A supply side story for a threshold model: Endogenous growth of the free and open source community

Rullani, Francesco; Zirulia, Lorenzo Working Paper A supply side story for a threshold model: Endogenous growth of the free and open source community Quaderni - Working Paper DSE, No. 781 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Rullani, Francesco; Zirulia, Lorenzo (2011) : A supply side story for a threshold model: Endogenous growth of the free and open source community, Quaderni - Working Paper DSE, No. 781, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4457 This Version is available at: https://hdl.handle.net/10419/159620 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ A supply side story for a threshold model: Endogenous growth of the free and open source community Francesco Rullani Lorenzo Zirulia Quaderni - Working Paper DSE N° 781 A supply-side story for a threshold model: Endogenous growth of open collaborative innovation communities Francesco Rullani Department of Economics and Business LUISS Guido Carli E-mail: [email protected] Lorenzo Zirulia Department of Economics, University of Bologna KITeS, Bocconi University RCEA E-mail: [email protected] 30 August 2011 Abstract The study of social institutions producing and disseminating knowledge has mainly concentrated on two main concepts: Science and Technology. This paper examines a recent institutional form that seems not to resemble either of the other two; that is, knowledge-intensive communities, where individuals freely exchange knowledge through information and communication technology. Using free and open source software as an example, we develop a model where this phenomenon is confronted with Technology with respect to its ability to attract researchers. JEL classification: O31, L86, L88 Key Words: free and open source software, science, technology, community, intellectual property rights Acknowledgement Francesco Rullani thankfully acknowledges the financial support of PRIN 2010-2011 CUP: B81J12002690008. 2 1. Introduction One of the main challenges the development of what has been called a “knowledge society” imposes to economic theory is the assessment of the changes that occurred in the institutions enabling knowledge production and diffusion. Moving from the production of physical goods to the production of knowledge, in fact, implies a reshaping of the structures upon which the economy has been constructed. The economic discourse on institutions connected to knowledge production has its modern origins in the work of Dasgupta and David’s (1987, 1994), who recognize two main institutional models, “Science” and “Technology”, whose real manifestations are the academic world for the former and markets for technology (Arora et al., 2001) for the latter. Science is based on disclosure, rewards from priority and peer recognition and, today, on the public funding of knowledge production. Technology is based on secrecy and/or intellectual property rights and is profit-motivated. However, in the “shaded areas” of this dual system, we observed the emergence of a series of examples of an open model of knowledge production—where most agents develop and distribute knowledge without direct external funding or rents assured by the Intellectual Property Rights (IPR). Collective invention (Allen, 1983, Nuvolari, 2004), or communities of user innovators (Jeppesen and Frederiksen, 2006; von Hippel, 1988), are just two examples of the specific forms this model can take, and represent a challenge to the explanatory power of the Science/Technology dual system. Nowadays, a particularly important and pervasive role is played by what David and Foray (2003) call knowledge-intensive communities. These communities are characterized by a significant number of members who produce and reproduce knowledge in a ‘public’ (often virtual) space, in which new information and communication technologies are intensively used to codify and transmit knowledge. The economic and social relevance of these communities is such that firms are in the need of understanding their principles to be able to relate to, and benefit, from them (Bonaccorsi et al., 2006). One of the most prominent examples of knowledge-intensive communities, both in terms of economic and social impact, is the free and open source software community. In this community, a large number of individuals spread all over the world (Gonzalez-Barahona et al., 2008) cooperate online to create 3 software, and release it freely and openly through the Internet. Anyone can enter the production process and report bugs, propose patches, cooperate with other developers on existing software, or launch new projects; while—thanks to the license scheme adopted by the community (mostly the General Public License, GPL) —no one can appropriate the software jointly developed. Firms have created business models to be able to leverage the capabilities of the community and create a positive coexistence with it (Dahlander and Magnusson, 2005; Dahlander and Wallin, 2006; Dahlander, 2007; Bonaccorsi et al., 2006; Fosfuri et al., 2008) Our paper develops a formal analysis that aims at capturing the essence of free and open source community, and of knowledge-intensive communities more in general, as institutions. In particular, we develop a model where “Community” confronts Technology with respect to the ability of attracting researchers. In the specific case of software, our interest is to understand the conditions under which voluntary, open source software development (Community) can co-exist in productive balance with proprietary software development (Technology), together with the determinants of the relative size of the two institutions and of the stability of the equilibrium configurations. In the modeling exercise, we represent the broad set of motivations that affects the functioning of communities and the individual choices between the two institutions, with a particular attention on the social dimension of communities. The abstraction of the model, which typifies both Community and Technology taking “a radical view” on its constituents, allows the identification of the determinants of different dynamics of these institutions, and allows economic actors, such as managers and practitioners, to better understand how the free and open source community works in an environment where it needs to compete for resources (i.e., developers) with other institutions traditionally related to the business sector. Moreover, it explicitly takes into account the role of each researcher’s externalities towards those who work in her same institution and in the other institution, a mechanism usually left in the background in the literature. Our first result show that Technology and Community may coexist. From an empirical point of view, this is consistent to what is observed in sectors like software, where similar, competing products are offered under proprietary and open regimes. What is notable in our discussion, and to the best of our knowledge 4 new in the literature on open source, is that this result has been obtained with developers identical in every the respect via endogenous mechanisms within each institutional regime. A second result is that multiple equilibria are pervasive. In particular, we identify a threshold that divides the realm of communities doomed to remain small from the set of communities that are able to grow endogenously fast and large. This threshold has been widely recognized in the literature about communities (e.g., Bonaccorsi and Rossi, 2003); what is new in our argument is that the threshold is not based on demand factors, but on the structure on developers’ motivation, i.e., on supply side factors. The paper is organized as follows. Section 2 elaborates the appreciative theory upon which the argument is based. Section 3 describes the model, which takes as its starting point the analysis made by Carraro and Siniscalco (2003) comparing Technology and Science. Section 4 derives the main results, while Section 5 discusses their properties in light of the discussion in Section 2. Finally, Section 6 concludes. 2. The free and open source community as a knowledge-related institution This section develops an appreciative theory the institutional status of knowledge-intensive communities, which will be formalized in Section 3. In particular, we will focus on the individual and social motivations prevailing in this institution, in order to characterize the main determinants of the payoff of the individuals acting in Community. In Section 3, we will confront them with motivations prevailing in Technology, whose characteristics are instead well known since the description of the dual system represented by Technology and Science in Dasgupta and David (1987, 1994). As a reference point, we use the existing literature on free and open source. It was, in fact, in this literature that the question was asked relative to whether, and to what extent, the community model seemed to just resemble the academic world, i.e. Science in the Dasgupta and David formulation, so that, indeed, a new theory of communities as institution, and an explicit analysis of the comparison between Science and Technology, would be unnecessary. In that respect, what we claim is that Science and communities are similar ab origine, in that they have the same nature: signaling, reputation, own-use, and social interaction are, in both cases, among the main 5 factors determining the payoff function. However, the relative importance of those factors differs, since Science can rely on State intervention assuring funds and reputation-based incentives, while endogenous mechanisms, notably social motivations, are most important in Community. 2.1 Individual motivations in Community: just a “fancy Science”? Bezroukov (1999a; 1999b) was among the first authors identifying a possible homomorphism between Community and Science in terms of the produced outcome, the involved incentives, the typology of teamwork and institution of collaboration, and the way in which the activity is financially supported. In particular, Bezroukov stresses the similar role of financial organizations, such as research institutes, universities, or private research labs, in providing the individuals with the funds to undertake their activities in the directions they desire; and the similarity between the rules upon which Science is based and the practices typical of the free and open source community, which are also based on a public debate where priority over solutions and peer review are the crucial mechanisms used to regulate and direct individuals’ activities (Dasgupta and David, 1987; Lee and Cole, 2003). Kelty (2001) stresses the same similarities. On the one hand he states that “[…T]he funding that supports many projects (in most cases indirectly) comes from those well-known scientific institutions” (Kelty, 2001, online). On the other hand, he also argues that the structure of incentives and the organization of the collaborative effort of developers and scientists are very close to one another, both based on rules connecting the openness of the results to the individual pursuit of recognition and reputation (see also Lerner and Tirole, 2002). Mustonen (2003) shares the same point of view: “The essential property of the copyleft licensing scheme [i.e. GPL] is that it creates a particular incentive structure… [that] has properties that are equivalent to the incentive structures of scientific communities” (Mustonen, 2003, p. 104). Following a similar path, Bonaccorsi and Rossi (2003) recall the origins of free and open source inside the university labs to claim that “Emerging as it does from the university and research environment, the movement adopts the motivations of scientific research” (Bonaccorsi and Rossi, 2003, p. 1245). Dalle and David (2003) also share a similar point of view, stressing the parallelism between the free and open source institutional 6 setting and the rules of “open science,” where “the norm of openness is incentive compatible with a collegiate reputational reward system based upon accepted claims to priority” (Dalle and David, 2003, pp. 3, 4). A similar point is made by Raymond (1998), who suggests that the correspondence between the two phenomena is just the outcome of the fact that the scientific and the free and open source enterprises had simply given the same answer to the same problem of collective knowledge production. In addition to reputation-based incentives that relate peer-judgement to possible psychological and also financial rewards (Lerner and Tirole, 2002), own-use has also been underlined as an important motivation both in Science and in free and open source communities. This relates to the literature, inspired by von Hippel (1988), that has highlighted the role of users as a source of innovation in a wide range of fields (e.g. sports equipment, as in Franke and Shah, 2003). In the software case, an individual who has the knowledge and the tools to develop software can easily customized the software she uses and even produce the one she needs (von Hippel, 2001). As Bessen (2006) showed, in fact, software is a complex good that can be personalized much more effectively by skilled users than by manufacturers. Once produced, the software is very inexpensive to exchange through the Internet, so that even a very small reward can push developers to exchange the codes they have written (von Hippel and von Krogh, 2003). Own-use has a relevant role also in the scientific environment, at least relative to software development. In the research fields where software is a fundamental instrument, as it happens in econometrics, for example, scientists often decide to develop the tools they need, and sometimes they decide to distribute their work widely and freely (Gambardella and Hall, 2006). Thus, the free and open source community and science also “overlap” with respect to the own-use incentive. So, signaling one’s talent, reputation and own-use, are the main individual incentives in action both in free and open source community and Science. However, similarities between Science and free and open source do not imply that the two systems simply coincide. Indeed, the point made by our paper is exactly that Science and free and open source do differ in some fundamental aspects, and those aspects matter in the ability of free and open source to attract researchers vis-à-vis Technology. The first difference we stress refers to elements characterizing the modern functioning of academia are 7 absent, or at least less relevant, in free and open source: the crucial role played by the State and the professionalization of the scientific career (Dalle and David, 2007). Even in a period of reduced bugdets, the public sector intervention in paying researcher wages and allocating funds is prominent. This direct involvement is much less relevant in free and open source. As long as professionalization is concerned, career advancements and access to funds in Science are strictly related to structured “reputational games” to which scientists must participate. As Dalle and David suggest “…one should observe that that the parallel [between Science and free and open source] is by no means exact: formal professional accreditation and institutional affiliation are salient de facto requirements for active participation in modern academic and public sector research communities, yet the computer programming and other software development tasks—whether in the commercial or the free and open-source spheres—remain activities that have resisted becoming ‘professionalized.’” (Dalle and David, 2007, p. 393n4). Until now, free and open source has attracted many firms, and its economic dimension has considerably grown (Ghosh, Haaland, and Hall, 2008; Henkel, 2006). However, activity is still generally characterized by a large number of volunteers performing a large amount of the coding and of the related activities, and it is still based on a very informal and unspecified set of rules changing from one open source project to another (O'Neil, 2009).1 The State intervention and the professionalization of the scientific world have made priority rule and signaling much more prominent in Science that it is in open source. As a consequence, Science and free and open source do not differ in terms of the components of the utility function they assure to their researchers; they differ in terms of the weights they assign to these components. Indeed, in free and open source, surveys and empirical studies, such as the FOSS-EU survey (Ghosh et al., 2002), the Boston Consulting Group survey (Lakhani et al., 2002) and many others (Bonaccorsi and Rossi, 2006; David and Shapiro, 2008; Elliott and Scacchi, 2003), confirm that own-use related incentives are among the most important motivations, but find that reputation and signaling play a role (e.g., Roberts et al 2006) that is 1 Indeed, “In an historical perspective, the current structure of the [free and open source] community in these dimensions still resembles the initial stages of the development of open science, the era of “the West’s ‘amateur’ and 14 0 *)(*)(     dn nd dn nd CT (4) which implies that there is a neighborhood of n* such that for any n in such a neighborhood the myopic (with respect to the choice of institution) best response dynamic adjustment process converges to n*. Informally, an allocation of researchers between Technology and Community is stable if (sufficiently small) exogenous shocks in institutions size do not move the equilibrium away (permanently) from the initial configuration. 3.1 Technology, Science and Community: our model and Carraro and Siniscalco (2003) As we previously mentioned, our framework is based on the model developed by Carraro and Siniscalco (2003), who describes the choice of researchers between Science and Technology. In that paper, the authors assume the following payoff function for participation to Science: )(),,,(Pr)( S i SSTSi S i SS ixckXXxNnF   (5) Some of differences we put forth between Community and Science in Section 2 are captured by differences in the elements constituting equation (2) and (5). The role of the State in Science lead to the presence of a fixed salary F(n) (increasing in n), which moves upward the payoff from Science. Firms involved in free and open source can also provide their employees with fixed salary to work on their projects, but as discussed in the previous sections, this will be true only for a minor fraction of the millions of free and open source developers and contributors. Equation (2), instead, includes the positive value attached to personal involvement and communitarian activity that is present in Community. Apart from the structural properties of the pay-off function, the discussion in Section 2 reflects also in the values reasonably taken by similar parameters in equations (2) and (5). It is the case of S k and C k, which measure the private value of innovation in Science and Community, and thus the own-use, signaling and reputational incentives. We showed that while own-use is likely to have similar values in both environments, signaling and reputational incentives do not appear as fundamental in free and open 15 source (in relative terms). We can thus assume that CS kk . Science and Community may also differ in terms of externalities towards Technology. In Science the produced knowledge can be easily adapted and translated into an IPR regime by someone other than the innovator6. In the free and open source community, however, the GPL protects the produced knowledge, thus limiting this possibility (Gambardella and Hall, 2006). This does not cancel out the benefit that Technology has from the community production of software (ideas can be reused because GPL is not a patent), but GPL does limit the effects of communitarian externalities on the Technology payoff function. This implies that we could assume that the marginal effect of total efforts in Science on the probability of innovation in Technology is higher than the marginal effect of total efforts in community on the same probability. 3.2 Our contribution to the formal literature on free and open source This paper contributes to a fast-growing literature developing formal models on diverse aspects of free and open source communities. A first stream of literature has looked at the conditions for developers (or user-developers) to contribute to free and open source communities, thus emphasizing supply-side and motivation issues. Bitzer and Schroder (2005), for instance, consider open source software as a public good, and develop a game-theoretic model of contribution by self-interested individuals, while Gambardella and Hall (2006) and Johnson (2006) considered the competition of the free and open source community and the IPR-based system in attracting developers. Our work is closely related to this literature. More specifically its contribution to it is twofold: on the one hand, it explicitly analyses the role of social motivations in explaining the relative attractiveness of the community model, and on the other hand it does that taking into account how spillovers link not only the members of the same institutions, but also those part of the competing one. A second stream of literature has looked at competition between proprietary and open source software more from the consumers’ point of view. Among others, Casadesus-Masanell and Ghemawat (2006) 6 The Bay-Dhole act and the recent increase in the importance of patents in the scientific world enhanced precisely this process easier. However, the investigation of the impact of the adoption of Technology-based practices by a 16 developed a dynamic, duopoly model between a profit-oriented firm and an open-source community; Economides and Katsamakas (2006) consider the two-sided competition between proprietary and open source platforms, with a particular attention to the incentives for complimentary good production; Lanzi (2009) jointly considers product differentiation, lock-in and network externalities, and consumers’ experience in software use and implementation; Dalle and Jullien (2003) and Bonaccorsi and Rossi (2003) take a technology diffusion perspective, studying the conditions under which open source software can overcome an existing and dominant proprietary software. Our model can also be seen a contribution to this literature, since it can explain the co-existence of the models of software production, and then their competition, on the basis of the structure of developers’ motivations. As a third contribution of our model is its capability to show that the two institutions can coexistence even when developers are identical. Another model tries to bridge the demand and the supply sides of literature (Mustonen, 2003), but it assumes that developers are heterogeneous in their productivity, and that more productive developers choose the open source model. 4. Results To solve the model, we first determine the equilibrium efforts in the second stage of game for given allocation of researchers in Technology and Community. Then we proceed backward by analyzing the first stage decision and determining equilibria and their stability properties. 4.1 Equilibrium efforts in the second stage In the second stage of the game, each researcher, both in Technology and Community chooses the effort that maximizes her payoff given n and the effort choices of the other researchers. The first order conditions for payoff maximization in Technology and Community are given by: 0 )(),,(Pr           T i T i T T T i CTi T i T T i T i x xc R x XXx x (6) Science environment is outside the scope of this paper. 17 0 ),,( )( )(),,(Pr              C i TC i C i C i C i C C T i TC i C i C C i C i x XXxY ne x xc k x XXx x (7) Since we are interested in symmetric Nash equilibria, equilibrium efforts in Technology and Community (as a function of n), denoted by )( ˆnxT and )( ˆnxC , are implicitly defined by: 0 ))( ˆ ())( ˆ )(),( ˆ )1(),( ˆ (Pr       T i TT T T i CTTT x nxc R x nxnNnxnnx (8) 0 ))( ˆ ),( ˆ ),( ˆ ( )( ))( ˆ ())( ˆ ),( ˆ )1(),( ˆ (Pr          C i TCC C i CC C T i TCCC x nxnxnxY ne x nxc k x nxnnxnNnx (9) Proposition 1, proved in the Appendix, characterizes the effect of n on the effort exerted by each researcher in Technology and Community. Proposition 1 An increase in group size reduces individual effort in Technology and increases it in Community. i.e. 0 )( ˆ   n nxT and 0 )( ˆ   n nxC . The intuition of this result is straightforward. In Technology, an increase in group size increases competition within the group and reduces spillovers from Community, both effects being detrimental to the productivity of individual effort. In Community, an increase in size leads to more efforts because of the complementarity among researchers’ investments and because of the lower negative externalities from Technology. When we look at total efforts in each institution, i.e. )( ˆ )( ˆnxnnX TT  and )( ˆ )()( ˆnxnNnX CC  , it is immediate to see that total efforts in Community is decreasing in n, i.e. increasing in its size. For Technology, instead, the effect is ambiguous. Following Carraro and Siniscalco (2003) we solve this ambiguity by assuming that the total effort is always increasing in group size also in Technology, i.e. 0 )( ˆ dn nXd T . Plugging equilibrium efforts in the payoff functions, we have the reduced-form payoff used for comparison in the first stage: 18 ) ˆ ())( ˆ )(),( ˆ )1(),( ˆ (Pr)( TTTCTTTT ixcRnxnNnxnnxn  (10) )(), ˆ )(),( ˆ ()() ˆ (), ˆ )(),( ˆ (Pr)( nCxnxnNnxYnexckxnxnNnxn TCCCCCTCCCC i   (11) 4.2 Equilibrium in the first stage In order to identify the equilibria and their stability properties it is useful to derive the first derivatives of )(n T i  and )(n C i . By use of the envelope theorem, we obtain: T C i C T Ti T T T iR dn dX X dn dX X dn nd i                  PrPr )( (12) dn dC dn dX X Y dn dX X Y n e k dn dX X dn dX X dn nd T T C i C C T T C C i C C C i ii                                    PrPr )( (13) In order to simplify the proofs, but without affecting the qualitative discussion that follows, we shall assume that 0 )( 2 2   dn nd T i (which is satisfied if the effort cost function is sufficiently convex) and 0 )( 2 2   dn nd C i (which is satisfied whenever the coordination costs are sufficiently convex.). These assumptions guarantee the existence of at most three equilibria, reducing in this way the number of cases to be considered. Next Proposition is proved in Appendix. Proposition 2 Payoff from Technology are always decreasing in the number of researchers in the group, i.e. dn nd T i)( is always positive. Instead, payoffs from Community are always increasing, always decreasing or first increasing and then decreasing in group size, i.e. dn nd C i)( can be i) always negative ii) always positive or ii) first positive and then negative. The intuition of the results in Proposition 2 closely mimics Proposition 1. Payoffs in Technology are 19 decreasing in the size of this group because, first of all, more researchers in Technology implies more competition in the “patent races” and, second, it implies less researchers active in Community, and then lower positive spillovers. In Community, size of the group has a positive effect on researchers’ payoff for three reasons: i) larger positive spillover within the group; ii) a positive impact on the communitarian activity; iii) a lower negative externalities from Technology. However, large communities incur in large coordination costs. This negative effect of group size can easily prevail for large groups. We are now ready to state our main proposition on equilibria existence and stability.7 Proposition 3 The equilibria of the game are characterized as follows: (Scenario I) If )0()0( C i T i and )()( NN C i T i , but )()( nn C i T i for some values of n, then there are two stable equilibria ),0(* 1Nn  (coexistence of Technology and Community) and Nn *(all researchers in Technology), and one unstable equilibria  Nn ,0 * 2(coexistence of Technology and Community), with * 2 * 1nn . (Scenario II) If )0()0( C i T i and )()( NN C i T i , then there are three equilibria: two stable equilibria, 0* n(all researchers in Community), and Nn  *(all researchers in Technology),, and one unstable equilibria  Nn ,0*  Nn ,0 * 2(coexistence of Technology and Community). (Scenario III) If )0()0( C i T i and )()( NN C i T i , then the equilibrium value  Nn ,0*(coexistence of Technology and Community) is unique and stable. A graphical representation of equilibria determined in Proposition 3 is shown in Figure 1 INSERT FIGURE 1 ABOUT HERE 7 From Proposition 3 we exclude the trivial cases in which   Nnnn CT ;0 )()(  , so that all researchers are in Technology as unique equilibrium, and   Nnnn CT ;0 )()(  , in which all the researchers are in Community as unique equilibrium. 20 5. Discussion 5.1 The social dimension of Community In this section we comment upon the different scenarios described in Proposition 3, in particular relating them to the social dimension of Community. Before entering into the discussion, we first argue how variations in the level of personal involvement, the value of communitarian activity and the coordination costs impact on the payoff function in Community An increased importance of personal involvement e(n) and of the value of communitarian activity has the effect of moving the payoff from the Community upwards, making the Community more attractive for any n. This effect is likely to be more significant for large size of Community, increasing the (positive) sensitiveness of the payoff of researcher belonging to Community to her group size, i.e. making dn nd C i)(more negative. As for coordination costs, their increase has the primary effect of reducing the Community payoff for all n. However, we could expect that any increase in coordination costs would have a greater impact for small n (large community). If this is the case we could expect )0( C i  to move down and dn nd C i)( to be increasing for small values of n (i.e., for large communities). In Scenario I, two stable equilibria exist, one in which all researchers choose Technology and one in which Community is “large” (while Technology is “small”); on the contrary, the equilibrium with a “small” Community is unstable. In this scenario, )0()0( C i T i and )()( NN C i T i : )(n C  must have an inverted-U shape. This is consistent with a situation where coordination costs, communitarian activity and personal involvement are all significant, i.e., all factors we identified as peculiar of knowledge intensive communities are present. High coordination costs would lead to )0()0( C i T i and to dn nd C i)( being increasing for low n ; important communitarian activity and personal involvement effects, making Community payoff highly (and positively) dependent on group 21 size, would make )(n C i  strongly decreasing for high values of n, inducing an inverted-U relationship in the Community payoff and )()( NN C i T i . As a first remark, we notice that in this Scenario there exists one stable equilibrium in which Technology and Community coexist, with groups size depending on parameters values. From an empirical point of view, this equilibrium is clearly consistent to what is observed in sectors like software, where similar, competing products are offered under proprietary and open regimes. As anticipated, a notable result is that this has been obtained with ex-ante symmetric researchers and it is the outcome of endogenous mechanisms within each institutional regime. While a large community is stable, the equilibrium where the community is small is an unstable equilibrium. As we suggested in the previous section, the model admits a dynamic interpretation, where individuals choose sequentially, and play a best response strategy to the current allocation of researchers between institutions. In this case, the unstable equilibrium constitutes a threshold that divides the realm of small communities from the set of communities that are able to grow fast and large. In each one of those spaces, the dynamics of the model shows a sort of bandwagon effects. If a community, for whatever reason, is able to grow enough and overcomes the threshold, then it grows endogenously until the large equilibrium, which in a sense expresses the full potential of a community. This appears the case of the free and open source community, as widely recognized in the literature (e.g. Bonaccorsi and Rossi, 2003, and Bitzer and Schröder, 2005).8 What is new in our “Critical Mass” argument for free and open source development is that it is not based on demand factors, (such as, for instance, in Bonaccorsi and Rossi, 2003), but instead it is based on the structure on developers’ motivation. It is the shape of the social forces we described and rooted in Wenger’s (1998) community of practice that determine the behavior we observe in the model. 8 Notice that this approach takes into account the quantitative aspect of the free and open source community growth, but not its qualitative side. When communities grow, their social space becomes more complex, and their forms of participation and governance structures are put under pressure. The case of Debian is a clear example of the radical transformation needed to make a growing project able to bear the challenges determined by its own growth (MateosGarcia and Steinmueller, 2008; O’Mahony and Ferraro, 2007; Sadowski et al., 2008). 22 Consider now Scenario II. It occurs when )0()0( C i T i and )()( NN C i T i . In this case, the stable equilibria correspond to the corner solutions, while an unstable interior equilibrium separates the two “basins of attraction”. This scenario corresponds to a situation where personal involvement and communitarian activity are important, but coordination costs are small.9 In a sense, this scenario is a special case of Scenario I: small coordination costs lead to )0()0( C i T i , instead of )0()0( C i T i , posing no limit to Community growth. What this scenario shows with clarity is that strong communitarian activity may create a large community, but this is not necessarily so: to be fully expressed, the self-reinforcing growth process needs a critical mass at the beginning. Finally, consider Scenario III, in which the unique and stable equilibrium is the coexistence TechnologyCommunity. This case requires )0()0( C i T i and )()( NN C i T i , and consequently the “absolute” value of dn nd C i)( is “small” (compared to dn nd T i)(). Therefore, this case is consistent with a situation where the value of communitarian activity, the degree of personal involvement and coordination costs are small. Low values of communitarian activity and personal involvement tend to induce low values of )0( C i 10; low values of communitarian activity, personal involvement and coordination costs tend to make )0( C i  relatively insensitive to n, i.e. dn nd C i)( “small”. In light of our previous discussion in Section 2, low coordination costs, low value of communitarian activity and low degree of personal involvement make communities resembling closely scientific community, where these elements are certainly present but not as crucial for the existence of the institutions. Indeed, this scenario is isomorphic to one of those identified in Carraro and Siniscalco (2003). However, differences between Science and Community are not limited to the importance of the 9 The symmetric case, where coordination costs are large, but the communitarian activity and the degree of personal involvement are low, would lead toward Scenario III, or a situations where all researchers choose Technology as unique equilibrium, in case they are very large. 10 Since low coordination costs tend to increase )0( C i , we are assuming that this effect is dominated by the other. 23 social dimension. In Section 3.1 we argued that the professionalization of modern Science leads to two other differences in payoffs. First of all, researchers in Science are paid by the State a fixed wage, which moves upwards their payoff. Second, we could expect that the individual return from innovation (k) is lower in Community than in Science. This implies that this type of equilibrium should be less likely to be observed when Technology faces Community than when it faces Science; or, if it is observed in both cases, the size of Community will be smaller than the size of Science. So, even when arguing that social motivations in the functioning of communities are not a necessary condition for their existence, it must be acknowledged that they clearly have a positive impact on their establishment and growth. Moreover, social motivations are crucial to generate the threshold level between small unstable communities and large stable communities, which is observed in the free and open source case. 5.1 Path dependency and the growth of communities In the dynamic interpretation, the basin of attraction of the two stable equilibria, in terms of initial condition for n, is determined by the unstable equilibrium, whose values depend on the parameters of the model. The path dependency revealed by the importance of initial condition for n in determining, as first, which equilibrium is selected, and then the size of Community, points at the fundamental role that the initial ability of attracting researchers has for the establishment and growth of this institutional mode. Considering the free and open source case, we can observe that communities become economically relevant when they fill an unfilled market, creating one ex novo or providing the conditions to fill an established one (Bonaccorsi and Rossi, 2003). The definition of “market,” of course, must be interpreted in a wide sense, so that not only the product is important, but also the model of production—in the free and open source case allowing users to be part of the process—.The simple existence of a community attracts all the individuals interested in that market (Green, 1999). Thus, the more the community responds to unfilled gaps, the more attractive it becomes to interested individuals. Moreover, communities, as other institutions, cover a particular space of social interaction. They provide If this is not case, we could expect to prevail the situation where all researchers choose Community. 30 commercialization of open source software products. Organization Science 19(2) 292-305. Franke, N., S. Shah. 2003. How communities support innovative activities: An exploration of assistance and sharing among end-users. Research Policy 32 (1) 157–178. Gambardella, A., B. Hall. 2006. Proprietary vs public domain licensing of software and research products. Research Policy 35 (6) 875–892. Ghosh, R.A., R. Glott, B. Kriege, G. Robles. 2002. Free/libre and open source software. Part iv: Survey of developers. International Institute of Infonomics, Berlecom Research GmbH, at http://www.infonomics.nl/free and open source/report/Final4.pdf. Giuri, P., G. Rocchetti, S. Torrisi. 2002. Open source software: from open science to new marketing models an enquiry into the economics and management of open source software. Working paper, Laboratory of Economics and Management, Sant’Anna School of Advanced Studies (LEM), Pisa, Italy, at: http://www.lem.sssup.it/WPLem/files/2002-23.pdf. Gonzalez-Barahona, J.M., G. Robles, R. Andradas-Izquierdo, R.A. Ghosh, R.A. 2008. Geographic origin of libre software developers. Information Economics and Policy 20 356–363. Granovetter, M.1978. Threshold models of collective behavior. American Journal of Sociology 83 (6) 1420-1443 Green, E.L. 1999. Economics of open source software. At http://badtux.org/home/eric/editorial/economics.php Henkel J., 2006. Selective revealing in open innovation processes: The case of embedded Linux. Research Policy 35, 953–969 Jeppesen, L.B., L. Frederiksen, L., 2006. Why do users contribute to firm-hosted user communities? The case of computer-controlled music instruments. Organization Science 17 (1) 45–63. Johnson, J.P., 2006. Collaboration, peer review and open source software. Information Economics and Policy 18 477–497. Kelty, C.M. 2001. Free software/free science. First Monday 6 (12) at http://firstmonday.org/issues/issue6_12/kelty/index.html 31 Lakhani, K.R., R. G. Wolf. 2005. Why hackers do what they do: understanding motivations and effort in free/open source software projects. J. Feller, B. Fitzgerald, S. Hissam, K.R. Lakhani, eds. Perspectives on Free and Open Source Software. MIT Press, Cambridge MA. Lakhani, K.R., R. Wolf, J. Bates, C. Di Bona. 2002. Hacker survey (release 0.73), the Boston Consulting Group. Presented at the O’Reilly Open Source Software Conference, 24 July, San Diego, CA. Available at http://www.osdn.com/bcg. Lanzi, D., 2009. Competition and open source with perfect software compatibility. Information Economics and Policy 21 (3) 192-200. Lave, J., E. Wenger 1991. Situated Learning: Legitimate Peripheral Participation. Cambridge University Press, Cambridge UK. Lee, G.K., R.E. Cole. 2003. From a firm-based to a community-based model of knowledge creation: the case of the Linux Kernel development. Organization Science 14 (6) 633–649. Lerner, J., J. Tirole. 2002. Some simple economics of open source. Journal of Industrial Economics 50 (2) 197–234. Lin, Y., 2003a. Culture, creativity, and innovation: practices of locally-crafted software. Presented at the joint seminar SATSU and HCI group at computer science and psychology departments on 20 May, University of York, UK. Available at http://www.york.lug.org.uk/talks/yuwei_satsu/satsu_paper.html (accessed May 5th, 2005). Lin, Y., 2003b. The institutionalisation of hacking practices. Ubiquity 4 (4) at: http://www.acm.org/ubiquity/views/y_lin_1.html Lin, Y., 2004a. Contextualising knowledge–making in Linux user groups. First Monday 9 (11) at: http://firstmonday.org/issues/issue9_11/lin/index.html Lin, Y., 2004b. Epistemologically multiple actor-centered systems: or, EMACS at work! Ubiquity 5 (1) at: http://www.acm.org/ubiquity/views/v5i1_lin.html Luo, J., 2002. Why people are willing to share knowledge: an open source account. Queen’s KBE Centre for Knowledge-Based Enterprises, Doctoral Consortium 2002 Papers, June. 32 Mateos-Garcia, J., W.E. Steinmueller 2008. The institutions of open source software: examining the Debian community. Information Economics and Policy 20 333–344. Muller, P., 2006. Reputation, trust and the dynamics of leadership in communities of practice. Journal of Management and Governance 10 381–400. Mustonen, M., 2003. Copyleft—the economics of Linux and other open source software. Information Economics and Policy 15 99–121. Nuvolari, A., 2004. Collective invention during the British Industrial Revolution: The case of the Cornish pumping engine. Cambridge Journal of Economics 28 (3) 347–363. O'Neil, M. 2009. Cyberchiefs: autonomy and authority in online tribes, Pluto Press. O’Mahony, S., F. Ferraro. 2007. The emergence of governance in an open source community. Academy of Management Journal 50 (5) 1079–1106. Raymond, E., 1998. Homesteading the noosphere. First Monday 3 (10) at http://www.firstmonday.dk/issues/issue3_10/raymond/#d12 Roberts, J.A., I. Hann, S.A. Slaughter. 2006. Understanding the motivations, participation, and performance of open source software developers: a longitudinal study of the apache projects. Management Science 52 (7) 984–999. Sadowski, B.M., G. Sadowski-Rasters, G. Duysters. 2008. Transition of governance in a mature open software source community: evidence from the Debian case. Information Economics and Policy 20 323–332. Shah, S.K. 2006. Motivation, governance, and the viability of hybrid forms in open source software development. Management Science 52 (7) 1000–1014. Torvalds, L., A. Cox. 2003. Open letter on software patents from Linux developers, 21 September 2003, http://www.effi.org/patentit/patents_torvalds_cox.html von Hippel, E. 1988. The Sources of Innovation. Oxford University Press, New York. von Hippel, E. 2001. Innovation by user communities: learning from open source software. Sloan Management Review 42 (4) 82–86. 33 von Hippel, E., G. von Krogh. 2003. Open source software and the ‘private-collective’ innovation model: Issues for Organization Science. Organization Science 14 (2) 209–223. Wenger, E., 1998. Community of Practice: Learning Meaning and Identity. Cambridge University Press, Cambridge UK. Williams, S., 2002. Free as in Freedom: Richard Stallman’s Crusade for Free Software. O’Reilly, Sebastopol CA. Appendix Proof of Proposition 1 By use of the implicit function theorem, we get:   0 Pr ˆ Pr ˆ Pr ˆ 2 2 2 2 T 2 T 2                             T T T T T T C C T Ti Ti T T x c R x R n X Xx n X Xx dn xd    0 )( Pr ˆˆ x )( ˆ Pr ˆ Pr ˆ 2 2 2 2 2 2 C 2 C 2 C 2 C 2                                                       C i C i C C C i C C C T TC i C C C C T T C C x Y ne x c k x n X Xx Y n X X Y ne x Y n e k n X Xx n X Xx dn xd whose signs are direct consequences of the assumptions made in the paper. Proof of Proposition 2 The sign of (12) comes directly from the assumptions made in the paper. On Equation (13) notice that C T T C C i C Ck dn dX X dn dX Xi                PrPr and                dn dX X Y dn dX X Y n eT T C i C i are negative for the assumptions made in the paper, while is n C   is positive. The overall sign is then ambiguous. If 34 n C dn dX X Y dn dX X Y n e k dn dX X dn dX X T T C i C C T T C C i C C ii                                    PrPr for any n, then it is always .0 )(   dn nd C iIf n C dn dX X Y dn dX X Y n e k dn dX X dn dX X T T C i C C T T C C i C C ii                                    PrPr for any n, then it is always .0 )(   dn nd C i Suppose now that there are some values n ~ for which 0 ) ~ (  dn nd C i. Since we assumed 0 )( 2 2   dn nd C i, n ~ is unique, and it is the global maximizer of )(n C i  in the relevant interval. Consequently, )(n C i  is increasing in n untiln ~ , and then decreasing. Proof of Proposition 3 Consider Scenario I. If )0()0( C i T i and )()( NN C i T i , with )()( nn C i T i for some values of n, then )(n C i  must be first increasing and then decreasing in n. Consequently )(n T i crosses )( n C i  twice, in ),0(* 1Nn  and   Nn ,0 * 2, with * 2 * 1nn . Since )0()0( C i T i * 1 n is stable ( )(n T i ”cuts” )( n C i  from below), while )( n C i  cuts )(n T i from above in * 2 n, and then * 2 n is unstable. Since * 2 n is unstable, also n=N is a stable equilibrium. Consider Scenario II. If )0()0( C i T i and )()( NN C i T i , then )(n T i and )( n C i  cross only once given our assumption. Since )0()0( C i T i , in   Nn ,0*  where )( )( nn C i T i )( n C i  cuts )(n T i from above, and then the equilibrium in unstable. Consequently 0* n and Nn  * are stable equilibria. Consider finally Scenario III. If )0()0( C i T i and )()( NN C i T i , )( and )( nn C i T i , which are continuous, must cross at least once. Given 0 )( 2 2   dn nd C i and 0 )( 2 2   dn nd T i, the value 35  Nn ,0* where )( )( nn C i T i must be unique. Since )0()0( C i T i , then )(n T i ”cuts” )( n C i from above, which guarantees stability. Figure 1 (Scenario I) (Scenario II) (Scenario III) )(n T  * 2 n CT  , n N * 1 n CT  , n N * 1 n )(n T  )(n C  )(n C  )(n T  )(n C  * 1 n CT  , n N 