Target Controllability and Time Consistency: Complement to the Tinbergen Rule

Transcription

1 Targe Conrollailiy and Time Consisency: Complemen o he Tinergen Rule Huiping Yuan Xiamen Universiy Sephen M. Miller Universiy of Nevada-Las Vegas Universiy of Connecicu Working Paper Decemer Fairfield Way, Uni 063 Sorrs, CT Phone: (860) Fax: (860) hp:// This working paper is indexed on RePEc, hp://repec.org

2 Targe Conrollailiy and Time Consisency: Complemen o he Tinergen Rule Huiping Yuan a,, Sephen M. Miller a Finance Deparmen, Xiamen Universiy, Xiamen, Fujian 36005, China hpyuan@xmu.edu.cn Economics Deparmen, Universiy of Nevada, Las Vegas, Las Vegas, NV , USA sephen.miller@unlv.edu Asrac: The Tinergen Rule saes ha achieving he desired arges requires an equal numer of insrumens. This paper shows ha ime inconsisency does no exis in he case of an equal numer of insrumens and arges. Targe unconrollailiy and ime inconsisency, however, emerge as prolems in he case of fewer insrumens han arges. In his case, we oain a neceary and sufficien condiion for join asympoic conrollailiy of arge values, which complemens he Tinergen rule. The condiion is idenical under commimen and under discreion. If he condiion does no hold, he seady-sae values of arge variales regre o heir respecive arge values. The paper solves oh prolems y deermining he cenral ank s arge values of inflaion and oupu as well as he relaive weigh eween sailizing inflaion and oupu. Inuiively, a proper arge value rade-off solves arge unconrollailiy, whereas a proper relaive weigh achieves opimal arge variailiy rade-off and solves ime inconsisency. As a resul, arge values are conrollale, esalishing moneary policy crediiliy. Discreionary policy under he designed lo funcion, which replicaes opimal policy under he social lo funcion, proves ime-consisen. In addiion, we idenify wo siuaions where he delegaed weigh equals he social weigh, providing addiional insigh ino ime inconsisency. J.E.L. Claificaion: E5, E58 Key words: Targe conrollailiy; Time inconsisency; Opimal policy; Discreionary policy; Trade-off This paper is a research resul of he Humaniies and Social Science Projec Theoreical and empirical research on China moneary policy rules in he open economy (Gran No. 0YJA79038) suppored y he Minisry of Educaion of China, as well as of he Key Projec Research on macro-prudenial policy, moneary policy, and governance arrangemens (Gran No. AJY0) suppored y he Naional Social Science Fund of China. Corresponding auhor. hpyuan@xmu.edu.cn. Pos ox 44, Xiamen Universiy, Xiamen, Fujian 36005, China.

3 . Inroducion The Tinergen Rule saes ha achieving he desired arges requires an equal numer of insrumens. This paper shows ha ime inconsisency does no exis in he case of an equal numer of insrumens and arges. Targe unconrollailiy and ime inconsisency, however, emerge in he case of fewer insrumens han arges. Time inconsisency also arises from expecaions (a game conex). This paper addrees oh iues, and argues ha policy makers mus addre arge unconrollailiy efore addreing ime inconsisency o avoid enanglemen of he wo iues when addreing ime inconsisency. Targe conrollailiy is a long-sanding policy iue (e.g., Kalman [960]; Tinergen [963]; Preson [974]; and Aoki [975]). Brocke and Mesarović (965) define hree ypes of conrollailiy (reproduciiliy): poin, pah, and asympoic conrollailiy. Poin conrollailiy means ha policy makers can achieve cerain arge values a a specified poin in ime. In pracice, policy makers proaly desire a sronger noion han poin conrollailiy, pah conrollailiy. Pah conrollailiy means ha policy makers can make arge variales follow some prescried rajecories over a cerain ime inerval (e.g., Aoki [975]). Oviously, pah conrollailiy implies poin conrollailiy, whereas he converse generally does no hold. Asympoic conrollailiy means ha policy makers can reach he arge values a infiniy. Pah. Brocke and Mesarović (965) use reproduciiliy insead of conrollailiy. The erminology reproduciiliy appears in he engineering lieraure, whereas he same concep of conrollailiy appears in he economics lieraure. Also, hey inroduce four raher han hree ypes: poin, locally pah, uniformly pah, and asympoic conrollailiy. We here refer o oh locally and uniformly pah conrollailiy as pah conrollailiy. In addiion, he lieraure also calls pah conrollailiy, funcional or perfec conrollailiy.

4 conrollailiy plays a growing role in dynamic models of economic policy. This paper considers oh pah and asympoic conrollailiy, which prove useful conceps in descriing he equilirium pahs of arge variales. Anoher long-sanding policy iue is he ime inconsisency of opimal policy (e.g., Kydland and Presco [977]; and Calvo [978]). Researchers ypically delegae a lo funcion o he cenral ank o solve he ime inconsisency of opimal policy, for insance, he conservaive cenral anker of Rogoff (985), he inflaion conrac of Walsh (995), he employmen conrac of Chorareas and Miller (003), he inflaion arge of Svenon (997), he nominal income growh arge of Beesma and Jensen (999) and Jensen (00), he price-level arge of Vesin (006), he consisen arge of Yuan, e al. (0), and so on. Though he various delegaion schemes osensily differ and lead o differen inerpreaions, several delegaions prove eenially idenical. 3 The delegaion approach solves he ime-inconsisency prolem in a sraighforward fashion. Under he delegaed lo funcion, he cenral ank operaes moneary policy wih discreion. As a resul, moneary policy is ime consisen. The main iue, however, is wheher discreionary policy can approximae or even reproduce opimal. See, for example, Nyerg and Vioi (978), Buier and Gersoviz (98, 984), Wohlmann (98, 985), Tondini (984), and Maas and Nijmeijer (994). Besides exending Tinergen s saic conrollailiy o dynamic conrollailiy (Preson [974]), he lieraure exends conrollailiy from one-decision-maker o mulipledecision-makers (game) conex. See Acocella and Di Barolomeo (006), Acocella e al. (006, 007), and Hughes Halle e al. (00, 0) for conrollailiy in a game conex. 3. Yuan, e al. (0) compare heir designed lo funcion wih he lo funcions in Svenon (997) wihou employmen persisence, Walsh (995), and Chorareas and Miller (003), and show ha he four lo funcions generae idenical resuls wih respec o he policy decision. Svenon (997) also oserves ha he inflaionarge-conservaive lo funcion wihou employmen persisence mimics he linear inflaion conrac in Walsh (995). 3

5 policy. Three oservaions lead o he conclusion ha policy makers mus addre arge unconrollailiy efore addreing ime inconsisency o avoid enanglemen of he wo iues when addreing ime inconsisency. Firs, his paper derives he proposiion wihin a new- Keynesian model srucure ha he desired arge values are joinly asympoically conrollale no maer wheher policies operae wih commimen or wih discreion, if and only if hey saisfy he seady-sae Phillips curve wha we call, he arge level rade-off equaion. In oher words, conrollale arge values mus relae o each oher hrough he macroeconomic srucure, and do no depend on he policy implemenaion syle -- commimen or discreion. Second, his paper disinguishes eween he arge value (level) and arge sailizaion (variailiy) rade-offs, which respecively correspond o arge unconrollailiy and ime inconsisency. The arge level rade-off does no depend on he policy implemenaion syle as jus noed, whereas he arge variailiy rade-off does and changes wih he relaive weigh eween arge sailizaions. Third, we oserve ha he consan average inflaion ias comes from fewer insrumens han arges, raher han from ime inconsisency as he lieraure srees. Boh he sae-coningen inflaion ias and he sailizaion ias, however, do arise from ime inconsisency. 4 The lieraure and we oh agree ha he inflaion ias arises from an overly amiious oupu arge value. Wih a moderae oupu arge value delegaed o he cenral ank, he consan average inflaion ias disappears. The lieraure misakes arge unconrollailiy for ime inconsisency. 4. See Svenon (997) for he hree iases he consan average inflaion ias, he sae-coningen inflaion ias, and he sailizaion ias. 4

6 The paper addrees oh iues y designing he appropriae cenral ank lo funcion ha deermines he cenral ank s long- and shor-run arge values of inflaion and oupu as well as he relaive weigh eween sailizing inflaion and oupu. Inuiively, delegaing proper arge values solves he iue of arge unconrollailiy. A proper weigh achieves opimal arge sailizaion rade-off and solves ime inconsisency. The delegaed arge values and weigh parameer play a role as follows. The long-run arge values guide moneary policy owards a correc end, resuling in asympoical conrollailiy and eliminaing he consan average inflaion ias. Though he cenral ank, using discreion, moves oward a correc end, i sill deviaes from he opimal equilirium pahs during he evoluion oward he correc end, causing a welfare lo. The shor-run arge values furher ind moneary policy o follow he opimal equilirium pahs, resuling in pah conrollailiy and removing sae-coningen inflaion ias. The relaive weigh eliminaes he sailizaion ias, making he arge sailizaion rade-off wih discreion under he cenral ank lo funcion replicae he arge sailizaion rade-off wih commimen under he social lo funcion. As a resul, he cenral ank achieves he delegaed arge values, esalishing moneary policy crediiliy. Discreionary policy under he cenral ank lo funcion, which removes he hree iases and replicaes opimal policy under he social lo funcion, proves opimal and ime-consisen. The paper unfolds as follows. Secions deermines oh opimal and discreionary equilirium pahs of arge variales in a hyrid new-keynesian model, given a social lo 5

7 funcion. 5 Secion 3 summarizes he resuls of Secion in a proposiion, he neceary and sufficien condiion for join asympoic conrollailiy of arge values wih commimen and wih discreion. This proposiion complemens he Tinergen rule. Secion 4 develops an approach o solving oh arge unconrollailiy and ime inconsisency y designing he cenral ank lo funcion. Secion 5 concreely designs cenral ank lo funcions for he hyrid new-keynesian model, using he approach in Secion 4. Secion 6 discues he delegaed weigh o deepen our undersanding of ime inconsisency. Secion 7 summarizes and concludes.. Opimal policy and discreionary policy We simply adop he hyrid new-keynesian model in Clarida e al. (999) wihou much descripion. The model comines he poiiliy of inflaion expecaions and inflaion ineria, and reduces o a purely forward-looking or a purely ackward-looking model y choosing he exreme values of he parameer ha indexes expecaions acro he forward- and ackwardlooking dimensions. See Clarida e al. (999) for more deails. The social ineremporal lo funcion equals E0 L, () 0 where (0 < < ) is he discoun facor and E is he expecaions operaor. The period lo funcion equals he following: L y y, () where is he inflaion rae, y is he log-level of oupu, is he socially desirale inflaion rae, 5. Yuan and Miller (03) carry ou similar analysis for a new-claical model. 6

8 y is he efficien, firs-es log-level of oupu, and is he social weigh on oupu sailizaion relaive o inflaion sailizaion around heir respecive arge values. For convenience, we aume a ime-invarian efficien, firs-es oupu level, y. Or we can inerpre y as he seady-sae level. Thus, y y denoes he welfare-relevan oupu gap. 6 Similarly, we aume a ime-invarian naural, second-es oupu level, y n, such ha he welfare-relevan oupu gap, y y, equals x x, where x y y n denoes he oupu gap eween he acual and naural oupu levels and x y y n denoes he gap eween he firs- and second-es oupu levels. Generally, imperfecions and/or disorions exis, and we aume ha x 0. Rewrie he lo funcion as follows: L x x. (3) Aggregae supply equals an expecaions-augmened Phillips curve wih forward-looking expecaions and endogenous inflaion x E u, (4) where ( >0) is he sensiiviy of he inflaion rae o he oupu gap, indexes he degree of lagged versus expeced fuure inflaion raes, and u is a cos-push shock ha follows an AR() proce, u ˆ u u, where 0, and u ˆ is a whie noise residual. We do no inroduce aggregae demand (IS curve), which involves a nominal ineres rae as he policy insrumen. Once we deermine he opimal pahs for, x 0 using he social lo funcion and he Phillips curve, oh of which do no involve he ineres rae, hen we can 6. Roemerg and Woodford (997) and Benigno and Woodford (005) jusify he lo funcion y deriving i from (a second-order approximaion of) he households uiliy funcion. 7

9 pin down he opimal pah of ineres raes hrough he IS curve. So he Phillips curve proves criical for policy... Opimal policy, ime inconsisency, and arge unconrollailiy... Opimal policy The consolidaed firs-order condiion of opimal policy under he social ineremporal lo funcion () wih period lo funcion (3) sujec o he Phillips curve in Eq. (4) equals 7 for 0, (5a) E0x x E0x x E0x x for. (5) E0 x 0 x E0x x 0 Comining he firs-order condiions (5a) and (5) and he Phillips curve (Eq. 4) leads o opimal equilirium pahs of arge variales 8 ˆ ˆ q u u for, ˆ ˆ (6a) x x x x qu for, (6) where is a roo of he characerisic equaion 4 3 a a 0 wih (7a) a, (7) a, (7c) q d a, (7d), (7e) d a 7. See Eqs. (A.3a) and (A.3) in Appendix A. 8. See Eqs. (A.35) and (A.4) and heir derivaion in Appendix A. See (A.39) and (A.4) for he soluion for =0. 8

10 ˆ x, and (7f) a xˆ ˆ. (7g) For he purely forward-looking, new-keynesian model (= 0), he soluion in Eqs. (6a) and (6) reduces o he soluion in Clarida e al. (999, ). 9 Eqs. (7a), (7), and (7c) deermine, wha we call, he sysem convergence persisence,. We canno deermine wheher a roo (0,) exiss for any (0,). Bu for he wo exreme cases of = 0 or, he characerisic Eq. (7a) reduces o 0, and a roo (0,) does exis. Tha is, he sysem can converge under opimal policy for he wo exreme cases. In addiion, wheher a roo (0,) exiss, does no relae o he persisence of he cos-push shock,. The irrelevance of he cos-push shocks reflecs he quadraic lo funcion and he linear Phillips curve wih addiive shocks. Oserving he opimal equilirium pah of inflaion rae, Eq. (6a), we find an ineresing value. Using he AR() proce of he cos-push shocks, ransform Eq. (6a) ino ˆ ˆ q u uˆ, where (8) c c, and 0 c. (9) The criical value, c, is meaningful. For convenience, we define he Phillips curve as principally ackward-looking (forward-looking) if c (0 c ). Wheher he Phillips 9. If = 0 or, hen a = 0, d, d, and Eq. (7a) reduces o 0. Thus,. Therefore, d or d. Thus, q d. 9

11 curve exhiis principally ackward- or forward-looking ehavior resuls in compleely differen (posiive or negaive) responses of policy makers o cos-push shocks and, we will see in Secion 5, very differen delegaions o he cenral ank. The criical value c depends on oh he discoun facor and he cos-push shock persisence. 0, and 0. 0 Tha is, a more imporan fuure and/or le c c persisen cos-push shocks lead o a higher criical value and, hus, he Phillips curve for a given inflaion persisence more likely exhiis principally forward-looking ehavior ( ). Accordingly, wheher he Phillips curve exhiis principally forward- or ackward-looking ehavior depends on he hree dynamic parameers,, and, raher han merely.... Time inconsisency The firs-order condiions (5a) and (5) sugges he ime inconsisency of he opimal soluion. Opimal policy requires ha he presen period follows condiion (5a) and ha fuure periods follow condiion (5). In pracice, however, he cenral ank re-minimizes he lo funcion each period and, hus, always follows condiion (5a)...3. Targe unconrollailiy In addiion o ime inconsisency, opimal policy sill faces arge unconrollailiy. We adop opimaliy from he imele perspecive, and analyze Eqs. (6a) and (6) for. The opimal equilirium pahs of he arge variales in Eqs. (6a) and (6) consis of deerminisic and c 0. 0 and c c 0 for 0.. Woodford (999) inroduces he concep of opimaliy from a imele perspecive, which means he policy he cenral ank o which i would have wished o commi iself o a a dae far in he pas. (93, ialics in original). 0

12 random componens. We denoe he deerminisic componens as follows ˆ ˆ ˆ ˆ, (0a) xˆ xˆ x xˆ x xˆ, (0) and random componens relaed o cos-push shocks as follows u ˆ q u u, and (a) x ˆu qu. () The deerminisic componens converge o ˆ, xˆ, if <, ˆ xˆ lim ˆ, and (a) lim xˆ. () Thus, ˆ and x ˆ are he seady-sae values of he inflaion rae and he oupu gap under opimal policy. As Eqs. (0) show, he deerminisic componen of he inflaion rae (he oupu gap) in each period, ˆ xˆ, equals a weighed average of is lag, x ˆ xˆ, and is seady-sae value,, wih weigh on is lag. In oher words, he wo arge variales evolve o heir respecive seady-sae values wih he sysem convergence persisence,. Targe conrollailiy does no depend on random shocks in our specific model wih a quadraic lo funcion, a linear Phillips curve, and addiive shocks. Accordingly, we ignore shocks and consider he deerminisic componens when we consider arge conrollailiy. The ideal social arge values of inflaion rae and oupu gap are and x whereas he realized arge values in he long run under opimal policy equal ˆ and x ˆ. Thus, he policy makers

13 generally canno asympoically conrol eiher arge, since ˆ and xˆ x...4. Neceary and sufficien condiion for join asympoic conrollailiy Though he social arge values are generally no conrollale, we oain a neceary and sufficien condiion for join asympoic conrollailiy of oh arge values. Tha is, oh arge values, and x, of he inflaion rae and he oupu gap are joinly asympoically conrollale under imele perspecive opimal policy if and only if hey saisfy he condiion x. (3) Proof. Sufficiency. Rearranging Eq. (3) gives x. (4) Susiuing Eq. (4) ino Eq. (7f) generaes ˆ. Using Eq. (7g), ˆ, and Eq. (4) in sequence produces conrollale. xˆ x. Tha is, he wo arge values, and x, are joinly asympoically Neceiy. Eq. (7g) equals he seady-sae Phillips curve ˆ xˆ ˆ ˆ. (5) If he wo arge values, and x, are joinly asympoically conrollale, hen ˆ and xˆ x. Replacing ˆ wih and x ˆ wih x in Eq. (5) produces Eq. (3). QED The neceary and sufficien condiion in Eq. (3) exhiis a rade-off eween arge values. Accordingly, we call Eq. (3) he arge level rade-off equaion. In addiion, oserving Eqs. (3) and (5), we conclude ha oh arge values are asympoically conrollale if and only if hey saisfy he seady-sae Phillips curve.. If = 0, hen he policy makers can asympoically conrol he inflaion rae ( ˆ ).

14 .. Discreionary policy and arge unconrollailiy... Discreionary policy The equilirium pahs of inflaion rae and oupu gap under discreion equal 3 q u, and (6a) x x x x qu u, (6) where is he roo of he characerisic equaion a 0, (7a) 3, (7) q, (7c) a x x, and (7d). (7e) Does a le-han-one roo exis for he characerisic equaion (7a)? Consider he wo exreme cases of = 0 and =. For he case = 0, he characerisic equaion (7a) reduces o 0, and is wo roos are 0, which is a degeneraive soluion, and. The sysem diverges under he discreionary policy when = 0, in conras o converging under he opimal policy. For he case =, he characerisic equaion 0. The larger roo exceeds one and he smaller (7a) reduces o roo lies eween zero and one. The sysem can converge under discreion when =. When he lag index goes from 0 o, he sysem changes from divergence o convergence. The sysem 3. See Eqs. (B.8) and (B.0) and heir derivaion in Appendix B. 3

15 may also go hrough divergence, convergence, divergence, convergence, and so on.... Targe unconrollailiy Similar o opimal policy, he discreionary equilirium pahs of he arge variales in Eqs. (6) consis of deerminisic and random componens. We denoe he deerminisic componens as follows, (8a) x x x x x x, (8) and random componens or reacions o cos-push shocks as follows u q u, and (9a) x q u u. (9) u The deerminisic componens converge o, x, if, x lim, and (0a) lim x. (0) Thus, and x are he seady-sae values of he inflaion rae and he oupu gap under discreionary policy. When we consider conrollailiy, we ignore shocks and consider he deerminisic componens. The ideal social arge values of inflaion rae and oupu gap are and x whereas he seady-sae values equal and x. Tha is, he policy maker generally canno asympoically conrol eiher inflaion rae or oupu gap ( and x x )...3. Neceary and sufficien condiion for join asympoic conrollailiy The same conclusion emerges under discreionary policy. Boh arge values, and x, of he 4

16 inflaion rae and he oupu gap are joinly asympoically conrollale under discreionary policy if and only if hey saisfy he condiion in Eq. (3). Proof. Sufficiency. If and x saisfy Eq. (3), hen Eq. (4) holds. Susiuing Eq. (4) ino Eq. (7d) generaes. Using Eq. (7e),, and Eq. (4) in sequence produces x x. Tha is, he wo arge values, and x, are asympoically conrollale. Neceiy. Eq. (7e) equals he seady-sae Phillips curve x. () If he wo arge values, and x, are joinly asympoically conrollale, hen and x x. Replacing wih and x wih x in Eq. () produces Eq. (3). QED 3. Complemen o he Tinergen rule We summarize he neceary and sufficien condiion for join asympoic conrollailiy of arge values in a proposiion. This proposiion complemens he Tinergen rule. We also show he regreive naure of seady-sae values of arge variales, and indicae ha he consan average inflaion ias comes from fewer insrumens han arges, raher han from ime inconsisency as he lieraure srees. In addiion, we claify rade-offs ino wo ypes -- arge level and arge variailiy rade-offs, which correspond respecively o arge unconrollailiy and ime inconsisency, and argue ha policy makers mus addre arge unconrollailiy efore addreing ime inconsisency. 3.. Neceary and sufficien condiion for join asympoic conrollailiy The conrollailiy lieraure usually considers he equaions of moion, he availale insrumens, and he iniial sae of he sysem. Raher, we find he neceary and sufficien condiion for he 5

17 join asympoic conrollailiy of arge values. Proposiion. If he sysem, wih period social lo funcion in Eq. (3) sujec o a linear Phillips curve in Eq. (4), converges wih commimen, and x are joinly asympoically conrollale if and only if hey saisfy he seady-sae Phillips curve in Eq. (3). The resul also holds wih discreion. Proof. We prove he proposiion for opimal policy in a more simple way. Sufficiency. If and x saisfy Eq. (3), hen comining Eq. (3) wih he Phillips curve produces he Phillips curve around he arge values as follows x x E u. () Now, he opimizaion prolem equals he minimizaion of he social ineremporal lo funcion wih period lo funcion (3) sujec o he Phillips curve around he arge values in Eq. (). Oviously, he equilirium pahs, x x under opimal policy will converge o (0, 0), if he sysem converges. Tha is,, x converge o, x. Therefore, and x are joinly asympoically conrollale. Neceiy. If and x are joinly asympoically conrollale under opimal policy, hen ˆ and xˆ x hold. The seady-sae Phillips curve under opimal policy in Eq. (5) leads o he condiion in Eq. (3). The proof for discreionary policy follows exacly he same as ha for opimal policy y using Eq. () insead of Eq. (5) and y noing ha he difference eween discreion and commimen merely reflecs he differen sequence of opimizaions of decision makers. QED 6

18 3.. Complemen o he Tinergen rule 3... The Tinergen rule () Le =0, and an equal numer of insrumens and arges exiss -- one insrumen and one arge. In his case, he cenral ank only cares aou inflaion. I always ses and achieves whereas oupu flucuaes wih shocks, x u. Time inconsisency does no exis Complemen o he Tinergen rule ( ) If 0, fewer insrumens han he numer of arges exis -- one insrumen and wo arges. In his case, arge values are generally unconrollale and ime inconsisency exiss as we discu in Secion. The proposiion suggess ha if he social arge values happen o saisfy he seady-sae Phillips curve, hey are joinly asympoically conrollale. Join asympoic conrollailiy reflecs he divine coincidence (Blanchard and Galí [007]). 4 Oviously, infinie pairs of arge values can lead o join asympoic conrollailiy. Consider he pair, n 0 and x 0 i.e., y y o he firs-es equilirium x 0. Tha is, a zero inflaion rae corresponds, and vice versa. Usually, monopolisic and/or ax disorions exis, resuling in he naural oupu elow he efficien oupu x 0. The inefficiency requires a posiive inflaion rae 0 according o Eq. (4). This explains o some exen why argeing a posiive inflaion rae prevails in he real world. 4. Blanchard and Galí (007) define divine coincidence o mean ha he sailizaion of he inflaion rae auomaically leads o he sailizaion of he oupu gap, join sailizaion. We orrow he concep here o mean join asympoic conrollailiy, focusing on he levels raher han he variailiy of inflaion and oupu. 7

19 3..3. The regreive naure of seady-sae values of arge variales The social arge values, and x, which come from he households uiliy funcion, generally do no saisfy he seady-sae Phillips curve, which reflecs he ehavior of he firm. Therefore, he arge values, and x, are generally unconrollale under oh commimen and discreion. The seady-sae values of arge variales, however, exhii regreion o he arge values, and x. To see his, ransform he expreions of seady-sae values under commimen and under discreion in Eqs. (7f), (7g), (7d) and (7e) as follows: ˆ x, (3a) a ˆ x x x, (3) a x, and (4a) x x x. (4) The expreions in he rackes equal he form of he seady-sae Phillips curve, and he negaive coefficiens efore he rackes reveal he regreive proce. Wih a high arge value of oupu gap relaive o ha of inflaion rae (i.e., x ), he seady-sae value of he oupu gap falls elow is arge value (i.e., xˆ x and x x ), whereas he seady-sae value of he inflaion rae rises aove is arge value (i.e., ˆ and ). The regreive naure of he seady-sae values of arge variales explains he inflaion ias. An overly amiious oupu arge value implies a relaively low inflaion arge value. A low inflaion arge value leads o he seady-sae value of inflaion higher han is arge value. 8

20 Boh he lieraure and we agree ha he inflaion ias resuls from an overly amiious oupu arge value. The reason for he inflaion ias, however, is differen. The lieraure srees ha he inflaion ias arises from ime inconsisency, whereas we argue ha he consan average inflaion ias comes from fewer insrumens han arges and ha he sae-coningen inflaion ias and he sailizaion ias do arise from ime inconsisency. When arge values happen o saisfy he seady-sae Phillips curve, he consan average inflaion ias disappears Targe level rade-off vs. arge variailiy rade-off The proposiion suggess ha a rade-off exiss eween arge values. A rade-off also exiss eween arge sailizaion, as Eqs. () and (9) show. Oviously, he arge sailizaion rade-off under commimen differs from ha under discreion, resuling in he sailizaion ias, which prevails in discreionary policy. We disinguish eween wo ypes of rade-offs arge level and arge variailiy rade-offs. They correspond respecively o arge unconrollailiy and ime inconsisency. Targe level rade-off occurs in he long run, independen of policy implemenaion syle -- commimen or discreion, as he proposiion reveals. The arge sailizaion rade-off occurs in he shor run and depends on he policy implemenaion syle, changing wih he relaive weigh eween arge sailizaions. In addiion, he oservaion ha he consan average inflaion ias comes from fewer insrumens han arges, raher han from ime inconsisency as he lieraure srees, implies ha he lieraure misakes arge unconrollailiy for ime inconsisency. As a resul, we ackle arge unconrollailiy efore ackling he iue of ime 5. We also discu he source of inflaion ias in Susecion

21 inconsisency, avoiding enangling he wo iues when ackling ime inconsisency. 4. An approach o solving oh arge unconrollailiy and ime inconsisency We develop an approach o solving oh arge unconrollailiy and ime inconsisency. Inuiively, a proper arge value rade-off solves arge unconrollailiy, whereas a proper relaive weigh eween arge sailizaions achieves he opimal arge variailiy rade-off and solves ime inconsisency. As a resul, we design he cenral ank lo funcion y deermining he cenral ank s long- and shor-run arge values of he inflaion rae and he oupu gap, as well as he weigh on sailizing he oupu gap relaive o sailizing he inflaion rae, given he social lo funcion and economic srucures. Aume ha he cenral ank operaes policy discreionarily. The long- and shor-run arge values and he weigh parameer play a role as follows. The long-run arge values guide moneary policy owards a correc end, resuling in asympoical conrollailiy and eliminaing he consan average inflaion ias. Though he cenral ank, using discreion, moves oward a correc end, i sill deviaes from he opimal equilirium pahs during he evoluion oward he correc end, causing a welfare lo. The shor-run arge values furher ind moneary policy o follow he opimal equilirium pahs, resuling in pah conrollailiy and removing sae-coningen inflaion ias. The relaive weigh eliminaes he sailizaion ias, making he arge sailizaion rade-off wih discreion under he cenral ank lo funcion replicae he arge sailizaion rade-off wih commimen under he social lo funcion. As a resul, discreionary policy under he cenral ank lo funcion removes he hree iases and replicaes opimal policy under he social lo funcion. 0

22 4.. Join asympoic conrollailiy wih long-run arge values We denoe x and, respecively, he long-run arge values of he inflaion rae, he oupu gap, and he relaive weigh on sailizing oupu relaive o sailizing inflaion. Tha is, he cenral ank period lo funcion equals L x x. (5) Now, we deermine he long-run arge values and x. As we know, he social arge values, and x, are generally unconrollale, and wha can es achieve in he long run is ˆ and x ˆ. Wih a pragmaic aiude, we should delegae he moderae arge values of ˆ and ˆ x, no he amiious values and x, o he cenral ank. Tha is, L ˆ ˆ x x. (6) Since ˆ and x ˆ saisfy he seady-sae Phillips curve, hey are joinly asympoically conrollale wih discreion according o he proposiion. Tha is,, x ˆ, xˆ, x, (7) where and x denoe, respecively, he seady-sae values of he inflaion rae and he oupu gap wih discreion under he cenral ank lo funcion. Acually, we can conrol ˆ and x ˆ y noing he lo funcion (6) and he Phillips curve around he opimal seady sae, using Eqs. (4) and (5), ˆ ˆ ˆ ˆ x x E u, (8) which is equivalen o he original Phillips curve. crediiliy. Moderae and achievale arge values, ˆ and x ˆ, esalish moneary policy

23 4.. Join pah conrollailiy wih shor-run arge values The desired rajecories of he arge variales equal he opimal equilirium pahs,, x 0 no merely he end poins ˆ, xˆ ˆ ˆ,. To achieve pah conrollailiy (o ind he cenral ank o follow he opimal equilirium pahs), aume a sae-coningen, shor-run arge values, denoed as and x. Three requiremens mus exis for meaningful shor-run arge values. One, lim and lim x x. Tha is, we achieve he long-run arge values sep y sep hrough shor-run arge values. Two, he shor-run arge values mus evolve wih persisence o ensure ha arge variales also evolve, on average, wih persisence. Three, we mus impose predeermined shor-run arge values. The predeermined arge values, hough sae-coningen, are feasile in pracice. We modify he cenral ank lo funcion in Eq. (5) as follows: L x x. (9) We specify and x so ha under he lo funcion (9), he deerminisic componens of he equilirium pahs of he inflaion rae and he oupu gap under discreion, denoed as, x 0, exacly replicae he desired rajecories,, x 0, x ˆ, xˆ ˆ ˆ. Tha is, for all. (30) 4.3. Eliminaing sailizaion ias wih a proper weigh Wih he shor-run arge values, discreionary policy is pah conrollale and, hus, eliminaes he consan average and sae-coningen inflaion iases. This leaves he sailizaion ias. We can eliminae his ias y deermining a proper weigh,, such ha u u u u, ˆ, ˆ x x, (3) u u where, x denoe he reacions o shocks wih discreion under he designed lo funcion.

24 Acually, we pin down he weigh hrough one of he wo equaions, u u u ˆ or x xˆ. Once u one equaion holds, he oher equaion also holds ecause he Phillips curve links he inflaion rae and he oupu gap. Now, we design cenral ank lo funcions using he aove approach. 5. Designing he cenral ank lo funcion 5.. Join asympoic conrollailiy wih long-run arge values, ˆ and x ˆ The equilirium pahs of arge variales wih discreion under period lo funcion (6) equal 6 ˆ ˆ q u, and (3a) ˆ ˆ x x x x q u u where is a roo of he characerisic equaion a 3, (3) 0, (33a), and (33) q a. (33c) For he case 0, 7 we can conclude ha ˆ and x ˆ are joinly asympoically conrollale, if he characerisic equaion (33a) poees a roo le han one,. Generally, he sysem is no pah conrollale under he discreionary policy since, where and are deermined, respecively, y Eqs. (33a) and (7a). Wih he change of he inflaion ineria index, he Phillips curve and he sysem exhii differen ehavior. We discu 6. See Eqs. (C.) and (C.) in Appendix C. The forms of he soluion are exacly he same as hose for discreionary policy wih social arge values, and x, in Eqs. (6) and (7). 7. When = 0, he sysem diverges. See Susecion... 3

25 he delegaions of he shor-run arge values and he relaive weigh ased on he parameer. 5.. Join pah conrollailiy wih long-run arge values and Eliminaion of sailizaion ias wih an idenical weigh when = The case = proves ineresing. Specifically, if he Phillips curve in Eq. (4) exhiis purely ackward-looking ehavior wih no expecaions as follows: x u or ˆ ˆ ˆ x x u, (34) hen we do no need o delegae shor-run arge values nor o change he relaive weigh. We only need o delegae moderae long-run arge values, ˆ and x ˆ. Tha is, he cenral ank adops he simple lo funcion as follows: L ˆ ˆ x x. (35) Wih he simple lo funcion (Eq. 35), he discreionary policy proves pah conrollale and eliminaes sailizaion ias! Tha is, (opimal sysem convergence persisence),, x ˆ, xˆ u u (pah conrollailiy), and ˆ (opimal sailizaion rade-off). 8 Inuiively, when he Phillips curve equals Eq. (34), which does no involve expecaions, he opimal prolem reduces o a conrol heory, raher han game heory, prolem. Tha is, only one decision maker exiss raher han muliple decision makers. Time inconsisency does no exis. The consolidaed firs-order condiions under opimal policy also reveal non-exisence of ime inconsisency. The condiion for = 0 (Eq. 5a) equals he condiion for (Eq. 5), when =, indicaing ha he opimal policy is ime-consisen. Though ime inconsisency does no exis, he iue of arge unconrollailiy does under he social lo funcion. The arge 8. Appendix C provides he deails of he derivaions. 4

26 variales' seady-sae values in Eqs. (7f) and (7g) wih = equal ˆ x,0. Thus, ˆ, xˆ, x and x, are unconrollale, if x 0., if x, xˆ 0. Tha is, he social arge values, In sum, when =, ime inconsisency does no exis and we only face arge unconrollailiy. As a resul, we do no need o delegae o he cenral ank he shor-run arge values, which commi or ind cenral ank ehavior. We also do no need o delegae o he cenral ank a differen weigh from ha of sociey o change he rade-off eween arge sailizaions. We solve arge unconrollailiy y delegaing moderae and achievale long-run arge values, ˆ and x ˆ, o he cenral ank Join pah conrollailiy wih lagged, shor-run arge values and eliminaion of sailizaion ias wih a proper weigh when c Join pah conrollailiy wih lagged, shor-run arge values Wheher ime inconsisency exiss, as well as wheher he sysem converges wih he long-run arge values under he discreionary policy, we consider delegaing shor-run arge values. The requiremens for shor-run arge values sugges ha we can aume he following values ˆ ˆ ˆ ˆ, and (36a) x ˆ ˆ ˆ ˆ x x x x x x. Oviously, he shor-run arge value, x, converges o is long-run arge value, ˆ xˆ (36), wih persisence, and is predeermined. We modify he corresponding cenral ank period lo funcion in Eq. (6) as follows: L ˆ ˆ ˆ ˆ x x x x. (37) 5

27 The consolidaed firs-order condiion of he prolem under discreion equals 9 ˆ ˆ E ˆ ˆ x ˆ ˆ x x x Ex x x x Ex ˆ ˆ x Ex x ˆ ˆ =0. (38) Comining he consolidaed firs-order condiion in Eq. (38) and he Phillips curve around he opimal seady sae in Eq. (8) produces he soluion. Because shocks do no affec conrollailiy, se u 0. If u 0, oviously, a soluion equals ˆ ˆ, and (39a) x ˆ ˆ x x x. (39) Tha is, here exiss a discreionary policy, which proves pah conrollale, and, hus, eliminaes he consan average and sae-coningen inflaion iases Eliminaion of sailizaion ias wih a proper weigh u u u u If u 0, he sailizaion ias arises. To eliminae his ias, ˆ and x ˆ x mus hold. Tha is, using Eqs. (6a) and (6) in Eq. (38) deermines as follows: u uˆ, u (40) where u ˆ is whie noise. Policy makers canno manage u ˆ even under commimen. Acually, we oain he opimal soluion under commimen y minimizing he ineremporal lo, forming 9. See Eq. (D.) in Appendix D. 6

28 raional expecaions a he eginning, E 0. Now, applying E o Eq. (40) produces c, (4) where c is defined in Eq. (9). Though he discreionary soluion slighly differs from he opimal soluion, he expecaions of he ineremporal loes equal each oher under he wo soluions. We see ha when. Tha is, when =, eiher he long-run or he shor-run arge values can achieve pah conrollailiy, and he weigh always equals he social weigh. Economical feasiiliy requires 0. Thus, c Join pah conrollailiy wih expeced, shor-run inflaion arge value and eliminaion of sailizaion ias wih a proper weigh when 0 c Join pah conrollailiy wih expeced, shor-run arge values For he case, he inflaion rae in he Phillips curve (Eq.4) exhiis principally c ackward-looking ehavior ( ), and he shor-run arge value for he inflaion rae in Eq. (36a) exhiis a lag. Now, for he case, we gue ha he shor-run arge value for he inflaion c rae in he principally forward-looking Phillips curve mus incorporae expeced inflaion, E. As a resul, we consruc he shor-run arge values as follows E E ˆ ˆ ˆ, and (4a) x xˆ x xˆ x xˆ. (4) Oviously, he shor-run arge value, x, converges o is long-run arge value, ˆ xˆ, wih persisence, and is predeermined. The shor-run inflaion arge value equals he weighed ( ) average of he expeced inflaion rae and he long-run arge value. Using an expecaion as a arge value 7

29 seems somewha srange, u we argue ha is raionaliy depends on he forward-looking Phillips curve. Also, he expeced inflaion arge value demonsraes he idea of implemening opimal policy hrough inflaion-forecas argeing (Svenon and Woodford [005]). We modify he corresponding cenral ank period lo funcion in Eq. (6) as follows: L ˆ E ˆ x xˆ x ˆ x The consolidaed firs-order condiion of he prolem under discreion equals 0 x ˆ ˆ x x x Ex x x x Ex ˆ x Ex xˆ. (43) E ˆ ˆ ˆ. (44) ˆ Comining he consolidaed firs-order condiion in Eq. (44) and he Phillips curve around he opimal seady sae in Eq. (8) produces he soluion. Once again, if u 0, a soluion equals Eqs. (39). Tha is, here exiss a discreionary policy, which proves pah conrollale, resuling in moving he consan average and sae-coningen inflaion iases Eliminaion of sailizaion ias wih a proper weigh u u u u If u 0, he sailizaion ias emerges. To eliminae his ias, ˆ and x ˆ x mus hold. Tha is, using Eqs. (6a) and (6) in Eq. (44) deermines as follows c. (45) Economical feasiiliy requires ha 0. Thus, c. For he case 0 c, he discreionary soluion exacly replicaes he opimal soluion. In sum, he shor-run inflaion arge value conforms o he macroeconomic srucure (i.e., 0. See Eq. (E.) in Appendix E. 8

30 Phillips curve). Tha is, i is lagged (expeced), if he inflaion rae in he Phillips curve exhiis principally ackward-looking (forward-looking) ehavior. We discu he delegaed weigh in nex secion. In shor, wih he delegaed lo funcion, a discreionary policy proves pah conrollale (removing he consan average and sae-coningen inflaion iases), and eliminaes sailizaion ias, resuling in opimaliy. 6. Discuion on he delegaed weigh The proposiion solves arge unconrollailiy. We discu he delegaed weigh o deepen our undersanding of ime inconsisency. 6.. Discuion on he delegaed weigh 6... The delegaed weigh for he purely ackward-looking Phillips curve (= ) When he Phillips curve exhiis purely ackward-looking ehavior wih no expecaions (= ), he ime-inconsisency prolem does no exis and neiher does he sailizaion ias. We, hus, do no need o delegae o he cenral ank a differen weigh from ha of sociey o change he rade-off eween arge sailizaions. Thus, =. When he Phillips curve involves an elemen of expecaions ( 0 ), he ime-inconsisency prolem and, hus, he sailizaion ias exis, and he delegaed weigh mus differ from he social weigh ( ) o make he arge sailizaion rade-off wih discreion under he delegaed lo funcion equal o ha wih commimen under he social lo funcion The delegaed weigh for he principally ackward-looking Phillips curve ( c ) When he Phillips curve exhiis principally ackward-looking ehavior ( ), he coefficien of he weigh depends only on he dynamic parameers (, and ), and does no c 9

31 relae o he sysem convergence persisence,. By Eq. (4), c. (46) Therefore, he cenral ank mus exhii conservaism ( ). Inuiively, if he inflaion rae persiss ( ), hen curren inflaion rae deviaion from is arge value will persis ino fuure and, hus, cause loes. To reduce hese loes, we c mus place more weigh on he inflaion rae sailizaion (i.e., ) The delegaed weigh for he principally forward-looking Phillips curve (0 < c ) When he Phillips curve exhiis principally forward-looking ehavior (0 ), he weigh in Eq. (45) depends no only on he dynamic parameers u also on he sysem convergence persisence. A weigh-lieral or weigh-conservaive cenral anker may emerge under differen circumsances. The evaluaion of he weigh ecomes more sule ecause of forward-looking ehavior. We see no analyical way o discu he relaionship of he weigh wih model parameers The delegaed weigh for he purely forward-looking Phillips curve (= 0) For he purely forward-looking case (= 0), he delegaed weigh in Eq. (45) reduces o c. (47) We repor he following condiions if c, (48a) if c, and (48) if c, (48c) 30

32 where c, and 0. c The cenral ank is weigh-conservaive (weigh-lieral), if cos-push shock persisence is 0. greaer (le) han he criical value, c. Acually, Tha is, a more persisen cos-push shock implies le weigh on oupu sailizaion. To see his, or i ieraing he Phillips curve (Eq. 4 or Eq. 8) forward produces E x u i ˆ ˆ i0 i i i0 i i E x x u wih = 0. Inflaion sailizaion depends enirely on curren and expeced fuure oupu gap sailizaion and cos-push shocks, ecause of he purely forward-looking naure of he Phillips curve. Thus, more persisen cos-push shocks imply ha more loes occur ecause more inflaion deviaion emerges, driven y curren and fuure cos-push shocks. To reduce he loes caused y curren and fuure inflaion deviaions, he policy makers mus place more weigh on inflaion sailizaion and, hus, le weigh on oupu sailizaion. 6.. Siuaions of he same weigh ( = ) We idenify wo siuaions where = -- one-decision-maker models and saic models. Though hese wo siuaions usually do no happen in policymaking, hey provide addiional insigh ino ime inconsisency One-decision-maker Models The case =, which requires delegaing an idenical weigh, can exend o one-decision-maker models. In such models, he opimal prolem is a conrol heory, raher han game heory, prolem, and he ime-inconsisency does no exis and neiher does he sailizaion ias. As a resul, =. Susecions 5.. and 6... provide exposiions. 3

33 6... Saic models We can implemen he approach in Secion 4 o gain more insigh ino delegaions using differen economic srucures, u we do no do i here due o he paper s lengh. We only presen and discu he delegaion wih a new-claical model. The new-claical model in Svenon (997) includes an expecaions-augmened Phillips curve wih oupu-gap persisence and raional expecaions e x x u, and (49) E, (50) e where (0 <) measures he degree of oupu-gap persisence, is he response of he oupu gap o unexpeced inflaion, e denoes inflaion expecaions in period l of he inflaion rae in period, and u is an i.i.d. supply shock wih mean 0 and variance. The cenral ank lo funcion, designed using he approach in Secion 4, equals L x x, where (5). (5) In conras o lagged or expeced arge value of inflaion rae in he new-keynesian model, he delegaed arge value of inflaion rae in he new-claical model equals ha of sociey as a resul of he privae secor s raional expecaions of inflaion. The delegaed shor-run arge value of he oupu gap equals he moderae, shor-run naural oupu gap, no he. We make a rivial change wih oupu replacing employmen.. See Yuan and Miller (03) for he derivaion of he lo funcion in Eqs. (5) and (5). Yuan and Miller (00) also derive he lo funcion using a differen approach. 3

34 amiious oupu gap, eliminaing oh he consan average and he sae-coningen inflaion iases. The delegaed weigh exhiis weigh-lieral ehavior ( > ), if oh and do no equal zero (i.e., he model is dynamic). Inuiively, since he oupu gap persiss, a curren oupu gap deviaion from is arge value will persis ino he fuure and, hus, cause loes. To reduce loes, he cenral ank mus place more weigh on he oupu gap sailizaion. Oviously, =, when = 0 and/or = 0. When he cenral ank does no care aou fuure loes ( 0 ), he model is eenially saic even if he Phillips curve exhiis oupu-gap persisence ( 0 ). When oupu-gap persisence does no ener he Phillips curve ( 0 ), he model is also eenially saic even if he cenral ank cares aou he fuure loes ( 0 ). In a word, he model is saic when 0 and/or 0. We argue ha he ime-inconsisency prolem does no exis in a saic model even wih raional expecaions. A model wih raional expecaions usually leads o ime inconsisency, u in a saic model, where he game complees in one... period.., he ime... -inconsisency prolem is acually non-exisen. Therefore, we only need o addre arge unconrollailiy while keeping he same weigh ( = ) Consideraion of Rogoff s (985) weigh-conservaive approach When = 0, he expecaions-augmened Phillips curve in Eq. (49) reduces o n e y y u, (53) and he delegaion in Eqs. (5) and (5) reduces o n L y y, (54) 3. The wo saic game models in Yuan, e al. (0), one of which equals here he case = 0, also demonsrae ha he delegaed weigh mus equal he social weigh. 33

35 which suggess naural oupu arge value and he same weigh as ha of sociey. The model, which includes he lo funcion (), he Phillips curve (53), and raional expecaions (50), is jus he model in Rogoff (985). In his saic model, we delegae a same weigh o he cenral ank whereas Rogoff (985) proposes a conservaive weigh. The poin is ha we do no hink he exisence of ime inconsisency whereas he, as well as he lieraure, does. An overly amiious oupu (employmen) arge value produces inflaion ias as he lieraure srees and as we explain in Susecion However, he lieraure argues ha inflaion ias arises from ime inconsisency whereas we argue ha i resuls from fewer insrumens han arges. To reduce inflaion ias, Rogoff (985) places a large weigh on inflaion sailizaion relaive o employmen sailizaion. Raher han adjusing he weigh, we ackle he prolem direcly y correcing he overly amiious oupu arge value and delegaing a moderae, naural level while keeping he same weigh. In sum, we idenify wo siuaions where = -- one-decision-maker models and saic models. The reason for = is ha ime inconsisency does no exis in hese siuaions. By conras, we conclude ha in a dynamic game model, where ime inconsisency exiss. Moreover, he coefficien of he delegaed weigh depends on dynamic parameers and he sysem convergence persisence as Eqs. (4), (45) and (5) show. 7. Summary and conclusions In his paper, we ackle wo iues in policymaking arge unconrollailiy and ime inconsisency in a hyrid new-keynesian model. Boh iues do no exis in he case of an equal 34

36 numer of insrumens and arges. They, however, emerge when fewer insrumens exis han arges. Time inconsisency also resuls from expecaions (muliple decision makers). We addre oh iues y delegaing a lo funcion o he cenral ank, deermining he cenral ank s long- and shor-run arge values as well as he weigh parameer. Inuiively, a proper arge value rade-off solves arge unconrollailiy, whereas a proper relaive weigh eween arge sailizaions achieves he opimal arge variailiy rade-off and solves ime inconsisency. We addre arge unconrollailiy efore ime inconsisency, avoiding enangling he wo iues when addreing ime inconsisency. The paper oains main resuls as follows. Firs, we oain a proposiion aou arge conrollailiy, which complemens he Tinergen rule. Tha is, if he sysem, which consiss of a quadraic lo funcion and a linear Phillips curve, converges wih commimen and wih discreion, arge values are joinly asympoically conrollale if and only if hey saisfy he seady-sae Phillips curve. The proposiion indicaes ha conrollale arge values exhii rade-offs, conform o he macroeconomic srucure, and do no depend on he policy implemenaion syle -- commimen or discreion. In oher words, infinie pairs of arge values can lead o join asympoic conrollailiy. The firs-es equilirium corresponds o a zero arge value of inflaion rae, and vice versa. An inefficiency (monopolisic and/or ax disorions) requires a posiive arge value of inflaion rae. This explains o some exen why argeing a posiive inflaion rae prevails in he real world. Second, if arge values do no saisfy he seady-sae Phillips curve, he seady-sae values of arge variales exhii regreion o heir respecive arge values. Tha is, wih a high 35

37 arge value of he oupu gap relaive o ha of he inflaion rae, he seady-sae value of he oupu gap falls elow is arge value, and he seady-sae value of he inflaion rae rises aove is arge value. The regreive naure explains he inflaion ias. An overly amiious oupu arge value implies a relaively low inflaion arge value. A low inflaion arge value leads o he seady-sae value of inflaion higher han is arge value. Therefore, oh he lieraure and we agree ha an overly amiious oupu arge value leads o inflaion ias. Bu we ariue he consan average inflaion ias o fewer insrumens han arges, raher han o ime inconsisency as he lieraure srees. Third, delegaion parameers have he following characerisics: () The cenral ank s long-run arge values jus equal he seady-sae values of arge variales wih commimen under he social lo funcion. They saisfy he seady-sae Phillips curve, and, hus, are joinly asympoically conrollale wih discreion, resuling in removing he consan average inflaion ias. The seleced long-run arge values achieve he es oucome, given he social lo funcion and he economic srucure. They guide moneary policy owards a correc end. () The shor-run inflaion arge value conforms o he macroeconomic srucure (i.e., Phillips curve). Tha is, i is lagged (expeced, or exacly he social arge value) if he inflaion rae exhiis principally ackward-looking (forward-looking, or raional expecaions) ehavior in he Phillips curve. The shor-run arge values of oh he inflaion rae and he oupu gap converge o heir respecive long-run arge values. Moreover, hey, hough sae-coningen, are predeermined and, hus, feasile in pracice. Wih he shor-run arge values, he discreionary policy proves pah conrollale and, hus, eliminaes he consan average and sae-coningen inflaion iases. The 36