Equilibrium Innovation Ecosystems: The Dark Side of Collaborating with Complementors
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Mantovani, Andrea; Ruiz-Aliseda, Francisco Working Paper Equilibrium Innovation Ecosystems: The Dark Side of Collaborating with Complementors Quaderni - Working Paper DSE, No. 825 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Mantovani, Andrea; Ruiz-Aliseda, Francisco (2012) : Equilibrium Innovation Ecosystems: The Dark Side of Collaborating with Complementors, Quaderni - Working Paper DSE, No. 825, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4189 This Version is available at: https://hdl.handle.net/10419/159664 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/
Equilibrium Innovation Ecosystems: The Dark Side of Collaborating with Complementors Andrea Mantovani Francisco Ruiz-Aliseda Quaderni - Working Paper DSE N° 825
Equilibrium Innovation Ecosystems: The Dark Side of Collaborating with Complementors Andrea MantovaniyFrancisco Ruiz-Alisedaz May 4, 2012 Abstract The recent years have exhibited a burst in the amount of collaborative activities among …rms selling complementary products. This paper aims at providing a rationale for such a large extent of collaboration ties among complementors. To this end, we analyze a game in which the two producers of a certain component have the possibility to form pairwise collaboration ties with each of the two producers of a complementary component. Once ties are formed, each of the four …rms decides how much to invest in improving the quality of the match with each possible complementor, under the assumption that collaborating with a complementor makes it cheaper to invest in enhancing match quality with such complementor. Once investment choices have taken place, all …rms choose prices for their respective components. Our main …nding in this setting is that …rms end up forming as many collaboration ties as it is possible, although they would all prefer a scenario where collaboration were forbidden, unlike a social planner. Key words: Systems Competition, Complementary Products, Interoperability, Collaboration Link, Co-opetition, Exclusivity. JEL code: L13, M21. The authors are very grateful to David Besanko, Francis Bloch, Rahul Kapoor, Joao Montez, Eduardo Perez, Regis Renault, Michael Ryall and Dongsoo Shin, as well as to seminar participants at the Kellogg School of Management, Ecole Polytechnique, University of Bologna, the 10th Annual International Industrial Organization Conference and London Business School, for helpful comments and suggestions. Financial support from the NET Institute (www.NETinst.org) is gratefully acknowledged. The usual disclaimer applies. yDepartment of Economics, University of Bologna, Strada Maggiore 45, 40125 Bologna, Italy, and Barcelona Institute of Economics (IEB), C/ Tinent Coronel Valenzuela 1-11, 08034 Barcelona, Spain; email: [email protected] zDepartment of Economics, Ecole Polytechnique, 91128 Palaiseau, France; email: [email protected]
1 Introduction The recent decades have witnessed a shift in the competitive paradigm in high-tech industries that is driven to a large extent by the increasing importance of product complementarity. Indeed, cooperation among …rms selling complementary products is playing a prominent role in industries such as consumer electronics, semiconductors or telecommunications. More generally, hardware-software industries have exhibited a surge in the extent of cooperation among producers of complementary goods with the aim of improving the interoperability of their respective products (see e.g. Moore 1996, Gawer and Cusumano 2002, Adner 2006, Adner and Kapoor 2010, Gawer and Henderson 2007).1Building such innovation ecosystems (Adner 2006) with the producers of complementary goods seems to be the key competitive weapon in most high-tech industries, in which the notion of competition has been displaced by that of co-opetition (Brandenburger and Nalebu¤ 1996). A noteworthy feature of collaboration with complementors (i.e., …rms selling products that complement each other from the point of view of consumers) is that it is not unusual for …rms to collaborate with several complementors that sell substitutes of each other.2 A natural question that arises in these settings is whether such extensive collaboration is desirable from the standpoints of …rms and consumers. Intuitively, one would be tempted to think that collaboration in improving the interoperability of complementary products is e¢ cient both for the …rms involved, and in fact for society as a whole. The purpose of this paper is to show that this intuition may be valid for society, but not for the …rms involved, which may be trapped in a prisonner’s dilemma. Collaboration may then result in equilibria in which …rms are worse o¤ than when …rms do not collaborate with complementors. This holds regardless of whether collaboration ties are exclusive or not, under the assumption that collaborating with a complementor makes it easier to enhance interoperability with such complementor. To formally analyze these issues, we consider a game played by two …rms X1and X2 that sell components that (perfectly) complement those sold by …rms Y1and Y2(both of which are also engaged in the game). In this mix-and-match setting (Matutes and Regibeau 1988 and Economides 1989), there are four systems that are contemplated by consumers 1The interoperability of the components of which a composite good consists refers to their coherence to work together with each other as a sole system. This is largely related to the absence of con‡icts arising from possible incompatibility issues. 2To give concrete examples, mobile phone manufacturer Nokia allied …rst with Intel to develop the MeeGo operating system for smartphones, and later signed an agreement with Microsoft to support the Windows Phone operating system. In addition, the Intel Architecture Lab (IAL) was formed to foster investment in components complementary to Intel’s microprocessors by …rms that many times competed against each other. 1
when they make their purchase decisions: X1Y1,X1Y2,X2Y1and X2Y2. The game that we study consists of three stages. In the …rst stage, each …rm decides whether to form a (pairwise) collaboration link with each of its possible complementors (collaboration among …rms selling substitute components of a system is not allowed). In the second stage, each …rm decides how much to invest in improving the interoperability of its component with each of its complementors.3It is assumed that a …rm that has formed a collaboration link with a complementor faces lower costs when enhancing interoperability with such complementor. In the third and …nal stage, each …rm decides independently on the price of its component, given past interoperability investments of all the …rms involved in the game. We …nd in this setting that the (unique) equilibrium collaboration network involves each …rm forming (pairwise) collaboration links with its two complementors. If collaboration ties can be formed only in an exclusive manner, then exactly the same forces (subject to the exclusivity restriction) imply that in equilibrium each …rm forms a collaboration link with just one of its complementors. In both the exclusive and non-exclusive settings, equilibria exhibit all …rms collaborating with at least one complementor, which seems to accord well with the empirical evidence on innovation ecosystems. These equilibrium outcomes seem quite intuitive, but it is worth noting that intuition may conceal the e¤ect of several forces working at the same time. Thus, two complementors that form a new collaboration link between them bene…t from cost sinergies and increase their investments in enhancing the interoperability with each other. This e¤ect conforms to the intuition that one may have on the impact of a new collaboration link. However, two …rms that form a new collaboration tie with each other must also bear in mind that the …rms not involved in such a tie will strategically react. This strategic e¤ect of collaboration turns out to be positive, and hence reinforces the e¤ect of the cost synergy that arises when two …rms start collaborating. Collaboration has a strategic e¤ect in that the …rms not involved in the new collaboration tie reduce their investments in each other as well as in the complementor involved in the new collaboration tie. From the viewpoint of the …rms that start collaborating, the latter reduction in interoperability investments is harmful, but its impact is lower than the former reduction, which is bene…cial, hence the positive strategic e¤ect of collaboration. Factoring all the incentives, we have that it is always desirable to form a new collaboration tie with a complementor with which a …rm does not have one. This rat race ends when no more ties are possible, and hence each …rm collaborates with as many complementors as it can. 3Greater investment in the interoperability of two components is modeled as an enhancement in the (perceived) quality of the system comprising both components (e.g., the investment by X1in improving interoperability with component Y2is speci…c to Y2, and has no e¤ect on the interoperability of components X1and Y1). 2
Although a …rm would bene…t from its competitor committing not to collaborate with any complementor, it holds that all the …rms would be better o¤ if each could make such commitment. Hence, the equilibrium outcome exhibits the features of a prisonner’s dilemma despite all …rms are more productive in enhancing interoperability between complementary components. Being more productive, each …rm invests more in interoperability than in the absence of any collaboration amongst complementors. The greater investment leads to higher investment costs, incurred with the aim of vertically di¤erentiating the systems in which a …rm participates. Because all other …rms act in the same way, …rms boost investments but do not manage to vertically di¤erentiate any system, and hence they attain the same pro…t in the product market as in the absence of collaboration. This growth in investment costs without greater product market pro…ts explains why the equilibrium outcome is ine¢ cient for …rms. As for consumers, all of them bene…t in equilibrium from the better functionality of every system relative to when …rms do not collaborate. This explains why collaboration arising as an equilibrium outcome enhances social welfare relative to the situation in which …rms do not collaborate with complementors. Our result that collaboration in R&D among complementors results in private ine¢ - ciencies is in stark contrast with the result that R&D collaboration among …rms selling substitute goods may be desirable both for …rms and society, as shown in the seminal papers by D’Aspremont and Jacquemin (1988) and Kamien, Muller and Zang (1992). These papers do not consider whether a …rm has incentives to collaborate with other …rms, a limitation that has been overcome by subsequent work by Bloch (1995) using a coalitions approach, and more recently by Goyal and Moraga-González (2001) using a bilateral link formation approach.4Both of these papers show that excessive collaboration may arise in equilibrium. Although we also contend that equilibria displaying collaboration may be ine¢ cient, it is worth noting that the results in Bloch (1995) and Goyal and Moraga-González (2001) are derived for substitute goods, not for complementary goods, as is our focus. Our paper also contributes to the literature analyzing strategic competition when there exists at least one complementor whose pricing activities interact with those of two …rms selling components that constitute substitutes for each other. This literature was pioneered by Economides and Salop (1992) as an extension of early work by Cournot (1838), who analyzed the e¤ect of a merger of two monopolists that produce complementary goods. The paper by Economides and Salop (1992) examines the e¤ect of cooperation in prices 4See Leahy and Neary (1997) for a generalization of the models in D’Aspremont and Jacquemin (1988) and Kamien, Muller and Zang (1992). See also Bloch (2005) for a comprehensive survey that covers strategic network formation games in settings with R&D activities. Finally, it is worth pointing out that Westbrock (2010) builds on Goyal and Moraga-González (2001) and Goyal and Joshi (2003) so as to analyze how asymmetric R&D networks may be socially e¢ cient if collaboration ties are somewhat costly to establish. 3
(i.e., a merger) between the two existing producers of one of the two components of which a system consists. They consider two scenarios, depending on whether or not the two producers of the complementary component are already cooperating in prices. In our work, we do not analyze price cooperation and, in fact, …rms always choose prices noncooperatively regardless of the structure of the collaboration network. The network architecture does have an e¤ect on cooperation in R&D activities, though.5Our paper is also related to recent work by Casadesus-Masanell, Nalebu¤ and Yo¢ e (2008). Their paper provides conditions under which a …rm may bene…t from having a new competitor enter with a substitute good whenever there exists a complementor for both the …rm under consideration and its new competitor. Our framework di¤ers in that it does not focus on the e¤ects of entry on co-opetive settings, as they do, but rather it examines the incentives to form collaboration links and to invest in enhancing interoperability among complementors. The remainder of the paper is organized as follows. Section 2 introduces the game we consider. Section 3 characterizes the e¢ ciency properties of the unique equilibrium of the game depending on whether or not collaboration is exclusive. Section 4 shows that results are robust to changes in the solution concept and the implications of collaboration ties. Section 5 deals with concluding remarks. 2 The model We de…ne a system as a pair of perfectly complementary goods such as hardware and software. The two perfect complements giving rise to a system are called components Xand Y. It is assumed that there are two …rms costlessly producing component X,X1and X2, and two …rms costlessly producing component Y,Y1and Y2.6As a result, there are n= 4 systems: X1Y1,X1Y2,X2Y1and X2Y2. System XiYj(i; j = 1;2) can be bought by any consumer at price pi;j =pXi+pYj, where pXiand pYjrespectively denote the prices at which components Xiand Yjare sold. Whenever there is no risk of confusion, we will write pij instead of pi;j for system XiYj. Also, …rms X1and X2are typically referred to as the complementors of …rms Y1and Y2, and vice versa. It is assumed that there exists a unit mass of consumers willing to buy at most one system. System XiYjis assumed to create a gross utility of vi;jto any consumer (again, 5There is a recent literature on (pure and mixed) bundling by …rms that produce two perfectly complementary components in competition with …rms that produce just one of these components (see e.g. Denicolò 2000 and Choi 2008). The reason why this stream of research building on Economides and Salop (1992) is not related to our work is that we do not consider bundling, an issue that certainly deserves a separate analysis beyond the scope of our paper. 6That production is costless is without loss of generality if the marginal cost of production is constant and the …xed costs of operation are not too large. 4
we will typically write vij instead of vi;j). The gross utility vij is largely the outcome of choices by …rms Xiand Yj. More speci…cally, for some given scalar v > 0, we have that vij =v+xj i+yi j, where xj iis …rm Xi’s R&D investment in improving the quality of the match with …rm Yj’s component and yi jis …rm Yj’s R&D investment in improving the quality of the match with …rm Xi’s component.7Thus, the investment variables xj iand yi ja¤ect the vertical attributes of system XiYj. Given their system-speci…city, they can be viewed as investments in improving the interoperability of components Xiand Yj, although other interpretations are possible and may be more appealing depending on the context. Besides (possibly) being vertically di¤erentiated, systems are perceived by consumers as being horizontally di¤erentiated in an exogenous manner. To model consumer preferences over horizontally di¤erentiated systems, we follow Chen and Riordan (2007) in using their "spokes" model of nonlocalized di¤erentiation. Thus, each of the N= 4 systems desired by consumers is represented by a point at the origin of a line of length 1=2, a line which is denoted by lXiYjfor system XiYj(i; j = 1;2). The other end of a line is called its terminal, and it is assumed that the terminals of all lines meet at a point called the center (see Figure 1). All the existing consumers are uniformly distributed along the four lines. A consumer who is located on line lXiYjat distance dXiYj2[0;1=2] from system XiYjmust incur a transportation/disutility cost of tdXiYjwhen buying XiYj, where t0is a unit transportation cost. The same consumer must incur transportation cost t(1 dXiYj)when purchasing any other system (since lXiYj= 1=2for all i; j = 1;2). It is assumed that XiYj is the preferred system for any consumer on lXiYj, and any other system has probability 1=(N1) = 1=3of constituting the benchmark against which XiYjis to be compared by a consumer on lXiYj. A system that is not deemed as preferred or as a benchmark for a consumer is assumed to yield no utility to such a consumer. This assumption completes the description of the spokes model we use for modeling the horizontal attributes of systems.8 7See Goyal, Konovalov and Moraga-González (2008) for another setting with relationship-speci…c actions. 8Note that although in the most general version of the spokes model there are Nnsystems over which preferences are de…ned, we have let N=nfor the sake of simplicity. This means that we have assumed that there is no uncommercialized system that is possibly the object of desire by (some) consumers. 5
Given these features of …rms and consumers, we study a three-stage game. In the …rst stage, …rms Xi(i= 1;2) simultaneously form pairwise collaboration links with …rms Yj (j= 1;2). We let gij = 1 if a (costless) collaboration link between Xiand Yjis formed and gij = 0 otherwise, with the convention that gji =gij. We denote the network (i.e., the set of collaboration links) by g, that is, g=fg11; g12; g21; g22g2f0;1g4. Note that in principle we allow a …rm to form more than one collaboration link with its complementors (e.g., it may be possible that gi1=gi2= 1 for some i2 f1;2g). In the second stage of the game we consider, we assume that …rm Xichooses xj iat the same time as …rm Yjchooses yi j(i; j = 1;2). Given network gand some parameter 2(0;1), investments of x1 iand x2 iby …rm Xiresult in an R&D cost equal to CXi(x1 i; x2 ijg) = gi1(x1 i)2+gi2(x2 i)2, whereas investments of y1 jand y2 jby …rm Yjresult in an R&D cost equal to CYj(y1 j; y2 jjg) = g1j(y1 j)2+g2j(y2 j)2.9Hence, collaboration between …rms Xiand Yjyields that it is easier/cheaper for any of them to enhance the quality of the match with the component provided by the complementor. This captures in a simple manner useful but costless information exchanges between …rms Xiand Yjwith the aim of improving the interoperability of system XiYj. For this reason, the inverse of parameter can be understood as representing the extent of information sharing and its economic relevance: lowering the value of represents in our model more exchange of technically useful information among collaborators. In the third and last stage, prices pXiand pYjare set simultaneously in the standard noncooperative manner, and consumers make their purchase decisions given pij for i; j = 1;2. The solution concept is the same as in Goyal and Moraga-González (2001). Thus, for 9For example, if g=f1;0;0;0g, then CX1(x1 1; x2 1jg) = (x1 1)2+ (x2 1)2,CX2(x1 2; x2 2jg)=(x1 2)2+ (x2 2)2, CY1(y1 1; y2 1jg) = (y1 1)2+ (y2 1)2and CY2(y1 2; y2 2jg) = (y1 2)2+ (y2 2)2. 6
is stronger, it then holds that …rm X2prefers to lower x1 2in such a way that x1 2+y2 1does not vary with respect to the level under g=g1. In an analogous fashion, total investment in system X1Y2does not vary because …rm Y2lowers y1 2in way that o¤sets the increase in x2 1. The strength of systems X1Y2and X2Y1is then una¤ected, but …rms X2and Y2end up respectively decreasing x2 2and y2 2because system X1Y1becomes stronger. Interestingly, …rm X2reduces x1 2and x2 2by the same amount (and analogously for …rm Y2with e1 2and e2 2). The reason why this happens is that …rm X2equally bene…ts from investing in the match with Y1or Y2, so we must have that both x1 2and x2 2are reduced by the same amount because the strict convexity of R&D costs implies that it is more e¢ cient to spread e¤ort over two complementors rather than just one. In short, g1is not a stable network because …rms X1and Y1would mutually bene…t from forming a link. This incentive to form a link arises because of the cost synergy that is fostered by their collaboration and the positive strategic e¤ect of such collaboration.14 Despite …rms X1and Y1bene…t from the fact that …rms X2and Y2reduce their investment in each other, …rms X2and Y2exploit the incentive that …rms X1and Y1have to invest more in systems X1Y2and X2Y1by cutting down their respective investments in such systems, which harms X1and Y1a bit. Overall, the strategic reaction of …rms X2and Y2bene…ts …rms X1and Y1, thus reinforcing the positive direct e¤ect of cost sinergies exploited by X1and Y1. We conclude this subsection by analyzing what happens if each …rm has one, and only one, collaboration link, i.e., g=g3 f1;0;0;1g. Under the assumption that t > (1 + )=(108), all payo¤ functions are strictly concave, and the unique equilibrium is symmetric, being characterized by the following investments in match quality: x1 1(g3) = x2 2(g3) = y1 1(g3) = y2 2(g3)=1=12and x2 1(g3) = x1 2(g3) = y2 1(g3) = y1 2(g3)=1=12. Equilibrium pro…ts under g=g3are Xi(g3) = Yj(g3) = 108t 1 144, which are positive for t > (1 + )=(108). This parametric assumption also yields that quantity sold of each system is positive in equilibrium. We then have all the elements to rule out g=g2as an equilibrium outcome. 14 We compute the direct (pro…t) e¤ect of collaboration between …rms X1and Y1through the following thought experiment. Upon collaborating, both of these …rms react to the change in their investment costs taking into account the reactions of each other in an optimal manner, but keeping the investments of …rms X2and Y2as in g=g1(i.e., X2and Y2do not react to the change in the network architecture). This yields some pro…t for …rms X1and Y1, which after respectively subtracting X1(g1)and Y1(g1), gives the direct e¤ect of collaboration for each of them. The di¤erence between X1(g2) X1(g1)and the direct e¤ect for …rm X1then gives the strategic (pro…t) e¤ect of collaboration for this …rm, that is, how its pro…ts change because of the reaction of …rms X2and Y2to collaboration between X1and Y1. One can compute the strategic (pro…t) e¤ect of collaboration for …rm Y1in an analogous manner. 13
Lemma 3 Network g=g2cannot arise in equilibrium for t > (1 + )=(108). Proof. Noting that g3=g2+g22, it holds that X2(g3) = Y2(g3)> Y2(g2) = X2(g2)for t > (1 + )=(108), and …rms X2and Y2would mutually bene…t from forming a link with each other, so g=g2cannot be a stable network. Starting from g=g2, let us consider the incentive for …rms X2and Y2to form a tie, an incentive that is somewhat similar to the one that …rms X1and Y1to form a link starting from g=g1. Of course, the sinergistic e¤ect of collaboration leads to higher x2 2and y2 2. In the light of Remark 1, though, the increases in x2 2and y2 2also create an incentive for …rms X2and Y2to respectively increase x1 2and y1 2. The higher x1 2has a negative impact on …rm X1’s marginal payo¤, whereas the higher y1 2has a positive impact on …rm X1’s marginal payo¤ (by Remark 1). Taking into account that both of these e¤ects cancel out and that system X2Y2is stronger, it follows from Remark 1 that …rm X1prefers to lower x2 1, and it does it in such a way that x2 1+y1 2remains unchanged with respect to the level under g=g2. Similarly, total investment x1 2+y2 1in system X2Y1does not vary because …rm Y1lowers y2 1so as to o¤set the increase in x1 2. Even though systems X2Y1and X1Y2are neither strengthened nor weakened, the fact that system X2Y2is stronger induces …rms X1and Y1to respectively decrease x1 1and y1 1. Again, collaboration by …rms X2and Y2results in a positive strategic e¤ect that reinforces the cost sinergies that arise because of their collaboration, and hence X2and Y2mutually bene…t from forming a link with each other. In the light of Lemmata 2-3, it is clear that the unique stable network that arises when collaboration is exclusive is g=g3. However, each …rm would be better o¤ if collaboration were forbidden or impossible. Gross pro…ts are the same under g=g1and g=g3because …rms do not change their pricing and end up selling the same (of course, systems X1Y1 and X2Y2are bought more under g=g3, but this is at the expense of X1Y2and X2Y1). However, total investment costs are greater under g=g3than under g=g1(more precisely, (1 + )=(144)vs. 1=72), which explains why a …rm’s payo¤ decreases when going from g=g1to g=g3. Firms are worse o¤, but consumers are much better o¤ under g=g3than under g=g1, and as a result social welfare increases when going from g=g1to g=g3. In particular, we have the following result. Proposition 4 Let t > 1=(27)and suppose that collaborating with a complementor precludes a …rm from collaborating with the complementor’s competitor. Then: (i) The unique (up to a relabeling of …rms) equilibrium network is g=f1;0;0;1g. (ii) In equilibrium, …rm X1chooses to invest x1 1(g)=1=(12)in improving the quality of its match with complementor Y1, whereas it chooses to invest x2 1(g) = 1=12 in improving the quality of its match with complementor Y2. In turn, …rm Y1chooses to invest y1 1(g) = 14
1=(12)in improving the quality of its match with complementor X1, whereas it chooses to invest y2 1(g) = 1=12 in improving the quality of its match with complementor X2. In addition, each …rm earns a payo¤ of (108t 1)=(144). (iii) The equilibrium network g=f1;0;0;1gresults in a payo¤ for each …rm smaller than that achieved when g=f0;0;0;0g, even though g=f1;0;0;1gis socially preferred over g=f0;0;0;0g. Proof. Both for g=g1and g=g3, it holds that p X1=p X2=p Y1=p Y2= 3t=2, so p 11 =p 12 =p 21 =p 21 = 3t. In addition, the number of consumers purchasing system XiYj(i; j = 1;2) under g=g1is Q ij(g1) = 1=4. However, the number of consumers purchasing systems X1Y1and X2Y2under g=g3is Q 11(g3) = Q 22(g3) = 1 4+1 36t , whereas the number of consumers purchasing systems X1Y2and X2Y1under g=g3is Q 12(g3) = Q 21(g3) = 1 41 36t . Taking into account that line lXiYj(i; j = 1;2) has a length of 1=2and that that there exists a unit mass of consumers uniformly spread all over the four existing lines, the aggregate consumer surplus under g=g1is CS(g1) = 4 "1 2 v+1 12 +1 12 3ttZ1 2 0 zdz!#, while the aggregate consumer surplus under g=g3is CS(g3)=2"1 2 v+1 12+1 123ttZ1 2+1 18t 0 zdz!#+ 2"1 2 v+1 12 +1 12 3ttZ1 21 18t 0 zdz!#. Because 54t > 27t > 1> implies that CS(g3)CS(g1) = (1 )(54t +1) 324t2>0 and 2 X i=1 [ Xi(g3) Xi(g1)] + 2 X j=1 [ Yj(g3) Yj(g1)] = 1 36<0, it follows from the fact that 45t > 27t > 1that social surplus increases by (1 )(45t +1) 324t2> 0when going from g=g1to g=g3despite …rms are worse o¤ than when collaboration is forbidden or impossible. 15
Given that the social welfare comparison is entirely driven by the increase in consumer surplus, it is worthwhile explaining why it happens. Note …rst that, under g=g3, the investment in enhancing the match quality with a complementor with which a …rm does not collaborate is exactly the same as under g=g1. However, the investment in enhancing the match quality with a complementor with which a …rm does collaborate increases relative to network g=g1. Taking into account that component prices are the same under g=g1and g=g3, it follows that some systems are more appealing in their vertical attributes when g=g3, and hence are bought more than when g=g1. However, no consumer is worse o¤ under g=g3than under g=g1. Those consumers who were already consuming one of the enhanced systems are obviously better o¤ given the enhancements. In turn, those new consumers attracted by any of the enhanced systems experience a greater transportation cost, but still prefer purchasing one of the enhanced systems. This revealed preference argument shows that these consumers achieve a greater utility under g=g3than under g=g1. Finally, those consumers who were already consuming one of the systems whose quality is not enhanced make the same utility under g=g3than g=g1given that prices and total investments in these systems do not change. 3.2.2 Network structures under non-exclusivity We now deal with network structures in which at least one …rm has more than one collaboration link. As with the previously considered network structures, in equilibrium, …rms try to collaborate with as many complementors as it is possible, which will rule out g=g3as a plausible equilibrium outcome. The underlying economic forces are quite similar to those behind Proposition 4, and it holds that collaboration between two …rms involves positive direct and strategic e¤ects. Thus, start from a network architecture in which …rms Xiand Yjare not linked and let us consider what happens if these …rms begin collaborating with each other. The cost sinergies that arise from collaboration between such …rms result in higher xj iand yi j. In turn, x3j iand y3i jalso augment (by Remark 1). The negative e¤ect of x3j ion …rm X3i’s marginal payo¤ is o¤set by the positive e¤ect of y3i j, so the fact that XiYjbecomes stronger leads …rm X3ito reduce xj 3iin such a way that xj 3i+y3i jdoes not change. Similarly, yi 3jis reduced by …rm Y3jin such a way that x3j i+yi 3jremains invariant, so neither system XiY3jnor system X3iYjbecome weaker or stronger. The fact that system XiYjis strengthened then implies that x3j 3iand y31 3jare reduced. The outcome of these economic forces leads to the following result. Proposition 5 Let t > bt(3 2+p426+ 3)=(54)and suppose that collaborating with a complementor does not preclude a …rm from collaborating with the complementor’s 16
competitor. Then: (i) The unique equilibrium network structure is the complete network, namely g = f1;1;1;1g. (ii) In equilibrium, …rm Xichooses to invest xj i(g) = 1=(12)in improving the quality of its match with complementor Yj, whereas …rm Yjchooses to invest yi j(g)=1=(12)in improving the quality of its match with complementor Xi(i; j = 1;2). In addition, each …rm earns a payo¤ of (54t 1)=(72). (iii) The equilibrium network g =f1;1;1;1gresults in a payo¤ for each …rm smaller than that achieved when g=f0;0;0;0g, even though g =f1;1;1;1gis socially preferred over g=f0;0;0;0g. Proof. We start by noting that neither g=g1nor g=g2can arise as stable networks in the light of Proposition 4. We proceed to show that neither g=g3nor g=g4nor g=g5 can arise as equilibrium network con…gurations, which requires that we compute payo¤s for each of them (note that this has already been done for g=g3). So we …rst compute equilibrium payo¤s under g=g4 f1;1;0;0gunder the assumption that t > (3 )=(54)so as to ensure the strict concavity of payo¤s and the non-negativity of equilibrium pro…ts, investment levels and quantities sold. Then we have that x1 1(g4) = x2 1(g4) = 54t + 1 3 12(54t 1),x1 2(g4) = x2 2(g4) = 54t 3 + 12(54t 1),y2 1(g4) = y2 2(g4) = 1 12 and y1 1(g4) = y1 2(g4) = 1 12. As for equilibrium pro…ts for g=g4, they are X1(g4) = (54t 1) (54t + 1 3)2 72(54t 1)2, X2(g4) = (54t 1) (54t 3 + )2 72(54t 1)2 and Yj(g4) = 108t 1 144,j= 1;2. We analyze now the cases in which g=g5 f1;1;0;1gunder the assumption that t > bt(3 2+p426+ 3)=(54), which guarantees that payo¤s are strictly concave and that equilibrium pro…ts, investment levels and quantities sold are all non-negative. Solving for an equilibrium then yields x1 1(g5) = x2 1(g5) = y1 2(g5) = y2 2(g5) = 27t1 6(54t 1), x1 2(g5) = y2 1(g5) = 27t 1 6(54t 1), and x2 2(g5) = y1 1(g5) = 27t 1 6(54t 1). As for pro…ts 17
in equilibrium under g=g5, they are X1(g5) = Y2(g5) = (54t 1)(27t1)2 18(54t 1)2 and X2(g5) = Y1(g5) = (108t 1)(27t 1)2 36(54t 1)2. Noting that g5=g3+g12, it holds that X1(g5) = Y2(g5)> Y2(g3) = X1(g3)for t > bt, and we have that …rms X1and Y2would mutually bene…t from forming a link with each other, so g=g3cannot be a stable network. Network g=g4can also be discarded as an equilibrium outcome. To show this, note that g5=g4+g22, so the fact that X2(g5)> X2(g4)and Y2(g5)> Y2(g4)for t > btimplies that …rms X2and Y2would mutually bene…t from forming a link with each other. We deal now with the …nal network that needs to be considered, namely g=g6 f1;1;1;1g. Under the assumption that t > 1=(54)(which ensures payo¤ concavity and non-negativity of the relevant variables), the unique equilibrium is symmetric and involves the following investment levels: xj i(g6) = yi j(g6)=1=(12)(i; j = 1;2). Equilibrium pro…ts under g=g6are Xi(g6) = Yj(g6) = 54t 1 72, which are positive for t > 1=(54). Using these pro…ts, we can rule out g=g5as an equilibrium network. Noting that g6=g5+g21, it holds that X2(g6)> X2(g5)and Y1(g6)> Y1(g5)for t > bt. It then follows that …rms X2and Y1would mutually bene…t from forming a link with each other, and hence g=g5cannot be a stable network. The fact that Xi(g6)> Xi(g5)for i= 1;2and Yj(g6)> Yj(g5)for j= 1;2implies that g=g6is indeed an equilibrium network for t > bt, which proves parts (i) and (ii). In order to examine the e¢ ciency properties of the equilibrium network in the absence of exclusivity constraints and thus prove (iii), note that it holds both for g=g1and g=g6 that p X1=p X2=p Y1=p Y2= 3t=2, so p 11 =p 12 =p 21 =p 21 = 3tand Q ij = 1=4for i; j = 1;2. Therefore, the aggregate consumer surplus under g=g1is CS(g1) = 4 "1 2 v+1 12 +1 12 3ttZ1 2 0 zdz!#, 18
while the aggregate consumer surplus under g=g6is CS(g6) = 4 "1 2 v+1 12+1 123ttZ1 2 0 zdz!#. It then holds that CS(g6)CS(g1) = (1 )=(3)>0. Because 2 X i=1 [ Xi(g6) Xi(g1)] + 2 X j=1 [ Yj(g6) Yj(g1)] = 1 18<0, we have that social welfare increases by 5(1)=(18)>0when going from g=g1to g=g6 despite …rms are worse o¤ than when collaboration is forbidden or impossible. Hence, collaboration cannot improve upon the case in which each …rm acts uncoordinatedly. In equilibrium, …rms engage in a futile …ght to vertically di¤erentiate the systems in which they participate by collaborating with as many complementors as possible and by boosting investments accordingly. The larger investments result in an increase in investment costs, and the greater investment costs end up being just a wasteful rent dissipation, since nothing is gained in return despite the (possibly substantial) downward shift in cost functions.15 All consumers greatly bene…t from the greater functionality of every existing system, though. This explains why society would be worse o¤without collaborative activities involving information sharing among complementors. When all …rms simultaneously start collaborating with their complementors, the greater investments of each complementor in the systems in which it participates do not a¤ect a …rm’s pro…t because these e¤ects cancel out. So the existence of complementors is irrelevant for the result that …rms prefer the empty network over the complete one. In particular, an envelope argument shows the result is simply driven by the positive direct e¤ect on a …rm’s pro…t of having lower investment costs and the negative strategic e¤ect of having the competitor invest more in all the systems in which it participates. The latter e¤ect dominates the former for any admissible value of < 1, which explains why any …rm is better o¤ when no collaboration at all takes place. In addition, the strategic e¤ect becomes more important than the direct e¤ect as decreases. As a result, a …rm’s preference for the empty network over the complete one is accentuated as is lowered, that is, as information sharing among complementors makes it cheaper to enhance the quality of the match with a complementor. 15 Thus, if …rms invested as much as under g=g1, their pro…ts would indeed augment because the cost function shifts downwards. But precisely this downward shift in the cost function of a …rm leads all of them to futilely invest more, thus dissipating the potential gains from the cost function shift. 19
Also, using Propositions 4 and 5 yields that …rms are better o¤ under exclusivity than under non-exclusivity. Proposition 6 Let t > bt(3 2+p426+ 3)=(54). Then equilibrium payo¤s decrease as decreases. Furthermore, the equilibrium network under formation of nonexclusive collaboration ties results in a payo¤ for each …rm smaller than that achieved under exclusive collaboration ties. 4 Robustness of the results 4.1 Stronger equilibrium concepts We recall that we have used the notion of pairwise stability as our solution concept for the strategic network formation game under consideration. In the context of our game, the main drawback of this solution concept has to do with the possibility that a …rm with several links may want to sever more than one link at a time.16 However, this criticism does not apply to the game just analyzed. Indeed, the fact that X2(g6)> X2(g4)implies that the complete network is stable even if the pairwise stability solution concept is augmented to allow for the deletion of several links at a time (the complete network is then said to be pairwise Nash stable). 4.2 Collaboration fostering partly cooperative investments The result that collaboration is ine¢ cient from the viewpoints of …rms also holds if freeriding by a …rm on a complementor’s investment e¤ort is partly mitigated if both …rms being collaborating. To this end, let = 1 and suppose that gij = 1 implies now that both …rms Xiand Yjcare in some sense about each other’s payo¤ when making investment choices.17 Speci…cally, given network architecture gand some parameter 2(0;1), suppose that …rm Xi(i= 1;2) chooses in the second stage x1 i0and x2 i0to maximize either Xi(x1 i)2(x2 i)2+ 2 X j=1 gij[ Yj(y1 j)2(y2 j)2] 16 See Jackson (2008, pp. 156 and 371-376) for a thorough discussion of the virtues and limitations of pairwise stability as a solution concept. 17 There are many formal or informal arrangements that may lead two complementors that collaborate with each other to make their investments in improving their match quality in a (somewhat) cooperative manner. Reasons range from research alliances (or collusive R&D cartels) to relational capital concerns in ongoing relationships between …rms that need each other to some extent because of their complementarity. 20
or Xi(x1 i)2(x2 i)2+ 2 X j=1 gij[i Yj(yi j)2], and similarly for …rm Yj(j= 1;2). Suppose also that, except in the second stage, …rms solely pay attention to their own payo¤s when making decisions of whether to form a collaboration tie or which price to set. It can be shown in these cases that our results go through, with the exception that the equilibrium networks are not only ine¢ cient for the …rms but also for consumers (and hence society).18 In the absence of pecuniary e¤ects of collaboration, one may then expect ine¢ ciencies to arise both at the …rm and the social levels. Collaboration with pecuniary e¤ects resolves the social ine¢ ciences but not the private ones, which remain, as Propositions 4 and 5 show. 5 Conclusion The locus of strategic interaction in many high-tech industries has broadened from the traditional competitive approach based on value capture towards one in which cooperative aspects with regards to value creation also play a critical role, as Brandenburger and Nalebu¤ (1996) emphasize. Not surprisingly, such "co-opetitive" settings display rich innovation ecosystems in which complementors collaborate with each other in R&D actitivities. This paper has shown that such rich innovation ecosystems may be an equilibrium phenomenon with disturbing properties for their members. In particular, we have shown that they may be an ine¢ cient outcome for competing …rms that can collaborate with complementors. They may also be ine¢ cient for society. These results hold under a variety of scenarios (e.g., regardless of whether or not collaboration exhibits exclusive features). In this paper, we have abstracted away from dynamics to clarify our points, but there are many issues that have to do with dynamic variables. For example, collaboration may refer to the timing at which complementary products are brought to the market. Exploring this kind of issues seems promising enough to warrant further work on this completely unexplored area. 18 See Mantovani and Ruiz-Aliseda (2011) for a full analysis of these situations, both under exclusivity and the lack thereof. 21
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