The Optimal Inflation Rate in New Keynesian Models: Should Central Banks Raise Their Inflation Targets in Light of the ZLB?


 Mervin Ward
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1 The Opimal Inflaion Rae in New Keynesian Models: Should Cenral Banks Raise Their Inflaion Targes in Ligh of he ZLB? Olivier Coibion Yuriy Gorodnichenko Johannes Wieland College of William and Mary U.C. Berkeley U.C. Berkeley and NBER March 3, Absrac: We sudy he effecs of posiive seadysae inflaion in New Keynesian models subjec o he zero bound on ineres raes. We derive he uiliybased welfare loss funcion aking ino accoun he effecs of posiive seadysae inflaion and solve for he opimal level of inflaion in he model. For plausible calibraions wih cosly bu infrequen episodes a he zerolower bound, he opimal inflaion rae is low, less han wo percen, even afer considering a variey of exensions, including opimal sabilizaion policy, price indexaion, endogenous and saedependen price sickiness, capial formaion, modeluncerainy, and downward nominal wage rigidiies. On he normaive side, price level argeing delivers large welfare gains and a very low opimal inflaion rae consisen wih price sabiliy. These resuls sugges ha raising he inflaion arge is oo blun an insrumen o efficienly reduce he severe coss of zerobound episodes. Keywords: Opimal inflaion, New Keynesian, zero bound, price level argeing. JEL codes: E3, E4, E5. We are graeful o Robero Billi, Ariel Bursein, Gaui Eggersson, Jordi Gali, Marc Gianonni, Chrisian Hellwig, David Romer, Eric Sims, Alex Wolman, and seminar paricipans a John Hopkins Universiy, Bank of Canada, College of William and Mary, and NBER Summer Insiue in Moneary Economics and Economic Flucuaions and Growh for helpful commens.
2 The crisis has shown ha ineres raes can acually hi he zero level, and when his happens i is a severe consrain on moneary policy ha ies your hands during imes of rouble. As a maer of logic, higher average inflaion and hus higher average nominal ineres raes before he crisis would have given more room for moneary policy o be eased during he crisis and would have resuled in less deerioraion of fiscal posiions. Wha we need o hink abou now is wheher his could jusify seing a higher inflaion arge in he fuure. Olivier Blanchard, February h, I Inroducion One of he defining feaures of he curren economic crisis has been he zero bound on nominal ineres raes. Wih sandard moneary policy running ou of ammuniion in he mids of one of he sharpes downurns in posworld War II economic hisory, some have suggesed ha cenral banks should consider allowing for higher arge inflaion raes han would have been considered reasonable jus a few years ago. We conribue o his quesion by explicily incorporaing posiive seadysae (or rend ) inflaion in a New Keynesian model as well as he zero lower bound (ZLB) on nominal ineres raes. We derive he effecs of nonzero seadysae inflaion on he loss funcion, hereby laying he groundwork for welfare analysis. While hiing he ZLB is very cosly in he model, our baseline finding is ha he opimal rae of inflaion is low, less han wo percen a year, even when we allow for a variey of feaures ha would end o lower he coss or o raise he benefis of posiive seadysae inflaion. Despie he imporance of quanifying he opimal inflaion rae for policymakers, modern moneary models of he business cycle, namely he New Keynesian framework, have been srikingly illsuied o address his quesion because of heir near exclusive reliance on he assumpion of zero seadysae inflaion, paricularly in welfare analysis. Our firs conribuion is o address he implicaions of posiive seadysae inflaion for welfare analysis by solving for he microfounded loss funcion in an oherwise sandard New Keynesian model wih labor as he only facor of producion. We idenify hree disinc coss of posiive rend inflaion. The firs is he seadysae effec: wih saggered price seing, higher inflaion leads o greaer price dispersion which causes an inefficien allocaion of resources among firms, hereby lowering aggregae welfare. The second is ha posiive seadysae inflaion raises he welfare cos of a given amoun of inflaion volailiy. This cos reflecs he fac ha inflaion variaions creae disorions in relaive prices given saggered price seing. Since posiive rend inflaion already generaes some inefficien price dispersion, he addiional disorion in relaive prices from an inflaion shock becomes more cosly as firms have o compensae workers Mos papers incorporaing posiive seadysae inflaion ino he New Keynesian framework have focused on he implicaions for dynamics and deerminacy issues. For example, Cogley and Sbordone (8) show ha accouning for posiive seadysae inflaion significanly improves he fi of he New Keynesian Phillips Curve. Kiley (7) and Ascari and Ropele (9) show ha he Taylor principle is no sufficien o guaranee a unique raional expecaions equilibrium in New Keynesian models for even moderae levels of inflaion. Coibion and Gorodnichenko () show ha once his feaure of New Keynesian models is incorporaed ino hisorical moneary policy analysis, he prevolcker moneary policy rule ensured he presence of selffulfilling expecaional flucuaions despie likely saisfying he Taylor principle, a reflecion of he high arge rae of inflaion over his ime period.
3 for he increasingly high marginal disuiliy of secorspecific labor. Thus, he increased disorion in relaive prices due o an inflaion shock becomes coslier as we increase he iniial price dispersion which makes he variance of inflaion coslier for welfare as he seadysae level of inflaion rises. In addiion o he wo coss from relaive price dispersion, a hird cos of inflaion in our model comes from he dynamic effec of posiive inflaion on pricing decisions. Greaer seadysae inflaion induces more forwardlooking behavior when sickyprice firms are able o rese heir prices because he gradual depreciaion of he relaive rese price can lead o larger losses han under zero inflaion. As a resul, inflaion becomes more volaile which lowers aggregae welfare. This cos of inflaion arising from he posiive relaionship beween he level and volailiy of inflaion has been welldocumened empirically bu is commonly ignored in quaniaive analyses because of quesions as o he source of he relaionship. As wih he pricedispersion coss of inflaion, his cos arises endogenously in he New Keynesian model when one incorporaes posiive seadysae inflaion. The key benefi of posiive inflaion in our model is a reduced frequency of hiing he zero bound on nominal ineres raes. As emphasized in Chrisiano e al. (9), hiing he zero bound induces a deflaionary mechanism which leads o increased volailiy and hence large welfare coss. Because a higher seadysae level of inflaion implies a higher level of nominal ineres raes, raising he inflaion arge can reduce he incidence of zerobound episodes, as suggesed by Blanchard in he opening quoe. Our approach for modeling he zero bound follows Bodensein e al. (9) by solving for he duraion of he zero bound endogenously, unlike in Chrisiano e al. (9) or Eggersson and Woodford (4). This is imporan because he welfare coss of inflaion are a funcion of he variance of inflaion and oupu, which hemselves depend on he frequency a which he zero bound is reached as well as he duraion of zero bound episodes. Afer calibraing he model o broadly mach he hisorical incidence of hiing he zero lower bound in he U.S., we hen solve for he rae of inflaion ha maximizes welfare. We show numerically ha he welfare loss funcion is generally concave wih respec o seadysae inflaion, such ha he opimal rae of inflaion is posiive as a resul of he zero bound. However, for plausible calibraions of he srucural parameers of he model and he properies of he shocks driving he economy, he opimal inflaion rae is quie low: less han wo percen per year. This resul is remarkably robus o changes in parameer values, as long as hese do no dramaically increase he implied frequency of being a he zero lower bound. In addiion, we show ha our resuls are robus if he cenral bank follows opimal sabilizaion policy, raher han he baseline Taylor rule. In paricular, if he cenral bank canno commi o a policy rule, hen he opimal inflaion rae is of similar magniude as in our baseline calibraion. Furhermore, we show ha all hree coss of inflaion he seady sae effec, he increasing cos of inflaion volailiy, and he posiive link beween he level and volailiy of For example, Mankiw s (7) undergraduae Macroeconomics exbook noes ha in hinking abou he coss of inflaion, i is imporan o noe a widely documened bu lile undersood fac: high inflaion is variable inflaion. Similar saemens can be found in oher prominen exs.
4 inflaion are quaniaively imporan: each cos is individually large enough o bring he opimal inflaion rae down o.7% or lower when he ZLB is presen. The key inuiion behind he low opimal inflaion rae is ha he uncondiional cos of he zero lower bound is small even hough each individual ZLB even is quie cosly. In our baseline calibraion an 8quarer ZLB even a % rend inflaion has a cos equivalen o a 9% permanen reducion in consumpion, above and beyond he usual business cycle cos. This is, for example, significanly higher han Williams (9) esimae of he coss of hiing he ZLB during he curren recession. However, in he model such an even is also rare, occurring abou once every years assuming ha ZLB evens always las 8 quarers, so ha he uncondiional cos of he ZLB a % rend inflaion is equivalen o.3% permanen reducion in consumpion. This leaves lile room for furher improvemens in welfare by raising he longrun inflaion rae. Thus, even modes coss of rend inflaion, which mus be borne every period, will imply an opimal inflaion rae below %, despie reasonable values for boh he frequency and cos of he ZLB. This explains why our resuls are robus o a variey of seings ha we furher discuss below, and suggess ha our resuls are no paricular o he New Keynesian model. Furhermore, while he New Keynesian model implies ha he opimal weigh on he variance of he oupu gap in he welfare loss funcion is small, we show ha pushing he opimal inflaion rae above he ypical inflaion arges of cenral banks would require his coefficien o be more han en imes larger han he weigh on he annualized inflaion variance. Such weighs would imply a welfare cos of being a he ZLB for 8 quarers roughly equal o a permanen decline in consumpion of 56% ($5.6 rillion per year), a magniude which srikes us as oo large o be plausible. Thus, i is unlikely ha augmening he baseline model wih mechanisms which could raise he welfare cos of oupu flucuaions (such as unemploymen or income dispariies across agens) would significanly raise he opimal arge rae of inflaion. Finally, while we use hisorical U.S. daa o calibrae he frequency of hiing he ZLB, an approach which can be problemaic when applied o rare evens, we show in robusness analysis ha even a ripling of he frequency of being a he ZLB (such ha he economy would spend 5% of he ime a he ZLB for an inflaion rae of 3%) would raise he opimal inflaion rae only o.7%, jus below he upper bound of mos cenral banks inflaion arges. To furher invesigae he robusness of his resul, we exend our baseline model o consider several mechanisms which migh raise he opimal rae of inflaion. For example, in he presence of uncerainy abou he rue parameer values, policymakers migh wan o choose a higher arge inflaion rae as a buffer agains he possibiliy ha he rue parameers imply more frequen and cosly incidence of he zero bound. We address his possibiliy in wo ways. Firs, we calculae he opimal inflaion rae aking ino accoun he uncerainy abou parameer values and find ha his raises he opimal inflaion rae only modesly, from.3% o.6% per year. Second, we repeaedly draw from he disribuion of parameers and calculae he opimal 3
5 inflaion rae for each draw. We find ha he 9% confidence inerval of opimal inflaion raes ranges from.3% o.5% a year, which closely mirrors he arge range for inflaion of mos modern cenral banks. Similarly, one migh be concerned ha our findings hinge on modeling price sickiness as in Calvo (983). Firs, because his approach implies ha some firms do no change prices for exended periods of ime, i could oversae he cos of price dispersion and herefore undersae he opimal inflaion rae. To address his possibiliy, we reproduce our analysis using Taylor (977) saggered price seing of fixed duraions. The laer reduces price dispersion relaive o he Calvo assumpion bu has no significan impac on he opimal inflaion rae. Second, he degree of price rigidiy in boh Calvo and Taylor pricing is commonly reaed as a srucural parameer, ye i is unlikely ha he frequency of price seing is compleely independen of he inflaion rae, even for low inflaion raes like hose experienced in he U.S. As a resul, we go beyond exising reamens of he opimal inflaion rae and consider wo modificaions ha allow for price flexibiliy o vary wih he rend rae of inflaion. In he firs specificaion, we le he degree of price rigidiy vary sysemaically wih he rend level of inflaion bu find ha his modificaion also does no qualiaively change he opimal inflaion rae. In he second specificaion, we employ he Dosey, King and Wolman (999) model of saedependen pricing, which allows he degree of price sickiness o vary endogenously boh in he shorrun and in he longrun, and hus we address one of he major criicisms of he previous lieraure on he opimal inflaion rae. However, even in his seing we coninue o find an opimal inflaion rae of less han wo percen per year. Tobin (97) suggess downward nominal wage rigidiy as an addiional facor which migh push he opimal inflaion rae higher. By faciliaing he downward adjusmen of real wages in he presence of downward nominal wage rigidiy, posiive inflaion can be beneficial. We incorporae his greasing he wheels effec by consraining changes in he aggregae nominal wage index o be nonnegaive. Srikingly, his addiion significanly lowers he opimal inflaion rae. The inuiion for his somewha surprising finding is ha downward wage rigidiy lowers he volailiy of marginal coss and hence of inflaion. In addiion, in he face of a negaive demand shock, marginal coss decline by less in he presence of downwardwage rigidiy, leading o a smaller decline in inflaion and hus a smaller change of ineres raes. Hence, he ZLB binds less frequenly, paricularly a low levels of inflaion, which furher reduces he benefis of posiive inflaion. Our analysis absracs from several oher facors which migh affec he opimal inflaion rae. For example, Friedman (969) argued ha he opimal rae of inflaion mus be negaive o equalize he marginal cos and benefi of holding money. Because our model is ha of a cashless economy, his cos of inflaion is absen, bu would end o lower he opimal rae of inflaion even furher, as emphasized by Khan e al. (3), SchmiGrohe and Uribe (7, ) and Aruoba and Schorfheide (). Similarly, a long lieraure has sudied he coss and benefis of he seigniorage revenue o policymakers associaed wih posiive inflaion, a feaure which we also absrac from since seigniorage revenues for counries like he U.S. are quie small, as 4
6 are he deadweigh losses associaed wih i. 3 Feldsein (997) emphasizes an addiional cos of inflaion arising from fixed nominal ax brackes, which would again lower he opimal inflaion rae. Furhermore, while our model includes an inflaion cos arising from he posiive link beween he level and he volailiy of inflaion, i is likely ha we sill undersae his cos of inflaion because we absrac from he possibiliy ha higher inflaion volailiy will raise risk premiums due o he increased risk of redisribuion among borrowers and lenders. In addiion, he relaionship beween he level and he volailiy of inflaion could be even sronger han in our model because higher seadysae inflaion moves he economy closer o he indeerminacy region where sunspo shocks would furher raise inflaion volailiy. Finally, because we do no model he possibiliy of endogenous counercyclical fiscal policy nor do we incorporae he possibiliy of nonsandard moneary policy acions during ZLB episodes, i is likely ha we oversae he coss of hiing he ZLB and herefore again oversae he opimal rae of inflaion. Neverheless, our finding ha he hrea of he ZLB coupled wih limied commimen on he par of he cenral bank implies posiive bu low opimal inflaion raes, goes some way in resolving he puzzle poined ou by SchmiGrohe and Uribe (), ha exising moneary heories rouinely imply negaive opimal inflaion raes, and hus canno explain he size of observed inflaion arges. This paper is closely relaed o recen work ha has also emphasized he effecs of he zero bound on ineres raes for he opimal inflaion rae, such as Walsh (9), Billi (9), and Williams (9). A key difference beween he approach aken in his paper and such previous work is ha we explicily model he effecs of posiive rend inflaion on he seadysae, dynamics, and loss funcion of he model. Billi (9) and Walsh (9), for example, use a New Keynesian model loglinearized around zero seadysae inflaion and herefore do no explicily incorporae he posiive relaionship beween he level and volailiy of inflaion, while Williams (9) relies on a nonmicrofounded model. In addiion, hese papers do no ake ino accoun he effecs of posiive seadysae inflaion on he approximaion o he uiliy funcion and hus do no fully incorporae he coss of inflaion arising from price dispersion. 4 SchmiGrohe and Uribe () provide an auhoriaive reamen of many of he coss and benefis of rend inflaion in he conex of New Keynesian models. However, heir calibraion implies ha he chance of hiing he ZLB is pracically zero and herefore does no quaniaively affec he opimal rae of inflaion, whereas we focus on a seing where cosly ZLB evens occur a heir hisoric frequency. Furhermore, none of hese papers consider he endogenous naure of price rigidiy wih respec o rend inflaion. An advanage of working wih a microfounded model and is implied welfare funcion is he abiliy o engage in normaive analysis. In our baseline model, he endogenous response of moneary policymakers o macroeconomic condiions is capured by a Taylor rule. Thus, we are also able o sudy he welfare effecs of 3 See for example Cooley and Hansen (99) and Summers (99). 4 Fuchi e al. (8) sudy he opimal inflaion rae for Japan allowing for he zerolower bound on ineres raes, price sickiness, nominal wage rigidiy and he opporuniy cos of holding money and find a range beween.5% and %. Ye, hey also do no explicily ake ino accoun he effecs of posiive seadysae inflaion on he dynamics of he model or on he uiliy funcion approximaion. 5
7 alering he sysemaic response of policymakers o endogenous flucuaions (i.e. he coefficiens of he Taylor rule) and deermine he new opimal seadysae rae of inflaion. The mos sriking finding from his analysis is ha even modes pricelevel argeing would raise welfare by nonrivial amouns for any seadysae inflaion rae and come close o he Ramseyopimal policy, consisen wih he finding of Eggersson and Woodford (3) and Wolman (5). In shor, he opimal policy rule for he model can be closely characerized by he name of price sabiliy as ypically saed in he legal mandaes of mos cenral banks. Given our resuls, we conclude ha raising he arge rae of inflaion is likely oo blun an insrumen o reduce he incidence and severiy of zerobound episodes. In all of he New Keynesian models we consider, even he small coss associaed wih higher rend inflaion raes, which mus be borne every period, more han offse he welfare benefis of fewer and less severe ZLB evens. Insead, changes in he policy rule, such as PLT, may be more effecive boh in avoiding and minimizing he coss associaed wih hese crises. In he absence of such changes o he ineres rae rule, our resuls sugges ha addressing he large welfare losses associaed wih he ZLB is likely o bes be pursued hrough policies argeed specifically o hese episodes, such as counercyclical fiscal policy or he use of nonsandard moneary policy ools. Secion presens he baseline New Keynesian model and derivaions when allowing for posiive seadysae inflaion, including he associaed loss funcion. Secion 3 includes our calibraion of he model as well as he resuls for he opimal rae of inflaion while secion 4 invesigaes he robusness of our resuls o parameer values. Secion 5 hen considers exensions of he baseline model which could poenially lead o higher esimaes of he opimal inflaion arge. Secion 6 considers addiional normaive implicaions of he model, including opimal sabilizaion policy and price level argeing. Secion 7 concludes. II A New Keynesian Model wih Posiive SeadySae Inflaion We consider a sandard New Keynesian model wih a represenaive consumer, a coninuum of monopolisic producers of inermediae goods, a fiscal auhoriy and a cenral bank.. Model The represenaive consumer aims o maximize he presen discouned value of he uiliy sream from consumpion and leisure max log / () where C is consumpion of he final good, N(i) is labor supplied o individual indusry i, η is he Frisch labor supply elasiciy and β is he discoun facor. The budge consrain each period is given by : / / / () where S is he sock of oneperiod bonds held by he consumer, R is he gross nominal ineres rae, P is he price of he final good, W(i) is he nominal wage earned from labor in indusry i, T is real ransfers and 6
8 profis from ownership of firms, q is a risk premium shock, and is he shadow value of wealh. 5 order condiions from his uiliymaximizaion problem are hen:, (3) / /, (4) / /. (5) The firs Producion of he final good is done by a perfecly compeiive secor which combines a coninuum of inermediae goods ino a final good per he following aggregaor / / (6) where Y is he final good and Y(i) is inermediae good i, while θ denoes he elasiciy of subsiuion across inermediae goods, yielding he following demand curve for goods of inermediae secor i / (7) and he following expression for he aggregae price level /. (8) The producion of each inermediae good is done by a monopolis facing a producion funcion linear in labor (9) where A denoes he level of echnology, common across firms. Each inermediae good producer has sicky prices, modeled as in Calvo (983) where is he probabiliy ha each firm will be able o reopimize is price each period. We allow for indexaion of prices o seadysae inflaion by firms who do no reopimize heir prices each period, wih ω represening he degree of indexaion ( for no indexaion o for full indexaion). Denoing he opimal rese price of firm i by B(i), reopimizing firms solve he following profimaximizaion problem max, Π () where Q is he sochasic discoun facor and Π is he gross seadysae level of inflaion. The opimal relaive rese price is hen given by, / /, / where firmspecific marginal coss can be relaed o aggregae variables using / / /. () Given hese priceseing assumpions, he dynamics of he price level are governed by () 5 As discussed in Smes and Wouers (7), a posiive shock o q, which is he wedge beween he ineres rae conrolled by he cenral bank and he reurn on asses held by he households, increases he required reurn on asses and reduces curren consumpion. The shock q has similar effecs as neworh shocks in models wih financial acceleraors (see Bernanke e al. 999 for a survey). Amano and Shukayev () documen ha shocks like q are essenial for generaing a binding zero lower bound. 7
9 Π. (3) We allow for governmen consumpion of final goods (), so he goods marke clearing condiion for he economy is. (4) We define he aggregae labor inpu as / /. (5). Seadysae and loglinearizaion Following Coibion and Gorodnichenko (), we loglinearize he model around he seadysae in which inflaion need no be zero. Since posiive rend inflaion may imply ha he seady sae and he flexible price level of oupu differ, we adop he following noaional convenion. Variables wih a bar denoe seady sae values, e.g. is he seady sae level of oupu. We assume ha echnology is a random walk and hence we normalize all nonsaionary real variables by he level of echnology. Lowercase leers denoe he log of a variable, e.g. log is he log of curren oupu. We le has on lower case leers denoe deviaions from seady sae, e.g. is he approximae percenage deviaion of oupu from seady sae. Since we define he seady sae as embodying he curren level of echnology, deviaions from he seady sae are saionary. Finally, we denoe deviaions from he flexible price level seady sae wih a ilde, e.g. is he approximae percenage deviaion of oupu from is flexible price seady sae, where he superscrip F denoes a flexible price level quaniy. Define he ne seadysae level of inflaion as log Π. The loglinearized consumpion Euler equaion is (6) and he goods marke clearing condiion becomes (7) where and are he seadysae raios of consumpion and governmen o oupu respecively. Also, inegraing over firmspecific producion funcions and loglinearizing yields. (8) Allowing for posiive seadysae inflaion (i.e., ) primarily affecs he seadysae and priceseing componens of he model. For example, he seadysae level of he oupu gap (which is defined as he deviaion of seady sae oupu from is flexible price level counerpar / ) is given by / /. (9) Noe ha he seadysae level of he gap is equal o one when seadysae inflaion is zero (i.e., Π ) or when he degree of price indexaion is exacly equal o one. As emphasized by Ascari and Ropele (7), here is a nonlinear relaionship beween he seadysae levels of inflaion and oupu. For very low bu 8
10 posiive rend inflaion, is increasing in rend inflaion bu he sign is quickly reversed so ha is falling wih rend inflaion for mos posiive levels of rend inflaion. Secondly, posiive seadysae inflaion affecs he relaionship beween aggregae inflaion and he reopimizing price. Specifically, he relaionship beween he wo in he seady sae is now given by / / () and he loglinearized equaion is described by () so ha inflaion is less sensiive o changes in he reopimizing price as seadysae inflaion rises. This effec reflecs he fac ha, wih posiive seadysae inflaion, firms which rese prices have higher prices han ohers and receive a smaller share of expendiures, hereby reducing he sensiiviy of inflaion o hese price changes. Indexaion of prices works o offse his effec however, wih fullindexaion compleely resoring he usual relaionship beween rese prices and inflaion. Similarly, posiive seadysae inflaion has imporan effecs on he loglinearized opimal rese price equaion, which is given by () where is a cospush shock, and / so ha wihou seadysae inflaion or full indexaion we have. When ω <, a higher increases he coefficiens on fuure oupu and inflaion bu also leads o he inclusion of a new erm composed of fuure differences beween oupu growh and ineres raes. Each of hese effecs makes priceseing decisions more forwardlooking. 6 The increased coefficien on expecaions of fuure inflaion, which reflecs he expeced fuure depreciaion of he rese price and he losses associaed wih i, plays a paricularly imporan role. In response o an inflaionary shock, a firm which can rese is price will expec higher inflaion oday and in he fuure as oher firms updae heir prices in response o he shock. Given his expecaion, he more forward looking a firm is (he higher ), he greaer he opimal rese price mus be in anicipaion of oher firms raising heir prices in he fuure. Thus, rese prices become more responsive o curren shocks wih higher. We confirm numerically ha his effec dominaes he reduced sensiiviy of inflaion o he rese price in equaion (), hereby endogenously generaing a posiive relaionship beween he level and he volailiy of inflaion. To close he model, we assume ha he log deviaion of he desired gross ineres rae from is seady sae value ( ) follows a Taylor rule 6 See Coibion and Gorodnichenko () for a discussion of each of hese effecs. 9
11 where,,, capure he srengh of he policy response o deviaions of inflaion, he oupu gap, he oupu growh rae and he price level from heir respecive arges, parameers and reflec ineres rae smoohing, while is a policy shock. We se,, and so ha he cenral bank has no inflaionary or oupu bias. The growh rae of oupu is relaed o he oupu gap by (3) where is he log level of echnology and is rend growh rae. Since he acual level of he ne ineres rae is bounded by zero, he log deviaion of he gross ineres rae is bounded by log log log and he dynamics of he acual ineres rae are given by max,. (4) We consider he Taylor rule a reasonable benchmark, because i is likely o be he closes descripion of he curren policy process, and because suggesions o raise he opimal inflaion rae are no commonly associaed wih simulaneous changes in he way ha sabilizaion policy is conduced. However, in secion 6., we also derive he opimal given opimal sabilizaion policy under discreion and commimen..3 Shocks We assume ha echnology follows a random walk process wih drif:. (5) Each of he risk premium, governmen, and Phillips Curve shocks follow AR() processes, (6), (7). (8) We assume ha,,,, are muually and serially uncorrelaed..4 Welfare funcion To quanify welfare for differen levels of seadysae inflaion, we use a secondorder approximaion o he household uiliy funcion as in Woodford (3). 7 We show our main resuls in a series of lemmas culminaing in Proposiion. All proofs are in Appendix A. Firs of all, we decompose uiliy described in equaion () ino uiliy due o consumpion and (dis)uiliy due o labor supply. Lemmas and provide second order approximaions for each componen. Lemma. Uiliy from consumpion in equaion () is given by... (9) where log / is he percen deviaion of consumpion from is flexibleprice level.i.p. sands for erms independen from policy, and h.o.. means higher order erms. 7 In our welfare calculaions, we use he nd order approximaion o he consumer uiliy funcion while he srucural relaionships in he economy are approximaed o firs order. As discussed in Woodford (), his approach is valid if disorions o he seady sae are small so ha he firs order erms in he uiliy approximaion are premuliplied by coefficiens ha can also be reaed as firs order erms. Since given our parameerizaion he disorions from imperfec compeiion and inflaion are small (as in Woodford 3), his condiion is saisfied in our analysis. Furhermore, we show in Appendix F ha he loglinear soluion closely approximaes he nonlinear soluion, which implies ha second order effecs on he momens of inflaion and oupu are small and can be ignored in he welfare calculaions.
12 Lemma. Using producion funcion (9), define /. Then... (3) where log / is he deviaion of firm i s oupu from he flexibleprice level of oupu. Correspondingly, he oal disuiliy from labor supply is... (3) Proof: See Proposiion 6.3 in Woodford (3). The key insigh from Lemmas and is ha welfare is diminished when consumpion is low relaive o is flexibleprice level and when he crosssecional dispersion of oupu is large. To undersand and assess he implicaions of crosssecional oupu dispersion, we need o examine he crosssecional dispersion of prices. Denoe he crosssecional dispersion of prices a ime wih var log and le be he crosssecional dispersion of prices in he nonsochasic seady sae. I is sraighforward o show ha where is increasing in price sickiness and seadysae inflaion and decreasing in he degree of indexaion. Define log as he average (across firms) log price of goods. Lemma 3. The difference beween he log price index P and he average log price across firms is given by log Δ... (3) where Δ Δ and Δ Δ. Lemma 3 is a manifesaion of Jensen s inequaliy. Noe ha since is quadraic in, he dispersion of prices is approximaely zero when and herefore, so ha log, which is he sandard resul. Again, since is quadraic in, one can show ha /, / when. Using Lemma 3, we describe he dynamic properies of he price dispersion in Lemma 4. Lemma 4. Le Ξ Δ Δ be he deviaion of crosssecion price dispersion from is nonsochasic seady sae level. Then Ξ Γ Γ Γ Γ Γ Γ Ξ... (33) where Γ, Γ Γ, Γ Γ, is he log of he opimal rese price in he nonsochasic seady sae. This lemma shows ha he crosssecional price dispersion is a funcion of is pas values as well as he deviaion of inflaion from is seady sae level. In he viciniy of, Γ, Γ, Γ and hus crosssecional price dispersion varies very lile over ime since i is only a funcion of he variance of
13 inflaion. This is he sandard resul for welfare calculaions in a zero seady sae inflaion environmen (see e.g. Proposiion 6.3 in Woodford (3)). However, Γ /, Γ /, Γ / locally a. Hence, deviaions of inflaion from is seady sae level have an increasingly srong effec on he crosssecional price dispersion as rises and, as a resul, he dynamics of price dispersion can become firsorder when is sufficienly high. However, given our parameer values and for he levels of rend inflaion ha we consider, Γ remains iny so ha price dispersion effecively remains of second order as in Woodford (3) and hus we can use a linear approximaion of he srucural relaionships in he economy and a second order approximaion of consumer uiliy for welfare calculaions. 8 Henceforh, we will rea Γ as negligibly small. Using he demand condiion (7), we can link he crosssecional dispersion of oupu o he crosssecional dispersion of prices: log log log log log log (34) and hence Υ var var var log Δ. (35) Le Υ be he crosssecional dispersion of oupu in he nonsochasic seady sae. The remaining piece in he secondorder approximaion of household s uiliy is, which is he average deviaion of oupu from flexible price level a he firm level. Using he insigh of Lemma 3, we can relae o he deviaion of oupu from is flexibleprice level a he aggregae level. Lemma 5. If he deviaion of oupu from is flexibleprice level a he aggregae level is defined as log /, hen Υ Υ... (36) where Υ Υ and Υ Υ. Similar o he crosssecional price dispersion, one can show ha, since Υ is quadraic in,, and /, / when. The cenral resul can be summarized wih he following proposiion. Proposiion. Given Lemmas 5, he nd order approximaion o expeced per period uiliy in eq. () is 9 Θ Θ var Θ var (37) where parameers Θ,,, depend on he seady sae inflaion and are given by Θ η log η η Υ 8 In our baseline calibraion he highes value for Γ is.44 which is reached a 6% annual inflaion. 9 The complee approximaion also conains wo linear erms, he expeced oupu gap and expeced inflaion. Since he disorions o he seady sae are small for he levels of rend inflaion we consider, he coefficiens ha muliply hese erms can be considered as firs order so we can evaluae hese erms using he firs order approximaion o he laws of moion as in Woodford (3). We confirmed in numeric simulaions ha hey can be ignored. Furhermore, second order effecs on he expeced oupu gap and expeced inflaion are likely o be quaniaively small since he linear soluion closely approximaes he nonlinear soluion o he model (see Appendix F).
14 log η Δ, Θ η /, Θ Γ log, Γ, Φlog. This approximaion of he household uiliy places no resricions on he pah of nominal ineres raes and hus is invarian o sabilizaion policies chosen by he cenral bank. The loss funcion in Proposiion illusraes he hree mechanisms via which rend inflaion affecs welfare: he seadysae effecs, he effecs on he coefficiens of he uiliyfuncion approximaion, and he dynamics of he economy via he second momens of macroeconomic variables. Firs, he erm Θ capures he seadysae effecs from posiive rend inflaion, which hinge on he increase in he crosssecional seadysae dispersion in prices (and herefore in inefficien allocaions of resources across secors) associaed wih posiive rend inflaion. Noe ha as approaches zero, Θ converges o zero. As shown by Ascari and Ropele (9), when, Θ /, bu he sign of he slope quickly reverses a marginally posiive inflaion raes. In our baseline calibraion, Θ is sricly negaive and Θ / when rend inflaion exceeds.4% per annum. Thus for quaniaively relevan inflaion raes, he welfare loss from seadysae effecs is increasing in he seadysae level of inflaion. This is inuiive since, excep for very small levels of inflaion, he seady sae level of oupu declines wih higher because he seady sae crosssecional price dispersion rises. The seadysae cos of inflaion from price dispersion is one of he besknown coss of inflaion and arises naurally from he inegraion of posiive rend inflaion ino he New Keynesian model. Consisen wih his effec being driven by he increase in dispersion, one can show ha he seadysae effec is eliminaed wih full indexaion of prices and miigaed wih parial indexaion. Second, he coefficien on he variance of oupu around is seady sae Θ does no depend on rend inflaion. This erm is direcly relaed o he increasing disuiliy of labor supply. Wih a convex cos of labor supply, he expeced disuiliy rises wih he variance of oupu around is seady sae. However, even hough Θ is independen of, his does no imply ha a posiive does no impose any oupu cos. Raher, rend inflaion reduces he seady sae level of oupu, which is already capured by Θ. Once his is aken ino accoun, hen log uiliy implies ha a given level of oupu variance around he (new) seady sae is as cosly as i was before. Furhermore, he variance of oupu around is seady sae depends on he dynamic properies of he model which are affeced by he level of rend inflaion. When, equaion (4) reduces o he sandard secondorder approximaion of he uiliy funcion as in Proposiion 6.4 of Woodford (3). There is a sligh difference beween our approximaion and he approximaion in Woodford (3) since we focus on he perperiod uiliy while Woodford calculaed he presen value. The parameer Ф measures he deviaion of he flexibleprice level of oupu from he flexibleprice perfeccompeiion level of oupu. See Woodford (3) for derivaion. 3
15 The coefficien on he variance of inflaion Θ capures he sensiiviy of he welfare loss due o he crosssecional dispersion of prices. One can also show analyically ha for, Θ / so ha he crosssecional dispersion of prices becomes ceeris paribus coslier in erms of welfare. Recall ha an inflaionary shock creaes disorions in relaive prices. Given ha posiive rend inflaion already generaes some price dispersion and hence an inefficien allocaion of resources, firms operaing a an inefficien level have o compensae workers for he increasingly high marginal disuiliy of secorspecific labor. Because of his rising marginal disuiliy, he increased disorion in relaive prices due o an inflaion shock becomes coslier as we increase he iniial price dispersion which makes he variance of inflaion coslier for welfare as he rend level of inflaion rises. Thus, his is a second, and o he bes of our knowledge previously unidenified, channel hrough which he price dispersion arising from saggered price seing under posiive inflaion reduces welfare. Finally, Θ increases in he Frisch labor supply elasiciy and decreases in he elasiciy of subsiuion across goods and he Calvo parameer. III Calibraion and Opimal Inflaion Having derived he approximaion o he uiliy funcion, we now urn o solving for he opimal inflaion rae. Because uiliy depends on he volailiy of macroeconomic variables, his will be a funcion of he srucural parameers and shock processes. Therefore, we firs discuss our parameer selecion and hen consider he implicaions for he opimal inflaion rae in he model. We invesigae he robusness of our resuls o parameer values in subsequen secions. 3. Parameers Our baseline parameer values are illusraed in Table. For he uiliy funcion, we se η, he Frisch labor supply elasiciy, equal o one. The seadysae discoun facor β is se o.998 o mach he real rae of.3% per year on 6monh commercial paper or asses wih similar shorerm mauriies given ha we se he seadysae growh rae of real GDP per capia o be.5% per year (.5. ), as in Coibion and Gorodnichenko (). We se he elasiciy of subsiuion across inermediae goods o, so ha seadysae markups are equal o %. This size of he markup is consisen wih esimaes presened in Burnside (996) and Basu and Fernald (997). The degree of price sickiness () is se o.55, which amouns o firms reseing prices approximaely every 7 monhs on average. This is midway beween he micro esimaes of Bils and Klenow (4), who find ha firms change prices every 4 o 5 monhs, and hose of Nakamura and Seinsson (8), who find ha firms change prices every 9 o monhs. The degree of price indexaion is assumed o be zero in he baseline for hree reasons. Firs, he workhorse New Keynesian model is based only on price sickiness, making his he mos naural benchmark (Clarida e al. 999, Woodford 3). Second, any price indexaion implies ha firms are consanly changing prices, a feaure srongly a odds wih he empirical findings of Bils and Klenow 4
16 (4) and more recenly Nakamura and Seinsson (8), among many ohers. Third, while indexaion is ofen included o replicae he apparen role for lagged inflaion in empirical esimaes of he New Keynesian Phillips Curve (NKPC; see Gali and Gerler 999), Cogley and Sbordone (8) show ha once one conrols for seadysae inflaion, esimaes of he NKPC rejec he presence of indexaion in price seing decisions. However, we relax he assumpion of no indexaion in he robusness checks. The coefficiens for he Taylor rule are aken from Coibion and Gorodnichenko (). These esimaes poin o srong longrun responses by he cenral bank o inflaion and oupu growh (.5 and.5 respecively) and a moderae response o he oupu gap (.43). The seadysae share of consumpion is se o.8 so ha he share of governmen spending is weny percen. The calibraion of he shocks is primarily aken from he esimaed DSGE model of Smes and Wouers (7) wih he excepion of he persisence of he risk premium shocks for which we consider a larger value calibraed a.947 o mach he hisorical frequency of hiing he ZLB and he rouinely high persisence of risk premia in financial ime series. 3 In our baseline model, posiive rend inflaion is cosly because i leads o more price dispersion and herefore less efficien allocaions, more volaile inflaion, and a greaer welfare cos for a given amoun of inflaion volailiy. On he oher hand, posiive rend inflaion gives policymakers more room o avoid he ZLB on ineres raes. Therefore, a key deerminan of he radeoff beween he wo depends on how frequenly he ZLB is binding for differen levels of rend inflaion. To illusrae he implicaions of our parameer calibraion for how ofen we hi he ZLB, Figure plos he fracion of ime spen a he ZLB from simulaing our model for differen seadysae levels of he inflaion rae. In addiion, we plo he seadysae level of he nominal ineres rae associaed wih each inflaion rae, where he seadysae nominal rae in he model is deermined by /. Our calibraion implies ha wih a seadysae inflaion rae of approximaely 3.5%, he average rae for he U.S. since he early 95 s, he economy should be a he ZLB approximaely 5 percen of he ime. This is consisen wih he poswwii experience of he U.S.: wih U.S. ineres raes a he ZLB since lae 8 and expeced o remain so unil he end of, his yields a hisorical frequency of being a he ZLB of 5 percen (i.e. around 3 years ou of 6). 4 In addiion, his calibraion agrees wih he hisorical changes in ineres raes associaed wih pos WWII U.S. recessions. For example, saring wih he 958 recession and excluding he curren recession, he Because empirical Taylor rules are esimaed using annualized raes while he Taylor rule in he model is expressed a quarerly raes, we rescale he coefficien on he oupu gap in he model such ha =.43/4 =.. 3 This calibraion is, e.g., consisen wih he persisence of he spread beween Baa and Aaa bonds which we esimae o be.945 beween 9: and 9: and.94 beween 95: and 9: a he quarerly frequency. 4 Of possible concern may be ha his calculaion includes he highinflaion environmen from Excluding hose years generaes a hisorical frequency a he ZLB of 3/45=6.66% bu now a a lower rend inflaion rae of 3% per year. Our baseline calibraion generaes precisely ha frequency a 3% rend inflaion. 5
17 average decline in he Federal Funds Rae during a recession has been 4.76 percenage poins. 5 The model predics ha he average nominal ineres rae wih 3.5% seadysae inflaion is around 6%, so he ZLB would no have been binding during he average recession, consisen wih he hisorical experience. Only he 988 recession led o a decline in nominal ineres raes ha would have been sufficienly large o reach he ZLB (8.66% drop in ineres raes), bu did no because nominal ineres raes and esimaes of rend inflaion over his period were much higher han heir average values. Thus, wih 33.5% inflaion, our calibraion (doed line in Figure ) implies ha i would ake unusually large recessions for he ZLB o become binding. In addiion, our calibraion indicaes ha a much lower levels of rend inflaion, he ZLB would be binding much more frequenly. For example, a a zero seadysae inflaion rae, he ZLB would be binding 7% of he ime. Given he hisorical experience of he U.S., his seems conservaive, as i exceeds he hisorical frequency of recessions. The model predics a seadysae level of ineres raes of less han.5% when, and six ou he las eigh recessions (again excluding he curren episode) were associaed wih decreases in ineres raes ha exceeded his value (specifically he 969, 973, 98, 98, 99 and recessions). Our calibraion is also largely in line wih he frequency of he ZLB we would have observed given hisorical declines in nominal ineres raes during recessions and counerfacual levels of rend inflaion (broken line in Figure ). Thus, we inerpre our parameerizaion as providing a reasonable represenaion of he likelihood of hiing he ZLB for differen inflaion raes given he hisorical experience of he U.S. 3. Opimal Inflaion Having derived he dynamics of he model, parameerized he shocks, and obained he secondorder approximaion o he uiliy funcion, we now simulae he model for differen levels of rend inflaion and compue he expeced uiliy for each. We use he Bodensein e al. (9) algorihm o solve he nonlinear model and verify in Appendix F ha his algorihm has very high accuracy, even afer large shocks leading o a binding ZLB. The resuls aking ino accoun he ZLB and in he case when we ignore he ZLB are ploed in Panel A of Figure. When he ZLB is no aken ino accoun, he opimal rae of inflaion is zero because here are only coss o inflaion and no benefis. 6 Figure also plos he oher exreme when we include he ZLB bu do no ake ino accoun he effecs of posiive seadysae inflaion on he loss funcion or he dynamics of he model. In his case, here are no coss o inflaion so uiliy is sricly increasing as seadysae inflaion rises and he frequency of he ZLB diminishes. Our key resul is he specificaion which incorporaes boh he coss and benefis of inflaion. As a resul of he ZLB consrain, we find ha uiliy is increasing a very low levels of inflaion so ha zero inflaion is no opimal when he zero bound is presen. 5 This magniude is calculaed by aking he average level of he Federal Funds rae (FFR) over he las 6 monhs prior o he sar of each recession as defined by he NBER and subracing he minimum level of he FFR reached in he afermah of ha recession. 6 We deermine opimal inflaion in a gridspace wih a widh of.4% per year. I is herefore possible, alhough quaniaively no imporan, ha he opimal inflaion rae wihou he ZLB is posiive bu less han.4% per year. 6
18 Second, he peak level of uiliy is reached when he inflaion rae is.3% a an annualized rae. This is a he boom end of he arge range of mos cenral banks, which are commonly beween % and 3%. Thus, our baseline resuls imply ha aking ino accoun he zero bound on ineres raes raises he opimal level of inflaion, bu wih no addiional benefis o inflaion included in he model, he opimal inflaion rae is wihin he sandard range of inflaion arges. Third, inflaion raes above he opimal level monoonically lower uiliy. Fourh, he coss of even moderae inflaion can be nonrivial. For example, a 4% annualized inflaion rae would lower uiliy by nearly % relaive o he opimal level, which given log uiliy in consumpion is equivalen o a permanen % decrease in he level of consumpion. As we show laer, he magniude of he welfare coss of inflaion varies wih he calibraion and price seing assumpions, bu he opimal rae of inflaion is remarkably insensiive o hese modificaions. Panel B of Figure quanifies he imporance of each of he hree coss of inflaion he seady sae effec, he increasing cos of inflaion volailiy, and he posiive link beween he level and volailiy of inflaion by calculaing he opimal inflaion rae subjec o he zero lower bound when only one of hese coss, in urn, is included. The firs finding o noe is ha allowing for any of he hree inflaion coss is sufficien o bring he opimal inflaion rae o.7% or below. Thus, all hree inflaion coss incorporaed in he model are individually large enough o preven he ZLB from pushing he opimal inflaion rae above he curren arge range of mos cenral banks. Second, he seadysae cos of price dispersion is he larges cos of inflaion ou of he hree, bringing he opimal inflaion rae down o.5% by iself. To ge a sense of which facors drive hese resuls, he op row of Figure 3 plos he coefficiens of he secondorder approximaion o he uiliy funcion from Proposiion. In shor, he resuls confirm he analyical derivaions in secion.4. Firs, rising inflaion has imporan negaive seadysae effecs on uiliy, as he increasing price dispersion inefficienly lowers he seadysae level of producion and consumpion. Second, he coefficien on he variance of oupu around is seady sae is independen of even hough he new seady sae level of oupu is lower. This reflecs our assumpion of loguiliy in consumpion. Third, he coefficien on inflaion variance is decreasing in, i.e., holding he inflaion variance consan, higher raises he uiliy cos of he variance in inflaion. This reflecs he fac ha when he seady sae level of price dispersion is already high hen a emporary increase in price dispersion due o an inflaion shock is even more cosly. Moving from zero inflaion o six percen inflaion raises he coefficien on he inflaion variance by over 4% in absolue value. Thus, as rises, policymakers should place an increasing weigh on he variance of inflaion relaive o he variance of he oupu gap. The middle row of Figure 3 plos he effecs of on he variance of inflaion and he oupu gap, i.e. he dynamic effecs of seadysae inflaion and he ZLB. In addiion, we plo he corresponding momens in he absence of he zerobound on ineres raes o characerize he conribuion of he zerobound on macroeconomic dynamics. A noable feaure of he figure is ha oupu volailiy rises much 7
19 more rapidly as falls when he ZLB is presen. Inuiively, he ZLB is hi more ofen a a low. Wih he nominal rae fixed a zero, he cenral bank canno sabilize he economy by cuing ineres raes furher and hus macroeconomic volailiy increases. As we increase, macroeconomic volailiy (especially for oupu) diminishes. This is he benefi of higher in he model. The effec of changes in, however, is nonlinear for he variance of inflaion when we ake ino accoun he zerobound on ineres raes. A low levels of inflaion, increasing reduces he volailiy of inflaion for he same reason as for oupu: he reduced frequency of hiing he zero bound. On he oher hand, higher also ends o make priceseing decisions more forwardlooking, so ha, absen he zero bound, inflaion volailiy is consisenly rising wih, a feaure emphasized in Kiley (7) and consisen wih a long lieraure documening a posiive relaionship beween he level and variance of inflaion (Okun 97, Taylor 98 and Kiley ). When rises pas a specific value, he laer effec sars o dominae and he variance of inflaion begins o rise wih. Given our baseline values, his swich occurs a an annualized rend inflaion rae of approximaely.5%. These resuls show he imporance of modeling boh he zerobound and he effecs of on he dynamics of he model. The boom row of Figure 3 hen plos he conribuion of hese differen effecs on he welfare coss of inflaion, i.e. each of he erms in Proposiion. These include he seadysae effecs of as well as he ineracion of he effecs of on he coefficiens of he uiliy funcion approximaion and he dynamics of he economy. The mos sriking resul is ha he welfare coss and benefis of posiive are essenially driven by only wo componens: he seadysae effec and he conribuion of inflaion variance o uiliy. In paricular, he Ushape paern of he inflaion variance combined wih decreasing Θ plays he key role in delivering a posiive level of he opimal inflaion rae, while he effecs of he ZLB on he conribuion of he oupu gap variabiliy are an order of magniude smaller and herefore play a limied role in deermining he opimal inflaion rae. 3.3 Are he coss of business cycles and he ZLB oo small in he model? The minor conribuion of oupu gap volailiy o he opimal inflaion rae migh be inerpreed as an indicaion ha he model undersaes he coss of business cycles in general and he ZLB in paricular. For he former, he implied welfare coss of business cycles in our model are approximaely % of seadysae consumpion a he hisorical rend inflaion rae, in line wih DeLong and Summers (988), Barlevy (4), and much larger han in Lucas (987). To assess he cos of hiing he ZLB, we compue he average welfare loss ne of seadysae effecs from simulaing he model under differen inflaion raes boh wih and wihou he zero bound. The difference beween he wo provides a measure of he addiional welfare cos of business cycles due o he presence of he ZLB. We can hen divide his cos by he average frequency of being a he zero bound from our simulaions, for each level of seadysae inflaion, o ge a perquarer average welfare loss measure condiional on being a he ZLB which is ploed in Figure 4. Wih a seadysae inflaion rae of 8
20 one percen, he average cos of a quarer spen a he ZLB is nearly equivalen o a permanen.5% reducion in consumpion. As seadysae inflaion rises, his perperiod cos declines because he average duraion of ZLB episodes ges shorer and he oupu losses during he ZLB are increasing nonlinearly wih he duraion of he ZLB (see Chrisiano e al. 9). A a seadysae inflaion rae of 3.5%, he average perquarer cos of he ZLB is approximaely equal o a permanen.5% reducion in consumpion. This implies ha he addiional cos of being resrained by he zero bound for 8 quarers exceeds ha of a permanen 4% reducion in consumpion, or $4 billion per year based on 8 consumpion daa. 7,8 For comparison, Williams (9) uses he Federal Reserve s FRB/US model o esimae ha he ZLB beween 9 and cos $.8 rillion in los oupu over four years, or roughly $3 billion per year in los consumpion over four years if one assumes ha he decline in consumpion was proporional o he decline in oupu. Thus, he coss of boh business cycles and he ZLB in he model canno be described as being uncharacerisically small. However, while he condiional coss of long ZLB evens are quie large, hey also occur relaively infrequenly. For example, if we assume ha all ZLB episodes are 8 quarers long, hen a 3.5% rend inflaion an 8quarer episode a he ZLB occurs wih probabiliy.7 each quarer, or abou 3 imes every years. This implies ha he expeced cos of he ZLB is a.8% permanen reducion of consumpion. Similar calculaions for % rend inflaion reveal ha while he condiional cos of an 8quarer ZLB even is abou a 9% permanen reducion of consumpion, he uncondiional cos of he ZLB is only a.3% permanen reducion in consumpion. Thus, while he model implies ha a higher inflaion arge can significanly reduce he cos of a given ZLB even, as suggesed by Blanchard, aken over a long horizon he expeced gain in miigaing he ZLB from such a policy is small. As a resul, even modes coss of inflaion, because hey mus be borne every period, are sufficien o push he opimal inflaion rae below %. 3.4 How does opimal inflaion depend on he coefficien on he variance of he oupu gap? Even hough he coss of business cycles are significan and ZLB episodes are boh very cosly and occurring wih reasonable probabiliy, one may be concerned ha hese coss are incorrecly measured due o he small relaive weigh assigned o oupu gap flucuaions in he uiliy funcion. A, he coefficien on he oupu gap variance in he loss funcion is less han onehundredh ha on he quarerly inflaion variance (or oneenh for he annualized inflaion variance), and his difference becomes even more pronounced as rises. The low opimal weigh on oupu gap volailiy is sandard in New Keynesian models (see Woodford 3) and could reflec he lack of involunary unemploymen, which inflics subsanial hardship o a small fracion of he populaion and whose welfare effecs may herefore be poorly 7 In his calculaion, we rea as negligible he fac ha in our model he presence of he ZLB can reduce he average level of oupu because he produc of he average reducion in oupu, he frequency of hiing he ZLB and he uiliy weigh on he firs order erm corresponding o he oupu gap is small. However, condiional on hiing he ZLB, he reducion in oupu could be subsanial. For example, he average perquarer reducion in oupu when he ZLB is binding a hree percen rend inflaion is abou one percen. Assuming he binding ZLB lass for eigh quarers, he fall in oupu is hen worh $. rillion dollars over years, or 33% more han Williams (9) esimae. 8 In our baseline calibraion, he 9% confidence inerval for he duraion of ZLB episodes is (,6) a 3.5% inflaion. 9
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