THE STRATEGIC TIMING OF R&D AGREEMENTS

Transcription

1 THE STRATEGIC TIMING OF R&D AGREEMENTS MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. Abstract. We present a model of endogenous formaton of R&D agreements among rms n whch also the tmng of R&D nvestments s made endogenous. The purpose s to brdge two usually separate streams of lterature, the noncooperatve formaton of R&D allances and the endogenous tmng lterature. Ths allows to consder the formaton of R&D agreements over tme. It s shown that, when both R&D spllovers and nvestment costs are su cently low, rms may nd d cult to mantan a stable agreement due to the strong ncentve to nvest noncooperatvely as leaders. In such a case, the stablty of an R&D agreement requres that the jont nvestment occurs at the ntal stage, thus avodng any delay. When nstead spllovers are su cently hgh, cooperaton n R&D consttutes a pro table opton, although rms also possess the ncentve to sequence ther nvestment over tme. Fnally, when spllovers are asymmetrc and the knowledge manly leaks from the leader to the follower, to nvest as follower becomes etremely pro table, makng R&D allances hard to sustan unless rms strategcally delay ther jont nvestment n R&D. Keywords: R&D nvestment, Spllovers, Endogenous Tmng JE: C7, D43, 11, 13, O30. Date: May 011. Correspondng author: Marco A. Marn, Department of Economcs, Unverstà d Urbno "Carlo Bo" and CREI, Unverstà Roma III. Address: va Sa, 4, 6019, Urbno (Italy). Tel ; Fa: E-mal: marco.marn@unurb.t. Mara usa Pett, Department of Computer and System Scences "Antono Rubert", Unverstà d Roma "a Sapenza". E-mal: pett@ds.unroma1.t. Roberta Sestn, Department of Computer and System Scences "Antono Rubert", Unverstà d Roma "a Sapenza". E-mal: sestn@ds.unroma1.t. We wsh to thank Mchael Kopel, uca ambertn, Gorgo Rodano, Francesca Sanna Randacco and the partcpants at the EARIE 010 Conference n Istanbul, the MDEF Workshop n Urbno and the II Workshop on Dynamc Games n Rmn for useful comments and dscussons. 1

2 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. 1. Introducton Research agreements among rms competng n the same markets have snce long become a farly wdespread form of ndustral cooperaton. Many countres have set up cooperatve R&D programmes for some well de ned research areas, both at a natonal and at an nternatonal level. The US, for eample, has created n the md-1980s the Natonal Cooperatve Research Act (NCRA) whch ams to protect and regulate R&D allances among rms. Also the EU encourages and subsdzes R&D cooperaton between partcpants belongng to EU countres wth ntatves amed at enhancng nter- rm research cooperaton, such as the Eureka and EU Framework Programmes. Important research agreements go back to the early eghtes, such as the Mcroelectroncs and Computer Technology Corporaton (MCC), formed n 198 to conduct research related to nformaton technology, and the Bell Communcatons Research created n 1984 by seven regonal US telephone companes. The economc lterature provdes a strong emprcal evdence on the estence of such arrangements also n more recent years, and the analyss of the e ects of cooperaton on nnovaton has emerged as an mportant research topc. A clear understandng of ths phenomenon s ndeed crucal for consderaton of technology and ndustral polces. As s well known, a research agreement s an allance between rms n order to coordnate ther research and development actvtes n a jont project, and to share, to some establshed degree, the knowledge obtaned from ths common e ort. Therefore, the creaton of such research agreements allows the rms not only to coordnate ther research e orts but also to mprove nformaton-sharng. Many reasons may push rms to form research cartels. Frst, nnovaton s epensve, and the possblty of cost sharng and avodance of duplcaton can strongly cut the epenses to each member. Second, the rsk for a rm that ts own nnovaton programmes wll not produce valuable results s reduced snce a research agreement has greater possbltes of dvers caton and each member can share rsks wth the other members. Thrd, the members of a research allance can acqure a greater compettve advantage than nonmembers, whch mples that there can be a strong danger n beng left out of such cartels (see on ths topc, Baumol 199; see also Katz and Ordover 1990, Hernan, Marn and Sots 003, and Alonso and Marn 004 for nterestng emprcal studes). Research coaltons may also have socally bene cal e ects, such as the nternalzaton of technologcal spllovers, whch n general produces an ncrease n the aggregate level of R&D, and the elmnaton of duplcaton e orts, whch clearly leads to a reducton n research ependtures. Departng from d Aspremont and Jacquemn (1988) poneerng work, a number of papers have analyzed the e ects of research allances n models wth endogenous R&D (see, among others, Katz and Ordover 1990, Kamen et al. 199, Suzumura 199, Pett and Tolwnsk, 1997, 1999). However, n these models, the creaton of research agreements s eogenously assumed. More recently, the endogenous coalton formaton lterature has attempted to endogenze the formaton of R&D cartels by applyng noncooperatve models of coalton formaton (see Bloch 003 and 004, Y and Shn 000 and Y 003). Usually, n these models, at a rst stage R&D coaltons are assumed to set ther nvestment to mamze ther jont pro t, and at a second stage rms compete ndvdually n the product market. A crucal aspect to assess the stablty of a gven structure of agreements among rms s the sgn of the eternaltes of R&D nvestments whch, n turn, depend on the level of spllovers. For su cently hgh spllovers, formng a research cartel reduces the undernvestment n R&D,

3 TIMING AND R&D AGREEMENTS 3 snce the eternaltes due to the nature of publc good of R&D nvestments are nternalzed. Thus, allances of rms can nvest more than small groups, and ths, n turn, may trgger some rms to stay out and free rde on the estng cartels. Moreover, d erent R&D allance formaton rules may yeld d erent outcomes n terms of stablty of cooperaton (see, for nstance, Y & Shn 000). In some cases, the whole ndustry allance of rms nvestng n R&D can be stable, especally f there are no synerges and f, after breakng the agreement, all rms end up nvestng as sngletons (see, for nstance, Y 003, Bloch 003 and Marn 008 for surveys). However, the stablty of allances s no longer guaranteed f rms are assumed to decde endogenously the tmng of ther nvestment The endogenous-tmng approach was rst ntroduced by Hamlton and Slutsky s (1990) wthn a duopoly game. In ther etensve game wth observable delay, the authors descrbe a two stage set-up n whch, at a preplay stage, two players (duopolsts) decde ndependently whether to move early or late n the basc game (e.g., a duopoly quantty game). If both players announce the same tmng, that s (early, early) or (late, late), the basc game s played smultaneously. If the players tme-announcements d er, the basc game s played sequentally, wth the order of moves as announced by the players. Hamlton and Slutsky s man results are that the two leader-follower con guratons (wth ether order of play) consttute pure subgame perfect equlbra of the etended game only f at least one player s payo as follower weakly domnates her correspondng payo n the smultaneous game. When, conversely, the payo of a follower s lower than n the smultaneous case, the only pure strategy subgame Nash equlbrum prescrbes that both players play smultaneously the basc game. In a symmetrc duopoly wth sngle-valued and monotone best-reples, f rms actons are strategc complements (wth ncreasng best-reples) the follower s payo domnates that of the leader, and therefore that of the smultaneous case. When nstead actons are strategc substtutes (wth decreasng best-reples) the opposte holds and a rst-mover advantage arses: 1 A few recent papers have ntroduced asymmetrc spllovers n a model à la d Aspremont & Jacquemn (1988) by assumng that rms sequence ther R&D actvtes. Whle some of these papers assume a gven eogenous tmng for the nvestment game (Goel 1990, Crampes and angner 003, Halmenschlager 004, Atallah 005, De Bondt 007) some other papers endogenze the tmng of nvestment (Amr et al. 000, Tesorere 008) by adoptng a framework à la Hamlton and Slutsky (1990). The degree of technologcal spllovers s thus shown to be crucal for these games to possess strategc substtutes vs. strategc complements attrbutes and gve rse to smultaneous vs. sequental endogenous tmng R&D equlbra (see Amr et al., 000). However, these models do not consder eplctly the possblty for rms to form research agreements. Our purpose n ths paper s to brdge these otherwse separate streams of lterature, the noncooperatve formaton of R&D agreements and the endogenous tmng lterature, wth the am to study the formaton of research allances when the tmng of R&D nvestments s endogenous. Our approach s novel n that t allows for a far more complete pcture of R&D agreements, by consderng the possble formaton of these agreements over tme. Frms may prefer to wat and enter a research coalton at a subsequent moment of tme. As observed by Duso et al (010), where an nterestng emprcal analyss s performed, rms at each perod 1 See also Dowrck (1985), Boyer and Moreau (1987), Amr (1995), Amr and Grlo (1995), Amr, Grlo and Jn (1999), von Stengel (004) and Currarn and Marn (003, 004) for varous leader-follower and smultaneous payo s comparsons.

4 4 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. n tme weght the bene ts aganst the costs of beng a research cartel member. For eample, a larger number of partcpants (.e. a larger pool of learnng) may ncrease the bene ts of enterng a research cartel (Bloch 1995, Veugelers 1998). Moreover, the ncentve to jon an R&D agreement s stronger n hgh-tech ndustres, due to hgher gans from cooperaton and knowledge transmsson (Cassman and Veugelers 00). These emprcally relevant ssues have been neglected n theoretcal studes. Our approach ams to ll ths gap. It makes t possble to analyze whether the possblty to cooperate n R&D across tme can change the results of the estng R&D lterature, n partcular, the results concernng the endogenous formaton of R&D agreements. Ths paper s organzed as follows. Secton ntroduces the setup adopted n the paper. Secton 3 and 4 apply the model to the game à la d Aspremont & Jacquemn (1988) wth symmetrc and asymmetrc R&D spllovers and present the man results. Secton 5 concludes.. The Model The typcal approach to R&D collaboraton among rms usually assumes that at a rst stage a rm can form an R&D allance wth ts compettors and at a second stage the formed allance decdes cooperatvely ts jont level of nvestment n R&D. At a thrd and nal stage, every rm sets noncooperatvely ts strategc market varable, typcally quantty or prce, to compete olgopolstcally wth all other rms. Our am s to ntroduce a varant of ths setup assumng that at the rst stage a rm decdes not only whether to form or not an R&D agreement, but also the tmng of ts nvestment n R&D. More spec cally, both the R&D agreement formaton process and the tmng of the nvestment are made endogenous. Introducng endogenous tmng bascally determnes at whch stage a sngle rm or an R&D cartel wll play ts nvestment n R&D..1. R&D Allances & Tmng Formaton Game. We magne that at a pre-play stage, denoted wth t 0, every rm sends smultaneously a message to ts rval announcng both ts ntenton to form or not an R&D allance as well as ts preferred tmng for the R&D nvestment. Every rm s message set M can be denoted as: (.1) M = [(f; jg ; t 1 ) ; (f; jg ; t ) ; (fg ; t 1 ) ; (fg ; t )] = 1; and j 6=. The message space contans 16 d erent message pro les m M 1 M, whch, n turn, may nduce the followng set of nonempty R&D tmng-parttons P (m), P = [ f1; g t 1 ; f1; g t ; f1g t 1 ; fg t 1 ; f1g t ; fg t ; f1g t 1 ; fg t ; f1g t ; fg t 1 ]: D erently from Hamlton and Slutsky s (1990) endogenous tmng game appled to a model à la d Aspremont and Jacquemn s (1988) (see Amr et al. 000), here t s assumed that the two rms may also form an R&D cartel at perod t 1 or t. 3 We assume that n order to form a research allance wth a gven tmng of nvestment n R&D requres the unanmty of rms decsons: when rms send messages ndcatng both the same R&D coalton and the same nvestment tmng, they wll sgn a bndng agreement to nvest at the prescrbed On average, four rms enter a research jont venture (RJV) yearly. The average entry decreases wth the age of these RJVs (Duso et al. 010). 3 Note that allowng the two rms to play ther cooperatve nvestment strategy at d erent stages, one at perod t 1 and the other at perod t, does not alter the basc results of the analyss.

5 TIMING AND R&D AGREEMENTS 5 tme; otherwse, they wll nvest as ndvdual rms wth the tmng prescrbed by ther own messages. Formally, for ; j = 1; and j 6= P (m) = f1; g f m = m j = (f; jg ; ) and P (m) = (fg ; fjg j ) f m 6= m j. The above rule prescrbes that f both rms agree to form the same allance and to nvest wth the same tmng, the allance wll be created and thus wll nvest at that gven tme. Conversely, f one rm dsagrees, ether on the allance or on the nvestment tmng, both rms wll play as sngletons the R&D nvestment game, wth the tmng dependng on ther message. The descrbed R&D agreement formaton rule re ects an eclusve membershp rule, where the consensus of all members s requred to complete the agreement. 4 In what follows, we formally ntroduce our model... The Investment Game. Once every rm has sent a message m and a tmngpartton, denoted P (m) P, has been nduced on the set of rms, every rm decdes ts cooperatve or noncooperatve nvestment accordng to the tmng prescrbed by P (m). At ths stage, as well as at the followng stages, t s assumed that a rm cannot manpulate ts level of nvestment n order to convnce ts rval to renegotate the tmng-partton decded at stage t 0. As n d Aspremont & Jacquemn (1988) every rm s assumed to set a nte level of nvestment X R + a ectng ts pro t functon va ts producton cost c ( 1 ; ) whch, n turn, n uences the nal market competton between ndvdual rms. Denotng wth q [0; 1) the nal market competton varable (here quantty), a rm pro t functon can be wrtten as (q ()). In a research agreement f1; g rms wll therefore set cooperatvely ther level of nvestment at stage = t 1 or t.e. (.) c = c 1 ; c such that, for every ; j = 1; and j 6= c = arg ma X (q ( ; j )) =1; gven the pro le of quanttes q = (q 1 ; q ) whch wll be optmally chosen n the nal market stage. If the rms play smultaneously as sngletons at stage = t 1 or t, the approprate equlbrum concept wll be the Nash equlbrum of the smultaneous nvestment game played at stage (.3) = ( 1 ; ) such that, for every = 1; and j 6= = arg ma (q ( ; j )) : 4 For a dscusson on whch coalton formaton rule may be more approprate accordng to the spec c contet, see the materal contaned n Hart & Kurz (1983), Y (003) and Ray (007).

6 6 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. When rms play sequentally, the relevant equlbrum nvestment wll be a Stackelberg (subgame perfect) equlbrum,.e. the pro le (.4) = ( ; g j ( )) such that, for the leader (henceforth rm ) = arg ma (q ( ; g j ( ))) and for the follower ( rm j), g j : X! X j s the best-reply mappng, g j ( ) = arg ma j j (q ( ; j )) : Note that for the nvestment game to be well-de ned, all equlbra n (.), (.3) and (.4) must est and be unque..3. The Market Game. Once the two rms have ether formed a research cartel or have chosen ther R&D nvestment as sngletons at t 1 or t, they set ther market varable n the last stage of the game (denoted wth t 3 ). We assume competton n quanttes and a unque Cournot equlbrum among rms, gven the equlbrum level of nvestment c, or or decded at stages 1, or both. In partcular, the Cournot quantty pro le s smply the vector q = (q 1; q ) such that, for every rm = 1; and j 6= q = arg ma q (q ; q j ):.4. Strateges. Frm strateges n the descrbed mult-stage game can formally be epressed as follows. When the nvestment game s played smultaneously, ether at stage t 1 or t, every rm N strategy set s a trple sm = (m ; ; q ) where, n turn, m s rm -th message, such that m = (S ; ) (fg ; f; jg) (t 1 ; t ) wth S beng a nonempty coalton selected by the rm, : M M j! X s the nvestment choce (a mappng from the message space to a gven nvestment level), and q : X X j! R + the output choce,.e. a mappng from the rm nvestments to a postve level of output. When the nvestment game s played sequentally, the strategy sets are trples seq = (m ; ; q ) and seq j = (m j ; g j ; q j ) ; for the -th leader and the j-th follower respectvely, where the follower nvestment choce s a mappng g j : M M j X! X j..5. Stable R&D Agreements. Gven the equlbrum quanttes of the market game played by rms at stage t 3, and gven the level of nvestment decded smultaneously or sequentally at stages t 1 and/or t ether by the research cartel or by ndvdual rms, all rms receve a pro t that, wth a slght abuse of notaton, can be denoted as (q ( (P (m))), where q ( (P (m)) ndcates the equlbrum quantty pro le when an nvestment pro le, as de ned by (.), or by (.3) or nally by (.4) s decded by the rms n a gven partton P (m) nduced by the message pro le m sent at stage t 0. At ths stage we need to make eplct a concept of equlbrum for the message game played at stage t 0. For ths purpose, we ntroduce two d erent equlbrum concepts. The rst s a standard Nash equlbrum of the R&D partton-tmng game. The second ntroduces a coaltonal stablty requrement, mplyng that a structure P (m) s stable f and only f the message pro le m s a strong Nash equlbrum,.e., cannot be mproved upon by an alternatve message announced by a rm or by a group of rms, here the grand coalton.

7 TIMING AND R&D AGREEMENTS 7 Formally, when a gven tmng-partton P s Nash stable, the pro le = (m ; q ; ) s a subgame perfect Nash equlbrum (SPNE) of the entre game. When, as addtonal requrement, the message pro le m played at t 0 s also strong Nash, s agan SPNE, wth the property to be Pareto-e cent for the two rms. De nton 1. (Nash stablty) A feasble R&D tmng-partton P P s Nash stable f P = P (m ), for some m wth the followng propertes: there ests no m 0 M for every rm = 1; and j 6= such that (q ( (P (m 0 ; m j))) > (q ( (P (m ))). De nton. (Strong Nash stablty) A feasble R&D tmng-partton P P s strongly stable f P = P ( bm), for some bm wth the followng propertes: for m 0 M and f thus for ; j = 1; and j 6= : (q ( (P ( bm))) (q ( (P (m 0 ; bm j ))) (q ( (P (m 0 ; bm j ))) > (q ( (P ( bm))) j (q ( (P (m 0 ; bm j ))) < j (q ( (P ( bm))) Note that a strong Nash equlbrum message pro le bm s both a Nash equlbrum and a Pareto-optmum. We are now ready to apply our framework to d Aspremont & Jacquemn s (1988) wellknown model. 3. Duopoly wth Symmetrc Spllovers Followng d Aspremont & Jacquemn (1988), we assume a lnear nverse market demand functon P (Q) = ma f0; a bqg ; wth Q = P =1 q and a lnear cost functon for every rm decreasng n R&D nvestment, (3.1) c ( ; j ) = (c j ) for j 6=, and c j. In ths setup, the learnng whch results from nvestment n R&D characterzes the producton process, mplyng that margnal and unt costs decrease as the nvestment n R&D ncreases. We allow for the possblty of mperfect approprablty (.e. technologcal spllovers between the rms), by ntroducng a spllover parameter [0; 1]. Obvously the case of no spllovers ( = 0) may only arse n a stuaton of strong ntellectual protecton. More frequently, however, nvoluntary nformaton leaks occur due to reverse engneerng, ndustral esponage or by hrng away employees of an nnovatve rm. The cases of partal to full spllovers can be modelled by settng 0 < 1. At ths stage the parameter n (3.1) s assumed to be dentcal for all rms. However, n Secton 4, ths parameter, though eogenously gven, wll d er due to the cooperatve versus noncooperatve nature and to the tmng propertes of the R&D nvestment game. Moreover, we assume a smple quadratc cost functon for the nvestment n R&D gven by I ( ) = ;

8 8 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. wth > 0. Ths guarantees decreasng returns to R&D ependture (see e.g. Cheng 1984; d Aspremont and Jacquemn 1988). As a result, under Cournot competton n the product market, and settng for smplcty b = 1, the last stage pro t functon for each rm = 1; can be obtaned as a functon of ( ; j ): (3.) (q ( ; j )) = (a c + ( ) + ( 1) j ) 9 : Note that n ths setup, for su cently hgh R&D spllover rates ( > 1=), there are postve R&D cartel-eternaltes, as the formaton of a cooperatve agreement a ects postvely all remanng rms. Only n ths case rms cooperatve choce mples socal e cency, whch nstead s not guaranteed for < 1= Man Assumptons. Some assumptons are now ntroduced to ensure the estence and unqueness of all stages equlbra as well as to smplfy comparatve statcs. A.1 (Quantty stage constrant). (a=c) >. A. (Pro t concavty and best-reply contracton property). > 4=3. A.3 (Boundares on R&D e orts) For every rm, X = [0; c]. Moreover, for < 1=: > a( )(+1) and for > 1=: > a(+1). 4:5c 4:5c As eplaned n detal n the Append, assumpton A.1 smply ensures that the last stage Cournot equlbrum quanttes for both rms are unque and nteror, wth assocated postve pro ts. Assumpton A. guarantees both the strct concavty of every rm non cooperatve payo (3.) n ts own nvestment (guaranteed for > 8 ) as well as a contracton property on 9 every rm best-reples g ( j ), whch requres that > 4. 3 Assumpton A.3 prescrbes a compact R&D nvestment set for every rm and mposes some Inada-type condtons to obtan nteror nvestment equlbra n all noncooperatve (smultaneous or sequental) and cooperatve R&D games (see Amr et al. 000, Amr et al. 011, Tesorere 008 and Stepanova and Tesorere 011). 5 Note that by assumpton A. every rm payo s strctly concave n ts own nvestment choce and thus best-reples are sngle-valued and contnuous. Investment spaces are compact by A.3 and therefore a Nash equlbrum ests by Brower ed-pont theorem. The contracton property mpled by A. ensures unqueness of the Nash equlbrum. The estence of a Stackelberg equlbrum - a subgame perfect Nash equlbrum (SPNE) of the sequental R&D game - s guaranteed by both rm contnuous payo s and contnuous best-reples, thus mplyng that a rm as leader faces a contnuous mamzaton problem over a closed set. Then, by the Weerstrass theorem, a SPNE equlbrum ests. Its unqueness s genercally ensured by the fact that rm best-reples are sngle-valued and monotone and all rm payo s are strctly monotone n ther rval nvestment. In fact, for every = 1; wth j 6=, by A.1 and by (q j = 9 ( 1) (a c + j + j )? 0 for? 1 : 5 For a detaled descrpton of the consequences occurrng to the smultaneous nvestment game when these boundares are volated, see, for nstance, Amr (011).

9 TIMING AND R&D AGREEMENTS 9 Hence, along the best-reply g j ( ) of ts rval s, no multple argma are possble for a -th rm actng as leader. The follower rm wll nstead act à la Nash and, by the property of best-reples, ts nvestment choce wll be unquely de ned. Moreover, the strct concavty of every rm pro t, under the addtonal constrant that the two rms select the same collusve nvestment, mples that also the jont R&D cartel pro t s strctly concave. In ths case, ths s mamzed by a unque nvestment pro le. In the net secton we characterze all stable R&D agreements wth endogenous tmng reached by the two rms. We then etend the symmetrc set-up to the case of asymmetrc spllovers. Ths can o er a broader vew on a recent stream of lterature concernng the endogenous tmng under asymmetrc spllovers (De Bondt and Vandekerckhove 008, Tesorere 008). 3.. Cooperatve R&D. The R&D cartel made of all rms N = f1; g (.e. the grand coalton) nvestng cooperatvely n R&D s assumed to mamze the sum of rms pro ts,.e. P (3.3) (q ( (f1; g P ))) = 1 9 [a c + ( ) + ( 1) j ] : =1 =1 where = ( ; j ) s any arbtrary pro le of R&D nvestment carred out smultaneously by the two rms ether at = t 1 or at = t, for = 1; and j 6=. Followng much of the lterature, we wll assume henceforth that the level of nvestment that mamzes (3.3) s equal for every rm,.e., s such that c = c j. 6 Mamzng the pro t of the R&D cartel n (3.3), and gven the constrant of symmetrc behavour, a rm cooperatve nvestment can be easly obtaned as (3.4) c (f1; g ) = wth an assocated (equal splt) equlbrum pro t (3.5) C q c (f1; g ) = (a c)(1 + ) 9 (1 + ) (a c) 9 (1 + ) : 3.3. Noncooperatve Smultaneous R&D. D erentatng (3.) and eplotng the symmetry of rm payo s, the noncooperatve level of nvestment can be obtaned as (3.6) (f1g ; fg ) = (a c)( ) 9 ( )(1 + ) for = 1;, wth the noncooperatve equlbrum pro t of every rm gven by: By the Pareto-e cency of c emma. N (q ( (f1g ; fg ))) = (a c) (9 ( ) ) (9 ( )(1 + )) : (f1; g ) (for the two rms) we can establsh the followng emma 1. Under hgh (low) spllover rate > 1 ( < 1 ) the cooperatve nvestment level s hgher (lower) than the smultaneous Nash nvestment level,.e. c > ( c < ). Proof. See the Append. 6 As shown by Salant and Sha er (1998,1999), for certan values of the parameters, the jont pro t mamzaton may easly mply unequal R&D nvestments for the two rms.

10 10 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI Sequental R&D Investment Game. Usng agan (3.) we can easly obtan the best-reply of the j-th rm playng as follower the nvestment game: (3.7) g j ( ) = ( )(a c (1 ) ) 9 ( ) : and Therefore the leader and the follower equlbrum nvestment levels are gven by where fg t 1 ; fjg t = ( ) (a c) (3 + ) ( ) j fg t 1 ; fjg t = ( ) (a c) = = (0 + 16) + 3 (64 64) wth assocated equlbrum pro ts gven by q fg t 1 ; fjg t = (a c) ( ) j F q fg t 1 ; fjg t = (a c) ( ) Comparng R&D equlbrum nvestment levels under assumptons A.1-A.3, we can state the followng: Proposton 1. () When rm R&D nvestments are strategc substtutes ( < 1 ) there ests a () and a such that, for < () and <, > > c > j : () When rm R&D nvestments are strategc substtutes ( < 1 ) and () or > > j c : () When rm R&D nvestments are strategc complements ( > 1 ), for = 1; and j 6=. Proof. See the Append. c > > j > The above proposton provdes a full rankng of rm equlbrum nvestment levels, as t combnes the well-known results by d Aspremont and Jacquemn (1988), who compare cooperatve and smultaneous noncooperatve R&D levels, wth Amr et al. (000) analyss, focussng on sequental vs. smultaneous noncooperatve outcomes. emma 1 has already proven that for low (hgh) spllover rates et al. (000) results, mples that > < c ( c > c > ( c ) whch, combned wth Amr s > and > j > ).

11 TIMING AND R&D AGREEMENTS 11 Proposton 1 completes ths rankng by also ncludng the cooperatve nvestment levels. It can be notced (see epresson (3.7)) that the level of spllover s crucal to determne the slope of the follower s best-reply n the nvestment game. Thus, when the spllover rate s very low (case ()), the follower s best-reply s etremely steep (and negatvely sloped) and ths strongly contracts ts equlbrum nvestment, whch s thus even lower than that resultng under a cooperatve agreement. A rm nvestng as leader at stage t 1 can therefore pro tably epand ts nvestment, and ths occurs n partcular when the cost of the nvestment (whch depends on ) s very low and an nvestor s very unlkely to be mtated (low ). Under such crcumstances, beng a leader may become more pro table than nvestng cooperatvely. When nstead spllover rates start to ncrease, the cooperatve nvestment overcomes that of the follower, although the leader s nvestment remans very hgh. Fnally, for > 1=, cooperaton mples the e cent and hghest level of R&D nvestment, regardless of the level of nvestment costs. In what follows we perform some comparsons of the rm payo s obtaned n the d erent nvestment games by combnng the results of emma 1 and Proposton 1 above, wth Amr s et al. (000) analyss. We recall that n Amr s et al. (000) paper, the followng rankng s establshed for smultaneous and sequental payo s n the symmetrc case: (3.8) ( ) > N ( ) > F j ( ) for < 1 (3.9) j F ( ) > ( ) > N ( ) for > 1 where, N and F denote the leader/nash smultaneous/follower roles, respectvely, n the d erent R&D nvestment games. 7 By the e cency of the pro le c, we also know that C c > N ( ). Moreover, the followng lemma proves that for < 1 ( > 1) a follower (leader) payo can never be greater than that of a rm n a cooperatve agreement. emma. Under hgh (low) spllovers > 1 ( < 1 ) the pro t of a rm n an R&D agreement s always hgher than the pro t of a leader (follower), namely, C c > ( ) ( C c > F j ( )). Proof. See the Append. The followng two propostons complete the full rankng of rm payo s n all d erent cases and for all levels of spllover rates. Proposton. When rm R&D nvestments are strategc substtutes ( < 1 ): () there ests a () and a such that, for < () and <, the pro t obtaned by a rm playng as leader n a sequental nvestment game s hgher than that obtaned n a cooperatve R&D agreement, and the followng rankng arses ( ) > C c > N ( ) > F j ( ). () When, nstead () or or both, the followng rankng arses: Proof. See the Append. C c ( ) > N ( ) > F j ( ) : 7 In what follows we mantan the conventon that rm ndcates the leader whle rm j ndcates the follower n the sequental nvestment game.

12 1 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. Fgure 1 and llustrate the e ect of on the nvestment levels and on C c ( ), the d erence between the pro t obtaned by a rm n a cooperatve agreement and that obtaned by the leader n a sequental game. When rm nvestments are strategc substtutes ( < 1=) there ests a narrow range of the spllover rate (between 0 and ) for whch beng leader, and thus epandng the nvestment, turns out to be etremely pro table. Ths occurs only when the cost to nvest n R&D s etremely low ( < ). [FIGURE 1 AND APPROXIMATEY HERE] The proposton that follows completes our ndngs for the model wth symmetrc spllovers by comparng the rms pro tablty under d erent arrangements when R&D nvestments are strategc complements. Proposton 3. When rm nvestments are strategc complements ( > 1 ) the pro t obtaned by a rm n a cooperatve R&D agreement s always hgher than the pro t obtaned by a rm nvestng as follower n the sequental nvestment game, and the followng rankng arses C c > F j ( ) > ( ) > N ( ) : Proof. See the Append. As t can be observed n gure 3, for > 1=, the hghest level of nvestment s selected by the research cartel. Under the sequental game the follower free-rdes on the leader nvestment and gans a hgher pro t. However, as shown n gure 4, the d erence between the cooperatve payo and the follower payo s postve and monotoncally ncreasng wthn the nterval for under analyss. [FIGURE 3 AND 4 APPROXIMATEY HERE] Fnally, the net two propostons characterze all Nash and strongly stable tmng-parttons accordng to De ntons 1 and. Proposton 4. (Nash stablty) () When the spllover rate < (), and <, the Nash stable tmng-parttons are gven by P (m ) = [ f1; g t 1 ; f1g t 1 ; fg t 1 ]: () When 1= > () or or both, the Nash stable tmng-parttons are, nstead, gven by P (m ) = [ f1; g t 1 ; f1; g t ; f1g t 1 ; fg t 1 ]: () Fnally, for (1=; 1], the Nash stable tmng-parttons are gven by Proof. See the Append. P (m ) = [ f1; g t 1 ; f1; g t ; f1g t 1 ; fg t ; f1g t ; fg t 1 ]:

13 TIMING AND R&D AGREEMENTS 13 It s obvous that, f we requre the strong stablty of tmng-parttons, by symmetry all noncooperatve parttons n whch rms nvest smultaneously à la Nash are Paretodomnated by the cooperatve parttons. Formng a cooperatve research agreement to coordnate costly nvestments n R&D s clearly more pro table than playng the symmetrc nvestment game à la Nash. If, however, < (), we have proven that beng leader n the nvestment game yelds a hgher pro t than playng cooperatvely, and therefore the only tmng-partton that remans strongly stable s the grand coalton nvestng at tme t 1. A cooperatve agreement, to be stable, requres that rms antcpate strategcally ther jont nvestments. Proposton 5. (Strong stablty) () when the spllover rate < () and <, the only strongly stable R&D tmng-partton s P( bm) = f1; g t 1 : () - () When 1 () or or both, the strongly stable R&D tmng-parttons are P( bm) = f1; g t 1 ; f1; g t : Proof. See the Append. Our results depart from those obtaned n the prevous lterature. In partcular, snce n our set-up rms can form a strategc allance to nvest cooperatvely n R&D, d erently from Amr et al. (000) the allance of rms always consttutes a SPNE of the whole game. However, our model suggests that n formng allances rms have to consder carefully the e ect of tmng. If a group of rms procrastnates ts cooperatve nvestment, t may rsk a defecton by a partner breakng the allance to nvest as leader. To avod ths problem, rms have to antcpate strategcally ther jont nvestment n R&D. As llustrated n detal, ths happens only when nvestng n R&D s not very costly and spllovers are very low. For hgher spllovers, to dscplne the stablty of a research cartel s easer and tme-constrants for the nvestment are no longer requred. Our model also shows that, wthout requrng Pareto-optmalty, the noncooperatve smultaneous (or sequental) con guratons are also stable under low (hgh) spllover rate, < 1= ( > 1=), as already establshed n Amr et al. (000) An Etenson to n-symmetrc Frms. The etenson of our model to n-symmetrc rms would allow to check the stablty of more comple allances between rms coordnatng ther nvestment n R&D. However, ncludng more than two rms n our analyss wth endogenous tmng makes the model hghly unmanageable. Only ntutve conclusons can be drawn wth the help of our prevous analyss and some well known estng results. A rst observaton concerns the whole ndustry R&D agreement (grand coalton of rms) nvestng at stage t,.e., usng the model notaton, the tmng-partton P = fng t whch s formed when at stage t 0 all rms = 1; ; ::; n send the message m = (fng ; t ). Ths partton can be strongly stable f every ndvdual rm nvestng as follower at stage t would be better o than any rm partcpatng to an R&D agreement nvestng at stage t 1 as leader. Thus, any coalton S N of rms that devates from the grand coalton fng t by sendng one of these alternatve messages, m 0 S = (fsg ; t ) or m 00 S = (fsg ; t 1), would nduce ether the smultaneous partton (3.10) P (m 0 S) = (fsg t ; fjg t jnns );

14 14 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. where all j-th rms outsde S are sngletons or, analogously, the sequental partton (3.11) P (m 00 S) = (fsg t 1 ; fjg t jnns ): However, f rms n coalton S cannot mprove upon partton fng t by playng as leaders n (3.11) they would not mprove a fortor by playng smultaneously n (3.11). Therefore, f we show that n the partton (3.11) all rms wthn the research cartel S (regardless of ts sze) do not mprove upon the cooperatve partton fng t, the stablty of the grand coalton agreement s proved as a result. When nvestment decsons are strategc complements, t can be proved that the payo of a symmetrc rm playng as sngleton follower aganst the coalton S playng as leader s always hgher than the payo of every rm n S. 8 Hence, gven the e cency of the grand coalton, t would be mpossble for any coalton S to mprove by devatng as leader, gven that followers would mprove even more ther payo s. Smlarly, t can be shown that when R&D nvestments are strategc substtutes ( < 1=) a coalton S N made of followers s beaten by ndvdual rms nvestng as leaders, and therefore the partton fng t 1 - made by the grand coalton of rms nvestng at stage 1- s strongly stable. The strong stablty of these two cooperatve tmng-parttons already observed n our duopoly model thus etends to an analogous endogenous tmng game played by n-symmetrc rms. 4. Duopoly wth Asymmetrc Spllovers Introducng asymmetrc spllovers equals to ntroducng a hgher degree of realsm nto the model. As s well known (see e.g. Atallah 005), asymmetres may derve from d erences n protecton practces, from geographcal localzaton (e.g. Pett et al. 009), from product d erentaton (Amr et al. 000), or from sequental moves n the R&D game, as n R&D models wth endogenous tmng (Tesorere 008). Other sources of asymmetry can arse from d erent technologcal capabltes, as n Amr and Wooders (1999, 000), where knowledge may leak only from the more R&D-actve rm to the rval, or from a better absorpton capacty n uencng the outcome of a technologcal race, as n De Bondt and Henrques (1995). The spllover asymmetry arsng n our model stems nstead from the cooperatve versus the non-cooperatve nature of the R&D game and from the tmng of the R&D nvestment process. The parameter, (0 1) wll represent henceforth the ncomng spllover for rm = 1;. Moreover, let N denote the rm spllover rate under smultaneous noncooperatve R&D, C the spllover rate under R&D cooperaton, and, j F the spllover rates for the leader and the follower n the sequental nvestment game, wth ; j = 1;, 6= j. Our assumptons on spllovers asymmetry are based on the followng consderatons: () When the two rms nvest smultaneously and noncooperatvely at stage one or two ther spllover rate s assumed to be symmetrc and lower than or equal to 0:5 (.e., 1 N = N 0:5). The dea s that the competton n R&D and the smultanety of rm decsons do not allow for a hgh amount of knowledge transmsson. () When a noncooperatve sequental nvestment n R&D takes place, the spllover rate can be though to be favorable to the rm playng as follower and unfavorable to the rm playng as leader (.e. j F > ). In partcular we shall set j F > 0:5 and 0:5. A sequental order of moves n the R&D nvestment game mples a greater amount of 8 See for a formal proof of ths fact, Currarn and Marn (003, 004).

15 TIMING AND R&D AGREEMENTS 15 knowledge leakng out from the leader to the follower than vce versa. The ratonale s that knowledge leaks also through mtaton, thus leadng to a strong advantage for the rm that s able to observe the rst mover nnovatve outcome. Therefore, bene ts from spllovers should be lower for a rst mover (see also Tesorere 008). Moreover, we assume that sectorspec c features determnng the ntensty of knowledge d uson a ect to the same etent the ncomng spllover for the leader n the sequental game (.e. ) and the ncomng spllovers for both rms n the smultaneous noncooperatve game (.e., N = 1; ). Therefore we wll set = N. () When the two rms play cooperatvely and form a research cartel, they generally also agree to share to some etent the knowledge obtaned from ther jont R&D e ort. It seems realstc to assume that they mght agree to fully share ther knowledge, and therefore ther spllover rates wll be symmetrc and su cently hgh (.e. 1 C = C close or equal to one). Moreover, we assume that knowledge leaks occurrng manly through mtaton and favourng the follower n a sequental game are less ntense f compared wth the voluntary echange of technologcal knowledge typcal of a research agreement. Thus, we mantan that C > j F, for ; j = 1;, 6= j: Takng nto account all the above nequaltes, our assumptons on the relatonshp among spllover values can be summarzed as follows: (4.1) 1 C > F j > = N 0 = 1;, j 6= wth = N 0:5 and F j > 0: Man assumptons. Also n ths secton we ntroduce some assumptons needed to ensure the estence and unqueness of equlbra at all stages: B.1 (quantty stage constrant). As n the case of symmetrc spllovers, a=c >. B. (Pro t concavty and best-reply contracton property). Agan, > 4=3. B.3 (Boundares on R&D e orts) For every rm = 1;, X = [0; c] and > a(c +1) 9c, for 0:5 < C 1. Assumpton B.1 does not vary wth respect to the symmetrc case. Assumpton B. deals wth the strct concavty of every rm s pro t (3.) wth respect to own nvestment. Gven our assumptons on spllovers n the noncooperatve smultaneous game,.e. N 0:5, the SOC requres that > 8. kewse, the SOC for pro t mamzaton n the case of a 9 research cartel, gven that C 1, requres, n the most strngent case, that > 8 as well. 9 The contracton propertes on both rms best-reples g ( j ) and g j ( ) ntroduced n secton 3.1 are stll vald and mpled by > 4=3 (see the Append). Assumpton B.3 follows the same logc of A.3. As prevously dscussed, from the strct concavty of pro ts t comes out that each best reply s sngle-valued and contnuous. The estence of a Nash equlbrum s therefore guaranteed. The contracton property n B. ensures unqueness of the Nash equlbrum. In order to have nteror equlbra t has to hold true that: for ; j = 1;, 6= j. ( (q )) = ( j )[a c(1 )] > It su ces assumpton B.1 for the above epresson to be strctly

16 16 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. 4.. Noncooperatve Sequental R&D wth Asymmetrc Spllovers. Snce only n the case of a game wth sequental moves at the nvestment stage our calculatons d er from the symmetrc case analyzed n the prevous sectons, we shall deal henceforth etensvely wth ths scenaro. Our man am s to nvestgate whether the asymmetry n the transmsson of knowledge between rms s relevant for the endogenous formaton of research allances. Usng an asymmetrc-spllover spec caton, every rm s objectve functon at the market game stage s gven by = (a (q + q j ))q (c j j ) q wth ; j = 1; and 6= j. Solvng the game by backward nducton, every rm s payo at the nvestment stage can be obtaned as: (4.) (q ( ; j )) = 1 9 [(a c) + ( j) + ( 1) j ] : D erentatng (4.) we get the best-reply for the follower n the nvestment game: g j ( ) = ( )[(a c) (1 F j ) ] 9 ( ) Hence, the sequental equlbrum nvestment levels for the two rms are gven by: fg t 1 ; fjg t = 9 a c+(a c)c + B F A A j j F +(F j 1)B 9 j fg t 1 ; fjg t = ( ) 8 A + F j where A = ; B = j F 1 1, C = 1 : et the assumptons on spllovers n equaton (4.1) as well as assumptons B.1-B.3 hold. Comparng rm R&D equlbrum nvestment levels under asymmetrc spllovers, we can state: Proposton 6. There ests a ~ (0; 1=) such that, f N =, ~ then j c >. If nstead N =, ~ then c j > >, for ; j = 1; 6= j. Proof. See the Append. An llustraton of ths result s shown n Fgure 5. To gve an ntuton, when spllovers are low, the asymmetry between the ncomng spllover of the leader ( ) and that of the follower ( F j ) s strong (snce F j s always greater than 0:5). Therefore, the leader has the lowest ncentve to nvest n R&D snce the hgh outgong spllover e ect overcomes the rst-mover advantage e ect. Conversely the follower takes advantage of a hgh learnng opportunty and of low knowledge leaks. Moreover, n ths case, the R&D nvestment of the follower overcomes that of the cooperatve rm, snce a competton e ect prevals. Conversely, when spllovers >

17 TIMING AND R&D AGREEMENTS 17 are hgh, the asymmetry between leader and follower decreases. In ths case a free-rdng e ect may preval for both players and the cooperatve outcome may become convenent, snce cooperaton between rms succeeds n nternalzng knowledge eternaltes. Frms pro ts could be compared only va numercal smulatons. In what follows the numercal values assgned to the parameters are as follows: a = 38, c = 18, =. In addton, we assume that n the case of cooperaton rms agree to share a hgh amount of technologcal knowledge. Thus we assgn a constant value C = 0:8. Moreover we set the ncomng spllover of the follower such that 1 C > j F > 0:5 (for nstance j F = 0:6 as n Fgures 5 and 6). As depcted n gure 6, there ests a value ^ (0; 1=), such that the followng payo rankng emerges: j F ( ) > C c > N ( ) > ( ) for = N ^. As a result, n ths case the Nash equlbrum tmng-parttons are P (m ) = [ f1; g t ; f1g t ; fg t ]; whle the strongly stable parttons are gven by P( bm) = f1; g t : When nstead 1 = N ^, the followng payo rankng comes out: and then C c > F j ( ) > N ( ) > ( ) P (m ) = [ f1; g t 1 ; f1; g t ; f1g t ; fg t ]; P( bm) = f1; g t 1 ; f1; g t : [FIGURES 5 AND 6 APPROXIMATEY HERE] These results can be eplaned by consderng that jont cooperatve agreements across tme or at tme t 1 are partcularly at rsk when there s a strong advantage to be follower n the R&D nvestment game. As a matter of fact, rms prefer to wat and observe the rval s move rather then tryng to reach an agreement. Ths happens n partcular when spllovers are etremely unbalanced (.e. when 1 = N ^) towards rms that wat before nvestng, thus conferrng a strong "follower advantage". Our ndngs complement the few estng results (Amr et al., 000; Tesorere, 008) on endogenous sequencng n R&D nvestment wth asymmetrc spllovers. In partcular, Tesorere (008) consders only the noncooperatve case wth etreme spllovers (1 = N = 0 and j F = 1). Under these values he proves that the only tmng con guraton whch s a SPNE nvolves smultaneous noncooperatve play at the R&D stage (wth zero spllovers). In contrast, n our setup the noncooperatve smultaneous con guraton may not be the only Nash stable tmng-partton and n addton s never strongly stable, as rms prefer to form an R&D cartel than playng (suboptmally) as sngletons the nvestment game.

18 18 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. 5. Concludng Remarks Ths paper consttutes a rst attempt to brdge two usually dstnct streams of the economc lterature, one dealng wth the endogenous formaton of R&D agreements, the other wth the endogenous tmng of R&D nvestments n a model wth spllovers à la d Aspremont and Jacquemn (1988). Ths s done by ntroducng a new set-up n whch rms epress both ther ntenton to form or not an allance as well as the tmng of ther e ort n R&D. Ths allows to assess the stablty of research cartels aganst devatons occurrng across tme. Every rm can epress ts wllngness to play cooperatvely or noncooperatvely as leader or follower accordng to the crcumstances. Our results show that the nature of the nteracton among the rms n the nvestment game plays an mportant role. In partcular, under symmetrc spllovers and when the level of spllovers s etremely low (and thus R&D nvestments are hghly substtutes) both rms want to play the nvestment game as leaders and, as a result, they may easly end up nvestng smultaneously ether cooperatvely or noncooperatvely. In ths case, any cooperatve agreement, to be stable, must contan a commtment to nvest at tme 1. A cooperatve agreement of ths sort would reman stable also aganst devatons by coaltons of rms, f we nclude n the model a number of symmetrc rms hgher than two. When nstead R&D nvestments are strategc complements, our model predcts that both sequental (noncooperatve) and smultaneous (cooperatve) R&D con guratons are stable aganst ndvdual devatons. However, only cooperatve agreements are strongly stable and, n ths case, the tmng of nvestment seems rrelevant for the stablty of cooperaton. Fnally, when spllovers are asymmetrc and favourable to the rm nvestng as follower, the model shows that an R&D agreement to be stable requres that the jont nvestment s strategcally delayed as to avod that a rm may break the agreement to eplot the estng "second-mover advantage". Ths occurs, n partcular, when the ncomng spllover of the follower s much hgher then that of the leader. Proofs of emmata and Propostons 6. Append Proof of emma 1. For every ; j = 1; wth j 6= c (q ) ; c j (q ) > (q ) ; j (q ) c (q ) ; j (q ) where the rst nequalty s due to the Pareto-e cency of c and the second by the Nash equlbrum property. Thus, by monotone negatve (postve) eternaltes for < 1 ( > 1 ), t follows that c < ( c > ). Proof of Proposton 1. ()-() For [0; 1=), the followng equaton (6.1) (a c) ( ) (a c)(1 + ) j c = 9 (1 + ) = 0 p can be solved for () = p 5 +, whch s strctly postve for, where = 16=9. Condton A.3 for < 1= requres that > a( )(+1) and snce ths constrant 4:5c reaches ts mamum for = 1=, t follows that for a ; 16 c 9 ; there ests a () [0; 1=) for whch j c < 0. It can be checked that the de ned nterval for s compatble wth a market sze-cost rato a=c 3=9. Moreover, by (6.1) for 1= > > ()

19 TIMING AND R&D AGREEMENTS 19 and/or for a > 16=9, j c > 0. Combnng these facts wth Amr s et al. (000) rankng on leader-follower and Nash smultaneous nvestments, the results follow. () For (1=; 1], by (6.1), t turns out that j c < 0: Moreover t can be easly checked that sgn j = sgn ( 1) > 0 whch holds for any and, thus, also for (1=; 1]. Agan, combnng the above fact wth Amr s et al. (000) results, the rankng between R&D nvestments s proven. Proof of emma 3. Suppose by contradcton that for > 1 and by (3.8) It follows that C c < ( ) C c < ( ) < F j ( ) : P C c < ( ) + j F ( ) =1 contradctng the e cency of pro le c (q ). Smlarly, let for < 1 and by (3.9) whch agan mples whch s a contradcton. C c < F j ( ) C c < F j ( ) < ( ) : P C c < ( ) + j F ( ) ; =1 Proof of Proposton. () By emma and 3 and by (3.8)-(3.9) we know that, under low spllover rates, ( < 1 ), ether (6.) ( ) > C c > N ( ) > F j ( ). or (6.3) C c > ( ) > N ( ) > F j ( ) : For [0; 1=) the followng equaton (6.4) C c ( ) = (a c) (a c) ( ) 9 (1 + ) p has only one root () = p 5 +, requrng that < 16=9 to be postve. Snce by a( )(+1) A.3 > and such constrant reaches ts mamum for = 1=, we conclude that 4:5c for a ; 16 c 9 there ests a () [0; 1=) ensurng that the nequalty C < 0 holds true. () For [0; 1=), when ether or > 16 or both, t can be assessed 9 that (6.5) C c ( ) = (a c) (a c) ( ) 9 (1 + ) The payo s rankng can therefore be completed as stated n (6.3)-(6.) usng emma and Amr s et al. (000) results. = 0 0:

20 0 MARCO A. MARINI, MARIA. PETIT, AND ROBERTA SESTINI. Proof of Proposton 3. By emma and 3 and by (3.8)-(3.9) we know that under hgh spllover rates ( > 1 ), ether (6.6) F j ( ) > C c > ( ) > N ( ) or (6.7) C c > F j ( ) > ( ) > N ( ) : For [1=; 1] and a(+1) ; 1, the equaton 4:5c C F j = (a c) 9 (1+) (a c) (9+8 8)( ) = 0 s solved only for = 1=: It can be checked that for any other spllover rate 1 > 1=, the d erence C j F s postve and ncreases monotoncally n. Only for! +1, t occurs that C j F! 0. Proof of Proposton 4. () By proposton, for [0; ()) < 1= and <, nvestng as leader at stage t 1 s more pro table for rms than formng a cooperatve agreement. As a result, the message m = f1; g t cannot be Nash-stable, because a rm can pro tably devates wth an alternatve message m 0 = (fg ; t 1 ), thus nducng the tmng-partton fg t 1 ; fjg t. Smlarly all sequental tmng-parttons fg t 1 ; fjg t can pro tably be objected by the j-th rm who, nstead of playng as follower, would prefer to nvest smultaneously. Ths s feasble f t sends the message m 0 j = (fjg ; t 1 ) ; and thus nduces the tmng-partton fg t 1 ; fjg t 1. Therefore we reman wth only two parttons f1; g t 1 and f1g t 1 ; fg t 1 that cannot be pro tably objected by any ndvdual rm. () We know by proposton that when [ (); 1=) the payo ganed n a cooperatve agreement s hgher than that obtaned by a leader (follower or smultaneous) rm, and therefore both cooperatve tmng-parttons f1; g t 1 and f1; g t are Nash-stable. Also the smultaneous partton f1g t 1 ; fg t 1 cannot be objected by ndvdual devatons. () For (1=; 1], by proposton 3 the payo ganed n a cooperatve agreement s the hghest obtanable by a rm, and thus, both cooperatve tmng-parttons f1; g t 1 and f1; g t are Nash-stable. Also the sequental parttons f1g t 1 ; fg t and f1g t ; fg t 1 cannot be pro tably objected nether by the leader nor by the follower (see proposton 3), and the result follows. Proof of Proposton 5. () Ths result easly follows from proposton and by the fact that all other tmng-parttons are Pareto-domnated by a cooperatve agreement, wth the ecepton of the sequental partton fg t 1 ; fjg t. However, snce by proposton, N ( ) > j F ( ) for < 1=; the sequental partton fg t 1 ; fjg t can pro tably be objected by the follower, who prefers to nvest smultaneously and, that, by sendng the message m 0 j = (fjg ; t 1 ) can nduce the smultaneous partton fg t 1 ; fjg t 1. However, the latter partton can, n turn, be objected by a message f1; g t 1 sent by both rms, and therefore, s not strongly stable. Fnally, also the partton f1; g t can be objected by a rm sendng an alternatve message m 0 = (fg ; t 1 ), hence nducng the relatvely more pro table sequental tmng-partton fg t 1 ; fjg t. () By proposton and 3 t follows that for [ (); 1] all sequental and smultaneous Nash tmng-parttons payo s are domnated by cooperatve agreements. As a result, the two message pro les m = (f; jg ; t 1 ) ; (f; jg ; t 1 ))