Flexible Inflation Targeting under a Non-Linear Phillipscurve

Transcription

1 Economeric Research and Special Sudies Deparmen Flexible Inflaion Targeing under a Non-Linear Phillipscurve W.H. Verhagen and S.C.W. Eijffinger Research Memorandum WO&E no. 663 Augus 001 De Nederlandsche Bank

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3 FLEXIBLE INFLATION TARGETING UNDER A NON-LINEAR PHILLIPSCURVE W.H. Verhagen and S.C.W. Eijffinger * * We would like o hank Lex Hoogduin and Peer van Els for useful commens. The usual disclaimer applies. Research Memorandum WO&E no. 663/0118 Augus 001 De Nederlandsche Bank NV Economeric Research and Special Sudies Deparmen P.O. Box AB AMSTERDAM The Neherlands

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5 ABSTRACT Flexible Inflaion Targeing under a Non-Linear Phillipscurve W.H. Verhagen and S.C.W. Eijffinger This paper analyses he opimal degree of flexibiliy under a Lucas ype convex Phillipscurve. As a benchmark, we firs analyse opimal moneary policy wih a linear Phillipscurve and persisen cospush shocks. As in Svensson (1997a, a cenral banker who possesses privae informaion and who inheris sociey s preferences will engage in oo much oupu sabilisaion because of which welfare will be improved by appoining an individual who is less flexible. Moreover, we are able o invesigae he deerminans of he opimal degree of flexibiliy. If he cenral banker has no privae informaion under a linear Lucas ype Phillipscurve, i will be opimal o absain from oupu sabilisaion enirely. Nex, we exend he symmeric informaion case by assuming he Phillipscurve is convex. In his respec, i is shown ha even under sric inflaion argeing, he opimal condiional inflaion forecas will be sae-dependen. Furhermore, if he degree of flexibiliy is zero, moneary policy will be subjec o a deflaionary bias which will exceed he one obained in a model where he expeced slope of he Phillipscurve is consan. We also show ha he long run average rae of inflaion will be sricly increasing in he degree of flexibiliy. Therefore some degree of flexibiliy will be socially opimal in his model because i will render he deflaionary bias obained under sric inflaion argeing less severe. Key words: convex Phillipscurve, uncerainy, opimal degree of cenral bank flexibiliy JEL codes: E5, E58 SAMENVATTING Een Flexibele Direce Inflaiedoelselling me een Nie-Lineaire Phillipscurve W.H. Verhagen en S.C.W. Eijffinger Di onderzoeksrappor analyseer de opimale mae waarin de cenrale bank de oupu gap moe sabiliseren wanneer er sprake is van een siuaie waarin he effec van de oupu gap op inflaie een oenemende funcie is van de oupu gap zelf. Als refereniekader analyseren we allereers he opimale beleid onder een lineaire Phillipscurve me exogene persisene inflaieschokken. Indien de cenrale bankier een informaievoordeel genie.o.v. he publiek kan de welvaar verbeerd worden door he benoemen van een cenrale bankier die.o.v. de maaschappelijke preferenies meer nadruk leg op inflaiesabilisaie (zie Svensson (1997a. Vervolgens worden de deerminanen van deze zogenoemde opimale mae van flexibiliei onderzoch. Indien de cenrale bankier geen informaievoordeel genie is he, onder een lineaire Phillipscurve, opimaal om helemaal van sabilisaie van de oupu gap af e zien. Vervolgens word he model uigebreid me een convexe Phillipscurve. Zelfs indien de mae van flexibiliei gelijk is aan nul zal in da geval de opimale condiionele inflaieverwaching afhankelijk zijn van de sae of he world. Verder zal er een deflaoire bias onsaan. He lange ermijn gemiddelde inflaieniveau blijk een oenemende funcie e zijn van de mae van flexibiliei. Daarom is in di model enige mae van flexibiliei opimaal omda di ervoor zorg da de deflaoire bias,die bij een beleid zonder flexibiliei onsaa, minder geprononceerd zal zijn. Trefwoorden: convexe Phillipscurve, onzekerheid, opimale mae van cenrale bank flexibiliei JEL codes: E5, E58

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8 - 1-1 INTRODUCTION Recenly, he lieraure has seen a revival of he idea ha he economy s aggregae supply relaionship may be non-linear. In a nushell, because of capaciy consrains booms may be more inflaionary han recessions are disinflaionary. If his is he case, i may have imporan implicaions for moneary policy. Recen conribuions in his respec include Clark, Laxon and Rose (1995, Bean (000 and Schaling (1998. Clark e al. analyse he implicaions of several exogenous back- and forward-looking policy reacion funcions under an acceleraionis Phillipscurve. They show ha a convex Phillipscurve implies ha he mean level of oupu will be inversely relaed o oupu variabiliy. This is because he increase in inflaion as a resul of a period of excess demand has o be compensaed by a period of greaer excess supply o disinflae he economy. In oher words, aemps o mainain oupu equal o poenial on average will be inconsisen wih a sable rae of inflaion. Therefore, here will be an addiional gain o oupu sabilisaion if he cenral bank is faced wih a convex Phillipscurve. Nex, hey show here will be a srong case for pre-empive srikes agains inflaion ( a sich in ime saves nine. These resuls are confirmed by Bean (000 who sudies he opimal endogenous policy rule in a model wih an acceleraionis Phillipscurve where he oupu gap eners linearly in he cenral banker s loss funcion and where she observes a noisy signal of he demand shock. In paricular, he shows ha, compared he linear case, opimal policy will be more conracionary and ha disinflaions will be implemened more gradually. Schaling (1998 reaches similar conclusions by inroducing a convex Phillipscurve in he Svensson (1997b inflaion forecas argeing framework for he case where he relaive weigh on oupu sabilisaion is equal o zero. This paper differs from previous lieraure in a number of ways: Firs of all, is main goal is o sudy he opimal degree of flexibiliy in an inflaion argeing regime when he ransmission mechanism feaures an expecaions-augmened (Lucas ype convex Phillipscurve. In conras o he acceleraionis Phillipscurve used in he papers menioned in he previous paragraph, he cenral bank s abiliy o reduce oupu variabiliy is no longer rivial bu depends on wheher or no i has privae informaion. In paricular, we will sudy he quesion o wha exen opimal moneary policy rule will feaure a reurn o oupu sabilisaion (compared o he linear case if here is no informaion asymmery. This enables us o idenify a possible second addiional source of oupu sabilisaion under a convex Phillipscurve.

9 - - Secondly, as a benchmark, we firs analyse he opimal degree of flexibiliy in a model wih a linear expecaions-augmened Phillipscurve. We assume ha he cenral bank cares abou he variabiliy of inflaion around he assigned arge and abou he variabiliy of oupu around poenial. The laer assumpion ensures ha moneary policy will no be burdened by an inflaionary bias of he Barro- Gordon ype (see Barro and Gordon (1983. We also assume ha inflaion is influenced by a cos-push shock ha is serially correlaed. In he linear model we make a disincion beween he case where he cenral bank has privae informaion abou he realisaion of he cos-push shock and he case where here is no informaion asymmery. If he cenral bank has privae informaion i will be able o sabilise oupu and herefore a policy of sric inflaion argeing will no be opimal. However, because he public can parially predic he cos-push shock we find ha he opimal degree of flexibiliy will be lower han sociey s preferences in his respec. This is because moneary policy will be subjec o he kind of sabilisaion bias described by e.g. Svensson (1997a and Clarida, Gali and Gerler (1999. However, we are also able examine he deerminans of he opimal degree of flexibiliy and conras hem wih he resuls obained in a siuaion where he cenral banker ries o push oupu above he long run naural rae sysemaically. Under symmeric informaion he cenral bank will no be able o affec oupu a all. In oher words, since he reurn o oupu sabilisaion is zero in his case we find i will be opimal o appoin a cenral banker whose relaive weigh on oupu sabilisaion is equal o zero as well. In Secion 3, we analyse he implicaions of a convex Phillipscurve under symmeric informaion. As shown earlier by Schaling (1998, we find ha a policy of sric inflaion argeing (which is opimal in he linear case will induce a deflaionary bias. Essenially, his is because he convexiy of he Phillipscurve causes he risks surrounding he opimal condiional inflaion forecas o be asymmeric. However, whereas in he Schaling model he opimal inflaion forecas will be consan over ime we find ha he presence of persisen cos-push shocks will cause his forecas o be sae-dependen. As a resul, in addiion o demand uncerainy, uncerainy abou inflaionary shocks will induce a furher downward bias in he long run average rae of inflaion. This bias is also he reason why here is a social reurn o oupu sabilisaion if he Phillipscurve is convex even if he cenral bank canno affec oupu a all. This is because he long run average rae of inflaion urns ou o be sricly increasing in he cenral bank s relaive weigh on oupu sabilisaion. As a resul, he deflaionary bias can be made less severe and welfare can be improved by he cenral bank s fuile aemps o sabilise oupu.

10 - 3 - THE OPTIMAL DEGREE OF FLEXIBILITY WITH AND WITHOUT PRIVATE INFORMATION: THE LINEAR CASE In his secion we presen an exension of one of he models sudied by Svensson (1997a. The aggregae supply relaionship is given by he expecaions-augmened Phillips-curve: π (1 e π y µ This equaion can be seen as he inverse of he Lucas supply funcion. Consequenly, wheher or no moneary policy can affec oupu in he shor-run depends crucially on he cenral bank s abiliy o generae a surprise inflaion. In his model, where we absrac from uncerainy on he par of he public concerning cenral bank preferences, his boils down o he quesion wheher or no he cenral bank has privae informaion abou he realisaion of he cos-push shock (µ. The laer follows a serially correlaed and saionary process where he innovaion o he cos-push shock (ν follows an i.i.d. normal wih mean zero and variance σ ν : µ µ 1 ν ( A his poin i is useful o make a disincion beween he naural rae of oupu and poenial oupu. As argued by Cukierman (000, hese wo conceps are ofen used inerchangeably bu are no necessarily idenical. The naural rae of oupu can be defined as he oupu level ha will prevail in he absence of inflaionary surprises. In his model he naural rae of oupu will herefore vary wih he realisaion of he cos-push shock. In paricular, a posiive cos-push shock will, in he absence of inflaionary surprises, induce a decrease in he shor-erm naural rae ha corresponds o a supply side induced decrease in he oupu gap.1 By conras, poenial oupu corresponds o he long-run capaciy 1 Noe ha equaion (1 can be rewrien as follows: π π e ( (y - y n (y n -y * µ where y n denoes he shor-run naural rae and y * denoes poenial oupu (he laer is implicily normalised o zero hroughou his chaper. If here is no surprise inflaion (i.e. if π π e acual oupu (y will by definiion be equal o he shor-erm naural rae which implies: y n - µ /. The oupu gap is defined as he difference beween acual oupu (y and poenial oupu (y *. Hence, acual oupu will be deermined by a combinaion of demand and supply (i.e. cos-push facors. However, since his is essenially a flex-price model (a leas as far as prices on he goods marke are concerned, demand facors will only be able o influence oupu (i.e. o generae a deviaion of y from y n o he exen ha hey will induce a deviaion of he acual rae of inflaion (π from he rae of inflaion expeced by he public (π e.

11 - 4 - of he economy o produce goods and services as deermined by facors such as he capial sock, he labour force and he sae of echnology. In oher words, his concep perains o he level of oupu ha will prevail in he absence of inflaionary surprises and ransiory cos-push shocks. In his model, poenial oupu is ime-invarian and normalised o zero. Furhermore, since he Phillipscurve is linear and since we assume ha cos-push shocks follow a saionary process, here will be no sysemaic deviaion of he naural rae of oupu from poenial. Nex, π e denoes he raional expecaion of inflaion in period condiional on all informaion available o he public a he end of period -1(i.e. π e E -1 π. Throughou we assume ha he public does no know he realisaion of he supply shock bu, due o he fac ha his shock is serially correlaed, can predic is realisaion parially. As far as he cenral bank s informaion se is concerned we will disinguish beween wo cases. In he firs case he cenral bank has an informaion advanage in he sense ha i knows he realisaion of he innovaion o he supply shock (ν when seing moneary policy. In he second case, which will serve as a benchmark for he analysis in Secion 3, we analyse he siuaion where here is no informaion asymmery. Aggregae demand (y is given by he following equaion 3: y ( π (3 i e Firs of all, demand is decreasing in he real ineres rae (i -π e. For analyical convenience we assume ha aggregae demand is influenced by he same expeced rae of inflaion ha feaures in he Phillipscurve. Nex, demand is also influenced by a demand shock ( which follows an i.i.d. normal disribuion wih mean zero and variance σ. To approximae he effec of lags in moneary policy we assume his shock is unobservable o boh he cenral bank and he public when hey respecively se he nominal ineres rae and he expeced rae of inflaion. According o Cukierman (000 he poenial level of oupu is he level of oupu ha will prevail in he absence of real business cycle effecs. In our model his real business cycle effec corresponds o he cos-push shock since his shock will generae supply side induced variaions in acual oupu. 3 Since he demand shock ( is by assumpion no incorporaed ino inflaionary expecaions, acual oupu will essenially always be demand deermined. Moreover, poenial oupu is fixed and normalised o zero. Hence hroughou his chaper he variable y will denoe hree conceps: aggregae demand, acual oupu and he oupu gap.

12 - 5 - The cenral bank s objecive is o sabilise inflaion around he arge assigned by he governmen and o sabilise oupu around poenial where he laer is normalised o zero 4: L j 1 (1 δ E δ π j y j (4 j.1 Cenral Bank has Privae Informaion Since we assume ha he cenral bank canno commi and since privae agens canno disill any addiional informaion from pas policy acions, he cenral bank s problem boils down o minimising he period loss funcion subjec o he Phillipscurve consrain. This yields he familiar firs order condiion: E y E π (5 1 1 To obain soluions for inflaion and oupu in he case where he cenral banker has an informaion advanage we ake expecaions condiional on he cenral bank s informaion se in period -1 across he Phillips curve equaion (5.1 and subsequenly subsiue he expression obained for E -1 π ino equaion (5.5 which yields he following expression for he cenral bank s opimal expeced value of he oupu gap: e [ π ] E µ 1 y (6 4 This assumpion is made in mos of he lieraure. Alernaively, we could assume he cenral bank ries o sabilise oupu around he shor-erm or long-erm naural rae. The laer will in he linear model be equal o poenial oupu. However, if he Phillipscurve is convex, he long-run naural rae will be below poenial oupu bu will sill be consan. For an analysis of a sae-coningen oupu arge which is equal o he shor-erm naural rae see Svensson (1997a.

13 - 6 - Plugging his expression ino he Phillipscurve equaion (5.1, aking raional expecaions across he resuling expression based on he public s informaion se in period -1 (i.e. excluding ν we obain he equilibrium expeced rae of inflaion 5: e π µ 1 (7 Since he supply shock is o some exen predicable o he public, par of he cenral bank s reacion o i is anicipaed and incorporaed ino inflaionary expecaions. However, since he cenral bank follows a discreionary policy i will no ake he effec of is sabilisaion effors on expeced inflaion ino accoun. In oher words, he cenral bank is faced wih a ime-inconsisency problem. Given he fac ha is effors o sabilise he effec of he anicipaed par of he cos-push shock (µ -1 on oupu will be fuile, i would be beer if he cenral bank could commi o no reacing o his par of he cospush shock. However, if he public were o believe his, he cenral bank would have an incenive o chea. Using equaion (6 and (7 in he aggregae demand equaion (3 yields he opimal ineres rae in period : i µ ν 1 (8 ϕ( The effec of he cenral bank s relaive weigh on oupu sabilisaion ( on he nominal ineres rae will be ambiguous. On he one hand, due is effec on inflaionary expecaions, a larger relaive concern for oupu sabilisaion will induce a sronger reacion o he realisaion of he supply shock in period -1 (µ -1. On he oher hand, he cenral bank s reacion o he innovaion of he supply shock (ν will be less srong if i aaches more weigh o oupu sabilisaion. Nex, subsiuing he expeced rae of inflaion back ino equaion (1 and realising ha y E -1 y we can compue he equilibrium oupu gap in period : 5 In he erminology of Svensson (1997a his equaion represens a sae-coningen inflaion bias.

14 - 7 - y µ 1 ν (9 The firs erm on he RHS of his equaion is simply a reflecion of he fac ha a posiive realisaion of he cos-push shock (or equivalenly a negaive supply shock which is fully anicipaed by he public will decrease he naural rae in period. The second erm on he RHS shows ha he effec of he innovaion o he supply shock on he shor-erm naural rae can be diminished by he cenral bank s sabilisaion effors by virue of he fac ν will no be incorporaed ino inflaionary expecaions. Finally, he hird erm reflecs he cenral bank s imperfec conrol over he oupu gap. Nex, by insering equaions (7 and (9 ino he Phillips-curve we obain he following equilibrium soluion for inflaion in period : π µ ν 1 (10 The firs expression on he RHS is equal o he expeced rae of inflaion. Obviously, he cenral bank would do beer no o reac o he par of he supply shock ha is also known o he public bu canno commi o doing so. The second par on he RHS represens he effec of he innovaion o he supply shock where we can see ha he degree o which his shock will feed hrough ino inflaion will be an increasing funcion of he cenral bank s relaive weigh on oupu sabilisaion (. Finally, he las expression on he RHS denoes he effec of he unexpeced demand shock on inflaion. The fac ha he cenral bank ries o sabilise he effec of he supply shock on oupu bu is parly prevened from doing so due o he fac ha he supply shock is parially predicable begs he quesion as o wha exen he cenral bank should seek o sabilise oupu in he firs place. To evaluae his quesion we need he uncondiional variances of inflaion and oupu which are equal o: Var ( π 4 (1 ( σ ν σ (11 Var ( y σ σ (1 ( ν

15 - 8 - The opimal value of can now be obained by insering he expressions in equaion (11 ino he uncondiional expeced value of he social loss funcion: [ ] ( L Var( π ( E( π ξ Var( y ( E( y E (1 Here he parameer ξ denoes he socially opimal relaive weigh on oupu sabilisaion. Minimising equaion (1 wih respec o we can derive he following proposiion: PROPOSITION 1: If he cenral bank is faced wih a linear expecaions-augmened Phillipscurve, if i has privae informaion concerning cos-push shocks and if he public can predic hese shocks parially, he opimal degree of flexibiliy will be sricly posiive bu lower han sociey s preferred degree of flexibiliy (i.e. i will be opimal o appoin a conservaive cenral banker for whom i holds ha: 0 < * < ξ. The cenral banker s opimal relaive weigh on oupu sabilisaion (* will increase if: 1 he slope of he Phillipscurve ( increases; he persisence of supply shocks ( decreases; 3 sociey s relaive weigh on oupu sabilisaion (ξ increases. Proof: see Appendix A Essenially, as in Svensson (1997a, he fac ha naural rae movemens are parially predicable o he public will induce a cenral banker who inheris sociey s preferences o induce a sabilisaion bias. This is because he cenral banker s fuile aemps o sabilise he effec of hese predicable movemens on oupu will only add o inflaion variabiliy. Therefore, welfare can be improved by appoining a conservaive cenral banker. The second par of his proposiion deals wih he deerminans of he opimal degree of flexibiliy. These deerminans have been examined by e.g. Eijffinger and Hoeberichs (1998 in he conex of a model in which he cenral banker s oupu arge exceeds he long-run naural rae. In ha case, here will be a rade-off beween credibiliy and flexibiliy. In oher words, by appoining a more

16 - 9 - conservaive (i.e. less flexible cenral banker, sociey will gain since his will lower he firs momen of he uncondiional disribuion of inflaion owards sociey s bliss poin. However, his will come a he cos of an increase in he second momen of he uncondiional disribuion of oupu which will be subopimal since oupu will no longer be sabilised in accordance wih social preferences. We emphasise ha in our model sociey is concerned wih a rade-off beween he second momens of boh disribuions. The comparison is summarised in Table 1. Table 1 Effec of Model Parameers on he Opimal Degree of Flexibiliy a Effec on Opimal Degree of Flexibiliy wih Oupu Targe Exceeding Long Run Naural Rae (see Eijffinger and Hoeberichs (1998 Effec on Opimal Degree of Flexibliy wih Oupu Targe equal o Long Run Naural Rae (his model a if ξ relaively high - if ξ relaively low r 0 - x - s n - 0 a Eijffinger and Hoeberichs (1998 absrac from persisence in supply shocks. However, based on he resuls obained by Svensson (1997a, i can be argued ha in he model wih an ambiious oupu arge he opimal degree of flexibiliy will be decreasing in he persisence of supply shocks as well. Firs of all, he parameer can be seen as he inverse of he slope of he Lucas supply funcion6. In he model wih an ambiious oupu arge, a decrease in would increase he effec of any given surprise inflaion on oupu supply. Eijffinger and Hoeberichs (1998 show ha his will increase he empaion for he cenral banker o generae a surprise inflaion if sociey already aaches a relaively high relaive weigh o oupu sabilisaion. Since his will aggravae sociey s credibiliy problem (i.e. i will increase he inflaionary bias he cenral bank needs o be more conservaive. However, in he model analysed in his paper here is no sysemaic empaion o spring surprise inflaion on he public. 6 e Noe ha he Lucas supply funcion corresponding o equaion (1 would be: y ( π π µ 1.

17 Consequenly, here is no long run inflaionary bias bu insead here will be excessive variabiliy in he expeced rae of inflaion. As can be seen from equaion (7 an increase in he slope of he Phillipscurve will diminish ha paricular source of inflaion variabiliy. As a resul here is more room for he cenral banker o sabilise he effec of supply shocks on oupu. Nex, an increase in he persisence of supply shocks ( will increase he volailiy of inflaionary expecaions and herefore also he variabiliy of inflaion caused by he cenral bank s fuile aemps o sabilise he effec of he predicable par of supply shocks on oupu. These aemps were earlier idenified o be he cause of he fac ha sociey gains by appoining a conservaive cenral banker. Hence, if he effec of he cenral bank s reacion o predicable supply shocks on he variance of inflaion increases, i will be opimal from sociey s poin of view o pay less aenion o oupu sabilisaion. Third, if sociey shows an increased concern for oupu sabilisaion (ξ in he model wih an ambiious oupu arge, a policymaker sharing sociey s preferences will produce a higher inflaionary bias. Hence, sociey s credibiliy problem will become more severe and his needs o be couneraced by an increase in he degree of conservaism. In he presen model, he effec of an increase in ξ is precisely he reverse. The reason is ha here is no empaion whasoever o push oupu above he long run naural rae sysemaically. The only relevan hing a sake here is he rade-off beween inflaion and oupu variabiliy. Hence, if sociey displays an increased dislike of oupu variabiliy relaive o inflaion variabiliy, his will spill over ino a decrease in he opimal degree of conservaism. Finally, an increase in he variance of supply shocks (σ ν will reduce he opimal degree of conservaism in he model wih an ambiious oupu arge because i aggravaes sociey s flexibiliy problem relaive o is credibiliy problem. In our model he variance of supply shocks drops ou of he firs-order condiion, and has consequenly no effec on *. Apparenly, given ha he degree of conservaism is opimal o sar wih, an increase in σ ν will no aler he balance beween he marginal benefi of an increase in he degree of conservaism (semming from lower inflaion variabiliy and he marginal cos of such an increase (semming from higher oupu variabiliy.

18 Cenral Bank has no Privae Informaion In he nex secion where we analyse he case of a non-linear Phillipscurve we will drop he assumpion ha he cenral bank has privae informaion in order o be able o obain analyical resuls. As a benchmark i is herefore insrucive o analyse he oucome in he linear model where he cenral bank has no privae informaion. In paricular we will assume ha he realisaion of he innovaion o he supply shock is no known o he cenral bank when seing moneary policy. Since he compuaions are in he same spiri as in he privae informaion case we will resric ourselves o presening he oucomes: π e µ 1 π µ 1 ν y µ 1 i µ 1 Var ( π 4 (1 1 σ ν σ Var ( y (1 σ ν σ (13 Examining he expressions for he variance of inflaion and oupu, we obain he following proposiion: PROPOSITION : If he cenral bank is faced wih a linear-expecaions augmened Phillipscurve, if here is no informaion asymmery and if boh he cenral bank and he public can parially predic supply shocks, i will be opimal o appoin an ulra-conservaive cenral banker who only cares abou inflaion sabilisaion (i.e. * 0 even if he cenral bank has an oupu arge which is equal o poenial oupu.

19 - 1 - The proof of his proposiion is relaively sraighforward since i can easily be seen ha he variance of inflaion is sricly increasing in he cenral bank s relaive weigh on oupu sabilisaion while he variance of inflaion is no affeced by. Obviously, minimising he social loss funcion (1 hen boils down o minimising he variance of inflaion wih respec o. If he cenral bank canno affec oupu while is aemps o do so will affec (expeced inflaion (and hence inflaion variabiliy, i will be opimal o absain from oupu sabilisaion enirely. In ha case, he only hing he cenral bank will do is o offse he impac of he predicable par of he supply shock on inflaion compleely by seing he ineres rae such ha he expeced value of aggregae demand equals he expeced shor-erm naural rae (-(µ -1 / a he poin where he inflaion forecas is equal o he arge. However, i should be noed ha he opimaliy of an ulra-conservaive cenral banker in his case criically depends on he fac ha he cos-push shock is parially predicable o he public. If his is no he case (i.e. if 0, he cenral bank s relaive weigh on oupu sabilisaion will become irrelevan in he sense ha is will no affec he variance of inflaion and oupu.

20 OPTIMAL MONETARY POLICY WITH A NON-LINEAR EXPECTATIONS-AUGMENTED PHILLIPSCURVE In his secion we will analyse he opimal degree of oupu sabilisaion when he cenral bank is faced wih a non-linear Phillipscurve. In paricular, we assume ha for given values of expeced inflaion and he supply shock, he inflaion-oupu rade-off (f(y becomes more unfavourable as he oupu gap increases and, secondly, ha an oupu gap which is equal o zero has no effec on inflaion (i.e. we assume f (y > 0, f (y > 0 and f(0 0. To derive some analyical resuls we will use he following funcional form which can be regarded as a local approximaion o any arbirary convex shor-erm inflaion-oupu rade-off 7: π π e e χy χ 1 µ (14 Here he cos-push shock is given by equaion (. The parameer χ indexes he curvaure of he inflaion-oupu rade-off. In paricular, his funcion has he desirable properies ha he curvaure does no depend on he oupu gap and ha we obain he linear specificaion as χ approaches zero: ( y χy '' e 1 y and χ 0 ' χ f ( y lim f χ (15 To ge some feel for he difference in he inflaion-oupu rade-off induced by he non-linear specificaion we have drawn equaions (1 and (14 in a diagram for 0.3 and χ 50 8: 7 Bean (000 uses his formulaion in empirical ess of he nonlineariy of he Phillipscurve. 8 These are he values esimaed by Bean (000 for he Unied Kingdom.

21 Figure 1 Linear and Non-Linear Inflaion-Oupu Trade-Off 3.1 Opimal Policy Again, he expeced value of he oupu gap condiional on he cenral bank s informaion se in period -1 (E -1 y can be seen as an indirec conrol variable. The firs-order condiion for he cenral bank s problem hen becomes: E χy [ π γ y ] where γ e 1 0 (16 y π Here we define γ as he shor-run inflaion oupu rade-off which, in conras o he cerainy equivalen case, depends on he oupu gap and which, from he poin of view of he cenral bank, is a sochasic variable. Solving for he expecaions operaor (see Appendix B we obain he cenral bank s opimal expeced value of he oupu gap as an implici funcion of expeced inflaion:

22 E 1γ χ χe ( χ 1 1 1π σ 1γ y E y E where E e (1 σ (17 In essence he cenral bank sill pursues a leaning agains he wind policy by conracing demand if he condiional inflaion forecas exceeds he arge. However, as indicaed by he second erm on he RHS, uncerainy abou he curren inflaion-oupu rade-off will cause he cenral bank o se he expeced value of he oupu gap so as o err on he side of cauion. In oher words, even if he inflaion forecas is equal o he arge (implying E -1 π 0 he cenral bank will conrac oupu. The reason is ha he risks surrounding he cenral forecas are no symmeric in he sense ha a given absolue value of he realisaion of he demand shock will be more inflaionary if i is posiive han deflaionary when i is negaive. Consequenly, he opimal expeced value of he oupu gap will be lower han in he cerainy-equivalen case (see also Schaling (1998 and Bean (000. Obviously, he cenral bank will be more cauious if he degree of uncerainy (as measured by he variance of demand shocks increases and if he curvaure of he inflaion-oupu rade-off increases. An increase in he relaive weigh on oupu sabilisaion ( will diminish he degree of cauion. In order o obain closed form soluions for E -1 y and E -1 π we use he Phillipscurve relaionship (14 and ake raional expecaions on boh sides of he equaion. From his i follows ha he condiional expeced value of he oupu gap is resriced o be equal o (see Appendix B 9: 1 ( χ E 1 y ln ( 1 χµ 1 ln 1 σ (18 χ This equaion represens he shor-erm expeced naural level of oupu and acs as an (expeced shor run supply consrain for moneary policy. This is because he cenral bank canno spring a surprise inflaion on he public 10. Therefore, he predicable par of he supply shock and he variance of demand shocks will deermine E-1y. Since he Phillipscurve is convex, a given posiive realisaion of lim E µ χ 0 9 Here i holds ha 1y 1, i.e. as χ approaches zero he model will collapse ino he linear symmeric informaion model (see equaion ( Of course, he acual level of oupu will be influenced by he demand shock (. However, he cenral bank does no know he realisaion of his shock when seing moneary policy.

23 he supply shock will decrease oupu by a larger amoun han a negaive realisaion of equal magniude will increase oupu. Nex, demand uncerainy will cause he risks surrounding he cenral inflaion forecas o be asymmeric because of which he exisence of a sable long run rae of inflaion will only be compaible wih an oupu gap which falls below poenial on average. Equaion (18 also allows us o pin down he value of he expeced inflaion-oupu rade-off. In Appendix B his is shown o be equal o: E π ( χµ y E γ (19 This equaion shows ha he fac ha cos-push shocks are parially predicable will cause he expeced slope of he Phillipscurve o be sae-coningen. In paricular, he condiional expeced inflaion-oupu rade-off will be more favourable (in he sense ha inflaion will be less sensiive o he oupu gap in he presence of posiive cos-push shocks. This is because he laer will induce a conracion in he expeced naural rae because of which he cenral bank expecs he economy o be on a flaer par of he Phillipscurve. Plugging equaion (19 ino equaion (17 and equaing he laer o he expeced naural rae given by equaion (18 yields he following soluion for he condiional expeced rae of inflaion 11: E 1 π χ ( χ ln 1 σ ln ( 1 χµ ( 1 χµ 1 1 χ σ (0 This equaion can be seen as he opimal condiional inflaion forecas. Firs of all, oupu gap uncerainy (capured by σ will have wo opposing effecs on his forecas. On he one hand, an lim E π µ χ 0 11 Where i holds ha 1 1 (see equaion (13.

24 increase in σ will induce he cenral bank o conrac demand because of he asymmeric risks surrounding he condiional inflaion forecas. This is capured by he las erm on he RHS of equaion (0. On he oher hand, an increase in oupu gap uncerainy will induce a decrease in he shor-erm naural rae. As in he case of a linear Phillipscurve, he fac ha he cenral bank seeks o sabilise oupu around poenial will induce i ceeris paribus o implemen a more expansionary policy o offse his effec. However, since his is fully anicipaed by he public he only resul will be an increase in he expeced rae of inflaion. The second ineresing feaure abou equaion (0 is ha even under sric inflaion argeing (0 opimal moneary policy will resul in a sae-coningen opimal condiional inflaion forecas. This resul sands in marked conras o earlier analyses of opimal moneary policy in he case where he cenral bank only cares abou sabilising inflaion around he assigned arge. In he case of a linear Phillipscurve, he opimal condiional inflaion forecas will hen simply be equal o he inflaion arge (see Svensson (1997b. If he Phillipscurve is convex and if here is no persisence in he process driving cos-push shocks, he opimal condiional inflaion forecas will also be consan over ime bu will be lower han he arge because of he asymmeric risks surrounding i (see Schaling (1998. In essence his is because he expeced slope of he Phillipscurve will be consan in ha case. This can be seen by seing equal o zero in equaion (0. In his model he expeced inflaion-oupu rade-off is a funcion of las period s cos-push shock. For insance, a posiive realisaion of his shock will reduce he expeced slope of he Phillipscurve resuling in an increase in he opimal condiional inflaion forecas. This is because he cenral banker now expecs demand shocks o have a smaller effec on inflaion. Because of his he need o hedge agains he asymmeric risks surrounding he cenral forecas will diminish. Equaions (18 and (0 also allow us o compue he opimal ineres rae in period 1: i 1 χ χ (1 χµ 1 ( χ ln( 1 χ σ σ ln(1 χµ 1 (1 (1 χµ 1 lim i 0 µ χ 1 Where 1 (see equaion (13.

25 Inuiively, a non-linear Phillipscurve will resul in a non-linear policy rule in which he reacion coefficiens o he deerminans of inflaion are no longer consan bu raher a funcion of hese deerminans hemselves. 3. The Opimal Degree of Flexibiliy Nex, we can compue he uncondiional expecaion of inflaion by applying he law of ieraed expecaions and aking a nd order Taylor expansion around E(µ -1 0 in equaion (0: E ( π χ χ σ ν ln 1 3 χ ( χ 4 σ ln 1 (1 σ χ σ ( χ σ ( One of he ineresing implicaions of equaion ( is ha he cenral bank s degree of flexibiliy ( will affec he firs momen of he long run (i.e. uncondiional disribuion of inflaion. I has long been recognised ha his preference parameer could be one of he deerminans of he well-esablished posiive correlaion beween he uncondiional mean and variance of inflaion (see e.g. Cukierman (199. However, heoreical explanaions for his usually rely criically on he assumpion ha he cenral bank has an ambiious oupu arge. As shown by he analysis in Secion, if his assumpion is dropped he degree of flexibiliy will cease o influence he uncondiional expeced rae of inflaion. Neverheless, as indicaed by equaion (, he inroducion of a non-linear Phillipscurve (or indeed, any deviaion from he cerainy equivalen framework will re-esablishes he link beween hese variables 13. In paricular, an increase in his preference parameer will also increase he long run average rae of inflaion (E(π. The reason for his is wofold: Firs of all, absracing from he 13 Cukierman (000 shows ha an inflaionary bias will also emerge under he realisic assumpion ha he cenral bank cares more abou negaive oupu gaps relaive o posiive ones even if he cenral bank does no seek o drive oupu above he naural rae sysemaically.

26 presence of cos-push shocks, demand uncerainy (σ will cause he naural rae o sysemaically fall below poenial oupu. Since he cenral bank ries o sabilise oupu around poenial, is aemps o counerac his will resul in a ceeris paribus higher long run average expeced rae of inflaion. Nex, for a given degree of demand uncerainy, he cenral bank also ries o sabilise he effec of he predicable par of he cos-push shock on oupu. However, because of he convexiy of he Phillipscurve, cos-push shocks ha are disribued symmerically around zero will on average resul in a decrease in oupu below poenial. Hence, he cenral bank s reacion o he cos-push shock will on average cause a furher ceeris paribus increase in he average expeced rae of inflaion. I is insrucive o firs consider he case where he cenral banks absains from oupu sabilisaion enirely (i.e. 0 since his was shown o be opimal in he case where he cenral bank also does no have an informaion advanage over he public bu is insead faced wih a linear expecaionsaugmened Phillipscurve. In ha case equaion ( simplifies ino he following expression: 3 χ σν σ E ( π χ σ 0 (3 (1 From his expression we can derive: PROPOSITION 3: If he expecaions-augmened Phillipscurve is non-linear and if here is no informaion asymmery, a cenral bank which engages in sric inflaion argeing (i.e. 0 will induce a deflaionary bias which will become more severe if: 1 he variance of demand shocks (σ increases; he variance of supply shocks (σ ν increases; 3 he curvaure indexaion parameer (χ increases; 4 he average slope of he Phillipscurve ( increases; 5 he persisence of supply shocks ( increases. The proof of his proposiion follows immediaely from equaion (3. The deflaionary bias in his equaion can be broken down in wo pars. The firs par, represened by he las erm on he RHS, reflecs he bias ha would obain if, due o he absence of persisence in cos-push shocks, he cenral

27 - 0 - bank were faced wih a consan expeced slope of he Phillipscurve. An increase in he curvaure of he Phillipscurve (χ will cause a decrease in he long run average rae of inflaion (E(π since i enhances he skewness of risks surrounding he cenral inflaion forecas. As a resul, he cenral bank will sep up he degree o which i hedges agains hese risks by aiming for a lower rae of inflaion. Similarly, he cenral bank will also become more cauious if (for a given curvaure of he Phillipscurve he variance of demand shocks increases. The firs erm on he RHS of equaion (3 indicaes ha he sae-coningen expeced inflaion-oupu rade-off will resul in a more severe deflaionary bias han would obain in he absence of persisen cos-push shocks. In paricular, a negaive realisaion of µ -1 will cause a decrease in he opimal condiional inflaion forecas (relaive o he consan value i would have in he absence of persisen cos-push shocks, i.e. relaive o E(π 0 χ σ which will be larger han he increase resuling from a posiive realisaion of µ -1 of equal magniude. The realisaion of he expeced slope of he Phillipscurve (E -1 γ is drawn from a symmeric (normal disribuion wih mean and variance (χ σ ν. However, because of he convexiy of he Phillipscurve he effec of his on he long run average rae of inflaion will be asymmeric. This also explains why an increase in he parameers and σ ν (alongside an increase in he curvaure of he Phillipscurve will cause he deflaionary bias o become more severe since boh will serve o increase he volailiy of he expeced slope of he Phillipscurve. In he previous secion i was shown ha i will be opimal o appoin a cenral banker who only cares abou inflaion sabilisaion if boh he cenral bank and he public can predic supply shocks parially. Given our resuls sofar i is ineresing o see wheher his claim sill holds in he case of a non-linear Phillipscurve. To assess he effec of on welfare we sar by noing ha from equaions (19 and ( we can see ha neiher he mean nor he variance of oupu will be affeced by his parameer. Hence, minimising he social loss funcion (1 wih respec o boils down o minimising he Mean-Squared Error of inflaion (MSE(π E(π Var(π (E(π. Due o he non-lineariy of he Phillipscurve, he value of E(π is very difficul o obain analyically. However, he acual rae of inflaion will only differ from is condiional forecas (given by equaion (0 because of shocks o which he cenral

28 - 1 - bank canno reac (i.e. because of ν and. Therefore, we will use he MSE of he condiional inflaion forecas (E -1 (π as an approximaion 14: E [( ] E 1π χσ 1 (1 ν χσ (1 ν ( χ ln(1 ( χ σ σ σ ( χ ln( 1 ( χ ( χ ln(1 σ σ ( χ σ (4 Taking he firs derivaive of equaion (4 wih respec o and seing he resuling expression equal o zero yields 15: 4 ( χ χ σ χσν (1 χσν ln(1 σ * (5 ( χ ( χ ln( 1 σ χσν (1 χσν ln(1 σ 14 This expression can be obained by aking he square of equaion (0 and compuing he uncondiional expecaion of he resuling expression using a nd order Taylor expansion around E(µ Alernaively, welfare could also be improved by selecing an opimal combinaion of he cenral banker s relaive weigh on oupu sabilisaion and her oupu arge. Our inuiion is ha hese wo would be inversely relaed. This can be seen by adding a consan non-zero oupu arge o he cenral bank s preferred value of he oupu gap in equaion (17. In ha case i can be shown ha he opimal condiional inflaion forecas in equaion (0 will be sricly increasing in his oupu arge. Hence, for insance choosing an oupu arge which exceeds poenial will resul in an opimal value of which is lower han would be he case if he oupu arge were equal o poenial oupu. This is because he ambiious oupu arge in iself will already make he deflaionary bias less severe.

29 - - PROPOSITION 4: If he expecaions-augmened Phillipscurve is non-linear and if here is no informaion asymmery, he cenral bank s opimal degree of flexibiliy will be sricly greaer han zero (* >0. The proof of his proposiion follows immediaely from equaion (0. Proposiion 4 can be illusraed graphically by ploing he MSE of he acual rae of inflaion (i.e. (E(π agains he cenral bank s relaive weigh on oupu sabilisaion using some plausible values of he oher parameers in he model (see Figure. Laer on we will use his baseline case o assess he effec of several parameers on * graphically. The parameer values are: 0.3, χ50, σ σ ν and 0.5. The firs wo correspond o he esimaes obained by Bean (000 for he Unied Kingdom while he laer wo correspond o a sandard deviaion of demand and supply shocks equal o %. Figure Opimal Degree of Flexibiliy The figure shows ha given our choice of parameers he MSE of inflaion (E(π will be decreasing (and hence welfare will be increasing in for relaively small values of his parameer. Hence, whereas in he linear model he appoinmen of a sric inflaion argeer (or, equivalenly, ulraconservaive cenral banker was shown o be opimal, sociey will gain if i appoins an individual who aaches some weigh o oupu sabilisaion if he Phillipscurve is convex. This resul holds even

30 - 3 - hough (due o he absence of an informaional asymmery beween he cenral banker and he public he cenral banker canno sabilise oupu a all in equilibrium (i.e. even hough her relaive weigh on oupu sabilisaion will no influence he Mean Squared Error of oupu. The inuiion is ha a policy of sric inflaion argeing will induce a deflaionary bias (see Proposiion 3. As argued before, his bias can be made less severe by he cenral banker s (fuile aemps o sabilise oupu since hese will cause he uncondiional expeced rae of inflaion o increase. Hence, moving from a siuaion of sric inflaion argeing o a relaively small degree of flexible inflaion argeing will reduce he deflaionary bias and will herefore increase social welfare 16. Finally, we can invesigae he effec of changes in several parameers on * by ploing he expression obained for MSE in equaion (4 agains and each of hese parameers separaely. Implicily, he relaionship beween he opimal level of and he parameer considered is hen given by he line which races ou he minimum value of MSE in he hree-dimensional plane. These Figures are displayed in Appendix D. Firs of all hese figures show ha he effec of changes in he variance of demand shocks (σ and he variance of he innovaion in supply shocks (σ ν on * is very small. Nex, for he baseline case here seems o be a clear posiive relaionship beween he average slope of he Phillipscurve ( and he exen o which he cenral bank should seek o sabilise oupu. This confirms he inuiion obained in Secion. The same holds for he persisence of cos-push shocks (. Finally, he parameer which indexes he curvaure of he Phillipscurve (χ seems o be negaively relaed o he opimal degree of oupu sabilisaion. 16 A simulaions exercise we underook suggess ha he variance of inflaion will be sricly increasing in he degree of flexibiliy which is exacly wha one should suspec. Apparenly, for small values of he effec of an increase in E(π on welfare will ouweigh he effec of an increase in Var(π.

31 - 4-4 SUMMARY AND CONCLUSION In his paper we analyse he opimal degree of flexibiliy in a world of flexible goods marke prices where he economy s aggregae supply relaionship is given by he expecaions-augmened (Lucasype Phillipscurve. In paricular, our main quesion is o ascerain wheher or no a convex Phillipscurve of his ype would enail an addiional reurn o oupu sabilisaion ha is no presen in a linear world. Such an addiional gain will be presen if price seing is purely backward-looking since he mean level of oupu can be increased by reducing he variance of oupu in ha case (see Clark e al. (1995. However, such an addiional gain from oupu sabilisaion is no immediaely obvious wihin a Lucas-ype ransmission mechanism if he cenral bank does no have an informaion advanage over he public. As a benchmark we firs analyse a linear model in which he cenral bank ries o sabilise inflaion around he assigned arge and o sabilise oupu around poenial. Hence, we absrac from he ype of Barro-Gordon (1983 credibiliy problem ha leads o a sysemaic inflaionary bias. However, on he assumpion ha cos-push shocks are persisen over ime here will be anoher ype of credibiliy problem in moneary policy. This is because he public can parially predic he realisaion of he cospush shock. As a resul, he cenral bank s reacion o his predicable par will be incorporaed ino inflaionary expecaions. This will lead o a subopimally high variabiliy of inflaion, which could be avoided if he cenral bank were able o commi o no reacing o he predicable par of he cos-push shock. In he absence of a credible commimen mechanism social welfare can improved by appoining a cenral banker whose relaive weigh on oupu sabilisaion is lower han sociey s relaive weigh. Moreover, if he cenral bank does no have privae informaion and hence canno affec oupu a all, i will be opimal o appoin an ulra-conservaive cenral banker who only cares abou inflaion sabilisaion. Nex, we exend he symmeric informaion model wih persisen cos-push shocks by assuming ha he Phillipscurve is convex. In his respec, we find ha a policy of sric inflaion argeing (which was shown o be opimal in he linear case will lead o a sysemaic deflaionary bias. Essenially, his is because he risks surrounding he condiional inflaion forecas are asymmeric. We also show ha, in he case of flexible inflaion argeing, he uncondiional expecaion of inflaion will be sricly increasing in he cenral bank s relaive weigh on oupu sabilisaion. This effec arises because of he credibiliy problem discussed earlier. In paricular, he cenral bank ries o sabilise oupu around

32 - 5 - poenial and will herefore ry o offse he effec of uncerainy abou demand shocks and he effec of he predicable par of cos-push shocks on oupu. In he long run hese aemps will on average cause a ceeris paribus increase in he expeced rae of inflaion since policy is always correcly and fully anicipaed by he public. As for demand uncerainy he posiive relaion beween he relaive weigh on oupu sabilisaion and he uncondiional expeced rae of inflaion arises because demand uncerainy in iself will cause oupu o sysemaically fall below poenial. Furhermore, since he Phillipscurve is convex, cos-push shocks ha are disribued symmerically around zero will be ranslaed ino decreases in oupu which will on average be larger han he increases in oupu. In a way his credibiliy problem can be used o improve social welfare compared o he case of sric inflaion argeing. In oher words, in his model here is an addiional reurn o oupu sabilisaion. This does no arise because i reduces he variance and increases he mean of oupu bu because a flexible inflaion argeer will induce a less severe deflaionary bias han an ulra conservaive cenral banker.

33 - 6 - APPENDIX A PROOF OF PROPOSITION 1 Plugging he expressions obained in equaion (11 ino he social loss funcion (1, aking he firsorder condiion of he resuling expression wih respec o and rearranging we obain: ( (1 (1 ( 0 (1 ( ( ( ζ ζ Φ L E (A.1 The funcion Φ( implicily defines *. In paricular, Φ ( has he following properies: 0 ( lim 0 (1 ( ( (1 ( 3 ( (1 ( Φ < Φ Φ ζ ζ (A. Using his equaion we can deermine * graphically in Figure 3 where he downward sloping line represens he funcion Φ(:

34 - 7 - Figure 3 The opimal value of in he linear model From his figure we can see ha proving ha * < ξ amouns o proving ha Φ (ξ < ξ: 6 6 ξ 1 (1 ( ( 3 < ξ ( ξ 3 ξ > 0 (A.3 Q.E.D. Finally, by compuing he following parial derivaives of Φ( we can deermine how he oher parameers in he model affec * 17: 17 The parameers and ξ will also influence F(0. However, since i holds ha F(0/ < 0 and F(0/ ξ > 0, hese effecs do no change he conclusion reached in Proposiion 5..

35 - 8-0 ( (1 (1 ( ( ( ( ( (1 ( 6 ( > Φ < Φ > Φ ξ ξ ξ (A.4

36 - 9 - APPENDIX B SOLVING THE FOC UNDER A NON-LINEAR PHILLIPSCURVE Solving ou for he expecaions operaor across equaion (15 yields: χy [ π, e ] E y 0 χy E 1 π E 1( e Cov 1 1 (B.1 As far as he firs expression on he LHS is concerned we use he following second order Taylorexpansion: χy χ ( χ E 1y E 1 ( e e (1 σ (B. The second expression on he LHS of equaion (B.1 can be approximaed as follows: χy χy 3 [ π e ] Var [ e ] χ σ Cov 1, 1 (B.3 χ Here we have used he fac ha e χy 1χy. Plugging equaions (B. and (B.3 back ino equaion (B.1 and rearranging we obain equaion (16.