Inflation and Economic Growth in a Schumpeterian Model with Endogenous Entry of Heterogeneous Firms

Transcription

1 MPRA Munich Personal RePEc Archive Inflaion and Economic Growh in a Schumpeerian Model wih Endogenous Enry of Heerogeneous Firms Angus C. Chu and Guido Cozzi and Yuichi Furukawa and Chih-Hsing Liao Fudan Universiy, Universiy of S. Gallen, Chukyo Universiy, Chinese Culure Universiy March 2017 Online a hps://mpra.ub.uni-muenchen.de/77543/ MPRA Paper No , posed 16 March :54 UTC

2 Inflaion and Economic Growh in a Schumpeerian Model wih Endogenous Enry of Heerogeneous Firms Angus C. Chu, Fudan Universiy Guido Cozzi, Universiy of S. Gallen Yuichi Furukawa, Chukyo Universiy Chih-Hsing Liao, Chinese Culure Universiy March 2017 Absrac This sudy develops a Schumpeerian growh model wih endogenous enry of heerogeneous firms o analyze he effecs of moneary policy on economic growh via a cash-in-advance consrain on R&D invesmen. Our resuls can be summarized as follows. In he special case of a zero enry cos, an increase in he nominal ineres rae decreases R&D, he arrival rae of innovaions and economic growh as in previous sudies. However, in he general case of a posiive enry cos, an increase in he nominal ineres rae affecs he disribuion of innovaions ha are implemened and would have an invered-u effec on economic growh if he enry cos is suffi cienly large. We also calibrae he model o aggregae daa of he US economy and find ha he growh-maximizing inflaion rae is abou 3%, which is consisen wih recen empirical esimaes. JEL classificaion: O30, O40, E41 Keywords: moneary policy, inflaion, economic growh, heerogeneous firms Chu: angusccc@gmail.com. China Cener for Economic Sudies, School of Economics, Fudan Universiy, Shanghai, China. Cozzi: guido.cozzi@unisg.ch. Deparmen of Economics, Universiy of S. Gallen, S. Gallen, Swizerland. Furukawa: you.furukawa@gmail.com. School of Economics, Chukyo Universiy, Nagoya, Japan. Liao: chihhsingliao@gmail.com. Deparmen of Economics, Chinese Culure Universiy, Taipei, Taiwan. 1

3 1 Inroducion This sudy develops a Schumpeerian growh model wih endogenous enry of heerogeneous firms o analyze he effecs of moneary policy on economic growh. The canonical Schumpeerian growh model in seminal sudies such as Segersrom e al. (1990), Grossman and Helpman (1991) and Aghion and Howi (1992) feaures an idenical sep size of qualiy improvemens across firms. In his sudy, we consider a Schumpeerian model wih random qualiy improvemens as in Minnii e al. (2013) bu wih he addiion of a fixed enry cos o generae endogenous enry of firms wih heerogeneous sep sizes of qualiy improvemens. To incorporae money demand ino his growh-heoreic framework, we impose a cash-in-advance (CIA) consrain on R&D invesmen. Berensen e al. (2012), Chu and Cozzi (2014) and Chu e al. (2015) provide exensive discussion on evidence for he presence of cash requiremens on R&D expendiures. 1 We capure hese cash requiremens using a CIA consrain on R&D. In his moneary growh-heoreic framework, we derive he following resuls. In he special case of a zero enry cos, an increase in he nominal ineres rae decreases R&D, he arrival rae of innovaions and economic growh as in previous sudies, such as Chu and Cozzi (2014) who consider a moneary Schumpeerian growh model wih an idenical sep size of qualiy improvemens, because he disribuion of innovaions ha are implemened is exogenous under a zero enry cos despie random qualiy improvemens. However, in he general case of a posiive enry cos, moneary policy affecs he disribuion of innovaions ha are implemened. Specifically, an increase in he nominal ineres rae decreases R&D and he arrival rae of innovaions, which increases he presen value of fuure profis. The resuling higher value of invenions leads o a lower hreshold of qualiy improvemens above which an innovaion is implemened generaing a posiive effec on economic growh due o more enries. Togeher wih he negaive effec on he arrival rae of innovaions, an increase in he nominal ineres rae would have an invered-u effec on economic growh if he enry cos is suffi cienly large. Because he Fisher equaion gives rise o a posiive longrun relaionship beween he nominal ineres rae and he inflaion rae ha is suppored by empirical sudies such as Mishkin (1992) and Booh and Ciner (2001), our resul also implies an invered-u relaionship beween inflaion and economic growh. This heoreical predicion on an invered-u relaionship beween inflaion and economic growh is suppored by empirical sudies such as Bick (2010) and López-Villavicencio and Mignon (2011). Finally, we calibrae he model o aggregae daa of he US economy o provide a quaniaive analysis and find ha he growh-maximizing inflaion rae is 2.9%, which is close o he empirical esimae in López-Villavicencio and Mignon (2011) who idenify a hreshold inflaion rae of 2.7% for indusrialized counries. 1 For example, early empirical sudies such as Hall (1992) and Opler e al. (1999) find a posiive and significan relaionship beween R&D and cash flows in US firms. More recenly, Baes e al. (2009) documen ha he average cash-o-asses raio in US firms increased subsanially from 1980 o 2006 and argue ha his is parly driven by heir rising R&D expendiures. Brown and Peersen (2011) provide evidence ha firms smooh R&D expendiures by mainaining a buffer sock of liquidiy in he form of cash reserves. Falao and Sim (2014) use firm-level daa in he US o show ha firms cash holdings increase (decrease) significanly in response o a rise (cu) in R&D ax credis. These resuls sugges ha due o financial fricions, firms need o use cash o finance heir R&D invesmen. 2

4 This sudy relaes o he lieraure on innovaion and economic growh. The R&Dbased growh model originaes from Romer (1990), who develops a variey-expanding growh model in which economic growh is driven by he developmen of new producs. Then, Segersrom e al. (1990), Grossman and Helpman (1991) and Aghion and Howi (1992) develop he Schumpeerian qualiy-ladder growh model in which economic growh is driven by he qualiy improvemen of exising producs. For simpliciy, hese sudies assume an idenical sep size for all qualiy improvemens. A recen sudy by Minnii e al. (2013) generalizes he Schumpeerian model by allowing for heerogeneous sep sizes of qualiy improvemens ha are randomly drawn from a disribuion. Our sudy exends he elegan framework of Minnii e al. (2013) by inroducing a fixed enry cos of implemening a developed invenion in order o generae endogenous enries of heerogeneous firms, 2 which urn ou o have imporan implicaions on he effecs of moneary policy. This sudy also relaes o he lieraure on inflaion and innovaion. In his lieraure, Marquis and Reffe (1994) is he seminal sudy ha analyzes he effecs of inflaion on innovaion in he Romer variey-expanding growh model. In conras, we analyze he effecs of inflaion in a Schumpeerian qualiy-ladder model as in Chu and Lai (2013), Chu and Cozzi (2014), Chu e al. (2015) and He and Zou (2016), whose models however feaure an idenical sep size of qualiy improvemens across firms. Subsequen sudies, such as Chu and Ji (2016) and Huang e al. (2015), consider moneary policy in a Schumpeerian growh model wih boh variey expansion and (idenical) qualiy accumulaion across firms. As in Marquis and Reffe (1994), hese sudies predic a monoonic relaionship beween inflaion and economic growh. 3 The presen sudy conribues o his lieraure by allowing for he endogenous enry of firms wih heerogeneous sep sizes of qualiy improvemens, which gives rise o a novel channel hrough which moneary policy affecs innovaion and growh. As a resul, he model generaes an invered-u relaionship beween inflaion and economic growh, which is suppored by recen empirical sudies. The res of his sudy is organized as follows. Secion 2 presens and solves he model. Secion 3 analyzes he effecs of moneary policy. The final secion concludes. 2 A Schumpeerian model wih heerogeneous firms The Schumpeerian qualiy-ladder growh model is based on Grossman and Helpman (1991). We exend heir model by (a) inroducing money demand via a CIA consrain on R&D o analyze moneary policy, (b) considering lab-equipmen innovaion and enry processes ha use final goods (insead of labor) as he inpu, (c) allowing for random qualiy improvemens as in Minnii e al. (2013), and (d) incorporaing a fixed enry cos o generae endogenous enry of heerogeneous firms as in Meliz (2003). In summary, when a firm invens a higher qualiy produc, he sep size of he qualiy incremen is randomly drawn from a Pareo 2 See also Baldwin and Rober-Nicoud (2008), Haruyama and Zhao (2008) and Gusafsson and Segersrom (2010) who adap his fixed enry cos ino he R&D-based growh model, bu hey do no consider random incremens on he qualiy ladder. 3 The relaionship beween he wo variables is usually found o be monoonically negaive, bu some of hese sudies also find ha he relaionship can be monoonically posiive under some condiions. 3

5 disribuion. If and only if he qualiy incremen is suffi cienly large, hen he firm would pay he fixed enry cos o implemen he invenion and ener he marke. 2.1 Household In he economy, here is a represenaive household which has he following lifeime uiliy funcion: U = 0 e ρ ln c d, (1) where he parameer ρ > 0 is he subjecive discoun rae and c denoes consumpion of final goods (numeraire) a ime. The household maximizes uiliy subjec o an asseaccumulaion equaion (expressed in real erms) given by ȧ + ṁ = r a π m + i b + w + τ c. (2) a is he real value of financial asses (in he form of equiy shares in monopolisic inermediae goods firms) owned by he household. r is he real ineres rae. π is he inflaion rae. m is he real money balance accumulaed by he household. b is he amoun of money borrowed by R&D enrepreneurs subjec o he following consrain: b m. i is he ineres rae on money b borrowed by R&D enrepreneurs, and i can be shown as a no-arbirage condiion ha i mus be equal o he nominal ineres rae such ha i = r + π from he Fisher equaion. To earn he wage rae w, he household inelasically supplies one uni of labor. 4 τ is a lump-sum ransfer from he governmen o he household. From sandard dynamic opimizaion, he familiar Euler equaion is ċ c = r ρ. (3) 2.2 Final goods Final goods are produced by perfecly compeiive firms ha employ labor and a composie of inermediae goods as inpus. The producion funcion of final goods is Y = L θ K 1 θ, where L = 1 is labor inpu. K is a composie of inermediae goods produced wih he following Cobb-Douglas aggregaor: { [ ] } 1 K = exp ln q (ω, j)y (ω, j) dω, (4) 0 j where he ineger j in q (ω, j) denoes he qualiy vinage of inermediae goods ω. Le j ω denoes he highes-qualiy vinage in indusry ω. Firms are indifferen beween he highesqualiy vinage and he second-highes-qualiy vinage if heir relaive price is p (ω, j ω ) p (ω, j ω 1) = q (ω, j ω ) q (ω, j ω 1) λ (ω), (5) 4 Given ha our model is already quie complex, we normalize he aggregae supply of labor o uniy in order o sidesep he issue of scale effecs; see for example, Pereo (1998, 2007) and Segersrom (1998) for imporan ways of removing he srong scale effec in he Schumpeerian growh model. 4

6 where λ (ω) > 1 is he qualiy incremen beween he wo consecuive vinages of inermediae goods ω a ime. As usual, whenever his equaliy holds, we focus on he case in which firms buy he highes-qualiy inermediae goods only. In equilibrium, only he highes qualiy inermediae goods are raded. From profi maximizaion, he condiional demand funcion for inermediae goods ω [0, 1] is given by y (ω, j ω ) = (1 θ)y p (ω, j ω ) (1 θ)k1 θ =. (6) p (ω, j ω ) Muliplying q (ω, j ω ) o boh sides of (6) and hen aggregaing he naural log of he resuling equaion wih respec o ω, we derive K = [(1 θ)q /P ] 1/θ, (7) [ ] [ 1 ] where Q exp ln q 1 0 (ω, j ω )dω and P exp ln p 0 (ω, j ω )dω denoe respecively he aggregae qualiy index and he aggregae price index of inermediae goods. 2.3 Inermediae goods There is a uni coninuum of indusries ω [0, 1] producing differeniaed inermediae goods. Each indusry is emporarily dominaed by a qualiy leader unil he arrival and implemenaion of he nex higher-qualiy produc. The owner of he new innovaion becomes he nex qualiy leader. 5 The curren qualiy leader in indusry ω uses one uni of final goods o produce one uni of inermediae goods y (ω, j ω ), so ha he marginal cos of producion is one. From Berrand compeiion, 6 limi pricing yields he equilibrium price given by p (ω, j ω ) = λ (ω). (8) Therefore, he amoun of monopolisic profi in indusry ω is [ ] λ (ω) 1 Π (ω, j ω ) = [λ (ω) 1] y (ω, j ω ) = (1 θ)y, (9) λ (ω) where he second equaliy uses (6) and (8). 2.4 R&D R&D is performed by a coninuum of compeiive enrepreneurs. If an R&D enrepreneur employs R (ω) unis of final goods o engage in innovaion in indusry ω, hen she is successful in invening he nex higher-qualiy produc in he indusry wih an insananeous probabiliy given by φ (ω) = R (ω)/α, (10) 5 This is known as he Arrow replacemen effec; see Cozzi (2007) for a discussion of he Arrow effec. 6 See Denicolò and Zanchein (2010) for an analysis of Courno compeiion in he Schumpeerian model. 5

7 where α αq (1 θ)/θ inversely measures R&D produciviy and is proporional o Q (1 θ)/θ o ensure balanced growh. To faciliae he paymen of R (ω), he enrepreneur needs o borrow cash from he household, and he cos of borrowing is deermined by he nominal ineres rae i. Therefore, he cos of R&D is (1 + i ) R (ω). Le v e (ω, j ω + 1) denoes he expeced value of an innovaion before he realizaion of is qualiy incremen. Then, he R&D free-enry condiion is given by v e (ω, j ω + 1)φ (ω) = (1 + i ) R (ω) v e (ω, j ω + 1)/α = (1 + i ). (11) 2.5 Random qualiy improvemens As in Minnii e al. (2013), when an R&D enrepreneur invens a higher-qualiy produc in indusry ω, he qualiy incremen λ (ω) > 1 is drawn from a saionary Pareo disribuion wih he following probabiliy densiy funcion: f(λ) = 1 1+ λ, (12) where he parameer (0, 1) deermines he shape of he Pareo disribuion. Given ha he expeced value of λ (ω) is equal across indusries, (9) implies ha he expeced value of Π (ω, j ω ) is also he same across indusries. Therefore, we will follow he sandard reamen in he lieraure o focus on he symmeric equilibrium in which he arrival rae of innovaions is equal across indusries, 7 such ha φ (ω) = φ for ω [0, 1]. 2.6 Endogenous firm enry To generae an endogenous disribuion of heerogeneous firms, we follow Meliz (2003) and ohers o consider a fixed enry cos. The enry cos is given by β βq (1 θ)/θ, 8 which is proporional o Q (1 θ)/θ o ensure balanced growh. Given he enry cos, a firm eners he marke if and only if v (λ) β, where v (λ) denoes he ex pos value of an innovaion (i.e., afer he realizaion of he qualiy incremen λ). 9 v (λ) is monoonically increasing in λ because Π (λ) = (1 θ)y (λ 1)/λ is increasing in λ. Given ha v (1) = 0 and v (λ)/q (1 θ)/θ is saionary in equilibrium, i can be shown ha here exiss a saionary hreshold value of λ, 10 denoed as, above which firms implemen heir innovaions and ener he marke generaing endogenous enry of firms wih heerogeneous qualiy improvemens. 7 Cozzi e al. (2007) provide a heoreical jusificaion for he symmeric equilibrium o be he unique raional-expecaion equilibrium in he Schumpeerian model. 8 We do no impose a CIA consrain on enry for he following reasons. Unlike R&D invesmen ha is subjec o uncerainy in innovaion success, he enry cos is incurred afer an innovaion is already developed and paened. Therefore, banks should be available o exend credis o he firm, which can use he paen as a collaeral. 9 In a symmeric equilibrium wih φ (ω) = φ, he value of innovaions does no depend on ω. 10 See Appendix A for he proof. 6

8 2.7 Asse prices The ex-ane value of an innovaion (i.e., before he realizaion of λ) is formally defined as v e (ω, j ω + 1) = 1 0 f(λ)dλ + [v (λ) β ]f(λ)dλ = v (λ)f(λ)dλ Pr(λ )β, where Pr(λ ) denoes he probabiliy of he innovaion being implemenable. In he symmeric equilibrium wih v e (ω, j ω + 1) v e, he no-arbirage condiion for he ex-ane value of innovaion can be derived as 11 r = Π e + v e + Pr(λ ) β Pr(λ )φ [ v e + Pr(λ )β ] v e + Pr(λ )β, (13) where φ is he arrival rae of innovaion. Pr(λ )φ is he insananeous probabiliy ha an innovaion is creaed and implemened in an indusry. The Pareo probabiliy densiy funcion implies ha Pr(λ ) = Subsiuing (14) ino (13) and rearranging erms yield f(λ)dλ = 1/. (14) Π e v e + = r 1/ + 1/ φ ve + 1/ β β v e +, (15) 1/ β where he ex-ane value of monopolisic profis can be shown o be [ ( ) ] [ ] λ 1 1/(1 + ) Π e = f(λ)dλ (1 θ)y = (1 θ)y. (16) λ Similarly, he no-arbirage condiion for he ex-pos value of an innovaion wih λ is 1+ Π (λ) v (λ) = r + 1/ φ v (λ) v (λ), (17) where he ex-pos value of monopolisic profis wih λ is given by ( ) λ 1 Π (λ) = (1 θ)y. (18) λ 11 See Appendix A for he proof. 7

9 2.8 Moneary auhoriy The moneary policy insrumen ha we consider is he nominal ineres rae i, which is exogenously se by he moneary auhoriy. Given i, he inflaion rae π is endogenously deermined according o he Fisher equaion such ha π = i r, where r is he real ineres rae and deermined from he Euler equaion in (3). Then, he growh rae of he nominal money supply is given by µ = π + ṁ /m, which becomes µ = i ρ on he balanced growh pah. 12 Finally, he moneary auhoriy reurns he seigniorage revenue as a lump-sum ransfer τ = ṁ + π m o he household. 2.9 Dynamics In his secion, we characerize he dynamics of he model. Lemma 1 shows ha given a consan nominal ineres rae i, he economy immediaely jumps o a balanced growh pah. On his balanced growh pah, each variable grows a consan (possibly zero) growh rae. Lemma 1 The economy jumps o a unique and saddle-poin sable balanced growh pah. Proof. See Appendix B Economic growh Recall ha he (log of) aggregae qualiy index is ln Q 1 ln q 0 (ω, j ω )dω. In indusry ω, he qualiy q (ω, j ω ) jumps o q (ω, j ω +1) = λ(ω)q (ω, j ω ) wih probabiliy Pr(λ )φ = 1/ φ. The coninuum of indusries shares his random process of qualiy improvemens. Therefore, he ime derivaive of ln Q is given by { 1 } [ 1 Q Q = 0 [ln q (ω, j ω + 1) ln q (ω, j ω )] dω Using he law of large numbers, we obain 13 1/ φ = 0 ] ln λ(ω)dω 1/ φ. (19) [ Q ] = (ln λ) Q f(λ)dλ 1/ φ = (ln + ) 1/ φ, (20) where ln + capures he average sep size of implemened qualiy improvemens and f(λ) is defined as f(λ) f(λ) f(λ)dλ = 1 f(λ). 12 I is useful o noe ha in his model, i is he growh rae of he money supply ha affecs he real economy in he long run, and a one-ime change in he level of money supply has no long-run effec on he real economy. This is he well-known disincion beween he neuraliy and superneuraliy of money. Empirical evidence generally favors neuraliy and rejecs superneuraliy, consisen wih our model; see Fisher and Seaer (1993) for a discussion on he neuraliy and superneuraliy of money. 13 Derivaions are available in an unpublished appendix; see Appendix C. 8

10 Finally, he growh rae of oupu Y and consumpion c is equal o g = 1 θ θ Q = 1 θ (ln Q θ + ) 1/ φ. (21) Equaion (21) shows ha he equilibrium growh rae depends on wo endogenous variables, he arrival rae φ of innovaions and he hreshold sep size. We can deermine φ using he R&D condiion v e = (1 + i)αq (1 θ)/θ, where he balanced-growh value of v e is given by v e = Π e /(ρ + 1/ φ) 1/ βq (1 θ)/θ using (15) and he Euler equaion. Then, subsiuing (16) ino he R&D condiion, we obain [ ] 1/(1 + ) Y [ (1 θ) = (1 + i)α + ] 1/ β (ρ + 1/ φ). (22) 1+ Q (1 θ)/θ In Appendix B, we show ha he producion funcion of final goods can be expressed as ( ) (1 θ)/θ 1 θ Y = Q (1 θ)/θ e. (23) Similarly, we can deermine using he enry condiion v () = βq (1 θ)/θ, where he balanced-growh value of v () is given by v () = Π ()/(ρ + 1/ φ) using (17) and he Euler equaion. Then, subsiuing (18) ino he enry condiion, we obain (1 θ) ( ) 1 Y Q (1 θ)/θ Combining (22) and (24), we have he condiion given by = β(ρ + 1/ φ). (24) ( 1) 1/ = 1 β 1 + i α 1 +, (25) where he lef-hand side is monoonically increasing in. Therefore, (25) implicily deermines he unique equilibrium value of. Using (23)-(25), we obain he φ condiion given by 1/θ (1 θ) 1/θ φ = (1 + i) 1 + αe ρ 1/. (26) (1 θ)/θ Given he equilibrium value of from (25), (26) deermines he unique equilibrium value of φ. 3 Moneary policy and economic growh In his secion, we explore he effecs of moneary policy on economic growh. In Secion 3.1, we analyically derive he effecs of he nominal ineres rae. In Secion 3.2, we calibrae he model o quanify he relaionship beween inflaion and economic growh. 9

11 3.1 Qualiaive analysis Here we firs derive he effecs of increasing he nominal ineres rae i on he innovaionarrival rae φ and he hreshold sep size. Lemma 2 shows ha φ is decreasing in i for a given. Lemma 3 shows ha is decreasing in i. The inuiion can be explained as follows. An increase in he nominal ineres rae i increases he cos of R&D and reduces he incenives for innovaion; as a resul, he innovaion rae φ decreases for a given. From he balanced-growh version of (15), we have v e = Π e /(ρ+ 1/ φ) 1/ βq (1 θ)/θ, which shows ha he decrease in φ, by reducing creaive desrucion, increases he presen value of he profi sream generaed by implemening an innovaion. This induces he implemenaion of innovaions associaed wih smaller profi margins, hereby reducing he hreshold mark-up for enry. Lemma 2 For a given, he innovaion rae φ is decreasing in he nominal ineres rae i. Proof. Use (26). Lemma 3 The hreshold sep size is decreasing in he nominal ineres rae i. Proof. Use (25). When he enry cos β is zero, he nominal ineres rae has no effec on he disribuion of innovaions ha are implemened because all firms ener he marke regardless of he size of qualiy incremens. In his case, = 1, and g = 1 θ φ is monoonically decreasing in i via θ φ. This resul is he same as in Chu and Cozzi (2014), who consider a Schumpeerian growh model wih an idenical sep size of qualiy improvemens across firms. However, when he enry cos β is posiive, he nominal ineres rae i affecs boh and φ. In his case, Pr(λ ) = 1/ is increasing in i. In oher words, an increase in he nominal ineres rae reduces he hreshold value for enry and leads o more innovaions being implemened for a given φ. When he enry cos β is suffi cienly large, he overall effecs of i on he composie innovaion rae 1/ φ and he equilibrium growh rae g = 1 θ(ln +) 1/ φ become nonmonoonic. Specifically, we find ha when he nominal ineres rae i increases, 1/ φ and θ g firs increase and evenually decrease. We summarize hese resuls in Proposiion 1. Proposiion 1 If he enry cos is suffi cienly large (small), an increase in he nominal ineres rae has an invered-u (negaive) effec on he composie innovaion rae 1/ φ and he equilibrium growh rae g Proof. See he Appendix B. Before we conclude his secion, we explore he relaionship beween inflaion and economic growh. The Fisher equaion gives rise o a posiive long-run relaionship beween he inflaion rae and he nominal ineres rae ha is suppored by empirical sudies such 10

12 as Mishkin (1992) and Booh and Ciner (2001). In our model, he inflaion rae is given by he Fisher equaion π = i r = i g(i) ρ, where he second equaliy follows from he Euler equaion. Therefore, so long as g(i)/ i < 1, we have π/ i = 1 g(i)/ i > Given his posiive relaionship, inflaion and economic growh would also exhibi an invered- U relaionship. Recen empirical sudies such as Bick (2010) and López-Villavicencio and Mignon (2011) provide evidence ha suppors an invered-u relaionship beween inflaion and economic growh. 3.2 Quaniaive analysis In his secion, we calibrae he model o aggregae daa of he US economy o provide a quaniaive illusraion on he effecs of moneary policy on economic growh. The model feaures he following srucural parameers {ρ, θ, α, β, } and policy variable i. For he discoun rae, we se ρ o a sandard value of For he labor share, we se θ o a value of 0.59; see Elsby e al. (2013) who documen ha he labor share in he US has fallen o less han 0.60 recenly. According o he Conference Board Toal Economy Daabase, he average growh raes of oal facor produciviy (TFP) in he US is abou 0.6% from 1990 o We calibrae he R&D cos parameer α by argeing he scenario in which domesic innovaion drives half of he TFP growh in he US (i.e., g = 0.3%). 15 For he cos of enry, we calibrae β by seing he ime beween arrivals of innovaion 1/φ o abou 3 years as in Acemoglu and Akcigi (2012). For he Pareo disribuion parameer, we follow Minnii e al. (2013) o consider = 0.21 as our benchmark, bu we also explore anoher value = 0.16 ha has ineresing implicaions. Finally, we calibrae he value of i by argeing he average inflaion rae π in he US, which is abou 2.5% in he pas wo decades. The parameer and variable values are summarized in Table 1. Table 1: Calibraion Targes r wl/y g φ π Parameers ρ θ α β i Under our benchmark parameer values, we find ha economic growh is an invered-u funcion of he nominal ineres rae. In Figures 1 and 2, we plo he equilibrium growh rae g agains he inflaion rae π, which is monoonically increasing in he nominal ineres rae i. Figure 1 presens our benchmark resul and shows ha he relaionship beween economic growh and inflaion follows an invered-u shape. Furhermore, he growh-maximizing inflaion rae is abou 2.9%, which is close o he empirical esimae in López-Villavicencio and Mignon (2011) who find a hreshold inflaion rae of 2.7% for indusrialized counries. 14 Under our calibraed parameer values, seady-sae inflaion is indeed increasing in he nominal ineres rae. 15 See Chu (2010) who finds ha domesic R&D drives less han half of he TFP growh in he US. 11

13 Figure 1: Inflaion and economic growh ( = 0.21) In he empirical lieraure, sudies someime find a monoonically negaive effec of inflaion on economic growh; see for example, Guerrero (2006) and Vaona (2012). Indeed, we find ha our model is flexible enough o deliver a negaive relaionship beween inflaion and economic growh under reasonable parameer values. When we decrease he value of o 0.16 and recalibrae he res of he parameers, we find ha he relaionship beween economic growh and inflaion becomes monoonically negaive. In his case, he smaller value of implies a smaller raio of β/α, such ha he negaive effec of inflaion dominaes he posiive effec. Figure 2: Inflaion and economic growh ( = 0.16) 12

14 4 Conclusion In his sudy, we have developed a moneary Schumpeerian growh model wih endogenous enry of firms wih heerogeneous qualiy improvemens. Given his moneary growhheoreic framework, we explore he effecs of moneary policy on economic growh and find ha inflaion could have an invered-u effec on economic growh. Furhermore, we calibrae he model o aggregae daa of he US economy o provide a quaniaive invesigaion. Under our benchmark parameer values, we find ha he growh-maximizing inflaion rae is abou 2.9%, which is consisen wih recen empirical esimaes. However, given ha we have a sylized model, he quaniaive analysis should be viewed as an illusraive exercise. References [1] Acemoglu, D., and Akcigi, U., Inellecual propery righs policy, compeiion and innovaion. Journal of he European Economic Associaion, 10, [2] Aghion, P., and Howi, P., A model of growh hrough creaive desrucion, Economerica, 60, [3] Baldwin, R., and Rober-Nicoud, F., Trade and growh wih heerogeneous firms. Journal of Inernaional Economics, 74, [4] Baes, T., Kahle, K., and Sulz, R., Why do U.S. firms hold so much more cash han hey used o?. Journal of Finance, 64, [5] Berensen, A., Breu, M., and Shi, S., Liquidiy, innovaion, and growh. Journal of Moneary Economics, 59, [6] Bick, A., Threshold effecs of inflaion on economic growh in developing counries. Economics Leers, 108, [7] Booh, G., and Ciner, C., The relaionship beween nominal ineres raes and inflaion: Inernaional evidence. Journal of Mulinaional Financial Managemen, 11, [8] Brown, J., and Peersen, B., Cash holdings and R&D smoohing. Journal of Corporae Finance, 17, [9] Chu, A., Effecs of paen lengh on R&D: A quaniaive DGE analysis. Journal of Economics, 99, [10] Chu, A., and Cozzi, G., R&D and economic growh in a cash-in-advance economy. Inernaional Economic Review, 55, [11] Chu, A., Cozzi, G., Lai, C., and Liao, C., Inflaion, R&D and growh in an open economy. Journal of Inernaional Economics, 96,

15 [12] Chu, A., and Ji, L., Moneary policy and endogenous marke srucure in a Schumpeerian economy. Macroeconomic Dynamics, 20, [13] Chu, A., and Lai, C., Money and he welfare cos of inflaion in an R&D growh model. Journal of Money, Credi and Banking, 45, [14] Cozzi, G., The Arrow effec under compeiive R&D. B.E. Journal of Macroeconomics (Conribuions), 7, Aricle 2. [15] Cozzi, G., Giordani, P., and Zamparelli, L., The refoundaion of he symmeric equilibrium in Schumpeerian growh models. Journal of Economic Theory, 136, [16] Denicolò, V., and Zanchein, P., Compeiion, marke selecion and growh. Economic Journal, 120, [17] Elsby, M., Hobijn, B., and Sahin, A., The decline of he U.S. labor share. Federal Reserve Bank of San Francisco Working Paper No [18] Falao, A., and Sim, J., Why do innovaive firms hold so much cash? Evidence from changes in sae R&D ax credis. FEDS Working Paper No [19] Fisher, M., and Seaer, J., Long-run neuraliy and superneuraliy in an ARIMA framework. American Economic Review, 83, [20] Grossman, G., and Helpman, E., Qualiy ladders in he heory of growh. Review of Economic Sudies, 58, [21] Guerrero, F., Does inflaion cause poor long-erm growh performance?. Japan and he World Economy, 18, [22] Gusafsson, P., and Segersrom, P., Trade liberalizaion and produciviy growh. Review of Inernaional Economics, 18, [23] Hall, B., Invesmen and R&D a he firm level: Does he source of financing maer?. NBER Working Paper No [24] Haruyama, T., and Zhao, L., Trade and firm heerogeneiy in a qualiy-ladder model of growh. Kobe Universiy Discussion Paper No [25] He, Q., and Zou, H., Does inflaion cause growh in he reform-era China? Theory and evidence. Inernaional Review of Economics & Finance, 45, [26] Huang, C., Chang, J., and Ji, L., Inflaion, marke srucure, and innovaiondriven growh wih various cash consrains. Manuscrip. [27] López-Villavicencio, A., and Mignon, V., On he impac of inflaion on oupu growh: Does he level of inflaion maer?. Journal of Macroeconomics, 33,

16 [28] Marquis, M., and Reffe, K., New echnology spillovers ino he paymen sysem. Economic Journal, 104, [29] Meliz, M., The impac of rade on inra-indusry reallocaions and aggregae indusry produciviy. Economerica, 71, [30] Minnii, A., Parello, C., and Segersrom, P., A Schumpeerian growh model wih random qualiy improvemens. Economic Theory, 52, [31] Mishkin, F., Is he Fisher effec for real? A reexaminaion of he relaionship beween inflaion and ineres raes. Journal of Moneary Economics, 30, [32] Opler, T., Pinkowiz, L., Sulz, R., and Williamson, R., The deerminans and implicaions of corporae cash holdings. Journal of Financial Economics, 52, [33] Pereo, P., Technological change and populaion growh. Journal of Economic Growh, 3, [34] Pereo, P., Corporae axes, growh and welfare in a Schumpeerian economy. Journal of Economic Theory, 137, [35] Romer, P., Endogenous echnological change. Journal of Poliical Economy, 98, S71-S102. [36] Segersrom, P., Endogenous growh wihou scale effecs. American Economic Review, 88, [37] Segersrom, P., Anan, T.C.A. and Dinopoulos, E., A Schumpeerian model of he produc life cycle. American Economic Review, 80, [38] Vaona, A., Inflaion and growh in he long run: A new Keynesian heory and furher semiparameric evidence. Macroeconomic Dynamics, 16,

17 Appendix A: The saionary qualiy hreshold In he symmeric equilibrium v e (ω, j ω + 1) v e, he ex-ane value of an innovaion is given by v e = 1 0 f(λ)dλ + [v (λ) β ]f(λ)dλ = v (λ)f(λ)dλ Pr(λ )β. (A1) Subsiuing he ex-pos no arbirage condiion condiion r v (λ) = Π (λ) + v (λ) Pr(λ )φ v (λ) ino (A1) yields r v e = Π e + v (λ)f(λ)dλ Pr(λ )φ [v (λ) β ] f(λ)dλ [Pr(λ ] )φ + r Pr(λ )β. Combining (A1) and he R&D condiion (11) and also using (14), we obain v (λ)f(λ)dλ = (1 + i ) α + (A2) 1/ β, (A3) where i is chosen exogenously by he moneary auhoriy. Differeniaing (A3) wih respec o, we use he Leibniz inegral rule o derive v (λ)f(λ)dλ v ( )f( ) = (1 + i) α + 1/ β 1 1+ We subsiue (12) and he enry condiion v ( ) = β ino (A4) o obain β. (A4) v (λ)f(λ)dλ = (1 + i) α + 1/ β. (A5) Subsiuing (A5) ino (A2), he ex-ane no-arbirage condiion for an innovaion can be expressed as [ ] [ ] Π e + v e 1/ 1/ + β φ v e 1/ + β r =, (A6) v e 1/ + β which uses (14) and he R&D condiion (11) again. Moreover, we make use of he R&D condiion (11), α = αq (1 θ)/θ and β = βq (1 θ)/θ o derive v e + v e + Wih his expression, (A6) becomes 1/ β = 1/ β ( 1 θ θ ) Q Q. r = v e + Π e 1/ β + ( 1 θ θ ) Q 1/ φ Q. (A7) 16

18 Meanwhile, he ex-pos no-arbirage condiion for he hreshold qualiy (λ = ) can be wrien as r = Π ( ) v ( ) + ( 1 θ By he R&D condiion (11), he enry condiion v ( ) = β, α, (A7) and (A8) imply βq (1 θ)/θ Π e (1 + i) α + θ 1/ β Given (16) and (18), (A9) can be rearranged as ( 1) 1/ = Equaion (A10) shows ha is always saionary. ) Q 1/ φ Q. (A8) = αq (1 θ)/θ and β = = Π ( ). (A9) β 1 β (1 + i) α 1 +. (A10) 17

19 Appendix B: Proofs Proof of Lemma 1. Using (8), we can express he aggregae price index of inermediae goods as 16 [ 1 ] [ ] P = exp ln λ (ω)dω = exp (ln λ) f(λ)dλ = e, (B1) 0 where f(λ) is defined as f(λ) f(λ) f(λ)dλ = 1 f(λ). (B2) Here we inroduce a modified densiy funcion f(λ) in summing λ on [, ] because he disribuion of λ in equilibrium is no on he original domain [1, ), bu insead on [, ), due o endogenous enry. Noe ha f(λ)dλ = 1. By (7) and (B1), we obain K = [ 1/θ. (1 θ)q /(e )] Incorporaing his condiion ino he producion funcion Y = L θ K 1 θ, we obain [ ] (1 θ)/θ (1 θ)q Y =, (B3) e noing L = 1. Recall ha final goods are used for consumpion, producion of inermediae goods, R&D and enry. Consumpion is given by c. By (6) and (8), he amoun of final goods used for he producion of inermediae goods is X m = 1 0 y (ω, j ω )dω = (1 θ)y 1 dω = (1 θ)y λ (ω) λ f(λ)dλ = (1 θ)y. (B4) (1 + ) Final goods for innovaion and enry are given by 1 X r = R (ω)dω = α φ and X e = β dω = β 1/ φ, ω Ω where Ω is he se of indusries in which innovaions ake place and are implemened a dae. Finally, we subsiue (B3), (B4) and (B5) ino he marke-clearing condiion Y = c + X m + X r + X e o derive [ (1 ) (1 θ)/θ ( 1 θ φ = 1 1 θ ) ] α + β 1/ e (1 + ) C, (B6) where C c /Q (1 θ)/θ is a ransformed variable ha is saionary. We subsiue (16) and he R&D condiion (11) ino (A7) o derive r = (1 θ)y (1 + i)α + 1/ β [ ] 1/(1 + ) We achieve his by applying inegraion by pars o 1/ ( (ln λ) f(λ)dλ d = (ln λ) dλ 1/ φ + 1 θ θ 1+ 1 λ 1 1+ ) dλ. Q Q. (B5) (B7) 18

20 Finally, subsiuing (B3) and (B7) ino (3) yields [ ] Ċ = r ρ 1 θ Q (1 θ) 1/θ 1/(1 + ) = C θ Q [(1 + i)α + 1/ φ 1/ β]e (1 θ)/θ (1/θ)+(1/) ρ, (B8) noing he definiions C c /Q (1 θ)/θ and α αq (1 θ)/θ. Subsiuing (B6) ino (B8), we have an one-dimensional differenial equaion in C. 17 Given ha φ decreases wih C in (B6), he righ-hand side of (B8) is increasing in C, so he dynamics of C is characerized by saddle-poin sabiliy, such ha C mus jump o is inerior seady-sae value. Given a saionary value of C, (B6) implies ha φ is also saionary. Proof of Proposiion 1. In his proof, we firs show ha he relaionship beween i and 1/ φ is eiher invered U-shaped or negaive. Combining (25) and (26), we have 1/ φ = (1 θ)1/θ βe (1 θ)/θ ( ) 1 ρ. By differeniaing he righ-hand side of (B9) wih respec o, we can easily show ha d( 1/ φ)/d > (<) 0 if < (>) 1/(1 θ), implying an invered-u relaionship beween and 1/ φ. In idenifying he relaionship wih respec o i, we naurally focus on a non-rivial range of, i.e., (λ, λ), where 1/ φ > 0 holds. 18 Given ha monoonically decreases wih i (Lemma 3), i 0 provides anoher naural upper bound of, say λ i, which is defined by (λ i 1) λ 1/ i = β α 1 +. (B10) When λ i is large enough (exceeding 1/(1 θ)), he relaionship beween i and 1/ φ is invered U-shaped on he non-rivial range (λ, λ i ); see Figure 3a. When λ i is small enough (falling below 1/(1 θ)), 1/ φ is monoonically decreasing in i on (λ, λ i ); see Figure 3b. Noe ha, by (B10), λ i increases wih β and, by (B9), λ decreases wih β. This implies ha for a larger (smaller) enry cos β, accompanied by a larger (smaller) λ i, he relaionship beween i and 1/ φ becomes invered-u (negaive). 17 Alhough is an endogenous variable, i is saionary and a funcion of parameers as shown in (A10). 18 The formal definiion of (λ, λ) is given by incorporaing 1/ φ = 0 ino (B9): λ and λ are equal o x such ha (x 1) /x 1/θ = ρβe (1 θ)/θ /(1 θ) 1/θ. This has he wo soluions such as x = λ and λ if and only if ρβ < θ (1 θ) 2 θ θ /e (1 θ)/θ. Oherwise, 1/ φ canno be posiive. 1/θ (B9) 19

21 In he res of his proof, we characerize he relaionship beween i and g. For < 1/(1 θ), i holds d( 1/ φ)/d > 0 as shown above. Given ha (ln + ) is also increasing in, his implies dg/d > 0 for < 1/(1 θ), by noing (21). To see he case where 1/(1 θ) <, using (21) and (B9), we can obain dg d = 1 θ θ 1+1/θ [ (1 θ) 1/θ ) ] ( 1 ρ 1/θ βe } (1 θ)/θ {{} ζ(): unimodal and concave in (ln (1 θ)1/θ + ) ( 1 ). 1 θ 1 θ βe (1 θ)/θ θ }{{} ξ(): increasing and convex in Noe he following properies: (a) ζ(1/(1 θ)) > 0 and ξ(1/(1 θ)) = 0; (b) ζ() is an uni-modal funcion 19 and ξ() is a sricly increasing funcion; (c) ζ(λ) = ζ(λ) = 0; and (d) ζ() is sricly concave and ξ() is sricly convex. Using hese properies, we can graphically show ha ξ() inersecs ζ() from below only once a some poin in (1/(1 θ), λ), below (above) which dg/d > (<) 0. This implies an invered-u relaion beween and g on (λ, λ). The res of Proposiion 1 sraighforwardly follows, noing ha λ i is increasing in β. 19 I is useful o noe ha ζ() is upward sloping a = 1/(1 θ). 20

22 Appendix C (no for publicaion) Equaion (20): Recall ha he equilibrium disribuion of λ is given by f(λ), which is defined by (B2) in Appendix B. Then we calculae 1 0 (ln λ(ω)) dω = (ln λ) f(λ)dλ = 1/ (ln λ) λ 1+ dλ, where he second equaliy uses (12) and (B2). Given ha we have 1 0 (ln λ(ω)) dω = λ 1+ = d dλ 1/ Applying inegraion by pars, we calculae 1/ [ ( )] 1+ 1 d λ (ln λ) dλ 1 1+ dλ = 1/ 1+ 1 λ 1 1+ (ln λ) { [, d dλ 1+ 1 λ 1 1+ ( )] 1+ 1 λ 1 1+ dλ. ln λ λ dλ }. From 1+ 1 λ 1 1+ ln λ we have 1 0 (ln λ(ω)) dω = ln +. = 1 ln and λ dλ = 2 1, 21