspecs of he analysis are bes illusraed wih a concree numerical example of he model. Therefore, we se β 0.99, λ 0.25, α 1.25, κ 0.8, r 1%, i LB 0.5%, and ˆµ 3.3%. The sandard deviaion of he demand shock is se o zero.3 Under he opimal discreionary policy, he probabiliy of hiing he lower bound is abou 27%. Inflaion expecaions are a 0.24% and hus below arge. Figure 1 shows he mechanism by which he various moneary policy frameworks affec inflaion expecaions. The lef panel plos he pahs of inflaion in response o a negaive supply shock a ime 0, µ 0 ˆµ, aking he uncondiional expecaion of all shocks for fuure periods. The righ panel shows he same calculaions for he pah of he real ineres rae, defined as he nominal ineres rae ne of expeced inflaion. The black lines show he responses under he opimal discreionary policy absen a lower bound on ineres raes. Under he opimal discreionary policy wih a lower bound (blue line), he shock causes inflaion o drop significanly furher below arge. In fuure periods, inflaion equals is uncondiional mean. The averageinflaion argeing policy is assumed o have an inercep of 0.90, which achieves an uncondiional mean inflaion of zero. Under his policy, he decline in inflaion is somewha smaller han under he opimal discreionary policy suppored by a sharper decline in he real ineres rae. Inflaion again rebounds o is mean immediaely. For his calibraion, he opimal mean inflaion rae is 0.10%, which implies an inercep of Excep for he slighly higher mean inflaion rae, he resuling simulaion is very similar o he one shown. The price-level argeing policy differs more significanly from he oher wo policies. For his exercise, we used a value for θ p of 0.36, he value ha minimizes he cenral bank loss for his calibraion of he model. Inflaion exceeds he arge rae in all fuure periods due o he real ineres rae saying lower han he naural rae. This lower-for-longer policy booss expecaions of fuure oupu gaps and inflaion (see Reifschneider and Williams (2000)). Figure 2 shows he social loss for he various policies associaed wih his model calibraion. As seen in his char, geing he average inflaion righ delivers benefis in erms of he cenral bank loss. Adding a moderae response o he price level produces addiional benefis owing o he sae-dependence of expecaions. 8

10 L = E: 2 + 6Ex 2 Discreion AIT PLT p Figure 2: The above graph shows he social loss for differen responses o he price-level arge. Noe ha he price-level arge does no appear in he ineres rae rule for policy under discreion and average-inflaion argeing. 2 Conclusion This paper applies a sandard New Keynesian model o analyze he effecs of moneary policy in he presence of a low naural rae of ineres and a lower bound on ineres raes. Under a sandard inflaion-argeing approach, inflaion expecaions will be anchored a a level below he inflaion arge, which in urn exacerbaes he deleerious effecs of he lower bound on he economy. Two key hemes emerge from our analysis. Firs, he cenral bank can correc for he downward bias in inflaion expecaions by following an average-inflaion argeing framework ha aims for above-arge inflaion during periods when policy is unconsrained. The resuling policy rule is equivalen o he one under discreion wih a lower naural rae of ineres han is rue value. Second, a dynamic sraegy such as price-level argeing ha raises inflaion expecaions when inflaion is low can boh anchor expecaions a he arge level and poenially furher reduce he effecs of he lower bound on he economy. Each of hese alernaive policy sraegies works hrough is effecs on expecaions of fuure ineres raes, he oupu gap, and inflaion. In addiion, each requires a commimen o ake fuure policy acions ha a fuure policymaker would prefer no o follow. Moreover, for inflaion-argeing and emporary price-level argeing policies o be successful in anchoring inflaion expecaions a he desired level requires knowledge of he effecs of he lower bound on he economy. Therefore, for any of hese frameworks o work as well in pracice as hey do in heory requires clear communicaion and consisen execuion of he policy and a belief by he public ha he policy is credible. 3The resuls for supply shocks presened here carry over o case of demand shocks. 9

11 References Benhabib, Jess, Sephanie Schmi-Grohé, and Marín Uribe The Perils of Taylor Rules. Journal of Economic Theory, 96(1): Bernanke, Ben S Moneary Policy in a New Era. Peerson Insiue for Inernaional Economics, Ocober Clarida, Richard, Jordi Gali, and Mark Gerler The Science of Moneary Policy: A New Keynesian Perspecive. Journal of Economic Lieraure, 37(4): Holson, Kahryn, Thomas Laubach, and John C. Williams Measuring he Naural Rae of Ineres: Inernaional Trends and Deerminans. Journal of Inernaional Economics, 108(S1): S59 S75. Laubach, Thomas, and John C. Williams Measuring he Naural Rae of Ineres Redux. Business Economics, 51: Merens, Thomas M., and John C. Williams Wha o Expec from he Lower Bound on Ineres Raes: Evidence from Derivaives Prices. Federal Reserve Bank of San Francisco, Working Paper Nakaa, Taisuke, and Sebasian Schmid Gradualism and Liquidiy Traps. Review of Economic Dynamics, forhcoming. Reifschneider, David L, and John C Williams Three lessons for moneary policy in a low-inflaion era. Journal of Money, Credi and Banking,

12 A Appendix A.1 Derivaion of inflaion equaions Combining equaions (1) and (2) from he New Keynesian model yields he equaion for inflaion π E π +1 µ + κ(ϵ α(i E π +1 r ))) + βe (π +1 π +2 ). Wih he ineres rae rules of he form (4), he final erm is zero and inflaion is deermined by π (1 + ακ)e π +1 + µ + κϵ ακ(i r ). Plugging in he ineres rae rule from equaion (4) for when he cenral bank is unconsrained, delivers inflaion of he form π ακ(r θ 0 ) + (1 + ακ(1 θ E ))E π +1 + (1 ακθ µ )µ + κ(1 αθ ϵ )ϵ. (5) In he case where he ineres rae rule would ask for nominal raes below he lower bound, he cenral bank ses he policy rae as low as possible, o i LB π µ + κϵ ακ(i LB r ) + (1 + ακ)e π +1. (6) These wo equaions in he appendix are used in he main body of he paper. Noe ha hey imply he exisence of wo seady-sae equilibria in he deerminisic version of he model. In boh equaion, inflaion expecaions and inflaion raes appear linearly. Hence each equaion may be associaed wih a seady-sae equilibrium. We refer o he equilibrium associaed wih he firs equaion o he arge equilibrium and he one associaed wih he second equaion as a liquidiy rap. A.2 Derivaion of inflaion expecaions Wih demand or supply shocks, he lower bound can become an occasionally binding consrain. In his case, boh equaions for inflaion (5) and (6) have o be used o deermine inflaion expecaions Eπ Prob ( ) [ ] ( ) [ ] i op < i LB E π i op < i LB + Prob i op i LB E π i op i LB. (7) 11

13 For his equaion, we drop he period subscrip from he expecaions operaor. In his model wih he specified moneary policy rule, here is no informaion a ime ha predics period + 1 inflaion and herefore condiional and uncondiional expecaions are idenical. For illusraive purposes, we drop he demand shock from he model, i.e., we se is variance o zero. Then he consrain on nominal ineres raes binds when µ falls below a cuoff value µ 1 θ µ (i LB θ 0 θ E Eπ +1 ). There are hree differen cases: The cuoff value µ can fall below, in, or above he suppor of he disribuion for he supply shock. The probabiliy of being consrained by he lower bound can hus be expressed as Prob 1 if µ ˆµ ( ) i op < i LB 1 2 ˆµ ( ˆµ + µ) if ˆµ < µ < ˆµ 0 if µ ˆµ. If we plug his expression in equaion (7) and compue condiional expecaions of inflaion from equaions (5) and (6) respecively, we ge inflaion expecaions as ακ(i LB r ) + (1 + ακ)eπ +1 if µ ˆµ Eπ ακ ( µ ˆµ) 2 4 ˆµ θ µ + (1 + ακ(1 θ E ))Eπ +1 ακ(θ 0 r ) if ˆµ < µ < ˆµ ακ(θ 0 r ) + (1 + ακ)eπ +1 if µ ˆµ. The expression in he middle where he consrain is occasionally binding is of paricular ineres. The cuoff for he supply shock ha depends linearly on inflaion expecaions appears quadraically. To find a seady-sae equilibrium for inflaion expecaions, we need o solve a quadraic equaion which resuls in wo soluions for a parameer range.4 A.3 Inflaion expecaions in he presence of an upper bound The derivaion of inflaion expecaions for he case where boh a lower and upper bound are presen follows he same seps as in Appendix A.2. Due o he addiional consrain, however, he lis of disinc cases increases. Wih various condiions C LB and C UB on he lower and upper bounds, respecively, we disinguish 4For cases of a single equilibrium or non-exisence, see Merens and Williams (2018). 12

14 he cases: (1 + ακ)e[π +1 ] ακ(i LB r ) if C LB c ακ ( i UB + i LB 2θ 0 2θ E E[π +1 ] ) ( (i UB i LB ) 2θ µ ˆµ ) + 4θ µ ˆµ + (1 + ακ(1 θ E ))E[π +1 ] + ακ(r θ 0 ) if C LB o and C UB o E[π ] ακθ µ 4 ˆµ ( ˆµ + 1 θ µ ( i LB θ 0 θ E E[π +1 ] )) (1 + ακ(1 θ E ))E[π +1 ] + ακ(r θ 0 ) if C LB o and C UB u (1 + ακ)e[π +1 ] ακ(i UB r ) if C UB c ακθ µ 4 ˆµ ( ˆµ 1 θ µ ( i UB θ 0 θ E E[π +1 ] )) (1 + ακ(1 θ E ))E[π +1 ] + ακ(r θ 0 ) if C LB u and C UB o (1 + ακ(1 θ E ))E[π +1 ] + ακ(r θ 0 ) if C LB u and C UB u The various condiions deermine wheher a consrain never binds, C u, always binds, C c, or occasionally binds, C o. The specific condiions on he lower bound are C LB u { } 1 (i LB θ 0 θ E E[π +1 ]) < ˆµ θ µ for he lower bound o never bind, C LB o { ˆµ 1 θ µ (i LB θ 0 θ E E[π +1 ]) ˆµ } for he lower bound o occasionally bind, and C LB c { } 1 (i LB θ 0 θ E E[π +1 ]) > ˆµ θ µ for he lower bound o always bind. 13

15 For he upper bound, he condiions are C UB u { } 1 (i UB θ 0 θ E E[π +1 ]) > ˆµ θ µ for he upper o never bind, C UB o { ˆµ 1 θ µ (i UB θ 0 θ E E[π +1 ]) ˆµ } for he upper bound o occasionally bind, and C UB c { } 1 (i UB θ 0 θ E E[π +1 ]) < ˆµ θ µ for he upper bound o always bind. When solving for inflaion expecaions, a hird equilibrium besides he arge equilibrium and he liquidiy rap emerges. This equilibrium is associaed wih he upper bound on nominal ineres raes. As in he case wih only a lower bound, we resric our analysis o he arge equilibrium. A.4 Comparison of policies Table 1 shows a comparison of various saisics from he simulaions using he differen policies. Discreion Dovish Policies AIT PLT No lower Symmeric bound upper bound Opimal θ µ Eπ 0 Opimal θ 0 Opimal θ p E(π ) V(π ) E(x ) V(x ) E(π 2 ) + λe(x2 ) P(i i LB ) E(π i i LB ) E(π i > i LB ) θ θ µ θ E θ p Table 1: The above able shows various saisics for he simulaions discussed in he main body of he paper. 14