R&D nvestment, asymmetrc costs, and research jont ventures Alejandro Montecnos Thomas Gresk July 6, 207 Abstract Ths paper nvestgates how an ntal asymmetry n producton costs affects the welfare dfferences between non-cooperatve R&D nvestment and a research jont venture, through cost-reducng R&D nvestment decsons. Usng a duopoly model n whch R&D nvestment generates spllovers and bulds absorptve capacty, we fnd that f there exsts a crtcal spllover value for whch welfare under the two regmes s the same, then ncreasng the ntal cost asymmetry has an ambguous effect on ths crtcal spllover value. Ths ambguty arses because the relatve sze of the welfare reducton due to an ncrease n the ntal cost asymmetry s determned by the rate at whch absorptve capacty s bult. Increasng ths rate amplfes the welfare losses from an ncrease n the cost asymmetry under non-cooperaton and mtgates the welfare losses under a RJV. JEL Classfcaton Codes: O3, L3, C72 Keywords: R&D nvestment, absorptve capacty, cost asymmetry, research jont venture. Escuela de Negocos, Unversdad Adolfo Ibáñez Department of Economcs, Unversty of Notre Dame We thank Rchard Jensen, Terrence Johnson, Kal Rath, and Thomas Jetschko for ther comments Contact: amontecnos@ua.clmontecnos); tgresk@nd.edugresk)
Introducton When a frm nvests n R&D, the nvestment may cause a postve spllover that benefts ts compettors, and the sze of the spllover can affect whether a research jont venture RJV) or non-cooperatve R&D nvestment NC) yelds hgher welfare. The early lterature on RJVs shows that ncreasng the spllover ncreases welfare under an RJV. Startng wth d Aspremont and Jacquemn 988), the theoretcal lterature has compared the benefts of RJVs and NC for dentcal frms n a duopoly that choose cost reducng R&D nvestment n the presence of a spllover. 2 Moreover, n comparng the benefts of RJVs versus NC, ths lterature has also tred to dentfy the crcumstances under whch RJVs generate more aggregate R&D nvestment than under NC. The man goal of ths paper s to determne how an ntal cost asymmetry affects the welfare comparsons of RJVs versus NC n the presence of a spllover. We extend the lterature ntated by d Aspremont and Jacquemn 988) by ntroducng an ntal cost asymmetry and by adoptng the Leahy and Neary 2007) focus on welfare dfferences to compare RJVs to non-cooperatve R&D nvestment. The ntroducton of a cost asymmetry s motvated by the emprcal lterature that documents the presence of cost dfferences among frms wthn an ndustry. 3 Yet the mpact of such ntra-ndustry cost heterogenety on R&D nvestments and ts welfare mplcatons cannot be analyzed wth the current theoretcal models, as they assume frms always have dentcal costs. In addton, wth asymmetrc ntal producton costs, the tradtonal rankngs of RJVs and NC based on aggregate R&D nvestment wll no longer concde wth broader welfare rankngs. In our model, we endow one frm wth a larger margnal cost of producton and analyze the equlbrum effects when the magntude of ths ntal asymmetry s close to zero. Followng See De Bondt 996) for a survey of the early lterature on spllovers n nnovatve actvtes. 2 López-Pueyo et al. 2008) fnds evdence that cost reducng R&D nvestment of one frm causes spllover benefts to other frms n the same ndustry. Examples of such spllovers occur n the food, textles, chemcal products, plastc products, and machnery ndustres. 3 Olley and Pakes 996) fnd evdence of heterogenety n productvty n the telecommuncaton ndustry. Roberts and Tybout 996) and Pavcnk 2002) also fnd evdence of ths same heterogenety n a sample of producton plants n Colomba and Chle, respectvely. 2
Kamen and Zang 2000), we assume that the exogenous spllover parameter consdered by d Aspremont and Jacquemn 988) s the maxmum possble fracton of R&D nvestment that splls over from one frm to the other. The actual spllover s endogenously determned by each frm s nvestment n absorptve capacty, where absorptve capacty captures the dea that a frm learns from a compettor s R&D only f the frm also conducts R&D. 4 Our analyss focuses prmarly on small ntal cost asymmetres. We then dscuss how several extensons to our model affect our resutls. We fnd that when the rate at whch absorptve capacty s bult s hgh, an ncrease of the ntal cost asymmetry reduces the crtcal spllover value. Ths occurs for two reasons. Frst under NC, when the compettor bulds absorptve capacty at a hgh rate, the ncentve for a frm to lmt the spllover benefts of ts own R&D nvestment ncreases, and aggregate equlbrum R&D nvestment s reduced. Second, the benefts that the compettor derves from the postve externalty under a RJV are nternalzed by the frms. Thus, when the rate of absorptve capacty acquston s hgh, an ncrease n the ntal cost asymmetry wll favor RJVs and lead to a reducton n the welfare-based crtcal spllover value. The effect s reversed when the rate at whch absorptve capacty s bult s low. Also, our analyss dentfes the key channels through whch cost asymmetres affect welfare comparsons between a RJV and NC, thereby provdng a useful framework for emprcal work. We show that n the presence of a small cost asymmetry and suffcently large demand, equlbrum welfare under a RJV s ncreasng n the exogenous spllover parameter 4 RJV and NC regmes under whch frms conduct cost reducng R&D nvestment wth spllovers have been studed snce d Apremont and Jacquemn 988). In ther model, d Apremont and Jacquemn assume that dentcal frms choose ther R&D nvestment and then they compete n a fnal good market. A key extenson to ths set up ntroduces endogenous spllovers Kamen, Müller, and Zang 992) and Kamen and Zang 2000)). More recently, the focused changed to demand condtons under whch endogenzng the spllover effect drves up the ncentves to nvest n R&D Grünfeld 2003)), the determnaton of R&D approprablty through the frm s choce of R&D approaches Wethaus 2005)), and dstngushng R&D nvestment orented towards nventve actvtes and R&D nvestment amed to learn from others Hammerschmdt 2009)). Untl Leahy and Neary 2007) ths early lterature studed the normatve mplcatons of dentcal frms R&D nvestment by focusng on aggregate R&D nvestment. However, Leahy and Neary use a model wth symmetrc costs to compare RJVs wth NC n terms of welfare dfferences. 3
whle equlbrum welfare under NC can be non-monotonc. 5 Under a RJV, as the exogenous spllover parameter ncreases, R&D nvestment ncreases, mplyng that equlbrum welfare under a RJV necessarly ncreases because the drect margnal cost reducton effect of the spllover s renforced by more R&D nvestment thereby ncreasng aggregate producton. Under NC, as the exogenous spllover parameter ncreases, R&D nvestment decreases and results n less equlbrum margnal cost reducton whch reduces aggregate producton. However, an ncrease of the exogenous spllover parameter also drectly reduces the margnal cost of producton and hence ncreases aggregate producton, because holdng R&D nvestment fxed, larger spllovers allow each frm to take more advantage of the compettor s R&D nvestment. Thus, the drect margnal cost reducton red effect ncreases welfare whle the decrease n R&D nvestment ncreases the margnal cost of producton thereby decreasng welfare. When the exogenous spllover parameter s small, the drect margnal cost reducton effect domnates and NC welfare s ncreasng n the spllover parameter. However, when the exogenous spllover parameter s large, the R&D nvestment effect domnates and NC welfare s decreasng n the the spllover parameter. The rest of the paper s organzed as follows. In secton 2, the model s ntroduced. Then n secton 3, the man effects of a cost asymmetry on welfare are analyzed. Fnally n secton 4, we dscuss the results and conclude. 2 The Model We buld upon the model poneered by d Aspremont and Jacquemn 988), and used more recently by Leahy and Neary 2007) and Hammerschmdt 2009), n whch two frms compete n a two-stage game. In the frst stage the two frms, j {, 2} and j) smultaneously choose R&D nvestment. We wll consder two scenaros: NC n whch the frms choose ther R&D nvestments non-cooperatvely and RJV n whch the frms choose ther R&D 5 Our analytc results formalze Leahy and Neary s 2007) numercal examples, n whch non-monotonc equlbrum welfare under NC can arse even wth symmetrc producton costs. 4
nvestments to maxmze ndustry profts. Then, n the second stage, the frms engage n Cournot competton n the market for an homogeneous product. The soluton concept we use n both scenaros s subgame perfecton. The nverse demand functon s p = a q q 2, where a > 0 s a demand parameter that determnes ts sze, and q s the quantty produced by frm {, 2}. Each frm has a constant return to scale producton functon, whch renders a total cost of producton C = c q, where c s frm s unt cost of producton. We ntroduce an ntal cost asymmetry by assumng n the absence of any R&D nvestment that c 2 c = ε, where ε 0. Thus ε represents the magntude of the ntal cost advantage that frm has over frm 2. 6 Each frm s able to reduce ts ntal unt cost of producton through the cost reducton functon f z ). Includng the cost-reducng effects of each frm s R&D nvestment we can wrte c = c f z ) and c 2 = c + ε f z 2 ), where z = x + γ x ) θx j s frm s effectve nvestment n R&D. Note that larger values of ε mply hgher average producton costs n the market. Frm can reduce ts margnal producton cost drectly by nvestng x 0 n R&D, and through an endogenously determned spllover effect γ x ) θx j. Ths term has three components. The frst component s frm j s R&D nvestment, x j. The second component s the exogenous spllover parameter, θ [0, ]. It represents the maxmum fracton of frm j s R&D nvestment, x j, that can spll over to frm. The thrd component s frm s absorptve capacty γ x ) : R + [0, ]. It determnes frm s ablty to take advantage of the maxmum spllover, θx j, from frm j s R&D nvestment by ncorporatng t nto ts effectve R&D nvestment, z. The absorptve capacty functon γ x ) s strctly ncreasng, strctly concave, and contnuously dfferentable. It s also assumed that the cost reducton functon f z ) : R + [0, s] s strctly ncreasng, strctly concave, and contnuously dfferentable. The number s s the maxmum possble level of cost reducton and 0 < s < c. Leahy and Neary 2007) consder a specfcaton of the endogenous spllover term that encompasses some specal cases consdered n the lterature, such as Kamen and Zang 2000). 6 We consder alternatve ways to formulate an ntal cost asymmetry n secton 3.5. 5
Our formulaton of the endogenous spllover dffers from that found n Leahy and Neary 2007) n two ways. Frst the Leahy and Neary formulaton uses a parameter that represents the dffculty of absorbng the rval s R&D nvestment. Ths absorbng dffculty parameter n Leahy and Neary 2007) s equvalent to θ. Second, the model presented here allows for an addtonal dmenson of absorptve capacty acquston related to the rate at whch absorptve capacty s bult. Ths new dmenson s ntroduced n secton 3.3 through a parameter λ by adoptng Hammerschmdt s 2009) absorptve capacty functon γ x ) = x / + x ) λ for λ [0, ]. Wth Hammerschmdt s functon lower values of λ correspond to a more concave functon, whch means that ntal R&D nvestments wll allow a frm to acqure absorptve capacty more quckly. Together these model components mply that frm earns a proft of π q, q j, x, x j, θ, ε) = pq +q j )q c x, x j, θ, ε)q x. The soluton of the second stage problem corresponds to a standard Cournot game. We denote frms s subgame equlbrum producton quantty by q x, x j, θ, ε) where q x, x j, θ, ε) = /3 a c + ε + 2f z ) f z 2 )) and q 2 x, x j, θ, ε) = /3 a c 2ε + 2f z 2 ) f z )), and frm s second-stage ndrect proft functon by Π x, x j, θ, ε) = p q x, x j, θ, ε) + q j x j, x, θ, ε)) q x, x j, θ, ε) c x, x j, θ, ε)q x, x j, θ, ε) x. Notce that a larger ntal cost asymmetry ncreases frm s output and decreases frm 2 s output n every subgame defned by x and x 2. The subgame equlbrum quanttes also reveal that an ncrease n one frm s effectve R&D nvestment ncreases ts subgame equlbrum quantty more than t decreases ts compettor s quantty. Wth the lnear demand specfcaton Π x, x j, θ, ε) = q 2 x, x j, θ, ε) x so that changes n θ affect Π only through changes n frm s output. That s Π /=0. The NC case s defned by the reduced form non-cooperatve game G RNC {X, X 2 ; Π, Π 2 }, where for all frm s strategy space s X = {x : x ɛr + }. 6
We assume that a unque, stable, pure strategy equlbrum to G RNC for each θ, ε) exsts, x j, θ, ε) = arg max Π and x 0 x j θ, ε), θ, ε ). Because Π = q 2 x n every subgame defned by x, x 2 ) and we refer to t as x θ, ε), x 2 θ, ε)), where for all j, x BR x θ, ε) = x BR the subgame perfect equlbrum R&D nvestment levels are defned by 2q q / = for =, 2. The RJV case s defned by the reduced form cooperatve problem max Π + Π 2 = p q x, x 2, θ, ε) + q 2 x 2, x, θ, ε)) q x, x 2, θ, ε) + q 2 x, x 2, θ, ε)) x,x 2 0 C + C 2 ) x + x 2 ) ) A soluton to ) s x c, x c 2 ). Snce q x, x j, θ, ε) s bounded and contnuous n x, x j, θ, ε), a soluton to ) for each θ, ε) exsts, and we assume t s unque. The non-cooperatve equlbrum producton and the cooperatve producton decsons are q θ, ε) = q x θ, ε), x j θ, ε), θ, ε ) and q c θ, ε), x c j θ, ε), θ, ε ) respectvely. To smplfy notaton we use x = x θ, ε), x c q c θ, ε) = q x c = x c θ, ε), q = q θ, ε), and = q c θ, ε) when convenent. We denote equlbrum frm profts by Π θ, ε) = Π x, x j, θ, ε) and Π c θ, ε) = Π x c, x c j, θ, ε). Aggregate R&D nvestment s defned by X = x + x 2. Welfare s defned by W = CSQ) + π q, q 2, x, x 2, θ, ε) + π 2 q 2, q, x 2, x, θ, ε), where aggregate producton Q s defned by Q = q + q 2, and consumer surplus CS Q) s defned Q by CSQ) = p t) dt p Q) Q. Evaluatng the aggregate R&D nvestment and welfare 0 functons n the non-cooperatve and cooperatve equlbra yelds the respectve equlbrum aggregate R&D nvestment and welfare levels: X = x + x 2, X c = x c + x c 2, W = CS Q ) + Π + Π 2 and W c = CS Q c ) + Π c + Π c 2, where Q = q + q 2 and Q c = q c + q c 2. 7 Fnally, we defne the set of welfare-based crtcal spllover values as 7 The assumptons on nverse demand, producton costs, the cost reducton functon, and the absorptve capacty functon, ensure that the equlbrum values of X, Π, CS, Q, and W are contnuous and dfferentable. 7
Θ ε) = {θ [0, ] W θ, ε) = W c θ, ε)}. Gven an ntal cost asymmetry ε, ths set of spllover values contans all the spllover values θ [0, ] for whch equlbrum welfare under a RJV s equal to equlbrum welfare under NC. At ths stage of the development of the model we defne Θ ε) as a set to allow for multple welfare-based crtcal spllover values. Proposton 3 provdes suffcent condtons under whch there exsts a unque crtcal spllover value. In order to compare our results to the prevous lterature we can also defne the set of nvestment-based crtcal spllover values Θ ε) = {θ [0, ] X θ, ε) = X c θ, ε)}. Lemma shows that n the absence of any cost asymmetry these two sets are dentcal when demand s suffcently large. Lemma There exsts a > 0 such that for all a > a and θ Θ 0), x θ, 0) = x c θ, 0) and Θ 0) = Θ 0). Our readng of the lterature suggests that RJVs and NC are compared n terms of aggregate R&D nvestment because the frst papers n the lterature also had a specfc focus on R&D nvestment. Lemma shows that n absence of any cost asymmetry, adoptng a welfare approach would mply the same crtcal spllover values as would an aggregate R&D nvestment approach. However, ths s not generally true n the presence of an ntal cost asymmetry, so n the rest of the paper we wll consder welfare dfferences between RJVs and NC. 3 R&D Investment and Welfare Three sets of results are presented n ths secton all focusng on ntal cost asymmetres close to zero. Frst, we show how welfare changes when the spllover parameter θ changes n the presence of a small asymmetry. Second, we show how welfare changes when ε changes. Thrd, we show that f a crtcal spllover value exsts, then t s unque and we show how t changes wth the sze of ε. All proofs are n the Appendx. We then dscuss these effects for larger cost asymmetres. 8
3. Margnal welfare effect of the exogenous spllover θ Snce d Aspremont and Jacquemn 988), the lterature has acknowledged that the welfare dfferences between a RJV and NC depends on the magntude of the spllover parameter θ. Usng the fact that Π = q 2 x, drect calculaton of W / and W c / yelds 8 { W = CS q + q2) q } + q 2 ) + 2q q 2 x 2 x x { + CS q + q2) q } + q 2 ) + 2q q x 2 x 2 x 2 + CS q + q2) q + q 2 ) + 2q q + q 2 2q 2 2) and W c = + { CS q c + q c 2 ) q + q 2 ) x { CS q c + q c 2 ) q + q 2 ) x 2 + CS q c + q c 2 ) q + q 2 ) + 2q c 2 + 2q c + 2q c q q 2 x q x 2 } x c } x c 2 + 2qc 2 q 2. 3) The frst two terms n 2) and 3) account for the ndrect welfare effect of θ through R&D nvestment decsons. The thrd lne n both equatons accounts for the ndrect effect of θ through changes n consumer surplus and frm profts caused by changes n output. Wrtng the output effect of θ on W and W c n more detal yelds ) q 2 3 5q c ) q 3 + 3 5q 2 c 2 3 4) and ) q c 2 3 5qc 3 c q c + ) 3 5qc 2 3 c 2 5) where c / = f z )γx )x j < 0. Expresson 4) s the welfare effect under NC from a change n θ through ts effect on frm output, whch s generated by a change n both frms 8 All dervatves are evaluated at q, q j, x, x j, θ, ε) or q c, q c j, xc, xc j, θ, ε) 9
margnal cost of producton. As θ ncreases, holdng x and x 2 fxed, each frm experences a reducton n ts margnal cost of producton. Wth lower margnal producton costs, both frms ncrease ther second-stage output decsons, whch ncreases consumer surplus and frm profts. An analogous nterpretaton holds for 5). 9 Substtutng 4) and 5) nto 2) and 3) respectvely shows that a change n θ affects welfare through two channels: R&D nvestment and margnal cost. Proposton shows how equlbrum welfare changes wth respect to the exogenous spllover parameter. For the NC case these welfare changes depend on the elastcty of R&D nvestment wth respect to θ whch we defne as η x,θ = θ/x θ, 0)) θ, 0)/. Proposton There exsts ε > 0, such that for all 0 ε < ε W c θ, ε) > 0 6) and f 2γ x ) x γ x ) < 0 for all x, there exsts a > 0, such that for all a > a W θ, ε) 0 f η x,θ 2 3 7) and W θ, ε) < 0 f η x,θ > 2 3. 8) Proposton establshes the sgn of the change n total welfare wth respect to a change n θ for the RJV and NC cases when the ntal cost asymmetry s small. Proposton shows that equlbrum welfare s monotoncally ncreasng under a RJV for all θ, and ncreasng for small θ under NC. Ths result formalzes and extends Leahy and Neary s 2007) numerc example for the symmetrc case. Equlbrum welfare under a RJV s monotoncally ncreasng because jont proft maxmzaton mples that R&D nvestment ncreases as θ ncreases. Therefore 9 It s easy to show that there exsts ε such that 4) and 5) are strctly postve for all ε < ε. 0
the welfare effect through R&D nvestment of an ncrease n θ renforces the postve welfare effect through margnal cost reducton. The techncal condton on the absorptve capacty functon n the second half of the proposton ensures that absorptve capacty s not bult too quckly. Ths upper bound on the rate at whch absorptve capacty can be bult, guarantees that each dollar of R&D nvestment benefts the compettor enough to reduce the ncentve to nvest n R&D under NC. Ths nvestment dsncentve nduces a countervalng welfare effect through the R&D channel that offsets the welfare effects through the margnal cost channel. Moreover, f R&D nvestment s suffcently elastc, the ncrease n θ reduces R&D nvestment enough to offset ts welfare-ncreasng effect through the reducton n margnal costs. If the techncal condton on γ ) s volated for some x, the method of proof used suggests that t may be more lkely that W θ, ε)/ 0 for all θ as n 7). Consder however the lmtng case n whch lm x γ x ) x =. Ths lmtng case approxmates the d Apremont and Jacquemn 988) model. Snce the sgn of W θ, ε)/ stll depends on the elastcty mplct n 7) and 8), we conjecture that the techncal condton on γ ) s not necessary. 3.2 Margnal welfare effect of the cost advantage ε Drect calculaton of W / and W c / mples { W = CS q + q2) q } + q 2 ) + 2q q 2 x 2 x x { + CS q + q2) q } + q 2 ) + 2q q x 2 x 2 x 2 + CS q + q2) q + q 2 ) + 2q q + q 2 2q 2 9)
and W c = + { CS q c + q c 2 ) q + q 2 ) x { CS q c + q c 2 ) q + q 2 ) x 2 + CS q c + q c 2 ) q + q 2 ) + 2q c 2 + 2q c + 2q c q q 2 x q x 2 } x c } x cc 2 + 2qc 2 q 2. 0) The frst two terms n 9) and 0) account for the ndrect welfare effect of ε through R&D nvestment decsons. The thrd lne n both equatons accounts for the drect effect through changes n consumer surplus and frm profts caused by changes n output. Performng an analogous decomposton of the output effect to that n 4) and 5) mples that the output effect under NC equals q /3 5q 2 and under a RJV equals q c /3 5q c 2. For ε close to zero, both terms are strctly negatve. An ncrease n ε also reduces welfare through the R&D effect. Proposton 2 thus shows that welfare decreases snce an ncrease n ε ncreases frm 2 s ntal cost dsadvantage. Proposton 2 There exsts ε > 0, such that for all 0 ε < ε, W θ, ε) / < 0 and W c θ, ε) / < 0 for all θ. Proposton 2 follows from the fact that the welfare effects through R&D nvestment and margnal costs are all decreasng. Ths occurs for two reasons. Frst, fxng frm s unt cost of producton, as frm 2 s unt cost of producton ncreases, R&D nvestment decreases, yeldng less cost reducton and welfare. Second, an ncrease n ε drectly reduces frm 2 s producton by more than frm s producton ncrease n the second stage also nducng a decrease of welfare. 2
3.3 The welfare rankng of RJV and non-cooperatve R&D nvestment In ths secton we study how the sze of the ntal cost asymmetry affects the dfference n welfare between a RJV and NC when the ntal cost asymmetry s small. Proposton 3 provdes suffcent condtons for the crtcal spllover value to be unque when t exsts. Then, we establsh how a change n ε affects the crtcal spllover value when the ntal cost asymmetry ε s small and demand s suffcently large. Fnally, usng Hammerschmdt s 2009) absorptve capacty functon, we show how the rate at whch absorptve capacty s bult plays a key role n determnng how a change n the cost asymmetry affects the crtcal spllover value. Proposton 3 Suppose Θ ε), and 2γ x ) x γ x ) < 0 for all x. Then there exsts a, ε > 0 such that for all 0 ε < ε and a > a, θ ε) s the unque crtcal spllover value. Moreover, for 0 θ < θ ε), W θ, ε) > W c θ, ε), and for θ ε) θ, W θ, ε) W c θ, ε). Defne Λ θ θ, ε) = W θ, ε)/ W c θ, ε) / where Λ θ θ, ε) measures the effect of a change n θ on the welfare dfference under NC and a RJV. The proof of Proposton 3 shows, f, when the ntal cost asymmetry s small there exsts a crtcal spllover value θ ε), then equlbrum welfare under NC crosses equlbrum welfare under a RJV from above,.e. Λ θ θ ε), ε) < 0. Then Proposton drectly mples that W and W c cross at most once. Suppose that for some ε, W c 0, ε) > W 0, ε). Under the condtons n Proposton 3, there cannot exst a crtcal spllover value snce W c θ, ε) > W θ, ε) for all θ. Therefore a necessary condton for Θ ε) s W 0, ε) > W c 0, ε). As a result W θ, ε) > W c θ, ε) for θ < θ ε) and W θ, ε) W c θ, ε) for θ θ ε). Fgure presents a graphcal representaton of Propostons and 3 based on Hammerschmdt s 2009) example, n whch frm s cost reducton functon s f z ) = s z / + z ) ρ, ts absorptve capacty functon s γ x ) = x / + x ) λ, a =.5, c = 0.25, s = 0.25, ρ = 0.47, and λ = 0.005. γ ) satsfes 3
the techncal condton of Proposton for all λ [ 0, 2). Consstent wth Proposton, the example shows that equlbrum welfare under a RJV W c ) s monotoncally ncreasng n θ and that equlbrum welfare under NC W ) s ncreasng for low values of θ. Fgure also confrms that W can be decreasng for larger values of θ. Consstent wth Proposton 3, ths example shows that for θ > θ ε), a RJV provdes hgher welfare than under NC, and when θ < θ ε) the reverse s true. Fgure : Welfare based crtcal spllover. To understand how a change n ε affects the crtcal spllover value, defne Λ ε θ, ε) = W θ, ε)/ W c θ, ε)/. Then drect calculaton yelds ε)/ = Λ ε θ ε), ε)/λ θ θ ε), ε). Under the condtons of Proposton 3, sgn ε)/) = sgn Λ ε θ 0), 0)) for ε close to zero. That s, when the cost asymmetry s small, the change n the welfare dfference wth respect to ε between a RJV and NC determnes the drecton of the change of the welfare-based crtcal spllover as ε changes. By Proposton 2, equlbrum welfare n both the NC and RJV cases the sold and the dashed lnes, as functon of θ, n Fgure respectvely) shft down as ε ncreases. Thus, the magntude of the reducton n equlbrum welfare under a RJV wth respect to an ncrease n 4
the ntal cost asymmetry, relatve to the non-cooperatve R&D nvestment case determnes whether the crtcal spllover value θ ε) decreases, stays at the ntal level, or ncreases. If θ ε) s decreasng,.e. the welfare loss due to an ncrease n ε s less under a RJV than under non-cooperatve R&D nvestment, a RJV wll generate hgher welfare than noncooperatve R&D nvestment for a larger set of exogenous spllovers. If θ ε) s ncreasng, welfare wll be larger under a RJV for a smaller set of exogenous spllovers. Drect calculaton of Λ ε θ 0), 0) shows that both outcomes are possble. Fgure 2 shows how the rate at whch absorptve capacty s bult can nfluence the drecton of the change of θ ε) wth respect to ε. 0 Increasng ε reduces θ ε) when λ < λ s and ncreases θ ε) when λ > λ s. When frms can buld absorptve capacty at a hgh rate λ < λ S ), ntroducng an ntal cost asymmetry by gvng frm a cost advantage, reduces the welfare under a RJV less than under NC. Under NC a hgher rate of absorptve capacty acquston reduces a frm s ncentve to nvest n R&D snce ts compettor can buld absorptve capacty wth very lttle R&D nvestment. Ths reduced ncentve results n lower equlbrum R&D nvestment and, therefore, less equlbrum welfare. However, under a RJV a hgher rate of absorptve capacty acquston lower λ) allows the frms to nternalze the spllover benefts at a lower cost, thereby reducng the welfare losses from an ncrease n the ntal cost asymmetry. These effects work n the opposte drecton when λ ncreases and absorptve capacty s bult more slowly. 3.4 Large ntal cost asymmetres Allowng for large ntal cost asymmetres does not appear to be amenable to analytc evaluaton. We can however consder the effect of ε n the context of the above example. If the ntal cost asymmetry s above 0.2, the cost dfference s drastc snce frm 2 wll not operate and frm wll be the monopolst. Wth such a drastc ntal cost dfference, the ssue of comparng RJV and NC becomes moot. Wthout an actve spllover mechansm, welfare 0 Decreases n λ make γ ) more concave snce γ x)/γ x) s decreasng n λ 5
Fgure 2: Dfference between the welfare-based crtcal spllover values for the cases wth cost asymmetry θ 0.00) and wthout a cost asymmetry θ 0) and the ablty to buld absorptve capacty λ. falls due to lower aggregate R&D nvestment, less aggregate cost reducton frm 2 s cost are unaffected), and lower producton due to frm s ablty to explot ts market power. For values of ε below 0.2, the cost dfference s non-drastc. We fnd that the qualtatve features and mplcatons of Propostons through 3 stll hold. Also, the qualtatve behavor of aggregate R&D nvestment wth respect to ε and θ n the RJV and NC cases s the same as n the small asymmetry case. Indvdual and aggregate R&D nvestment contnue to be decreasng n ε. However, we also fnd that each frm s R&D nvestment can be nonmonotonc n θ, as opposed to the ncreasng behavor they exhbt n the small asymmetry case. For large enough ε frm s R&D nvestment decson can be decreasng for large θ, and frm 2 s R&D nvestment can be decreasng for small values of θ. Fnally, our smulatons suggest that movng from symmetry to a large ε reduces welfare under a RJV by less than welfare under NC, regardless of the rate at whch absorptve capacty s bult. That s, when the ntal cost asymmetry s large but non-drastc), assumng symmetry n producton costs underestmates the range of values of θ for whch a RJV generates hgher equlbrum welfare than under NC. 6
3.5 Alternatve cost asymmetry formulatons In ths secton we consder a more general cost asymmetry formulaton n whch c = c αε fz ) and c 2 = c+ α)ε fz 2 ) for 0 α 0.5. Our man model corresponds to α = 0 whle α = 0.5 mples that changes n ε do not change the ntal average ndustry margnal producton cost. One can also consder ntal cost asymmetres that mply lower average ntal costs wth ths model by allowng for negatve values of ε. Wth ths more general formulaton, the welfare changes n Propostons and 3 are qualtatvely the same as n the man model because these propostons descrbe welfare propertes for ε close to zero. Wth regard to Proposton 2, for the case n whch α = 0 and ε < 0, t remans the case that W / and W c / are both negatve for ε close to zero. For the case n whch α = 0.5 and ε > 0 ), the sgns of W / and W c / are ndetermnate. Ths ndetermnancy arses because the effect of ε on welfare through aggregate R&D nvestment and through the margnal cost of producton s zero n the lmt as ε 0. Thus, we can no longer use these channels to sgn the margnal welfare changes. However, numercal analyss ndcates that f the ntal cost asymmetry s smaller than 0.005, then both W and W c are ncreasng n ε when α = 0.5. Wth regard to the effect of λ on θ ε), note that the postve slope of the curve n Fgure 2 mples that dθ ε)/dε s ncreasng n λ. Ths contnues to be true n our more general model for α [0, 0.5] although the mplcatons of an ntal negatve cost shock ε > 0) and an ntal postve cost shock ε < 0) are somewhat dfferent. Wth an ntal negatve cost shock as would occur when frms n an ndustry are affected dfferentally by ncreased regulatory costs), the relatve welfare benefts of RJVs ncrease when frms can buld absorptve capacty at a hgher rate low λ ). Under NC, the lower-cost frm s dscouraged from nvestng n R&D W θ, ε)/ = Γθ, ε) + 2q q2) and W c θ, ε)/ = Γ c θ, ε) + 2q c q2 c ). The functons Γθ, ε) and Γ c θ, ε) are the welfare effects of R&D nvestment under NC and RJV n 9) and 0) respectvely. The term 2q q2) s the welfare effect of a small change of ε through the margnal cost channel under NC. Analogous analyss and nterpretaton hold for 2q c q2 c ) n the RJV case. 7
as dong so works to reduce ts compettor s cost dsadvantage. As λ ncreases, aggregate R&D nvestment ncreases under NC and decreases under RJV, thus moderatng the relatve welfare benefts of an RJV and causng θ ε) to ncrease. Wth an ntal postve cost shock as would occur when frms n an ndustry are affected dfferentally by decreased regulatory costs), an ncrease n λ causes θ ε) to fall faster or ncrease less slowly. Whle an ncrease n ε > 0 causes W c θ, ε) to shft down along W θ, ε), a decrease n ε < 0 causes W c θ, ε) to shft up along W θ, ε). The latter shft wll generate a smaller change n NC welfare due to the concavty of W wth respect to θ Proposton ), and thus necesstate a smaller value of θ ε). Ths result favors the formaton of RJVs. 4 Concluson The lterature on strategc R&D nvestment followng the semnal work of d Aspremont and Jacquemn 988), and advanced by Kamen and Zang 2000) to nclude endogenous spllover effects, has looked at the effect of spllovers on aggregate R&D nvestment n a symmetrc framework. The predomnant focus has been to compare non-cooperatve R&D nvestment wth RJVs based on aggregate R&D nvestment. However, the emprcal lterature supports the exstence of producton cost asymmetres among frms, and comparsons based on aggregate R&D nvestment are equvalent to welfare-based comparsons only under symmetry. In ths paper, we extend ths work by ntroducng a cost asymmetry and by evaluatng the effect of the magntude of ths cost asymmetry and the exogenous spllover n terms of welfare. Ths paper presents four key theoretcal results for a small ntal cost asymmetry. Frst, we show that n the presence of an ntal cost asymmetry t s no longer suffcent to compare RJVs and NC solely n terms of aggregate R&D nvestment. Wth an ntal cost asymmetry, the crtcal spllover value s determned by drect margnal cost dfferences n addton to dfferences n aggregate R&D nvestment, whereas wth symmetrc costs the crtcal spllover value can be determned exclusvely by comparng aggregate R&D nvestment levels. Second, 8
the effect of a change n the sze of the ntal cost asymmetry on the crtcal spllover s ambguous because the change n the crtcal spllover value s shown to depend on the rate at whch a frm can buld absorptve capacty. If an ncrease of the ntal cost asymmetry ncreases welfare under NC and RJV, then when the frms can buld absorptve capacty at a hgh rate, an ncrease of the cost asymmetry renforces the welfare ncrease under a RJV and reduces the welfare ncrease under NC. Analogously, f an ncrease of the ntal cost asymmetry reduces welfare under NC and RJV, then for frms wth hgh absorptve capacty buldng rate, an ncrease of the cost asymmetry renforces the welfare reducton under NC and weaken the welfare reducton under a RJV. These effects reverse for frms wth low rate of absorptve capacty buldng. Thrd, wth small ntal cost asymmetres, welfare s monotoncally ncreasng n the exogenous spllover parameter under a RJV, whereas under NC welfare can be non-monotonc n the sze of the exogenous spllover parameter ncreases. Ths occurs because, whle an ncrease of the exogenous spllover parameter drectly reduces margnal costs thereby ncreasng aggregate producton and welfare under a RJV and under NC, the effect through the R&D nvestment channel ncreases welfare under a RJV and reduces welfare under NC. Fnally, fxng the unt cost of producton of the low-cost frm, as the hgh-cost frm becomes less productve, equlbrum welfare decreases because of lower R&D nvestment, hgher producton costs, and less aggregate producton. These results expand our understandng about when RJVs provde hgher welfare than non-cooperatve R&D nvestment n a model that can ncorporate the emprcal realty that frms n an ndustry exhbt heterogeneous costs. In partcular, our model provdes an nsght about how the nteracton between frms ablty to learn from ther compettors R&D and the frm heterogenety affect welfare dfferences between a RJV and NC. Moreover, our analyss shows that ntroducng an ntal cost asymmetry generates rcher welfare dfferences between a RJV and NC. Therefore the crtera commonly used to compare RJVs and NC, whch solely focuses on aggregate R&D nvestment, and reles on the absence of ntal cost asymmetres can be msleadng for polcy purposes. 9
References [] d Aspremont, C., Jacquemn, A., 988. Cooperatve and Noncooperatve R&D n Duopoly wth Spllovers. Amercan Economc Revew 78, 33-37 [2] De Bondt, R., 996. Spllovers and nnovatve actvtes. Internatonal Journal of Industral Organzaton 5, -28 [3] Grünfeld, L.A., 2003. Meet me halfway but don t rush: absorptve capacty and strategc R&D nvestment revsted. Internatonal Journal of Industral Organzaton 2, 09-09 [4] Hammerschmdt, A., 2009. No pan, No gan: An R&D Model wth Endogenous Absorptve Capacty. Journal of Insttutonal and Theoretcal Economcs 65, 48-437 [5] Kamen, M., Muüller, E., Zang, I., 992. Research jont Ventures and R&D Cartels. Amercan Economc Revew 82, 293-306 [6] Kamen, M., Zang, I., 2000. Meet me halfway: research jont ventures and absorptve capacty. Internatonal Journal of Industral Organzaton 8, 995-02 [7] Leahy, D., Neary, P., 2007. Absorptve capacty, R&D spllovers and publc polcy. Internatonal Journal of Industral Organzaton 25, 089-08 [8] López-Pueyo, C., Barcenlla-Vsús, S., Sanaú, J. 2008. Internatonal R&D spllovers and manufacturng productvty: A panel data analyss. Structural Change and Economcs Dynamcs 9, 52-72 [9] Olley, S., Pakes, A., 996. The Dynamcs of Productvty n the Telecommuncatons Equpment Industry. Econometrca 64, 263-297 [0] Pavcnk, N., 2002. Trade Lberalzaton, Ext, and Productvty Improvements: Evdence from Chlean Plants. The Revew of Economc Studes 69, 245-276 20
[] Roberts, M., Tybout, J., 996. Industral Evoluton n Developng Countres. Oxford: Oxford Unversty Press. [2] Wethaus, L., 2005. Absorptve Capacty and Connectedness: Why Competng Frms also Adopt Identcal R&D Approaches. Internatonal Journal of Industral Organzaton 23, 467-48 5 Appendx 5. Proof of Lemma Frst, we show that ε = 0, mples that R&D nvestments are equal among frms. Frm s FOC n G RNC s 2q x, x j, θ, ε) q x, x j, θ, ε)/ =, where j. Then the subgame perfect nvestment levels are descrbed by 2q x, x 2, θ, ε) q x x, x 2, θ, ε) = ) and Gven the demand and costs assumptons, 2q 2 x 2, x, θ, ε) q 2 x 2 x 2, x, θ, ε) =. 2) q x, x 2, θ, ε) = 3 a c + ε + 2f z ) f z 2 )) and q 2 x 2, x, θ, ε) = 3 a c 2ε + 2f z 2) f z )). When ε = 0, ) and 2) are symmetrc. Therefore x θ, 0) = x 2 θ, 0) = x θ, 0). Analogously x c θ, 0) = x c 2 θ, 0) = x c θ, 0). Now we show that f θ Θ 0), then θ Θ 0). Suppose θ Θ 0), then X θ, 0) = 2x θ, 0) and X c θ, 0) = 2x c θ, 0). 2
By the defnton of Θ ε), θ Θ 0) mples x θ, 0) = x c θ, 0). Substtutng x θ, 0), x c θ, 0), and θ nto ) and 2) and nto the defntons of CS, Π, and W yelds W θ, 0) = W c θ, 0). Next we prove that f θ Θ 0), then θ Θ 0). By the defnton of Θ ε), θ Θ 0) mples W θ, 0) = W c θ, 0). By the defnton of W and W c, W θ, 0) W c θ, 0) = W x, x, θ, 0) W x c, x c, θ, 0) where x = x θ, 0) and x c = x c θ, 0). Defne Ŵ x, θ, 0) = W x, x, θ, 0) and q x, θ, 0) = q x, x, θ, 0). Wthout loss of generalty assume x x c then W θ, 0) W c θ, 0) = x θ,0) Ŵ t, θ, 0)/ xdt. t=x c θ,0) If Ŵ t, θ 0), 0)/ x > 0 for all t, then W θ, 0) = W c θ, 0) f, and only f, x θ, 0) = x c θ, 0). To show ths s true for A suffcently large, note that Ŵ / x = 8q x, θ, 0) q x, θ, 0)/ x 2. Drect calculaton yelds q x, θ, 0)/ x = f z) + γ x) θx + γ x) θ) > 0 and q x, θ, 0) = a c f z)). Be- 3 cause q x, θ, 0) s lnear and ncreasng n a, then there exsts a > 0 such that Ŵ x t, θ 0), 0) > 0 for all t. Then x θ, 0) = x c θ, 0) and θ Θ 0). 5.2 Proof of Proposton Lemmas 2, 3, and 4 provde the key steps for the proof of 7) and 8). Lemma 2 states that, n the lmt as ε 0, the frms strateges and equlbrum decsons converge n both the non-cooperatve R&D nvestment and RJV cases. Lemma 3 states that when the ntal cost asymmetry s small, θ affects the equlbrum R&D nvestment level under NC n the same drecton as t affects the ncentves to nvest n the symmetrc case. Lemma 4 states that, for large enough demand and an absorptve capacty functon that does not buld absorptve capacty too quckly, an ncrease n θ necessarly reduces the ncentves to nvest under NC. The rest of the proof refers to the change n equlbrum welfare under a RJV and only uses mplcatons from the maxmzaton of jont profts. Lemma 2 For all and j,.lm ε 0 x θ, ε), q x, x j, θ, ε)) = x θ, 0), q x, x j, θ, 0)), 2.lm ε 0 x c θ, ε), q x, x j, θ, ε)) = x c θ, 0), q x, x j, θ, 0)), 22
3.lm x θ, ε) = x θ, 0) and lm q x θ, ε), x j θ, ε), θ, ε ) = q θ), and ε 0 ε 0 4.lm x c ε 0 θ, ε) = x c θ, 0) and lm q c ε 0 x c θ, ε), x c j θ, ε), θ, ε ) = q c θ). Proof: The proof follows drectly from the fact that q x, x j, θ, ε) and q x, x j, θ, ε)/ are contnuous n x, x j, θ and ε, that x θ, 0) = x 2 θ, 0), and that x c θ, 0) = x c 2 θ, 0). Lemma 3 For any stable equlbrum of G RNC, for all j {, 2}, there exsts ε such that for all 0 ε < ε ) x sgn θ, ε) 2 Π x, x j, θ, 0 ) ) = sgn 3) and ) x c θ, ε) sgn = sgn 2 Π x c, x c j, θ, 0 ) Proof: To prove 3), totally dfferentatng ) and 2) yelds + 2 Π j x c j, x c, θ, 0 ) ). 4) x = q qj q + q q j ) 2 q 2 q j + qj ) 2 q x + q 2 q x 2 q j q j + q j 2 q j q q 2 q + q ) ) 2 qj + q 2 q j j x 2 j q q + q 2 q x ) j 2 qj + q 2 q j j x 2 j. 5) The stablty condton for the equlbra of G RNC at each θ, ε) mples the denomnator of 5) s strctly postve. Thus, ) x sgn θ, ε) [ = sgn + qj q q + q q j + q j 2 q j ) ) 2 2 q qj + q 2 q j j x 2 j ) q q + q )] 2 q ) 6) where the rght-hand sde of 6) s the sgn of the numerator of 5). The contnuty of q x, x j, θ, ε), q x, x j, θ, ε)/, q x, x j, θ, ε)/, 2 q x, x j, θ, ε)/ and 23
2 q x, x j, θ, ε)/ n x, x j, θ, ε) for all and j mples, that the numerator 5) s contnuous. Takng the lmt as ε 0 of the rght-hand sde of 6), Lemma 2 and contnuty of the numerator of 5) mply that there exsts ε > 0 such that for 0 < ε < ε, ) x sgn θ, ε) [ ) ) 2 qj = sgn + q 2 q j j x 2 j q q + q + 2 q ) ) q q + q 2 q )] 7) where each term on the rght-hand sde of 7) s evaluated at θ, 0). The stablty condton for the equlbrum of G RNC at θ, 0) has the form A 2 B 2 > 0, where A = q x θ, 0), x j θ, 0), θ, 0 ) ) 2 + q 2 q x θ, 0), x j θ, 0), θ, 0 ), x 2 B = q x θ, 0), x j θ, 0), θ, 0 ) q x θ, 0), x j θ, 0), θ, 0 ) 2 q +q x θ, 0), x j θ, 0), θ, 0 ) ), and A + B s the term n the square brackets on the rght-hand sde of 7). The SONC of frm s proft maxmzaton problem for θ, 0) s equvalent to A < 0. If B > 0, then A > B. If B < 0, then A > B. Therefore A + B > 0 and the rght-hand sde of 7) s equal to sgn q / q / + q 2 q / ) evaluated at θ, 0), whch can also be wrtten as sgn Π x, x j, θ, 0 ) / ) for all j. To prove 4), totally dfferentatng the FOC of ) yelds x c = M M 22 M 2 M 2 22 2 2 8) where q q M = + q c 2 q + q ) j q j + q c 2 q j j, 24
M 2 = 2 = q q + q c 2 q + q j q j + qj c q q M 2 = + q c 2 q + q j q j + q c 2 q j j q M 22 = 22 = ) 2 + q c 2 q qj + x 2 j ) 2 q = + q c 2 q + x 2 qj ) 2 + q c j ) 2 + q c j 2 q j, x 2 2 q j, ), 2 q j, x 2 j and 2 = q q + q c 2 q + q j q j + q c 2 q j j. Here the SONC of the reduced form cooperatve problem n ) at each θ, ε) mples that the denomnator of 8) s non-negatve. Moreover the contnuty of q x, x j, θ, ε), q x, x j, θ, ε)/, q x, x j, θ, ε)/, 2 q x, x j, θ, ε)/ and 2 q x, x j, θ, ε)/ n x, x j, θ, ε) for all and j mples that the numerator of 8) s contnuous. By Lemma 2) and contnuty of the numerator of 8) there exsts ε > 0 such that for 0 ε < ε ) x c θ, ε) sgn = sgn [ [ q ) 2 + q c 2 q x 2 j )] q q +2 + q c 2 q [ ] [ q q + q c 2 q qj + ] + [ qj ) 2 + qj c 2 q j x 2 j ])) q j + q c 2 q j j ]) 9) where the rght hand sde of 9) s evaluated at θ, 0). Fnally the SONCs for a local maxmum of the reduced form cooperatve problem ) at θ, 0) take the form C < 0 and where q C = ) 2 + q c 2 q qj + x 2 j ) 2 + q c j 2 q j, x 2 j ) q q D = 2 + q c 2 2 q, and C + D s the sum of the two frst lnes n 9). Therefore the SONC of problem ) 25
for θ, 0) mples that f D > 0, then C > D, and f D < 0, then C > D. Therefore C + D > 0 and the rght-hand sde of 9) s equal to q q sgn + q c 2 q + q )) j q j + q c 2 q j j evaluated at θ, 0), whch we can wrte as sgn Π x c, x c j, θ, 0 ) + Π j x c j, x c, θ, 0 ) ) for all j, completng the proof of Lemma 3. Lemma 4 Suppose 2γ x ) x γ x ) < 0. Then for ε = 0, there exst a > 0 such that for all a > a, Π x, x j, θ, 0 ) / < 0. Proof: By defnton γ ) s bounded between 0 and, strctly ncreasng and strctly concave. Therefore, f 2γ x ) x γ x ) < 0, then 2γ x) x γ x) 0. Under symmetry z = z, x = x, and q x, x, θ, 0) = q x, θ) for all, where q x, θ) = φ z) = a c + f z)) and z = x + γ x) θx. Therefore 3 2 Π x, x, θ, 0) = 2 9 f z) γ x) x) f z) [2 + θ 2γ x) x γ x))] + q 2 3 f z) γ x) x [2 + θ [2γ x) x γ x)]] + f z) [2γ x) x γ x)]). 20) Then 2γ x) x γ x) 0 mples that 2 + θ [2γ x) x γ x)] > 0. Because f z) s concave n z and because q x, θ) s contnuous and lnear n a for all x, θ), there exsts a > 0 such that for all a > a, q x, θ) s large enough so that 20) wll be negatve for all x, θ). Ths completes the proof of Lemma 4. To complete the proof 7 of and 8 n Proposton, substtutng the equlbrum R&D 26
nvestment and producton decsons nto 2) mples that W = { [ q + q2) q + q ) 2 c q + + q ) ] 2 c2 c c x c 2 c 2 x [ q2 q c + q ) c 2 c ]} 2 x c x c 2 x x { [ + q + q2) q + q ) 2 c q + + q ) ] 2 c2 c c x 2 c 2 c 2 x [ 2 q q2 c + q ) 2 c 2 c ]} x 2 c x 2 c 2 x 2 x 2 [ ) ) ] + q q + q2 q 2 c c c + q q + q2 q 2 c2 c 2 c 2 q c + q 2 c 2 ). 2) The contnuty of q x, x j, θ, ε), q x, x j θ, ε)/, q x, x j, θ, ε)/, 2 q x, x j, θ, ε)/ and 2 q x, x j, θ, ε)/ n x, x j, θ, ε) for all and j mples the contnuty of W /. Rearrangng terms, substtutng equlbrum decsons, and takng the lmt of 2) as ε 0, t follows from Lemma 2 that lm ε 0 W θ, ε)/ = 4q f z ) γ x ) [ 2 3 η x,θ]. By Lemma 3 and the assumptons on f ) and γ ), there exsts ε > 0 such that for 0 ε < ε, f 2 Π x, x j, θ, 0 ) / > 0, then x θ, ε)/ > 0, and f 2 Π x, x j, θ, 0 ) / < 0, then x / < 0. By Lemma 4 f 2γ x ) x γ x ) < 0, there exsts a > a such that 2 Π x, x j, θ, 0 ) / < 0. Therefore f 2γ x ) x γ x ) < 0, by the contnuty of W θ, ε)/ and Lemmas 3 and 4, then there exsts a, ε > 0 such that for 0 ε < ε, and a > a, 7) and 8) n Proposton hold. Now we turn to prove 6) n Proposton. We want to show that there exsts ε > 0 such that for 0 ε < ε, W c θ, ε)/ > 0. Snce W c = CS c + Π c + Π c 2, frst we show that there exst ε > 0 such that for 0 ε < ε, Π c + Π c 2 ncreases n θ, and then we show that there exst ε 2 > 0 such that for 0 ε < ε 2, CS ncreases n θ. We can then defne ε = mn {ε, ε 2 } Let x c θ, 0), x c j θ, 0) ) be the unque soluton to ) when ε = 0. In ths case x c θ, 0) = x c j θ, 0) = x c θ). Defne the ndrect jont proft functon under symmetry 27
by Π x c θ), θ) = 2 q x c θ), θ)) 2 2x c θ). Let θ > θ. Then by proft maxmzaton Π x c θ ), θ ) > Π x c θ), θ ). Snce q x, θ) = a c + f x + γ x) θx)), f ) ncreasng 3 mples that Π x c θ), θ ) > Π x c θ), θ). Therefore Π x c θ ), θ ) > Π x c θ), θ). By Lemma 5 and contnuty of the jont profts Π x, x j, θ, ε) + Π j x j, x, θ, ε) n x, x j, θ and ε, there exsts ε > 0 such that for 0 < ε < ε, Π x c θ ), θ ) > Π x c θ), θ) mples Π x c > Π x c θ, ε), x c j θ, ε), x c j θ, ε), θ, ε ) + Π j x c j θ, ε), x c θ, ε), θ, ε ) θ, ε), θ, ε ) + Π j x c j θ, ε), x c θ, ε), θ, ε ). In the symmetrc case the cooperatve problem s max x,x 2 0 Π + Π 2 = q 2 x, x 2, θ, 0) + q 2 2 x 2, x, θ, 0) x + x 2 ). 22) The FOC s of 22) are 2q x, x 2, θ, 0) q x, x 2, θ, 0) x + 2q 2 x 2, x, θ, 0) q 2 x, x 2, θ, 0) x = 0 23) and 2q x, x 2, θ, 0) q x, x 2, θ, 0) x 2 + 2q 2 x 2, x, θ, 0) q 2 x, x 2, θ, 0) x 2 = 0. 24) By the defnton of q x, x 2, θ, ε) and q 2 x 2, x, θ, ε), q x, x 2, θ, 0) = q 2 x 2, x, θ, 0). Therefore 23) and 24) form a symmetrc system wth soluton x = x 2. Therefore solvng 22) s equvalent to solvng max x 0 Π x, θ) = 2q2 x, θ) 2x 25) whch has the unque soluton x θ). Monotoncty of γ ) mples that there exsts an nverse functon χ ω, θ) : R + R + that maps the effectve and symmetrc R&D nvestment ω nto 28
R&D nvestment x, such that χ ω, θ) = x. Therefore 25) can be wrtten as max Π ω, θ) = 2φ 2 ω) 2χ ω, θ). 26) ω 0 The FOC of 26) mples that 2φ ω) φ ω) ω χ ω, θ) ω = 0. Hence ω θ) = 2 [ φω) ω χω,θ) ω ] ω 2 ) 2 + 2 φω) 2 χω,θ) ω 2. 27) The SONC of 26) mples that the denomnator of 27) s non-postve and drect calculaton yelds 2 χ ω, θ)/ ω < 0, hence ω θ)/ > 0. Moreover, by the defnton of χ ω, θ), n equlbrum χ ω θ), θ)/ > 0. By the defnton of φ ω), ω θ)/ > 0 mples φ ω θ))/ > 0 whch s equvalent to q c θ, 0)/ > 0. By the defnton of CS Q), q c θ, 0)/ > 0 s equvalent to CS Q c )/ > 0. By Lemma 2 and the contnuty of CS Q) n x, x j, θ and ε, there exsts ε > 0 such that for 0 < ε < ε, f CS Q )/ > 0, then CS Q c )/ > 0, completng the proof. 5.3 Proof of Proposton 2 Lemmas 2 and 5 provde the key steps of the proof. Lemma 5 states that for a small ntal cost asymmetry, aggregate nvestment s decreasng n ε under a RJV and under NC. Defne q x, θ) q x, x, θ, 0), q θ) q x θ, 0), θ), and q c θ) q x c θ, 0), θ). Lemma 5 Gven the assumptons on demand, the unt cost of producton, f ), and γ ), 29
there exsts ε > 0 such that for 0 ε < ε, f the equlbrum of G RNC s locally stable, x θ, ε) + x 2 θ, ε) < 0 28) and x c θ, ε) + xc 2 θ, ε) < 0. 29) Proof: Gven the assumptons on the demand and the unt cost of producton drect calculaton yelds, n the NC case for every j, x θ, ε) = q q q j q j ) 2 q + q 2 q x 2 q j q j + q j 2 q j q q 2 q + q ) 2 qj + q 2 q j j x 2 j q q 2 q + q ) 2 qj + q 2 q j j x 2 j 30) and n the RJV case, x c θ, ε) = N N 22 N 2 N 2 D D 22 D 2 D 2 3) where N = N 2 = D 2 = q N 2 = q q + q j q j ), q q q N 22 = D 22 = D = q D 2 = q q q + q j ) 2 + q c + q c q j ), ) 2 + q c + q c 2 q x 2 + 2 q 2 q 2 q x 2 j qj + + q j q j qj ) 2 + q c j + q j q j + q c j ) 2 + q c j + q c j 2 q j, and x 2 2 q j. 2 q j, 2 q j, x 2 j Local stablty of an equlbrum of G RNC at θ, ε) mples that the denomnator of 30) s postve. The SONC for a local maxmum of the reduced form cooperatve problem mples 30
that the denomnator of 3) s postve. Defne S j θ, ε) q x, x j, θ, ε ) q x, x j, θ, ε ) 2 q + q x, x jθ, ε ) and L j θ, ε) q x c, x c + q j x c j, x c j, θ, ε ) q x c, x c j, θ, ε ), θ, ε ) q j x c j, x c, θ, ε ) 2 q + q c x c, x c j, θ, ε ) 2 q + qj c j x c j, x c, θ, ε ) for all, j {, 2}. Then ) x sgn θ, ε) = sgn q x, x j, θ, ε ) q x, x j, θ, ε ) S j j θ, ε) + q j x j, x, θ, ε ) q j x j, x, θ, ε ) ) S j θ, ε). Gven the SONC of the cooperatve problem, ) x c θ, ε) sgn [ q x c, x c j, θ, ε ) q x c, x c j, θ, ε ) = sgn + q j x c j, x c, θ, ε ) q j x c j, x c, θ, ε ) ] L j j θ, ε) [ q x c, x c j, θ, ε ) q x c, x c j, θ, ε ) + + q j x c j, x c, θ, ε ) q j x c j, x c, θ, ε ) ] ) L j x θ, ε). j By Lemma 2, lm S = lm S j j, lm ε 0 ε 0 ε 0 Sj = lm S ε 0 j, and lm L = lm L j j. The contnuty of ε 0 ε 0 q x, x j, θ, ε), q x, x j, θ, ε)/, and 2 q x, x j, θ, ε)/ n x, x j, θ, ε) for all and j, mples that S j and L j are contnuous n x, x j, θ, ε) for all and j. By contnuty of S, 3
S j, L, and L j there exsts ε > 0 such that for all 0 ε < ε, and ) x sgn θ, ε) + x 2 θ, ε) q θ, 0) = sgn + q ) 2 θ, 0) q θ, 0) ) ) S x + S 2 32) x c θ, ε) sgn + xc ) 2 θ, ε) [ L ] [ = sgn + L 2 q θ, 0) [ q θ, 0) x 2 + q θ, 0) x + q ] 2 θ, 0) ]). 33) The stablty condton of G RNC at each θ, ε) mples that S θ, ε) + S 2 θ, ε) > 0 and the SONC at each θ, ε) of the reduced form cooperatve problem mples that L θ, ε) + L 2 θ, ε) > 0. The FOC of the ndvdual frm problem n G RNC at each θ, ε) mples that q x, x 2, θ, ε)/ x > 0. The FOCs of ) at each θ, ε) mply that q x c θ, 0), x c 2 θ, 0), θ, 0) + q x c θ, 0), x c 2 θ, 0), θ, 0) > 0. x 2 x Fnally the demand and costs assumptons mply that q x, x 2, θ, ε) + q 2 x 2, x, θ, ε) = q x c, x c 2, θ, ε) + q 2 x c 2, x c, θ, ε) = 3 for all θ and ε. Hence the sgn of the left-hand sde of equatons 32) and 33) are negatve, whch completes the proof of Lemma 5. Returnng now to the proof of Proposton 2, substtutng equlbrum R&D nvestment and producton decsons nto W, drect calculaton of W / and W c / allows one to 32