Insrumenal rules and argeing regimes Giovanni Di Barolomeo Universiy of Teramo
Preview Definiions Par one Insrumenal rules 1. Taylor rule 2. The problem of insabiliy (Taylor principle) Par wo Targeing regimes (basic) 1. Definiion of a argeing regime 2. Inflaion argeing Par hree Opimal policy implemenaion 1. Commimen 2. Timeless perspecive 3. Discreion 2
Definiions Possible sources of confusion 1. Acivis vs. Non-Acivis Rule (Friedman) 2. Rule (coningen plan) vs. Discreion (opimizaion period by period) 3. Targeing vs. Insrumenal Rules Here we focus on he laer 3
More in deail: 3 approaches The sandard approach The policy-maker behaves by minimizing an approximae social loss derived from he micro-founded model (nex seminar). Insrumenal rule The policy-maker commis o a simple feedback rule. Targeing rule The policy-maker behaves by minimizing a differen loss based on arge deviaions differen from hose derived from he micro-founded model. 4
Par one Insrumen rules 5
Insrumenal rule implemenaion If a rule ha describes how he moneary auhoriy ses he nominal ineres rae is added o he model, his mus be done wih care o ensure ha he policy rule does no render he sysem unsable or inroduce muliple equilibria. 6
Insabiliy due o self-fulfilling expecaions Consider wha would happen if expeced inflaion were o rise. E π(+1) [i() E π(+1)] x() π() Since here is no endogenous feedback from his rise in expeced inflaion o he nominal ineres rae, he real ineres rae mus fall. This decline in he real rae is expansionary and he oupu gap increases. The rise in oupu increases acual inflaion. 7
Taylor rule (Taylor, 1993) The mos famous insrumen rule is he Taylor one: i() = i + π() + 0.5 x() + 0.5 [ π() π T ] where π T was he arge level of average inflaion (Taylor assumed i o be 2%) and i was he equilibrium nominal rae of ineres (Taylor assumed he real rae was equal o 2%); x() is he poenial oupu gap (linear deerminisic rend). Please noice (Taylor rule): i() = 1.5 π() + 0.5 x() + 0.01 8
A formal analysis in a NK model The Taylor Rule i = aπ + a x 1 2 A simple New Keynesian (Neo Wicksellian) model x = E x σ i Eπ ( ) 1 + 1 + 1 9 π = βeπ + κx + 1
Rearranging Eβπ = 1 π + κx 1 a2 a1 Ex + 1+ Eπ + 1 = 1+ x + π σ σ σ 10 1 1 1 1 σ Ex + 1 1+ a x 2σ a1σ 0 β Eπ = + 1 κ 1 π
Sabiliy analysis 11 AEY EY = + 1 BY = + 1 MY M = [ ] Y = x π 1 A B 1 1 1 1 σ Ex+ 1 1+ a2σ a1σ x 0 Eπ = 1 1 π β + κ 1 1 1 1 Ex + 1 1 σ 1+ a x 2σ a1σ = Eπ + 1 0 β κ 1 π M a2β + γ a1β 1 1+ θβ θβ = γ 1 β β
The wo eigenvalues of Marix M λ 1 ( M ) a+ b 2β = λ ( M ) 2 = a b 2β. γ + β a a = + β + > θ 2 1 0 2 γ a2 1 β γ β 2 2 b= + β + + β + ( a1γ + a2) θ θ θ θ 12 λ ( M ) λ ( M ) 1 2
We look a he second eigenvalue only. a b 2β ( ) 2 > 1 a b > 2β a 2β > b a 2β > b 2 2 γ + βa2 γ a2β γ β 1 1 2 2 + β + > + β + + β + ( a1γ + a2) θ θ θ θ θ 13 Sabiliy requires 1 β 1 + 1 a > 0 a1+ a2 > 1 γ ( a ) γ ( β) 1 2
Taylor Principle Seing a 1 > 1 is referred o as he Taylor Principle, because John Taylor was he firs o sress he imporance of ineres rae rules ha called for responding more han one-for-one o changes in inflaion. Wih Taylor s original coefficiens, 1.5, so ha he nominal rae is changed more han one-for-one wih deviaions of inflaion from arge. Thus, he rule saisfies he Taylor Principle. 14
Taylor Rule sabiliy: Summary I Suppose he cenral bank responds o boh inflaion and he oupu gap according o: i() = a1 π() + a2 x() + v() Wih his policy rule, he condiion o ensure he economy has a unique, saionary equilibrium becomes: a1 + (1 β) a2 / κ > 1 Sabiliy depends on boh he policy parameers. 15
Taylor Rule sabiliy: Summary II Self-fulfilling expecaions: E π(+1) [i() E π(+1)] x() π() This suggess ha a policy which raised he nominal ineres rae enough o increase he real ineres rae when inflaion rose would be sufficien o ensure a unique equilibrium he Taylor Principle. A feedback rule saisfies he Taylor Principle if i implies ha in he even of a susained increase in he inflaion rae by k percen, he nominal ineres rae will evenually be raised by more han k percen. 16
Taylor Rule wih ineres-rae ineria A similar resul is obained in he case of a rule ha incorporaes ineres-rae ineria of he kind characerisic of esimaed Fed reacion funcions (e.g., Judd and Rudebusch, 1997): i() = i () + a1 [π() π ] + a2 [x() x ] + ρ [i( 1) i ( 1)] Wih his policy rule, he condiion o ensure he economy has a unique, saionary equilibrium becomes: a1 + (1 β) a2 / κ > 1 ρ Once again i corresponds precisely o he Taylor principle. 17
Taylor Rule & NK approach The Taylor rule resuls from he Social Loss minimizaion. Zero inflaion arge. Flexible price oupu gap (uni labor cos) insead of de-rended oupu. Non-consan inercep. 18
19 Real GDP (y) vs. uni labor cos (ulc)
20 Inflaion predicion
Esimaed Taylor rules & co In general, he basic Taylor Rule, when supplemened by he addiion of he lagged nominal ineres rae, does quie well in maching he acual behavior of he policy ineres rae. Bu since hey a mos explain 2/3 of he empirical insrumenrae changes, cenral banks in pracice deviae subsanially from such a rule. Moreover, Ophanides (2000) ha when esimaed using he daa on he oupu gap and inflaion acually available a he ime policy acions were aken (i.e., using real-ime daa), he Taylor Rule does much more poorly in maching he U.S. funds rae. 21
The advanages The rule can easily be verified by ouside observers and a commimen o he rule would herefore be echnically feasible. Varians of he Taylor rule have been found o be relaively robus o differen models, in he sense ha hey perform reasonably well in simulaions wih differen models and rarely resul in very bad oucomes 22
The disadvanages The rule will no resul in an opimal oucome, for several reasons (sub-opimal). No cenral bank has made a commimen o follow i (never observed). Empirical esimaes of Taylor-ype reacion funcions show ha hey a mos explain 2/3 of he empirical insrumen-rae changes. Thus, cenral banks in pracice deviae subsanially from such a reacion funcion. 23
Money rules 24 The quaniy of money is no oally absen from he underlying model, since equaion money demand mus equal money supply in equilibirum (LM curve). In a linearized form: m() p() = 1/(σb) x() 1/σ i() Given he nominal ineres rae chosen by he moneary policy auhoriy, his equaion endogenously deermines he nominal quaniy money. Alernaively, if he policymaker ses he nominal quaniy of money, he he model can be solved joinly for x(), π(), and i(). E.g. he Taylor rule (i() = a1 π() + a2 x()) becomes: m() = 1/σ a1 π() + 1/σ (1/b + a2) x() + p()
Par wo Targeing regimes 25
Targeing regimes Definiion. A argeing regime is defined by (a) he variables in he cenral bank s loss funcion (he objecives), and (b) he weighs assigned o hese objecives, wih policy implemened under discreion o minimize he expeced discouned value of he loss funcion. Operaionally. Nex par. 26
Issues Who assigns arges? Who assigns weighs? Is he arge public? Wha happens if arges are missed? 27
Some argeing regimes Inflaion argeing (he mos famous). Price level argeing (Svensson 1997, Vesin 2002), Nominal income growh argeing (Jensen 2002), Hybrid price level/inflaion argeing (Baini and Yaes 2001). Average inflaion argeing (Nessén and Vesin 2003) Regimes based on he change in he oupu gap or is quasi-difference (Walsh 2003, Jensen and McCallum 2002). 28
Inflaion argeing Asymmery beween he inflaion and oupu arge, consisen wih he inflaion arge being he primary objecive. (inflaion arge is subjec o choice bu no he oupu arge) There is in fac general agreemen ha inflaionargeing cenral banks do normally no have overambiious oupu arges, ha is, exceeding poenial oupu. Thus, discreionary opimizaion does no resul in average inflaion bias, couner o he case in he sandard Kydland-Presco-Barro-Gordon seup 29
How should we define inflaion argeing? 1. There is a numerical inflaion arge, in he form of eiher a poin arge (wih or wihou a olerance inerval) or a arge range. This numerical inflaion arge refers o a specific price index. Achieving he inflaion arge is he primary objecive of moneary policy, alhough here is room for addiional secondary objecives. There is no oher nominal anchor, like an exchange-rae arge or a moneygrowh arge. 30
How should we define inflaion argeing? 2. The decision-making process can be described as inflaion-forecas argeing. In he sense ha he cenral bank s inflaion forecas has a prominen role and he insrumen is se such ha he inflaion forecas condiional in he insrumen-seing is consisen wih he arge. This does no exclude ha oupu and oupu-gap forecass also ener in an essenial way. 31
How should we define inflaion argeing? 3. There is a high degree of ransparency and accounabiliy. The cenral bank is accounable for achieving he inflaion arge and provides ransparen and explici moneary-policy repors presening is forecass and explaining and moivaing is policy. 32
Inflaion forecas argeing (inuiion) Moneary policy affecs inflaion wih a long lag. Thus, cenral banks need o focus on heir forecass of fuure inflaion. If inflaion is forecas o rise above arge, policy should be ighened. If he forecas falls below arge, policy should be loosened. The cenral bank should arge is forecas of inflaion. Unforunaely, such an approach o policy can lead o muliple equilibria. 33
Insiuional commimen A clear (preferably legislaed) mandae for a moneary policy direced owards low inflaion, cenral-bank independence ( insrumen independence, meaning independence in seing he moneary-policy insrumen, in some cases also independence in formulaing an operaional inerpreaion of he low-inflaion mandae), and accounabiliy of he cenral bank for achieving he mandae. 34
Price level argeing Vesin (2002) shows price level argeing can replicae he imeless pre-commimen soluion if he cenral bank is assigned a loss funcion defined on zero price level deviaions insead of inflaion deviaions. Walsh (2003) adds lagged inflaion o he inflaion adjusmen equaion and shows ha he advanages of price level argeing over inflaion argeing decline as he weigh on lagged inflaion increases. 35
Par hree Opimal policy implemenaion 36
The wo-sep problem The problem can be divided ino wo subproblems: Choose {x(+i), π(+i)} o minimize he lossfuncion subjec o he Phillips curve; Using he focs in he IS-curve o find he opimal ineres rae rule. 37
38 The Lagrangian focs wih respec o imply: [ ] [ ] { } = + + + + + + + + 0 1 2 2 2 1 (*) i i i i i u kx E x E π π β λ α π β ( ) ( ) 0 0 (3) 0 0 (2) 0 (1) 1 = > = + = + + + + + + i k x E i E i i i i i λ α λ λ π λ π { } i i x + + π,
Dynamic inconsisency From equaion (1) and (2): 1. a ime, i is opimal o se π() = - λ() and o se π(+1) = - (λ(+1) - λ() ); 2. When ime +1 arrives and he CB re-opimizes i will be opimal o se π(+1) = - λ(+1) (condiion (1) updaed a ime +1) 39
The opimal rule 40 Combining focs (1), (2) and (3): ( 4) x + i x + i 1 = π+ i (5) x k = π α k α using (4) and (5) ogeher wih he IS, we find he opimal ineres rae rule: k iˆ 1 ( 6) 1 Eπ 1 + g ασ + σ =
Indeerminacy Given ineres-rae rule (6), 41 The coefficien k iˆ 1 = 1 E π 1 g ασ + + σ k 1 < 1, ασ hus, he CB adjuss demand only parially in response o increase in expeced inflaion. Such a rule involves indeerminacy.
Timeless Perspecive Commimen 42 To avoid ime inconsisency he cenral bank implemens condiions (2) and (3) for all periods, including he curren period. Combining he wo, he implici policy rule, α ( 7) π + i = + i + i 1 i k (8) ( x x ) for 0 afer some algebra, he equilibrium inflaion is: π α ( 1 a ) x + = x 1 2 k α x [ ( )] 1+ β 1 ρ a + k The pre-commimen policy inroduce ineria ino he oupu gap and inflaion policy. α u
The disadvanage Dennis 2001 The policy rule represen a sub-opimal. π π + i = = ( λ λ ) + i ( λ λ ) 1 + i 1 In he fully opimal commimen λ(-1) = 0, alernaive value of λ(-1) = 0 lead o alernaive policy choices consisen wih imeless perspecive. Hence he policy rule (7) may be dominaed by oher rules. 43
Discreion Wihou an explici commimen Cenral bank will choose is decision variables x(), π(), each period o maximize is objecive (*) subjec o he aggregae supply curve (2) and he IS curve (1), and aking privae secor's expecaions as given. The discreion problem can be rewrien as: 44
45 Discreion problem [ ] i i i u E f x E F where f x s F x + = + = + = + + + + = + 1 2 1 2 2 2.. 2 1 π β π α λ π π α min
Discreion 46 The opimal soluion of he problem is: in +1 (9) x k = π α = αqu ( 10) kπ + + αx + 1 1 = while, in he case of imeless perspecive commimen, ( 11) kπ + α + 1 0 ( x x ) 0 + 1 =
The opimal ineres rae rule 47 Subsiuing (9) in he Phillips Curve, solving forward, and hen using he IS equaion, he opimal ineres rae rule, consisen wih he Taylor Principle, is: ˆ 1 (13) i = Φ Eπ + 1 + g where Φ π = π 1 + k ( 1 ρ ) αρσ σ > 1
Equilibrium inflaion Under opimal discreion, he equilibrium inflaion is given by: (14) π α k α α = x = 2 u ( ) 1 βρ + k 48 he uncondiional expeced value of inflaion is zero, here is no average inflaion bias. There is a sabilizaion bias, (he response o inflaion o a cos-push shock differs from he response under commimen).
Discreion and he Phillips rade-off Cenral bank references Inflaion sabilizaion Variances rade-off A 49 O Oupu sabilizaion
Inflaion and oupu gap response o a cos-push shock Commimen (solid line) Discreion (dashed line) Inflaion Oupu gap 50
Sabilizaion bias Under imeless pre-commimen he cenral bank keeping oupu below is poenial for more han one period, lows expecaions of fuure inflaion improving he rade-off beween inflaion and oupu gap. Under discreion here is no ineria. Oupu gap and inflaion reurn o heir seady-sae value in he period afer he shock occurs. 51