Col lecció d Economia

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1 DOCUMENTS DE TREBALL DE LA FACULTAT DE CIÈNCIES ECONÒMIQUES I EMPRESARIALS Col lecció d Economia E7/179 Cos-Based Models of Economic Growh* Daniel Cardona Universia de les Illes Balears Fernando Sánchez-Losada Universia de Barcelona Adreça correspondència: Deparamen de Teoria Econòmica Facula de Ciències Econòmiques i Empresarials Universia de Barcelona. Av. Diagonal Barcelona (Spain) Phone: Fax fernando.sanchezlosada@ub.edu *We acknowledge he useful commens of Salvador Origueira, Xavier Raurich, and Anonio Manresa. Fernando Sánchez-Losada acknowledges he financial suppor from he Generalia de Caalunya hrough gran 25SGR984 and Miniserio de Ciencia y Tecnología hrough gran SEJ Daniel Cardona acknowledges he financial suppor from he Miniserio de Ciencia y Tecnología hrough gran SEJ This paper was begun before, sopped while, and finished afer my faher was in he hospial. In memoriam of José Manuel Sánchez-García.

2 Absrac: In his paper we highligh he imporance of he operaional coss in explaining economic growh and analyze how he indusrial srucure affecs he growh rae of he economy. If here is monopolisic compeiion only in an inermediae goods secor, hen producion growh coincides wih consumpion growh. Moreover, he paern of growh depends on he paricular form of he operaional cos. If he monopolisically compeiive secor is he final goods secor, hen per capia producion is consan bu per capia effecive consumpion or welfare grows. Finally, we modify again he indusrial srucure of he economy and show an economy wih wo differen growh speeds, one for producion and anoher for effecive consumpion. Thus, boh he operaional cos and he paricular srucure of he secor ha produces he final goods deermines ulimaely he paern of growh. JEL Classificaion: O4, O41, O47 Keywords: monopolisic compeiion, operaional cos, growh Resum: En aques aricle, es fa èmfasi en la imporància dels cosos operacionals en explicar el creixemen econòmic i s analiza com l esrucura indusrial afeca la axa de creixemen de l economia. Si només hi ha compeència monopolísica en secors de béns inermedis, aleshores el creixemen de la producció coincideix amb el creixemen del consum. A més, l esquema de creixemen, endogen o semi-endogen, depèn de la forma paricular del cos operacional. Si el secor on es produeix la compeència monopolísica és la de béns finals, aleshores la producció per càpia és consan però el consum per càpia efeciu o benesar creix. Finalmen, si es modifica un alre cop l esrucura indusrial de l economia, es é una economia amb dos velocias de creixemen diferens, una per la producció i l alre pel consum efeciu. Es conclou doncs que an els cosos operacionals com l esrucura paricular del secor que produeix els béns finals deerminen en úlim erme l esquema de creixemen.

3 1 Inroducion In his paper we analyze growh in an economy wih monopolisic compeiion characerized by he presence of operaional coss. To his end, we firs presen a reduced model o show ha he popular growh models based on R&D are specific cases of his reduced model. The growh source is simply he growh of he number of firms or goods in he economy, which is direcly relaed o he opporuniy of having profis. Second, we inroduce microfoundaions ino he reduced model and analyze he relaionship beween he firms growh mechanism and he paern of growh. In his par, and as he curren models of economic growh, here is monopolisic compeiiononlyinaninermediaegoodssecor,andproduciongrowhcoincides wih consumpion growh. Third, we modify he indusrial srucure of he economy by considering monopolisic compeiion in he final goods secor, and describe an economy where per capia producion is consan bu per capia effecive consumpion or welfare experiences growh. The economy seems o be a a sandsill bu individuals are beer off. Finally, we show an economy wih wo differen growh speeds, one for producion and anoher for effecive consumpion. The growh models based on R&D have a common propery: here is an inermediae goods secor. A firm of his secor is he only producer of one inermediae good, wha confers o i a cerain degree of marke power and, hence, posiive profis. These profis are used o ren a paen, which in urn guaranees o he firm ha i can produce and, moreover, i is he only producer of he good. Therefore, he exisence of a secor wih posiive profis is necessary for R&D o arise. Oherwise, no economic agen would devoe any amoun of resources o engage in an R&D aciviy. This mechanism reveals ha a requiremen for growh o arise is no necessarily R&D, bu profis. A paen is a ype of operaional cos, i.e., a cos ha does no depend on he quaniy produced bu, a he same ime, producion canno be engaged if his cos is no suppored. Moreover, he operaional cos is necessary o fix he number of inermediae firms. Therefore, any economy wih a secor in which firms have boh marke power and an operaional cos can experience growh. The paern of growh will be deermined by he specific ype of operaional cos. The raionale is ha he exisence of a free enry condiion causes ha he operaional cos ulimaely deermines boh he number of firms and he quaniy of capial and labor per firm and, hus, growh. In secion 2, we show he influence of he operaional cos and he number of firms on growh hrough a reduced model of monopolisic compeiion. The reduced model allows o compare growh raes by considering an ad-hoc rule ha fixes he number of inermediae firms. In paricular, by assuming a consan operaional 3

4 cos and fixing he evoluion of firms as a linear funcion of boh labor and he number of firms, we obain he resuls of Romer (199): endogenous growh wih scale effecs. Consan populaion is required for a balanced growh pah o exis. Insead, by assuming an operaional cos ha is linear in oal populaion and an evoluion of firms ha does depend non linearly on boh labor and he number of firms, we obain he resuls of Jones (1995): semiendogenous growh, i.e., here is growh if and only if here is populaion growh. The reduced model sheds ligh on he necessiy in Jones (1995) of he operaional cos o depend on oal populaion insead of he workers employed by a firm, which denoes some ype of scale effec. We endogeneize he number of firms in order o deepen ino he relaionship beween growh and operaional coss in secion 3. We depar from he model proposed by Coo-Marínez, Garriga and Sánchez-Losada (27), where he number of firms or varieies is endogenous. The economy has wo secors: an inermediae goods secor wih monopolisically compeiive firms, and a compeiive final goods secor where he firms combine he inermediae goods à la Dixi-Sigliz. However, i is aken ino consideraion he formulaion proposed by Ehier (1982) or Benassy (1996) ha separaes he reurns o specializaion from he monopolisic mark-up. Moreover, and in conras wih Romer (199) and Jones (1995), he monopolisically compeiive secor uses labor as a producion inpu. In his ype of economy, he enrance of a new firm in he marke has wo opposie effecs on he incumben firms: a complemenary effec and a business-sealing effec. When a new firm eners he marke, i does no ake ino accoun he posiive effec on aggregae produciviy, which increases he demand of he incumben firms. This is he complemenary effec. Moreover, a he same ime he presence of a new inermediae firm means ha he final good producers can choose among a greaer variey of (parially subsiuable) inpus, which decreases he demand of he incumben firms. This is he business-sealing effec. The naure and evoluion of he operaional cos plays a crucial role in he deerminaion of each effec and, hus, on he incenives for new firms o enry. In his paper we consider an operaional cos funcion ha depends on he pas hisory by assuming ha hese coss depend on how producion has been organized in he pas. This implies ha since he producion funcion does no vary, no firm will change is operaional behavior if i implies a higher operaional cos. 1 We assume an operaional cos funcion such ha only more capial inensive firms may improve he operaional echnology and induce growh. Wih his cos srucure, we show ha endogenous growh is characerized by some knife-edge condiion: 2 he 1 Fuure research should address he case where a higher operaional cos is associaed wih an improvemen of he oal facor produciviy of he producion funcion. 2 Growiec (27) defines a knife-edge condiion as a condiion imposed on parameer values 4

5 economy experiences endogenous growh only when he operaional cos is asympoically linear in he capial per firm. Oherwise, populaion growh is required o grow. We pay special aenion o wo paricular cases ha illusrae he imporance of he operaional coss. Firs, we assume ha he operaional cos is consan and independen of he quaniy produced, as in Masuyama (1995). Examples are fixed mainenance coss, managerial coss, or simply enry barriers as adverising. In his case, here is no mechanism o induce grow oher han populaion growh. Wih consan populaion he complemenary effec compensaes he business-sealing effec and, herefore, here is no incenive for a new firm o ener he marke. Insead, populaion growh ranslaes ino economic growh hrough he growh of he number of firms. When labor supply grows, he wage decreases in he shor run, which alleviaes he business-sealing effec and causes he complemenary effec o reinforce enry, implying ha in he long run labor demand increases. As a consequence, firms end up o become more capial inensive. The second case assumes ha he operaional cos varies along he ime. In paricular, we assume ha he operaional cos is relaed o boh he own capial and he (pas) average level of capial per firm. 3 The idea is ha he operaional echnology may be endogenously improved in economies where each firm accouns for and conrols only a small amoun of capial, and here is a process of diffusion of such operaional echnology improvemens. Now, a posiive populaion growh enhances economic growh bu i is no necessary o experience growh. Posiive economic growh only needs improvemens in he echnology associaed o operaional coss in order o compensae he negaive effecs of enry due o he business-sealing effec. The number of firms grows posiively bu each one becomes more capial inensive and a he same ime smaller by hiring less capial and labor. 4 The firm has always he elecion beween coninuing wih he old operaional echnology or adaping ohenewonewihadifferen operaional cos. Since firmshirelesscapial,he operaional cos echnology is improved and, herefore, firms end up choosing he new echnology. Assuming ha he operaional cos is no only posiively relaed o he average level of capial per firm, bu also o he mean of he labor growh used in a represenaive firm (wha means ha he operaional cos depends on henumberofworkers)orohegrowhofhenumberoffirms (wha allows o sudy eiher when he difficuly o exploi a new produc increases wih he number of producs or he opposie, when echnology spreads wih he number of firms), such ha he se of values saisfying his condiion has an empy inerior in he space of all possible values. 3 In Pereo (1999), firms only use labor and direcly engage in R&D expendiures in order o improve he marginal produciviy (or lower he marginal coss) of labor. 4 A firm mus be inerpreed as a producive process. 5

6 gives he same qualiaive conclusions han in he case of he operaional cos only relaed o he average level of capial per firm. The resuls are consisen wih he finding of Jones (22) abou he inexisence of a relaionship beween growh and he number of researchers. Tha he economy becomes more capial inensive in he growh process has been recenly sressed, among ohers, by Givon (26), Zulea (26) or Lis and Zhou (27). However, and in conras wih hese auhors, our economy does no need any inended R&D expendiure made by firms o save labor in he producion process. I is he evoluion of he operaional coss wha makes he economy o become more capial inensive. In he models of endogenous growh ypically producion growh coincides wih consumpion growh. In secion 4 we show ha a change in he secor ha is monopolisically compeiive changes he resuls abou he growh rae. We consider an economy wih differeniaed consumpion and invesmen goods. In paricular, we assume ha he individual buys several differeniaed final goods and derive uiliy hrough a love of variey parameer from a mix of hem, wha we call effecive consumpion. Invesmen goods are produced by compeiive firms hrough a mix of final goods. There is no aggregae reurns o specializaion for he invesmen. In his economy per capia producion does no grow regardless of he assumed operaional cos. However, having he same amoun of final goods per capia does no mean ha per capia effecive consumpion (or welfare) does no grow. In oher words, real per capia producion is he same bu he subjecive value he consumer gives o he producion grows. In paricular, wih consan operaional coss we have effecive consumpion semiendogenous growh whereas wih operaional coss posiively relaed o he average capial per firm populaion growh is no necessary o have effecive consumpion growh. Finally, we modify again in secion 5 he indusrial srucure of he economy and show ha assuming boh love of variey and aggregae reurns o specializaion in he invesmen secor gives a differen posiive final goods growh rae han he effecive consumpion one. We analyze when populaion growh is needed o have boh producion and effecive consumpion growh. The exercise made in his paper shows ha boh he operaional cos and he paricular indusrial srucure deermines ulimaely he paern of growh. Firs, a echnological change affecing he operaional coss (ex. he promoion of echnology diffusion hrough a Marshallian indusrial disric) can have dramaic consequences on economic growh by moving an economy from a semiendogenous o an endogenous growh paern. Second, he indusrial or secorial srucure can explain a leas par ofheeconomicgrowh 5 and, a he same ime, i can hide par of his growh, i.e., 5 An example, from a differen poin of view, is Alonso-Carrera and Raurich (26). 6

7 he effecive consumpion growh. Hence, accouning for growh should go beyond a Solow residual decomposiion exercise. 2 Profis, number of firms and economic growh In his secion, we show ha monopolisic compeiion can generae endogenous (semiendogenous) growh when he marke srucure evolves and becomes more (remains equally) compeiive as a consequence of an increase in he number of firms. Assume here is a unique finalgoodwhichisproducedbycompeiivefirms using a coninuum of inermediae goods. Toal populaion bl grows a a consan rae, so ha bl /bl 1 = n. A final goods firm maximizes Y Z z p i x i di, (1) where x i and p i are he quaniy and price of he inermediae good i in period, respecively, and z is he oal number of inermediae goods in period, which is aken as given by he (compeiive) finalgoodssecor. Wehavenormalizedhe final goods price o one. The producion funcion Y depends in a symmeric way on he available inermediae goods, which are no perfec subsiues. Therefore, from his profis maximizaion problem we can recover a demand funcion for each inermediae good. In each period, new inermediae goods producers may ener and produce a new variey. Each firm produces a mos one inermediae inpu x i. In order o operae, firms have o pay an operaional cos ψ. An inermediae goods firm i maximizes π i = p i x i w L i (1 + r ) K i ψ, (2) where π i is he profis funcion, x i = Ki 1 α L α i, K i and L i are he capial and labor used by firm i, respecively, w is he wage, and r is he ineres rae, so ha here is complee depreciaion. We have assumed ha he operaional cos is measured in erms of he final good. Since inermediae goods are no perfec subsiues, firms in he inermediae goods secor face a downward slopping demand curve which confers hem some degree of marke power. Then, he profis funcion can be simplified o 6 π i = ηp i K 1 α i L α i ψ, (3) 6 We assume ha he operaional cos is compleely ou of he scope of he firm decisions. In he nex secion, we sudy a more general case where he operaional cos can parially depend on he firm decisions. 7

8 where η is he inverse of he elasiciy of he demand for each inermediae good and measures he degree of marke power. Since here is perfec compeiion in he final goods secor, in a symmeric equilibrium where all firms produce he same oupu level x i = x,wihhesame quaniy of inpus K i = K and L i = L, se he same price p i = p,andhavehe same profis π i = π,wehaveha Y = z p K 1 α L α. (4) There is free enry in he inermediae goods secor. Thus, he oal number of inermediae goods z is deermined by he zero profi condiion and, hence, Eq.(3) is equal o zero. Applying his condiion o Eq.(4), he per capia final goods growh can be wrien as 7 g y+1 = g z +1 g ψ+1, (5) n where y = Y /bl is per capia final goods producion, and g h+1 = h +1 /h denoes he growh beween +1 and of he variable h. FromEq.(5)iisclearha he paricular paern of growh depends on he assumed operaional cos funcion. Moreover, we could idenify differen operaional cos funcions for eiher differen hisoric imes or differen degrees of developmen. Therefore, if we idenify he funcional form of he cos funcion and he evoluion of he number of firms, we would be able o also idenify he growh rae of he economy. Obviously, he number of firms in he economy is he resul of he free enry condiion. Then, depending on he naure of he operaional cos, associaed eiher o he same inermediae secor or o anoher secor, we have differen growh mechanisms. Nex, we show wo of he mos popular cases of his mechanism when he operaional cos is associaed o a hird secor. In he endogenous growh model of Romer (199), he operaional cos is assumed o be consan and equal o he price of a paen, and he number of inermediae firms coincides wih he number of paens, which are produced in anoher secor. Hence, by assuming ha he number of paens evolves as z +1 z = δl z, z, (6) where L z, is labor used o produce (or search) a new paen a, andδ is a posiive consan, he per capia final goods growh becomes g y+1 = 1+δL z,. (7) n 7 In naional accouning, final goods producion corresponds o he Gross Domesic Produc, i.e., Y z ψ. Since in his paper he dynamics of Y is he same han he dynamics of Y z ψ, hereinafer we concenrae on Y. 8

9 We need consan populaion for a balanced growh pah o exis, i.e., L z, mus be fixed. Obviously, L z, comes from he free enry condiion. If here is populaion growh, hen growh is no balanced, since L z, in Eq.(7) is permanenly growing whereas n is a consan. Moreover, wihou populaion growh, he counry wih he bigges populaion would have he bigges L z, and, hen, he bigges per capia final goods growh. This is he reason why his model is said o have scale effecs. In he endogenous growh model of Jones (1995), he operaional cos is also assumed o be equal o he price of a paen, bu no necessarily consan, and he number of inermediae firms coincides wih he number of paens, which are produced in anoher secor. In paricular, i is assumed ψ = ψbl, where ψ is a posiive consan. This in urn means ha he price of a paen increases if and only if populaion grows. 8 The number of paens is assumed o evolve as z +1 z = δl z, z φ l λ 1 z,, (8) where l z, is he mean of he labor used in a represenaive firm ha produces paens, i.e., an exernaliy accruing from he duplicaion of R&D, and φ and λ are consans. The parameer φ capures he fac ha he sock of curren paens can affec eiher posiively or negaively he producion of new paens. In a symmeric equilibrium, where L z, = l z,, he growh of he number of firms is ³ g z+1 =. (9) 1+δL λ z,z φ 1 Populaion canno be consan in a balanced growh pah, since herefore he growh of he number of firms z does no allow he facor 1+δL λ z,z φ 1 o be consan. Such facor remains consan if g z = n λ/(1 φ).rewriingeq.(5),wehave g y = n λ/(1 φ). (1) Hence, we need populaion growh for a balanced growh pah o exis. This is he reason why his model is said o be a semiendogenous growh model. When populaion does no grow, he operaional cos for he inermediae goods firms collapses ino a consan and, hen, growh is no possible. In order o realize he role of he operaional cos in he deerminaion of growh, assume insead ha ψ = ψl z,. Then, in a symmeric equilibrium and a balanced growh pah, as boh z L z, and z L are consan proporions of he oal populaion, we have from Eq.(5) ha g y =1regardless of he number of paens accumulaion law given by Eq.(8). In his sense, we can say ha he necessiy in Jones (1995) of he operaional cos o depend on oal populaion insead of he labor mean employed by a firm denoes some ype of scale effec. 8 I can be inferred from equaions (A2) and (A15) in Jones (1995). 9

10 Obviously, g z depends on he assumed ψ. We deail his relaionship in he nex secion. 3 A cos-based model of endogenous growh We consruc an economy wih aggregae reurns o scale o analyze how he free enry condiion ogeher wih he paricular evoluion of he operaional coss deermine he growh rae of he economy. In sandard models (see Mankiw and Whinson, 1986), free enry reduces welfare in he sense ha he increase in he aggregae operaional coss is no compensaed by he posiive benefis arising from increasing compeiion or business-sealing effec: new firms enering he marke have a negaive impac on he incumben firms demand. Hence, he marke equilibrium can generae excessive enry. However, in he presence of increasing reurns o specializaion he enrance of a new inermediae firm has a complemenary effec: i increases he aggregae produciviy and, hus, he incumbens demands. Therefore, when considering he complemenary effec he free enry may yield an inefficienly low number of firms. The evoluion of he operaional coss will deermine he new enry in he marke and, hus, he aggregae produciviy growh, and a he same ime he growh of he aggregae losses due o he presence of he operaional coss. Final goods producion: There is a unique final good which is produced by compeiive firms hrough a coninuum of inermediae goods, wih he following echnology (as in Benassy, 1996): 9 Y = µ Z z z v(1 η) η 1 x 1 η 1 η i di, η (, 1), v (, 1). (11) In a symmeric equilibrium, all he firms in he inermediae goods secor produce hesameoupulevelx and, hus, aggregae oupu is Y = z v+1 x. Then, he elasiciy of oupu wih respec o he number of firms z is given by he degree of reurns o specializaion v, as in Ehier (1982). This parameer measures he degree o which sociey benefis from specializing producion beween a large number of inermediae goods z. As a resul, an increase in he number of inermediae goods improves he oal facor produciviy of he final goods echnology. This formulaion allows o separae he effec of he mark-up from he economies of scale. 1 Since 9 We mainain he same definiion of he variables as in he previous secion. 1 The convenional formulaion esablished by Dixi and Sigliz (1977) corresponds o he case v = η/ (1 η) < 1, where here exiss a one-o-one relaionship beween he marke power and he degree of reurns o specializaion. 1

11 hereisfreeenryinheinermediaegoodssecor,aheaggregaelevelhenumber of inermediae goods z is deermined by he zero profis condiion. However, he represenaive firm in he final goods secor akes his value as given. From he profis maximizaion problem, given by v(1 η) η max z 1 η {x i } µz z 1 x 1 η 1 η i di Z z p i x i di, (12) we obain he inverse demand funcion for each inermediae inpu, x i =(p i ) 1 v (1 η) 1 η z η Y. (13) Inermediae goods producion: Each inermediae goods firm solves max π i = p i x i (1 + r ) K i w L i ψ, (14) {p i,k i,l i } subjec o he final goods secor demand, Eq.(13), and where x i = Ki 1 α L α i. The operaional cos can depend parially on he firm decisions, bu no compleely. Noe ha an operaional cos means ha here are goods neiher consumed direcly by individuals nor invesed in capial. We have assumed here is complee capial depreciaion. 11 The associaed firs-order condiions of he firm problem yield 1+r = p i (1 η)(1 α) K α i L α i ψ Ki, (15) w = p i (1 η) αki 1 α Li α 1 ψ Li, (16) where ψ h is he parial derivaive of ψ wih respec o h. In a symmeric equilibrium final oupu is equal o and he price, by subsiuing Eq.(17) ino Eq.(13), is Y = z v+1 K 1 α L α, (17) p = z v. (18) The free enry condiion (each inermediae firm makes zero profis, i.e., π =) deermines he number of firms. Formally, and using Eq.(18), we have ηz v K 1 α L α = ψ ψ K K ψ L L. (19) Since he final cos is definedinermsofhefinal oupu, he enry of any firm reduces he relaive price beween final oupu and inermediae goods 1/p = z v and, 11 Capial depreciaion does no vary he qualiaive resuls. 11

12 hus, i makes enry more profiable. However, individual firms do no inernalize his (complemenary) effec. Noe ha in our model we obain he sandard formulaion where p =1when v =. In his case, aggregae reurns o specializaion are absen and, hence, here would be only one (normalized) firm. Consumers: We assume Solow individuals: a any period, each individual j saves a consan fracion of her income R j and is endowed wih one uni of labor ha she supplies inelasically. Therefore, savings for he individual j are where s (, 1) is he consan propensiy o save. 12 S j = sr j, (2) Labor marke clearing condiion: In equilibrium, labor demand and supply coincides, i.e., z L = L b. (21) As populaion grows a a consan rae, we have g z+1 g L+1 = g L +1 = n. (22) Noe ha assuming anoher secor employing labor would modify he labor marke clearing condiion and, hus, he paricular indusrial srucure of he economy would become crucial in deermining he growh rae of he economy. Capial marke clearing condiion: The amoun saved by individuals a equals he sock of physical capial a +1;i.e., Z L S j dj = s Z L R j dj = z +1 K +1. (23) Noing ha R L R j dj = w L b +(1+r ) z K = z [w L +(1+r ) K ]= (1 η) z v+1 K 1 α L α z ψk K + ψ L L, where we have used he definiion of he individuals income and Eqs (21), (15), (16) and (18), he previous equaion becomes or s (1 η) z v+1 K 1 α L α z ψk K + ψ L L = z+1 K +1, (24) s (Y z ψ )=z +1 K +1. (25) Balanced growh pah: The dynamics of he model can be reduced o he capial accumulaion Eq.(24) and he free enry condiion Eq.(19), which using Eqs (21) and (22) can be wrien as ηn g K +1 g L+1 = s (1 η) ψ L ψ K K ψ L, (26) K 12 An infinie horizon consumer wih CES preferences gives he same qualiaive resuls. 12

13 and ηbl v K α L α v = ψ L ψ K K ψ L. (27) K Thus, for a balanced growh pah o exis, from Eq.(26) we have ha in equilibrium he operaional cos funcion mus be asympoically of he form ψ = 1 HK + ψ (1 η) K K + ψ L L where H denoes a posiive consan. 13 In ha case, from Eq.(27) we have and combining i wih Eq.(26) gives g L = g α α v K (28) n v α v, (29) ηn α v α v g α v K = sh. (3) From Eqs (17), (21) and (29), we obain µ n g y = g K v α v. (31) For α>v,hisequaionclearlyinformsusabouhenegaive(posiive)effec of he capial per firm growh (populaion growh) on he growh of per capia oupu; i.e., small firms (or producive processes) are he source of growh. Thus, any mechanism whose effecs are ranslaed ino g K < will induce posiive growh. 14,15 Nex, in order o complee he analysis, we assume a paricular funcional form for he operaional cos. 3.1 Homogeneous operaional coss Operaional coss consis of resources ha he firm consumes, i.e., hey consiue final goods ha are neiher consumed direcly by individuals nor invesed in capial. We assume he following specificaion of he operaional cos: ψ = ψk γξ L γ(1 ξ) Φ ({k τ } τ=1 ), (32) 13 Noe ha Eq.(27) informs ha ψ canno be homogeneous of degree one in K and L, since hen here would be no equilibrium. 14 Noe ha g K < is compaible wih more capial inensive firms. Also noe ha his resul depends crucially on he fac ha he producion funcion does no vary. 15 Alhough capial per firm growh has he opposie sign han he oupu growh, we have a balanced growh pah because aggregae capial growh and oupu growh coincide. 13

14 where k τ is he average level of capial per firm a period τ, ψ>, γn [, o1) and ξ [, 1]. The funcion Φ is defined as Φ ({k τ } τ=1 )=min τ=1,..., k β τ, wih β>. This means ha he operaional cos depends on how producion has been organized in he pas. In paricular, large producive srucures are less able o improve he operaional echnology han small ones. Moreover, a any ime he firm has always he elecion beween coninuing wih he old echnology or adaping ohenewonewihadifferen operaional cos. However, in order o simplify noaion and since he economy will evolve such ha k 1 is decreasing, we direcly wrie Φ ({k τ } τ=1 )=kβ 1. Noe ha his operaional cos funcion is homogenous of degree γ<1in K and L, bu in aggregae i exhibis homogeneiy of degree γ + β. This formulaion guaranees he exisence of a balanced growh pah as i saisfies Eq.(28). Noe ha he cases in which γ 6= mean ha he firm realizes ha he operaional cos parially depends on producion. In he Appendix we show boh he exisence and he sabiliy properies of he balanced growh pah for any parameer configuraion. Endogenous growh is obained if and only if γ + β =1and ξ =1; i.e., he operaional cos funcion is homogeneous of degree one in aggregae wih respec o k 1 and K. Thus, he usual knife-edge condiion for endogenous growh o arise is reduced o he propery ha operaional coss mus be linear on he average capial per firm. We nex illusrae he imporance of he operaional coss on economic growh hrough wo special cases: ψ = ψ and ψ = ψk γ k 1 γ γ = β = This parameer configuraion gives a consan operaional cos, i.e., ψ = ψ for all, as in Masuyama (1995). This means ha he operaional cos is no only independen of he produced quaniy, bu also on how he echnology has been used in he pas. We need o assume ha v<αin order o have convergence. This means ha if he aggregae reurns o specializaion are oo high, he complemenary effec can never be compensaed by he business-sealing effec and, hen, muliplying he number of firms simply by reducing he size of each firm makes growh o be explosive. In his case, from Eq.(28) we have H =(1 η) ψ/k and, as i mus be asympoically consan, he unique possible balanced growh pah mus saisfy ha K is asympoically consan, so ha g K =1. From Eqs (29), (22) and (31), 16 Oher differen operaional cos funcions, as ψk γ k β 1 (L /l 1 ) ε, where l 1 is he average level of labor per firm a period 1, ψk γ k β 1 (l /l 1 ) ε or ψk γ k β 1 (L /l 1 ) ε (z /z 1 ) ϕ give hesamequaliaiveconclusions. 14

15 we obain g K =1, g L = n v α v, g z = n α α v, gy = n v α v. (33) Finally, Eq.(3) yields K = s (1 η) ψ/ηn α α v. The economy experiences semiendogenous growh. When populaion grows, he number of firms grows a a posiive rae and labor demand per firm grows a a negaive rae. Thus, firms become smaller and more capial-inensive, implying a higher labor produciviy, which in urn makes per capia growh o increase. Noe ha no only he produciviy of labor increases, bu also he produciviy of capial can increase because of he complemenary effec. If here is no populaion growh, hen any variable grows regardless of he reurns o specializaion ξ = 1 and β = 1 γ This parameer configuraion means ha he operaional cos is only relaed o he capial per firm, i.e., ψ = ψk γ k 1 γ 1. In equilibrium, where k 1 = K 1, 17 we have H = ψg γ 1 K (1 η γ), which in urn does impose he resricion γ + η<1o have a posiive balanced growh pah. We need o assume ha v<α(1 γ) / (2 γ) in order o have convergence. Eqs (3), (29), (22) and (31) yield g K = D (α v) n α α(1 γ) v(2 γ), gz = D α n α(2 γ) α(1 γ) v(2 γ), g L = D α n [v(2 γ)+α] α(1 γ) v(2 γ), 1 v(2 γ) gy = D v n α(1 γ) v(2 γ), (34) where D = [η/sψ (1 γ η)] α(1 γ) v(2 γ). The economy experiences a combinaion of endogenous growh and semiendogenous growh. In case of zero populaion growh, we have posiive growh if ψ<η/s(1 γ η). We need a sufficienly low uni operaional cos in order ha firms ener he marke. Oherwise, he operaional cos canno be compensaed by profis. Wih posiive growh, he number of firms increases bu each firm hires less capial and labor, such ha hey become more capial inensive. The complemenary effec due o he growh in he number of firms dominaes he business-sealing effec (less producion per firm), which in urn makes producion per capia o grow. Populaion growh reinforces he magniude of each growh rae. The fac ha labor and capial per firm grows a a negaive rae regardless of he populaion growh means ha he operaional cos decreases and, herefore, firms choose he new echnology. 18 However, noe ha 17 If insead of k 1 we have k by assuming Φ ({k τ } τ= ) hen here would be no dynamics. 18 Alernaively, we could hink ha echnology becomes more sandard or, wha is he same, ha knowledge diffusion is faser and, hen, he operaional cos associaed o creae a new firm decreases. 15

16 g z g ψ > 1, which means ha he aggregae operaional cos increases. 4 Monopolisic compeiion in he final goods secor By he problem of economic developmen I mean simply he problem of accouning for he observed paern, across counries and across ime, in levels and raes of growh of per capia income. This may seem oo narrow a definiion, and perhaps i is, bu hinking abou income paerns will necessarily involve us in hinking abou many oher aspecs of socieies oo, so I would sugges ha we wihhold judgmen on he scope of his definiion unil we have a clearer idea of where i leads us. (R.E. Lucas, Jr., 1988, p. 3) In his secion, we show ha a differen indusrial srucure dramaically changes he inerpreaion of he resuls obained regarding growh. In paricular, we show ha accouning for he observed paern in levels and raes of growh of per capia income does no necessarily explain economic developmen. We consider a formulaion wih differeniaed consumpion and invesmen goods. In paricular, we assume ha he individual j buys several differeniaed final goods and derives uiliy from he following mix, ha we call effecive consumpion, c j = µ Z z 1 z v(1 η) η x j 1 η 1 η ci di, η (, 1),v (, 1), (35) where x j c i is he final good produced by firm i and consumed by individual j, and now v is a love of variey parameer. Invesmen goods, I, are produced by compeiive firms hrough he following echnology: I = µ Z z z η 1 x 1 η 1 η I i di, η (, 1), (36) where x Ii is he final good produced by firm i used o produce invesmen goods. Noe ha in order o concenrae on he love of variey we have assumed ha here is no aggregae reurns o specializaion for he invesmen. Also noe ha he same varieies are used o consume and o produce invesmen goods and ha boh share he inverse of he elasiciy of he demand for each inermediae good η. 16

17 Consumers: Since individuals save a consan fracion of heir income, each individual solves X µ Z z 1 max β z v(1 η) η {{x ci } z x j 1 η 1 η ci di (37) i= } = = The firs order condiion is s.o Z z p i x j c i di =(1 s) R j. (38) β c j η z v(1 η) η x j η ci = λ p i, i, (39) where λ is he Lagrange muliplier associaed o Eq.(38). From Eq.(39) we have x j c m = µ pi p m 1 η x j ci, i, m. (4) In order o recover each individual good demand, combine Eqs (38) and (4) o ge Z z Z z Z p m x j c m dm = p 1 1 η m p 1 η i xj c i dm = p 1 z η i xj c i p 1 1 η m dm =(1 s) R, j (41) where he second equaliy comes from he fac ha good i is infiniesimal. From his equaion we have he good i demand as Z L x j c i dj = (1 s) R L R j dj R p 1 η i. (42) z dm 1 p1 η m Savings are used by he individuals o buy invesmen goods ha are rened o he final goods producion firms. Invesmen goods producion: Each firm solves max P {x Ii } z I I p i x Ii di (43) i= subjec o Eq.(36), and where P I is he price of he invesmen good. The inverse demand funcion for each inermediae good is µ 1 pi η x Ii = z 1 I. (44) P I Z z Final goods producion: Each firm solves max {p i,k i,l i } Ã Z L! p i x j c i dj + x Ii P I (1 + r ) K i w L i P I ψ, (45) 17

18 s.o Z L x j c i dj + x Ii = K 1 α i L α i, (46) and Eqs (42) and (44), where K i denoes he invesmen goods rened o he firm by he individuals. 19 The associaed firs-order condiions of he firm problem yield P I (1 + r )=p i (1 η)(1 α) K α i L α i P I ψ Ki, (47) w = p i (1 η) αki 1 α L α 1 i P I ψ Li. (48) In he symmeric equilibrium, x j c i = x j c,x Ii = x I and p i = p for all i. Noe ha his implies ha x ci = x c, i.e., all he aggregae demands are equal. Subsiuing Eq.(36) ino Eq.(44) gives he relaive prices P I = p. (49) As in he previous secion, we consider one final good p i as he numéraire and normalize is price o one. 2 Hence, we have ha p = P I =1, and final oupu is equal o Y = z K 1 α L α. (5) Thefreeenrycondiionyields ηk 1 α L α = ψ ψ K K ψ L L. (51) Capial marke clearing condiion: Since now he invesmen goods have a price, he condiion becomes Z L Z L S j dj = s Rdj j = P I z +1 K +1. (52) Noing ha R L R j dj = w L b + P I (1 + r ) z K = z [w L + P I (1 + r ) K ]= (1 η) z K 1 α L α z ψk K + ψ L L, where we have subsiued for he prices, he condiion becomes s (1 η) z K 1 α L α z ψk K + ψ L L = z+1 K +1. (53) Consumpion: From Eqs (35) and (38) we have c j = z v+1 x j c = z v (1 s) R j. (54) 19 We have measured he operaional cos in erms of he invesmen good in order o analyze he cases where he operaional cos is compleely ou of he scope of he firm. 2 Alernaively, we could assume ha he numéraire is he invesmen good price, P I =1. Therefore, in a symmeric equilibrium monopolisic firms choose heir prices such ha hey are also one. 18

19 Aggregaing, and using Eq.(53), we have bl c = Z L Z L c j dj = z v (1 s) where c is he effecive consumpion per capia. Rdj j = (1 s) zv z +1 K +1, (55) s Balanced growh pah: As in he previous secion, boh he free enry and he capial accumulaion condiions deermine he dynamics of he economy. Combining Eqs (22), (53) and (51), hese condiions can be wrien as ηn g K +1 g L+1 = s (1 η) ψ L ψ K K ψ L, (56) K and ηk 1 α L α = ψ ψ K K ψ L L. (57) These condiions are he same as Eqs (26) and (27), evaluaed wih a differen price for he inermediae goods. Hence, he properies regarding exisence and sabiliy are he same han hose of he previous secion. Therefore, combining Eqs (51) and (22) give g K = g L, (58) g z g K = n. (59) From Eqs (55) and (59) we obain µ v n g c =. (6) And Eqs (5), (51) and (59) yield g K g y =1. (61) Hence, he economy does no grow regardless of he assumed operaional cos. However, having he same amoun of final goods per capia does no mean ha per capia effecive consumpion does no grow. We illusrae his asserion hrough he same wo operaional coss of he previous secion. γ = β = : Ifψ = ψ for all, henwehave g K = g L =1, g z = n, g c = n v. (62) Eq.(56) deermines he level of capial per firm. The economy experiences semiendogenous effecive consumpion growh. Once capial per worker has been fixed, he size of he firms does no vary wih he populaion growh, bu i does he number of 19

20 firms. As a resul, alhough he amoun of consumpion of each good decreases, he oal amoun of goods consumed remains unchanged and he effecive consumpion increases due o he love of variey parameer. ξ = 1 and β = 1 γ: Ifψ = ψk γ k 1 γ 1,henwehave g K = g L = G 1 n 1 1 γ, gz = Gn 2 γ 1 γ, g c = G v n v(2 γ) 1 γ, (63) where G = [η/sψ (1 η γ)] 1/(1 γ). The economy experiences endogenous and semiendogenous effecive consumpion growh. Discussing prices and oupu growh: In his economy, growh of final goods (zero) does no coincide wih growh of he individual effecive consumpion (posiive). In oher words, real producion is he same bu he subjecive value he consumer gives o he producion grows. In he marke, per capia expendiure in consumpion R z p ix j c i di remains unchanged. The reason is ha in adding differen goods up, we need o use relaive prices and, hen, o measure oupu in unis of one specific good. Inhelieraure,i isypically assumedaneffecive consumpion price index P c. 21,22 In our case, we would have P c = p z v, so ha if we measure in unis of effecive consumpion hen g y = g c. The main problem wih his uni of measure is ha i does no exis as a good in he economy and, hen, here is no economic reasonwhygoodsshouldbemeasuredinermsofagoodhadoesnoexisin he economy. Moreover, if we consider a differen love of variey parameer for each individual, hen each individual has her own price index, and i would be impossible o find ou an aggregae price index. On he oher hand, since he prices of all he goods of his economy are he same, a direc esimaion of he naional produc would give no growh a all. Neverheless, his paricular economic srucure shows ha he way producion is measured by he curren governmens underesimaes real (effecive consumpion) growh. Thus, accouning for he observed paern in levels and raes of growh of per capia income does no necessaily explain economic developmen. 5 Love of variey and aggregae reurns o scale Nex, we show ha assuming boh love of variey and aggregae reurns o scale in he invesmen secor gives a differen posiive final goods growh rae han he ³ R 21 In paricular, in our economy i would be P c = z v+η/(1 η) z p(η 1)/η i di η/(η 1). 22 An example of effecive consumpion wihou love of variey is Pereo (1998). 2

21 effecive consumpion one. In paricular, we assume ha individuals effecive consumpion is µ Z z 1 c j = z vc(1 η) η x j 1 η 1 η ci di, η (, 1),vc (, 1), (64) and ha invesmen goods are produced hrough he following echnology: µ Z z 1 I = z v I(1 η) η x 1 η 1 η I i di, vi (, 1), (65) where v c and v I areheloveofvarieyandheaggregae reurns o specializaion parameers, respecively. This model ness he economy of he previous secions. For his reason we skip all he seps. Balanced growh pah: If one final good is he numéraire, hen p =1and P I = z v I. In he balanced growh pah, we have he following relaionships: ηn α v I α v α v I g I K = s (1 η) ψ L ψ K K + ψ L, (66) K α g z = g α v I K αvc v I (1 α) α v g c = g I K n α α v I, (67) n vi (1 α)+αvc α v I, (68) g y = g v I (1 α) α v I K n vi (1 α) α v I. (69) γ = β = : Ifψ = ψ for all, henwehaveg K =1, and g z = n α α v I, g L = n v I α v I, g y = n v I (1 α) α v I, g c = n v I (1 α)+v cα α v I, (7) where v I <αin order o have convergence. The economy experiences semiendogenous final goods growh and semiendogenous effecive consumpion growh, bu he las is greaer han he final goods growh. ξ = 1 and β = 1 γ: Ifψ = ψk γ k 1 γ 1,henwehave g K = M v I α n α, g z = M α n α(2 γ) α(1 γ) v I (2 γ), g L = M α n α v I (2 γ) α(1 γ) v I (2 γ), g y = M vi(1 α) n v I (1 α) 2 α v I, g c = M v I(1 α)+v c α n v I (1 α)2 +α(1 α)vc α v I, (71) 1 α(1 γ) v where M = [η/sψ (1 η γ)] I (2 γ) and v I < α(1 γ) / (2 γ) in order o have convergence. The economy experiences endogenous and semiendogenous growh for boh final goods and effecive consumpion, bu again effecive consumpion growh is greaer han final goods growh. 21

22 6 Concluding remarks The main conribuion of his paper is o sress he imporance of he evoluion of operaional coss, as hey aler he marke srucure of he economy, which in urn deermines he growh paern. Moreover, also he indusrial srucure has been found o be crucial when inerpreing growh. In paricular, we show ha a real per capia producion growh can be differen han he per capia effecive consumpion growh or he subjecive value he consumer gives o producion. We sill know no much abou growh. We need o invesigae more on he relaionship beween echnology, marke srucure and preferences. Also, facors as human capial or public infrasrucures could accelerae an indusrial srucure change. Obviously, he endogeneizaion of he mark-up, differen mark-ups for differen secors, or differen economic srucures, would shed more ligh abou his relaionship. 22

23 References [1] Alonso-Carrera, J. and X. Raurich, 26, Growh, Secorial Composiion, and he Wealh of Naions, Barcelona Economics Working Paper Series 278, CREA. [2] Benassy, J.P., 1996, Tase for Variey and Opimum Producion Paerns in Monopolisic Compeiion, Economics Leers 52, [3] Chrisiaans, T., 24, Types of Balanced Growh, Economics Leers 82, [4] Coo-Marínez, J., Garriga, C. and F. Sánchez-Losada, 27, Opimal Taxaion wih Imperfec Compeiion and Aggregae Reurns o Specializaion, forhcoming in Journal of he European Economic Associaion. [5] Dixi, A.K. and J.E. Sigliz, 1977, Monopolisic Compeiion and Opimum Produc Diversiy, American Economic Review 67, [6] Ehier, W.J., 1982, Naional and Inernaional Reurns o Scale in he Modern Theory of Inernaional Trade, American Economic Review 72, [7] Givon, D., 26, Facor Replacemen versus Facor Subsiuion, Mechanizaion and Asympoic Neuraliy, mimeo, Hebrew Universiy. [8] Growiec, J., 27, Beyond he Lineariy Criique: he Knife-Edge Assumpion of Seady-Sae Growh, Economic Theory 31, [9] Jones, C.I., 1995, R&D-Based Models of Economic Growh, Journal of Poliical Economy 13, [1] Jones, C.I., 22, Sources of U.S. Economic Growh in a World of Ideas, American Economic Review 92, [11] Lis, J.A. and H. Zhou, 27, Inernal Increasing Reurns o Scale and Economic Growh, NBER Working Paper [12] Lucas, R.E., Jr., 1988, On he Mechanics of Economic Developmen, Journal of Moneary Economics 22, [13] Mankiw, N.G. and M.D. Whinson, 1986, Free Enry and Social Inefficiency, Rand Journal of Economics 17, [14] Masuyama, K., 1995, Complemenariies and Cumulaive Processes in Models of Monopolisic Compeiion, Journal of Economic Lieraure 33,

24 [15] Pereo, P.F., 1998, Technological Change and Populaion Growh, Journal of Economic Growh 3, [16] Pereo, P.F., 1999, Cos Reducion, Enry, and he Inerdependence of Marke Srucure and Economic Growh, Journal of Moneary Economics 43, [17] Romer, P.M., 199, Endogenous Technological Change, Journal of Poliical Economy 98, S [18] Zulea, H., 26, Facor Saving Innovaions and Facor Income Shares, mimeo, Universidad de Rosario. 24

25 Appendix Exisence and sabiliy of he balanced growh pah Applying he operaional cos Eq.(32) o he free enry condiion Eq.(19) we have ηz v K 1 α L α =(1 γ) ψ. (A.1) Combining Eqs (24), (32) and (A.1) gives µ 1 γ η z s ψ η = z +1 K +1. (A.2) Using Eqs (32) and (21), Eqs (A.1) and (A.2) ransform in equilibrium ino bl v K 1 α γξ L α v γ(1 ξ) K β 1 = (1 γ) ψ, (A.3) η µ 1 γ η sψ bl L γ(1 ξ) 1 K γξ K β 1 = L η b +1 L 1 +1K +1. In view of Eq.(A.3), he following resul is immediae. (A.4) Proposiion 1 When α v γ (1 ξ) =and v 6= 1 γ β, hen 1. g K = n v/(1 v γ β). 2. The balanced growh pah is sable if β/(1 v γ) ( 1, 1). In case ha α v γ (1 ξ) 6=, subsiuing L from Eq.(A.3) ino Eq.(A.4), and afer applying growh raes, yields F g K+1,g K,g K 1 = n E 1 g βe 2 K 1 g E 3+β K g E 4 K +1 =1, (A.5) where E 1 = vγ (1 ξ),e 2 =1 α+v, E 3 =(1 γ)(1 α) vγξ, and E 4 =1 v γ. We concenrae on g K since if i exiss, Eq.(31) informs ha g y exiss, oo. Defining E 5 = βe 2 +E 3 +β E 4, noe ha Eq.(A.5) allows o calculae he balanced growh pah only when E 5 6=.IncasehaE 5 =, he balanced growh pah may eiher no exis or i canno be inferred from Eq.(A.5). 25

26 Proposiion 2 If E 5 =and E 1 6=and n 6= 1,hennobalancedgrowhpah exiss. Proposiion 3 If E 5 6=hen g K = n E 1/E 5. Noe ha he case γ = β =, implying ha E 1 =, belongs o he las Proposiion. Nex, we show sufficien condiions for he balanced growh pah o be sable when E 5 6=.Thefirs order Taylor s expansion around he balanced growh pah is P 1 (g, g, g) =F (g, g, g)+f gk+1 (g, g, g) g K+1 g + + F gk (g, g, g)(g K g)+f gk 1 (g, g, g) g K 1 g, (A.6) where F gk+1 (g, g, g) = E 4 F (g, g, g) g 1 ; F gk (g, g, g) =[E 3 + β] F (g, g, g) g 1 ; and F gk 1 (g, g, g) = βe 2 F (g, g, g) g 1. Thus, he linearized differenial equaions can be wrien as P 1 (g, g, g) F (g, g, g) == E 4 g K+1 +[E 3 + β] g K βe 2 g K 1 ge 5. (A.7) For he case E 4 6=,defining N = K 1 yields " # " gk+1 ge 5 E3 +β βe 2 = E 4 + E 4 E 4 1 g N+1 which eigenvalues saisfy p(λ) =λ 2 + bλ + c =, # gk g N, (A.8) (A.9) where b = (E 3 + β) /E 4 and c = βe 2 /E 4. Since E 2 >, iisimmediaeha E 4 < implies ha p(λ) has real soluions. In his case, sabiliy is assured as far as p( 1) > and p(1) >, sincep() = c<; i.e., 1 b + c> and 1+b + c>. As E 4 <, he previous inequaliies can be wrien as E 4 + E 3 + β (1 + E 2 ) < and E 4 E 3 β (1 E 2 ) <. Thus, he following proposiion holds. Proposiion 4 If E 6= and E 4 <, a balanced growh pah is sable whenever β< (E 4 + E 3 ) / (1 + E 2 ) and eiher 1. E 2 < 1 and β>(e 4 E 3 ) / (1 E 2 ),or 2. E 2 > 1 and β<(e 4 E 3 ) / (1 E 2 ),or 3. E 2 =1and E 4 E 3 <. 26

27 When E 5 6=and E 4 >, asc>, p(λ) may have eiher real or complex soluions. If soluions are complex, hey are λ =( b/2) ± i (c b 2 /4) 1/2 and, herefore, he square of he absolue value of he module is c. Hence, sabiliy occurs whenever c<1. Forhecaseofrealsoluions,sincep() = c>, he requiremen is eiher 23 < b/2 < 1 and 1+b + c>, or 1 < b/2 < and 1 b + c>. Proposiion 5 If E 6= and E 4 >, a balanced growh pah is sable whenever 1. The soluions of p(λ) are complex, i.e., (E 3 + β) 2 < 4βE 2 E 4,andβ<E 4 /E The soluions of p(λ) are real, i.e., (E 3 + β) 2 > 4βE 2 E 4,and (a) β> E 3 and β<2e 4 E 3 and eiher i. E 2 < 1 and β<(e 4 E 3 ) / (1 E 2 ). ii. E 2 > 1 and β>(e 4 E 3 ) / (1 E 2 ). iii. E 2 =1and E 4 E 3 >. (b) β< E 3 and β> 2E 4 E 3 and β> (E 4 + E 3 ) / (1 + E 2 ). Proposiion 6 If E 5 6=and E 4 =we disinguish wo siuaions: 1. E 3 + β =or βe 2 =, in which cases we have sabiliy (no ransiion). 2. E 3 + β 6= and βe 2 6= and hus he balanced growh pah is sable if βe 2 / (E 3 + β) ( 1, 1). When E 5 =and eiher E 1 =or n =1, we have o check each possible case. We concenrae on E 5 =and E 1 =. This happens when (1 β)(v α)+γ (α vξ) = and eiher ξ =1or γ =, which gives six possible cases. However, he cases ξ =1, v = α and β 6= 1 γ; ξ =1,v= α and β =1 γ; γ =,v= α and β 6= 1;and γ =,v= α and β =1belong o Proposiion 1. Thus, we analyze he oher wo cases. If ξ =1,β=1 γ and v 6= α, combining Eqs (A.2) and (32), evaluaing in equilibrium, and applying growh raes, gives µ 1 γ η sψ g γ 1 K η = g z+1 g K+1. (A.1) 23 These condiions guaranee ha p(λ) aains a minimim in (, 1) and p (1) >, respecively, or he minimum is in ( 1, ) and p ( 1) >. 27

28 And from Eqs (A.1), (21) and (32), evaluaing in equilibrium, and applying growh raes, we obain gz v α +1 n α = g γ+α 1 K +1 g 1 γ K. (A.11) From hese wo equaions we have g K+1 = sψ µ 1 γ η η α v 1 γ v n α (1 γ)(1+v α) 1 γ v g 1 γ v K, (A.12) from where (α v) η α(1 γ) v(2 γ) α g K = n α(1 γ) v(2 γ), sψ (1 γ η) and he economy converges o a unique sable balanced growh pah when (A.13) i.e., < (1 γ)(1+v α) 1 γ v v (2 γ) <α(1 γ). < 1, (A.14) (A.15) If γ =,β=1and v 6= α, combining Eqs (A.1) and (A.11) wih γ =gives α v sψ (1 η) 1 v α g K+1 = n η 1+v α 1 v g 1 v K, (A.16) from where which is sable if α v sψ (1 η) α 2v α g K = n α 2v, η 1 < 1+v α 1 v (A.17) < 1. (A.18) Proposiion 7 When ξ =1,β=1 γ and v 6= α, hen 1. g K =[η/sψ (1 γ η)] (α v)/[α(1 γ) v(2 γ)] n α/[α(1 γ) v(2 γ)]. 2. The balanced growh pah is sable if v (2 γ) <α(1 γ). Proposiion 8 When γ =,β=1and v 6= α, hen 1. g K =[sψ (1 η) /η] (α v)/(α 2v) n α/(α 2v). 28