Endogenous Rule-of-Thumb Price Setters and Monetary Policy

Transcription

1 Endogenous Rule-of-Thumb Price Seers and Moneary Policy Rober Amano y, Rhys Mendes z, and Sephen Murchison x Bank of Canada PRELIMINARY AND INCOMPLETE November 9, 2009 Absrac Sudies evaluaing he e cacy of moneary policy rules and regimes are ofen based a benchmark new Keynesian model where he parameers are assumed o be policy invarian. I is possible, however, ha some key parameers may no be invarian o changes in moneary policy. In his paper, we use a hybrid new Keynesian Phillips curve o examine he in aion versus price-level argeing debae when he proporion of rule-ofhumb price seers is allowed o change endogenously wih he moneary regime. Alhough here are oher facors ha may also be endogenous, we focus on in aion ineria since i has been ideni ed in he lieraure as a crucial parameer a ecing he performance of moneary policy. This paper was prepared for he Bank of Canada s annual conference eniled "New Froniers in Moneary Policy Design," o be held November 2 and 3, The views expressed are he auhors and do no necessarily re ec hose of he Bank of Canada or is sa. y RAmano@BankofCanada.ca z RMendes@BankofCanada.ca x SMurchison@BankofCanada.ca

2 . Inroducion The new Keynesian Phillips curve wih lagged in aion has become a workhorse model for applied moneary policy analysis. Is wide-spread use re ecs he abiliy of he canonical new Keynesian Phillips curve o provide a srucural framework for undersanding in aion dynamics and is lagged exension o accoun for he degree of in aion persisence observed in he daa. This hybrid in aion model has been used o sudy, among oher issues, he e cacy of di eren moneary policies. Seinsson (2003), for example, follows Gali and Gerler (999) by assuming he presence of rule-of-humb price-seers and uses his ype of hybrid new Keynesian Phillips curve o sudy opimal moneary policy. He nds a purely forward-looking new Keynesian Phillips curve admis price-level sabiliy as opimal moneary policy, whereas opimal policy in he same model augmened wih rule-of-humb behaviour allows for price-level drif ha increases wih he proporion of rule-of-humb price-seers. This is an especially ineresing resul for moneary auhoriies as i suggess ha he case for price-level argeing is weakened as he proporion of rule-of-humb price-seers increases. Oher research comparing he e cacy of in aion versus price-level argeing in a new Keynesian framework exended o include rule-of-humb price-seers ofen conclude ha price-level argeing dominaes in aion argeing. One reason for his resul is ha he proporion of rule-of-humb price-seers is ypically se a a relaively low value (0. o 0.3), and as Seinsson (2003) shows his value has imporan implicaions for comparisons beween in aion and price-level argeing. While his line of research has improved our undersanding of he e eciveness of di eren moneary policies, i has he poenial drawback ha i reas he imporance of rule-of-humb behavior as xed across policy experimens. The loss in pro s associaed wih following a simple rule-of-humb is likely o be sensiive o he policy regime so policy experimens ha hold he proporion of rule-of-humb price-seers xed are vulnerable o he Lucas criique. In an e or o examine he relevance of his concern, we examine he in aion versus price-level argeing quesion when he proporion of rule-of-humb priceseers is endogenous. We work in he conex of a hybrid new Keynesian Phillips developed by Gali and Gerler (999) and allow he proporion of rule-of-humb The new Keynesian Phillips curve wih rule-of-humb price seers has been used o sudy oher imporan moneary issues such as opimal moneary policy rules, zero bound on nominal ineres raes, welfare coss of in aion, desirabiliy of in aion argeing, opimal policy under discreion and commimen, ec.

3 price-seers o be a funcion of he moneary policy regime. Our focus on ruleof-humb behaviour and in aion persisence re ecs is imporance in in uencing moneary policy. Rudebusch (2002), for insance, nds he e eciveness of nominal income argeing o be an inverse funcion of he degree of in aion ineria. Similarly, Walsh (2003) nds he performance of price-level argeing deerioraes signi canly as in aion becomes more persisen. Levin and Williams (2003) show ha a policy rule ha is opimal in a model wih a low degree of in aion persisence can lead o disasrous resuls if he rue model is characerized by a high degree of in aion ineria. To preview our resuls, we nd he relaive performance of price-level argeing deerioraes or, even reverses in favour of in aion argeing, when he proporion of rule-of-humb price-seers is allowed o be endogenous. The inuiion for his resul is ha he e eciveness of price-level argeing in sabilizing in aion causes rule-of-humb behaviour o become more aracive. This leads o an increase in he proporion of rule-of-humb price-seers and, in urn, makes in aion more inrinsically persisen and hereby undermines he e eciveness of price-level argeing. The remainder of he paper is organized as follows. Secion 2 oulines he main feaures of our model and Secion 3 describes is calibraion. Secion 4 presens resuls comparing in aion versus price-level argeing when he proporion of rule-of-humb price-seers is endogenously deermined. Concluding remarks are provided in Secion Model The model consiss of a coninuum of households wih an in nie planning horizon, a collecion of monopolisic compeiive rms ha produce di ereniaed inermediae goods, sluggish price adjusmen, and a moneary auhoriy ha ses he shor-erm ineres rae according o a prespeci ed rule. The derivaions in he forhcoming four subsecions follow closely hose in Seinsson (2003) so we will provide only a high-level skech of he benchmark rule-of-humb price seer model. The fh secion provides deail on our exension of he benchmark model o an environmen where he proporion of rule-of-humb price-seers is endogenous.

4 2.. Households Households maximize expeced uiliy given by: E X s= [u (C s ; s ) v (y s (z) ; s )] where 2 (0; ) is a discoun facor, and is a vecor of shocks o household preferences and producion. Each household is assumed o specialize in he producion of one di ereniaed good denoed y (z), and consume a composie consumpion good represened by C. The laer is combined via a Dixi-Sigliz consumpion index given by, Z C 0 c (z) dz and z indexes di ereniaed goods and is he elasiciy of subsiuion beween di ereniaed goods in period. I is sandard pracice o assume ha is consan bu we follow Seinsson and allow he parameer o be sochasic. Noe ha we have suppressed he household index on consumpion since he assumpion of complee markes will eliminae consumpion heerogeneiy in equilibrium. Uiliy maximizaion subjec o a sandard budge consrain leads o household demand for individual varieies of: p (z) c (z) = C 8z 2 [0; ] where he aggregae price index is given by: P Z P = p (z) 0 di () In equilibrium, markes mus clear so c (z) = y (z) and C = Y for all and z. Combining hese marke clearing condiions wih he household s rs-order condiions for consumpion and asse holdings yields he familiar Euler equaion: ( UC Y + ; + E U C (Y ; ) P P + where i is he risk-free shor-erm nominal ineres rae. ) = + i (2)

5 Household pricing decisions are based on Calvo (983) nominal price conracs. Following Calvo, we assume ha in any given period each household has a xed probabiliy ha i may adjus is price and, hence, a probabiliy ha he household mus leave is price unchanged. In an e or o beer mach he persisence found in in aion daa, we depar from he sandard Calvo srucure by having wo ypes of price-seers wihin he cohor allowed o change heir price 2. One ype, which is a fracion! of price changing households, behave like sandard Calvo price-seers in he sense hey se prices opimally given he consrains of he model and use all available informaion. We refer o hese price-seers as "forward looking." The remaining! households wihin, which we refer o as "rule-of-humb" price-seers, use a simple rule o adjus heir prices. Forward-looking (FL) price-seers choose prices o solve:! ) max E p f X s= () s (u C (Y s ; s ) ( ) s p f P s y s (z) v (y s (z) ; s ) subjec o,! y s (z) = s p f s Y s P s where = is a ax rae and p f is he opimal price which sais es he rs-order condiion: 8 9 X! E () s s p f s s >< u C (Y s ; s ) ( ) p f P s >= Y s ( s ) P s= s >: s s s v p f s y Ys P s ; s >; = 0 (3) Noe ha we suppress he z index on he rese price since all reseing FL rms choose he same price in a given period. Also noe ha, in he case of fully exible prices, (3) collapses o he sandard saic condiion: p v y Y P ; p = P u C (Y ; ) 2 See Seinsson (2003) and Gali and Gerler (999) for a deailed descripion of his ype of pricing model. (4)

6 where p is he price chosen under exible prices. So, under exible prices, he rm will se opimally is relaive price as a markup over real marginal cos (adjused for he ax rae). We assume ha rule-of-humb (RT) price-seers se heir price according o: p b = & p (5) where P =P 2 is gross in aion, = ( ) is he gross markup, and governs he degree of indexaion. 3 An index of prices se a ime is given by log p = (!) log p f +! log p b (6) Given ha all FL price-seers will choose he same price and all RT priceseers will choose he same price in any given period, we can wrie he overall price index () as: P = (P ) + ( ) (!) p f + ( )! p b (7) 2.2. Log-linearizaion of he model The log-linear approximaion of (2) yields he IS curve: x = E x + b i E + r n where x = log(y =Y n ) and Y n is he level of oupu consisen wih an economy wih fully exible prices, perfecly compeiive markes and no disorionary axes. Moreover, bi = log + i + i = log ( ) u C = u CC Y > 0 Y r n = E n log + Y n uc u + C 3 Noe ha his is a generalizaion of he Gali and Gerler (999) rule-of-humb rule: I allows he weigh on lagged in aion o be di eren from uniy and he weigh on he markup o be di eren from zero.

7 Nex, consider he supply-side of he economy, where he log-linear approximaions of (5)-(7) are: bp b = &b + bp + (8) bp = (!) bp f +!bp b = (!) bp f (9) +!bp b (0) where (as in Seinsson) bp f, bp b and bp denoe percen deviaions of p f =P, p b =P and p =P, respecively, from heir seady-sae values of uniy. Combining he rs-order condiion (3) wih equaions (8), (9) and (0) yields a Phillips curve of he form 4 : = f E + + b + x + ' b () where, f b '! [ + ( + ( ))] +! ( + ( ))! [ + ( + ( ))] + (!) ( ) ( ) ( + ) (! [ + ( + ( ))] + ) + " (!) ( ) &! +! [ + ( + ( ))] +! The log-linear approximaion o (4) is: where, # ( ) + bp = b + x x (2) x We have used he fac ha he markup shock is calibrared o be i.i.d., o eliminae he lead of he shock from he Phillips curve.

8 2.3. Moneary Policy As a baseline we assume moneary policy is conduced according o a simple rule of he form: i = i i + ( i ) + " mp (3) where i is a parameer governing he persisence of ineres rae movemens, re ecs he aggressiveness of moneary policy o in aion ucuaions and " mp is an i.i.d. shock wih zero mean and consan variance Welfare A quadraic approximaion o social welfare in his economy is proporional o: X L + :i:p: + O kk 3 (4) =0 where :i:p. represens he so-called "erms independen of policy," and L = 2 + x [ ( + ( )) ] b ( ) + 4 b 2 (5) Y u C 2 " + ( ) ( ) ( ) ( ) ( + ) +! 2 (!)! ( ) & 3 2 (!)! ( ) & 4 ( ) (!) Noe ha his social loss funcion is idenical o ha in Seinsson (2003) when = and & = 0. # 2

9 2.5. Endogenizing he Proporion of RT Price-Seers In he preceding secion, we followed sandard pracice and reaed he proporion of RT price-seers,!, as a xed parameer wih respec o changes in policy regime. This is poenially problemaic because he coss of following a simple rule-of-humb (or, equivalenly, he bene s of being raionally forward-looking) are likely o change wih he policy regime. If price-seers re-evaluae heir decision procedures in ligh of changes o he regime, hen he proporion of RT price-seers will be regime-dependen. In our analysis, we assume ha priceseers incur a xed cos each ime hey raionally re-opimize heir price. They can avoid paying his xed cos by insead using he simple rule-of-humb in (5). We rea he decision o be RT or FL as regime-dependen: Price-seers make a raional decision o be RT or FL so as o maximize he uncondiional expecaion of pro s, given he policy regime. Tha is, price-seers condiion his decision on he policy regime, bu no on he sae of he economy. Before explaining how we de ne he equilibrium proporion of RT price-seers, i will be helpful o noe some preliminary relaionships. As in Romer (990), we can show ha a quadraic approximaion o he loss in pro s from having a subopimal price is given by: bq (bp (z) ; Y ; ; ) jbq j bp (z) bp 2 + erms independen of bp (z) 2 where, bq = + Y Le bp f (z) denoe he lagged relaive price of a FL price-seer and bp b (z) denoe he lagged relaive price of a RT price-seer. Then we can wrie he value funcions for FL price-seers as: V nr (z) = bq bp f (z) ; Y ; ; + E V nr + (z) + ( ) E V r + (z) V r (z) = bq bp f ; Y ; ; c + E V nr + (z) + ( ) E V r + (z) where c is he xed of raional re-opimizaion. expeced value of a FL price seer: W E [V nr (z)] + ( ) E [V r (z)] De ne W as he uncondiional and de ne he FL price gap as: bp f;gap bp f (z) (z) = bp in he "nr" sae bp f bp in he "r" sae

10 Then we can wrie, W = jbq j 2 var bp f;gap (z) c (6) The value funcions for RT price-seers akes he form: n o n ev nr (z) = bq bp b (z) ; Y ; ; + E ev nr + (z) + ( ) E ev r n o n o ev r (z) = bq bp b ; Y ; ; + E ev nr + (z) + ( ) E ev r + (z) De ne W f as he uncondiional value of a RT price-seer: h i h i fw E ev nr (z) + ( ) E ev r (z) + (z) and he RT price gap as: bp b;gap bp b (z) = (z) bp in he "nr" sae bp b bp in he "r" sae Then, fw = jbq j var bp b;gap (z) (7) 2 We can now urn o he de niion of he equilibrium value of!, which we will denoe by!. Noe ha in our model, and wih he calibraed values we use below, he condiion W 0 (!) W f 0 (!) > 0 holds for all! 2 [0; ]. Tha is, he marginal bene of being an FL price-seer is increasing in!. This ensures ha we always have a unique equilibrium value of!. However, we have no reason o believe ha his condiion will hold in oher models or for oher calibraions of our model. Thus, in oher environmens i may be possible o nd siuaions wih muliple equilibria. Given ha W 0 (!) W f 0 (!) > 0, an equilibrium will saisfy one of he following condiions: (i) If W (! ) = f W (! ) for some! 2 [0; ] hen! is an equilibrium. (ii) If W (!) > f W (!) 8! 2 [0; ] hen! = 0 is an equilibrium. (iii) If W (!) < f W (!) 8! 2 [0; ] hen! = is an equilibrium. Type (i) equilibria require ha he values of being FL and RT are equaed a!. Figures illusraes his ype of equilibrium. Type (ii) equilibria arise when rms are beer o being FL for all permissible values of!. This is illusraed in Figure 2. Finally, ype (iii) equilibria occur when rms are beer o being RT for all values of!. Figure 3 provides examples of ype (iii) equilibria. o

11 3. Calibraion In order o generae resuls, we need o assign numerical values o he parameers under consideraion. For he benchmark model, we follow Seinsson and se = 0:99; = 5; = 0:7; = 2; and = 5: These values are consisen wih hose repored in Roemberg and Woodford (997) and Gali and Gerler (999). On he moneary policy side, we se i = 0:8 and = :5 which is broadly in line wih he Taylor rule esimaion resuls repored in Orphanides (2003). Gali and Gerler (999) and Gali, Gerler and Lopez-Salido (200) provide a range of empirical esimaes for he proporion of RT price-seers bu he average fullsample value appeared cenered on 0.3; hus we se! = 0:3 as a baseline. We se he weigh on lagged in aion in he rule-of-humb, equal o uniy. In addiion, we will se & = = +, so ha he RT price-seers reac o markup shocks as if hey were solving a saic problem. The shock process are calibraed using he pos-982 resuls repored in Rabanal and Rubio-Ramirez (2005), viz., = 0, r;n = 0:7, = 0:44, r;n = 0:0244 and mp = 0:002. We view he curren calibraion as capuring he midpoin of he ranges for he parameers under consideraion. In order o calibrae he cos of raional forward-looking reopimizaion, c, we ask wha value of c would be consisen wih he benchmark calibraion described earlier (ha is, wha value of c would raionalize! = 0:3 given he calibraion for he oher parameers). This procedure yields c = 0: IT versus PLT wih an Endogenous Proporion of RT Price-Seers A general resul of models wih lagged in aion is ha some degree of pricelevel drif is opimal, even if he moneary auhoriy can commi o is fuure policies. Seinsson (2003) demonsraes ha he opimal degree of price-level drif is relaed posiively o he proporion of RT price-seers. Thus, when sudies use values of! ha are relaively low (0. o 0.3), i is no surprising ha hey ofen conclude ha price-level argeing dominaes in aion argeing. The low value of!, however, has been based on samples where cenral banks have explicily or implicily argeed in aion so i is possible ha he proporion of RT price-seers may change wih a shif o an explici price-level argeing regime. We explore his possibiliy and is implicaions in his secion. We consider wo simple moneary policy rules of he form:

12 i = i i + ( i ) (r n + ) i = i i + ( i ) r n + p p where he rs rule is referred o as an in aion argeing (IT) rule and he second as a price-level argeing (PLT) rule. We choose he parameers of hese rules o minimize he uncondiional expecaion of he welfare loss (4). Table repors resuls for he case in which! is held consan a is benchmark value while he policy rule is opimized over i and (or p ). The parameer changes only lile relaive o is benchmark calibraion. For boh IT and PLT, he policy ineria parameer is zero, re ecing he relaive imporance of shocks o he naural rae of ineres and he absence of an ineres rae erm in he loss funcion. As he benchmark calibraion of! is relaively low, PLT dominaes IT as expeced (he welfare loss associaed wih PLT relaive o IT is 0.8). The able also repors several oher saisics ha help us beer undersand he subsequen resuls. In paricular he value of he loss funcion weigh on quasi-di erenced in aion, 2, is repored as i is a funcion of!. In Table, herefore, he value of 2 remains consan across he wo policy experimens as he proporion of RT price seers is xed. Neverheless, changes in 2 will play an imporan role in he subsequen resuls. In addiion, we repor he sandard deviaions of in aion and he oupu gap. In Table, hese saisics are repored relaive o he benchmark policy rule calibraion, so we see reducions in boh in aion and oupu gap variabiliy from he "opimized" policy rule parameers. These uncondiional momens will play an imporan role in households decisions beween pricing in a FL or RT manner. To gain some inuiion ino he role played by hese uncondiional momens, i is useful o hink abou he deerminans of he variance of bp b;gap (z). Recall ha if bp b;gap (z) is more volaile, hen he losses associaed wih being a RT price-seer are greaer, ceeris paribus. The de niion of bp b;gap (z) implies: var bp b;gap (z) = var bp b (z) bp + ( ) var bp b We can gain furher insighs by focusing on he second erm on he righ-hand side. Using he expressions for bp b and bp we can wrie: & var bp b bp = var b + bp + b + x x bp

13 Using he fac ha = &= and ha bp is proporional o we can rewrie his as: var bp b bp = var + x x or, var bp b bp " 2 = + # var ( ) + 2 xvar (x ) (8) 2 2x cov ( ; ) + cov (x ; ) 2 x cov ( ; x ) Thus, var bp b bp depends on a number of variances and covariances. The wo uncondiional momens ideni ed in he ables are hose ha we found o be mos imporan for he resuls. In paricular, i is possible o show ha var bp b;gap (z) is increasing in var ( ) and var (x ). Tha is, as in aion and he oupu gap become more volaile, i becomes more cosly o be an RT price-seer, ceeris paribus. As will become apparen from he ables, oupu gap volailiy is he single mos imporan facor in deermining he equilibrium proporion of RT price-seers. This is because he oupu gap erm in (8) is generally an order of magniude greaer han he in aion erm, so i ends o be he primary deerminan of var bp b bp. Turning again o Table, noe ha boh repored sandard deviaions decline relaive o he benchmark calibraion. This implies ha he performance of he rule-of-humb has improved. On his basis, one migh conjecure ha, were we o hold he policy parameers consan a he values repored in Table while endogenizing!, he equilibrium proporion of RT price-seers would exceed 0.3. This exercise is repored in Table 2. The resuls in Table 2 can be inerpreed as an exercise where he cenral bank chooses is policy parameers assuming ha! will remain a 0.3 regardless of wha i does o he policy rule, bu, in pracice,! reacs endogenously. As expeced he proporion of rule-of-humb price-seers rises subsanially relaive o Table. Noe ha he relaive sandard deviaions repored in Table 2 do no provide informaion on he sources of he increase in!. These sandard deviaions are oucomes of an equilibrium in which! is endogenous, so hey are consisen wih price-seers being indi eren beween he RT and FL opions.

14 Raher, he increase in! can be raced o he decline in he sandard deviaions repored in Table (which make RT behaviour relaively more aracive). The increase in! causes he performance of boh he PLT and IT rules o deeriorae. However, he PLT rule deerioraes more severely, reversing is dominance over IT. The primary source of his relaive deerioraion is he direc e ec of he change in! on he weigh ( 2 ) on he squared quasi-di erence of in aion in he loss funcion. Alhough he values of! repored in Table 2 may no appear o be very di eren across IT and PLT, he small di erence ranslaes ino a large di erences in 2 (because 2 =!= ( (!))). In paricular, 2 increases from 0.6 in Table o 7.5 under IT and o 2.86 under PLT in Table 2. Thus, failing o accoun for he impac of policy on he proporion of RT price-seers can lead o oucomes ha are signi canly worse han anicipaed. The fac ha IT ouperforms PLT when he proporion of RT price-seers is high, as in Table 2, is no surprising. As noed earlier, Seinsson has shown ha he degree of price-level drif ha is opimal is increasing in!. In ligh of his, one may be emped o conclude ha our assumpion of full indexaion ( = ) in he rule-of-humb is driving our resuls, as indexaion a ecs he weigh on lagged in aion in he new Keynesian Phillips curve. Tha is, one migh presume ha RT price-seers inroduce opimal price-level drif primarily hrough he lagged in aion erm in heir price-seing rule. Amano, Mendes and Murchison (2009) nd, however, ha he degree of indexaion does no play an imporan role in deermining opimal moneary policy. Even for exreme indexaion values of = 0 and =, he changing weigh on lagged in aion in he new Keynesian Phillips curve plays only a small par in deermining opimal price-level drif and in aion undershooing. In oher words, in aion persisence via lagged in aion plays only a small role in deermining he e cacy of moneary policy. Raher, Amano, Mendes and Murchison show ha he new Keynesian Phillips curve wih RT price-seers implies an overdiscouning e ec: Fuure oupu gaps are discouned a a rae greaer han he discouning of fuure losses in he social welfare funcion. To see his, noe ha he forward soluion of he new Keynesian Phillips curve in () is: where, = + f = X i=0 2 f f f + E x +i + q 4 b f ' f b

15 If! = 0, hen f =, b = 0 and herefore = 0. In his case, f = f =. Tha is, he discoun facor in he forward soluion is equal o he discoun facor in he welfare funcion. Bu, if! > 0, hen f = f <. This overdiscouning reduces he impac of he expecaions channel and, herefore, leads o greaer opimal drif. In numerical experimens, Amano, Mendes and Murchison nd he overdiscouning e ec o be more imporan in deermining opimal moneary policy han he degree of indexaion in he RT. Finally, we consider he case where he cenral bank adjuss he parameers of is policy rule o minimize he social loss funcion while aking ino accoun he reacion of!. These resuls are repored in Table 3. The values of i increase subsanially o 0.9 and 0.8 under IT and PLT, respecively. However, he aggressiveness parameers do no behave similarly under IT and PLT. The value of remains largely unchanged, bu he value of p increases from 0.2 o.72. The hree parameers ha change (boh i s and p ) induce greaer undershooing of in aion as hey increase (his is no rue of he one parameer ha remains unchanged, ). When! is endogenous, several facors in uence he cenral bank s incenive o induce undershooing. Firs, as! approaches uniy he coe cien on he markup shock in he Phillips curve increases by a facor of = ( ). This occurs because RT price-seers reac o markup shocks in a saically opimal manner hey ignore he fac ha hey migh no have he opporuniy o change heir price for several periods. Second, as we describe in more deail below, he fac ha 2 has increased relaive o he benchmark makes a greaer degree of in aion undershooing desirable. Third, increases in undershooing end o increase oupu gap volailiy which reduces he incenive o be RT. This leads o a welfare-improving reducion in he proporion of RT price-seers, ceeris paribus. Noe ha he performance of PLT relaive o IT does no improve. However, he performance of boh he PLT and IT rules improve relaive o he case in which he cenral bank erroneously reas! as xed (under boh IT and PLT he losses are smaller by a facor of roughly 0.55). Mos of hese welfare gains come from he reducions o 2 and in aion volailiy. Since he quasi-di erence of in aion erm in he loss funcion plays an imporan role in he resuls, i is worh discussing i in more deail. The presence of 2 6= 0, in isolaion, will reduce he opimal level of price drif since sabilizing in aion also helps o sabilize changes in in aion. In he limi as 2 goes o in niy, he cenral bank will fully sabilize in aion and a mark-up shock will only be re eced in he oupu gap, jus as i would if he level of in aion were he only

16 argumen in he loss funcion. The blue (dark) line relaive o he green (ligh) line in Figure 4 shows he marginal impac under he opimal sae-coningen policy from using he correc loss funcion, wih 2 =!, raher han zero. (!) Noe ha having a non-zero value for 2 causes here o be less price-level drif. This can be explained by expanding he hird argumen in he loss funcion: 2 ( ) 2 = and in uncondiional expecaion: 2 E ( ) 2 = 2 2 [var ( ) cov ( ; )] The inroducion of he 2 2 cov ( ; ) erm implies ha he cenral bank will have an incenive o induce some posiive auocorrelaion in in aion. This explains he somewha more drawn ou undershooing of in aion relaive o he case wih 2 = 0. When! is very high, as i is in Table 2, he cenral bank has a very srong incenive o increase he exen of in aion undershooing. Wih an IT rule, his is bes achieved by increasing he value of he smoohing parameer, i. This is in fac wha happens in Table 3. This change also causes he sandard deviaion of in aion o fall and he sandard deviaion of he oupu gap o rise. As discussed above, oupu gap volailiy plays a criical role in he decision o be RT of FL. In his case, he increase in oupu gap volailiy is a main driver of he reducion in!. 5. Concluding Remarks In his paper we examined he in aion versus price-level argeing debae when he proporion of rule-of-humb price seers is allowed o vary wih he moneary policy regime. Two enaive resuls emerged from his research. Firs, we found ha if a moneary auhoriy fails o accoun for he impac of policy changes on he proporion of rule-of-humb price-seers, oucomes ha are signi canly worse han anicipaed can occur. Second, in conras o earlier sudies in his lieraure, we nd ha allowing he fracion of rule-of-humb price seers o respond endogenously o he moneary policy framework can lead us o conclude ha in- aion argeing dominaes price-level argeing in minimizing welfare losses. We would like o close by emphasizing ha our resuls are quie preliminary and ha a full round of sensiiviy analysis is required before rmer conclusions can be drawn.

17 References. Amano, Rober, Rhys Mendes and Sephen Murchison "Deerminans of Opimal Moneary Policy in a New Keynesian Model." Manuscrip. Bank of Canada. 2. Calvo, G. A "Saggered Conracs in a Uiliy-Maximizing Framework." Journal of Moneary Economics 2: Gali, Jordi and Mark Gerler "In aion Dynamics: A Srucural Economeric Analysis." Journal of Moneary Economics 44: Gali, Jordi, Mark Gerler and J. David Lopez-Salido "Robusness of he Esimaes of he Hybrid New Keynesian Phillips Curve." Journal of Moneary Economics 52: Levin, Andrew T. and John C. Williams "Robus Moneary Policy wih Compeing Reference Models." Journal of Moneary Economics 44: Seinsson, Jon "Opimal Moneary Policy in an Economy wih In- aion Persisence." Journal of Moneary Economics 50: Romer, David "Saggered Price-Seing wih Endogenous Frequency of Adjusmen." Economics Leers 32(3): Rudebusch, Glen "Assessing Nominal Income Rules for Moneary Policy wih Model and Daa Uncerainy." Economic Journal 2: Walsh, Carl "Implicaions of a Changing Economic Srucure for he Sraegy of Moneary Policy." Moneary Policy Under Uncerainy: Adaping o a Changing Economy. Federal Reserve Bank of Kansas Ciy, Jackson Hole Symposium,

18 Table : Fixed! = 0:3 Relaive Loss Relaive o Benchmark Case Case i! (PLT/IT) 2 sd ( ) sd (x ) IT PLT Table 2: Fixed Policy Parameers, Endogenous! Relaive Loss Relaive o Table Cases Case i! (PLT/IT) 2 sd ( ) sd (x ) IT PLT Table 3: Opimized Policy Parameers wih Endogenous! Relaive Loss Relaive o Table 2 Case i! (PLT/IT) 2 sd ( ) sd (x ) IT PLT

19 Figure : Excess Value of Forward Looking Behaviour W (!) ~W (! ) Equilibrium ω 0 0 ω

20 Figure 2: Excess Value of Forward Looking Behaviour W (!) ~W (! ) Equilibrium ω 0 0 ω

21 Figure 3: Excess Value of Forward Looking Behaviour W (!) ~W (! ) 0 Equilibrium ω 0 ω

22 Figure 4: Loss Funcion E ec