The Optimal In ation Rate with Heterogeneous Firms

Transcription

1 The Opimal In aion Rae wih Heerogeneous Firms Klaus Adam, Universiy of Mannheim & CEPR Henning Weber, Deusche Bundesbank January 9, 207 Absrac We augmen a sandard sicky price model by idiosyncraic shocks ha rigger a change in rm level produciviy and allow he rm o adjus is price. In economic erms, hese shocks may capure he inroducion of new producs or produc qualiy levels, or he replacemen of an exising rm by a new rm. We derive closedform expressions for he opimal in aion rae in he resuling sicky price economy wih heerogeneous rms and show ha () he opimal seady sae in aion rae generically di ers from zero, and (2) he opimal in aion rae persisenly responds o produciviy innovaions. Using rm level daa from he U.S. Census Bureau, we nd ha he opimal U.S. in aion rae implied by he model is posiive, bu approximaely dropped by one half over he period 986 o 203. Keywords: opimal in aion rae, sicky prices, rm heerogeneiy JEL Class. No.: E52, E3, E32 We would like o hank Mahias Kehrig for generously supporing us wih access o he LBD daabase. We also hank Pierpaolo Benigno, Saki Bigio, V.V. Chari, Michael Woodford, seminar paricipans a he 206 CSEF-CIL-UCL conference in Alghero, he 2h Dynare conference in Rome, 206, 206 EUI Workshop on Economic Policy Challenges, Florence, and he 206 German Economic Associaion (VfS) meeing for commens and suggesions. The opinions expressed in his paper are hose of he auhors and do no necessarily re ec he views of he Deusche Bundesbank, he Eurosysem, or is sa. Any remaining errors are our own. 0

2 Inroducion This paper makes progress by inroducing rm level heerogeneiy ino an oherwise sandard sicky price economy. I shows ha he resuling generalized sicky price framework yields raher di eren implicaions for he opimal in aion rae and can raionalize - amongs oher hings - posiive in aion raes in seady sae. Due o he echnical di culies associaed wih aggregaing heerogeneous rm models, i is sandard in he sicky price lieraure o absrac from all rm level heerogeneiy beyond ha generaed by price adjusmen fricions hemselves. As is well known, price adjusmen fricions hen ighly anchor he opimal seady sae in aion rae a zero, e.g., Woodford (2003). This raher robus bu somewha puzzling implicaion of sandard sicky price models arises because he homogeneiy assumpion implies ha he produciviy of price adjusing rms is equal o ha of non-adjusing rms. Wih economic e ciency requiring relaive prices o re ec relaive produciviies, his feaure srongly calls for price adjusing rms o charge he same price as non-adjusing rms, i.e., for zero in aion. 2 The presen paper exends he basic sick price seup by inroducing idiosyncraic rm level produciviy adjusmens ha arrive in conjuncion wih a price adjusmen opporuniy. This gives rise o a seing wih heerogeneous rm level produciviies in which he produciviy of price adjusing rms is no necessarily equal o ha of nonadjusing rms. In economic erms, one can inerpre he combined shock ha we inroduce in a number of meaningful ways. I may represen, for example, an even in which a rm inroduces a new produc ha is produced wih a new echnology and ha requires he rm o choose a price for his new produc. Alernaively, rms may inroduce a new model of heir exising produc ha o ers new qualiy feaures. Again, his would require he rm o choose a price for he new model. Such produc or model subsiuions ake place raher frequenly in he daa and are ypically associaed wih price changes, bu are rouinely absraced from in sandard sicky price seups. 3 I is also possible o inerpre he shocks we inroduce as capuring rm urnover, whereby an exising rm exis he economy and is replaced by a new rm. The produciviy adjusmen hen represens he gap beween he echnology used by he old and he new rm. Again, i is naural o Secion 2 discusses a range of exensions of he basic framework considered in he lieraure and heir implicaions for he opimal in aion rae. 2 Yun (2005) shows, using a seing wih homogenous rms, ha if iniial prices do no re ec iniial produciviies, he opimal in aion rae can display ranisory deviaions from zero. 3 Secion III.C in Nakamura and Seinsson (2008) provides evidence on he rae of produc subsiuion in he U.S. CPI.

3 assume ha he new rm can freely choose is produc price upon enry. To illusrae he implicaions of such combined rm level shocks for he opimal in aion rae, we inroduce hem ino an oherwise sandard sicky price seing wih Calvo price adjusmen fricions. The choice of a baseline seing wih ime dependen price adjusmen fricions is hereby purely for convenience and we show ha our main resuls equally apply when inroducing hese shocks insead ino menu cos seups. For simpliciy, we shall refer o he arrival of a price adjusmen opporuniy ha occurs in conjuncion wih an adjusmen of rm level produciviy as a -shock and we le 0 denoe he idiosyncraic probabiliy ha such a shock arrives a a given rm a any poin in ime. The -shocks di er from he idiosyncraic price adjusmen shocks presen in Calvo price sickiness models, which feaure no produciviy adjusmen. The main poin of he paper is ha he opimal in aion rae disconinuously jumps when moving from a siuaion where = 0 o one where > 0. To model produciviy dynamics, we consider a seing feauring hree sysemaic produciviy rends, each of which has di eren implicaions for he opimal in aion rae. Firs, here is a common rend in oal facor produciviy (TFP), which a ecs all rms equally, as in a sandard homogeneous rm model. This common TFP rend capures general purpose innovaions ha are adoped by all rms simulaneously. Second, here is an experience rend in rm TFP, which deermines how rms accumulae experience since hey las received a -shock. Depending on he economic inerpreaion of -shocks, he experience rend capures experience accumulaion in he producion of a speci c produc or a produc qualiy, or experience accumulaion over he lifeime of he rm. One can inerpre he experience rend more broadly as capuring learning-by-doing e ecs. Third, here is a cohor produciviy rend, which deermines he produciviy (or qualiy) level of he cohor of rms ha receives a -shock a a given poin in ime. This rend seeks o capure he fac ha new producs/qualiies/ rms bring ino he economy new echnologies ha are no (ye) used by oher rms. Firms ha receive a -shock receive his cohor produciviy (in addiion o he common TFP componen) and hen gradually accumulae experience over ime. We show ha he opimal seady sae in aion rae in his seing depends on he srengh of he experience rend relaive o he srengh of he cohor rend, whenever > 0, bu is independen of, provided > 0. The opimal in aion rae is also independen of he common TFP process. This di ers noably from a seing wihou idiosyncraic shocks ( = 0), where he opimal in aion rae is always equal o zero. To provide economic inuiion for hese ndings, consider wo polar seings. The rs seing absracs from he presence of a cohor rend and considers a seing where 2

4 he only rend is ha rms accumulae experience over ime. 4 Firms ha receive a -shock lose heir exising experience sock and become in his seing less producive han he remaining rms. From a welfare sandpoin, he opimal price of rms ha receive a -shock should herefore exceed he average price of he oher rms, so as o accuraely re ec relaive produciviies. Achieving his requires eiher ha rms wih - shocks choose higher prices or ha rms wihou -shocks reduce prices, or a combinaion hereof. In he presence of sicky prices, price reducions by rms wihou -shocks are cosly. In ime dependen adjusmen models, hey lead o ine cien price dispersion due o asynchronous price adjusmen; in sae-dependen pricing models, hey require rms o pay adjusmen coss. Therefore, i is opimal o implemen he e cien relaive price exclusively by having rms wih -shocks charge higher prices, while all oher rms keep heir prices. Clearly, his implies ha he aggregae in aion rae mus be posiive in seady sae. As menioned before, he opimal seady sae in aion rae urns ou o be independen of he probabiliy > 0. A lower value for decrease he share of rms receiving -shocks, which - ceeris paribus - reduces he opimal rae of in aion. Ye, lower values for increase he produciviy of rms ha do no receive a -shock relaive o hose ha do, as he former had - on average - more ime o accumulae experience. This increases he opimal relaive price for rms wih a -shock and - ceeris paribus - increases he opimal in aion rae. In ne erms, hese wo e ecs exacly o se each oher. In he second polar seing, here is no experience e ec and he only rend is one where rms ha receive a -shock become as described by he cohor rend. Firms receiving a - shock are hen become more producive han he remaining rms, hus should opimally charge lower prices. This makes negaive raes of in aion opimal. 5 As before, he srengh of his e ec is independen of, as long as > 0. A lower value for increases he produciviy of rms wih -shocks relaive o he remaining rms, as he laer have received heir -shocks and associaed produciviy increase a longer ime ago. This e ec exacly o ses he fac ha here are fewer rms wih -shocks. Combining he experience and he cohor e ecs wihin a common seing, he opimal seady sae in aion rae depends on he srengh of he experience e ec, which pushes owards posiive in aion raes, relaive o he srengh of he cohor e ec, which pushes owards negaive in aion raes. We also deermine in closed-form he dynamic response of he opimal in aion rae 4 As menioned before, we can absrac from he common TFP rend, as i does no a ec he opimal in aion rae. 5 Due o price seing fricions, i is again no opimal ha rms wihou -shocks adjus prices. 3

5 following shocks o experience and cohor produciviy. We show ha such shocks have fairly persisen e ecs on he opimal in aion rae, especially in seings in which - shock inensiy is low ( > 0 bu close o zero). A low value for causes any persisen level shock o experience or o cohor produciviy o disseminae only gradually in he economy. This requires ha in aion persisenly moves along he ransiion unil he produciviy disribuion has reached is new seady sae. For he limi! 0 opimal in aion becomes a random walk. To obain a plausible framework for empirical analysis, we exend our resuls o a muli secor economy. The muli secor seup allows for secor-speci c experience and cohor rends, secor speci c common TFP rends, as well as secor speci c rm urnover raes and degrees of price sickiness. We hen show ha he in aion rae ha maximizes seady sae welfare is a weighed average of he in aion raes ha would achieve e cien relaive prices wihin each secor individually. The laer depend again on he secor speci c cohor and experience rends. Based on his nding, we devise a model-based empirical sraegy ha allows esimaing hese secor speci c cohor and experience rends and hus he opimal in aion rae from rm level daa. As we show, rm level daa are required for esimaing he in aion relevan rends implied by he model, as hese canno be ideni ed from aggregae daa. 6 To esimae he relevan rm level rends, we inerpre -shocks as enry and exi shocks for producive esablishmens. We hen use he Longiudinal Business Daabase (LBD) of he U.S. Census Bureau, which covers all privae secor esablishmens in he Unied Saes, and esimae he relevan cohor and experience rends and heir evoluion over ime. Our regression resuls show ha he opimal in aion rae implied by our model is posiive bu approximaely halved over he period 986 o 203. Depending on he precise value of he elasiciy of produc demand assumed, he level of he opimal in aion rae varies. For our preferred demand elasiciy speci caion, he opimal in aion rae declined from around 2% in 986 o approximaely % in 203. The remainder of he paper is srucured as follows. Secion 2 discusses he relaed lieraure. Secion 3 presens our heerogeneous rms model wih sicky prices. Secion 4 analyically aggregaes he model and secion 5 shows ha he exible price equilibrium is rs bes when a Pigouvian oupu subsidy correcs rms monopoly power. The main resul on he opimal rae of in aion is presened in secion 6. I derives in closed form he sae-coningen in aion rae ha replicaes he rs bes allocaion in a seing wih sicky prices and -shocks. The resul does no rely on any approximaions and apply for a fully sochasic seup. Secion 7 discusses he implicaions of he main resul for he opimal seady sae in aion rae and seady sae welfare. Secion 8 shows how he opimal in aion rae jumps disconinuously when moving from a sandard sicky price 6 This holds rue even for he baseline seup wih a single economic secor. 4

6 economy wihou -shocks o one including such shocks. Secion 9 deermines he uiliy coss of implemening sric price sabiliy in an economy where he opimal in aion rae according o our model is posiive. Secion 0 discusses he opimal response of he in aion rae o economic disurbances. Secion exends he baseline seup o a muli-secor economy, allowing for secoral degrees of price sickiness, rm urnover and secor speci c produciviy rends. Secion 2 presens a model-consisen approach for esimaing he opimal in aion rae and repors our empirical resuls regarding he opimal rae of in aion in he U.S. economy. Finally, secion 3 discusses he robusness of our ndings owards various exensions. A conclusion brie y summarizes. Proofs and echnical maerial is relegaed o a series of appendices. 2 Relaed Lieraure A small se of papers discusses he relaionship beween he opimal in aion rae and produciviy rends focusing on aggregae or secoral produciviy rends. All of hese papers nd ha he opimal in aion rae is negaive. Amano e al. (2009) consider an economy wih aggregae produciviy growh in which nominal wages and prices are sicky. They show how moneary policy a ecs wage and price mark-ups and ha his can make i opimal o implemen de aion, so as o reduce average wage mark-ups. Wolman (20) considers a wo secor sicky price economy wih secoral produciviy rends. He shows ha - even in he absence of moneary fricions - he opimal in aion rae is eiher negaive or close o zero, depending on he precise modeling of price adjusmen fricions. Golosov and Lucas (2007) and Nakamura and Seinsson (200) consider sicky price seups wih heerogeneous rms and sudy moneary non-neuraliy wihin hese seups wihou considering he issue of he opimal in aion rae. Firms in heir seings are subjec o random idiosyncraic produciviy shocks. This di ers from he presen seup, where he idiosyncraic -shocks give rise o sysemaic produciviy adjusmens as implied by he cohor and experience rends and also allow for exible price adjusmens. These feaures ogeher give rise o a siuaion where he (average) produciviy of price adjusing rms di ers from he (average) produciviy of non-adjusing rms, which ulimaely makes non-zero in aion raes opimal. In Golosov and Lucas (2007) and Nakamura and Seinsson (200), i is mainly he rms wih very posiive or very negaive idiosyncraic produciviy shocks ha adjus prices, which suggess ha he produciviy of price adjusing rms is on average similar o ha of non-adjusing rms. Alhough seling his issue needs furher sudy, i suggess ha zero in aion is close o opimal in seady sae, as in seings wih homogeneous rms. The presen paper is also relaed o a large lieraure sudying he deerminans of 5

7 opimal in aion, mos of which nd ha he opimal in aion rae is eiher negaive or close o zero. None of hese papers makes a connecion beween he opimal in aion rae and rm level produciviy dynamics. In classic work, Kahn, King and Wolman (2003) explore he rade-o beween price adjusmen fricions, which call for price sabiliy, and moneary fricions ha call for a Friedman-ype de aion. They documen how a sligh rae of de aion is opimal in such frameworks. In a comprehensive survey, Schmi-Grohé and Uribe (200) documen he robusness of hese ndings o a large number of naural exensions. They show ha axaion moives, including he presence of unaxed income, foreign demand for domesic currency (Schmi-Grohé and Uribe (202a)), as well as a poenial qualiy bias in measured in aion raes (Schmi-Grohé and Uribe (202b)), are all unable o raionalize signi canly posiive raes of in aion. Adam and Billi (2006) and Coibion, Gorodnichenko and Wieland (202) explicily incorporae a lower bound on nominal ineres raes ino sicky price economies. Boh papers nd ha opimal moneary policy is consisen wih close o zero average raes of in aion. While zero lower bound episodes make i opimal o promise in aion in he fuure, hese promises should only be made condiionally on being a he lower bound, which happens raher infrequenly. A number of papers nds posiive in aion raes o be opimal on average when inroducing downward nominal wage rigidiies ino he sandard seup. Kim and Ruge-Murcia (2009) argue ha such rigidiies allow jusifying opimal in aion raes of approximaely 0.35% on average when using a model wih aggregae shocks only. Looking a a seing wih idiosyncraic shocks, Benigno and Ricci (20) also nd a posiive seady sae in aion rae o be opimal. 7 Carlsson and Wesermark (206) consider a seing wih nominal wage rigidiies and search and maching fricions in he labor marke. They show how a sandard U.S. calibraion of he model implies failure of he Hosios condiion and jusi es an in aion rae of abou.6% annually. Schmi-Grohé and Uribe (203) analyze he case for emporarily elevaed in aion in he Euro Area due o he presence of downward rigidiy of nominal wages. Brunnermeier and Sannikov (206) show ha he opimal in aion rae can also be posiive in a model wihou nominal rigidiies. They presen a model wih undiversi able idiosyncraic capial income risk in which he opimal in aion rae increases wih he amoun of idiosyncraic risk. There is also a lieraure sudying endogenous rm enry decisions in homogenous rm economies, focusing on he e ec of in aion on he rm enry margin, e.g., Corsei and 7 Since posiive in aion has no welfare coss in heir seup, hey do no quanify he opimal in aion rae. 6

8 Bergin (2008), Bilbiie e al. (2008) and Bilbiie, Fujiwara and Ghironi (204). Bilbiie e al. (204) documen - amongs oher hings - ha he welfare opimal in aion rae is posiive, whenever he bene of addiional varieies o consumers falls shor of he marke incenives for creaing hese varieies. In aion hen reduces he value of creaing varieies and brings rm enry closer o is e cien (lower) level. The presen paper absracs from endogenous rm enry decisions and hus from he implicaion of moneary policy for he enry margin, insead considers a seing wih heerogeneous rms and exogenous enry and exi. A se of empirical papers decomposes he observed U.S. in aion rae ino a rend and cyclical componen and show ha rend in aion displays subsanial low-frequency variaion over ime, e.g., Cogley and Sargen (200), Cogley, Primiceri and Sargen (200). The sicky price lieraure has reaced o hese facs by incorporaing rend in aion ino heir workhorse models, see Ascari and Sbordone (204) and Cogley and Sbordone (2008). Trend in aion emerges in hese seups because he cenral bank pursues an exogenous in aion arge, which is non-zero and poenially ime-varying. Primiceri (2006), Sargen (999), Sargen, Williams and Zha (2006) pesen seings in which policymakers are learning abou he Phillips Curve rade-o and show how his can endogenously give rise o he observed low-frequency movemens in U.S. in aion. The presen paper does no explore o wha exen changes in rm level produciviy can conribue o explaining he observed U.S. in aion hisory, as i focuses on he normaive implicaions of hese rends. Exploring he posiive conen of he heory presened in his paper appears o be an ineresing avenue for furher research. 3 Economic Model We consider a cashless economy wih nominal rigidiies and monopolisically compeiive rms. The model is enirely sandard, excep for he more deailed modeling of rm level produciviy and price adjusmen dynamics. For simpliciy, we derive our resuls wihin a ime-dependen price adjusmen model à la Calvo (983). As we show in secion 3., our main ndings remain unalered when considering insead a seing where price adjusmen fricions ake he form of menu coss. 3. Technology, Prices and Price Adjusmen Opporuniies We consider a seing wih a uni mass of monopolisically compeiive rms indexed by j 2 [0; ] and discree ime ( = 0; ; : : :). Each rm j produces oupu Y j, which eners as an inpu ino he producion of an aggregae consumpion/invesmen good Y according 7

9 o R Y = Y 0 j dj ; () where < < denoes he price elasiciy of produc demand. Le P j denoe he price charged by rm j in period. As in a sandard Calvo seup, rms can adjus prices wih probabiliy each period (0 < < ). The arrival of a Calvo price adjusmen opporuniy is hereby idiosyncraic and independen of all oher exogenous random variables in he economy. We augmen his sandard seing by a second idiosyncraic price adjusmen opporuniy ha arrives wih probabiliy 0 each period. This second adjusmen opporuniy is also idiosyncraic bu arrives in conjuncion wih a rm level echnology change, as described in deail below. In paricular, le j 2 f0; g denoe he idiosyncraic i.i.d. random variable governing his second price and echnology adjusmen and le j = indicae he arrival of such an adjusmen even for rm j in period (Pr( j = ) = ). We shall informally refer o he even j = as he arrival of a -shock. Leing K j and L j denoe he amoun of capial and labor used by rm j, respecively, rm oupu Y j is given by Y j = A Z j K j L j F ; (2) where A capures aggregae produciviy, Z j rm-speci c produciviy, and F 0 he poenial presence of xed coss for operaing he rm. To be consisen wih balanced growh, we assume F = f ( e ) (3) for some f 0, where e capures he growh rend in he balanced growh pah, as de ned in equaion (20) below. 8 Aggregae produciviy evolves according o A = a A ; rm speci c produciviy according o ( g Z j Z j = Q if j = 0 if j = ; where Q is given by Q = q Q ; (4) wih a ; g ; q > 0 being saionary produciviy growh processes wih uncondiional mean a; q; g > 0, respecively. Produciviy dynamics hus feaure hree rends: () he aggregae growh rend a ; (2) he rm level growh rend g, which applies in he absence of 8 Absen aggregae produciviy growh, he formulaion of xed coss in equaion (3) corresponds o ha used in Meliz (2003). 8

10 -shocks; and (3) he produciviy rend q, which deermines he e ecs of -shocks on echnology. Each growh rend will have a di eren implicaions for he opimal in aion rae wihin our seing. To undersand he produciviy dynamics implied by he previous seup, consider rs he special case wih = 0. In he absence of idiosyncraic -shocks o rm echnology, all rms experience he same produciviy growh rae a g. Such a seing wih homogeneous produciviy growh across all rms is he one rouinely considered in he sicky price lieraure. 9 Nex, consider he case > 0 and le s j denoe he number of periods ha has elapsed since he rm las experienced a -shock (i.e., j; sj = and j e = 0 for e = s j + ; :::; ). Firm-speci c echnology can hen be wrien as Z j = G j Q sj ; where G j = ( for s j = 0 g G j oherwise, and where Q follows equaion (4). This alernaive formulaion illusraes ha all rms ha receive a -shock in upgrade idiosyncraic echnology o Z j = Q, so ha Q can be inerpreed as capuring a cohor e ec of produciviy dynamics, where cohors are deermined by he arrival ime of he las -shock. Following any -shock, he rm experiences furher produciviy gains, as described by he process G j, as long as no furher -shocks arrive. Since he produciviy gains G j are los wih he arrival of he nex -shock, one can inerpre he process G j as capuring experience or learning-bydoing e ecs associaed wih he cohor producion echnology Q sj. Following a -shock in period, our speci caion hereby implies ha rm echnology increases (emporarily decreases), if Q has been growing faser (slower) han G since he ime of arrival of he las -shock prior o period. Noe ha he long-erm growh rae of rms produciviy is deermined by he process a q, as he experience growh raes g generae - due o he occasional reses - only a level e ec for produciviy. As usual, we de ne he aggregae price level as Z P 0 (P j ) dj : (5) 9 For he case = 0, our seing sill allows for a non-degenerae iniial disribuion of relaive rm produciviies. Typically, his iniial disribuion is also assumed degenerae in he sicky price lieraure. As we show below, however, he addiional assumpion of a degenerae iniial disribuion does no a ec he conclusions regarding he opimal in aion rae, as long as iniial prices re ec iniial produciviies, see Yun (2005) for a discussion of his and relaed issues in a homogeneous rms seing. 9

11 Cos-minimizaion in he producion of nal oupu Y implies Z Yj P = P j dj, 0 so ha he price level is an expendiure weighed average of he prices in he di eren expendiure caegories, in line wih he pracice a saisical agencies. The in aion rae is de ned as Y P =P : We shall furhermore assume ha a = a a, q = q q, and g = g g wih a ; q ; g 0 being saionary shocks wih an arbirary conemporaneous and ineremporal covariance srucure, saisfying E[ a ] = E[ q ] = E[ g ] =. Finally, o obain a well-de ned seady sae, we assume ( ) (g=q) < : (6) This condiion insures ha relaive prices in he exible price economy remain bounded. 3.2 Alernaive Inerpreaions of he Seup The previous secion de ned -shocks ( j = ) as an idiosyncraic change in rm level produciviy ha is associaed wih a price adjusmen opporuniy. This secion presens hree alernaive inerpreaions of -shocks ha clarify why such produciviy changes may plausibly be associaed wih price exibiliy a he rm level. Produc subsiuion. The even j = can be inerpreed as an even in which he produc previously produced by rm j is no longer demanded by consumers. Firm j reacs o his by inroducing a new produc, which - for simpliciy - is assigned he same produc index j. The variable Q hen capures he produciviy level associaed wih producs ha are newly inroduced in and G j capures experience accumulaion in producing he new produc. Produc subsiuions, e.g., in he form of new produc versions or models, ake place raher frequenly in he daa and are also prevalen in he CPI baskes of saisical agencies. Evidence provided in Moulon and Moses (997), Bils (2009) and Syed and Myers (206) shows ha he prices of new producs are ypically higher han hose ha hey replace, even when accouning for qualiy improvemens. 0 I hus appears reasonable o assume price exibiliy for new producs. Enry and exi. The even j = can also be inerpreed as an even in which rm j becomes unproducive and exis he economy. I is hen replaced by a rm o which we 0 Evidence provided in Bils (2009) shows ha in aion for durables ex compuers over he period averaged 2.5% per year, bu when only including only mached iems, he in aion rae was -3.7% per year. 0

12 assign - for simpliciy - he same index j. The variable Q hen capures he produciviy level of he cohor of rms ha eners in period and G j he experience accumulaed over he lifeime of he rm. Firm enry and exi raes are high in he Unied Saes and range in he order of 8-2% per year, see gure 3 in Decker e al. (204). I also appears plausible o assume ha newly enering rms can choose prices exibly. We shall use an inerpreaion of our model along hese lines in our empirical exercise in secion 2. Qualiy improvemens. rm j in period. De ning Q j = Q Le Q j denoe he qualiy of he produc produced by sj, he even j = capures he fac ha rm j upgrades he qualiy of is produc from level Q sj; o level Q. Le aggregae oupu produced wih inermediae inpus of di eren qualiy be given by Y = R Q 0 jyj e dj ; le rm oupu of a given qualiy level Q j be given by ey j = A G j K j L j F ; where G j now capures experience e ecs associaed wih producing qualiy Q j, and le ep j denoe he price of a uni of good j of qualiy level Q j. Assuming ha saisical agencies can perfecly adjus he price level for qualiy, i.e., P is given by 0 P Z 0 ep j Q j! dj A his seup wih qualiy improvemens is mahemaically idenical o he one wih produciviy improvemens spelled ou in he previous secion. ; As wih new producs, i appears naural o assume ha rms can exibly price goods wih new qualiy feaures. 3.3 Opimal Price Seing Firms choose prices, capial and hours worked o maximize pro s. While price adjusmen is subjec o adjusmen fricions, facor inpus can be chosen exibly. Leing W denoe he nominal wage and r he real renal rae of capial, rm j chooses he facor inpu mix so as o minimize producion coss K j P r + L j W subjec o he consrains imposed by he producion funcion (2). Le I j F + Y j =(A Q sj G j ) The qualiy-adjused price P e j =Q j and he qualiy adjused quaniies Y e j Q j hen correspond o he price P j and quaniy Y j, respecively, in he previous secion.

13 denoe he unis of facor inpus (K j L j ) required o produce Y j unis of oupu. As appendix A. shows, cos minimizaion implies ha he marginal coss of I j are given by MC = W = P r = : (7) Now consider a rm in period ha can freely choose is price because i has eiher received a -shock or a Calvo adjusmen shock. Leing denoe he probabiliy implied by he Calvo process ha he rm has o keep is price (0 < < ), he rm will no be able o reopimize is price wih probabiliy ( receives neiher a - shock nor a Calvo adjusmen shock. 2 a rm ha can opimize is price in period is hus given by max E P j X (( i=0 ) a any fuure dae, i.e., whenever i The price seing problem of )) i ;+i P +i [( + )P j+i Y j+i MC +i I j+i ] (8) s:: I j+i = F + Y j+i =A +i Q sj G j+i ; Y j+i = (P j+i =P +i ) Y +i ; P j+i+ = +i+;+i P j+i : where denoes a sales ax/subsidy and ;+i denoes he represenaive household s discoun facor beween periods and +i. The rs consrain capures rm s echnology, he second consrain capures he demand funcion faced by he rm, as implied by equaion (), and he las consrain capures how he rm s price is indexed over ime (if a all) in periods in which prices are no rese opimally. We consider general price indexaion schemes and allow +i+;+i o be a funcion of aggregae variables up o period + i. 3 Appendix A.2 shows ha he opimal price P? j can be expressed as P j? Q P sj G j Q = + N D : (9) where he variables N and D are funcions of aggregae variables only and evolve recur- 2 In any period, he rm can adjus is price wih probabiliy due o he arival of a -shock and wih probabiliy ( )( ) due o he arrival of a Calvo price adjusmen shock. 3 We only require ha price indexaion is such ha he price seing problem remains well de ned, ha price indexaion does no give rise o mulipliciies of he opimal in aion rae and ha indexaion is such ha +; = in a seady sae wihou in aion. For insance, when indexing occurs wih respec o lagged in aion according o + ; = ( ) wih 0, we rule ou > o avoid non-exisence of opimal plans and rule ou = o avoid mulipliciies of he seady sae in aion rae. 2

14 sively according o N = MC P A Q + ( D = + ( )E " )E " ;+ Y + Y ;+ Y + Y ( ;+ ) ( ;+ ) P+ P P+ P # q+ N + g + (0) D +# : () Equaion (9) shows ha he opimal rese price of a rm depends only on how is own produciviy (A Q sj G j ) relaes o he produciviy of a rm hi by a -shock in period (A Q ), as well as on aggregae variables. I is precisely his feaure, which will permi aggregaion of he model in closed form. Equaion (9) furhermore shows ha more producive rms opimally choose lower prices. For he special case wih homogeneous rms, where relaive produciviy is always equal o one (Q sj G j =Q = ), equaions (9)-() reduce o hose capuring price dynamics in a sandard homogeneous rm model. 3.4 Household Problem There is a represenaive household wih balanced growh consisen preferences given by! X E 0 [C V (L )] ; =0 where C denoes privae consumpion of he aggregae good, L labor supply, a preference shock wih E[ ] = and 2 (0; ) he discoun facor. We assume > 0 and ha V () is such ha period uiliy is sricly concave in (C ; L ) and ha Inada condiions are sais ed. The household faces he ow budge consrain C + K + + B P = (r + d)k + W P L + Z 0 j P dj + B P ( + i ) T ; where K + denoes he capial sock, B nominal governmen bond holdings, i he nominal ineres rae, W he nominal wage rae, r he real renal rae of capial, d he depreciaion rae of capial, j nominal pro s from ownership of rm j, and T lump sum axes. Household borrowing is subjec o a no-ponzi scheme consrain. The rs order condiions characerizing opimal household behavior are enirely sandard and are derived in Appendix A.3. To insure exisence of a well-de ned balanced growh pah, we assume hroughou he paper ha < (aq) : 3.5 Governmen To close he model we consider a governmen which faces he budge consrain B = B ( + i ) + R Pj 0 P P P Y j dj T ; 3

15 where denoes a sales subsidy, which will be used o correc for he monopoly power disorions in produc markes. The governmen levies lump sum axes T, so as o implemen a bounded sae-coningen pah for governmen deb B =P. 4 Since we consider a cashless limi economy, here are no seigniorage revenues, even hough he cenral bank conrols he nominal ineres rae. We furhermore assume ha moneary policy is no consrained by a lower bound on nominal ineres raes. 3.6 Equilibrium We are now in a posiion o de ne he marke equilibrium: De niion An equilibrium is a sae-coningen pah for f(p j ; L j ; K j ) for j 2 [0; ], W ; r ; i ; C ; K + ; L ; B ; T g =0 such ha. he rms choices fp j ; L j ; K j g =0 adjusmen fricions, maximize pro s for all j 2 [0; ], given he price 2. he household s choices fc ; K + ; L ; B g =0 maximize expeced household uiliy, 3. he governmen ow budge consrain holds each period, and 4. he markes for capial, labor, nal and inermediae goods and governmen bonds clear, given he iniial values B ( + i ); K 0 ; P j;, and A Q sj; G j;, j 2 [0; ]. 4 Analyical Aggregaion wih heerogeneous Firms This secion oulines he main seps ha allow us o aggregae he model in closed form. In a rs sep, we derive a recursive represenaion describing he evoluion of he aggregae price level P over ime. In a second sep, we derive a closed form expression for he aggregae producion funcion. In a las sep, we show how o appropriaely derend aggregae variables, so as o render hem saionary. Evoluion of he aggregae price level. Le P? s; k denoe he opimal price of a rm ha las received a -shock in s and ha has las rese is price in k (s k 0). In period, his rm s price is equal o k; P? s; k, where k; = Q k j= k+j ; k+j capures he cumulaive e ec of price indexaion (wih k; = in he absence of price indexaion). Le (s) denoe he weighed average price in period of he cohor of rms 4 Household s ransversaliy condiion will hen auomaically be sais ed in equilibrium. 4

16 ha las received a -shock in period, i.e., s, where all prices are raised o he power of Xs (s) = ( ) k ( k; P? s; k) + s ( s; P? s; s) : (2) k=0 There are s rms ha have no had a chance o opimally rese prices since receiving he -shock and ( ) k rms ha have las adjused k < s periods ago. From equaion (5) i follows ha one can use he cohor average prices (s) o express he aggregae price level as P = X ( ) s (s); (3) where is he mass of rms ha receive a -shock each period and ( of hose rms ha have no received anoher -shock since s periods. s=0 ) s is he share To express he evoluion of P in a recursive form, consider he opimal price P? a rm ha received a -shock s > 0 periods ago, bu can adjus he price in due o he s; of he arrival of a Calvo shock. Also, consider he price P ;? of a rm ha receives a -shock in period. The opimal price seing equaion (9) hen implies P ;? = P? s; g g s+ q q s+ : (4) The previous equaion shows ha a sronger cohor produciviy rend (higher values for q) causes he rm ha receives a -shock in period o choose lower prices relaive o rms ha received -shocks furher in he pas, as a sronger cohor rend makes his rm relaively more producive. Conversely, he experience e ec (higher values for g) increases he opimal relaive price of he rm ha received a -shock in. The ne e ec depends on he relaive srengh of he cohor versus he experience e ec. Appendix A.4 shows how one can combine equaions (2), (3), and (4) o obain a recursive represenaion for he evoluion of he aggregae price level given by P = (P? ;) + ( )( ) (pe ) (P? ;) + ( )( ; P ) ; (5) where p e depends on he hisory of shocks o cohor and experience produciviy and evolves recursively according o The las erm on he r.h.s. share ( keep heir old price (P (p e ) = + ( ) p e g =q : (6) of equaion (5) capures he price level e ecs from he ) of rms ha neiher received a Calvo shock nor a -shock. These rms on average), adjused for possible e ecs of price indexaion, as capured by he indexaion erm ;. The rs erm on he r.h.s. of equaion (5) 5

17 capures he price e ecs of he mass of rms ha received a -shock in period ; hese rms opimally charge price P? ;. The second erm capures price reseing by rms ha received a Calvo shock in period ; heir share is ( )( ) and hey se a price ha on average di ers from he price charged by rms hi by a -shock, depending on he value of p e. This laer aspec in equaion (5) is he key di erence relaive o he sandard model wihou rm heerogeneiy in produciviy. A sronger experience rend (a higher value for g ), for insance, increases (p e ), and - ceeris paribus - causes rms ha receive a Calvo shock o choose a lower value for he opimal rese price. A sronger cohor rend (a higher value for q ) has he opposie e ec. Overall, he ineresing new feaure is ha price dynamics now depend on he produciviy rends. In a seing where all rms have idenical produciviy, e.g., where he cohor e ec is as srong as he experience e ec (q = g for all ), equaion (6) implies ha p e converges o one so ha he price level evenually evolves according o P = [ + ( )( )] (P? ;) + ( )( ; P ) ; which is independen of produciviy developmens. If in addiion here are no -shocks ( = 0), he previous equaion simpli es furher o P = ( )(P? ;) + ( ; P ) ; which describes he evoluion of he aggregae price level in he sandard Calvo model wih homogenous rms. Aggregae producion funcion. can be wrien as Y = A Q In appendix A.5 we show ha aggregae oupu Y K L F ; (7) where K denoes he aggregae capial sock, L aggregae hours worked and = evolves recursively according o Z 0 Q G j Q = + ( )( ) (pe ) P? ; + ( ) P sj Pj dj (8) P q g ; : (9) TFP in he aggregae producion funcion (7) is a funcion of he TFP of he laes cohor hi by -shock, A Q, and of he adjusmen facor. The laer is de ned in equaion (8) and capures a rm s produciviy relaive o ha of he laes cohor, Q = Q sj G j, and weighs his relaive produciviy by he rm s producion share 6

18 (P j =P ). Equaions (7) and (8) hus show how relaive price disorions may lead o oupu losses by negaively a ecing aggregae echnology, e.g., by allocaing more demand o relaively ine cien rms. The evoluion of he adjusmen facor over ime is described by equaion (9) and depends on produciviy rends - amongs oher ways - hrough he variable p e. In he limi wih homogeneous rm rends (i.e., q = g ), p e converges o one and he evoluion of becomes independen of produciviy realizaions. If - in addiion - here are no -shocks ( = 0), hen equaion (9) simpli es furher o P? ; = ( ) + ; P ; which is he equaion capuring he poenial disorions from price dispersion wihin he sandard homogeneous rm model. Balanced Growh Pah. hem by he aggregae growh rend One can obain saionary aggregae variables by rescaling e = (A Q = e ) ; (20) where e denoes he e cien adjusmen facor chosen by he planner, de ned in equaion (24) below. Speci cally, he rescaled oupu y = Y = e and he rescaled capial sock k = K = e are now saionary and he aggregae producion funcion (7) can be wrien as e y = In he deerminisic balanced growh pah, k L f : (2) e grows a he rae (aq) and hours worked are consan, whenever moneary policy implemens a consan in aion rae. Appendices A.6 and A.7 wrie all model equaions using saionary variables only and appendix A.8 deermines he resuling deerminisic seady sae. 5 E ciency of he Flexible Price Equilibrium This secion derives he e cien allocaion and provides condiions under which he exible price equilibrium is e cien. Appendix B shows ha he e cien consumpion, hours and capial allocaion fc ; L ; K + g =0 solves! X max E 0 [C V (L )] fc ;L ;K + g =0 s:: C + K + = ( d)k + A Q e (22) (K ) (L ) F ; (23) 7

19 where which evolves according o e Z 0 Q G j Q sj dj! ; (24) ( e ) = + ( ) e q =g : (25) Consrain (23) is he economy s resource consrain, when expressing aggregae oupu using he aggregae producion funcion (7). The e cien produciviy adjusmen facor e showing up in he planner s producion funcion is de ned in equaion (24); is recursive evoluion is described by equaion (25). The rs order condiions of problem (22)-(23) are necessary and su cien condiions characerizing he e cien allocaion. Decenralizing he e cien allocaion requires ha rms prices, which ener and hus in he aggregae producion funcion (7), saisfy cerain condiions. In paricular, equaion (8) implies ha one achieves = e if prices saisfy P j P = e Q G j Q sj ; (26) which requires relaive prices o accuraely re ec relaive produciviies. Furhermore, as in models wihou rm heerogeneiy, one has o eliminae rm s monopoly power by a Pigouvian subsidy o obain e ciency of he marke allocaion. We hus impose he following condiion: Condiion The sales subsidy correcs rms marke power, i.e., =. + Appendix C hen proves he following resul: Proposiion The exible price equilibrium ( = 0) is e cien, if condiion holds. The proof of he proposiion shows ha condiion (26) holds under exible prices, so ha one achieves = e and hereby producive e ciency. In he presence of he assumed sales subsidy, consumer decisions are also undisored, so ha he consumpion, hours and capial in he exible price equilibrium are idenical o he values ha hese variables assume in he e cien allocaion. 6 Opimal In aion wih Sicky Prices This secion deermines he opimal in aion rae for an economy wih sicky prices ( > 0). I derives he opimal rae of in aion for he nonlinear sochasic economy wih heerogeneous rms in closed form and shows how in aion opimally depends on 8

20 he produciviy growh raes a ; q and g. As i urns ou, he opimal in aion rae implemens he e cien allocaion (he exible price benchmark). To esablish our main resul in he mos sraighforward manner, we impose an assumpion on iniial condiions, in paricular on how rms iniial prices and iniial produciviies are relaed. Similar condiions are imposed in sicky price models wih homogenous rms, where i is rouinely assumed ha iniial dispersion of prices has reached is saionary oucome. We impose: Condiion 2 Iniial prices in = P j; / re ec rms relaive produciviies, i.e., Q sj; G j; all j 2 [0; ]: We discuss he e ecs of relaxing his condiion below. saes our main resul: The following proposiion Proposiion 2 Suppose condiions and 2 hold. The equilibrium allocaion in he sicky price economy is e cien, if moneary policy implemens he gross in aion rae? =? ; ( e )! for all 0; (27) where? ; capures price indexaion beween periods and (? ; in he absence of indexaion) and e is de ned in equaion (24) and evolves according o equaion (25). In he absence of price indexaion (? ; ), he opimal in aion rae is only a funcion of he variable e, which capures he disribuion of relaive produciviies beween newly enering rms and exising rms, see equaion (24). Since hese relaive produciviies are independen of he common TFP growh rae a, i follows ha he opimal in aion rae does no depend on he realizaions of a. In conras, he cohor produciviy growh rae q and he experience growh rae g do a ec e, see equaion (25). Ye, hese rends a ec he opimal in aion rae in opposie direcions: a sronger cohor produciviy growh rae q decreases he opimal in aion rae, while a sronger experience growh rae g increases he opimal in aion rae. For he special case in which all rms have idenical produciviy rends ( = 0) or even idenical produciviies ( e = ), he opimal gross in aion rae is equal o one in he absence of price indexaion, as in a sandard homogenous rm model. Perfec price sabiliy is hen opimal a all imes. Price indexaion by non-adjusing rms (? ; 6= ), say because of indexaion o he lagged in aion rae, inroduces addiional componens ino he opimal aggregae 9

21 in aion rae. In paricular, i requires ha price-adjusing rms, i.e., rms ha receive eiher a -shock or a Calvo shock, also adjus heir price by he indexaion componen. This way prices coninue o accuraely re ec relaive produciviies a all imes. This explains why indexaion a ecs he opimal in aion rae one-for-one. Alhough proposiion 2 assumes ha rms iniial prices accuraely re ec he iniial relaive produciviies, he iniial produciviy disribuion iself is unresriced. We conjecure ha for a seing where condiion 2 fails o hold, one would obain addiional ransiory and deerminisic componens o he opimal in aion rae, as in he homogeneous rm seing sudied by Yun (2005). The in aion rae saed in proposiion 2 would hen become opimal only asympoically. The proof of proposiion 2, which is conained in appendix D, esablishes ha wih he opimal in aion rae rms choose relaive prices as in he exible price equilibrium. This resul is esablished by showing ha () rms ha receive a -shock choose he same opimal relaive price as in he exible price economy, and ha (2) rms ha receive a Calvo shock opimally choose no o adjus heir price, which avoids he emergence of price dispersion beween oherwise idenical rms. The opimal in aion rae hus insures ha boh () and (2) are sais ed. This, ogeher wih he fac ha (3) iniial prices re ec iniial produciviies, ensures ha all relaive prices are idenical o he ones in he exible price equilibrium. Under he assumed oupu subsidy, i hen follows ha household allocaions are also idenical o he exible price equilibrium, which has been shown o be e cien, see proposiion. Ineresingly, i follows from he proof of proposiion 2 ha he in aion rae (27) coninues o insure producive e ciency (bu no full e ciency) in seings where condiion fails o hold. From he heory of opimal axaion i hen follows ha i coninues o be opimal o implemen he in aion rae (27), as i is subopimal o disor inermediae producion as long as (disorionary) axes on nal goods are available. 7 The Opimal Seady Sae In aion Rae This secion discusses he opimal seady sae in aion rae implied by he model. To simplify he discussion, we absrac from price indexaion, unless oherwise saed. Proposiion 2 makes i clear ha in he case in which he produciviy of all rms grows a he same rae ( = 0), which includes as a special case a seing wih homogeneous rms, we obain ha he opimal in aion rae is? =, independenly of all shock processes. For = 0, he opimal (gross) seady sae in aion rae is hus rivially equal o one. For he case > 0, he opimal in aion rae jumps disconinuously away from? =, 20

22 bu urns ou o be independen of he value of. The following lemma summarizes his resul: Lemma Suppose condiions and 2 hold, here are no economic disurbances, here is no price indexaion (? ; ) and > 0. The opimal in aion rae hen sais es lim!? = g=q: (28) Proof. From equaions (6) and (25) i follows ha ( e )! [ ( ) (g=q) ]=. I hen follows from proposiion 2 ha lim!?! g=q. Since we allow for arbirary iniial produciviy disribuions, he absence of shocks does no necessarily imply ha he opimal in aion rae is consan from he beginning. This only happens asympoically, once he produciviy disribuion converges o is saionary disribuion (in derended erms). 5 The lemma provides he in aion rae ha is asympoically opimal as his saionary disribuion is reached. 6 Ineresingly, he opimal long-run in aion rae is compleely independen of he inensiy of -shocks, which may appear surprising. A higher value for implies ha a larger share of rms receive a -shock. In a seing where g > q, his implies ha many rms will - upon arrival of he -shock - loose more in erms of accumulaed experience han hey gain in erms of cohor produciviy, i.e., heir produciviy iniially falls. Since less producive rms should charge higher prices, one may conjecure ha higher in aion should be opimal, as higher values for imply ha here are more of hese relaively unproducive rms. Ye, his argumen ignores, ha a higher inensiy of -shocks also shorens experience accumulaion wih old cohor echnologies. This makes rms ha do no receive a -shock relaively less producive han in a seing wih lower -shock inensiy. This second e ec suggess lower in aion raes o be opimal. As i urns ou, he wo e ecs exacly o se each oher and he opimal seady sae in aion rae is independen of. I appears empirically plausible o assume ha indeed g > q, i.e., he experience e ecs is sronger han he cohor e ec. Inerpreing he seing as one in which -shocks represen produc subsiuions, g > q implies ha new producs are more expensive han old producs and ha heir relaive price is falling over he life cycle of he produc, in line wih evidence provided by Syed and Myers (206). Likewise, inerpreing he seing as one in which -shocks represen a rm exi shock, wih exiing rms being replaced 5 This is no an issue when = 0 as he iniial disribuion remains hen unchanged in derended erms. 6 The ransiional dynamics can easily be derived from proposiion 2 using he iniial produciviy disribuion and equaion (25). 2

23 by new rms, g > q implies ha old rms are larger and more producive han young rms and ha rms oupu grows over heir lifeime, in line wih empirical observaions. Given his, lemma likely implies posiive values for he opimal seady sae in aion rae. Ineresingly, aggregae produciviy dynamics are no informaive abou wha is he opimal in aion rae in an economy wih -shocks. The aggregae seady sae growh is equal o (aq) and is driven by a facor ha does a ec he opimal in aion rae (q) and one ha does no (a). Moreover, he experience e ec (g) has no growh rae implicaions in he presence of -shocks bu does a ec he opimal in aion rae. Deermining he opimal in aion rae hus requires sudying rm level produciviy rends, as aggregae produciviy rends fail o be informaive. We shall come back o his issue in secion 2. Finally, we discuss he e ecs of price indexaion. For > 0 he opimal long-run in aion rae is hen given by? ; (g=q). For he case where prices are indexed o lagged in aion according o? ; = ( ) for some 2 [0; ), one obains lim!? = (g=q) : The presence of price indexaion hus ampli es he divergence of he opimal gross in aion rae from one. The resuls in his secion show ha he opimal in aion rae disconinuously jumps when moving from a seing wihou -shocks o one wih > 0. Ye, he e cien allocaion also disconinuously jumps when moving from = 0 o > 0, as in he former case e cien aggregae growh is equal o ag and in laer case equal o aq. For his reason, he nex secion discusses he disconinuiy of he opimal seady sae in aion rae in greaer deail. 8 Disconinuiy of he Opimal In aion Rae This secion compares he opimal in aion rae in an economy wih -shocks ( > 0) o he one in he absence of such shocks ( = 0). We refer o he rs economy as he -economy and o he laer as he 0-economy. Comparing hese wo economies is no as sraighforward as migh iniially appear: even if boh economies are subjec o he same fundamenal shocks (a ; q ; g ), he e cien allocaion displays a disconinuiy when considering he limi! 0. 7 To properly deal wih his issue, we consruc a -economy whose e cien aggregae allocaion (consumpion, hours, capial) is idenical o he e cien aggregae allocaion 7 This has o do wih he fac ha aggregae produciviy growh in he -economy is equal o a q while i is equal o a g in he 0-economy. 22