KNOWLEDGE CREATION AND TECHNOLOGY DIFFUSION: A FRAMEWORK TO UNDERSTAND ECONOMIC GROWTH

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1 Revisa KNOWLEDGE de Análisis CREATION Económico, AND Vol. TECHNOLOGY 20, Nº 2, pp DIFUSSION:... (Diciembre 2005) 41 KNOWLEDGE CREATION AND TECHNOLOGY DIFFUSION: A FRAMEWORK TO UNDERSTAND ECONOMIC GROWTH ORLANDO GOMES* Escola Superior de Comunicação Social and UNIDE/ISCTE Absrac The mos influenial explanaions of economic growh along he pas five decades rely on wo main iems: human capial accumulaion and he disseminaion of knowledge/echnological diffusion. These iems radiionally appear as separae growh sources. In his paper an inegraed perspecive is adoped. We begin by building a growh model where wo goals regarding echnological achievemens are considered; economic agens simulaneously wan o expand he heoreical knowledge fronier and o reduce he gap beween ready-o-use echniques and poenially available knowledge. Considering an objecive funcion ha capures he wo poined goals, one develops an ineremporal opimizaion seup concerning a wo secor scenario. The firs secor adaps exisen echnology o producive uses, while he second is an educaion secor. In his way, we can sudy he close relaionship beween echnical progress and human capial generaion decisions under an ineremporal perspecive. Keywords: Technology, Human Capial, Economic Growh, Opimal Conrol, Transiional Dynamics. JEL Classificaion: C61, O33, O41. * Escola Superior de Comunicação Social, Campus de Benfica, Insiuo Poliécnico de Lisboa (IPL), and Unidade de Invesigação em Desenvolvimeno Empresarial-ISCTE, Lisboa, Porugal. ogomes@escs.ipl.p. I would like o acknowledge he helpful commens of wo anonymous referees and he kind aenion of he journal s edior.

2 42 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 I. Inroducion In mos economic growh models a human capial variable and a echnology variable alernaively arise as he engine of growh. They acquire such qualiy when inroduced in an aggregae producion funcion alongside wih physical capial and labor. Solow (1956) and Swan (1956) have demonsraed ha under a neoclassical producion funcion, hese wo las inpus are unable o produce, wihou anyhing furher, long run susained growh. Uzawa (1965), Lucas (1988), Caballé and Sanos (1993), Mulligan and Sala-i-Marin (1993), Xie (1994), Bond, Wang and Yip (1996) and Ladrón-de-Guevara, Origueira and Sanos (1999), among ohers, have modelled human capial formaion and hey have included such an inpu in he aggregae producion funcion in order o suppor he concep of endogenous growh. Romer (1986, 1990), Aghion and Howi (1992), Jones (1995), Evans, Honkapohja and Romer (1998) and Young (1998) have chosen o pu echnology in he cenre of economic growh explanaions, and a ime dependen echnological index arises in he producion funcion as he vehicle o susained growh. All he referred work and all he discussion around aggregae growh models pu he several inpus o producion a a same analysis level. This means ha in boh, he physical capial-human capial approach and he physical capial-echnology approach, all he inpus ha are relevan o growh have o appear as argumens in he final goods producion funcion. An early aemp on he inerpreaion of economic growh, Nelson and Phelps (1966), avoids such a sraighforward view. For hese wo auhors, i makes sense o assume ha human capial serves as a means o generae and spread echnology and ha echnology is hen usable in he physical goods producion funcion. Recognizing ha he Solow- Swan paradigm is valid when assuming a fixed level of echnology, we may coninue o use such a framework bu replacing he consan level of echnology by a ime dependen variable, where he respecive ime pah is deermined by wo facors: invesmen in humans and echnological diffusion. Recen lieraure poins in his direcion. For example, Aghion, Howi and Violane (2003) sae ha skills are no jus an inpu o he producion of goods and services bu also a facor in he creaion and absorpion of new echnological knowledge (page 444). and Acemoglu (2003) reemphasizes ha Nelson and Phelps (1966) posulaed ha human capial is essenial for he adopion of new echnologies. This view has a leas wo imporan implicaions: Firs, he demand for skills will increase as new echnologies are inroduced, and second, economies wih high human capial will effecively possess beer echnologies (page 465). Also Benhabib and Spiegel (2002) recover he Nelson-Phelps approach, presening a generalized model, where differen possibiliies abou echnology diffu-

3 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: sion paerns are assumed. Boh, he echnology fronier - curren produciviy level relaion and he impac of human capial accumulaion over echnology resuls, are considered in his analysis. In his paper he Nelson-Phelps approach is revisied. We sudy growh giving paricular aenion o echnology and human capial and aribuing a supporing role o he capial accumulaion consrain of Solow and Swan, and o he consumpion uiliy ineremporal maximizaion framework of Ramsey (1928), Cass (1964) and Koopmans (1964). Our main argumen is ha since i is he human capial and he echnology variables ha deermine growh, we can find he basic growh resuls and dynamics hrough he analysis of he facors ha influence he ime evoluion of hese variables. The Solow-Ramsey framework becomes accessory and i may be used solely o jusify ha main economic aggregaes (per capia oupu, physical capial per labor uni, per capia consumpion) follow a same long run growh rae as he one ha is found for he echnology variable. Modelling human capial a a differen level han i is usually modelled in economic growh models will allow for wo new resuls. Firs, we will be able o define a human capial producion funcion where decreasing reurns are compaible wih long run consan per capia economic growh, wha implies purging a counerfacual aspec of wo secor growh models: he lineariy problem raised by Solow (1994) and Jones (2003) is in his way eliminaed. We do no have o assume any kind of arificial knife-edge lineariy o encouner a consan long run growh rae. The second resul will seem, a leas a a firs glance, a more awkward one. The way in which we will model human capial will lead o a long run seady sae where all relevan per capia variables (physical capial, oupu, consumpion, echnology) grow a a same rae while human capial will exhibi a zero growh rae in he long run soluion. Growh models generally presen human capial as growing wih he oher variables. Here, we arrive o he resul of convergence of he human capial variable o a consan long run value. This may be suppored on he idea ha we may expand echniques and physical goods indefiniely bu here is a limi o he expansion of human capabiliies. In wha concerns he modelling of echnology we follow he Nelson-Phelps approach in disinguishing wo conceps of echnology. Firs, here is a heoreical level of echnology or a echnological possibiliies fronier; second, we assume a level of echnology in pracice, which is a fracion of he firs and ha can be applied direcly o producive uses when embodied in an aggregae final goods producion funcion. Solow (2003) also poins o a same kind of disincion: Endogeneizing echnological progress is a fine idea, provided ha i can acually be done in a sensible way ( ) he growh lieraure proceeds as if all improvemens in oal facor produciviy originae in research and developmen aciviies. Tha is wha ges modelled. Bu i is cerain ha an appreciable fracion of echnological progress has nohing o do wih R&D. I originaes on he shop (or office) floor and is invened by workers and

4 44 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 managers who see ha here are improvemens o be made in he locaion of aciviies, he flow of work, he way par A is fasened o par B, and so on (page 548). The disincion concerning heoreical knowledge and applied echniques has been widely discussed in he lieraure, from several poins of view. For insance, Bresnahan and Trajenberg (1995), Helpman and Trajenberg (1994, 1996) and Helpman (1998) make reference o he pervasiveness of general purpose echnologies (GPTs). The imporan argumen advanced by hese auhors is ha GPTs, like elecriciy or microelecronics, are generaed disconinuously in ime and in an exogenous way relaively o marke mechanism incenives. A new GPT (a kind of scienific no direcly applied discovery) generaes wo subsequen phases on he evoluion of he economic sysem. The firs phase, ha his group of auhors calls ime o sow, is characerized by a ransfer of resources o he producion of applied echniques and ools ha make use of he new GPT. This firs sage will almos cerainly imply low or negaive produciviy growh. In he second phase, called ime o reap, here is already a generalized use of he GPT under he form of complemenary inpus, making i possible, under nework effecs, o boos produciviy growh. This view of knowledge creaion and echnology adopion suppors a business cycles perspecive. Growh declines or acceleraes as a resul of he dynamics regarding he creaion, developmen and spreading of echnologies. This is also he view of several oher auhors, namely Andolfao and MacDonald (1998), Boldrin and Levine (2001), Hall and Khan (2003), Mukoyama (2003) and Siver (2003). These auhors develop concepual frameworks where aggregae flucuaions end o arise from he discovery and diffusion of new echnologies and, hus, exogenous shocks can be excluded from he analysis and from he explanaion of growh dynamics. Under his perspecive, boh expansions and recessions are he resul of good news, ha is, periods of high growh mean he generalized adopion of a given new scienific / echnological knowledge, while periods of low or even negaive growh will correspond o periods where he echnological fronier is expanding bu he direc produciviy gains of such expansion are no ye visible in he economic sysem. Furhermore, he cied auhors call he aenion o he fac ha invenions ofen occur as sudden evens, while diffusion is a slow coninuous process which involves a grea number of individual decisions. Neverheless, here is he clear conscience ha i is diffusion and no creaion ha ulimaely deermines he pace of produciviy change and economic growh. Also Keller (2002, 2004) discusses echnical change and diffusion, in his case focusing he aenion on inernaional paerns of knowledge disseminaion. While he cied auhors analyze in deail he naure and linkages of adoping secors, he speed and he exen of he diffusion process, and he social and privae benefis of he innovaion - diffusion mechanism, hese do no consiue he main concern in our presen analysis. Insead, we jus focus on he logical observaion ha wihou heoreical knowledge generaion, here is no possible

5 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: applicaion, and hence, he absence of he use of new echnologies in producion implies ha growh will no occur. Anoher difference in approach relaes o he fac ha we consider he creaion of basic knowledge as endogenous he model o be developed is a social planner problem ha aribues o he governmen a very meaningful role in he managemen of knowledge creaion and diffusion. The governmen has a fundamenal word o say wih respec o R&D allocaion of resources. If auhoriies choose o expand he knowledge fronier hey concenrae public resources wih such a purpose. If, insead, governmen is mainly concerned wih giving o he sociey he possibiliy o diffuse knowledge in order o creae wealh, fewer resources will be allocaed o public R&D expendiures, and hus he knowledge fronier does no move ouwards. Therefore, our model furnishes a parial explanaion of he growh process, neglecing he role of capial accumulaion, savings, populaion growh and consumer preferences. I focus he aenion on he group of feaures ha in any sociey, even in he ones which have more liberal economic regimes, are under he conrol (a leas parially) of a social eniy. The social planner has he responsibiliy o allocae more or less resources o innovaion and diffusion (he resources available for diffusion are he ones ha he sae does no wihdraw o creaion, and ha simulaneously are no divered o oher privae or public uses). The goal of he decenralized economy may consis mainly or solely in having he higher level of applied knowledge i can ge, in order o accelerae growh, bu he auhoriy has an allocaion of resources responsibiliy a wo levels: firs, i has o have conscience ha if no knowledge is creaed, he sociey canno apply i in any way; second, he generaion of scienific knowledge is also an end in iself, ha is, any sociey, in order o progress and sand on he inernaional arena, has o be able o display culural and scienific poenial. These poins will suppor he inroducion, along he nex secion, of a social uiliy funcion where knowledge adopion and knowledge generaion boh appear as argumens. The reamen ha we will give in he following secions o echnology choices will imply an opimal conrol problem, where he goals (of a social planner) are simulaneously o amplify he fronier of echnological knowledge and o approximae he level of ready o use echnology o such fronier. This opimizaion problem is consrained precisely by he Nelson-Phelps moion equaion ha characerizes he relaion beween he wo echnology conceps. The seup under consideraion relies on a rade-off respecing echnology choices: o ge new invenions o be available o produce final goods one has o renounce o some of he growh of he basic science fronier and vice-versa. Economic resources are scarce in he sense ha hey canno be used in a simulaneous way o give birh o new ideas and o make hese ideas immediaely ready o producive ends. We will end up wih a wo secor growh model similar o he ones in he lieraure already referred along his inroducion, neverheless wih some imporan new feaures: physical capial and consumpion are no nuclear in our analysis, he wo secors generae embodied (human capial) and disembod-

6 46 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 ied (echnology) knowledge and he fundamenal choices are no abou consumpion and savings bu abou echnology producion and echnology diffusion. 1 In shor, we rely on he Nelson-Phelps approach in considering a echnology producion funcion where human capial and a echnology gap appear as inpus and we make a deeper analysis by considering he process hrough which human capial is produced and by considering as well, in alernaive o an exogenous echnology growh rae, a conrollable echnology growh rae. The model ha we will be faced wih is an opimal conrol growh model where he only variables ha are considered are he fundamenal growh sources: human capial and echnology. The remainder of he paper has he following conens. Secion II develops he srucure of he model and refers o he seady sae resuls. Secion III discusses he naure of he balanced growh pah and characerizes he ransiional dynamics in he neighbourhood of he seady sae. Secion IV concludes. II. A Two Secor Model of Opimal Technology Choices Take echnology and human capial as he engines for economic growh. The firs variable, echnology, may be decomposed in wo. The echnology possibiliies fronier will be variable T() and a ready o use in producion echnology variable is represened by A(). In every ime momen A() T(), represening boh variables posiive quaniies for all >0. We will be concerned wih undersanding he emporal evoluion of a gap variable φ( ) A()/ T(), φ() [0,1], ha has a sraighforward inerpreaion: he higher he value of φ() he smaller is he gap beween he values of he echnology variable represening knowledge immediaely available o produce and of he echnology variable represening he scienific sae of he ar. Oher fundamenal variable is he growh rae of he echnology fronier: τ( ) T ( )/ T(), T(0)=T0 given. In our model his growh rae is assumed as a conrol variable for he represenaive agen (as discussed in he inroducion, his is he public auhoriy); ha is, in wha concerns echnology decisions, he auhoriy is able o choose he rae a which scienific progress occurs. 2 Neverheless, obviously his mus be a consrained decision process because o allocae more or less resources o research aciviies implies an opporuniy cos respecing o he resources ha remain available for oher economic aciviies. In paricular, in our framework we assume ha he choice in erms of basic echnology progress is consrained by he necessiy of using economic resources o apply echnology o he goods producion process. A rade-off emerges beween our firs wo endogenous variables: φ() and τ(). In wha concerns echnology choices, he basic economic goal of he social planner is o maximize he ineremporal sream of v(φ(),τ()) funcions, being a funcion v defined as follows. 3

7 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: Definiion 1. Funcion v. The represenaive agen ha makes echnology choices faces a real valued objecive funcion v: R 2 + R ha obeys he following properies: i) Coninuiy, concaviy and smoohness. 4 ii) v φ φ() τ() = θ ( 01, ); vτ = µ ( 01., ) v v The second condiion in definiion 1 defines consan elasiciy parameers. This condiion ensures ha he uiliy of sraighening he echnology gap and he uiliy of increasing he pace of echnological progress are, for he represenaive agen, boh posiive and diminishing. Noe ha such an assumpion follows from he naure of he represenaive agen, as discussed in he inroducion. Public policy mus be oriened o he expansion of he knowledge fronier, as well as o he simulus o apply echnology o producion because: (i) he governmen has he responsibiliy o guaranee an everlasing flow of new knowledge in order o be possible o apply i, and (ii) enlarging he knowledge fronier is a vial process o he survival and developmen of any organized sociey or civilizaion. Therefore, here is a double concern of he represenaive agen: o expand he echnology fronier and o guaranee ha such knowledge is applied o generae wealh. Le us remark, once again, ha in his process he public secor has conrol over he amoun of knowledge ha is creaed, hrough he resources i allocaes o such purpose, and herefore he growh rae of echnology appears as a conrol variable. The amoun of knowledge ha arrives o he direc producive aciviy is no under he governmen conrol; as we presen bellow, a sae consrain will characerize such relaion. To simplify our reamen of he model we will work wih a specific funcional form of he above defined v funcion. We ake a Cobb-Douglas ype of funcion (for insance, a CES funcion would also be a candidae funcional form). v( φ( ), τ( )) = φ( ) τ( ) 1/( 1+ σ) σ /( 1 + σ) (1) The correspondence beween parameers θ and µ in definiion 1 and parameer σ in equaion (1) is he following: σ =µ/θ and µ=1-θ. The rade-off beween he wo endogenous echnology variables, φ() and τ(), becomes explici by considering he Nelson-Phelps echnology consrain: A ( ) = gh ( ()) ( T () A ( )), g( 0) = 0, g' > 0, g'' < 0, A( 0) = A 0 given (2) Wih (2) we sae ha he rae of increase of he index A() is a funcion of human capial [h() is a human capial per uni of labor variable or a human capial efficiency index] and of he gap ha exiss beween he wo echnology variables. Firs and second derivaives of g indicae ha here are posiive bu

8 48 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 diminishing reurns of human capial in he producion of echnology. The gap erm ranslaes he idea ha he level of echnology in pracice will evolve faser when here is a large gap beween echnology possibiliies and he sock of knowledge insananeously available o produce. Recovering variable φ(), from equaion (2) we arrive o he final form of he firs resource consrain of he opimal conrol problem: ( φ ) = g( h()) ( 1 φ( )) τ( ) φ(), φ( 0) = φ (3) 0 To complee he presenaion of he model i is necessary o define a rule for he ime evoluion of he human capial variable. This can have he sandard form in mos growh models, i.e., human capial evolves in ime hrough a producion process ha involves a producion funcion, f, and a consan depreciaion rae, δ. h ( ) = f( h ()) δ h (), δ > 0, h( 0) = h 0 given (4) The human capial variable is, alongside wih variable φ(), a sae variable of he ineremporal conrol problem. Equaions (3) and (4) are he moion equaions ha consiue he resource consrains o which he opimizaion problem is subjec o. The analyical racabiliy of he model demands ha we ake explici funcional forms for funcions f(h()) and g(h()). Imagining ha i is possible o choose a each momen in ime he shares of human capial o allocae o each of he wo economic secors (echnology and educaion secors), we define a new variable u() ha represens precisely he share of human capial allocaed o he generaion of echnology. Obviously u() 1,. The properies of funcion g were se forh in equaion (2). Posiive and diminishing reurns of human capial in he producion of echnology imply a funcion wih he following shape: η gh ( ( )) = a ( u ( ) h ( )), a> 0, η ( 0, 1 ) (5) Funcion f may be defined in a similar way: [ ] > β f( h( )) = b ( 1 u( )) h ( ), b 0 (6) For equaion (6) we find i essenial o impose β (0,1) in order o encouner a long run balanced growh pah. Thus, one assumes ha he educaion secor exhibis diminishing reurns in he accumulaion of human knowledge. This poin was referred earlier in he inroducion and should be sressed since i consiues an innovaion relaively o convenional growh models. In our model, where human capial conribues o he generaion of physical goods solely in an indirec man-

9 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: ner, we mus assume diminishing reurns in he accumulaion of human capial in order o ge a long erm consan seady sae growh. As a resul of his assumpion, he human capial per uni of labor variable will have a differen naure relaively o oher inpus: is level will no grow in he long run in opposiion o wha happens o he several per capia variables, namely oupu, consumpion and physical capial. Remember ha h() defines he skills of an average individual, and so i makes sense o find a resul where hese end o a consan value, ha is, inuiively i is hard o suppor he concep ha human skills may be improved furher and furher indefiniely a a consan rae. Technology and machines can be expanded wihou limi, individual skills canno - his is a fundamenal argumen of our analysis. The opimal conrol growh problem has now all he necessary ingrediens. Definiion 2 saes wha we undersand by an opimal soluion. Definiion 2. Conrol problem opimal soluion. An opimal soluion is a se of pahs φ(), h(), τ(), u() { } ha solve he maximizaion problem τ (), u() + ρ. Max 0 v( φ( ), τ()) e d subjec o consrains (3) and (4) and where funcions v, f and g are defined respecively by (1), (5) and (6). All variables assume non negaive values, iniial values for sae variables are given and shares φ() and u() remain always below uniy. Furhermore, i is clear from he problem ha we ake an infinie horizon and ha fuure echnology accomplishmens are discouned. A consan discoun rae, ρ>0, is considered. Le p φ () and p h () be co-sae variables. We synhesize our model s informaion ino a curren value Hamilonian funcion: H( φ( ), h( ), τ( ), u( ), pφ ( ), ph( )) η v( φ( ), τ( )) + [ a ( u() h ()) ( 1 φ()) τ() φ() ] pφ () + β { b [ ( 1 u( )) h( ) ] δ h() } ph() (7) Applying he Ponryagin s maximum principle, he firs order opimaliy condiions can be compued: H τ 1/( 1+ σ ) σ φ = pφ φ + σ () 0 τ = () () 1 () (8) ( 1 η) η Hu = 0 η a u() h () ( 1 φ()) pφ () = ( 1 β) β (9) β b ( 1 u( )) h() p () h

10 50 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 H = ρ p () p () φ φ φ σ + σ η τ pφ = [ ρ+ a u h + τ ] pφ σ () (10) () ( () ()) () () 1 φ() H = ρ p () p () h h h β ( 1 β) p h() = [ ρ+ δ β b ( 1 u()) h() ] ph() η ( 1 η) η a u() h() ( 1 φ()) p () φ (11) η Hp φ = ( φ ) ( φ ) = a ( u () h ()) ( 1 φ ( )) τ () φ () (12) β Hp h = h ( ) h ( ) = b [ ( 1 u ()) h () ] δ h () (13) lim pφ () e φ() = 0 (14) + ρ. lim ph() e h() = 0 (15) + ρ. Condiions (8) o (15) are sufficien condiions of opimaliy given ha he opimal Hamilonian is concave in [φ(),h()]. The opimaliy condiions are he relaions necessary o prove he following proposiion. Proposiion 1. Exisence and uniqueness of a balanced growh equilibrium. Under he condiion β+ρ/δ<1 here exiss a unique balanced growh pah or unique seady sae four dimensional poin ha saisfies (8) o (15). Proof: To prove he exisence of a unique seady sae poin one has o solve he sysem [ φ() h () τ () µ () ]= 0. The soluion for his sysem consiss on a se { φ, h, τ, µ } of consan values. To solve he sysem one is compelled o find equaions of moion for he wo conrol variables. Le us sar wih τ(). Differeniaing (8) in order o ime, he following growh raes relaion is obained: 5 γ = σ γ ( 1 + σ) γ p (16) τ φ φ

11 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: Replacing γ ρφ and γ φ in (16) by he corresponding expressions aainable from (10) and (12), he moion equaion for τ() comes ( τ ) () ( ) ( () ()) ( / ()) () σ τ σ ρ η = 1 1+ a u h 1 + σ φ τ (17) The ime evoluion of u() is derived from he following growh raes relaion ha is rue under (9): γ 1 u () φ() = γ γ γ η β γ φ h φ + η β η u ( ) ( 1 ) ( ) ( ) 1 φ() (18) u p p h We arrive o 1 u () () u ( ) = φ 1 σ () ( ) ( ) u ( ) () τ 1 η β η + 1 φ σ ( ) [ + ( ) u ( )] b [( u ( )) h ()] 1 β η β η 1 ( 1 β + η) δ} u () (19) Equaions (17) and (19), alongside wih he wo resource consrains, consiue he sysem from which we derive he seady sae soluion. The sysem has in fac a unique soluion, which is ( 1 σ ) g σ ρ+ g φ b β h β + ρ δ = ( / ) δ τ σ u ( σ ρ + g) 1 1 β ρ/ δ 1/( 1 β) (20) ( ) η. If β+ρ/δ<1 hen we guaranee ha u (0,1) and his is he wih g = a u h only boundary condiion ha mus be imposed in order o (20) o be a feasible four dimensional seady sae poin The seady sae resuls deserve some commens. Firs, we noice ha he share u depends upon parameers β, ρ and δ. The higher he elasiciy parameer

12 52 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 β and he discoun rae he lower he value of he share of human capial allocaed o echnological developmen. The faser he depreciaion of human capial he more his form of capial is allocaed o is own producion relaively o a echnological use. Second, he human capial efficiency index is indeed a consan amoun on he balanced growh pah. The beer he secor echnical condiions (he higher he value of b), he higher he elasiciy parameer β, he higher he discoun rae and he lower he depreciaion rae, he larger will be he accumulaed human capial sock under a long run perspecive. Third, we observe ha φ obeys he boundary condiion φ 1, because ρ>0 and g > 0. Fourh, he echnology growh rae arises in he seady sae depending on a mulipliciy of facors, namely: (i) he objecive funcion parameer, (ii) he ineremporal discoun rae, (iii) boh secors educaion funcions parameers (a and b), (iv) he elasiciy parameers (β and η) and (v) he depreciaion rae. For he seady sae growh rae resul, ( ) σ τ ρ β ρ δ δ β ρ δ β β = + 1/( 1 ) a (( 1 ) / ) ( b/ ) ( + / ) 1 σ η (21) we highligh ha he parial derivaives τ / σ > 0, τ / a> 0, τ / b > 0 and τ / η > 0 have unquesionable signs. The same is no rue for β, ρ and δ because when changes in hese facors benefi he accumulaion of human knowledge (make h o rise) hey injure he ransference of human capial o innovaion purposes (make u o fall). A his sage, found he long erm echnology growh rae, we could ransform he model by adding o i a physical capial accumulaion consrain ha should include a labor augmening aggregae producion funcion; hen, we would ge an economic growh framework. This framework will no be developed, bu he resuls ha is inroducion would bring can be saed in some deail. Under such a scenario he growh rae in (21) would be he seady sae economic growh rae for he various per capia aggregaes: physical capial, consumpion and oupu. In such a model we have endogenous growh hrough echnology choices, ha is, long run consan per capia growh is obained and his is a funcion of several parameers of our analysis. These parameers concern he way in which he economy is able o creae and diffuse echnology and o inves in human capabiliies. The accomplished growh rae is precisely he resul of he way he economy handles he wo rue engines of growh: human capial and echnology. III. Sabiliy and Dynamics To make a meaningful analysis of he sabiliy and dynamics of he model we have o ake some simplifying assumpions over he problem proposed in he previous secion. Under he wo following assumpions we prove ha he model

13 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: is saddle-pah sable and we proceed o he characerizaion of he dynamic behavior of he various endogenous variables. Assumpion 1. Le φ() = φ ( 01,, ) for all ime momens. Assumpion 2. Le he following consrain over parameer values hold: σ = β/( η β). Assumpion 1 eliminaes he variable φ() of he model, and from (3) we realize ha under his assumpion he oher hree variables are no independen from each oher any more, ha is, for every momen of ime is now rue ha 1 φ() τ() = a ( u ( ) h ( )) φ() η (22) Assumpion 2 implies he condiion η > 2 β if we wan σ (0,1) o coninue o hold. The opimizaion problem is hereafer he one in definiion 3. Definiion 3. Opimal conrol problem wih a fixed echnology gap. Le assumpions 1 and 2 hold. The goal of he represenaive agen is now o + ( maximize φ η 2 β )/ η φ β / η β ρ. [ a ( 1 )] [ u( ) h() ] e d subjec o 0 h ( ) = b [ ( u ()) h ()] β 1 δ h (), h(0)=h 0. To prove he exisence of a unique seady sae one has o proceed as in he previous secion. Necessary opimaliy condiions are: φ [ ] = [ ] ( η 2 β)/ η β / η ( 1 β) ( 1 β) wih p() a shadow-price variable. And a ( 1 φ) u( ) b 1 u() p() (23) ( 1 β) p ( ) = { ρ+ δ β b [ ( 1 u ()) h ()] } p () (24) lim p() e h() = 0 (25) + ρ. The consrain (4) is also an opimaliy condiion. From he opimaliy condiions we arrive o he expression for he ime movemen of u(),

14 54 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 1 ( 1 β) u ( ) = b [ ( u ()) h ()] ( + ) u () u () { β 1 ρ δ } 1 1 β [ ] (26) Equaions (4) and (26) urn possible he presenaion of he seady sae under he new assumpions, β /( 1 β) 1/( 1 β) β b ρ δ δ h + ρ+ ( 1 β) δ u = ρ+ δ τ φ ρ+ ( 1 β) δ β b a φ ρ+ δ ρ+ δ δ β /( β) η /( β) (27) All he equilibrium values in (27) obey he necessary boundary condiions. In he viciniy of he balanced growh pah a wo-dimensional linearized sysem may be compued concerning he problem in definiion 3: h h ( ) ( 1 β) δ β δ u h () h = 1 u ( ) u u β δ β δ u () u h 1 u (28) Sysem (28) allows o prove proposiion 2. Proposiion 2. Saddle-pah sabiliy. Under assumpions 1 and 2 he model ha relaes he quaniy and he allocaion of human capial exhibis saddle-pah sabiliy. Proof: For a wo-dimensional Jacobian marix as he one in sysem (28), saddlepah sabiliy means ha one of is eigenvalues is posiive (respecing o he unsable componen of he sysem) while he oher is negaive (respecing his o he sable pah). Noicing he algebra resul Tr(J)=λ 1 +λ 2 and De(J)=λ 1.λ 2, one has jus o obain he signs Tr(J)>0 and De(J)<0 o prove he expeced sabiliy resul. Compuing he race we realize ha Tr(J)=ρ; he deerminan is a negaive amoun given ha β<η/2 and η<1: De(J)= ( 1 2 β) δ [ ρ+ 11 ( β) δ ]. We confirm in his way he exisence of a saddle-pah sable equilibrium

15 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: To undersand how he endogenous variables joinly evolve owards he seady sae we make use of a graphical analysis hrough he consrucion of a phase diagram. This analysis allows o prove proposiion 3. Proposiion 3. Dynamic adjusmen. Along he sable rajecory, he adjusmen o he seady sae implies ha a growing sock of human capial is accompanied by a shif of human capial from he educaion secor o he echnology secor. On he oher hand, if he sock of human capial per labor uni grows negaively hen his form of capial ends o be reallocaed from he echnology generaion o he educaion aciviies. Proof: The proof of proposiion 3 is made hrough he consrucion of a phase diagram. To illusrae graphically he dynamics of he model we need o represen wo auxiliary lines: h ( ) = 0 and u ( ) = 0 ; hese can be aken from he linear sysem (28): ρ ( ) ( u) h ( ) = u () = h () β 1 0 ρ δ β h (29) ρ+ ( 1 2 β) δ ( 1 u) u ( ) = 0 u () = + h () ρ+ δ h (30) Under he assumpions of he model, line (29) has a larger slope han (30), and he inercep of (30) is a higher value han he inercep of (29). Given ha h ( )/ h () < 0 and u ( )/ h () < 0, he following vecor fields apply: o he righ of line (29) he value of h() decreases and o he righ of he same line ha value increases; o he righ of line (30) he value of u() decreases and o he righ of (30) i increases. The corresponding direcional arrows are drawn below in he phase diagram of Figure 1. The direcions o which he referred arrows poin allow for an exac noion abou he sable rajecory locaion. The sable rajecory, as he unsable arm, is upward sloped, being he slope of he firs less accenuaed han he one concerning he unsable pah. Drawing he phase diagram we confirm he saddle-pah sable naure of he relaion beween h() and u() and we find ha he variables evolve in he same direcion owards he equilibrium poin. If h0 < h, hen h() and u() grow posiively unil he poin ( hu, ) is aained. The phase diagram is skeched in Figure 1.

16 56 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 u() FIGURE 1 PHASE DIAGRAM U h ( ) = 0 u ( ) = 0 u S h h() To conclude he dynamic analysis we have ineres in knowing how he echnology growh rae evolves o he seady sae given he sable pah found for he relaion beween h() and u(). The growh rae is no an independen variable; i relaes o h() and u() hrough (22). Our dynamic analysis becomes complee wih proposiion 4. Proposiion 4. Growh rae dynamics. In a echnology choices model where he gap variable is a given consan value, he dynamic adjusmen o he seady sae where none of he variables h(), u() and τ() grows is of he following kind: if h0 < hin he neighbourhood of he seady sae hen h(), u() and τ() all grow a posiive raes o he equilibrium posiion; if h0 < h hen τ() follows he same negaive growh behavior of h() and u(). Proof: Assume several possible τ() values, τ0 < τ1 <... < τ <... < τn. From equaion (22) follows ha for any τ i, i=1,, n, 1 φ u () = τ i a 1 φ 1/ η 1 h() (31) Equaion (31) represens a concave funcion in he referenial [h(),u()] ha is as much far from he origin as he higher is he value of τ i. Thus, we may draw

17 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: in such referenial a se of funcions (31) for he various levels of τ i. We regard ha as we approach he seady sae, he value of τ() will be higher if we sar from he lef of h, or lower if we sar from he righ of h. In his way, variable τ() has a same qualiaive behavior in he adjusmen process as he conrol variable u() Figure 2 elucidaes he argumen in he proof of proposiion 4. FIGURE 2 GROWTH RATE DYNAMICS In his paper we have argued ha physical capial accumulaion plays jus a subsidiary role in erms of long run growh. The analysis focused on he wo cenral engines of growh, which are human capial accumulaion and echnological diffusion. The way in which echnological diffusion is modelled relies on he Nelson-Phelps approach: he echnology index o include in an aggregae producion funcion evolves in ime according o a echnological gap ha relaes o a reference value ha may be undersood as he scienific fronier or sae of he ar knowledge capabiliies of he economy. The major new modelling assumpion is ha he economy has he abiliy o conrol he growh of he knowledge fronier, in he exen ha public auhoriies choose he level of basic R&D ha is pursued in universiies and sae laboraou() τ τ n τ 0 τ 1 S u h h() IV. Final Remarks

18 58 REVISTA DE ANALISIS ECONOMICO, VOL. 20, Nº 2 ries. This assumpion makes sense if one hinks abou a rade-off ha is clear from he echnology resource consrain: o creae knowledge less resources will be available o apply knowledge, o apply knowledge i is necessary o devoe relaively less resources o is creaion. In his way, he conrol of decisions abou echnology creaion is condiioned by he economic non conrollable rules of echnology adopion. The way we have chosen o sudy simulaneously he behavior of human capial and echnology variables leads o a wo-secor opimal conrol problem, from which we have wihdrawn some meaningful resuls: (i) In he seady sae, sae variables echnology gap and human capial efficiency index and conrol variables growh rae and human capial share, hey all display consan values. (ii) The consan seady sae resul is possible only if one assumes human capial decreasing reurns in boh he echnology and he educaion secors. In his way decreasing reurns become compaible wih susained growh. (iii) Assuming a consan gap in erms of echnology adopion, he model gives a saddle-pah sable resul ha implies ha variables human capial index, human capial share and echnology growh rae will evolve qualiaively in he same way owards he seady sae. In wha concerns welfare implicaions, he model makes a clear saemen abou he role of he public secor in research and more generally in economic growh. Wihou he incenive of he auhoriies in promoing scienific knowledge, here would be no knowledge o apply, and herefore he economy would sagnae. From a welfare poin of view, our framework indicaes he absence of long run growh under a fully decenralized seup (wihou governmen inervenion) and posiive growh, according o (21), when here is a managemen of resources beween public basic knowledge creaion and privae innovaion and diffusion processes. One migh argue ha basic research is no a monopoly of he public secor; many general purpose invenions are jus he frui of brillian minds ha ook he ime o arrive o new resuls, wihou imporan financial incenives. In fac, his is no quesionable; however, i is also rue ha from a macroeconomic poin of view susained economic growh became possible only when some imporan public insiuions were creaed. Today s economic srucures are no conceivable wihou a basic R&D secor conrolled and financed publicly; and herefore he knowledge fronier does no move a a pace deermined by a few alruisic geniuses. Science is financed by public resources and governmens decide in wha exen he science froniers are simulaed. A las word o he endogenous naure of he growh process. As we have saed, o ranslae our analysis o an economic growh seup i is necessary jus o include a capial accumulaion consrain and, evenually, an ineremporal consumpion uiliy opimizaion framework. In his way, he rae ha describes he echnology variables evoluion in ime would simulaneously be he rae of economic growh, under a seady sae perspecive. Thus, i is possible o alk

19 KNOWLEDGE CREATION AND TECHNOLOGY DIFUSSION: abou endogenous economic growh because he growh rae is deermined endogenously (in fac, i is conrollable under some resource consrains) and because he several parameers ha appear in is long run expression may be influenced by he economic agens decisions, including he policy measures ha he governmen underakes. Neverheless, he endogenous naure of our growh model is differen from he naure of he convenional growh models. Firs, only echnology and educaion decisions influence long run growh; facors as he eagerness o consume, savings decisions or he rae of populaion growh are absen from he seady sae growh rae expression. Second, despie oupu, he physical capial sock and echnology all grow in he seady sae, he human capial average efficiency does no - under our assumpions he model is pessimisic abou human capabiliies, ha is, he way in which we may acquire more skills is bounded. Noes 1 To highligh he innovaion - diffusion process, we do no consider capial accumulaion and privae consumpion uiliy maximizaion. The inroducion of hese feaures would no consiue any analyical obsacle; combining a Ramsey framework wih he Nelson-Phelps seup, we would have all of he mos common ingrediens on growh analysis. The poin is ha under a Solow perspecive of decreasing marginal reurns in capial accumulaion, his variable does no add imporan informaion o long run growh analysis, which Solow indeed associaed wih echnical progress. 2 More accuraely, he social planner is able o choose he amoun of resources allocaed o basic research; i is expeced ha here will be a direc macro relaion beween such effor and he obained resuls, despie he uncerainy associaed wih he creaive aciviy. 3 The social planner objecive funcion reflecs micro-level economic and social goals ha individual agens pursue. On one hand, bridging he gap beween available and aainable echniques is an insrumenal objecive of each individual firm. On he oher hand, an ever-expanding scienific fronier has no as single purpose o provide he condiions for a wider ready-o-use echnology developmen; i is also a way hrough which households acquire human capial ha provides individual saisfacion (scienific knowledge can be hough as an argumen of he represenaive agen uiliy funcion) and unique capabiliies ha offer o hem a sronger posiion when facing labour markes. 4 By smoohness we mean ha funcion v is differeniable and ha he corresponden firs order derivaive is a coninuous funcion. 5 The symbol γ represens he growh rae of he variable referred in index. References ACEMOGLU, D. (2003). Facor Prices and Technical Change: From Induced Innovaions o Recen Debaes. in P. Aghion, R. Frydman, J. Sigliz and M. Woodford (eds.) Knowledge, Informaion and Expecaions in Modern Macroeconomics (in honor of Edmund S. Phelps). Princeon, New Jersey: Princeon Universiy Press, pp AGHION, P. and P. HOWITT (1992). A Model of Growh Through Creaive Desrucion. Economerica, 60 (2), pp AGHION, P.; P. HOWITT and G. VIOLANTE (2003). Wage Inequaliy and Technological Change: A Nelson-Phelps Approach. in P. Aghion, R. Frydman, J. Sigliz and M. Woodford (eds.) Knowledge, Informaion and Expecaions in Modern Macroeconomics (in honor of Edmund S. Phelps). Princeon, New Jersey: Princeon Universiy Press, pp

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