Endogenous Indexing and Monetary Policy Models

Transcription

1 Endogenous Indexing and Moneary olicy Models Richard Mash 1 Deparmen of Economics and New College Universiy of Oxford June 2007 Absrac Models in which firms use rules of humb or parial indexing in heir price seing have become prominen in he recen moneary policy lieraure. The exen o which hese firms adjus heir prices o lagged inflaion has been aken as fixed. We consider he implicaions of firms choosing he opimal degree of indexaion so hese simple pricing rules deliver prices as close as possible o hose which would be chosen opimally. We find ha he degree of indexaion depends on he exen of persisence in he economy such ha models wih consan indexaion are vulnerable o he Lucas criique. We also sudy he ineracions beween firms price seing and he macroeconomic environmen finding ha, for he models which appear mos plausible on microeconomic grounds, he Nash equilibrium beween firms and he policy maker is characerised by zero indexaion and zero macroeconomic persisence. JEL E52, E58, E22 Key Words: Indexing, Moneary olicy, hillips curve, Inflaion persisence, Microfoundaions 1 Correspondence: New College, Oxford, OX1 3BN, UK; Tel ; Fax ; richard.mash@economics.ox.ac.uk. 1

2 Inroducion Models in which decision or opimisaion coss inroduce fricions o price changing and give rise o (parial) indexing or rule of humb behavior by firms have been a cenral feaure of he recen moneary policy lieraure. This conrass wih earlier models which emphasised price sickiness while assuming ha price seing was opimal subjec o consrains on when prices could be changed. The wo mos prominen models wih indexing behaviour are Gali and Gerler (1999) and Chrisiano, Eichenbaum and Evans (2005), he pricing mechanisms in which have also been used in Eichenbaum and Fisher (2003), Seinsson (2003) and Smes and Wouers (2003) amongs many ohers since hese have become workhorse models (Woodford, 2006). They differ as o wheher indexing behavior is combined wih price sickiness or no bu boh give rise o he sandard hybrid New Keynesian hillips curve wih forward looking expeced inflaion and lagged inflaion. A feaure of he exising indexing and rule of humb models is ha he degree o which firms index o pas inflaion, which we refer o as heir indexing parameer, has been reaed as an exogenous consan. The conribuion of his paper is o explore he implicaions of varying ha assumpion. I is naural o suppose ha firms migh consider he opimal or a leas near opimal value of heir indexing parameer so as o achieve higher profis by more closely maching heir prices o he prices which would be se in he absence of decision or opimisaion coss and hus come closer o consrained opimal behavior. I may no be appropriae o assume ha firms would necessarily opimise heir indexing parameer coninuously, since infrequen re-opimisaion is a mainained assumpion of hese models, bu i seems plausible ha hey may review heir indexing parameer periodically. We sudy he implicaions of endogenous indexing behavior of his kind wihin he wo esablished models while adding a hird, a more effecive varian of he Chrisiano, Eichenbaum and 2

3 Evans (2005) model. We follow he lieraure in assuming a Calvo (1983) consan hazard srucure, he Calvo signal being inerpreed eiher as an opporuniy o change price or as an opporuniy o reopimise prices depending on he model. The Gali and Gerler (1999) model assumes opimisaion coss for a proporion of firms, who apply a rule of humb in he seing of a new price when hey may do so, while reaining he Calvo model s assumpion ha he firm s price remains fixed unil he nex Calvo signal. The rule of humb deermined price depends on a measure of lagged aggregae prices plus he produc of an indexing parameer and lagged inflaion. We find ha his srucure is consrained opimal and hence focus on he size of he indexing parameer wihin he rule of humb. The scope of he Chrisiano, Eichenbaum and Evans (2005) model is differen in ha firms may change price every period, he fricion being ha hey do so fully opimally only when a Calvo signal arrives. Chrisiano e. al. assume an indexing srucure in which he firm akes is own lagged price as a base and adds he produc of an indexing parameer and lagged inflaion. We refer o his as lagged own price indexing (LOI) and find ha his srucure is no consrained opimal and a varian in which he firm uses he lagged aggregae price as a base is generally superior. We refer o his as lagged aggregae price indexing (LAI) and find ha his apparenly minor change in specificaion maers a grea deal for he resuls obained.. A resul common o all he models is ha he opimal value of he indexing parameer depends on firms beliefs abou he degree of persisence in he economy. Inuiively if inflaion is srongly persisen i is consrained opimal for an indexed or rule of humb price o be srongly influenced by lagged inflaion whereas weak persisence means ha lagged inflaion should no feaure so prominenly. This has wo immediae implicaions for he sandard versions of hese models in which he indexing parameers are assumed fixed. Firsly hey are vulnerable o he Lucas criique since he value of he indexing parameers influences he coefficiens in he hillips curve and hence a change in moneary policy regime which changes he degree of persisence will lead o changes in hose coefficiens under endogenous indexing. Secondly, he derivaion of microfounded loss 3

4 funcions for he indexing/rule of humb models as a guide o opimal moneary policy on he assumpion of fixed indexing parameers is quesionable for he same reason ha changes o he moneary policy regime may aler hem. A furher poin is simply ha he models wih endogenous indexing allow one o check he plausibiliy of he paricular values of he indexing parameers ypically assumed in he lieraure. The second se of resuls concerns he sabiliy of he models and heir abiliy o predic inflaion persisence once we combine firm s choices wih a model of policy. Here we focus on he Gali and Gerler (1999) and LAI models which appear o be he mos plausible from he micro analysis. Under endogenous indexing one may hink of a firm s choice of indexing parameer in relaion o perceived persisence as a reacion funcion. Collecively hey deermine he hillips curve which in urn acs as a consrain on he policy maker. olicy in par deermines persisence given he hillips curve (even if he degree of persisence is no a primary objecive) so i may also be characerised as a reacion funcion specifying realised persisence as a funcion of he hillips curve and hus indirecly firms beliefs abou persisence. Hence a naural quesion is he naure of he Nash equilibrium a which firms behavior is opimal given he degree of persisence a he macro level and policy is opimal given he hillips curve which resuls from he behavior of firms. In paricular i is ineresing o ask a wha level of persisence he models are sable in he sense ha reacion funcions inersec and a fixed poin beween firm and policy behavior is achieved. We find srong resuls wih very mild resricions on policy behavior ha amoun o ensuring sabiliy. For boh he Gali and Gerler (1999) and LAI models he fixed poin is zero persisence. Secion 1 analyses he choice of indexing/rule of humb srucures and indexing parameers a he level of a single firm. Secion 2 aggregaes hese o give he hillips curves and Secion 3 sudies he ineracions beween firm and policy behavior. Secion 4 concludes. 4

5 1. Consrained Opimal Indexing Behavior We consider he opimal choices of indexing/rule of humb srucures and indexing parameers for individual firms who ake he macroeconomic environmen as given. The framework is sandard wih monopolisically compeiive firms and a Calvo consan hazard srucure. The opimisaion problem for he firm is given in general form by (1) where V is he firm s expeced ne presen value of profis over he relevan decision horizon, $ is he discoun facor, " is one minus he (consan) probabiliy of he Calvo signal (and hence he probabiliy of he price no changing if prices are sicky or he probabiliy of no reopimising if hey are flexible), X is he firm s price in levels, Q is oupu and C(Q) is cos funcion. The opimisaion problem (1) is subjec o he usual consrain (2) which is he firm s demand curve given sandard Dixi-Sigliz preferences in which is he aggregae price index and D an index of aggregae demand. j j MaxV = E βα[ X + jq+ j C( Q+ j)] (1) j= 0 Q + j X + j = ( ) η D (2) + j + j The models differ according o he resricions placed on he evoluion of he firm s price in (1). In a sandard Calvo model i is simply fixed and deermined fully opimally a ime. Wih Gali and Gerler (1999) firms i is also fixed bu se according o a rule of humb applied a ime. In he Chrisiano, Eichenbaum and Evans (2005) model and he LAI varian derived below he iniial price, X is se opimally and hen varies according o an indexing funcion. We refer o hese mechanisms collecively as simple pricing rules and follow he lieraure in assuming ha a simple rule uses lagged informaion only and hence canno respond o he curren value of any shock variables. In he framework ha follows, lagged inflaion is he only pas daed sae variable and hence condiioning a pricing rule on i remains simple in he sense of using only a small 5

6 informaion se. In more general models wih many pas daed sae variables an issue would arise of wheher a simple rule can be condiioned on all of hem or only a subse. If all pas daed sae variables may be included as here (and cerainy equivalence applies) he opimal simple pricing rule will deermine a price equal o he ime -1 raional expecaion of he ime opimal price since he only consrain is he inabiliy of he simple rule o reac o conemporaneous (and assumed zero mean) shocks. Before urning o he individual models we place furher srucure on he firms decision problem by assuming ha inflaion, B (=ln -ln -1 ), and he oupu gap in log deviaion form, y, follow he processes (3) and (4) in which, is a shock o marginal cos inroduced below. For he ime being we assume ha he firms know he D parameers in (3)-(4) wih cerainy. π = ρ π + kε y π 1 (3) = ρ π 1 + cε (4) y A his sage he assumpion of hese processes is arbirary bu laer we show ha hey correspond o he reduced form of hese variables in hese models under plausible resricions on policy. A remaining common elemen o he models is he assumpion ha he log deviaion of real marginal cos from is seady sae, m, is linear in he oupu gap and he cos push shock o give (5). m = φy + ε (5) 1.1 Decision Coss wih Sicky rices We consider he soluion o (1) when price sickiness means ha he firm s price remains fixed over he decision horizon a X bu decision or opimisaion coss give rise o ha price being se by a rule of humb. This corresponds o he scope of he Gali and Gerler (1999) model. As a baseline we repor he soluion o (1) for a fully opimising Calvo firm and hen derive he opimal rule of humb which delivers a price equal in expecaion o he opimised price given he consrain of no using 6

7 conemporaneous informaion. For a sandard Calvo firm, subsiuing (2) ino (1) and imposing he resricion ha he price remains fixed gives he problem (6) wih firs order condiion (7). j j 1 η η + j η Max V = E βα[ X D C([ ] D )] (6) X j= 0 + j + j X + j + j j j η + j 0 = E βα D [( 1 η) X + η ] j= 0 + j + j CQ ( ) Q + j (7) Following Goodfriend and King (1997), for example, (7) log linearises o (8) where lower case implies he log of he variable. j j x = ( 1 βα) E β α [ p + m ] (8) j= 0 + j + j Following he argumen above, he opimal rule of humb will efficienly use pas daed informaion o achieve a price as close as possible o ha in (8) and hence will amoun o he ime -1 raional expecaion of (8). An inerim sep is given by (9) which makes use of (3)-(5) and subsiuing (9) ino he -1 daed expecaion of (8) and summing gives (10) where x r indicaes he opimal rule of humb price and ( r he indexing parameer wih value shown by (11). j+ 1 1 ρπ j E 1[ p+ j + m+ j] = p 1 + π 1[ ρπ ( ) + φρyρπ ] 1 ρ π (9) x r r = p + γ π (10) 1 1 ρπ + ( 1 βα) φρ r y γ = 1 βαρ π (11) 7

8 We nex relae (10)-(11) o Gali and Gerler (1999) framework. In heir model he rule of humb formula is expressed slighly differenly bu is equivalen o (10). A generalised version of he Gali- Gerler formula is (12), in which n -1 are prices ha were newly se he previous period, which reproduces heir (23) excep ha we inroduce he parameer ( G which is implicily se o uniy in he original form. x G G = n + γ π (12) 1 1 New prices may be relaed o aggregae prices by (13) given he Calvo pricing rule. p = ( 1 α) n + α p 1 (13) Lagging (13) one period and subsiuing ino (12) gives (14). x G G α = p + ( γ + α ) π (14) Comparison of (14) wih (10) confirms ha he Gali-Gerler rule of humb srucure (12) is consrained opimal in he sense of corresponding o he efficien forecasing srucure of (10), while (10) and (11) permi an assessmen of he appropriae size of ( G. Seing his parameer o uniy implies ha he coefficien on lagged inflaion in (14) is 1/(1-"). This will be opimal if he D parameers in (11) are such ha he righ hand side of ha expression equals 1/(1-") which implies he following condiion. ρ π = 1 ( 1 α)( 1 βα) φρ 1 α( 1 β) y Wih $<1 and D y <0 (as we argue below) his implies D B >1 so inflaion would have o be unsable for ( G =1 o be opimal. A value of D B in he more plausible range below uniy would imply a lower value of ( G and, as will be seen below, a smaller coefficien on lagged inflaion in he hillips curve for his model. 8

9 1.2 Decision Coss wih Flexible rices We urn o he siuaion where decision/opimisaion coss coninue o moivae simple pricing rule behavior bu prices are flexible. The Calvo signal is now inerpreed as an opporuniy o opimize he firm s price bu prices may sill change (according o he simple pricing rule) each period in he absence of ha signal. This corresponds o he scope of he Chrisiano, Eichenbaum and Evans (2005) model bu we derive he consrained opimal soluion firs before conrasing i wih ha framework. If prices are flexible he opimisaion (1) can reduce o a saic problem since wih flexible prices here is no inheren reason for he choice of price in one period o affec fuure periods. Hence a baseline resul is he fully opimal flexible price using full informaion which, in log linear form, is given by x f in (15). This is a sandard resul and may be derived following he seps above or simply aken from (8) wih "=0. f x = p + m (15) Given (15) he consrained opimal indexed price, x a, will be he -1 expecaion of (15) given by (16) and (17) which make use of (3)-(5). x a a = p + γ π (16) 1 1 a γ = ρ + φρ (17) π y Hence firms will se he fully opimal flexible price in (15) when he Calvo signal permis hem o opimise wih conemporaneous informaion and he indexed price in (16) beween signals. 9

10 We noe ha he consrained opimal indexing formula (16) involves he use of he lagged aggregae price as a base o which he indexing erm is added and hence refer o his model as lagged aggregae price indexing (LAI). This conrass wih he indexing formula assumed in Chrisiano, Eichenbaum and Evans (2005) in which he firm ses a price, X c, when he Calvo signal arrives and subsequenly indexes o is own lagged price raher han he aggregae lagged price unil he nex signal, hence he erm lagged own price indexing (LOI). The indexing formula akes he form (18) in levels or (19) in logs as in Woodford s (2003, chaper 3, eqn. 3.4) version of he model, also used in Smes and Wouers (2003), which generalises he original Chrisiano e. al. formulaion (2005, eqn 8) in which ( c was implicily se o uniy. X x c + j 1 c = X j 1( ) γ (18) c + j + + j 2 c c = x + γ π (19) c + j + j 1 1 The LOI firm s problem is hence (1) subjec o (2), plus he need o use a simple pricing rule based on -1 informaion, and (18) where (18) is an addiional consrain no presen in deriving he LAI resuls immediaely above. Here (18), or (19) in logs, is imposed as a consrain whereas in he LAI model he indexing formula (16) emerged as par of he soluion. This consrain binds and hence we find resuls for he LOI model which differ from he LAI framework. Given ha he laer maximised (1) subjec o (2) and he need o use a simple pricing rule, he LOI oucome for firms will be inferior due o he presence of he exra consrain. This quesions he use of he LOI model if we seek consrained opimal behavior unless i is argued ha he firm s own lagged price is more readily observable han he lagged aggregae price and hence simpler o use. Tha simpliciy, however, comes a he cos boh of he inferior performance already noed and a considerably more complex opimisaion problem. Using (18) wih (2), he LOI firm s opimisaion problem (1) is given by (20). 10

11 Max V E X c j j γ 1 η η D C Q X = βα[( ( ) ) j j ( j)] (20) c, γ c j= 0 c + j 1 1 The firs order condiion for he iniial price se when he Calvo signal is received, X c a ime, is given by (21) whch linearises o (22) = E + D + 1 X j j j c c j c η γ η γ βα j j( ) [( η) ( ) + η+ jm+ j] (21) j= 0 1 j j c c 0 = ( 1 βα) E β α [ x + p j + m j γ ( p j p )] (22) j= The firs order condiion for he indexing parameer, ( c, is given by (23) which linearises o (24) = E + D + 1 X j j j j c c j c η γ η γ βα j j ln( )( ) [( η) ( ) + η+ jm+ j] (23) j= j j c c 0 = ( 1 βα) E β α ( p j p )[ x + p j + m j γ ( p j p )] (24) j= Solving his model amouns o deriving he simulaneous soluion o (22) and (24) which may readily be done using (3)-(5). The soluion clearly involves he D parameers in (3)-(4), which is imporan for roposiion 1 below, bu we do no presen i since i is clear ha he soluion will generally differ from he LAI model resuls of (15)-(17). Hence in addiion o adding complexiy o he firm s opimisaion problem, he binding naure of he exra consrain (18) means ha (19) will give rise o inferior oucomes compared wih (16), making his model arguably less plausible wihou srong moivaion for he need o index o he firm s own price. Inuiion for he worse performance under own price indexing may be seen by comparing (16) wih (19). In (16) he indexed price will have auomaically caugh up wih aggregae price developmens up o -1 by he firs erm, and he second indexing erm plays a forecasing role of 11

12 developmens beween -1 and (necessiaed by he simple rule making use of informaion up o -1 only). If he D parameers in (3)-(4) are zero lagged inflaion is no helpful for forecasing and he ( parameer is zero by (17). In (19) he use of he firm s own lagged price as a base in he firs erm implies ha pricing errors from he pas may sill be presen, here is no auomaic error correcion, and we find ha he ( parameer is posiive even if he D parameers are non-zero since i parly allows he firm s price o cach up wih aggregae price developmens o -1. The second erm also plays he forecasing role if inflaion is persisen (he ( parameer increases) and hence his erm has o balance he cach-up and forecasing roles whereas hese are separaed in (16). roposiion 1. Under endogenous indexing, he consrained opimal price seing behavior of firms wih significan decision or opimisaion coss depends on heir percepion of he degree of persisence in inflaion and he oupu gap/marginal cos. In he models above, significan decision or opimisaion coss leads firms o employ a rule of humb or indexing formula when no reseing heir prices opimally. From (11), (17) and (22)-(24) he indexing parameers depend on he coefficiens in (3)-(4). 2. hillips Curves This secion presens he hillips curves for he hree simple pricing models considered above. Firs, he Gali and Gerler (1999) hillips curve is given by (25) which is simply heir equaion (24) wih noaion adjused o allow for he more general ( r in (10) raher han heir value of uniy for ( g in (12) which would imply ( r =1/(1-") as above. This is a noaional change and generalisaion, he difference in he framework above being he poenial endogeneiy of he indexing parameer. In he Gali-Gerler model, rule of humb firms are mixed wih sandard Calvo firms in he proporions T and (1-T) and hence if T=0, (25) reduces o he sandard New Keynesian hillips curve. 12

13 r βαe[ π + 1] + ωγ ( 1 α) π 1 + ( 1 ω)( 1 α)( 1 βα)( φy + ε) π = r α + ω( 1 α)( 1+ βαγ ) (25) We briefly noe wo properies of (25). Firs i fails he weak form of he naural rae hypohesis, in he sense of he sum of coefficiens on inflaion on each side of he equaion no being equal, even if $=1 unless ( r equals 1/(1-") which is he value obained from Gali and Gerler s assumpion of ( G =1 in (12). I was argued ha his value was implausible under endogenous indexing unless inflaion was believed o be exremely persisen. Second, if empirical esimaes of he coefficien on lagged inflaion are used o infer possible values of he srucural parameers as in Gali and Gerler (1999), a lower value of ( r (reflecing more plausible beliefs abou inflaion persisence) would require a larger implied proporion of rule of humb firms, T. The LAI model is new bu is hillips curve may readily be derived. All prices may change each period and are se opimally according o (15) if he Calvo signal occurs (wih probabiliy 1-") and se according o he indexing funcion (16) if no (wih probabiliy "). If we assume ha here are a large number of firms hese probabiliies ranslae ino proporions and hence he price level is given by (26) which, using (16), gives rise o he hillips curve (27). f a p = ( 1 α) x + α x (26) a 1 α π = γ π + ( φy + ε ) (27) 1 α We noe ha he very simple form of (27) wihou any forward looking expecaions reflecs he absence of any price saggering when all prices are flexible and backward looking firms use he lagged aggregae price as an indexing base. I also violaes he weak form of he naural rae hypohesis unless ( a =1 which is implausible from (17) given D y <1 unless inflaion is again unsable wih D B >1. 13

14 For he LOI model wih an indexing parameer no resriced o uniy, he hillips curve is presened by Woodford (2003, chaper 3, eqn 3.6) and may be expressed as (28) using he noaion above. c 1 α βe[ π+ 1] + γ π 1 + ( α )( 1 βα)( φy + ε) π = c 1 + βγ (28) roposiion 2. Under endogenous indexing he srucure and coefficiens of he hillips curve depend on firms percepions of he degree of persisence in inflaion and he oupu gap. From roposiion 1, percepions of he degree of persisence deermine he indexing parameers in (11), (17) and (24). These appear in he hillips curves respecively (25), (27) and (28) and hence percepions of he degree of persisence parly deermine he hillips curve coefficiens. In he special case where he ( parameers are zero, lagged inflaion no longer appears in he hillips curve. Corollary 1. Under endogenous indexing, he indexing or rule of humb models are vulnerable o he Lucas criique. A change in moneary policy regime which affecs he persisence parameers in (3)-(4) will change he ( parameers by roposiion 1 and hence he hillips curve by roposiion 2 in which case he hillips curves for hese models are no invarian o moneary policy. Remark 1. Under endogenous indexing, he derivaion of appropriae microfounded loss funcions for hese models would be more complex han if he indexing parameers are fixed as has ypically been assumed (Seinsson 2003, Woodford 2003). We leave his issue o fuure research. 3. olicy and Sabiliy Analysis We consider he ineracion beween firms price seing behavior from Secion 1, which depends on macroeconomic persisence, and he degree of persisence which depends on he hillips curve and 14

15 hence he underlying price seing behavior as shown in Secion 2. In paricular we derive he fixed poin beween he D parameers in (3)-(4) which guide price seing and he values which mus obain given he hillips curve. This is done for he Gali and Gerler (1999) and LAI models only since he LOI framework appears o be less compelling a he microeconomic level. A prior sep is o noe ha from he hillips curves he relevan sae variables are lagged inflaion and he curren value of he cos push shock. We assume ha policy iself does no inroduce addiional sae variables. This will be valid if a simple Taylor rule combined wih an IS relaionship wihou addiional lagged variables is used, or a simple quadraic loss funcion in inflaion and he oupu gap is minimised under discreion. This assumpion suppors he original assumpion of he processes for inflaion and he oupu gap in (3)-(4). I may be noed ha i would be violaed if policy is implemened under commimen since he presence of he forward looking inflaion erm in he hillips curves will give rise o he lag of he oupu gap in he reduced form of he sysem. Efficien simple pricing rules would hen incorporae he lagged oupu gap in firms indexing formulae so ha variable would appear in he hillips curve which would in urn affec he naure of opimal policy and so on. This example suggess ha here may be rich ineracions beween he se of variables included in simple pricing rules and he se of sae variables in he reduced form of he sysem as policy ineracs wih hose pricing rules via he hillips curve. We focus on he simples fixed poin of ha ineracion in which firms perceive he reduced form processes (3)-(4) and policy does no add addiional sae variables. This case is he simples wih which o explore he dynamics of he models under endogenous indexing and corresponds o he sandard indexing/rule of humb models. Beyond he assumpion concerning he relevan sae variables we find ha resuls may be derived by placing very lile addiional srucure on he policy process. In paricular wo resricions are required. Firs ha if he coefficien on lagged inflaion in he hillips curve is zero he D parameers 15

16 in (3)-(4) will also be zero. This may be jusified on sandard MSV grounds (McCallum 1983, 1999). Second we assume ha if he coefficien on lagged inflaion is posiive, D B will be posiive and D y negaive. The former simply follows from inflaion generally being brough back o arge gradually when lagged inflaion appears in he hillips curve and he laer is necessary o ensure ha. Given ha hese mild resricions are sufficien for our resuls we do no presen a formal represenaion of he policy process. Hence we proceed by seeking he simulaneous soluion o he relevan hillips curve from Secion 2 ogeher wih he relevan ( parameer from Secion (1) and he processes (3)-(4). As a firs sep we subsiue (3)-(4) ino he relevan hillips curve unil only erms in he wo sae variables remain. Since ha equaion mus be saisfied for any values of he sae variables he coefficien on each of hem mus be zero which gives wo equaions of which ha from he lagged inflaion erm is informaive for he D parameers. This may hen be compared wih he deerminans of he ( parameers from he micro analysis. This procedure is mos easily inroduced using he LAI model. Subsiuing (3)-(4) ino (27) unil only erms in he wo sae variables remain gives (29) in which he wo square brackeed expressions mus be zero for a soluion o obain. a 1 α 1 α 0= π [ ρ γ ( ) φρ ] + ε [ k ( )( 1+ cφρ )] (29) 1 π α y α y The firs square bracke in (29) is informaive abou he relaionship beween he D parameers in (3)- (4) and he D parameers in (17) which mus be equal a a fixed poin beween he price seing behavior of firms and he degree of persisence a he macro level. Subsiuing (17) ino he firs erm in (29) shows ha he only values of he D parameers a which his is he case is if boh are zero. In his case he ( parameer in (17) is zero so lagged inflaion disappears from he hillips 16

17 curve (27) and here is no macroeconomic persisence by (3)-(4). Furhermore if firms use incorrec values of he D parameers in (17) such ha ( is oo high versus he full informaion case he firs erm in (29) may be facorised o show ha he rue value of ( will always be lower han he incorrecly perceived value. Hence if firms observe he rue value of ( over ime heir indexing parameer will end o fall under learning so he zero persisence equilibrium is likely o be sable under imperfec informaion and learning. For he Gali-Gerler framework, subsiuing (3)-(4) ino (25) unil only erms in he sae variables remain yields (30). (30) Furhermore we may facorise he second line of (30) using (11) o give (31) in which ( r is he firm s indexing parameer and ( d is he same expression bu wih he empirical values in (3)-(4) raer han he values used by he firm in (11). In equilibrium hese mus be equal in which case (31) shows ha D y and D B mus be zero also. Hence his replicaes he zero persisence resul found for he LAI model above and lagged inflaion will once again disappear from he hillips curve. (31) The facorisaion in (31) is also informaive abou he poenial sabiliy of his equilibrium under imperfec informaion and learning becase if he D parameers are non-zero, D y. will be negaive and hence ( d <( r so a oo high value of he firms indexing parameer is likely o be pulled down under learning as firms observe is smaller empirical counerpar 17

18 roposiion 3. In he Gali and Gerler (1999) and LAI models he unique Nash equilibrium beween he pricing behavior of firms based on he degree of macroeconomic persisence and he laer resuling from ha pricing behavior is one wih zero persisence and lagged inflaion is no longer presen in he hillips curve. This follows from he discussion above wih he sabiliy resricions imposed on he D parameers. 4. Conclusion The paper has analysed he implicaions of endogenising he exen o which firms adjus heir prices o lagged inflaion when applying a rule of humb or indexing formula. The moivaion for his behavior by firms simply being o achieve a simple pricing rule which would deliver oucomes as close as possible o hose which would obain under full opimisaion. The wider moivaion of he paper was o explore he suiabiliy of hese models for moneary policy analysis when firms behavior is consrained opimal given he need o employ a simple pricing rule raher han hem simply choosing arbirary values for heir indexing parameers. The key resuls were i) ha consrained opimal rules of humb/indexing formulae depend on he degree of persisence in h macroeconomic environmen, ii) following ha hese models are vulnerable o he Lucas criique (and microfounded loss funcions which assume consan indexing parameers appear quesionable), iii) he indexing srucure in Chrisiano, Eichenbaum and Evans (2005) appears less plausible on microeconomic grounds han he simpler lagged aggregae price indexing framework, and iv) he Gali and Gerler and LAI models are unable o reproduce inflaion persisence under endogenous indexing when a fixed poin is reached beween pricing behavior and macroeconomic persisence. These resuls appear o srongly quesion he suiabiliy of hese models for moneary policy analysis, a leas in he long run when learning dynamics may have worked hemselves ou. 18

19 References Calvo, G. A. (1983), Saggered prices in a uiliy maximizing framework, Journal of Moneary Economics 12, Chrisiano, L. J., M. Eichenbaum and C. Evans (2005), Nominal rigidiies and he dynamic effecs of a shock o moneary policy, Journal of oliical Economy, Vol. 113 No. 1., Eichenbaum, M. and J. Fisher (2003), Evaluaing he Calvo model of sicky prices, Federal Reserve Bank of Chicago Working aper Gali, J. and M. Gerler (1999), Inflaion dynamics: a srucural economeric analysis, Journal of Moneary Economics 44, Goodfriend, M. and R. G. King (1997), The new neoclassical synhesis and he role of moneary policy, NBER Macroeconomics Annual 12, MIT ress, Cambridge. McCallum, B. T. (1983), On non-uniqueness in raional expecaions models: An aemp a perspecive, Journal of Moneary Economics 11, McCallum, B. T. (1999), Role of he minimal sae variable crierion in raional expecaions models, Inernaional Tax and ublic Finance 6, Smes, F. and R. Wouers (2003), An esimaed sochasic dynamic general equilibrium model of he Euro area, Journal of he European Economic Associaion Vol 1 No. 5, Sepember, Seinsson, J. (2003), Opimal moneary policy in an economy wih inflaion persisence, Journal of Moneary Economics 50(7), Ocober, Woodford, M (2003), Ineres and rices: Foundaions of a Theory of Moneary olicy, rinceon, rinceon Universiy ress. Woodford, M. (2006), Inerpreing inflaion persisence: Commens on he conference on Quaniaive evidence on price deerminaion, mimeo, available from 19