Optimal Policy When the In ation Target is not Optimal

Transcription

1 Opimal Policy When he In aion Targe is no Opimal Sergio A. Lago Alves March 7, 0 Absrac This paper derives he opimal policy o be followed by a welfare-concerned cenral bank when assigned an in aion arge ha is no necessarily welfare-opimal. In his regard, he paper reas he in aion arge as he rend in aion, and have four main conribuions: (i) i derives welfare-based loss funcion under rend in aion and shows how wo ine ciency sources a ecs he loss funcion, i.e. monopolisic compeiion and non-opimal in aion arges; (ii) as opposed o Coibion e al (00) resuls, i shows ha he rend in aion does a ec he relaive weigh of he oupu gap: hey are inversely relaed; (iii) under rend in aion, i derives he ime consisen opimal policies wih boh uncondiional and imeless commimen, and shows how o ranslae he pursui of he in aion arge ino an addiional consrain in he minimizaion sep; (iv) i highlighs a fac ha has no been given proper aenion in he lieraure on rend in aion: as i depars from zero, cos shocks naurally appears in he Phillips curve when he oupu gap is de ned as log-deviaions from he naural oupu ( exible prices). This holds even in he absence of ine cien shocks as ime varying ax raes, of ime-varying markups. This fac alone is relevan, for i implies ha he only siuaion in which he moneary auhoriy faces no policy con ic in sabilizing in aion and oupu gap is under he seady sae wih zero in aion. Keywords: Opimal policy, rend in aion, in aion argeing JEL codes: E3, E5, E58 Inroducion As of 00, 7 cenral banks have adoped in aion argeing (IT, henceforh) as heir framework for moneary policy, as able shows (based on Rogers (00)). In general, he cenral governmen se posiive in aion arges ( rg, henceforh), accepance bands, and he horizons in which he arges are supposed o be me. The cenral banks are assigned he ask o pursui he arges by means of heir moneary policy decisions. However, he in aion arges are no guaraneed o be welfare-opimal. Indeed, here can be many facors leading o ha resuls. Issues regarding poliical economy (median voer preferences, lobbies, ec.) may play an imporan role when he arges are se. I may also be he case of uncerainy on he model form or on is parameers, which may include a mismach beween he preferred models used by he governmen and he ones used by he cenral bank. Ph.D. suden a UCSC (inernaional economics), and Banco Cenral do Brasil (Brazilian Cenral Bank). slagoalv@ucsc.edu. I am hankful o Fulbrigh, Capes and Banco Cenral do Brasil for funding he research leading o his work. The views expressed here are of my own and do no represen he ones of he Banco Cenral do Brasil.

2 Table : In aion argeers Ausralia 3 Mexico 3 Brazil 4:5 N ew Zealand 3 Canada Norway :5 Chile 3 P eru Colombia 4 P hilippines 4:5 Czech Republic 3 P oland :5 Ghana 4:5 Romania 3:5 Guaemala 5 Serbia 4 8 Hungary 3 Souh Af rica 3 6 Iceland :5 :5 Sweden Indonesia 4 6 T hailand 0:5 3 Israel T urkey 6:5 Korea 3 Unied Kingdom European Cenral Bank % over he medium erm indicaes he olerance band around he cenral arge, indicaes he olerance band wih no cenral arge Source: Roger(00) and websie of he European Cenral Bank Dynamic sochasic general equilibrium (DSGE, henceforh) models gradually became he cenral bank s preferred models during he las weny years. They have been exensively designed, improved and esimaed. And mos imporanly, hey currenly are no only used as a policy evaluaion ool. DSGE models are indeed gauging policy decisions in cenral banks such as he Bank of England, he Bank of Canada and he European Cenral Bank, among ohers. The boom line is ha while a grea majoriy of DSGE models, wih parial price indexaion, have zero (e.g. Woodford s (003) cashless economy) or negaive (e.g. Friedman s (969) rule) in aion as he opimal level o be pursued by he moneary auhoriy, in aion arges around he world have always been posiive. Recenly Coibion e al (00) have embedded he zero lower bound for he nominal ineres raes, and found he opimal level of in aion o be smaller han he ypical arges adoped by developed economies, even afer considering di eren ype of fricions. Therefore, a consensus model jusifying he observed levels of in aion arges is ye o come. In he meanime, cenral banks have been using DSGE models wih full price indexaion or changing he opimal policy rule, derived for a zero in aion rae, o incorporae he non-zero arges (see below). Wha is imporan o noe is ha, even hough he arges are exogenously given o he cenral banks, hey sill have exibiliy o choose wha hey perceive as he bes policy o pursue hem. Since December 005 he Bank of Canada has been using he Terms-of-Trade Economic Model (ToTEM) as he "Bank s principal projecion and policy-analysis model for he Canadian economy," as saed in Murchison and Rennison (006). As for he Unied Kingdom, he Bank of England (BoE) Quarerly Model (see Harrison e al (005)) is "he main ool in he suie of models employed by Bank sa and he MPC in he consrucion of he projecions conained in he quarerly In aion Repor," as repored in he BoE s websie. In he same line, he European Cenral Bank uses he New-Area-Wide Model (NAWM) for forecasing and policy analysis (see Chriso el e al (008)). See Tovar (008) and Sbordone e al (00)) for reviews on how DSGE models have been used by cenral banks for policy analysis and decision.

3 And hence one quesion arises: if a welfare-concerned cenral bank is no seeking o bring in aion back o is welfare opimum (zero in he sandard DSGE framework wih Calvo (983) ype rigidiies), which ool should be used o evaluae policies and pick he bes one from he policy family ha brings he in aion back o he arge? Woodford (003) derived he sandard welfare-based (SWeB, henceforh) loss funcion as a second-order approximaion of he negaive welfare funcion (True loss funcion, henceforh), around seady sae equilibrium wih zero in aion (Zero SS, henceforh). He also showed ha i can be used for any policy evaluaion using linearized srucural equaions, including policies leading o non-zero in aion raes. Bu i raises anoher quesion: is i he bes possible approximaion for he rue loss funcion (negaive welfare) when he in aion arge is no zero? Answering he second quesion rs, his paper nds srong evidence ha i does no bes approximae under he sandard DSGE framework. The main reason is ha, as shown here, he curvaure of he True loss funcion sharply increases as rend in aion 3 (, henceforh) rises above zero, while i is relaively a for negaive rend in aion. Since he True loss funcion has a minimum a he zero rend in aion and is much aer a his poin han under any posiive rend in aion, he SWeB loss funcion underesimaes welfare losses under posiive rend in aion. This is due o he fac ha i is a second order approximaion of he True loss funcion abou he zero rend in aion, where he curvaure is relaively small. This paper ideni es wo main wedges driven beween he True loss funcion and is approximaed assessmen, which I call he saic and he sochasic wedge, respecively. The saic wedge arises when boh loss funcions (True and approximaed) are evaluaed in equilibria where he in aion remains xed a he rend in aion, he remaining endogenous variables are xed a he levels consisen wih he rend in aion, and he exogenous shocks are xed a heir means. When assessing he SWeB loss funcion wih linearized srucural equaions around he Zero SS, his condiion implies ha he policy maker mus adjus he nominal ineres rae in order o keep in aion a he rend in a way such ha he linearized equaions are sais ed. As a second order approximaion of he rue loss funcion, he SWeB is a (facing up) quadraic form in, and has consan and (relaively) small curvaure compared o he True loss funcion evaluaed a each seady sae equilibrium wih rend in aion. This fac implies ha boh measures agree only a he Zero SS. Bu as depars from zero, he SWeB loss increasingly underesimaes he True loss. The sochasic wedge arises due o in aion volailiy around any posiive rend in aion in sochasic equilibria. I is a direc applicaion of he Jensen s inequaliy. Since he SWeB loss funcion is less convex han he rue loss funcion, he underesimaion of he expeced loss is even more criical. This paper derives he rend in aion welfare-based (TIWeB, henceforh) loss funcion (Proposiion ), as a second order approximaion of he rue loss funcion around he seady sae equilibrium wih rend in aion (Trend SS, henceforh). In an imporan aspec, i di ers from he rend in aion approximaion obained by Coibion e al (00), he closes paper in his regard. For he reasons shown below, he auhors approximaed he preference funcions (consumpion uiliy and he average labor disuiliy over he di ereniaed rms) of he Broadly speaking, he seady sae equilibrium is de ned as he one achieved when all disurbances are xed a heir means. 3 The rend in aion, as i came o be known, is he level of he in aion rae in he seady sae equilibrium. 3

4 represenaive household abou he seady sae equilibrium wih exible prices (Flex SS, henceforh), which does no depend on he rend in aion per se and fas depars from he Trend SS when i is no a he narrow viciniies of zero. As a consequence, hey conclude ha he coe cien on he variance of he log-deviaion of oupu around is seady sae does no depend on he rend in aion. The dependence on he rend in aion is hen re eced only in he inercep of he approximaion and in he coe cien of he variance of he log-deviaion of in aion from is rend. This las conclusion re ecs he fac ha par of he approximaion was around he Trend SS, as he losses from relaive price dispersion abou he acual rend in aion. More commens are made in subsecion 4. The reason for ha, I conjecure, is ha hey closely followed Woodford s approach in deriving he SWeB loss funcion abou he Zero SS. Among oher feaures, his approach depends on all producion levels of he di ereniaed rms o converge o he same seady sae. While he Flex SS has his propery, i is no he case under he Trend SS. While here is a seady sae level for all aggregae real variables, he Calvo price seing implies ha he relaive prices of individual rms z (0; ) never converge o a xed seady sae level in he Trend SS. Thus here is always dispersion of he relaive prices, even when he aggregae shocks remain a heir means. This also implies ha individual producion levels and individual supplied labor h (z) never converge as well. Thus he aggregae disuiliy 4 R 0 (h (z)) dz of labor canno be direcly approximaed around he Trend SS equilibrium. Finally, he suiabiliy of he Flex SS as an approximaion poin is also problemaic. Even hough i has he nice convergence propery, i does no depend on he rend in aion. Moreover, is implied levels of aggregae consumpion and producion fas depar from he corresponding levels under he Trend SS. The aggregae oupu level under he laer seady sae fas deerioraes compared o is level in he Flex SS. More deails are shown in subsecion.3. This paper avoids his approximaion pifall when deriving he TIWeB loss funcion. As shown in proposiion, i derives a pair of equaions describing he evoluion of aggregae disuiliy funcion ha only depend on aggregae variables. This resul allows for deriving he TIWeB loss funcion direcly as a second order approximaion of all componens of he rue welfare funcion around he Trend SS equilibrium. I am aware ha he degree of price rigidiy a la Calvo is likely o endogenously decrease as he rend in aion rises. I assume, however, ha he parameer remains consan for all values of rend in aion as long as is su cienly small (less han 5% year, for insance). Tha being said, wo ine ciency sources are shown o a ec he loss funcion, i.e. monopolisic compeiion and non-opimal in aion arges 5. And in order o cope wih he increasing curvaure of he welfare funcion, he TIWeB loss funcion suggess ha i is opimal for he moneary auhoriy o worry even less abou he oupu gap sabilizaion han abou sabilizaion of he in aion rae around he rend. The weigh of he oupu gap volailiy decreases as he rend in aion rises. 4 I isproperly de ned in secion. 5 Since Coibion e al (00) used he seady sae wih exible prices abou wich o approximae preference funcions, hey did no need o worry abou he linear erms arising when he rend in aion sead sae is chosen insead. More informaion on why i is exremely imporan o cope wih he linear erms is found in Woodford (003). 4

5 Recall he rs quesion his paper poses: if a welfare-concerned cenral bank is no seeking o bring in aion back o is welfare opimum, which ool should be used o evaluae policies and pick he bes one from he policy family ha brings he in aion back o he arge? To answer ha, his paper uses he TIWeB loss funcion o derive he ime consisen opimal policies under boh uncondiional (e.g. Damjanovic e al. (005)) and imeless (e.g. Woodford (999 and 003)) commimen (proposiions 3 and 4), and shows how o ranslae he pursui of he in aion arge ino an addiional consrain in he loss-minimizaion problem. I is known ha he uncondiionally opimal policy slighly dominaes he imeless one (e.g. Jensen (00), Jensen and McCallum (00)). As shown here, i is well-suied o cope wih he pursui of he in aion arge, if de ned as he uncondiional expecaion of he in aion rae. Deriving he imeless perspecive opimal policy ha is conformable wih his de niion of he in aion arge requires an approximaion assumpion, i.e. ha he subjecive discoun rae is close enough o uniy. Noneheless, he assumpion does no seem o be srong, since his parameer (in quarerly frequency) seems o vary in [0:98; ) in mos economies. An ineresing resul of boh ime consisen opimal policies under rend in aion is ha he argeing rules are more hisory-dependen han he ones derived for he Zero SS. Indeed, he argeing rules derived in he realm of he Zero SS depend on he rs lag of (log-deviaion) oupu gap and curren (log-deviaion) in aion, discouned by he indexaion rae (e.g. Woodford (003) and Damjanovic e al. (005)). In he rend in aion case, he paper shows how he rules are augmened o also depend on he second lag of he (log-deviaion) oupu gap and on he rs lag of (log-deviaion) in aion, discouned by he indexaion rae. The inerial propery of he argeing rules is shown o increase wih he rend in aion. Lacking he knowledge of such opimal policies, consisen wih he Trend SS, many cenral banks and he academia have gone around his problem eiher by: (i) imposing complee indexaion 6 o he in aion arge when rms do no opimally readjus heir prices (e.g. Yun (996)); or (ii) changing he opimal policy based on he SWeB loss funcion o have he gap of curren in aion from he arge (e.g., Alsadheim e al (00)). Those approaches, even ad-hocly imposed, are imporan for allowing beer ing o daa (he former), and o jusify he pursui of a non-zero in aion arge when he cenral bank has welfare concerns (he laer) as i is he case of he Norges Bank. Concerning he former case, however, imposing full indexaion is no likely o be a good assumpion. Analysis from micro daa suggess here is very small or no indexaion a all on individual prices (e.g. Klenow and Kryvsov (008) and Bils and Klenow (004)). Many auhors have been working wih rend in aion models, no only because a number of economies adop he in aion argeing framework, bu due o he fac ha posiive in aion averages have been observed in developed and emerging counries 7. Some auhors embed he rend in aion in he indexaion rules he agens follow when no opimizing (e.g. Yun (996), Alves and Areosa (005)), while he recen approach is o embed 6 Indexaion is known o o se he role of he in aion rend. Wih complee indexaion, he exra dynamics creaed by he rend in aion disappears. In his case, he argeing rule and he linearized equaions derived under he Zero SS urn ou o mach heir peers derived in he Trend SS. Under full indexaion o he in aion arge, i becomes he opimal in aion rae. 7 In aion has sysemaically been posiive since he World War II. Indeed, average annual in aion ranged beween 4% o 0% in European counries, abou 4-5% in he US, and higher in developing counries even afer heir disin aionary plans succeeded. Japan had a long period of negaive in aion, from 999 o 005. Even hough he CPI annual in aion rae averaged 3.4% from 97 o 008, he average from 999 o 005 was -0.46%. Source: Japan s Saisics Bureau. 5

6 he rend in aion deep in he linearizaion of he srucural equaions. Good reference is found in Amano e al (006), Ascari (004), Ascari and Ropele (007 and 007b), Blake and Fernandez-Corugedo (006), Cogley and Sbordone (008), Coibion and Gorodnichenko (0), Coibion e al (00), Fernandez-Corugedo (007), Kichian and Kryvsov (007), and Sahuc (006). Such an approach allows for ineresing conclusions, specially on wha concerns he dynamics of he supply curve under non-zero rend in aion. Models using Calvo-ype price seing sars delivering inerial dynamics, even when here is no indexaion rule. Moreover, he coe cien of he oupu gap in he Phillips curve decreases as he rend in aion rises. The analysis also highlighs a fac ha has no been given proper aenion in he lieraure on rend in aion. I is a well known resul ha he New Keynesian Phillips Curve (NKPC), when derived as a log-linearizaion of he rms rs order condiion around he Zero SS, and wrien as a funcion of he oupu gap from he naural ( exible prices) oupu, has no cos-push shock. Suppose ha he shocks are all e cien, i.e. he naural and he e cien (cenral planner) oupu move in andem (e.g. preference and produciviy shocks). Thus he cenral bank is able o sabilize boh he oupu gap and in aion when minimizing he SWeB loss funcion, wrien in erms of he e cien oupu gap, subjeced o he NKPC. A sandard way o give a rade-o problem o he cenral bank is o add non-srucural shock erms direcly o he NKPC or o assume he exisence of ine cien shocks, which makes he naural oupu gap no o move in andem wih he e cien oupu gap (e.g. ime-varying axes on rms income or ime-varying markups over he marginal cos). Ineresingly, as rend in aion depars from zero, he Phillips curve has an addiional aggregae shock erm which is a combinaion of he same srucural shocks ha would no, under he zero rend in aion linearizaion, a ec he sandard NKPC. This holds even in he absence of non-srucural or ine cien shocks. In his case, he oupu gap from he naural oupu does no enirely summarize he way he srucural shocks ( rs order) a ec he Phillips curve. This addiional shock can explain why he sandard NKPC seems o have cos-shocks componens. Moreover, he impac of his aggregae cos shock increases as he rend in aion rises. Thus he only siuaion in which he moneary auhoriy faces no policy con ic in in aion and oupu gap is when he seady sae in aion is zero. In ligh of his resul, he analyses presened in his paper are only dependen on shocks ha also a ec he e cien oupu, i.e. preference shocks on consumpion uiliy and labor disuiliy, and produciviy shocks. No ad-hoc cos push shocks are added. Also in order o highligh he role of rend in aion, he indexaion degree is mainained a zero. The paper also predics an ineresing resul achieved by he opimal policies: in he absence of ine cien shocks, in aion and oupu gap volailiy iniially grows as he rend in aion rises. Bu afer some in aion hreshold, oupu volailiy sars o fall while he in aion volailiy keeps on rising. In he impulse responses o he srucural shocks, he responses on impac of he oupu gap increases (in absolue value) as he rend in aion rises. Afer he hreshold is achieved, he dynamics of he oupu gap changes dramaically. I behaves as if i has a roo close o uniy, and he response on impac sars o fall as he in aion rend rises. The responses of 6

7 he in aion rae, on he order hand, seems no o be in uenced by he hreshold. The paper is organized as follows. Secion shows he model, makes welfare analysis and presens proposiion on how o compue he aggregae disuiliy as funcion of only aggregae variables. Secion 3 shows he loglinearized equaions under rend in aion. Secion 4 derives he TIWeB loss funcion, de ned by proposiion, and inroduces he conceps of he saic and sochasic wedges o compare he performances of he SWeB and he TIWeB loss funcions as merics for policy evaluaion. Secion 5 derives he ime consisen opimal policies under imeless and uncondiional commimen wih rend in aion, as de ned by proposiions 3 and 4. The srucural model The srucural model used in his paper is a sandard DSGE model wih Calvo price-seing and exible wages, as presened in Woodford (003). The main conribuions of he paper are he welfare analysis under rend in aion, he derivaion of he Trend In aion Welfare-Based loss funcion, and he derivaion of he implied opimal policy using he rend in aion NKPC. In addiion o he already known condiions for he exisence of he equilibrium under rend in aion (as posed in he lieraure menioned before), anoher condiion is shown. To he bes of my knowledge, his addiional condiion lacked in he curren rend in aion lieraure for hey did no focused on welfare analysis. This resricion is necessary for he disorion creaed by dispersion of he relaive prices o converge under rend in aion. The curren lieraure has focused on he convergence condiions of he rms rs order condiion under rend in aion. The economy consiss of a represenaive in nie-living household, ha consumes an aggregae bundle (de ned furher on) and supplies di ereniaed labor o a coninuum of di ereniaed rms indexed by z (0; ) ha produce and sell goods in a monopolisic compeiion environmen. Firms follows Calvo ype price-seing behavior and maximize heir expeced ow of pro s, subjeced o heir own demand curves.. Households The represenaive household from his closed economy has insananeous uiliy u (C) from consuming he aggregae good C and disuiliy (h (z)) from labor supplied a each rm z. Preferences, also subjeced o shocks, are such ha u (C ) = C exp ( u ) (h (z)) = h(z)+ + exp ( ) The household solves max fc T ; h T (z)g fa T + g P E = 0 0 [u(c) R 0 (h(z))dz] ; s:: Eq +A + A + R 0 w(z)h(z)dz + d + T PC, for 0 where A is he wealh a he beginning of period, w (z) is he nominal wage rae for work ype z, d denoes nominal dividend income, and q + is he sochasic discoun facor from ( + ) o, and i is he riskless shorerm (one-period) nominal ineres rae, which sais es E q + = ( + i ). 7

8 Aggregaion is via Dixi and Sigliz (977): C = R 0 c (z) dz, P = R 0 p (z) dz and c (z) = C P (p (z)), where > is he elasiciy of subsiuion beween goods. In equilibrium 8,he opimal real wage sais es w(z) P = h(z) e ( u ) Y, and he opimal consumpion plan is de ned by he Euler equaion: q = Y Y exp u u, I = E q + () where I = + i and = + are he gross ineres and in aion raes a period.. Firms The assumpions are he sandard ones of a simple DSGE model wih Calvo price seing, wih probabiliy ( ) of opimally readjusing each period. If no opimally readjusing, rms index heir prices wih he rule p (z) = p rend, i.e. ind (z) ind, where ind is a funcion of he pas (gross) aggregae in aion rae and he (gross) in aion =, where (0; ) and (0; ). The producion funcion of each rm z (0; ) is y (z) = h (z) " exp ( a ), and he following law of moion drives he dynamics of he aggregae price: = ( ) pc where p c is he opimal readjusing price a period : P ind + () pc P +! = E P T = (+!) T q ;T G ;T ;T ;T = ind ;T (XT ) (!+ ) E P T T = q ;T G ;T ind ;T ;T = ind ;T where! " ( + ) and = = ( ) is he consan markup in he alernaive equilibrium wih exible prices. Moreover, ;T P T =P, q ;T = Q T =+ q, and G ;T Y T =Y sand for he cumulaed gross in aion rae, discoun rae and gross oupu growh from o T, respecively. Finally X Y =Y n he naural ( exible prices) oupu Y n : " Y n = exp (( +!) a + u (!+ ) ) is he gross oupu gap from In order o make he Firs order condiion neaer, le N and D denoe he numeraor and he denominaor on he righ hand side of he rms rs order condiion, so ha i is wrien as (p c =P ) +! = N =D. As shown in nex subsecion, hese variables are very helpful in deriving a recursive equaion for he aggregae disuiliy funcion ha depends only on aggregae variables. This is one of he conribuions of his paper, for i allows us 8 Equilibnrium is de ned as he equaions describing he rs order condiions for he represenaive household and rms (shown in he following subsecion), in adiion o he marke clearing condiions: C = Y, and c (z) = y (z) (8z), where Y and y (z) are he aggregae and individual producion levels. 8

9 o derive he TIWeB loss funcion. Thus he su cien supply equaions are = ( ) N D +! ind + N = (X ) (!+ ) (+!) + E n + n = q G N ind D = + E d + d = q G ind D ind where G Y =Y sands for he -period gross oupu growh..3 Welfare compuaion Following, I presen one of he conribuions of his paper: I derive a pair of equaions describing he evoluion of he average disuiliy funcion ~ R 0 (h (z)) dz from labor supplied o all rms, evaluaed a he equilibrium wih Calvo price seing, as a funcion of aggregae variables only. This resul is imporan because i allows he derivaion of he TIWeB loss funcion as a second order approximaion of he (negaive) value funcion of he represenaive household. The approximaion is done around he Trend SS, and i allows for avoiding he problems faced by Coibion e al (00). I is imporan o sress ha he resul presened in his secion holds for he non-linear srucural model, i.e. i is independen of any rend in aion. And o he bes of my knowledge, his approach was no done before for welfare analysis in he rend in aion lieraure. ~ = Noe ha, in equilibrium 9, he average disuiliy ~ from labor supplied o all rms can be rewrien as follows R 0 h(z)+ + exp ( ) dz = exp( ) + = exp( (+!) a ) + Y (+!) P (+!) R 0 R 0 y (z) + " exp " a dz = exp( (+!) a ) + p (z) (+!) dz Le now /P denoe he aggregae price relevan for labor disuiliy concerns: R 0 Y p(z) P (+!) dz /P (+!) Z 0 p (z) (+!) dz, whose dynamics is described he he following equaion (implied by he Calvo price seing): /P (+!) = ( ) p (+!) (+!) c + /P ind Muliplying by he las resul by P (+!),and de ning P /P P o be he relevan relaive price for disuiliy issues, I sae he following proposiion ha describes he evoluion of he aggregae disuiliy funcion as a funcion of only aggregae variables. I is imporan for allowing he derivaion of he TIWeB loss funcion wihou having o worry abou he fac ha he producion levels of individual rms and he corresponding individual relaive 9 The equilibrium equaions are he ones implied by he households and rms rs order condiions, added o he marke clearing condiions such as C = Y, 8. 9

10 prices do no have a seady sae when he rend in aion is posiive. This problem was probably faced by Coibion e al (00), when hey decided o approximae he aggregae disuiliy funcion around he seady sae equilibrium wih exible prices, insead. Proposiion The aggregae disuiliy is compued as ~ = + Y (+!) P (+!) where he relevan relaive price for disuiliy issues evolves according o exp ( ( +!) a ) (3) P (+!) = ( ) N D (+!) +! ind (+!) + P (+!) (4) Noe ha he reciprocal of P is responsible for he welfare losses due o dispersion on he relaive prices. Indeed, is rs order approximaion abou he seady sae is zero. Thus =P moves in andem wih he cross secion variance of prices as an appreciable approximaion. In he sandard approach of loglinearizing he equaions around he zero rend in aion, he second order (log) approximaion of =P is direcly proporional o he second order (log) approximaion of R 0 (p (z) =P ) dz, he sandard measure of relaive price disorion (e.g. Yun (005)). However, as shown in his secion, he endogenous (welfare-based) measure of relaive price dispersion should be Z =P = 0 (p (z) =P ) (+!) (+!) dz Le W denoe he welfare funcion, compued as he discouned ow of preferences evaluaed a he equilibrium variables. In his case, W evolves according o he Bellman equaion. Thus, he economy s welfare can be compued (and simulaed) by he following su cien equaions: W = (u ~ ) + E W + u Y exp ( u ) ~ = + Y (+!) P (+!) P (+!) (+!) = ( ) N +! ind (+!) (+!) D + P exp ( ( +!) a ) (5) A poin ha is very imporan o sress is relaed o he concaviy of he welfare funcion as he rend in aion increases. The welfare funcion decreases fas and becomes highly concave in in aion as i increases in he posiive side. However, he welfare funcion is relaively aer for negaive in aion raes. For insance, he following parameer se was consruced o be in line wih he curren mosly acceped values 0 for he USA. The Calvo parameer, for insance, is consisen wih he micro evidence such as he ones repored in Klenow and Kryvsov (008) and Bils and Klenow (004). From Cogley and Sbordone (008): = 0:6, = 0:99 (quarerly), = 0, = = 0, " = 0:75. From Smes and Wouers (007): = :50, = :50. Wih hose values, he limiing annual rend in aion rae (laer explained) is 6:3%. Using he seady sae resuls shown in Appendix A, gure shows he behavior of he seady sae welfare 0 There are di eren parameer ses esimaed in he lieraure, such as he ones repored in Roemberg and Woodford (997): = 0:65, = 0:99 (quarerly), = 8, " = 0:75, = 0:0 and = 0:5. 0

11 funcion, is second derivaive, and he seady sae oupu ( Y ) and naural oupu ( Y n, normalized o ) as he rend in aion rises from 0% o 4.5%. As expeced, he welfare funcion his is maximum when he rend in aion is 0%. However, i is much more concave for posiive values. For insance, he concaviy of he rue welfare funcion a % is 4.97 as large as i is a he Zero SS. If he rend in aion is 4%, he concaviy is 0.6 as large as i is a he Zero SS. I implies ha: (i) second order approximaions around he Zero SS will underesimae he appropriae curvaure of he welfare funcion when he in aion rae is acually oscillaing around a posiive value; and (ii) and such second order approximaions will no inernalize he large welfare loss achieved when he rend in aion is posiive. When policy is mean o keep in aion a a posiive level, as in an in aion argeing framework, hose properies sugges ha beer policy evaluaion is obained when he approximaion is done around he rend in aion. Figure : Welfare Funcion, is second derivaive and he seady sae oupu Finally, noice ha he seady sae oupu Y decreases fas below he naural oupu Y n as he rend in aion rises above zero. Since he naural oupu is invarian wih he rend in aion, is level was normalized o. I implies ha he graphic is acually picuring he gross oupu gap Y = Y n. When in aion is %, he oupu gap should be evolving around %. And a he 4% in aion rae, he gap opens o 7%. This e ec is he clear picure of he disorion caused by high rend in aion..3. Disorions This subsecion expands Wordford s (003) analysis on he e cien oupu under he Zero SS and he disorion caused by he monopolisic compeiion. The conribuion is a way o cope wih he addiional disorion caused by he rend in aion. Consider a cenral planner ha chooses he prices and he oupu level o maximize he welfare. The opimal soluion clearly imposes every rm o produce he same e cien level Y ef, which implies ha all prices are he same, i.e. P =. Thus he soluion (see Woodford (003)) o max Y W =@Y =, which implies ef =@Y

12 following equaion for he e cien oupu: Y ef " = exp (( +!) a + u (!+ ) ) Appendix A shows he equilibrium seady sae values. In paricular, he seady sae consumpion uiliy and he labor disuiliy under rend in aion depends on he seady sae aggregaes as follows: Y u = ~ = +! Y (+!) Y ef (!+ ) P (+!) where barred variables sand for seady sae levels, and Y ef (!+ ) = =". Ignoring he indirec of Y on P, he seady sae value of he marginal rae of subsiuion under he equiibrium wih sicky prices can be roughly approximaed by he raio of he derivaiaves of he seady sae levels ~ and u wih respec o Y, i.e. ~ Y =u Y. Noe ha i is no he same as he seady sae level of he raio of he derivaiaves of ~ and u wih respec o Y. However, his approximaion makes i easier o undersand he rs disorion componen ha exiss in his model economy: ~ Y u Y = Y (!+ ) Y ef (!+ ) P (+!) = ( #) ( ) ( ) ( #) (6) where! " ( + ), = ( ) is he markup, ( + ) is he gross rend in aion # (+!)( ( +)) is a posiive ransformaion of he rend in aion ( )( ( +)) is he e ecive degree of price sickiness Noe ha # and if, i.e. he e ecive degree of price sickiness is greaer han he Calvo degree when he rend in aion is posiive. Moreover, since # is a posiive ransformaion of he rend in aion, i is a convenien variable o re ec he e ecs of he rend in aion. Parallel o Woodford (003), le y ~ Y =u Y denoe he ine ciency degree of he seady sae oupu. He assumed a Zero SS, and hence he ine ciency degree was only y = =. The rs erm (=) of ~ Y =u Y is driven by he monopolisic compeiion disorion alone, while he second ( rend in aion. Noe ha he second erm collapses o one when # =. #) ( ) ( ) ( #) is driven by he non-zero The second componen of disorionary e ecs of he non-zero rend in aion is explained as follows. Under Calvo price seing, he dispersion of relaive prices does no disappear in he Trend SS, i.e. when > and a = u = = 0 8. Whenever a paricular rm does no readjus, is price lags behind because he aggregae price is sill rising. As a consequence, here is sill oupu dispersion in he seady sae. In a nushell, he indirec e P=@ Y omied in he previous naive compuaion of of ~ Y =u Y he in aion rend disorion. would capure an addiional componen of

13 Insead of indirecly capuring he rend in aion ine ciency P=@ Y, i is beer o keep rack of wo ine ciency parameers for reasons explained laer on. The rs one parallels Woodford s variable y ~ Y =u Y, and he second is inroduced here as # (# greaer han one. ). Thus # capures how much of he gross in aion rend is E ciency requires boh parameers o be zero. In order o proceed wih he approximaion of he welfare funcion abou he non-zero rend in aion, i will be ineresing o assume boh parameers o be small enough, as rs order disurbance erms. Wih such an assumpion, linear erms muliplied by y and # become of second order, allowing he welfare-based loss funcion o be useful for policy analysis using he loglinearized srucural equaions of he model. 3 The loglinearized model The Euler equaion and he supply rs order condiions can be log-linearized as he IS curve, and New Keynesian Phillips Curve (NKPC) under rend in aion. Haed variables, including ^ and ^{, are log-deviaions from heir seady sae levels. The seady sae levels under rend in aion are shown in Appendix A. Before showing he linearized equaions, recall he meaning of he srucural parameers: coe cien of risk aversion reciprocal of he Frisch labor supply elasiciy Scale parameer on labor disuiliy Elasiciy of subsiuion beween goods coe cien of lagged in aion on indexaion rule coe cien of in aion rend on indexaion rule " labor coe cien in he producion funcion Calvo degree of price sickiness The following resricions are necessary for convergence and consisency resuls in he seady sae, i.e. posiive relaive prices in he seady sae (p (z) =P and P): # < (i su ces if ) and < (7) The resricions imply # < and <. And hence, here is a maximum level for he rend in aion o exis under he premises of he model: min (+!)( ( +)) ;! ( )( ( +)) The paper does no show he log-linearizaion seps, as similar ones are well documened in he lieraure on rend in aion. This secion also poins some aspecs of he NKPC under rend in aion ha was no sressed Woodford (003) was he rs auhor o model he disorion variable y as a rs order disurbance erm, in order o derive he SWeB loss funcion. See, for insance, Ascari (004), Ascari and Ropele (007 and 007b), Cogley and Sbordone (008), Coibion and Gorodnichenko (008). 3

14 in he lieraure. In hose equaions, haed variables ^{ are always de ned as log-deviaions from heir levels in he Trend SS, e.g. ^{ log (^{ ={). The loglinearized IS curve is: where ^x = ^y ^y n is he (log-deviaion) oupu gap, ^r n (!+ ) E (log-deviaion) real ineres rae under exible prices, and ^y n = (!+ ^x = E ^x + E (^{ ^ + ^r n ) (8) ) (a a + a!e u + u is he + u ) is he (log-deviaion) naural oupu. Moreover, a ( +!) a is a shock compound ha aggregaes he only way he echnology and disuiliy shocks a ec he model. As for he NKPC under rend in aion, wih indexaion erm ^ ind = ^, I chose a represenaion ha makes i easier o see he e ec of : ^ ^ ind = E ^ + ^ ind + + ^x (# ) + ( #L ) E (# ) ^x + + ( #L ) ' E ^ + ^ ind + (# ) + ( #L ) ' E (^x + ^x ) (# ) + ( #L ) ' 3E + (9) where L is he lead operaor, ( )( #) (!+ ) (+!), ' ( )(+!) (+!), ' ( )( ) (+!), ' 3 (!+ ) ', and (# ) ' 3 E + is he inernally consisen cos push shock under rend in aion, and is he aggregae shock: = +! a + u As menioned before, # = under he Zero SS. In his case, all erms muliplied by (# ) disappear and he sandard form is obained. I chose his way o wrie he he NKPC, for being a nea way o represen i wih only one equaion and no in nie sum, as opposed o he way commonly presened in he lieraure. Since he denominaors of mos of he erms in he righ hand side are funcion of he lead operaor L, i implies ha we could eiher have erms wih in nie sums, or erms wih he expecaion of in aion wo periods ahead. Moreover, no only he level of he oupu gap, bu also is expeced growh rae (^x + ^x ) is imporan. Finally, he rend in aion speci caion allows for an inernally consisen cos push shock, as long as he rend in aion is no zero. This cos shock is inernally consisen because i is generaed by he same srucural shocks ha a ec boh he naural and he e cien oupus. In general his feaure is no observed in he lieraure, i.e. cos shocks are usually hough as proporional o he componens of he naural oupu ha do no a ec he e cien one. Mos ofen hey are modelled as ime varying ax raes on rms income or ime varying markups (e.g. Clarida e al (999), Galí (003), Smes and Wouers (003, 005, 007), Ascari and Ropele (007)). Noneheless, cos shocks play an imporan role in opimal policy analyses. If he only rigidiy is on he price seing a la Calvo, here is no policy con ic under he zero rend in aion paradigm if he only source of shocks are he ones considered in his paper. As a consequence, he opimal policy is able o obain he unrealisic resul of sabilizing 4

15 boh he in aion and he oupu gap. In order o obain he realisic rade o beween boh objecives, cos shocks are more han necessary (e.g. Walsh (003), Woodford (003)). Noe ha (# ) is he shock ampli er. Wih larger rend in aion, in aion dynamics are naurally more volaile. Ascari and Ropele (007) nd similar resul also using a rend in aion model wih Calvo price seing. Bu hey sress ha heir nding sems from he opimal response of he paricular moneary policy hey assess (discreion), using an addiional cos push shock o induce he policy con ic. This paper, on he oher hand, highlighs ha his resul is independen of any policy srucure and does no require any addiional cos shock o hold. Ineresingly, anoher feaure mus be highlighed. The sign of coe cien on he growh of he oupu gap depends on being less or greaer han one. Since is he ineremporal elasiciy of subsiuion, i means ha he ension beween he subsiuion and income e ec direcly a ecs he in aion dynamics under rend in aion. And again, (# ) has a muliplier role. Noe also ha he subsiuion e ec also a ecs he aggregae shock. If =, he cos shock does no respond o echnology and disuiliy shocks. The exercises presened in his paper compares he resuls for less han, equal, and greaer han one. 4 The welfare-based loss funcion under rend in aion This secion presens he one of he main conribuions of his paper: he rend in aion loss funcion, derived as he second order log-approximaion of he (negaive) welfare funcion. The proof, and deails are shown in Appendix B. Before showing he main resul, some clari caion are needed in order o answer why his paper considered a di eren derivaion approach from he one adoped by Woodford (003). Under rend in aion wih Calvo price seing, he seady sae is sill dynamic. Indeed, while srucural shocks are consrained o remain a heir means ( a = u = = 0 8), he posiive in aion rend forces a saionary dispersion of relaive prices. This resul is beer undersood if one noes ha here is an addiional family of exogenous shocks: he Calvo "green ligh" Bernoulli random variables for each rm. Those green lighs are sill working in he seady sae, and rend in aion causes a fracion of rms prices o lag behind he opimum. This pariculariy makes he individual producion levels o be dispersed, even under he seady sae. Ineresing, aggregae oupu converges o a xed seady sae. The individual oupu dispersion is such ha i cancels ou when aggregaing, and Y is ime invarian. A side e ec of his dynamic seady sae is ha he second order approximaion of he welfare funcion mus be done wih a di eren sraegy han he one chosen by Woodford (003). If one makes he second order logapproximaion of each labor disuiliy, i will be a funcion of each individual secor producion level in he seady sae. Bu hose levels are di eren from each oher and are sill varying in he seady sae. As a consequence, inegraing he approximaed disuiliies mus deal wih he oupu dispersion in he seady sae, along wih he log deviaion dispersion of he equilibrium oupu from he seady sae levels. Under zero in aion in he 5

16 seady sae, his problem disappears, for each producion level converges o he same ime-invarian level, i.e. he aggregae oupu. And hence, hey are no par of he inegrand. I conjecure Coibion e al (00) mus have faced he same problem, and hence hey decided o approximae he preference funcions around he seady sae wih exible prices insead. As previously menioned, he auhors conclude ha he coe cien on he variance of he oupu gap does no depend on he rend in aion. They also assumed log-uiliy on consumpion, which i no innocuous under rend in aion. This fac could be he source of heir resul. However, even assuming a more general form for he uiliy funcion U (C) of he aggregae consumpion bundle C, he coe cien on he variance of (log-deviaion) oupu would sill be independen of he rend in aion when approximaed abou he Flex SS. Indeed, regardless of chosen seady sae abou which o approximae he rue loss funcion, some ine ciency sources 3 can be assumed o be su cienly small and hence his coe cien can be shown o be proporional o CU 0 C muliplied by he reciprocal of he ineremporal elasiciy of subsiuion evaluaed in he seady sae. Here, C sands for he seady sae level of he aggregae consumpion. If U (C) = log (C), hen CU 0 C = and he dependency clearly disappears. If i is no he case, he coe cien depends on he characerisics of he chosen seady sae equilibrium. Therefore he dependency on he rend in aion disappears if his funcion is no approximaed around he Trend SS. Since he auhors chose he Flex SS, he dependency was ruled ou. Adoping a log-uiliy funcion, i.e. U (C) = log (C), implies CU 0 C = and so he e ec of he rend in aion on he seady sae level of consumpion would be o se anyway. The approximaion around he Flex SS is also no a good alernaive if he in aion rend is no in he viciniy of zero. As depiced in gure, he consumpion level under he Trend SS fas deerioraes compared o he Flex SS level. Therefore, one mus nd a way o approximae he rue loss funcion around he Trend SS, avoiding he pifalls previously described. To ge around his problem, I derived he resuls shown in proposiion. The resuls allow he evoluion of he aggregae disuiliy, and hence he evoluion of he rue welfare funcion, o be described by he evoluion equaion of he su cien saisics P, whose reciprocal accouns for he welfare loss due o he dispersion of relaive prices under he Calvo price seing framework. Those equaions play an imporan role in deriving he TIWeB loss funcion, described as follows: Proposiion The rue welfare funcion is (second-order) approximaed as W = V X #E L + + ip W (0) =0 where L ^ ^ ind ( + + ) ( #) ^x ef x is he rend in aion welfare-based (TIWeB) loss funcion, ^x ef ^Y ef ^Y is he (log-deviaion) gap beween he aggregae oupu and he e cien oupu (regarding heir own seady sae levels), and x are consans ha 3 Monopolisic compeiion and non-zero rend in aion. 6

17 depend on he ine ciency degrees # and y and ac as correcing he levels abou which he log-deviaions of in aion and oupu gap are evaluaed in order o capure dynamic shifs in he welfare funcion, and V # correcs for he aggregae reducion in he welfare when he rend in aion increases. Those parameers are de ned as follows: ( ) ( #)(+!) # x (!+ ) y V # Y ( #)(!+ ) ( ) The proof, and deails are shown in Appendix B. Since ^x ef is equal o ^x ^Y ef ^Y n, he e cien gap ^x ef will be di eren from he sandard oupu gap measure if and only if he levels of he naural and he e cien oupus do no move in andem jus as shown in Woodford (003) for he Zero SS case. This holds even when hey have di eren seady sae levels. However, he naure of he shocks assumed in he model implies ha ^x ef = ^x 8. The absence of sandard sources of cos-push shocks (e.g. ime-varying ax raes on rms income) implies ha he di erence beween boh oupu de niions is ime-invarian. The assumpions ha y and # are of rs order were necessary o cope wih he wo linear erms in he approximaions. Finally, ip W sands for erms independen of policy a period. Noe ha, under he rend in aion approximaion, he coe cien on he oupu gap volailiy is, where ( ) ( #). Since >, he coe cien is greaer han. In he Zero SS, on he oher hand, he opimal coe cien is only. This fac could induce us o hink ha he cenral bank should be more "dovish" if she inends o maximize he economy welfare under rend in aion. However, his rs hough is misleading. Recall ha <, i.e. he coe cien of he oupu gap in he Phillips curve is smaller in he Trend SS han he one in he Zero SS. And as shown below, he laer e ec more han dominaes he former. Indeed, he whole coe cien is a decreasing funcion of he rend in aion. This resul was expeced, due o he resuls shown before on he concaviy of he welfare funcion in he seady sae. Acually, he second order approximaion under rend in aion bene s from inernalizing he big welfare loss due o in aion volailiy under posiive in aion rend. This e ec is ranslaed ino he decreasing coe cien on he oupu gap volailiy. Figure : Coe cien Raio (R): = 0:50 (doed blue), = 0:60 (black), = 0:65 (dashed red). 7

18 Le R denoe he raio of he coe ciens under rend in aion and under he Zero SS: R = = = = ( ) ( ) ( ) ( #) ( #) () Since # (+!)( ( +)) and ( )( ( +)), he elasiciy of R o he rend in aion is R = ( ) ( ) ( ) ( ) ( + ) ( ) ( +!) # ( ) ( ) ( #) ( + ) ( #) () Even hough he second erm in he brackes can be greaer han when he rend in aion rises close o is criical value, reasonable paramerizaion has been shown o be consisen wih a negaive elasiciy when he rend in aion is no high enough. For insance, le us use again our benchmark parameer se: = 0:6, = 0:99 (quarerly), = 0, = = 0, " = 0:75, = :50, = :50, wih limiing annual rend in aion rae 6:3%. Figure plos R as a funcion of he annual rend in aion rae for his parameer se, as benchmark, and for = 0:50 and = 0:65. Noe ha he weigh of he oupu gap volailiy decreases fas when he rend in aion is higher and when he degree of price rigidiy is larger. Indeed, for = 0:65 and = 4% (annual), he oupu gap weigh is abou 67% of he coe cien suggesed by he SWeB loss funcion. The main message is ha he moneary policy mus increase he srengh wih which i reacs o in aion volailiy when he in aion rend is larger. 4. The saic and he sochasic wedges This secion inroduces he conceps of wha I called saic and sochasic wedges and uses hem o compare he performances of he SWeB and he TIWeB loss funcions as merics for policy evaluaion. The sandard approach considers approximaions around he Zero SS. In his case, as shown in Woodford (003), he linearized aggregae supply and demand curves of he DSGE model used in his paper, and he SWeB loss funcion are: where r n = ~V = " (!+ ~x = E ~x + E (~{ ~ + r n ), ~ ~ ind = E ~ + ~ ind + + ~x W = ~ VE P =0 ) E ( +!) a + a (!+ ) (!+ ) L+ ~ + fip W, L ~ = ~ ~ ind + (~x x) +! u + u, and here is no cos shock in he Phillips curve., = ( )( ) (!+ ) (+!), x = (!+ ), I represened he log-deviaions wih he ilde mark (~) in order o make i clear ha hey refer o he Zero SS. Noice ha he real rae of ineress under exible prices (r n ) is he same as before. I is due o he fac ha he equilibrium wih exible prices is independen of rend in aion. As previously menioned, he saic wedge is he one arising when boh loss funcions (he rue one and he SWeB) are evaluaed in equilibria where he in aion remains xed a he rend in aion, he remaining 8

19 endogenous variables are xed a he levels consisen wih he rend in aion, and he exogenous shocks are xed a heir means. Considering linearized srucural equaions around he Zero SS, his condiion implies ha he policy maker mus adjus he nominal ineres rae in order o keep in aion a he rend in a way such ha he linearized equaions are sais ed. Noe ha ~x = log (X ), ~ = log ( ), ~ ind = log ( ) and ~{ = log (I ) + log (). Therefore, if he cenral bank wans o keep he in aion a he arge in his conex, he approximaed equaions imply: r n = 0, ~ = log, ~{ = ~, ~x = ( )( ) ~ ~L = ( ) ~ + (~x x), ~ W = W~=0 ~V L ~ Figure 3: Loss funcions: T rue (black), SW eb (doed blue), Saic W edge (doed red). Picure 3 compares he True and he SWeB loss funcions by means of heir (negaive) discouned ows for each in aion arge, i.e. W and ~ W. For simpli caion, heir values a he zero in aion rae were normalized o zero. The parameer se was again xed a = 0:6, = 0:99 (quarerly), = 0, = = 0, " = 0:75, = :50, = :50. Noe ha boh merics agree prey well boh in level and curvaure a he viciniy of he zero rend in aion (up o abou = %). Afer ha, he curvaure of he rue loss funcion increases fas, while he SWeB curvaure remains consan. The (red) verical doed lines depic he saic wedges, which increase fas as he rend in aion rises. As a second order approximaion of he rue loss funcion, he SWeB has a consan and (relaively) small curvaure relaively o he curvaure of he rue loss funcion evaluaed a each seady sae equilibrium. Indeed, he curvaure raio of boh merics jumps from.6 a = % o.4 a = T SW The sochasic wedge arises due o in aion volailiy around any posiive rend in aion in sochasic equilibria. I is a direc applicaion of he Jensen s inequaliy. Since he SWeB loss funcion is less convex han he rue loss funcion, he underesimaion of he expeced loss is even more criical. There is no closed form expression for he expecaion of he rue loss funcion. One mus consider simulaing higher order approximaions of he model in 9

20 order o obain a numerical value of his loss. Therefore he sochasic wedge depends on he whole disribuion of he shocks and mus be compued numerically. Tha is he reason i is no shown in picure 3. Figure 4: The TIWeB loss funcion a a = and a = 4 Regarding he TIWeB loss funcion, i is buil o be angen o he rue loss funcion a any rend in aion. And so is saic wedge is zero a each rend in aion. However, ha is no he case wih he sochasic wedge. The TIWeB loss funcion also has a consan curvaure. Even hough i maches he slope a he in aion rend, i also underesimaes he rue loss whenever he in aion rae oscillaes around he rend. In a neighborhood of he rend, however, he loss underesimaions are smaller when using he TIWeB loss funcion compared o he SWeB loss funcion. In a nushell, he saic wedge of he TIWeB loss funcion is always zero and is sochasic wedge would be smaller han he one from using he SWeB loss funcion. If he TIWeB loss funcion for a speci c rend a were o be used o evaluae he loss under a di eren rend level b, hen he saic wedge would be di eren from zero. Indeed he rs case we assessed can be hough as using he TIWeB loss funcion a a = 0 for evaluaions a b > 0. Figure 4 depics he performances of wo TIWeB loss funcions, a a = and a = 4, o measure he rue loss around di eren rends. Noice ha he saic wedges increase fas as he he gap of a from b increases. 5 Opimal policies when he rend in aion is no opimal Many cenral banks are assigned he duy of pursuing in aion arges ha are de ned by he cenral governmens. Even hough speci c values are assigned, cenral banks are ofen auonomous and exible o choose he bes way o mee he arges. The main reason behind his fac is ha he arges are generally de ned as he cumulaed in aion rae over long enough ime horizons, and he realized in aion rae is allowed o deviae from he cenral arges as long as hey remain inside olerance bands. The key aspec of he olerance bands is ha he arge will be considered equally me in any poin in is inerior. This feaure gives room for welfare-concerned cenral banks o search for ime-consisen opimal policies conformable wih he cenral arges and he olerance bands. However, he assigned arges are no guaraneed 0