[621] Working Paper Meet Me Halfway but don t Rush Absorptive capacity and strategic R&D investment revisited Leo A. Grünfeld

Transcription

1 [61] Workng Paper Meet Me Halfway but don t Rush Absorptve capacty and strategc R&D nvestment revsted Leo A. Grünfeld No. 61 December 001 Norsk Utenrkspoltsk Insttutt Norwegan Insttute of Internatonal Affars

2 Utgver: Copyrght: ISSN: Besøksadresse: Addresse: Internett: E-post: Fax: Tel: NUPI Norsk Utenrkspoltsk Insttutt Alle synspunkter står for forfatternes regnng. De må kke tolkes som uttrykk for oppfatnnger som kan tllegges Norsk Utenrkspoltsk Insttutt. Artklene kan kke reproduseres - helt eller delvs - ved trykkng, fotokoperng eller på annen måte uten tllatelse fra forfatterne. Any vews expressed n ths publcaton are those of the authors. They should not be nterpreted as reflectng the vews of the Norwegan Insttute of Internatonal Affars. The text may not be prnted n part or n full wthout the permsson of the authors. Grønlandsleret 5 Postboks 8159 Dep Oslo pub@nup.no [+ 47] [+ 47]

3 Meet Me Halfway but don t Rush Absorptve capacty and strategc R&D nvestment revsted Leo A. Grünfeld [Abstract] In ths paper, we analyse how R&D nvestment decsons are affected by R&D spllovers between frms, takng nto consderaton that more R&D nvestment mproves the ablty to learn from competng frms - the so-called absorptve capacty effect of R&D. The model n ths paper s an extenson of d Aspremont and Jacquemn (1988), where they show that exogenous R&D spllovers reduce the ncentve to nvest n R&D when frms compete n a Cournot duopoly. Our model treats R&D spllovers as endogenous, beng a functon of absorptve capacty effects. Contrary to earler studes, we show that absorptve capacty effects do not necessarly drve up the ncentve to nvest n R&D. Ths only happens when the market sze s small or the absorptve capacty effect s weak. Otherwse frms wll actually chose to cut down on R&D. Furthermore, absorptve capacty effects also ncrease the crtcal rate of spllovers that determnes whether partcpatng n research ont ventures leads to lower or hgher R&D nvestment. Fnally, we show that strong learnng effects of own R&D are not necessarly good for welfare. Moreover, f the market sze s large, welfare wll be at ts hghest when the learnng effect s small. JEL classfcaton code: L13, O31, O3, O38 Key words: R&D nvestment, technology spllovers, absorptve capacty, RJV Correspondng address: NUPI, POBox 8159 Dep, 0033 Oslo, Norway. Tel: Fax: E-mal: leoa.grunfeld@nup.no URL:

4 1. Introducton The ttle of ths paper s a modfcaton of the ttle used by Kamen and Zang (000) where t s emphassed that n order to be able to take advantage of the R&D and nnovatons produced by other frms, you wll have to nvest n R&D yourself (the absorptve capacty effect). Here, we clam that although ths s true, t s only half the story. As ponted out by Cohen and Levnthal (1989), the fact that R&D nvestment both ncreases a frm s nnovatve abltes as well as ts ablty to learn from others, has two effects on R&D nvestments n strategc games. On the one hand, t ncreases the ncentve to nvest n own R&D. But on the other hand, t gves your compettors a ds-ncentve to nvest n R&D, mplyng that there s less to learn from. As opposed to the conclusons made by Cohen and Levnthal, we clam that the stronger the absorptve capacty effect of R&D s, the smaller s the ncentve to nvest n R&D. It s only when the absorptve capacty s weak or the market sze s small that you wll actually observe hgher equlbrum R&D nvestment than n the case wth no absorptve capacty effects. Ths s why frms are told to meet me half way but don t rush. A growng amount of emprcal evdence n the economc lterature ndcates that frms whch devote a large amount of resources to R&D, ncrease ther ablty to approprate the knowledge and technology possessed by other frms. 1 The story behnd ths mechansm s rather smple. In order to understand and mplement deas and concepts of others, you need to posses the competences that enable you to decodfy and utlse these deas. Also, n order to undertake effcent survellance of external knowledge and technology development, a rgorous understandng of the feld of actvtes s necessary. In other words, n order to know what knowledge you are lookng for, you must hold a certan amount of knowledge your self, see Levn et al. (1987) for more on ths subect. The dea that knowledge spllovers are a functon of the frm s technology and knowledge ntensty was already conceptualsed n the early seventes (see Tlton (1971), but the dea was not dscussed rgorously before Cohen and Levnthal (1989) presented a ont theoretcal and emprcal nvestgaton of the subect. Ther econometrc analyss of US frms gves support to the concept of own R&D

5 dependent approprablty, a phenomenon whch the authors called the two faces of R&D. The theoretcal lnk between technology spllovers and the economc behavour of frms has prmarly been analysed wthn the school of ndustral organsaton. Studes that ask whether such spllovers tend to affect the ncentve to nvest n costreducng or demand-ncreasng technology represent the man body of ths lterature, and game theory provdes a framework n whch you can nvestgate the strategc response of frms under alternatve R&D spllover mechansms. After the semnal paper by Brander and Spencer (1983), economsts workng wth IO models have been able to study nvestment games where decsons are taken n two steps. Ths allows us to model behavour where frms frst decde upon the optmal nvestment level, and thereafter compete on the product markets. The vast maorty of studes that dscuss the effect of technology spllovers n strategc two stage games are based on the model developed by d Aspremont and Jacquemn (1988) where spllovers are treated as a lnear functon of the opponents R&D actvtes. 3 Hence, they do not take nto account the dea that spllovers are dependng on the R&D actvty of the knowledge absorbng frm. When R&D spllovers are modelled ths way, d Aspremont and Jacquemn (from here on abbrevated to DJ) mplctly assume that frms learn from external R&D wthout puttng any efforts nto the learnng process. Such a feature resembles weaknesses snce external R&D comes to the frms as some knd of manna from heaven (an expresson ntroduced n ths settng by Kamen and Zang (000)). Symmetrc models wth lnearly dependent spllovers provde the well known predcton that an ncrease n R&D spllovers dscourages R&D nvestment n the frst stage of the game. Suzumura (199) and Smpson and Vonortas (1994) provde comparatve statcs results based on general cost and demand functons, whch mplctly nclude the case where the frm s own R&D actvtes affect the frm s absorptve capacty. However, none of these studes undertake an explct analyss of equlbrum R&D nvestment where the case wth exogenous R&D spllover rates are compared wth the case where R&D nvestment mproves the absorptve capacty. Cohen and Levnthal (1989) present a formal model that takes account of ths effect, 1 See e.g. Cohen and Levnthal (1989), Kogut and Chang (1991) and Neven and Sots (1996). At the macro level, the mperfect approprablty of knowledge forms one of the man ngredents n the study of endogenous growth processes. See for nstance Romer (1986) for more on ths. 3 d Aspremont and Jacquemn (1988), Vonortas (1994), De Bondt, Sleuwagen and Veugelers (1988) 3

6 however, the authors are not able to provde analytcal results based on the full strategc effects, and hence confne the analyss to the frst order effects n a stage duopoly game 4. As mentoned above, they conclude that the ntroducton of absorptve capacty reduces and possbly removes the dsncentve effect of spllovers. Josh and Vonortas (1996) present an elastcty charactersaton of how alternatve knowledge dffuson or R&D spllover mechansms affect the decson to nvest n R&D. Among the studed mechansms, the case wth own R&D effects on absorptve capacty s ncluded. Yet, once agan, we are not presented wth analytcal results that allow us to compare the equlbrum R&D outcomes of games wth and wthout ths knd of effect. More recently, Kamen and Zang (000) ntroduced a 3 stage game where frms frst decde upon the R&D approach, mplyng that frms are able to control the degree of knowledge dffuson stemmng from ts own R&D actvtes. Thus, the exogenous spllover parameter ntroduced by DJ s endogensed. Ths approach may be relevant n cases where e.g. frms may decde upon degrees of patent protecton or the degree of frm specfc nnovatons. Based on a symmetrc noncooperatve Cournot duopoly model, the authors show that frms may fnd t optmal to chose a R&D approach that lmts the dffuson of knowledge to other frms. Hammerschmdt (1998) presents a model where frms undertake two knds of nvestment, one that ncreases the ablty to approprate external knowledge and one that reduces the margnal cost of producton. She shows that although an ncrease n spllovers reduces the optmal nvestment n cost-reducng R&D, such an ncrease may also result n hgher nvestment n the R&D component that s desgned to mprove the absorptve capacty. Thus the effect of hgher spllovers on total R&D nvestment n the frm may actually be postve. Snce both of these last two studes ntroduce new features to the R&D nvestment process, ther predctons can not be drectly compared to the lnear model developed by DJ. In ths paper, we ntroduce a new spllover mechansms n the symmetrc Brander- Spencer model that allows for the presence of absorptve capacty effects n R&D. The mechansm s a generalsaton of the DJ model, allowng both exogenously gven R&D spllover rates s as well as spllovers that depend on the own R&D nvestment of frms. The R&D spllover functon enables us to drectly compare the results 4 Cohen and Levnthal (1989) drect the reader to a techncal note whch shows that the outlned conclusons are also relevant when the full model s analysed usng numercal smulatons. 4

7 stemmng form absorptve capacty effects wth the conclusons provded by the DJ model as well as the Brander and Spencer (1983) model where spllovers are gnored. The model shows that contrary to earler studes, absorptve capacty effects do not necessarly drve up the ncentve to nvest n R&D. Ths only happens when the market sze s small or the absorptve capacty effect s weak. Otherwse frms wll actually chose to cut down on R&D. Furthermore, when absorptve capacty effects are ncluded, we show that the crtcal value on the spllover rate that determnes whether onng a research ont venture (RJV) wll provde hgher R&D nvestment than the non cooperatve equlbrum, s hgher than n the DJ model. Ths mples that the lkelhood of observng hgher R&D nvestment n a RJV than n the non-cooperatve game falls when we allow for absorptve capacty effects. Fnally, we show that strong learnng effects of own R&D s not necessarly good for welfare. Moreover, f the market s large, welfare wll be at ts hghest when the learnng effect s small. However, we fnd that welfare wll always be hgher n a model wth absorptve capacty effects than n a model wth no spllovers at all. In secton we present the spllover mechansm that allows for absorptve capacty effects and provde equlbrum R&D nvestment solutons to the non-cooperatve symmetrc game. In secton 3 we dscuss the mpact of absorptve capacty on R&D nvestment n RJVs. In secton 4, we analyse the welfare mplcatons of absorptve capacty effects n ths knd of models. Secton 5 concludes and gves some prospects for further research. 5

8 . Absorptve capacty effects n the non-cooperatve game The pont of departure n ths model s the two stage Cournot duopoly model frst descrbed by Brander and Spencer (1983) and used as the model framework n DJ. Here, frms choose ther nvestment levels x (=1,) n the frst stage. The game structure sets no lmt to what knd of nvestment the frm undertakes n the frst stage as long as the nvestment s ether cost-reducng or demand-ncreasng. Most often, however, nvestment s nterpreted as R&D actvtes. At the second stage frms play a regular Cournot game n outputs q. In our set-up, we specfcally nvestgate process enhancng nvestments, reducng costs as opposed to ncreasng demand. Hence, R&D nvestment only enters the unt cost functon and not the demand functon. The equlbrum output and nvestment level n a mult-stage game s calculated by usng backwards nducton, dentfyng the subgame prefect equlbrum R&D nvestment levels. Usng general cost and demand functons, both frms maxmse profts: [ ] π = p q q c x x q u x (, ) (, ) ( ) = 1, (1) Here, the prce p s a decreasng functon of quantty produced, unt cost c s a decreasng functon of your own as well as the opponent s R&D nvestments and u s the R&D nvestment cost functon. Maxmsng (1) wth respect to output n the second stage yelds the followng expresson: π q p = q + p c q = 0 () Based on the optmum output levels (q *, q *) derved n ths stage from (), frm wants to maxmse profts wth respect to x n the frst stage, whch results n the followng frst order condton: max x * π p q c * u π = = q = 0 * (3) q snce π = 0 n optmum from (). The frst element wthn the parenthess q represents the strategc effect of R&D decsons workng through the opponent s output reacton. The rest of the expresson represent the drect effect of own R&D 6

9 nvestment. In order to study the effect of alternatve spllover mechansms, t s necessary to specfy the unt cost functon. To keep the analyss smple, assume that the unt cost for frm s composed by a gven margnal cost component b and the technology element g. ( x, x, θ ) = b x θ ( x x c = b g ) (4) The functon g represents the effectve R&D nvestment n the frm, and s a composte of the frm s own R&D nvestment and the opponent s R&D nvestment 5. The varable θ descrbes the proporton of R&D results that spll over from frm to frm. As outlned n the ntroducton, ths varable has tradtonally been treated as a lnear exogenous parameter (θ = γ), where γ vares between 0 and 1 as n DJ. In that case, a value of 0 mples that no R&D results leak to the competng frm, whle a value of 1 mples that all developed knowledge wthn the frm s shared wth the compettor,.e. full R&D spllovers. In the model presented here, we ntroduce a mechansm where the R&D spllover rates are treated as a functon of the frms absorptve capacty, measured n terms of ther R&D nvestments. Frst, we wsh to satsfy the condton statng that 0 θ 1. If θ > 1, a frm has a stronger economc gan from ts compettors R&D than ts own R&D nvestments. Ths specfcaton could be relevant f frms nvest n complementary R&D actvtes, such that external and somewhat dfferent R&D output add strongly to the effects of own R&D actvtes. 6 However, such a specfcaton ntroduces a new dmenson to the model whch makes t mpossble to undertake drect comparsons wth the DJ model. If θ < 0, we have a case where the compettors R&D nvestment not only affect the frm s profts negatvely through the output market as the compettor s costs are reduced, but t also has a drect negatve effect on the frm s cost functon. Ths could be relevant f frms e.g. nvolve n some knd of patent race where the lkelhood of loosng the race s an ncreasng functon of the compettor s R&D nvestments. However, such a specfcaton removes all 5 Accordng to Amr (000), ths cost functon s assocated wth a weakness snce t may be proftable for one frm to gve a R&D dollar to the compettor nstead of nvestng tself. However, snce the model s symmetrc, the Amr crtque wll not apply. 6 Katsoulacos and Ulph (1998) develop a model where strategc R&D nvestment s allowed to dsplay degrees of complementarty. 7

10 learnng effects from the model, and ntroduces a negatve externalty as opposed to the postve externalty we usually relate to the term R&D spllovers. Second, we are searchng for a functonal form that allows the margnal absorptve capacty effect to be decreasng n the frm s own R&D nvestments. In other words, the margnal ncrease n the ablty to learn from the R&D undertaken by the compettor shall be larger when you nvest one more dollar at a low R&D level as compared to one more dollar nvested at a hgh R&D level 7. A rather smple functonal form that satsfes these two requrements s gven by: γ + ax θ ( γ, x, a) = 0 γ 1, a 0, x > 0 (5) 1+ ax where γ s the exogenous spllover rate used n the DJ model. The parameter a s a scalng parameter that regulates the sze of θ. If a=0, we are back to the tradtonal exogenous spllover mechansm used n DJ where the frm s own R&D does not affect the ablty to learn from the compettor. In other words, R&D spllovers enter the frms cost functon as manna from heaven. If both a and γ are set equal to zero, there are no spllovers at all, hence, we are back to the Brander and Spencer (1983) model. The hgher a s, the easer wll the frm learn from external R&D through own R&D nvestment. Thus, the parameter says somethng about the effcency of own R&D n promotng absorptve capacty. That s, a s a learnng parameter that tells us how much the frm s R&D helps learnng from the R&D undertaken by the compettor 8. Frst, observe that the absorptve capacty functon (5) has the followng lmt propertes: lm θ ( γ, x, a) = 1 and lm θ ( γ, x, a) = γ x x 0 7 A possbly more realstc learnng functon s based on the logstc learnng curve, see Kashenas and Stoneman (1995). However, such a specfcaton would vastly complcate the dervaton of strategc responses n the game. 8 Cohen and Levnthal (1989) apply a related procedure where ther parameter β descrbes the characterstcs of outsde knowledge that makes R&D more or less crtcal to absorptve capacty. The dfference however, les n modellng of absorptve capacty on the one hand and spllovers on the other. In our model, we treat these two effects as ntegral parts of the effectve R&D, whereas Cohen and Levnthal explctly separate them. 8

11 Hence, the absorptve capacty functon satsfes the outlned restrcton on spllovers. Furthermore, f the exogenous spllover parameter γ s set to zero, the specfcaton allows no manna from heaven,.e. a frm that does not nvest n R&D has no ablty to learn from external R&D 9. Notce also that f γ =1, there s no openng for further ncreases n the spllover rate through own R&D nvestment, and the functon θ(x ) takes the value 1 for any sze of R&D nvestment (x ). In Fgure 1, we llustrate how the absorptve capacty functon vares n a and x when there s no manna from heaven. Fgure 1 shows that hgher R&D nvestments mply that the frm gans a stronger absorptve capacty, and the absorptve capacty s an ncreasng functon of the scalng parameter a. If we allow manna from heaven n the absorptve capacty functon, the functon wll smply start at γ nstead of 0 when x =0. Fgure 1: The absorptve capacty functon wth no manna from heaven Spllover rate (theta) a= a=1 a=0.5 a= R&D nvestment 9 Ths aspect s also dscussed by Kamen and Zang (000). 9

12 Usng (5), the effectve R&D nvestment functon outlned n (4) now takes the followng form: g γ + ax ( γ, x, x, a) = x + θ x = x + x (6) 1+ ax where we have abbrevated θ ( λ, x, a) θ for expostonal smplcty. Throughout the analyss, we assume that the products are homogenous and that the frms face the followng lnear demand functon p=α-q - q, (7) mplyng that frms confront the same market prce. Fnally, to ensure equlbrum, t s assumed that the nvestment cost functon n (1) s quadratc, u ( ) 1 x = x, whch guarantees decreasng returns to R&D. Substtuton for the general expresson n () usng the cost functon n (4), gves the followng Cournot-Nash equlbrum output levels: ( m + ( ) x + ( ) x ) * 1 q = θ θ 1 (8) 3 where m α b + b > 0 n order to ensure postve outputs. In lne wth standard = Cournot duopoly analyss, f frm has a cost advantage (b <b ), the frm wll capture a hgher market share than the compettor, ceters parbus. In the followng dscusson of the model, the varable m plays a pvotal role. In the prevous lterature, the varable has usually been named the demand cost margn as t says somethng about how large varable costs are relatve to the market sze. A larger m can ether be nterpreted as a larger market sze or lower margnal costs of producton, yet n the followng analyss we wll focus on market sze. The more R&D that splls over from frm to frm descrbed by θ, the hgher wll frm s output and proft be relatve to the compettors output and proft 10. Also, the larger the leakage of frm s own R&D results, descrbed by θ, the lower wll ts output and proft be. We now use (8) and the quadratc nvestment cost functon to 10 Remember that proft n a Cournot duopoly of ths knd s gven by π = q u ( x ) 10

13 derve the frst order condton for optmal R&D nvestment n the frst stage of the game from (3), whch yelds: π = 9 θ ( m + ( θ ) x + (θ 1) x ) θ + x x = 0 (9) If we further assume that frms are symmetrc, we can drop all frm specfc subscrpts and provde the followng frst order condton n the symmetrc game: π θ = 4(1 + θ ) x θ + 4m + ( θ)(1+ θ ) 9 x + m( θ ) = 0 (10) Furthermore, nsertng our endogenous R&D spllover functon (5) nto (10) gves the followng frst order condton for a frm n the symmetrc game wth absorptve capacty effects: π = 9 γ + ax 1+ ax a(1 γ) (1 + ax) ( m(1 + ax) + x(1 + γ + ax) ) + x x(1 + ax) 0 = (11) Notce that n the case wth no absorptve capacty effects (a=0) as n the DJ model, we have that θ = γ and θ = 0, and as descrbed by DJ, (11) has an explct and unque soluton for symmetrc non-cooperatve equlbrum R&D nvestment (x*) n both frms: m( γ ) x * = (1) 9 ( γ)(1+ γ ) Notce that a larger market sze (m) gves a stronger ncentve to nvest n R&D, ndependently of the sze of the spllover parameter. 11 Ths mples that the gans from nvestng n R&D s growng n m snce the cost reducng effect of R&D affects a larger volume of sales n the second stage of the game, drvng profts to a hgher level. To map the effect of R&D spllovers on equlbrum R&D nvestment n the DJ model, we take the dervatve of (1) wth respect to γ, whch yelds: * = γ m(1 + 8γ γ ( 9 ( γ )(1 + γ )) ) 0 snce 0 γ 1 (13) 11 If there are no spllovers at all (γ=0), equlbrum R&D nvestment becomes x*=4m/5. 11

14 Thus, a hgher spllover rate n a symmetrc game wthout absorptve capacty effects leads to lower equlbrum R&D nvestment. Ths s due to well known ds-ncentve effect of spllovers n symmetrc games hghlghted by DJ. Frms are less wllng to nvest n R&D snce the postve effect on profts through a cost reducton s outweghed by the negatve strategc effect drven by the cost reducng effect of spllovers on the compettor s costs. Now, let us return to our case wth absorptve capacty effects. Rearrangng (11) gves the followng 4. order polynomal n R&D nvestment, under the assumpton that there s no manna from heaven (γ=0): π = m * * 3 *3 * * *3 3 *4 ( ax + 1a x + a x ) 5x 9ax 5a x 5a x = 0 (14) The case wth manna from heaven s dscussed later n ths secton. Notce that the equlbrum R&D nvestment based on the absorptve capacty functon (5) becomes solely an mplct functon of the market sze (m) and the learnng parameter (a). Thus we can study the equlbrum R&D level as a functon of these varables only. There exst no explct soluton for the equlbrum R&D nvestment n ths game as outlned by (14), yet t s possble to analyse the behavour for all combnatons of a and m usng numercal smulatons 1. The second order condton and Tatônnement requrement for local stablty are satsfed for all parameter combnaton. A dscusson of these condtons s presented n Appendx 1. In fgure, we smulate the equlbrum R&D nvestment n 4 dfferent games, all wthout manna from heaven,.e. γ=0. The frst game s llustrated by the thck full lne, whch descrbes R&D nvestment n the game allowng for absorptve capacty effects wth the learnng parameter a=1. The second game s llustrated by the thn and lnear lne that descrbes R&D nvestment n a game where we have no absorptve capacty effects (a=0). Thus, ths s equvalent to the Brander and Spencer game wth no spllovers at all, where frms over-nvest n R&D. Here, the equlbrum R&D nvestment s smply gven by x * = 4m 5. In the thrd game (the thn dotted lne), the DJ model s smulated wth a spllover rate γ=0.5. The reason why we present ths 1 The expresson n (14) has four soluton: Two of them have complex roots, one s always negatve and one s always postve. Hence, we focus on the real and non-negatve soluton snce negatve R&D nvestments gve no clear meanng n the game. 1

15 game s that the DJ model predcts that frms wll nether under-nvest nor over-nvest n R&D at ths spllover rate. In the fourth game (the thck dotted lne), we translate the spllovers generated by the absorptve capacty effects (a=1) n the frst game nto exogenous spllover rates. In mathematcal terms we set: γ = θ * * ax = 1+ ax * In other words, we study how equlbrum R&D nvestment n the DJ model compares to our model, usng the same spllover rate. Ths spllover rate θ* vares wth the sze of m and s represented by the marked lne that converges to 1 as m grows (see the rght vertcal axs). Thus, the generated spllover rate apples to both the thck full lne and the thck dotted lne. Fgure : Equlbrum R&D nvestment n 3 dfferent games Equlbrum R&D nvestment spllover rate (theta) m' m R&D wth absorptve capacty R&D wthout absorptve capacty but the same spllover rate R&D wth exogenous spllover rate (gamma =0.5) R&D wth no spllovers Spllover rate (theta)- rght axs 0 13

16 The smulatons n Fgure suggest that R&D nvestment n the absorptve capacty game (the thck full lne) wll always be hgher than n a DJ game where the exogenous spllover rate s set equal to the spllovers generated n the absorptve capacty game, (.e. γ=θ*) (the thck dotted lne). It s actually possble to prove that ths s true for any absorptve capacty mechansm where own R&D mproves the ablty to learn form others. The proof s based on closer nvestgaton of the frst order condton (10) where the absorptve capacty mechansm s expressed n general terms. Proposton 1: If the exogenous spllover rate (g) n the symmetrc DJ game, 0 γ 1, s the same as the spllover rate (θ*) generated by the symmetrc game wth absorptve capacty effects but no manna from heaven, equlbrum R&D nvestment n the DJ game wll always be lower than n the absorptve capacty game. Proof of Proposton 1: Let x represent equlbrum R&D nvestment when we have absorptve capacty effects, and xˆ be the equlbrum R&D nvestment when no such effects are present. Furthermore, defne θ θ A = 4 (1 + θ ) > 0, B = 4m > 0, C = 9 ( θ )(1 + θ ) > 0, D = m( θ ) > 0 n the frst order condton n (10). Then, we know from (10) that A x + ( C B) x = D and C xˆ = D snce γ = θ and θ = 0 n the case wthout absorptve capacty effects. Thus, Ax + ( C B) x = D = Cxˆ. If equlbrum R&D n the game wth absorptve capacty effects s to be smaller than n the DJ case, we must have that ˆ C x Cx levels. x < xˆ. For ths to be the case, the followng devaton must be satsfed: = Ax Bx > 0, but ths s not possble for non-negatve R&D nvestment QED. Proposton 1 tells us that spllovers work dfferently n the absorptve capacty model as compared to the DJ model where spllovers are exogenous. Although we compare two games based on the exactly same spllover rate, we stll get hgher R&D n the absorptve capacty game. Ths extra equlbrum R&D nvestment stems from what we name the postve learnng effect. It smply states that f we separate out the 14

17 negatve tradtonal effect of spllovers on R&D nvestment n the model wth absorptve capacty, we are left wth a pure learnng effect of own R&D that drves up the ncentve to nvest n R&D. In Fgure, we see that the postve learnng effect s growng n m but that the margnal contrbuton of m s decreasng. It s mportant to notce that when we now go on wth comparng the equlbrum R&D nvestment level n the absorptve capacty model wth the DJ model, t s the nterplay between these two effects that drves the conclusons. Table 1: Equlbrum R&D nvestment under alternatve combnatons of a and m m a=0 a=0 a=0.1 a=0.5 a=1 a= a=10 a=1000 a=1bn g=0 g=0.5 g=0 g=0 g=0 g=0 g=0 g=0 g=0 No spllovers Manna from heaven E E-05 4E For values of m larger than m n Fgure, R&D nvestment s lower n the game wth absorptve capacty effects than the game wthout spllovers. At even hgher values of m, R&D nvestment n the absorptve capacty game actually also undercuts the nvestment level generated by the DJ model wth a spllover rate γ=0.5. In Table 1, we calculate the equlbrum R&D nvestment as a functon of m and a, spannng out the range of a from 0.1 to 1 bllon n order to ensure the reader that the patterns depcted n Fgure are representatve for all possble values of the learnng parameter a. The shaded area n Table 1 represents all those combnatons of m and a where R&D 15

18 nvestment s lower n the absorptve capacty game than n the game wth no spllovers. The lghtly shaded areas represents the cases where R&D nvestment s also lower than n the DJ game wth an exogenous spllover rate set to γ=0.5. The reported fgures clearly show that as the learnng parameter (a) ncreases, the crtcal sze of m=m unambguously falls. Ths pattern s vald for all values of the learnng parameter a and provdes us wth the followng results: Result 1: For a suffcently large market sze (m>m ), equlbrum R&D nvestment s lower n the absorptve capacty game (no manna from heaven) than n the game wth no spllovers. For an even larger market sze, R&D nvestment n the absorptve capacty game wll also fall below the R&D nvestment generated n the DJ game wth exogenous spllovers. Result : If the learnng parameter a s ncreased, the crtcal sze of the market (m ) falls. Ths mples that the range of the market sze for whch R&D nvestment s lower n the absorptve capacty game than the game wthout spllovers, s wdened. On the other hand, t s mportant to notce that wth a suffcently small m (m<m ), the absorptve capacty game actually generates hgher R&D nvestment than a game wth no spllovers. Thus, n a case where the market sze s small or margnal costs are hgh, our model predcts that spllovers wll gve an extra ncentve to nvest n R&D. Ths result also contrasts the earler theoretcal lterature on spllovers. The mechansm drvng result 1 s drectly lnked to the fndngs n proposton 1 where we separate the postve learnng effect from the negatve tradtonal spllover effect on R&D nvestment n the absorptve capacty model. From (13), we know that for any exogenously gven spllover rate γ n the DJ model, there s a postve and lnear relatonshp between the sze of the market (m) and the strength of the negatve tradtonal effect of spllovers on R&D nvestment. Ths effect s best llustrated by the ncreasng gap between equlbrum R&D nvestment n the game wth no spllovers and the DJ game wth γ=0.5 as m grows n Fgure. Although the postve learnng effect also grows wth the sze of m, the growth rate s decreasng. Ths s due to the way we model the absorptve capacty mechansm n (5). A larger market drves up equlbrum R&D nvestment, but as equlbrum R&D nvestment ncreases due to a hgher m, the margnal capacty to absorb external R&D falls. Therefor, the ncrease 16

19 n the postve learnng effect of own R&D s fallng wth the sze of m. Consequently, at a suffcently large market sze (m>m ), the negatve tradtonal effect of spllovers outweghs the postve learnng effect, drvng equlbrum R&D nvestment n the game wth absorptve capacty effects below R&D nvestment n the game wth no spllovers. Alternatvely, snce the endogenously determned spllover rate θ (the marked lne n Fgure ) grows towards 1 as m ncreases, the equlbrum R&D nvestment n the absorptve capacty game wll eventually be lower than the R&D nvestment n the DJ game where γ<1. The ntuton behnd result can also be related to the spllover mechansm n (5). We know from proposton 1 that there exsts a postve learnng effect on R&D nvestment n the game wth absorptve capacty effects. When the learnng parameter a s ncreased, the postve learnng effect s also strengthened. However, a hgher value of a also drves up the negatve tradtonal spllover effect snce the spllover rate θ grows. If we now ncrease the sze of the market, the relatve mportance of the negatve tradtonal spllover effect s enlarged, drvng down the equlbrum R&D nvestment level faster. Consequently, we wll observe that the crtcal level m s reduced as the learnng parameter a s ncreased. Fgure 3: Equlbrum R&D nvestment for varyng learnng parameter values Equlbrum R&D nvestment a=0.1 a=0.5 a=1 a= a= m 17

20 The mpact of changng the learnng parameter a on equlbrum R&D nvestment s llustrated n Fgure 3, where we descrbe the same exercse as n Table 1, but for expostonal purposes only report for a selecton of values of a. The thck marked lne n Fgure 3 s once agan the Brander and Spencer game where a=0 (no spllovers). The case where a s large s llustrated by the thck unmarked lne (a=). When we compare ths case wth the cases based on lower parameter values, we once agan observe the nterplay between the two effects. The postve learnng effect domnates when (m) s small, mplyng that a large a generates the hghest equlbrum R&D nvestment level. But n larger markets, the negatve tradtonal spllover effect s magnfed by a, drvng down R&D nvestment. The thn dotted lne represents the case wth a very small learnng parameter a=0.1. Here, for small values on m, the equlbrum R&D level wll be only margnally hgher than n the case wthout spllovers (a=0) but sgnfcantly smaller than n the case wth large learnng effects, bascally due to the smaller postve learnng effect. However, f we ncrease m n ths case, the negatve mpact on equlbrum R&D nvestment wll be moderated snce the spllover rate θ grows slower n own R&D nvestment when a s small. Consequently, for large values on m, the equlbrum R&D nvestment level s hgher the smaller the learnng parameter s 13. Allowng both endogenous and exogenous spllovers We now turn to the case where there exst both R&D spllovers that depend on the absorptve capacty of the frms and exogenous spllovers,.e. γ>0. In other words, there s manna from heaven n the model. Why should one be concerned wth such a case. One may clam that a proporton of the R&D results or knowledge that s generated wthn an ndustry s wdely understood by the general publc, thus, rval frms do not need to nvest further n absorptve capacty n order to take advantage of ths knowledge. For nstance, f the R&D results are avalable to the publc through the school or unversty system, the cost of acqurng ths knowledge s low. 13 In Fgure 3, t looks lke the lne representng a=0.1 always stays above the lne representng the case wthout absorptve capacty effects, but ths s not correct (see Table 1). If we extend the graphs along the horzontal axs, the thn dotted lne wll eventually fall below the thck marked lne. 18

21 When we also allow exogenous R&D spllovers n the model, the frst order condton for optmal R&D nvestment becomes slghtly more complex: m * * *3 * * ( x + 1x + x 8γx 6γx γ ) 5x * 9x * 5x *3 5x *4 + γx * 10γx *3 6γ x * θ x * = 0 (15) In (15) we have set a=1, and the effect of changng γ s llustrated n Fgure 4. Clearly, a hgher γ contrbutes to lower R&D nvestment, ust as descrbed n DJ. Ths llustrates that the ntroducton of exogenous R&D spllovers only works through the tradtonal negatve spllover effect on R&D nvestment as n DJ, although the γ parameter actually enters the absorptve capacty functon n (5). Hence, when the games nclude manna from heaven, the crtcal value of m=m s reduced snce the negatve spllover effect on R&D nvestment out-competes the postve learnng effect of own R&D at a smaller market sze. Ths leads us to the followng remark: Result 3: In a game wth both absorptve capacty effects and spllovers ndependent of own R&D nvestment ( manna from heaven ), γ>0, the crtcal value m falls wth a hgher γ. Fgure 4: Equlbrum R&D nvestment n the game wth absorptve capacty (a=1) and varyng degrees of manna from heaven 1 Equlbrum R&D nvestment gamma=0.5 gamma=0.5 gamma= m 19

22 3. Optmal R&D nvestment n research ont ventures. A well-known property of the Brander Spencer model s the so-called overnvestment effect whenever there are no spllovers present n the ndustry. Snce frms have to pre-commt to the R&D nvestment level before the second stage, they are forced nto a prsoner s dlemma stuaton where over-nvestment n R&D becomes the best response to the possblty that the opponent may nvest more and capture some of the frm s proft n the output game. As shown by e.g. d Aspremont and Jacquemn (1988), the over-nvestment effect s not necessarly vald n a game wth R&D spllovers snce spllovers force down the equlbrum R&D nvestment level. When frms on together n a research ont venture (RJV), but compete aganst each other on the output market n the second stage of the game, frms nternalse the external effect of R&D spllovers n the frst stage of the game. Hence, the optmal R&D nvestment level n a RJV s consstent wth cost mnmsaton for any gven output level, see Brander and Spencer (1983). The RJV seeks to maxmze the sum of profts wth respect to R&D nvestment: max ( π +π ) = max Π = q + q u( x ) u( x ) (16) x, x x, x Mnmzng costs wth respect to R&D nvestment for a gven output level gves the followng condton: c q u + = 0 (17) Snce the second order condton for the optmsaton problems n (17) has the c u followng property: + 0, a frm wll be under-nvestng n the non- cooperatve equlbrum f the expresson on the left hand sde of (17) s negatve. Ths s so, snce an ncrease n nvestment wll cut unt costs more than t contrbutes to ncrease nvestment costs. Over-nvestment, on the other hand, s assocated wth a postve value. Usng the frst order condton n (3) and the assumpton of symmetry, the condton (17) specfed n the non-cooperatve case wth no absorptve capacty (a=0) gves the followng expresson: 0

23 c q u π q 1 + = q = q(1 γ ) q 3 (18) where q s non-negatve. Ths provdes us wth the well known result from the DJ model wth exogenous spllovers, statng that frms wll under-nvest n R&D as long as the spllover rate s hgher than γ**=0.5. If t s lower than 0.5, frms wll overnvest n R&D. Next, we analyse the same crtera n the case wth absorptve capacty effects, but wth no manna from heaven (γ=0). c q u π q 1 1+ ax(1 ax) + = q = q q 3 (1 + ax) (19) Snce (1+ax) s always postve, the crtcal spllover value θ** for whether frms over or under-nvest depends on the sgn of (1+ax(1-ax)). Solvng ths expresson wth respect to ax gves the followng condton for when the sgn shfts: 1+ ax = θ **(a,x, γ = 0) = 0,618 (0) Ths gves a clear nterpretaton of the consequence of mplementng absorptve capacty effects n a Cournot duopoly wth R&D spllovers. Proposton : The crtcal rate of spllovers (θ**) where equlbrum R&D nvestment s the same n the RJV game as n the non-cooperatve game, s hgher when we take nto consderaton the absorptve capacty effect of R&D as compared to the case wth exogenous R&D spllovers. Proof: The proposton s based on drect the comparson of (19) and (0). The logc behnd Proposton relates drectly to Proposton 1 and the fndngs n Fgure. Snce frms n the game wth absorptve capacty effects always nvest more than n the game wthout such effects but the same R&D spllover rate (γ=θ*), we know that the nvestment level wth absorptve capacty effects wll be hgher when θ*=γ=0.5. Thus, the ntroducton of absorptve capacty effects ncreases the range of spllover rates where frms over-nvest n R&D. Wth the functonal forms studed here, the queston of whether frms over-nvest or alternatvely under-nvest can be studed by lookng at the dfference between the soluton to the obectve functon n 1

24 (3) and the soluton to the cost mnmzaton problem for a gven output level. Notce that a consequence of Proposton s that frms wll over-nvest n R&D for a wder range of R&D spllovers n the game wth absorptve capacty effects as compared to the game wth exogenous R&D spllovers. 4. The welfare effects of absorptve capacty In order to assess how welfare s affected by the ntroducton of absorptve capacty effects n Cournot duopoles, we need to take nto consderaton both frms proft as well as consumer surplus. Usng the symmetry assumpton and the nverse demand functon n (7), consumer surplus s gven by: 1 S ( q) = ( p(0) p(q*) ) q* = q * (1) Thus, welfare s smply gven by: W = π + S( q) = 4q * x * () In the DJ model where the R&D spllover rate s gven exogenously, we fnd that the spllover rate that provdes the hghest welfare s gven by 14 : W γ q* q* * * = 8q* + x * = 0 γ * γ γ γ 1γ 1γ = 0 (3) Usng Cardanos formula for a cubc equaton leaves us wth γ w = as the only soluton to the optmsaton problem n (3) that satsfes the condton 0 γ 1. From (18) we know that frms wll under-nvest n R&D at ths spllover rate. Hence, gven that frms compete n a Cournot duopoly, the socally optmal R&D nvestment level s below the frms cost mnmsng R&D nvestment level e.g. obtaned through a RJV. 14 The second order condton requres that 148γ+4γ 4-7γ -3γ 3-5>0 whch s satsfed for all values of γ>

25 In Table we report the results from numerc smulatons of welfare outcomes under alternatve market szes (m) and absorptve capacty effects (a). The shaded observatons represent the value on the learnng parameter a=a w (m) that provde the hghest welfare outcome for alternatve market szes. It s mportant to notce that n contrast to the DJ model, there does not exst a unque R&D spllover rate that maxmses welfare n the game wth absorptve capacty effects. The smulatons provde the followng result: Result 4: When we nclude absorptve capacty effects n the Cournot duopoly model, the relatonshp between welfare and absorptve capacty becomes a functon of the market sze. Hghest welfare n a small market s reached when the absorptve capacty effect of R&D (a) s large, whle welfare s hghest n a large market when the absorptve capacty effect of R&D s small. The ntuton behnd result 4 s strongly related to the fndngs n Fgure 3. We know from secton, that when the market sze (m) s small, the postve learnng effect of R&D has a relatvely strong mpact on R&D nvestment as compared to the negatve tradtonal spllover effect. If welfare s mproved through hgher R&D nvestment and output, then welfare wll be hgh f the value of the learnng parameter (a) generates hgh equlbrum R&D nvestment. In Fgure 3, we see that as the market sze grows, the value of the learnng parameter that provdes the hghest R&D nvestment s fallng, explanng the welfare results n Table. Furthermore, accordng to our numerc smulatons, the welfare level wll never be lower n the model wth absorptve capacty effects (a>0) than n the model wthout spllovers (a=0). Ths result mmcs the result based on the DJ model. The logc relates drectly to how R&D spllovers affect equlbrum output. In the DJ model, the hghest output s reached when the R&D spllover rate γ=0.5, and the equlbrum output declnes symmetrcally around ths pont 15. Smlarly, snce the absorptve capacty mechansm generates spllovers n the model, t s only when a= that output gets as low as when a=0. 15 Ths can be found by maxmsng output wth respect to the R&D spllover rate. 3

26 Table :Welfare as a functon of market sze (m) and absorptve capacty (a) E+09 m a=0 a=0 a=0 a=0.1 a=0.5 a=1 a= a=10 a=1000 a=1bn g w = g=0.5 g=0 g=0 g=0 g=0 g=0 g=0 g=0 g=0 Manna from heaven No spllovers E E E-09 8E-09 8E-09 8E-09 8E E E E E E E-07 8E E E E E E E E E E E E E E E E E E E E E E E E E+08 Result 5: For any market sze (m), there always exsts a learnng parameter value a w (m), such that welfare n the model wth absorptve capacty effects s hgher than the welfare obtaned n the DJ model wth the optmal spllover rate γ w. Result 5 s based on the smulatons n Table and hghlghts the mportance of the postve learnng effect of absorptve capacty. By contnuty, there wll always exst a learnng parameter value that generates the spllover rate γ w n equlbrum, but snce the postve learnng effect of own R&D s always present for a>0, ths specfc value wll provde hgher R&D nvestment than n the DJ game based on γ w. 4

27 5. Conclusons and prospects for further research. The man message n ths paper states that results derved from the study of optmal R&D nvestment wth R&D spllovers depend strongly on how we model the R&D spllover mechansm. More specfcally, t has been shown that f we treat the absorptve capacty of frms as a functon of ther own R&D actvty, the queston of whether equlbrum R&D nvestment wll ncrease or decrease as compared to the case wth exogenous R&D spllovers, s predomnantly a queston of market sze. If the market sze s small, the absorptve capacty effect wll drve up R&D nvestment, whle the opposte s true when the market sze s large. We explan ths result trough two opposng effects of absorptve capacty generated through own R&D nvestment. The frst effect works smlar to the tradtonal negatve spllover effect on R&D outlned n the prevous lterature. It states that ncludng absorptve capacty effect ncreases the spllover rate n a symmetrc R&D game whch unambguously drves down R&D nvestment. The other effect whch we call the learnng effect of own R&D nvestment relates to the postve mpact of absorptve capacty on the frms own cost functon. We show that the same spllover rate n a game wth absorptve capacty effects always provdes hgher R&D nvestment as compared to a game wthout such effects. The model presented n ths study, has the advantage of beng drectly comparable wth the model developed by d Aspremont and Jacquemn (1988) (DJ). Our conclusons mply that the prevously outlned relatonshp between R&D spllovers and R&D nvestment s altered when we allow for absorptve capacty effects. Furthermore, the predctons outlned by Cohen and Levnthal (1989) where absorptve capacty effects unambguously ncreases the ncentve to nvest n R&D, s questoned n ths study. The conclusons from ths paper also add new nsght nto the theory of research ont ventures (RJVs). We show that for any gven spllover rate, frms n the absorptve capacty game wll fnd t optmal to nvest more n R&D, mplyng that the R&D spllover rate that provdes cost mnmsng R&D nvestment levels s hgher n the absorptve capacty model. Broadly speakng, ths means that ceters parbus, more frms wll over-nvest n R&D as compared to what s predcted n the models wth 5

28 exogenous R&D spllover rates. Ths mples that the ntroducton of a RJV wll force up R&D nvestments n fewer cases. Fnally, the model shows that strong learnng effects of own R&D s not necessarly good for welfare. Moreover, f the market s large, welfare wll be at ts hghest when the learnng effect s small. However, we fnd that welfare wll always be hgher n a model wth absorptve capacty effects than n a model wth no spllovers at all. The conclusons derved n ths study are solely based on the assumpton of symmetrc frms. In the real world, frms are equpped wth vastly dfferent technologes and abltes to learn from external knowledge. Thus, future studes should devote resources to the mpact of absorptve capacty effects n asymmetrc games, where the outlned effects may be modfed. However, studyng asymmetrc games of ths knd s a complex analytcal task, yet numercal smulatons may also provde valuable nsghts to the R&D nvestment response of frms. Appendx 1 The second order condton π < 0 usng (14), gves the followng condton for a global maxmum: m (14a + 4a x + 6a x ) 5 18ax 15a x 0a x < 0 (1A) Notce that as opposed to the case wth no absorptve capacty effects (a=0 )where the demand cost margn (m) does not affect the curvature of the proft functon (the frst expresson on the left hand sde falls out), n the case wth such effects, ths varable does play a role. Numercal smulatons based on equaton (14) shows that all combnatons of m and a, satsfy the second order condton locally around the R&D equlbrum. An mportant requrement n the analyss of Cournot games, whch s much to often gnored, s the stablty of the equlbrum. A small devaton from the equlbrum R&D strateges may ether brng the game back to the equlbrum outcome or generate unstable patterns. The commonly used Tatônnement requrement for local stablty of an equlbrum s gven by (see Vves (1999)): 6