COMPLEMENTARY RELATIONSHIPS BETWEEN EDUCATION AND INNOVATION

Transcription

1 Discussion Paper No. 731 COMPLEMENTARY RELATIONSHIPS BETWEEN EDUCATION AND INNOVATION Kasuhiko Hori and Kasunori Yamada February 2009 The Insiue of Social and Economic Research Osaka Universiy 6-1 Mihogaoka, Ibaraki, Osaka , Japan

2 Complemenary Relaionships beween Educaion and Innovaion Kasuhiko Hori Kasunori Yamada February, 2009 Absrac This paper combines wo prooype endogenous growh models: he Schumpeerian endogenous growh model developed by [Howi(1999)] and human capial growh models developed by [Uzawa(1965)] - [Lucas(1988)]. While sandard Schumpeerian growh models sugges ha a subsidy o R&D has long-run effecs, we show ha a subsidy o human capial invesmen has a posiive impac on R&D effors as well as on human capial accumulaion. Because in our model, he per capia oupu growh rae depends on boh echnology improvemens and human capial accumulaion, he model bridges he gap beween he lieraure concerning Schumpeerian growh model and ha concerning growh empirics. JEL classificaion: O11; O31; O41 Keywords: Schumpeerian endogenous growh model; R&D; human capial; subsidy The firs auhor would like o acknowledge he financial suppor provided by he Global Cener of Excellence (GCOE) program eniled Raising Marke Qualiy Inegraed Design of Marke Infrasrucure of Keio Universiy. The second auhor is graeful for he research gran provided by he GCOE program eniled Human Behavior and Socioeconomic Dynamics of Osaka Universiy. Corresponding auhor: Insiue of Economic Research (KIER), Kyoo Universiy. Address: Yoshida Hon-machi, Sakyo-ku, Kyoo , Japan. address: hori@kier.kyoo-u.ac.jp Insiue of Social and Economic Research (ISER), Osaka Universiy. Address: Mihogaoka 6-1, Ibaraki , Japan. address: kyamada@econ.osaka-u.ac.jp

3 1 Inroducion This sudy aims o inegrae wo major srands of endogenous growh models: Schumpeerian endogenous growh models developed by [Young(1998)] and [Howi(1999)] and human capial accumulaion models developed by [Uzawa(1965)] and [Lucas(1988)]. We explain how a subsidy o human capial invesmen posiively affecs no only R&D effors bu also human capial accumulaion. A significan meri of he model below is ha he per capia oupu growh rae depends on boh echnology improvemens and human capial accumulaion. We argue ha when a Schumpeerian endogenous growh model mees a human capial accumulaion model, endogenous growh heory mees growh empirics. Ever since Jones criique ([Jones(1995)]), wo differen ypes of endogenous growh heories, namely, he Schumpeerian endogenous growh heory and he semi-endogenous growh heory, have sruggled for supremacy. The main proposiion of he former heory is ha a subsidy on R&D invesmen has long-run effecs, while he laer argues oherwise. In his paper, afer he recen finding by [Ha and Howi(2007)] ha he Schumpeerian endogenous growh heory accommodaes o he U.S. experience, we augmen Howi s Schumpeerian endogenous growh model o consider he human capial accumulaion process. 1 The moivaion is sraighforward: while in [Howi(1999)], human capial accumulaion is disregarded, here is a srong consensus among economiss ha human capial is also an engine of growh in addiion o echnology improvemens. In paricular, from he viewpoin of growh empirics lieraure since [Mankiw e al.(1992)mankiw, Romer, and Weil], if we disregard human capial, hen he esimaors in growh regressions will be biased because he per capia oupu growh rae is assumed o be aribuable o capial accumulaion, echnology progress, and human capial accumulaion. The model below provides a srucural basis for including human capial as a deerminan of long-run growh by using a Schumpeerian growh model. In his sense, his sudy can be aken as providing a microfoundaion for reduced form analyses in growh empirics lieraure, where he producion funcion of final oupu is assumed. Wih respec o policy implicaions, we obain he following resuls. Firs, a subsidy o R&D invesmen acceleraes he oupu growh rae as [Howi(1999)]. Second, i is shown ha a subsidy o human capial in- 1 A echnical feaure of his sudy is ha here are hree engines of growh: horizonal R&D, verical R&D, and human capial accumulaion. To he bes of our knowledge, his is he firs sudy feauring an endogenous growh model wih hree engines of growh. 1

4 vesmen has a direc posiive effec on oupu growh via he promoion of human capial accumulaion and has an indirec posiive effec hrough improvemens in echnology. 2 To he bes of our knowledge, his is he firs sudy ha demonsraes complee general equilibrium policy implicaions by aking ino accoun he endogenous deerminaions of boh echnology improvemens and human capial accumulaion. This paper is organized as follows. In Secion 2, we se up he model. The equilibrium is analyzed in Secion 3. Secion 4 focuses on a seady growh pah and derives chief proposiions, and Secion 5 concludes he paper. 2 The Model The basic seup of our model follows [Howi(1999)]. A significan difference beween [Howi(1999)] and our model is ha we consider human capial as he sole producion inpu for he inermediae goods secor, whereas [Howi(1999)] assume ha exogenously growing row labor force is inelasically supplied o his secor. In his sudy, human capial can be used for he producion of inermediae produc and for invesmen in human capial creaion. The populaion growh rae n is exogenously deermined, and he populaion size is denoed as L. The final goods oupu, which is he numeraire in he model, can be used for consumpion (C ), invesmen in verical R&D (Z A ), and invesmen in horizonal R&D (Z N ). 2.1 Producion srucure The final goods secor is under perfec compeiion. The echnology for final goods producion is specifically given by Y X 1 N 0 A i, xi, di, (1) where Y is he final goods oupu, X is a fixed resource such as land, N measures he varieies of inermediae goods a ime, A i, is a produciviy parameer aached o he incumben version of inermediae produc 2 See [Arnold(1998)] for his poin. In [Arnold(1998)], a subsidy o R&D invesmen has no long-run effecs since he model is semi-endogenous. In addiion, while a subsidy o human capial invesmen has posiive effecs according o [Arnold(1998)], he did no delve on he issue. 2

5 i, x i is he amoun of inermediae produc i used in he economy, and ¾ 0, 1µ is he capial share. Since his sudy assumes ha he oal endowmen of he fixed resource in he economy is equal o 1, we henceforh abbreviae variable X from he producion funcion. Under perfec compeiion, he firs-order condiion for he final goods secor wih respec o x i, is given as A i, x 1 i, p i,, (2) where p i, is he price of inermediae produc i. The inermediae goods secor is under monopolisic compeiion. In his sudy, we assume ha one uni of inermediae good is made from one uni of human capial. By his assumpion, wo sources of endogenous growh-r&d and human capial accumulaion-are ineraced. The profi in creaing inermediae produc i is given by È i, p i, x i, w x i,, where w is he real wage for human capial. Wih he demand funcion of x i, from (2), he firs-order condiion wih respec o x i, is hen given by w 2 A i, x 1 i,. Hence, he demand of x i, is deermined as x i, A 1 1 i, w 1 and he profis of an inermediae goods firm are given by 2.2 Innovaions È i 1 µ A 1 i,, (3) w 1. (4) Following [Howi(1999)], we consider wo ypes of innovaions. Verical innovaions improve produciviy in each inermediae goods secor i, A i,, and horizonal innovaions bring new varieies ino he economy, N Verical Innovaions The Poisson arrival rae of verical innovaions in each secor is defined as Z A ÐA ÐAz A, A 3

6 where ÐA 0 is he produciviy parameer of verical innovaions, and Z A is he amoun of resource devoed o verical R&D for each secor i. Here, z A Z A /A is he per secor produciviy adjused expendiure on verical R&D, and A can be regarded as he leading-edge produciviy parameer, wih A maxa i, i ¾ 0, N. From he zero-profi condiion for he verical R&D secor, he marke clearing condiion is given by ÐAz A V 1 s R µ Z A, (5) where V is he expeced presen value of a verical innovaion a ime from he sream of fuure profis, and s R ¾ 0, 1µ is he general subsidy rae o R&D. Because a every ime, he innovaion will be replaced by he nex innovaor wih he Poisson arrival rae, V is deermined as Ø exp r s ÐAz A µds ÔØ,dØ, (6) V where r s is he ineres rae, and ÔØ, is he profi of he incumben on dae Ø for any secor wih vinage echnology a ime. Furher, we can define he qualiy adjused value of a verical innovaion as v V /A. Finally, he inensiy of he qualiy improvemen for each verical innovaion is capured by a parameer 0, wih which he growh rae in he leading-edge produciviy is given as Horizonal Innovaions Ȧ A Ð Az A. (7) The variey of inermediae goods can be augmened by horizonal innovaions and he evoluion of varieies is specified as Ṅ Z ÐN NY1 A, where ÐN 0 and ¾ 0, 1µ are he parameers, and Z N is he amoun of numeraire devoed o horizonal innovaions. To guaranee he inner soluion, we impose he following assumpion. Assumpion 1. Ð A ÐN. 4

7 Each horizonal innovaion resuls in a new inermediae produc whose produciviy is randomly drawn from he disribuion of exising inermediae producs. Furher, from he definiion of he value of he leadingedge inermediae good A given by (6) and from he definiion of he profis of inermediae good firm wih qualiy A i, given by (4), he expeced value of a horizonal innovaion is derived as E A i, /A µ 1/ 1 µ V. Hence, from he zero-profi condiion in he horizonal R&D secor, we obain he nex condiion. Z ÐN NY1 A E Ai, A 1 1 V 1 s R µ Z N. (8) From he srucure described, he disribuion of relaive produciviy A i, /A converges o he ime-invarian disribuion funcion F qµ q 1/, where 0 q 1. 3 Hence, in he long run, we obain E Ai, A Households problem 1 1 / 1 µ 1 1. The maximizaion problem of a represenaive household is given by max exp Ö nµ log C d 0 where Ö n is he subjecive discoun rae, and C is he per capia consumpion. The laws of moion for financial asses and human capial in per capia erms are respecively given as Ẇ r nµw w u h C 1 s h µz h T, ѵ and ḣ Zh A 1 u µh 1 nh, Òµ (9) where W denoes he per capia financial asse, h is he per capia amoun of human capial, u ¾ 0, 1 is he raio of human capial devoed o he inermediae goods secor, Z h is he expendiure on human capial accumulaion, ¾ 0, 1µ, s h ¾ 0, 1µ is he subsidy rae o expendiure on human capial accumulaion, and Ñ and Ò are he co-sae variables aached 3 See [Howi(1999)] and [Segersrom(2000)] for he proof. 5

8 o he respecive consrains. We divide he amoun of expendiure (Z h ) by A because (1) he higher he leading-edge qualiy in he economy, he more difficul is he acquisiion of new skills o handle he cuing-edge echnology and (2) he more he economy has human capial, he more difficul i will be o obain addional human capial. Finally, he expendiure of he governmen is financed by lump sum ax (T ), and he budge of governmen is balanced a all imes. From he above specificaions, he firs-order condiions of he problem are obained as C 1 Ñ (10) and Ñw u Ò Ñw Ò 1 µ 1 u µ Ñ 1 s h µòa h 1 Z 1 h A h Z h (11) 1 u µ 1 (12) Ñ r nµ Ö nµñ Ñ (13) 1 µ 1 u µ 1 A h Z h n Ö nµò Ò. (14) In addiion, he usual oal variable coss (TVCs) are imposed on a and h. From (10) and (13), we obain From (11) and (12), we obain Z h Ċ C r Ö. (15) 1 u µ 1 µ 1 s h µ w h. (16) Subsiuing he above equaion back ino (11), we obain Ñ Ò 1 µ 1 1 s h µ A w 1. (17) Therefore, subsiuing (16) and (17) ino (14), we obain Ò Ò Ö 1 µ 1 1 s h µ A w. (18) Furher, (17) implies Ñ Ñ Ò Ò 1µẇ w Ȧ. A 6

9 (13) and (18) ogeher wih he above condiion lead o 1 µẇ w r 1 µ 1 1 s h µ A w. (19) Finally, subsiuing (16) ino (9), we obain ḣ h 1 µ 1 s h µ 1 u µa w n. (20) 3 Equilibrium In his secion, we derive he equilibrium pah of he model. Nex, as usual, i is convenien o define new variables ha are consan along he seady growh pah. Specifically, we define qualiy adjused human capial wage (Û w /A ), qualiy and human capial adjused per capia consumpion (c C / A h µ), and human capial per variey (l h L /N ). From (7) and (19), he evoluion of qualiy adjused human capial wage can be depiced as 4 Û Û r 1 µ 1 1 s h µ Û Ð Az A. (S1) In addiion, from (15) and (20), he evoluion of he qualiy and human capial adjused per capia consumpion is derived as ċ c r Ö n ÐAz A 1 µ 1 s h µ 1 u µû. (S2) Finally, from he definiion of l, (8) and (20), he following is derived. l l ḣ h L L Ṅ N 1 µ 1 s h µ 1 u µû Ð Nz N y, (S3) where y Y / A N µ is he qualiy and human capial adjused per capia oupu, and z N Z N /Y is he share of expendiure on horizonal R&D from oal oupu. In his model, he marke clearing condiions are obained as follows. For he final goods secor, we obain Y C L Z h L Z A N Z N. 4 Noice here ha he per secor produciviy adjused expendiure on verical R&D (z A ) is also saionary along he seady growh pah. 7

10 Dividing he above equaion by A N, we obain he inensive form resource consrain as From his, we obain y c l z h l z A z N y, z A 1 z N µy c z h µl. (21) A each ime, he human capial marke clears such ha N Using (3), (22) can be rewrien as 0 N A 1 1 i, w 1 0 x i, di u h L. (22) di u h L. (23) Wih he definiions of Û and l, we divide (23) by N o obain he inensive form version of (23), deermining u as u Û Nex, using he definiions of y, (1), and (3), we obain Nex, from (21) and (24), we obain z A 1 z N µ 2 y Û Û 1 l. (C1). (24) c z h µl. (C2) From he marke clearing condiion of verical R&D (5), we obain v 1 s R, (25) ÐA and from he marke clearing condiion of horizonal R&D (8) wih he definiion of, we obain ÐN 1 v 1 s R µz 1 N. Combining hese wo equaions, we obain z N Þ 1 Ð N 1 ÐA 8 1. (26)

11 Since 1 and ÐA ÐN are assumed, Þ hus derived saisfies he inner soluion condiion of 0 Þ 1. Finally, from he definiion of v, we obain v v V V Ȧ A. In he long run, v /v 0. We can derive V /V by using he definiions of V in (6), (7) and (25). Hence, a lile algebra leads o r ÐA 1 s R µ 1 1 µ Û 1 1 z A. (C3) The equilibrium dynamics of he model consiss of hree variables, c, Û, l µ. The dynamical sysems are given by (S1) (S3), ogeher wih hree insananeous variables z A, u, r µ given by (C1) (C3). 4 Seady Growh Pah In his secion, we focus on he seady growh pah, where ĉ, Û, and l are consan over ime. Hereafer, we add subscrip o any variable whenever i is consan in he seady growh pah. From (S1) and (S2), we obain u 1 µ 1 s h µ Ö nµû. (27) Nex, subsiuing (24), (26), and (27) ino (S3), we obain 1 µ 1 s h µ Û Ö Þ Û n. (28) Since he lef-hand side of he above equaion is increasing in Û, while he righ-hand side is decreasing, (28) uniquely deermines Û in he seady sae. Similarly, subsiuing (24), (26), and (27) ino (C1), we obain l Û u 1, (29) respecively. Therefore, subsiuing (C3) and (27) (29) ino (S1), we obain z A s R µ 1 1 µ 9 Þ Û ÐA Ö n ÐA (30).

12 Finally, from (21), (16), (24), (26), and (27) (30), we obain c 1 Þµ Û l 1 u µ 1 µ 1 s h µ Û z A l. (31) Therefore, (28), (29), and (31) give he candidaes of he seady sae values of he dynamical sysem concerning he equaions characerizing he economy, (S1) (S3). Therefore, we have he following proposiion. Proposiion 1. There exis Ö ¾ n, µ and Ð A ¾ 0, µ such ha he sysem of differenial equaions, (S1) (S3) wih (C1) (C3), has a unique seady-sae equilibrium, where u ¾ 0, 1µ and z A ¾ 0, µ, if Ö ¾ n, Öµ and ÐA ¾ Ð A, µ. The se of seady-sae values u, z A, c µ is given by (28), (29), and (31), ogeher wih (27) and (30). Proof. Since he candidaes of he seady-sae values are given by (28), (29), and (31), i is sufficien o show he exisence of Ö and Ð A. From (27), in order o saisfy u ¾ 0, 1µ, i mus hold ha Û Û Ö nµ 1 1 µ 1 s h µ. Therefore, i follows from he fac ha he righ-hand side of (28) is increasing in Û and ha he lef-hand side is decreasing, and he condiion for u ¾ 0, 1µ is given by 1 µ 1 s h µ Û Ö Þ 2 1 Û 1 n, 1 or 1 Þ 2 1 Ö nµ 1 1 µ 1 s h µ 1 Ö nµ. Since he righ-hand side of he above inequaliy diverges o as Ö n, and i is obvious ha u 0 from (27), we find ha here exiss a sufficienly small value Ö ¾ n, µ such ha u ¾ 0, 1µ if Ö ¾ n, Öµ. Finally, if follows from (30) ha z A s R µ 1 1 µ Û 0 as ÐA. Therefore, we find ha here exiss a sufficienly large value Ð A ¾ 0, µ such ha z A 0ifÐA ¾ Ð A, µ. 10

13 In his proposiion, he condiion ha Ö ¾ n, Öµ guaranees a posiive invesmen on human capial and he condiion ha Ð A ¾ Ð A, µ guaranees a posiive R&D expendiure. Therefore, we have he following proposiion. Proposiion 2. Suppose ha Ö ¾ n, Öµ and ÐA ¾ Ð A, µ, where Ö and Ð A are given in Proposiion 1. Then, he growh rae in he seady sae is given by g Ẏ g A g Y h n, where g A and g h are he respecive growh raes of A and h in he seady sae, given as g A Ȧ A Ð A z A (32) and g h ḣ Û Ö. (33) h 1 µ 1 s h µ Proof. From he definiion of y and he fac ha y is consan over ime in a seady sae equilibrium, he growh rae can be wrien as g g A g N, where g N is he growh rae of he variey of inermediae goods: g N Ṅ /N. From (7) and (30), he growh rae of he leading-edge qualiy of echnology, g A, is given by (32) in a seady sae. Moreover, from he definiion of l and he fac ha l is consan over ime in a seady sae, we obain g N g h n. We obain (33) by subsiuing (27) ino (20), which complees he proof. As is discussed in he inroducion, proposiion 2 bridges he gap beween he lieraure concerning Schumpeerian growh models and ha concerning growh empirics. Wih our specificaion, he growh rae of oupu depends on boh echnological improvemens and human capial accumulaion. Our model provides a srucural basis for including human capial as a deerminan of long-run growh by using a Schumpeerian growh model. Hence, his sudy can be aken as providing a microfoundaion for reduced form analyses in growh empirics lieraure. I should also be noed ha his is he firs sudy ha demonsraes complee general equilibrium policy implicaions by aking ino accoun he 11

14 endogenous deerminaions of boh echnology improvemens and human capial accumulaion. Nex, we provide he policy implicaions of he model as heorems. Theorem 1. The subsidy o R&D has a posiive effec on he growh rae of he leading-edge echnology, bu does no have any effec on he growh rae of human capial. òg A 0 òs R and òg h 0. òs R Proof. The differeniaion of (30) wih respec o s R gives dz A ds R s R µ 2 1 µ Û. (34) Here, i should be noed ha Û is deermined by (28), independen of s R. Since he righ-hand side of (34) is posiive, we obain he heorem. Therefore, our modificaion does no provide significan new insighs in erms of he effec of R&D subsidy. 5 On he conrary, he following heorem provides a new insigh ino he subsidy o human capial invesmen. Theorem 2. The subsidy o human capial invesmen has posiive effecs on he growh raes of he leading-edge echnology and human capial. òg A òs h 0 (35) and Proof. See Appendix A. òg h òs h 0. (36) Figure 1 represens he effec of subsidy o human capial invesmen on Û. In Figure 1, he RHS curve depics he righ-hand side of (28). Noe ha i is decreasing in Û. Two LHS curves in Figure 1 respecively depic he lef-hand side of (28) corresponding o he subsidy raes of human capial invesmen, s h and s ¼ h, where s¼ h s h. I should be noed ha he 5 The effecs of subsidy o R&D are clarified, for example, in [Howi(1999)]. 12

15 g h RHS LHS (s ¼ h ) LHS (s h ) g ¼ h g h O Û ¼ Û Û Figure 1: Effec of subsidy o human capial invesmen on Û lef-hand side of (28) is equal o g h and is increasing in Û. Moreover, since i is increasing in s h, he LHS curve shifs up as he subsidy rae on human capial invesmen increases from s h o s ¼ h. Therefore, Figure 1 shows ha an increase in s h raises he growh rae of human capial. Moreover, since i reduces he adjused wage rae Û and z A are decreasing in Û from (30), he increase in he subsidy rae on human capial also raises he growh rae of he leading-edge produciviy g A. Inuiively, he reason for he indirec posiive impac of subsidy o human capial invesmen on produciviy growh is explained as follows. Since human capial accumulaion is labor augmening, an increase in he growh rae of human capial reduces he adjused wage rae. This raises he adjused profi and he expeced value of firms (6), moivaing he enhancemen of horizonal and verical innovaions. This indirec effec on he subsidy of human capial invesmen, given by Theorem 2, provides a new insigh ino he subsidy policy, which is he main resul of his sudy. Hence, human capial invesmen augmens he growh rae of qualiy, which in urn acceleraes he growh of per capia oupu. Furher, if we consider only he relaionship beween he growh rae of oupu and R&D invesmen, as in [Howi(1999)], i misses he effec coming hrough human capial invesmen (and subsidy). The omission of his general equilibrium effec was a drawback in he excellen model of [Howi(1999)] 13

16 and is remediaed wih our specificaion. 5 Conclusion In his paper, afer he recen finding by [Ha and Howi(2007)] ha he Schumpeerian endogenous growh heory accommodaes o he U.S. experience, we augmen Howi s Schumpeerian endogenous growh model o consider he human capial accumulaion process. We show ha a subsidy o human capial invesmen has a posiive impac no only on R&D effors bu also on human capial accumulaion. Because in our model, he per capia oupu growh rae depends on boh echnology improvemens and human capial accumulaion, he model bridges he gap beween he lieraure concerning Schumpeerian growh model and ha concerning growh empirics. Appendix A Proof of Theorem 2 Differeniaing boh he sides of (28), we obain dû ds h 1 1 µ 1 s h µ 1 Û 1 1 µ 1 s h µ Û 1 Þ Û 1 0. Therefore, noing ha he lef-hand side of (28) is equal o g h and he righhand side is decreasing in Û, we obain (36). Similarly, since from (30), z A is decreasing in Û, i gives (35). References [Arnold(1998)] L. Arnold. Growh, Welfare, and Trade in an Inegraed Model of Human-Capial Accumulaion and Research. Journal of Macroeconomics, 20(1):81 105, [Ha and Howi(2007)] J. Ha and P. Howi. Accouning for Trends in Produciviy and R&D: A Schumpeerian Criique of Semi-Endogenous Growh Theory. Journal of Money, Credi and Banking, 39(4): ,

17 [Howi(1999)] P. Howi. Seady Endogenous Growh wih Populaion and R& D Inpus Growing. Journal of Poliical Economy, 107(4):715, [Jones(1995)] C.I. Jones. R & D-Based Models of Economic Growh. Journal of Poliical Economy, 103(4):759, [Lucas(1988)] R. Lucas. On he Mechanisms of Economic Developmen. Journal of Moneary Economics, 22(1):3 42, [Mankiw e al.(1992)mankiw, Romer, and Weil] N. G. Mankiw, David Romer, and David Weil. A Conribuion o he Empirics of Economic Growh. The Quarerly Journal of Economics, 107(2):407 37, [Segersrom(2000)] P. S. Segersrom. The Long-Run Growh Effecs of R&D Subsidies. Journal of Economic Growh, 5(3): , [Uzawa(1965)] H. Uzawa. Opimum Technical Change in an Aggregaive Model of Economic Growh. Inernaional Economic Review, 6(1):18 31, [Young(1998)] A. Young. Growh wihou Scale Effecs. Journal of Poliical Economy, 106(1):41 63,