Opimal Research and Developmen Expendiure: a General Equilibrium Approach Galo Nuño y Banco de España May 7, 2 Absrac How much should be spen in research and developmen (R&D)? How should R&D vary over he business cyle? In his paper we answer boh quesions in he conex of a calibraed dynamic general equilibrium model wih Schumpeerian endogenous growh. Firsly, we demonsrae ha, alhough he exisence of disorions in a decenralized economy produces underinvesmen in R&D, a simple proporional subsidy o R&D spending alone canno resore he rs bes allocaion. The opimal proporional R&D subsidy aains a second bes allocaion in which R&D spending exceeds is rs bes level. Secondly, we show how he observed procyclicaliy of R&D is socially ine cien. However, he welfare loss due o his dynamic ine ciency is much smaller han he loss due o underinvesmen in R&D. Keywords: Schumpeerian growh, echnology adopion, opimal subsidy. JEL E32, O38, O4 Inroducion How much should be spen in research and developmen (R&D)? How should R&D invesmen vary over he business cycle? In his paper we ry o provide an answer o boh quesions. The rs quesion refers o he socially opimum average amoun of resources devoed o innovaion aciviies. Romer (99), Aghion and Howi (992) or Grossman and Helpman (99, ch. 4) underline how in a decenralized economy several disorions may produce an ine cien allocaion of resources o R&D. These disorions include he exisence of monopolisic pro s, he presence of knowledge spillovers, and he redisribuion of rens from pas innovaors o curren ones hrough a process of creaive desrucion. Jones and Williams (2) analyze hese e ecs in he conex of a endogenous growh model calibraed wih U.S. daa and conclude ha here exiss underinvesmen in R&D. The second quesion refers o how R&D spending should respond o di eren shocks. The noion ha macroeconomic shocks migh a ec R&D aciviies implies ha such shocks can have long-lived consequences, far beyond any paricular cyclical episode, as suggesed by Comin and Gerler (26). The radiional Schumpeerian view implies ha recessions should promoe innovaion and resrucuring aciviies, as in Caballero and Hammour (994) or Aghion and Sain-Paul (998). This view ress on he idea ha he opporuniy cos of R&D are lower in recessions, providing incenives o underake such aciviies in downurns. Barlevy (27), however, shows how R&D is procyclical in he Unied Saes. He explains his procyclicaliy as he resul of he dynamic exernaliies ha make enrepreneurs concenrae heir innovaion in booms. He also concludes ha opimal R&D would be less procyclical. Nowihsanding, his model is silen abou he magniude of he welfare loss of his dynamic ine ciency, compared o he saic loss due o underinvesmen in R&D. I am graeful for commens and suggesions by Alessio Moro, Anon Nakov and paricipans a he 28 EEA meeing in Milano and he 29 CEMLA Conference in Salvador de Bahia. The views expressed are hose of he auhor and do no represen he views of he Bank of Spain or he Eurosysem. y Galo Nuño is a Banco de España, Economía y Asunos Inernacionales, Alcalá 48, 284 Madrid. There are oher complemenary explanaions for he procyclicaliy of R&D such as credi fricions, as in Aghion e al. (25) or endogenous labor supply (Faas, 2).
In his paper we presen a dynamic sochasic general equilibrium (DSGE) model ha inegraes endogenous growh ino an oherwise sandard real business cycle (RBC) model. Endogenous growh is based on a Schumpeerian heory of verical innovaions à la Aghion and Howi (998, ch. 2). This heory provides a robus descripion of he hisorical growh process, as discussed in Howi(2) or Howi and Mayer-Foulkes (25). To analyze how much R&D is socially opimum, we calibrae he seady sae soluion of he decenralized economy model wih U.S. daa for he pos-war period, and compare i wih he counerfacual e cien allocaion soluion in he case of a social planner, a similar approach o Jones and Williams (2). We follow a similar approach o analyze how R&D invesmens should be allocaed over he cycle. Firsly we show ha our calibraed model is able o replicae he main dynamic feaures of he daa, especially he procyclicaliy of R&D spending and he high persisence of he responses o he shocks, and hen we compare i wih he counerfacual e cien allocaion. Our ndings can be summarized as follows. The exisence of disorions in a decenralized economy produces underinvesmen in R&D, which generaes lower business urnover and produciviy growh han under he e cien allocaion. A proporional subsidy o R&D does no allow he economy o reach is Pareo-e cien level. This is due o he fac ha he economy is facing several disorions, so miigaing a single disorion may lead o a second-bes siuaion. We compue he subsidy ha maximizes social welfare under he consrains of a decenralized economy and show how i produces a level of R&D invesmen higher han under a social planner. We also nd how he observed procyclicaliy in R&D expendiure may be explained as he response of raional enrepreneurs facing exogenous shocks. We exend he resuls of Barlevy (27) by showing ha alhough his procyclicaliy is ine cien, he associaed welfare loss is more han one order of magniude smaller han he loss due o he saic inef- ciency. Therefore, even if hese ndings suppor he necessiy of counercyclical policies, hey underline he imporance of srucural policies ha encourage R&D expendiure. Finally, we show how even in he case of an opimal R&D subsidy ha parially miigaes seady sae ine ciencies, equilibrium R&D is sill oo procyclical: a resul ha jusi es he necessiy of counercyclical policies. In secion 2 we lay ou he decenralized model and he counerfacual e cien allocaion. In secion 3 we calibrae he model and derive resuls regarding he long-run behavior of he economy. In secion 4 we analyze he dynamic properies of he model. Finally, in Secion 5 we conclude. 2 The Model We presen a model ha inegraes endogenous growh in an oherwise convenional RBC model. Endogenous growh is based on verical innovaions as in Aghion and Howi (998, ch. 2) and Howi (2). The skech of he model is as follows. Final goods producers use labor and a coninuum of inermediae goods as inpus. These inermediae goods di er in heir relaive produciviy and each of hem is produced by a monopolisic rm using capial. The amoun of capial necessary o produce each inermediae good is proporional o is produciviy, hus re ecing ha more advanced producs require increasingly capial-inensive echniques. Each period, here is a probabiliy ha he produciviy of an inermediae good jumps o he echnology fronier due o he innovaion aciviies of enrepreneurs. Enrepreneurs borrow resources and inves hem in an aemp o increase heir probabiliies of making a discovery. If a discovery happens, he successful enrepreneur inroduces a new enhanced inermediae produc in her secor and becomes he new monopolis unil he momen ha she is replaced by anoher enrepreneur. The echnology fronier, ha is, he produciviy level of he mos advanced secor, evolves endogenously as he resul of posiive spillovers from innovaion aciviies. We inroduce he model, characerize is equilibrium condiions and presen a counerfacual model where decisions are aken by a benevolen social planner. 2. Final Goods Oupu In he model, a counry economy produces a nal good under perfec compeiion by using labor and a coninuum of inermediae producs. Final goods rms maximize heir pro s Z max Y W l p j; m j; dj ; m j;;l 2
subje o Y = z l Z A j; m j;dj ; () where m j; is he ow oupu of inermediae produc j 2 [; ], l is labor supply 2, and A j; is a produciviy parameer aached o he laes version of inermediae produc j. z is an aggregae produciviy shock ha follows a saionary AR() process wih persisence z and variance of he innovaion 2 z. 3 The model displays decreasing marginal producs in each of he inermediae producs and in labor. The rs-order condiions are p j; = A j; z l m j; ; (2) and he wages 2.2 Inermediae Goods Firms W = ( ) Y l : (3) Final oupu can be used inerchangeably as a consumpion or capial good, or as an inpu o innovaion. Each inermediae produc is produced by an incumben monopolis using capial, according o he producion funcion: m j; = K j; =A j; ; (4) where K j; is he capial in secor j a ime, insalled in period. Division by A j; indicaes ha successive vinages of he inermediae produc are produced by increasingly capial-inensive echniques. The incumben monopolis of each secor solves he problem max (p j; m j; q K j; ) ; m j; subjec o (2) and (4), where where q is he renal cos of capial. Marginal coss and marginal revenues are proporional o A j;. Therefore all inermediae producers choose o supply he same =( ). amoun of inermediae produc m = The aggregae capial in he economy is q 2 z l K = R K j; dj = m A, where A = R A j;dj is he average produciviy across all secors in nalgoods producion. As a resul, he aggregae producion funcion of he economy () can be reduced o he sandard consan reurns o scale one Y = z K (A l ) : The cos of capial can be expressed as a funcion of he aggregae level of capial and he ow of pro s ha each incumben earns is Y q = 2 ; (5) K j; (A j; ) = ( ) Y A j; A ; (6) so ha a share ( ) of nal oupu is allocaed o wages, 2 o capial coss and ( ) o pro s. 2.3 Produciviy Innovaions resul from enrepreneurship ha uses echnological knowledge. A any dae here is a echnology fronier ha represens he mos advanced echnology across all he secors: A max maxfa j; j j 2 [; ]g: (7) 2 We absrac from populaion growh by assuming ha variables are scaled by he working age populaion. This may be seen as a special case of Howi (2) where he number of secors is no consan, bu grows asympoically a he same rae as he populaion so ha he model does no exhibi he sor of scale e ec ha Jones (995) argues is conradiced by poswar rends in R&D spending and produciviy, 3 Throughou he paper we denoe wih capial leers S he non saionary variables whereas we reserve lowercase leers s for saionary variables. Variables in seady sae are denoed wihou ime subscrip s. 3
Each period, produciviy in secor j evolves according o A max ; wih probabiliy n A j;+ = j; : (8) A j; ; wih probabiliy n j; Once an innovaion happens, i creaes an improved version of he exising produc by raising is produciviy A j;+ o he echnology fronier A max : The enrepreneur hen eners ino Berrand compeiion wih he previous incumben in ha secor, who by de niion produces a good of inferior qualiy. Raher han facing a price war wih a superior rival, he incumben exis. Having exied, he former incumben canno hreaen o reener. Therefore, in + he former enrepreneur has become he new incumben. The probabiliy n j; is a funcion of he quaniy of nal oupu devoed o R&D in his secor X j; : Xj; (+) n j; = ; > : (9) A max Equaion (9) displays decreasing reurns o scale in innovaion 4. The parameer accouns for he produciviy of resources devoed o R&D. The amoun of resources is adjused by he echnology fronier variable A max o represen he increasing complexiy of progress: as echnology advances, he resource cos of furher advances increases proporionally. 2.4 Enrepreneurs The value of being he incumben in period in a secor wih produciviy A; V j; ( A); is he discouned ow of pro s ha he incumben may obain by aking ino accoun he probabiliy of obsolescence due o he arrival of a new innovaion in his secor, so V j; ( A) = j; ( A) + ( n j;) r E Vj;+ ( A) ; () where r is he risk-free ineres rae. The rs erm re ecs he ow of pro s of he monopolis whereas he second erm is he discouned value of sill being he incumben a +. We consider ha each period here is a single enrepreneur in each secor. Her problem can be expressed as max E [V j;+ (A max )] ( ) X j; ; () n j; X j; r subjec o (9), where X j; is he amoun of governmen-subsidized R&D. I means ha he enrepreneur maximizes he discouned value of becoming he incumben he nex period, weighed by he probabiliy of doing so, which is a funcion of he amoun of R&D spending. The rs order condiion is ha he marginal coss of an exra uni of goods allocaed o research ( ) equal he discouned marginal expeced bene E[Vj;+(Amax )]mc A max r, where mc A max dn j; dx j; = n j; (+): Since he value of becoming he incumben in he nex period V j;+ (A max ) is he same for all secors (as all of hem jump o he echnology fronier if an innovaion happens), he inpu invesed in R&D in each inermediae secor is he same: X j; = X and n j; = n. Proposiion In a decenralized economy, he aggregae business urnover n is given by n ( ) Y + = E + ( n +)n + : (2) ( ) ( + ) A + r Proof. See Appendix A.. Growh in he leading-edge parameer A max occurs as a resul of he knowledge spillovers produced by innovaions, as in Aghion and Howi (992). A any momen in ime, he echnology fronier is available o any successful innovaor, and his publicly available knowledge grows a a rae proporional o he aggregae rae of innovaions. Therefore we have where is he spillover coe cien. r g Amax A max = + n ; (3) 4 Previous sudies have found decreasing reurns in R&D expendiure, such as Korum (993). 4
2.5 Households The represenaive household solves wih < <, subjec o max E C ;I ;B ;K ;l = X "log (C ) l + + # ; (4) C + I + B r + T = W l + q K + D + B ; (5) K = I + ( )K ; (6) where C is consumpion, I is invesmen, B is he amoun of sae-coningen bonds, T is a governmen ax and D = R ( j; ( ) X j; ) dj are he dividends from an invesmen fund ha nances enrepreneurs invesmens and collecs he pro s from he ownership of he monopolis rms The soluion of he households problem yields he sandard Euler equaions for he risk-free ineres rae and he cos of capial and he relaionship of wages wih he marginal rae of subsiuion beween consumpion and labor: C = E r ; (7) 2.6 Equilibrium C + C = E (q + + ( )) ; (8) C + W = l C : (9) A compeiive equilibrium for his economy is a se of prices and allocaions so ha given prices households, nal and inermediae rms and enrepreneurs solve heir maximizaion problems and markes clear. The capial renal marke clears when he demand for capial by inermediae good producers equals he supply by households. The labor marke clears when rms demand for labor equals labor supply by households. The governmen always runs a balanced budge so ha axes are equal o governmen subsidies T = X : Finally, he nal goods marke clears if producion equals demand for consumpion, capial accumulaion and enrepreneurship Y = C + I + X : (2) In equilibrium, he evoluion of he average produciviy of he economy is given by he number of secors ha experience an innovaion: A = Z nj; A max + ( n j; )A j; dj = n A max A + A ; (2) which describes how he produciviy increases due o he disance o he echnology fronier A max A muliplied by he enry rae of new rms n (he number of secor where a new incumben appears). 2.7 The E cien Allocaion In he model presened above he compeiive equilibrium may no be socially opimal. This is due o he exisence of monopolisic compeiion in he inermediae goods secor and o he spillovers associaed wih he decenralized innovaion process, which enrepreneurs do no inernalize when making heir R&D decisions. The exisence of hese disorions may produce an ine cien allocaion of resources so ha he equilibrium is no Pareo opimal, as discussed in Aghion and Howi (992). 5
To see how he economy behaves in he Pareo opimal case we assume ha he economy is managed by a benevolen social planner who maximizes # X max E "log l + (C ) + C ;I ;K ;l ;X ;A ;A max = subjec o he aggregae budge consrain (2), he capial accumulaion law (6), he producion funcion of enrepreneurs (9), he evoluion of aggregae produciviy (2) and he spillover e ec of innovaion ino echnology growh (3). The rs order condiions are C E C ;+ n + 2;+ g + = X + 2; ; + C ( ) Y E C ;+ ( n ) + 2;+ g + = + + A ; ; C E C ;+ ( a ) + 2;+ = ( + ) X : + n C E Y + + ( ) = C + K and equaion (9), where ; and 2; are he Lagrange mulipliers of (2), and (3), respecively. In his case, here are no governmen subsidies ( = ). 3 How much R&D Expendiure? 3. Calibraion In his secion we use a calibraed version of he model o sudy he long-run implicaions of R&D expendiure. The model has a deerminisic seady sae ha displays a balanced-growh pah, where variables Y ; C ; I ; X ; A ; A max ; W ; ; and K grow a rae g = + n, whereas n ; q ; l ; and g are saionary. In order o solve he model, we divide Y ; C ; I ; X ; A ; W ; ; and K and by A max o make hem saionary. The parameers of he model are calibraed o mach key empirical evidence in he Unied Saes for he pos-war period. Informaion abou daa sources is provided in Appendix B. The value of is se o.35 so ha he share of oupu devoed o labor compensaion is 65%. The average growh of GDP per working-age populaion is.9% so ha assuming a real ineres rae of 4%, we obain a value of = g r of.984. Capial depreciaion is se o % per year and Frisch labor supply elasiciy = o, all sandard values in he lieraure. We mach he values of business urnover and public and privae R&D spending.wih he parameers, and. The average business urnover of U.S. rms n (he rae of creaion/desrucion of rms in he economy) in he las wo decades has been %. As shown in Figure, his value is also consisen wih he empirical evidence abou average survival raes for he 963 and 976 cohors of U.S. manufacuring rms. The oal share of R&D spending X has been roughly sable a abou 2.6 percen for mos of he pos-war period. However, he shares of privae and public R&D have signi canly changed during his period. We concenrae on he las hree decades so ha he average share of public R&D is around % and herefore we se o :4, log() o 25.73 and o 2, respecively. Finally, given a business urnover of %, o replicae he value of GDP growh of.9% we se he spillover coe cien o.9. Table displays he comparison beween he benchmark calibraion and he daa. 3.2 Pareo-e cien R&D Invesmen To analyze e cien R&D expendiures, we solve he social planner s problem for he se of calibraed parameers presened above. Table presens he resuls. In his case, oal R&D spending (public plus 6
privae) represens 4.8 percen of he GDP, which resuls in an increase in long-erm growh (2.%) and in business urnover (.2%). The raio beween observed R&D invesmen and is e cien is close o 2. This is in line wih he calibraion exercise of Jones and Williams (2), who nd raios beween and 3 for plausible paramerizaions. 5 To compue how his ranslaes o social welfare, we compue he seady sae uiliy as Proposiion 2 Normalizing A max = ; he seady sae value of he represenaive household s uiliy is U = log(c) ( ) Proof. See Appendix A.2. Welfare depends on seady sae e ecive consumpion (c l + ( ) ( + ) + log(g) ( ) 2 : (22) C A max ), leisure and growh. Naurally, he impac of growh is quie signi caive as i is weighed by which is higher han ( ) 2 ( ) for he calibraed value of. This is so as a higher growh rae allows households o consume more in he fuure. As presened in Table, he social welfare in he Pareo opimal case is more han 9 imes higher han in a decenralized equilibrium. 3.3 R&D Subsidies To miigae he welfare loss due o he subopimal allocaion of resources in he decenralized equilibrium, governmens may subsidize R&D expendiures so ha innovaors inernalize some of he spillovers derived from heir aciviies. The change in seady sae uiliy U as a funcion of he value of is shown in Figure 2. The scale is normalized so ha U ( = :4) is se o (he benchmark case). Toal uiliy grows as subsidies increase unil a maximum is reached (a = :78): We de ne his value as he opimal subsidy, ha is, he subsidy ha maximizes welfare under he consrains of a decenralized economy. Table displays he resuls for he case of no subsidies and for he opimal subsidy. When he opimal subsidy is applied, oal R&D expendiures rise o 6.9% of he GDP, a higher value han un he Pareoe cien allocaion. Mos of his increase is due o public R&D (5.3%) whereas privae R&D increases slighly (only an addiional.%). The increase in R&D spending raises growh and business urnover. The ne impac on welfare is a wo-fold increase wih respec o he benchmark. Nowihsanding, social welfare wih he opimal subsidy is sill below he level wih a benevolen social planner. Why is oal R&D invesmen di eren wih he opimal subsidy han under a social planner? This is due o he fac ha he economy is facing several disorions, so miigaing one disorion may lead o a second-bes siuaion. Figure 3 shows how public subsidies a ec he di eren componens of welfare. An increase in public R&D redirecs resources from consumpion and invesmen o innovaion, hus reducing e ecive oupu and consumpion. This fall in oupu forces agens o work more hours, wih he consequen loss in leisure uiliy. A he same ime, he increase in public R&D increases growh and simulaes privae R&D. The growh e ec is more signi caive for subsidies below.78 whereas he consumpion and leisure e ecs prevail above his subsidy level. Finally, and for he maer of comparison, we include in Table he he case of no subsidies. In his siuaion growh and welfare fall in comparison o he benchmark, as enrepreneurs decide o inves slighly less (he di erence is in he second decimal) and here is no public R&D. 3.4 R&D Subsidies versus Corporae Subsidies R&D subsidies allow enrepreneurs o inernalize some of he spillovers of innovaion by reducing heir R&D sunk coss. An alernaive approach would be o increase heir prospecive pro s so ha he value of becoming he incumben in a secor V j; increases. I can be done by subsidizing pro s (or cuing corporae axes, if hey are presen). Proposiion 3 In a decenralized economy, a subsidy o corporae pro s nanced by lump-sum axes is equivalen o a proporional R&D subsidy = + : 5 Esimaing he social reurn o R&D, Jones and Williams (998) conclude ha a conservaive lower bound o he raio beween opimal R&D invesmen and acual invesmen would be around 4. 7
Proof. See Appendix A.3. This is so in he conex of he model due o he fac ha incumbens do no research. Thus, corporae pro s a ec exclusively he prospecive value of becoming he nex monopolis (Aghion and Howi, 998). We leave for furher research he sudy of how his resul would change in he case of R&D aciviies by incumbens. 3.5 Robusness The main resuls presened in his secion are robus o reasonable changes in parameer values. Nowihsanding, i is ineresing o explore wha happens in a exreme case where curren observed public R&D invesmen canno be considered as R&D subsidies and herefore i has no e ec on innovaion ( = ). We hen recalibrae parameers, and o explain he sylized facs. The values of log() and are 25.32 and 2 (as in he sandard case), respecively. Table 2 suggess how he resuls change very lile. R&D expendiure under a social planner is now 5.% and i produces a 2.2% growh rae. Opimal R&D subsidies are roughly he same as wih he sandard calibraion, now producing a higher e ec on business urnover and growh. In conclusion, he main resuls of his secion (he subopimaliy of R&D spending and he possibiliy of parially miigaing i by increasing public subsidies) remain unalered. 4 How Should i Vary over he Cycle? 4. Procyclical R&D Barlevy (27) nds a posiive correlaion beween he growh rae of R&D expendiure and GDP growh of.39. As our model is in per capia erms, 6 we recompue his correlaion in per capia erms for he growh raes of R&D dx and GDP dy o nd i o be.26, which con rms he procyclicaliy of R&D. To check wheher his procyclicaliy may be reporduced in he conex of our model, we calibrae he parameers z and z of he emporary aggregae produciviy shock z so ha he model replicaes he volailiy and rs auocorrelaion coe cien of he GDP growh ( z = :96; z = :2). 7 Resuls in able 3 show ha in his case he benchmark model produces a correlaion beween GDP and R&D of.997, higher han he one observed in he daa. Addiionally, he model generaes a volailiy of R&D growh of.9%, roughly half of he one observed in he daa. Compared o Barlevy (27), we do no need o inroduce any xed cos o generae procyclicaliy due o he endogeneiy of labor supply and he use of nal oupu in R&D aciviies. Figure 4 shows he impulse responses o a z shock of GDP, he R&D share, business urnover and employmen. GDP is expressed as log-deviaions from he seady sae linear growh rend. When a posiive shock increases oupu, he size of he poenial marke for enrepreneurs increases, hus encouraging innovaion. The rise in he prospecs of higher pro s also induce enrepreneurs o expec higher business urnover in he fuure, which disencourages innovaion. This rade-o beween higher pro s and shorer monopolies produces a emporary increase in R&D expendiure accompanied by a emporary increase in he growh rae of produciviy, which generaes a permanen increase in he deviaion of he GDP from is rend. The increase in R&D is slighly less procyclical han he GDP, so ha he R&D share decreases by almos. percenage poins during he rs years, o remain consan hereafer. We exend he analysis o oher ypes of shocks. Firsly, we se a labor shock.so ha W = l C This is he shock considered in Comin and Gerler (26) and can be consider as a shock o he labor disuiliy or a wage markup shock. We consider he shock o follow an AR() and calibrae i as in he previous case. Resuls in able 3 show how his shock produces quaniaively similar resuls o he aggregae produciviy shock. 8 This con rms he resuls by Comin and Gerler (26), who show how his non-ecnological shock may drive business cycles a he high frequencies and generae he srong medium frequency movemens in produciviy observed in he daa. The main di erence is ha Comin 6 More precisely, in erms per working age person erms, as we do no model any kind of populaion dynamics. 7 Compuaions are performed in dynare. Daa frequency is annual. We always refer o he GDP and R&D expendiure per person aged 5-64. Addiional informaion abou daa sources may be found in Appendix B. 8 We omi a gure as he impulse response is similar o ha of a produciviy shock. 8
and Gerler consider endogenous echnological change à la Romer (99), so ha he produc survival rae is consan hroughou he cycle, whereas in our model i is also endogenous via creaive desrucion. Boh he produciviy shock and he labor shock generae correlaions beween GDP and R&D higher han he observed in he daa. This resul does no apply o all he shocks. For example in able 3 we also consider an i.i.d. shock o he spillover parameer ha we calibrae o replicae he variance of he GDP growh. In his case he model is no able o replicae he oupu auocorrelaion, 9 bu i generaes a volailiy of R&D higher han he empirical one and a negaive (and small) correlaion beween GDP and R&D growh. Figure 5 displays he impulse responses o his shock. When a shock increases a year, i immediaely raises he growh rae of he echnology fronier, given (3). The rise in A max increases he expeced value of becoming he incumben in a secor and he amoun of resources necessary o achieve an innovaion in equaion (9). Therefore, he iniial e ecs of he shocks are a fall in he business urnover and an surge in R&D expendiure, which reduces consumpion, invesmen and labor supply. In he coming years, hese conracionary e ecs are compensaed by an increase in he produciviy of he economy as he increased R&D invesmens begin o pay resuls. The nal e ec is ha an i.i.d. shock generaes a permanen increase in he GDP level ha reaches is plaeau afer a leas a couple of decades. This ype of response resemble he models of general purpose echnologies such as Helpman and Trajenberg (994), where he arrival of a new echnology ha raises oupu and produciviy in he long run can also cause cyclical ucuaions while he economy adjuss o i, a fac documened in cases such as he seam engine or he dynamo by Lipsey, Carlaw and Bekar (25). The level of procyclicaliy of R&D observed in he daa could be explained as he resul of he ineracion of shocks ha produce srong procyclicaliy, such as shocks o labor supply or produciviy, and counercyclical shocks, such as he spillover shock. 4.2 Pareo-e cien Dynamic R&D Invesmen According o Barlevy (27), socially opimal R&D spending can be counercyclical under very resricive assumpions, such as consan labor supply. When hese consrains are relaxed, socially opimal R&D seems o be procyclical, alhough less han decenralized one. To analyze whher his conlcuion sill holds in he conex of our model we compare he momens and impulse responses beween he benchmark and he e cien allocaion under he same shocks. Resuls in able 3 show how he correlaion beween GDP and R&D growh for produciviy and labor shocks is.97. This value exend Barlevy s conclusion ha socially opimal R&D invesmen is less procyclical han he decenralized one. The di erence is larger in he case of he spillover shock, where he socially opimal response is quie counercyclical, wih a correlaion coe cien of -.28. Figures 4 and 5 display a comparison beween he benchmark and he response wih a social planner. In boh he cases he socially opimal response induces a more pronounced change in he R&D share in he rs years afer he shock. In addiion, he auocorrelaion of oupu is smaller under a social planner han in a decenralized economy. The conclusion is ha he presence of economic disorions ine cienly prolong he e ecs of exogenous shocks. How imporan is his dynamic e ec compared o he saic one, ha is, o he welfare loss in seady sae?. To analyze his, we compue numerically he uncondiional expeced uiliy of he represenaive agen (4) by MoneCarlo mehods. Resuls in able 4 show how he divergence beween his simulaed approach and he heoreical seady sae uiliy equaion (22) is less han %. We employ his approach o assess he welfare loss in he case of produciviy shocks z wih he calibraion presened above. Table 4 shows how he presence of shocks generaes a welfare loss of 5 percen wih respec o he benchmark seady sae uiliy. In he opimal case, his welfare loss would be of percen of he benchmark uiliy. Therefore, he di erenial impac of he shocks in boh cases is around a 5 percen of he benchmark seady sae uiliy. This amoun is small compared o he poenial gains of miigaing he seady sae 9 This oupu auocorrelaion could have been replicaed by considering his shock o be AR() insead of i.i.d.. We leave his for furher research. In each simulaion, we generae a vecor of T = 5 years of shocks! i = [! i ; :::;!i T ], simulae he corresponding endogenous variables C (! i ) and l (! i ), and compue he cumulaive discouned uiliy U(! i ) = T P log l (C (! i )) (! i ) +. + = P We hen perform he average over N = ; simulaions o compue he expeced uiliy N U(! i ): i= 9
welfare di erences. The conclusion is ha, even if a counercyclical policy is welfare-enhancing, srucural policies aimed a increasing R&D expendiures have a higher pay-o in erms of uiliy. 4.3 The Case wih Opimal Consan R&D Subsidies In secion 2 we have shown how he saic welfare loss is parially miigaed by inroducing a proporional subsidy o R&D expendiure or o corporae pro s. Does his subsidy also helps o improve he dynamic response of R&D?. To check i, we simulae he decenralized model under he shocks assuming a subsidy rae of = :78: Resuls are shown in able 3 and gure 6. In general here is no a considerable change in second order momens. Nowihsanding, he simulaed welfare analysis of gure 4 shows how he dynamic welfare loss in his case (3 percen of he benchmark) represens a midpoin beween he losses in he benchmark and opimal cases. The conclusions is ha he subsidy helps o miigae no only he saic losses, bu he dynamic ones. However, as commened above, he saic welfare di erences are one order of magniude larger han he dynamic ones. 5 Conclusions This paper examines which is he opimal R&D expendiure from a social welfare poin of view. The main resuls are wo. Firsly, he use of a proporional R&D subsidy is no able o compleely miigae he disorions inroduced by monopolisic compeiion, creaive desrucion and echnology spillover. The opimal R&D subsidy induces a level of R&D higher han in he e cien allocaion, which for he U.S. seems o be around 7% of he GDP. Secondly, alhough opimal R&D invesmen is procyclical, i is less procyclical han in he case of a decenralized economy. Nowihsanding, from a welfare perspecive, here are more gains in srucural policy inervenions, such as an increase in he R&D subsidy, han in counercyclical policies. Given he poenial gains from increasing R&D expendiure and he subopimaliy of a proporional subsidy, an ineresing quesion is how a ax sysem should be designed in order o incenivize innovaion. In paricular, i is no clear wheher his opimal ax sysem would allow he economy o reach he rs bes opimum. We leave his quesion for fuure research. References Aghion, P. and Howi, P.: 6(2), 323 5. 992, A model of growh hrough creaive desrucion, Economerica Aghion, P. and Howi, P.: 998, Endogenous growh heory, he MIT Press. Aghion, P. and Sain-Paul, G.: 998, VIRTUES OF BAD TIMES Ineracion Beween Produciviy Growh and Economic Flucuaions, Macroeconomic Dynamics 2(3), 322 344. Audresch, D. B.: 99, New- rm survival and he echnological regime, The Review of Economics and Saisics 73(3), 44 5. Barlevy, G.: 27, On he cyclicaliy of research and developmen, American Economic Review 97(4), 3 64. Caballero, R. J. and Hammour, M. L.: 994, The cleansing e ec of recessions, American Economic Review 84(5), 35 68. Comin, D. and Gerler, M.: 26, Medium-erm business cycles, American Economic Review 96(3), 523 55. Dunne, T., Robers, M. J. and Samuelson, L.: 988, Paerns of rm enry and exi in u.s. manufacuring indusries, RAND Journal of Economics 9(4), 495 55.
Faas, A.: 2, Do business cycles cas long shadows? shor-run persisence and economic growh, Journal of Economic Growh 5(2), 47 62. Grossman, G. and Helpman, E.: 99, Innovaion and growh in he world economy, Cambridge, ma: mi Press. Helpman, E. and Trajenberg, M.: 994, A ime o sow and a ime o reap: Growh based on general purpose echnologies, CEPR Discussion Papers 8, C.E.P.R. Discussion Papers. Howi, P.: 2, Endogenous growh and cross-counry income di erences, American Economic Review 9(4), 829 846. Howi, P. and Mayer-Foulkes, D.: 25, R&d, implemenaion, and sagnaion: A schumpeerian heory of convergence clubs, Journal of Money, Credi and Banking 37(), 47 77. Jones, C. I.: 995, R&d-based models of economic growh, Journal of Poliical Economy 3(4), 759 84. Jones, C. I. and Williams, J. C.: 998, Measuring he social reurn o r&d, The Quarerly Journal of Economics 3(4), 9 35. Jones, C. I. and Williams, J. C.: 2, Too much of a good hing? he economics of invesmen in r&d, Journal of Economic Growh 5(), 65 85. Korum, S.: 993, Equilibrium r&d and he paen-r&d raio: U.s. evidence, American Economic Review 83(2), 45 57. Lipsey, R., Carlaw, K. and Bekar, C.: 25, Economic Transformaions: General Purpose Technologies and long-erm economic growh, Oxford Universiy Press, USA. Romer, P. M.: 99, Endogenous echnological change, Journal of Poliical Economy 98(5), S7 2. Appendix A: Proofs Proposiion Proof. We can wrie equaion () as V ( A) = E " X so he rs order condiion of he enrepreneur resuls in ( ) r mc i=! # iy ( n l ) Y i ( ) A; r l A i l= " = E [V + (A max )] X A max = E i=+ We can use his condiion o express V (A max ) as V (A max ) = ( ) Y A max + ( n ) A = ( ) Y A max r E " X i=+ + ( n ) ( ) A mc and aking expecaions of he nex period iy l=+ A max ;! # ( n l ) Y i ( ) : r l A i iy l=+! # ( n l ) Y i ( r l A i E [V + (A max )] = E ( ) Y+ A + + ( n +) ( ) mc + ha can be re-inroduced in he rs order condiion of he enrepreneur o ge ( ) Y+ E + ( n +) ( ) = ( ) r : A + mc + mc A max ; ) A max
Proposiion 2 Proof. Given equaion (4), a household s expeced uiliy is given by # " X E "log l + X (C ) = E log (c ) + log(a max ) + log(g ) + = = l + + # : In seady sae and wih A max = ;his expression can be simpli ed o X log (c) + log(a max ) + log(g ) = l + + = = log (c) l + + h log (c) l + + i X = + log(g) ( ) 2 :! + log(g)! X = Proposiion 3 Proof. A subsidy o corporae pro s modi es equaion () as V j; ( A) = ( + ) j; ( A) + ( n j;) r E Vj;+ ( A) ; so ha proceeding like in proposiion, he rs order condiion for enrepreneurs resuls in E ( + ) ( ) Y + + ( n +) = r ; A + mc + mc and herefore if ( + ) = ( ) hen = + : Appendix B: Daa Sources GDP daa for he period 95-27 comes from he Bureau of Economic Analysis. Daa abou civilian noninsiuional populaion aged 6 and older 95-27 comes from he Bureau of Labor Saisics. Informaion abou business urnover is provided for he period 99-23 by he U.S. Small Business Adminisraion. Evidence abou survival raes for he 963 and 976 cohors is obained from Dunne, Robers and Samuelson (988) and Audresch (99), respecively. Daa abou R&D expendiures 953-27 comes from he Naional Science Foundaion. Appendix C: Tables and gures Table. Comparison beween daa, benchmark, social opimum and opimal subsidy % GDP growh Welfare Turnover R&D Subsidies Privae R&D Toal R&D Daa -.9 -...5 2.6 Benchmark 4.9...5 2.5 Social Planner - 2. 927.2 - - 4.8 No subsidy.8 5 9.7.5.5 Opimal subsidy 78 2. 54.9 5.3.6 6.9 Welfare refers o he seady sae uiliy compared o ha of he benchmark GDP growh refers o he growh of GDP per working age populaion (civilian noninsiuional populaion 6 years and older) 2
Table 2. Robusness check when here are no public subsidies in he benchmark % GDP growh Welfare Turnover R&D Subsidies Privae R&D Toal R&D Benchmark.9..5.5 Social Planner - 2.2 939.6 - - 5. Opimal subsidy 79 2. 25.3 5.3.6 7.2 The benchmark refers o he new calibraion Table 3. Second momens of GDP and R&D growh % (dy ) (dy ; dy ) (dx ) (dy ; dx ) Daa - 2.2 8.8 4.4 26.6 Benchmark z shock 2.2 8.2.9 99.7 Benchmark shock 2.2 8.2.9 99.7 Benchmark shock 2.2 9.7 5.8 -.9 Social Planner z shock 2.4 4..5 97.2 Social Planner shock 2.4 4..5 97.2 Social Planner shock 2.7 8.4.6-27.9 Opimal Tax z shock 2.2 8.2.9 99.7 Opimal Tax shock 2.2 8.2.9 99.7 Opimal Tax shock 2.4 92.7 6.3. GDP and R&D growh per working age person () denoes volailiy and () is he correlaion coe cien Table 4. Dynamic welfare analysis wih produciviy shocks % Benchmark Social Planner Opimal subsidy Seady sae welfare 927 54 No shocks welfare 99 926 53 Dynamic welfare 84 96 4 Dynamic welfare loss 5 3 Welfare refers o uiliy compared o ha of he seady sae benchmark Mean values afer. Mone Carlo simulaions of lengh 5 years 3
surviving firms as a fracion of age cohor.9.8.7.6.5.4.3.2 Model, n = % U.S. manufacuring firms, 976 cohor U.S. manufacuring firms, 963 cohor. 2 4 6 8 2 4 6 8 2 years since enry Figure : Survival raes of rms in he economy 2.5.5 -.5 - -.5-2 -2.5-3..2.3.4.5.6.7.8.9 τ Figure 2: Impac of a general R&D subsidy on welfare 4
.5 C - u(c)=log(c)/(-β) -.95-2 -3.9-4.85-5.8..2.3.4.5.6.7.8.9-6..2.3.4.5.6.7.8.9.26 G 25 u(g)=β*log(g)/(-β)^2 24.24 23.22 22 2.2 2.8..2.3.4.5.6.7.8.9 9..2.3.4.5.6.7.8.9.25 L -7 u(l)=-l^(+ψ)/((-β)(+ψ)).2-8.5-9..5 - -.95..2.3.4.5.6.7.8.9 τ -2..2.3.4.5.6.7.8.9 τ Figure 3: Impac of a R&D subsidy on consumpion, growh and labor 2.6 GDP. R&D share 2.4 2.2 2 -..8 -.2.6.4 -.3.2 -.4.8 5 5 2 25 3 35 4 -.5 5 5 2 25 3 35 4.2 Business Turnover.6 Employmen Benchmark.5 Opimum.5.4.3.2...5 5 5 2 25 3 35 4 years -. 5 5 2 25 3 35 4 years Figure 4: Impulse responses o a produciviy shock z : Decenralized economy ( Benchmark ) versus social planner ( Opimum ). 5
2 GDP.6 R&D share.5 8.4 6.3 4.2 2. -2 5 5 2 25 3 35 4 -. 5 5 2 25 3 35 4 Business Turnover.6 Employmen -.5.4 -..2 -.5 -.2 -.2 -.25 -.3 -.4 -.35 -.4 -.6 -.8 Benchmark Opimum -.45 5 5 2 25 3 35 4 years - 5 5 2 25 3 35 4 years Figure 5: Impulse responses o a spillover shock :Decenralized economy ( Benchmark ) versus social planner ( Opimum ). 2.6 GDP. R&D share 2.4 2.2 2 -..8 -.2.6.4 -.3.2 -.4.8 5 5 2 25 3 35 4 -.5 5 5 2 25 3 35 4.2 Business Turnover.6 Employmen Benchmark.8.5 Opimum.6.4.4.3.2.2...8.6 5 5 2 25 3 35 4 years -. 5 5 2 25 3 35 4 years Figure 6: Impulse responses o a produciviy shock z in he case of opimal consan ax ( = :78). Decenralized economy ( Benchmark ) versus social planner ( Opimum ). 6