Monetary Policy and Business Cycles with Endogenous Entry and Product Variety

Transcription

1 Moneary Policy and Business Cycles wih Endogenous Enry and Produc Variey Florin O. Bilbiie Universiy of Oxford, Nuffield College Fabio Ghironi Boson College, EABCN, and NBER Marc J. Meliz Princeon Universiy, CEPR, and NBER March 8, 27 In preparaion for he NBER Macroeconomics Annual 27 Preliminary Absrac This paper sudies he role of endogenous producer enry and produc creaion for moneary policy analysis and business cycle dynamics in a general equilibrium model wih imperfec price adjusmen. Opimal moneary policy sabilizes producer prices, bu les he consumer price index vary o accommodae changes in he number of available producs. The free enry condiion links he price of equiy (he value of producs) wih marginal cos and markups, and hence wih inflaion dynamics. No-arbirage beween bonds and equiy links he expeced reurn on shares, and hus he financing of produc creaion, wih he reurn on bonds, affeced by moneary policy via ineres rae seing. This new channel of moneary policy ransmission hrough asse prices resores he Taylor Principle in he presence of capial accumulaion (in he form of new producion lines) and forward-looking ineres rae seing, unlike in models wih radiional physical capial. We also sudy he implicaions of endogenous variey for he New Keynesian Phillips curve and business cycle dynamics more generally, and we documen he effecs of echnology, deregulaion, and moneary policy shocks, as well as he second momen properies of our model, by means of numerical examples. JEL Codes: E3; E32; E52. Keywords: Business cycles; Moneary policy; Producer enry; Produc variey. We hank Daron Acemoglu for helpful commens, and Massimo Giovannini, Margaria Rubio, and Frank Virga for excellen research assisance. Remaining errors are our responsibiliy. Bilbiie hanks he NBER for hospialiy as a Visiing Fellow while his paper was wrien. Ghironi and Meliz hank he NSF for financial suppor hrough a gran o he NBER. Nuffield College, New Road, Oxford, OX NF, UK or florin.bilbiie@nuffield.oxford.ac.uk; URL: hp:// Deparmen of Economics, Boson College, 4 Commonwealh Avenue, Chesnu Hill, MA , U.S.A. or Fabio.Ghironi@bc.edu. URL: hp://fmwww.bc.edu/ec/ghironi.php. Deparmen of Economics, Princeon Universiy, Fisher Hall, NJ 8544, U.S.A. or mmeliz@princeon.edu. URL: hp://

2 Inroducion A large body of lieraure has grown since he mid 98s ha analyzes moneary policy in microfounded, dynamic, sochasic, general equilibrium (DSGE) models of he business cycle wih monopolisic compeiion and nominal rigidiy. The imporance of his New Keynesian lieraure (summarized, for insance, by Woodford, 23) for policymaking is esified by he curren use of such models by many cenral banks or inernaional insiuions as inpu for policy decisions. Mos of his lieraure, however, relies on monopolisic compeiion merely as a vehicle o inroduce price (or wage) seing power and hen assume ha price (or wage) seing is no fricionless, resuling in nominal rigidiy and a role for moneary policy. The overwhelming majoriy of models absracs from free enry and assumes a consan number of producers. The join assumpion of monopolisic compeiion and no enry raises wo quesions: one peraining o modeling and he oher o empirical evidence. Firs, absen eiher properly designed markup-offseing subsidies or increasing reurns of appropriae degree, monopolisic compeiion in hese models resuls in permanen (i.e., seady-sae) posiive profis, casing doubs on he heoreical appeal of he zero-enry assumpion. 2 Furhermore, recen empirical evidence for he U.S. has subsaniaed he endogenous flucuaions in he number of producers and he range of available goods ha ake place over he ypical lengh of a business cycle. A previous lieraure documened he srong procyclical behavior of ne producer enry (measured eiher as incorporaed firms or as producion esablishmens). 3 Bernard, Redding, and Scho (26) documen how exising esablishmens devoe a subsanial porion of heir producion o goods ha hey did no previously produce. Axarloglou (23) and Broda and Weinsein (27) direcly measure he inroducion of new varieies in he U.S. economy and documen a srong correlaion wih he business cycle. These heoreical and empirical observaions sugges ha here is scope for inroducing producer enry and produc creaion in models wih monopolisic compeiion and imperfec price adjusmen, and sudying he consequences of endogenous produc variey for business cycle propagaion and policy in hese models. This paper akes an iniial sep in his direcion by re-inroducing he endogenous link beween produc creaion (firm enry) and monopolisic compeiion in a DSGE model wih imperfec price See, for insance, he IMF s GEM model (illusraed by Laxon and Peseni, 23, among ohers) and he Federal Reserve Board s SIGMA model (illusraed by Erceg, Guerrieri, and Gus, 25, among ohers). 2 Roemberg and Woodford (995) addressed he implausibiliy of posiive seady-sae profis by assuming increasing reurns o scale induced by fixed, per-period coss. However, under his assumpion, any shock ha causes profis o fall below zero should generae exi and induce a non-lineariy in firm decisions. 3 See Campbell (998), Chaerjee and Cooper (993), and Devereux, Head, and Lapham (996a,b). We illusrae similar evidence in Bilbiie, Ghironi, and Meliz (25).

3 adjusmen. We explore he posiive and normaive consequences of endogenous producer enry and produc variey over he business cycle in a version of he model developed by Bilbiie, Ghironi, and Meliz (25 henceforh, BGM) feauring nominal rigidiy of a sandard form ofen used in he recen New Keynesian lieraure a quadraic cos of price adjusmen as in Roemberg (982). 4. Producer enry and produc creaion subjec o sunk enry coss are a he hear of he model, wih producer enry under monopolisic compeiion ied o household saving decisions via he household s choices of share holdings in he porfolio of firms ha operae in he economy. 5 In BGM, we show ha a benchmark version of he flexible-price model wih C.E.S. preferences and labor as he only facor of producion performs similarly o he sandard real business cycle (RBC) framework wih regard o he cyclicaliy of U.S. macroeconomic aggregaes ha are normally he focus of RBC sudies, bu i addiionally allows us o explain such feaures of he daa as he procyclicaliy of firm enry and profis. When preferences are of he ranslog form, he model addiionally generaes a counercyclical markup, owing o he posiive relaion beween subsiuabiliy and he number of producs wih such preferences, and i maches he ime profile of he correlaion beween he markup and GDP hanks o he slow adjusmen of he sock of producers over ime. Imporanly, he model generaes a counercyclical markup while sill preserving he procyclicaliy of profis a well known challenge for he benchmark New Keynesian model wih sicky prices. 6 The inroducion of endogenous produc variey in a sicky-price model of he business cycle allows us o address issues ha are absen in exising, fixed-variey models, as well as o qualify some of he resuls of hose models in he presence of his new margin. To sar wih, onesecor, fixed-variey models usually do no conain a meaningful disincion beween producer and consumer price indices (and inflaion raes). Therefore, hose models are silen as o wha is he inflaion rae ha he cenral bank should sabilize. In a model wih endogenous variey such as ours, a meaningful disincion beween he wo price indices arises, because he welfare-relevan 4 We choose he Roemberg model over he familiar Calvo (983)-Yun (996) seup o avoid heerogeneiy in prices wihin and across cohors of price seers ha enered a differen daes. Earlier flexible-price, business cycle models wih monopolisic compeiion and endogenous enry include also Ambler and Cardia (998) and Cook (2). Comin and Gerler (26), Jaimovich (24), Jovanovic (26), and Sebunovs (26) are more recen conribuions o he heoreical lieraure. See BGM for a discussion of he relaion wih our model. 5 There is a one-o-one idenificaion beween a produc, a producer, and a firm in our model. For consisency wih much lieraure, we rouinely use he word firm o refer o an individual producive uni. Bu he laer is bes hough of as a producion line associaed o he specific good i produces, poenially inroduced a a pre-exising, muli-produc firm whose managers are each in charge of a specific producion line. The seup of our model is such ha we can remain agnosic as o where he boundaries of firms are. 6 When we augmen he model o include physical capial in producion of exising goods and creaion of new producion lines, he model does beer han he sandard RBC framework a maching volailiy and persisence of U.S. GDP. However, a high rae of capial depreciaion is required for local deerminacy and sabiliy. 2

4 consumer price index varies wih he number of varieies (i is cheaper o saisfy a given level of demand wih more varieies) for given producer price level. Oherwise pu, he relaive price of each good is increasing wih he number of varieies. We show ha, when price rigidiy concerns price seing for individual goods, opimal policy should sabilize producer prices (he average price of oupu) raher han he welfare-consisen consumer price index. The issue of wha inflaion rae should be argeed by policy is also relaed o an empirically relevan measuremen problem ha occurs because CPI daa do no accoun for he inroducion of new goods in he welfare-consisen manner prescribed by he model. As a consequence, he observed CPI is a biased measure of he welfare-based cos-of-living index, as documened by a recen and growing lieraure see e.g. Broda and Weinsein (26). Any implemenaion of normaive prescripions needs o address his issue: If he cenral bank were o arge he cos of living as measured by CPI inflaion and here were subsanial bias in is measuremen, ineres rae seing should reflec ha bias (his poin is made for example by Broda, 24). However, since we find ha in he presence of endogenous produc creaion (precisely of he form ha would induce a bias in CPI daa) he cenral bank should no aim o sabilize CPI inflaion, he measuremen bias seems o be raher innocuous from a normaive perspecive as long as he cenral bank sabilizes he appropriae price index unlike in Broda (24), who argues for sabilizing CPI inflaion as a policy prescripion. Moreover, we show ha even if he cenral bank were obliviously argeing measured CPI inflaion, he bias would paradoxically work in he righ direcion of inducing he cenral bank o arge an index ha is in fac closer o accouning for producer prices han o welfare-consisen cos of living. Our framework also sugges a new moive for price sabiliy as a desirable policy prescripion. Since, as in Roemberg (982), price adjusmen coss are deduced from firm profis, and hese coss are proporional o (squared) producer price inflaion, he laer acs as a disorionary ax on firm profis in our model. This ax disors he allocaion of resources o produc creaion (versus producion of exising varieies) and induces a subopimal amoun of produc variey in each period. This is an inuiive explanaion for why he cenral bank should pursue producer price sabiliy in our model, and an exra argumen for price sabiliy absen from fixed-variey models. Turning o implicaions of our model ha qualify resuls from fixed-variey models, bu remaining in he area of policy prescripions, i is by now convenional wisdom from he benchmark fixed-variey model wihou physical capial ha he cenral bank should follow wha has become known as he Taylor Principle. This policy prescripion requires ha he cenral bank be acive, in he sense of increasing he nominal ineres rae more han one-o-one in response o increases in 3

5 inflaion. 7 Perhaps surprisingly, however, he inroducion of physical capial in he fixed-variey model changes his prescripion dramaically, as shown by Dupor (2) in a coninuous-ime model and furher developed by Carlsrom and Fuers (25) in discree ime and in he presence of adjusmen coss. Dupor shows ha passive ineres rae seing (a less han proporional response o inflaion) is necessary and sufficien for local deerminacy and sabiliy, while Carlsrom and Fuers conclude ha i is essenially impossible o achieve deerminacy wih forward-looking ineres rae seing. In conras o hese resuls, he Taylor Principle holds in our economy in which capial accumulaion akes he form of creaing new producion lines, regardless of wheher he moneary auhoriy responds o expeced or curren producer price inflaion. 8 The Taylor Principle is resored wih our form of capial accumulaion precisely because our framework feaures an endogenous price of capial ha plays a crucial role in moneary policy ransmission. Indeed, we show ha free enry implies ha he price of equiy shares (he value of he firm) appears in he New Keynesian Phillips curve ha governs he dynamics of inflaion. Moreover, a no-arbirage condiion links he real reurn on bonds (which he cenral bank affecs by seing he nominal ineres rae) o he real reurn on equiy he raio of nex period s dividends and share price o he curren price of equiy. This idenifies a novel channel of moneary policy ransmission ha links ineres rae seing o equiy prices and, hrough he free enry condiion, inflaion. In a nushell, a emporary ineres rae cu reduces he real reurn on bonds, inducing he household o consume more oday and he price of equiy o rise oday relaive o omorrow. Since he price of equiy (he value of he firm) is relaed o marginal cos by he free enry condiion in our model, marginal cos also rises, inducing a fall in he markup and, by he Phillips curve, an increase in inflaion. This ransmission of moneary policy hrough he price of equiy is absen in sandard, fixed-variey models even when hose models do feaure an endogenous price of capial due o adjusmen coss (see Carlsrom and Fuers, 25). The validiy of he Taylor Principle wih curren or forward-looking ineres rae seing is an imporan resul (in he ligh of findings by Dupor and Carlsrom and Fuers), which could be inerpreed as a relaive advanage of modeling capial accumulaion as invesmen oward he creaion of new produc lines associaed o he inroducion of new varieies (he exensive margin), raher han reproducing he same good (inensive margin). 7 Kerr and King (996) and Clarida, Galí, and Gerler (2) were he firs o derive his resul in he now sandard New Keynesian framework. Leeper (99) has a relaed discussion. 8 The same holds for welfare-consisen CPI inflaion, subjec o he cavea implied by our normaive analysis ha moneary policy should no arge welfare-consisen consumer prices in our model. 4

6 Furher implicaions of explicily modeling endogenous produc creaion perain o inflaion and markup dynamics. As in he sandard fixed-variey model, a New Keynesian Phillips curve relaing producer price inflaion o is expeced value and he curren markup holds in our model. However, endogenous produc creaion has imporan consequences for empirical exercises ha esimae Phillips curves. Firs, in he presence of endogenous variey, he markup is no simply he inverse of he labor share of income, as in Sbordone (22) or Galí and Gerler (999). In our model, he markup can be expressed as he inverse of a labor share in consumpion oupu, conrolling for labor used o se up new producion lines (labor which is overhead from an aggregae perspecive). A close proxy for his labor share has been esimaed by Roemberg and Woodford (999), and iisherelevanvariablehashouldbeusedo esimae he Phillips curve in he presence of endogenous variey. Moreover, we propose an alernaive proxy for he markup based on he inverseofheshareofprofis in consumpion, which is model-free, in he sense ha i could be used regardless of one s sand on produc creaion. Furhermore, we idenify an endogeneiy bias in he idenificaion of wha he lieraure commonly labels cos-push shocks (see e.g. Clarida, Galí, and Gerler, 999): In he presence of endogenous variey, he Phillips curve feaures an exra erm ha depends on he number of available varieies. This erm would be aribued o cos-push shocks by a researcher using a markup proxy ha does no accoun for variey when esimaing he Phillips curve. Finally, i has been poined ou (e.g. Fuhrer and Moore, 995) ha one of he main drawbacks of he forward-looking New Keynesian Phillips curve is is failure o generae endogenous inflaion persisence. We show ha our version of he Phillips curve can poenially alleviae his problem, because he number of varieies feaured in he Phillips curve is a sae variable, and hence i induces exra persisence in inflaion. Numerical examples show ha he responses o aggregae produciviy and deregulaion shocks under simple, bu plausible specificaions of ineres rae seing are very close o he flexibleprice responses. Exogenous ineres rae cus induce he economy o expand, bu reduce enry because of he unfavorable effec of inflaion on firm profis. Wih produciviy shocks as he source of flucuaions and an empirically plausible, simple rule for ineres rae seing involving ineres rae smoohing and a response o expeced producer price inflaion, he cyclical properies of endogenous variables are very close o hose of he flexible-price counerpar and, in urn, o hose of he benchmark RBC model, as documened by BGM. In conras o he flexible-price model wih ranslog preferences sudied in BGM, sicky prices wih C.E.S. preferences yield oo much markup counercyclicaliy and a highly counerfacual ime profile of his cyclicaliy. This 5

7 happens because he markup is no longer ied o he number of producers as in BGM-ranslog. On he brigh side, aggregae profis remain procyclical (consisen wih sylized facs) even in he presence of a very counercyclical markup, and he model remains able o explain he procyclicaliy of business creaion. Producer enry and produc creaion pose an ineresing quesion for he modeling of nominal rigidiy. In our model, a new enran in period sars producing (and seing prices) only in period +, capuring a realisic ime-o-build lag in he developmen of new producion lines. When a new enran makes is firs price seing decision, we mus ake a sand on wheher i operaes as all pre-exising producers do, subjec o he same nominal rigidiy hus preserving he symmery across producers ha is a feaure of he Roemberg (982) model, or wheher i ses is price in flexible fashion, bu knowing ha i will face a cos of adjusing is price in all subsequen periods. We begin our analysis by assuming ha new enrans inheri he same price rigidiy as pre-exising firms. This considerably simplifies he model and allows us o obain an iniial se of analyical and numerical resuls. We hen urn o he model in which new enrans se prices in flexible fashion, bu knowing ha hey will be subjec o a cos of price adjusmen from he following period on. 9 In his case, nominal rigidiy resuls in heerogeneiy in price levels across cohors of producers ha enered he economy a differen poins in ime, and he aggregae degree of nominal rigidiy is endogenous: Expansions are associaed wih lower aggregae rigidiy because he number of new enrans whose decision is no influenced by pas price seing increases. We show ha he log-linear version of his exended model can sill be solved in racable fashion, and we explore he consequences of endogeneiy in aggregae rigidiy by means of numerical examples. Plausible parameer values imply responses o shocks ha are virually indisinguishable from hose of he benchmark model. Since we assume ha average produc urnover is realisically small a quarerly frequency, small changes in he fracion of firms ha se prices in more flexible fashion riggered by shocks have negligible aggregae consequences, and he benchmark model in which new enrans inheri he same price adjusmen cos as incumbens yields robus conclusions. Finally, we explore he consequences of non-c.e.s. preferences by replacing he familiar Dixi- Sigliz (977) variey aggregaor wih a general, homoheic specificaion of symmeric preferences paramerized in ranslog form for model soluion purposes. This implies ha he elasiciy of subsiuion across producs increases wih he number of producers, inroducing an addiional 9 For compleeness of comparison, we also consider a version of he model in which new price seers simply se heir iniial price as a consan markup over marginal cos. 6

8 effec of he number of available goods on inflaion in he New Keynesian Phillips curve. In our numerical examples, his exension yields conclusions ha are similar o hose of he benchmark model, alhough i furher improves he performance of he model on he inflaion persisence fron. Lewis (26) and Mancini-Griffoli and Elkhoury (26) develop models wih nominal rigidiy ha are closes o he one sudied here. Lewis inroduces monopoly power in he labor marke and sicky wages in familiar Calvo (983)-Yun (996) fashion ino he BGM model. She documens VAR evidence ha moneary policy expansions resul in increased firm enry by boosing aggregae demand, and she shows ha he sicky-wage model reproduces his evidence. Mancini-Griffoli and Elkhoury assume ha enry coss in he BGM model ake he form of fees paid o lawyers wih monopoly power. Under nominal rigidiy, he lawyers se he enry fees in Calvo-Yun fashion and, as in Lewis, a moneary expansion ha booss he economy resuls in increased firm enry. The posiive effec of inflaionary shocks on firm enry in hese models follows from he assumpion ha inflaion has no direc, disorionary effec on enry decisions. Bergin and Corsei (25) documen VAR evidence on he consequences of exogenous changes in moneary policy for enry similar o ha in Lewis paper. They se up a model wih enry and one-period price rigidiy ha replicaes his evidence, and hey characerize opimal moneary policy and he properies of shock ransmission in heir model. As in Lewis and Mancini-Griffoli and Elkhoury s models, Bergin and Corsei s resul ha moneary expansions induce increased firm enry sems from a modeling of nominal rigidiy ha implies no direc coss of inflaion o he firms, oher han he inefficiency of pre-se prices. In our seup wih coss of price adjusmen, flexible wages, and exogenous enry coss, inflaion axes firm profis, and an exogenous decrease in he ineres rae generally reduces producer enry., The res of he paper is organized as follows. Secion 2 presens our benchmark model. Secion 3 obains he resuls on opimal moneary policy in he benchmark seup. Secion 4 discusses he implicaions of endogenous enry and produc variey for he New Keynesian Phillips curve. Secion 5 sudies moneary policy hrough ineres rae seing in our model. Secion 6 illusraes he business cycle properies of he model. Secion 7 discusses he main resuls of he exensions we explore: he alernaive assumpion on iniial price seing by new enrans and non-c.e.s. Depending on he response of policy o endogenous variables, his negaive effec of an inflaionary shock can be limied o he impac period if labor supply is sufficienly elasic, bu a negaive conemporaneous effec is a robus feaure of he scenarios we consider. The models in Lewis (26) and Mancini-Griffoli and Elkhoury (26) are subjec o one of he main problems ha our approach aims o address: They rely on monopoly power as a sepping sone for nominal rigidiy, bu hey absrac from enry (by workers or lawyers) in he presence of monopoly profis. 7

9 preferences. Secion 8 concludes. 2 The Model Household Preferences and he Inraemporal Consumpion Choice We consider a cashless economy as in Woodford (23). The economy is populaed by a uni mass of aomisic, idenical households. The represenaive household supplies L hours of work in each period in a compeiive labor marke for he nominal wage rae W and maximizes P expeced ineremporal uiliy E s= βs U (C s,l s ),wherec is consumpion and β (, ) he subjecive discoun facor. The period uiliy funcion akes he form U (C,L )=lnc χ (L ) +/ϕ / ( + /ϕ), χ>, whereϕ is he Frisch elasiciy of labor supply o wages, and he ineremporal elasiciy of subsiuion in labor supply. A ime, he household consumes he baske of goods C,defined over a coninuum of goods Ω: ³ R C = ω Ω c (ω) θ /θ θ/(θ ) dω, where θ>is he symmeric elasiciy of subsiuion across goods. A any given ime, only a subse of goods Ω Ω is available. Le p (ω) denoe he nominal price of a good ω Ω. The consumpion-based price index for he home economy is ³ R hen P = p ω Ω (ω) θ /( θ) dω, and he household s demand for each individual good ω is c (ω) =(p (ω) /P ) θ C. Firms There is a coninuum of monopolisically compeiive firms, each producing a differen variey ω Ω. Producion requires only one facor, labor. Aggregae labor produciviy is indexed by Z, which represens he effeciveness of one uni of labor. Produciviy is exogenous and follows an AR() process in percen deviaion from is seady-sae level. Oupu supplied by firm ω is y (ω) =Z l (ω), wherel (ω) is he firm s labor demand for producive purposes. The uni cos of producion, in unis of he consumpion good C,isw /Z,wherew W /P is he real wage. Prior o enry, firms face a sunk enry cos of f E, effecive labor unis, equal o w f E, /Z unis of he consumpion good. There are no fixed producion coss. Hence, all firms ha ener he economy produce in every period, unil hey are hi wih a deah shock, which occurs wih probabiliy δ (, ) in every period. We assume ha he enry cos f E, is exogenous and rea changes in f E, as changes in marke regulaion. Firms face nominal rigidiy in he form of a quadraic cos of adjusing prices over ime (Roem- 8

10 berg, 982). Specifically, he real cos (in unis of he composie baske) of oupu-price inflaion volailiy around a seady-sae level of inflaion equal o facing firm ω is: PAC (ω) κ 2 µ p (ω) 2 p (ω) p (ω) y D (ω). P This expression is inerpreed as he amoun of markeing maerials ha he firm mus purchase when implemening a price change. We assume ha his baske has he same composiion as he consumpion baske. The cos of adjusing prices is proporional o he real revenue from oupu sales, (p (ω) /P ) y D (ω), wherey D (ω) is firm ω s oupu demand. Firms face demand for heir oupu from consumers and firms hemselves when hey change prices. In each period, here is a mass N of firms producing and seing prices in he economy. When a new firm ses he price of is oupu for he firs ime, we appeal o symmery across firms and inerpre he price in he expression of he price adjusmen cos for ha firm as he noional price ha he firm would have se a ime if i had been producing in ha period. An inuiion for his simplifying assumpion is ha all firms (even hose ha are seing he price for he firs ime) mus buy he bundle of goods PAC (ω) when implemening a price decision. 2 I should be noed, however, ha his assumpion is enirely consisen boh wih he original Roemberg (982) seup and wih our iming assumpion. Specifically, new enrans behave as he (consan number of) price-seers do in Roemberg s framework, where an iniial condiion for he individual price is dicaed by naure. In our framework, new enrans a any ime who sar producing and seing prices a +are subjec o precisely he same assumpion as price seers in Roemberg s original seup. Moreover, he assumpion ha a new enran, a he ime of is firs price seing decision, knows he average marke price las period is consisen wih he iming assumpion ha an enran sars producing only one period afer enry, hence being able o learn he average marke price during he enry period. The oal demand for he oupu of firm ω is hus y D µ p (ω) θ (ω) (C + PAC ), P where PAC N PAC (ω), and we used symmery across firms in he definiion of he aggregae demand of he consumpion baske for price adjusmen purposes PAC. Le ρ (ω) p (ω) /P denoe he real price if firm ω s oupu. Then, firm ω s profi inperiod 2 We relax his assumpion below. 9

11 (disribued o households as dividend) can be wrien as d (ω) =ρ (ω) y D (ω) w l (ω) κ 2 µ p (ω) 2 p (ω) ρ (ω) y D (ω). The real value of he firm a ime (in unis of consumpion) is he expeced presen discouned value of fuure profis from + on, discouned wih he household s sochasic discoun facor (see below): v (ω) =E X s=+ Λ,s d s (ω), () where Λ,s [β ( δ)] s U C (C s,l s ) /U C (C,L ) is he discoun facor applied by households o fuure profis from firm ω (which faces a probabiliy δ of being hi wih he deah shock in each period). A ime, firm ω chooses l (ω) and p (ω) o maximize d (ω)+v (ω) subjec o y (ω) =y D (ω), aking w, P, C, PAC,andZ as given. Leing λ (ω) denoe he Lagrange muliplier on he consrain y (ω) =y D (ω), hefirs-order condiion wih respec o l (ω) yields: λ (ω) = w Z. The shadow value of an exra uni of oupu is simply he firm s marginal cos, common across all firms in he economy. The firs-order condiion wih respec o p (ω) yields: p (ω) =μ (ω) P λ (ω). Firm ω ses he price as a markup (μ (ω)) over nominal marginal cos, where he markup μ (ω) is given by μ (ω) Υ y (ω) (θ ) y (ω) κ 2 θy (ω) p µ (ω) p (ω) p (ω) p (ω) ³ p(ω) p (ω) 2 + κυ, " E Λ,+ y + (ω) P P + µ p+ (ω) 2 µ p+ (ω) #. p (ω) p (ω) As expeced, he markup reduces o θ/ (θ ) in he absence of nominal rigidiy (κ =)orifhe price p (ω) is consan.

12 Firm Enry and Exi In every period, here is an unbounded mass of prospecive enrans. These enrans are forward looking, and correcly anicipae heir fuure expeced profis d (ω) in every period as well as he probabiliy δ (in every period) of incurring he exi-inducing shock. We assume ha enrans a ime only sar producing a ime +, which inroduces a one-period ime-o-build lag in he model. The exogenous exi shock occurs a he very end of he ime period (afer producion and enry). A proporion δ of new enrans will herefore never produce. Prospecive enrans in period compue heir expeced pos-enry value given by he presen discouned value of heir expeced sream of profis v (ω). This also represens he average value of incumben firms afer producion has occurred (since boh new enrans and incumbens hen face he same probabiliy δ of survival and producion in he subsequen period). Enry occurs unil firm value is equalized wih he enry cos, leading o he free enry condiion v (ω) =w f E, /Z. This condiion holds so long as he mass N E, of enrans is posiive. We assume ha macroeconomic shocks are small enough for his condiion o hold in every period. Finally, he iming of enry and producion we have assumed implies ha he number of producing firms during period is given by N =( δ)(n + N E, ). Symmeric Firm Equilibrium In equilibrium, all firms make idenical choices. Hence, λ (ω) =λ, p (ω) =p, μ (ω) =μ, ρ (ω) =ρ, l (ω) =l, y (ω) =y, d (ω) =d,andv (ω) =v. The aggregae oupu of he consumpion baske (used for consumpion and o pay price adjusmen coss) is Y C C + PAC = N ρ y = N ρ Z l. The expression of he price index P implies ha he relaive price ρ and he number of producing firms N are ied by he variey effec equaion ρ = p /P =(N ) θ. Le π denoe inflaion in producer prices: π p /p. Then, we can wrie: θ μ = h (θ ) κ 2 (π ) 2i n h + κ ( + π ) π β ( δ) E C C + N Y+ C N + Y C ( + π + ) π + io. h i This can be simplified furher by noing ha PAC = κ (π ) 2 Y C /2,sohaC = κ (π ) 2 /2 Y C,

13 o obain: μ = h (θ ) κ 2 (π ) 2i + κ θ ½( + π ) π β ( δ) E κ 2 (π)2 κ 2 (π +) 2 (2) N N + ( + π + ) π + ¾. Log-linearizaion of his equaion delivers our model s New Keynesian Phillips curve incorporaing he effec of endogenous produc variey, which we discuss in deail in Secion 4. Household Budge Consrain, Saving, and Labor Supply Households hold wo ypes of asses: shares in a muual fund of firms and bonds. Le x be he share in he muual fund of firms held by he represenaive household enering period. The muual fund pays a oal profi in each period (in unis of currency) ha is equal o he oal profi of all firms ha produce in ha period, P N d.duringperiod, he represenaive household buys x + shares in a muual fund of N H, N + N E, firms (hose already operaing a ime and he new enrans). Only N + =( δ) N H, firms will produce and pay dividends a ime +.Since he household does no know which firms will be hi by he exogenous exi shock δ a he very end of period, ifinances he coninuing operaion of all pre-exising firms and all new enrans during period. The dae price of a claim o he fuure profi sream of he muual fund of N H, firms is equal o he average nominal price of claims o fuure profis of home firms, V P v. The household eners period wih nominal bond holdings B N, and muual fund share holdings x. I receives gross ineres income on bond holdings, dividend income on muual fund share holdings and he value of selling is iniial share posiion, and labor income. The household allocaes hese resources beween purchases of bonds and shares o be carried ino nex period and consumpion. The period budge consrain (in unis of currency) is: B N,+ + V N H, x + + P C =(+i )B N, +(D + V ) N x + +τ L W L + T L, where i denoes he nominal ineres rae on holdings of bonds beween and, D denoes nominal dividends (D P d ), τ L lump-sum ax saisfying he consrain T L is a labor subsidy whose role we discuss below, and T L is a = τ L W L in equilibrium. Dividing boh sides by P and denoing holdings of bonds in unis of consumpion wih B + B N,+ /P,wecanwrie B + + v N H, x + + C =(+r )B +(d + v ) N x + +τ L w L + L, (3) 2

14 where + r is he gross, consumpion-based, real ineres rae on holdings of bonds beween and, defined by +r ( + i ) / +π C,wihπ C P /P, and L T L /P. The home household maximizes is expeced ineremporal uiliy subjec o his budge consrain. The Euler equaions for bond and share holdings are: (C ) = βe " +i +π C + # (C + ) and " µc+ v = β ( δ) E (v + + d +)#. C As expeced, forward ieraion of he equaion for share holdings and absence of speculaive bubbles yield he asse price soluion in equaion (). 3 The firs-order condiion for he opimal choice of labor effor requires ha he marginal disuiliy of labor be equal o he marginal uiliy from consuming he real wage received for an addiional uni of labor: χ (L ) ϕ Aggregae Accouning and Equilibrium = +τ L w C. Aggregaing he budge consrain (3) across households and imposing he equilibrium condiions B + = B =and x + = x =,, yields he aggregae accouning ideniy Y C + N E, v = w L + N d, where we defined GDP, Y : Consumpion plus invesmen (in new firms) mus be equal o income (labor income plus dividend income). Labor marke equilibrium requires N l + N E, f E, /Z = L :Theoalamounoflaborused in producion and o se up he new enrans plans mus equal aggregae labor supply. course, his condiion is redundan once equilibrium in goods and asse markes is imposed.) The equilibrium condiions of our benchmark model are summarized in Table. 3 We omi he ransversaliy condiions for bonds and shares ha mus be saisfied o ensure opimaliy. (Of 3

15 Table. Benchmark Model, Summary Pricing ρ = μ w Z θ Markup μ = (θ )[ κ 2 (π ) 2 ]+κ Variey effec ρ =(N ) θ Profis d = ³ 2 κ μ 2 (π Y ) C N Free enry v = w f E, Z Number of firms N =( δ)(n + N E, ) Inraemporal opimaliy χ (L ) ϕ = +τ L Euler equaion (shares) w C (+π )π β( δ)e κ 2 (π ) 2 v = β ( δ) E ³ C+ C (v+ + d + ) h Euler equaion (bonds) (C ) = βe +i (C +π+ C + ) i h Oupu of consumpion secor C = κ 2 (π ) 2i Y C Aggregae accouning CPI inflaion C + N E, v = w L + N d +π +π C = ρ ρ κ 2 (π +) 2 N (+π N + )π + + The model is closed by specifying a rule for nominal ineres rae seing by he moneary auhoriy, he seing of he labor subsidy τ L, and processes for he exogenous enry cos f E, and produciviy Z. 3 Price Sabiliy wih Endogenous Enry and Produc Variey The flexible-price analysis of BGM leaves inflaion in consumer and producer prices indeerminae because moneary policy has no real effec in he flexible-price, cashless economy, and we need no concern ourselves wih he pahs of nominal variables in order o solve for real ones. When prices are sicky, his is no longer he case. Specifically, given a change in he number of producers N and he associaed movemen in he relaive price ρ implied by he variey effec equaion ρ = p /P =(N ) θ, he allocaion of his relaive price movemen o changes in producer or consumer prices is imporan for he dynamics of real variables and welfare. In urn, producer price inflaion is a deerminan of firm enry and hus N via is impac on firm profis. This secion sudies opimal moneary policy in our model and he opimal allocaion of variey effecs o producer versus consumer prices. 4

16 Our analysis of opimal moneary policy builds on resuls in Bilbiie, Ghironi, and Meliz (26). We show here ha he flexible-price version of he economy described above is efficien he compeiive equilibrium coincides wih he social planner s opimum if labor supply is inelasic (ϕ =)andl =. The reason is ha, wih C.E.S. Dixi-Sigliz preferences, he profi desrucion exernaliy generaed by producer enry (which reduces demand for each individual firm) is exacly mached by he consumer s love for variey boh deermined by he elasiciy of subsiuion θ. The flexible-price economy is inefficien if ϕ> because here is a misalignmen of markups across he iems he consumer cares abou (consumpion, priced a a markup over marginal cos, and leisure, priced compeiively), bu efficiency is resored if he labor subsidy τ L is equal o he ne markup of pricing over marginal cos, / (θ ) in all periods. This subsidy aligns markups across consumpion goods and leisure while preserving he expeced profiabiliy of firm enry, hus inducing he efficien equilibrium. We assume ha τ L =/ (θ ) below. Sicky prices imply a ime-varying markup whenever producer prices are changing over ime. As shown in Bilbiie, Ghironi, and Meliz (26), markup non-synchronizaion across periods (as well as across saes and argumens of he uiliy funcion) generaes inefficiency compared o he planner s opimum. Since in his paricular model ime variaion of he markup in he compeiive equilibrium is due o producer price inflaion, we expec a zero rae of inflaion in producer prices o be he opimal moneary policy chosen by a planner. The following proposiion confirms ha his is indeed he case. To isolae our main resul, we prove he proposiion for he case of inelasic labor and hen briefly discuss he elasic labor case. Proposiion The opimal rae of producer price inflaion π chosen by a social planner is zero. Proof. See he Appendix. The inuiion for Proposiion is sraighforward: Producer price inflaion acs as a ax on firm profis in our model. I disors firm enry decisions and he allocaion of labor o creaion of new firms versus producion of exising goods, resuling in subopimal consumpion and lower welfare. Opimal policy, herefore, aims o sabilize producer price inflaion a zero. Imporanly, however, while producer prices mus be sabilized, he opimal rae of consumer price inflaion mus move freely o accommodae changes in he number of varieies: µ ρ +π C µ = N θ ρ =, N where a sar denoes variables in he efficien equilibrium. Given he evidence of bias in he 5

17 measuremen of CPI inflaion (precisely due o poor accouning for new varieies) convincingly documened by Broda and Weinsein (26a), we view his normaive implicaion of our model as good news. The cenral bank should arge inflaion in producer prices raher han (mismeasured) CPI inflaion. When labor supply is elasic, he subsidy τ L =/ (θ ) ensures ha he flexible-price equilibrium is efficien, removing he wedge oherwise presen beween he marginal raes of subsiuion and ransformaion beween consumpion and leisure. In his case, price sickiness disors boh he oal amoun of labor supplied and is allocaion o creaion of new firms and producion of exising goods. I is easy o verify ha a zero rae of inflaion in producer prices is sill he opimal moneary policy. The opimaliy of producer price sabiliy wih inelasic labor supply highlighs a new argumen for price sabiliy (a he producer level) implied by endogenous enry and produc variey. In a model wih exogenously fixed number of firms and inelasic labor supply, ime variaion in he markup would have no impac on he equilibrium pah of consumpion and welfare: Consumpion would be simply deermined by he exogenous produciviy and labor supply regardless of markup dynamics. Endogenous enry and produc variey imply ha markup variaion reduces welfare by disoring enry decisions and he allocaion of he fixed amoun of labor o firm creaion versus producion of exising goods. This inroduces a role for moneary policy in welfare maximizaion by sabilizing producer price inflaion a zero and he markup a is flexible-price level. We discuss implemenaion of he opimal moneary policy by seing he nominal ineres rae below. 4 The New Keynesian Phillips Curve and he Log-Linear Model This secion describes he implicaions of endogenous enry and produc variey for he New Keynesian Phillips curve and presens he key log-linear equaions of he model. The New Keynesian Phillips Curve To sudy he propagaion of shocks and compue second momens of he endogenous variables implied by assumpions on he processes for exogenous shocks, we log-linearize he model around he efficien seady sae wih zero inflaion under assumpions of log-normaliy and homoskedasiciy, and denoe percen deviaions from seady sae wih sans serif fons. Our model s version of he 6

18 New Keynesian Phillips curve follows from log-linearizing equaion (2): π = β ( δ) E π + θ κ μ, (4) where π and μ now denoe percen deviaions from seady sae (of gross inflaion in he case of π ). Since ρ = p /P =(N ) θ ha μ =(N ) θ Z /w, or, in log-linear erms: and opimal firm pricing implies μ = ρ /λ = ρ Z /w,ifollows μ = θ N (w Z ). (5) (Wih a consan number of firms, his relaion reduces o he familiar negaive relaion beween markup and marginal cos of he benchmark New Keynesian model.) Subsiuing (5) ino (4) yields: π = β ( δ) E π + + θ κ (w Z ) κ N. (6) Equaion (6) is a New Keynesian Phillips curve relaion ha ies firm-level inflaion dynamics o marginal cos in a sandard fashion. Imporanly, he effec of marginal cos is adjused o reflec he number of producers ha operae in he economy. This is a predeermined, sae variable, which inroduces direcly a degree of endogenous persisence in he dynamics of firm-level inflaion in he Phillips curve. Furhermore, our model links he dynamics of inflaion o asse prices in an endogenous way, as can be seen by combining (6) wih he log-linear free enry condiion o obain: π = β ( δ) E π + + θ κ (v f E, ) κ N. (7) This equaion ies inflaion dynamics o he relaive price of invesmen in new firms. I sipulaes ha, for given expeced inflaion and number of firms, inflaion is posiively relaed o equiy prices. Togeher wih he no-arbirage condiion beween bonds and equiy implied by opimal household behavior, his connecion beween inflaion and equiy prices (and hus capial accumulaion in our model) plays a crucial role for he deerminacy and sabiliy properies of ineres rae seing ha we discuss below. Finally, using he definiion of CPI inflaion, we can wrie he New Keynesian Phillips curve 7

19 for consumpion-based inflaion: π C = β ( δ) E π C + + θ κ (w Z ) κ N θ [N N β ( δ)(n + N )]. (8) Consumpion-based inflaion displays an addiional degree of endogenous persisence relaive o firm-level inflaion in ha i depends direcly on he number of firms ha produced a ime, which was deermined in period 2. Implicaions for Empirical Exercises Exising empirical sudies dealing wih esimaion of he New Keynesian Phillips curve (4), such as Sbordone (998) and Galí and Gerler (999), proxy he (unobservable) markup variable wih he inverse of he labor share. This is an approximaion ha holds exacly in a model wihou endogenous variey. In our model wih endogenous variey, however, his relaionship no longer holds. Indeed, if one believes produc variey o be imporan for business cycles, he proxy for he markup ha one should use is he inverse of he share (in consumpion oupu) of labor beyond he overhead quaniy (from an aggregae perspecive) used o se up new produc lines, μ = Y C / [w (L L E, )]. This markup measure corresponds closely o he labor share measure used by Roemberg and Woodford (999) ha akes ino accoun overhead labor (repored in column 2 of heir Table 2, page 66), which we reproduce in Figure 4 below. Log-linearizaion of his equaion, when replaced ino (4), delivers a relaionship ha is esable empirically. Alernaively, exploiing he equaion for profis, one could use as a proxy for markups as he inverse of (one minus) he profi share,μ = D G /Y C, where D G d N + κ 2 (π ) 2 Y C are profis gross of he coss of price adjusmen. Noe ha since when log-linearizing around a zero-inflaion seady-sae, hese coss are zero (and hence consumpion is equal o consumpion oupu and gross profis are equal o ne profis), he empirically usable equaion will feaure only observable variables, i.e., consumpion and oal profi receips (or dividends). 4 A furher implicaion of our framework for empirical exercises comes from he naural disincion beween consumer and producer price inflaion in our model: Our framework implies ha, in order o overcome measuremen issues inheren in using CPI inflaion, empirical sudies of he Phillips curve should concenrae on producer price inflaion (which is also he relevan objecive for moneary policy). Consrucion of CPI daa by saisical agencies does no adjus for availabil- 4 We leave esimaion of Phillips curve using hese alernaive proxies for he markup for fuure research. 8

20 iy of new varieies in he specific funcional form dicaed by he welfare-consisen price index. Furhermore, adjusmen for variey, when i happens, cerainly does no happen a he frequency represened by periods in our model. Acual CPI daa are closer o p (he average price level in our economy) han P. For his reason, when invesigaing he properies of he model in relaion o he daa (for insance, when compuing second momens below or in he specificaion of policy rules ha allow for reacion o measured real quaniies), one should focus on real variables deflaed by a daa-consisen price index. For any variable X in unis of he consumpion baske, such daa-consisen counerpar is obained as X R, P X /p = X /ρ. Relaed o his measuremen issue, our framework implies an endogeneiy bias in cos-push shocks in much empirical lieraure on he New Keynesian Phillips curve. An endogenous erm ha depends on N (due he measuremen bias from no accouning for variey) is aribued o exogenous cos-push shocks when esimaing he Phillips curve equaion (6) using a proxy for marginal cos wihou variey. When he variey effec is removed from he welfare-consisen equiy price, he Phillips curve (7) becomes: π = β ( δ) E π + + θ κ (v R, f E, ), (9) where v R, is he value of he firm/price of shares ne of he variey effec. For given expecaions of fuure inflaion, acual inflaion is increasing in he daa-consisen price of equiy. The Log-Linear Model The log-linear counerpars of he equaions in Table are in he appendix. The sysem can be reduced o he following equaions (plus he New Keynesian Phillips curve (4)): N + =[+r + ψ] N (r + δ + ψ)(θ ) C ψ (θ ) μ +((r + δ + ψ)(θ ) + δ) Z δf E,, C = δ + δ +r E C + δ +r r + δ (θ ) +r () +r θ r + δ N + + +r θ N () μ + μ δ +r E f E,+ + f E,, E C + = C + i E π + + θ N + θ N, (2) 9

21 wherewedefined ψ ϕ [(r + δ)(θ ) + δ] / (θ ), which is zero when labor supply is inelasic. The model is closed by specifying he conduc of moneary policy (via he seing of he nominal ineres rae i ) over he business cycle, which we discuss below. 5 Moneary Policy over he Business Cycle In his secion, we discuss deerminacy and sabiliy properies of simple rules for nominal ineres rae seing over he business cycle and he implemenaion of he opimal policy of producer price sabiliy. Simple Policy Rules We consider he following class of simple inflaion-argeing rules for ineres rae seing: i = τ i i + τe π +s + τ C E π C +s + ξ i, >τ i, τ, τ C, s=,. (3) where ξ i is an i.i.d. shock capuring he non-sysemaic componen of moneary policy. We assume ha τ C =when τ>and vice versa, resricing he cenral bank o reacing o eiher producer or consumer price inflaion. For he reasons we discussed above, a response o welfare-based CPI inflaion is clearly subopimal (and no feasible in realiy due o he measuremen problems we menioned). In considering his scenario below, we absrac from he clear normaive prescripion of no argeing welfare-based CPI inflaion and from measuremen issues; raher, we ask he quesion: Wha would he response of he economy o various shocks be if he cenral bank could monior movemens in welfare-consisen CPI inflaion and followed a rule involving he laer? Deerminacy and Sabiliy In his secion we sudy he deerminacy and sabiliy properies of our model under differen moneary policy rules. To analyze local deerminacy and sabiliy of he raional expecaion equilibrium, we can focus on he perfec foresigh version of he sysem formed by (4), (), () and he equaion obained by subsiuing he moneary policy rule (3) ino he Euler equaion for bonds (2). To begin wih, consider he simple rule in which he cenral bank is responding o expeced producer price inflaion wih no smoohing: i = τe π +. The following Proposiion esablishes ha he Taylor Principle holds in our model economy for all plausible combinaions of parameer values. 2

22 Proposiion 2 Le γ [ β ( δ)] / [β ( δ)]. Assume ha ϕ =,andβ, δ, andθ are such ha γ (θ ) >, β>/2, θ>2, andτ< τ =(κ + θ ) / (θ ). Thenτ> is necessary and sufficien for local deerminacy and sabiliy. Proof. See he Appendix. We remark ha he parameer resricions in he Proposiion 2 are sufficien condiions for he Taylor Principle o hold, and hey are exremely weak condiions. For insance, he values of κ and θ ha we consider below (κ =77and θ =3.8) imply τ =28.5: Thesufficien condiion τ<28.5 is saisfied by any realisic paramerizaion of ineres rae seing. Moreover, while we canno prove i analyically, we verify numerically ha deerminacy and sabiliy hold for values of τ well above he hreshold τ for he parameer values we consider. Validiy of he Taylor Principle is an imporan resul given he debae on he Taylor Principle in models wih physical capial accumulaion. Dupor (2) shows ha passive ineres rae seing (τ <) is necessary and sufficien for local deerminacy and sabiliy in a coninuous-ime model wih physical capial. Carlsrom and Fuers (25) sudy he issue in a discree-ime model wih capial and conclude ha i is essenially impossible o achieve deerminacy wih forward-looking ineres rae seing. Our resul shows ha he sandard Taylor Principle is resored when capial accumulaion akes he form of he endogenous creaion of new producion lines. Inuiion: The Role of Asse Prices in Moneary Policy Transmission Since he validiy of he Taylor principle in our seup is in sriking conras o resuls of models wih radiional physical capial, an inuiive explanaion of his difference is in order. Indeed, he mechanism for his resul in our model is cenered precisely on he role of he endogenous price of equiy he value of he firm in our New Keynesian model wih free enry. As we anicipaed, he explanaion relies on one hand on he Phillips curve (9) above ha relaes inflaion and asse prices (ne of he variey effec) v R, and on he oher hand on he no-arbirage condiion implied by he Euler equaions for bonds and shares. This condiion can be wrien as: i E π + = v R, + δ +r E v R,+ + r + δ +r E d R,+. (4) Focus firs on he policy rule sudied in Proposiion 2, where he relevan inflaion objecive is expeced PPI inflaion, and run he following menal experimen. Suppose ha a sunspo shock unrelaed o any fundamenals his he economy, and ha (wihou losing generaliy) i is locaed 2