Contributions to Macroeconomics

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1 Conribuions o Macroeconomics Volume 6, Issue 26 Aricle Inflaion Ineria in Sicky Informaion Models Olivier Coibion Universiy of Michigan, OCOIBION@UMICH.EDU Copyrigh c 26 The Berkeley Elecronic Press. All righs reserved.

2 Inflaion Ineria in Sicky Informaion Models Olivier Coibion Absrac This paper considers wheher he sicky informaion model of Mankiw and Reis (22) can robusly deliver inflaion ineria. I find ha four feaures of he model play a key role in deermining inflaion ineria: he frequency of informaion updaing, he degree of real rigidiies, he naure and persisence of moneary policy, and he presence or no of informaion sickiness elsewhere in he economy. Real rigidiies serve o dampen firms desired price changes and are a criical elemen in delivering inflaion ineria. The ype of moneary policy, money-growh vs. ineres rae rules, also maers, wih Taylor rules making inflaion ineria less likely han under money growh rules. Adding sicky informaion in consumpion o he model yields a more gradual adjusmen of oupu, hereby decreasing he incenive for firms o change prices on impac and increasing he ineria of inflaion. I also explore he implicaions of using random versus fixed duraions of informaion rigidiy and argue ha wih he laer, he choice of he policy rule has a smaller effec on he qualiaive response of inflaion. These resuls allow us o sor ou some conflicing conclusions on inflaion ineria in sicky informaion models and sugges ha ineria is more sensiive o parameer choices han previously hough. KEYWORDS: Inflaion Ineria, Sicky Informaion I am graeful o Bob Barsky, Yuriy Gorodnichenko, Chris House, Peer Ireland, Peer Morrow, Ricardo Reis, and hree anonymous referees for helpful commens and o he Rober V. Roosa Disseraion Fellowship for financial suppor. Conac Informaion: Olivier Coibion, 23 Lorch Hall, Deparmen of Economics, 6 Tappan S., Ann Arbor, MI ocoibion@umich.edu.

3 Coibion: Inflaion Ineria in Sicky Informaion Models Inroducion Sylized facs serve as useful benchmarks wih which o evaluae models. The observed hump-shape response of inflaion o moneary shocks has received paricular aenion in recen years. This is due o he fac ha his resul is no only empirically robus bu also surprisingly difficul o reproduce in simple moneary models. Mos famously, he New Keynesian Phillips Curve (NKPC), based on he assumpion ha firms face coss o changing prices, is incapable of generaing such a response wihou he addiion of numerous (and frequenly adhoc) propagaion mechanisms. Mankiw and Reis (22, henceforh MR) presen an alernaive sicky informaion model, based on he assumpion ha firms face coss o updaing heir informaion, which can deliver inflaion responses o moneary shocks ha are srikingly similar o hose observed in he empirical lieraure wihou resoring o addiional exernal mechanisms. In his paper, I consider he robusness of heir resuls o more general aggregae demand seings and how o reconcile he conflicing findings of previous sudies also concerned wih applying sicky informaion o DSGE models. The failure of he basic NKPC o mach he observed response of inflaion o moneary shocks has led many auhors o propose soluions wihin he conex of sicky price models. However, hese ypically rely upon he addiion of ruleof-humb firms who use ad-hoc rules for changing prices. While his ype of behavior helps sicky price models yield more plausible heoreical responses o shocks and receives srong empirical suppor, he lack of microfoundaions renders he use of such models problemaic for policy analysis. The appeal of MR s approach is ha i relies upon he single alernaive assumpion ha firms updae heir informaion according o a Poisson process. 2 They argue ha his one deviaion from a flexible-price and informaion model can generae inflaion ineria and reproduce he posiive correlaion beween oupu and changes in inflaion ha is observed in he daa. 3 Several papers invesigae he robusness of hese resuls by inegraing he sicky informaion Phillips Curve (SIPC) ino dynamic general equilibrium models and draw very differen conclusions. Keen (24), Collard and Dellas (24), and Andres e al (25) conclude ha he SIPC fails o live up o he claims of MR in more sophisicaed models. Traband (25) and Korenok and Gali and Gerler (999), Woodford (23) and Chrisiano, Eichembaum and Evans (25) are well-known examples. 2 Reis (25) shows ha when firms face a fixed cos o updaing informaion, aggregaion across firms will lead o a Poisson process for he arrival rae of informaion, hereby providing srikingly srong microfoundaions for he sicky informaion model. 3 I will use he erm inflaion ineria o mean he hump-shape response of inflaion o moneary shocks, raher han jus persisen inflaion responses (in he sense of high auocorrelaion). Produced by The Berkeley Elecronic Press, 26

4 2 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle Swanson (24) conclude insead ha he findings of MR are robus, bu ha a hybrid sicky price model performs equally well. In his paper, I seek o deermine wheher he resuls of MR are indeed robus and why hese differen papers draw opposie conclusions. I firs invesigae he sources of inflaion ineria in he sylized seup of MR and place paricular emphasis on he role of real rigidiy. While real rigidiy (or sraegic complemenariy in price seing) has been idenified as an imporan mechanism in generaing persisen real effecs from nominal shocks in New Keynesian models, i also plays a role in he sicky informaion model. I show ha wihou sufficien amouns of real rigidiy, he sicky informaion model canno deliver inflaion ineria. I hen consider how more realisic demand seings affec he likelihood of observing inflaion ineria in response o moneary shocks. In paricular, I focus on he effec of forward-looking consumers and he choice of a money growh or a Taylor rule o capure he effecs of moneary policy. The laer choice implies an endogenous response of policymakers o deviaions of inflaion and oupu, which makes inflaion ineria more difficul o achieve in an oherwise idenical model. I argue ha adding sicky informaion o consumers helps produce an inerial response of oupu and hus makes inflaion ineria easier o achieve, bu he resuls remain sensiive o parameer values. Following Dupor and Tsuruga (25), I also consider he implicaions of using a fixed duraion of informaion rigidiy raher han a Poisson process for he arrival of informaion as assumed in MR. In his case, given enough sraegic complemenariy in price seing, he peak response of inflaion occurs a he ime when all firms have learned abou he shock, regardless of wheher a money growh or Taylor rule is used. The srucure of he paper is as follows. In secion 2, I describe he sources of inflaion ineria in he model of MR. Secion 3 presens a more general model ha allows us o compare alernaive assumpions abou policy rules, consumer informaion delays, and he iming assumpions of informaion rigidiy. In Secion 4, I presen he main resuls hrough impulse responses and sensiiviy analysis o parameer values. Secion 5 considers how his relaes o he findings of previous auhors. Secion 6 concludes. 2 Sources of inflaion ineria in he sicky informaion model The sicky informaion model inroduced by MR consiss of a quaniy equaion y = m! p () where y is he log of real oupu, m is he log of nominal income, and p is he log price level. The exogenous process for nominal income is $ m = " $ m + # (2) m! hp://

5 Coibion: Inflaion Ineria in Sicky Informaion Models 3 where " m %[,]. Equaions () and (2) describe aggregae demand in he model. Aggregae supply uilizes wo equaions. The firs describes firm j s insananeously opimal price condiional on informaion daed -k # p ' E p + & y (3) [ ],! k! k where smaller values of & correspond o greaer degrees of real rigidiy (or sraegic complemenariy in price seing). A firm seeking o deermine wha price o charge will look a he aggregae price level (as a measure of wha oher firms are doing) and he level of aggregae demand. The coefficien of real rigidiy deermines he relaive imporance of aggregae demand in he price seing decision. Low values of & (high real rigidiy) indicae ha he firm cares relaively more abou he pricing decisions of oher firms han abou aggregae demand. The second equaion is he sicky informaion price level, which relies upon a Poisson process applied o informaion updaing, where µ is he probabiliy ha a firm will no be able o updae is expecaions each period. The price level can hen be wrien as p j # ( µ ) µ p,! j j= ( =! ). (4) Combining (3) and (4) yields he Sicky Informaion Phillips Curve (SIPC) (! µ ( ) j * = & y + (! µ )) µ E!! j (* + & $ y ) (5) µ j= in which inflaion depends on curren oupu, as well as pas expecaions of curren inflaion and changes in oupu. MR show ha, for µ=.75 and &=., he sicky informaion model can generae a hump-shape response o shocks o nominal income, gradual and cosly disinflaions, and a posiive correlaion beween oupu and inflaion. Given he inabiliy of he basic sicky price model o reproduce any of hese phenomena, his is a sriking resul. To undersand why inflaion is inerial in he sicky informaion model, consider he response of he model o a moneary shock ou of he seady sae. For simpliciy, I normalize all expecaions daed before he shock o be zero. Assuming he shock occurs a ime zero, he price level equaion reduces o # p =! µ + p (7) ( ) so ha he price level is simply he fracion of firms which have updaed heir informaion imes he price hey all se. To have inflaion ineria, we firs need changes in he price level o be relaively small iniially. This will occur if he fracion of firms who updae heir informaion in he firs few periods is small (>µ>>) and hese firms have opimal prices ha are largely unaffeced by he shock. Significan real rigidiies, Produced by The Berkeley Elecronic Press, 26

6 4 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle (<&<<) are criical here so ha firms care relaively more abou he price level, which will be largely unchanged if few firms know abou he shock, han abou he shock o aggregae demand. We subsequenly need large movemens in he price level in laer periods. While he fac ha more firms will be aware of he change plays an imporan role here, i is criical ha all firms sill have an opimal price sufficienly differen from he original price level. For his, we need shocks o be sufficienly long-lived. If he nominal income shock is shor-lived, by he ime many firms become aware of he shock, hey will no longer change heir prices because he effec of he shock will have largely dissipaed. Thus, hree elemens play criical roles in generaing inflaion ineria in his model: he size of nominal and real rigidiies, as well as he persisence of he shock. The imporance of hese elemens has long been emphasized in he New Keynesian lieraure. Ball and Romer (99) and Kimball (995), for example, characerize he imporance of having boh real and nominal rigidiies o generae persisen real effecs from nominal shocks. Woodford (23) shows ha he persisence of oupu can be replicaed in New Keynesian models if here is enough sraegic complemenariy in price seing. The same inuiion naurally exends o he sicky informaion model. However, real rigidiy now also plays a new role. Specifically, i is necessary o generae inflaion ineria. The paricular role played by sraegic complemenariy in price seing can bes be seen in he response of he sicky informaion model o a permanen shock o he level of money. If here is neiher sraegic complemenariy nor subsiuabiliy in price seing, hen he response of he sicky informaion model is idenical o ha of he basic sicky price model. 4 In his case, here is no inflaion ineria. As sraegic complemenariy in price seing rises (& falls), he qualiaive response of inflaion changes and becomes hump-shaped for a low enough &. 5 This is illusraed in panel A of Figure, in which µ=.75 as in MR. For &=.7, inflaion peaks in he period of he shock, as would be rue in he sicky price model. When &=.4, inflaion peaks one quarer afer he shock. Finally, wih &=. as assumed in MR, inflaion peaks nearly wo years afer he shock. Panel B illusraes he response of inflaion o a permanen shock o he level of nominal GDP for differen levels of informaion rigidiy (assuming &=.). As firms updae heir informaion more frequenly, inflaion ineria becomes increasingly hard o generae. Similarly, panels C and D show he response of inflaion o emporary shocks o he level and he growh rae of 4 This is formally demonsraed in he appendix. 5 In a coninuous ime version of he model, one can show ha &</2 is a necessary and sufficien condiion for inflaion ineria in he sicky informaion model in response o a permanen shock o he level of nominal income if here is a posiive level of informaion rigidiy (µ>). hp://

7 Coibion: Inflaion Ineria in Sicky Informaion Models 5 Figure : Inflaion ineria in he sicky informaion model of Mankiw-Reis. Noe: The baseline parameer values are µ=.75, &=., and a permanen shock o he level of nominal income (" m =). Panels A and B consider variaions in & and µ separaely where oher parameers are a baseline values. Panels C and D consider alernaive shock processes for differen values of &. nominal income respecively for differen levels of & (assuming µ=.75). Again, wheher inflaion ineria is presen is sensiive o parameer values. If here is no enough sraegic complemenariy given a shock process, inflaion will jump on impac jus as i would in he sicky price model. The evidence above indicaes ha for he SIPC o be able o generae a hump-shaped response of inflaion o moneary policy shocks, high enough levels of informaion and real rigidiies are necessary, as well as ha he shock be sufficienly persisen. The qualiaive response of inflaion is hus sensiive o parameer values. Mos empirical esimaes of he degree of informaion rigidiy Produced by The Berkeley Elecronic Press, 26

8 6 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle yield µ approximaely equal o.75, as assumed by MR and subsequen papers. 6 In addiion, Woodford (23) argues ha a value of & of beween. and.5 is boh empirically and heoreically plausible. Finally, ypical esimaes of he ime series process of nominal income are consisen wih nominal income growh following an AR() process wih persisence parameer of abou.5. 7 Thus, since he parameers used by MR are no paricularly conroversial, one could view he sensiiviy of he SIPC o less realisic parameer values as a moo poin. However, given he sylized demand side assumpions of MR, i is noneheless worhwhile pondering how oher demand side assumpions would affec heir resuls. MR correcly argue ha heir simpler approach bes draws ou he implicaions of differen supply-side assumpions. Bu if one is ineresed in maching he response of inflaion o ha observed in he daa, which embodies he endogenous response of consumers and policymakers, more sophisicaed demand side approaches are warraned. I now urn o how such approaches affec he resuls of MR. 3 Model In his secion, I apply he noion of sicky informaion o a model wih firms, consumers and moneary policy-makers. Because sicky-informaion is a leas as likely o apply o consumers as firms, I allow for sicky-informaion in consumpion in a way ha ness he sandard consumer maximizaion problem. 8 The model consiss of hree secors: firms, consumers, and he cenral bank. Producion is composed of an inermediae goods secor of monopolisic compeiors, whose producs are combined ino a final good by a perfecly compeiive indusry. The consumer s problem is a sandard represenaive agen problem, excep for he fac ha uiliy depends upon an index of consumpion levels chosen by subagens. These agens ac o maximize he expeced uiliy of he represenaive agen, bu only conrol heir individual consumpion of he final 6 See Mankiw and Reis (23), Carroll (23), Khan and Zhu (22), Andres e al (25), and Kiley (25) for such esimaes, hough Coibion (25) presens a conrarian view. In addiion, Reis (25 and 24) shows ha for reasonable coss of informaion, inaeniveness for firms and consumers could plausibly exceed one year. 7 See MR and Woodford (2) among ohers. 8 Reis (24) provides rigorous microfoundaions for consumers facing coss of informaion updaing. I mus be emphasized ha Reis approach yields differen implicaions for he ime series properies of consumpion han ha assumed here. I follow a simpler approach designed o be highly racable, direcly comparable o he sicky informaion Phillips Curve, and easily applicable in a DSGE model. In addiion, his approach could easily be exended o sicky informaion in wage-seing and invesmen. While inconsisen wih Reis (24), his seup is consisen wih he informaion diffusion mechanism described in Carroll (23). Mankiw and Reis (25) also consider an alernaive approach o inegraing sicky informaion in consumpion. hp://

9 Coibion: Inflaion Ineria in Sicky Informaion Models 7 good. Finally, he cenral bank ses ineres raes or he money supply according o a pre-specified rule. 3. Consumers The consumer s problem is wrien in a sandard represenaive agen seup! v (.! + z Z! + z! 2 +, 3. M // + z max E! k) - 4! N ( ) + z j 6 j + 4 { N+ z ( j) },{ B+ z},{ M+ z},{ C + z ( i! 7 8! 55 )} z= 4! + +, =! v P 49 + z ; : 5 < 5 ; < subjec o C M B N j W ( j) j B R M T + z + + z + z + z + z! + z! + > z ( ) z z P z P = ! + +? z P + z P + z P + z @ ( ). / Z = 4 C i 6i5 4; 5< C = C ( i) 6i = where M represens money holdings, N (j) and W (j) are labor supplied o and wage earned in inermediae goods secor j respecively, B are riskless bonds held which earn he nominal gross ineres rae R, P is he price of final goods a ime, C (i) is consumpion of he final good by subagen i, C is he sum of subagen consumpion levels, Z is he consumpion aggregaor ha maers for uiliy,? are profis, and T are exogenous ransfers. The maximizaion problem is separaed ino wo pars. Firs, a coninuum of sub-agens indexed from o each choose C (i) o maximize expeced aggregae uiliy condiional on heir (possibly oudaed) informaion se aking all oher choices as exogenously deermined. 9 The firs order condiion for a subagen wih informaion daed -k is!! E.! k Z C ( i) /! = E! k A 4 5 (8) ; < where! is he Lagrangian muliplier on he budge consrain. If all subagens had he same informaion (daed ), hen equaion (8) would reduce o he sandard FOC wih respec o consumpion. I will hus be convenien o define he gap beween he marginal uiliies of consumpion and wealh, log-linearized around he seady-sae as c x '! + c! B (9) 9 The sub-agens are all purchasing he same final good. Produced by The Berkeley Elecronic Press, 26

10 8 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle which will be referred o as he consumpion gap. One can hen wrie he loglinearized perceived opimal level of individual consumpion by a subagen wih informaion daed -k as # c c =, k E.!! k ; c x / < () An ineresing feaure of his seup is he presence of sraegic complemenariy or subsiuabiliy in consumpion, deermined by is greaer or less han one. Suppose he marginal uiliy of consumpion is greaer han he shadow value of wealh, or equivalenly he consumpion gap is posiive. is greaer han one, hen individuals will wish o raise heir relaive consumpion; his is he case of sraegic subsiuabiliy. is less han one, individuals will no raise heir consumpion much because hey care more abou ohers consumpions han abou he marginal uiliy incenive of consumpion. This feaure can help generae persisence in consumpion, jus as i does wih price seing in sicky price models. The represenaive agen conrols he remaining decisions over individual labor supplies, saving, and money holdings. I assume ha he represenaive agen always has complee and up-o-dae informaion. His log-linearized firsorder condiions are hen n ( ) c i + + c + x = w ( i)! p C i (), c $ + =!* +! $ + E c E. i x / + ; < (2) + c m! p = c! i! x (3) v v( R! ) v The firs is he sandard labor supply condiion, augmened by he consumpion gap. The second condiion, drawn from opimal choice of bond holdings, is he radiional consumpion Euler equaion, which now includes he expeced change in he consumpion gap. Finally, equaion (3) is he money demand equaion. I follow MR and assume ha he adjusmen of subagens expecaions follows a Poisson process, as in Calvo (983). Specifically, each period here is a consan probabiliy ha an individual will be able o updae his informaion se. When one is allowed o do so, complee informaion is acquired and raional expecaions apply o all fuure ime periods. The consumer bases all subsequen decisions on hese expecaions unil he is allowed o updae heir informaion once again. The probabiliy of updaing is independen of he amoun of ime I use lower-case leers o denoe log-deviaions from he non-sochasic seady-sae and he fac ha, once linearized, c =z.. Noe ha i is he log-deviaion of he gross nominal ineres rae from is seady-sae value. hp://

11 Coibion: Inflaion Ineria in Sicky Informaion Models 9 since he las updaing occurred. Leing -D denoe he probabiliy of updaing informaion each period, log-linearized aggregae consumpion is c ( j # ( D ) D c =! ). (4),! j j= The appendix shows ha, given he opimal consumpion levels deermined by () and he bond condiion (2), he Euler equaion for aggregae consumpion can be rewrien as E c = FEc + + (! F ) c!! E ( i!* + ) + E3!. (5) + ( )( ) 2 ;. +@ ( D ) D < / ( D ) ( +@ ) ( i * ),! ( ) ( ) +@! D + D + D. / +@! D where F ' %,, E ' %[, ],! ; 2 5 < +@! D + D G =! $ +! D. ( j / and 3! = E! G! D D ( E!! j$g + + $ E!! jg ). 7 +@ ( D ) 8 4 ) 5 9! : ; j= < There are several aspecs of his relaionship worh noing. Firs, when D=, i.e. when here are no old informaion ses, he expression reduces o he sandard consumpion Euler equaion. Second, he assumpion of sicky informaion naurally generaes a backward-looking componen o he consumpion funcion, bu he weigh on he forward-looking par of consumpion is bounded below by one-half, while he backward-looking componen is bounded above by one-half. Thus, even if one allowed for significan delays in informaion updaing, aggregae consumpion would sill mainain a srong forward-looking componen. The reason is ha even wih informaion delays, consumers are choosing heir consumpion in a forward-looking manner. Third, he coefficien on he real ineres rae will be less han he ineremporal elasiciy of subsiuion. Esimaes of he IS curve frequenly find near-zero esimaes of he IES and have difficuly explaining such a phenomenon 2. In his model, as he degree of sicky informaion approaches one, he coefficien on he expeced real ineres rae goes o zero regardless of he IES. Finally, old expecaions, as represened by 3 -, play a role in deermining curren consumpion. This occurs in hree ways. Firs, he previous period s expecaion of he curren change in consumpion as well as he previous period s real ineres rae end o raise curren consumpion. The second erm is a weighed average of old expecaions of he fuure change in he growh rae of consumpion as well as he fuure change in he real ineres rae. The hird and 2 See Fuhrer and Rudebusch (24) for recen IS curve esimaes. Produced by The Berkeley Elecronic Press, 26

12 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle final erm capures he revision of pas expecaions of he curren real ineres rae and change in consumpion. 3.2 Firms Final goods are produced in a compeiive indusry ha combines a coninuum of inermediae goods using a CES aggregaor H H H H =! ( )! Y = 7 Y j 6j 8 9 : where H is he elasiciy of subsiuion beween inermediae goods. Since here is no capial or governmen in he model, he goods marke clearing condiion is simply Y =C. The price level mus be! H! H P = 7= Pj ( j) 6j 8. 9 : Each inermediae variey is produced monopolisically using a producion funcion linear in labor supplied Y (j)=a N (j) where A denoes he aggregae level of echnology and follows a =" a a - +e such ha e is iid (, + 2 a ). Each monopolis mus hire a specific ype of labor supply, assumed wihou loss of generaliy o be indexed by he same number as ha firm s index, bu does so as a price-aker. The demand faced by firm j depends upon is relaive price as well as aggregae demand P ( j) Y ( j) = 7 8 Y. 9 P : A firm s desired price a any momen in ime is a consan markup over marginal cos. Log-linearizing around a zero-inflaion seady sae, he desired price can hen be wrien as # +,+ +,, c p = p + y! a + x. (6) H +, H +, H +, If all firms are free o se prices and have he same informaion, hen p # = p. Le such an oucome deermine he firm flexible-informaion naural level of oupu y f. We can wrie i as C f ( +, ) a!, x y =. +,+ The flexible informaion equilibrium for firms hus depends on echnology as well as on he consumpion gap ha arises when some consumers have oudaed f f expecaions. Define he firm oupu gap x ' y! y as he difference beween acual oupu and oupu when all firms have flexible informaion. We can hen!h hp://

13 Coibion: Inflaion Ineria in Sicky Informaion Models rewrie a firm s perceived insananeously opimal price, condiional on informaion daed -k, as # f p =, k E.!! k ; p + & x / < (7) where &=(+,+)/(H+,) is he degree of sraegic complemenariy in price seing. Imposing anoher Poisson process on informaion updaing of firms (wih -µ being he probabiliy of updaing informaion in any period), he loglinearized price level is p ( s # ( µ ) µ p =! ) (8) which can be rearranged o yield he sicky informaion Phillips Curve (! µ ( ) f j f * = & x + (! µ )) µ E!! j (* + & $ x ). (9) µ j= Curren inflaion hus depends on he curren oupu gap as well as on pas expecaions of he curren inflaion rae and changes in he oupu gap Moneary policy I allow for wo ypes of moneary policy. Firs, he cenral bank may follow an exogenous money growh process given by $ m = " $ m + #. (2) s= m! where # is iid(,+ 2 m ). Alernaively, he cenral bank is assumed o se ineres raes using a Taylor rule wih ineres smoohing as follows N i = (! " i ). ; J ** + J yx / < + " ii! + I (2) where I is iid(,+ 2 N N N +, i ), x ' y! y, and y ' 7 8 a is he level of oupu 9+,+ : when neiher firms nor consumers have sicky informaion. 3.4 Parameer Values I le, he Frisch labor supply elasiciy, be /3. +, he inverse of he ineremporal elasiciy of subsiuion, is se o 2. I assume H =, which is consisen wih a seady sae markup of abou en percen. This implies ha he degree of sraegic complemenariy in price seing is approximaely.6. For he ineres rae rule, I follow Taylor (993) and le J * =.5 and J y =.5 and consider differen levels of ineres smoohing (" i ). The discoun facor - is se o,! s 3 Noe here ha he correc measure of he oupu gap in he Phillips Curve is he difference beween acual oupu and oupu wih flexible informaion for firms, bu no necessarily for consumers. Produced by The Berkeley Elecronic Press, 26

14 2 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle.99. To ge a uni income elasiciy of money demand, I se v=+. Finally, he degree of informaion rigidiy for firms is se o µ=.75, while ha for consumers (D) is iniially assumed o be zero. 4 4 Resuls I urn now o he impulse responses of inflaion and oupu o moneary shocks. I firs consider he implicaions of using money growh versus Taylor rules for he cenral bank. In his case, I limi he analysis o he case wih purely forwardlooking consumers. I hen examine he role played by sicky informaion in consumpion, in paricular is implicaions for inflaion ineria. Finally, I briefly consider a similar analysis when one replaces he Poisson process for he arrival of informaion wih a fixed duraion of informaion rigidiy. 4. Money growh vs. Taylor rules In his secion, I focus on he implicaions of wo alernaive formulaions for he cenral bank s acions. The firs is o assume ha he money supply follows an auoregressive process in he growh rae, for which I assume an AR() coefficien of.5. The second approach follows Taylor (993) and assumes ha he cenral bank ses ineres raes in response o he curren sae of he economy, as described by equaion (2), wih a baseline degree of ineres rae smoohing of.8. The resuls are presened in Figure 2. The op panel shows he response of inflaion under each policy seing. The difference is sark: whereas inflaion is inerial wih a money growh rule, i peaks on impac when using he Taylor rule. The criical difference beween he wo models is he endogenous response of he cenral bank o higher inflaion and oupu wih a Taylor rule. Because he cenral bank seeks o reduce inflaion and oupu, posiive inflaion and oupu lead o higher real ineres raes han is he case wih a money growh rae rule. This causes oupu o fall rapidly, despie he fac ha he ineres smoohing parameer is relaively high a.8. As discussed in secion 2, o have inflaion ineria, i is criical ha he oupu gap no go o zero oo rapidly, else firms have lile incenive o change prices when hey learn abou he shock. Figure 3 considers he ime of he peak response of inflaion wih a Taylor rule for differen levels of ineres smoohing and oher parameer values. Wih he benchmark se of parameers, inflaion ineria can occur if he degree of ineres smoohing is sufficienly high. Increasing he persisence of he shock yields a more gradual adjusmen of oupu, making inflaion ineria a more likely oucome. Similarly, higher values of + (or lower values of he ineremporal 4 When D is subsequenly allowed o be posiive, I will consider differen values hp://

15 Coibion: Inflaion Ineria in Sicky Informaion Models 3 Figure 2: Taylor rules vs. money growh rules Noe: The figure plos impulse responses of oupu and inflaion o a one-uni shock o he money growh rae and he Taylor rule in he model described in secion 3. All parameer values are a he baseline levels. elasiciy of subsiuion) lead o slower responses of oupu, hereby allowing inflaion ineria for sufficienly persisen shocks o he ineres rae. Lower levels of J* also help generae inflaion ineria, since he cenral bank reacs less o inflaion. Finally, higher levels of H, by raising he sraegic complemenariy in price seing, allow for inflaion ineria o appear a lower levels of ineres smoohing. Noneheless, even when all of hese elemens are added (+=, H=3, J * =.), inflaion ineria is no presen unless he degree of ineres smoohing is abou.9. A he roo of he problem is he cenral bank s endogenous response combined wih forward-looking consumers. The laer, by fronloading he response of oupu o a moneary shock, make i difficul for inflaion o respond slowly o a shock since aggregae demand is highes on impac. Bu his is also counerfacual. Empirically, he response of oupu o moneary shocks is delayed, hough less so han ha of inflaion. The lack of inflaion ineria for ypical parameer values in his seing may hus simply reflec he failure of he model o generae sufficien ineria in oupu. Produced by The Berkeley Elecronic Press, 26

16 4 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle Figure 3: Time of peak inflaion wih a Taylor rule for alernaive parameer values Noe: Figure 3 displays he ime a which inflaion achieves is maximum value for alernaive raes of ineres smoohing (" i ) in he Taylor rule. The shock is assumed o occur a =. The firs (Baseline) line represens he baseline parameer values. The following hree lines represen a single deviaion from baseline values. The final line includes he hree previous deviaions from baseline parameer values. 4.2 Forward-looking vs. sicky informaion consumers I now urn o he implicaions of including sicky informaion in consumpion when moneary policy is characerized by a Taylor rule. To do so, one mus now calibrae he degree of informaion rigidiy for consumers as well as he degree of sraegic complemenariy in For informaion rigidiy, I follow Carroll (23) who uses survey daa and finds ha he diffusion of informaion from professional forecasers o consumers approximaely follows a Poisson process wih D=.75. I hen assume as a baseline value so ha sraegic complemenariy in consumpion is approximaely he same as ha in price seing. While admiedly ad-hoc, a low value is qualiaively similar o he high esimaed values of exernal habis. I will laer consider alernaive values in sensiiviy analysis. The op wo panels of figure 4 presen he response of inflaion and oupu o a shock o he Taylor rule (" i =.8) when here is eiher no sicky informaion in hp://

17 Coibion: Inflaion Ineria in Sicky Informaion Models 5 Figure 4: Response of he sicky informaion model o a shock o he Taylor rule wih sicky informaion for consumers Noe: This figure shows impulse responses of economic variables o a uni shock o he ineres rae for baseline parameer values, wih he addiion of sicky informaion consumers (D=.75) and sraegic complemenariy in consumpion assumed o The Consumpion Gap and Firm Oupu Gap are described in he ex. consumpion (D=) or equal amouns of informaion rigidiy for firms and consumers (µ=d=.75). Noe firs ha adding sicky informaion o consumpion has he desired effec of generaing ineria in oupu. The impac effec on oupu of an ineres rae shock is small, and oupu follows a hump-shaped response ha peaks 2 quarers afer he shock. However, here is sill no inflaion ineria, hough inflaion does respond more slowly han when here is no informaion rigidiy for consumers. To see why his is he case, one mus consider he effec of sicky informaion for consumers on he firm s price seing decision. Specifically, we can rewrie equaion (7) as. # N, c / p,! k = E! k 4 p + & 7 x x 85 (22) 4; 9 9+,+ : : 5< Produced by The Berkeley Elecronic Press, 26

18 6 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle Figure 5: Sensiiviy of inflaion ineria wih Taylor rule and sicky informaion for consumers Noe: The figures displays he ime a which inflaion peaks afer a shock o he ineres rae for differen levels of ineres smoohing and varying parameer values wih sicky informaion in consumpion. The shock occurs a =. In response o a moneary shock, he naural oupu gap simply follows oupu. However, he firm oupu gap, which is wha maers for price seing decisions, is also a funcion of he consumpion gap. This is because he consumpion gap affecs he opimal labor supply decision. The consumpion gap depends on he marginal uiliy of wealh B relaive o ha of consumpion. Since he MU of wealh is a purely forward-looking process (because he represenaive agen has full informaion) whereas consumpion is sicky (because of he informaion rigidiy of subagens), he consumpion gap jumps on impac and follows an AR() ype ime pah, as illusraed in he boom lef panel of Figure 4. The firm oupu gap x, being he weighed sum of he naural oupu gap and he f consumpion gap, is dominaed by he movemens in he consumpion gap and so also follows an AR() ype ime pah. Because i converges more slowly o he seady sae han when here is no informaion rigidiy for consumers, inflaion also adjuss more slowly. However, inflaion ineria requires a slower adjusmen sill. Figure 5 presens some sensiiviy analysis for he degree of inflaion ineria as measured by he ime of he peak response of inflaion. Noe firs ha simply including sicky informaion for consumers does make inflaion ineria a hp://

19 Coibion: Inflaion Ineria in Sicky Informaion Models 7 Figure 6: Sensiiviy of inflaion ineria o sraegic complemenariy in consumpion Noe: The figure displays he ime a which inflaion peaks afer a shock o he ineres rae for differen levels of ineres smoohing and varying degrees of sraegic complemenariy in consumpion (@) assuming D=.75. more common response o ineres rae shocks, bu is no enough o delay he response of inflaion for " i =.8. Figure 5 also shows ha lower values of he IES or he Frisch labor supply elasiciy make inflaion ineria a more likely oucome. The reason is found in equaion (22). Lower values of, or higher values of + will slow down he adjusmen of he firm oupu gap by lowering he weigh placed on he consumpion gap. Figure 6 illusraes he effec of differen values Overall, lower levels by yielding slower responses of consumpion, help raise he probabiliy of observing inflaion ineria. sicky informaion for consumers has lile effec on he likelihood of inflaion ineria occurring relaive o he case wih informaion rigidiy for firms only. 4.3 Saggered informaion updaing vs. Poisson arrival raes Dupor and Tsuruga (25) argue ha he resuls of MR are sensiive o he assumpion abou he iming of informaion updaes. Replacing he Poisson arrival rae wih a fixed duraion scheme (a la Fischer (977)), hey find ha he persisence of oupu is lower and he response of inflaion less realisic han hose found in MR. To see his, consider he price level when firms updae heir expecaions every N periods Produced by The Berkeley Elecronic Press, 26

20 8 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle p N! # p,! j N j= = ). (23) Assuming ha he shock occurs a ime =, hen all firms have he same # informaion a ime N. Thus he price level is p = p which implies y = when aggregae demand is represened by a quaniy equaion. Thus nominal income shocks have no real effecs beyond he duraion of he informaion rigidiy. We mus also have pn = mn. If he price level adjuss slowly in he firs N- periods, hen he price level mus jump up o m N a ime N, leading o a spike in inflaion raher han he gradual hump-shape response in MR. Because of his difference in resuls, I also examine he implicaions of using fixed duraions of informaion rigidiy in he model presened in secion 3. Thus equaion (23) replaces equaion (8) for he price level. In addiion, I also allow for fixed duraions of informaion rigidiy in consumpion, so ha equaion (4) is replaced by M! c # = 7 8 ) c,! j (24) 9 M : j= where M is he duraion of informaion rigidiy in consumpion. The choice of values for M and N is open o inerpreaion. If we seek o mach he average duraion of informaion rigidiy o ha of he Poisson process, hen N=4 is equivalen o µ=.75. On he oher hand, if one wishes o mach he crosssecional average age of informaion, hen his would imply N=7. I consider boh values for comparison. Figure 7 presens he responses of inflaion and oupu for he case in which he cenral bank follows a money growh rule. All parameers are a heir baseline levels. The wo panels on he lef have informaion rigidiy for firms only, whereas hose on he righ have informaion rigidiy for consumers as well. In all cases, inflaion peaks a =N and oupu goes o zero a ha ime. When only firms have sicky informaion, he peak response of oupu is insananeous because of he forward-looking behavior of consumers. Adding sicky informaion o consumers provides a more gradual response of oupu o moneary shocks, bu does no aler he qualiaive response of inflaion. Figure 8 presens he same responses in he case of Taylor rule for ineres raes. Again, inflaion peaks a =N and oupu goes o zero a ha ime. Sicky informaion for consumers is again effecive a yielding a more hump-shape response of oupu han wih forward-looking consumers. Thus, whereas he choice of a money growh or Taylor rule could aler he qualiaive response of inflaion wih Poisson arrival raes of informaion, in he case of fixed duraions of informaion rigidiy he sicky informaion model appears o generae inflaion ineria, he peak response of which is deermined by he value of N. N N, N N hp://

21 Coibion: Inflaion Ineria in Sicky Informaion Models 9 Figure 7: Response of he sicky informaion model wih saggered informaion updaing o a money growh shock Noe: This figure displays he impulse response of he sicky informaion model of secion 3 in which he Poisson process for he arrival of informaion is replaced by fixed inervals. The shock is assumed o occur a =. N and M are he number of quarers firms and consumers, respecively, mainain he same informaion se. The shock is o he money growh rule. Sraegic complemenariy in consumpion All oher parameers are a heir baseline values. Noneheless, here remains a difference in he impac effec on inflaion. Wih sicky informaion for firms only, inflaion is rising immediaely afer he shock wih a money growh rule, whereas i is falling wih a Taylor rule. This occurs hrough he same mechanisms as wih Poisson arrival raes, bu hese effecs are overwhelmed by he jump in he price level when all firms acquire he same informaion and oupu is driven o zero. This will be he case as long as sraegic complemenariy in price seing is sufficienly high o keep iniial changes in prices small relaive he change in prices ha mus occur when informaion rigidiy ends. Figure 9 plos he response of inflaion when N=7 and M= wih a Taylor rule for differen levels of & (generaed by changing he value of H as necessary). For he baseline value of &=.5, we have a very small impac effec on inflaion, followed by a gradual decline due o he endogenous response of he cenral bank. This is followed by a large spike in inflaion a =N. Wih &=.5, he impac effec is larger, hough sill somewha less han he =N spike. Produced by The Berkeley Elecronic Press, 26

22 2 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle Figure 8: Response of he sicky informaion model wih saggered informaion updaing o a shock o he Taylor rule Noe: This figure displays he impulse response of he sicky informaion model of secion 3 in which he Poisson process for he arrival of informaion is replaced by fixed inervals. The shock is assumed o occur a =. N and M are he number of quarers firms and consumers, respecively, mainain he same informaion se. The shock is o he Taylor rule. Sraegic complemenariy in consumpion All oher parameers are a heir baseline values. Finally, wih neiher sraegic complemenariy nor subsiuabiliy in price seing (&=), he peak response of inflaion occurs on impac. While inflaion sill rises a =N, his spike is now smaller han he impac effec. 5 Discussion and relaion o he previous lieraure This paper has focused on he deerminans of inflaion ineria in he sicky informaion model. Paricular emphasis has been placed on he role of sraegic complemenariy in price seing, which maers no jus for he persisence of real effecs as in sicky price models, bu also for he qualiaive hump-shaped response of inflaion. The resuls have shown ha inegraing he sicky informaion approach ino more developed aggregae demand seings can affec he model s abiliy o deliver inflaion ineria. In paricular, he assumpion of hp://

23 Coibion: Inflaion Ineria in Sicky Informaion Models 2 Figure 9: Sensiiviy of inflaion response under fixed duraion of informaion rigidiy for firms and a Taylor rule. Noe: This figure shows he response of inflaion o a one-uni shock o he Taylor rule in he model of secion 3, bu where he Poisson process for he arrival of informaion is replaced wih a fixed duraion beween informaion updaes (he lengh of which is N=7). There is no sicky informaion for consumers and all parameers are a heir baseline levels. forward-looking consumers and a Taylor rule o capure he endogenous response of consumers and policy-makers makes inflaion ineria hard o deliver for ypical parameer values. Including sicky informaion for consumers can help yield oupu ineria and herefore makes i easier o achieve inflaion ineria, bu he resuls remain very sensiive o parameer values. These resuls are useful o sor he conflicing resuls found in previous papers regarding he robusness of MR s original findings. Closes o MR is Reis (25) whose primary purpose is o develop microfoundaions for he ime-dependen sicky informaion model of MR. Afer showing ha aggregaing across firms facing coss o he acquisiion of informaion leads o a Poisson process for he arrival of informaion a he aggregae level, he presens a model in which labor is supplied individually o firms and applies he SIPC o his seup. As emphasized by Woodford (23), his approach generaes sraegic complemenariy in price seing for sandard parameer values. In his case, his leads o &=., nearly he same value as in MR. Since aggregae demand is modeled as in MR, his resuls closely replicae hose of MR. Produced by The Berkeley Elecronic Press, 26

24 22 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle Keen (24) considers a more general model wih uiliy-maximizing consumers, a money demand equaion, and shocks o he growh rae of money. Unlike Reis (26), he assumes ha labor and capial are supplied o aggregae facor markes, from which all firms do heir hiring and rening. This yields, for mos parameer values, sraegic subsiuabiliy in price seing (&>). In response o money growh shocks, Keen finds ha he sicky informaion model leads o immediae jumps in inflaion unless he persisence of money growh shocks is much higher han esimaed values. Wih Taylor rules, he finds ha he sicky informaion model is incapable of generaing inflaion ineria. His resuls are hus largely due o he absence of enough sraegic complemenariy in price seing, which follow from his assumpion of economy-wide inpu markes. The assumpion of purely forward-looking consumers also makes i harder for his model o deliver inflaion ineria. Traband (25) and Korenok and Swanson (24) each consider a similar model o Keen (24), bu one in which labor is individually supplied o firms, hereby generaing significan sraegic complemenariy in price seing. Their models also include forward-looking consumers and a money demand equaion combined wih an exogenous AR() process for he growh rae of money. As argued in secion 4., his combinaion can generae inflaion ineria in response o a money growh shock despie he presence of forward-looking consumers, as conclude boh papers. Hence, he differen assumpions abou facor markes, and herefore sraegic complemenariy in price seing, lead hese auhors o reach a very differen conclusion han Keen (24). Andres e al (25) esimae a dynamic model wih a sicky informaion Phillips Curve by full informaion maximum likelihood. They use heir parameer esimaes o consider he response o a shock o he Taylor rule. Given heir parameer esimaes, real marginal coss adjus very rapidly in response o he shock, a resul equivalen o having lile sraegic complemenariy in price seing. Despie having subsanial inernal habi formaion in he consumpion Euler equaion, he response of inflaion o a emporary shock o he ineres rae is immediae. This resul is similar o ha found in secion 4.2, where even he presence of sicky informaion consumers is insufficien o yield inflaion ineria in response o an ineres rae shock unless here are significan sraegic complemenariies in boh consumpion and price seing, a feaure absen in heir parameer esimaes. Collard and Dellas (24) follow Dupor and Tsuruga (25) in using fixed duraions of informaion rigidiy. Aggregae demand consiss of forward-looking consumers and a money demand equaion wih moneary policy described by a money growh rule. Like Keen (24), hey assume economy-wide labor and capial markes, implying &> for heir parameer values. They find ha he peak response of inflaion o moneary shocks is immediae. As shown in secion 4.3, hp://

25 Coibion: Inflaion Ineria in Sicky Informaion Models 23 Table : Summary of wheher previous sudies found inflaion ineria using sicky informaion models Policy Response Exogenous Process Endogenous Response Price Seing Behavior Sraegic Complemens Sraegic Subsiues Mankiw Reis (22): YES Reis(24): YES Traband(25): YES Korenok Swanson (24): YES Keen (24): NO Keen (24): NO Collard Dellas (24): NO Andres e al (25): NO Noe: Yes/No indicaes wheher auhor(s) found inflaion ineria in response o moneary shock. Exogenous process for policy response includes money growh rules and exogenous nominal income growh processes. Endogenous response means Taylor rule. his resul hinges again upon he implied level of sraegic complemenariy in price seing. I summarize he resuls of hese previous sudies in Table. The criical role played by sraegic complemenariy in price seing is clear in he resuls found. Unforunaely, all of he papers ha find an inerial response of inflaion in response o moneary shocks do so in response o money growh or nominal income growh shocks bu do no explicily model he endogenous response of moneary policy-makers. Secion 4. shows ha he remaining box in he able has no clear-cu answer and he resuls are highly sensiive o parameer esimaes. 6 Conclusion This paper considers wheher he sicky informaion model of MR can robusly deliver inflaion ineria. I idenifies sraegic complemenariy in price seing as a criical deerminan for observing gradual hump-shape responses of inflaion in response o moneary shocks. In addiion, inegraing he SIPC ino more developed models of aggregae demand leads o conradicory conclusions, even in he presence of real rigidiies. Whereas money growh rules combined wih forward-looking consumers yield responses ha closely resemble he original resuls of MR, he inclusion of a Taylor rule designed o capure he endogenous response of moneary policy-makers generally produces a ime pah of inflaion ha peaks in he same period as he shock occurs. The addiion of sicky informaion in consumpion yields an inerial response of oupu and increases he Produced by The Berkeley Elecronic Press, 26

26 24 Conribuions o Macroeconomics Vol. 6 [26], No., Aricle parameer space in which inflaion can be inerial. However, under he baseline parameers considered in he model, inflaion coninues o peak on impac in response o an ineres rae shock. These resuls allow us o sor ou he conflicing conclusions drawn by previous sudies regarding he robusness of MR s findings. Because of he sensiiviy of he sicky informaion model for firms o he assumpions of he res of he model, fuure work should consider more carefully he implicaions of sicky informaion for oher secors of he economy. One of he aracive feaures of he sicky informaion model is ha if informaion delays exis for firms, hey should maer for oher secors of he economy, such as facor markes. This paper considers one such exension for consumers, bu he approach used here could feasibly be exended o wage seing decisions. 5 More ineresing sill would be applying sicky informaion o he invesmen decisions of firms, since invesmen is ypically considered o be he primary mechanism hrough which ineres rae changes affec he economy. 5 See Mankiw and Reis (25) for a model wih sicky informaion in price and wage-seing as well as consumpion. hp://