Long-run growth effects of taxation in a non-scale growth model with innovation

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1 Deparmen of Economics Working Paper No hp:// Long-run growh effecs of axaion in a non-scale growh model wih innovaion (Forhcoming in he Economics Leer) Jinli Zeng Naional Universiy of Singapore Jie Zhang Universiy of Queensland and Vicoria Universiy of Wellingon 22 Ocober 2001 Absrac: In previous sudies, axing income or consumpion hinders long-run growh. Incorporaing saving and leisure ino he non-scale Schumpeerian model of Howi (1999), we show ha he usual growh effecs of axing consumpion and labor income do no exis. JEL classificaion: E62; H20; O40 Keywords: Scale effecs; Taxaion; Long-run growh 2001 Jinli Zeng and Jie Zhang. Jinli Zeng, Deparmen of Economics, Naional Universiy of Singapore, 1 Ars Link, Singapore , Republic of Singapore; Telephone: (65) ; Fax: (65) ; ecszjl@nus.edu.sg. Views expressed herein are hose of he auhors and do no necessarily reflec he views of he Deparmen of Economics, Naional Universiy of Singapore

2 1. Inroducion Growh effecs of axaion have been sudied exensively in he recen lieraure on axaion (e.g., Judd, 1985; Chamley, 1986; King and Rebelo, 1990; Rebelo, 1991; Jones e al., 1993; Pecorino, 1993; Devereux and Love, 1994; Sokey and Rebelo, 1995; Milesi-Ferrei and Roubini, 1998), using eiher he neoclassical growh model wih physical capial only, or he wo-secor growh model wih boh human and physical capial. The ypical view in he previous sudies is ha he hree commonly used axes (on consumpion, labor or capial income) have in general negaive effecs on long-run growh. Income axes creae disincenives for invesmen and work by reducing he afer-ax raes of reurn on capial and labor, while consumpion axes creae disincenives for work by reducing he price of leisure relaive o consumpion. The objecive of his paper is o reexamine he growh effecs of axaion by using a very recen growh model of Howi (1999) where innovaion is he engine of long-run growh. In his model as well as in a few ohers (e.g., Segersrom, 1998; Young, 1998), here is no scale effec, in erms of he effec of he size of he populaion on growh ha was presen in earlier R&D endogenous growh models resuling from spillovers of knowledge or invesmen (e.g., Romer, 1990; Grossman and Helpman, 1991; Aghion and Howi, 1992). 1 Using he non-scale growh models in he analysis of he growh effecs of axaion is appealing, because innovaions are viewed by many as he main conribuor o long-run growh, and because here is lack of clear evidence for he presence of such a scale effec. We exend he non-scale R&D growh model by considering wo imporan facors ha have been used in he lieraure on axaion and growh: saving and he rade-off beween labor and leisure. We show ha alhough axes on consumpion and labor income change he labor-leisure choice and hence he size of he effecive labor force, hey do no influence long-run growh in his non-scale R&D model. The reason is ha he scale effec of changes in he size of he effecive labor force is nullified by produc proliferaion. A capial-income ax, as in oher models, is harmful for growh by discouraging saving and capial invesmen. 1 Barro and Sala-i-Marin (1995, p.442) found only a weak and minor scale effec in a cross-counry panel daa se. 1

3 The res of his paper is organized as follows. The nex secion inroduces he model. Secion 3 characerizes he equilibrium and derives he resuls. The las secion concludes. 2. The model We exend he Schumpeerian framework of Howi (1999) by considering saving and leisure. A firs glance he framework seems o be an overly complicaed model for sudying he growh effecs of axaion. The raionale for using such a model will become clear when we see he channels hrough which axes affec long-run growh, however. In his exended model, boh saving and labor supply are deermined by ineremporal uiliy maximizaion of a represenaive household as in he lieraure on axaion vs. growh. There is a final-good secor, using a variey of inermediae goods whose range expands and qualiy improves over ime hrough innovaion Households We assume ha he represenaive household is endowed wih one uni flow of ime which is allocaed beween leisure l and producion 1 l. We also assume ha he represenaive household s preferences are given by ] 0 ) 1 ɛ [( C e ρ l η 1 ɛ L d, (1) where C is per capia consumpion; L is he size of populaion, growing a a consan rae g L > 0; ρ is he consan rae of ime preference; ɛ is he elasiciy of marginal uiliy; and refers o ime. Suppose ha he governmen imposes a consumpion ax (a a fla rae) τ c, a labor-income ax τ L, and a capial-income ax τ k o collec revenue for lump-sum ransfers and subsidies on R&D. Having in mind ha ax raes are relaively sable over ime in he real world, we only consider a case where ax raes are saionary, oherwise echnical complexiy would be subsanial. The represenaive household s budge consrain is (1 + τ c ) C = W (1 τ L )(1 l ) + [r (1 τ k ) g L ] K + T K, (2) where K is per capia capial asse; T is per capia lump-sum ransfer; W is he wage rae; and r is he ineres rae. A do above a variable represens he ime change rae of ha variable. Noe ha he final good is used as he numeraire. 2

4 The represenaive household chooses consumpion C and leisure l o maximize (1) subjec o (2). Solving his maximizaion problem gives he opimal ime pah of per capia consumpion (see Appendix A) C = r (1 τ k ) ρ, C ɛ (3) and he relaionship beween leisure and consumpion: l = η(1 + τ c) C (1 τ L )W. (4) In (3), he capial-income ax has a direc negaive effec on consumpion growh by reducing he afer-ax rae of reurn o capial, while all axes may affec consumpion growh hrough he ineres rae. In (4), a higher labor-income ax, or a higher consumpion ax, ends o raise leisure relaive o consumpion by lowering he afer-ax wage, or by raising he price of consumpion Technologies There are five ypes of producion aciviies in his economy: final-good producion, inermediaegood producion (Q secors), physical capial accumulaion, verical and horizonal innovaions. I is assumed ha perfec compeiion prevails in all secors excep he inermediae secors where here exiss emporary monopoly power. We briefly describe each ype of he aciviies below; for more deails, see Howi (1999) and Aghion and Howi (1992, 1996) Final-good producion Using a coninuum of inermediae goods and labor as inpus, he final-good producion has he echnology 2 Q Y = L 1 α Y 0 A i x α idi, 0 < α < 1, (5) where Y is final oupu; L Y and x i are, respecively, labor and he flow of inermediae good i. The parameer α measures he conribuion of an inermediae good o he final-good producion 2 In he original model of Howi (1999), labor is used only in he inermediae-good producion. We assume ha boh he final secor and he inermediae secors use labor as heir inpus o allow he full srengh of ax disorions on he labor-leisure rade-off. 3

5 and inversely measures he inermediae monopolis s marke power. The parameer A i is he produciviy coefficien of inermediae good i. Final oupu is allocaed among verical R&D expendiures (N v ), horizonal R&D expendiures (N h ), oal consumpion (C ), and invesmen in capial ( K ): Y = N v + N h + C + K. (6) In (6), we absrac from capial depreciaion for simpliciy. The compeiive final secor yields he demand for boh labor and he inermediae good i Q W = (1 α)l α Y 0 A i xidi, α (7) p i = αl 1 α Y A i x α 1 i, i [0, Q ], (8) where p i is he price of inermediae good i in erms of he final good Inermediae-good producion Each inermediae good, i, is produced using labor, L i, and capial, K i, according o ( Ki ) 1 γ, 0 < γ < 1. (9) x i = L γ i A i In (9), we deflae he capial inpu K i by he produciviy parameer A i o reflec he fac ha more recen innovaions are more capial inensive. 3 Given he wage rae W, he ineres rae r, and he final secor s demand for inermediae goods (8), each inermediae-good producer chooses a monopolisic price, p i, o maximize is profi π i = p i x i W L i r K i = αa i L αγ i ( Ki A i ) α(1 γ) L 1 α Y W L i r K i. (10) The soluion for his gives he profi flow a dae s for an inermediae-good producer who uses a echnology of vinage (see Appendix B) { [ ] } αγg π s = A max α(1 α)γ 1 y exp 1 α g L (s ), (11) 3 Technically, his is a necessary assumpion ha makes echnological progress purely labor augmening and generaes a seady sae wih a consan ineres rae. 4

6 where A max max{a i, i [0, Q ]} is he leading-edge produciviy parameer; y Y/(Q A max ) is he produciviy-adjused oupu; g is he seady-sae growh rae of per capia oupu; and Γ is a parameer deermined by he disribuion of he relaive produciviy of inermediae goods a A i /A max. Following Howi (1999), we have Γ ( 1 0 a 1 α(1 γ) 1 α 1 σ a 1 1) { σ da = 1 + σ[1 α(1 γ)] } 1 1 α where σ > 0 is a parameer o be explained laer Verical R&D A verical R&D innovaion, once successful, improves an exising inermediae produc, and replaces he exising one in he final-good producion. The successful innovaor becomes he emporary monopolis unil he arrival of he nex successful innovaion in ha secor. Assume ha verical innovaions follow a Poisson process, wih a common arrival rae given by φ = λn, n = N v /(Q A max ), λ > 0, (12) where λ is he produciviy parameer of verical R&D, and n is he produciviy-adjused expendiure on verical R&D in each secor. Deflaing verical R&D expendiures by he leading-edge produciviy parameer means ha he complexiy of innovaion increases proporionally o he echnological progress. Since he expeced reurn on invesmen in verical R&D is he same in each inermediae secor, he amoun of expendiure on verical R&D is also he same in each secor. A verical R&D firm chooses is R&D expendiure N v /Q o maximize is profi {φ V v (1 s v )N v /Q }, where V v is he expeced value of a verical innovaion and s v is he subsidy rae on verical R&D. Assume ha he successful innovaor in each inermediae secor eners ino Berrand compeiion wih he previous incumben in ha secor and ha he previous incumben exis and canno hreaen o reener. So he successful innovaor can charge a monopolis price. As a resul, he expeced value of a verical innovaion is given by V v = exp [ s (r τ + φ τ )dτ] π s ds. Subsiuing (11) and he seady-sae equilibrium condiions r = r, n = n, and y = y ino he value funcion gives V v = Amax α(1 α)γ 1 y r + φ + αγg 1 α g. (13) L 5

7 In (13), he discoun rae in he denominaor includes four erms: he ineres rae r, he arrival rae of verical innovaions φ, he rae of gradual crowding ou αγg/(1 α) due o he coninual rise in wages, and he growh rae of profis ( g L ) due o he populaion growh. The las erm ( g L ) comes from he fac ha labor is an inpu in he final-good producion; his erm does no appear in Howi (1999) because labor is no used in he final secor in his model. For n > 0, he firs-order condiion for a verical innovaor s maximizaion problem is λv v A max = λα(1 α)γ 1 y r + λn + αγg 1 α g L = 1 s v. (14) This equaion says ha he afer-subsidy marginal cos of R&D (he far righ-hand side) equals he expeced marginal benefi of R&D Horizonal R&D A horizonal R&D innovaion aims a a new inermediae produc. A successful innovaor becomes he monopolis of his newly creaed produc unil he produc is improved by a verical innovaion. Assuming diminishing reurns o expendiures on horizonal R&D, we can specify he rae of new produc innovaion as Q = ψ(n h, Y ) A max, (15) where ψ(, ) is increasing and concave, and has consan-reurns o scale. Similar o verical R&D, he inpus N h and Y in horizonal R&D are deflaed by he leading-edge produciviy parameer. Equaion (15) implies ha he average produc Q /N h is a decreasing funcion of he fracion h N h /Y of final oupu allocaed o horizonal R&D. Also assume ha he produciviy of a newly creaed inermediae good is drawn randomly from he disribuion of exising inermediae goods. I follows from his assumpion ha he expeced value of a horizonal innovaion is V h = E ( Ai A max ) 1 α(1 γ) 1 α V v. (16) Similar o verical R&D firms, a horizonal R&D firm chooses is R&D expendiure N h o maximize 6

8 is profi {[ ] } ψ(nh,y ) A max V h (1 s h )N h, where s h is he subsidy rae on horizonal R&D. The firs- order condiion for his maximizaion problem is ψ (h )V h A max = 1 s h, (17) where ψ(h ) ψ(h, 1) wih ψ > 0 and ψ < 0. Equaion (17) saes ha he afer-subsidy marginal cos of R&D (he righ-hand side) equals he expeced marginal benefi of R&D (he lef-hand side) Knowledge spillovers Following Caballero and Jaffe (1993) and Howi (1999), we assume ha growh in he leadingedge produciviy A max resuls from knowledge spillovers of verical innovaions. More specifically, he leading-edge produciviy A max is assumed o grow a a rae proporional o he aggregae rae of verical innovaions; and he facor of his proporionaliy is assumed o equal σ/q > 0, which measures he marginal impac of each innovaion on he sock of public knowledge, where σ > 0 is a parameer. This marginal impac of each innovaion depends negaively on he number of inermediae goods because as he number of inermediae goods rises, an innovaion of a given size in any inermediae secor will have a smaller impac on he aggregae economy. Since he aggregae flow of verical innovaions equals he number of inermediae secors Q imes he flow of verical innovaions in each secor λn, he growh rae of he leading-edge produciviy is g A A max A max = ( σ Q ) (λn Q ) = σλn, σ > 0. (18) From (18), we can see ha he poenial supply-side scale effec of increasing R&D expendiures is nullified by he rise in he number of inermediae goods, which reduces he marginal spillover effec σ/q of each innovaion. As he produciviy of a newly creaed inermediae good is randomly drawn from he disribuion of he exising inermediae goods, he produciviy disribuion of new inermediae goods is idenical o he produciviy disribuion of exising inermediae goods. As a resul, 7

9 he disribuion of relaive produciviy a i A i /A max converges o he invarian disribuion Prob{a i a} = F (a) = a 1/σ, where 0 < a 1. I follows ha in he long run E ( Ai A max ) 1 α(1 γ) 1 α 2.3. Governmen budge consrain = Γ. (19) Suppose ha he governmen s budge is balanced a each poin in ime τ c C + τ k r K + τ L W (1 l )L = T + s v N v + s h N h, (20) where C, K and T are aggregae variables. In (20), he lef-hand side is he governmen s ax revenue from consumpion τ c C, capial income τ k r K and labor income τ L W (1 l )L, and he righ-hand side is he governmen s expendiures on lump-sum ransfer T, subsidies on verical R&D s v N v and subsidies on horizonal R&D s h N h. 3. Seady-sae equilibrium and resuls In a seady-sae balanced growh equilibrium, saionariy is imposed on he allocaion of ime (l ), and on he raios of oupu, consumpion, and capial sock o produciviy in erms of Q A max, such as y, c = C /(Q A max ), and k = K /(Q A max ). Saionariy is also imposed on he amoun of verical R&D expendiure per produc (n ), he fracion of final oupu allocaed o horizonal R&D (h ), and he ineres rae (r ). In addiion, he wage rae (W ), he number of inermediae goods (Q ), and he leading-edge produciviy (A max ) grow a consan raes g, g Q, and g A, respecively. From equaion (A.13), he produciviy-adjused oupu is [ y = Γ Y Γ 1 α k α(1 γ) (1 l )L Q αγ α(1 γ) 1 ] 1 α(1 γ), (21) where Γ Y α 2αγ (1 α) 1 α γ αγ [1 α + α 2 γ] α(1 γ) 1. Since (y, k, l ) is saionary, L Q αγ α(1 γ) 1 mus be saionary by (21). Thus, he growh rae of he number of inermediae goods mus saisfy [ ] 1 α(1 γ) g Q = g L. αγ From (22), he growh rae of he number of inermediae goods is proporional o, and greaer han, he growh rae of he populaion. Unlike he original model of Howi (1999), here he number (22) 8

10 of inermediae goods Q grows faser han he populaion. The reason is as follows. On he one hand, by consrucion, he equilibrium labor employmen in all inermediae secors, Q 0 L i di, ( ) equals he average labor employmen in each inermediae secor, Q 0 L i di /Q, imes he number of inermediae secors, Q. On he oher hand, he labor marke clearing condiion requires ha he equilibrium labor employmen in all inermediae secors mus grow a he same rae as he populaion, g L. We can easily verify ha he average labor employmen in each inermediae secor grows more slowly han he populaion. As a resul, he number of inermediae secors mus grow faser han he populaion. Per capia oupu is Ȳ = Q A max y /L, and is growh rae, g, mus saisfy ( ) 1 α g = g Q + g A g L = g A + g L. (23) αγ From (23), we can see ha he growh rae of per capia oupu depends posiively on boh he growh rae of he leading-edge produciviy and he populaion growh rae. Also differen from Howi (1999), per capia oupu grows faser han he leading-edge produciviy because he number of inermediae goods increases faser han he populaion. Now he producion funcion of horizonal R&D (15) implies ψ(h)y = g Q, which gives he equilibrium produciviy-adjused oupu [ ] 1 α(1 γ) y = g Q /ψ(h) = g L /ψ(h). (24) αγ In (24), he produciviy-adjused oupu relaes posiively o he populaion growh rae bu negaively o he horizonal R&D inensiy. The saionary equilibrium ineres rae resuls from boh (3) and C / C = g, ha is r = ρ + ɛg 1 τ k. (25) Equaion (25) shows he usual posiive relaionship beween he equilibrium ineres rae and he equilibrium growh rae of per capia oupu. To see he effecs of axaion on he equilibrium growh rae more clearly, we furher simplify he seady-sae equilibrium condiions. Subsiuing (13) and (16), ogeher wih (12), (18), (19) 9

11 and (23)-(25), ino (14) and (17), we have he following wo equilibrium condiions ha deermine per capia oupu growh, g, and he proporional allocaion o horizonal R&D, h: 4 Horizonal R&D condiion: ψ (h) = λ(1 s h) 1 s v Verical R&D condiion: { 1 + } σ[1 α(1 γ)] ; (H) 1 α λ {(1 α) + σ [1 α(1 γ)]} [1 α(1 γ)] g L /[γψ(h)] ( ) ( ) ρ 1 τ k + ɛ 1 τ k + 1 σ + = 1 s αγ v. (V) 1 α g α σαγ g L According o equaion (H), he horizonal R&D inensiy h is independen of he growh rae of per capia oupu g because here is only one value of h a which he marginal values of horizonal and verical R&D are equal. According o equaion (V), he growh rae of per capia oupu g depends negaively on horizonal R&D inensiy h. This negaive influence of he horizonal R&D inensiy on he growh of per capia oupu arises because an increase in he horizonal R&D inensiy reduces he produciviy-adjused oupu as shown in (24), which in urn lowers he profi flow of a successful verical innovaor, and hus discourages invesmen in verical R&D. Also noe ha he populaion growh rae, g L, in (V) is exogenously given in his model. From hese wo equilibrium condiions, we can see ha here exiss a unique seady-sae equilibrium as long as ρ is small enough (bu ρ > [1 + (1 α)/(σαγ)]g L ). The comparaive-saic resuls are similar o hose in Proposiion 1 of Howi (1999). Wha are he implicaions of he axes for long-run growh? Since he consumpion ax τ c and he labor-income ax τ L are absen in he wo equilibrium condiions (H) and (V), hey do no affec long-run economic growh, ha is, g/ τ L = g/ τ c = 0. In our model, he only channel hrough which he consumpion and labor-income axes can poenially affec long-run growh is he size of he effecive labor force since hey increase leisure in (4). These ax disorions on he effecive labor force (1 l )L affec he produciviy-adjused oupu y in (21), and hus ener he innovaors profi in (11) and he values of innovaion in (13) and (16) from he demand side 4 The soluion for oher variables is given in Appendix C. 10

12 hrough y. Wihou produc proliferaion (from horizonal innovaion), hese axes would hinder long-run growh. When produc proliferaion arises from horizonal R&D invesmen and final oupu hrough a consan-reurns-o-scale echnology as in (15), he produciviy-adjused oupu y is deermined by he rae of produc proliferaion g Q and he horizonal R&D inensiy h as in (24). The wo deerminans of y are independen of he size of he effecive labor force and he axes (τ c, τ L ). Firs, he rae of produc proliferaion is proporional o he rae of populaion growh and independen of he axes in (22). Second, he horizonal R&D inensiy h may be affeced by he size of he effecive labor force and he ax disorions as menioned above. However, because of he proporional relaion beween he values of he wo ypes of innovaion V h = ΓV v by (13), (16) and (19), he scale effec and he ax disorions hrough y on he marginal values of he wo ypes of innovaion are proporional and cancel ou enirely in (H) where he marginal values of innovaion are equalized. As a resul, he labor income ax and he consumpion ax do no affec he growh rae of per capia oupu. By conras, he effec of he capial-income ax on growh is negaive by (V), since in equilibrium h/ τ k = 0 and sign g/ τ k = sign [ (ρ + ɛg)] < 0 for all g 0, when holding subsidy raes s h and s v consan, and leaving he ask of balancing he governmen budge o residual changes in lump-sum ransfers T. We illusrae he case wih a rise in he capial-income ax rae in Figure 1, where such a ax rise does no affec he horizonal line (H) bu i shifs he downward-sloping curve (V) o he lef, and hereby reduces he long-run growh rae. 5 Summing up he discussion, we have [Figure 1 abou here] Proposiion 1. The long-run growh rae of per capia oupu depends negaively on he capialincome ax. However, i is independen of he consumpion ax and he labor-income ax. Alhough axes on consumpion and labor income affec he labor-leisure choice and hence he size of he effecive labor force, hey do no influence long-run growh in his non-scale R&D model 5 From he equaions in Appendix C, we can see ha all he hree axes have level effecs. Tha is, hey affec he magniudes of labor supply, oupu, consumpion and ohers. 11

13 in conras o he convenional view. The reason is ha he scale effec on long-run growh of changes in he size of he effecive labor force, which exised in earlier Schumpeerian models, is nullified by produc proliferaion, keeping he reward o any specific innovaion unchanged. As in he lieraure, one he oher hand, a capial-income ax is harmful for growh, since i depresses saving and capial invesmen. 4. Conclusions By incorporaing endogenous saving and labor-leisure choices ino he non-scale R&D growh model of Howi (1999), his paper showed ha long-run growh is independen of boh consumpion axes and labor-income axes, and is only affeced negaively by capial-income axes. This resul sands in sharp conras wih he convenional conclusion ha in general all he hree ypes of axaion have negaive effecs on long-run economic growh. This differen conclusion is mainly reached by our use of he R&D growh model ha has no scale effec regarding he size of he effecive labor force. While emphasizing innovaion, we absraced from oher growh deerminans like human capial invesmen used in he lieraure. In his sense, our resuls complemen hose in he lieraure. The policy implicaion of our resuls is ha in he echnology-leading counries where innovaion is imporan for long-run growh, more reliance on consumpion axes or labor-income axes han on capial-income axes for public ransfers or subsidies may benefi he economy. Acknowledgemens We would like o hank he Edior and especially one anonymous referee for helpful commens and suggesions. Any remaining errors or omissions are our own. 12

14 Appendix A: The soluion o he represenaive household s opimizaion problem. The curren-value Hamilonian funcion and he firs-order condiions for he represenaive household s uiliy maximizaion problem are H = L ( C l η ) 1 ɛ { + θ W (1 τ L )(1 l ) + [r (1 τ k ) g L ] 1 ɛ K + T (1 + τ c ) C }, H C = L C ɛ l η(1 ɛ) θ (1 + τ c ) = 0, (A.1) H = ηl l C1 ɛ l η(1 ɛ) 1 θ W (1 τ L ) = 0, (A.2) H K = θ [r (1 τ k ) g L ] = ρθ θ, (A.3) lim e ρ θ K = 0, (A.4) where θ is he co-sae variable. Solving he above firs-order condiions gives he opimal ime pah of per capia consumpion (3) and he relaionship beween leisure and consumpion (4). Appendix B: Derivaion of equaion (11). The firs-order condiions for inermediae monopolis i s profi maximizaion problem are W = α 2 γa i x α il 1 α Y /L i, (A.5) r = α 2 (1 γ)a i x α il 1 α Y /K i. (A.6) Solving equaions (A.5) and (A.6) gives he inermediae secor i s demand for labor and capial and is opimal oupu L i = Γ L [ K i = Γ K [ A 1 α(1 γ) i W α(1 γ) 1 A 1 α(1 γ) i W αγ [ x i = Γ x A γ i W γ r (1 γ) ] 1 1 α L Y, r α(1 γ) ] 1 1 α L Y, r (1 αγ) ] 1 1 α L Y, (A.7) (A.8) (A.9) where Γ L [α 2 γ 1 α(1 γ) (1 γ) α(1 γ)] 1 1 α, Γ K [α 2 γ αγ (1 γ) 1 αγ] 1 1 α, 13

15 Γ x [α 2 γ γ (1 γ) 1 γ] 1 1 α. Using equaions (5), (7), (9), (A.5) and (A.6), along wih he facor marke equilibrium condiions Q 0 L i di + L Y = (1 l )L and Q 0 K i di = K, we have he following soluion L Y = (1 α)(1 l )L 1 α + α 2, (A.10) γ W = (1 α + α2 γ)y (1 l )L, (A.11) r = α2 (1 γ)y K, (A.12) Y = Γ Y (ΓQ ) 1 α (A max ) 1 α(1 γ) K α(1 γ) [(1 l )L ] 1 α(1 γ), (A.13) where Γ Y α 2αγ (1 α) 1 α γ αγ [1 α + α 2 γ] α(1 γ) 1 and A max max{a i, i [0, Q ]}. In order o derive he expeced values of verical and horizonal innovaions, we calculae he inermediae producer i s profi flow π i = A i α(1 α)x α il 1 α Y = Γ π [ A 1 α(1 γ) i W αγ r α(1 γ) ] 1 1 α L Y, (A.14) where Γ π (1 α) [α 1+α γ αγ (1 γ) α(1 γ)] 1 1 α. Therefore, he profi flow a dae s for an inermediae-good producer who uses a echnology of vinage is π s = Γ π [ (A max ) 1 α(1 γ) W αγ s ] rs α(1 γ) 1 1 α L Y s. (A.15) From (A.10)-(A.13) and he seady-sae equilibrium condiions, he ineres rae r is consan and he wage rae W grows a he same consan rae g as per capia oupu. Thus, we can rewrie he expression for he profi flow (A.15) as (11) in he ex. Appendix C: Seady-sae soluion for oher variables. The saionary equilibrium soluions ( ) } for oher variables are: c = 1 1+τ c {(1 τ k )α 2 (1 γ) 1 g+g L ɛg+ρ + (1 τ L )(1 α + α 2 γ) + β y [ ( ) ] η(1+τ Γ c y from (2), where β T /Y ; l = c )Γ c 1 η(1+τ c )Γ c+(1 α+α 2 γ)(1 τ L from (4); n = ) λσ g 1 α αγ g L from ] (18) and (23); and k = y from (A.12). [ (1 τk )α 2 (1 γ) ɛg+ρ 14

16 References Aghion, P. and P. Howi, 1992, A model of growh hrough creaive desrucion, Economerica 60, Aghion, P. and P. Howi, 1996, The observaional implicaions of Schumpeerian growh heory, Empirical Economics 21, Barro, R.J. and X. Sala-i-Marin, 1995, Economic growh (McGraw-Hill, New York). Caballero, R.J. and A.B. Jaffe, 1993, How high are he gians shoulders: An empirical assessmen of knowledge spillovers and creaive desrucion in a model of economic growh, in: O. Blanchard and S. Fischer, eds., NBER macroeconomics annual (MIT Press, Cambridge) Chamley, C., 1986, Opimal axaion of capial income in general equilibrium wih infinie lives, Economerica 54, Grossman, G. and E. Helpman, 1991, Innovaion and growh in he global economy (MIT Press, Cambridge). Devereux, M. and D.R.F. Love, 1994, The effecs of facor axaion in a wo-secor model of endogenous growh, Canadian Journal of Economics 27, Howi, P., 1999, Seady endogenous growh wih populaion and R&D inpus growing, Journal of Poliical Economy 107, Jones, L.E., E.E. Manuelli and P.E. Rossi, 1993, Opimal axaion in models of endogenous growh, Journal of Poliical Economy 101, Judd, K.L., 1985, Redisribuive axaion in a simple perfec foresigh model, Journal of Public Economics 28, King, R.E. and S. Rebelo, 1990, Public policy and economic growh: developing neoclassical implicaions, Journal of Poliical Economy 98, S126-S151. Milesi-Ferrei, G.M. and N. Roubini, 1998, Growh effecs of income and consumpion axes, Journal of Money, Credi and Banking 30, Pecorino, P., 1993, Tax srucure and growh in a model wih human capial, Journal of Public Economics 52, Rebelo, S., 1991, Long-run policy analysis and long-run growh, Journal of Poliical Economy 99, Romer, P.M., 1990, Endogenous echnological change, Journal of Poliical Economy 98, S71-S102. Segersrom, P.S., 1998, Endogenous growh wihou scale effecs, American Economic Review 88, Sokey, N.L. and S. Rebelo, 1995, Growh effecs of fla-rae axes, Journal of Poliical Economy 103, Young, A., 1998, Growh wihou scale effecs, Journal of Poliical Economy 106,

17 h (V) τ k (H) ** g * g g Figure 1: Growh Effec of Capial Income Tax