The Nonlinear Phillips Curve and Inflation Forecast Targeting - Symmetric Versus Asymmetric Monetary Policy Rules Schaling, E.

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1 Tlburg Unversy The Nonlnear Phllps Curve and Inflaon Forecas Targeng - Symmerc Versus Asymmerc Moneary Polcy Rules Schalng, E. Publcaon dae: 998 Lnk o publcaon Caon for publshed verson (APA: Schalng, E. (998. The Nonlnear Phllps Curve and Inflaon Forecas Targeng - Symmerc Versus Asymmerc Moneary Polcy Rules. (CenER Dscusson Paper; Vol Tlburg: Macroeconomcs. General rghs Copyrgh and moral rghs for he publcaons made accessble n he publc poral are reaned by he auhors andor oher copyrgh owners and s a condon of accessng publcaons ha users recognse and abde by he legal requremens assocaed wh hese rghs. Users may download and prn one copy of any publcaon from he publc poral for he purpose of prvae sudy or research You may no furher dsrbue he maeral or use for any prof-makng acvy or commercal gan You may freely dsrbue he URL denfyng he publcaon n he publc poral Take down polcy If you beleve ha hs documen breaches copyrgh, please conac us provdng deals, and we wll remove access o he work mmedaely and nvesgae your clam. Download dae: 4. jul. 05

2 The Nonlnear Phllps Curve and Inflaon Forecas Targeng -Symmerc versus Asymmerc Moneary Polcy Rules Erc Schalng Deparmen of Economcs, RAU, PO Box 54, 006 Auckland Park, Johannesburg, Republc of Souh Afrca December 998 Absrac We exend he Svensson (997a nflaon forecas argeng framework wh a convex Phllps curve. We derve an asymmerc arge rule, ha mples a hgher level of nomnal neres raes han he Svensson (997a forward lookng verson of he reacon funcon popularsed by Taylor (993. Exendng he analyss wh uncerany abou he oupu gap, we fnd ha uncerany nduces a furher upward bas n nomnal neres raes. Thus, he mplcaons of uncerany for opmal polcy are he oppose of sandard mulpler uncerany analyss. Keywords: nflaon arges, nonlneares, asymmeres, sochasc conrol JEL Codes: E3, E4, E5, E58 Correspondence o: Professor Erc Schalng, Deparmen of Economcs, RAU, PO Box 54, 006 Auckland Park, Johannesburg, Republc of Souh Afrca. Phone ; fax ; e-mal ESC@EB.RAU.ac.za.

3 Inroducon ( The 990s saw he nroducon of explc nflaon arges for moneary polcy n a number of counres vz. New Zealand, Canada, he Uned Kngdom, Sweden, Fnland and Span. Inflaon argeng has been nroduced as a way of furher reducng nflaon and o nfluence marke expecaons, afer dsapponmen wh moneary argeng (New Zealand and Canada or fxed exchange raes (Uned Kngdom, Sweden and Fnland. The relaon beween nflaon arges and cenral bank preferences has been horoughly nvesgaed. On he one hand here s a heorecal leraure Walsh (995, Svensson (997 ha concludes ha nflaon arges can be used as a way of overcomng credbly problems because hey can mmck opmal performance ncenve conracs. 3 On he oher hand here s an emprcal leraure ha looks wheher nflaon arges have been nsrumenal n reducng he polcy-mpled shor-erm rend rae of nflaon [Lederman and Svensson (995. Broadly speakng he evdence s ha nflaon arges have ndeed brough abou a change n polcy maker s nflaon preferences. Unlke he relaon beween nflaon arges and cenral bank preferences, a relavely underexplored ssue s how o ranslae nflaon arges no shor-erm neres raes. Ths s he ssue of how o map explc arges for moneary polcy no moneary polcy nsrumens, or how o mplemen an nflaon argeng framework. An excepon s a recen, and mporan conrbuon by Svensson (997a. Svensson shows ha - because of lags n he ransmsson process of shor-erm neres raes o nflaon - nflaon argeng mples nflaon forecas argeng. In hs analyss he cenral bank s forecas becomes an explc nermedae arge and s opmal reacon funcon has he same form as he Taylor rule ( Recenly, Clarda, Gal and Gerler (997b have shown ha hs ype of reacon funcon does que a good job of characersng moneary polcy for he G3. The knd of rule ha emerges s wha hey call sof-heared nflaon argeng. In response o a rse n expeced nflaon relave o arge, each cenral bank rases nomnal neres raes suffcenly enough o push up real raes, bu here s a modes pure sablsaon componen o each rule as well. (An earler verson of hs paper was wren whle Schalng was an Economs a he Moneary Assessmen and Sraegy Dvson of he Bank of England. However, he vews are hose of he auhor and no necessarly hose of he Bank of England. The auhor s graeful for helpful commens by Marco Hoeberchs, Allson Suar, Tony Yaes, Andy Haladane, Mke Joyce, Douglas Laxon, Lavan Mahadeva, Peer Wesaway, Jagj Chadha, Paul Tucker, Alsar Mlne, Peer Pauly and Semnar Parcpans a he Bank of England, he Souh Afrcan Reserve Bank, CenER, RAU, he Unversy of he Wwaersrand and aendans a he hrd Economercs Conference a he Unversy of Preora. Bruce Devle and Marn Cleaves helped prepare he paper. 3 Ths leraure s surveyed n Schalng (995. Also by ncreasng he accounably of moneary polcy nflaon argeng may reduce he nflaon bas of dscreonary polcy. See Svensson (997 and Nolan and Schalng ( For an neresng recen sudy of he Taylor rule n a UK conex see Suar (996.

4 3 Also, he 990s have seen he developmen of he leraure on he so-called nonlnear Phllps curve. [Chada, Masson and Meredh (99, Laxon, Meredh and Rose (995, Clark, Laxon and Rose (995,996, and Bean (996. Ths recen leraure pus he me-honoured nflaon oupu radeoff debae n a fresh perspecve by allowng for convexes n he ransmsson mechansm beween he oupu gap and nflaon. More specfcally, accordng o hs leraure posve devaons of aggregae demand from poenal (he case of an upswng or boom are more nflaonary han negave devaons (downswngs are dsnflaonary. 5 Ths paper marres boh srands of leraure. We exend he Svensson (997a nflaon forecas argeng framework wh a convex Phllps curve. Usng opmal conrol echnques we derve an asymmerc polcy rule, ha mples hgher nomnal neres raes han he Svensson (997a forward lookng verson of he reacon funcon popularsed by Taylor (993. Ths means ha - f he economy s characersed by asymmeres - he Svensson (997a lnear arge rule may underesmae he correc level of neres raes. The paper s organsed no four remanng secons followed by one appendx. In Secon we presen he model. In Secon 3 we presen he asymmerc polcy rule n he deermnsc case. In Secon 4 we exend he analyss wh uncerany abou he oupu gap. Secon 5 confrons he mplcaons of mulplcave parameer uncerany for polcy wh hose of he classc Branard (967 analyss. Secon 6 concludes, he appendx provdes proofs behnd key resuls. A Nonlnear Phllps Curve As saed by Laxon e al (995, pp he broad accepance of he expecaons-augmened Phllps curve - and he assocaed naural rae hypohess - led o he mporan concluson ha a long-run rade-off beween acvy and nflaon dd no exs. Subsequen research on oupu-nflaon lnkages has focused on how expecaons are formed and he reasons for prce sckness ha causes real varables o respond o nomnal shocks. Almos all of hs work, however, has been predcaed on he assumpon ha he rade-off beween acvy and nflaon s lnear, ha s, he response of nflaon o a posve gap beween acual and poenal oupu s dencal o a negave gap of he same sze. Though analycally convenen, he lnear model gnores much of he hsorcal conex underlyng he orgnal spl beween classcal and Keynesan economcs: under condons of full employmen, nflaon appeared o respond srongly o demand condons, whle n deep recessons, was relavely nsensve o changes n acvy. 6 5 In addon here s he vew ha he Phllps curve s concave Sglz (997. I can be modeled by changng he sgn of ϕ n equaon (.. Obvously, all polcy concluson are reversed. 6 Indeed, as poned ou by Laxon e. al (995, he orgnal arcle by Phllps emphassed such an asymmery, wh excess demand havng had a much sronger effec n rasng nflaon han excess supply had n lowerng.

5 4 Many of he ess ha have been performed o es for nonlneary have been unnformave because he flers ha people have chosen have smply been fundamenally nconssen wh he exsence of convexy. However, when properly esed, here s some evdence for asymmeres. Laxon e al. (995 fnd ha by poolng daa from he major seven OECD counres he Phllps curve s nonlnear. Clark e al. (996 fnd ha he US nflaon-oupu rade-off s also nonlnear, usng quarerly daa from Debelle and Laxon (997 fnd ha he unemploymen-nflaon rade-off s nonlnear n he Uned Kngdom, he US and Canada. Fnally, recen research a he Bank of England [Fsher e al. (997 also fnds ha a Phllps curve ha embodes a mld asymmery s conssen wh UK daa.. Nonlnear Oupu Inflaon Dynamcs The man purpose of hs Secon s o combne a convex Phllps curve along he lnes of Laxon, Meredh, and Rose (995 wh he Svensson (997a model of nflaon argeng o allow for lags n he ransmsson process of shor-erm neres raes. Nex, we use hs model o analyse he effecs of delayng moneary polcy measures on he fuure levels of nflaon and nomnal neres raes. The funconal form we employ o represen he nonlneary n he nflaon-oupu relaonshp s α y f ( (. α ϕy where s p p s he nflaon (rae n year, p s he (log prce level, y s an endogenous varable oupu, and α > 0 and 0 ϕ < are parameers, and s he backward dfference operaor. We normalse he naural rae of oupu n he absence of uncerany o zero 7. Ths means ha y s he (log of oupu relave o poenal.e. he oupu gap. Equaon (. s graphed n Fgure.. Is relevan properes can be derved by lookng a he frs dervave of f ( - ha s, he slope of he oupu nflaon rade-off: 7 Wh uncerany he naural rae of oupu n he nonlnear model, wll always be below ha of he lnear model. See for nsance Clark e al (995. The reason s ha f oupu were mananed, on average, equal o he naural rae of he lnear model, hen he asymmery n he response of nflaon o demand shocks would make mpossble o manan nflaon a a consan nflaon arge. To see hs formally lead he Phllps curve one perod and ake expecaons a me, hs yelds E E[ α y α ϕy. In a susanable equlbrum wh a consan rae of nflaon equal o he nflaon arge, E 0. Takng accoun of Jensen s nequaly we ge 0 f ( E y ϕ f'' ( E y σ ε. Ths equaly hen (mplcly defnes E y, he average level of oupu n he presence of shocks. Wh he convexy parameer value used n hs paper (ϕ 0. 5 hs level les abou 0. percen below he correspondng level of oupu n he absence of shocks. Snce several emprcal papers - see for nsance Debelle and Laxon (997 - sugges a larger gap beween he sochasc and deermnsc equlbrum.

6 α f ' ( [ α ϕ y 5 (. Folowng Laxon, Meredh and Rose (995, pp , s useful o consder he lmng values of f ( and s dervave for some specfc values of ϕ and y ' lm f ( α ϕ 0 (.3.a lm f' (, f ( y α ϕ (.3.b Tle: Creaor: MASS- Draw V5. CreaonDae:. lm f' (, f ( 0 y ϕ (.3.c f' ( 0 α, f ( 0 0 (.3.d Equaon (.3.a shows ha, as he parameer ϕ becomes very small, he Phllps curve approaches a lnear relaonshp, hence as n Bean (996, he parameer ϕ ndexes he curvaure. Equaon (.3.b ndcaes ha he effec on nex year s nflaon rses whou bound as oupu approaches α ϕ. Hence, as n Chada, Masson and Meredh (99 - henceforh CMM - α ϕ represens an upper bound (henceforh y max beyond whch oupu canno ncrease n he shor run. Havng descrbed he Phllps curve remans o specfy he evoluon of oupu. Followng Svensson (997a, p. 5, we

7 6 assume ha oupu s serally correlaed, decreasng n he shor -erm neres rae and ncreasng n an exogenous demand shock x. y β y ( x (.4 Where 0 < β < As can be seen from equaons (. and (.4, he real base rae affecs oupu wh a one-year lag, and hence nflaon wh a wo year lag, he conrol lag n he model. 8 The exogenous varable s also serally correlaed and assumed o be subjec o a random dsurbance ε no known a me. x 9 β x ε (.5. Opmal Moneary Polcy As n Svensson (997a moneary polcy s conduced by a cenral bank wh an nflaon arge (say.5 percen per year. We nerpre nflaon argeng as mplyng ha he cenral bank s objecve n perod s o choose a sequence of curren and fuure neres raes { τ } τ such ha MnE { } τ δ τ [ ( τ (.6 where he dscoun facor δ fulflls 0 < δ < and he expecaon s condonal on he cenral bank s nformaon se, Ω, ha conans curren (predeermned oupu and nflaon, s forecas of he demand shock and s percepon of he asymmery n he economy ϕ. 0 Thus he cenral bank wshes o mnmse 8 In case of raonal expecaons w.r.. nflaon n equaon (.4, wha happens s he followng. Through he Phllps curve (. can be seen ha nflaon a me depends on he f ( funcon. Ths means ha - wh model conssen expecaons - expeced nflaon responds n a nonlnear fashon o he oupu gap as well. More specfc, a posve oupu gap wll ncrease expeced nflaon by more han a negave gap wll reduce. Of course, hs mples ha ex ane real raes now also respond asymmercally. Ths add-on effec wll hus renforce he ransmsson effecs of he asymmery of he Phllps curve. 9 I s no really necessary o specfy a dsrbuon as long as s assumed ha hs has fne suppor. Ths s necessary because by nverng he Phlps curve can be seen ha oupu wll h he consran f nflaon goes o nfny. Now, wh nflaon argeng (ha serves as a naural brake on he expanson of oupu and ( appropraely specfed fne suppor of shocks nflaon wll always be close enough o he arge o preven oupu hng he capacy consran. 0 Noe ha here he cenral bank s conducng moneary polcy from a clear forward lookng perspecve. Ths means ha - as eleganly saed by Greenspan n hs Congressonal esmony on February 'moneary polcy wll have a beer chance of conrbung o meeng he naon's macroeconomc objecves f we look forward as we ac, however ndsnc our vew of he road ahead. Thus, over he pas year [994 we have frmed polcy o head off nflaon pressures no ye evden n he daa.' Now, an neresng parallel can be drawn. If polcy akes accoun of he curvaure, (as an nformaon varable say nflaon wll be closer o he arge and smlarly oupu wll be closer o rend. Ths means ha under opmal polcy he observed (reduced form Phllps curve wll almos ceranly be eher

8 7 he expeced sum of dscouned squared fuure devaons from he nflaon arge.ths s conssen wh he UK s New Moneary Framework, where he operaonal arge for moneary polcy s an underlyng nflaon rae (measured by he -monh ncrease n he RPI excludng morgage neres paymens of.5 per cen. For smplcy we focus on he nflaon objecve and absrac from oupu sablsaon and monorng ssues. Followng Bean (996, s convenen o formulae hs opmsaon problem usng dynamc programmng. Le V( be he mnmsed expeced presen value n (.6 (he value funcon. Then: V( Mn { E[ ( δe[ V( } { } (.7 Usng (. hs can be wren as V( Mn { E[ ( δe[ V( f ( } { } (.8 subjec o (.4 and (.5. Noe ha f ϕ 0 we oban he Svensson (997a model exacly. Snce he neres rae affecs nflaon wh a wo year lag, s praccal o express n erms of year and varables. Leadng he Phllps curve one perod and subsung for oupu from (. yelds α y α y α ϕy α ϕy (.9 As n Svensson (997a he neres rae n year does no affec he nflaon rae n year and, bu only n year, 3 ec, smlarly he neres rae n year wll only affec he nflaon rae n year 3, 4 lnear or non-exsen. Thus, he more he cenral bank akes accoun of possble asymmerc (ex ane nflaon rsks because of perceved nonlneares n he nflaon oupu relaon, he less vsble hey wll be n he daa as a resul. Ths problem has been suded formally by Laxon, Rose and Tambaks (997. Svensson (997a, pp shows ha he wegh on oupu sablsaon deermnes how quckly he nflaon forecas s adjused owards he nflaon arge. Ths s he mos realsc case and also relevan for he UK suaon. The reason s ha s recognsed by he Chancellor ha sckng o he nflaon arge - n he case of exernal evens or emporary dffcules - may cause undesrable volaly n oupu. However, n he more complcaed case of mulpler uncerany, Svensson (997b also focuses on src nflaon argeng. In order o keep our (already farly complcaed analyss racable, we focus on src nflaon argeng. Moreover, hs faclaes comparson wh he Svensson (997b resuls. Svenson (997a, p. 3 saes ha: Cenral banks have a srong radon of secrecy mosly for no good reasons I beleve. For an alernave vew where cenral bank secrecy may be benefcal because of a posve effec on oupu sablsaon see Ejffnger, Hoeberchs and Schalng (997.

9 8 ec 3. Therefore we can fnd he soluon o he dynamc programmng problem by assgnng he neres rae n year o he nflaon arge for year, he neres rae n year o he nflaon arge for year 3 ec. Thus, we can fnd he opmal neres rae n year as he soluon o he smple perod by perod problem 4 Mn Eδ [ ( (.0 The frs order condon for mnmsng (.0 wh respec o s E δ L δ ( E E [ ( δ α E 0 ( α ϕ[ β y ( β x ( (. 5 where we have used ha by (.9 he effec of neres rae ncremens on expeced nflaon wo years ahead s E E E y α. f '( E y E y ( α ϕ[ β y ( β x (..a I follows ha he frs-order condon can be wren as E (. Hence, as n Svensson (997a, p. 8 he ners rae n year should be se so as ha he nflaon forecas for, he mean of nflaon condonal upon nformaon avalable n year, equals he nflaon arge. The one-o-wo-year nflaon forecas s gven by E f ( E f ( (.3 3 Ths se-up s a sylsed verson of he Bank of England s forecasng model where hree quarers of he effec of neres rae changes on nflaon occurs n wo year s me. 4 For a proof see Appendx A of Svensson (997a. 5 For analycal racably n hs Secon we do no analyse he mplcaons of uncerany abou he oupu gap. Ths makes he analyss farly complcaed, as mples solvng a nonlnear sochasc conrol problem ha excludes closed form soluons for neres raes. We analyse hs ssue n Secon 4.

10 9 The las erm s he forecas of he nflaonary presssure as mpled by nex year s oupu gap. Usng (. and (.4 hs forecas s α[ βy r x α[ βy r β x E ( ( [ f E f E y α ϕ β y r x αϕ[ βy r β x (.4 6 where r s he real base rae. Subsung (. and (.4 no (.3 and seng he one-o-wo year nflaon forecas equal o he nflaon arge leads o he cenral bank s opmal polcy rule r ( α ϕ[ βy r β x b y x r ( ( [ α ϕβ α ϕ β y α [ α ϕy β x (.5 where b ( β Accordng o hs equaon he opmal shor -erm neres rae s a nonlnear funcon of he devaon from he nflaon arge ( on he one hand, and he oupu gap ( y, on he oher. Ths s n conras o Bean (996, who ges a lnear polcy rule. Ths s due o he fac ha he employs a specfc funconal form for he nonlnear Phllps curve. 7 An mporan lmng case of (.5 s when ϕ becomes very small. In he laer case he Phllps curve approaches he sandard lnear funconal form and he polcy rule collapses o r r a ( b y (.6 8 where a, r β x α whch - as n Svensson (997, p.9 - s essenally a forward lookng verson of he smple backward lookng reacon funcon popularsed by Taylor (993. In wha follows for brevy s sake I 6 Because we absrac from he mplcaons of uncerany abou he oupu gap here s no Jensen s nequaly effec n (.4.We address hs exenson n Secon 4. 7 In fac hs specfcaon s probably he only specfcaon ha (ogeher wh sandard quadrac preferences over nflaon and oupu mples a lnear polcy rule as he soluon o he assocaed dynamc programmng problem. 8 Noe ha hs soluon does ake accoun of uncerany abou he oupu gap. The reason s ha because of cerany equvalence, he opmal conrol rajecory for he sochasc problem s dencal wh he soluon o he deermnsc problem when he error erms ake her (zero expeced values.

11 0 wll refer o (.6 as he Taylor rule. 9 The nonlnear rule (.5 wll be analysed n deal n he nex Secon. 3 A Nonlnear Polcy Rule In hs Secon we focus on he properes of he nonlnear rule. We show ha nomnal neres raes accordng o hs rule are hgher han under he Svensson (997a forward lookng verson of he Taylor rule. Ths means ha - f he economy s characersed by asymmeres - he Svensson rule may underesmae he correc level of neres raes. To recap we focus on our nal resul,.e. equaon (.5 Rearrangng and usng ha β x r we ge α { ϕ[( f ( } α f ( { ϕ[( f ( } r r ( βy (3. Equaon (3. s he cenral resul of hs Secon and shows ha he real neres-rae penaly r a nonlnear funcon of he devaon of he nflaon rae from s arge r s and he oupu gap y. In order o make progress s useful o focus on he nflaon argumen n he rule. Tha s, for he momen we se y 0 n (3.. Ths yelds α r r { ( } ( ϕ (3..a The mos neresng feaure of (3..a s ha he elascy of he neres rae penaly wh respec o devaons from he nflaon arge s sae-conngen. Meanng ha hs elascy depends on he level of nflaon. To gve a numercal example, consder he effecs of a 0.5% and a % devaon of nflaon from arge. We analyse he mplcaons of hese nflaon gaps for shor-erm neres raes under he followng parameer values α 0. 5, ϕ 0. 5 and r Then, he approprae neres rae penales are.33 and -0.80% respecvely. In he lnear case (Taylor rule we ge.00 and -.00% 9 Also, should be emphassed ha he orgnal Taylor rule s an nsrumen rule: drecly specfes he reacon funcon for he nsrumen n erms of curren nformaon. In conras a arge rule resuls n an endogenous opmal reacon funcon expressng he nsrumen as a funcon of he avalable relevan nformaon. For hs dsncon see Svensson (997a, p. 36. We call (.6 forward-lookng because - alhough neres raes feed off curren-daed varables only - he laer are leadng ndcaors of fuure nflaon. For more deals see Svensson (997a.

12 Hence he neres rae response s asymmerc; posve devaons from he nflaon arge mply hgher (absolue values of real neres rae penales han negave devaons. 0 The nuon behnd hs resul s he followng. If nflaon s above arge, shor-erm real neres raes wl be below her equlbrum level. The resul of hs s ha here are nflaonary pressures n he economy ha - f lef o her own devces - wll ncrease omorrow s oupu gap. Snce he Phllps curve s nonlnear hs posve oupu gap a me wll ncrease he nflaon rae a me by more han f he world was lnear. To offse hs he cenral bank needs o ncrease nomnal neres raes a me furher han n he Svensson model. Of course, n case of a negave devaon from he nflaon arge, he reverse s rue. Tha s, hen real neres raes are above her equlbrum level. The assocaed dsnflaonary pressures wll depress omorow s oupu gap. However, hs wll now cause less dsnflaon han n he lnear case. Hence, he cenral bank does no need o cu raes as much. Now, we focus on he oupu gap argumen; hence we look a he oppose case of he one analysed above. Seng n (3. yelds α f ( r r β { ϕf ( } y (3..b I can be shown ha (3..b has characerscs smlar o (3..a. In parcular he elascy of he neres rae wh respec o oupu depends on he level of he oupu gap. To gve a numercal example, consder he efecs of a 0.50 % and % oupu gap on he real neres rae penaly. Usng he same parameers as n he nflaon example, and seng β 0. 7, we ge.0 % and -0.75% respecvely. In he lnear case (Taylor rule we ge 0.85% and -0.85% respecvely. Thus, also wh respec o he oupu gap he neres rae response s asymmerc. Posve oupu gaps mply hgher (absolue values of real neres rae penales han negave oupu gaps. The nuon s smlar o he one of he nflaon argumen n he rule. If oupu s above rend a me, hen because of seral correlaon n oupu, omorow s oupu gap wll be hgher as well. Then, because of he asymmery he nflaon rae a me wll ncrease by more han f here were no asymmeres. In order o preven all hs from happenng, he cenral bank needs o pu up nomnal neres raes by more han acordng o he forward lookng verson of he Taylor rule. Smlarly, n 0 Noe ha applyng Svensson s dsncon beween offcal verus mplc nflaon arges - and for ease of exposon seng y 0 - s possble o reformulae he nonlnear polcy rule (3..a as a lnear response o a nonnear (sae-conngen mplc nflaon arge b. Afer some algebra can be shown ha hen (.6 can be b reformulaed as r r a( where b ϕ(. ϕ( Clarda and Gerler (997a have found ha s possble o represen Bundesbank polcy acons n erms of an neres rae reacon funcon whch maps back no a Taylor-ype rule. Ther specfcaon allows a modfed Taylor rule wh lnear responses o expeced nflaon and asymmerc responses o he oupu gap.

13 case of a negave oupu gap he danger of dsnflaon s less severe, callng for a less subsanal cu han accordng o he lnear rule. The above analyss sheds some lgh on he mechancs of our polcy rule (3.. However, hs was done by focusng on he nflaon gap, gven a zero oupu gap and vce versa. In he real world s no very lkely ha hose are he only relevan cases. Therefore now we drop hs resrcon and allow boh gaps o vary smulaneously. To ge a feel for wha happens hen consder Table 3.. Table 3. Implcaons of Polcy Rules for Shor Term Ineres Raes Inflaon mnus Targe Oupu Gap Real Rae Penaly Idem Taylor Rule Nom Ineres Rae 'Taylor' Idem Non Lnear Rule Idem wh Unc Abou Ineres Rae Bas n Bass Pons 4 'Branard' Effec n Bass Pons Rule (.6 3 (3. Oupu Gap ( ( ( ( ( ( ( ( ( (8 Ths Table maps oupu and nflaon gaps no real neres rae penales (columns 3 and 4, and no nomnal neres raes (he shaded columns 5, 6 and 7. Please noe ha hs Table s no compued by sochasc smulaons. All ha s necessary o oban he numbers n hs Table s o sar wh ceran oupu - and nflaon gaps, and plug hese no he polcy rule (3. (and (4.4 for column 7, gven he parameer values used earler. Also noe ha our prevous numercal examples are repored n rows 8 and, (for he nflaon example and rows 7 and 3 (for he oupu gap example. Noe ha whle Taylor prescrbes coeffcens of one half on boh he nflaon and oupu gaps under plausble parameer values he Svensson rule responds o nflaon and oupu gaps wh elasces of and.7 respecvely. In hs respec see Broadben (996, who fnds numbers of 5 and 3.5. Also, as poned ou by Svensson (997a, p. 33 wh a posve wegh on oupu sablsaon, he coeffcens n he opmal reacon funcon - and consequenly he numbers n he Table - wll be smaller. 3 Nomnal neres rae r ( r r r, where (. 4 Frs number (3. -- (.6. Bas due o uncerany ( ( 3. n brackes. The effecs of uncerany wll be explaned n Secon 4.

14 3 Consder frs he shaded row. Ths row corresponds wh he case of neural moneary condons. Meanng ha he economy s operang a full poenal (zero oupu gap and nflaon on course (equal o he nflaon arge. Thus boh gaps are zero and real neres raes are a her equlbrum level. Noe ha n hs case he lnear and nonlnear polcy rules mply he same level of shor-erm neres raes. However, by lookng a he oher rows n hs Table becomes mmedaely clear ha n all oher cases shor-erm neres raes are always hgher under he nonlnear rule. To see hs consder he frs se of numbers n column 8. The dfference n nomnal raes s zero for neural moneary condons bu ranges from abou 40 o 00 bass pons oherwse. Hence, he numbers sugges ha neres raes are hgher n a nonlnear han n a lnear world. In order o nvesgae hs conjecure formally consder equaon (3. r α [( f ( r α ( y (3. ϕ[( f ( NL L Ths s he algebrac equvalen of he frs se of numbers n column 8 of Table 3.. I s obaned by subracng he level of neres raes accordng o he Taylor rule r L (gven by equaon (.6 from ha under he nonlnear rule r NL (equaon (3.. From equaon (3. we conlude ha he level of shor erm neres raes as mpled by he nonlnear polcy rules s hgher han under he Taylor rule. For a proof see he Appendx A, where we show ha (3. has a local mnmum a (, y ( 0, 0. Hence, under non-neural moneary condons neres raes accordng o he nonlnear rule are hgher han under he Taylor rule. The nuon s he followng. If he Phllps curve s nonlnear, hen posve shocks o demand - n he form of posve oupu andor nflaon gaps- are more dangerous for nflaon hen f he world s symmerc. Ths means ha he cenral bank wll need o rase raes by more han n he Svensson model. Smlarly, negave gaps wll be less dsnflaonary urgng he cenral bank o cu by less. Of course he ne resul s ha nomnal neres raes are hgher on average. Noe ha here s one neresng nermedae case whch we dd no nvesgae. 5 Ths s he scenaro where he model s nonlnear, bu he polcy rule remans lnear (.e. of he form gven by (.6. Usng sochasc smulaons can hen be shown ha neres raes wll be hgher han under a lnear Phllps curve wh a lnear polcy rule. Moreover, can hen be analysed how much furher neres raes need o rse under he opmal (asymmerc polcy rule compared o he lnear rule. The level of neres raes n he nonlnear model under he nonlnear polcy rule can hen be decomposed no wo pars: ( he jump n raes caused by he change from a lnear o a nonlnear model (where he polcy rule remans lnear, and ( he furher change n raes (n he nonlnear model caused by he swch 5 I owe hs suggeson o Peer Wesaway.

15 4 from a lnear o a nonlnear polcy rule. The sochasc smulaons show ha boh he effecs under ( and ( are posve, where he effec under ( s quanavely he mos mporan Uncerany abou he Oupu Gap Now we analyse he effecs of uncerany abou he oupu gap on he seng of shor-erm neres raes. Ths means ha we now analyse random shocks o he oupu gap. Ths effec s capured n he model by he erm ε n equaon (.5. Thus, from he perspecve of he cenral bank, he nflaon rae becomes a random varable ha can only be mperfecly conrolled. 7 More specfc, because of he nonlneary of he economy, uncerany abou he rue value of nex year s oupu gap mples ha he slope of he Phllps curve - and hence he effec of neres rae ncremens on nflaon wo years ahead - also becomes a random varable. Hence, he combnaon of addve uncerany abou he economy combned wh a nonlnear srucure gves rse o ssues of mulpler or model uncerany. However, here he mplcaons for opmal polcy are que dfferen from eher he sandard Branard (967 analyss, or Svensson s (997b exenson of hs nflaon forecas argeng framework wh model uncerany. We now exend he analyss of Secon 3. As saed n Secon we can fnd he opmal base rae n year as he soluon o he problem Mn [ ( ( (.0 Eδ MnEδ L subjec o (., (.4 and (.5 We can rewre he expeced value of he dscouned loss as 8 E E Eδ L( Eδ [ ( ( δ [ Var ( E (4..a and we can defne 6 The resuls are avalable from he auhor upon reques. 7 Ths s also rue n he lnear sochasc model bu here he forecas error does no depend on he neres rae. 8 Usng E ( E.

16 5 E ( E E d,.e. he one o wo year nflaon forecas equals he deermnsc (or cerany equvalen nflaon forecas f ( f ( E y (4..b where f ( E y α[ βy α ϕ[ β r βx y r β x plus he expeced devaon E d of he one o wo year nflaon forecas from he cerany equvalen forecas: E d E E f ( f ( E y ( f ( E f ( ( f ( f ( E y (4..c Ths spl s mporan because wll enable us o denfy one of he wo channels hrough whch he uncerany affecs nflaon forecas argeng. Subsung he decomposon of he one o wo year nflaon forecas no (4..a gves δ Eδ L( [ Var ( Ed ( ( E d (4. The advanage of (4. over (.0 s ha he sochasc elemens of he soluon have been solaed n he erms E d and Var. I s precsely hrough hese wo erms ha he uncerany abou he oupu gap affecs nflaon forecas argeng. We wll now derve he polcy rule n he presence of boh asymmeres and uncerany. Because he rule s hghly nonlnear, unlke he prevous Secon s no possble o derve an explc funcon ha maps oupu and nflaon gaps no he approprae level of neres raes. Insead we resor o numercal mehods. However, we are able o derve robus qualve analycal resuls. The punch lne s ha no maer wha parameer values, nomnal neres raes wll be hgher he hgher he uncerany abou he oupu gap The frs order condon s

17 6 0 [ ( E d Var E d (4.3 9 Where he frs erm s he dfference beween he cerany equvalen nflaon forecas (4..b and he nflaon arge, he second erm s he expeced devaon of he one o wo year nflaon forecas from s cerany equvalen value (4..c, and he las erm capures he effec of nomnal neres raes on he condonal varance of nflaon, ha s on he varably or 'rsks' surroundng he cenral forecas. Subsung (4..b and (4..c no (4.3 and rearrangng leads o he cenral bank's opmal polcy rule y E d Var E d f E d Var E d E d Var E d f f E d Var E d f r r } [ ( [( { [ ( } [ ( [( { ( ( } [ ( [( { β ϕ α ϕ α ϕ α (4.4 where 0 '''( ( 0 '( 0 ''( '( 0 ''( ( < < < > ε ε ε σ ϕ σ σ ϕ y E f E d y E f y E f y E f Var y E f E d (4.5 Accordng o equaon (4.4 he opmal shor-erm neres rae s deermned by he devaon from he nflaon arge ( on he one hand and he oupu gap y (hrough he erms y β and ( f on he oher. 9 Noe ha n he deermnsc case E d Var 0 and we ge whch s he frs order condon n he cerany equvalence case as n Svensson (997a, p. 8.

18 7 An mporan lmng case of (4.4 s when σ becomes very small. In he laer case he sochasc elemens of he rule, E d rule collapses o, ε E d and Var become very small as well, and he polcy { ϕ[( α ( ( } { α ( ϕ[( ( } β (3. whch s he asymmerc polcy rule for he case where ϕ > 0, ha s he cerany equvalen rule he nonlnear model. Of course, f we se ϕ 0 n, he asymmerc cerany equvalen rule collapses o he (.6 Svensson resul. Table 4. summarses he cases dscussed above. Table 4. Classfcaon of Polcy Rules Lnear ϕ 0 Nonlnear ϕ > 0 Uncerany abou he Oupu Gap No Uncerany 0 σ ε Svensson Resul (.6 Nonlnear Cerany Equvalen Rule (3. Uncerany > 0 σ ε Svensson Resul (.6 Nonlnear Rule (4.4 Turnng o he case where boh ϕ and σ ε are posve, from equaons (4.4 and (4.5 can be seen ha he sochasc elemens of he rule E d,e and Var depend on he level of he neres rae. Thus, boh he lef-hand sde and he rgh-hand sde of equaon (4.4 depend on he neres rae. Therefore, s no possble o derve an explc funcon ha maps oupu and nflaon gaps no he approprae level of neres raes. Insead, we have o resor o numercal mehods o fnd he level of he real neres rae ha s mplcly deermned by equaon (4.4. Seng σ ε a and keepng he he real neres rae a he cerany equvalen level accordng o rule (3. we can compue he effec of he uncerany on he nflaon forecas and on he rsks surroundng he forecas. 30 Beng he MSE of ONS revsons o real GDP n he lae 980s. For more deals see Dcks (997. Obvously hs s a crude way of paramersng he model, bu n he lnear case here s a one o one correspondence beween he condonal varance of he oupu gap a me and he varance of shocksσ ε. Also, hs hghlghs anoher aracve feaure of he model. We have a naural mappng of nosy daa (whch s very much a real lfe problem no ssues of mulpler uncerany.

19 8 We fnd ha he nflaon forecas s adjused upwards. Ths forecas now overshoos he.5% arge level ha would be aaned n wo year s me wh neres raes accordng o (3. and no uncerany. Moreover, he same s rue for he condonal varance of nflaon. A he level of neres raes mpled by he cerany equvalen rule (3. we ge a varance ha goes up o 86% of he varance of he shock o he oupu gap. Ths means ha only a very small amoun of he demand shock s dampened before passes hrough and causes sgnfcan nflaon rsks. Clearly n he presence of uncerany neres raes accordng o (3. are a a subopmal level. To fnd he approprae level we numercally compue he real neres rae ha solves he frs order condon. The resuls can be found n column 7 of Table 3.. I becomes mmedaely clear ha shor-erm neres raes accordng o rule (4.4 are hgher han under he cerany equvalen nonlnear rule (3.. To see hs consder he numbers n brackes n colum 9. The dfference due o he uncerany s abou 5 bp for neural moneary condons and rages from abou 0 o 40 bp oherwse. 3 Ths means ha uncerany nduces a furher upward bas n nomnal neres raes on op of he effec of he nonlneary per se as analysed n Secon 3. In order o nvesgae hese resuls more formally consder (4.4. In hs equaon he sochasc elemens of he soluon have been solaed n he erms E d,ed and Var. Now, he sgn of E d n (4..c wll always be posve, mplyng ha he one o wo year nflaon forecas wll be hgher hen he cerany equvalen nflaon forecas as derved n Secon 3. The reason s ha posve shocks o he oupu gap are more nflaonary hen negave shocks are dsnflaonary, hence wh equal probables of posve and negave shocks, he nflaon forecas wll adjused upwards, and he more so he hgher he varance of shocks hng he oupu gap σ ε. Ths can be resaed n a more echncal way by nong ha he forecas of omorrow s nflaonary pressures, E f (, nvolves he expecaon of a convex funcon whch wll always be hgher hen he value of he f funcon a he expeced value, f ( E y. Hence, he frs channel hrough whch he uncerany affecs nflaon forecas argeng s he Jensen s nequaly effec. Noe ha from (4.5 hs effec becomes smaller he hgher he neres rae,.e. E. < 0 The second channel hrough whch he uncerany affecs nflaon forecas argeng s hrough s effecs on he condonal varance of he one o wo year nflaon forecas Var. Ths s mporan because mples ha n case of mperfec conrol of he nflaon rae he polcymaker should also ake accoun of he rsks surroundng he cenral nflaon projecon. 3 Noe ha srcly speakng he defnon of neural moneary condons needs o be changed n he nonlnear model. The reason s ha now he naural rae of oupu les below he naural rae of oupu n he lnear model. Wh he parameer values n he paper hs dfference amouns o abou -0.% of GDP. Therefore neural moneary condons now means nflaon a arge and oupu a he adjused naural rae. Indeed can be shown ha wh nflaon on arge and oupu a -0.3 he neres rae bas dsappears and he approprae level of he real neres rae (as defned by he polcy rule (4.4 s equal o r 3.8.

20 I can be shown ha hs varance s 9 Var [ f' ( E y σ ε (4.6 Now, from (4.5 can be seen ha by ncreasng neres raes hs varance can be reduced. The reason s ha by pung up raes, oday s forecas of omorrow s oupu gap goes down. Ths means ha nex year s Phllps curve wll be flaer whch n urn mples ha he effecs of demand shocks a me on nflaon n wo year s me wll be smaller. Hence, he varably of nflaon around he cenral projecon can be reduced by ncreasng shor-erm neres raes. For nsance, reurnng o our earler numercal example, by pung up raes o her approprae level he condonal varance of nflaon s reduced from 86% o abou 5% of he nal varance of demand shocks. The mplcaon for polcy s ha wh uncerany abou he oupu gap (and asymmeres n he oupu nflaon rade-off, cauous polcymakng mples a more acvs (more aggressve raher han a less acvs (more passve neres rae polcy. To recap, he nuon s ha a hgher varance of shocks hng he oupu gap mples a hgher nflaon forecas (hrough he Jensen's nequaly effec and a hgher condonal varance of nflaon. Boh can be reduced by ncreasng nomnal neres raes above her cerany equvalen level. To see he benefs of hs polcy from a dfferen perspecve, consder he mplcaons of sablsaon for he level of oupu. Wh a convex Phllps curve, he mean level of oupu s nversely relaed o he varably of nflaon around he cenral projecon. Therefore a moneary sraegy ha reduce hs varably (by respondng correcly o he mulpler uncerany ssue does no only keep he nflaon rae closer o he arge, bu also has he mporan added bonus of pushng up he level of oupu Branard Uncerany and Nonlneares Noe ha he resuls wh respec o he condonal varance of nflaon are he oppose of hose of he sandard Branard (967 mulpler uncerany analyss. 33 There uncerany abou he effecs of polcy calls for a less acvs polcy. The reason s ha - accordng o Branard's analyss - he varance of he arge varable s a lnear funcon of he varance of he polcy mulpler. Moreover, he laer s posvely relaed o he level of he nsrumen. I follows ha polces ha are oo 3 I owe hs nsgh o Clark e al (995, p. 8. They n urn quoe Mankw (988, p The resul can be verfed by nverng he Phllps curve (..Ths yelds y. Now leadng hs equaon one perod and akng α ( ϕ expecaons a me of he resulng concave funcon yelds ha mean oupu, E y, s nversely relaed o he condonal varance of nflaon Var.

21 0 acvs ncrease he varance of he arge varable hereby deerorang he performance of sablsaon polcy. In hs Secon we show ha n he nonlnear sochasc model uncerany abou he effecs of polcy calls for more a more acvs polcy. Thus, he Branard resul s reversed. In hs (967 paper Branard denfed wo ypes of uncerany ha may face a polcymaker. Frs, a he me he mus make a polcy decson he s unceran abou he mpac of he exogenous varables whch affec he goal varable. Ths may reflec he polcymaker's nably o forecas perfecly eher he value of exogenous varables or he response of he goal varable o hem. Second, he polcymaker s unceran abou he response of he goal varable o any gven polcy acon. He may have a cenral esmae of he expeced value of he response coeffcen, bu he s aware ha he acual response of he goal varable o polcy acon may dffer subsanally from he expeced value. Le us now rephrase he above n he conex of nflaon forecas argeng. To make hngs comparable wh Branard, for he momen we focus on he lnear verson ( ϕ 0 of he sochasc model presened earler. Type uncerany means ha when he cenral bank ses s nsrumen varable, he nomnal neres rae a me, s unceran abou he realsaon of he exogenous shock o he oupu gap a me. Here he cenral bank's nably o forecas nex year's oupu gap perfecly, mples ha s also unable o forecas nflaon perfecly. As a consequence nflaon n wo year's me wll dffer from s forecas a me (whch s he bass for s neres rae polcy. More specfc, f he oupu gap s hgher han expeced nflaon overshoos s arge and vce versa. The second ype of uncerany means ha he cenral bank may have a cenral esmae of he expeced value of he response coeffcen of nflaon n wo year's me wh respec o he nomnal neres rae a me, bu ha s aware ha hs cenral esmae s subjec o error. More specfc, assume ha he cenral esmae s α - beng he produc of he neres elascy of oupu (whch s and he slope of he Phllps curve (whch s α n he lnear sochasc model - and ha he varance of hs cenral esmae s σ τ. Branard shows ha boh ypes of uncerany mply ha he polcymaker canno guaranee ha he arge varable wll assume s arge value. Bu hey have que dfferen mplcaons for polcy acon. The frs ype of uncerany, f presen by self has nohng o do wh he acons of he polcymaker; s - as Branard (967, p. 43 descrbes - 'n he sysem' ndependen of any acon he akes. He hen saes ha f all of he unceranes are of hs ype, opmal polcy behavour s cerany equvalence behavour. Tha s, he polcymaker should ac on he bass of expeced values as f he were ceran hey would acually occur. Moreover, snce n hs case he varance and hgher momens of he dsrbuon of he goal varable do no depend on he polcy acon aken, he polcymaker's 33 Throughou he paper f we refer o he Branard resul, we mean Branard's resul for he one nsrumen and one

22 acons only shf he locaon of he arge varable's dsrbuon. In he presence of he second ype of uncerany, however, he shape as well as he locaon of he dsrbuon of he arge varable depends on he polcy acon. In hs case he polcymaker should ake no accoun hs nfluence on he varably of he arge varable. In hs analyss 34 Branard assumes ha he varance of he arge varable s a lnear and ncreasng funcon of he level of he polcy nsrumen. I follows ha polces ha are 'oo acvs' ncrease he varance of he arge varable hereby worsenng he performance of economc polcy. Branard hus shows ha uncerany abou he response coeffcen, ha s abou he polcy mulpler, leads o an opmal polcy ha s less acve. As he varance of he mulpler rses, he polcy of ryng o mnmse he varance of he arge varable ends owards lowerng he opmal amoun of polcy. Le us now rephrase he above n he conex of nflaon forecas argeng. An example of cerany equvalence behavour s he Svensson (.6 forward lookng polcy rule. Ths rule s opmal n he lnear sochasc model. Because shocks o he oupu gap have a zero expeced value a me, s opmal for he cenral bank o ac as f hese zero values would acually occur. An example of uncerany abou response coeffcens s Svensson s (997b exenson of hs nflaon forecas argeng framework wh mulpler uncerany. Indeed he fnds ha mulpler uncerany calls for a more gradual adjusmen of he condonal nflaon forecas oward he nflaon arge. Ths means ha - smlar o Branard - opmal moneary polcy wll be less acvs n he sense ha he response coeffcens n he opmal polcy rule for shor-erm neres raes declne wh he uncerany. 35 arge case where he random response coeffcen s uncorrelaed wh he exogenous dsurbances. 34 Here we focus on Branard's mos smple case; ha s he one nsrumen and one arge case where he random response coeffcen s uncorrelaed wh he exogenous dsurbances. The reason for dong hs s ha hs case has he closes correspondence o nflaon forecas argeng. There we also have one arge, nflaon, and one nsrumen, he nomnal neres rae. 35 The above can be derved by resorng o he lnear model (seng ϕ equal o zero and modfyng equaon (.4 as y β y τ ( x wh E ( τ E ( τ σ τ and E ( τ ε 0 (.4' Ths means ha now he effecs of neres rae changes on omorrow s oupu gap are unceran because he neres elascy of oupu s a random varable. If σ τ 0 he cenral esmae s no subjec o error and he equaon reduces o (.4. Now can be shown ha Var α ( r σ τ σ ε so ha Var > 0 and we oban he sandard Branard (967 resul. I can be shown ha he opmal (lnear polcy rule hen becomes r a b r y ( ( ( σ σ ( σ τ τ τ (.6' So as n Svensson (997b he response coeffcens declne wh he uncerany, callng for more cauous polcy makng. If σ τ 0 hs rule reduces o (.6.

23 Le us now focus on nflaon forecas argeng n he nonlnear model and relae he effecs of uncerany abou he oupu gap o he Branard paper. Here - followng Branard's ermnology - appears ha we only have ype uncerany. Tha s, because of an addve (whe nose shock o omorrow's oupu gap he cenral bank s unable o forecas nflaon perfecly. If he model were lnear, cerany equvalence would hold and ha would be he end of he sory. However n a nonlnear model hs uncerany has very dfferen mplcaons. Smlar o he lnear model he uncerany eners he sory hrough addve shocks o he oupu gap a me. Suppose now ha he oupu gap s hgher han expeced. In he nonlnear model he slope of he Phllps curve, y, depends on he level of he oupu gap. Because he Phllps curve s convex s slope s ncreasng n he level of he oupu gap (see equaon (. and Fgure.. Thus, f he oupu gap urns ou o be hgher han expeced (because of a posve shock, he slope of he Phllps curve s also hgher han expeced. Smlarly, f we have a negave shock he slope of he Phllps curve wll be lower han expeced. Ineresngly, he above mples ha he cenral bank becomes unceran abou he response of nflaon o any gven polcy acon. Ths response coeffcen s equal o he produc of he neres elascy of oupu (whch s - and he slope of he Phllps curve (whch now depends on he realsaon of he addve shock o oupu. 36 Thus f he slope of he Phllps curve s hgher han expeced (because of a posve realsaon of he demand shock, he response coeffcen of nflaon n wo year's me wh respec o he nomnal neres rae a me s lower han expeced. Because he response coeffcen s negave (ncreasng nomnal neres raes reduces nflaon, n hs case moneary polcy urns ou o be more effecve han expeced. Smlarly, f he slope of he Phllps curve s lower han expeced he response coeffcen s hgher (less negave han expeced. In hs case moneary polcy s less effecve han expeced. To conclude, n he nonlnear model addve shocks o he oupu gap generae uncerany abou he polcy mulpler; ha s ype uncerany has ype mplcaons. From he prevous paragraph we learn ha he slope of he Phllps curve depends on he realsaon of he shock o he oupu gap. Wh a posve realsaon moneary polcy was shown o be more effecve han expeced a me and vce versa. Meanng ha he dampenng effec of a gven nomnal neres rae a me, on nflaon n wo year's me s proporonal o he realsaon of he shock. However, n hs paper we are concerned wh opmal polcy and s herefore of some neres o relax he assumpon of a gven nomnal neres rae. Suppose he cenral bank decdes o ncrease he nomnal neres rae. From equaon (.4 follows ha a hgher nomnal neres rae - ceers parbus - lowers he level of omorrow's oupu gap. Moreover, n he nonlnear model he slope of he Phllps curve s ncreasng n he level of he oupu 36 Ths can be seen by adapng (..a. The algebrac expresson for he response coeffcen of nflaon n wo year's me wh respec o he nomnal neres rae a me s y α.. f '( y y ( α ϕ[ β y ( β x ε

24 3 gap. Thus, a hgher nomnal neres rae lowers he slope of he Phllps curve. Ths n urn mples ha any posve oupu shock ha may h he economy a me wll be less nflaonary. Smlarly, by lowerng he slope of he Phllps curve, a hgher neres rae wll also dampen he dsnflaonary effecs of negave shocks. Thus, n he nonlnear model a hgher nomnal neres rae causes posve demand shocks o nduce less nflaon and negave shocks o cause less dsnflaon. Of course, f he cenral bank decdes o cu he nomnal neres rae, he reverse apples. By ncreasng he slope of he Phllps curve, a lower neres rae amplfes he nflaonary effecs of posve oupu shocks, and enhances he dsnflaonary effecs of negave shocks. Thus, nomnal neres raes can dampen or amplfy he second round effecs of oupu shocks on nflaon. To be more precse, can be shown ha he condonal varance of he one o wo year nflaon forecas, Var, s a decreasng funcon of he nomnal neres rae. Ths can be seen from equaons (4.5 and (4.6. As explaned above, he reason s ha by pung up raes, oday s forecas of omorrow s oupu gap goes down. Ths means ha nex year s Phllps curve wll be flaer whch n urn mples ha he effecs of demand shocks a me on nflaon n wo year s me wll be smaller. Hence, he varably of nflaon around he cenral projecon can be reduced by ncreasng shor-erm neres raes. A hs sage s useful o summarse resuls so far. Frs, we have shown ha n he nonlnear model addve shocks o he oupu gap mply uncerany abou polcy; ha s ype uncerany has ype mplcaons. Second, n he nonlnear model he varance of he arge varable (nflaon s a decreasng funcon of he level of he polcy nsrumen (he nomnal neres rae. Please noe ha he second resul s he oppose of Branard's (967 analyss. Branard assumes ha he varance of he arge varable s a lnear and ncreasng funcon of he level of he polcy nsrumen. I follows ha polces ha are 'oo acvs' ncrease he varance of he arge varable hereby worsenng he performance of economc polcy. In conras here he varance of he arge varable s a nonlnear and decreasng funcon of he level of he polcy nsrumen (hs can be seen from equaons (4.5 and (4.6. I follows ha polces ha are 'oo acvs' from a Branard perspecve may acually decrease he varance of he arge varable, hereby mprovng he performance of polcy. Thus, n he nonlnear model uncerany abou he polcy mulpler leads o an opmal polcy ha s more acve. To be more precse, as he varance of he mulpler rses, he polcy of ryng o mnmse he varance of he arge varable ends owards ncreasng he opmal amoun of polcy. Ths means ha he Branard resul s reversed. To see hs we focus on he cenral bank's opmal polcy rule (4.4. As saed before he sochasc elemens of hs rule are solaed n he erms E d,e and Var. Here he frs wo erms relae o he effecs of he uncerany on he nflaon forecas, and hence capure he Jensen's nequaly effec. The second channel hrough whch he uncerany affecs nflaon forecas

25 4 argeng s hrough s effecs on he condonal varance of he one o wo year nflaon forecas Var. Now we can solae he mplcaons of he second channel for he amoun of opmal polcy by absracng from he Jensen's nequaly effec. Ths can be done by seng E d equal o zero n he cenral bank's opmal rule (4.4. Ths yelds E and r r { ϕ[( { ϕ[( α Var f ( [ α f ( Var f ( [ Var α( [ Var { ϕ[( f ( [ } ( } βy } (5. Equaon (5. mplcly defnes he opmal level of he nomnal neres rae n he nonlnear sochasc model where he uncerany s only allowed o affec he varance of he arge varable. Thus, hs s as close as we can ge o he lnear-quadrac Branard framework. Snce boh he lefhand sde and he rgh-hand-sde depend on he nomnal neres rae, agan we have o resor o numercal mehods o fnd he opmal level of he cenral bank's polcy nsrumen. The resuls can be found n column 9 of Table 3.. Ths column gves he dfference beween he level of nomnal raes mpled by he rule (5. and he nonlnear cerany equvalen rule (3.. As can be seen from he numbers n hs Table he dfference s posve, mplyng ha n he nonlnear model uncerany abou polcy calls for a hgher raher han a lower opmal amoun of polcy. 6 Summary and Concludng Remarks In hs paper we exended he Svensson (997a nflaon forecas argeng framework wh a convex Phllps curve. Usng opmal conrol echnques we derved an asymmerc polcy rule. We found ha nomnal neres raes accordng o hs rule were hgher han under he Svensson forward lookng verson of he Taylor rule. Exendng he analyss wh uncerany abou he oupu gap we found ha our earler resuls became even sronger. We found ha he uncerany nduced a furher upward bas n nomnal neres raes on op of he effec of he nonlneary per se. Also we found ha he mplcaons of uncerany for opmal polcy are que dfferen from eher he sandard Branard (967 analyss, or Svensson s (997b exenson of hs nflaon forecas argeng