GROWTH AND COMPETITION IN A MODEL OF HUMAN CAPITAL ACCUMULATION AND RESEARCH

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1 Volume IV/ Issue 3(9)/ Fall 2009 GROWT AND COMPETITION IN A MODEL OF UMAN CAPITAL ACCUMULATION AND RESEARC Domiique Biaco Uiversity of Nice-Sophia-Atipolis, GREDEG (CNRS), Frace domiique.biaco@gredeg.crs.fr Abstract: The aim of this paper is to aalyze the relatioship betwee competitio ad growth i a model of huma capital accumulatio ad research by disetaglig the moopolistic mark-up i the itermediate goods sector ad the returs to specializatio i order to have a better measure of competitio. We fid that the steady-state output growth rate depeds o the parameters describig prefereces, huma capital accumulatio techology ad R&D activity. We also show that the relatioship betwee competitio ad growth is iverse U shaped. This result that seems to be i lie these empirical results (Aghio ad Griffith, (2005)) is explaied by the resource allocatio effect. Keywords: edogeous growth, horizotal differetiatio, techological chage, imperfect competitio, huma capital JEL Classificatio: D43, J24, L16, 031, Itroductio Ecoomists have log bee iterested i the relatioship betwee competitio ad growth, but ecoomic theory seems to be cotradicted by the evidece. Ideed, the most importat growth models i which there exists a imperfect competitio show a decreasig relatioship betwee competitio ad growth [Romer, (1990), Grossma ad elpma, (1991) ad Aghio ad owitt, (1992)]. owever, a recet empirical work (Aghio ad Griffith, (2005)) which disputes the form of this relatioship, fids a iverted-u relatioship betwee competitio ad growth that is robust to may alterative specificatios ad remais true i the data for may idividual idustries. I order to recocile theory with evidece, Aghio ad Griffith (2005) ad Bucci (2005a) exted the basic Schumpeteria edogeous growth model. Aghio ad Griffith (2005) itroduce a escape competitio effect i the Aghio ad owitt (1992) model whereas Bucci (2005a) itroduces a resource allocatio effect i the Romer (1990) model. O the other had, recet theoretical works aalyze the relatioship betwee R&D ivestmet ad huma capital accumulatio ad its impact o growth [(Ziesemer, (1991); Eicher, (1996); Reddig, (1996); Arold, (1998); Blackbur et al. (2000); Sögre, (1998) ad Llyod-Ellis ad Roberts (2002))]. owever, all these studies do't focus o the effect of competitio o growth. More recetly, Bucci (2003) focuses explicitly o the relatioship betwee competitio ad growth. The purpose of this paper is to exted these works. Ideed, cotrary to all these models, first, by followig Beassy (1998) ad Biaco (2006), we itroduce a distictio betwee the returs to specializatio ad the market power parameter which allows us to have a better measure of the competitio. Ideed, i our model the market power parameter is ot strogly related to the returs to specializatio but it is completely idepedet. Secodly, ulike Bucci (2003), we itroduce two differet assumptios cocerig the kowledge accumulatio. I a first time, the degree of R&D difficulty i the sese that higher values of skilled labor force require more skilled labor allocated i the research sector to achieve the same level of the growth rate of kowledge, is assumed. I the secod time, by itroducig a exterality of kowledge, we itroduce the possibility to have two sources of growth, huma capital ad iovatio, ulike Bucci (2003) who assumes oly oe source of growth, huma capital. Our paper is structured as follows. Sectio 2 presets our model. Sectio 3 aalyses the market equilibrium. Sectio 4 discusses the relatioship betwee competitio ad growth i a geeral case. Fially, sectio 5 cocludes. 341

2 Joural of Applied Ecoomic Scieces 2. The model The model developed is based o Bucci (2003) 1 model i which we disetagle the returs to specializatio from the market power. The ecoomy is structured by three sectors: fial good sector, itermediate goods sector ad R&D sector The fial good sector I this sector atomistic producers egage i perfect competitio. The fial good sector produces a composite good by usig all the th type of itermediate goods x ad skilled labor 2. Productio is give by: = A 1 α γ 1 +α α x 0 Where: α ad γ ] 0, 1[ d, (1) ad A are techological parameters. This productio fuctio allows us to disetagle the degree of market power of moopolistic 1 competitors i the itermediate sector ( 1 ) ad the degree of returs from specializatio γ 3. I this α sese, this model is a geeralizatio of the Bucci (2003) ad the Arold (1998) models 4. Uder perfect competitio i the fial output market ad the factor iputs markets, the represetative firm chooses itermediate goods ad skilled labor i order to maximize its profit takig prices as give ad subect to its techological costrait. The first order coditios are the followigs: π x π x =αa = 1 α γ 1+α x α 1 p α γ 1+α α 1 ( 1 α) A x w = 0 = 0, (2), (3) where W is the wage rate i the fial good sector ad p is the price of the th itermediate good. Equatio (2) is the iverse demad fuctio for the firm that produces the th itermediate good whereas equatio (3) characterizes the demad fuctio of skilled labor. 2.2 The itermediate goods sector I the itermediate goods sector, producers egage i moopolistic competitio. Each firm produces oe horizotally differetiated itermediate good ad has to buy a pateted desig before producig it. Followig Grossma ad elpma (1991), Bucci (2003), Bucci (2005b) ad Bucci (2005c), we assume that each local itermediate moopolist has access to the same techology employig oly skilled labor h : x = B. (4) h Usig the first order coditio, we obtai the price of the th itermediate good: w p =, (5) B α where w is wage rate i the itermediate goods. At the symmetric equilibrium, all the firms produce the same quatity of the itermediate good, face the same wage rate ad by cosequece fix the same price for their productio. The price is equal to a costat mark up α 1 over the margial cost B w. 1 We use the otatios of Bucci (2003) i order to have a direct compariso with his model. 2 Time subscripts are omitted wheever there is o risk of ambiguity. 3 Beassy (1996) made a simple modificatio of the Dixit ad Stiglitz (1977) model which clearly disetagles taste for variety ad market power. At the same time, Beassy (1998) ad de Groot ad Nahuis (1998) show that the itroductio of this modificatio i a edogeous growth model with expadig product variety à la Grossma ad elpma (1991) affects the welfare aalysis. 4 Ideed, we obtai the Bucci (2003) model by itroducig the followig costrait i our model. I the same way, we obtai the Arold (1998) model by itroducig the followig costraits ad A=1 i our model. 342

3 Volume IV/ Issue 3(9)/ Fall 2009 Defiig by = h d, the total amout of labor employed i the itermediate goods sector, 0 we ca rewrite the equatio (4): B x =. (6) Fially, the profit fuctio of the firm 5 which produces the th itermediate good is: α γ 1 α 1 π = A α α B. (7) ( ) α The R&D sector There are competitive research firms udertakig R&D. Followig Diopoulos ad Thompso (1999), we assume that ew blueprits are produced usig old blueprit, a amout of R&D skilled labor ad the skilled labor force : C =, (8) Where C is a productivity parameter. Ulike Bucci (2003), we explicitly assume that positive spillover effect is attached to the available stock of disembodied kowledge (approximated by the existig umber of desigs, ) i discoverig a ew product variety. Aother importat differece is the existece of a dilutio effect 6 i our model. Ideed, the equatio (8) captures i a very simple way the idea that R&D difficulty grows with the labor force 7. Accordig to this effect, the ivetio of a ew itermediate requires a share of R&D skilled labor i the skilled labor force =, which chages over time because of both C iovatio ad skilled labor growth. While iovatio geerates a positive iter-temporal exterality, the skilled labor growth teds to reduce iovatio via a fall i the R&D productivity. Because of the perfect competitio i the R&D sector, we ca obtai the real wage by usig the zero profit coditio: w = V, (9) where w represets the real wage eared by R&D skilled labor ad th is the real value of such a blueprit which is equal to: r( t ) V = τ π e dτ, τ> t, (10) t sice the research sector is competitive, the price of the th desig at time t will be equal to the discouted value of the flow of istataeous profits that is possible to make i the itermediate goods sector by the th firm from t owards. Give V, the free etry coditio leads to: V w =. (11) 2.3 The cosumer behavior The demad side is characterized by the represetative household who holds asset i the form of owership claims o firm ad chooses plas for cosumptio (c), asset holdigs (α) ad huma capital (h) 8. Followig Lucas (1988), we assume that the household is edowed with oe uit of time ad optimally allocates a fractio u of this time edowmet to productive activities (fial good, itermediate goods ad research productio) ad the remaiig fractio (1 u) to o productive 5 I order to have a egative relatioship betwee competitio ad profit, we assume that. 6 Diopoulos ad Segestrom (1999) have itroduced this kid of effect i a edogeous growth model. I our model, ulike Diopoulos ad Segestrom (1999), this effect is also liked to the huma capital accumulatio. 7 For more details about this issue, see Diopoulos ad Thompso (1999), Diopoulos ad Seer (2006) ad Joes (2005). 8 Like Bucci (2003), Bucci (2005b) ad Bucci (2005c), for the sake of simplicity, we assume that there is o populatio growth. 343

4 Joural of Applied Ecoomic Scieces activities (educatio). Followig Romer (1990), we assume that the utility fuctio of this cosumer is 9 : U = 0 e ρt log( c ) dt, (12) where c is private cosumptio, ρ is the rate of pure time preferece. The flow budget costrait for the household is: a= wuh= ra c, (13) where w is the wage rate per uit of skilled labor services. The huma capital supply fuctio is give by: h= ( 1 u )h, (14) where δ > 0 is a parameter reflectig the productivity of the educatio techology. From the maximizatio program of the cosumer 10, the first order coditios are: 1 ρt λ1 = e, (15) c λ1 = λ 1 r, (16) λ2 =λ1wu+λ 2δ( 1 u ), (17) δ λ1 =λ2, (18) w Equatio (15) gives the discouted margial utility of cosumptio which satisfies the dyamic optimality coditio i equatio (16). Equatio (18) gives the static optimality coditio for the allocatio of time. The margial cost of a additioal uit of skills devoted to workig evolves optimally as i equatio (17). Coditios (15) through (18) must satisfy the costraits (13) ad (14), together with the trasversality coditios: lim t λ 1 at = 0, (19) lim t λ 2 ht = 0, (20) 3. The equilibrium ad the steady state I this sectio, we characterize the equilibrium ad give some aalytical characterizatio of a balaced growth path. 3.1 The equilibrium It is ow possible to characterize the skilled labor market equilibrium i the ecoomy cosidered. O this market, because of the homogeeity ad the perfect mobility across sectors, the arbitrage esures that the wage rate that is eared by employees who work i the fial good sector, itermediate goods sector or R&D sector is equal. As a result, the followig three coditios must simultaeously be checked: * u = S + S + S, (21) w = w, (22) w = w. (23) Equatio (21) is a resource costrait, sayig that at ay poit i the time the sum of the skilled labor demads comig from each activity must be equal to the total available supply. Equatio (22) ad equatio (23) state that the wage eared by oe uit of skilled labor is to be the same irrespective of the sector where that uit of skilled labor is actually employed. We ca characterize the product market equilibrium i the ecoomy cosidered. Ideed, o this market, the firms produce a fial good that ca be cosumed. Cosequetly, the followig coditio must be checked: = C. (24) 9 This specificatio of the utility fuctio does ot alter our results. 10 The cotrol variables of this problem are ad whereas ad are the state variables. ad deote the shadow price of the household's asset holdigs ad huma capital stock. 344

5 Volume IV/ Issue 3(9)/ Fall 2009 Equatio (24) is a resource costrait o the fial good sector. We ca describe the capital market equilibrium i our ecoomy. Because the total value of the household's assets must be equal to the total value of firms, the followig coditio must be checked: a= V, (25) where V is give by the equatio (10) ad satisfies the followig asset pricig equatio: V = rv π. (26) 3.2 The steady state At the steady state, all variables as, c,, a, 11,,, grow at a positive costat rate. Propositio : If u is costat the all the other variables grow at strictly positive rate :, (27), (28). (29) Proof. From the equilibrium o the skilled labor market, give by the equatio (21), it easy to show that if u is costat. Because of the assumptio o the size of the represetative household ad the populatio growth rate, it is obviously that. Combiig these two coditios, we get. From the defiitio of the firm research process, give by the equatio (8), we obtai that. From the equilibrium o the product market, give by the equatio (24), it is easy to fid that. The equatios (7, 10, 25, 26) implies that. By substitutig equatio (6) ito equatio (1), the by logdifferetiatig the equatio (1), we obtai. By combiig the previous equatios, we fid the equatio (28). Usig the previous equatios, we ca demostrate the followig steady state equilibrium values for the relevat variables of the model 12 :, (30), (31), (32), (33), (34), (35), (36). (37) Accordig to the equatio (30), the real iterest rate (r) is costat. Equatios (31), (32) ad (33) give the amout of skilled labor i each sector at the equilibrium. Equatio (34) represets the optimal ad costat fractio of the household's time edowmet that it will decide to devote to work (u * ) at the equilibrium. Equatio (35) states that the growth rate of huma capital depeds o techological ad preferece parameters (δ ad ρ ) 13. Ulike the Lucas (1988), Blackbur et al. (2000) ad Bucci (2003) papers, the equatio (36) shows that the growth rate of the iovative activity depeds ot oly o preferece ad techological parameters but also o competitio. Ulike the Lucas (1988), Blackbur et al. (2000) 14 ad Bucci (2003) 15 papers, the equatio (37) shows that the 11 Give the assumptios o the size of the represetative household ad the populatio growth rate, which implies that we ca use istead of. 12 See Biaco (2008) for the demostratio of these results. 13 The coditio also assures that. 14 I these models, output growth depeds oly o huma capital accumulatio ( ). 345

6 Joural of Applied Ecoomic Scieces growth rate depeds ot oly o the competitio (α ) ad huma capital accumulatio ( ) but also o the degree of returs from specializatio (γ ) ad a productivity parameter (C). As i the recet Lucas (1988), Blackbur et al. (2000) ad Bucci (2003) papers, the market power eoyed i the moopolistic sector does ot play ay role o the cosumers decisio about how much time to ivest i educatio ad traiig (such a decisio beig solely drive by the parameters describig prefereces ad huma capital techology). 3.3 The relatioship betwee product market competitio ad growth I this sectio, we study the log ru relatioship betwee competitio ad growth i the model preseted above. Followig most authors (Bucci (2003); Aghio et al. (2005); Aghio ad Griffith (2005); Aghio ad owitt (2005); Bucci (2005a); Bucci (2005b); Bucci (2005c); Biaco (2006) ad Biaco (2007)), we use the so-called Lerer Idex to gauge the itesity of market power withi a market. Such a idex is defied by the ratio of price (P) mius margial cost (Cm) over price. Usig P P Cm the defiitio of a mark up ( Mark up= ) ad Lerer Idex ( Lerer Idex= ), we ca use (5) Cm P to defie a proxy of competitio as follows: (1 Lerer Idex ) =α. (38) We show that i this model which cosists i havig the moopolistic mark-up i the itermediate goods sector ad the returs to specializatio treated separately, the relatioship betwee competitio ad growth is iverse U shaped. This theoretical result is i lie with empirical results [Aghio et al. (2005)]. Propositio: The relatioship betwee competitio ad growth is iverse U shaped. Proof. The proof is obtaied by differetiatig (37) with respect to α :. (39) As C, δ, ρ> 0 ad 0 < γ< 1$, the we the sig of the derivative is give by the opposite sig of 2α 1. Fially, we obtai that if ad oly if α< ad if ad oly if < α To elighte Propositio 2, remark that ay icrease i competitio has a o liear effect o the skilled labor allocated to the research sector ( ). This meas that the resource allocatio effect seems to predict a iverted-u relatioship betwee product market competitio ad growth. Moreover, the huma capital accumulatio which is the secod source of growth i our model has a positive ad liear effect o growth. Fially, due to the compositio of these two effects, the relatioship betwee product market competitio ad growth is iverse U shaped. 4. Coclusio I this paper, we preseted a geeralizatio of the Bucci (2003) model i which we disetagle the moopolistic mark-up i the itermediate goods sector ad the returs to specializatio i order to have a better measure of competitio. Ideed, i our model the market power parameter is ot strogly related to the returs to specializatio but it is completely idepedet. Moreover, we itroduce the degree of R&D difficulty i the sese that higher values of skilled labor force require more skilled labor allocated i the research sector to achieve the same level of the growth rate of kowledge. We also itroduce aother importat differet. Ideed, ulike Bucci (2003), we explicitly assume that positive spillover effect is attached to the available stock of disembodied kowledge i discoverig a ew product variety. The results of the model ca be summarized as follows. First of all, ulike Bucci (2003), the steady-state output growth rate depeds o the parameters describig prefereces, huma capital 15 I this model, output growth depeds ot oly o huma capital accumulatio ( ) but also o competitio ( ). 346

7 Volume IV/ Issue 3(9)/ Fall 2009 accumulatio techology ad R&D activity. Secodly, ulike Bucci (2003), we fid that the relatioship betwee competitio ad growth is iverse U shaped. This result that seems to be i lie these empirical results (Aghio et al. (2005)) is explaied by the resource allocatio effect. 5. Refereces [1] Aghio, P.; Bloom, N.; Bludell, R.; Griffith, R. ad owitt, P., (2005), Competitio ad Iovatio: A Iverted-U Relatioship, i: Quarterly Joural of Ecoomics, Vol. 120, No. 2, pp [2] Aghio, P. ad Griffith, R., (2005), Competitio ad Growth: Recocilig Theory ad Evidece. MIT Press, Cambridge ad Lodo. [3] Aghio, P. ad owitt, P., (1992), A Model of Growth through Creative Destructio, i: Ecoometrica, Vol. 60, No. 2, pp [4] Aghio, P. ad owitt, P., (2005), Growth with Quality-Improvig Iovatios: A Itegrated Framework, i: adbook of Ecoomic Growth, vol. 1, pp Elsevier Sciece, Amsterdam, North ollad. [5] Arold, L.G., (1998), Growth, Welfare, ad Trade i a Itegrated Model of uma Capital Accumulatio ad Research, i Joural of Macroecoomics, Vol. 20, No. 1, pp [6] Beassy, J.-P., (1996), Taste for Variety ad Optimum Productio Patters i Moopolistic Competitio," i: Ecoomics Letters, Vol. 52, pp [7] Beassy, J.-P., (1998), Is There Always Too Little Research i Edogeous Growth with Expadig Product Variety? i: Europea Ecoomic Review, Vol. 42, pp [8] Biaco, D., (2006), Competitio ad Growth i a Edogeous Growth Model with Expadig Product Variety without Scale Effect, i: Brussels Ecoomic Review/Cahiers Ecoomiques de Bruxelles, Vol. 49, N 3, pp [9] Biaco, D., (2007), A iverted-u Relatioship betwee Product Market Competitio ad Growth i a Exteded Romeria Model: A Commet, i: Rivista di Politica Ecoomica, September- October, pp [10] Biaco, D., (2008). Growth ad Competitio i a Model of uma Capital Accumulatio ad Research, MPRA, Workig Paper [11] Blackbur, K.; ug, V. T.. ad Pozzolo, A. F., (2000), Research, Developmet ad uma Capital Accumulatio, i: Joural of Macroecoomics, Vol. 22, N 2, pp [12] Bucci, A., (2003), R&D, Imperfect Competitio ad Growth with uma Capital Accumulatio, i Scottish Joural of Political Ecoomy, Vol. 50, N 4, pp [12] Bucci, A., (2005a), Iverted-U Relatioship betwee Product Market Competitio ad Growth i a Exteded Romeria Model, i: Rivista di Politica Ecoomica, Vol. 95, pp [13] Bucci, A., (2005b), Moopoly Power i uma Capital based Growth, i: Acta Oecoomia, Vol. 55, N 2, pp [14] Bucci, A., (2005c), Product Market Competitio, R&D Effort ad Ecoomic Growth, UNIMI - Research Papers i Ecoomics, Busiess, ad Statistics 1011, Uiversita degli Studi di Milao. [15] de Groot,. L. ad Nahuis, R., (1998), Taste for Diversity ad the Optimality of Ecoomic Growth, i: Ecoomics Letters, Vol. 58, pp [16] Diopoulos, E. ad Segestrom, P., (1999), A Schumpeteria Model of Protectio ad Relative Wages, i: America Ecoomic Review, Vol. 89, N 3, pp [17] Diopoulos, E. ad Seer, F., (2006), New Directios i Schumpeteria Growth Theory, i: Edgar Compaio to Neo-Schumpeteria Ecoomics. Edward Elgar, Chelteham, UK. 347

8 Joural of Applied Ecoomic Scieces [18] Diopoulos, E. ad Thompso, P. (1999), Scale Effects i Schumpeteria Models of Ecoomic Growth, i: Joural of Evolutioary Ecoomics, Vol. 9, N 2, pp [19] Dixit, A. K. ad Stiglitz, J. E., (1977), Moopolistic Competitio ad Optimum Product Diversity, i: America Ecoomic Review, Vol. 67, N 3, pp [20] Eicher, T. S., (1996), Iteractio betwee Edogeous uma Capital ad Techological Chage, i: Review of Ecoomic Studies, Vol. 63, N 1, pp [21] Grossma, G. M. ad elpma, E., (1991), Iovatio ad growth i the global ecoomy. MIT Press, Cambridge ad Lodo. [22] Joes, C. I., (2005), Growth ad Ideas, i: adbook of Ecoomic Growth, Vol. 1, pp Elsevier Sciece, Amsterdam, North ollad. [23] Lloyd-Ellis,. ad Roberts, J., (2002), Twi Egies of Growth: Skills ad Techology as Equal Parters i Balaced Growth, i:" Joural of Ecoomic Growth, Vol. 7, N 2, pp [24] Lucas, R. J., (1988), O the Mechaics of Ecoomic Developmet, i: Joural of Moetary Ecoomics, Vol. 22, N 1, pp [25] Reddig, S., (1996), The Low-Skill, Low-Quality Trap: Strategic Complemetarities betwee uma Capital ad R&D, i: Ecoomic Joural, Vol. 106, N 435, pp [26] Romer, P. M., (1990), Edogeous Techological Chage, i: Joural of Political Ecoomy, Vol. 98, N 5, pp. S71-S102. [27] Sögre, A., (1998), uma Capital, R&D ad Ecoomic Growth, Stockholm School of Ecoomics, Workig Paper Series i Ecoomics ad Fiace 238. [28] Ziesemer, T., (1991), uma Capital, Market Structure ad Taxatio i a Growth Model with Edogeous Techical Progress, i: Joural of Macroecoomics, Vol. 13, N 1, pp

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