Abstract This article analyzes the conditions under which open collaborative innovation communities can be generated and are able to grow and prosper in the long run. We build a formal model of the interaction between such communities and the alternative institutional setting based on intellectual property rights that we call Technology. The model accounts for both the communitarian social process taking place inside the community and for the presence of spillovers both across and within each institution. We prove that communities grow endogenously above a certain size threshold, and that such growth depends on supply-side mechanisms centered on the developers’ social motivations and interactions. In a comparative statics analysis, we show that the community’s innovative performance rests not only on its own characteristics (strength of social motivations and/or of protection of the community-produced knowledge) but also on factors outside its scope, namely, the institutional characteristics of Technology. We discuss the managerial and policy implications of our findings. 1. Introduction The open model of knowledge production has been recently gaining momentum in market economies. Knowledge-intensive communities (David and Foray, 2003) are typically characterized by a large 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. In this article, we focus on Baldwin and von Hippel’s (2011) version of knowledge-intensive communities, referred to as Open Collaborative Innovation Communities. In such communities, agents collaboratively develop and openly distribute knowledge without direct external funding or rents assured by the usual intellectual property rights (IPRs) environment (O’mahony, 2003). One of the most prominent examples of open collaborative innovation communities, in terms of economic and social impact, is the free and open-source software community (FOSS). In this community, a large number of individuals, spread all over the world, cooperate online to create software and release it openly through the Internet (David and Rullani, 2008, Gonzalez-Barahona et al., 2008). Anyone can enter the production process and report bugs, propose patches, cooperate with other developers on existing software, or launch new projects. Moreover, thanks to the license scheme adopted by the community (mostly the General Public License, GPL), no one can appropriate the jointly developed software. Openness in this case is preserved via copyleft, i.e., via licenses that prevent the appropriation and force the subsequent developers of the original code to redistribute the improved software under the original open terms.1 An extensive literature has investigated the motivations that induce researchers to join open communities through different methodologies. A first, quantitative, approach has collected survey data and then clustered community members around their core motivations. The empirical evidence produced by this stream of research suggests that, together with individual motives, social aspects—such as social identification with the community as well as sharing the community’s ideology—are key drivers for the formation and growth of open communities (Lakhani and Wolf, 2005; Stewart and Gosain, 2006).2 The importance of social motivations has also been emphasized in empirical studies focusing on specific communities [such as that of Linux User Groups (LUGs); Hertel et al., 2003; Bagozzi and Dholakia, 2006].3 Finally, ethnographic research has provided additional evidence in favor of the community’s social dimension, stressing in turn the importance of its ideological component (Elliott and Scacchi, 2008), as well as the development of a social fabric among the community members that fosters cohesion and collective learning (Lin, 2004a,b,, 2006). In this article, we incorporate the social dimension in a formal model that explicitly studies the interaction between such communities and the alternative, competitive, institutional setting based on IPRs that we call Technology. The model, which features knowledge externalities both within and across institutions, allows us to analyze the conditions under which open collaborative innovation communities can be generated and are able to grow endogenously and prosper in the long run. The modeling strategy to capture such interaction is inspired by Carraro and Siniscalco (2003). In the model, N researchers have to take two sequential decisions. First, they choose between a closed mode of knowledge production based on copyright and patenting (what Dasgupta and David, 1987, 1994, call Technology or T), and an open collaborative innovation community (more simply, a Community or C). While in Technology researchers gain from economic rents, in Community openness hinders the possibility to obtain directly a monetary reward. However, benefits of different nature may attract the researcher, ranging from individual motives (e.g., signaling one’s ability, reputation, fun, own-use of the produced knowledge) to social motives (being involved in the social dimension of Community). Second, once inside either T or C, members decide how much research effort to exert. Solving these two decision problems (entry and effort choice), we are able to characterize the equilibrium distribution of the N researchers across T and C. In the model, we consider both the intergroup and intragroup knowledge externalities that participation in either T or C generates. We suppose that Community generates (both intergroup and intragroup) positive externalities, while Technology generates (both intergroup and intragroup) negative externalities (we further discuss these issues in Sections 2 and 3). This complex interaction across the two institutional settings is responsible for equilibrium multiplicity. In particular, there is a threshold in terms of the number of Community members below which communities are doomed to fail and above which communities are able to grow endogenously and establish themselves as leading actors in the production of innovation (Granovetter, 1978). As a result, initial conditions may turn out to be crucial in determining the prosperity of the open model of knowledge production. These findings are useful to inform managers and project leaders on the strategies to create communities around their projects. Initial steps to generate a social environment and an organization able to attract and motivate participants are more important than any future efforts, as remaining below the threshold at the beginning means being doomed to remain small and eventually disappear, while being above means triggering an endogenous and self-reinforcing growth. We return to this issue in the conclusions. While the presence of a threshold size has been already recognized in the literature about open communities (Bonaccorsi and Rossi, 2003), the explanation has always been based on demand-side factors. This article is the first one that concentrates on supply-side factors (namely, the structure of the researchers’ motivations) as key determinants for the existence of the threshold. In particular, we formally show that the strength of social motivation is a key determinant of the Community’s development. The analysis of the interaction between the two alternative modes of generating innovation also allows us to focus on a second supply-side factor affecting the evolution of communities: the level of openness protection associated with the innovation produced in C, which governs the strength of positive knowledge externalities from C to T. Communities whose ability to defend the openness from the appropriation of Technology researchers is stronger are more likely to grow and successfully endure. This happens because protection mechanisms (such as licenses) preserving openness limit Technology’s capability to exploit the positive Community’s spillovers, thus reducing its attractiveness while triggering endogenous Community growth. Finally, after defining the concept of innovativeness in terms of the total flow of innovations generated by each institutional setting, we carry out a comparative statics analysis to study the impact of changes in the strength of social motivation and in openness protection policy on both C’s and T’s innovativeness. We find that an increase in the attractiveness of Community in terms of a stronger social motivation may even reduce its innovativeness—the reason being that larger positive spillovers from Community increase the level of investments of Technology researchers, which in turn negatively affects incentives to join the Community and thus may, in principle, more than offset the direct positive effect on Community’s innovativeness. An analogous reasoning applies when investigating the effect of the Community’s ability to protect openness: a stronger openness protection policy has an ambiguous effect on Community’s innovativeness—in particular, the effect is positive only if total efforts in Technology decrease. Such ambiguities allow us to highlight an interesting and nonobvious point: in both cases, the overall effect on Community’s innovativeness crucially depends on the institutional features of the alternative setting, Technology. We return to this point in the concluding remarks. The remainder of this article is organized as follows. Section 2 reviews the related literature. Section 3 describes the formal model. Section 4 presents the main findings. Section 5 concludes with a few managerial and policy implications. 2. Related literature Our article directly relates to two distinct streams of literature focusing on the open collaborative innovation model. A first stream focuses on what drives participation in open collaborative communities. Benefits from being part of a community are of both individual and social type (Nuvolari and Rullani, 2007). Among the main individual benefits, the existing literature has emphasized (i) reputation and peer regard (Bezroukov, 1999a, 1999b; Dalle and David, 2005), (ii) status and signaling one’s talent (Roberts et al., 2006; Lerner and Tirole, 2002, Bitzer et al., 2016), (iii) own-use (von Hippel, 1988; von Hippel, 2001; Franke and Shah, 2003; von Hippel and von Krogh, 2003; Bessen, 2006), (iv) fun (Raymond, 1998a,b; Torvalds and Diamond, 2001; Lakhani and Wolf, 2005), and (v) desire to learn from others (Ghosh et al., 2002; von Hippel and von Krogh, 2003).4 The social dimension has been analyzed along several dimensions, such as gift economy (Raymond, 1998a), epistemic communities (Amin and Cohendet, 2004; Mateos-Garcia and Steinmueller, 2008), and Communities of Practice (CoPs, Wenger, 1998; Lin, 2003).5 Our formal approach is inspired by the CoP perspective (see Subsection 3.2).6 This article is closely related to this literature in that it places a key role to the structure of the researchers’ motivations. Differently from the extant literature, however, ours is the only article incorporating such structure into a formal model of the interaction between T and C and, hence, analyzing the implications of the knowledge spillovers within and across the two groups. This is important because, as we will see, it allows us to understand that, however strong individual and social motives may be, communities may still fail to arise endogenously. A second stream of related literature analyzes the interaction between the proprietary and the open-source model in the production of knowledge through the lens of formal models. Particularly relevant for our purposes is the work of Carraro and Siniscalco (2003, CS henceforth), which builds a theoretical framework in which the two alternative institutional settings (Technology and Science) compete to attract new researchers and studies the conditions under which they can coexist. Other related papers include Gambardella and Hall (2006) and Johnson (2006), who consider how the competition of the free and open-source community with IPR-based system attracts developers. In particular, Gambardella and Hall (2006) also include positive externalities from researchers operating in an open environment and allow the benefit from participation in the community to depend on its size. Finally, Dalle and Jullien (2003) and Bonaccorsi and Rossi (2003) take a technology diffusion perspective to study the conditions under which open-source software can overcome an existing and dominant proprietary software.7 Our work is close in the spirit to this line of research. It, however, extends it in at least two respects. First, we take full account of both intergroup and intragroup spillovers in both institutional settings, which allows us a richer comparative statics analysis (Subsection 4.4). Second, we build on the CoP literature (Wenger, 1998) to analyze the inner mechanisms through which social motivations affect the relative attractiveness of the Community Model, thus casting a bridge between this class of formal models and the “supply-side” literature presented above. 3. The theoretical framework A population of N researchers is active in a given field of research. Researchers are assumed to be risk-neutral and identical in terms of both preferences and productivity. They exert effort to produce knowledge, and they can do this in two institutional settings: Technology (T) or Community (C). We assume that researchers, before choosing how much effort to exert, choose which institution they intend to belong to, based on an expected payoff comparison. Participation in one institution is exclusive. Technology and Community differ both in their payoff structures, capturing different motivations, and in the nature of externalities within and between institutions. We now characterize the payoff functions in the two institutional settings, starting with Technology. 3.1 The payoff function in Technology Economic return (RT) constitutes the main source of motivation to join Technology: new knowledge is kept secret or protected by patent and copyright law (Dasgupta and David, 1994). Moreover, the knowledge produced by a researcher that chooses this institution has a negative impact on the probability of any others’ success in knowledge creation, both in Technology and in Community, since the limits imposed by property rights or secrecy reduce the space for further innovations. This hypothesis captures the classical idea that patents or other intellectual property protection mechanisms shield innovators from potential competitors and thus may hamper the innovation opportunities. Formally, we define the payoff function from participating in Technology as: ΠiT≡PriT(xiT,X−iT,βXC)RT−ciT(xiT), (1) where xiT is the individual effort of researcher i in institution T, X−iT and XC represent, respectively, the sum of efforts of all researchers in T (excluding i) and in C, and PriT(·) is the probability of innovation (successful production of knowledge) in T. The product PriT(xiT,X−iT,βXC)RT is the expected revenue associated with the entrepreneurial activity (or the expected wage for employed software developers). The probability of innovation is increasing in effort at decreasing rate, i.e., ∂PriT/∂xiT>0 and ∂2PriT/∂(xiT)2<0 (that is to say, researchers’ marginal productivities are positive and decreasing in effort). The individual cost of effort is, instead, increasing and convex in effort, i.e., ∂ciT/∂xiT>0 and ∂2ciT/∂(xiT)2>0. These assumptions are standard and guarantee that each researcher always exerts positive (and finite) effort in equilibrium. In line with the previous literature, spillovers from Community are assumed positive ( ∂PriT/∂XiC>0). On the other hand, spillovers from Technology are assumed negative ( ∂PriT/∂X−iT<0). β∈[0,1] is a policy parameter capturing the strength of spillovers from Community to Technology. Its size (inversely) depends on how strong openness is protected from exclusive appropriation (O’mahony, 2003). In open source, for instance, licenses may limit the rights of others to use the code as inputs for their productions of “closed software,” thus reducing spillovers from Community to Technology. Creative Commons licenses may have the same effect. Finally, we assume that individual efforts in Technology are strategic complements with total efforts in Community ( ∂PriT/∂xiTXiC>0) and strategic substitutes with total efforts in Technology ( ∂PriT/∂xiTX−iT<0). Intuitively, the larger is the total effort in Community, the larger are the opportunities for a researcher in Technology to recombine and build upon the knowledge produced in Community, while the larger is the total effort in Technology, the smaller are these opportunities due to stronger technological competition, and so the lower is the marginal return from individual effort. 3.2 The payoff function in Community In line with the literature cited above, we must incorporate both individual and social benefits in the Community’s payoff function. Individual benefits (such as reputation, own-use, fun, status, salary, or benefits from FOSS firms) are not at the core of our analysis, which is why in the model we group them all in a single parameter (kC).8 The social dimension of the researchers’ motivations is, instead, central to our model. The CoP perspective can be particularly useful to describe in detail the social processes at work in the free and open-source community. Following Wenger (1998), the payoff structure must account for two factors directly related to socially based motivations: the communitarian activity and the degree of personal involvement in it. The communitarian activity (denoted by Y(·)) is the production activity undertaken by the Community. In the case of the free and open-source software, for instance, the communitarian activity is software development. The subjective effect of that activity on the individual’s payoff function is then mediated by his or her degree of personal involvement (denoted by function e(·)). For example, in the free and open-source case, the development of GNU/Linux (the most famous open-source operating system) has a greater effect on the payoff of a developer who “believes” in the GNU/Linux project compared to the payoff based on the simple usefulness of GNU/Linux as software program. Personal involvement is endogenous to the development of the Community. Shah (2006) describes the evolution in developers’ motivations as follows: “… a need for software-related improvements drives initial participation. The majority of participants leave the community once their needs are met, however, a small subset remains involved. For this set of developers, motives evolve over time and participation becomes a hobby” (p. 1000). Among the possible explanations for this process, the author also identifies the hypothesis that the “interaction with the community leads to a shift in the individual’s identity and self-perception” (p. 1011). This is the perspective taken by Bagozzi and Dholakia (2006), who write: “Initial participation by novice users is driven by specific task-oriented goals … . But over time, as the user comes to form deeper relationships with other [free and open source community] members, the community metamorphosizes into a friendship group and a social entity with which one identifies” (p. 1111). Therefore, when a community grows, it becomes stronger not only “quantitatively” (e.g., it produces more software) but also “qualitatively,” determining a higher average rate of personal involvement of its core members: this reasoning would suggest that the degree of personal involvement should be increasing in the size of the Community. On the other hand, abundant experimental evidence, partly inspired by Social Identity Theory (Tajfel and Turner, 1985), suggests that larger groups are more prone to loosen identitarian ties among their members (see, among others, Brewer and Kramer (1986), Hornsey and Jetten (2004), Hogg et al. (2017), and references therein). In fact, other things equal, an ever larger Community might translate into weaker commitment and participation by its members. This reasoning points to a relation between Community’s size and members’ identification of opposite sign. To capture both aspects, we will consider a nonmonotonic—inverted U-shaped—function e(·), increasing for small communities and decreasing when communities grow large enough.9 A third aspect related to the social dimension of open collaborative innovation communities refers to scale costs (denoted by function C(·)), that is, to costs related to the scale of the Community. First of all, C(·) includes coordination costs, i.e., capturing the increasing difficulties of organizing the work of an ever larger group of collaborators (Comino et al., 2007). Moreover, as any group of people who collaborate, the Community is expected to be subject to free riding episodes. The group must then create some rules and enforcing mechanisms to sustain cooperation and avoid free riding (O’mahony, 2003). Monitoring others’ behavior, spreading information about it, discovering rule violation, and punishing free riding are costly activities (Rullani and Haefliger, 2013), which probably increase with the scale of the Community. Taking all these aspects into account, we can define the payoff function from participating in Community as:10 ΠiC≡PriC(xiC,X−iC,XT)kC−ciC(xiC)+αe(n)Y(xiC,X−iC,XT)−C(n), (2) where n (N−n) denotes the endogenous number of researchers in Technology (Community), while xiC is the individual effort of researcher i in institution C. X−iC and XT represent, respectively, the sum of efforts of all researchers in C (excluding i) and in T. PriC(·) is the probability of innovation in C. kC is the “prize” obtained from successful innovating, and captures all the different motivational dimensions at the individual level, while α measures the relative strength of social motivations. The same hypotheses as those under Technology hold about the effect of effort on the probability of innovation and on individual costs: (i) ∂PriC/∂xiC>0, ∂2PriC/∂(xiC)2<0 and (ii) ∂ciC/∂xiC>0, ∂2ciC/∂(xiC)2>0. As for spillovers within and across institutions, following our previous discussion we assume that Technology’s spillovers are always negative, i.e., ∂PriC/∂XT<0, while Community’s spillovers are always positive, i.e., ∂PriC/∂X−iC>0. Moreover, individual efforts in Community are strategic complements with total efforts in Community and strategic substitutes with total efforts in Technology, i.e., ∂PriC/∂xiXT<0 and ∂PriC/∂xiX−iC>0: in words, the larger is the total effort in Community (Technology), the larger (smaller) are the opportunities for an individual researcher in Community to recombine and build upon them. The other terms in (2) capture the social dimension as defined above. The product of Y with e is meant to represent the two main processes described by CoP theory, Engagement and Legitimate Peripheral Participation (Lave and Wenger, 1991; Wenger, 1998). Community members, while collaborating to undertake the communitarian activity, get involved in a process of reciprocal influence which alters their personal involvement.11 We assume that the communitarian activity (i) responds positively to the total effort in Community and negatively to the total effort in Technology, i.e., ∂Y/∂X−iC>0, ∂Y/∂XT<0; (ii) is weakly increasing (and concave) in the individual effort (so that we even allow it to have a negligible impact value), that is, ∂Y/∂xiC≥0, ∂2Y/∂(xiC)2≤0. Moreover, in producing communitarian value, individual efforts are supposed to be weak strategic complements with total efforts in Community ( ∂Y/∂xiX−iC≥0) and weak strategic substitutes with total efforts in Technology ( ∂Y/∂xiXT≤0). In line with the previous discussion suggesting that the members’ degree of involvement should increase with communities’ size when they are small and decrease when they grow large, we suppose that function e(n) is inverted U-shaped in Community’s size (N–n), and thus also in Technology’s size (n). Finally, and as usual, scale costs are assumed increasing and convex in the size of Community (N–n), that is, ∂C(n)/∂n<0, ∂2C(n)/∂n2>0 (with C(N) = 0). 3.3 Definition of equilibrium and stability Researchers’ interaction is represented as a two-stage noncooperative game: in the first stage, each researcher decides whether to enter into Technology or Community, while in the second stage, after observing n, each agent decides simultaneously his or her effort level. The game is solved backward, computing the optimal effort of each researcher given N and n. Then, the analysis moves to the first stage, where researchers choose the institution predicting correctly the outcome and reap the payoffs associated with their choices. We restrict our attention to pure strategy Nash equilibria in which n∗ researchers choose Technology and the remaining N−n∗ choose Community. Furthermore, we consider only symmetric equilibria in terms of efforts within each institution. Consequently, we define ΠT(n) and ΠC(n), the reduced-form payoffs in the first stage for a researcher choosing Technology or Community, as a function of the number of researchers in Technology. Following CS and established coalition formation theory (D’Aspremont et al., 1983; Yi, 1997), we define the Nash equilibrium as the size of Technology n∗∈(0,N) that satisfies the following two conditions:12 ΠiT(n∗)≥ΠiC(n∗−1). (3) ΠiC(n∗)≥ΠiT(n∗+1). (4) Condition (3) implies that, at equilibrium, researchers in Technology do not have the incentive to move to Community, and (4), symmetrically, implies that researchers in Community do not have the incentive to move to Technology. If N is large enough, so that n can be in fact treated as a continuous variable, the determination of an interior equilibrium n∗ can be approximated by the condition: ΠT(n∗)=ΠC(n∗), (5) which we will use on in the next section. We interpret (Nash) equilibria as the stationary states of a dynamic adjustment process in which individuals may switch over time the institution they belong to, taking as given the choice of others, that is, assuming given institutions’ size. Keeping in mind such adjustment process, we are able to discuss the steady-state community size and its stability properties. We use the standard notion of equilibrium stability. An equilibrium is (locally) stable if there is a neighborhood of n∗ such that any n in such a neighborhood converges to n∗. In other words, an allocation of researchers between Technology and Community is stable if (sufficiently small) exogenous shocks in institution size do not move the equilibrium away (permanently) from the initial configuration. Formally, an equilibrium n∗ is stable if and only if:13 dΠT(n∗)dn−dΠC(n∗)dn<0. (6) 4. Findings We solve the game through backward induction. We first determine the equilibrium efforts in the second stage of the game for a given allocation of researchers in Technology and Community (Subsection 4.1). We then proceed backward to analyze the first-stage decision (Subsection 4.2). We finally characterize equilibria and their stability properties (Subsection 4.3), and carry out a comparative statics analysis (Subsection 4.4). 4.1 Decision in the second stage In the second stage of the game, each researcher, either in Technology or Community, chooses the effort that maximizes his or her payoff given n and the effort choices of the other researchers. The first-order conditions for payoff maximization in Technology and Community are, respectively, given by: ∂ΠiT∂xiT=∂PrT(xiT,X−iT,βXC)∂xiTRT−∂ciT(xiT)∂xiT=0 (7) and ∂ΠiC∂xiC=∂PrC(xiC,X−iC,βXT)∂xiCkC−∂ciC(xiC)∂xiC+αe(n)∂Y(xiC,X−iC,XT)∂xiC=0. (8) Since we are interested in symmetric Nash equilibria, the equilibrium efforts in Technology and Community (as a function of n), denoted by x^T(·) and x^C(·), are implicitly defined by: ∂PrT(x^T(n∗),(n∗−1)x^T(n∗),β(N−n∗)x^C(n∗))∂xiTRT−∂cT(x^T(n∗))∂xiT=0. (9) ∂PrC(x^C(n∗),(N−n∗−1)x^C(n∗),n∗x^T(n∗))∂xiCkC−∂ciC(x^C(n∗))∂xiC+αe(n)∂Y(x^C(n∗),x^C(n∗),x^T(n∗))∂xiC=0. (10) As proven in Appendix A, an increase in the size of Technology reduces individual effort in both Technology and in Community, i.e., ∂x^T(n∗)/∂n<0 and ∂x^C(n∗)/∂n<0. The intuition is straightforward. An increase in the Technology’s size (i.e., a higher n) is associated with higher negative spillovers from Technology (a stronger competition effect) and lower positive spillovers from Community (a weaker cooperation effect), which in turn negatively affects both Technology’s and Community’s payoffs and thus lowers the equilibrium research efforts in both institutions, x^T(n∗), x^C(n∗). When we look at total efforts in each institution, it is clear that total efforts in Community, defined as X^C(n)≡(N−n)x^C(n), are decreasing in n, i.e., dX^C(n)/dn<0. As for total efforts in Technology, defined as X^T(n)≡nx^T(n), the effect is ambiguous. Following CS, we focus on the case in which total effort is increasing in group size in Technology ( dX^T(n)/dn>0), that is, in which the positive effect on the extensive margin—the increase in n—dominates the negative effect on the intensive margin—the decrease in x^T(n). We restrict our attention to this case for a number of reasons. First, our primary objective is to study the determinants of open collaborative communities in a framework as close as possible to our benchmark theoretical framework (CS). Second, as we will see in the next subsection, this condition guarantees that the payoff function in Technology is always decreasing in its size, which is exactly what the most genuine economic intuition strongly suggests. 4.2 Decision in the first stage We are now ready to analyze the decision in the first stage. Substituting for the effort functions, x^T(n∗) and x^C(n∗) into the payoff functions, we obtain the reduced-form payoffs (depending on n only) used for comparison in the first stage: ΠiT(n)=PrT(x^T,X^−iT(n),βX^C(n))RT−cT(x^T). (11) ΠiC(n)=PrC(x^C,X^−iC(n),X^T(n))kC−cC(x^C)+αe(n)Y(x^C,X^−iC(n),X^T(n))−C(n). (12) In Appendix B we show that (i) (11) is decreasing in n; (ii) (12) has an inverted-U shape in n. Both results have a clear intuition. The payoff function in Technology is decreasing in the size of this group because more researchers in Technology (and thus fewer researchers in Community) imply more competition within Technology and lower positive spillovers from Community, and thus a lower probability for individual innovation. As for (12), when the Community is small, an increase in its size (lower n) has a positive effect on researchers’ payoff for three reasons: (i) larger positive intragroup spillovers; (ii) smaller negative intergroup spillovers from Technology; and (iii) higher value of the communitarian activity. On the other hand, when communities grow larger and larger, they (i) incur ever higher-scale costs (as these are increasing and convex in Community’s size) and (ii) deplete their power of social identification. This negative effect of group size will ultimately prevail for large enough communities. 4.3 Equilibrium analysis Given the reduced-form payoffs (11) and (12), and the notions of equilibrium and stability (5) and (6), we are now ready to characterize the equilibria, together with their stability properties. Along the lines of CS, the equilibrium analysis is performed through a graphical representation (Figure 1). Figure 1. View largeDownload slide Equilibria. Figure 1. View largeDownload slide Equilibria. In Figure 1, we represent the most interesting and natural situation in which ΠiT(0)>ΠiC(0) and ΠiT(N)>ΠiC(N). In words, when either everybody works in Community and no one in Technology (n = 0), or viceversa (n = N), the payoff associated with joining Technology is higher than in Community. The first inequality— ΠiT(0)>ΠiC(0)—is totally intuitive and due to the presence of scale costs in Community and of decreasing marginal returns in joining Technology. The second inequality— ΠiT(N)>ΠiC(N)—captures the idea that, when Community does not exist (or is very small), the incentives to found (and/or join) one are modest because its crucial social elements—that is, the communitarian activity and the degree of personal involvement—tend to increase and self-reinforce with the size of the Community itself (footnote 15 below discusses the two alternative situations in which either of the two inequalities does not hold). Recalling that an interior equilibrium n∗ satisfies ΠT(n∗)=ΠC(n∗), Figure 1 shows the existence of two interior equilibria, n1∗ and n2∗, with n1∗<n2∗. Since dΠT(n1∗)/dn−dΠC(n1∗)/dn<0 and dΠT(n2∗)/dn−dΠC(n2∗)/dn>0, it turns out that n1∗ is stable, while n2∗ is unstable. The instability of n2∗ implies that any shock that “perturbates” the system by increasing n moves it toward n∗=N, which is then also a stable equilibrium (corner solution). As a result, there are two possible stable equilibria, one in which Technology and Community coexist and Community is relatively large, and one in which all researchers are in Technology and Community does not exist. The “small Community” equilibrium, instead, is unstable. From an empirical point of view, equilibrium n1∗ is consistent with the evidence in the software industry, where similar competing products are offered under proprietary and open regimes. Notably, this result has been obtained with exante symmetric researchers, and it is the outcome of the endogenous mechanisms within and across the two alternative institutions.14 In the dynamic interpretation we previously suggested, the unstable equilibrium ( n2∗) constitutes a threshold that divides the realm of small communities, which are doomed to disappear over time, 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 effect. If a Community, for whatever reason, is able to grow enough and overcomes the threshold, then it grows endogenously up to size N−n1∗, which in a sense expresses the full potential of a Community. The system is then characterized by path dependence (David, 1985).15 The importance of the initial conditions has been already recognized in the literature on the free and open-source community (Bonaccorsi and Rossi, 2003; Bitzer and Schröder, 2005). The novelty in our “Critical Mass” argument for free and open-source development is that it is not based on demand factors (as, for instance, in Bonaccorsi and Rossi, 2003) but, instead, on the structure of the developers’ motivation and thus on, among others, the social forces described in the CoP literature (Wenger, 1998). Also note that the relevance of the initial Community’s size to predict its subsequent development finds some indirect empirical support in the literature on free and open-source community. On the one hand, larger projects/communities act as more powerful attractors for new members (David and Shapiro, 2008); on the other hand, smaller communities have a higher chance of becoming inactive (see the regressions reported in Zirpoli et al., 2013). This empirical evidence seems to support the narrative proposed in this article, where the growth of relatively larger communities tends to be strong and self-reinforcing, while small communities are more likely to vanish. Finally, our equilibrium analysis also provides insights into the role of intergroup spillovers in the comparative development of the two alternative institutions. In the case of open source, for instance, the role of licenses has been at the center of a lively debate for a long time (Lerner and Tirole, 2005; Comino et al., 2007). In the model, tools that protect openness and prevent appropriation such as strict licenses (e.g., the GPL) are captured by lower values of β, since they reduce the positive externalities from Community to Technology without affecting intragroup externalities. The consequence of a lower β is that of decreasing directly the Technology’s payoff ( dΠiT(n)/dβ=RT·(∂PrT/∂XC)·XC>0) and of increasing indirectly (through weaker negative spillovers) the Community’s payoff. The resulting equilibrium is characterized by a larger Community. Therefore, instruments that protect openness (e.g., the GPL) are fundamental for enhancing the sustainability of Community, favoring the conditions for its endogenous growth (Gambardella and Hall, 2006). Furthermore, when set at the initial stage of Community, such instruments may signal attention to openness protection and can help attract individuals that care about it and about the ideological component of the Community. 4.4 Comparative statics analysis on innovativeness Thus far, we have limited our attention to the size of groups that choose each institution at equilibrium. However, from a social point of view, it is the performance of institutions that matters. We now assess the performance in terms of the expected number of innovations in the institution, i.e., its innovativeness. We define innovativeness for Technology as: IT(n∗)≡n∗·PrT(x^T(n∗),(n∗−1)x^T(n∗),β(N−n∗)x^C(n∗)) (13) and innovativeness for Community as IC(n∗)≡(N−n∗)·PrC(x^C(n∗),(N−n∗−1)x^C(n∗),n∗x^T(n∗)). (14) We are interested in the effects on innovativeness in Technology and Community of changes in the two key dimensions shaping the Community’s development, i.e., α—measuring the strength of social motivations—and β—measuring the strength of knowledge externalities from Community to Technology, under the control, at least in part, of the policy maker regulating the degree of openness protection of the knowledge produced in open communities. Let us consider them in order. Previous literature on open innovation communities has emphasized the crucial role of social motivation for the Community’s development (refer to Sections 1 and 2 for the references on this literature). In the first comparative statics exercise, we prove that the strength of social motivation is, however, no guarantee of Community’s success once we measure success in terms of innovativeness. In particular, an increase in α has an ambiguous effect on Technology’s innovativeness, while it increases Community’s innovativeness if total efforts in Technology decrease (otherwise the effect is ambiguous), that is to say: dIT(n∗)dα≶0 anddIC(n∗)dα>0 if dXT(n∗)dα<0. While the formal proof is contained in Appendix C, the intuitive argument can be explained in the following terms. An increase in α makes Community more attractive: a larger Community increases positive intragroup externalities, with a positive effect on innovativeness. On the Technology side, its size reduction has an ambiguous effect on innovativeness because on one side, there are fewer researchers; on the other side, the effort exerted in Technology by each individual increases. Therefore, the effect on total effort is ambiguous. From the point of view of Community, if the total effort in Technology decreases, this reinforces the direct positive effect of the increase in α. Otherwise, the overall impact of this increase on Community’s innovativeness is ambiguous. This result hinges upon two crucial features of our formal model. The first is that the researchers’ effort is endogenously chosen (and, hence, it “reacts” to changes in the numerosity of the two alternative groups). The second feature is that our model incorporates both the intragroup and the intergroup spillovers working within and across the two alternative modes of innovation. As a result, when evaluating the effect of α on C, the model is able to capture both the direct effect and the indirect effect running through T. Let us now turn to β. Previous literature on open innovation communities has stressed the positive link between openness protection and Community’s development. Is it true that a stronger openness protection policy also fosters Community’s innovativeness in the context of our model? Again, rather surprisingly, we prove that a decrease in β (i.e., a stronger policy of openness protection) has an ambiguous effect on Technology’s innovativeness, while it raises Community’s innovativeness if total efforts in Technology decrease (otherwise the effect is ambiguous), that is to say: dIT(n∗)dβ≶0 anddIC(n∗)dβ<0 if dXT(n∗)dβ>0. Intuitively, a decrease in β makes Technology less attractive, and thus contributes to reduce its size. The impact on Technology’s innovativeness is, however, ambiguous because the effort exerted in Technology by each individual may increase. On the other hand, the size of the Community increases, each individual exerts higher effort, and intragroup positive externalities are stronger. If total effort in Technology decreases, then the expected number of innovations in Community unambiguously increases; otherwise, the overall effect on Community is ambiguous (again, the formal argument is developed in Appendix C). The reason why these two comparative statics exercises are interesting is exactly their ambiguity. In fact, the ultimate effect on the Community’s degree of innovativeness of the two parameters α and β (which both concern the functioning of Community) crucially depends on Technology, whose institutional characteristics then contribute to determine the overall level of innovativeness of the system. Needless to say, this result can only arise in a theoretical framework that explicitly models the interaction between the two alternative modes of innovation. We come back to this point in the concluding remarks. 5. Concluding remarks In this article, we have developed a model where Open Collaborative Innovation Communities are compared with Technology in their ability to attract researchers. In particular, attention is paid to the social nature of the community institution—as captured by the degree of personal involvement, the product of the communitarian activity, and the scale costs—and to the role of knowledge externalities within and across institutions. We confirm the presence of a size threshold for community, below which it can only remain small and eventually disappear, and above which it is pushed by internal forces to grow large. However, in contrast to all previous literature that focused on the final market for knowledge products (demand side), we highlight the economic forces working on the supply side (the input market where institutions compete for knowledge workers). This perspective enriches the debate around the effectiveness of the two alternative institutional environments (Community vs. Technology) to promote the innovative performance of the system. 5.1 Managerial implications This conclusion is important for firms. It suggests that, when firms decide to initiate a community around their innovation processes, they cannot adopt a step-by-step procedure. Such a “real option-like” approach, where firms gradually increase their investments by deciding in each subsequent step whether and how to foster community growth and solidity, does not square with our findings that emphasize the importance of the initial conditions and of overcoming the initial threshold size. In contrast, we describe the community’s development as an endogenous and self-reinforcing process. The lesson is that firms need to invest a lot in planning and realizing the initial phase of community development, and in gathering an initial core group of members that is large enough to place the community beyond the threshold size. These initial crucial steps will then trigger an endogenous growth process, attracting researchers from outside and enlarging the payoffs for those already in the community. Firms can then compensate for the larger expenses that this strategy calls for in the initial phase, with lower control and support costs in the takeoff stage. This mechanism has wide implications for managers and project leaders because it speaks against the diffused wisdom that community growth (i) can be treated as a gradual process and (ii) should be closely attended by the firm. We claim here instead that an important and careful investment at the beginning would be enough to generate endogenous growth later on. For instance, Spaeth et al. (2010) show that, in the case of the Eclipse development process, not only did IBM release the source code but its employed contributors played a fundamental role in fueling the growth of the community in its starting phase. This sort of “preemptive generosity,” as the authors call it, is the strategy our model indicates as the most effective one. 5.2 Policy implications Our model has also allowed us to evaluate the role of the two main drivers of community development—the strength of social motivation (α) and the openness protection policy (β)—in affecting its overall level of innovativeness. We have shown that a deeper social motivation (higher α) and/or a stronger openness protection policy (lower β) both foster innovativeness in Community only if the research effort in Technology is not too sensitive to the number of individuals in the institution, i.e., to the level of competition. The policy implication is worth emphasizing: to spur the Community’s level of innovativeness, the policy maker also needs to take into account the institutional design of the alternative institution, that is, of Technology. For instance, if the regulatory background that describes IPR and related markets for technology (Arora et al., 2001) is designed in ways that protect the effort of researchers from high-level competition, then Community’s innovativeness is more likely to respond positively to openness protection and to stronger social motivations. While this article does not aim at investigating how Technology should be designed to maximize innovativeness in the overall system, it however indicates to policy makers that (i) the interrelations between Community and Technology are tight and subtle and hence that, (ii) to obtain positive results on one side, action is also needed on the other side. In other words, designing markets crucially affects nonmarket social bodies, and viceversa. Neglecting such links risks causing unintended policy effects. 5.3 Further research While the model is suggestive of several forms of interaction between the two institutional modes of Community and Technology, its stylized form gives several opportunities for potentially useful extensions. First, product market competition (including the issue of pricing and product differentiation) could be modeled explicitly, both in Technology and in Community. Second, the role of firms in Community could be considered, removing explicitly the assumption that participation in one institution is exclusive. Finally, the value of innovation (rather than the probability of innovation) could be made endogenous. We leave these model extensions for future research. Footnotes 1 Recently, firms have developed business models featuring a close cooperation with the FOSS community, and have started populating it with their employees and projects. The relationship between private corporations and open communities is outside the scope of this article. The interested reader can refer to Dahlander and Wallin (2006) and Stam (2009) among the others. 2 For instance, Lakhani and Wolf (2005) show that 19% of their entire sample “consists of people motivated primarily by obligation/community-based intrinsic motivations. A majority of this cluster report group-identity-centric motivations derived from a sense of obligation to the community and a normative belief that code should be open” (Lakhani and Wolf, 2005: 16). 3 For instance, in Bagozzi and Dholakia (2006), 402 participants, representing 191 different LUGs, completed the survey. In their own words, “taken together, our results acknowledge the social nature of OSS and lead us to conclude that LUGs are indeed influential and cohesive communities of Linux users” (Bagozzi and Dholakia, 2006: 1107). 4 Surveys and empirical studies that measure the relative importance of individual motivations—such as the FOSS-EU survey (Ghosh et al., 2002) and the Boston Consulting Group survey (Lakhani et al., 2002), Bonaccorsi et al. (2006), and David and Shapiro (2008)—show that own-use-related incentives and psychological motivations such as fun are the most important drivers, together with learning, while reputation, signaling, and possible monetary gains are marginal (Lakhani and Wolf, 2005). 5 The previous section has cited empirical evidence emphasizing the importance of the social dimension as a chief motivating factor in community development. 6 Other relevant papers that study, with a game-theoretic approach, the conditions under which individuals are more likely to contribute include Johnson (2002), Baldwin and Clark (2006), Bitzer and Schroder (2005), and Reisinger et al. (2014). 7 The list of related papers that focus on the relation between the open and the closed mode of innovation is long and growing. Among the most recent contributions, we recall Mustonen (2003), Economides and Katsamakas (2006), Casadesus-Masanell and Llanes (2011), Landini (2012), Llanes and de Elejalde (2013), and Di Gateano (2014). 8 The reader interested in this dimension of analysis may further refer to Krishnamurthy and Tripathi (2009), Sauermann and Cohen (2010), Krishnamurthy et al. (2014), and Belenzon and Schankerman (2015). 9 Our assumption is in line with Fosfuri et al. (2016), who posit a nonmonotonic relationship between the size of the consumer base and the willingness to pay, for firms whose products are associated with social values and identity. 10 In reality, the motivational differences between Technology and Community can be much more blurred than those implied by the comparison between (1) and (2). A researcher that works for a firm but embedded in the scientific debate with his or her colleagues from other firms can reach the same social motivation as an open-source developer. Likewise, the latter can find a job in an open-source-based company and receive a monetary incentive similar to that of the former. However, we seek to grasp the inner difference between the two institutions, and thus, we magnify the differences in the payoff functions they offer to the researchers. 11 Y represents the research activity (say, software development): by construction, even if a valuable innovation does not materialize (with chance 1−PriC), the social content of that activity may still be positive (Y > 0), and the higher the greater the degree of involvement (e) of the researcher into the community’s activities. While this hypothesis captures the idea of the social dimension that we have in mind, the model and the findings would go through even without it. 12 In addition, it must be that ΠT(n∗)>0 and ΠC(n∗)>0: each researcher prefers to join one of the two institutions rather than getting an outside option normalized to 0. We shall assume that this is always true in our model. 13 To see why (6) must hold, suppose that the adjustment process (in continuous time) is represented by the differential equation dn/dt=F(n)≡g(ΠT(n)−ΠC(n)), with g being a positive constant affecting the speed of adjustment. A state n∗ is stationary if F(n∗)=0, while local stability requires F′(n)<0, which corresponds to g(dΠT(n)dn−dΠC(n)dn)<0. We also observe that, given our notion of stability, all equilibria are stable in CS setup. 14 A related alternative framework in which researchers are assumed heterogeneous in their productivity is developed by Mustonen (2003). 15 Let us hint at two special cases, alternative to that depicted in Figure 1. The first (probably unrealistic) case occurs when ΠiT(0)<ΠiC(0), which implies the existence of a single interior, unstable, equilibrium n∗ and of two corner solutions ( n∗=0 and n∗=N) as stable equilibria: here the large Community equilibrium involves all researchers. This scenario occurs when scale costs are low enough to allow Community to grow unbounded. The other special case is the one in which ΠiT(N)<ΠiC(N). In this situation, there exists a unique interior, stable, equilibrium n∗. Coexistence of Technology and Community emerges as the only possible configuration. This scenario coincides with that identified in CS, where the social side of Science (the institution “competing” with Technology in their model) is not considered. A more detailed analysis of these scenarios is available upon request. Acknowledgments The authors wish to thank the Editor and an anonymous referee for their precious comments and suggestions. All errors are authors’ responsibility. Francesco Rullani gratefully acknowledges financial support from the Italian Ministry of University Research [Project CUP B81J12002690008]. References Amin A. , Cohendet P. ( 2004 ), Architectures of Knowledge: Firms, Capabilities and Communities . Oxford University Press : Oxford, UK . 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( 2013 ), Coordination of joint search in distributed innovation processes. Lessons from the effects of initial code release in Open Source Software development. Presented at The Academy of Management, August 9-13, 2013, Orlando, FL. Appendix A Group size and individual/total effort in T and C Let us start with individual effort. Applying the implicit function theorem on (9) and (10), we obtain, respectively: ∂x^T∂n=−(∂PrT∂xT∂X−iTx^T−∂PrT∂xTXCx^C)RT∂2PrT∂(xT)2RT−∂2cT∂(xT)2,∂x^C∂n=−(∂PrC∂xC∂X−iTx^T−∂PrC∂xCXCx^C)kc+∂e∂n∂Y∂xC+e(n)(∂Y∂xC∂X−iTx^T−∂Y∂xCXCx^C)∂2PrC∂(xC)2kC−∂2cC∂(xC)2+e(n)∂2Y∂(xC)2. Concerning the former, we notice that the denominator is negative, since ∂2PrT/∂(xT)2 is negative, while ∂2cT/∂(xT)2 is positive by assumption. As for the numerator, ∂PrT/∂xT∂X−iT is negative and ∂PrT/∂xTXC is positive by assumption. Therefore ∂x^T/∂n is negative. Concerning the latter, the denominator is negative, since ∂2PrC/∂(xC)2 and ∂2Y/∂(xC)2 are negative by assumption, while ∂2cC/∂(xC)2 is positive. As for the numerator, ∂e/∂n is negative, while ∂Y/∂xC is positive; ∂PrC/∂xC∂X−iT and ∂Y/∂xC∂X−iT are negative by assumption, while ∂PrC/∂xCXC and ∂Y/∂xCXC are positive. Therefore ∂x^C/∂n is negative. We now turn to total efforts. Total efforts in Technology are defined as X^T(n)≡nx^T(n). Hence, it is ∂X^T(n)/∂n≡x^T(n)+n∂x^T(n)/∂n, whose sign is ambiguous given that ∂x^T/∂n<0. Proving that X^C(n)/n<0 is straightforward given that, in X^C(n)≡(N−n)x^C(n), the two terms (N−n) and x^C(n) are both decreasing in n. B Group size and payoff function in T and C Differentiating (11) w.r.t. n, we obtain: dΠiT(n)dn=(∂PrT∂X−iTdX−iTdn+∂PrT∂XCdXCdn)RT, which is lower than 0 given that ∂PrT/∂X−iT<0, dX−iT/dn>0, ∂PrT/∂XC>0, and dXC/dn<0. On the other hand, differentiating (12) w.r.t. n, we obtain: dΠiC(n)dn=(∂PrC∂X−iCdX−iCdn+∂PrC∂XTdXTdn)kC+αe·(∂Y∂X−iCdX−iCdn+∂Y∂XTdXTdn)+αdednY−dCdn. Our hypotheses ensure that the first two addends are both negative: in fact, an increase in the Technology’s size implies both a lower economic value (the first addend) and a lower social value of the research endeavor in Community. However, de/dn is positive for low values of n, while dC/dn is always negative by hypothesis: in fact, a decrease in the Community’s size ( n↑) implies lower-scale costs ( dC/dn↓) and, possibly, higher degree of personal involvement ( de/dn↑) if Community was large enough. The overall sign is then ambiguous. We now prove that, if there exists a unique solution to the equation dΠiC(n)/dn=0, then that solution is a global maximizer, denoted by nmax . When n∈(nmax ,N), Community’s (convex) scale costs are relatively low, and hence, net benefits from its size increase are positive, that is, dΠiC(n)/dn<0. When n∈(0,nmax ), Community’s scale costs are relatively high and predominant, and hence, net benefits from its size decrease are positive, that is, dΠiC(n)/dn>0. As a result, ΠiC(n) is increasing up to nmax and then decreasing. That is to say, function ΠiC(n) has an inverted-U shape. C effects of α and β on innovativeness in T and C To determine the effect of the variation of a parameter j=α, β on the expected number of innovations, we proceed as follows. We calculate (i) the derivative of n∗ w.r.t. parameter j; (ii) the derivative of the individual and total efforts in each institution w.r.t. parameter j; (iii) the derivative of the expected number of innovations w.r.t. parameter j. Furthermore, we observe that, since we focus on stable equilibria, condition (6) must hold. Start with α. Applying the implicit function theorem on the equilibrium condition, we obtain the effect of α on the equilibrium size of the Technology group as: dn∗dα=−e(n)YdΠT(n∗)dn−dΠC(n∗)dn<0. The effect of α on x^C is obtained as the sum of the direct impact of the parameter variation through the first-order condition, and the indirect impact due to the variation in the number of individuals in the institution. Then, by applying the implicit function theorem to Equation (8), we obtain: dx^Cdα=−e(n)∂V∂x^C∂2PrC∂(x^C)2−∂2cC∂(x^C)2+∂x^C∂ndn∗dα>0. In the first term (the direct effect), our hypotheses ensure that the denominator is negative and the numerator is positive. The second term (the indirect effect) is negative (see Appendix A). The effect of total investment in Community is given by: dX^Cdα=(N−n∗)dx^Cdα−dn∗dαx^C>0. As for the effect on the Technology side, we note that only an indirect effect exists. By applying the implicit function theorem to (7), we obtain: dx^Tdα=−∂x^T∂ndn∗dα∂2PrT∂(x^T)2−∂2cT∂(x^T)2, which is positive, since the denominator is positive and ∂x^T/∂n is negative (Appendix A). In terms of total effort, we obtain: dX^Tdα=n∗dx^Tdα+dn∗dαx^T, whose sign is ambiguous, since dx^T/dα is positive, while dn∗/dα is negative. Considering the results so far, we finally get: d(N−n∗)PrCdα=−dn∗dαPrC+(N−n∗)dPrCdα=−dn∗dαPrC+(N−n∗)(∂PrC∂x^Cdx^Cdα+∂PrC∂X^−iCdX^−iCdα+∂PrC∂X^TdX^Tdα). While the first term is positive, the term within brackets has an ambiguous sign, since dX^T/dα has an ambiguous sign (the other terms are positive). Then, the effect of α on the innovativeness in the Community group is overall ambiguous, unless dX^T/dα is negative, which would imply d(N−n∗)PrC/dα>0, since ∂PrC/∂X^T<0. As for the effect on Technology, we obtain: dn∗PrTdα=dn∗dαPrT+n∗dPrTdα=dn∗dαPrT+n∗(∂PrT∂x^Tdx^Tdα+∂PrT∂X^−iTdX^−iTdα+∂PrT∂X^CdX^Cdα), which has an ambiguous sign, since dn∗/dα<0 and, within brackets, the first and the third terms are positive, while the second term is ambiguous. Now consider β. Applying the implicit function theorem to (5), we obtain the effect of β on the equilibrium size of the Technology group as: dn∗dβ=RTXC∂PrT∂βdΠT(n∗)dn−dΠC(n∗)dn>0, since the denominator is negative in a stable equilibrium, while the numerator is positive given that ∂PrT/∂β>0 by hypothesis. As for the effect of β on x^T, this is obtained as the sum of the direct impact of the parameter variation through the first-order condition, and the indirect impact due to the variation in the number of individuals in the institution. Therefore, by applying the implicit function theorem to Equation (7), we obtain: dx^Tdβ≡∂x^T∂β+∂x^T∂ndn∗dβ=−RTXC∂2PrT∂x^T∂β∂2PrT∂(x^T)2−∂2cT∂(x^T)2+∂x^T∂ndn∗dβ. In the first term (direct effect), the denominator is positive, since, by assumption, ∂2PrT/∂(x^T)2 is negative and ∂2cT/∂(x^T)2 is positive. RTXC(∂2PrT/∂x^T∂β) is positive and (∂x^T/∂n)·(dn∗/dβ) (the indirect effect via n∗) is negative, since dn∗/dβ>0 and ∂x^T/∂n<0 (Appendix A). Therefore, the overall sign is ambiguous, with dx^T/dβ being positive if the direct effect prevails. Computing dX^T/dβ we obtain: dX^Tdβ=n∗dx^Tdβ+dn∗dβx^T, which is positive if dx^T/dβ>0, while it has an ambiguous sign in the opposite case. On the Community side, we note that only an indirect effect exists. Applying the implicit function theorem to Equation (8), and computing afterward the derivative of total effort in Community, yields: dx^Cdβ=−∂x^C∂ndn∗dβ∂2PrC∂(x^C)2−∂2cC∂(x^C)2<0,dX^Cdβ=(N−n∗)dx^Cdβ−dn∗dβx^C<0. since ∂x^C/∂n<0, ∂2PrC/∂(x^C)2<0 and ∂2cC/∂(x^C)2>0 by assumption. Considering the results so far, we finally get the impact of β on the expected number of innovations in Technology as: dn∗PrTdβ=dn∗dβPrT+n∗dPrTdβ=dn∗dβPrT+n∗(∂PrT∂x^Tdx^Tdβ+∂PrT∂X^−iTdX^−iTdβ+β∂PrT∂X^CdX^Cdβ+XC∂PrT∂β), which has an ambiguous sign. The first term is positive. Within brackets, the first two terms, capturing the effect of a change in β on Technology, have ambiguous signs, which turn out to be positive if the direct effect prevails. The third term is positive and the fourth negative, capturing the idea that an increase in β increases the spillovers toward Technology for given total effort in Community, but also reduces such effort via a reduction in N−n∗. As for the impact of β on the expected number of innovations in Community, we obtain: d(N−n∗)PrCdβ=−dn∗dβPrC+(N−n∗)dPrCdβ=−dn∗dβPrC+(N−n∗)(∂PrC∂x^Cdx^Cdβ+∂PrC∂X^−iCdX^−iCdβ+∂PrC∂X^TdX^Tdβ), whose sign is ambiguous. However, if dXT/dβ is positive, the expression above is unambiguously negative, since all the addends are negative. © The Author(s) 2018. Published by Oxford University Press on behalf of Associazione ICC. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)
Industrial and Corporate Change – Oxford University Press
Published: Mar 7, 2018
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