A General Schema for Optimal Monetary Policymaking: Objectives and Rules

Transcription

1 Univrsiy of Conncicu Economics Working Paprs Dparmn of Economics 3--7 A Gnral Schma for Opimal Monary Policymaking: Ojcivs and Ruls Huiping Yuan Xiamn Univrsiy Sphn M Millr Univrsiy of Conncicu and Univrsiy of Nvada, Las Vgas Follow his and addiional works a: hp://digialcommonsuconndu/con_wpaprs Rcommndd Ciaion Yuan, Huiping and Millr, Sphn M, "A Gnral Schma for Opimal Monary Policymaking: Ojcivs and Ruls" (7) Economics Working Paprs Papr 79 hp://digialcommonsuconndu/con_wpaprs/79 This is rough o you for fr and opn accss y h Dparmn of Economics a DigialCommons@UConn I has n accpd for inclusion in Economics Working Paprs y an auhorizd adminisraor of DigialCommons@UConn For mor informaion, plas conac digialcommons@uconndu

2 Dparmn of Economics Working Papr Sris A Gnral Schma for Opimal Monary Policymaking: Ojcivs and Ruls Huiping Yuan Xiamn Univrsiy Sphn M Millr Univrsiy of Conncicu and Univrsiy of Nvada, Las Vgas Working Papr 7-9 March 7 34 Mansfild Road, Uni 63 Sorrs, CT Phon: (86) Fax: (86) hp://wwwconuconndu/ This working papr is indxd on RPEc, hp://rpcorg/

3 Asrac This papr xamins four quivaln mhods of opimal monary policymaking, commiing o h social loss funcion, using discrion wih h cnral ank long-run and shor-run loss funcions, and following monary policy ruls All lad o opimal conomic prformanc Th sam prformanc mrgs from hs diffrn policymaking mhods caus h cnral ank acually follows h sam (similar) policy ruls Ths ojcivs (h social loss funcion, h cnral ank long-run and shor-run loss funcions) and monary policy ruls imply a compl rgim for opimal policy making Th cnral ank long-run and shorrun loss funcions ha produc h opimal policy wih discrion diffr from h social loss funcion Morovr, h opimal policy rul mrgs from h opimizaion of hs diffrn cnral ank loss funcions Journal of Economic Liraur Classificaion: E4, E5, E58 Kywords: Opimal Policy, Cnral Bank Loss Funcions, Policy Ruls Profssor Yuan grafully acknowldgs financial suppor from h Naional Social Scinc Foundaion of China and h China Scholarship Council

4 Inroducion In his papr, w considr h dsign of monary policy ihr a cnral ank ojciv funcion or a cnral ank policy rul o opimiz social wlfar W assum ha h social wlfar funcion corrsponds o h rprsnaiv houshold s uiliy funcion, which capurs h final ojcivs of monary policy As a pracical mar, howvr, h cnral ank should no adop h social wlfar funcion as is gnral arging rul (i, ojciv funcion) undr discrion Tha is, if h social wlfar funcion incorporas h arg valus for h arg varials ha prov muually inconsisn wih h srucur of h conomy, hn opimizing h social wlfar funcion gnras im inconsisncy undr raional xpcaions Rahr, h cnral ank s gnral arging rul mus incorpora arg valus ha prov muually consisn wih h srucur of h conomy If so, hn opimal policy ha opimizs h social wlfar funcion will prov consisn as wll Th cnral ank s gnral arging rul (i, ojciv funcion) implis a spcific arging rul (i, policy rul), which also nsurs ha opimal policy provs consisn W find, howvr, ha an infini numr of gnral arging ruls associa wih a spcific arging rul Onc again, h spcific arging rul (i, cnral ank policy rul) nsurs ha w opimiz h social wlfar funcion and ha w adop an opimal and consisn policy Svnsson (3) dfins a gnral arging rul as incorporaing h ojcivs o achivd, for insanc, y lising h arg varials, h args (arg lvls) for hos varials, and h (xplici or implici) loss funcion o minimizd (p 49) Morovr, Svnsson (3) argus gnral arging ruls ssnially spcify opraional ojcivs for monary policy (p 43) Svnsson (3) dfins a spcific arging rul spcifis condiions for h arg varials (or forcass of h arg varials), for insanc, lik h rul of hum of h Bank of England and h Riksank (p 49) Morovr, Svnsson (3) argus spcific arging ruls ssnially spcify opraional Eulr condiions for monary policy (p 43)

5 Rogoff (985), Walsh (995, 3), Svnsson (997), and ohrs adop h noion ha h cnral ank wih discrionary policy mploys an ojciv funcion (i, gnral arging rul) diffring from h social wlfar funcion o nsur opimal policy provs consisn For insanc, Walsh (3, p 76) sas, I hav assumd h rlvan policy rgim is on of discrion, and h prolm facd in dsigning policy is o assign a loss funcion o h cnral ank Similarly, Svnsson (999, p 636) sas Boh mony-growh arging and nominal-gdp arging ar inrprd as inrmdia-arging ruls, ha is, h assignmn or adopion of an inrmdia loss funcion wih mony growh or nominal GDP as a arg varial Sinc h purpos of an inrmdia-arging sragy is o fulfill som final loss funcion, h prformanc of h inrmdia-arging rul mus valuad according o ha final loss funcion rahr han o h inrmdia loss funcion In our conx, h inrmdia loss funcion rfrs o h cnral ank ojciv funcion (i, gnral arging rul), which aks h sam form as h social loss funcion u wih diffrn paramrs Yuan and Millr (7a, 6) and Yuan, Millr and Chn (7) argu ha h social wlfar cririon (loss funcion) provs inappropria for h dirc cnral ank ojciv funcion in monary policymaking, caus h arg lvls in h social loss funcion ar inconsisn wih ach ohr and h cnral ank undr discrion facs a dilmma if dlgad h social loss funcion This papr xnds Yuan, Millr and Chn (7) wih saic Barro-Gordon modl y using a dynamic ackward-looking modl wih mploymn prsisnc In sum, w xplor h dlgaion of cnral ank ojciv funcions and policy ruls o produc opimal and consisn policy oucoms Th rs of his inroducion provids 3

6 ackground informaion on hs issus Cnral Bank Inrmporal (long-run) Loss Funcion Dsigning a scinific and dirc ojciv for monary policy is on of h main asks of his papr W will show ha h quilirium (i, discrion or consisn) policy undr h dsignd and dlgad cnral ank ojciv funcion rplicas h opimal policy undr h social wlfar funcion wih commimn Tha is, h consisn policy undr h dsignd and dlgad ojciv funcion provs opimal undr h social wlfar cririon So dsigning h dirc cnral ank ojciv funcion for monary policy srvs as a mans, no an nd in islf, o opimizing h social wlfar Th dsignd and dlgad monary policy (cnral ank) ojciv funcion posssss a sraighforward inrpraion I xhiis four characrisics (propris) Firs, h quilirium (consisn) policy undr h dsignd and dlgad ojciv funcion rplicas h opimal policy undr h social wlfar funcion wih commimn Thus, consisncy and opimaliy rconcil undr h dsignd and dlgad ojciv funcion Scond, h arg lvls in h dsignd and dlgad ojciv funcion prov modra, in ha hy ar aainal, on avrag, ach priod Thus, h cnral ank can asily arn crdiiliy and accounailiy Third, h arg lvl of mploymn (oupu) quals h naural (ponial) on This wll-known oucom rquirs ha h cnral ank no adop an oupu ias in is dirc ojciv funcion Morovr, as Svnsson () argus Thr is gnral agrmn ha inflaion-arging cnral anks do normally no hav ovramiious oupu args, ha is, xcding ponial oupu (p 774) Our papr arrivs a his rsul from a diffrn dircion Fourh, h rlaiv wigh placd on h wo arg varials -- 4

7 inflaion and oupu (mploymn) -- rflcs h social prfrnc, as wll as h conomic srucur Ths four propris xhii rousnss caus hy also hold in Yuan and Millr (7a, 6) and Yuan, Millr and Chn (7) Svnsson (3) wris: Wha ar h prolms wih a commimn o a gnral arging rul? On prolm is ha h ojcivs may sill no sufficinly wll spcifid no o opn o inrpraion For insanc, h rlaiv wigh on oupu-gap sailizaion in flxil inflaion arging is no dircly spcifid y any inflaion-arging cnral ank A scond ponial prolm is h ponial consquncs of h discrionary opimizaion undr a commimn o a gnral arging rul, mor prcisly ha such discrionary opimizaion is no fully opimal in a siuaion wih forward-looking varials 3 (p 454) Our papr o som xn solvs h aov wo prolms in hory -- how o dsign a dirc cnral ank ojciv funcion for monary policy and how o implmn opimal discrionary (consisn) policy In addiion, similariy xiss wn our findings and hos of Clarida, Gali and Grlr (999) Th soluion undr commimn in his cas prfcly rsmls h soluion ha would oain for a cnral ank wih discrion ha assignd o inflaion a highr cos han h ru social cos (p 68) Wha diffrs? Thy plac mor wigh on inflaion variailiy; w pu mor wigh on mploymn (oupu) gap variailiy Th diffrnc rsuls from diffrn modls Inuiivly, wih mploymn prsisnc, any mploymn gap no liminad oday prsiss ino h fuur and, hus, inducs mor loss To rduc loss, w plac mor wigh on mploymn 3 Th ackward-looking modls in Svnsson (997a, 3) acually conform o dynamic programming prolms, no gams Inconsisncy issus of opimal policy do no xis in hs ackward-looking modls, u do xis in h forward-looking modl of Svnsson (3) Though our modl is ackward-looking wih mploymn prsisnc, h inconsisncy issu of opimal policy xiss W us h ackward-looking modl of Svnsson (997) in our papr o illusra our idas Using a forward-looking modl maks our idas lss ransparn caus of mor complicad mahmaics 5

8 Monary Policy Ruls Th propry of opimal and consisn policy provs a good oucom Bu, why? In conmplaing h answr, w find ha h rason provs surprisingly simpl Tha is, h firs-ordr condiion of h valu funcion undr h dsignd and dlgad cnral ank ojciv funcion wih discrion xacly quals h firs-ordr condiion of h valu funcion undr h social wlfar funcion wih commimn As a rsul, discrionary policy undr h dsignd and dlgad ojciv funcion rplicas h commimn policy undr h social ojciv funcion From his viw, monary policy ruls appar mor asic han h ojcivs funcions W also xamin such policy ruls in his papr Diffrn manings for monary policy ruls xis in liraur Whn ruls appar in h phras ruls vrsus discrion, ruls man commimn Tha is, McCallum (4) sas, for xampl, ha monary policy is conducd in a rul-asd mannr ha viws policy as an ongoing procss (p 367), rahr han on a priod-y-priod asis Whn ruls appar in Taylor ruls or McCallum ruls, ruls man insrumn ruls in rspons o currn conomic condiions Svnsson (3) argus, h concp of monary-policy ruls should roadnd yond h narrow insrumn ruls and also includ arging ruls (p 466) W dfin a monary policy rul as a cnral ank s havior quaion, which quals an xplici or implici funcion of insrumns or arg varials in rlaion wih prdrmind varials and srucural shocks So, w diffrnia wn a rul vrsus an ojciv, as dos Ccchi () In addiion, w also assum wih Ccchi () ha a rul rsponds o conomic varials as wll as dmand and supply shocks, if h rul 6

9 prforms a is s 4 Morovr, monary policy ruls in our dfiniion prov roadr han insrumn ruls Thy can rflc spcific arging ruls or insrumn ruls in Svnsson s rms, dpnding on h assumpions and h conomic srucur ha w us If h conomic srucur involvs only an aggrga supply funcion and h cnral ank dircly conrols a arg varial, hn h opimal monary policy rul is a spcific arging rul, which includs arg varials, prdrmind varials, and srucural shocks If h conomic srucur involvs oh aggrga dmand and aggrga supply funcions and h cnral ank dircly conrols an insrumn (no a arg varial), hn a monary policy rul is an insrumn rul, which includs arg varials, prdrmind varials, srucural shocks, as wll as h insrumn varial This papr idnifis four ways o oain opimal policy ruls -- drivd from h firs-ordr condiion of h valu funcion of h social wlfar funcion undr commimn, drivd from h firs-ordr condiion of h valu funcion of h cnral ank long-run and shor-run ojciv funcions undr discrion, and drivd from opimizing h social wlfar funcion using our dfiniion of a monary policy rul To concnra on h main issus, w considr only spcific arging ruls W can asily oain opimal insrumn ruls, howvr, y comining spcific arging ruls wih h aggrga dmand funcion Givn our dfiniions of monary policy ojcivs and ruls, our papr concluds ha monary policy ojcivs and ruls can horically play h sam rol in monary policymaking Spcifically, h four opimal policymaking mhods -- commimn o h social wlfar ojciv, discrion and h dsignd and dlgad long-run and shor-run 4 Brnank and Mishkin (997) and Svnsson (997a,, 3) apparnly rgard monary policy ruls as mchanical rsponss o currn and forcas varials, xcluding rsponss o shocks 7

10 cnral ank ojciv funcions, 5 and jus follow h dsignd and dlgad policy rul -- yild quivaln oucoms Thy all produc h sam opimal and consisn oucoms This conclusion smingly conradics Svnsson s (997a) argumn Commimn o arg ruls may r han commimn o insrumn ruls (p ) 6 Th conradicion, howvr, occurs caus of diffrn assumpions concrning knowldg aou h conomic srucur as wll as a diffrnc in h undrsanding of h ruls No ssnial conflics xis in h da on whhr arging ruls prov suprior o insrumn ruls Wih imprfc knowldg aou h conomic srucur, arging ruls (gnral arging ruls and spcific arging ruls) may domina insrumn ruls Mor spcifically, wih imprfc knowldg aou h aggrga supply funcion, gnral arging ruls may domina spcific arging ruls Wih prfc knowldg aou h aggrga supply funcion, howvr, using gnral arging ruls or spcific arging ruls maks no diffrnc Wih prfc knowldg aou h aggrga supply funcion and imprfc knowldg aou h aggrga dmand funcion, spcific arging ruls may domina insrumn ruls Sill, wih prfc knowldg aou h conomic srucur (aggrga supply and dmand funcions), gnral arging ruls, spcific arging ruls, and insrumn ruls prov ssnially h sam Assuming imprfc knowldg aou h conomic srucur is mor ralisic and pracical Sinc our horical papr only addrsss h issus of h consisncy and opimaliy of monary policy (no h pracical implmnaion of monary policy, such as in Svnsson, 997a), w assum prfc knowldg aou h conomic srucur y h 5 W discuss h cnral ank shor-run ojciv funcion in h nx suscion 6 As nod for, Svnsson s (3) arg ruls includ gnral arging ruls and spcific arging ruls Gnral arging ruls spcify an opraional ojciv for monary policy wih a commimn o ha ojciv Monary policy ruls in our conx prov roadr han, and hus includ, insrumn ruls 8

11 cnral ank and h pulic Undr his assumpion and our dfiniion of monary policy ruls, i maks no diffrnc whhr h cnral ank opras monary policy y opimizing policy ojcivs 7 or y following policy ruls Cnral-Bank Priod (Shor-Run) Loss Funcion In his papr, w also considr myopic policy Gnrally an quilirium of an infini-priod dynamic gam rquirs srong assumpions, including ha all playrs possss high inllignc and mak no misaks Accordingly, w assum ha a ounddly raional cnral ank opras policy myopically, minimizing only h currn priod loss W sill hop, howvr, ha h myopic quilirium policy rplicas h opimal policy For convninc, w dfin h shor-run ojciv funcion, whr a oundd-raional cnral ank opimizs h priod ojciv, and h long-run ojciv funcion, whr an unoundd cnral ank opimizs h inrmporal ojciv W find ha h opimal shor-run ojciv in ach priod quals h priod ojciv of h dsignd long-run ojciv W inrpr his rsul roughly as follows Rcall ha on characrisic of h dsignd and dlgad long-run ojciv funcion is ha arg lvls ar ralizd ach priod, on avrag Tha is, h cnral ank minimizs h loss ach priod a zro, rsuling in h opimizaion of h inrmporal ojciv funcion In ohr words, as long as h cnral ank currnly minimizs ach priod s loss funcion of h inrmporal ojciv, h myopic quilirium policis of all priods rplicas h opimal policy pah In ohr words, minimizing h inrmporal loss funcion also minimizs h priod loss funcion In shor, no inrmporal loss susiuion occurs Similarly, h firs-ordr 7 Opimizing policy ojcivs mans ihr opimizing h social wlfar ojciv wih commimn, or opimizing h dirc cnral ank ojciv funcion wih discrion 9

12 condiion of h opimal shor-run ojciv funcion rplicas h opimal policy rul W organiz h papr as follows Scion prsns h modl and is commimn (opimal) and discrion (consisn) policy Consisn policy dos no prov opimal Scion 3 dsigns h cnral-ank inrmporal (long-run) loss funcion W find h discrionary policy undr h dsignd and dlgad cnral-ank inrmporal (long-run) loss funcion rplicas opimal policy and h loss funcion posssss a sraighforward inrpraion Discrionary policy undr h dsignd and dlgad cnral-ank loss funcion rplicas h opimal policy caus h firs-ordr condiions of hir valu funcions (h dsignd and dlgad cnral-ank inrmporal loss funcion and h social inrmporal loss funcion) prov idnical As a rsul, Scion 4 sudis monary policy ruls, providing hr ways of dsigning opimal monary policy ruls Scion 5 dsigns h cnral-ank priod (shor-run) loss funcion W oain inuiiv rsuls Th dsignd and dlgad cnral-ank priod loss funcion coincids wih h priod loss funcion of h dsignd and dlgad cnral-ank inrmporal loss funcion, implying ha h firs-ordr condiion of h dsignd and dlgad cnral-ank priod loss funcion also rplicas h opimal policy rul Scion 6 concluds Opimal and Consisn Policy in a Simpl Modl Th modl follows h analysis of Svnsson (997) 8 Sociy minimizs h following inrmporal (long-run) loss funcion: () E β L, = whr h discoun facor quals β, < β <, E quals h mahmaical xpcaions 8 S foono 3 for h rasons ha w us h modl in Svnsson (997)

13 opraor, and h priod (shor-run) loss funcion quals L Th priod loss funcion quals h following: L L π π λ = π π + λ ), () (, ;,, ) ( ) ( whr π quals h inflaion ra, quals h naural logarihm of h mploymn ra (i, h shar of mploymn in full mploymn), sarrd valus idnify sociy s arg valus for h inflaion and mploymn ras, and λ masurs h rlaiv imporanc of mploymn and inflaion dviaions from hir arg ras Th conomic srucur quals h following wo-quaion sysm, which incorporas mploymn prsisnc: = ρ + α π π + ε and (3) ( ) (4) π E ( π ) =, whr ε quals h whi nois random shock wih varianc qual o σ No ha h sady-sa valu of quals zro, sinc a full mploymn, h mploymn ra quals on and is naural logarihm quals zro Thus, h cas wihou prsisnc (i, ρ = ) corrsponds o h sandard Barro-Gordon (983) modl Opimal Policy (Bnchmark) Assum ha h govrnmn dircly conrols h cnral ank and ha h govrnmn can commi o a sa-coningn rul on h inflaion ra Th Bllman quaion for drmining h opimal policy and oucoms from h opimizaion quals h following: (5) V ( ) min ( ) ( ) ( ) = E π π λ βv π, π + +

14 W minimiz his quaion sujc o h conomic srucur givn in quaions (3) and (4) 9 Th soluion for V ( ) mus qual a quadraic form, sinc w minimiz h quadraic ojciv funcion sujc o linar consrains Thus, h hypohsizd soluion quals h following quaion: (6) V ( ) = γ + γ + γ, whr w nd o drmin h unknown cofficins in quaion (6) Th soluion quals h following: (7) γ λρ = βρ and λρ γ = βρ Th soluion for h opimal policy producs h following: (8) whr (9) opimal π π ε =, = λα βρ + λα Th opimal mploymn ra quals h following: opimal βρ = ρ + ε βρ + λα Consisn (Discrionary) Policy Now, assum ha h govrnmn sill dircly conrols h cnral ank, u i canno commi o a sa-coningn rul on h inflaion ra As such, h dcision prolm of h govrnmn aks h xpcd inflaion ra as a givn Tha is, no longr dos h govrnmn inrnaliz h ffcs of is dcisions on h xpcd inflaion ra Carrying ou h opimizaion producs h following consisn inflaion and mploymn ras: 9 Lockwood al (995) and Svnsson (997) provid mor dails of h drivaion Svnsson (997) also rpors wo addiional xisnc condiions in his Appndix

15 discrion () π = a ε c and discrion () = ρ + ( α) ε, whr λα λα + βαc a= π +, =, and βρ βαc βρ + α λα + βαc ( ) c = βρ ( βρ ) 4 λα βρ αβρ Comparing quaions (8) and (), h inflaion ias in h consisn inflaion ra undr discrion quals h following: () discrion opimal π π ( ) = a π c ε Th inflaion ias includs an avrag inflaion ias ( a π ), a sa-coningn inflaion ias ( c ), and a sailizaion inflaion ias [ ( ) ε ] In sum, consisn policy dos no prov opimal 3 Dsign of Cnral Bank Long-Run Loss Funcions W show in h prior scion ha discrion producs a consisn, u non-opimal, policy Tha finding implicily assums ha h cnral ank adops h social loss funcion as is own loss funcion Our papr firs considrs dlgaing a loss funcion o h cnral ank ha diffrs from h social loss funcion, u ha dlivrs h opimal oucoms whn h cnral ank adops a consisn policy asd on h dlgad loss funcion Tha is, can w find a loss funcion ha whn dlgad o h cnral ank yilds opimal oucoms? Whn h cnral ank minimizs h inrmporal xpcd loss from h currn and all fuur priods, w call ha ojciv funcion h long-run cnral-ank loss funcion This Scion xamins his prolm Corrspondingly, whn h cnral ank only minimizs h 3

16 currn-priod xpcd loss, w call ha ojciv funcion h shor-run cnral-ank loss funcion Scion 5 considrs h currn-priod xpcd-loss minimizaion prolm Equaions () and () rprsn h long-run (inrmporal) social loss funcion and is shor-run (priod) componn Th proposd dlgad cnral ank shor-run (priod) loss funcion quals h following xprssion: (3) L = ( π π ) + λ ( ) whr, π and qual sa-coningn args (i, π = g + g and = h + h ) Tha is, h proposd dlgad shor-run cnral ank loss funcion mirrors h shor-run social loss funcion, u wih ponially diffrn paramrs Morovr, h proposd shor-run loss funcion allows sa-coninn args o rflc h prsisnc of mploymn in h conomy Basd on h discussion in h Inroducion ha cnral-ank loss funcion srvs as a mans o h nd minimizaion of h social long-run loss funcion, w dsign h cnral ank loss funcion hrough h following modl: (4) min g, g, h, h, λ E = β π π + λ ( ) ( ) min E β ( π g g ) λ ( h h + ) { π} = = s = ρ + α( π π ) + ε s π = E ( π) W solv his modl in wo sps and oain an infini numr of opimal cnral ank loss funcions Wih mor assumpions, w pin down h uniqu rasonal cnral ank loss funcion from h infini numr of opimal candidas Spcifically, Sp I solvs h following parial modl: 4

17 (5) min E g g h h = = { π } β ( π ) + λ ( ) ( ) = ρ + α π π + ε s π = E ( π) Tha is, Sp I proposs a class of loss funcions, involving paramrs ha dfin h prcis funcion chosn from h class of funcions, and drivs a consisn policy whn h cnral ank minimizs ha class of loss funcions sujc o h conomic srucur Clarly, h consisn policy will dpnd on h paramrs ha dfin h prcis loss funcion chosn from h class of loss funcions Th consisn (quilirium) inflaion and mploymn ras qual: ( q) π = α λ βγ λ α αρ λ βγ ε and α (6) g ( h ) ( g h ) ( ) (7) = ρ + qε, whr q = + α λ + βγ, [ ( )] (8) ( ) ( ) γ = λαh αρ λ + βγ + λ h ρ + βγρ (9) ( ) ( ) ( ) or γ γ( λ, h) γ = α λ h βγ λ αh αρ λ + βγ + λ h h ρ + βγρ or (,, ) γ γ λ h h, and Givn h soluion for h consisn or quilirium oucoms for inflaion and mploymn ha dpnd on h paramrs of h cnral ank loss funcion (i, g, g, h, h, and λ ), Sp II choos valus for hos paramrs o minimiz h xpcd inrmporal social loss Th analyical prolm quals h following: Equaions (6) and (7) qual, rspcivly, quaions (A-8) and (A-9) in Appndix A Equaions (8) and (9) qual, rspcivly, quaions (A-) and (A-) in Appndix A 5

18 () min E β ( π ) ( ) π λ g, g, h, h, λ + = ( q) π = g α( λ h βγ) ( g λ αh) αρ( λ βγ) ε s α = ρ + qε Th prolm yilds h following firs-ordr condiions: 3 g + h = () α( λ βγ ) π ( ) ( () g + λαh αρ λ + βγ )=, and, (3) βρ q = = + α ( λ + βγ) βρ + λα By using quaions () o (3), h consisn policy oucoms for h inflaion and mploymn ras (6) and (7) qual: (4) (5) = λα βρ + λα and π π ε βρ = ρ + ε, βρ + λα which qual hir opimal oucoms S quaions (8) and (9) Gnrally, modl (4) lads o an infini s of possil cnral ank loss funcions, sinc h class of loss funcions conains mor han h minimum numr of paramrs ndd o lad o a soluion of h minimizaion Spcifically, as long as h 7 paramrs, g, g, h, h,,, and λ γ γ, of h cnral ank loss funcion, saisfy quaions (8), (9), (), (), and (3), h consisn policy will prov opimal Among h infini numr of opimal candidas for h cnral ank loss funcions, dos on appar mor rasonal? Ys W argu ha h chosn paramrs should also minimiz h dsignd and dlgad cnral ank loss funcion islf This ida pins down a 3 Equaions (), () and (3) qual, rspcivly, quaions (A-6), (A-7) and (A-8) in Appndix A 6

19 uniqu cnral ank loss funcion Th prolm quals h following: (6) min g, g, h, h, λ { ( + ) + ( + ) } E β π g g λ h h = λα π = π ε βρ + λα s βρ = ρ + ε βρ + λα Th opimizaion yilds h following soluion: 4 (7) g = π, g =, h = and h = ρ Equivalnly, (8) π = π and = ρ Morovr, if ρ quals zro, hn quals zro or full mploymn, givn ha h quals zro Viwing h prolm somwha diffrnly, u lading o h sam conclusion, modrn cnral anks mus accoun for hir acions How can w mak cnral anks accounal? W do so y dlgaing achival arg lvls Th cnral ank, consraind y h conomic srucur, will fac a dilmma if i canno achiv h dlgad arg lvls Tha is, w assum h dlgad arg lvls ar avragly aainal: (9) π = E ( π ) and = E ( ) Thus, sinc E ( ) = and E ( ) π π = ρ from quaions (4) and (5) hold for ach of h opimal cnral ank loss funcions, quaion (8) also holds Finally, w drmin h opimal cnral ank long-run loss funcion as follows: 5 (3) L = ( π π ) + λ ( ) whr, π π ρ λ λ βρ,, and = = = [ /( )] 4 Equaion (7) quals quaion (A-5) in Appndix A 5 S Appndix A for furhr dails 7

20 Thr osrvaions mrg from hs findings Firs, h paramr valus ha minimiz oh h social and cnral ankr loss funcions (i, h uniqu soluions) simulanously imply ha h args in h cnral ankr loss funcion prov raional, in h sns ha h xpcd inflaion and mploymn ras qual h cnral ankr args Tha is, (3) E ( π ) π and E ( ) = = In addiion, h rsul of = ρ mans ha h opimal mploymn arg should qual h ponial mploymn lvl ach priod This rsul provs consisn wih gnral agrmn ha inflaion-arging cnral anks do normally no hav ovramiious oupu args, ha is, xcding ponial oupu (Svnsson,, p 774), and also conforms o Blindr s inuiion I can assur you ha i would no surpris my cnral ankr frinds o larn ha conomic horis ha modl hm as sking o driv unmploymn low h naural ra imply ha hir policis ar oo inflaionary Thy would no dou rply, Of cours ha would inflaionary Tha s why w do no do i (Blindr, 998, p 4-43) Scond, Rogoff s (985) consrvaiv cnral ankr proposal (i, λ < λ ) provs inconsisn wih our finding To compar wih Rogoff s modl, which dos no incorpora mploymn prsisnc, w s ρ = Thn, λ = λ 6 Wih mploymn prsisnc, w find ha a lss-consrvaiv cnral ankr han sociy provs opimal, sinc (3) λ > λ Inuiivly, mploymn prsisnc in our modl mans ha any mploymn gap no liminad oday prsiss ino fuur, and hus inducs loss To rduc loss, mor wigh gos 6 Yuan, Millr and Chn (7) discuss h rsul of λ = λ 8

21 on h mploymn arg Finally, our spcificaion ha achivs h opimal oucoms involvs dlgaing o h cnral ank an inflaion arg qual o h social inflaion arg (i, π = π ), a sa-coningn mploymn arg qual o h shor-run naural ra of mploymn (i, = ρ ), and an mploymn wigh in h cnral ank shor-run (priod) loss funcion - grar han h social wigh (i, wigh-liral cnral ank, = [ /( )] > ) This λ λ βρ λ offrs a soluion ha diffrs from ha proposd in Svnsson (997) 4 Dsign of Monary Policy Ruls Unil now, w prsn wo mhods of implmning an opimal monary policy -- commimn o h social loss funcion and discrion o h dsignd and dlgad cnral ank loss funcion Th wo mhods achiv h sam opimal oucoms, sinc h firs-ordr condiion of h valu funcion of h social loss funcion wih commimn quals h firs-ordr condiion of h valu funcion of h dsignd and dlgad cnral ank loss funcion wih discrion Tha is, alhough i appars ha policy opras in diffrn ways, h cnral ank acually follows h sam havioral quaion, invialy rsuling in h sam conomic prformanc As a rsul, monary policy ruls can horically play h sam rol in policymaking as monary policy ojcivs As dfind in h Inroducion, a monary policy rul spcifis a havioral quaion for h cnral ank I can includ prdrmind varials, arg varials, insrumn varials, as wll as srucural shocks No insrumn varials appar in our monary policy ruls caus w assum ha h cnral ank dircly conrols h inflaion ra, h arg varial So w sudy only spcific arging ruls in Svnsson s rminology W can asily 9

22 driv insrumn ruls, howvr, from spcific arging ruls and an aggrga dmand funcion W now prsn hr mhods of driving opimal monary policy ruls Two rflc h firs-ordr condiions mniond aov Th hird mhod mirrors h mhod of dsigning and dlgaing a cnral ank loss funcion, y choosing paramrs from policy ruls ha minimiz h social loss Acually, h hird mhod frqunly appars in h liraur Firs-Ordr Condiion of Commimn o h Social Loss Funcion Rpa h Bllman quaion (5) as follows: (5) V ( ) min ( ) ( ) ( ) = E π π λ βv π, π + + Th firs-ordr condiion quals: (33) ( π π ) λα ( ) αβv ( ) E λα ( ) αβv ( ) = Susiuing V ( ) = γ + γ, E ( ) = ρ and λρ γ = ino quaion βρ (33) vnually rducs o: λα βρ (34) ( π π ) ( ρ ) + = This dfins h spcific arging rul To xprss h policy rul as an xplici funcion of prdrmind varials ( π and ) and srucural shocks ( and rarrang rms o giv: ε ) susiu = + ( ) + ρ α π π ε ino quaion (34) (35) βρ λα λα π = π + π ε βρ + λα βρ + λα βρ + λα

23 This dfins h opimal monary policy rul Firs-Ordr Condiion of Discrion wih h Dsignd and Dlgad Cnral Bank Loss Funcion Th Bllman quaion for his prolm quals h following: 7 (36) V ( ) = E L + βv ( ), π min { } L π π λ ρ whr = ( ) + ( ) Th firs-ordr condiion quals: (37) ( π π ) λ α( ρ ) αβ ( ) + + V = No ha V ( ) γ γ = + Also, Appndix A dmonsras ha γ γ = = 8 Thus, quaion (37) rducs o: ( ) ( ) (38) π π + λ ρ =, whr λ = [ λ/( βρ )] Policy rul (38) rplicas h rul (34) Policy Ruls ha Minimiz h Social Loss Funcion Th mhod of dsigning policy ruls in his suscion is acually usd frqunly in h xising liraur (g, Clarida, Gali and Grlr 999) Rsuls, howvr, dpnd on h corrc dfiniion of monary policy ruls As mniond in h Inroducion, monary policy ruls in our conx imply ha h conrol arg varial ( π ) dpnds on prdrmind varials ( π and (39) π = a+ π + c + dε ) and srucural shocks ( ε ) as follows: Th cnral ank jus follows h dlgad policy rul (4) Thus, h modl usd o dsign h opimal policy rul quals h following: 7 S Appndix A, quaion (A-4) 8 S Appndix A, quaions (A-6) and h rlad discussion

24 (4) min E acd,,, β π π + λ = ( ) ( ) π = a+ π + c + dε s = ρ + α( π π ) + ε π = E ( π) Th dsign of an opimal policy rul procds in wo sps Sp I drivs h quilirium oucoms, givn h policy rul Sp II chooss h paramrs from h policy rul ha minimiz h social loss funcion Th opimal policy qual: 9 (4) ( ) λα π = π + π βρ + λα ε, whr (4) (43) < Th quilirium inflaion and mploymn ras qual h following: = λα βρ + λα and π π ε βρ = ρ + ε, βρ + λα which qual h opimal oucoms S quaions (8) and (9) Thr characrisics of policy rul (4) mrg Firs, i provs opimal caus h quilirium oucoms rmain opimal as long as h cnral ank follows h rul Scond, h shock cofficin quals ha of rul (35) Tha is, a uniqu way xiss o rspond opimally o h supply shock Third, a class of opimal policy ruls xiss, which includ h spcial cas of rul (35) Wihou considring shocks, any inflaion ra can do as long as i quals h wighd avrag of social arg valu and h priva scor s inflaion xpcaion, or in h simpls cas, h inflaion ra quals h social arg π π (, ε ) = = = 9 S Appndix B for dails

25 5 Dsign of Cnral Bank Shor-Run Loss Funcions From our prior discussion, w know ha h cnral ank can implmn opimal policy y commiing o h social inrmporal loss funcion, using discrion o h dsignd and dlgad cnral ank inrmporal loss funcion, or jus following h dlgad opimal policy rul Supporing h quilirium of an infini-priod dynamic gam rquirs srong assumpions and making no misaks, including xrmly inllign playrs Shuik (998) argus i can provd ha chss is an inssnial gam, i, if on could do all h calculaions hr would no rason o play chss as ach sid would hav an opimal sragy (Zrmlo, 9) (p 3) Accordingly, w assum in his scion ha h cnral ank posssss oundd raionaliy and only minimizs h currn priod s cnral ank loss funcion Tha is, h cnral ank implmns is policy myopically W will drmin whhr h cnral ank wih oundd raionaliy can rplica opimal policy W assum ha h cnral ank shor-run (priod) loss funcion quals h following rlaionship ( ) L = π π + λ ( ), whr h cnral ank uss sa-coningn inflaion and mploymn ra args (i, π = g + g and = h + h ) Sinc w dsign (choos) h paramrs of h cnral ank loss funcion, w do no opimiz myopically, vn hough h cnral ankr dos Tha is, w minimiz h infini horizon social loss funcion wih h knowldg ha h cnral ankr, who acually implmns policy, only opimizs myopically Tha is, h opimizing prolm is xprssd as follows: 3

26 (44) min g, g, h, h, λ E β ( π π ) λ( ) + = min ( π ) λ ( s = ρ + α( π π ) + ε s π = E ( π) g g + h h π ) As for, w solv his modl in wo sps and oain an infini numr of opimal cnral ank priod loss funcions Wih addiional assumpions, w pin down h uniqu rasonal cnral ank priod loss funcion from h infini numr of opimal candidas Spcifically, Sp I compus h consisn or quilirium oucoms for h inflaion and mploymn ras, givn ha h cnral ank maks policy myopically from h cnral ank priod loss funcion Sp II chooss h paramrs, g, g, h, h, and λ, for h cnral ank loss funcion o minimiz h xpcd social loss Onc again, an infini cominaion of paramrs xis ha minimizs h xpcd social loss In ordr o drmin a uniqu soluion, w choos h sam paramr s as in Sp II o minimiz h cnral ank loss funcion Through h aov sps, w drmin h opimal cnral ank shor-run loss funcion as follows: (45) L = ( π π ) + λ ( ), whr π = π, = ρ, and λ = [ λ/( βρ )] This oucom provs idnical o h long-run cnral ank priod loss funcion in quaion (3) Now, w can guss ha h firs-ordr condiion of (45) mus qual quaion (34) or (38), caus monary policy undr h shor-run loss funcion (45) rplicas opimal policy S Appndix C for dails 4

27 Oviously, h firs-ordr condiion of (45) quals: ( ) ( ) (46) π π + λ ρ =, whr λ = [ λ/( βρ )] Policy rul (46) rplicas h rul (34) or (38) Why dos h cnral ank shor-run loss funcion coincid wih h priod loss funcion of h long-run loss funcion? Noic from Appndix A, quaion (A-6) ha γ = = This mans ha h cnral ank minimizs h inrmporal loss a zro wih γ ε = for all, implying ha h minimizaion of ach priod s loss occurs a zro, oo Convrsly, if h cnral ank minimizs ach priod s loss a zro, hn h minimizaion of h inrmporal loss also occurs a zro 6 Conclusion This papr prsns four ways of opimal policymaking, commiing o h social loss funcion, using discrion and h cnral ank long-run and shor-run loss funcions, and following monary policy ruls Thy all lad o opimal conomic prformanc Th sam prformanc mrgs from hs diffrn policymaking mhods caus h cnral ank acually follows h sam (similar) policy ruls Basd on h rsuls, w conclud ha wha mars in opimal policymaking is h way of policymaking Spcifically, h opimal policy nchmark coms from commiing o h inrmporal social loss funcion sujc o h conomic srucur Thn h nchmark opimal policy provids h goal for dsigning h cnral ank long-run and shor-run loss funcions, as wll as h opimal policy rul In shor, h dsignd cnral ank long-run and shor-run loss funcions, as wll as h opimal policy rul, mrg from opimizing h W assum a prfc-informaion and, hus, h pulic knows how h cnral ank implmns is policy 5

28 inrmporal social loss funcion and h conomic srucur W rconcil consisn and opimal oucoms in our modl srucur In our papr, h implmnaion of opimal monary policy rlis on prfc knowldg of h conomic srucur Th ffcs of conomic modl uncrainy on policymaking li yond our scop Wha implicaions mrg from hs horical rsuls for monary policy? Th hr ojcivs (h social wlfar cririon, cnral-ank long-run and shor-run ojcivs) and opimal policy ruls oghr consiu a rgim for opimal policy making Th social inrmporal wlfar funcion informs h pulic aou h final ojciv of monary policy I provids h ulima ojciv for monary policy Th cnral ank long-run and shor-run ojciv funcions provid h mans for achiving h ulima social wlfar ojciv Th pulic can undrsand h inrmdia ojcivs of monary policy, ha is, how h cnral ank opimizs is ojciv funcion o achiv h social wlfar opimum In addiion, monary policy gains crdiiliy and accounailiy wih h inrmdia and aainal ojcivs Th policy ruls mak monary policy opraional for h cnral ank In ohr words, h social wlfar cririon salishs h ulima ojciv, cnral-ank long-run and shor-run ojcivs provid h mans o h nd, and policy ruls provid an opraional shor cu In conclusion, w advoca assigning ojcivs, no ruls, for h cnral ank Firs, w argu ha a cnral ank should hav insrumn indpndnc, u should no hav goal indpndnc (Fischr, 995, p ) Scond, h opraions of h cnral ank coms clar wih xplici ojcivs Spcifically, h social wlfar cririon and h cnral ank long-run and shor-run ojcivs, as undrsood y h pulic, incrass policy crdiiliy 6

29 and accounailiy Spcifying xplici ojcivs, oghr wih opraional indpndnc and ffciv accounailiy srucurs is righly considrd ssnial in an ffciv monary-policy sup (Svnsson, 3, p 454) Third, compn cnral ankrs do no nd insrucions on how o opra monary policy, if dlgad clar ojcivs Wih insrumn indpndnc, cnral ankrs can opimiz h dlgad ojcivs Finally, wih xplici policy ojcivs, h pulic can asily undrsand monary policy and, hus, mak good choics Th pulic mus infr, which hy can do, h implid policy ojcivs, if monary policy ruls ar dlgad o h cnral ank 7

30 Appndix A Dsign of Cnral Bank Long-Run Loss Funcions Th modl for drmining h long-run cnral ank loss funcion is givn as follows: (A-) min g, g, h, h, λ E = β π π + λ ( ) ( ) min E β ( π g g ) λ ( h h + ) { π} = = s = ρ + α( π π ) + ε s π = E ( π) As nod in Scion 3, w solv his modl in wo sps and oain an infini numr of opimal cnral ank loss funcions Wih addiional assumpions, w pin down h uniqu rasonal cnral ank loss funcion from h infini numr of opimal candidas W dno his las sp as Sp III Spcifically, Sp I compus h consisn, or quilirium, policy whn givn h following shor-run cnral ank loss funcion: { } (A-) L = π ( g + g ) + λ ( h + h ) Th consisn, or quilirium, policy coms from minimizing h long-run discound cnral ank loss funcion y choosing h pah for h inflaion ra (i, { } π = ) Sp II minimizs h long-run xpcd social loss funcion y choosing h paramrs, g, g, h, h, and λ, which com from h shor-run cnral-ank loss funcion Finally, Sp III pins down h prcis cnral-ank loss funcion y choosing h paramrs, g, g, h, h, and λ, o minimiz h long-run cnral ank loss funcion Sp I: Drivaion of Consisn Policy, Givn h Cnral Bank Loss Funcion W firs minimiz h long-run cnral-ank loss funcion y choosing h pah of h inflaion ra Th following spcifis h opimizaion prolm: 8

31 (A-3) min E g g h h = = { π } β ( π ) + λ ( ) ( ) = ρ + α π π + ε s π = E ( π) Th Bllman quaion for drmining h opimal policy and oucoms from h opimizaion is givn as follows: (A-4) V ( ) = E L + βv ( ), π min { } whr L is givn in quaion (A-) W minimiz his rlaionship sujc o h srucur of h conomy, givn in quaions (3) and (4) Th soluion for V ( ) mus qual a quadraic form, sinc w minimiz a quadraic ojciv funcion wih linar consrains Thus, h hypohsizd soluion quals h following quaion:, (A-5) V ( ) = γ + γ + γ whr w nd o drmin h unknown cofficins in quaion (A-5) Th firs-ordr condiion for h minimizaion of quaion (A-4) yilds h following: (A-6) π ( g g ) λ α ( h h ) αβ( γ γ ) = Rarranging h rms and susiuing ( ) quaion producs h following: = ρ + α π π + ε ino h aov (A-7) ( g h ) ( g h) ( ) π + λ α αβγ + λ α + α λ + βγ ρ ( )( ) ( ) = α λ βγ π π α λ βγ ε Finding h xpcd valu of quaion (A-7) and solving for h soluions for h inflaion ra and mploymn givs h following rsuls: ( q) π = α λ βγ λ α αρ λ βγ ε and α (A-8) g ( h ) ( g h ) ( ) 9

32 (A-9) = ρ + qε, whr { q q( λ, γ) [ α ( λ βγ)] } = = + + Now, compu V ( ) = E L + βv ( ) wih h soluions for π and in π min { } quaions (A-8) and (A-9) and compar h cofficins wih quaion (A-5) Thus, (A-) ( ) ( ) γ = λαh αρ λ + βγ + λ h ρ + βγρ (A-) ( ) ( ) ( ) or γ γ( λ, h) γ = α λ h βγ λ αh αρ λ + βγ + λ h h ρ + βγρ or (,, ) γ γ λ h h, and In sum, h consisn or quilirium oucoms for h inflaion ra and mploymn appar in quaions (A-8) and (A-9) wih q= q( λ, γ ), γ γ( λ, h) and (,, ) γ γ λ h h Sp II: Drmining h Cnral Bank Loss Funcion ha Minimizs h Expcd Social Loss Givn our soluion for h consisn or quilirium oucoms for inflaion and mploymn ha dpnd on h paramrs of h cnral ank loss funcion (i, g, g, h, h, and λ ), w now choos valus for hos paramrs o minimiz h xpcd inrmporal social loss Th analyical prolm quals h following: (A-) min g, g, h, h, λ E β π π + λ = ( ) ( ) ( q) π = g α( λ h βγ) ( g λ αh) αρ( λ βγ) ε s α = ρ + qε To solv his prolm rquirs h rcursiv susiuion for ack o in h consisn oucoms in quaions (A-8) and (A-9) and hn susiuing h soluions ino h social inrmporal wlfar (loss) funcion Carrying ou h algra lads o h following 3

33 soluion for h xpcd social loss: (A-3) L E β ( π π ) λ( ) + L() I + L( II), = whr (A-4) L() I g + α( λ h βγ) π ( β ) ( ) ( ) ( g λαh αρ λ βγ ) βρ ( βρ) ( ) ( ) + g α( λ h βγ ) π + ( g + λ αh ) αρ( λ + βγ ( ) ( ) λ λρ λρ + + β βρ βρ ) and (A-5) ( II) ( + ) ( + ) λ α αρ λ βγ σ L q g h q + σ + λq ( β) α ( β)( βρ ) β ( β)( βρ ) σ L(I) and L(II) qual h componns of h social wlfar (loss) funcion ha incorporas non-sochasic and sochasic rms, rspcivly Choosing h valus for h following paramrs, g, g, h, h, and λ, o minimiz h xpcd social loss yilds h following condiions: (A-6) g α( λ h βγ ) π (A-7) + = ( ) ( g + λαh αρ λ + βγ)=, and, (A-8) βρ q = = + α ( λ + βγ) βρ + λα As long as h 7 paramrs, g, g, h, h, λ, γ, and γ, of h cnral ank loss funcion saisfy quaions (A-), (A-), (A-6), (A-7), and (A-8), h consisn policy 3

34 will prov opimal No also ha y using quaions (A-6) and (A-7), w can rwri quaions (A-) and (A-) as follows: (A-9) [ g ] ( h ) γ = + λ ρ + βγ ρ or γ = [ g ] + λ ( h ρ ) βρ, and (A-) ( g )[ g ] h ( h ) γ = π + λ ρ + βγ ρ or γ = ( g π )[ g] + λ h( h ρ) βρ By using quaions (A-6) o (A-8), h consisn policy oucoms for h inflaion and mploymn ras (A-8) and (A-9) qual: (A-) (A-) = λα βρ + λα and π π ε βρ = ρ + ε, βρ + λα which qual hir opimal oucoms S quaions (8) and (9) Sp III: Drmining h Cnral Bank Loss Funcion ha Minimizs h Is Expcd Loss An infini numr of soluions for h paramrs g, g, h, h, λ, γ, and γ saisfy quaions (A-6) o (A-) and minimiz h social loss funcion Only on s of hos paramr valus, howvr, will also minimiz h cnral ank loss funcion Tha is, w wan o choos paramr valus for g, g, h, h, and λ o solv h following prolm: (A-3) min g, g, h, h, λ { ( + ) + ( + ) } E β π g g λ h h = λα π = π ε βρ + λα s βρ = ρ + ε βρ + λα Th soluion of his prolm firs rquirs h rcursiv susiuion for ack o ino h mploymn ra quaion Thn susiu h valus of h opimal inflaion and 3

35 mploymn ras ino h cnral ank loss funcion and calcula h valu of h xpcd cnral ank loss as follows: (A-4) () I L ( II) { ( ) ( ) } β π + + λ + = L E g g h h, L + whr L g + g + g g () I ( ) ( ) ( ) ( ) ( ) ( π π ) β βρ βρ + λ + λ ρ + λ h h h h ( ) ( ) ( ) β βρ ( βρ) ( ρ) and L ( II) βρ β λα σ σ g + βρ + λα ( β)( βρ ) ( β ) βρ + λα βρ β ( h ) + λ ρ σ + λ σ βρ + λα ( β)( βρ ) ( β ) L (I) and L (II) qual h componns of h social wlfar (loss) funcion ha incorporas non-sochasic and sochasic rms, rspcivly Choosing h valus for h following paramrs, g, g, h, h, and λ, o minimiz h xpcd cnral ank loss yilds h following rsuls: (A-5) g = π, g =, h = and h = ρ Susiuing hs condiions ino quaions (A-9) and (A-) yilds: (A-6) γ = γ = Using quaion (A-8) and h dfiniion of q ha follows quaion (A-9) producs: (A-7) λ λ βρ = [ /( )] Finally w drmin h opimal cnral ank long-run loss funcion as follows: (A-8) L = ( π π ) + λ ( ) whr, π = π, = ρ, and λ = [ λ/( βρ )] - 33

36 Appndix B Dsign of Monary Policy Ruls Th modl for drmining h monary policy rul quals h following: (B-) min E acd,,, β π π + λ = ( ) ( ) π = a+ π + c + dε s = ρ + α( π π ) + ε π = E ( π) Solv h modl in wo sps Sp I compus h quilirium policy undr h givn policy rul, π = a+ π + c + dε Sp II chooss h paramrs, a,, c and d for h policy rul o minimiz h xpcd oal social loss Sp I: Drminaion of Consisn Policy, Givn h Policy Rul Whn h cnral ank jus follows h rul in quaion (39) and inracs wih h priva scor, h prolm quals: π = a+ π + c + dε (B-) ( ) = ρ + α π π + ε π = E ( π) Solving for h quilirium oucoms for h inflaion and mploymn ras yilds h following: (B-3) π a c = + d ε + and (B-4) = + ( + ) ρ α ε d Sp II: Drmining h Policy Rul ha Minimizs h Expcd Social Loss Now, choos h paramrs a,, c, and d o minimiz h xpcd oal social loss, givn h quilirium oucoms Tha is, solv h following prolm: 34

37 (B-5) min E acd,,, β π π + λ = a c π = + + dε s = ρ + ( + αd ) ε ( ) ( ) W nd firs o susiu rcursivly ino h quilirium oucoms for h inflaion and mploymn ras for ack o and hn calcula h xpcd valu of h social loss funcion as follows: (B-6) L E β ( π π ) λ( ) + L() I + L( II), = whr (B-7) ( ) c π a c a L() I π + + β βρ βρ λ λρ λρ + + β βρ βρ and (B-8) c β L αd σ λ αd σ ( II) ( + ) + ( + ) ( )( ) ( )( β βρ β βρ ) + d σ β L(I) and L(II) qual h componns of h social wlfar (loss) funcion ha incorpora non-sochasic and sochasic rms, rspcivly Choosing h valus for h following paramrs, a,, c, and d, o minimiz h xpcd social loss yilds h following condiions: (B-9) a ( ) = π, c (B-), ( ) = 35

38 (B-) d = λα βρ + λα As a rsul, h opimal policy quals h following: (B-) ( ) λα π = π + π βρ + λα ε, whr (B-3) (B-4) < Th quilirium inflaion and mploymn ras qual h following: = λα βρ + λα and π π ε βρ = ρ + ε, βρ + λα which qual h opimal oucoms S quaions (8) and (9) Appndix C Dsign of Cnral Bank Shor-Run Loss Funcions Th opimizing prolm quals h following sysm: (C-) Sp I: min g, g, h, h, λ E β ( π π ) λ( ) + = min ( π ) λ ( s = ρ + α( π π ) + ε s π = E ( π) g g + h h π Drivaion of Consisn Policy, Givn h Cnral Bank Loss Funcion ) To drmin h consisn or quilirium oucoms, w solv h following prolm: (C-) min π ( π ) λ ( g g + h h ( ) = ρ + α π π + ε s π = E ( π) ) Th quilirium inflaion and mploymn oucoms qual h following: (C-3) g h ( g h ) π = + λα + + λα λαρ λαqε (C-4) = ρ + ε, q and 36

39 whr Sp II: q [/( + λ α )] Drmining h Cnral Bank Loss Funcion ha Minimizs h Expcd Social Loss Now, w choos h paramrs, g, g, h, h, and λ, o minimiz h xpcd oal social loss Th prolm quals h following sysm: (C-5) min g, g, h, h, λ E β π π + λ = ( ) ( ) ( ) π = g + λαh + g+ λαh λαρ λαqε s = ρ + qε W nd firs o susiu rcursivly ino h quilirium oucoms for h inflaion and mploymn ras for ack o and hn calcula h xpcd valu of h social loss funcion as follows: (C-6) L E β ( π π ) λ( ) + L() I + L( II), = whr (C-7) L = g + h + g + h () I ( ) ( ) ( ) ( ) λα π λα λαρ β βρ ( ) ( )( ) λ ( ) g λαh π g λαh λαρ βρ λρ + λρ ( βρ) ( βρ ) ( β ) and (C-8) ( II) ( ) ( )( ) β βρ ( β) ( β)( βρ ) β q L = q g + λ αh λ αρ σ + σ α λq + σ L(I) and L(II) qual h componns of h social wlfar (loss) funcion ha 37

40 incorpora non-sochasic and sochasic rms, rspcivly Choosing h valus for h following paramrs, g, g, h, h, and λ, o minimiz h xpcd social loss yilds h following condiions: (C-9) g + λαh π =, (C-) (C-) g + λαh λαρ=, and βρ λ q = λ = = λ α βρ + λα βρ As a rsul, h quilirium inflaion and mploymn oucoms qual h following: (C-) (C-3) = λα βρ + λα and π π ε βρ = ρ + ε, βρ + λα which qual h opimal oucoms S quaions (8) and (9) As long as h four paramrs, g, g, h, and h, of h cnral ank loss funcion saisfy quaions (C-9) and (C-), h consisn policy undr h myopic cnral ank provs opimal Sp III: Drmining h Cnral Bank Loss Funcion ha Minimizs Is Expcd Loss An infini numr of soluions for h paramrs g, g, h, and h saisfy quaions (C-9) and (C-) and minimiz h xpcd social loss Only on s of hos paramr valus will also minimiz h cnral ank shor-run (priod) loss funcion, whr h cnral ank opras myopically Tha is, w wan o choos paramr valus for g, g, h, and h o solv h following prolm: Equaion (C-) pins down h valu of λ 38

41 (C-4) g, g, h, h { π ( + ) + λ ( + ) } min E g g h h λα π = π ε βρ + λα s βρ = ρ + ε βρ + λα Sinc h cnral ank opras myopically, w do no nd o rcursivly susiu for ack o Thus, w mrly susiu h consisn or quilirium oucoms ino h cnral ank loss funcion and minimiz wih rspc o g, g, h, and h Th xpcd valu of h cnral ank loss funcion wih h consisn or quilirium oucoms rducs o h following: (C-5) q E ( L ) = ( g π + g ) + σ + λ h ( h ) α + ρ + λ q σ Choosing h paramrs g, g, h, and h o minimiz h xpcd valu of h cnral ank loss funcion producs h following rsuls: (C-6) g π + g = and (C-7) ( ) h + h ρ = To mak h inflaion arg in h cnral ank loss funcion indpndn of h sa varial, a rasonal assumpion, w s h paramr valus as follows: (C-8) g = π, g =, h = and h = ρ Finally, his producs h opimal cnral ank loss funcion as follows: (C-9) L = ( π π ) + λ ( ) whr, π = π, = ρ, and λ = [ λ/( βρ )] This loss funcion machs h loss funcion drivd for h long-run inrmporal opimizing cnral ank S quaion (3) 39

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43 Svnsson, L EO Inflaion Targing as a Monary Policy Rul Journal of Monary Economics, 999, 43(3), pp Svnsson, Lars E O "Inflaion Targing: Should I B Modld As An Insrumn Rul Or A Targing Rul?," Europan Economic Rviw,, 46 (4-5,May), Svnsson, Lars EO (3) Wha is Wrong wih Taylor Ruls? Using Judgmn in Monary Policy hrough Targing Ruls, Journal of Economic Liraur 4, Svnsson, Lars EO (5)"Targing Ruls vs Insrumn Ruls for Monary Policy: Wha Is Wrong wih McCallum and Nlson?" Fdral Rsrv Bank of S Louis Rviw 87 (5) Walsh, C E, (995) Opimal Conracs for Cnral Bankrs Amrican Economic Rviw 85 (March), 5-67 Walsh, C, (3) Spd Limi Policis: Th Oupu Gap and Opimal Monary Policy Amrican Economic Rviw, 93(), (March), Yuan H, and S M Millr, (7a) Consisn Targs and Monary Policy: A No hp://idasrpcorg//pmi6hml Yuan, H, and S M Millr, (6) Th Making of Opimal and Consisn Policy: An Implmnaion Thory Framwork for Monary Policy hp://idasrpcorg//pmi6hml Yuan, H, S M Millr and L Chn (7) Th Making of Opimal and Consisn Policy: An Analyical Framwork for Monary Modls hp://idasrpcorg//pmi6hml 4