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5 working paper deparmen of economics A MODEL OF GROWTH THROUGH CREATIVE DESTRUCTION Philippe Aghion Peer Hewi No. 527 May 1989 massachuses insiue of echnology 50 memorial drive Cambridge, mass

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7 A MODEL OF GROWTH THROUGH CREATIVE DESTRUCTION Philippe Aghion Peer Howi No. 527 May 1989

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9 A Model of Growh hrough Creaive Desrucion by Philippe Aghion* and Peer Howi** May, 1989 * Deparmen of Economics, M.I.T. ** Deparmen of Economics, Universiy of Wesern Onario The auhors wish o acknowledge he helpful commens and criicisms of Roland Benabou, Olivier Blanchard, Parick Bolon, Mahias Dewaripon, Dick Eckaus, Zvi Griliches, Rebecca Henderson, Louis Phaneuf, and Parick Rey. The second auhor also wishes o acknowledge helpful conversaions wih Louis Corriveau.

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11 1. INTRODUCTION This paper presens a model of endogenous sochasic growh, based on Schumpeer's idea of creaive desrucion. We begin from he belief, suppored by many empirical sudies saring wih Solow (1957), ha a large proporion of economic growh in developed counries is aribuable o improvemen in echnology raher han he accumulaion of capial. The paper models echnological progress as occurring in he form of innovaions, which in urn resul from he aciviies of research firms. We depar from exising models of endogenous growh (Romer, 1986, 1988; and Lucas, 1988) in wo fundamenal respecs. Firs, we emphasize he fac ha echnological progress creaes losses as well as gains, by rendering obsolee old skills, goods, markes, and manufacuring processes. This has boh posidve and normaive implicaions for growh. In posiive erms, he prospec of a high level of research in he fuure can deer research oday by hreaening he fruis of ha research wih rapid obsolescence. In normaive erms, obsolescence creaes a negaive exernaliy from innovaions, and hence a endency for laissez-faire o generae oo much growh. Obsolescence does no fi well ino exising models of endogenous growh. Those models have only posiive exernaliies, in he form of echnology spillovers, and hus end o generae oo lile growh. The closes in spiri o our model is ha of Romer (1988). In ha model innovaions consis of he invenion of new inermediae goods, neiher beer nor worse han exising Ones. Once invened a good remains in producion forever. Growh akes place because of he lenghening of he lis of available inermediae goods. This model does a good job of capuring he division of labor aspec of growh. Bu adding obsolescence, by allowing old goods o be displaced by he inroducion of new goods, may eliminae growh in his kind of model (see Deneckere and Judd, ). Our second deparure is ha we view he growh process as disconinuous. We share he view aken by many economic hisorians ha individual echnological breakhroughs have aggregae effecs, ha he uncerainy of innovaions does no average ou across indusries. 1

12 This view is suppored by recen empirical evidence of auhors such as Nelson and Plosser (1982) and Campbell and Mankiw (1987) ha he rend componen of an economy's GNP includes a subsanial random elemen. 2 Exising models of endogenous growh do no produce a random rend, unless exogenous echnology shocks are added o he model, as is done by King and Rebelo (1988). We ake he view ha echnology shocks should no be regarded as exogenous in an analysis ha seeks o explain he economic decisions underlying he accumulaion of knowledge. Insead, we suppose ha hey are enirely he resul of such decisions. Saisical innovaions in GNP are produced by economic innovaions, he disribuion of which is deermined by he equilibrium amoun of research. This makes he disribuion of echnology shocks endogenous o he model. 3 In making boh of hese deparures from recen models of endogenous growh we ake our inspiraion from Schumpeer (1942, p. 83, his emphasis): The fundamenal impulse ha ses and keeps he capialis engine in moion comes from he new consumers' goods, he new mehods of producion or ransporaion, he new markes,... [This process] incessanly revoluionizes he economic srucure from wihin, incessanly desroying he old one, incessanly creaing a new one. This process of Creaive Desrucion is he essenial fac abou capialism. In Schumpeer's view, capialis growh is inherenly uncerain. Fundamenal breakhroughs are he essence of he process, and hey affec he enire economy. However, he uncerainy is endogenous o he sysem, because he probabiliy of a breakhrough depends upon he level of research, which in urn depends upon he he monopoly rens ha consiue "he prizes offered by capialis sociey o he successful innovaor." (1942, p. 102) Thus, "economic life iself changes is own daa, by fis and sars." (1934, p. 62). The presen paper develops a simple model ha ariculaes hese basic Schumpeerian elemens. Alhough i is quie special in some dimensions we believe i is also simple and flexible enough o serve as a prooype model of growh hrough creaive desrucion. 4

13 2. THE BASIC MODEL a. Assumpions There are four classes of xadeable objecs: land, labor, a consumpion good, and a coninuum of inermediae goods i 6 [0,1]. There is also a coninuum of idenical infiniely lived individuals, wih mass N, each endowed wih a one uni flow of labor, and each wih idenical ineremporally addiive preferences defined over lifeime consumpion, wih he consan rae of ime preference r > 0. Excep in secion 5 below, we assume a consan marginal uiliy of consumpion a each dae; hus r is also he unique rae of ineres in he economy. There is no disuiliy from supplying labor. There is also a fixed supply H of land. The consumpion good is produced using land and he inermediae goods, subjec o consan reurns. Since H is fixed we can express he producion funcion as: (2.1) y=\ (F[x(i)]/c(i)}di 1 where F' > 0, F" < 0, y is he flow oupu of consumpion good, x(i) he flow of inermediae inpu i, and c(i) a parameer indicaing, for given facor prices, he uni cos of producing he consumpion good using he inermediae inpu i. Each inermediae good i is produced using labor alone, according o he linear echnology: (2.2) x(i) = L(i) where L(i) is he flow of labor used in inermediae secor i. Labor has an alernaive use o producing inermediae goods. I can also be used in research, which produces a random sequence of innovaions. The Poisson arrival rae of innovaions in he economy a any insan is Xn, where n is he flow of labor used in research and X, a consan parameer given by he echnology of research. There is no memory in his

14 echnology, since he arrival rae depends only upon he curren flow of inpu o research, no upon pas research. 5 Time is coninuous, and indexed by x > 0. The symbol = 0,1... denoes he inerval saring wih he innovaion (or wih x = in he case of = 0) and ending jus before he s +1. The lengh of each inerval is random. If he consan flow of labor n is applied o research in inerval, is lengh will be exponenially disribued wih parameer Xn. Each innovaion consiss of he invenion of a new line of inermediae goods, whose use as inpus allows more efficien mehods o be used in producing consumpion goods. We have in mind such "inpu" innovaions 6 as he seam engine, he airplane, and he compuer, whose use made possible new mehods of producion in mining, ransporaion, and banking. An innovaion need, no, however, be as revoluionary as hese examples, bu migh consis insead of a new generaion of inermediae goods, similar o he old ones. Specifically, use of he new line of inermediae goods reduces he cos parameers c(i) in (2.1) by he facor y e (0,1). We assume away all lags in he diffusion of echnology. 7 The mos modern inermediae good is always produced in each secor, and he uni cos parameer during inerval is he same for all secors; hus: (2.3) c (i) 1 = c = CqY Vie [0,1], = 0,1,..., where c~ is he iniial value given by hisory. (Of course, i is always possible o produce he consumpion good using an old echnology, wih a correspondingly old line of inermediae goods.) 8 A successful innovaor obains a coninuum of paens, one for each inermediae secor, each one graning he holder he exclusive righ o produce he newly invened inermediae good in ha secor. We assume, however, ha ani rus laws prohibi anyone from reaining ha righ in a posiive measure of secors. So he innovaor sells each paen o a firm ha becomes he local monopolis in ha secor unil he nex innovaion occurs. (We assume perfec compeiion in all markes oher han hose for he inermediae goods and for paens.)

15 The innovaor offers each paen for sale a a price equal o he expeced presen value of he monopoly rens accruing o he paen. exchange for unconsrained use of he paen. The buyer pays ha price as a lump sum in Thus we implicily rule ou royalies and oher coningen conracs beween he innovaor and he local monopolis. The force of his resricion will be discussed in (d) below where i is argued ha wihou he resricion he wo paries would wan a conrac wih negaive royalies. The resricion migh herefore be raionalized eiher by noing ha negaive royalies are no in fac observed, or by referring o coss of monioring he oupu of he inermediae good. No paen covers he use of a new inermediae good in he consumpion good secor. The idea behind his assumpion is ha poenial uses of he new good are oo obvious, even before is invenion, o be paened. The assumpion migh also be raionalized by he observaion ha imporan innovaions like he ones menioned above have had widespread uses oo numerous and diffuse for anyone o monopolize hem. b. The Inermediae Monopolis's Decision Problem The local monopolis's objecive is o maximize he expeced presen value of he flow of profis over he curren inerval. When he inerval ends so do he rens. The only uncerainy in he decision problem arises because he lengh of he inerval is random. Because no one operaes in a posiive measure of secors he monopolis akes as given he amoun of research a each ime, and hence also akes as given he lengh of he inerval. Because of his and because he cos of he paen is sunk, he monopolis's sraegy will be simply o maximize he flow of operaing profis K a each insan. In equilibrium all prices and quaniies are consan hroughou he inerval. Also, by symmery, he same quaniy x will be chosen by each monopolis. This quaniy will also be he oal oupu of inermediae goods in period, which by (2.2) equals he oal employmen of labor in manufacuring; i.e.:

16 1 1 x = J x. (i) di = J L (i) di 1 l l We can hen re express (2.1) as: y F(x ) = -^- The inverse demand curve facing a monopolis charging he price p is he marginal produc schedule: (2.4) p = F'(x )/c. Thus he decision problem is o choose x so as o maximize [F'(x )/c w ]x, aking c and w as given. The necessary firs order condiion is (2.5) c w = F'(x ) + x F"(x ). Assume ha he monopolis's marginal revenue schedule is downward sloping: Assumpion A.l : 2F"(x) + x F"(x) < V x > This condiion holds auomaically when F" < 0; we show in Appendix 1 ha i also holds when F comes from a CES producion funcion. I follows from (2.5) and A.l ha here is a unique soluion o he decision problem: (2.6) x = x(c w f ) where x is sricly decreasing. Thus he demand for labor in he inermediae indusry is a decreasing funcion of he cos (in erms of he consumpion good) of producing one efficiency uni of an inermediae good. Likewise we can express he monopolis's price and flow of profis as: (2.7) p = p(c w )/c if'(x(c w ))/c, and (2.8) 7C = 5r(c w )/c = [(P(c ) - w c w ) x(c w )]/c = -[x(c w )] F"(x(c w ))

17 where p is sricly increasing and % is sricly decreasing. For fuure use, we also assume: Assumpion A.2 : x(0) = <*>, x( ) = 0. We have no allowed he local monopolis's decision problem o be consrained by poenial compeiion from he holder of he previous paen. This is because if he consrain were poenially binding; i.e. if he innovaion were non-drasic in he usual sense (see, for example Tirole, 1988, ch.10) hen he curren paen would be of greaes value o he holder of he previous paen, who would face no such compeiion. Thus our model predics ha non drasic innovaions would be implemened by incumben producers, whereas drasic innovaions would generally be implemened by new firms. This is in accordance wih several empirical sudies o he effec ha incumben firms implemen less fundamenal innovaions han do new enrans. (See, for example, Scherer, 1980.) Wheher or no he innovaion is drasic is deermined by wheher or no a compeiive producer of he consumpion good would incur a loss buying from he previous monopoliss, a a price equal o he uni cos w. C(p^,w ) > ^c" 1, Thus he condiion for a drasic innovaion is where C( ) is he uni cos funcion uniquely associaed wih he producion funcion F, and p is he equilibrium price of land. The laer is deermined by he condiions for compeiive equilibrium in he markes for land and he consumpion good: C(p^,p ) = c" 1 where p is he equilibrium marke price of all oher inermediae goods, given by (2.7). The special case of a Cobb Douglas producion is defined by: (2.9) F(x ) = x^ I is sraighforward o verify ha in his case: (2.10) x^cc^a 2 ) 17^" 1 ^

18 (2.11) P^Wj/a, (2.12) 7C = [(1 - a)/a]w x, and innovaions will be drasic if and only if: (2.13) y<a a. c. Research There are no conemporaneous spillovers in he research secor; ha is, a research firm employing z will experience innovaions wih a Poisson arrival rae Xz, and hese arrivals will be independen of oher firms' research employmen n = n z. 9 Le W be he value of a research firm afer he innovaion, and le V be he value of paens from he innovaion. If he firm makes an innovaion i will receive V,. The densiy of his even a he curren insan is Xz. Thus he expeced payoff per uni ime from successful innovaion is Xz V i.. The firm will realize a capial gain on W W when any firm makes an innovaion. Thus is expeced rae of capial gain is Xn (W. W ). Is flow of labor cos is w z. I akes w and n as given. Thus we have he Bellman equaion: 10 (2.14) rw, = max X z V. + X(n + z)(w.,, - W.) - w.z { z > o) +i c +i Assume free enry ino research. Then W = 0, so (2.14) can be expressed as: (2.15) 0= max XzV,. - w z [z > 0} The Kuhn-Tucker condiion is: +i C (2.16) w > XV,, z > 0, wih a leas one equaliy (waloe). The value V is he expeced presen value of he consan flow K over an inerval whose lengh is exponendally disribued wih parameer Xn : \ (2.17) V= -V-. v ' r + An

19 Noe ha here is an imporan ineremporal spillover in his model. An innovaion reduces coss forever. I allows each subsequen innovaion o reduce he uni cos parameer c by he same fracion y, and wih he same probabiliy An, bu from a saring value c, ha will be lower by he fracion y han i would oherwise have been. The producer of an innovaion will capure (some of) he rens from ha cos reducion, bu only during he nex inerval. Afer ha he rens will be capured by oher innovaors, building upon he basis of he presen innovaion, bu wihou compensaing he presen innovaor. 11 This ineremporal spillover will play a role in he welfare analysis of secion 4 below. The model hus embodies Schumpeer's idea of "creaive desrucion". Each innovaion is an ac of creaion aimed a capuring monopoly rens. Bu i also desroys he monopoly rens ha moivaed he previous creaion. Creaive desrucion accouns for he erm An in he denominaor of V in (2.17). More research his period will reduce he expeced enure of he curren monopoliss, and hence reduce he expeced presen value of heir flow of rens. ]2 / / d. Wage Deerminaion. Because n is aggregae research employmen, (2.16) implies: (2.18) - w > W +1, n > 0, waloe. "!' '. Combining (2.8), (2.17), (2.18) and he equilibrium condiion: (2.19) x + n = N; = 0,l,... yields he condiion: ^gj#p^:; : (2.20) w "^Vifc +i w +i )/c +i <M, >, =, x < N ; ^r^wmk r + An +i (ra) + n - x (c +1 w +1 ) r waloe ^:: ", Condiion (2.20) describes he supply of labor o he inermediae secor. I says ha unless ha secor absorbs all he economy's labor, he wage will equal labor's opporuniy cos in research, XV~[7.- This condiion, ogeher wih he labor-demand schedule (2.6), joinly

20 10 deermine w and x in erms of w, and x,, as shown in Figure 1. Figure 1 X.=X(W.c ) N x I is clear now why a successful innovaor would wan o offer a conrac wih negaive royalies. Such a conrac would induce each local monopolis o produce more han x(c w ). This would have wo effecs on V, as given by (2.17). Firs, i would reduce he numeraor below he maximal value given by (2.8). Second, i would also reduce he denominaor by reducing n, hrough (2.19). In he neighborhood of he zero royaly soluion analyzed above he former effec would be a second order small by he envelope heorem, bu he laer would no. Thus he seller of he line of paens would wan negaive royalies so as o discourage creaive desrucion during inerval. The local monopoliss would no aemp o discourage creaive desrucion on heir own by producing more han x(c w ) in he absence of negaive royalies, because each one is oo small o affec n.

21 11 PERFECT FORESIGHT EQUILIBRIA (PFE) Using (2.6) and he fac ha x is an inverible funcion, we can wrie w = x (x )/c. Muliplying boh sides of (2.20) by c and using he ideniy: c, = yc we obain:, rc(x_1 (x ))/ Y (3.1) x h* )Z Trk+ H l l\ +] -. x <N waloe ; which we can re-express by he following firs order difference equaion: (3.2) x = ^(x +1 ) = min(g(x +1 ), N) where: 13 G(x +1 ) = x c(x 1 (x +1 ))/y r/x + N - x +1 We define a perfec foresigh equilibrium (PFE) as a sequence {x },-. in [0, N] saisfying (3.2) for all > 0. In PFE everyone can predic he fuure evoluion of he endogenous variables (w, x, y, %, n, V ) wih cerainy. 14 However, he lengh of each inerval (in real rime %), as well as he ideniy of each innovaor, is random. The characerizaion of PFE is simplified by he following: Lemma 1 : The mappine x i» G(x) is decreasing in x. Lemma 1 follows from he fac ha boh x and k are decreasing funcions. The economic inerpreaion is as follows: A foreseen increase in x, will raise he reward X V, o he nex innovaor: on he one hand i raises he flow of monopoly rens.(x (x + i)) and on he oher hand i reduces he amoun of research and hence he amoun of creaive desrucion nex inerval (he effec in he denominaor). The increase in X V, will raise he equilibrium wage w his period (if x < N), which in urn will induce he inermediae monopoliss o reduce heir demand for labor his period, x.

22 12 Because (3.2) is forward looking, hisory does no deermine a unique value of x n in PFE. Typically, here will be a coninuum of PFE indexed by he iniial value x^. However here will only be a finie number of periodic rajecories o which any PFE could converge asympoically. More precisely: Proposiion 1 : All PFE are, or converge asympoically o one of he following : a saionary equilibrium (SE) wih posiive growh. SE wih zero growh. a "real" 2 cycle, or a "no growh rap" (NGT). Furhermore here always exiss a unique SE. We define SE as a seady sae x = ^(x). In SE he economy experiences balanced growh in he sense ha he allocaion of employmen beween manufacuring (x) and research (n = N x) remains consan. Growh is posiive if x < N and zero if x = N, because here will be innovaions if and only if labor is allocaed o research. We define real 2 cvcle as a pair (x, x ) such ha: (3.3) x = ^(x 1 ), x 1 = ^(x ), x 1 * x and N {x, x 1 }. Then a real 2 cycle corresponds o PFE in which manufacuring employmen oscillaes beween wo differen values wih each succeeding innovaion. High manufacuring employmen in odd inervals raises he reward o research during even inervals, and hence depresses manufacuring employmen during even inervals, hrough he mechanism discussed above. Likewise, low manufacuring employmen in even inervals raises manufacuring employmen in odd inervals. A "no growh" rap is a pair (x, x ) such ha he firs hree condiions of (3.3) hold, bu N e {x, x }. As shown in Figure 2 below, NGT exiss when G(N) < G (N) = x.

23 13 Figure 2 ** = #0 Even hough NGT defines an infinie sequence (x } ~, he oscillaion of he economy will sop afer period 1. From hen on he economy will perform as if in SE wih zero growh. The inerpreaion of NGT is ha he prospec of low manufacuring employmen in even periods so depresses he incenive o research in odd periods ha research sops. As we shall see, his can happen even in an economy ha possesses a posidve balanced growh equilibrium. Proposiion 1 rules ou complicaed periodic rajecories such as k cycles wih k > 3, or chaoic PFE. I follows immediaely from Lemma 1, which implies ha x in odd periods follows a monoonic pah in [0, N]. Therefore he sequence converges. By coninuiy, he... limi poin is a fixed poin of he second ierae map 3 f ", which corresponds o eiher SE or a "> 2 cycle (eiher real or NGT). Since ^(x) is non increasing in x, here is a unique inersecion xe [0,N] beween he graph of J?and he 45 line; in oher words, here is a unique SE. 15 We conclude his secion wih a brief discussion of each ype of asympoic PFE.

24 14 a. Saionary Equilibrium (SE) In he Cobb-Douglas case, SE wih posiive growh is deermined by:. 1 a - a x x = G(x) = a y _r/x + N - X - 1/(<X-1) i.e. by he simple equaion: 1-oc (3.4) Y = - a X rfx + N - x For growh o be posiive, i is necessary and sufficien o have: (3.5) Y <^.1^.N In paricular, for a fixed value of Y, X, r, N, a necessary and sufficien condiion for posiive * growh is ha he parameer a be sufficienly small: a < a = XN/(kN + yr) < \. To inerpre his resul noe ha a is an inverse measure of monopoly power in each inermediae secor. Specifically, 1 a is he Lerner (1934) measure of monopoly power (price minus marginal cos divided by price), (1 a) is he elasiciy of demand faced by an inermediae monopolis, and 1 a is he fracion of equilibrium revenue in an inermediae n secor accruing o he monopolis, -. Thus, if monopoly power is oo weak (a > a ) * hen he flow of monopoly rens from he nex innovaion would no be enough o induce posiive research aimed a capuring hose rens even if hey could be reained forever, wih no creaive desrucion in he nex inerval. b. "No growh" Trap As we have seen, NGT will exis iff:

25 15 (3.6) G(N)<x C = G! (N) ^c\j-\ 6(H) N6J / X + 2. Figure 3 In paricular NGT will always exis when he ineres rae r is sufficienly small. 16 Noe ha he smaller r (or r/x), he easier i is o saisfy (3.5): in oher words, "no growh rap" equilibria are more likely o exis in an economy ha possesses a posiive balanced growh equilibrium! c. Real 2 cycle A sufficien condiion for a real 2-cycle o exis is ha: G(N) < x (i.e. (3.6)) and dj? 2 -j (x) < 1, where x is he unique SE. 17 (Figure 4 below).

26 16 = &C*lJ) s"l.fc k' 5 ^ aa Figure 4 Proposiion 2 : When R = r/x. is sufficienly small and he elasiciy of demand for inermediae inpu (V) - ) 18 i s sufficienly close o 1. here exiss a real 2 cycle. The assumpion ha R is small guaranees ha G(N) < x. (See (b) above). Also: (3.7) ^'(X)=(T1-1) 7(1 - Tl) 1 -I XT] Tl Tl 1 (See Appendix 2). This expression approaches zero as r\ approaches 1. Proposiion 2 hen follows from he fac ha: jf-(x) = >'(x) ^'(^(x)) = (^'(x)) 2 In he Cobb Douglas case we have: (3.8) 1 Then, from Proposiion 2, we know ha here exiss a real 2 cycle whenever a is small and G(N) < x (ra small). We can summarize he conclusions obained in he Cobb Douglas case

27 17 by he following picure. Again a can be inerpreed as an inverse measure of he degree of monopoly power in he inermediae indusry. \ -X. * \ 1 < Pofivc W.I«nUi L^ c#) '.II ZCT [r*<»~uj growh i (uur i r l^wt) NH1. Figure 5 The reason for cycles in his model is similar, bu no idenical, o he reason for cycles in he model of Shleifer (1986). 19 In he Shleifer model innovaions are exogenous, and here are muliple equilibrium sraegies for implemening hem, because of an aggregae demand exernaliy. Tha exernaliy is presen in his model in an ineremporal raher han inersecoral form. As (3.1) makes clear, higher manufacuring oupu nex period raises he flow of monopoly rens nex period. As we have seen, i is his effec, ogeher wih he effec of creaive desrucion, ha makes x depend inversely upon x,, and which herefore makes high manufacuring employmen in odd inervals ogeher wih low manufacuring employmen in even inervals a possible equilibrium configuraion. There are several quesions ha would need o be addressed before judging he likely empirical relevance of hese 2 cycles. One is heir observabiliy under realisic expecaional assumpions; equilibria ha are sable under perfec foresigh (as in Figure 4 above) are ofen unsable under learning (Grandmon and Laroque, 1986). Anoher is he size of he resuling oupu flucuaions; modern economies ypically allocae no more han 2 or 3 percen of heir

28 18 labor force o research. We regard hese quesions as open. Learning raises many unresolved issues, and even if he movemen of labor beween manufacuring and research would cause small proporional flucuaions in he level of oupu i could produce large proporional flucuaions in research, and hence in he growh rae of oupu. Raher han pursue hese quesions furher we shall focus he res of he paper on balanced growh equilibria. Accordingly, he main ineres of he above analysis is no o propose a heory of cycles bu o describe he logic of he model of creaive desrucion. 4. BALANCED GROWTH a. Time Series Properies of Oupu One of he benefis of endogenizing echnical change is ha we can endogenize he average rae of growh (AGR) of he economy. Raher han have AGR depend upon exogenous populaion growh and/or exogenous echnical progress as in he neoclassical growh model, we have made i depend upon various facors ha affec he incenive o do research and he fruifulness of research in reducing producion coss. In his sense we are following he seminal conribuions of Romer (1986), Lucas (1988) and ohers. Anoher benefi is ha we can endogenize he variabiliy of he growh rae (VGR). The variance of wha appear in he model as echnology shocks depends upon he same economic facors ha deermine AGR. In he exreme case where he incenive o do research is so small as o resul in no growh, VGR will also be zero. We shall now proceed under he assumpion ha he economy is always in a siuaion of posiive balanced growh. Thus: 20 (4.1) x = x X/y c(x r + X(N - x) l (x)) = G(x) (more precisely G(x, r, X, y, N)) To see how a change in he parameers r, X, y, N affecs he equilibrium level of research (n = N x), i suffices o deermine he sign of he parial derivaives J&', J&, <-%^, <%\r, c^

29 19 where <^fis defined by: <#(x, r, X, y, N) = G(x) - x. The following Lemma is an immediae consequence of Lemma 1 and he fac ha j and x are decreasing: Lemma 2 : ^ = G^ - 1 < 0; <#. > 0; ^ < 0; <%. ' > 0; ^ + Jj^ < 0. We hen obain: Proposiion 3 : The amoun of research performed in a posiive balanced growh equilibrium will: fa) decrease wih he rae of ineres r. (b) increase wih he arrival parameer X, (c) increase wih he size of each innovaion (i.e. decrease wih y), and (d) increase wih he oal labor endowmen N. This proposiion is inuiive: (a) an increase in he ineres rae r means an increase in he required rae of reurn on research, whose effec will be o reduce he oal invesmen in R & D. (b) an increase in he arrival rae X will have boh he posiive effec of raising he speed a which research pays (he effec on he numeraor of 4.1), and he negaive effec of increasing creadve desrucion (effec on he denominaor). The firs effec dominaes. 21 (c) an increase in he size of each innovaion (decrease in y) also increases research by increasing he size of nex inerval's monopoly rens relaive o oday's research coss. (d) an increase in oal labor endowmen N increases research by increasing, for a given fi, he size of he marke ha can be "monopolized" by a successful innovaor.

30 20 We now complee our comparaive saics analysis by sudying he effec of he degree of monopoly power in he inermediae indusry. In he Cobb Douglas case, we saw ha a good (inverse) measure of monopoly power was he parameer a. In he general case we can proceed in a similar way and define: (4.2) F(x) = (f(x)) a, where < a < 1. The elasiciy r\ again depends posiively on a. 22 We can hen show: Lemma 3 : d% > 0, where <#(x, c) = G(x, a) x. Proof: Sfa! (x)) x (x) T](x,a) - 1 This expression decreases wih a. The lemma hen follows from (4.1). We hen obain he following inuiive resul: Proposiion 4 : An increase in monopoly power (reducion in a) increases he amoun of research in a balanced growh equilibrium. We now derive an explici expression for boh he average growh rae (AGR) and hevariabiliy of he growh rae (VGR) in a balanced growh equilibrium. The volume of real oupu (i.e. he flow of consumpion goods) during inerval is: ia i\ F(x) (4.3) y = J^ whirh imnlies: (4-4) y +1 =y _1 y. Thus he ime pah of he log of real oupu b\ y(x) where x is real ime will be a random

31 21 sep funcion saring a in y«= mf(h,x) (n c~, wih he size of each sep equal o he consan in y > 0, and wih he ime beween each sep {A,,A~,...} a sequence of iid. variables exponenially disribued wih parameer Xn. This saemen, ogeher wih (4.1), fully specifies he sochasic process driving oupu, as a funcion of he parameers of he model. No surprisingly, his sochasic process is nonsaionary. Suppose observaions were made a discree poins in ime 1 uni apar. Then from (4.4): (4.5) m y(x+l) = m y(x) + e(x); x = 0,1,... where e(x) is -in y imes he number of innovaions beween x and x+1. I follows from he above discussion ha 1 \. \ ',... is a sequence of iid. variables disribued Poisson wih parameer Xn. Thus (4.5) can be rewrien as: (4.6) in y(x+l) = in y(x) -lnlhy+ e(x); x = 0,1,... wih (4.7) e(x) iid., E(e(x)) = 0, var e(x) = Xn(in y) 2 where e(x) = e(x) + Xn in y. From (4.6) and (4.7), he discree sequence of observaions on log oupu follow a random walk wih consan posiive drif. I also follows ha: (4.8) AGR = -Xn in y, VGR = Xn(in y) 2. Combining (4.8) wih Proposiions 3 and 4 we can now sign he impac of parameer changes on AGR and VGR. Increases in he arrival parameer, he size of innovaions, he size of labor endowmen and he degree of monopoly power all raise AGR. Increases in he rae of ineres lower i. These parameer changes have he same qualiaive effec on VGR as on AGR. The effecs are inuiive and sraighforward. The effec of monopoly power, combined wih our earlier finding (in he Cobb Douglas case) ha a minimal degree of monopoly power is needed before growh is even possible, underline he imporance of imperfec compeiion for he growh process.

32 22 The effecs also highligh anoher Schumpeerian heme: he radeoff beween curren oupu y = F(x)/c and growh. Any parameer change (oher han N) ha raises AGR also reduces y (aking c as predeermined), by drawing labor ou of manufacuring. (In he case of a ha effec is amplified by he aendan shif of he producion funcion in he consumpion good secor.) However, he loss of curren oupu is no, as Schumpeer argued, a saic efficiency loss of monopoly. Our assumpion of inelasic labor supply prevens imperfec compeiion from puing he economy inside is curren fronier of manufacuring and research. Wheher he economy picks a good poin on ha fronier will be examined in he nex subsecion. The posiive effec of N on AGR has he unforunae implicaion, which Romer (1988) has noed in a similar conex, ha larger economies should grow faser. In fac, (4.1 ') in foonoe 20 above implies ha, in he Cobb Douglas case, a doubling of populaion should more han double he growh rae. (These comparisons mus be ineremporal raher han inernaional, since we have no deal wih he quesion of ransboundary echnology flows.) We accep he obvious implicaion ha his class of models has lile o say, wihou considerable modificaion, abou he relaionship beween populaion size and growh rae. The variabiliy of oupu in his model should be disinguished from is variabiliy in exising real business cycle models, for he following reasons. Firs, he variabiliy of shocks has been endogenized here. An innovaion in he saisical sense in real oupu is he effec of an innovaion in he economic sense, he disribuion of which depends as we have seen upon economic decisions ha respond o parameer changes. Second, oupu never falls in his model, conrary o wha is observed in recessions, and conrary o wha happens in exising real business cycle models. This is because all innovaions are incremens in knowledge. A negaive e(x) indicaes no a decrease in knowledge bu a smaller han average increase. We accep he implicaion ha he random flucuaions of oupu around he balanced growh pah of our model ell a seriously

33 Y 23 incomplee sory of he business cycle. Insead hey are bes hough of as porraying he sochasic rend of oupu. (Oupu can fall, however, in a 2 cycle.) b. Welfare We now compare he laissez-faire AGR derived above wih he AGR ha would be chosen by a social planner whose objecive was o maximize he expeced presen value of uiliy of consumpion y(). The maximized value of he planner's objecive funcion, Z, when c is hisorically given, is deermined by he following Bellman equaion: (4.9) rz = max 1 (x <N) PF(xJ C + X(N-x ) (Z +1 -Z ) The firs order condiion an inerior soluion for his problem is: (4.10) F'(x )/c = ^(Z +1 -Z ) I is easily checked ha he soluion o (4.9) is: * Z. F(x )/c (4.11) Z =-V = *-* zp 1 * where x solves: ' r + X(N - x )(1 - y ) (4.12). Hi-y) [$$ - x*] = y- r + XN ( 1 - y) l Thus he social opimum is a balanced growh pah, wih real oupu growing according o he same sochasic process as before, bu wih innovaions arriving a he rae * * X(N x ) = Xn. insead of X(N x) = A.n. Accordingly laissez-faire produces an AGR more (less) han opumal if x < (>) x. * Which way hese inequaliies go can be checked by comparing (4.1) and (4.12). The former can be rewrien as: y 7u(x~ (x)) : i (4.13) Y= ^~ r + X(N - x)

34 24 which, from he definiion of % and x, can be rewrien as: X(l-y)( f^ - 1) x + X y x (4.14) y= ^- C ' / *" \ where *- = is wc w he consan raio beween price and marginal cos in he monopolisic inermediae indusry on he balanced growh pah. The righ side of (4.14) is he privae reward o research, deflaed by nex period's real wage. The righ side of (4.12) is he corresponding social reward, deflaed by he shadow cos of research nex period. There are hree differences beween (4.12) and (4.14): The firs difference is he presence of he erm XNy in he denominaor of (4.12). This corresponds o he ineremporal spillover effec discussed in secion 2(c). A privae innovaor will capure ren from his innovaion for one inerval only, whereas he social ren coninues forever. Formally i comes from he presence of he erm X Z, in he definiion (4.9) of he social planner's value funcion Z, whereas no such erm' appears in he definiion (2.17) of he privae value of an innovaion, V. This effec leads he laissez-faire economy o underinves in research: i.e. X>X.2 The second difference is he presence of he erm X y *- x in he numeraor of (4.14). This corresponds o a "business sealing" effec. The privae innovaor does no inernalize he loss of surplus o he monopolis whose rens he desroys. I comes formally from he presence of X(N x )Z in he social planner's maximand in (4.9), whereas no corresponding subracion appears in he problem (2.15) solved by he privae research firm. This effec leads privae firms o overinves in research: i.e. i < x. d - F(x*) * The hird difference is he presence of he erm (-- l)x insead of [p ') x *p x ] in he numeraor of (4.14). This corresponds o he monopolisic disorion effec. I arises because he flow of reurns whose capialized value he social planner aemps o maximize is oal oupu, whereas he pnvae value V is liie capialized value of profis, 7. This disorion can work in eiher direcion. In he Cobb Douglas case i is non-exisen, since in ha case

35 F/F' x = ( )x = (-2- l)x. (In he Cobb Douglas case oupu and profis are proporional o one anoher: K = (l-c)y, so he effec amouns o nohing more han a muliplicaive consan in he social planner's decision problem.) The ineremporal spillover effec will dominae when he size of innovaions is large (i.e. y is small) in his case he privae reward o research will be small compared o he big social reward and laissez-faire will generae an AGR less han opimal. 24 On he oher hand when here is much monopoly power (a close o zero in he Cobb Douglas case) and innovaions are no oo large (i.e. y is no oo small), he business sealing effec will dominae, leading o an AGR under laissez-faire which exceeds he opimal level of average growh. 25 (This case canno arise in Romer's (1988) model where here is no desrucion of exising monopolisic aciviies). 26 c. Inroducing Learning by Doing We now inroduce a second source of growh besides innovaions: namely, he accumulaion of learning by-doing (lbd) in he inermediae indusry. A naural way o formalize lbd is o assume ha in he ime inerval beween wo successive innovaions, he inermediae indusry can sill improve upon he qualiy of is inermediae goods hrough manufacuring hem: more formally, if c (x) is he value of he uni cos parameer a ime x in inerval, and if x is he amoun of labor devoed o manufacuring inermediae goods a ha insan, we assume: (4.15) = -gx where g > 0. Thus, learning by doing in he inermediae indusry will increase produciviy in he consumpion good secor a he rae g x beween wo innovaions. When a new innovaion occurs, produciviy will jump as before by he facor y. Following A. Young (1928) and more recenly Arrow (1962), Romer (1986), and Dasgupa Sigliz (1988), we assume ha he reurns from learning by doing are shared by all

36 26 firms in he economy. In paricular he inermediae firms experience a complee spillover of heir lbd, which also spills over ino he research secor. Because of his exernaliy, each inermediae firm will choose is opimal level of inermediae oupu x(i) as before, relying for lbd on he oher inermediae producers; ha is, aking as given he economy wide average x in (4.15). In Romer (1986), his ype of exernaliy leads privae firms o underinves in knowledge, hence o an average growh rae under laissez-faire which is less han he social opimum. In his paper, however, he spillover of lbd can generae he opposie resul; i.e. ha privae economies grow oo fas. 27 Le V (x) denoe he value o a research firm of making he innovaion a ime x. Define x (x), w (x) and (x) analogously. We confine aenion o balanced growh equilibria, in which x (x) = x (consan). Then x = x(c (x)w (x)) as before, so c (x)w (x) = cw (consan). Likewise 7U (x) = T(cw)/c (x). As before, V (x) is he expeced presen value of k evaluaed over inerval. Since K (x) grows a he consan exponenial rae gx over his inerval, i follows ha: (4.16) V (x) = Vx) r + An - gx Therefore: (4.17) V (x) = (x -1 (x))/c (x) f r + XN - (X + g)x The firs order condiion for posiive research o occur is, as before, w (x) = XV, (x). Thus he level of oupu x in a (non degenerae) SE is given by: (4.18) y ~ rc(* ( *))/x (x) r + XN - (?+g)x

37 27 This equaion is idenical o (4.1) excep for he erm gx in he denominaor. Thus he previous analysis of saionary saes was a special case, wih g = 0. The AGR under laissez-faire is given by: AGR =Umi- E(my(x)-my ) = limi(gx-e()my) T- eo X- (where is he number of innovaions ha occurred unil dae x.) Given our assumpion ha innovaions follow a Poisson process wih arrival rae X h, we have: lim E(-) = ln. X X- oo Hence: (4. 19) AGR = g(n - n) - Xn&ry. In words, he average growh rae now derives from wo componens: a deerminisic componen due o Ibd (g(n n)) and a random componen corresponding o he innovaion process ( Xnmy). On he oher hand, VGR is given by he same formula as before since he addiional growh generaed hrough lbd is deerminisic: (4.20) VGR = Xfi(foy) 2 By conras wih Secion 4(a) above, parameers may affec AGR and VGR in opposie direcions: for example, an increase in r will reduce he amoun of research n performed in equilibrium, hereby reducing VGR. The effec on AGR is ambiguous: he random par will

38 28 be reduced as before bu he deerminisic par will increase ogeher wih he amoun of manufacuring labor x = N h. In paricular AGR will increase if g > Xfiry. An increase in he speed of learning by doing (i.e. in g) will have a posiive direc effec on AGR (equal o Ag(N - n)). I will also indirecly affec AGR by increasing he level of research ri. 28 When he iniial value of g is sufficienly small ( Xlny g > 0), he indirec effec will increase AGR. When he iniial g is large however, a furher increase in g may have a perverse effec on growh by inducing oo much research a he expense of learning by doing. The endency for lbd o induce oo much research is brough ou more clearly by he following welfare analysis. Le Z (x) = Z(c (x)) be he social planner's value funcion. I saisfies he following Bellman equaion, analogous o (4.9) above: (4.21) rz(c (x)) = max (x<n) ^ F(x) -1 + X(N-x) [Z(y x c ()) - Z(c (T))] - Z'(c (x))gxc (T) The addiional erm Z'(c (x))gx c (x) is he social reurn from learning by doing a ime x, according o (4.15). I is sraighforward o verify ha if research is posiive hen he soluion o (4.21) is: (4.22) Z(cfx)) = 1 yf'(x*) VL "-c^xt >(l -y) SY where x solves he equaion: (4.23) y: (X(1_y) - gyxpwo- x * } -i; r + XN(1 -y L ) The comparison beween he equilibrium level x under laissez-faire and he corresponding level x a he social opimum (i.e. beween (4.18) and (4.23)) involves he same hree effecs as before, i.e. he ineremporal spillover effec, he business sealing effec and he monopolisic disorion. Bu here is now an addiional source of disorion due o he inroducion of learning by doing as a second facor lbd; erms in ( g) appear in he

39 29 denominaor of (4.18) and in he numeraor of (4.23): in oher words inroducing lbd increases research under laissez-faire, bu reduces i in he social opimum! This effec reinforces he "business sealing effec" menioned earlier. The economics of his resul can be summarized as follows: learning by doing raises he reward o research aciviies, by raising he ne presen value of rens from a successful innovaion. Bu i does no raise he marginal profiabiliy of hiring manufacuring workers, since he benefis of lbd are exernal o he firm (our spillover assumpion). This explains why a laissez-faire economy will respond o an increase in g by invesing more in research a he expense of manufacuring. Wih he social planner hings will go he oher way around: inernalizing he effec of lbd in his decisions, he social planner will respond o an increase in g by puing more workers ino manufacuring, hus fewer workers ino research. To see ha learning by doing can make a laissez-faire economy grow oo fas, consider he following example. Sar wih an economy wihou learning by doing, where x = x(g = 0) * * is almos equal o x = x (g = 0). 29 Then: ^ (AGR* - AGR) = j g(g + X in y) * = x*-x + (g + \my)-(g--g) (x*(g) - x(g)) dx dx ^ As we have seen, -- -r- n > 0. Therefore, if x (0) is sufficienly close o x(0), inroducing learning by doing will generae oo much growh a he margin: AGR*(dg) < AGR(dg). 5. CONSUMPTION - SMOOTHING: MAKING INNOVATIONS LARGER CAN DETER RESEARCH In his secion we relax he assumpion of consan marginal uiliy. Insead, he insananeous uiliy funcion has a consan elasiciy of marginal uiliy equal o a > 0. Thus

40 o" u(c) = c. The pure rae of ime preferences is sill he consan r > 0. Thus he elasiciy of ineremporal subsiuion is l/o". The preceding analysis deal wih he special case of a = 0. Wih a > 0, consumers will wan o smooh consumpion over ime, which will affec he equilibrium. We confine aenion o siuaions of non degenerae balanced growh. All households are idenical, so all will have equal consumpion in equilibrium, equal o y/n. Thus each innovaion will reduce he marginal uiliy of consumpion by he facor u ' (y, /N)/u ' (y 7N) = y. Each shareholder of a research firm will hus have a marginal rae of subsiuion equal o y beween he consumpion good jus before an innovaion and he consumpion good jus afer. Accordingly, a research firm will discoun he payoff o a successful innovaion by he facor y, and is marginal condiion for posiive research will be (5.1) w = Xy CT V +1 s where V. is he marke value in erms of he consumpion good of he + 1 innovaion. Tha value coninues o be deermined by (2.17). 30 From (5.1) and (2.17):,, * l Vi The local monopolis's choice problem will be he same as before. Thus in a balanced growh equilibrium (5.2) implies: (5.3) y 1^7 = X ff*" 1 (*))/*"* (*> r + X(N - x) Equaion (5.3) deermines x, and hence also he level of research ri = N x. The only difference beween (5.3) and he previous equaion (4.13) is he presence of C on he lef side. The average growh rae and is variabiliy coninue o be deermined by (4.8). Thus all he comparaive saics effecs on n, AGR and VGR remain unchanged excep for he effec of he size of innovaions. Specifically, an increase in he size of innovaions (decrease in y) sill increases research, and hence increases he average growh rae and is variabiliy, if he elasiciy of

41 31 ineremporal subsiuion exceeds uniy (a < 1). lef side of (5.3), which causes a decrease in x. In his case he decrease in y decreases he Bu if he elasiciy of ineremporal subsiuion is less han uniy (a > 1), hen an increase in he size of innovaions leads o a reducion in research. If he elasiciy is sufficienly close o zero his could lead o such a large reducion in research as o reduce AGR and VGR. The economic inerpreaion of his new possibiliy is sraighforward. A decrease in y raises he payoff o research as measured in unis of consumpion good. Bu i also reduces he marginal uiliy of consumpion afer he innovaion relaive o before. When a > 1 he laer effec is so srong as o reduce he uiliy rae of reurn o research. The effec is similar o he negaively sloped savings schedule ha occurs when a > 1 in he sandard wo period overlapping generaions model wih all endowmens accruing o he young. An increase in he ineremporal elasiciy of subsiuion (decrease in o~) increases he amoun of research, and hence also increases he average growh rae and is variabiliy, because i reduces he lef side of (5.3). Inuiively, i raises he uiliy rae of reurn o research by reducing he rae a which marginal uiliy falls afer an innovaion. The welfare analysis of balanced growh equilibria is almos he same as before. The maximized value of he social planner's objecive funcion Z is deermined by he Bellman equaion: (5.4) rz = max [j-^ [F(x )/c.] 1-CT + X(N - x.)(z. -Z)}. (x <N) l ~ a l l +l I is easily verified ha if research is posiive hen he soluion o (5.4) is z* = z^/y 1^ = FCx*)" F'(x*)A(y a_1 - Dc -. where x* solves: (5 5) y l~ - ^(1-Y 1^) F(x*) * * x F'(x ) r + 7iN(l - y a-1 )

42 32 Comparison of (5.5) wih (5.3) reveals he same hree differences as before beween x and x. So, as before, laissez-faire may produce eiher oo much or oo lile research. 6. ENDOGENOUS SIZE OF INNOVATIONS: INNOVATIONS ARE TOO SMALL UNDER LAISSEZ-FAIRE This secion generalizes he analysis of balanced growh developed in secion 4 by allowing research firms o choose no only he frequency bu also he size of innovaions. We show ha he equilibrium size of innovaions will be independen of everyhing in he model excep he echnology of research firms. Thus all he comparaive saics effecs previously analyzed remain unchanged. There is, however, a change in he welfare analysis. Specifically, under laissez-faire innovaions will be oo small. This new effec will reinforce he ineremporal spillover effec in ending o make he economy grow oo slowly. Raher han assume a linear research echnology a he individual firm level, i is helpful o assume an infiniely elasic supply of idenical research firms wih U shaped cos curves. Assume ha o experience innovaions wih a cos reducion facor y a he rae Xz a firm mus hire v(z,y) unis of labor, wih v, > 0, v~ < 0. We look for a saionary equilibrium wih a consan aggregae level of research employmen n, wih z and y equal o he consans z and y wihin each research firm, and wih c w = cw = consan for all. Since n/v(z,y) is he number of firms, he aggregae arrival rae of innovaions is [n/v(z,y)]?.z = (Vk)n, where k = v(z,y)/z is a consan. In a saionary equilibrium, he inermediae demand for manufacuring labor, i.e. x = N n, is given by: (2.5) c w = cw = F ' (x) + x F"(x), s and he value of he +1 innovaion is: ^(CW)/C r*n v +1 k(cw)/cw (6J) V A +1 = = r+(a/k)n r+( Uk)n w = Aw +l +1

43 If a firm choosing y makes he (+1) innovaion, i will make he wage rae rise o: /z: ->\ cw,cw>~+i c c ~ (6.2) w = = ( )y =wy. + 1 During inerval, he research firm akes as given ha V. will be deermined by (6.1) and (6.2), and akes A and w as given. Thus is decision problem is: (6.3) max - w v(z,y) + A,zAy~ w (z.y) Assume ha (6.3) has an inerior soluion. By free enry, he maximized expression in (6.3) mus equal zero. (6.4) Vj = vz _1 From his and he firs order condiions: (6.5) v 2 = -vy _1 From (6.3) he oupu of research is proporional o z/y. Therefore a research firm's average cos is proporional o: K(z,y) = y v(z,y)/z. Our assumpion of U-shaped cos is: Assumpion A. : K(z,y) is sricly convex. I follows from (A. 3) ha (6.4) and (6.5) define a unique equilibrium combinaion (z,y); specifically, he combinaion ha minimizes average cos: (6.6) (z,y) = arg min K(z,y). Since none of he parameers (r,x,n) ener (6.6), i follows ha y and z are independen of hem. Similarly, he size of each innovaion is independen of he degree of monopoly power. Thus all comparaive saics resuls go hrough as before excep for hose describing

44 34 he effecs of varying he size of innovaions, which are no longer relevan since he size is endogenous. To conduc he welfare analysis, noe ha he social planner's maximized objecive funcion Z(c ) is given by he Bellman equaion: (6.7) rz(c)= max ( F(N ~ mv ( z '7» + Xmz(Z( T c ) - Z(c.))) 1 {z,y,m} c l l where m is he number of firms. 31 The firs order condiions for his problem are: -F' vc; 1 + Xz(Z(yc )-Z(c )) = - F' c" 1 m V][ + A.m(Z(7C )-Z(c )) = F' c" mv~ + Xzm c Z'(yc ) = I is easily checked ha he soluion o (6.7) is: 7r ^ J-l-7, \ F(N -n*), Z(c ) = y Z(c, ) = *-, -4 1 l l r - (k/k )(y -l)n w /c C where k = v(z,y )/z, n = m v(z,y ) and: (6.8) Vj = vz*" 1 (6.9) v 2 = -<l-y*r 1 vy*~ 1 Our main resul is ha innovaions are oo small in equilibrium: Proposiion 5 : Y > Y * Proof: Define a s y v(z*,y*)/z*(l-y*) > 0. From A.3, (6.8) and (6.9): (6.10) (z,y ) = arg min{k(z,y) + ay}.

45 35 From (6.6) and (6.10): K(z,y) < K(zV). K(z,y) + ay > K(z,y ) + ay. From hese inequaliies: ay > ay. Since a > 0, herefore y > y. a This resul is anoher manifesaion of he business sealing effec. The smaller he innovaion he larger he losses hrough obsolescence relaive o he gains of an innovaion. The privae firm ignores his cos of raising y. By free enry he maximand in (6.3) equals zero. This implies: i i (6.11) y = W rc(g (x))/* kra + N - x where k = v(z,y)/z. The soluion o (6.7) yields: * (1-Y*) (6.12) y = F'(x*) F'(x ). x * k*ra + (l-y* _1 ) N Comparison of (6.11) wih (6.12) reveal he same hree disorions as before on he equilibrium amoun of research. In addiion, he fac ha y > y will end o creae oo lile research. The fac ha k * k will also generae a disorion. As before, he oal level of research may be oo much or oo lile. 7. RANDOM ARRIVAL PARAMETER: CREATIVE DESTRUCTION CAN DESTROY GROWTH In his secion we allow he arrival parameer X o vary randomly. This generalizaion illusraes he force of creaive desrucion by allowing an increase in X in some saes of he world o deer research in oher saes. In fac, we presen an example where in he limi as he value of X in one sae becomes infinie, he average growh rae falls o zero. Le {X.,..,X } be he finie se of possible values of X. A he momen of any innovaion a new X is drawn, according o he ransiion marix A, and all research firms learn

46 36 his value. Transiion ino a high X sae could represen a fundamenal breakhrough ha leads o a Schumpeerian wave of innovaions, whereas ransiion o a low sae could represen he exhausion of a line of research. The sochasic equivalen of balanced growh equilibrium is an equilibrium in which manufacuring employmen depends only on he sae of he world, no on ime. Le V./c be he value of making he innovaion arid moving ino sae j. In any sae i, he marginal expeced reurn o research in inerval is X- S a-. V./c r 1 m : _ j U J + * This will equal he wage if posiive research occurs in sae i. If research occurs in all saes, he V-'s mus saisfy he Bellman equaions: (7.1) vy [ = Sfl.jIa.jVj/y) - ^[N-x^Ia-.V./yXIV. ; i=l,..,m j J J j J J The AGR equals f /h y, where f is he asympoic frequency of innovaions. Define n. =N-x(X. Za.. V./y). Then: v l y J m (7.2) f= I X. n. q. i_i.. i=l q» where q- is he asympoic fracion of ime spen in sae i. Our example has m = 2 and a.. = 1/2 V... I is easily verified ha in his case: (7.3) q 1 = l_ q2 = X~n~ ;) 7 _

47 37 From (7.2) and (7.3): (? - 4) ' = 2 ^1^-9 n i n 9 Vl + ^2 To complee he example, ake he Cobb Douglas case (F(x) = x ) and suppose a = y = 1/2, and r = N = X, = 1. Using (2.10) and (2.12), (7.1) can be rewrien as: (7.5) fv 1 = [16(V 1+ V 2 )] ^(l-^vj+v^] 2 }V 1 V 2 = [\6X 2 (W l+ W 2 )\- 1 -\ 2 {l -[4L,(V 1+ V 2 )r 2 )V 2 When X~ = 1, he soluion o (7.5) is V. = V~ = «7 /8, which implies n, = n~ = 1/3, and f = 1/3. When Xj =, he soluion is V, = 1/4, V? = 3/4, which-implies n, = 0, n^ = 1, and 2n n~ 1 f = jj = 0. Thus raising he produciviy of research in one sae can cause research o be discouraged in he oher sae, hrough he hrea of creaive desrucion, o such an exen ha growh is eliminaed. 8. CONCLUSION We have presened a model of economic growh based on Schumpeer's process of creaive desrucion. Growh resuls exclusively from echnological progress, which in urn resuls from compeiion among research firms ha generae innovaions. Each innovaion consiss of a new line of inermediae goods ha can be used o produce final oupu more efficienly han before. Research firms are moivaed by he prospec of monopoly rens ha can be capured when a successful innovaion is paened. Bu hose rens in urn will be desroyed by he nex innovaion, which will render obsolee he exising line of inermediae goods.

48 38 The model possesses a unique balanced growh equilibrium, in which here is a consan allocaion of labor beween research and manufacuring. In ha equilibrium he log of GNP follows a random walk wih drif. The size of he drif is he economy's average growh rae, and he variance of he incremens is he variabiliy of he growh rae. Boh hese parameers of growh are endogenous o he model, and depend upon he rae of ime preference and elasiciy of ineremporal subsiuion of he represenaive household, and upon he size and likelihood of innovaions resuling from research. As Schumpeer argued, he degree of marke power available o someone implemening an innovaion will also be an imporan deerminan of growh. We have paramerized he degree of marke power, and shown ha i has a posiive effec upon boh he average growh rae and is variabiliy. The average growh rae and is variabiliy are also affeced by he exen of learning by doing in he manufacuring secor of he economy. The greaer he coefficien of learning by doing, he more research will be underaken. This will raise he average rae of growh aribuable o research, bu i may decrease he growh aribuable o learning bydoing, because he only way for sociey o engage in more research is o ake labor ou of manufacuring. The average growh rae may be more or less han socially opimal, because here are wo couneracing disorions. The firs is a echnology spillover. A successful innovaion produces knowledge ha oher researchers can use wihou compensaion o he innovaor. The couneracing disorion is a "business sealing" effec. The privae research firm does no inernalize he loss o ohers due o obsolescence resuling from his innovaions. We also show ha innovaions are oo small under laissez-faire, again because of he business sealing effec. Learning by doing inroduces a furher disorion because i is exernal o he individual manufacuring firm. This disorion ends o produce oo lile manufacuring, and hence oo much research. Wheher i produces oo much or oo lide growh depends upon he relaive

49 39 efficacy of learning by doing and research as sources of growh. Under some circumsances he model possesses equilibria in addiion o he unique one wih balanced growh. One is a "no growh rap" in which he economy sops growing in finie ime because of he (raional) expecaion ha if one more innovaion were o ake place i would be followed by a flurry of research aciviy. This expecaion makes i so likely ha he rens accruing o he producer of he nex innovaion will be desroyed quickly ha i is no worh rying o creae he innovaion, and research sops. Anoher possible equilibrium is a real 2 cycle, in which manufacuring employmen oscillaes beween wo values. These cycles, as well as hose arising from random imiaion or a random arrival parameer of innovaions, have he ineresing implicaion of a negaive correlaion beween he cyclical componens of produciviy (y c/n) and real wages (w c ), because y cvn = F(x(w c ))/N. Relaively high real wages arise from a relaively high incenive o research, and discourage he oupu of goods. Two final resuls are worh menioning here. Firs, if he represenaive household in he model has a sufficienly low ineremporal elasiciy of subsiuion hen an increase in he size of innovaions can acually reduce he amoun of research, and hence reduce he average rae of growh in he economy. Second, when he Poisson arrival rae of innovaions is a random funcion of he amoun of research, an increase in he likelihood of innovaions in one sae of he world will resul in exra creaive desrucion in ha sae; his can discourage research in oher saes o such an exen ha he economy's average growh rae falls. We conclude by nodng direcdons for fuure research. I would be useful o allow he size of innovadon evenually o fall, o allow for he possibiliy ha echnology is ulrimaely bounded. Also o include a richer inersecoral srucure o sudy boh he posiive dynamics of diffusion and he normaive effecs of inersecoral spillover. The model would also gain

50 40 richness and realism if capial were inroduced, eiher physical or human capial embodying echnical change, or Rand D capial ha affecs he arrival rae of innovaions. 32 Allowing unemploymen by inroducing search exernaliies ino he labor marke, and changing he srucure of demand for inermediae goods so as o allow for a conemporaneous aggregae demand exernaliy, migh also generae muliple equilibria and allow us o sudy he reciprocal ineracion beween echnical change and he business cycle. All hese exensions seem feasible because of he simpliciy and ransparency of he basic model.

51 41 APPENDIX 1 Assumpion A.l is saisfied in he CES case. Le F(x) = (x p + H P 1/p ), where < p < 1. We have: F' = l/p(xp + HP)!/ P- V" 1 = xp- 1 (xp + HP) 1^ 1 Hence: F" = (p - 1)xP- 2 (xp + hp) 1^" 1 + xp" 1 (1/p - 1)( X P + np) 1^2^1 and 1- = (p - l)(xp + hp) 1/ P- 2 [xp" 2 xp + xp- 2 RP - 2 x P- 2 ] = (P -D(xP + hp) 1/p-2 xp- 2 hp<o F" = (p - l)(l/p - 2)(xP + HPj^P^pxP-^P- 2 H p + (p-1)(xp+hp) 1/ P- 2 (P -2)xP-3 = (p - l)(xp + HP)!/ P- 3 [(1-2p)x2 P" 3 + (p - 2)x 2P~ 3 + HPxP" 3 (p - 2)] = (p - l)x P-3 (xp + HP) 17 P- 3 [(p - 2)HP - (1 + p)xp]. Therefore: 2F" + xf" = (x p + hp) 1/p_3 hp(p - l)[2xp~ 2 (x p + H p ) + xp- 2 [(p-2)hp-(l + p)xp]] = (p - 1) hp <0 (xp + n p ) l/p- 3 [J- 2 ][xp(i - p) -f p hp]

52 42 APPENDIX 2 Derivaion of (3.7) We have: r/x + N- x Using (2.5), (2.8), and he fac ha x = x(wc), we ge: or' /-N X' [ WC] J? (x) = ' TI^WC)/! 7 (X (x))/y + rfk + N?7X + - x(wc) L x ' (wc) N -x 1/y /X + N - x(wc) (5ic '(wc) + ywc x'(wc)) wc 7(wc) [ x(wc) + ywc x'(wc)] X(WC) 'WC,,m x 1N = ( YH 1) ~, N 7(wc) wc x ' (wc) v ' 'wc = (r, _ i)(-yn; c - 1),. x where ri = ^ -. 'wc ~, N x(wc) By he firs order condiion (2.5): ' F'^Xl-^y)=wc- - Differeniaing his w.r.. (wc), we can reexpress r\ as: 'wc 1-1/T xv Hence: ^'(X) = (T1-1) Y(l- ri).1-1/t XT]'

53 43 References Aghion, Philippe, and Peer Howi, "Growh and Cycles hrough Creaive Desrucion," Unpublished, Universiy of Wesern Onario, Arrow, Kenneh J., "The Economic Implicaions of Learning by Doing," Rev. Econ. Sud. 29 (June 1962): Azariadis, Cosas, "Self Fulfilling Prophecies," Journal of Economic Theory 25 (December 1981): Campbell, J. and N.G. Mankiw, "Are Oupu Flucuaions Transiory?" Quar. J. Econ. 102 (November 1987): Cochrane, John H., "How Big is he Random Walk in GNP?" Journal of Poliical Economy 96 (Ocober 1988): Corriveau, Louis, "Enrepreneurs, Growh, and Cycles," Unpublished, Universiy of Wesern Onario, Dasgupa, Parha, and Joseph Sigliz, "Learning by Doing, Marke Srucure, and Indusrial and Trade Policies, Oxford Economic Papers 40 (June 1988): Deneckere, Raymond J., and Kenneh L. Judd, "Cyclical and Chaoic Behavior in a Dynamic Equilibrium Model, wih Implicaions for Fiscal Policy," unpublished, Norhwesern Universiy, Grandmon, Jean Michel, and Guy Laroque, "Sabiliy of Cycles and Expecaions," Journal of Economic Theory 40 (Ocober 1986): Griliches, Zvi, "Issues in Assessing he Conribuion of Research and Developmen in Produciviy Growh," Bell Journal of Economics 10 (Spring, 1979): Griliches, Zvi, "Paens: Recen Trends and Puzzles," unpublished, Harvard Universiy, Hausman, Jerry A., Bronwyn Hall, and Zvi Griliches, "Economeric Models for Coun Daa wih an Applicaion o he Paens R and D Relaionship," Economerica 52 (July 1984):

54 44 King, Rober G., and Sergio T. Rebelo, "Business Cycles wih Endogenous Growh," unpublished. Universiy of Rocheser, Lerner, Abba P., "The Concep of Monopoly and he Measuremen of Monopoly Power," Review of Economic Sudies 1 (June 1934): Lucas, Rober E. Jr., "On he Mechanics of Economic Developmen," Journal of Moneary Economics 22 (July 1988): 3^2. Nelson, Charles, and Charles Plosser, "Trends and Random Walks in Macroeconomics," Journal of Moneary Economics 10 (Sep. 1982): Pakes, Ariel, "Paens as Opions: Some Esimaes of he Value of Holding European Paen Socks," Economerica 54 (July, 1986): Reinganum, Jennifer, "Innovaion and Indusry Evoluion," Quarerly Journal of Economics 100 (February 1985): , Jennifer, "The Timing of Innovaion: Research, Developmen and Diffusion," unpublished, Universiy of Iowa, Romer, Paul M., "Increasing Reurns and Long Run Growh," Journal of Poliical Economy 94 (Ocober 1986): , "Endogenous Technical Change," unpublished, Universiy of Chicago, Scherer, F.M., Indusrial Marke Srucure and Economic Performance. 2nd ed. Chicago: Rand McNally, 1980., Innovaion and Growh: Schumpeerian Perspecives. Cambridge, MA: MIT Press, Schumpeer, Joseph A., The Theory of Economic Developmen. New York: Oxford Universiy Press, 1934., Capialism, Socialism and Democracy. New York: Harper and Brohers, Shleifer, Andrei, "Implemenaion Cycles," Journal of Poliical Economy 94 (December 1986):

55 45 Solow, Rober M., "Technical Change and he Aggregae Producion Funcion, Review of Economics and Saisics 39 (Augus, 1957): Tirole, Jean, The Theory of Indusrial Organizaion. Cambridge, MA: M.I.T. Press, Woodford, Michael, "Saionary Sunspo Equilibria," unpublished, Universiy of Chicago, Young, Allyn, "Increasing Reurns and Economic Progress," Economic Journal 38 (Dec. 1928):

56 . 46 FOOTNOTES! In he Deneckere Judd model a consan proporion of exising inermediae goods disappear each period. The equilibrium of he model exhibis no growh. 2More recen work by Cochrane (1988) shows ha his evidence is no, however, conclusive. 3Thus policy analysis based on he King Rebelo model would be subjec o he Lucas criique because i reas he variabiliy of growh as invarian o he policy deerminans of growh. 4 Menion should also be made of he preliminary work of Corriveau (1988), who is developing a similar model of growh and cycles in a discree ime seing. The discreeness of ime In Corriveau's model inroduces complicaions from he possibiliy of simulaneous innovaions, which we avoid by our assumpion of coninuous ime. In Corriveau's framework all innovaions are non drasic, and i is no possible o parameerize he degree of monopoly power in he economy as we have done. 5 Some consequences of inroducing memory are discussed in secion 8 below. 6 Scherer (1984) combines process and inpu oriened R and D ino a measure of "used" R and D, which he disinguishes from pure produc R and D. He esimaes ha during he period in U.S. indusry he social rae of reurn o "used" R and D lay beween 71% and 104%, whereas he reurn o pure produc R and D was insignifican. 7 Gradual diffusion could be inroduced by allowing he cos parameer afer each innovaion o follow a predeermined bu gradual pah asympoically approaching he limi c, and hen o jump o c upon he nex innovaion and follow a gradual pah approaching c, This would produce a cycle in research wihin each inerval, as he gradual fall in coss induces manufacuring firms o hire more and more workers ou of research unil he nex innovaion occurs. 8 Noe ha his descripion of echnology implies increasing reurns in he producion of he consumpion good, he ulimae inpus of which are land, labor (as embodied in he inermediae goods) and knowledge (as embodied in he c's). If land and labor alone were doubled, hen he flow of inermediae inpu would also double, and hence, by he assumpion of consan reurns, final oupu would jus double. Since here are consan reurns in hese wo facors holding he hird consan, here are increasing reurns in all hree. The echnological specificaion of he economy follows ha of Griliches (1979). exis empirical sudies casing doub on our assumpions ha here are consan reurns in research in ha a doubling of research will double he rae of innovaions (Griliches, 1989), ha here is no variabiliy in he size of innovaions (Pakes, 1986), and ha here is no There variabiliy in he Poisson arrival rae of innovaions (Hausman, Hall, and Griliches, 1984). The firs wo assumpions can be relaxed wih only noarional difficulies. The hird is relaxed in secion 7 below. Finally, we could modify he inermediae echnology (2.2) o allow a specialized fixed facor in addiion o labor. We could inerpre he specialized facor as unskilled labor, and, inerpre he labor ha can be used in eiher manufacuring or research as skilled labor, hus adding plausibiliy o he model. The work of Romer (1988) suggess ha no subsanive difficulies would be creaed. 9The alernaive assumpion ha research firms have idenical U shaped funcions wih a small efficien scale produces idenical resuls. This alernaive is employed in secion 6 below. 10Bellman equaions like (2.14) are sandard in he paen race lieraure (see Tirole,

57 ' , ch. 10). Several more are inroduced below. This foonoe presens an alernaive derivaion. Le T, T', and T" be, respecively, he ime of he nex innovaion, he ime a which he oher firms would experience heir firs innovaion if hey held employmen consan a n forever, and he ime a which his firm would experience is firs innovaion if i held is employmen consan a z forever. Under he no spillover assumpion, T' and T" are independen random variables, expoenially disribued wih respecive parameers An and Xz, and T = min (T", T"). I follows ha, as already menioned, T is exponenially disribued wih parameer X(n + z) and ha he probabiliy ha he nex innovaion will be made by his firm, condiional on he dae of ha innovaion, Prob {T = T' T) is he consan z/(n + z). Thus, condiional on W (T) -rt ', he value of he research firm is: V/ + 1+ Prob{T=T' T}V +1 e w.zdx ' o Thus: W (T) = e -rt W + (z/(n + z))v n - - rt (1-e N ) w z By definiion: W = Max EWf (T). 1 {z >0} l Since E(e -rt ") = X(n f + z)/(r + A.(n + z)): W = 1 Max X(n + z)w + 1 [z > 0} r. +?L(n + XzV - i+{ w z.+ z) The las equaion is equivalen o (2.14). n h I is sraighforward o relax he assumpuon of complee spillover by allowing he s innovaor o have an advanage in finding he +1 innovaion; o have an arrival parameer X' > X. We have worked ou his analysis under he assumpuon of U shaped cos wih small efficien scale (see foonoe 9), wih no imporan change in resuls.- 12Our analysis of research is derived from he paen race lieraure surveyed by Reinganum (1987) and Tirole (1988, ch. 10). Wihin ha lieraure he paper closes o ours is Reinganum (1985), which emphasizes he affiniy o creaive desrucion. Mos oher papers do no have he sequence of races necessary for creaive desrucion. Our paper goes beyond ha lieraure by embedding he sequence of races in a general equilibrium seing. Thus (a) he flow of profis J, resuling from an innovaion will depend upon he amoun of research during he nex inerval, (b) our welfare crierion is he expeced lifeime uiliy of he represenaive consumer, (c) we characerize he ime series properies of GNP, and (d) in secion 5 we find ha he size of innovaions affecs he rae of discoun applied o research. In addiion, he analysis in secdon 6 of he endogenous deerminaion of he size of

58 ^(x) > 48 innovaions, and he analysis of cycles in secion 3, go beyond anyhing we have found in he paen race lieraure. 13 G is well defined on (0,N), by A The mehods of Azariadis (1981) or Woodford (1986) could also be used o consruc raional expecaions sunspo equilibria, a leas in some cases. 15 This disinguishes our model from he growh model wih exernaliies developed in Romer (1986), which may have no seady sae. n 16 When r=0, x is defined by: \~, l 1, 7(x (x c ))/y N = N; whereas G(N) = x 5(x l (N)Vj (T = x(+») = < x, by A.2. By coninuiy, (3.6) will hold for r (or r/x) posiive bu small. 17 A real 2 cycle also exiss if G(N) > x and 1. However, his can never occur in he Cobb Douglas case (see Aghion and Howi, 1988). 18T1(X) H F'OQ xf'(x) 19 Likewise, he reason for cycles is similar o ha underlying he chaoic flucuaions in he model of Deneckere and Judd (1986). They model innovaions as Romer (1988) does, wih no creaive desrucion. Their dynamics are backward looking, because he incenive o innovae depends upon he exising number of producs, which depends upon pas innovaive aciviy, whereas heir assumpion ha monopoly rens are no desroyed by fuure innovaions makes he incenive independen of fuure innovaive aciviy. Their cycles arise because he only res poin of heir backward looking dynamics is unsable. 20 In he Cobb Douglas case, (4.1) can be expressed by: (4.1'): Y X 1-q a 1-a a r + X(N - x) /X (N - h) + n 21 In he exension of he model considered in secion 7 he second effec may dominae. 22r (x,a) = -F'(x)/xF"(x) = f(x) C (x). (l^x)x[f'(x)] Z -xf(x)f'(x) 23Two addiional kinds of spillover can easily be included. Firs, researchers could benefi from he flow of ohers' research, so ha an individual firm's arrival rae would be a consan reurns funcion X(z,n) of is own and ohers' research. Second, here could be an exogenous Poisson arrival rae \i of imiaions ha coslessly circumven he paen laws and clone he exising line of inermediae goods. Boh would have he effec of lowering AGR. Also as we showed in Aghion and Howi (1988), he inclusion of fi would inroduce anoher source of cycles in he economy, since each imiaion would make he inermediae indusry perfecly compeiive, which would raise manufacuring employmen, unil he nex innovaion

59 a 49 arrives. *.. XN 24 More formally, as y falls o he lower limi - -rrr, x approaches zero, whereas x approaches a sricly posiive limi. 1 ^In he Cobb-Douglas case, if > ran, hen n > 0, for all y (see 3.5) whereas n* = when y^- > XN/cr. 26 There seems o be a presumpion among empirical sudens of R and D ha spillovers will dominae business sealing in a welfare comparison. See, for example, Griliches (1979, p. 99). 27The formal source of hese differen resuls is ha in our model research is a separae aciviy from he producion of he goods embodying knowledge. The aciviy whose doing creaes exernal learning is he producion. Doing i more inensively requires less research o be underaken. Romer follows Arrow (1962) in assuming ha he wo aciviies are inseparably bundled, hereby eliminaing his radeoff. 28 This follows immediaely from equaion (4.18), using he fac ha boh c and x are decreasing funcion. 29Tn innecobb he Cobb-Douelas Douglas case- case. (1 T? )N ~ ayv/x ~ ( *T y)n " "T 1^ (i_a ) + y " (l-y)(l oc) 30 Someone who gave up a marginal uni of consumpion o buy shares in a local paen would receive a consan dividend of ctv unil T, he dae of he nex innovaion. The expeced uiliy gain of he ransacion would be: ' T U'(C)7C/V Equaing his o he expeced uiliy cos u'(c ) yields (2.17) We assume ha no non negaiviy consrains bind. 32 We have examined he consequences of a limied kind of R and D capial by allowing for he arrival parameer X o increase randomly in he middle of an inerval, and hen o remain high unil he nex innovaion, so ha he probabiliy of an innovaion depends no jus upon he flow of research bu also upon he lengh of ime over which he innovaion has been sough. The analysis is formally like ha of secion 7 above, and mos of he comparaive saics resuls of secion 4 go hrough.

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