VARIETY, SPILLOVERS AND MARKET STRUCTURE IN A MODEL OF ENDOGENOUS TECHNOLOGICAL CHANGE *

Transcription

1 VARIETY, SPILLOVERS AND MARKET STRUCTURE IN A MODEL OF ENDOGENOUS TECHNOLOGICAL CHANGE * Petro F. Peretto Department of Economcs Duke Unversty Revsed: September 1995 ABSTRACT I study an economy where olgopolstc frms establsh n-house R&D programs to produce a contnuous flow of cost-reducng (ncremental) nnovatons. The scale of frms' R&D operatons determnes the rate of productvty growth. I frst study the role of concentraton, frm sze, and demand, approprablty, and opportunty condtons, when the number of frms s exogenous. I fnd hump-shaped relatonshps of knowledge approprablty and the number of frms wth steady-state growth that gve a frst ntuton about the forces at work n the model. In symmetrc equlbrum, statc economes of scale due to varety of nput supply and knowledge spllovers yeld ncreasng returns to the number of frms. A larger number of frms leads to hgher output, hgher (aggregate) R&D, and faster growth. Offsettng ths force, market fragmentaton leads to smaller frms, smaller R&D programs, and slower growth. Next, I let the number of frms be endogenous and study the balanced-growth path of the economy n two dmensons: productvty growth and the number (varety) of goods. The prce, nvestment, entry, and ext decsons are nterdependent. In partcular, R&D s a fxed (sunk) cost and n zero-proft equlbrum s negatvely related to the number of frms. Ths addtonal feed-back renforces the market fragmentaton effect. More mportantly, many parameters have no longer the effects predcted by the model wth an exogenous number of frms (and the standard models n the lterature). For example, the scale effect of populaton sze may be negatve. Large markets have many frms, hgh output, and hgh aggregate R&D. Market fragmentaton may offset ths force and result n small frms and slower growth. Keywords: Growth, Market Structure, Product Varety, Spllovers, Endogenous Technologcal Change. JEL Classfcaton Numbers: E10, L16, O31, O40 * Address: Department of Economcs, Duke Unversty, Durham, NC Phone: (919) Fax: (919) E-mal: Peretto@lews.econ.duke.edu. I thank Rchard Levn and Xaver Sala--Martn for helpful dscussons. Ths s a revsed verson of the paper presented at the Internatonal Conference on Increasng Returns and Economc Theory n honor of Kenneth J. Arrow held at Monash Unversty, Clayton, Vctora, Australa. I thank dscussant Alfred Guender for useful comments. All errors are my responsblty.

2 1. Introducton Growth theorsts have produced a number of nterestng models nvestgatng the dea that technologcal progress, the engne of growth n ncome per capta, s endogenous to the economc system and drven by market forces. These models are radcally dfferent from the tradtonal theory of economc growth based on captal accumulaton and emphasze the ncentves for proft-seekng agents to undertake R&D amed at developng new products and processes, or ncrementally mprovng old ones. The key problem that these models must tackle s the approprablty problem. That s, they must specfy an economc or nsttutonal devce that allows nnovators to approprate the returns from ther R&D actvty. Such a devce must be at the heart of any theory of endogenous nnovaton because wthout the expectaton that the (ex post) economc returns from nnovaton are approprable, proft-seekng agents would not ncur the (ex ante) sunk costs that R&D requres. Models of endogenous nnovaton characterze R&D as the development of blueprnts for a new (consumpton or ntermedate) good. Anyone wth access to the blueprnt can produce the new good. Innovatons are essentally new peces of knowledge and knowledge, by ts very nature, s non-rval and can be reproduced at wll and at zero cost by anyone wth access to t. To nduce proft-seekng agents to undertake R&D the nformaton contaned n the blueprnts has to be protected. Accordngly, the nsttutonal devce that generates the ncentves to nnovate s the patent law that restrcts the rght to use the product-specfc knowledge to the patent holder. 1 These models have two mportant features. Frst, nfntely lved and perfectly enforceable patents generate local monopoles by assgnng to the nnovator the exclusve rght to manufacture and sell the new good. In addton, producton frms conductng n-house R&D to develop ther own nnovatons have no 1 The varety-expanson model (Romer 1990, Grossman and Helpman 1991, Ch. 3) formalzes nnovaton as the expanson of the range of ntermedate (consumpton) goods that enter symmetrcally the producton of fnal output (consumers' utlty). R&D s determnstc and the contnuous ntroducton of horzontally dfferentated goods, each one produced by a new local monopolst, drves growth. The qualty-ladder model (Segerstrom, Anant and Dnopulous 1990, Grossman and Helpman 1991, Ch. 4, Aghon and Howtt 1992) formalzes nnovaton as the ntroducton of hgher-qualty versons of exstng ntermedate (consumpton) goods. R&D s stochastc and the arrval at (random) dscrete ntervals n tme of vertcally dfferentated goods drves growth. Goods of dfferent qualty are perfectly substtutable and product obsolescence from nnovaton generates frm turnover over tme. Therefore, each generaton of the good s produced by a local monopolst whose tenure s temporary. 1

3 specfc advantage over ndependent research frms that sell n the patent market the rghts to use ther nnovatons. The emphass on the non-rvalry of knowledge and the patent law makes clear that mperfect markets and monopoly power are necessary for proft-seekng agents to undertake R&D and nnovaton. However, t focuses too much on monopolstc competton and neglects that most nnovatons are carred out by establshed producers. 2 In so dong, the models neglect the central role of the modern olgopolstc corporaton and ts n-house ntegraton of manufacturng and R&D. As a consequence, the theory does not address those components of the market structure, lke concentraton and frm sze, emphaszed by Schumpeter (1942) as key determnants of the R&D actvty of proft-seekng frms. In ths paper, I extend the theory beyond the case of local, temporary or permanent, monopoles and consder olgopoles wth an endogenous number of frms undertakng n-house R&D. Busness and economc hstorans stress the central role of large corporatons wth ntegrated manufacturng and R&D actvtes n generatng nnovaton and growth (see, e.g., Freeman 1982, Mowery and Rosenberg 1989, Chandler 1990). To begn wth, R&D generates, n addton to specfc nnovatons, knowledge that s not nnovaton-specfc and that can be used n subsequent R&D actvty. Moreover, the patent market for the exchange of nnovaton-specfc knowledge contaned n blueprnts s far from perfect and suffers from nformatonal problems that do not allow to wrte perfect contracts between users and developers of nnovatons. Frms, therefore, have strong ncentves to nternalze the R&D functon and to keep knowledge secret. 3 Evdence n IO shows that frms buld up ther compettve poston by consstently nvestng n R&D over tme and accumulatng n-house knowledge protected by secrecy, patents and other approprablty devces (see, e.g., Levn et al and Mowery and Rosenberg 1989). 2 Segerstrom (1991) developed a qualty-ladder model wth nnovaton and mtaton that relaxes the emphass on monopolstc competton by allowng nnovators and mtators to coexst as collusve duopolsts. Ths model, however, stll neglects the role of n-house R&D and does not allow to study olgopolstc markets wth more than two frms. Barro and Sala--Martn (1995, Ch. 7) developed a verson of the qualty-ladder model where nnovaton s carred out by the leader-ncumbent and there s no frm turnover. Ths model takes nto account the role of n-house R&D by establshed producers but lke many others t consders monopolstc competton only. 3 In contrast, models based on (ntertemporal) knowledge spllovers (see footnote 1) post that the knowledge generated as a by-product of R&D s publc and can be used by any subsequent nnovator for free. Snce developng an nnovaton does not gve any advantage beyond the patent relatve to the new product, there s no ncentve to ntegrate n-house manufacturng and R&D actvtes. 2

4 The Industral Organzaton (IO) lterature, theoretcal and emprcal, has typcally assumed that R&D s undertaken by establshed frms and focused on those features of the busness envronment, the number and sze of frms, barrers to entry, dversfcaton, that determne market rvalry and, therefore, that determne the opportuntes and constrants that frms face when plannng R&D actvtes. 4 The IO lterature has also dentfed a number of addtonal features of the ndustry envronment that are at least as mportant as the concentraton rate and the average sze of frms. These features have proven so mportant emprcally that n recent work the term market structure has come to denote the entre vector of ndustry characterstcs that are relevant for the R&D decsons of frms: concentraton, frm sze, characterstcs of demand (e.g., volume, growth, degree of product substtuton), and approprablty and opportunty condtons (see, e.g., the dscusson n Cohen and Levn 1989). In the general equlbrum (GE) model that follows, I study an economy where olgopolstc frms establsh n-house R&D facltes to produce over tme a contnuous flow of cost-reducng (ncremental) nnovatons. In GE, frms compete not only for sales but also for resources. In symmetrc equlbrum, the number of frms determnes concentraton and frm sze. These determne the scale and the effcency of frms' R&D operatons and, therefore, the rate of nnovaton. If the number of frms s exogenous, one can study the relatonshp between number of frms, elastcty of product substtuton, approprablty and opportunty condtons, and the rate of growth of the economy. The number of frms, however, must be endogenous and determned, jontly wth the rate of growth of the economy, by the zero-proft condton. Ths generates some new nterestng effects. To solate these effects from those dscussed above, I study the mplcatons of the model under two assumptons: exogenous and endogenous number of frms. In secton 2, I descrbe the structure of the model. In secton 3, I derve the prce and R&D strateges of the typcal olgopolst and the propertes of the symmetrc Nash Equlbrum (NE). In secton 4, I construct the GE of the economy and argue that, when the number of frms s exogenous, the model 4 See, e.g. the surveys by Kamen and Schwartz (1982), Baldwn and Scott (1987), Dos (1988), Trole (1988, Ch. 10), Cohen and Levn (1989), and Scherer and Ross (1990, Ch. 17). The emphass on rvalry s mportant. Scherer and Ross (1990, Ch. 1), n partcular, defne market rvalry as "a strvng for potentally ncompatble postons... combned wth a clear awareness by the partes nvolved that the postons they seek to attan may be ncompatble" (p.16). Monopolstc competton models, wth ther emphass on atomstc frms, do not capture very well the noton of rvalry. 3

5 does not have transtonal dynamcs and the economy s n steady state at all tmes. I characterzes ths steady state and dscuss how opportunty, approprablty, and demand condtons, and the number of frms shape the balanced-growth path (BGP) of the economy. Not surprsngly, most results at hs stage resemble those of standard models of endogenous nnovaton. Interestng, however, are the effects of the number of frms and the degree of approprablty on steady-state growth. These depend on the elastcty of product and knowledge substtuton. When the elastcty of knowledge substtuton s hgh, growth s hghest f approprablty s perfect (spllovers are zero). When the elastcty of knowledge substtuton s low, a hump-shaped relatonshp between approprablty and growth arses. Smlarly, dependng on the values of the elastcty of product and knowledge substtuton, the relatonshp between number of frms and growth can be ether monotoncally ncreasng or hump-shaped. These results gve a frst ntuton about the economc forces at work n ths class of models. There are macroeconomc complementartes arsng from two sources: statc economes of scale due to varety of nput supply and knowledge spllovers yeld ncreasng returns to the number of frms. A larger number of frms leads to hgher aggregate output, hgher R&D n GE, and faster growth. Offsettng ths force, the fragmentaton of the market due to a larger number of frms leads to smaller frms and slower growth. Market fragmentaton and macroeconomc complementartes work n opposte drecton snce n ths model the sze of the frm determnes the scale of the frm's R&D program and ths determnes the frm's rate of nnovaton. To see n detal how these forces nteract, n secton 7, I study the endogenous determnaton of the number of frms n zero-proft equlbrum. There are no exogenous fxed costs and the only factor that checks the number of frms s R&D expendtures. These, however, are endogenous to the market equlbrum and are not necessarly postve. Provded approprablty s above a threshold level, when the economy s large enough and/or the productvty of R&D s hgh enough, a zero-proft equlbrum wth postve growth and a fnte number of frms exsts. Indeed, two such zero-proft equlbra exst and I dscuss some alternatve selecton mechansms. The most nterestng mplcaton of the endogenety of the market structure s that the model now represents the BGP of the economy n two dmensons, the rate of growth of productvty n the ntermedate sector and the number of dfferentated nputs t supples. In addton, the endogenety of the number of frms leads to nterdependent prce, 4

6 nvestment, entry, and ext decsons. The effects of many parameters may be reversed wth respect to standard models and the case where the number of frms s exogenous. For example, the scale effect, the effect on growth of an ncrease n the sze of populaton, s no longer unambguously postve and n some cases t s actually negatve. 2. Structure of the Model I consder an economy wth a two-tered manufacturng sector. One compettve frm produces output wth a constant returns to scale (CRS) technology defned over a contnuum of ntermedate goods. Olgopolstc frms produce ntermedate goods wth a technology exhbtng ncreasng returns to scale (IRS) to labor and (labor-augmentng) knowledge. Intermedate olgopolsts accumulate knowledge to ncrease factor productvty, reduce producton costs, and wn market share by offerng lower prces. 5 Manufacturng and research have the same factor ntensty and output s ether consumed or nvested to accumulate knowledge captal. Ths secton descrbes n detal the producton sde of the economy Fnal Producer One representatve frm produces and sells a homogenous fnal good, Y, n a compettve market. N ( z 0 The producton technology s defned over a contnuum of dfferentated, non-durable ntermedate goods, =L N M ε 1) ε (2.1) Y X d O QP ε ( ε 1), where ε>1 s the elastcty of substtuton between ntermedate goods, X s the fnal producer's use of each ntermedate good, and N s the number of ntermedate goods suppled at a moment n tme. The producton technology s defned over the nterval [ 0, ) because the lst of ntermedate goods s nfntely long, all of these goods are known, and ntermedate frms have the ablty to produce them. 6 However, at 5 Qualty-mprovng and cost-reducng technologcal progress are formally equvalent when the qualty of the good determnes the cost at whch the good delvers ts servces (see, e.g., Spence 1984 and Trole 1988). I focus on cost-reducng technologcal progress only, but the model can easly be rewrtten n terms of qualty-mprovng technologcal progress wth the same qualtatve results. 6 Thus, there s no research nvolved n expandng the range of ntermedate goods used n fnal producton. All of these goods and the technology to produce and use them are known. The number of avalable ntermedates, however, s an endogenous varable because of the entry and ext decsons of the ntermedate producers (see below). The focus here s not on how new ntermedates are ntroduced nto the economy. Rch models addressng ths ssue already exst (see the papers cted n footnote 1). The focus s on how, at any moment n tme, rvalry n the ntermedate sector determnes the varety of nput supply 5

7 a moment n tme only N of such goods are suppled. The fnal producer, therefore, faces the producton technology (2.1) defned over the nterval [ 0N, ]. Facng N avalable nputs, he uses all of them because each ntermedate's margnal product approaches nfnty as ts use approaches zero. The producton technology (2.1) exhbts economes of specalzaton. Suppose, for example, that the fnal producer uses all the avalable ntermedate goods n equal quantty. One can wrte (2.2) Y = N ε ( ε 1 ) X, where X s the quantty of each ntermedate nput. Holdng constant the use of each ntermedate good, output ncreases by ncreasng the number of ntermedate goods. These are not dynamc gans, lke those emphaszed by Romer (1990). Snce all possble ntermedate goods are already known, these are statc economes of specalzaton, smlar to those dscussed by Adam Smth n hs famous argument that the sze of the market determnes the extent of the dvson and specalzaton of labor. To close ths descrpton of the fnal output sector, I now solve for the optmal behavor of the fnal producer. Let the prce of the fnal good be normalzed to one, P Y 1. The fnal producer acts as a prce-taker n the nput and output markets and maxmzes profts (2.3) Π F Y PX d = z0 N subject to the producton technology (2.1). The fnal producer allocates total expendture on ntermedates accordng to (see, e.g., Dxt and Stgltz 1977) ε N 1 ε (2.4) X Y P P do, = L N M z 0 QP where total expendture on ntermedate goods s equal to total revenues that accrue to the fnal producer because n a compettve market he just breaks even. Equaton (2.4) s the nstantaneous demand schedule faced by each ntermedate producer Intermedate Olgopolsts Each ntermedate producer faces the producton technology (2.5) X = A L, where and, therefore, specalzaton n the fnal sector. I augment the model and ntroduce research costs for the ntroducton of new ntermedates n Peretto (1995b). 6

8 L NM zn j 0 ( η 1) η ( η 1) η (2.6) A αz + ( 1 α) Z dj, O QP η ( η 1) s a labor productvty ndex determned by the frm's (labor-augmentng) knowledge captal (herenafter knowledge), Z, and by spllovers of knowledge from the other frms, Z j for all j, X s output, and L s labor employment. 7 Knowledge s non-rval wthn one frm and s partally excludable between frms. The parameter η>1 measures the elastcty of knowledge substtuton. The parameter 0 α 1 measures the approprablty condtons. 8 Knowledge s a publc good when α =0, whle t s a prvate good when α=1. Snce knowledge s non-rval, the producton technology exhbts CRS to labor alone, the rval nput, and IRS to labor and knowledge. I assume CRS to knowledge to ensure that constant endogenous growth be feasble n steady state. Equaton (2.6) captures two dfferent effects of knowledge accumulaton. The frst s the drect effect of ncreasng the frm's labor productvty for gven external opportuntes, as measured by the knowledge of the rvals. The second s the ndrect effect of ncreasng the margnal contrbuton to the frm's productvty of each unt of outsde knowledge. Knowledge accumulaton, thus, enhances the "absorptve capacty" of the frm, the frm's capacty to learn from the surroundng envronment and apply to ts producton process deas and methods developed elsewhere (Cohen and Levnthal 1989). Equaton (2.6), moreover, exhbts economes of specalzaton arsng from the varety and substtutablty of knowledge across frms. Consder a symmetrc equlbrum where frms have the same knowledge stocks, (2.7) A= Z 1+ ( 1 α)( N 1) η ( η 1). Average productvty, A, s proportonal to average knowledge, Z. Ths reflects that one frm's knowledge contrbutes to the pool of knowledge spllovers and mproves the technologcal opportunty condtons that all frms face. The rate of proportonalty, s decreasng n the elastcty of knowledge substtuton, η, 7 I abstract from physcal captal for smplcty. Physcal captal has very nterestng mplcatons for the transtonal dynamcs of the model but does not alter sgnfcantly the propertes of the balancedgrowth path that I dscuss n ths paper. See Peretto (1995a) for an example. 8 The IO lterature has developed ths concept to denote technologcal and nsttutonal factors lke the protecton granted to ntellectual property rghts, the effectveness of ndustral secrecy, or some measure of the "natural" transferablty of technology across frms (e.g., tact vs. formal knowledge). Levn et al. (1987) show that there s a varety of means of excludng the other frms from beneftng from one frm's technologcal advance. Thus, the parameter α s a summary measure of the effectveness of all these means (see, e.g., Cohen and Levnthal 1989). 7

9 because the more substtutable are the dfferent knowledge sources, the lower s the margnal contrbuton to the productvty of the ndvdual frm of the pool of knowledge spllovers. The rate of proportonalty s decreasng n the approprablty parameter, α, and ncreasng n the number of frms, N, because the less knowledge s a prvate good (the lower s α) and the more knowledge sources there are (the larger s N), the larger s the pool of knowledge spllovers and the hgher s each frm's productvty. Knowledge s accumulated accordng to the smple R&D technology (2.8) Z & =θ I, where Z & s the ncrement n knowledge obtaned by frm from nvestng I Z unts of fnal output for an nterval of tme dt and θ>0 s a parameter of technologcal opportunty n research, the extent to whch nsttutonal and other envronmental factors determne R&D productvty. 9 Equatons (2.5) through (2.8) emphasze local, ncremental technologcal progress and formalze t as accumulaton of labor-augmentng knowledge. The focus s on those costly learnng actvtes, not necessarly formal R&D, that frms undertake to mprove the productvty of the labor force (see, e.g., Malerba 1992). 10 Dfferently form most models n the lterature, n equatons (2.5) through (2.8) IRS to knowledge accumulaton apply at the frm level, not at the ndustry level. Ths assumpton s mportant. By ts very nature, dsemboded knowledge s non-rval across frms as well as wthn the frm. It s certanly excludable, however, often to a substantal degree (see, e.g., Levn et al. 1987), due to patent laws, protecton of ntellectual property rghts, and secrecy or tactness of knowledge tself. In addton, a relevant component of knowledge s emboded n the frm and ts members as part of an organzaton and s rval across frms (see, e.g., the dscusson n Nelson and Wnter 1982 and Dos 1988). Thus, knowledge 9 In IO lterature ths concept denotes, among others, factors lke the contrbutons of unversty and government research to ndustral R&D. See, e.g., Levn et al. (1987), Cohen and Levn (1989) and Klevorck et al. (1993) for a detaled dscusson. 10 In all recent models that rely on aggregate ntertemporal spllovers to sustan growth n the long run (e.g., Romer 1990, Grossman and Helpman 1991, Ch. 3), knowledge accumulaton s formally equvalent to labor-augmentng, dsemboded technologcal progress. Recently, Lucas (1993) has stressed that, as an explanaton of aggregate growth, there s only a neglgble dfference of nterpretaton between models of human captal accumulaton and models of techncal change. In hs vew, the crucal queston s whether to emphasze human captal accumulaton at school or by frms. Models of technologcal change emphasze accumulaton of human captal by frms. In partcular, they emphasze accumulaton of human captal through research rather than as a by-product of experence n producton (learnng-by-dong). Wth the dfferences dscussed n the text, I follow ths tradton. 8

10 does not dffuse mmedately and at no cost across frms both because t s partally excludable, n ts dsemboded and emboded parts, and because t s partally emboded n the frm as an ndvdual entty. It s mportant to emphasze that excludablty and rvalry are related but not the same. If knowledge were totally emboded n the members of the organzaton, t would be totally rval across frms and therefore excludable. If knowledge were totally dsemboded, t would be totally non-rval across frms but stll partally excludable. In both cases, however, knowledge would be non-rval at the frm level. When the frm undertakes an R&D project that develops a new pece of knowledge, all members of the organzaton beneft from the project and learn the pece of knowledge. Ths vew of technologcal change s dfferent from the one formalzed by Romer (1990) and Grossman and Helpman (1991, Ch. 3-4). It s ndeed more smlar to the one that Romer (1986) proposed n the semnal paper that started the theory of endogenous growth. 11 Technology does not consst of blueprnts that are taken off the shelf and put to use n producton. New technology and ts commercal applcaton must be learned at a substantal cost by all those nvolved n ts use. An R&D project does not consst of the research dvson of the frm, or an outsde contractor, developng a new blueprnt that then s passed along to the producton dvson and put to commercal use. Rather, t conssts of a set of ntegrated actvtes, formal research n a lab beng only one of these, that accumulate useful knowledge as human captal emboded n the sklls of the labor force. When new products and processes are developed, or old ones ncrementally mproved, a blueprnt s only one step toward commercal mplementaton. The labor force has to be traned and develop new sklls, old facltes have to be upgraded or new ones set up, managers have to learn to run a dfferent organzaton of labor and handle dfferent problems. These actvtes develop the body of knowledge that ultmately determnes what and how effcently the frm can do. Ths knowledge for the most part cannot be formalzed and artculated n blueprnts and techncal manuals and s emboded n the human captal of the frm and ts people. 11 There are two key dfferences. Romer assumes that knowledge accumulates as a by-product of nvestment n physcal captal (learnng-by-dong or learnng-by-nvestng), whle I assume that knowledge accumulates through specfc R&D actvty. In addton, Romer assumes that knowledge s a publc good, whle I assume that t s partally approprable. 9

11 3. Knowledge Accumulaton n Olgopoly Equlbrum Intermedate olgopolsts behave non-cooperatvely and maxmze ther value. In ths secton, I defne the equlbrum wth free entry and free ext for the economy's ntermedate sector and study ts propertes. All frms face the same producton and R&D technologes and demand schedules. I restrct the analyss to symmetrc equlbra to smplfy the model and focus on the key propertes of the BGP Defnton of Equlbrum The typcal frm chooses prce and nvestment polces that maxmze the present dscounted value of net cash flow, (3.1) V( t) = z R( τ) P( τ) X( τ) wl( τ) I( τ) dτ, t where wl s total producton cost and I s R&D expendture. Wth perfect foresght, V s the stock market value of the frm, the prce of the ownershp share of an equty holder. I consder a symmetrc Nash Equlbrum (NE) n open loop strateges. Let a =[ P ( τ), I ( τ )] for τ t be frm 's strategy. Ths strategy s a tme-path of prce and nvestment. To smplfy the analyss, I assume that entry and ext nvolve zero costs and frms do not have any scrappng value. The number of frms s free to jump to ts equlbrum level. I construct an equlbrum where at tme t frms commt to tme-paths of prce and nvestment, whle smultaneously free entry and ext determne the number of frms. The prce and nvestment strategy nduces tme-paths of producton, sales, and knowledge accumulaton. Followng Dasgupta and Stgltz (1980), I defne the equlbrum as follows. Let A N be the strateges of the N frms. At tme t, [ NA, N ] s an nstantaneous equlbrum wth free entry and free ext f for all (3.2) V [ N, A ] V [ N, A ] 0 and for all N > N N N (3.3) V [ N, A ] 0, N where [ A N ] denotes the strateges when frm devates from ts optmal tme-path of prce and nvestment whle the other frms do not devate. Condton (3.2) requres that the actve frm maxmze the present value of net cash flow, takng as gven the compettors' strateges, and that ths maxmzed value be nonnegatve. The latter nequalty s the free-ext condton snce the scrappng value of an actve frm, the 10

12 opportunty cost of ncumbency, s zero. Condton (3.3) s the standard free-entry condton that the value of the margnal frm be non-postve. I study the propertes of ths NE n two steps. I frst study the propertes of the GE of the economy treatng the number of frms as a parameter, and then study the endogenous determnaton of the number of frms and ts nteracton wth the rate of growth. The frst step of ths procedure determnes the value of ncumbency for a gven number of frms. To determne the NE wth free entry and free ext, one compares ths value to the opportunty cost of ncumbency and the opportunty cost of entry. These are both zero Knowledge Accumulaton Dervng the cost functon from the producton technology (2.5), the typcal frm solves (3.4) max m V ( ) ( ) P, I R P t w A X I d r = z τ τ, subject to the research technology (2.8), the demand schedule (2.4), Z ()= t Z>0 (ntal knowledge s gven), Z j ( τ ) for all τ t gven for all j (the frm takes as gven the R&D paths of ts rvals), and Z& ( τ) 0 for all τ t (knowledge accumulaton s an rreversble nvestment). There are dfferent approaches to ths problem. One s to observe that the objectve functon (3.4) s weakly separable n the prce and nvestment decson and two-stage budgetng s feasble for the frm. Consder stage-one proft maxmzaton. Gven the frms' knowledge stocks, the optmal Bertrand-Nash prce strategy s (3.5) P = ξ w A ( ξ 1 ), where ξ = ε ( ε 1) S > 1 s the prce elastcty of demand that frm faces and S P P z 1 ε N 1 0 frms follow the optmal strategy n prce and quantty, S s the frm's market share, defned as the output of the frm dvded by the total output of the ndustry. A Bertrand-Nash equlbrum wth postve output exsts f the prce elastcty of demand faced by each frm s larger than one. Ths requres 0 < 1. In symmetrc equlbrum, ths condton s met f N > 1. In stage-two proft maxmzaton, frms hold constant the prce elastcty of demand they face, ξ. Substtuton of the prce polcy nto the demand schedule (2.4) determnes the demand for each varety as a functon of the N labor productvty ndexes. Ths yelds the nstantaneous proft functon, S ε. If 11

13 (3.6) π Y A ξ ξ = z [ ( 1) ] N ξ [ A( ξ 1) ξ] 0 ε 1 ε 1, d expressed as the frm's nstantaneous revenues gross of R&D costs. 12 Maxmzng equaton (3.5) subject to equaton (2.8) requres maxmzng the Current Value Hamltonan (3.7) CVH = π I + q θ I, where the costate varable, q, s the shadow value of knowledge, the frm's knowledge, Z, s the state varable and R&D nvestment, I, s the control varable. The Hamltonan s lnear n R&D nvestment, thus gvng the optmal polcy I = when CVH I >0, and I = 0 when CVH I <0. The former case volates the GE condtons and s ruled out. The latter case corresponds to an nstantaneous equlbrum wth zero R&D. The frst order condtons when CVH I =0 and I > 0 are gven by the equalty between the margnal revenue from one unt of R&D and ts margnal cost, 1 θ=q, the transversalty condton that at the end of the plannng horzon the frm's knowledge has no value, lm t Rt () q() t Z() t =0, the constrant on the state varable gven by the R&D equaton (2.8), and a dfferental equaton n the costate varable q& π 1 (3.8) r = +. q Z q Snce the shadow value of knowledge s the nverse of R&D productvty, the transversalty condton s satsfed as long as the dscount factor s postve. In the steady state wth constant endogenous growth, ths condton requres that knowledge does not grow at a rate faster than the nterest rate (see below). The frst order condtons reduce to (3.9) r = θ π. Z Equaton (3.9) s a perfect-foresght, no-arbtrage condton whch requres that there are no unexploted proft opportuntes n R&D: the margnal proft from one R&D project must be equal to the cost of the 12 Ths procedure s equvalent to the followng reasonng. Frst, frms choose to charge a markup over margnal cost. They treat ths mark-up as an unknown to be determned by the soluton of the system of optmalty condtons and thnk of prces as proportonal to margnal costs. Gven ths prcng rule, they realze that ther knowledge affects ther compettors' prces va the spllovers. The optmal knowledge accumulaton decson must take nto account ths effect. Dfferentatng profts wth respect to the frm's knowledge requres dfferentatng the demand schedule wth respect to the compettors' prces as well. To compute the optmal mark-up, frms take the usual frst order condton wth respect to ther prce (see, e.g., Beath and Katsoulacos 1991). 12

14 R&D project fnanced by borrowng at the rate r (drect cost of R&D), and ths must be equal to the return from a rskless loan, at the rate r, of the resources requred for the R&D project (opportunty cost of R&D). The two nterpretatons of the cost of R&D are equvalent under perfect foresght and mply that the frm can fnance ts own R&D ether through debt or through equty. The NE of the R&D game s gven by the set of frst order condtons and dynamc constrants for all the actve frms. Snce I have assumed that all frms n the ndustry start out wth the same amount of knowledge, I have ndeed assumed that the model s symmetrc not only n the fundamentals that frms face, but also n the startng values of the state varables. Snce the dynamc behavor of each frm s dctated by the same equatons and by the same ntal condtons, frms accumulate knowledge at the same rate and the NE s symmetrc at all tmes The Returns to R&D n Bertrand-Nash Equlbrum Theory and evdence n IO support the dea that frms account for the partal approprablty of knowledge when plannng ther nvestment (see, e.g., Spence 1984, Levn et al. 1987, Cohen and Levnthal 1989, Cohen and Levn 1989). Frms realze that knowledge spllovers accrung to compettors do not contrbute to mprovng ther market poston because they do not gve any advantage over the compettors. Thus, what matters for market competton s the knowledge that can be approprated. (3.10) Consder the margnal gross revenue from knowledge accumulaton N π π j Z = A A Z dj. z0 j The returns to R&D are determned by the effect of labor productvty on proft and by the effect of knowledge on labor productvty. The frst effect s captured by the dervatves of the nstantaneous proft 13 An addtonal argument s needed here. The soluton dscussed n the text follows form the fact that the Hamltonan s lnear n nvestment. The correspondng bang-bang strateges yeld mmedate convergence to the frm's steady state only f there are no constrants to nvestment. Although I dd not specfy t formally, n GE frms are constraned by the savng behavor of households. Thus, an addtonal assumpton s needed to smplfy the analyss. Specfcally, the NE of the R&D game s gven by the set of frst order condtons from the ntertemporal maxmzaton problem. If nvestment s unconstraned all frms converge nstantaneously to ther symmetrc steady state. To avod problems dervng from the constrants nduced by the GE structure of ths model, I assume that the N frms start out wth the same level of the knowledge stock. For an ntroductory dscusson of the propertes of ths type of dynamc (captal accumulaton) games see Fudenberg and Trole (1986) and Trole (1988, Ch. 8). Fudenberg and Trole (1991, Ch. 13) provde a formal textbook treatment. 13

15 functon wth respect to the frms' productvty ndexes. The second effect captures the role of the partal excludablty and partal substtutablty of knowledge. Takng dervatves and rearrangng for symmetry equaton (3.10) yelds, after substtuton nto equaton (3.9), 1 ( η 1) Y N N (3.11) r = N + θα( ε 1)( 1) 1+ ( 1 α)( 1) 1. ε( 1) 1 N A The equaton captures a number of effects that derve from the nteracton between frms. The ( η ) terms Y [( ε N 1) + 1 ] and [( ε 1)( N 1) N] α 1+ ( 1 α)( N 1) 1 1 measure, respectvely, the grossproft and the busness-stealng effects. Gven sze of the market Y (the volume of demand), the NE rate of return to R&D s gven by the gross profts earned for a gven market share and by the ncrease n the market share acheved by the R&D project. The gross-proft effect s decreasng n the elastcty of product substtuton, ε, because the olgopoly mark-up s lower the hgher s ε. The busness-stealng effect, on the other hand, s ncreasng n the elastcty of product substtuton, ε, because the effectveness of costreducng R&D n stealng customers from other frms s hgher the hgher s ε. The busness-stealng effect s a hump-shaped functon of the approprablty parameter, α. Spllovers have two opposte effects on the ncentves to undertake R&D (Spence 1984). If approprablty s zero, f knowledge s a publc good, the margnal contrbuton of knowledge to the rvals' productvty s equal to ts margnal contrbuton to the frm's productvty. The frm does not gan any margnal advantage over the compettors and the R&D project s worth zero. Ths s the ncentve effect of spllovers and reduces the frms' R&D effort for gven market and technology condtons. However, f approprablty s low, large spllovers from the rvals' knowledge ncrease the margnal contrbuton of the frm's own knowledge to productvty and ncrease the rate of return to R&D. Ths effcency effect of spllovers ncreases the productvty of knowledge for gven market and technology condtons. The gross-proft effect s decreasng n the number of frms, N, because n the symmetrc NE both the market share and the olgopoly mark-up are lower the larger s N. The busness-stealng effect has two components. The frst, the market nteracton between frms, captured by the term ( ε 1)( N 1) N, s ncreasng n the number of frms, N, because t captures the frm's potental gan of market share, 14

16 measured by the rvals' total market share. 14 The second, the technologcal nteracton between frms, s η captured by the term α 1 1 α 1 1 ( + ( )( 1 ) N and s ncreasng n the number of frms, N, because the margnal value of knowledge ncreases wth the pool of spllovers. 4. Balanced-Growth wth an Exogenous Number of Frms To construct the GE of the model, I mpose three condtons: labor and output market clearng, and equalty between the rate of return to nvestment and the rate of return to savng. I assume that there exsts a fxed populaton, L, of dentcal ndvduals endowed wth L unts of labor that they supply nelastcally n a compettve labor market. Indvduals consume and save. The captal market s perfect Labor and Output Market Equlbrum Substtutng the prce strategy (3.5) nto the condtonal labor demand obtaned from the producton functon (2.5), and usng the frm's demand schedule (2.4), yelds L = Y( ξ 1 ) S wξ. Integraton over frms yelds aggregate employment, and snce labor s n fxed supply equal to L, the labor N market clearng condton, z 0 Ld = L, yelds the equlbrum wage rate, w= ( ξ 1 ) Y Lξ. The fnal producer breaks even and Π F = 0, where profts for the fnal producer have been defned n equaton (2.3). In the symmetrc NE the producton technologes (2.1) and (2.5) and L = L N yeld Y (4.1) A = LN 1 ( ε 1), that can be plugged nto (3.11) to peg the rate of return to R&D, 1 ( ε 1) 1 ( η 1) Lαθ( ε 1)( N 1) N 1+ ( 1 α)( N 1) (4.2) r =. [( ε N 1) + 1] N Wth an exogenous number of frms, there are no transtonal dynamcs and the economy s always n steady state (see below). Output market clearng yelds the economy's resource constrant (4.3) Y = LE + NZ&. Usng equatons (2.7) and (4.1), ths can be wrtten (4.4) c = 1 ( ε 1) N N + N η ( η 1) 1 ( 1 α)( 1) Lθ g, 14 Borrowng termnology from Baldwn and Scott (1987), I call ths the "cannbalzaton" effect snce t captures the dea that cost-reducng nvestment transfers surplus across frms. Ths effect s closely related to the noton of "creatve destructon" proposed by Schumpeter (1942, Ch. 7). 15

17 where g Z & Z s the average rate of growth of knowledge and c E Z s consumpton per effectve worker. Takng logs and tme dervatves of equatons (4.1) and (2.7) yelds YY & = AA & = ZZ & snce the number of frms, N, s constant Captal Market Equlbrum and Dynamcs The typcal household maxmzes lfetme utlty 1 σ ρτ ( t) E( τ) 1 (4.5) Ut () = z e dτ, t 1 σ subject to the ntertemporal budget constrant that the present dscounted value of expendture cannot be z greater than the present dscounted value of wages and dvdends plus ntal wealth, t z t (4.6) R( τ) E( τ) dτ R( τ)[ w( τ) + d( τ)] dτ+ B( t), where σ>0 s the nverse of the constant elastcty of ntertemporal substtuton, ρ>0 s the ndvdual τ rsds () t dscount rate, R( τ) e z s the cumulatve dscount factor from tme t to tme τ, B s the ndvdual's assets holdng, w s the wage rate, and d s dvdends. 15 The household chooses the optmal plan &E r (4.7) = ρ E σ. Takng logs and tme dervatves of consumpton per effectve worker, and usng the expresson L NM for the rate of return to R&D n equaton (4.2), one obtans ε c& 1 Lαθ( ε 1)( N 1) N 1+ ( 1 α)( N 1) (4.8) = M c σ [ ε( N 1) + 1] N 1 ( 1) 1 ( η 1) O P Q ρ g P. Fgure 1 shows the dynamcs of the GE of the economy n ( g, c ) space. Paths that eventually yeld negatve consumpton per effectve worker are not equlbra snce they yeld negatve utlty. Paths that eventually yeld nfnte consumpton per effectve worker volate the GE condtons snce aggregate consumpton, LE, wll exceed output, Y, n fnte tme after productvty growth drops to zero. The unque GE s the saddle pont where per capta consumpton, E, and average productvty, A, grow at the same constant rate L 1 ( ε 1) 1 ( η 1) 1 Lαθ( ε 1)( N 1) N 1+ ( 1 α)( N 1) (4.9) g = M O ρp. σ [( ε N 1) + 1] N NM QP 15 Profts n the ntermedate sector are dstrbuted as dvdends n equal shares to households. 16

18 The economy grows f the rate of return to nvestment and savng s greater than the ntertemporal dscount rate. Snce R&D expendtures must be non-negatve, the rate of growth s zero whenever ths condton s not satsfed Comparatve Statcs In the rest of ths secton, I dscuss how opportunty, approprablty and demand condtons, and the exogenous number of frms shape the BGP of the economy. Populaton Sze and the Scale Effect The sze of the populaton, L, affects postvely the rate of growth, g. R&D nvestment s a sunk cost producng a non-rval good. The value of knowledge depends on the scale of producton to whch t s appled. Ths ntuton s captured by the gross-proft effect (see secton 3). The larger demand, the hgher the rate of return to R&D, and the hgher the rate of growth of the economy. 17 Ths effect s related to the noton of "demand pull" (see, e.g., Schmookler 1966 and Dos 1988) and to the scale effect present n most endogenous growth models. Wth an exogenous number of frms, therefore, the role of populaton sze s the same as n standard models of endogenous nnovaton. R&D Productvty The effect of R&D productvty, θ, on the rate of growth, g, s postve because the margnal product of R&D s hgher the hgher s θ. Ths effect s related to the noton of "technology push" (see, e.g., Rosenberg 1976 and 1982, and Dos 1988). The Intertemporal Dscount Rate and Elastcty of Substtuton The ntertemporal dscount rate, ρ, and the nverse of the ntertemporal elastcty of substtuton, σ, have the standard negatve effects on g. The more mpatent socety or the lower the propenson to smooth consumpton over tme, the lower the savng rate and the lower the rate of growth. The Elastcty of Product Substtuton 16 The condton ( 1 σ) g < ρ ensures that the consumer's lfetme utlty s bounded. 17 See Backus, Kehoe, and Kehoe (1992) and Barro and Sala--Martn (1995) for a dscusson of scale effects n endogenous growth models. The scale effect n ths model can also be nterpreted as a relatonshp between growth and the sze of the labor force because labor market clearng yelds that the labor force determnes the scale of actvty n the ntermedate goods sector. 17

19 The effect of the elastcty of product substtuton, ε, has two dstnctve aspects whch are captured by the gross-proft and busness-stealng effects (see secton 3). A hgher elastcty of product substtuton mples that the scope for competton based on productvty mprovement s larger, but t also mples that the mark-up over margnal cost s lower. The balance between these two effects s captured by the term ( ε 1) [ ε( N 1) + 1] n equaton (4.2). In GE there s an addtonal effect. Equaton (4.1) states that there s a constant rato between fnal output, Y, and average manufacturng productvty, A. Two deas are captured by the term N 1 ( ε 1). On one hand, the larger s the varety of ntermedate goods, the hgher s productvty n the fnal output sector, and the larger s total demand for ntermedates. Ths effect ncreases the returns to each ntermedate good. On the other hand, the hgher s the elastcty of product substtuton between ntermedate goods, the less demand for ntermedate goods concentrates one each ntermedate, and the lower are the returns to each ntermedate producer. To sgn the net effect of the elastcty of product substtuton, ε, on steady-state growth, g, I need to sgn the term ( ε 1) N [ ε( N 1) + 1]lnN. Therefore, I have a U-shaped relaton wth a mnmum at ε * ( N + ln N) ( N + lnn Nln N ). For ntermedate values of ε, growth may be zero. The Elastcty of Knowledge Substtuton An ncrease n the elastcty of knowledge substtuton, η, decreases steady-state growth, g. The more substtutable (homogeneous) s knowledge across frms, the lower s the margnal contrbuton to the frm's productvty of the pool of knowledge spllovers (see secton 2). The effcency effect of spllovers falls, and the busness-stealng effect falls (see secton 3). The rate of return to R&D n equaton (4.2) falls, and the rate of growth falls. Approprablty There s a lower bound α 0 on the degree of approprablty below whch steady-state growth, g, s zero. Frms nvest n R&D only f they can approprate a large enough fracton of the benefts from nnovaton. When α =0 and knowledge s a publc good, the ncentves to undertake R&D dsappear, frms do not nvest, and the economy does not grow. In addton, there s a value of the approprablty * parameter, α, that maxmzes steady-state growth. Let ths value be α mn l1,( η 1) N η( N 1) q. 18

20 Ths yelds α = 1 when η N, and α = ( η 1) N η( N 1) < 1 when 1 < η< N. The former case arses when knowledge substtuton s hgh and/or there are few frms. The latter case arses when knowledge substtuton s low and/or there are many frms. To nterpret these results, consder the condton 1 < η< N. The frst element of the nequalty measures the substtutablty of knowledge and the second element measures two related but opposte effects of the number of frms: the varety of knowledge sources and the fragmentaton of the market. Each frm s both a source of technologcal opportunty and a compettor n the product market. As a source of learnng, each frm mproves the effcency of the other frms. As a compettor n the product market, each frm reduces the market shares of the other frms. The condton 1 < η< N compares the substtutablty of knowledge wth the number of sources. Ths suggests, that when knowledge becomes more homogeneous across frms, the benefts from spllovers decrease and rval frms are more compettors n the product market than sources of productvty advance n the producton process. Thus, when the substtutablty of knowledge s lower than the number of frms, the trade-off n the approprablty level occurs. Too hgh approprablty prevents the explotaton of knowledge spllovers more than t nduces hgher R&D nvestment by excludng the rvals from beneftng from one frm's knowledge accumulaton. Conversely, when the elastcty of knowledge substtuton s larger than the number of frms, η N, the degree of approprablty that maxmzes steady-state growth s the one correspondng to knowledge beng a prvate good. The trade-off n the approprablty level now does not occur because spllovers allow too many rvals to beneft from one frm's knowledge accumulaton, whle the benefts that the frm derves from spllovers from the compettors are not sgnfcant snce knowledge s hghly substtutable. 18 These results emphasze the trade-off between the role of approprablty as a prvate ncentve to nvest and the role of spllovers n creatng externaltes (Dos 1988, Levn et al. 1987). It s nterestng that whle for α =0 the argument nsde equaton (4.6) s always negatve, for α =1 t can be ether postve or negatve. In the latter case there exsts an upper bound on approprablty above whch the economy 18 Ths ntuton s further confrmed by the result dscussed above that the hgher s knowledge substtutablty, the lower s steady-state growth. The mportant effect, n fact, s that the more homogenous s knowledge across frms, the less frms learn from each other, and the less spllovers contrbute to hgher productvty. 19

21 exhbts zero growth. The absence of spllovers from other frms makes R&D and the creaton of new knowledge too costly an enterprse for the ndvdual frm. The Number of Frms There exsts a value of the number of frms, N 0, below whch the economy does not grow. Too few frms are n the market and the scope for nnovaton and the advantage from nteracton through technologcal spllovers are too low to nduce the frms to nvest n R&D. When there are too few frms, although average profts are hgh, the busness-stealng effect s so small that the rate of return to R&D becomes very small and drves to zero the ncentves to save and nvest. Frms do not pursue market expanson through R&D but proft by chargng hgh mark-ups on margnal costs. Recall from the prevous dscusson that each frm s both a source of technologcal opportunty and a compettor n the product market. As a source of learnng, each frm mproves the effcency of the other frms. As a compettor n the product market, each frm reduces the market shares of the other frms. Two parameters drve the balance between these effects: the elastcty of knowledge substtuton and the elastcty of product substtuton. The substtutablty of knowledge determnes the effect of the rvals' knowledge on the frm's margnal return to R&D. The elastcty of product substtuton has three man effects. A hgher elastcty of product substtuton mples a lower olgopolstc mark-up and lower profts for a gven market share (gross-proft effect), but t also ncreases the scope for R&D competton because t ncreases the potental of each prce reducton to steal customers from the compettors (busness-stealng effect). Moreover, n GE the hgher the elastcty of product substtuton, the lower the rato between fnal output producton and average manufacturng productvty n the ntermedate sector (specalzaton effect), and the lower the rate of return to R&D. Consder now an ncrease n the number of frms, N. The ndustry pool of knowledge spllovers becomes larger and the rate of return to R&D rses. The scope for R&D competton ncreases because the market share of the compettors ncreases and the potental for stealng busness from the compettors through prce reductons s larger. In addton, an ncrease n the number of frms ncreases the rato between total demand for ntermedate goods and average productvty n the ntermedate sector, and ncreases sales per unt of knowledge and the returns to R&D. On the other hand, an ncrease n the number 20