Stephen M. Miller University of Nevada, Las Vegas and University of Connecticut

Transcription

1 The Opimaliy and Conrollailiy of Discreionary Moneary Policy Huiping Yuan Xiamen Universiy Sephen M. Miller Universiy of Nevada, Las Vegas and Universiy of Connecicu Working Paper 0-7 Augus 0

2 The Opimaliy and Conrollailiy of Discreionary Moneary Policy Huiping Yuan Xiamen Universiy, Xiamen, Fujian 36005, China and Sephen M. Miller Universiy of Nevada, Las Vegas, Las Vegas, NV , USA Asrac: This paper addresses wo issues -- he ime-inconsisency of opimal policy and he conrollailiy of arge variales wihin new-classical and new-keynesian model srucures. We can resolve oh issues y delegaion. Tha is, we design cenral ank loss funcions y deermining he wo arge values and he weigh eween he wo arges. Wih a single decision maker, he ime-inconsisency issue does no exis; he arge conrollailiy issue does. Delegaing he long-run arge values (arge variales equiliriums under he Ramsey opimal policy and he same weigh as sociey o he cenral ank can achieve Ramsey opimaliy and pah conrollailiy. Wih muliple decision makers (game, oh issues of ime-inconsisency and arge conrollailiy exis and he delegaion ecomes more complicaed. The long-run arge values can only achieve asympoic, no pah, conrollailiy. Pah conrollailiy requires he delegaion of shor-run arge values, which commis or inds he cenral ank o follow exacly he Ramsey opimal pahs. The shor-run inflaion arge value conforms o he macroeconomic srucure (i.e., Phillips curve. Wih pah conrollailiy, he consan average and sae-coningen inflaion iases are removed. To eliminae he sailizaion ias, he delegaed weigh mus differ from sociey in a dynamic game. When he Phillips curve exhiis oupu (inflaion persisence, he cenral ank mus place more weigh on oupu (inflaion sailizaion. When he Phillips curve exhiis principally forward-looking ehavior, he delegaed weigh can require a conservaive or lieral cenral ank. In sum, delegaing cerain shor-run arge values and a differen weigh can cause discreionary moneary policy o prove Ramsey opimal and pah conrollale in a dynamic game. J.E.L. Classificaion: Key words: E40, E50, E580 Opimal Policy, Conrollailiy, Policy Rules Professor Yuan graefully acknowledges he financial suppor from he Minisry of Educaion of China. This paper is a research resul of he Humaniies and Social Science Projec 0YJA Corresponding auhor.

3 . Inroducion Moneary economiss coninue o deae wo long-sanding policy issues -- he imeinconsisency of opimal policy (e.g., see Kydland and Presco, 977 and Calvo, 978 and arge conrollailiy (e.g., see Kalman, 960; Tinergen, 95, 963; Preson 974; and Aoki, 975. This paper ackles oh issues, which usually receive separae reamen, making discreionary (ime-consisen policy Ramsey opimal and arge conrollale in dynamic models. Researchers ypically delegae a loss funcion o he cenral ank o solve he imeinconsisency 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 Svensson (997, he nominal income growh arge of Beesma and Jensen (999 and Jensen (00, he price-level arge of Vesin (006, and he consisen arge of Yuan, e al. (0, and so on. Though he various delegaion schemes osensily differ and lead o differen inerpreaions, many delegaions prove idenical in effec. The delegaion approach solves he ime-inconsisency prolem in a sraighforward fashion. Under he delegaed loss funcion, he cenral ank operaes moneary policy wih discreion. As a resul, moneary policy is ime-consisen. The main issue, however, is wheher discreionary policy can approximae or even reproduce he Ramsey policy. Yuan, e al. (0 sudy oh issues in saic models. Yuan, e al. (0 compare heir designed loss funcion wih he loss funcions in Svensson (997 wihou employmen persisence, Walsh (995, and Chorareas and Miller (003, and show ha he four loss funcions generae idenical resuls wih respec o he policy decision. Svensson (997 also oserves ha he inflaionarge-conservaive loss funcion wihou employmen persisence mimics he linear inflaion conrac in Walsh (995.

4 Brocke and Mesarović (965 define hree ypes of conrollailiy (reproduciiliy: poin, pah, and asympoic conrollailiy. 3 Poin conrollailiy means ha he policy makers can achieve cerain arge values a a specified poin in ime. In pracice, policy makers proaly desire more han poin conrollailiy, ha is, he sronger noion of pah conrollailiy. Pah conrollailiy means ha policy makers can make arge variales follow some prescried rajecories over a cerain ime inerval (see 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 conrollailiy plays a growing role in dynamic models of economic policy. 4 To he es of our knowledge, he economics lieraure does no consider asympoic conrollailiy. This paper considers oh pah and asympoic conrollailiy, which prove useful conceps in descriing he pahs of arge variales. 5 Our analysis proceeds in several seps. Firs, given a social loss funcion, we deermine he Ramsey opimal policy pah. This opimal rajecory deermines he design of he cenral ank loss funcion. Second, we design he cenral ank loss funcion y deermining hree key 3 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 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 muliple-decisionmakers (game conex. See Acocella and Di Barolomeo (006, Acocella e al. (006, 007, Hughes Halle e al. (00 for conrollailiy in a game conex. 5 Phillips (954 considers he ime pahs of economic variales as well as he saic analysis of he final equilirium. Tha is, his analysis involves he idea of pah and asympoic conrollailiy. 3

5 parameers he wo arge values (i.e., he inflaion rae and oupu gap, and he weigh eween he wo arges. Third, we deermine he wo arge values. We find ha he arge variales equiliriums under he Ramsey policy prove joinly asympoically conrollale. We call hese equilirium values he long-run arge values. The discreionary policy wih hese long-run arge values, if he sysem converges, can only achieve asympoic raher han pah conrollailiy. Fourh, o oain pah conrollailiy, we adop sae-coningen, shor-run arge values. Inuiively, he exisence of ime-inconsisency lures he cenral ank o deviae from he Ramsey opimal pahs each period. As a resul, we mus delegae sae-coningen, shor-run arge values each period. They commi and ind he cenral ank o follow exacly he Ramsey opimal pahs. The shor-run arge values converge o he long-run arge values in a sep-y-sep process. Also, hey, hough sae-coningen, are predeermined and, hus, feasile in pracice. Wih he shor-run arge values, discreionary policy proves pah conrollale and, hus, eliminaes he consan average and sae-coningen inflaion iases (Svensson, 997, p04. Fifh, when shocks occur, sailizaion ias arises under discreionary policy. We can eliminae he sailizaion ias y delegaing o he cenral ank a delierae weigh, which differs from he social weigh. The cenral ank wih a differen weigh makes a differen rade-off eween he inflaion rae and he oupu gap and, hus, sailizes he shocks in exacly a Ramsey opimal fashion. In sum, delegaion of sae-coningen, shor-run arge values and he correc weigh parameer cause discreionary policy o follow he Ramsey opimal pah. We deermine he necessary and sufficien condiions for asympoic conrollailiy of consan arge values. In conras, he conrollailiy lieraure generally considers he equaions 4

6 of moion, he availale insrumens, and he iniial sae of he sysem. We demonsrae ha he consan arge values, if he sysem converges, are joinly asympoically conrollale if and only if hey saisfy he macroeconomic srucure (i.e., he Phillips curve in our models in equilirium. The resul holds under commimen and under discreion. We sae our findings in Proposiions and. Inuiively, he macroeconomic srucure (i.e., he Phillips curve consrains he arge variales and as a resul, proper arge values will exhii he Phillips curve rade-off. The arge variales equiliriums under he Ramsey policy, as an example of consan arge values, saisfy he Phillips curve in equilirium and, herefore, are joinly asympoically conrollale. The paper unfolds as follows. Secion deermines he Ramsey opimal pahs of arge variales, using he social loss funcion wih commimen, wihin new-classical and new- Keynesian models. We illusrae he ime-inconsisency and arge unconrollailiy of Ramsey opimal policy, which leads o Proposiion. Secion 3 exends Proposiion o Proposiion. We design he cenral ank loss funcions y deermining he long-run and shor-run arge values as well as he weigh parameer in he wo models. Secion 4 discusses wo siuaions where he delegaed weigh equals he social weigh. Secion 5 concludes.. Time-Inconsisency and Targe Conrollailiy of Ramsey Opimal Policies New-Classical Model wih Oupu Persisence The model follows Svensson (997. We simply adop i wihou much descripion. See Svensson (997 for more deails. Sociey minimizes he following ineremporal loss funcion 5

7 ( L = E0 β L, = where β (0 < β < is he discoun facor and E is he expecaions operaor. The period loss funcion equals he following ( L ( π π ( x, = + x where π is he inflaion rae, x is he oupu gap, π is he socially desirale inflaion rae, x is he socially desirale oupu gap, and is he social weigh on oupu sailizaion relaive o inflaion sailizaion around heir respecive arges. The economic srucure includes an expecaions-augmened Phillips curve wih oupu persisence and raional expecaions e (3 η α( π π x = x + + u, and e (4 π = E π, where η (0 η< measures he degree of oupu gap persisence, α is he response of oupu gap o unexpeced inflaion, e π 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 σ. Ramsey Opimal Policy. The cenral ank minimizes he social ineremporal loss funcion ( wih period loss funcion ( sujec o Eqs. (3 and (4. The socially opimal pahs of arge variales under commimen equal 6 (5a (5 π = π + du, and η ( α x = x + + d u, where 6 See Svensson (997. 6

8 (6 d = α βη + α. Time-Inconsisency of Ramsey Opimal Policy. Ramsey opimal policy equals an ex ane plan made y a social planner, who conrols all insrumens -- he insrumens of he policy makers and he privae secor. Such an ex ane plan, however, is no implemenale. Wih muliple decision makers (game, policy makers canno conrol oher players insrumens, and implemens policies using discreion. 7 Discreionary policies are ime-consisen, u non- opimal. Targe Conrollailiy of Ramsey Opimal Policy. In addiion o ime-inconsisency, Ramsey opimal policy sill faces arge conrollailiy. The Ramsey opimal pahs of he arge variales in Eqs. (5a and (5 consis of sysemaic and random componens. We denoe he sysemaic componens as follows (7 ( π, x ( π, ηx =, and random componens or reacions o supply shocks as follows (,,( α (8 π ( ( ( u x u = d u + d u. Conrollailiy is irrelevan o exernal shocks in our specific models wih quadraic loss funcions, linear Phillips curves, and addiive shocks. Accordingly, we ignore shocks and consider he sysemaic componens when we consider conrollailiy. The sysemaic componens converge o( π,0 (9 ( π, x lim ( π, x ( π, = 0, 7 For discreionary policy, refer o Svensson (997. 7

9 where π (=π and x (=0 denoe he equiliriums (limis of he inflaion rae and he oupu gap, respecively. In addiion, he convergence persisence of he inflaion rae and he oupu gap equals 0 and η, respecively. Tha is, he inflaion rae his is equilirium, π π =, immediaely wih zero persisence, and he oupu gap converges o is equilirium, x = 0, wih persisence η. Consider arge conrollailiy. Oviously, he policy maker can poin conrol he inflaion rae each period ( π = π and, hus, conrol is pah. The policy maker canno poin conrol he oupu gap each period ( x x and canno even asympoically conrol he oupu gap ( x x, if x 0. To undersand why he policy maker canno asympoically conrol he social oupu gap arge value, if x 0, consider he Phillips curve in Eq. (3 and he raional expecaions assumpion in Eq. (4. In equilirium, ignoring shocks, π = E π = π, π e = π, and he Phillips curve equals o (0 x = η x, i.e., x = 0. Tha is, he oupu gap mus equal zero in equilirium. If we require asympoic conrollailiy of he oupu gap (i.e., x x =, hen we mus oserve ( x = 0. On he conrary, if x = 0, hen he policy maker can asympoically conrol he zero arge value ecause x = 0 = x. In sum, Eq. ( is he necessary and sufficien condiion for asympoic conrollailiy of he oupu gap arge value, x. 8 8 In he simpliciy of he new-classical model, he asympoic conrollailiy condiion in Eq. ( proves rivial. This condiion is nonrivial, however, in he new-keynesian model. 8

10 The arge variales equiliriums, π (=π and x (=0, saisfy he necessary and sufficien condiion and, hus, are joinly asympoically conrollale. They are he long-run arge values according o he definiion in he inroducion. A Hyrid New-Keynesian Model Researchers developed and applied new-keynesian models in he pas decade. We adop a hyrid new-keynesian model, which comines he possiiliy of forward- and ackward-looking inflaion. This model reduces o a purely forward-looking or a purely ackward-looking model y choosing he exreme values of he parameer ha indexes expecaions across he forwardand ackward-looking dimensions. See Clarida e al. (999 for more deails. The social ineremporal loss funcion equals 9 ( L = E0 β L = 0 wih he same period loss funcion as in Eq. (. Aggregae supply equals an expecaions-augmened Phillips curve wih forward-looking expecaions and endogenous inflaion (3 ( π = x + φπ + φ βeπ + u, + where π, x, and β are defined as efore, ( >0 is he sensiiviy of he inflaion rae o he oupu gap, φ indexes he degree of lagged versus expeced fuure inflaion raes, u is a cos-push shock ha follows an AR( process 9 The expecaion operaor, E (, slighly differs from ha in he new-classical model in ha he privae secor forms raional expecaions in he new-keynesian model a he eginning of he presen period and in he new-classical model from he previous period. We jus follow he usual definiions in lieraure for convenience. 9

11 (4 u ˆ = ρu + u, where 0 ρ <, and u is a whie noise residual. ˆ We do no inroduce aggregae demand (IS curve, which involves a nominal ineres rae, he policy insrumen. Once we deermine he opimal pahs for { π, x} = 0 using he social loss funcion and he Phillips curve, oh of which do no involve he ineres rae, hen we can pin down he opimal pah of ineres raes hrough he IS curve. So he Phillips curve proves criical for policy. Ramsey Opimal Policy. The consolidaed firs-order condiion of opimal policy under he social ineremporal loss funcion ( wih period loss funcion ( sujec o he Phillips curve in Eq. (3 equals 0 (5a ( x0 x φβ ( E0x x = ( π 0 π for = 0, and (5 ( Ex 0 x φβ ( Ex 0 x ( ( Ex 0 x + φ = ( E0π π f or. Comining he firs-order condiions (5a and (5 and he Phillips curve (Eq. 3 leads o Ramsey opimal pahs of arge variales (6a ( π π = δ π φ( π π ( φ ( x x + h0u0 for = 0, (6 ( x x δ ( x x ( π ( a φ β φ π = + + f u 0 x0 0 0 for = 0, (7a ( π π = δ ( π π + ( φβρ u ( φ ( π π δ ( π π δρ u, or δρd a ( φ φc( βρ δρ u = + for ; and δρd a + ( φβρ uˆ 0 See Eqs. (A.3a and (A.3 in Appendix A. See Eqs. (A.4, (A.4, (A.35 and (A.39 and heir derivaion in Appendix A. 0

12 (7 δρ = δρ ( x x δ ( x x ( d a u for, where δ, δ x, h, f are defined in Appendix A, and δ is a roo of he characerisic equaion π (8a aβδ βδ + δ δ+ a = 0 wih ( (8 a φ φ, (8c + + φ β + ( φ β = + β + aβ, (8d β ( δ δρ ρ β( δ ρ d a , (8e (8f ( ( π + φ β x π, + a β x ( φ( β π, and ρ (8g φc, and 0 < φ c. βρ For he purely forward-looking, new-keynesian model (φ = 0, he soluion in Eqs. (7a and (7 reduces o he soluion in Clarida e al. (999, p y noing ha (9 d δ =, when φ = 0 and. δβρ Eqs. (8a, (8, and (8c deermine, wha we call, he sysem convergence persisence, δ. We canno deermine wheher a roo δ (0, exiss for any φ (0,. Bu for he wo exreme cas es of φ = 0 and φ =, he characeris ic Eq. (8a reduces o βδ δ + = 0, and a roo δ (0, does exis. Tha is, he sysem can converge under he Ramsey policy for he wo If φ = 0 or, hen a = 0, d = β ( δ + ρ, βδ δ + = 0. Thus, δ βδ =. Therefore, dδ d δ δ βδ δβρ =, and he characerisic equaion reduces o = δβρ or d = δ ( δβρ.

13 exreme cases. Wheher a roo δ (0, exiss, δ depends on he model srucural parameers (i.e., φ, β,, and, and does no relae o he persisence of he cos-push shock. We ariue he ir relevance of he cos-push shocks o he quadraic loss funcion and he linear Phillips curve wih addiive shocks. The criical value, φ c, is meaningful. For convenience, we define he Phillips curve as principally forward-looking (ackward-looking if 0 φ φ ( φ φ. Wheher he Phillips curve exhiis principally forward- or ackward-looking ehavior resuls in differen (negaive or posiive responses of he inflaion rae o cos-push shocks and, we will see in nex secion, differen shor-run arge values and weigh parameer ha are delegaed o he cenral ank. The criical value φ c depends on oh he discoun facor β and he cos-push shock persisence ρ, and responds o hem as follows 3 (0a φ β > 0, and (0 φ ρ < 0. c c Tha is, a more imporan fuure and/or less persisen cos-push shocks lead o a higher criical value and, hus, he inflaion rae for a given inflaion rae persisence φ more likely exhiis principally forward-looking ehavior ( φ φ c. Accordingly, wheher he Phillips curve exhiis principally forward- or ackwardlooking ehavior depends on he hree parameers, φ, β, and ρ, raher han merely φ. We refer o he parameers, φ, β, and ρ, as dynamic parameers, since seing hem equal o zero produces a c c 3 φ c ρ ( ρ = > 0 β ( βρ and ( + ( φ c β β ρ = < 0 for 0 < ρ <. ρ ( βρ

14 saic model. As such, he dynamic parameers are β and η in he new-classical model, where supply shocks equal whie noise. We will see ha he delegaed weigh depends on dynamic parameers and/or he sysem convergence persisence in a dynamic game model and equals he social weigh in a saic model. Time-Inconsisency of Ramsey Opimal Policy. The firs-order condiions (5a and (5 sugges he ime-inconsisency of he Ramsey opimal soluion. Ramsey 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 loss funcion each period and, hus, always follows he firs-order condiion (5a. Targe Conrollailiy of Timeless Perspecive Opimal Policy. We adop opimaliy from he imeless perspecive, 4 and analyze Eqs. (7a and (7 for. The opimal pahs of he arge variales in Eqs. (7a and (7 consis of sysemaic componens denoed as ( ( π x = ( π + δ ( π π x + δ ( x x,,, and random componens relaed o cos-push shocks denoed as δρ δρ u = u u, u δρd a. ( δρd a ( ( π ( u, x ( ( φβρ ( φ Similarly, conrollailiy does no depend on exernal shocks, and we consider he sysemaic componens. The sysemaic componens converge o ( π, x, if δ <, (3 ( π, x lim ( π, x =, 4 Woodford (999 inroduces he concep of opimaliy from a imeless 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. 3

15 where Eqs. (8e and (8f define π and x. So π and x are he equiliriums (limis of he inflaion rae and he oupu gap. The wo arge variales evolve o heir respecive equiliriums wih he same sysem convergence persisence, δ. Now, consider arge conrollailiy. The policy makers canno poin conrol oh he inflaion rae and he oupu gap each period, since π π and x x. They also generally canno asympoically conrol he arges, since π π and x x. 5 Consider Eq. (8f. The equiliriums of he arge variales saisfy he Phillips curve in equilirium. Tha is, (4 x ( π = + φπ + φ βπ. The wo arges are joinly asympoically conrollale, if he wo arge values, π and x, saisfy he Phillips curve in equilirium. Tha is, ( (5 βπ π = x + φπ + φ. We can easily verify his resul. Rearranging Eq. (5 gives (6 x ( φ( β = π. Susiuing Eq. (6 ino Eq. (8e generaes ( ( ( ( φ ( β π + φ β π π + φ β x (7 π = = π. + a β + a β Using Eqs. (8f, (7, and (6 in sequence produces ( 5 If φ = 0, hen he policy makers can asympoically conrol he inflaion rae ( π = π. 4

16 ( φ( β ( φ( β (8 x π = π = x. Tha is, he wo arge values, π and x, are asympoically conrollale. Conversely, if he wo arge values, π and x, are asympoically conrollale (join asympoic conrollailiy implies π = π and x = x, hen hey saisfy Eq. (5. In sum, Eq. (5 provides he necessary and sufficien condiion for join asympoic conrollailiy of he social arge values, π and x. The arge variales equiliriums, π and x defined in Eqs. (8e and (8f, saisfy he necessary and sufficien condiion and, hus, are joinly asympoically conrollale. They are he long-run arge values according o he definiion in he inroducion. 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 necessary and sufficien condiion for he join asympoic conrollailiy of consan arge values. We summarize he aove resuls in Proposiion. Proposiion. If he sysem converges under he Ramsey policy wih period social loss funcion in Eq. ( sujec o a linear Phillips curve, π and x are joinly asympoically conrollale if and only if hey saisfy he Phillips curve in equilirium. Proof. We prove he proposiion in a more simple way. Take he hyrid new-keynesian model as an example. Sufficiency. If π and x saisfy he Phillips curve in Eq. (5, hen comining Eq. (5 5

17 and Eq. (3 produces he Phillips curve around is equilirium as follows (9 ( π ( ( ( ( π x x φ π π φ β Eπ+ π = u. Now, he opimizaion prolem equals he minimizaion of he social ineremporal loss funcion ( wih period loss funcion ( sujec o he Phillips curve (9. Oviously, he opimal pahs ( π π, x x will converge o (0, 0, if he sysem converges. Tha is, ( π, x converge o ( π, x. Therefore, π and x are joinly asympoically conrollale. Necessiy. If π and x are joinly asympoically conrollale, π = π and x = x hold. The Phillips curve in equilirium, which involves π and x, leads o he condiion ha π and x saisfy he Phillips curve in equilirium. 3. Designing Cenral Bank Loss Funcions Though we deermine he necessary and sufficien condiion for joinly asympoically conrollale arge values, he social arge values generally do no saisfy his condiion ecause he social loss funcion (e.g., he represenaive household s uiliy, Arrow s (95 social welfare funcion, or Rawls s (97 maximin crierion reflecs a normaive prolem in philosophy and usually does no relae o economic models. Proposiion, however, provides a way o design he cenral ank loss funcion. We exend Proposiion o discreionary policy under a cenral ank loss funcion. As noed in he inroducion, we design he cenral ank loss funcion y deermining he hree key parameers he wo arge values of he inflaion rae and he oupu gap denoed as (, x π and he weigh on oupu sailizaion relaive o inflaion sailizaion denoed as. Thus, he cenral ank period loss funcion equals 6

18 . (30 ( L ( = π π + x x Proposiion. If he sysem converges under he discreionary policy wih period cenral ank loss funcion in Eq. (30 sujec o a linear Phillips curve, π and x are joinly asympoically conrollale if and only if hey saisfy he Phillips curve in equilirium. The proof is similar o Proposiion y noing ha he difference eween Ramsey policy and discreionary policy merely reflecs he differen sequence of opimizaions of decision makers. For convenience, denoe he equiliriums of he inflaion rae and he oupu gap under he discreionary policy wih he cenral ank loss funcion, respecively, as π and x, which are he co unerpars of π and x, he equiliriums under he Ramsey policy wih he social loss funcion. On he one hand, we require ha he discreionary policy is Ramsey opimal (i.e., a leas π = π and x = x ; on he oher hand, we require ha he consan arge values are asympoically conrollale (i.e., π = π and x = x. As a resul, (3 ( π, x ( π, x =. Since π and x saisfy he Phillips curve in equilirium, π and x are joinly asympoically conrollale under he discreionary policy if he sysem converges. Thus, x are also he long-run arge values under he discreionary policy. π and We will see ha he discreionary policy wih he long-run arge values, however, is generally no pah conrollale in he wo example models, where he desired rajecories equal h eir respecive Ramsey opimal pahs. Pah unconrollailiy leads o social losses. The loss funcion in Eq. (30 sill needs improvemen. 7

19 The long-run arge values canno achieve pah conrollailiy (or poin conrollailiy each period. Inuiively, he exisence of ime-inconsisency lures he cenral ank o deviae from he Ramsey opimal pahs each period. As a resul, we mus delegae shor-run arge values each period. They commi and ind he cenral ank o follow exacly he Ramsey opimal pahs. Pah conrollailiy requires ha (, x π, which denoe he sysemaic pahs of he inflaion rae and he oupu gap under he discreionary policy wih he designed loss funcion, replicae (, x denoed as π for each period. A he same ime, we require ha he shor-run arge values, π and (3 ( π, x ( π, x. As a resul, x, are conrollale. Tha is, (, ( π x = π, x =. The corresponding loss funcion equals (33 L = ( + ( x x π π. Though he shor-run arg e values are sae-coningen, hey are predeermined ecause π ( x is he weighed average of lagged values, π ( x, and long-run arge values, π ( π x, and he weigh on ( x equals he convergence persisence under he Ramsey opimal policy. The predeermined arge values are feasile in pracice. 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 (34 where ( π ( u, x ( u = ( π ( u, x ( u ( π ( u, x ( u, denoe he reacions o shocks under he discreionary policy wih he designed loss funcion. Acually, we pin down he weigh hrough one of he wo equaions, 8

20 ( u = π ( u or x ( u = x ( u π. Once one equaion holds, he oher equaion also holds ecause he inflaion rae and he oupu gap are linked hrough he Phillips curve. Now, we design cenral ank loss funcions using he aove approach for he wo differen models. Designing he Cenral Bank Loss Funcion in he New-Classical Model wih Oupu Persisence Asympoic Conrollailiy under Discreion wih he Long-Run Targe Values. By Eqs. (9 and (3, (35 π = π = π, and x = x = 0. The corresponding cenral ank period loss funcion equals = π π + x. (36 ( L The oucomes under he discreionary policy wih period loss funcion (36 equal 6 (37a π = π cx + du, and (37 x = ηx + ( + αd u, where ( (38a c = βη βη 4 α β αβη η, and (38 c d α+ βα = βη + α +βα c. (39 Denoe he sysemaic pahs as ( π, x = ( π cx, ηx. Oviously, 6 See he discreionary policy in Svensson (997. Our noaion differs slighly from his. Seing he employmen gap (oupu gap arge value o zero and replacing wih and ρ wih η creaes he resuls in Eqs. (37 and (38. 9

21 (40 ( π, x lim ( π, x = ( π,0. Tha is, π and 0 are joinly asympoically conrollale under he discreionary policy. By Eqs. (7 and (39, generally. (4 ( π, x ( π, x Tha is, he discreionary policy is pah unconrollale, where he desired rajecories equal he Ramsey opimal pahs. Pah Conrollailiy wih he Sae-Coningen, Shor-Run Targe Values. By Eqs. (7 and (3, (4 π π π = = η. = =, and x x x The corresponding cenral ank period loss funcion equals π π η. (43 L = ( + ( x x Inuiively, wih he shor-run naural oupu gap ηx as he oupu gap arge, he cenral e ank does no possess an incenive o produce surprise inflaion ( π π o raise he oupu gap aove he shor-run naural gap η x. As a resul, he cenral ank can joinly realize he arge values π and ηx. The formal calcu laion of discreionary policy wih he arge values verifies he inuiion. The discreionary oucomes equal 7 (44a (44 π = π + du, and x = ηx + ( + αd u, where wih he same noaion d wihou causing confusion (45 d α =. + α 7 See Eqs. (B.a and (B. and heir derivaion in Appendix B. Muliple equiliriums exis. We repor one here. 0

22 Denoe he sysemaic pahs and he reacion pahs respecively as ( π, x = ( π, ηx (46, and ( u, x u = ( du,( + αd u (47 π ( (. Thus, (48 ( π, x = ( π, x, which means pah conrollailiy and, hus, eliminaes he consan average and sae-coningen inflaion iases. Eliminaion of Sailizaion Bias. To eliminae sailizaion ias, we se π ( u = π ( u. Tha is, d = d, which produces 8 (49 =. βη The coefficien of he cenral ank s weigh depends only on he dynamic parameers, β and η. 9 If β and/or η equal zero (i.e., he model is saic, he cenral ank mus exhii he same weigh as sociey; 0 oherwise, i mus exhii weigh-lieral ehavior, conrary o he usual recommendaion of appoining a (weigh- conservaive cenral anker. Inuiively, since he oupu gap persiss, a curren oupu gap deviaion from is arge value will persis ino he fuure and, hus, cause losses. To reduce losses, he cenral ank mus place more weigh on he oupu gap arge (i.e., >. 8 Yuan and Miller (00 also oain he resul in Eq. (49. 9 Here, we assume ha supply shocks equal whie noise. Thus, he weigh does no reflec he characerisic of he supply shocks. We discuss he persisence of shocks in he new-keynesian model. 0 Yuan, e al. (0 ariue he idenical weigh o he saic models and/or he whie-noise shocks. Policy makers do no face a rade-off eween arges and/or eween periods if he governmen delegaes he correc arge values o he cenral ank in saic models wih whie-noise shocks. (p. 84. We will see ha a weigh-lieral or weigh-conservaive cenral anker may emerge under cerain circumsances in he new-keynesian model.

23 In sum, he discreionary policy proves pah conrollale (removing he consan average and sae-coningen inflaion iases, and eliminaes sailizaion ias under he loss funcion in Eq. (43 wih defined in Eq. (49, resuling in Ramsey opimaliy. Designing Cenral Bank Loss Funcions in a Hyrid New-Keynesian Model Divergence/Convergence under Discreion wih he Long-Run Targe Values. By Eq. (3, (50 π = π, and x = x, where π and x are defined in Eqs. (8e and (8f. The corresponding cenral ank period loss funcion equals. (5 L = ( π π + ( x x The oucomes under discreionary policy wih period loss funcion (5 equal (5a ( δ ( x x = x x + fu, and, u (5 ( π π = δ ( π π f ( φβρ where (53a (53 δ is a roo of he characerisic equaion 3 aβδ βδ φβ δ φ= 0, and f = δ { ( }. a + β δ ρ δβρ φ Does a less-han-one roo exis for he characerisic equaion (53a? Consider he wo exreme cases of φ = 0 and φ =. For he case φ = 0, he characerisic equaion (53a reduces o ( + = 0, and is wo roos are δ = 0, which is a degeneraive soluion, and βδ δ See Eqs. (C.9, (C.3, (C.4 and (C.6 and heir derivaion in Appendix C.

24 ( δ = + β >. The sysem diverges under he discreionary policy when φ = 0, in conras o converging under he Ramsey policy. For he case φ =, he characerisic equaion (53a reduces o (54 βδ + + β δ + = 0. We can show ha he igger roo exceeds one and he smaller roo lies eween zero and one. The sysem can converge under discreion when φ =. Acually, ime-inconsisency does no exis in he case φ =, and we discuss his special case furher in he nex susecion. When he lag index φ goes from 0 o, he sysem changes from divergence o convergence. The sysem may also go hrough divergence, convergence, divergence, convergence, and so on. For he case 0<φ<, we can conclude ha π and x are joinly asympoically conrollale, if he characerisic equaion (53a does have a roo ha less han one, δ <. Generally, he sysem is no pah conrollale under he discreionary policy wih he long-run arge values since δ δ where δ and δ are deermined respecively y Eqs. (53a and (8a. Pah Conrollailiy wih he Long-Run Targe Values When φ=. For he case φ =, we oain an ineresing resul. Tha is, when he Phillips curve in Eq. (3 exhiis purely ackward-looking ehavior wih no expecaions (55 π = x + π + u, we delegae o he cenral ank a simple loss funcion wih he long-run arge values ( π = π and x = x and he same weigh ( = as follows 3

25 (56 L = ( π π + ( x x. Wih he simple loss funcion (Eq. 56, he discreionary policy proves pah conrollale and eliminaes sailizaion ias! When he Phillips curve equals Eq. (55, 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. We demonsrae his hrough he consolidaed firs-order condiions under he Ramsey opimal policy. The condiion for =0 (Eq. 5a equals he condiion for (Eq. 5, when φ=. Tha is, he Ramsey policy is ime-consisen. 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. In addiion, he sailizaion ias does no exis eiher. Accordingly, he cenral ank adops he same weigh ( =. When =, 3 (57 δ = δ φ =, and (58 ( π ( u x ( u = ( π ( u x ( u,,. Tha is, he policy under he delegaion of he long-run arge values ( π = π and x he same weigh ( = proves pah conrollale and of no sailizaion ias, when φ =. = x and Though ime-inconsisency does no exis, he issue of arge conrollailiy does. The arge variales' equiliriums in Eqs. (8e and (8f wih φ = equal (59 ( π x = π + ( β, x,0. Thus, 3 See Eqs. (C.5 and (C.30 and heir derivaion in Appendix C. 4

26 , x, x if x 0. (60 ( π ( π Tha is, he social arge values, π and x, are unconrollale, if x 0. The long-run arge values, π and x, however, prove asympoically conrollale, since δ < from Eq. (54. In sum, when φ =, ime-inconsisency does no exis and we only ackle he issue of conrollailiy. We solve i y delegaing he long-run arge values and he same weigh o he cenral ank. Pah Conrollailiy wih Lagged, Shor-Run Targe Values When φ c φ. Wheher or no he sysem converges wih he long-run arge values under he discreionary policy, we consider delegaing shor-run arge values. By Eqs. ( and (3, π = π = π + δ π π = δπ + δ π, and (6a ( ( (6 = = + δ ( = δ + ( δ x x x x x x x. The shor-run arge value, ( x π, equals he weighed average of he lagged value, ( x and he long-run arge value, π ( x. The weigh on he lagged value is he sysem persisence, δ. The corresponding cenral ank period loss funcion equals (6 { ( ( ( ( } π π δ π π δ L = + x x x x. π, convergence Because shocks do no affec conrollailiy in he model wih a quadraic loss funcion, a linear he Phillips curve, and addiive shocks, we se u = 0 and deermine he oucomes under a discreionary policy wih period loss funcion (6 as follows 4 (63a ( π π δ( π π =, and 4 See Eqs. (D.0 and (D. and heir derivaion in Appendix D. 5

27 (63 ( x x = δ ( x x. Tha is, here exiss a discreionary policy, which proves pah conrollale and, hus, eliminaes he consan average and sae-coningen inflaion iases. When 0 and x ( u = x ( u (64 u, he sailizaion ias arises. To eliminae iases, using π ( u = π ( u produces 5 ( ( βρ = φ φc ρ where φ c is defined in Eq. (8g. ( φβρ, = when = φ. Tha is, when φ=, eiher he long-run or he shor-run arge values can achieve pah conrollailiy, and he delegaed weigh always equals he social weigh. Economical feasiiliy requires 0. Thus, (65 φ φ c. Pah Conrollailiy wih Expeced, Shor-Run Inflaion Targe Value When 0 φ φc. Now, deermine he arge values, if φ φc. Consider more carefully he loss funcion and he Phillips curve. In he new-classical model wih oupu persisence, he cenral ank s opimal and conrollale loss funcion involve a erm, ηx, which is he imporan characerisic of he Phillips curve. Accordingly, we guess ha he cenral ank loss funcion in he principally forward-looking, new-keynesian model mus involve a erm, E π +, which is he main characerisics of he Phillips curve. A he same ime, we require pah conrollailiy. Tha is, we require ha he arge variales evolve a he persisence, δ. As a resul, we consruc he 5 See Eq. (D.3 and is derivaion in Appendix D. 6

28 shor-run arge values as follows (66a ( E E ( + + π = π + δ π π = δ π + δ π, and (66 = + δ ( = δ + ( δ x x x x x x. The shor-run inflaion arge value equals he weighed ( δ > average of he expeced inflaion rae and he long-run arge value. Using an expecaion as a arge value 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 (Svensson and Woodford, 005. The corresponding cenral ank period loss funcion equals { + } (67 L = ( π π δ ( Eπ π + ( x x δ ( x x. Once again, we do no consider shocks and se u = 0 discreionary policy wih period loss funcion (67 equal 6. The oucomes under a (68a ( π π = δ ( π π, and (68 ( x x = δ ( x x. Tha is, here exiss a discreionary policy, which proves pah conrollale. u π ( u = π ( u When 0, he sailizaio n ias arises. To eliminae iases, using and x ( u x ( u (69 = produces 7 ( φc φ( βρ =. δ δβρ φβρ ( ( Economical feasiiliy requires ha 0. Thus, 6 See Eqs. (E.8 and (E.0 and heir derivaion in Appendix E. 7 See Eq. (E. and is derivaion in Appendix E. 7

29 (70 φ φ c. In sum, he shor-run inflaion arge value conforms o he macroeconomic srucure (i.e., Phillips curve. Tha is, i is lagged (expeced, if he inflaion rae in he Phillips curve exhiis principally ackward-looking (forward-looking ehavior. Wih he delegaed loss funcion, a discreionary policy proves pah conrollale (removing he consan average and sae- coningen inflaion iases, and eliminaes sailizaion ias, resuling in Ramsey opimaliy. The Delegaed Weigh for 0 φ. When he Phillips curve exhiis purely ackward-looking ehavior wih no expecaions (φ=, he ime-inconsisency prolem does no exis and neiher does he sailizaion ias. Thus, =. When he Phillips curve exhiis principally ackward-looking ehavior wih an elemen of expecaions ( φ φ <, he ime-inconsisency prolem and, hus, he sailizaion ias exis, c and he delegaed weigh mus differ from he social weigh (. The coefficien of he weigh depends only on he dynamic parameers (φ, β, and ρ, and does no relae o he sysem convergence persisence, δ. By Eq. (64, (7 ( c ( ( ( φ φ βρ φ = = <. ρ( φβρ ρ φβρ Therefore, he cenral ank mus exhii conservaism ( <. Inuiively, if he inflaion rae persiss ( φ φ <, hen curren inflaion rae deviaion f rom is arge value will persis ino c fuure and, hus, cause losses. To reduce hese losses, we mus place more weigh on he inflaion rae arge (i.e., <. In he new-classical model wih oupu persisence, we mus place more weigh on he oupu gap ecause of oupu gap persisence. 8

30 When he Phillips curve exhiis principally forward-looking ehavior ( 0 < φ φ, he weigh in Eq. (69 depends no only on he dynamic parameers u also on he sysem convergence persisence. The weigh ecomes sule ecause of forward-looking ehavior. No analyical way exiss o discuss he relaionship of he weigh wih model parameers. For he purely forward-looking case (φ=0, he cenral ank exhii weigh-conservaive (weigh-lieral ehavior, if he cos-push shock more (less persiss. The delegaed weigh in Eq. (69 reduces o c (7 = δ ( ρ ( δβρ. We repor he following condiions (73a (73 (73c < if ρ > ρ, = if ρ = ρ, and > if ρ < ρ, where c c c δ (74 ρc =, and 0 < ρ c <. βδ The criical value, ρ c, depends on model parameers. The cenral ank is weighconservaive (weigh-lieral, if cos-push shock persisence is greaer (less han he criical value. Acually, ρ < 0. 8 A more persisen cos-push shock implies less weigh on oupu sailizaion. To see his, ierae he Phillips cu rve (Eq. 3 wih φ=0 forward as follows (75 i π = E β ( x + u. + i + i i= 0 8 ( ( ρ = βδ δ δβρ < 0 9

31 Inflaion depends enirely on curren and expeced fuure oupu gaps and cos-push shocks, ecause of he purely forward-looking naure of he Phillips curve. Thus, more persisen cos-push shocks imply ha more losses occur ecause more inflaion increases emerge, driven y curren and fuure cos-push shocks. To reduce he losses caused y curren and fuure inflaion, he policy makers mus place more weigh on inflaion sailizaion and, hus, less weigh on oupu sailizaion. 4. Siuaions of he Same Weigh ( = A summary of he siuaions of he same weigh ( = provides more insigh aou delegaion. We deermine wo siuaions where = -- saic models in a game conex and models in a one-decision-maker conex. Saic Models in a Game Conex Consider he delegaed weigh, ( βη =, in he new-classical model wih oupu persisence. Oviously, =, when β=0 and/or η=0. When he cenral ank does no care aou fuure losses (β=0, he model, which includes he loss funcion, he Phillips curve, and raional expecaions, is essenially saic even if he Phillips curve exhiis oupu persisence (η 0. When persisence does no ener he Phillips curve (η=0, he model is also essenially saic even if he cenral ank cares aou he fuure losses (β 0. In a word, he model is saic when β=0 and/or η=0. The ime-inconsisency prolem, however, always exiss ecause of he assumpion of raional expecaions. We know ha he sailizaion ias arises, which requires a differen weigh o eliminae i, when he ime-inconsisency prolem exiss. Why does he delegaed weigh equal he social 30

32 one? We argue ha discreionary policy need no rade off over ime wih a saic model and need no aler he alance eween inflaion and oupu-gap sailizaion when we delegae he correc shor-run arge values ( π π = and x x = η. As a resul, =. The correc arge values are imporan as he lieraure sresses. An overly amiious oupu (employmen arge value produces inflaion ias. Rogoff s (985 weigh-conservaive approach ackles his overly amiious employmen arge value y placing less (more weigh on he employmen (inflaion sailizaion. Raher han adjusing he weigh, we ackle he prolem direcly y correcing he overly amiious emp loymen arge value. Wih correc arge values, he cenral ank does no need a differen weigh o alance wo arge variales sailizaion in a saic model. The wo saic game models in Yuan, e al. (0, where he ime-inconsisency prolem exiss, also demonsrae ha he delegaed weigh equal he social weigh when we delegae he cenral ank he correc arge values. Models in a One-Decision-Maker Conex Consider he hyrid new-keynesian model wih φ=. In his siuaion, =. When φ=, he Phillips curve conains no expecaions and, hus, he ime-inconsisency prolem and he sailizaion ias do no exis. As a resul, he cenral ank adops he same weigh as sociey. As an exercise, we consider he hyrid new-keynesian model wih β=0. When β=0, he Phillips curve conains no expecaions and, hus, he ime-inconsisency prolem does no exis. 3

33 Thus, = in his siuaion. 9 Based on he wo siuaions of = -- saic models in a game conex and models wih one decision maker, we conclude ha in a dynamic game model. Moreover, he coefficien of he delegaed weigh depends on dynamic parameers and/or he sysem convergence persisence. 5. Conclusion In his paper, we ackle wo issues in policymaking ime-inconsisency and arge conrollailiy in wo dynamic models. Time-inconsisency resuls from expecaions (muliple decision makers. Targe conrollailiy arises ecause he arge variales mus conform o he consrain imposed y he economic model (he Phillips curve. We can resolve oh issues y delegaing a loss funcion, which differs from he social loss funcion, o he cenral ank. The delegaion scheme is simple wih one decision maker whereas i is relaively complicaed in a game conex. We oain proposiions aou arge conrollailiy. Tha is, if he sysem, which consiss of a quadraic loss funcion and a linear Phillips curve, converges under he Ramsey policy or under discreionary policy, consan arge values are joinly asympoically conrollale if and only if hey saisfy he Phillips curve in equilirium. We find ha he arge variales equiliriums under he Ramsey policy saisfy he Phillips curve in equilirium and, hus, are joinly asympoically conrollale. We call hese equilirium values he long-run arge values. Wih one decision maker as in he hyrid new-keynesian model wih φ=, he model 9 See Eq. (F.9 in Appendix F for he derivaion. 3

34 involves no expecaions and exhiis purely ackward-looking ehavior. In his conex, he ime-inconsisency prolem does no exis, and we only need o ackle he arge conrollailiy prolem. Delegaing he long-run arge values (he arge variales equiliriums under he Ramsey policy o he cenral ank can produce no only asympoic u also pah conrollailiy. Delegaing shor-run arge values proves unnecessary ecause ime-inconsisency does no exis. Moreover, he cenral ank adops he same weigh as sociey ecause sailizaion ias does no exis. Wih muliple decision makers where he model involves expecaions and, hus, imeinconsisency exiss, he long-run arge values can only achieve asympoic raher han pah conrollailiy. As a resul, we delegae shor-run, sae-coningen arge values o he cenral ank o oain pah conrollailiy. They commi and ind he cenral ank o follow exacly he Ramsey opimal pahs. The shor-run inflaion arge value conforms o he macroeconomic srucure (i.e., Phillips curve. Tha is, i is lagged (or expeced, or exacly he social arge value if he inflaion rae exhiis principally ackward-looking (or forward-looking, or raional expecaions ehavior in he Phillips curve. 30 Specifically, he shor-run inflaion arge value equals he weighed (or weighed in form average of he lagged (or he expeced value and is long-run arge value, if he inflaion rae exhiis principally ackward-looking (or forward- looking ehavior in he Phillips curve as in he new-keynesian model wih φ φ < (or c φ φ. The weigh on he lagged value equals he sysem convergence persisence under he 0 c 30 In a model (Phillips curve wih principally forward-looking ehavior, he delegaed inflaion arge value is expeced, demonsraing he idea of implemening opimal policy hrough inflaion-forecas argeing (Svensson and Woodford,

35 Ramsey policy whereas he weigh on he expeced value equals he reciprocal of he persisence. The inflaion arge value always equals he social arge value, if he privae secor forms raionally expecaions aou he inflaion as in he new-classical model. Wih regards o he oupu gap, he shor-run arge value equals he weighed average of he lagged value and is long-run arge value. In a word, he shor-run arge values converge o he long-run arge values and, hough sae-coningen, are predeermined. Wih he shor-run arge values, he discreionary policy proves pah conrollale and, hus, eliminae he consan average and sae-coningen inflaion iases. Delegaing a differen weigh o he cenral ank can eliminae sailizaion ias in a dynamic game model. When he oupu gap exhiis persisen, he cenral ank mus place more weigh on he oupu gap sailizaion as in he new-classical model. When he inflaion rae exhiis more persisence han expecaion, he cenral ank mus place more weigh on inflaion rae sailizaion as in he hyrid new-keynesian model wih φ φ <. The inuiion is sraighforward. When he oupu gap (inflaion rae exhiis persisen, curren deviaion from is arge value will persis ino fuure, and induce losses. To reduce he losses, he cenral ank mus place more weigh on oupu-gap (inflaion sailizaion. More over, he coefficiens of he weighs in hese wo siuaions depend only on he dynamic parameers, and do no relae o he sysem convergence persisence. When he inflaion rae exhiis principally forward-looking ehavior as in he hyrid new-keynesian model wih 0 φ φ, he delegaed weigh ecomes sule and he cenral c ank may adop a lieral or conservaive weigh, depending on oh he dynamic parameers and 34 c

36 he sysem convergence persisence. We deermine wo siuaions where he delegaed weigh equals he social weigh ( = -- saic models in a game conex and models wih one decision maker. As a resul, we conclude ha he delegaed weigh mus differ from sociey ( in a dynamic game model. The coefficien of he delegaed weigh depends on dynamic parameers and/or he sysem convergence persisence. In sum, we can solve oh issues ime-inconsisency and arge conrollailiy y delegaion. Through delegaion, discreionary policy proves pah conrollale (removing he consan average and sae-coningen inflaion iases and eliminaes sailizaion ias, resuling in Ramsey opimaliy. Appendix A: Ramsey Opimal Soluion in a Hyrid New-Keynesian Model The opimizaion prolem minimizes he social ineremporal loss funcion ( wih period loss funcion ( sujec o he Phillips curve (Eq. 3. Is Lagrangian expression equals he following (A. ( π π + ( x x L = E0 β. = 0 + ψ x + φπ + ( φ βπ+ + u π The firs-order condiions equal { β ( ψ } L (A.a = E0 x x + 0 x = for 0, (A. L = E 0 ( 0 0 π π π ψ + βφψ = 0 0 ( L (A.c E β π π ψ = 0 = 0 π + + β φψ+ + β ( φ βψ for = 0, and for. 35

37 Eliminaing he mulipliers from Eqs. (A. gives he consolidaed firs-order condiions as follows (A.3a = π for = 0, and Ex 0 x φβ Ex 0 + x φ Ex 0 x = E0π π for. ( ( x0 x φβ E0x x ( π0 (A.3 ( ( ( ( ( Comining Eqs. (A.3a and (A.3 wih he Phillips curve (Eq. 3 produces (A.4a (A.4 (A.5a aβ E0x βe0x ( φ β x0 0 ( + u φ π π ( φ( β π + φ( β ( φ + φ ( β = = x x 0 for 0, and β β + + a Ex 0 + Ex 0 + Ex 0 Ex 0 aex 0 ( φ( β π + φ( β + = ( a φ φ x E0u 0 for, where (A.5 + φ β ( φ β β a + + = + + β, and (A.5c x 0. equaion Solve Eqs. (A.4a and (A.4 ackwards. Assume ha δ equals a roo of he characerisic (A.6 aβδ 4 βδ 3 + δ δ+ a = 0. Assume ha he soluion of Eq. (A.4 for akes he following form: (A.7 x = δ x + e+ fu. Using Eq. (A.7 and Eu 0 = ρeu 0 leads o (A.8a Ex 0 = δ Ex 0 + e+ fρeu 0, 36

38 (A.8 ( ( E x = δ E x + δ + e+ δ + ρ f ρe u, and (A.8c E x+ = δ E + ( δ + + e+ ( δ + + (A.9 δ δρ ρ ρ 0 0x f E0u. Susiuing Eqs. (A.8a, (A.8, (A.8c, and Eu 0 = ρeu 0 ino Eq. (A.4 resuls in a + E x + ae x + df + ρ E u + ce ( φ( β π + φ( β = 3 ( βδ βδ δ (A.0a β ( δ δ β( δ x 0, where c a , and ( ( (A.0 + (A.6 produces d aβ δ + δρ + ρ β δ + ρ. 3 Transforming Eq. (A.9 and noing ha ( aβ δ βδ + δ = δ a from Eq. (A. δ ( ( ( E0x = δex 0 + ce φ β π φ β x a + δ + df + ρe0u. a Comparing Eq. (A. wih Eq. (A.7 implies (A.a (A. ( ( δ φ β e = π + φ ( β x δc f δρ = δρ d a a, and (. To inerpre he consan e in Eq. (A.7, we compue (A.3 x ( ( ( c a( e δ φ β = π + φ ( β x. δ δ δ Wih he noaion x, he soluion of he oupu gap for equals (A.4 ( x x δ ( x x = + fu, 37