The Risky Steady State and the Interest Rate Lower Bound

Transcription

1 The Risky Seady Sae and he Ineres Rae Lower Bound Timohy Hills New York Universiy Taisuke Nakaa Federal Reserve Board This Draf: December 5 Sebasian Schmid European Cenral Bank Absrac Even when he policy rae is currenly no consrained by is effecive lower bound ELB), he possibiliy ha he policy rae will become consrained in he fuure lowers oday s inflaion by creaing ail risk in fuure inflaion and hus reducing expeced inflaion. In an empirically rich model calibraed o mach key feaures of he U.S. economy, we find ha he ail risk induced by he ELB causes inflaion o undershoo he arge rae of percen by as much as 45 basis poins a he economy s risky seady sae. Our model suggess ha achieving he inflaion arge may be more difficul now han before he Grea Recession, if he recen ELB experience has led households and firms o revise up heir esimae of he ELB frequency. JEL: E3, E5, E6, E6, E63 Keywords: Deflaionary Bias, Disinflaion, Inflaion Targeing, Risky Seady Sae, Tail Risk, Zero Lower Bound. We would like o hank Oliver de Groo, Jean-Philippe LaFore, John Robers and seminar paricipans a he Universiy of Tokyo for useful commens. Paul Yoo provided excellen research assisance. The views expressed in his paper, and all errors and omissions, should be regarded as hose solely of he auhors, and are no necessarily hose of he Federal Reserve Board of Governors, he Federal Reserve Sysem, or he European Cenral Bank. Sern School of Business, New York Universiy, 44 Wes 4h Sree New York, NY ; hills.imoeo@gmail.com Division of Research and Saisics, Federal Reserve Board, h Sree and Consiuion Avenue N.W. Washingon, D.C. 55; aisuke.nakaa@frb.gov European Cenral Bank, Moneary Policy Research Division, 664 Frankfur, Germany; sebasian.schmid@ecb.in.

2 Inroducion This paper characerizes he risky seady sae in an empirically rich sicky-price model wih occasionally binding effecive lower bound ELB) consrains on nominal ineres raes. The risky seady sae is he poin where agens choose o say a a given dae if hey expec fuure risk and if he realizaion of shocks is a his dae Coeurdacier, Rey, and Winan )). The risky seady sae is an imporan objec in dynamic macroeconomic models: This is he poin around which he economy flucuaes. This is he poin where he economy evenually converges o when all headwinds and ailwinds dissipae. We firs use a sylized New Keynesian model o illusrae how, and why, he risky seady sae differs from he deerminisic seady sae. We show ha inflaion and he policy rae are lower, and oupu is higher, a he risky seady sae han a he deerminisic seady sae. This resul obains because he lower bound consrain on ineres raes makes he disribuion of firm s marginal coss of producion asymmeric; he decline in marginal coss caused by a large negaive shock is larger han he increase caused by a posiive shock of he same magniude. As a resul, he ELB consrain reduces expeced marginal coss for forwardlooking firms, leading hem o lower heir prices even when he policy rae is no currenly consrained. Reflecing he lower inflaion rae a he risky seady sae, he policy rae is lower a he risky seady sae han ha a he deerminisic seady sae. In equilibrium, he expeced real rae is lower a he risky seady han a he deerminisic seady sae, and he oupu gap is posiive as a resul. These qualiaive resuls are consisen wih hose in Adam and Billi 7) and Nakov 8) on how he ELB risk affecs he economy near he ELB consrain under opimal discreionary policy. We hen urn o he main exercise of our paper, which is o explore he quaniaive imporance of he wedge beween he risky and deerminisic seady saes in an empirically rich DSGE model calibraed o mach key feaures of he U.S. economy. We find ha he wedge beween he deerminisic and risky seady saes is non-rivial in our calibraed empirical model. Inflaion is abou 5 basis poins lower han he arge inflaion of percen a he risky seady sae, wih 8 basis poins aribuable o he ELB consrain as opposed o oher nonlineariies of he model. The policy rae and he oupu gap are 5 basis poins lower and.3 percenage poins higher a he risky seady sae han a he deerminisic seady sae. The magniude of he wedge depends imporanly on he frequency of hiing he ELB, which in urn depends imporanly on he level of he long-run equilibrium policy rae. If he policy rae a he deerminisic seady sae is 4 basis poins lower han our baseline of 3.75 percen, hen he deflaionary bias would increase o more han 5 basis poins, wih he ELB risk conribuing 45 basis poins o he overall deflaionary bias. The observaion ha inflaion falls below he inflaion arge in he policy rule a he risky seady sae is differen from he well-known fac ha he average inflaion falls below he arge rae in he model wih he ELB consrain. The decline in inflaion arising from

3 a conracionary shock can be exacerbaed when he policy rae is a he ELB, while he rise in inflaion arising from an expansionary shock is empered by a corresponding increase in he policy rae. As a resul, he disribuion of inflaion is negaively skewed and he average inflaion falls below he median. This fac is inuiive and has been well known in he profession for a long ime Coenen, Orphanides, and Wieland 4) and Reifschneider and Williams )). The risky seady sae inflaion is differen from he average inflaion; i is he rae of inflaion ha would prevail a he economy s seady sae when agens are aware of risks. I is worh menioning ha he average inflaion falls below he arge even in perfec-foresigh models or backward-looking models where he inflaion rae evenually converges o is arge. On he oher hand, for he risky seady sae inflaion o fall below he inflaion arge, i is crucial ha price-seers are forward-looking and ake ail risk in fuure marginal coss ino accoun in heir pricing decisions. Our resul ha he ELB consrain has enduring effecs on he economy even afer lifoff provides a cauionary ale for policymakers aiming o overcome he problem of persisenly low inflaion. In paricular, according o our model, inflaion a he risky seady sae is ighly linked o how frequenly he policy rae will be consrained by he ELB in he fuure. Thus, our model suggess ha achieving he inflaion arge may be more difficul now han before he Grea Recession, if he recen lower bound experience, ogeher wih he recen downward assessmen of he long-run growh rae of he economy and long-run equilibrium policy rae, have made he privae secor o increase is assessmen of he likelihood of hiing he ELB in he fuure. The quesion of how he possibiliy of reurning o he ELB affecs he economy has remained largely unexplored. The majoriy of he lieraure adops he assumpion ha he economy will evenually reurn o an absorbing sae where he policy rae is permanenly away from he ELB consrain, and analyzes he dynamics of he economy, and he effecs of various policies, when he policy rae is a he ELB Eggersson and Woodford 3), Chrisiano, Eichenbaum, and Rebelo )). While an increasing number of sudies have recenly depared from he assumpion of an absorbing sae, he focus of hese sudies is mosly on how differenly he economy behaves a he ELB versus away from he ELB, insead of how he ELB risk affecs he economy away from he ELB. Wih he federal funds rae finally raised from he ELB consrain afer saying here for seven years, and wih he pace of he policy ighening expeced o be gradual, he quesion of how he possibiliy of reurning o he ELB affecs he economy is as relevan as ever. Our paper builds on he work by Adam and Billi 7) and Nakov 8) who firs observed ha he possibiliy of reurning o he ELB has consequences for he economy even when he policy rae is currenly away from he ELB. Our work differs from hese papers in For example, Gavin, Keen, Richer, and Throckmoron 5) and Keen, Richer, and Throckmoron 5) ask how differenly echnology and anicipaed moneary policy shocks affec he economy when he policy rae is consrained han when i is no, respecively. Schmid 3) and Nakaa 3a) ask how differenly he governmen should conduc fiscal policy when he policy rae is a he ELB han when i is no. 3

4 wo subsanive ways. Firs, while hey poined ou he anicipaion effec of reurning o he ELB on he economy when he policy rae is near he ELB and he economy is away from he seady sae, our work shows ha he possibiliy of reurning o he ELB has consequences for he economy even when he policy rae is sufficienly above he ELB and he economy is a he seady sae. Second, and more imporanly, while hey sudied he effecs of he ELB risk in a highly sylized model, we quanify he magniude of he effecs of he ELB risk in an empirically rich, calibraed model. Our work complimens a body of work ha explores he so-called deflaionary seady sae in sicky price models. A seminal work of Benhabib, Schmi-Grohe, and Uribe ) shows he exisence of a deflaionary seady sae wih he policy rae suck a he lower bound and inflaion below arge in a sandard sicky-price model wih a Taylor rule. Some have recenly sudied deflaionary seady saes wih zero nominal ineres raes in oher ineresing models wih a nominal fricion and a Taylor rule Benigno and Fornaro 5); Eggersson and Mehrora 4); Schmi-Grohe and Uribe 3)). Bullard ) argues ha hese deflaionary seady saes are relevan in undersanding he Japanese economy in he las wo decades as well as wha may happen in oher advanced economies. The seady sae we focus on is similar o he deflaionary seady sae in ha boh enail below-arge inflaion. However, unlike in he deflaionary seady sae, he nominal ineres rae is above he ELB in he risky seady sae. Finally, his paper is relaed o recen papers ha have emphasized he imporance of he effec of risk on seady saes in various nonlinear dynamic models. de Groo 4) and Gerler, Kiyoaki, and Queralo ) discuss how he degree of risk affecs he balance shee condiions of financial inermediaries a he seady sae. Coeurdacier, Rey, and Winan ), Devereux and Suherland ) and Tille and van Wincoop ) sudy opimal porfolio choices a he risky seady sae in open-economy models. While our work is similar o heirs in analyzing he effec of risk on he seady sae, we differ from hem in a fundamenal way: While he wedge beween he deerminisic and risky seady saes is driven by he nonlineariy of smooh differeniable funcions in heir models, he wedge is driven by an inequaliy consrain in our model. Reflecing he difference in he ypes of nonlineariy involved, he soluion mehods employed are differen as well. While hey all solve heir models by using local approximaion mehods ha ake advanage of differeniabiliy of policy funcions, we use a global mehod o solve he model. The res of he paper is organized as follows. Afer a brief review of he concep of he risky seady sae in secion, Secion 3 analyzes he risky seady sae in a sylized New Keynesian economy. Secion 4 quanifies he wedge beween he deerminisic and risky seady saes in an empirically rich DSGE model. Afer puing our analysis in he conex of he curren policy debae in Secion 5, secion 6 concludes. See de Groo 3) and Meyer-Gohde 5) for recen progress in compuing he risky seady sae in nonlinear differeniable economies. 4

5 The Risky Seady Sae: Definiion The risky seady sae is defined generically as follows. 3 Le Γ and S denoe a vecor of endogenous variables and a vecor of exogenous variables in he model under invesigaion. Le f, ) denoe a vecor of policy funcions mapping he values of endogenous variables in he previous period and oday s realizaions of exogenous variables ino he values of endogenous variables oday. 4 Tha is, Γ = fγ, S ) ) The risky seady sae of he economy, Γ RSS, is given by a vecor saisfying he following condiion. Γ RSS = fγ RSS, S SS ) ) where S SS denoe he seady sae of S. 5 Tha is, he risky seady sae is where he economy will evenually converge o as he exogenous variables sele a heir seady sae. In his risky seady sae, he agens are aware of he fac ha shocks o he exogenous variables can occur, bu curren realizaions of hose shocks are zero. On he oher hand, he deerminisic seady sae of he economy, Γ DSS, is defined as follows: Γ DSS = f P F Γ DSS, S SS ) 3) where f P F, ) denoes he vecor of policy funcions obained under he perfec-foresigh assumpion. 3 The Risky Seady Sae in a Sylized Model wih he ELB 3. Model We sar by characerizing he risky seady sae in a sylized New Keynesian model. Since he model is sandard, we only presen is equilibrium condiions here. The deails of he model are described in he Appendix A. C χc = βδ R E C χc + Π + 4) w = N χn C χc 5) 3 Our definiion is idenical o ha found in he lieraure on he risky seady sae, hough our noaion is somewha uncommon. 4 Noe ha he policy funcion does no need o depend on he enire se of he endogenous variables in he prior period. I may no depend on any endogenous variables in he prior period a all, as in he sylized model presened in he nex secion. 5 There is no disincion beween deerminisic and risky seady saes for S because S is exogenous. 5

6 [ ) ] Y Π ϕ C χc Π Π Π θ) θw ) Y + Π+ = βδ E ϕ C χc + Π Π+ Π 6) R = max Y = C + ϕ [ [ Π Π ] Y 7) Y = N 8) ) Π φπ ) ] φy Π Y R ELB, 9) β Π Ȳ δ ) = ρ δ δ ) + ɛ δ ) C, N, Y, w, Π, and R are consumpion, labor supply, oupu, real wage, inflaion, and he policy rae, respecively. δ is an exogenous shock o he household s discoun rae, and follows an AR) process wih mean one, as shown in equaion. Equaion 4 is he consumpion Euler equaion, Equaion 5 is he inraemporal opimaliy condiion of he household, Equaion 6 is he opimaliy condiion of he inermediae good producing firms forward-looking Phillips Curve) relaing oday s inflaion o real marginal cos oday and expeced inflaion omorrow, Equaion 7 is he aggregae resource consrain capuring he resource cos of price adjusmen, and Equaion 8 is he aggregae producion funcion. Equaion 9 is he ineres-rae feedback rule where Π and Ȳ are he cenral bank s objecives for inflaion and oupu. A recursive equilibrium of his sylized economy is a given by a se of policy funcions for {C ), N ), Y ), w ), Π ), R )} saisfying he equilibrium condiions described above. The model is solved wih a global soluion mehod described in deail in he Appendix C. Table liss he parameer values used for his exercise. Table : Parameer Values for he Sylized Model Parameer Descripion Parameer Value β Discoun rae χ c Inverse ineremporal elasiciy of subsiuion for C. χ n Inverse labor supply elasiciy θ Elasiciy of subsiuion among inermediae goods ϕ Price adjusmen cos 4 Π ) Annualized) arge rae of inflaion. φ π Coefficien on inflaion in he Taylor rule.5 φ y Coefficien on he oupu gap in he Taylor rule R ELB The effecive lower bound ρ AR) coefficien for he discoun facor shock.8.4 σ ɛ The sandard deviaion of shocks o he discoun facor *The implied prob. ha he policy rae is a he lower bound % 6

7 Figure : Policy Funcions from he Sylized Model Annualized % Nominal Ineres Rae Annualized % 3 Inflaion % Deviaion from he De. Seady Sae Oupu δ δ δ Real Wage 3 Exp. Real Rae Annualized % Annualized % Wih uncerainy Wihou uncerainy Deerminisic S.S. Risky S.S δ δ *The dashed black lines Wihou uncerainy case) show policy funcions obained under he perfec-foresigh assumpion i.e., σ ɛ = ). 3. Dynamics and he risky seady sae Before analyzing he risky seady sae of he model, i is useful firs o look a he dynamics of he model. Solid black lines in Figure show he policy funcions for he policy rae, inflaion, oupu, and he expeced real ineres rae. Dashed black lines show he policy funcion of he model obained under he assumpion of perfec foresigh. Under he perfec-foresigh case, he agens in he model aach zero probabiliy o he even ha he policy rae will reurn o he ELB when he policy rae is currenly away from he ELB. Under boh versions of he model, an increase in he discoun rae makes he households wan o save more for omorrow and spend less oday. Thus, as δ increases, oupu, inflaion, and he policy rae decline. When δ is large and he policy rae is a he ELB, an addiional increase in he discoun rae leads o larger declines in inflaion and oupu han when δ is small and he policy rae is no a he ELB, as he adverse effecs of he increase in δ are no 7

8 counered by a corresponding reducion in he policy rae. When he policy rae is a he ELB, he presence of uncerainy reduces inflaion and oupu. This is capured by he fac ha he solid lines are below he dashed lines for inflaion and oupu in he figure. The non-neuraliy of uncerainy is driven by he ELB consrain. If he economy is buffeed by a sufficienly large expansionary shock, hen he policy rae will adjus o offse some of he resuling increase in real wages. If he economy is hi by a conracionary shock, regardless of he size of he shock, he policy rae will say a he ELB and he resuling decline in real wages will no be empered. Due o his asymmery, an increase in uncerainy reduces he expeced real wage, which in urn reduces inflaion as price-seers are forward-looking. Wih he policy rae consrained a he ELB, a reducion in inflaion leads o an increase in he expeced real rae, pushing down consumpion and oupu oday. These adverse effecs of uncerainy a he ELB are sudied in deail in Nakaa 3b). When he policy rae is away from he ELB, he presence of uncerainy reduces inflaion and he policy rae, bu increases oupu. If he economy is hi by a sufficienly large conracionary shock, he policy rae will hi he ELB and he resuling decline in real wages will no be empered. If he economy is hi by an expansionary shock, regardless of he size of he shock, he policy rae will adjus o parially offse he resuling increase in real wages. Thus, he presence of uncerainy, by generaing he possibiliy ha he policy rae will reurn o he ELB, reduces he expeced real wage and hus oday s inflaion. When he policy rae is away from he ELB, is movemen is governed by he Taylor rule. Since he Taylor principle is saisfied i.e., he coefficien of inflaion is larger han one), he reducion in inflaion comes wih a larger reducion in he policy rae. As a resul, he expeced real rae is lower, and hus consumpion and oupu are higher, wih uncerainy han wihou uncerainy. Table : The Risky Seady Sae in he Sylized Model Inflaion Oupu Policy rae Deerminisic seady sae 3.75 Risky seady sae Wedge).9).3).43) Risky seady sae w/o he ELB Wedge).).).3) *Oupu is expresed as a percenage deviaion from he deermisic seady sae. While hese effecs are sronger he closer he policy rae is o he ELB, hey remain nonrivial even a he economy s risky seady sae. In he sylized model of his secion in which he policy funcions do no depend on any of he model s endogenous variables from he previous period, he risky seady sae is given by he vecor of he policy funcions evaluaed a δ =. Tha is, inflaion, oupu, and he policy rae a he risky seady sae are given by he inersecion of he policy funcions for hese variables and he lef verical axes. As shown in Table, inflaion and oupu are 9 basis poins lower and.3 percenage 8

9 poins higher a he risky seady sae han a he deerminisic seady sae, respecively. The risky seady sae policy rae is 43 basis poin lower han is deerminisic counerpar. In our model, he ELB consrain is no he only source of nonlineariy. Our specificaions of he uiliy funcion and he price adjusmen cos also make he model nonlinear, and hus explain some of he wedge beween he deerminisic and risky seady saes. To undersand he exen o which hese oher nonlineariies maer, Table also repors he risky seady sae in he version of he model wihou he ELB consrain. Overall, he differences beween he deerminisic and risky seady saes are small were i no for he ELB consrain. Inflaion and he policy rae a he risky seady sae are only and 3 basis poins below hose a he deerminisic seady sae, respecively. Oupu a he risky seady sae is abou basis poins below ha a he deerminisic seady sae. Thus, he majoriy of he overall wedge beween he deerminisic and risky seady saes is aribued o he nonlineariy induced by he ELB consrain, as opposed o oher nonlineariies of he model. 3.3 The risky seady sae and he average Figure : Uncondiional Disribuion of Inflaion in he Sylized Model Inflaion Targe: RSS Inflaion:.7 Average Inflaion:.65 Skewness:.338 ELB Binds *RSS sands for he Risky Seady Sae. I is imporan o recognize ha he risky seady sae is differen from he average. Le s ake inflaion as an example. The risky seady sae inflaion is he poin around which inflaion flucuaes and coincides wih he median of is uncondiional disribuion in he model wihou any endogenous sae variables like he one analyzed here. On he oher hand, he average inflaion is he average of inflaion in all saes of he economy. Provided ha he probabiliy of being a he ELB is sufficienly large, he uncondiional disribuion of inflaion is negaively skewed and herefore he risky seady sae inflaion is higher han 9

10 he average inflaion, as depiced in Figure. The observaion ha he ELB consrain pushes down he average inflaion below he median by making he disribuion of inflaion negaively skewed is inuiive and has been well known for a long ime Coenen, Orphanides, and Wieland 4) and Reifschneider and Williams )). This holds rue even when price-seers form expecaions in a backward-looking manner. The resul ha he ELB risk lowers he median of he disribuion below he arge is less inuiive and requires ha price-seers are forward-looking in forming heir expecaions. The magniude of he wedge beween he deerminisic and risky seady saes depends imporanly on he probabiliy of being a he ELB. The blue line in he lef panel of Figure 3 illusraes his poin for inflaion. In his figure, we vary he sandard deviaion of he discoun rae shock o induce changes in he probabiliy of being a he ELB. According o he blue line, a higher probabiliy of being a he ELB is associaed wih a larger deflaionary bias a he risky seady sae. In his sylized model, he risky seady sae increases from 9 basis poins o 38 basis poins when he probabiliy of being a he ELB increases from percen o percen. Similarly, he average inflaion is lower when he probabiliy of being a he ELB is higher, as shown by he red line. As discussed earlier, when he ELB probabiliy is sufficienly high, he average inflaion is below he risky seady sae inflaion. However, when he ELB probabiliy is sufficienly low, he average inflaion is above he risky seady sae inflaion, which reflecs he fac ha oher nonlinear feaures of he model make he uncondiional disribuion of inflaion slighly posiively skewed. Figure 3: Condiional and Uncondiional Averages of Inflaion in he Sylized Model..4 E[Π R = ].6 Inflaion Annualized %) Inflaion Annualized %) E[Π] E[Π R > ] RSS Inflaion Prob[R = ] in %) Prob[R = ] in %) Figure 3 also plos he condiional averages of inflaion away from he ELB he black

11 line in he lef panel) and he condiional average of inflaion a he ELB he dashed black line in he righ panel). No surprisingly, he condiional average of inflaion away from he ELB is higher han he uncondiional average of inflaion, which in urn is higher han he condiional average of inflaion a he ELB. The condiional average of inflaion a he ELB monoonically declines wih he probabiliy of being a he ELB, as he risky seady sae inflaion and he uncondiional average inflaion do. In conras, he condiional average of inflaion away from he ELB is non-monoonic; I increases when he ELB probabiliy is low and declines when he probabiliy is high. As a resul, he condiional average of inflaion away from he ELB is above he arge rae of percen when he ELB probabiliy is sufficienly low. This happens because, when he ELB probabiliy is sufficienly low, he condiional disribuion of inflaion away from he ELB excludes he lower ail of he uncondiional disribuion which is cenered around a level ha is only slighly below percen. However, he condiional average of inflaion away from he ELB is below he arge rae when he lower bound risk is sufficienly high and he uncondiional disribuion of inflaion is cenered around a poin sufficienly below percen. As described in he Appendix H, he condiional average of inflaion away from he ELB is always above he arge rae of percen in he perfec-foresigh version of he model wih he ELB. Thus, he imporance of he ELB risk manifess iself in he below-arge condiional average of inflaion away from he ELB. While he risky seady sae inflaion canno be measured in he daa, he condiional average of inflaion away from he ELB can be. Thus, he imporance of he lower bound risk in he daa manifess iself in he exen o which he condiional average of inflaion away from he ELB falls below he arge rae of inflaion. We will laer examine he average inflaion away from he lower bound in several advanced economies in Secion The risk-adjused Fisher relaion One way o undersand he discrepancy beween deerminisic and risky seady saes is o examine he effec of he ELB risk on he Fisher relaion. Le R DSS and Π DSS be he deerminisic seady sae policy rae and inflaion. In he deerminisic environmen, he consumpion Euler equaion evaluaed a he seady sae becomes R DSS = Π DSS β ) afer dropping he expecaion operaor from he consumpion Euler equaion and eliminaing he deerminisic seady-sae consumpion from boh sides of he equaion. This relaion is ofen referred o as he Fisher relaion. In he sochasic environmen, he consumpion Euler equaion evaluaed a he risky)

12 Figure 4: The Risk-Adjused Fisher Relaion and he Taylor Rule R DSS Sandard Fisher Relaion Risk Adjused Fisher Relaion RSS Taylor Rule DSS sands for deerminisic seady sae, and RSS sands for risky seady sae. Π seady sae can be wrien as R RSS = Π RSS β E RSS [ CRSS C + ) χc Π RSS Π + ] ) where R RSS, Π RSS, and C RSS are he risky seady-sae policy rae, inflaion, and consumpion. E RSS [ ] is he condiional expecaion operaor when he economy is a he risky seady sae oday. In he sylized model wih one shock and wihou any endogenous sae variables, E RSS [ ] := E [ δ = ]. We will refer o Equaion as he risk-adjused Fisher relaion. Relaive o he sandard Fisher relaion, here is an adjusmen erm ha reflecs he discrepancy beween oday s economic condiions and he expeced economic condiions nex period. Noice ha he adjusmen erm is less han one, E RSS [ CRSS C + ) χc Π RSS Π + ] <, 3) because of he fa ail on he lower end of he disribuions of fuure inflaion and consumpion induced by he ELB consrain. In equilibrium, he seady sae is given by he inersecion of he line represening he Fisher relaion and he line represening he Taylor rule. Since he risk-adjusmen erm is less han one, he line represening he risk-adjused Fisher relaion crosses he line represening he Taylor rule a a poin below he line for he sandard Fisher relaion crosses i, as shown

13 in Figure 4. Thus, inflaion and he policy rae are lower a he risky seady sae han a he deerminisic seady sae. 6 4 The Risky Seady Sae in an Empirical Model wih he ELB We now quanify he magniude of he wedge beween he deerminisic and risky seady saes in an empirically rich model calibraed o mach key feaures of he U.S. economy. 4. Model Our empirical model adds four addiional feaures on op of he sylized New Keynesian model of he previous secion. The four addiional feaures are a non-saionary produciviy process, consumpion habis, sicky wages, and an ineres-rae smoohing erm in he ineres-rae feedback rule. Since hese feaures are sandard, we relegae he deailed descripion of hem o he Appendix B and only show he equilibrium condiions of he model here. Le Ỹ = Y A, C = C A, w = w A, and λ = λ be he saionary represenaions A χc of oupu, consumpion, real wage, and marginal uiliy of consumpion, respecively. The saionary equilibrium is characerized by he following sysem of equaions: N w λ λ = [ ) Π w ϕ Π w w Π w Π w θw ) θ w N ] χn λ w Ỹ Π p ) Π p [ϕ λ p Π p β a χc δ R E λ+ Π p +) 4) λ = C ζ a C ) χc 5) Π p θp ) θ p w ] Ỹ = C + ϕ p = βϕ w a χc δ N + w + Π w ) + Π w E λ + Π w + Π w 6) Π w = w w Π p 7) = βϕ p a χc δ Ỹ + E λ + Π p ) + Π p + Π p Π p 8) [ Π p ] Π p Ỹ + ϕ [ ] w Π w Π w w N 9) Ỹ = N ) 6 The oher inersecion of he risk-adjused Fisher relaion and he Taylor-rule equaion indicaes ha, in a deflaionary equilibrium, inflaion is higher a he risky seady sae han a he deerminisic seady sae. We have confirmed ha his is indeed he case using a semi-loglinear model wih a hree-sae discoun rae shock. We will leave an in-deph examinaion of he risky seady sae in a deflaionary equilibrium o fuure research. 3

14 and where R = max [, R ] ) R R ) ρr R = Π p ) ρr)φπ R Π p ) ρr)φy Ỹ ) Ȳ and he following processes for he discoun rae: δ ) = ρ δ δ ) + ɛ δ 3) ζ is he degree of consumpion habis in he household s uiliy funcion and a is he rend growh rae of produciviy. ϕ p and ϕ w are he price and wage adjusmen coss. ρ R is he weigh on he lagged shadow policy rae in he runcaed ineres-rae feedback rule. Πp and Π w are price and wage inflaion raes in he deermisic seady sae. In he runcaed ineres-rae feedback rule, R and Ȳ are he policy rae and oupu normalized by A ) a he deerminisic seady sae and are funcions of he srucural parameers. Ỹ /Ȳ is he deviaion of he saionarized oupu from is deerminisic seady sae and will be referred o as he oupu gap in his paper. 7 In he Appendix G, we also sudy he risky seady sae in several exended models which add more fricions and shocks o he empirical model of his secion. 4. Calibraion We calibrae our model o mach key feaures of he oupu gap, inflaion, and he policy rae in he U.S. over he pas wo decades, which are shown in Figure 5. We focus on his relaively recen pas because long-run inflaion expecaions were low and sable and he ELB was eiher a concern or a binding consrain o he Federal Reserve during his period. As shown in he boom-righ panel of Figure 5, he median of CPI inflaion forecass 5- years ahead in he Survey of Professional Forecasers, a commonly used measure of long-run inflaion expecaions, declined o.5 percen in lae 99s and has been relaively sable since hen. 8 Also, he concern for he ELB surged in he U.S. in he second half of 99s when he Bank of Japan lowered he policy rae o he lower bound for he firs ime in he Pos WWII hisory among major advanced economies. 9 We se he ime discoun rae o so ha he conribuion of he discoun rae o 7 Noe ha our oupu gap is no he flexible-price oupu gap ha is he deviaion of he normalied) oupu from is flexible-price counerpar. 8 The long-run inflaion expecaions measured by PCE inflaion is available only from 7. The average differenial beween CPI and PCE inflaion raes over he pas wo decades is abou 5 basis poins. Thus, he sabiliy of CPI inflaion expecaions a.5 percen can be inerpreed as he sabiliy of PCE inflaion expecaions a percen. 9 Some of he earlies research on he ELB were iniiaed wihin he Federal Reserve Sysem in his period. See, for example, Clouse, Henderson, Orphanides, Small, and Tinsley 3), Reifschneider and Williams ), and Wolman 998). 4

15 Figure 5: Oupu Gap, Inflaion, Policy Rae and Long-Run Inflaion Expecaions 5 Oupu Gap %) 3 Inflaion Annualized %) Year Year 8 Policy Rae Annualized %) 4 Long Run Inflaion Expecaions Year Year The measure of he oupu gap is based on he FRB/US model. The inflaion rae is compued as he annualized quarerly percenage change log difference) in he personal consumpion expendiure core price index. The quarerly average of he annualized) federal funds rae is used as he measure for he policy rae. Verical lines mark he year when ELB sared binding. Horizonal lines represen arge values for he respecive variables. he deerminisic seady sae real rae is 5 basis poins. We se he arge inflaion in he ineres-rae feedback rule o percen as his is he FOMC s official arge rae of inflaion. We se he rend growh rae of produciviy o.5 percen so ha he policy rae is 3.75 percen a he economy s deerminisic seady sae. In he household uiliy funcion, he degree of consumpion habis, he inverse Frisch labor elasiciy, and he inverse ineremporal elasiciy of subsiuion are se o.5,.5 and, respecively. These are all wihin he range of sandard values found he lieraure. Following Erceg and Lindé 4), he parameers governing he seady-sae markups for inermediae goods and he inermediae labor inpus are se o and 4 and he parameers governing he price adjusmen coss for prices and wages o and 3. In a hypoheical log-linear environmen, hese values would correspond o 9 and 85 percen probabiliies ha prices and wages canno adjus each quarer in he Calvo version of he model, respecively. High degrees of sickiness in prices and wages help he model o mach he moderae decline in inflaion in he daa while he federal funds rae was consrained a he ELB. The coefficiens on inflaion and he oupu gap in he ineres-rae feedback rule are 5

16 Table 3: Parameer Values for he Empirical Model Parameer Descripion Parameer Value β Discoun rae a Trend growh rae of produciviy 4 χ c Inverse ineremporal elasiciy of subsiuion for C. ζ Degree of consumpion habis.5 χ n Inverse labor supply elasiciy.5 θ p Elasiciy of subsiuion among inermediae goods θ w Elasiciy of subsiuion among inermediae labor inpus 4 ϕ p Price adjusmen cos ϕ w Wage adjusmen cos 3 Ineres-rae feedback rule 4 Π ) Annualized) arge rae of inflaion. ρ R Ineres-rae smoohing parameer in he Taylor rule.8 φ π Coefficien on inflaion in he Taylor rule 3 φ y Coefficien on he oupu gap in he Taylor rule.5 4R ELB ) Annualized) effecive lower bound.3 Shocks ρ d AR) coefficien for he discoun facor shock σ ɛ,δ The sandard deviaion of shocks o he discoun facor se o 3 and.5. The coefficien on he oupu gap,.5, is sandard. The coefficien on inflaion is a bi higher compared o he values commonly used in he lieraure. A higher coefficien serves wo purposes. Firs, i reduces he volailiy of inflaion, relaive o he volailiy of he oupu gap. Second, a higher value makes he exisence of he equilibrium more likely. Erceg and Lindé 4) argue ha an inflaion coefficien of his magniude is consisen wih a IV-ype regression esimae of his coefficien based on a recen sample. The ineres-rae smoohing parameer for he policy rule is se o.8. This high degree of ineres-rae smoohing helps in increasing he expeced duraion of he lower bound episodes, improving he model s implicaion in his dimension. The ELB on he policy rae is se o.3 percen, he average of he annualized federal funds rae during he recen lower bound episode from 9:Q o 5:Q). The persisence of he discoun rae shock is se o.85. This is a bi higher han he common value of.8 used in mos exising sudies of models wih occasionally binding lower bound consrains. A higher persisence of he shock helps in increasing he expeced duraion of being a he ELB, as he higher ineres-rae smoohing parameer in he policy rule does. The sandard deviaion of he discoun rae shock is chosen so ha he sandard deviaion of he policy rae in he model maches wih ha in he daa. Table 4 shows he key saisics for he oupu gap, inflaion and he policy rae in he model and in he daa. The measure of he oupu gap is based on he esimae of poenial Richer and Throckmoron 5) show ha he model wih occasionally binding ELB consrains may no have minimum-sae-variable soluions when his coefficien is low even if he Taylor-principle is saisfied. See, for example, Adam and Billi 7), Nakov 8), and Fernández-Villaverde, Gordon, Guerrón- Quinana, and Rubio-Ramírez 5). 6

17 oupu based from he FRB/US model. As for he measure of inflaion, we use core PCE Price Index inflaion as his is he measure U.S. policymakers focus on. Table 4: Key Momens Momen Variable Model S.Dev. ) EX ELB) ELB Daa 995Q3 5Q) Oupu gap 3..9 Inflaion.3.5 Policy rae Oupu gap Inflaion..48 Policy rae.3.3 Frequency 3.8% 36% Expeced/Acual Duraion 8.6 quarers 6 quarers The measure of he oupu gap is based on he FRB/US model. Inflaion rae is compued as he annualized quarerly percenage change log difference) in he personal consumpion expendiure core price index. The quarerly average of he annualized) federal funds rae is used as he measure for he policy rae. The sandard deviaion of he oupu gap in he model is 3., which is in line wih he sample sandard deviaion from he daa. The condiional mean of he oupu gap a he ELB in he model is -3.7 percen, which is above he esimae from he daa. The sandard deviaion of inflaion in he model is.3 percen, which is lower han wha s observed in he daa, while he ELB condiional mean of inflaion in he model is. percen, which is somewha lower han wha s observed in he daa. I should be noed ha, given ha inflaion was remarkably sable during he recen lower-bound episode in he U.S., here is a ension in maching he uncondiional sandard deviaion of inflaion and he condiional average of inflaion a he ELB. Since he condiional mean of inflaion is he average of he lef-ail of he uncondiional disribuion of inflaion, an increase in he sandard deviaion of inflaion would necessarily imply a decrease in he condiional mean of inflaion a he ELB. In our case, if we increase he sandard deviaion of inflaion in our model o ge closer o wha s observed in he daa, he condiional mean of inflaion a he ELB would decline furher away from is empirical counerpar. Our calibraion reflecs a compromise of saying reasonably close o he daa in hese wo dimension. I is worh noing ha our esimae of he deflaionary bias would be higher if we ignored he observed sabiliy of inflaion a he ELB and adjused our calibraion o mach he sandard deviaion of inflaion, as a lower condiional average of inflaion a he ELB induces firms o lower prices by more when away from he ELB. Thus, our esimae of he deflaionary bias can be seen as a conservaive esimae. As previously menioned, he sandard deviaion of he discoun rae shock was chosen A similar ension exiss for maching he sandard deviaion of he oupu gap and he condiional mean of he oupu gap a he ELB, bu o a lesser exen. 7

18 so ha he sandard deviaion of he policy rae in he model maches wih ha in he daa, which is.34 percen. The model-implied uncondiional probabiliy of being a he ELB and he expeced ELB duraion are abou 4 percen and years, respecively. While hese numbers are subsanially higher han hose in oher exising models wih occasionally binding ELB consrains, hey are subsanially lower han he empirical counerpars over he pas wo decades in he U.S. In paricular, he duraion of he recen ELB experience is seen by he model as surprisingly long. Consisen wih his inerpreaion, he daa on lifoff expecaions shows ha marke paricipans have underprediced how long he policy rae will be kep a he ELB hroughou he recen ELB episode, as described in he Appendix E. 4.3 Resuls Table 5: The Risky Seady Sae in he Empirical Model Inflaion Oupu gap Policy rae Deerminisic seady sae 3.75 Risky seady sae Wedge).6).3).49) Risky seady sae w/o he ELB Wedge).8).5).9) E[ R > R ELB ] Table 5 shows he risky and deerminisic seady sae values of inflaion, he oupu gap, and he policy rae from our empirical model. For his model, he risky seady sae is compued by simulaing he model for a long period while seing he realizaion of he exogenous disurbances o zero. All saionarized) endogenous variables evenually converge in ha simulaion, and ha poin of convergence is he risky sae of he economy. consrucion, he deerminisic seady sae of inflaion is given by he arge rae of inflaion and he oupu gap is zero a he deerminisic seady sae. As explained earlier, parameer values β, χ c and a) are chosen so ha he deerminisic seady sae of he policy rae is 3.75 percen. Consisen wih our earlier analyses based on a sylized model, inflaion and he policy rae are lower, and he oupu gap is higher, a he risky seady sae han a he deerminisic seady sae. Inflaion falls 6 basis poins below he arge rae of inflaion a he risky seady sae. This is large given he small sandard deviaion of inflaion. The policy rae a he risky seady sae falls 49 basis poins below is deerminisic counerpar. While his is a small number relaive o is sandard deviaion, i is neverheless significan in ligh of recen discussions among economiss and policymakers regarding he long-run equilibrium policy rae. 3 Finally, he oupu wedge beween he deerminisic and risky seady saes is small, wih he oupu gap sanding a.3 percenage poin a he risky seady sae. By 3 See, for example, Hamilon, Harris, Hazius, and Wes 5). 8

19 As explained in he previous secion, he discrepancy beween he deerminisic and risky seady saes is no only driven by he lower bound consrain on policy raes, bu is also affeced by oher nonlinear feaures of he model. To isolae he effecs of he lower bound consrain, he fourh line of Table 5 shows he risky seady sae of he model wihou he lower bound consrain. Inflaion, he oupu gap, and he policy rae are.9,.5, and 3.56 percen, respecively. Thus, he mos of he wedge beween he deerminisic and risky seady saes in he model wih he ELB consrain is aribued o he nonlineariy associaed wih he ELB consrain, as opposed o oher nonlinear feaures of he model. For inflaion, he ELB risk accouns for 8 basis poins of he overall deflaionary bias. 4.4 Sensiiviy Analyses 4.4. Long-Run Ineres Raes Figure 6: Long-Run Ineres Raes and he Risky Seady Sae 6 Probabiliy of being a he ELB RSS Inflaion Wih ELB Consrain Wihou ELB Consrain DSS Policy Rae DSS Policy Rae.7 RSS Oupu Gap 4.5 RSS Policy Rae DSS Policy Rae DSS Policy Rae DSS sands for deerminisic seady sae, and RSS sands for risky seady sae. Verical lines mark he DSS policy rae in he baseline calibraion. There are subsanial uncerainies surrounding he level of he long-run real equilibrium ineres rae. Many economiss have recenly argued ha various srucural facors including a lower rend growh rae of produciviy, demographic rends, and global facors 9

20 have conribued o a persisen downward rend in he long-run equilibrium ineres rae. 4 A lower long-run equilibrium ineres rae means ha he probabiliy of hiing he ELB is higher, which ceeris paribus increases he magniude of he undershooing of he inflaion arge a he risky seady sae. Figure 6 shows how sensiive he risky seady sae of our empirical model is o alernaive assumpions abou he deerminisic seady sae ineres rae. In his exercise, we vary he long-run deerminisic seady sae policy rae by varying he rend growh rae. As shown in he op-lef panel, he probabiliy of he policy rae being a he ELB increases as he deerminisic seady sae policy rae declines. Wih he deerminisic seady sae policy rae a 3.35 percen, he probabiliy of being a he ELB is approximaely 5 percen. A higher probabiliy of being a he ELB increases he wedge beween he deerminisic and risky seady saes. Wih he deerminisic seady sae policy rae a 3.35 percen, he risky seady sae inflaion, oupu, and he policy raes are.47 percen,.64 percen, and.39 percen. 5 Since he risky seady sae does no depend on he deerminisic seady sae policy rae in he model wihou he ELB consrain, as shown in he dashed lines, a large fracion of he overall wedge is explained by he ELB risk when he deerminisic seady sae policy rae is lower. For inflaion, he ELB risk accouns for 45 basis poins of he overall deflaionary bias of 53 basis poins Policy Parameers We have shown ha, a he risky seady sae, inflaion falls below he arge rae of percen by a nonrivial amoun in our empirical model. In our model where he prices and wages are indexed o he arge rae of inflaion, such undershooing of he inflaion arge is undesirable. A naural quesion o ask is wha he cenral bank can do o miigae he deflaionary bias. Figure 7 showd how he probabiliy of being a he ELB and he risky seady-sae inflaion rae depends on parameers governing he ineres-rae feedback rule. According o he op-lef panel, a higher coefficien on inflaion reduces he probabiliy of being a he ELB, bringing he risky seady sae closer o he inflaion arge. On he oher hand, a higher coefficien on he oupu gap increases he probabiliy of being a he ELB and reduces he RSS inflaion. These resuls are consisen wih he analyical resuls from a wo-sae shock model in Nakaa and Schmid 4). The op-righ panel shows ha a higher ineres-rae smoohing parameer reduces he probabiliy of being a he ELB, hus reducing he deflaionary bias a he risky seady sae. This makes sense as higher ineria in he policy rule limis he response of he policy rae o flucuaions in he demand shock, lowering he sandard deviaion of he policy rae. 4 See, for example, he Council of Economic Advisers 5) and IMF 4). 5 Noe ha an increase in he oupu gap does no necessarily mean an increase in he level of oupu because oupu measures are saionarized by he rend growh rae.

21 Figure 7: Alernaive Policy Parameers 7 4 RSS Inflaion Prob[R =R ELB ] Inflaion Coefficien φ π ) Oupu Gap Coefficien φ y ) Ineres Rae Smoohing Parameer ρ r ) RSS Inflaion Prob[R =R ELB ] RSS Inflaion Lef Axis) ELB Frequency Righ Axis) Inflaion Targe Π p ) Effecive Lower Bound R ELB ) In each panel, he verical line marks he baseline parameer value. No surprisingly, a higher inflaion arge reduces he probabiliy of being a he ELB and reduces he difference in inflaion beween deerminisic and risky seady saes, as shown in he boom-lef panel. Wih he inflaion arge a.8 percen, he deflaionary bias a he risky seady sae is abou 35 basis poins. Wih he inflaion arge a 3 percen, he deflaionary bias a he risky seady sae is abou basis poins. This exercise demonsraes he imporance of aking ino accoun he lower bound risk in he cos-benefi analysis of raising he inflaion arge. 6 Finally, he las panel demonsraes ha, if he effecive lower bound is lower, he probabiliy of being a he ELB is lower and hus he deflaionary bias is lower a he risky seady sae. 5 Discussion Why should we care abou wha a heoreical model has o say abou he ELB risk and he resuling deflaionary bias? In his secion, we argue ha we should care abou i for 6 The compuaion of he opimal inflaion arge is ofen conduced under he assumpion of perfecforesigh. See, for example, Williams 9) and Coibion, Gorodnichenko, and Wieland ).

22 wo reasons. Firs, he model wih he ELB risk is consisen wih he undershooing of he inflaion arge observed in some economies even before he policy rae became consrained by he ELB consrain. Second, he model wih he ELB risk provides a cauionary ale for policymakers aiming o achieve heir inflaion objecives in he curren environmen of low long-run equilibrium real ineres raes. 5. Low inflaion before he Grea Recession Figure 8 shows he condiional averages of inflaion over he pas wo decades when policy raes were no consrained by he ELB in six advanced economies economies ha have recenly faced he ELB consrain for he firs ime since WWII and arge % inflaion. According o he figure, he condiional averages of inflaion away from he ELB are below he arge rae of percen in all hese economies. In he U.S and Canada, inflaion averaged abou and 4 basis poins below percen while he policy rae was no consrained by he ELB. In he euro area and he UK, he condiional averages of inflaion away from he ELB are 5 and 6 basis poins below percen. In Sweden and Swizerland, he average inflaion raes are below percen while he policy rae was above he ELB. In he Appendix F, we show ha he undershooing of he inflaion arge is robus o alernaive saring daes of sample and he difference beween he condiional average and he arge rae is saisically significan for all economies. While here are many explanaions for his sysemaic undershooing of he inflaion arges before he ELB became a binding consrain such as posiive supply shocks from emerging economies and persisen slack in he economies i is ineresing o noe ha his undershooing is consisen wih he predicion of he model wih he ELB risk. As argued earlier, one key predicion of he model wih he ELB risk is ha he condiional average of inflaion away from he ELB falls below he arge rae of percen, provided ha he probabiliy of being a he ELB is sufficienly high. In our empirical model, he deflaionary bias is indeed large enough o push he condiional average of inflaion away from he ELB below he inflaion arge, as shown in he las row of Table 5. 7 Since he risk of hiing he ELB was widely seen as unlikely before he Grea Recession, we do no hink ha his ELB risk is a good explanaion for he arge undershooing in pre- ELB era in realiy. However, we hink he abiliy of our model o generae he undershooing in pre-elb era is aracive. As explained in Secion 3.3 and Appendix H, a version of our model wih he ELB consrain, bu wihou he ELB risk, does no have his feaure. More research is cerainly needed o beer undersand he sources of low inflaion in advanced economies and wha policymakers can do o address his problem. 7 Noe ha his undershooing of he inflaion arge while he policy rae is above he ELB is no consisen wih he deflaionary seady sae of he sicky-price economy Benhabib, Schmi-Grohe, and Uribe ) and Armener 4)). In he deflaionary seady sae, inflaion is below he arge, bu he policy rae is a he ELB.

23 Figure 8: Condiional Average of Inflaion Away From he ELB.5.5 Unied Saes Canada Euro area 9 counries) Unied Kingdom Sweden Swizerland The figure shows he average of he annualied quarerly inflaion rae over he pas wo decades since 995Q3) for each counry during non-elb quarers when he counry s policy rae was no consrained by he ELB. For he U.S, he measure of inflaion is based on core PCE Price Index. For he Euro Area, he measure of inflaion is based on core HICP. For oher four economies, he measure of inflaion is ased on core CPI Index. ELB-binding quarers are from 9Q o presen in he U.S., from 9:Q o :Q in Canada, from 4:Q o presen in Euro Area, from 9:Q o presen in UK, from 9:Q4 o :Q and from 4:Q4 o presen in Sweden, and from 9:Q o presen in Swizerland. Daa for inflaion are obained from OECD, excep from he U.S. where we obained he daa from S. Louis Fed s FRED. 5. An implicaion for he fuure I is quie likely ha he perceived probabiliy of hiing he ELB is higher now han before he Grea Recession. The Grea Recession made many of us o realize ha he economy can be hi by shocks ha are subsanially larger han he macro shocks ha hi he economy during he Grea Moderaion. Also, several years of disappoining oupu growh in he afermah of he Grea Recession have led many analyss o revise down heir esimaes of he rend growh rae of produciviy and long-run nominal ineres raes. A lower long-run equilibrium ineres rae would imply a higher ELB frequency. As our earlier sensiiviy analyses demonsraed, he size of he deflaionary bias a he risky seady sae increases wih he probabiliy of being a he ELB. Thus, our model provides a cauionary ale for policymakers: Achieving he arge rae of inflaion may have become more difficul now han before he Grea Recession if he recen ELB experience has led he privae secor o revise is assessmen of he likelihood of ELB evens. 3