Implementing Technology

Transcription

1 Implemening Technology Diego Comin New York Universiy and NBER Bar Hobijn Federal Reserve Bank of New York January 2007 Absrac We inroduce a racable model of endogenous growh in which he reurns o innovaion are deermined by he echnology adopion decisions of he users of new echnologies. Technology adopion involves an implemenaion invesmen ha deermines he iniial produciviy of a new echnology. Afer implemenaion, learning increases he produciviy of a echnology o is full poenial. In his framework, implemenaion enhances growh, while growh increases obsolescence and reduces implemenaion. In a calibraed version of our model, he opimal policy involves a subsidy o capial and o implemenaion and a R&D ax. This policy would lead o a welfare improvemen of 7.6 percen. Ou of seady-sae analysis yields ha he ransiional dynamics of he derended variables afer a shock o capial are very similar o he dynamics of he neoclassical growh model, bu ransiory shocks have permanen e ecs on he level of produciviy. keywords: endogenous growh, implemenaion, learning, echnology adopion, echnology di usion. JEL-code: O3. We would like o hank Frédéric Dufour, Sefano Eusepi, and Chloe Tergiman for heir commens and suggesions. We are graeful o he NSF (Gran # SES-05790) and he C.V. Sarr Cener for Applied Economics for heir nancial suppor. Par of his research was conduced while Bar Hobijn was a visiing scholar a he Graduae Cener of he Ciy Universiy of New York. The views expressed in his paper solely re ec hose of he auhors and no necessarily hose of he Naional Bureau of Economic Research, Federal Reserve Bank of New York, or hose of he Federal Reserve Sysem as a whole.

2 Inroducion Adoping a echnology requires an acive engagemen of he adoper beyond he selecion of which echnology o adop. The iniial produciviy of a newly adoped echnology depends on a series of invesmens underaken by he adoper. We refer o hese invesmens as he echnology implemenaion process. There is ample evidence on he imporance of echnology implemenaion. Lienz and Rea (998) recognize ha when implemening a new echnology, he curren business process is ofen sreamlined and reengineered o ake advanage of he echnology. Bikson e al. (987) sudy he deerminans of success in implemening mulifuncion ineracive compuer sysems 2. Their ndings indicae ha sie-o-sie variaions in he success of implemening new echnologies are more fully explained by di erences in he implemenaion process han by di erences in he sysems or in he organizaions. Moreover, Brynjolfsson, Hi, and Yang (2002) nd evidence of subsanial organizaional invesmens o accommodae he adopion and enhance he produciviy of new IT echnologies. Despie is empirical signi cance, sae of he ar models of endogenous innovaion and growh ignore he echnology implemenaion process. In his paper, we develop a racable model of endogenous developmen and implemenaion of echnologies. In our model, research and developmen (R&D) aciviies deermine he poenial produciviy of new echnologies. The iniial produciviy of a new echnology and is inensiy of adopion are deermined, however, by he invesmens in implemening he echnology underaken by he adoper. Over ime, he adoper learns how o use he echnology e - cienly and he acual produciviy converges smoohly o he poenial. This generaes a Based on he experience of implemening Plao, an educaional sofware, a over 5000 sies, Plao Learning Inc. has wrien an eigh-poin echnology implemenaion guide o maximize he educaional bene s from he echnology. The di eren sages of implemenaion involve (i) assigning roles prior o adoping echnology, (ii) deciding on program goals, (iii) deciding how echnology will be inegraed in producion process, (iv) deermining who does wha once he echnology is adoped, (v) planning he deploymen of echnology, (vi) planning echnical suppor, (vii) raining professionals and (viii) planning he evaluaion of he program goals and he implemenaion process. 2 They use daa covering 530 employees from 55 di eren deparmens in 26 privae secor organizaions 2

3 smooh di usion of new echnologies ha resembles empirical di usion curves. Including he implemenaion margin in a heory of endogenous innovaion and growh has imporan consequences for he posiive and normaive implicaions of he model. On he posiive side, beer implemenaion makes new echnologies more producive and leads o a faser adopion of echnologies. innovaion and, consequenly, economic growh. Implemenaion, herefore, increases he reurns o Growh, in urn, a ecs he echnology implemenaion decision. In paricular, since higher growh increases he rae of obsolescence of echnologies, growh decreases he reurn o implemening new echnologies a a higher produciviy level and hus leads o lower implemenaion invesmens. I is crucial for his ineracion ha boh echnology di usion as well as obsolescence are endogenously deermined in our model. 3 On he normaive side, he consideraion of an implemenaion margin provides an addiional growh policy insrumen. Besides simulaing growh hrough R&D, he policy maker can also consider simulaing growh by enhancing he inensiy of adopion. As i urns ou, hese wo policy alernaives have very di eren consequences. R&D subsidies simulae innovaion and growh, bu, a he same ime, increase endogenous obsolescence and hus reduce implemenaion. Simulaing implemenaion increases he rae of di usion of echnologies which in urn increases he reurn o R&D and hus growh 4. The implemenaion decision in our model deermines how far below heir poenial produciviy echnologies are operaed. When aggregaing his gap over he echnologies available for producion, our model yields an endogenous level of TFP, saisfying Presco s 3 Endogenous growh and creaive desrucion do yield an endogenous rae of obsolescence bu sandard models generally ake he rae of di usion of echnologies as exogenous. This is he case, for example, in Aghion and Howi s (992) model of creaive desrucion in which echnology di usion is immediae and in Dinopoulos and Waldo (2005) where di usion occurs gradually bu a an exogenous rae. Comin (2000) and Comin and Gerler (2006) inroduce models of boh endogenous di usion and endogenous growh bu hese models di er from ours in ha he inensiiy of adopion is no deermined by he implemenaion invesmens. 4 Kok (2004) and Sapir (2003) emphasize he imporance of policy measures beyond R&D subsidies and ax credis o simulae growh in he European Union. imporance of speeding up echnology di usion. Many of he alernaive policies emphasize he 3

4 (997) reques. Ours is no he rs model in which he disance from he echnological fronier is endogenously deermined. There are basically wo oher mechanisms proposed ha creae such a disance. The rs is knowledge spillovers, which are considered, for example, by Barro and Sala-i-Marin (997) as well as Eeckhou and Jovanovic (2002). Knowledge spillovers will make cerain agens or counries decide o fall behind he fronier and wai for he opporuniy o imiae raher han innovae. The second is heerogenous adopion coss, eiher in he form of enry barriers as in Parene and Presco (994), or because newly invened echnologies are no appropriae for he skill endowmens in he economy, as in Basu and Weil (998) and Acemoglu and Ziliboi (200). The concep of implemenaion is similar in spiri o he idea of appropriae echnology. However, where in he appropriae echnology case produciviy is exogenously deermined by he endowmens of is users, he implemenaion decision ha we consider here endogenously deermines he produciviy of echnologies, he inensiy wih which hey are used, and he rae a which hey di use. One imporan virue of our model is ha i is quie racable and yields a parsimonious represenaion of he equilibrium dynamics of he relevan aggregaes as well as closed form soluions for he underlying di usion curves ha drive he aggregae dynamics. Our model conains eigh preference and echnology parameers. Five of hem are common o sandard business cycles models and only hree are new o he calibraion lieraure. We calibrae hem exploiing informaion on he R&D share in he economy, he produciviy growh rae and micro esimaes of learning by doing. This calibraion permis a reasonable quaniaive analysis of our model economy. We ake advanage of he racabiliy of our model by no only analyzing is seady sae properies bu also is ransiional dynamics. Two of our mos signi can ndings on his fron are; Firs, ransiory perurbances o he model parameers and/or sae variables have permanen e ecs on he level of produciviy. Second, despie having hree sae variables, he responses of he derended variables in our model o shocks o he capial sock are very similar o he responses of hese variables in he neoclassical growh model. 4

5 The seady sae welfare cos of he disorions in he economy is 7.6 percen. The policy ha resores he rs bes involves a subsidy o capial and o implemenaion and a ax on R&D. This is in sharp conras o sandard endogenous growh models ha do no include he implemenaion margin. The srucure of he res of his paper is as follows. We inroduce our model of endogenous growh wih echnology implemenaion in Secion 2. In Secion 3, we presen he posiive resuls of our analysis by considering he dynamic properies of he decenralized marke equilibrium allocaion implied by our model. In Secion 4, we inroduce he normaive par of our analysis. In i, we consider he resource allocaion ha a social planner would choose, compare i o he decenralized economy, and discuss policies ha can suppor he planner s seady sae allocaion in he decenralized equilibrium. We conclude in Secion 5. We leave he mahemaical deails behind he main resuls presened in he ex for he Appendix. 2 The model The model economy ha we consider consiss of four secors: a household secor, a nal goods secor, an inermediae goods secor, and an R&D secor. In his secion, we inroduce each of hese secors separaely. We consider he equilibrium oucome, when hey inerac, in he nex secion. 2. Household secor The represenaive household in our model economy is endowed wih one uni of ime ha i inelasically supplies a each insan. I also owns he capial sock ha i rens o rms a a ne rae r : The household selecs he pah of consumpion c o maximize he presen discouned value of he uiliy ow () Z e (s ) c s ds subjec o he sandard ow budge consrain. 5

6 The resuling opimal consumpion pah is characerized by he Euler equaion 5 (2) _c c = (r ) 2.2 Final goods secor The nal good consumed by households is produced using a se of inermediae goods. Le y denoe he nal good oupu a ime and le y i be he number of unis of he i h inermediae good used in producion a ime. A new inermediae good is inroduced each insan. Therefore, he range of inermediae goods used in producion a ime equals ( ; ]. The nal goods producion funcion is given by Z = (3) y = yidi where 0 < < The marke for he nal good is perfecly compeiive. Throughou, we use he nal good as he numeraire good and will normalize is price o uniy. Le p i be he price of a uni of he i h inermediae good. Given his price, he demand for inermediae good i a ime is given by (4) y i = 2.3 Inermediae goods producers p i y Each inermediae good is provided by a single producer who owns a paen ha ensures his monopoly over he producion of he good. Inermediae goods suppliers make wo ypes of decisions. Firs, a every insan in ime, hey decide on facor demand decisions, he level of oupu, and se he price of he inermediae good hey supply. 5 Caselli and Venura (2000) have shown ha, because he growh rae of consumpion for a household is he same independen of is level of wealh, he disribuion of pro s among households does no maer for aggregae household behavior. Therefore, we ignore he issue of disribuion of pro s across households in our model. 6

7 Second, a he ime hey obain he monopoly righ o he supply of he inermediae good, hey decide a wha iniial produciviy level o implemen he inermediae good s echnology. Over ime, he producer will learn o produce his inermediae good e cienly and he gap beween acual and poenial produciviy will evenually disappear. We consider hese wo choices sequenially. We rs solve for he opimal facor demands and pricing rule aking as given he pahs for produciviy, facor prices, and nal goods demand. Then, we solve for he opimal implemenaion decision. Facor demands and price seing: Inermediae goods are produced using capial and labor ha are combined using a Cobb- Douglas echnology of he form (5) y i = a i k il i where 0 < < The produciviy level wih which he i h inermediae good is produced, a i, is ime-varying. Labor and capial are homogenous and each inermediae goods producer hires hem a he compeiive ne raes w and r. The ow pro accrued by he i h inermediae goods producer a ime, i, is (6) i = p i y i w l i (r + )k i where denoes he capial depreciaion rae. The pricing and producion decisions are saic. Given he producion and demand funcions, inermediae goods producers, herefore, se p i and demand l i and k i o maximize (6). This yields facor demands ha saisfy (7) w = ( ) p i y i l i and r + = p i y i k i and an opimal price equal o a consan gross markup facor, = > ; imes he marginal cos of producion, mc i. Tha is (8) p i = mc i The resuling level of ow pro s in each period is given by (9) i = ( ) p i y i ; 7

8 and he value of he rm equals (0) V i = ( ) Aggregaion over inermediae goods: Z e R s r jdj p is y is ds As we show in he Appendix, he rms decisions in he nal and inermediae goods secor aggregae o a producion funcion of he form () y = z k l = z k where he aggregae capial and labor inpus are given by (2) k = Z k i di and l = Z l i di = Furhermore, he aggregae level of oal facor produciviy is given by a CES aggregae of he produciviy levels of he inermediae goods: Z (3) z = a i di since his is a power mean of he produciviy levels of all inermediaes available, we will refer o z as average produciviy. (4) Firm i s share in aggregae oupu as well as he inpus is given by y i y = ai z, k i k = l i l = ai z p i z, and = p i = p a i The aggregae facor demands urn ou o saisfy he same opimaliy condiions as he facor demands of he individual rms in he sense ha (5) w = ( ) y l and r + = y k This aggregae producion funcion represenaion allows us o rewrie he value of he rm as (6) V i = ( ) Z e R s r ais jdj ys ds z s 8

9 which depends on he pahs of he rm s produciviy level a i, aggregae produciviy, z, he real ineres rae, r, oupu, y. Technology implemenaion: Prior o saring producion, inermediae goods producers decide a which produciviy level o implemen he echnology hey use o produce heir inermediae good. Afer ha momen, hey learn o produce he inermediae good more e cienly and he produciviy level a i increases unil i reaches he poenial produciviy level of inermediae i, which we denoe by a i. Jus like in Parene (994) and Basu and Weil (998), learning occurs exogenously. We de ne he implemenaion level of inermediae i a ime as (7) x i = (a i =a i ), where xi 2 [0; ] We assume ha learning leads x i o evolve according o (8) _x i = ( x i ) where > 0 is he learning rae. This learning paern implies ha (9) a is = e (s ) a i + e (s ) a i for s > : Given his pah of produciviy, he value of he rm can be wrien solely as a funcion of is curren produciviy level and he pahs of aggregae produciviy, he real ineres rae, and oupu. Tha is, he value funcion simpli es o (20) V i (x i ) = ( ) a i Z e R s r jdj ( ) a i ( x i ) Z z s ys ds e R s r jdj e (s ) z s ys ds which is linear in he implemenaion level, x i. This value consiss of wo erms. The rs erm is he presen discouned value of he rm s ow pro s if i would have implemened is echnology a full poenial, i.e. x i =. 9

10 The second erm is he loss in value due o he rm s produciviy being below poenial on is learning curve. Firms only decide where on heir learning curve o sar. The subsequen pah of he implemenaion level is deermined by he exogenous learning process. We assume ha new inermediaes are adoped a he momen ha hey are invened. This implies ha, a each momen in ime, he rm ha obains he monopoly righs o supply inermediae has o decide on is iniial implemenaion level, x. Figure depics he produciviy pah, a i, of inermediae i, implemened a ime i a level x ii and subsequenly converging o a i due o he exogenous learning process. The implemenaion cos of good a ime akes he form (2) C implemen (x ) = ( ) z a z [ ln ( x ) x ] Here, > 0 is he implemenaion cos parameer and x is he iniial implemenaion level. In addiion, z a is a scaling facor ha guaranees balanced growh, while z capures he fac ha i is more cosly o implemen echnologies whose poenial produciviy level, a, is furher from he average produciviy level in he economy, z. The opimal implemenaion level, x, maximizes he di erence beween he value of he rm and he implemenaion coss (22) V (x ) C implemen (x ) As we show in he Appendix, x is equal o (23) x where (24) b x = Z a a = b x y e R s r jdj e (s z + b x y ) z ys ds z s y Inuiively, he marginal bene of implemenaion is increasing in he erm b x y while he marginal cos is increasing in z in he former and decreasing in he laer. : The opimal implemenaion level is herefore increasing 0

11 The resuling value a of he rm ha produces he las good inroduced ne of implemenaion coss, i.e. he value of (22) in he opimal implemenaion level, equals "!# (25) V = ( ) z a y (b 0 + b x ) ln + y b x where (26) b 0 = 2.4 R&D secor Z z e R s r jdj z e (s ) z ys ds z s y Following Reinganum (989) and Tirole (988), we model he R&D process as a paen race. One paen is awarded o he innovaor who, wihin he insan, develops he inermediae good wih highes poenial produciviy. z Le g denoe he growh rae of poenial produciviy beween he inermediae good developed a (27) d and : Tha is: a a = g The inensiy of research and developmen deermines g : In paricular, he cos of invening an inermediae good whose poenial produciviy represens a growh in he echnological fronier of g is equal o 8 < (28) C R&D (g ) = : ( ) z 0 for g < 0 a z g for g 0 where > 0 is he R&D cos parameer, z is a scaling facor ha guaranees balanced a growh, and z re ecs he fac ha i is more cosly o develop echnologies whose poenial produciviy is furher from he average produciviy in he economy. The R&D paen race beween innovaors brings he growh rae of he poenial produciviy induced by inermediae good o he poin where he value of he inermediae good equals he R&D cos. Tha is, (29) V = C R&D (g )

12 which can be rewrien as (30) g = y 3 Equilibrium z (b 0 + b x ) ln + b x The dynamic equilibrium allocaion of resources in his economy can be described by a se of di erenial equaions. In his secion, we presen hese equaions, we ransform hem o y z allow us o de ne a balanced growh pah and ransiional dynamics. Crucial for he equilibrium pah of he economy are hree measures of produciviy. The rs is he poenial produciviy level of he newes inermediae ha is inroduced, a. The second is he average poenial produciviy level a which all currenly available inermediae goods echnologies can be operaed, which equals Z (3) z = a i di The hird is he average produciviy level, z. Noe ha boh he average produciviy and average poenial produciviy levels are aggregaes ha summarize he poenial and acual produciviies across inermediae goods. De niion: The equilibrium of his economy is a pah of he variables (32) fy ; c ; i ; k ; x ; b x ; b 0 ; g ; z ; z ; a g ha sais es he following equaions.! (i) The resource consrain ha implies ha he nal good oupu is eiher consumed, saved, used for implemenaion purposes, or used for R&D (33) y = c + i + ( ) z (ii) The consumpion Euler equaion 6, (2). a z ( [ ln ( x ) x ] + g ) 6 The opimal capial inpu condiion, (5), is wha pins down he real ineres rae in he Euler equaion as a funcion of he marginal revenue of capial. 2

13 (iii) The aggregae producion funcion, (). (iv) The capial accumulaion equaion (34) _ k = i k (v) The opimal implemenaion level, x, given by (23). (vi) Equilibrium R&D condiion ha deermines he growh rae g ; (30). (vii + viii) The expressions for he presen discouned value coe ciens, b x and b 0, given by (24) and (26). (ix) The law of moion for a, (27). (x) The law of moion for average poenial produciviy, (35) z = a (xi) The law of moion for average produciviy, derived in he Appendix, which is (36) z = z z + a x Since c ; y ; k ; and i are no saionary on he equilibrium pah, we ransform hem ino saionary variables before exploring he ransiional dynamics and seady sae. We do so by scaling hem by he average produciviy rend z are (37) c = c z, y = y z, k = k =( ) z. The resuling derended variables, and i = i z In addiion, we de ne ransformed presen discouned value coe ciens as (38) (39) b x = b x b 0 = b 0 + b x = Z e R s r z ys jdj ds z s y 3

14 Because our aim is o obain a saionary represenaion of he equilibrium pah, we derend our hree produciviy measures, a, z, and z. produciviy measures (40) = Here, a z and = This yields he following wo derended z z represens he gap beween he poenial produciviy of he las inermediae good invened and he average poenial produciviy level. We herefore refer o i as he poenial produciviy gap. The second measure,, re ecs he gap beween average poenial produciviy and he average acual produciviy level a which he curren inermediaes are implemened. We call his he implemenaion gap. In he Appendix we de ne a saionary equilibrium in erms of he 0 ransformed variables (4) fy ; i ; c ; k ; x ; g ; b 0; b x; ; g The hree sae variables of he 0 equaion sysem are k,, and. 3. Seady sae A seady sae of his economy is an equilibrium pah on which he en ransformed variables are consan. Le he seady sae values of hese variables be n (42) ey ;ei ; ec ; e k ; ex; eg; e b 0; e o b x; e; e As we prove in he Appendix, he seady sae exiss and is unique whenever (43) = + > 0 which is he case under he empirically plausible su cien condiion ha >.7 The seady sae is deermined by he following equilibrium R&D zero pro condiion! (44) eg = y e b 0 ln + y e b x 7 We also derive he condiions under which he parameers are such ha he household s and inermediae goods producer s objecives are bounded on he balanced growh pah. 4

15 where he seady sae level of oupu, y, is deermined by he law of moion of aggregae capial and he aggregae producion funcion and equals (45) ey = + + eg ( )! and he seady sae presen discouned value erms are (46) e b 0 = + eg and e b x = ( + + eg) The lef hand side of (44) re ecs he normalized seady sae R&D expendiures, while he righ hand side corresponds o he normalized presen discouned value of he pro s ne of he coss of implemening a new inermediae goods echnology. Given eg, he opimal implemenaion level sais es (47) ex ex = ey ( + + eg), and hus ex = + (++ eg) ey Noe ha his expression is decreasing in eg. This is because an increase in he growh rae increases he endogenous rae of obsolescence of new echnologies. I does so hrough wo channels. Firs, increased obsolescence reduces he seady sae level of (derended) capial and hus of oupu, ey. This reduces he size of he marke and he value of he rm. Second, eg increases he e ecive discoun rae boh hrough a higher ineres rae, er; and hrough a higher replacemen rae of demand by fuure, more producive, inermediae goods. 8 Boh of hese e ecs reduce he marginal value of implemening a a higher level and herefore lead o a lower ex. The seady sae poenial produciviy gap equals (48) e = eg; 8 There is a hird e ec of ~g on he e ecive discoun rae. Namely, he posiive e ec ha ~g has on he growh of aggregae demand. Our assumpion ha > 0 implies ha his e ec is dominaed by he previous wo and herefore he e ecive discoun rae is increasing in ~g: 5

16 while he implemenaion gap is (49) e = + e + ex e on he balanced growh pah. For compleeness, he scaled seady sae levels of capial, invesmen and consumpion are given by he following expressions: (50) e k = (ey ) = (5) ei = + eg e k (52) ec = ey ei ( ) ee [ ( ln ( ex) ex) + eg] Comparaive saics of growh and implemenaion in seady sae Since we focus on he ineracion beween innovaion and echnology implemenaion, we consider nex how he seady sae growh rae, eg, and implemenaion level, ex, vary as a funcion of he R&D coss,, he implemenaion coss,, as well as he learning rae,. Table summarizes he signs of hese comparaive saic exercises. The deails underlying hese resuls are presened in he Appendix. The inuiion behind hese resuls is bes undersood hrough Figure 2. This gure depics he seady sae R&D free enry condiion, (44), as he g-locus and he seady sae opimal implemenaion condiion, (47), as he x-locus. The laer is downward sloping because of he negaive e ec ha increased obsolescence has on he implemenaion level. The former is a verical line because we have subsiued he opimal implemenaion level ino he value of he inermediae good rm ha appears in he R&D free enry condiion. This gure reduces he comparaive saics of eg and ex o shifs in he g- and x-loci. An increase in he R&D cos parameer,, reduces he R&D e ors of he innovaors, for a given x, causing an inward shif in he g-locus, and has no e ec on he x-locus. As a resul, an increase in he R&D cos reduces he seady sae growh rae, increases he 6

17 presen discouned value of demand faced by an inermediae good producer and leads o an increase in he seady sae implemenaion level. An increase in he learning rae,, makes rms implemen less and depend more on heir subsequen learning for a given g. Thus, he x-locus shifs inward in response o an increase in he learning rae. On he oher hand, increases he produciviy growh rae for he inermediae good rm hus increasing is value and he reurn o R&D invesmens. As a consequence, an increase in shifs he g-locus ouward. The resul is ha, in response o an increase in he learning rae, he seady sae growh rae increases while he implemenaion level decreases. An increase in he implemenaion cos reduces he reurn o implemenaion for a given g causing an inward shif in he x-locus. I also reduces he value of an inermediae good rm and he reurn o innovaion causing an inward shif in he g-locus. This means ha our diagram, in principle, does no su ce o deermine he sign of he e ec of an increase of on eg and ex. In he Appendix we show, however, ha he downward movemen in he x-locus dominaes he lefward shif of he g-locus in his case and ha boh he seady sae growh rae as well as he implemenaion level are decreasing in he implemenaion cos. These comparaive saics acually provide an ineresing insigh ino he e ecs of wo alernaive policies aimed a simulaing long-run growh: (i) subsidizing R&D, by reducing, and (ii) subsidizing implemenaion and speeding up di usion hrough reducing. Boh of hese policies will increase he long-run, seady sae, growh rae. R&D subsidies, however, come a he cos of reducing implemenaion and slowing down di usion because of he obsolescence cos i imposes. This is no he case for a reducion of implemenaion coss. Such a reducion will also increase he implemenaion level and di usion. Technology di usion Underlying he seady sae is a coninuous process of di usion of new inermediae goods. Conrary o oher models of endogenous growh, he rae of di usion of (inermediae good) echnologies is endogenously deermined in he model here. This rae of di usion can be considered in wo ways. 7

18 The rs way is o consider he marke share of an individual inermediae good. We will denoe he marke share of inermediae i a ime as 8 (53) s i = p < iy i = y : p i In he Appendix, we show ha in he seady sae 0 if < i a = i z oherwise (54) s i = ( ex) e ( i) e eg( i) ee As shown in equaion (55), wo opposie forces drive he dynamics of he marke share for i h inermediae good: On he one hand, learning o produce e cienly he inermediae good induces a produciviy gain ha raises he marke share. On he oher, he increase in he average produciviy due o he developmen of new (more producive inermediae goods) and o learning makes obsolee he i h inermediae good reducing is marke share. Learning z } { (55) _s i = e eg( i) ee Obsolescence z } { + eg s i When he implemenaion coss are large he marke iniial share is so small ha he learning e ec dominaes and he marke share is iniially increasing. As he marke share increases, he endogenous obsolescence of he inermediae good dominaes and he marke share declines o zero. An iniially increasing and subsequenly decreasing marke share is consisen wih evidence on produc life cycles documened in Jovanovic and McDonald (994), Aizcorbe, Corrado, and Doms (2000), and Koler (2005). The second way o characerize he rae of di usion is o consider he share of expenses in inermediae goods ha are newer han good i: Given he facor demands (7), his is equivalen o he fracion of workers employed in he producion of an inermediae good inermediae good newer han i: This fracion is given by (56) S i = Z i p y y d = Z i l d 8

19 As we show in he Appendix, on he balanced growh pah, his adopion share follows a di usion curve of he form (57) S i = e eg( i) e (e ) e ( i) for i 3.2 Transiional dynamics The equilibrium dynamics of he model are deermined by hree sae variables. Thus, analyical resuls abou he dynamic properies are beyond he scope of our analysis. We resor, insead, o a numerical approximaion of he ransiional dynamics for a speci c se of calibraed parameer values. The numerical approximaion of he ransiional dynamics is based on he log-linearizaion derived in he Appendix. Calibraion: The parameers ha need o be calibraed are lised in Table 2. We calibrae our model such ha is measured in years. For he preference parameers, i.e. he discoun rae,, and he ineremporal elasiciy of subsiuion,, as well as for he capial depreciaion rae,, we use he parameer values from Cooley and Presco (995). Given our model s emphasis on R&D and implemenaion, i seems paricularly appropriae o use he evidence from Corrado, Hulen, and Sichel (2006, CHS in he following) o calibrae mos oher parameers from our model. CHS provide an analysis of he sources of growh of he U.S. business secor ha includes exensive measures of inangible capial, including R&D. We calibrae he demand elasiciy parameer,, and he capial elasiciy of oupu,, o mach U.S. income shares repored by CHS. Firs, labor coss represen 60% of corporae income. Second, reurns o inangible capial represen 5% of he price. In our model hese are he pro s ha ow o he implemenaion and R&D coss. The remaining parameers are he learning rae,, and he implemenaion and R&D cos parameers, and. These parameers are chosen o mach hree observaions, he rs wo of which are based on CHS. Firs, invesmen in R&D by he U.S. business secor 9

20 is approximaely 5.7% of corporae income. Second, adjused for inangible capial, labor produciviy in he U.S. grew a an average rae of.9% a year over he period. Finally, we use evidence from Bahk and Gor (993) on learning by doing in U.S. manufacuring plans. In paricular, we choose our parameers o mach heir empirical resul ha a % increase in a rm s cumulaive oupu leads o a 0.028% increase in is TFP level. The way in which we speci cally mach he parameers wih hese facs is described in Appendix A. The resuling values of he parameers are lised in he las column of Table 2. Seady sae: The seady sae values of he equilibrium variables are given in he equilibrium column of Table 3. The resuling implemenaion level is abou 4.3%, while he implemenaion gap is 4.6. The relaively low implemenaion level in seady sae induces he marke share of new inermediaes o increase a rs and hen o sar decreasing afer abou 0 years. This can be seen from he equilibrium curve in he op panel of Figure 3. Tha is, 0 years ino he life cycle of an inermediae in his economy he endogenous obsolescence sars o dominae he learning e ec. The implied di usion curve is ploed as he equilibrium curve in he boom panel. In he decenralized equilibrium, 50% of he workers produce inermediaes ha were invened less han 8 years ago. Transiional dynamics: We compare he ransiional dynamics of our model wih hose of he sandard Neoclassical growh model, explained for example in Barro and Sala-i-Marin (2004). Tha is, if growh is exogenous, consan and equal o he seady sae growh rae, eg, and if implemenaion coss are zero, such ha = 0, hen our model boils down o he Neoclassical growh model wih a markup disorion 9. In paricular, we compare he dynamics in response o a % deviaion of capial above is seady sae derended level. Figure 4 conains he impulse responses for he model wih implemenaion, he implemenaion line, and he Neoclassical benchmark, he NC benchmark line. In he benchmark model he excess capial is used for curren and fuure consumpion 9 The dynamic equilibrium equaions of his resriced model are provided in he Appendix. 20

21 hrough ineremporal subsiuion. The same is rue in he model wih implemenaion. However, in he model wih implemenaion here is no jus ineremporal subsiuion of consumpion. An above rend capial sock increases he size of he marke and hus he presen discouned value of he sream of fuure pro s. This raises he bene s from innovaion and implemenaion. As a resul, he above seady sae level of capial leads o a ransiory increase in he growh rae of poenial produciviy and in he implemenaion level and o a permanen increase in he level of produciviy. 0 This is he main deparure of our model from he dynamics of he Neoclassical benchmark. Relaive o he Neoclassical model, he addiional implemenaion and R&D expendiures seem o come mainly a he cos of invesmen and no of consumpion. Wha is remarkable is ha, in erms of he derended variables, he impulse responses in he model wih implemenaion are very similar o ha of he neoclassical growh model. Our model has wo sae variables in addiion o he capial sock: he implemenaion gap and he poenial produciviy gap. Figure 5 plos he impulse responses o a % deviaion of he hese gaps from heir seady sae level. For comparison purposes, we have also included he impulse response o capial. Increases in hese wo gaps increase implemenaion and R&D coss and herefore reduce boh implemenaion and innovaion. The increase in he implemenaion and R&D coss induce a shif in resources from implemenaion and innovaion o invesmen in physical capial. As a resul, hese changes induce very signi can ransiory declines in he implemenaion level and he growh rae of poenial produciviy and an imporan permanen decline in he level of produciviy. There seems, however, o be lile or no e ec on derended consumpion and derended oupu. 4 Social planner soluion So far, we have focused on he decenralized equilibrium oucome of our economy. Nex, we explore he opimal innovaion and implemenaion decisions from he social planner s perspecive. 0 This may no be obvious from he impulse response funcions because oupu is derended by z : 2

22 In his secion we derive he rs order necessary condiions for he social planner s problem. We sudy he seady sae implemenaion level, ex sp, and growh rae, eg sp, chosen by he planner, and compare hem wih hose resuling from he decenralized equilibrium. Finally, we show how he planner s seady sae resource allocaion can be suppored hrough axes and subsidies in he decenralized equilibrium. The social planner in his economy chooses a pah for (58) fc s ; y s ; i s ; k s ; x ss ; g s ; z s ; a s ; z s g s= o maximize he presen discouned value of he represenaive household s sream of uiliy, (), subjec o he resource consrain, (33), he nal goods producion funcion, (), he capial accumulaion consrain, (34), he law of moion of poenial produciviy of he newes inermediae, (27), he law of moion of average poenial produciviy, (35), and he law of moion of average produciviy, (36). The curren value Hamilonian associaed wih his problem is: (59) H = c ( + r y c i ( ) z + y " + z y a z k + z # z a z ) [ ( ln ( x ) x ) + g ] + k [i k ] + a a z + a x g where r is he cosae variable associaed wih he resource consrain, y is he cosae variable associaed wih he aggregae producion funcion, a is he cosae variable associaed wih he law of moion of poenial produciviy of he las inermediae good, z is he cosae variable associaed wih he law of moion of average poenial produciviy, and z is he cosae variable associaed wih he law of moion of average produciviy. A any insan along he e cien resource allocaion pah he planner equaes he marginal uiliy cos of implemenaion o he shadow value of he marginal average produciviy ha his implemenaion generaes. This is represened by equaion (60). The planner also 22

23 equaes he marginal uiliy cos of a beer innovaion, i.e. of g, o he shadow value of he addiional poenial produciviy his innovaion generaes hrough a. Mahemaically, his corresponds o equaion (6). (60) (6) x ( ) c = x z; z ( )c = a;z 4. Disorions in decenralized equilibrium We characerize he full dynamics of he planner s opimal resource allocaion in he Appendix and focus here on he resuling seady sae, also derived in he he Appendix. To disinguish he planner s seady sae allocaion from ha of he decenralized equilibrium, we denoe he planner s allocaion wih a superscrip sp. Thus, ex sp and eg sp are he planner s seady sae implemenaion level and growh rae respecively. The planner s seady sae level of derended oupu is (62) ey sp = " + + egsp ( ) # = " + + egsp ( ) # For a given growh rae, i is higher han he oupu level in he decenralized equilibrium, because he monopolisic compeiion beween he inermediae goods producers leads o an ine cienly low level of oupu in he decenralized equilibrium. The seady sae implemenaion level sais es " ex sp (63) ex sp = # e e + ( ) ey sp [ ( ln ( exsp ) ex sp ) + eg sp ey sp ] ( + + eg sp ) We call his he x sp -locus. I is he social planner s counerpar o (47). This implies ha, given he growh rae, he planner s implemenaion level is higher han ha in he decenralized equilibrium for hree reasons. Firs, oupu in he planner s allocaion is higher, herefore increasing he marginal bene of implemenaion. Second, he monopolisic compeiion also in uences he marginal bene of implemenaion because he rms ha implemen in he decenralized equilibrium equae he marginal revenue produc of implemenaion o is marginal cos. The planner insead equaes he marginal produc of implemenaion 23

24 o is marginal cos. Because of he downward sloping demand curves ha he monopolisically compeing inermediae goods producers face in he decenralized marke, he marginal revenue produc is less han he marginal produc and, hence, hese rms underimplemen relaive o he social planner. This di erence is re eced in he =, erm ha pre-muliplies he squared bracke in he RHS of equaion (63). The nal reason ha, a a given growh rae, he planner chooses a higher implemenaion level han is realized in he decenralized equilibrium is ha he planner inernalizes he e ec of he implemenaion level on he implemenaion gap. A higher implemenaion level decreases he implemenaion gap and, hrough ha, reduces he implemenaion and R&D coss. This e ec is re eced by he second erm in he above equaion. The comparison of he seady sae oupu and implemenaion levels above was done condiional on he growh rae, eg sp. Given ex sp and ey sp, he e cien seady sae growh rae is deermined by he opimal innovaion condiion (64) = Mg. Soc. value of a z h } i{ ++ eg sp ex sp + ln ( ex sp ) eg sp + eg sp ex sp + = egsp We call his he g sp -locus. The decenralized equilibrium counerpar of his equaion, which is a rewrien version of (44), reads + + eg ex (65) 0 = + ln ( ex) + eg ex For a given implemenaion level, here are hree di erences beween he opimal innovaion condiion in he decenralized equilibrium, (65), and ha of he social planner, (64). Firs, he paen race ha deermines he R&D inensiy in he decenralized economy equalizes he value of he inermediae good producer ne of innovaion and implemenaion coss o 0. The social planner, insead, equalizes he marginal value and he marginal cos of innovaion. The (scaled) marginal cos of innovaion is > 0: Hence, he paen race leads o oo much innovaion in he decenralized equilibrium relaive o he planner s. Second, he planner inernalizes he fac ha g leads permamenly o a higher a : As a resul, he marginal social value of innovaing is higher han he marke value of he 24 eg

25 inermediae good porducer, for a given x, for wo reasons. On he one hand, a higher g increases poenial produciviy levels in all fuure imes. Hence he discouning in he RHS of (64). On he oher hand, g increases he growh rae of a by g: Hence, he erm ha pre-muliplies he marginal social value of a in he righ hand side of equaion (64). Because he x sp -locus is above he x-locus for all growh raes, he planner s seady sae can no be such ha boh he growh rae and he implemenaion level are below ha of he decenralized seady sae. Figure 6 adds he planner s g sp - and x sp -loci o Figure 2. I shows he hree possible cases of he planner s seady sae compared o he decenralized one: (i) eg sp < eg and ex sp > ex; (ii) eg sp > eg and ex sp > ex; and (iii) eg sp > eg and ex sp < ex. 4.2 Supporing he planner s soluion We do no formally characerize hese hree cases, bu insead describe he policies ha suppor he social planner s seady sae equilibrium in he decenralized economy. Such a policy analysis serves wo main purposes. Firs, i allows us o consider how privae coss and bene s can be correced o coincide wih hose in he e cien resource allocaion. Second, i helps us undersand how opimal R&D policies inerac wih policies aimed a a ecing he implemenaion and adopion of echnologies. The decenralized and social planner s seady saes generally di er in he R&D inensiy, in he implemenaion level, and in he saving rae. To align hese hree margins in he decenralized equilibrium wih hose in he planner s allocaion, i is necessary o use hree insrumens. The insrumens we consider o decenralize he social planner s seady sae are an R&D subsidy, s g, a subsidy o implemenaion, s x, and a subsidy for capial, s k. These subsidies can poenially be negaive, and hus be a ax. We assume ha hese subsidies are nanced by a lump-sum ax or subsidy ha balances he governmen s budge. Because of he nondisorionary naure of his lump-sum ax, we do no focus on i in our analysis. We derive he opimaliy condiions in he decenralized equilibrium under he subsidies in he Appendix. In order o see how he disorionary subsidies can be used o suppor he planner s soluion, we consider each of he opimaliy condiions ha hey a ec in he 25

26 decenralized equilibrium sequenially and compare hem wih hose of he planner s seady sae. Condiional on he oher subsidies supporing he planner s growh rae and implemenaion level, he opimal capial inpu subsidy is such ha " (66) ey sp = s k + + egsp ( ) # Hence, he opimal capial inpu subsidy is s k =. This correcs for he disorion induced by he monopolisic compeiion beween he inermediae goods suppliers. In he decenralized equilibrium, his disorion leads o an undersupply of oupu, a less han e cien level of capial, and hus a lower han opimal saving rae. The capial subsidy increases he privae marginal reurn o capial o o se his. To suppor he opimal implemenaion level, condiional on supporing ey sp and eg sp, he implemenaion subsidy has o saisfy (67) ex sp ex sp = ( s x ) ( + + eg sp ) eysp which, combined wih (63), yields ha he opimal implemenaion subsidy should solve " (68) ( s x ) = # e e + ( ) ey sp [ ( ln ( exsp ) ex sp ) + eg sp ] > s x correcs for he wo e ecs ha are no inernalized in he decenralized implemenaion decision: (i) The e ec of monopolisic compeiion on equilibrium pro s, re eced by (=); (ii) he e ec of he implemenaion decision on he implemenaion gap and hus on he coss of implemening and invening. Because boh of hese e ecs lead o underimplemenaion in he decenralized equilibrium relaive o he e cien allocaion, he opimal policy in his conex always involves a subsidy o implemenaion. To suppor he planner s growh rae in he decenralized equilibrium, he R&D subsidy has o saisfy (69) eg sp = ( s x) ( s g ) + + eg sp + eg sp which, combined wih (64), allows us o wrie (70) ( s g ) = ( s x ) s 0 g ex sp ex + ln ( sp exsp ), where s 0 g = 26 eg sp +

27 Thus he opimal R&D subsidy consiss of wo pars. The rs par correcs for he wedge beween privae and social implemenaion coss induced by he implemenaion subsidy s x. The second par is given by (7) s 0 g = eg sp Inuiively, he social bene s from innovaion are, in large par, deermined by he ineremporal elasiciy of subsiuion of he represenaive consumer because he ineremporal elasiciy of subsiuion ogeher wih he discoun rae,, and he growh rae of he economy, is wha deermines he e ecive discoun rae a which he planner discouns he fuure gains from curren innovaion invesmens. When he represenaive consumer has a low ineremporal elasiciy of subsiuion, i.e. <, he planner heavily discouns fuure gains from curren R&D invesmens. Hence, he e cien allocaion involves devoing fewer resources o innovaion han he decenralized economy. In ha case, he opimal R&D subsidy would be lower han he implemenaion subsidy and poenially involve a ax on innovaive aciviies. Conversely, when he agen is more willing o ineremporally subsiue consumpion oday for consumpion omorrow, i.e. >, he presen discouned value of he social payo s from R&D are larger and he social planner prescribes more growh han in he decenralized economy. Besides he insighs ha we obained on how he disorions in he decenralized equilibrium can be correced using he opimal choice of hree policy insrumens, we obained anoher imporan insigh. The opimal R&D and implemenaion policies are inherenly inerwined. This implies ha any policy discussion abou simulaing or reducing he incenives o innovae should also ake ino accoun he incenives o implemen and adop he innovaions. 4.3 Quaniaive evaluaion To illusrae he planner s seady sae, we consider i for he calibraed parameer values of Table 2. The planner column of Table 3 shows he e cien allocaion of resources for 27

28 he same parameers as he decenralized equilibrium oucome we discussed in Subsecion 3.2. For he calibraed parameer values, he decenralized equilibrium would have oo lile growh and implemenaion. A.38%, he e cien growh rae is half a enh of a percenage poin higher han ha in he marke equilibrium. Moreover, he e cien implemenaion level is 6.75%, which is almos 2.5 percenage poins higher han in he decenralized equilibrium. These disorions in he marke implemenaion level and growh rae means ha echnologies di use a an ine cienly low rae. To see his, compare he e cien planner di usion curves wih he decenralized equilibrium ones in Figure 3. The e cien 50% di usion ime in he boom panel of he gure is abou 6 years, while he 50% di usion ime in he marke equilibrium is closer o 8 years. The misallocaion of he resources in he decenralized economy cause a signi can welfare loss. Welfare in he seady sae of he decenralized equilibrium is 7.6% lower han ha resuling from he planner s allocaion. A a consan growh rae, his would amoun o 0.36% of seady sae consumpion. This is more han 40 imes larger han he welfare cos for log preferences of consumpion ucuaions around a linear rend esimaed by Lucas (987). The opimal policy derived above can o se he resource misallocaion caused by he model disorions in he decenralized equilibrium. For our calibraed parameer values, he opimal policy involves a 5% capial inpu subsidy, a 35% implemenaion subsidy, and a 7% R&D ax. These policies are in sharp conras wih he policy prescripions of sandard endogenous growh models ha do no include he implemenaion margin. These generally yield an opimal R&D subsidy raher han a ax, as in Jones and Williams (998, 2000). If, in our economy, he governmen ried o implemen he opimal policy ignoring implemenaion, i.e. assuming = 0, i would choose a capial inpu subsidy of 5% and an R&D subsidy of 59%. This, however, would lead o a 5% loss in welfare relaive o he policies derived above, which represens only a 2% improvemen over doing nohing. The opimal policies for an economy wihou implmenaion are derived in he Appendix. 28

29 5 Conclusion We have inroduced he echnology implemenaion decision in a heory of endogenous growh. In our model, he implemenaion decision deermines he iniial produciviy of he echnology o produce a new inermediae good. The gap beween he poenial and he acual produciviy of he new echnology is closed over ime hrough exogenous learning. Our model is su cienly racable o analyze no only is seady sae bu also is ransiional dynamics. The addiion of a echnology implemenaion decision o an, oherwise, sandard model of endogenous growh leads o wo imporan insighs. Firs, he equilibrium e ec of growh on implemenaion is he opposie of he e ec of implemenaion on growh. An increase of he growh rae increases he rae of endogenous obsolescence of echnologies. I reduces he presen discouned value of pro s and herefore he bene and level of implemenaion. An increase in implemenaion leads o a more inensive adopion of new echnologies and raises he marke value of he rms ha produce hem. This raises he reurn o R&D, and hus leads o an increase in long-run growh. Second, opimal policy in our model does no only involve R&D axes or subsidies bu also requires inervening in he cos of implemenaion. This suggess ha any discussion of policies o simulae long run growh should no only consider subsidizing he aciviies of innovaors, bu, jus as imporanly, consider subsidizing he implemenaion, and hrough i he di usion, of he echnologies ha hese innovaors creae. The model ha we analyzed here is basically a sylized model of he world echnology fronier and, as such, he normaive resuls in his paper could be inerpreed as applying o a world growh policy. In pracice, such policies are no decided on a a global level, bu, insead, are chosen by naional governmens. The consideraion of opimal naional implemenaion policies versus R&D ax credis becomes even more relevan when i is done in a muli-counry conex wih R&D spillovers. This is he subjec of our fuure research. 29

30 References [] Acemoglu, Daron and Ziliboi, Fabrizio (200), Produciviy Di erences Quarerly Journal of Economics, 6, [2] Aghion, Philippe, and Peer Howi (992), A Model of Growh hrough Creaive Desrucion, Economerica, 60, [3] Aizcorbe, Ana, Carol Corrado, and Mark Doms (2000), Consrucing Price and Quaniy Indexes for High Technology Goods, mimeo, Federal Reserve Board of Governors. [4] Bahk, Byong-Hyong and Michael Gor, (993) Decomposing Learning by Doing in New Plans The Journal of Poliical Economy, Vol. 0, No. 4. (Aug.), pp [5] Barro, Rober J. and Xavier Sala-i-Marin (997), Technological Di usion, Convergence, and Growh, Journal of Economic Growh, 2, 26. [6] Barro, Rober J. and Xavier Sala-i-Marin (2003) Economic Growh, 2nd Ediion, MIT Press. [7] Basu, Susano and David N. Weil (998), Appropriae Technology and Growh, Quarerly Journal of Economics, 3, [8] Bikson, Tora K., Don Mankin, Barbara A. Guek (987), Implemening Compuerized Procedures in O ce Seings: In uences and Oucomes Rand Repor. [9] Brynjolfsson, Erik, Lorin M. Hi, and Shinkyu Yang (2002), Inangible Asses: How he Ineracion of Compuers and Organizaional Srucure A ecs Sock Marke Valuaions, Brookings Papers on Economic Aciviy: Macroeconomics,, [0] Caselli, Francesco, and Jaume Venura (2000), A Represenaive Consumer Theory of Disribuion, American Economic Review, 90, [] Cooley, Thomas F. and Edward C. Presco (995), Economic Growh and Business Cycles, in Froniers of Business Cycle Research, Thomas F. Cooley (ed.). 30