Economics of Open Source Technology: A Dynamic Approach

Transcription

1 Economcs of Open Source Technology: A Dynamc Approach Jeongmeen Suh Soongsl Unversty Murat Yılmaz Boğazç Unversty February 8, 2018 Abstract We analyze open source lcensng and ts effects on frms decsons whether to use the open source technology or not and on the ncentves for nnovaton, through a dynamc model of nnovaton and competton n an envronment wth a ladder type technology. We model the basc features of the General Publc Lcense (GPL), one of the mt popular open source lcenses and study how frms behave under ths lcense when competton s present. Under the GPL, any nnovatve fndngs usng open source technology must also be open source n the subsequent perods, and ths oblgaton creates a trade-off. We focus on how ths trade-off affects ncentves to use and buld up the open source technology. Keywords : open source technology, General Publc Lcense, olgopolstc competton, nnovaton. We are grateful to Barton Lpman for hs advce and support. Dlp Mookherjee and Chng-to Albert Ma for ther useful feedback. We would also lke to thank Andrew Newman, We also thank the edtor, assocate edtor and two anonymous referees for ther constructve comments. We also thank the partcpants at Boğazç Unversty-CED Mcroeconomcs Occasonal Workshop. Mare Cure Career Integraton Grant, FP7PEOPLE-2011CIG s acknowledged. Soongsl Unversty, Department of Global Commerce, 369 Sangdo-Ro, Dongjak-Gu, Seoul, Korea. Emal: jsuh@ssu.ac.kr. Phone: Boğazç Unversty, Department of Economcs, Bebek, Istanbul, Turkey Emal: muraty@boun.edu.tr. Phone: Web ste: 1

2 1 Introducton Open software development nvolves a major devaton from the prvate nvestment model of nnovaton; open source nnovators freely share the propretary software that they have developed at ther prvate expense. For example, Lnux, a computer operatng system, s evolvng wth many ndependent developers revealng the code to develop and refne t. Its source code s open n the sense that anyone has free access to t. One of the mt popular web servers has always been an open source software. 1 The success of open source software rases many questons about nnovaton polces wth nontradtonal property rghts. 2 We partcularly pay attenton to the fact that, although the source code of open source software s freely avalable, open source programs are dstrbuted under very precse lcensng agreements. The broad purpe of ths paper s to mprove our understandng on how a certan form of lcense affects frms ncentves of both nnovaton and partcpaton n an open source communty. Specfcally, we provde a dynamc model that captures the mportant characterstcs of open source lcensng and explan, to some extent, the open source development phenomenon. To capture mportant features open source nnovaton and ts lcensng have, we focus on a concrete example, the GNU General Publc Lcense (GPL) whch s one of the mt common lcenses. 3 Among a number of features GPL has, we can summarze some of the crucal ones as follows. Frst, whle every user has the freedom to use and modfy programs subject to the GPL, such modfcatons must be dstrbuted under the terms of the lcense tself f they are to be dstrbuted at all. Second, the GPL does not preclude the commercal explotaton of the software, at any stage. That s, the program users have to mantan the free access to the source, but they do not need to share any proft they make. It s evdent that there s strong competton n ths feld. Once open source code s mproved by a frm, by ts nature, t s accessble to ts customers or even to ts compettors. However, due to ts complexty of programmng, the nventor can enjoy advantageous pton as the frst mover for a span of tme. In ths paper, we study these major features of open nnovaton and ther effects on ncentves for nnovaton and usage of the open source technology. In order to capture these key characterstcs, namely the dynamc nature of the open source development and the GPL restrctons (successful 1 For example, t s well known that Apache, an open source software web server has been domnatng the publc Internet web server market ever snce Aprl Another successful open source software example mght be Perl, a robust scrptng language whch s used for hgh-traffc webstes such as Prcelne.com, Tcketmaster, and IMDb. 2 See Lerner and Trole (2002) for a broad dscusson of ssues concernng economcs of open source technology. 3 GNU s a recursve acronym whch stands for GNU s Not Unx. 2

3 nnovatons beng subject to GPL) wth allowance of open source technology frms makng proft, we use a T-perod 3-stage model where, n each perod, frms decde whether to use the open source technology n the frst stage, pursue technology advancement through ct-reducng nvestment n the second stage, and engage n Cournot competton n the thrd stage. Wthn the framework of our model, whch we beleve reflects these mportant features of the open source nnovaton n a more drect way than the exstng lterature does, we study the open source technology usage decsons of the frms as well as ther nvestment decsons n nnovaton and competton quanttes. We also extend the model nto an nfnte horzon case. We characterze when t s optmal for a frm to jon the open source communty and when t s not, together wth the nvestments n nnovaton. The man tradeoff n our analyss s stemmng from the very nature of the GPL lcensng. When a frm s decdng to jon the open source communty, t wll be able to use the open source technology at no drect ct, and ths wll potentally remove a ct dsadvantage n the competton stage aganst other frms. However, f ths frm, after jonng the open source communty, succeeds n nnovaton or n ct reducton, then n later perods t has to make t publc due to lcensng, and thus, mght le a potental ct advantage, whch may not be the case f nstead t stayed out of the open source communty. Ths may decrease the expected future revenue of the frm, when joned to the open source communty. On the other hand there s also a drect beneft of jonng the open source communty. We nvestgate how ths trade-off nfluences frms open source technology use decson dependng on ther technology level relatve to the open source technology level. We fnd that ths tradeoff s resolved n a way that when a frm s at the same level wth (or at a hgher level than) the open source technology, t does not prefer to jon the open source communty. If, however, t s behnd the open source n the technology ladder, t optmally choes to use the open source. Thus, f the open source communty succeeds consstently n nnovaton (ct reducton), then the open source communty wll sweep out the propretary frms, otherwse there wll be a set of frms at a hgher level than the open source technology. The lterature on economcs of open source technology has been growng snce early 2000s. Lerner and Trole (2002, 2005) provde a general dscusson of the economcs of open source development and lay out a broad lterature revew. They pont out that the open source developers receve a drect effect n the form of mproved open source technology, snce they drectly beneft from t, and an ndrect effect through sgnallng ther abltes and through reputatonal gans. They show that the lterature mtly consders ndvdual motves, ncentves to adopt open source softwares and the effect 3

4 of competton wthn an open source communty. 4 The man contrbuton of our study s to present a smple model whch s both dynamc and tractable, and t s capable of capturng the essence of the GPL lcensng. A good amount of the lterature focuses on the open source development as a publc good and uses a statc approach (e.g., Johnson 2002, Llanes and Elejalde 2013, Atal and Shankar 2014). Our model and approach dffers from these studes n the way t captures the essence of the GPL lcensng n open source nnovaton envronments. Snce these studes use a statc model, they do not fully capture the man characterstc of the GPL, whch s bascally get t for free now, pay back when/f you succeed. We beleve, our model captures the essence of the GPL n a more drect way, when a frm uses the open source technology, then ts successful nnovaton s made avalable n later perods, yet n the current perod wth the nnovaton success, the frm can enjoy an advantage n the competton stage. Among the studes whch use a dynamc approach, Athey and Ellson (2014) use a dynamc model where the open source user/programmers are motvated by recprocal altrusm. The evoluton of the open source technology depends on the qualty and the altrustc developers. Btzer, Schrettl and Schröder (2007) provde a dynamc model of prvate provson of a publc good and focus on the ntrnsc motvaton of the programmers to explan the open source development. We are not modellng the frms/developers as altrustc or ntrnscally motvated players, rather they are strategc agents who wsh to maxmze expected profts. Fnally, among frms decson problems related to open source technology, we focus on whether a frm jons an avalable open source communty or not. Caulkns et al. (2013) study the queston of how long does a frm keep ts software propretary and when does t release t to be open source technology, over a contnuous tme dynamc model where frms nvest n qualty and pck own prce for ts software and complementary product. Kort and Zaccour (2011) study a smlar problem through a 3 stage duopoly game, where they characterze the condtons under whch t s optmal for a frm to open ts code. 5 Though these questons are also mportant, our model addresses a more fundamental queston, thus may complement ther fndngs. Secton 2 depcts the model. Secton 3 solves the model and provdes the man results for fnte horzon and Secton 4 extends t to the nfnte horzon. In Secton 5, we dscuss some relevant ponts and extensons. Secton 6 concludes. 4 See Lerner, Pathak and Trole (2006) for an emprcal study on the dynamcs of contrbutons to open source software projects. 5 Also see Haruvy, Seth and Zhou (2008). 4

5 2 The Model There are M 2 frms nteractng over T 3 many perods. Each perod t, each frm produces a good at a frm-perod specfc unt ct c(k t ), whch depends on the frm s technology level kt and s stochastcally determned by frm s nvestment n ct-reducng nnovaton. There s also a publc producton technology, called open source technology, whch can produce the good n perod t at a unt ct c(k) t > 0, whch depends on the open source technology level, k, t whch s also stochastcally determned by the open source technology usng frms nvestments n ct-reducng nnovaton. Chan of events wthn a perod: In each perod, there are three stages: (1) each frm decdes whether to adopt the open source technology or not, (2) each frm nvests n ct-reducng nnovaton, (3) frms compete n quanttes n a Cournot fashon. To capture the effect of open source technology under GPL, we make the followng assumptons. Assumpton 1 Each frm s free to use the open source technology at no drect ct. Assumpton 2 Any nnovaton made by a frm whch uses the open source technology n perod t, wll be open source technology from perod t + 1 on. Producton ct: To be more precse, let k t Z + denote the producton technology level for frm at the begnnng of perod t. The unt ct of frm s gven by the functon c(k t ), wth c ( ) < 0 and lm k c(k) = 0. That s, the unt ct s strctly decreasng as the producton technology level ncreases and n the lmt as the technology level k goes to nfnty, the unt ct goes to zero. Let k t Z + denote the producton technology level of the open source technology at the begnnng of perod t. That s, before perod t starts, the publc producton technology level s able to produce the good at a unt ct c(k t ). Lkewse, k 1 s the ntal technology level of the open source technology at the begnnng of perod 1. We keep ths general form for the ct functon through out the model and through our results. 6 Open source technology use decsons: At the frst stage of each perod t, each frm who has not used the open source technology before decdes whether to use the open source technology or not. 6 In case of open source software, usng more advanced or upgraded software may reduce the unt ct by lessenng the repettve efforts to fx bugs n the prevous verson or by processng the same volume of data at a faster rate than before. Thus, although havng a hgher technology level may reduce the fxed cts, we also beleve that a unt ct functon that decreases wth the technology level s also plausble. For nstance, open source based commercal softwares n the Project Management Software product market such as Project-open, Collabtve, and egroupware may serve as a good example wth such a ct functon. 5

6 Let d t {0, 1} denote ths open source technology use decson of frm at the frst stage of perod t, where 1 stands for use decson and 0 stands for not use decson. We call a frm wth d t = 0 a non-user frm and a frm wth d t = 1 a user frm. When a frm s ndfferent between usng and not usng the open source technology, we assume that t choes to use t. If a frm j has already used the open source technology at some perod t, then d t j = 1 for each t t. 7 Also, a frm can freely use the open source technology at no drect ct (Assumpton 1 above). Let κ t (d, k ) denote the technology level of frm after ts open source technology use decson d t, before whch t had a technology level k. That s, κ t (d, k ) = d t max{k t, k t } + ( 1 d t ) k t. Note that κ t = k t for each t, snce the open source technology has no use-not use decson. Investment n nnovaton: At the second stage of each perod t, each frm decdes how much to nvest n nnovaton. We model nvestment n nnovaton through probablty of success: frm wth a technology level k t and open source technology use decson dt, pcks a probablty of success, p(d t, kt ),8 at a ct C(p(d t, kt )) wth C > 0 and C > 0. Wth probablty p(d t, kt ) there s success and the frm advances one level n the technology ladder, that s, acheves a new level, κ t + 1, and reduces ts unt ct. Wth probablty 1 p(d t, kt ) the frm fals to advance one level and stays at the current technology level κ. We denote the realzaton of the new technology level wth K t. If a frm s usng the open source technology at perod t, then ts technology level depends on the open source s current technology level and ts own success/falure outcome. More precsely, K t κ t = + 1 wth probablty p(dt, kt ) κ t wth probablty 1 p(d t, kt ) If d t = 1, then K t max{k t =, kt } + 1 max{k t, kt } f success f fal The open source technology level depends on ts prevous perod technology level and the prevous perod s success/falure outcome of the frms that were usng the open source technology. That s, the 7 Here we assume that a frm who has already joned the open source communty cannot leave t. A more general way to model t would be to allow the frms to leave the open source communty whenever they want, and show that they wll not leave t n the equlbrum. We dscuss ths n the Secton 5. 8 We wrte p(d t, k t ) nstead of p(κ t ), snce two frms wth the same κ t mght choe dfferent probabltes f they have dfferent d t. 6

7 successful nnovaton by a user frm s reflected on the open source technology wth exactly one perod lag (Assumpton 2 above). More precsely, K t = k t = k t wth probablty 1 [1 p(1, k t 1 k t 1 wth probablty [1 p(1, k t 1 where n 1 (k t 1, t 1) s the number of frms wth d t 1 = 1 n perod t 1. ) n 1(k t 1,t) ) n 1(k t 1,t) Cournot competton: At the thrd stage of each perod t, frms engage n quantty competton a la Cournot. Each frm smultaneously decdes how much to produce, q K t, when t has a technology level K t at the begnnng of the thrd stage. A frm wth Kt has an expected nverse demand gven by P K t P (Q K t ) = A Q K t, where A > 0 s suffcently large, P s the market prce, and Q K t s the total quantty demanded that a frm wth K t expects. Ths total quantty can be decomped nto two parts, ts own quantty, whch s known, and expected total quantty of all other frms, that s, Q K t = q K t + Q t. At the end of ths stage, each frm realzes ts proft level πt K n the Cournot competton. Overall payoff of a frm: Frm s overall payoff s ts dscounted sum of wthn perod Cournot profts and ct of nvestment. That s, Π = where δ s the dscount factor of frm. T δ t 1 [πk t C(p(d t, k)) t t=1 Dstrbuton of frms: At the begnnng of the frst stage of perod t = 1, there are n(k, 1) frms wth unt ct c(k) where k n(k, 1) = M and k {0, 1, 2}. Here, 1 n n(k, 1) refers to the frst perod. We assume that the ntal technology level of the open source s k 1 = 1. Thus, there are frms whch are at a lower technology level than the open source technology, frms whch are at a hgher technology level than the open source technology, and frms whch are at the same technology level as the open source technology. The number of frms that have the technology level k 1 = 1 at perod t = 1, n(1, 1), s comped of user and non-user frms: n(1, 1) = n 0 (1, 1)+n 1 (1, 1), where n 0 (1, 1) and n 1 (1, 1) denote the number of non-user and user frms, respectvely. We assume that each of n(0, 1), n 0 (1, 1), n 1 (1, 1) and n(2, 1) are publcly observed at the begnnng of the frst stage. At the begnnng of the frst stage n perod t > 1, there are n(k, t) frms wth unt ct c(k) where k {0, 1,..., t + 2}. For each k, n(k, t) = n 0 (k, t) + n 1 (k, t), where n 0 (k, t) and n 1 (k, t) denote the 7

8 number of non-user and user frms, respectvely. 9 We assume that for each k, n 0 (k, t) and n 1 (k, t), thus n(k, t), are all publcly observed at the begnnng of each perod t. Note that t s psble that n(k, t) = 0 for some k, for nstance, when k t > k and there s no non-user frm. At the begnnng of the second stage of perod t 1, after the open source use decsons have been made, the number of frms wth technology level κ s denoted by η(κ, t). At ths stage, let η 1 (κ, t) and η 0 (κ, t) denote the number of user frms and non-user frms respectvely, that s, η(κ, t) = η 0 (κ, t) + η 1 (κ, t). Note that whenever κ κ, η 1 (κ, t) = 0. At the begnnng of the thrd stage of perod t 1, after the success/falure outcomes are realzed, the number of frms that have technology level K s denoted by N(K, t). Smlarly, N(K, t) = N 0 (K, t) + N 1 (K, t), where N 0 (K, t) and N 1 (K, t) denote the number of non-user and the number of user frms, respectvely. To sum up, n a gven perod t, the technology level of a frm s k t at the begnnng of the frst stage, κ t at the begnnng of the second stage, and Kt at the begnnng of the thrd stage, where = denotes these technology levels for the open source technology. And, at any gven perod t, n(k, t) s the number of frms wth technology level k at the begnnng of the frst stage, η(κ, t) s the number of frms wth the technology level κ at the begnnng of the second stage, and N(K, t) s the number of frms wth the technology level K at the begnnng of the thrd stage. Note that, η(, t) n(, t), n(, t + 1) = N(, t) and n d (, t + 1) = N d (, t) for each d {0, 1} and t 1. 3 Equlbrum Analyss: Fnte Horzon To solve ths model, we study the Subgame Perfect Nash equlbrum of the 3 stage - T perod game. We focus on symmetrc equlbra, that s, we assume that n the thrd stage all frms wth the same technology level pck the same quantty, n the second stage all frms wth the same use decson and technology level pck the same success probablty, and fnally n the frst stage all non-user frms wth the same technology level make the same open source technology use decson. 3.1 Last Perod: t=t Before we start the equlbrum analyss wth the last stage of the last perod, we provde a lemma frst, whch deals wth the frst stage of the last perod. 9 Note that f k < k or k > k +1, then n 1(k, t) = 0. When k = k +1, there may be user frms who have succeeded n nnovaton n the prevous perod. 8

9 Lemma 1 In the last perod, usng the open source technology s a best response for any hstory of the game, for every frm, that s, d T = 1 for all. Proof. The proof s straghtforward. Any frm wth k T < k T n the begnnng of perod T wll be strctly better off usng the open source technology snce t wll strctly lower ts unt ct at no extra ct. Any frm wth k T = k T wll be ndfferent between usng the open source technology and not usng t. Ths s because, the unt ct wll not depend on the use decson and snce there s no future perod, there s no future affect, thus all that matters s todays unt ct. Any frm wth k T > k T has no beneft from usng the open source technology, snce the technology level wll not change wth and wthout the open source, thus, the unt ct wll not be affected. Agan, snce there s no future perods, there s no negatve future effect of usng the open source technology as well. Thus, such frms wll also be ndfferent between usng and not usng the open source technology. Thus, all frms at T wll be weakly better off usng the open source technology. Lemma 1 says that n the very last perod, t s optmal for every non-user frm to start usng the open source technology. Ths s ntutve snce there s no further perod where usng the open source technology would result n other frms free rde on the success of other frms. For frms wth a technology level at least as hgh as the open source technology level, however, not adoptng the open source technology s another best-response. We take the best response to be adoptng the open source technology n ths last perod T. However, even f such frms choe not to adopt the open source technology (maybe due to an exstng salvage value or maybe they just choe not adoptng the open source technology f they are ndfferent between adoptng or not), the dstrbuton of frms accordng to the technology levels wll be the same as n the case where such frms choe to adopt the open source technology. An adoptng frm wll acheve a technology level that s the maxmum of the current level and the level of the open source. Thus, for a frm wth a technology level at least as hgh as the open source technology level, ts technology level does not depend on ts decson to adopt the open source. Snce the unt ct of a frm depends only on ts technology level, for such a frm, the unt ct s not affected by ts decson to adopt the open source. Thus the dstrbuton of frms accordng to ther unt cts wll not be affected even f these frms do not adopt the open source technology n the last perod. Thus, t does not affect the rest of the analyss for the frst T 1 perods. Now we turn to the last stage of the last perod. We denote the equlbrum number of frms of a certan technology level and thus a frm s expectaton of the equlbrum number of frms of that technology level wth the boldface counterparts. For nstance, N(K, t) denotes the equlbrum number 9

10 of frms wth technology level K at the begnnng of the thrd stage of perod t and also the expected number of such frms Thrd Stage: Cournot Competton For each perod t, a frm observes own unt ct K t but does not observe the nnovaton outcome of the other frms. Thus, each frm has an expectaton of the technology dstrbuton n the market, {N(K, t)} K. The expected nverse demand wth K s P K t P (Q K t ) = A Q K t. The expected total quantty can be decomped; Q K t = q K t +Q t where Qt = (N(Kt, T) 1)q K t + K K t N(K, T)q K t. When a frm wth K t choes ts quantty qt K, t solves the followng problem max E[π qk t K t = (P K t c(k t ))qk t Below, we characterze the Cournot equlbrum quanttes and the expected Cournot profts for any perod t. Let K and ˆK be any two technology levels. Lemma 2 The Cournot equlbrum quanttes must satsfy q K q K = c( ˆK) c(k) for any K, ˆK n each t. The expected Cornout equlbrum profts are E[π t K = (q K t ) 2, where q K t = 1 M + 1 A c(k t ) 1 + ˆK K t N( ˆK, t) + ˆK K t N( ˆK, t)c( ˆK) Proof. The proof s n the Appendx, n Secton 7.1. The equlbrum Cournot quantty levels depend on the expected dstrbuton of frms accordng to ther technology levels, N( ˆK, t). If we wrte the quantty, q K t, n Lemma 2 as q K t = 1 A c(k t ) M + 1 ˆK K t [ N( ˆK, t) c(k t ) c( ˆK) one can see that, for a frm wth technology level K t, the larger the expected number of frms wth a hgher technology level ˆK (hgher than K t ), the lower frm s equlbrum quantty, snce n that case c(k t ) c( ˆK) > 0. Ths s ntutve snce f the more advanced frms are expected to be larger n number, t wll be harder to compete wth them and thus quantty wll be smaller. More mportantly, f K > ˆK, then q K > q ˆK, snce c( ˆK) c(k) > 0, as c( ) s a decreasng functon. Snce the expected 10

11 Cournot proft levels at ths stage are E[πK t = (q K t ) 2 for each K t Kt, the more advanced a frm s at the technology ladder, the more t produces and the hgher expected Cournot proft t gets. Ths s ntutve because the more advanced frms have lower unt cts, thus they have ct advantage n the Cournot competton. Thus, they produce more n the equlbrum and end up wth hgher profts n the Cournot competton. In the last perod T, by Lemma 1, there s no frm wth K T < K T after nnovaton realzatons. Thus, n the equlbrum, t must be that K T K T for all. Recall that n the frst perod, k 1 = 1 and there are frms wth k = 0, k = 1 and k = 2 n ths frst perod. Thus, t s psble that a non-user frm n the frst perod wth a technology level k = 2 succeeds n every perod, and when such a frm arrves at perod T, t s technology level becomes T + 2. Thus, the technology level of frms n the last perod may range between K T and T Thus, we have K T {K, T K T + 1,..., T + 2} for each Second Stage: Investment n Innovaton In the second stage, frms decde ther nvestment levels by pckng probablty of success to advance a level n the technology level, that s, to decrease the unt ct. Each frm knows own unt ct at the begnnng of ths stage. Frms, after observng n(k, t) for each k, do not observe the open source technology use decsons of other frms, d t j. They, however, know the expected number of frms for each k, η(k, t). Lke the thrd stage, n ths nnovaton stage, the technology level a frm has s at least κ T = k T by Lemma 1. Thus, κ T {k T, k T + 1,..., T + 1}. Frm wth a technology level k T and open source technology use decson d T pcks a probablty of success of nnovaton, p(d T, kt ) (0, 1), by maxmzng ts expected proft. That s, max p pe[π T κ +1 + (1 p)e[π T κ C(p) Note that by Lemma 1, d T = 1 for each. Thus, κ T (d T, k T ) = d T max{k T, k T } + (1 d T )k T = max{k T, k T } s frm s (potentally) new technology level. The equlbrum probablty of frm wth k T and d T = 1 s obtaned from the frst order condton of the objectve functon n the above maxmzaton problem. 10 Note that K T T + 2, snce even f open source communty succeeds (wth a frm wth k = 2 jonng the communty n the frst perod) every perod, the maxmum technology level t can acheve s T + 2, as there are only T perods. 11

12 Then, we get C (p(d T, k T )) = E[π T κ +1 E[π T κ = q 2 κ T +1 q2 κ T (1) The optmal success probablty (nvestment level) s ctly and t brngs some expected beneft as well, thus, to get the optmal success probablty, one must solve ths trade-off: The optmal success probablty should be such that ts margnal ct must equal ts margnal beneft. Equaton 1 summarzes ths condton, as the left hand sde s the margnal ct of the success probablty, and the rght hand sde s ts expected beneft (the margnal expected proft of a one step hgher technology level). Here we provde a specfcaton for the ct functon, whch we use to get a cled form for the condton n Equaton 1. The specfcaton we use here s c(k t) = 1 k t+1 wth c(kt ) = Ths specfcaton does not affect any of the results we provde, but t s useful to see the relevant cled form of the condton above. 11 When usng ths specfcaton, ths condton bols down to k t C (p(d T, k T )) = (2q K T + 2 K T κ T κ T + 1 )( 1 κ T κ T + 1 ) When d T = 1 (by Lemma 1), we have κ T = max{k T, k T }. Also, kt = K T snce the nnovaton success by a user frm s not reflected untl next perod (n ths case, t s never reflected because T s the last perod). Then, for a frm wth k T k T, we get, C (p(1, k T )) = 2 (2q K T + K T K T K T + 1 )( 1 K T K T + 1 ) = 1 [2q K T + (K T + 1)(K T + 2) 1 (K T + 1)(K T + 2) For a frm wth k T k T, the margnal expected proft from success s affected adversely by the open source technology level, K T, whch s ntutve snce the hgher the level K T, the lower the motvaton to nvest n success probablty, because such a frm wll be adoptng ths open source technology level anyways by Lemma An alternatve specfcaton, for nstance, s the one used n Modca (2012): when there s nnovaton the new unt ct s c new = (1 b)c 0, where b (0, 1) and c 0 beng the ntal unt ct. Also, Aghon, Harrs, Howtt and Vckers (2001), lookng at the effect of producton market competton and mtaton on growth, assume that a frm s unt ct depends on ts technology level and when a frm advances ts technology level by one step, ts unt ct decreases by some factor. However, snce our model s dynamc, we want to have a tractable ct reducton process as a functon of the technology level. Wth c new = (1 b)c type of reducton process, we would need to have c(k) = (1 b)c(k 1) = (1 b) 2 c(k 2) = (1 b) k c 0, whch s not analytcally tractable n our model. 12

13 3.1.3 Frst Stage: Usng Open Source Technology The frst stage n perod T s already dscussed n the begnnng of ths secton. And the result s summarzed n Lemma 1: each frm (weakly) prefers to use the open source technology n the last perod. 3.2 Next to Last Perod: t=t Thrd Stage: Cournot Competton Snce a frm s quantty choce q K t only affects perod-specfc proft πk t for each perod t, the thrd stage problem n T 1 s genercally the same as n perod T. Thus, the quantty choce n ths next to last perod, T 1, s also gven by Lemma 2. For the detals of whch, please see the proof of Lemma 2 n the Appendx, Secton Second Stage: Investment n Innovaton For frm wth a technology level k T 1 and open source technology use decson d T 1, κ T 1 (d T 1, k T 1 ) = d T 1 max(k T 1, k T 1 ) + (1 d T 1 )k T 1 s ts (potentally) new technology level. p(d T 1, k T 1 ) (0, 1), by maxmzng ts expected proft. That s, Such a frm pcks a probablty of success of nnovaton, max p [ p E[πκ T δw S(d T 1, κ T 1 + 1) [ + (1 p) E[πκ T 1 + δw F (d T 1, κ T 1 ) C(p) where W S (d T 1, κ T 1 + 1) s the expected contnuaton payoff from perod T on, when at T 1 the open source technology use decson s d T 1 κ T 1 and the technology level at the end of perod T 1 s + 1, wth a success n nnovaton n the nvestment stage. Smlarly, W F (d T 1, κ T 1 ) s the expected contnuaton payoff from perod T on, when at T 1 the open source technology use decson s d T 1 and the technology level at the end of perod T 1 s κ T 1, wth a falure n nnovaton n the nvestment stage. Note that n ths nvestment stage, psble technology levels are κ T 1 {0, 1,..., T }. Smlar to perod T, the equlbrum nvestment probablty of frm wth k T 1 and d T 1 s then 13

14 gven by C (p(d T 1, k T 1 )) = E[π T 1 κ +1 [ 1 E[πT κ + δ W S (d T 1, κ T 1 + 1) W F (d T 1, κ T 1 ) (2) [ = q 2 κ T 1 +1 q2 + δ W κ T 1 S (d T 1, κ T 1 + 1) W F (d T 1, κ T 1 ) As n perod T, the condton for the success probablty takes both the margnal ct and margnal beneft nto account. The left hand sde of Equaton 2 s the margnal ct of success probablty and the rght hand sde s ts expected margnal beneft (n terms of Cournot profts) whch takes today s change n Cournot profts (wth a success outcome) nto account, as well as the future expected margnal change n future Cournot profts (when there s a sucess today), whch s the term followng the dscount factor, n brackets Frst Stage: Usng Open Source Technology In ths stage, each frm decdes whether to use the open source technology or not. depends on the current own technology level k T 1 The decson, the current open source technology level, k T 1, and the dstrbuton of non-user and user frms for each technology level k, n 0 (k, T 1) and n 1 (k, T 1). The man trade-off for a non-user frm wth k T 1 < k T 1 s as follows. Startng to use the open source n ths perod wll be benefcal for the current perod through lower unt ct thus hgher Cournot proft. Also, n case of a falure, success of a user frm wll carry the technology level of one step hgher and the frm wll be able to use t from next perod on. However, a potental nnovaton by such a frm wll decrease the unt ct of other frms n the next perod, because of the structure mped by the GPL, and that wll provde the other frms wth the same ct advantage n the Cournot competton n the current perod. Let V (d T 1, k T 1 ) denote the expected payoff of frm from perod T 1 on. That s, V (d T 1, k T 1 ) = p(d T 1, k T 1 ) + (1 p(d T 1, k T 1 )) [ E[πκ T δw S(d T 1 [ E[π T 1 κ, κ T 1 + 1) + δw F (d T 1, κ T 1 ) C(p(d T 1, k T 1 )) We show that the optmal open source technology use decson for a frm wth k T 1 the open source technology, and the optmal open source use decson for a frm wth k T 1 < k T 1 k T 1 s to use s not to use the open source technology. These are summarzed, respectvely, n the two proptons below. 14

15 Propton 1 In the next to last perod, f the technology level of a frm s at least as hgh as the open source technology level, then not usng the open source technology s optmal for ths frm, that s, f k T 1 k T 1, then d T 1 = 0. Proof. See the Appendx, Secton 7.1. The ntuton for behnd Propton 1 s that whenever a frm starts the game wth the same unt ct of the open source technology, there s no drect beneft from usng the open source technology. However, usng the open source technology makes the frm oblged to share ts potental frst perod nnovaton, wth other frms, n the second perod, who choe to use the open source technology n the second perod. Ths removes any potental ct advantage the frm could have n the second perod quantty settng game. Hence any such frm wll avod usng the open source technology. Our second result says that any frm that produces the good at a hgher unt ct than the unt ct of the open source technology wll choe to use the open source technology even though GPL makes the frm oblged to share ts potental nnovaton n the frst perod wth the frms n the second perod. Propton 2 In the next to last perod, f the technology level of a frm s one step behnd the open source technology level, then usng the open source technology s optmal for ths frm, that s, f k T 1 = k T 1 1, then d T 1 = 1. Proof. See the Appendx, Secton 7.1. The ntuton for Propton 2 s as follows. For a frm wth a lower technology level (that s, a hgher unt ct) than the open source technology level, usng the open source technology drectly mproves the frm s producton technology hence ts expected proft n perod T 1. However, usng the open source technology now and succeedng n ct reducton n the current perod wll provde the other users who have not succeeded wth the ct reducton n the next perod. Not usng the open source technology now, and succeedng both now and n the next perod may result n a ct advantage n the next perod f the open source communty fals both now and later. Thus, there s a tradeoff. The drect effect domnates the negatve effect of usng the open source technology, snce the negatve effect s of second degree. Therefore, the ncentve to use the open source technology domnates the ncentve not to use t. 15

16 Corollary 1 In the next to last perod, f the technology level of a frm s lower than the open source technology level, then usng the open source technology s optmal for ths frm, that s, f k T 1 then d T 1 = 1. k T 1 < k T 1, Corollary 1 s a drect mplcaton of Propton 2. The reason s as follows. Compared to the case = k T 1 one for case k T 1 1, the ptve/mmedate effect of usng the open source technology s larger than the < k T 1 1. However, the negatve/future effect s the same n both cases. Thus, any frm who has a technology level less than the open source technology level, wll prefer to start usng the open source technology. 3.3 Any perod t < T Now we argue that the Propton 1 s vald for any perod t < T. Propton 3 In any perod t < T, f the technology level of a frm s at least as hgh as the open source technology level, then not usng the open source technology s optmal for ths frm, that s, f k t kt, then d t = 0 for all. Proof. The proof s smlar to the proof of Propton 1. We also argue that both the Propton 2 and Corollary 1 are vald for any perod t < T. Thus we prove the followng propton. Propton 4 In any perod t < T, f the technology level of a frm s lower than the open source technology level, then usng the open source technology s optmal for ths frm, that s, f k t < kt, then d t = 1. Proof. The proof s smlar to the proof of Propton 2. Snce we know that n perod T 1 each frm wth a lower technology level than the open source s technology level, wll use the open source technology. Thus, at perod T 1 each frm wll have at least open source technology level. Thus, at T 2, the reasonng to use the open source technology wll be smlar to the one n the proof of Propton 2. An mmedate mplcaton of the propton above s gven below. Corollary 2 At each perod t T, after the open source technology use decsons are made, there s no frm left wth a technology level lower than the open source technology level, that s, for all t T, we have η(k, t) = 0 for all k < k t. 16

17 Proof. By Proptons 1-4, we know that at any perod after the open source technology use decsons are made, a frm has a technology level ether same as the open source technology level or hgher. Ths s because, f a frm s behnd the open source technology level, t s optmal for ths frm to start usng the open source technology. Thus, puttng all these together we get the followng result. Propton 5 At the begnnng of the frst stage of any perod 1 t < T, there s no frm wth a technology level that s behnd the open source technology level by more than one level, that s, k t kt 1 for all. At any perod t, f the technology level of a frm s at least as hgh as the open source technology level, then the frm does not use the open source technology, that s, f k t kt, then d t = 0. At any perod t, f the technology level of a frm s lower than the open source technology level, then the frm uses the open source technology, that s, f k t = kt 1, then d t = 1. Proof. The frst argument for t > 1 follows from Corollary 2, whch mples that n any perod after the open source technology use decsons are made, there s no frm who s behnd the open source technology. Thus, at the end of the perod, after success/falure outcomes are realzed, a non-user frm may end up fallng behnd the open source technology by at mt one level, when t fals and the open source technology succeeds. Therefore, at the end of any perod, that s, at the begnnng of any perod t > 1, all frms have a technology level at least as hgh as one level less than the open source technology level, whch s k t 1. Note that, for t = 1, t follows from our assumpton that the ntal technology level of the open source s k 1 = 1, and any frm can at worst be k 1 = 0. The second and the thrd arguments are summarzng Propton 1-4. Thus, each frm stays out of the open source communty as long as t s not behnd the open source technology level. A frm whch has the same technology level as the open source technology level may fal whle the open source communty may succeed, thus next perod the frm s one step behnd the open source technology level. At that pont the frm starts usng the open source technology. In terms of the evoluton of the open source communty, n the lght of the above results, as long as the open source technology users are successful n nnovaton, the open source communty wll grow and sweep the non-users, and the set of non-user frms wll shrnk. As long as, the open source communty s not always successful n nnovaton, and the non-user frms that are ahead of the open source technology level succeed, there wll be a set of propretary frms wth a hgher technology level. 17

18 4 Equlbrum Analyss: Infnte Horzon We have assumed that the tme horzon s fnte, and thus we were able to fnd the Subgame Perfect Nash equlbrum through a backward nducton dea, startng wth the very last perod, T. However, t s also mportant to consder the nfnte horzon case and see f our results stll hold. To extend the model nto an nfnte horzon model, we need to consder how the equlbrum s defned by a sequence of both technology level dstrbuton profle K t = ( k t 1,, kt m), where k t j s a technology level at least one frm has at perod t, and ts correspondng dstrbuton of frms N t = ( N t k 1,, N t k m ) where km t N t k=k1 t k = M for each t. Our results extend to the nfnte horzon case, where a key ntuton n ths case s as follows. For any frm, the perod-specfc proft π t k s ncreasng n the number of compettors wth an nferor technology to ts own, and t s decreasng n the number of compettors wth a superor (or same) technology level, that s, π t k N t k > 0 and πt k N t k+ < 0, where k < k k +. Suppe that all other frms follow a strategy, d t = 0 when kt kt, and d t = 1 when kt < kt. Then, for a frm wth k t kt, d t = 1 may rase N t k whle t lowers N t k for some t > t when ths frm s successful n the nvestment stage. Also, usng and not usng the open source technology level wll not affect the current technology level of ths frm. Thus, ths frm s better off by not usng the open source technology. Smlarly, for a frm wth k t < kt, usng the open source technology level hardly rases N t k + for any t > t, and not usng t does not change the dstrbuton. However, by usng t ths frm jumps up n the technology ladder at no drect ct. Thus, such a frm s better off by usng the open source technology. Under a mld condton, we prove the followng result below. Propton 6 In the nfnte horzon case, () f the technology level of a frm s hgher than the open source technology level, then not usng the open source technology s optmal for ths frm, that s, f k t > kt, then d t = 0 for all, and () f the technology level of a frm s lower than the open source technology level, then usng the open source technology s optmal for ths frm, that s, f k t < kt, then d t = 1 for all, f ether δ s small enough, δ < δ, or the ntal number of open source users s large enough, n(k, 1) > n, or both, where δ (0, 1) and 0 < n < M. Proof. See the Appendx (Secton 7.2) for the proof and for the detaled analyss of the nfnte horzon case. 18

19 Whle the non-user frm does not need to share ts successes wth other open source frms, the user frm has to share them wth other user frms wth one perod lag, whch may worsen the dstrbuton of the technology levels aganst the favor of ths frm. However, the contnuaton proft s dscounted by δ, and f δ s small enough, ths effect wll be smaller relatve to the ptve effect due to the mmedate proft dfference. The fnte horzon case n prevous secton reflects ths pont. Alternatvely, f the current sze of the open source communty, n(k, t), s large enough, the lkelhood of the open source technology to advance n the technology wll more lkely and ths wll decrease the negatve effect of the oblgaton to share the success snce the open source technology wll advance regardless. Thus, agan the negatve effect wll be smaller relatve to the ptve effect of the larger mmedate proft. Note that, f the ntal number of open source frms, n(k, 1) s large enough, then n the equlbrum n(k, t) wll also be large enough. Snce n(k, 1) s a model parameter, but n(k, t) s an equlbrum object, we mpe the restrcton on n(k, 1). 5 Dscusson and Extensons In ths secton, we dscuss several relevant ponts and psble extensons. Intal ct dstrbuton: The ntal unt ct dstrbuton we have assumed s a specfc one; k s equal to ether 0, 1 or 2, where the open source technology level s at a level k = 1, to allow to have frms both at a lower and a hgher technology levels relatve to the open source technology level. Instead, we can assume a more general dstrbuton. The ntal technology levels of the frms can be k {0, 1,, K} and k {0, 1,, K 1}, where k K ensures that there are frms wth an ntal technology level hgher than the open source s technology level. Under ths generalzaton, we beleve that the dea behnd our results wll stll be vald, that s, the frms that have a hgher unt ct (lower technology level) than the source wll use the open source, the frms that have the same or lower unt ct wll choe not to use the source. Leavng the open source communty: Also, we assumed that the frms who start usng the open source technology they do not leave the open source communty. However, one could relax ths assumpton and let the users of open source technology leave t whenever they want. Under ths specfcaton, f, n the symmetrc equlbrum, at any perod f the user frms choe to stay n the open source communty, then the outcome s equvalent to what we have provded n our equlbrum analyss above. However, f the user frms decde to leave the open source communty the followng 19

20 perod, then the number of user frms wll be less relatve to the specfcaton n our model. Thus, the open source technology wll evolve slower, n expectatons, relatvely. But, we beleve that the use/not use decsons of the behnd and ahead frms wll not be affected. Producton ct: We have used a general unt ct functon, whch s decreasng n the technology level, and n the lmt as the technology level goes to nfnty, the unt ct goes to zero. Wth any such unt ct functon, our results, Proptons 1-5, together wth the lemmas and the corollares are all vald, as we dd not use any ct specfcaton n the proofs of any of these results. However, we also employed a specfc ct functon, c(k) = 1 k+1, just to see the cled forms of Cournot quanttes and profts. Contnuous technology levels: When the technology levels are contnuous, a frm wth a prvate technology level slghtly below the open source technology level, may prefer not to use the open source and stay prvate. However, f the technology level of the prvate frm s below a threshold, or say, f t s suffcently lower than the open source technology level, then t wll stll be optmal for ths frm to start usng the open source technology, just lke a frm, whch s one level behnd the open source technology, n our model wth ladder-type technologes. That s, f k < k k, for some k > 0, then frm uses the open source. Thus, even wth contnuous technology level, frms wth relatvely lower technology levels wll keep jonng the open source communty. And among prvate frms wth technology level lower than that of the open source, only the wth slghtly below the open source level wll stay prvate. So at any perod t, as n our Propton 5, whch says at the begnnng of the frst stage of any perod 1 t < T, there s no frm wth a technology level that s behnd the open source technology level by more than one level, there wll be no prvate frm that has a technology level that s consderably lower than the open source level. Also, over tme, as the open source communty grows, there wll be more open source frms nvestng n nnovaton, relatve to the prvate frms that are just slghtly below the open source level, thus, these prvate frms may, at some pont, fall behnd the open source level more than k and they may end up usng the open source next perod. Thus, we beleve that our predcton that the open source wll grow and number of prvate frms wll shrnk overtme wll stll be the case wth contnuous technology levels, and we also beleve that solvng ths threshold, k wll not gve us qualtatvely a dfferent result. Adopton ct: Assumpton 1 essentally says that there s no adopton ct to acqure the open source technology, whch s new to the frm. However, there may be compatblty ssues, ct of nstallng the new software and necessary tranng for usng the new technology. Thus ths adopton ct 20

21 may not be zero. In ths paper, we kept zero adopton ct assumpton as a smplfyng assumpton and we leave t as an extenson to see what happens when there s some ptve adopton ct to acqure the new open source technology. We guess that the hgher the adopton ct of the open source technology, the lower the ncentves to jon the open source communty. However, the forces that derve our man result wll stll be there, and thus as far as the evoluton of the open source communty s concerned, the open source communty wll stll grow, but at a smaller rate relatve to the case wth no adopton ct. 6 Concluson In ths paper, we analyzed a smple dynamc model of nnovaton n ct reducton wth an open source producton technology present for the frms to freely use. We assumed, n the sprt of the GPL, that whenever a user succeeds n ct reducton nnovaton, t has to share ths new technology wth other users, n the subsequent perods. Because of ths aspect of the GPL, we used a dynamc model, where n each perod the frms decde whether to use the open source technology or not, nvest n nnovaton, and engage n quantty competton. We characterzed the optmal open source technology use decson of a frm as a functon of ts technology level relatve to the open source technology level. A frm that has the same (or hgher) technology as the open source fnds t optmal not to use the open source technology. A frm that has a lower producton technology level fnds t optmal to use the source. These results show what prncpal effects of open source lcense on ncentves for nnovaton and usage of the open source technology are. We beleve that our model can be used as a tractable tool for further studes n open nnovaton. 7 Appendx 7.1 Fnte Horzon Proof of Lemma 2. Recall that the expected nverse demand wth K s P K t P (Q K t ) = A Q K t. The expected total quantty can be decomped; Q K t = q K t +Q t where Qt = K K t N(K, t)q K t + ( N(K t, t) 1)q K t. Under Cournot competton, a frm wth Kt solves the followng problem max E[π t q K = (P K t c(k t ))q 21

22 The frst order condton s E[π K t q = P (Q K t ) c(k t ) + P (Q K t) q q = A Q K t c(k t ) q = A c(k t ) Q t 2q = 0 Usng symmetry, we get the followng equlbrum condton; 2q K t = A c(k t ) K K t N(K, t)q K + (N(K t, t) 1)q K t That s, (N(K t, t) + 1)q K t = A c(kt ) N(K, t)q K K K t To see q K q ˆK = c( ˆK) c(k) for any K, ˆK, frst note that (N(K, t) + 1)q K = A c(k) N(k, t)q k = A c(k) N( ˆK, t)q ˆK k K and smlarly, (N( ˆK, t) + 1)q ˆK = A c( ˆK) N(k, t)q k = A c( ˆK) N(K, t)q K k ˆK Subtractng the two equatons we get, N(k, t)q k k K, ˆK N(k, t)q k k K, ˆK (N(K, t) + 1)q K (N( ˆK, t) + 1)q ˆK = c( ˆK) c(k) + N(K, t)q K N( ˆK, t)q ˆK Thus, we get, q K q ˆK = c( ˆK) c(k), that s, for any K, ˆK, we have q ˆK = q K + c(k) c( ˆK). Now, substtutng q ˆK = q K t + c(k t ) c( ˆK) nto ( N(K t, t) + 1 ) q K t = A c(kt ) N( ˆK, t)q ˆK (3) ˆK K t 22