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Fundación de Estudios de Economía Aplicada

Compatibility with Firm Dominance by 米 María Fernanda Viecens DOCUMENTO DE TRABAJO 2009-12

February 2009

FEDEA.

Jorge Juan, 46 28001 Madrid -España Tel.: +34 914 359 020 Fax: +34 915 779 575 infpub@fedea.es

María Fernanda Viecens FEDEA^†

This Version: February, 2009 First Version: November, 2007

Abstract

This paper analyzes the efect of firm dominance on the incentives to become compatible and how compatibility decisions afect investment incentives. We will consider compatibility in two dimensions: compatibility of the complementary good and inter-network compatibility. We show that if products are substitutes, compatibility tends to be welfare decreasing with the potential negative consequences of increasing compatibility being more likely when asymmetries are strong. We also find that in many instances the dominant firm’s interests regarding compatibility are in line with those of users, and are opposite to those of the weak firm, which will always demand more compatibility to be enforced. Finally we show that compatibility may harm innovation, particularly for the dominant firm.

Keywords: Firm dominance, compatibility, incentives to innnovate, competition policy, technology platforms, Microsoft case.

JEL Classification: L10, L15, L41, L86

^∗The starting point for this paper was chapter 4 of my Ph.D. dissertation at Universidad Carlos III de Madrid. I thank M. Ángeles de Frutos for her advise, support and encouragement. I am grateful to Natalia Fabra, Andrea Fosfuri, Juanjo Ganuza, Doh-Shin Jeon, Gerard Llobet, Martin Peitz and Konrad Stahl for very helpful suggestions. The usual disclaimers apply.

^†Mailing address: Jorge Juan 46, 28001, Madrid, Spain; E-mail: fviecens@fedea.es; tel.: +34 91 435 9020, fax: +34 91 577 9575.

1 Introduction

"Customers have mandated that the companies, long arch-rivals, must ensure that their infrastructure work together well. But beyond that, they will continue to compete tooth and nail" (Ron Hovsepian, Novell’s president, BrainShare 2007).¹

"We’re evolving to a point where there are a couple of platforms. You can’t standardize everything ... They (customers) pushed on us a lot about interoperability, but also about continuing innovation" (Craig Mundie, Microsoft’s chief research and strategy oficer, BrainShare 2007).²

Several dominant firms are being scrutinized by the European Commission in cases where interoperability/compatibility issues have been one of the main arguments in the authorities’ concerns. For instance, Sun Mycrosystems charged that Microsoft was refusing to share information that would allow interoperability between its servers and the equipment produced by the software giant. In March 2004 the Commission ordered Microsoft to disclose confidential computer code to competitors and in September 2007, Europe’s Second-Highest Court reafirmed the decision. In January 2008, the European Commission began two new antitrust investigations into Microsoft focussed on the compatibility of its Ofice package with other companies’ software. Then, the company made an announcement to improve interoperability with competitors. Similarly, in March 2007 the European Commissioner for Consumers complained about the fact that iPod (Apple’s device) was the only portable device that will play iTunes.³ Given the importance of these antitrust cases and of these industries in modern economies, compatibility/ interoperability appears as a relevant issue for competition policy and regulation.

Discussions about compatibility and market power are certainly nothing new. However, the recent evolution of technological industries has brought forth a new set of issues, in particular the implications of market dominance in terms of strategic decisions on compatibility/interoperability and the impact of the latter on innovation incentives. After the EU resolution on the Microsoft case, some voices believe that the Commission will now go after other technology firms with large market shares, which will force companies to give up intellectual property and will curb the incentives to innovate.⁴ Among these voices, Thomas O. Barnett, the assistant attorney general for the Justice Department’s Antitrust Division has declared that "rather than helping consumers, it may have the unfortunate consequence of harming consumers by chilling innovation

¹ See http://www.crn.com/software/198100037 ² Ibidem.

³ Norway has threatened to take action against Apple if it does not open up its digital rights management (DRM) system to other companies. France proposed a law requiring digital music retailers to make all downloaded songs compatible with various MP3 players. Other music download sites use a variety of DRM technologies that are incompatible with the iPod. (see Reuters, April 3, 2007 and http://www.crmbuyer.com/story/ipod/56255.html).

⁴ See www.economist.com, September 20, 2007.

and discouraging competition".⁵

The impact of firms’ decisions on compatibility issues and/or on innovation goes beyond antitrust analysis. On one hand, the extent to which a firm will be compatible with other firms is one of its key strategic decisions. Note that vertical arrangements lead to a situation of incompatibility. In contrast, it may be in the firms’ interest to promote compatibility, as the agreement signed by Microsoft and Novell to enhance interoperability between Linux and Windows illustrates.⁶ On the other hand, consumers will clearly be afected by firms decisions. A console firm that impedes its video-game developers to provide their products to its competitors is de facto refusing compatibility, and it is hence afecting to consumers of the multi-billion video game industry.

In this paper we will analyze firms’ attitudes towards compatibility and how compatibility decisions afect incentives on investments in producing a product improvement. In particular, we study how diferent levels of firm dominance, measured by a premium in consumer valuations, influence these incentives. Thus, the questions we address include (a) is there a relationship between market dominance and compatibility incentives?, (b) does compatibility enhance or discourage innovation?, (c) is a policy of mandatory compatibility socially desirable?.

To address these issues we propose a model of platform competition. We assume that users buy a platform and its compatible applications. We allow for applications to be substitutes, complements or independent. We consider compatibility in two dimensions. First, compatibility of the complementary good, to which we will refer as compatibility in applications. Typical examples are software that can be either run or not with diferent hardware. Second, we consider inter-network compatibility. In this case, direct network externalities are present in the sense that one user’s value for a good is higher when another user buys the compatible good, as in the case of personal computers that allow users to exchange e-mails. Literature has largely ignored the diference between both types of compatibility, however our model yields diferent results for each of them (see Section 1.2 for more details about these two forms of compatibility and about related literature).

We find that the dominant firm will never promote compatibility in applications. In contrast, both firms find inter-network compatibility profitable.

Compatibility in applications is not always beneficial for consumers. Moreover, we find that the dominant firm’s incentives to be compatible may coincide with those of users while being against those of the weak firm. When this is the case, imposing compatibility would harm both consumer surplus and total welfare. In particular, compatibility in applications decreases both consumer surplus and welfare if applications are close substitutes. Furthermore, the efect of compatibility on welfare and on consumer surplus tends to be negative when firm dominance is important. We also find that the presence of strong direct network efects strengthen the potential negative incidence of compatibility in applications. Regarding inter-network compatibility, it is always consumers’ surplus reducing and has a negative impact on total welfare as long as the market is very asymmetric.

⁵ The New York Times, September 18, 2007. See also Nicholas Economides in his "Commentary of the EU Microsoft Antitrust Case" (September 2007, www.NETInst.org) where he remarks: "By requiring full disclosure at a nominal price, the EU decision in efect reduces the value of intellectual property for dominant firms. Additionally, in the particular case, full understanding of internal Windows functions is valuable to Sun beyond interoperability (...). That is, the Commission’s vertical remedy gives an advantage to Sun in horizontal competition".

⁶ The Novell and Microsoft Collaborate relates to both interoperability and innovation. Note that the companies expressed the intention of creating a joint research facility to pursue new software solutions for virtualization, management, and document format compatibility (for details, see http://www.novel.com/linux/microsoft/faq.html).

⁷ See Lee (2007) for an empirical analysis on the impact of vertical integration and exclusive contracting in the US video game industry during the period 2000-2005.

We also show that any type of compatibility often reduces the incentives of the firms to invest, particularly those of the dominant one. When compatibility in applications is present a free-riding problem arises: an investment made by one firm that adds value to the compatible good is shared by the other firm, thus compatibility may induce lower incentives to invest. Additional reasons for the lack of interest in compatibility (besides the free-riding problem) arise from the substitutability between the applications.

The main antitrust implication of our results is to advise that decisions on compatibility and/or on exclusivity must be tailored to each particular case as the degree of substitutability, direct network efects, and firm dominance may alter the desirability or not of enforcing compatibility.

The paper is organized as follows. The rest of this Section presents the related literature and a discussion on compatibility and its possible modeling strategies. Section 2 presents the basic framework. In Section 3 we show the results for compatibility in applications and inter-network compatibility. Section 4 analyses investment incentives related to compatibility. Section 5 studies the robustness of the model proposed in Section 2. Section 6 concludes. Finally, proofs are relegated to the Appendix.

1.1 Related literature

This paper borrows modelling strategies from, and contributes to, two strands of literature. First, to the literature on compatibility in industries with network efects. The seminal papers of Katz and Shapiro (1985) and Farrell and Saloner (1985) predict that incentives for incompatibility will difer across firms and will be greater for firms with larger networks, since under compatibility these firms will lose the competitive advantage that their networks confer. Adopting the Katz and Shapiro-model, Crémer, Rey and Tirole (2000) analyze the competition between Internet backbone providers with asymmetric installed bases. They show that a firm with a large installed base may have incentives to reduce the degree of compatibility towards its smaller rivals. In the same vein, Malueg and Schwartz (2006) analyze the conditions under which the firm with the largest market share of installed-base customers will prefer incompatibility with smaller rivals that are compatible among themselves. They find that the largest firm is more likely to prefer incompatibility over compatibility if with the latter its market share rises above fifty percent or if the potential to add consumers falls. Chen, Doraszelski and Harrington (2007) consider product compatibility with market dominance in a dynamic setting. They find that if firms have similar installed bases they make their products compatible. But, if a firm gets a larger installed base then it may make its product incompatible.

Similar to these papers, we find that the dominant firm is against compatibility related to applications. However, this firm would indeed promote inter-network compatibility.

In the absence of consumption externalities, some papers have analyzed firms’ incentives to make components of diferent systems compatible, in what is known as the mix-and-match literature.⁸ Economides (1989) finds that profits are higher under compatibility, so that a fully compatible regime is the unique perfect equilibrium. However, symmetry of system demands is crucial for compatibility to occur.

Economides (2006) argues that it is socially eficient to move towards compatibility. In the same vein, Katz and Shapiro (1985) show that a move to complete compatibility would raise consumers’ surplus. In contrast, these results do not always hold in our framework. In particular, we conclude that a move towards a larger degree of compatibility in the applications may be harmful for users and social welfare, particularly when asymmetries are strong. Moreover, in the case of inter-network compatibility our model predicts that it should not be promoted by consumers.

The second strand of the literature which this paper relates to, is devoted to studying incentives to innovate when firms are asymmetric. Cabral and Polak (2007) analyze the efect of firm dominance on the incentives for R&D. They find that an increase in firm dominance increases the dominant firm’s incentives while decreasing the other firm’s incentives for R&D. They also show that total research efort decreases when firm dominance increases, and that firm dominance is good for innovation only when property rights are strong.

Few papers combine the two aforementioned strands of the literature. Cabral and Salant (2007), in a model of R&D competition and cooperative standards setting, argue that standardization leads to a free riding problem and thus to a decrease in marginal incentives for R&D investment. Our model also identifies the free-riding problem and a decreasing marginal incentive for the dominant firm, but an increasing one for the weak firm. Hannan and Borzekowski (2006) investigate whether incompatibility across rival systems may influence firms’ incentives to invest in product changes that are beneficial to the consumer in the case of bank ATM networks. They consider the number of ATM locations as the measure of product quality and the surcharge fees as an index of incompatibility. They find that an increase in incompatibility for Iowa banks caused a substantial increase in the number of ATM locations ofered to customers. Besides, the efect is greater for larger banks than for smaller ones. In a symmetric setting Choi (2003) analyzes the efect of compatibility on R&D incentives. He finds that a firm that makes two components incompatible increases its R&D level and outside firms reduce theirs in a linear demand model with quadratic R&D cost functions.

⁸ The mix-and-match literature does not assume a priori network externalities; however, it is clear that demand in mix-and-match models exhibits network externalities. The mix-andmatch approach was originated by Matutes and Regibeau (1988), and has been used in several papers including Economides (1989), Matutes and Regibeau (1989, 1992).

1.2 On the modeling of compatibility in the literature

Standards specify properties that a product must have in order to work (physically or functionally) with complementary products within a product or service system. Compatibility refers to "the possibility of costlessly combining various links and nodes on the network to produce demanded goods. Two complementary components A and B are compatible when they can be combined to produce a composite good or service" (see Economides (2006)). Interoperability refers to "the ability of two or more systems or components to exchange information and to use the information that has been exchanged" (as defined by the IEEE Standard Computing Dictionary). Both compatibility and interoperability are hence related to the possibility of sharing standards.

The economic literature has modelled compatibility (or interoperability) in two ways. First, as in the seminal paper of Katz and Shapiro (1985), compatibility afects the consumers’ surplus of buying one unit of a good, making it depend on the number of other agents who join the network associated with that product. If denotes the number of customers that a consumer expects firm i to have and denotes the consumers’ prediction of the size of the network with which firm i is associated, then brands are incompatible whenever . In contrast, if m firms products are compatible then

\[y _ {i} ^ {e} = \sum_ {j = 1} ^ {m} x _ {j} ^ {e} \text { for } i = 1, 2,..., m.\]

Crémer, Rey and Tirole (2000) adapted Katz and Shapiro (1985) to incorporate installed-base customers where firms difer in their locked-in, installed bases of customers.⁹ Without compatibility, the size of the network of each firm is its installed base and the mass of new customers. With compatibility, the networks include all new customers and the installed bases of all the firms. In all of these papers network efects are direct since the main purpose of the consumers is to contact other users.

There is a second set of papers that also studies compatibility issues in a mixand match framework (Economides (1989), Matutes and Regibeau, (1988, 1992), Choi (2003)). They consider systems that are composed of two components (e.g. hardware and software) produced by two firms A and B (Economides (1989) generalizes to n firms). If the components sold by the two firms are not compatible, only two systems are available for consumers, systems and . If the components are compatible, consumers have the additional options of and . In these papers consumers derive utility from the system itself.

⁹ Other papers that consider similar models include Malueg and Schwartz (2006) or Chen, Doraszelski and Harrington (2007).

We will analyze here platforms that together with a set of applications compose a system so that, in the context of the hardware-software paradigm, we relate compatibility to the number of applications that can be used with the platform.¹⁰ Because of this, we say that there is compatibility in applications if the users have the possibility to construct a wider variety of systems (e.g. and .

In contrast, we talk about inter-network compatibility when consumers are concerned with exchanging information and files with other consumers. Specifically, consumers may exchange information with consumers of other platforms and not only with consumers that bought the same platform. The modeling of compatibility in Katz and Shapiro (1985) seems to us the appropriate model to address this type of compatibility as direct network efects are more implicated.¹¹

2 The Model

We consider a market in which consumers derive utility from consuming a platform and its compatible applications. There are two platforms that compete à la Hotelling and are located at the two extremes of the unit line. We assume that they cover the market.

Each platform has a fully compatible application . Application 1 is fully compatible with platform A and its value is . This application is compatible with platform B in a degree δ, so that its value for a user of platform B is δW₁.¹² Analogously, application 2 is fully compatible with platform B with value W (δW ) for users of platform B (A). Platforms have also a stand-alone value to consumers so that the value of platform l is either or

Applications are produced by third party developers. The consumers pay a price for the platform and for each application that they buy from the developers. We consider that platform firms follow a two-sided strategy, setting a fee to developers per unit of application sold to be used with the platforms. We allow platforms to discriminate by type of application. Developers behave competitively, consequently, the price they charge to consumers for their applications is their marginal cost given here by the per unit fee set by the platforms.¹³ Note that the model resembles the traditional models of telephony (Lafont, Rey and Tirole (1998)).

^{1 0} Corts and Lederman (2007) consider the existence of both exclusive (non-compatible) and non-exclusive (compatible) software in an empirical model of indirect network efects. They argue that over the last 20 years - over succesive technological generations - software has become less likely to be exclusive to a particular platform, and that this trend and its consequences have largely been ignored.

^{1 1} In some sense our interpretation of the diference between "compatibility in applications" and "inter-network compatibility" is similar to the one employed by Clements (2004) to distinguish between direct and indirect network efects. Similarly, Economides and White (1994), diferentiates between two-way networks and one-way network.

^{1 2} Alternatively, we may interpret the application fully compatible for platform l as a set of applications and δ as a measure of the subset of applications that can be used by a user of the other platform.

We assume that the market structure is asymmetric with a dominant and a weak firm. There are two potential sources of asymmetry: an asymmetry in the stand-alone value of the platform and/or an asymmetry in the value of the fully compatible application. Note that under full compatibility the latter asymmetry disappears.¹⁴ We assume that and initially, we also assume that

Applications side A representative consumer of platform A maximizes the following net utility from consuming the applications,

\[u _ {A} = W _ {1} q _ {A 1} + \delta W _ {2} q _ {A 2} - \frac {1}{2} b \left(q _ {A 1} ^ {2} + 2 \sigma q _ {A 1} q _ {A 2} + q _ {A 2} ^ {2}\right) - p _ {A 1} q _ {A 1} - p _ {A 2} q _ {A 2},\]

where is the quantity of application i consumed by a consumer of platform is its price, and . Negative values of σ would make the model one of demand for complementary applications. the applications are independent in demand. As σ approaches 1, the applications become closer substitutes.¹⁵ The representative consumer of platform B maximizes an analogous net utility . These utilities yield demands

\[\begin{array}{r c l} q _ {A 1} & = & \frac {1}{b (1 - \sigma^ {2})} (W _ {1} - \sigma \delta W _ {2} - p _ {A 1} + \sigma p _ {A 2}), \\ q _ {B 2} & = & \frac {1}{b (1 - \sigma^ {2})} (W _ {2} - \sigma \delta W _ {1} - p _ {B 2} + \sigma p _ {B 1}), \\ q _ {A 2} & = & \left\{ \begin{array}{c c c} 0 & \mathrm{if} \delta = 0 \\ \frac {1}{b (1 - \sigma^ {2})} (\delta W _ {2} - \sigma W _ {1} - p _ {A 2} + \sigma p _ {A 1}) & \mathrm{if} \delta > 0 \end{array} \right., \\ q _ {B 1} & = & \left\{ \begin{array}{c c c} 0 & \mathrm{if} \delta = 0 \\ \frac {1}{b (1 - \sigma^ {2})} (\delta W _ {1} - \sigma W _ {2} - p _ {B 1} + \sigma p _ {B 2}) & \mathrm{if} \delta > 0 \end{array} \right.. \end{array}\]

Note that although the market for the platforms is fixed and covered, the size of the applications market is increasing in both δ and the value of the applications.

Users side Consumers are uniformly distributed on the unit line with respect to their preferences for the platforms. A consumer with a most preferred platform type (or location) x derives utility from the use of the platform she buys and from its compatible applications. In particular, the net utility derived by a consumer who buys platform l located at the beginning of the unit line is given by

^{1 3} This assumption is not innocuous. We introduce it for simplicity to concentrate on the consequences of changes in compatibility. However, we conjecture that, although quantitative results will change if we introduce some market power on the sellers’ side there is a (i.e., double marginalization in the price), they will not change qualitatively.

^{1 4} The way we introduce asymmetries in the platforms is similar to Carter and Wright (2003).

^{1 5} Examples of substitute applications are text processors written for diferent operating systems, or football games written for diferent video consoles. Complementary applications are a text processor and a spreadsheet, while independent applications are a role-play game and a boxing game.

\[U _ {l} = w _ {l} (\delta) + V - t x - s _ {l}.\tag{1}\]

The first and second term reflect the value of owning platform l. The term is the indirect utility that the applications compatible with platform l generate. Thus, is a function of and , where the level of δ is exogenously determined, for instance, by a regulator. The third term is the disutility stemming from not consuming the most preferred platform type (the transportation cost or the degree of diferentiation in the standard Hotelling model). Finally, is the price charged by platform l. Note that in (1) users utility do not depend on the number of platform users, i.e., there are no direct network efects.

If direct network efect exist then (1) becomes

\[U _ {l} = w _ {l} (\delta) + V + \eta \theta_ {l} - t x - s _ {l},\tag{2}\]

where measures the extent of the network efect. We will model consumers’ net utility as in (2) to analyze inter-network compatibility where direct network efects among users are present. These efects could be important in markets where users want to exchange or share the applications (for instance users of Word can share files, gamers of the same video console can exchange the games, etc ).

The problem of the platforms Each platform has two sources of income, one from the consumers and another from the developers of applications. Thus, under the proviso that costs are null, platform profits are given by

\[\Pi_ {l} (\delta) = \left(s _ {l} (\delta) + \pi_ {l} (\delta)\right) \theta_ {l} (\delta),\]

where represents the profit that the platform gets, per consumer, from the developers side.¹⁶

The timing of the game is the following: first, the platforms set prices to consumers who then decide which platform to buy. Then, the platforms set fees to developers. Finally, the developers set competitive prices to the consumers for the applications they sell. We compute the subgame perfect equilibria, which implies that consumers form rational expectations to determine both the size of each network and the prices that developers will set for the applications, while knowing the prices set by the platforms.

3 Compatibility in Applications

To study the impact of compatibility on platform competition we first analyze the market for applications, then analyze the stage at which platforms set prices.

Π_l = s_l (δ) θ_l (δ)

^{1 6} Our model is quite similar to the model in Church and Gandal 1992 and 2000. However, they only consider platforms that follow a one-sided strategy with profits .

Since developers behave competitively, the prices that they set coincide with the fees that platforms charged to them so that we can rewrite the problem faced by platform l in the second stage as follows

\[\max _ {p _ {l 1}, p _ {l 2}} \pi_ {l} = p _ {l 1} q _ {l 1} + p _ {l 2} q _ {l 2}.\tag{3}\]

Note that at this stage each platform’s market share is given so that each platform sets prices for applications as if it were a monopolist. Lemma 1 summarizes some properties of the indirect utilities and profits which arise from the applications.

Lemma 1

i) If applications are complementary or independent, then and for all l.

ii) If applications are substitutes, there exist and such that

if then for all

if then for all l,

\[i f \bar {\sigma} _ {2} < \sigma < \bar {\sigma} _ {1} t h e n \frac {\partial w _ {A} (\delta)}{\partial \delta} < 0, \frac {\partial \pi_ {A} (\delta)}{\partial \delta} < 0, \frac {\partial w _ {B} (\delta)}{\partial \delta} > 0, \frac {\partial \pi_ {B} (\delta)}{\partial \delta} > 0.\]

Proof. see the Appendix.

Results in Lemma 1 resemble the standard results in the literature on multiproduct monopolist (see Tirole (1987)). The price and the demanded quantity of the non-fully compatible applications are increasing in for each platform and for any value of . However, how the quantity of the fully compatible applications reacts to changes in δ depends on the sign of (the price is neutral). As expected, this quantity increases with δ if applications are complementary, it decreases if applications are substitutes and it is neutral if they are independent.

Let us define w (δ) (π (δ)) as the diference in consumer surplus (profits from developers’ side) generated by the two platforms, i.e.,

\[w (\delta) = w _ {A} (\delta) - w _ {B} (\delta) = \frac {1}{4 (1 - \sigma^ {2}) b} \left(W _ {1} ^ {2} - W _ {2} ^ {2}\right) (1 - \delta^ {2}) \left(1 - \frac {1}{b}\right)\tag{4}\]

\[\pi (\delta) = \pi_ {A} (\delta) - \pi_ {B} (\delta) = \frac {1}{4 (1 - \sigma^ {2}) b} (W _ {1} ^ {2} - W _ {2} ^ {2}) (1 - \delta^ {2}).\tag{5}\]

Two facts that trivially follow from (4) and (5) are next stated:

Fact 1. and

Fact 2. and π if and w (δ) = π (δ) = 0 if δ = 1.

The rationale behind the above facts has an intuitive reasoning. The difference between the two platforms in the indirect utility that consumers derive from the applications decreases with the degree of compatibility. In the limit, when , this diference is zero since both platforms will provide both cations with values and . Similarly, the platform with the most valuable application receives higher profits from the developers side, a diference that disappears if there is full compatibility (δ = 1) .

To avoid platform A tipping the market, an additional assumption is introduced:

Assumption

This implies that the transportation cost parameter is greater than the term that reflects the asymmetries between the platforms. Note that if there is no asymmetry in the applications or if there is full compatibility, i.e., w (δ) = , Assumption 1 is trivially satisfied.

Platform competition At the first stage, platforms compete by setting prices to consumers. We assume that consumers have beliefs about network sizes which lead them to make purchase decisions that in equilibrium confirm their beliefs. Moreover, they anticipate that monopoly prices for applications will be set at the last stage. Since platform profits are given by

\[\Pi_ {l} = \left(s _ {l} + \pi_ {l} (\delta)\right) \theta_ {l} (\delta),\]

the degree of compatibility afects price competition as it has an impact on platforms’ market shares. Moreover, compatibility makes platforms softer or tougher competitors depending on whether applications are complementary or substitutes. The problem faced by platforms at this stage is similar to that of a multimarket oligopoly.¹⁷ An increase in the level of compatibility acts as a positive (negative) shock in the markets of applications when as Lemma 1 shows. Consequently, a change in δ has a direct efect on the profits made from the applications side and an indirect efect on the platforms market share.

Consider first the case of independent or complementary applications. An increase in δ moves platforms’ reaction curves, i.e., and , inwards. Furthermore, this move is larger for the platform A which is stronger in the applications market as compatibility "reduces diferentiation" which afects this platform negatively. Recall that as compatibility increases the platforms tend to ofer the same surplus in terms of applications. Consequently, both firms will be more aggressive in the platform market. This is explained by two facts: first, the complementarity between the applications and the platform, and second, by the fact that the reaction curves and are upward sloping as prices are strategic complements. Assume now that applications are close substitutes. In this case both firms will be less aggressive in the platform’s market since an increase in δ moves platforms’ reaction curves outwards.

These results are the content of the next lemma.

Lemma 2

i) If applications are complementary or independent, platform price competition is more aggressive the higher the degree of compatibility between the firms.

^{1 7} See Bulow, Geanakoplos and Klemperer (1985).

ii) If applications are close substitutes, platform price competition is less aggressive the higher is the degree of compatibility.

iii) There is a degree of substitution for which the platform with the dominant application becomes less aggressive whereas the platform with the weak application becomes more aggressive as the degree of compatibility increases.

Proof. see the Appendix.

In our framework, incompatibility generates a similar situation to the one in which developers single-home. In contrast, when compatibility takes place, it leads to partial (or full) multihoming.¹⁸ Lemma 2 shows that if applications are complementary or independent, platform price competition (referred to the price set to users) is more aggressive the higher the degree of compatibility, a result that is consistent with a competitive bottleneck equilibrium. In contrast, platform price competition is less aggressive, as compatibility increases, when applications are close substitutes. It follows that the typical outcome related to a competitive bottleneck equilibrium (that the platforms strongly attend the interests of the single-homing group) may not be robust to introducing some substitution on the sellers side.

3.1 Welfare analysis

We have seen that the degree of compatibility afects the prices that platforms charge to users. We next analyze its impact on platforms’ profits. We will say that firm l has incentives to promote compatibility if its profits increase when the degree of compatibility increases, otherwise we will say that firm l does not have incentives to promote compatibility.

A priory, one may think that both platforms would earn more as the degree of compatibility increases since the market for applications expands and at the same time both platforms will ofer a better product. However this is only true for the platform that is weaker for which an increase in compatibility positively afects its total profits. Regarding the other platform, compatibility has a positive and a negative impact on its profits. Consider complement or independent applications. Thus, on the one hand, platform A increases its profits in the applications market. But, on the other hand, the strong competition for the users in the first stage in a covered and fixed market, leads to a price reduction. Platform A uses the gains obtained on the applications side to ofer lower prices to consumers in the first stage. The reduction in overtakes the gains in due to the fight to maintain the market share. In contrast, for platform B the reduction in price is indeed compensated by the gains in profits from the applications, In this case, platforms are less aggressive as compatibility increases but since the "reducing diferentiation" efect remains, the profits of the strong platform in the market of applications still decrease whereas the profits of the weaker one increase. Finally, note that if we consider platforms with symmetric applications, their profits would not depend on the degree of compatibility. The following lemma summarizes the above discussion.

^{1 8} Incompatibility can also be interpreted as exclusive developers and compatibility as nonexclusive ones. Although we focus on compatibility, decisions on standards and/or on exclusivity arrangements can be considered forms of compatibility.

^{1 9} Note that if applications are close substitutes, profits in the applications market are decreasing in δ. Under these conditions platforms charge more to the users side and less to the applications side when compatibility increases. More precisely, there is a range of σ for

Lemma 3

i) If both firms are indiferent about compatibility

ii) If the dominant firm in the applications market does not benefit from an increase in the degree of compatibility whereas the weaker one does

Proof. see the Appendix.

Lemma 3 shows that whenever applications are symmetric, the higher profits that the platforms can make on one side due to a greater compatibility get discounted in the price of the other side in such a way that in equilibrium total profits are not afected by δ. The lemma also shows that, with asymmetric applications, the dominant firm has lower profits under compatibility but the weak firm has larger ones. Consequently, the strong platform has no incentive to promote compatibility whereas the weak one would like compatibility to be enforced. This implies that if compatibility were an outcome of a coordinated decision between firms, it would never arise.

Compatibility, consumer surplus and welfare

From (1) it follows that total consumer surplus denoted by is given by

\[\int_ {0} ^ {\theta_ {A} (\delta)} \left(w _ {A} (\delta) + V - t x - s _ {A} (\delta)\right) d x + \int_ {\theta_ {A} (\delta)} ^ {1} \left(w _ {B} (\delta) + V - t (1 - x) - s _ {B} (\delta)\right) d x,\]

which equals

\[V + \theta_ {A} (\delta) (w _ {A} (\delta) - s _ {A} (\delta)) + (1 - \theta_ {A} (\delta)) (w _ {B} (\delta) - s _ {B} (\delta)) - \frac {1}{2} t (\theta_ {A} ^ {2} + (1 - \theta_ {A}) ^ {2}).\]

Total welfare denoted by can also be written using a similar decomposition as above as

\[V + \theta_ {A} (\delta) (w _ {A} (\delta) + \pi_ {A} (\delta)) + (1 - \theta_ {A} (\delta)) (w _ {B} (\delta) + \pi_ {B} (\delta)) - \frac {1}{2} t (\theta_ {A} ^ {2} + (1 - \theta_ {A}) ^ {2}).\]

Next proposition shows that the overall efect on consumer surplus and on welfare of an increase in compatibility may turn negative. In particular, it shows that there is a critical degree of substitutability among applications above (below) which compatibility is bad (good) for consumers.

Proposition 1

which the weak platform charges more to the applications side whereas the dominant charges more to users for the platform as a response to a higher compatibility (see lemmas 1 and 2).

There exist and , which are decreasing in and increasing in such that

\[i f \sigma > \sigma^ {*} t h e n \frac {\partial T S (\delta)}{\partial \delta} < 0 a n d \frac {\partial T W (\delta)}{\partial \delta} < 0,\]

\[i f \sigma^ {* *} < \sigma < \sigma^ {*} t h e n \frac {\partial T S (\delta)}{\partial \delta} > 0 a n d \frac {\partial T W (\delta)}{\partial \delta} < 0,\]

\[i f \sigma < \sigma^ {* *} t h e n \frac {\partial T S (\delta)}{\partial \delta} > 0 a n d \frac {\partial T W (\delta)}{\partial \delta} > 0.\]

Proof. see the Appendix.

Note that the impact of an increase in depends on both the level of substitution between the applications and on their asymmetry. In particular, an increase in the degree of compatibility is more likely to yield a reduction in the consumers surplus and in the total welfare if there is a high level of substitution and a low value of δ. The fact that welfare is more likely to decrease than consumers surplus is explained by the fact that the total profits of the industry are decreasing in the degree of compatibility, i.e., ∂δ Moreover, based on Lemma 3 and Proposition 1 we can conclude that a move towards a larger degree of compatibility in the applications may be harmful for users and social welfare, particularly when asymmetries are strong. Consequently, it may be undesirable to force a larger degree of compatibility not only because it is against the dominant firm interests but also because it may lower consumers’ surplus. Based on these arguments, next corollary follows.

Corollary 1

There is a range of parameter values for which the dominant firm, the users and the social planner share the same interests regarding compatibility that are against those of the weak firm. Furthermore, this range is higher the larger the dominance of the strong firm.

Note finally that firms can difer in another dimension, namely in the stand alone values of their platforms. Denoting by to it is easy to see that ∆ has the same impact on equilibrium market outcomes than has. Consequently, all the potential negative efects on consumers’ surplus and on welfare of increasing compatibility get reinforced when this second source of asymmetry is present, i.e., the critical values and in Proposition 1 are decreasing in

3.2 Direct network efects and Inter-network compatibility

We have assumed so far that there are no direct network efects which may be considered as a limiting assumption. We next introduce direct network efects among users with a twofold objective: on one hand, we want to study if they make compatibility in applications more or less desirable, on the other hand, by introducing direct network efects we can analyze inter-network compatibility. Note that since consumers derive value from the existence of consumers using the other platform, inter-network compatibility might turn desirable. Furthermore, it may be in the interest of firms to enforce it even though they might be against compatibility in applications.

When there are direct network efects users’ utility is given by (2) , i.e., by

\[U _ {l} = w _ {l} (\delta) + V + \eta \theta_ {l} - t x - s _ {l},\]

where η measures the extent of the network efects. Since compatibility makes the platforms more symmetric, it reduces the positive direct network externalities. Thus, in markets where direct network efects are important, the potential harming efect that an increase in compatibility has on consumers surplus and on welfare is more likely to occur. In other words, strong direct network efects make more likely the negative consequence of forcing compatibility in applications. It hence follows that the presence of network efects make compatibility in applications less desirable.²⁰

Regarding inter-network compatibility, it allows consumers who buy platform l to get an extra net-utility of from the existence of users of the other platform, so that consumers overall utility becomes

\[U _ {l} = w _ {l} + V + \eta [ \theta_ {l} + \beta (1 - \theta_ {l}) ] - t x - s _ {l}\]

where measures the level of inter-network compatibility.

We find that inter-network compatibility would indeed be promoted by the strong firm. Although an increase in reduces the platform diferentiation, it also allows platforms to set higher prices to consumers. This last positive efect compensates the negative impact on the dominant firm’s market share of the reduced diferentiation. Furthermore, inter-network compatibility reduces consumers surplus and its efect on total welfare might turn negative if the level of asymmetries are large. Firms’ incentives to become inter-network compatible are opposite to those of consumers. Furthermore, as the market dominance of the strong platform (either in platforms or applications values) grows larger it is more likely that an increase in inter-network compatibility will decrease welfare. These results are the content of next Proposition.

Proposition 2

i) Both firms benefit from an increase in inter-network compatibility.

(ii) Inter-network compatibility has a negative impact on consumers’ surplus, which is larger the stronger the asymmetries are.

(iii) If asymmetries are strong, inter-network compatibility has a negative impact on welfare.

Proof. see the Appendix.

σ*

σ**

^{2 0} More precisely, the critical values σ^∗ and σ^∗∗ in Proposition 1 are decreasing in η, as the market share of the dominant firm is now given by

1 1 θ_A (δ) = 2 + (w (δ) + π (δ)) , 6 (t − η)

η.

that is increasing in η.

4 Incentives to innovate

A platform can invest in the stand-alone value of the platform itself in the value of the fully compatible application. Think of the video-game industry, a sector characterized by huge levels of investment. We observe that firms invest in developing new applications for the platform (console),²¹ but also invest in producing more advanced technology.²²

We will consider here that the result of an innovation is an increase in a firm’s value, so that the steeper the slope of the firm’s profit function with respect to its own value level, the larger the firm’s incentive to increase this value.²³ We will determine the efect of compatibility on a firm’s incentives to undertake an innovation by the sign of . Based on the nature (positive or negative) of the incentives to undertake these innovations we will propose a four type taxonomy of firms:

\[\begin{array}{r l r} & & {\frac {\partial | \frac {\partial \Pi_ {l} (\delta)}{\partial W _ {i}} |}{\partial \delta} > 0 \quad \frac {\partial | \frac {\partial \Pi_ {l} (\delta)}{\partial W _ {i}} |}{\partial \delta} \leq 0} \\ & & {\frac {\partial | \frac {\partial \Pi_ {l} (\delta)}{\partial V _ {l}} |}{\partial \delta} > 0 \qquad \mathrm{TypeI} \qquad \mathrm{TypeII}} \\ & & {\frac {\partial | \frac {\partial \Pi_ {l} (\delta)}{\partial V _ {l}} |}{\partial \delta} \leq 0 \qquad \mathrm{TypeIII} \qquad \mathrm{TypeIV}} \end{array}\]

Table 1: Firm taxonomy

We will say that a firm is of type I if compatibility increases the marginal incentive to invest in both values. In contrast, a firm is of type IV if compatibility decreases both marginal incentives. Finally, a firm is of type II or III if the marginal incentive to invest is increasing in compatibility for one value but decreasing for the other one. By analyzing firm types we will state the relationship between increasing firms’ compatibility and getting more valuable products in the market.

We next study the efect of compatibility in applications on the incentives to invest. The next proposition presents the results regarding the typology introduced in Table 1 and the efect on total marginal incentives.

Proposition 3

When we consider an increase in

i) The weak firm is type I if the asymmetries (any of them) are very strong, otherwise it is type II.

ii) The dominant firm in the market of applications is type IV.

"By

^{2 1} Satoru Iwata, the president of Nintendo declared "By designing products for existing gamers and neglecting non-gamers, it undermines the prospects to future growth"...."by providing non-gaming functions such as news and weather too, the aim is to overcome nongamers’ aversion to consoles, so that eventually they might flick to the "games" channel and give something a try" (see The Economist, 26/10/2006). try"

^{2 2} Sony is pushing the PS3 as the most advanced console, with a powerful new processor chip and a high definition "Blu-ray" optical drive ( see The Economist 26/10/2006 and 10/11/2006).

^{2 3} We introduce investment incentives as in Valletti and Cambini (2005).

iii) The total marginal incentive to invest in the stand-alone value does not change, however the corresponding to the value of the fully compatible applications decreases.

Proof. see the Appendix.

As the degree of compatibility increases, the strong platform in the market of applications has lower marginal incentives to invest in either type of investment. In contrast, compatibility positively afects the marginal incentive of the weak platform to invest in its stand-alone value. The efect is also positive regarding its incentive to invest in the fully compatible application when existing asymmetries are high.

The rationale is as follows. Consider first the incentives to invest in the applications. When a firm invests in the value of its fully compatible application, this investment is shared with the other firm, through δ, which free-rides from this increase in its value. Consequently, the marginal benefit of this investment is decreasing in δ. In the case of the dominant platform, this efect is reinforced by the fact that an increase in δ also leads to a loss in its relative advantage, so that ii) follows. Consider now the incentives of the weak platform. When asymmetries are very strong, the benefit due to its gain in the relative advantage compensates the loss due to the free-rider efect, which explains i). The opposite occurs when asymmetries are weak.

In the case of investment in the stand-alone value, the free-rider efect does not exist but the efect of δ on the relative advantages between firms remains. It reduces the incentives of the strong firm, while increasing those of the weak firm. These efects balance each other out and this explains first part in point (iii). The free-rider efect explains the second part of the last point.

Consider now the efect on the incentives to invest that inter-network compatibility yields.

Proposition 4

When internetwork compatibility increases, i.e., when β increases,

i) The dominant firm is type IV

ii) The weak firm is type I

iii) The total marginal incentive to invest in the stand-alone value does not change, however the corresponding to the value of the fully compatible applications decreases.

Proof. see the Appendix.

When considering the efect of β on the incentives to invest the problem of the free-rider problem disappears, and only the efect on the relative advantage is present. It explains that the weak firm is always type I, whereas the rest of the results are similar to those under compatibility in the applications, stated in proposition 4.

5 Robustness

We have shown that whenever the applications to be made compatible are close substitutes, users’ surplus and welfare are decreasing in the degree of compatibility. Moreover, the dominant firm will never promote compatibility in applications as both its profits and its incentives to invest will decrease.

Throughout the analysis we have assumed that for a given compatibility level the two platforms compete à la Hotelling in attracting customers. Note that two assumptions are hidden behind this mode of competition. First, that the market size is fixed, and second, that although the model allows for applications to be substitutes, complements or independent, the systems as a whole are assumed substitutes.²⁴ Either of these assumptions is not innocuous for the aforementioned results. On the one hand, the incentives of the strong platform may change if we allow for a non-covered market. An increase in the value of its product due to compatibility would expand the market and the profits generated by this expansion may compensate the reduction in the advantage of the dominant firm. If this were the case, the dominant firm might also be interested in making applications compatible. On the other hand, complementarities among systems do exist (there are complementarities among computer operating systems and mobile phones, as well as between video consoles and mobile phones),²⁵ and they may alter firms’ attitudes towards compatibility. Our purpose here is to study the robustness of the results by examining the extent to which they hold true for a potential expandable market and for systems that are not necessarily substitutes.

To do so we analyze a market in which two systems provide a service to consumers.²⁶ System , is composed of platform i, application i which is fully compatible for this system and application j which is fully compatible for system j but compatible in a degree δ for system i. Consumers’ valuations are given by

\[\begin{array}{r c l} \alpha_ {1} & = & V _ {H} + W _ {1} + \delta W _ {2}, \\ \alpha_ {2} & = & V _ {L} + W _ {2} + \delta W _ {1}. \end{array}\]

where are the stand alone values of the platforms, is the value of the application that is fully compatible for system 1, W₂ the value of the application that is fully compatible for system 2 and measures the degree of compatibility. We assume and

The price that consumers pay for the system can be decomposed in the following way:

\[\begin{array}{r c l l} p _ {i} & = & s _ {i} + x _ {i i} & \text {if} \delta = 0 \\ p _ {i} & = & s _ {i} + x _ {i i} + x _ {i j} & \text {if} 0 < \delta \leq 1, \end{array}\]

^{2 4} A system is the good composed by the platform plus the applications.

^{2 5} For instance, the Wii of Nintendo has a game that is played with photographs, so that it can be played with a camera or a mobile phone.

^{2 6} We here present the main results and their intuition. All the details are relegated to Appendix B.

where is the price that the platform of the system i sets to users, and is the price that users pay for an application to be used in system Firms compete in prices and the representative consumer maximizes a net utility based on Bowley’s (1924) mode and given by

\[U = \alpha_ {1} q _ {1} + \alpha_ {2} q _ {2} - \frac {1}{2} b \left(q _ {1} ^ {2} + 2 \sigma q _ {1} q _ {2} + q _ {2} ^ {2}\right) - p _ {1} q _ {1} - p _ {2} q _ {2},\tag{6}\]

where is the quantity of system i consumed, and . Negative values of σ make the model one of demand for complementary systems. If the two systems are independent in demand. As σ approaches 1, the systems become closer substitutes.

Since the market is two-sided, the profits of platform i depend on the price set to consumers, as well as on the prices that it sets to the developers so that

\[\Pi_ {i} = (s _ {i} + d _ {i i} + d _ {i j}) q _ {i}\]

where is the price that platform i sets to the developer of its fully compatible application (of application

The timing of decisions is as follows. First, platforms set prices , and . Then developers of the complementary product set their prices competitively, . Because of this and since the platform and the applications are perfect complements, the model becomes equivalent to that of a one-sided market in which firms compete by setting the prices for their systems

The best response of firm i to a price by its rival is given by

\[p _ {i} \left(p _ {j}\right) = \frac {1}{2} \left(\alpha_ {i} - \sigma \alpha_ {j} + \sigma p _ {j}\right).\tag{7}\]

Note that price systems are strategic complements, with , if systems are substitutes. If they are complementary , prices are strategic substitutes. From (7) equilibrium prices are derived, with

\[\begin{array}{r c l} p _ {1} ^ {*} & = & \frac {1}{4 - \sigma^ {2}} \left(\left(2 - \sigma^ {2}\right) \alpha_ {1} - \sigma \alpha_ {2}\right), \\ p _ {2} ^ {*} & = & \frac {1}{4 - \sigma^ {2}} \left(\left(2 - \sigma^ {2}\right) \alpha_ {2} - \sigma \alpha_ {1}\right). \end{array}\tag{8}\]

The impact of an increase in the degree of compatibility on the equilibrium prices depends, on one hand, on the nature of the strategic interaction between firms’ choices, and on the other hand on the interplay between two efects which are brought about by compatibility: a market expansion efect and a reducing diferentiation efect.

When prices are strategic substitutes, so that , both firms respond to an increase in the degree of compatibility by increasing their price. Demanded quantities also increase, thus when systems are complementary or independent, both firms benefit from compatibility. When they are strategic complements, so that , the price of system 2 also increases and the efect is always positive for the weak firm’s profits. Regarding system 1 two outcomes can occur: i). The equilibrium price and quantity of system 1 increase as compatibility increases. This outcome emerges in either of the following two circumstances. First, if the market expansion efect dominates the reducing diferentiation efect, if , which happens when , then, in equilibrium, the positive efect from is reinforced by the price strategic complementarity (since Compatibility shifts system 1’s reaction function outwards and in equilibrium its price and quantity sold are higher (see Figure 1 (a)). Second, if the reducing diferentiation efect dominates the market expansion efect, but this negative efect is compensated by the price increase of the weak platform, the strategic complementarity still leads to in equilibrium. This is the case when and the final efect looks as depicted in Figure 1 (b). ii). The equilibrium price and quantity of system 1 decrease as compatibility increases. This outcome emerges whenever so that the negative impact brought by the reducing diferentiation efect dominates overall and in equilibrium (see Figure 1 (c)).

^{2 7} See Martin (2001).

P₁

Figure 1 P₁

Based on the above discussion it follows that compatibility will hurt the dominant platform as long as . The reducing diferentiation efect tends to dominate as long as σ and the diference between the values of the complements are high. In contrast to the Hotelling case, the dominant platform might now be interested in inducing compatibility with the weak platform. However, a strong market expansion is crucial for this result.

Regarding users’s surplus and welfare, Proposition B1 in Appendix B shows that whenever systems are complements or independent, they are both increasing in the degree of compatibility in applications. However, if the systems are substitutes the opposite result can hold.

Finally, when considering the marginal incentives to invest, the net efect will depend on whether or not the gains coming from ofering a higher product value compensate the losses coming from the free-rider efect and from the reduction in the relative advantage the dominant system enjoys (the formal result is presented in Proposition B2). If this happens, the dominant firm can be of type I (i.e., compatibility encourages investment incentive in the platform and the fully compatible application), something that never occurs under the Hotelling price competition. Again this will only occur if a very important market expansion takes place.

The incentives to invest in the stand-alone value only depend on the diference . The change, due to , in the dominant platform’s marginal incentive to invest is positive as long as . Note that this is the condition under which and are increasing in . Thus, the marginal benefit of investing in when δ increases is negative if the efect of δ on profits is also negative. Since this efect is always positive for the weak platform, its marginal incentive to invest in is always positive.

The incentives to invest in the value of the applications depend on both sources of asymmetries and . When asymmetries are strong, compatibility increases the weak platform’s incentives to invest. For the dominant firm, the efect of on incentives goes in the opposite direction, and when asymmetries are important it is negative (the losses coming from the free-rider efect and from decreasing its relative advantage are larger than the gains from ofering a higher product value).

In sum, even though some of our basic findings could change if a larger degree of compatibilityo brings about the possibility of a market expansion, we have shown that a large expansion is needed to compensate the prevailing (negative) forces identified in the basic model.

6 Final remarks

We have developed a model where firms are horizontally diferentiated á la Hotelling and are asymmetric in the value of their fully compatible application. We have analyzed firms’ compatibility decisions and investment incentives to assess the benefits to users and to welfare of imposing a larger degree of compatibility among firms. We have stressed that compatibility is not always beneficial for consumers. Furthermore, in many instances the dominant firm’s interests regarding compatibility decisions are in line with those of users, and are opposite to those of the weak firm, which will always demand more compatibility to be enforced. When this is the case, imposing a higher degree of compatibility will damage total welfare and consumer surplus.

Let us first recap the driving forces that underlie our main results. We have shown that the gap or the level of asymmetry given by determines firms attitudes towards compatibility in applications. In particular, if they are indiferent about the degree of compatibility that is enforced. When applications have diferent values so that then the weak firm is always interested in promoting compatibility in applications. In contrast, the dominant one is never interested in doing so. The reason behind this result is that compatibility mainly decreases the market power of the dominant firm by reducing its advantage in product diferentiation. However, regarding inter-network compatibility we have shown that both firms find profitable to promote it.

We have also shown that the welfare results strongly depend on the level of complementarity/substitutability of the applications, with compatibility decreasing both consumer surplus and welfare if applications are close substitutes. The level of the asymmetries in applications and/or in platform value, may also shape the efect of compatibility on welfare and consumer surplus. In particular, it tends to be negative when the asymmetries are important. We have also found that strong direct network efects strengthen the potential negative incidence of increasing the degree of compatibility in the applications. Moreover, our results show that a marginal increase in inter-network compatibility is always consumers’ surplus reducing and it may have a negative impact in total welfare if asymmetries are strong. Finally, when systems are independent or complements, compatibility should indeed be promoted by any sector of the society.

Regarding the efects on the incentives to innovate, we have found that the marginal incentives to invest in the stand-alone value for the weak platform is increasing in δ, and so are the marginal incentives to invest in the application provided that asymmetries are important. For the dominant one, the marginal incentives to invest in either the stand-alone value or in the value of the fully compatible application are decreasing in the degree of compatibility. Changes in the level of inter-network compatibility positively afect the weak platform’s marginal incentives to invest and negatively afect those of the dominant one.²⁸

In sum, two salient aspects of the compatibility-welfare puzzle follow. First, substitutes goods are "bad" for compatibility. If products are substitutes, even if the market is not covered, compatibility tends to be welfare decreasing. Second, the negative potential consequences of compatibility are more likely when asymmetries are strong. To the best of our knowledge these are two novel results in the literature, as we ofer a study of compatibility issues from the perspective of complementarities/substitutabilities among the goods to be made compatible.

Our analysis has several implications for evaluating "real world" policy decisions. In particular, we believe that these results shed light on some issues related to the EC vs. Microsoft case that we mentioned in the introduction. We find that when deciding about forcing compatibility/interoperability, the existence of a dominant firm does not provide enough arguments for more compatibility to be desirable. Moreover, network efects, almost always present in this kind of industries, may act as a countervailing reason to not force more compatibility in applications. Related to the incentives to innovate, when the Commission took the decision of imposing interoperability, it stated that the remedy would be good for innovation. We have shown that interoperability generates a free-rider efect, so that more interoperability may not encourage innovation either for the dominant firm or for the weak firm.

^{2 8} We have considered how compatibility afects the marginal incentives to invest of both platforms. We acknowledge that the study of the efect of compatibility on innovation demands analysing its impact on equilibrium investment levels. We consider this as a future extension.

Appendix A

Proof of Lemma 1

Solving the maximization problem in (3) gives equilibrium prices

\[\begin{array}{r c l} p _ {A 1} & = & \frac {1}{2} W _ {1}, p _ {A 2} = \frac {1}{2} \delta W _ {2}, \\ p _ {B 1} & = & \frac {1}{2} \delta W _ {1}, p _ {B 2} = \frac {1}{2} W _ {2}, \end{array}\]

so that the corresponding indirect utilities and the profits from applications are given by

\[w _ {A} (\delta) = \frac {1}{4 (1 - \sigma^ {2}) b} \left(W _ {1} ^ {2} + (\delta W _ {2}) ^ {2} - 2 \sigma \delta W _ {1} W _ {2}\right) \left(1 - \frac {1}{b}\right),\tag{9}\]

\[w _ {B} (\delta) = \frac {1}{4 (1 - \sigma^ {2}) b} \left(W _ {2} ^ {2} + (\delta W _ {1}) ^ {2} - 2 \sigma \delta W _ {2} W _ {1}\right) \left(1 - \frac {1}{b}\right),\tag{10}\]

\[\pi_ {A} (\delta) = \frac {1}{4 (1 - \sigma^ {2}) b} \left(W _ {1} ^ {2} + (\delta W _ {2}) ^ {2} - 2 \sigma \delta W _ {1} W _ {2}\right), \mathrm{and}\tag{11}\]

\[\pi_ {B} (\delta) = \frac {1}{4 (1 - \sigma^ {2}) b} \left(W _ {2} ^ {2} + (\delta W _ {1}) ^ {2} - 2 \sigma \delta W _ {2} W _ {1}\right).\tag{12}\]

By inspecting expressions above it trivially follows that

\[\begin{array}{r c l} \frac {\partial w _ {A} (\delta)}{\partial \delta} & > & 0, \frac {\partial \pi_ {A} (\delta)}{\partial \delta} > 0 \Leftrightarrow \delta W _ {2} - \sigma W _ {1} > 0, \\ \frac {\partial w _ {B} (\delta)}{\partial \delta} & > & 0, \frac {\partial \pi_ {B} (\delta)}{\partial \delta} > 0 \Leftrightarrow \delta W _ {1} - \sigma W _ {2} > 0, \end{array}\]

inequalities that always hold if applications are complementary or independent, so that i) trivially follows. In contrast, if applications are substitutes 1 the sign of equals the sign of and it depends on the relationship between σ and the ratio so that ii) follows. Similar arguments apply to the other platform.

Proof of Lemma 2

By looking for the indiferent consumer as is usual in the Hotelling model, the market share of platform A is given by the following expression,

\[\theta_ {A} = \frac {1}{2} + \frac {w (\delta) - (s _ {A} - s _ {B})}{2 t},\]

and the market share of B by . Note that Assumption 1 is suficient to ensure that the market share equation above is well behaved.

Each platform will set a price to maximize

\[\Pi_ {l} = \left(s _ {l} + \pi_ {l} (\delta)\right) \theta_ {l}.\]

By solving maximization problem above we derive the platform’s reaction functions which are given by

\[\begin{array}{r c l} s _ {A} (s _ {B}) & = & \frac {1}{2} (t + w (\delta) + s _ {B} - \pi_ {A} (\delta)) \\ s _ {B} (s _ {A}) & = & \frac {1}{2} (t - w (\delta) + s _ {A} - \pi_ {B} (\delta)). \end{array}\]

Since they are increasing in the choice of the rival, prices are strategic complements. From reaction functions above it is straightforward to derive the equilibrium platform prices, which are given by

\[\begin{array}{r c l} s _ {A} (\delta) & = & t + \frac {1}{3} w (\delta) - \frac {2}{3} \pi_ {A} (\delta) - \frac {1}{3} \pi_ {B} (\delta) \\ s _ {B} (\delta) & = & t - \frac {1}{3} w (\delta) - \frac {2}{3} \pi_ {B} (\delta) - \frac {1}{3} \pi_ {A} (\delta). \end{array}\tag{13}\]

If applications are either complements, independent or substitutes with we observe that . Note that for platform B, an increase in compatibility has a positive efect on via a reduction in w (δ) . However, this is compensated by the increase in and , so that it also holds for platform B that

If applications are substitutes and , then holds for both platforms. Finally, if then and and the statements in the lemma follow.

Proof of Lemma 3

Platform profits are given by

\[\Pi_ {l} (\delta) = \left(s _ {l} (\delta) + \pi_ {l} (\delta)\right) \theta_ {l} (\delta),\]

where, using (13) , (11) and (12) , platform’s revenues per consumer are given by

\[\begin{array}{r c l} s _ {A} (\delta) + \pi_ {A} (\delta) & = & t + \frac {1}{3} (w (\delta) + \pi (\delta)) \\ s _ {B} (\delta) + \pi_ {B} (\delta) & = & t - \frac {1}{3} (w (\delta) + \pi (\delta)), \end{array}\tag{14}\]

and equilibrium market shares by

\[\begin{array}{r c l} \theta_ {A} (\delta) & = & \frac {1}{2} + \frac {1}{6 t} (w (\delta) + \pi (\delta)) \\ \theta_ {B} (\delta) & = & \frac {1}{2} - \frac {1}{6 t} (w (\delta) + \pi (\delta)). \end{array}\]

If occurs, then it happens that and point i) in the lemma follows. To prove ii) note that because of lemma 1 we have that and are decreasing in so that is strictly decreasing in δ. In contrast, and are all increasing in δ functions so that is strictly increasing in δ.

Proof of Proposition 1

Consider first the analysis for the consumers’ surplus. For expositional simplicity, let us define A and B as

\[\begin{array}{r c l} {A} & {=} & {w _ {A} (\delta) + V - s _ {A} (\delta) = \frac {2}{3} (w _ {A} (\delta) + \pi_ {A} (\delta)) + \frac {1}{3} (w _ {B} (\delta) + \pi_ {B} (\delta)) + V,} \\ {B} & {=} & {w _ {B} (\delta) + V - s _ {B} (\delta) = \frac {2}{3} (w _ {B} (\delta) + \pi_ {B} (\delta)) + \frac {1}{3} (w _ {A} (\delta) + \pi_ {A} (\delta)) + V.} \end{array}\]

To study the efect of compatibility on total consumers surplus we need to compute

\[\frac {\partial \left(\theta_ {A} (\delta) A + (1 - \theta_ {A} (\delta)) B - \frac {1}{2} t \left(\theta_ {A} (\delta) ^ {2} + (1 - \theta_ {A} (\delta)) ^ {2}\right)\right)}{\partial \delta}\]

that is

\[\theta_ {A} \frac {\partial A}{\partial \delta} + (1 - \theta_ {A}) \frac {\partial B}{\partial \delta} + (A - t \theta_ {A}) \frac {\partial \theta_ {A}}{\partial \delta} + (B - t (1 - \theta_ {A})) \frac {\partial (1 - \theta_ {A})}{\partial \delta}.\]

By definition of the indiferent consumer in equilibrium we have that

\[\left(A - t \theta_ {A}\right) \frac {\partial \theta_ {A}}{\partial \delta} + \left(B - t \left(1 - \theta_ {A}\right)\right) \frac {\partial \left(1 - \theta_ {A}\right)}{\partial \delta} = 0,\]

Consequently, the efect of compatibility on total consumers surplus does only depend on the sign of

\[F (\delta , \sigma) = \theta_ {A} \frac {\partial A}{\partial \delta} + (1 - \theta_ {A}) \frac {\partial B}{\partial \delta}.\tag{15}\]

We next show that is decreasing in . First, note that from the expressions for , and given respectively in (9) , (10) , (11) and (12) it follows that their cross partial derivatives are negative. Consequently, is negative if

\[\frac {2 \sigma}{(1 - \sigma^ {2})} (\bar {\sigma} _ {2} - \sigma) < 1 \Leftrightarrow \bar {\sigma} _ {2} < \frac {1 + \sigma^ {2}}{2 \sigma},\]

and is negative if

\[\frac {2 \sigma}{(1 - \sigma^ {2})} (\bar {\sigma} _ {1} - \sigma) < 1 \Leftrightarrow \bar {\sigma} _ {1} < \frac {1 + \sigma^ {2}}{2 \sigma},\]

where and are defined in lemma 1 and they both belong to the interval . Since we can conclude that both and are negative.

Since and it follows straightforwardly that stated in (15) is decreasing in σ.

By Lemma 1 we have that for all it holds that and for all it holds that . Appealing to the mean value theorem there is a critical degree of substitutability among applications that makes (15) zero. Furthermore since is decreasing in σ, this value is unique which shows our claim.

Consider now total welfare. We first show . To do so we only need to show that the cross derivative of the total profits of the industry with respect to δ and σ is also negative, given that we already shown that this is the case for consumers’ surplus. Total industry profits are given by

\[\Pi_ {A} + \Pi_ {B} = \left(t + \frac {1}{3} (w (\delta) + \pi (\delta))\right) \theta_ {A} + \left(t - \frac {1}{3} (w (\delta) + \pi (\delta))\right) (1 - \theta_ {A}).\]

Since profits can be rewritten as

\[\Pi_ {A} + \Pi_ {B} = t + \frac {1}{9 t} \left(w (\delta) + \pi (\delta)\right) ^ {2}\]

so that

\[\frac {\partial (\Pi_ {A} + \Pi_ {B})}{\partial \delta} = \frac {2}{9 t} (w (\delta) + \pi (\delta)) \frac {\partial (w (\delta) + \pi (\delta))}{\partial \delta} < 0,\]

and

\[\begin{array}{r c l} \frac {\partial^ {2} \left(\Pi_ {A} + \Pi_ {B}\right)}{\partial \delta \partial \sigma} & = & \frac {2}{9 t} \frac {\partial (w (\delta) + \pi (\delta))}{\partial \sigma} \frac {\partial (w (\delta) + \pi (\delta))}{\partial \delta} + \\ & & \frac {2}{9 t} (w (\delta) + \pi (\delta)) \frac {\partial^ {2} (w (\delta) + \pi (\delta))}{\partial \delta \partial \sigma} \end{array}\]

and it follows that holds.

Now, to set the efect of a change in δ on welfare we need to sign

\[\theta_ {A} \frac {\partial A ^ {\prime}}{\partial \delta} + (1 - \theta_ {A}) \frac {\partial B ^ {\prime}}{\partial \delta} + \left[ (A ^ {\prime} - t \theta_ {A}) \frac {\partial \theta_ {A}}{\partial \delta} + (B ^ {\prime} - t (1 - \theta_ {A})) \frac {\partial (1 - \theta_ {A})}{\partial \delta} \right],\tag{16}\]

where and Since and we have that . Moreover, since , we can conclude that the term in brackets in (16) is always negative and it is decreasing in the level of asymmetry measured by

Appealing again to Lemma 1 it follows that and if then If the expression can be positive or negative. However, at we have that . Consequently, there is that makes (16) zero.

Given that,

\[\theta_ {A} \left(\frac {\partial A}{\partial \delta} - \frac {\partial A ^ {\prime}}{\partial \delta}\right) + (1 - \theta_ {A}) \left(\frac {\partial B}{\partial \delta} - \frac {\partial B ^ {\prime}}{\partial \delta}\right) < 0\]

it follows that . Moreover, by Lemma 1, we also have that and . Thus, the critical values are decreasing in

\[\theta_ {A} = \frac {1}{2} + \frac {1}{6 t} (w (\delta) + \pi (\delta)),\]

and consequently they are decreasing in and increasing in the level of δ as the proposition states.

Proof of Proposition 2

With inter-network compatibility, equilibrium prices, market share and profits are given by

\[\begin{array}{r c l} {s _ {A} (\delta , \beta)} & = & {t + \frac {1}{3} w (\delta) - \frac {2}{3} \pi_ {A} (\delta) - \frac {1}{3} \pi_ {B} (\delta) - \eta (1 - \beta),} \\ {s _ {B} (\delta , \beta)} & = & {t - \frac {1}{3} w (\delta) - \frac {2}{3} \pi_ {B} (\delta) - \frac {1}{3} \pi_ {A} (\delta) - \eta (1 - \beta),} \\ {\theta_ {A} (\delta , \beta)} & = & {\frac {1}{2} + \frac {1}{6 (t - \eta (1 - \beta))} (w (\delta) + \pi (\delta))} \\ {\theta_ {B} (\delta , \beta)} & = & {\frac {1}{2} - \frac {1}{6 (t - \eta (1 - \beta))} (w (\delta) + \pi (\delta))} \\ {\Pi_ {A} (\delta , \beta)} & = & {\frac {1}{2 (t - \eta (1 - \beta))} (t + \frac {1}{3} (w (\delta) + \pi (\delta)) - \eta (1 - \beta)) ^ {2},} \\ {\Pi_ {B} (\delta , \beta)} & = & {\frac {1}{2 (t - \eta (1 - \beta))} (t - \frac {1}{3} (w (\delta) + \pi (\delta)) - \eta (1 - \beta)) ^ {2}.} \end{array}\]

To prove statement i) note that and are positive if

\[t > \eta (1 - \beta) + \frac {1}{3} (w (\delta) + \pi (\delta)),\]

a condition which is implied by the analogous condition to Assumption 1, namely Assumption

We next study its impact on consumers’ surplus and on welfare. To do so, note that total surplus is given by

\[\begin{array}{l} V + \int_ {0} ^ {\theta_ {A} (\delta , \beta)} \left(w _ {A} (\delta) + V - t x + \eta \left[ \theta_ {A} (\delta , \beta) + \beta (1 - \theta_ {A} (\delta , \beta)) \right] - s _ {A} (\delta , \beta)\right) d x + \\ \int_ {\theta_ {A} (\delta , \beta)} ^ {1} \left(w _ {B} (\delta) + V - t (1 - x) + \eta \left[ (1 - \theta_ {A} (\delta , \beta)) + \beta \theta_ {A} (\delta , \beta) \right] - s _ {B} (\delta , \beta)\right) d x. \end{array}\]

To study how changes in afect note that

\[\begin{array}{r c l} \frac {\partial T S (\delta , \beta)}{\partial \beta} & = & \theta_ {A} \frac {\partial A (\beta)}{\partial \beta} + (1 - \theta_ {A}) \frac {\partial B (\beta)}{\partial \beta} + \\ & & (A (\beta) - t \theta_ {A}) \frac {\partial \theta_ {A}}{\partial \delta} + (B (\beta) - t (1 - \theta_ {A})) \frac {\partial (1 - \theta_ {A})}{\partial \delta}, \end{array}\]

where

\[\begin{array}{r c l} A (\beta) & = & (w _ {A} (\delta) + V + \eta [ \theta_ {A} + \beta (1 - \theta_ {A}) ] - s _ {A} (\delta , \beta)), \text {and} \\ B (\beta) & = & (w _ {B} (\delta) + V + \eta [ (1 - \theta_ {A}) + \beta \theta_ {A} ] - s _ {B} (\delta , \beta)) \end{array}\]

definition of indiferent user we know that

\[\left(A (\beta) - t \theta_ {A}\right) \frac {\partial \theta_ {A}}{\partial \delta} + \left(B (\beta) - t (1 - \theta_ {A})\right) \frac {\partial (1 - \theta_ {A})}{\partial \delta} = 0.\]

Therefore to study how afects we only need to analyze the sign of

\[\theta_ {A} \frac {\partial (\eta [ \theta_ {A} + \beta (1 - \theta_ {A}) ] - s _ {A})}{\partial \beta} + (1 - \theta_ {A}) \frac {\partial (\eta [ (1 - \theta_ {A}) + \beta \theta_ {A} ] - s _ {B})}{\partial \beta}.\tag{17}\]

Given that it follows

\[\eta \left(\theta_ {A} \frac {\partial [ \theta_ {A} + \beta (1 - \theta_ {A}) ]}{\partial \beta} + (1 - \theta_ {A}) \frac {\partial [ (1 - \theta_ {A}) + \beta \theta_ {A} ]}{\partial \beta}\right) - \eta .\]

Taking into account that , computing the derivatives and rearranging terms lead to

\[\frac {\partial T S (\delta , \beta)}{\partial \beta} = \eta \left(2 \theta_ {A} (1 - \theta_ {A}) + (2 \theta_ {A} - 1) \frac {\partial \theta_ {A}}{\partial \beta} (1 - \beta)\right) - \eta .\]

Since and it follows that the impact of on is negative as claimed. Furthermore, he first term is decreasing in so that the negative impact of is higher, the higher the asymmetries as stated in ii).

We can obtain the efect on total welfare by signing

\[\frac {\partial T W (\delta , \beta)}{\partial \beta} = \frac {\partial T S (\delta , \beta)}{\partial \beta} + \frac {\partial \Pi_ {A} (\delta , \beta)}{\partial \beta} + \frac {\partial \Pi_ {B} (\delta , \beta)}{\partial \beta}.\]

Straightforward computations yield that

\[\frac {\partial \Pi_ {A} (\delta , \beta)}{\partial \beta} + \frac {\partial \Pi_ {B} (\delta , \beta)}{\partial \beta} = \eta \left(1 - \frac {(w (\delta) + \pi (\delta)) ^ {2}}{(3 (t - \eta (1 - \beta))) ^ {2}}\right) < \eta ,\]

Note that is decreasing in the size of the asymmetries. With no asymmetries the total gain of the industry would be However, given the existence of asymmetries, this gain is lower. Moreover, the higher the asymmetries in applications, the lower the gain of the industry due to Thus, given that both components of total welfare are decreasing in the asymmetries, point iii) follows.

Proof of Proposition 3

To prove the first statement note that

\[\frac {\partial \left(\frac {\partial \Pi_ {A} (\delta)}{\partial V _ {A}}\right)}{\partial \delta} = \frac {1}{9 (t - \eta)} \frac {\partial (\pi (\delta) + v (\delta))}{\partial \delta} < 0.\]

If we consider investment in the fully compatible application, straightforward computation shows that

\[\frac {\partial \Pi_ {A} (\delta)}{\partial W _ {1}} = \left(\frac {(t + \frac {1}{3} (\Delta + w (\delta) + \pi (\delta)) - \eta)}{(t - \eta)}\right) \left(\frac {(2 - \frac {1}{b}) (1 - \delta^ {2}) W _ {1}}{6 (1 - \sigma^ {2}) b}\right).\]

Since is the product of two decreasing and positive functions, it trivially follows that

\[\frac {\partial \left(\frac {\partial (\Pi_ {A} (\delta))}{\partial W _ {1}}\right)}{\partial \delta} < 0,\]

and the first statement follows.

To prove the second statement we note that

\[\frac {\partial \left(\frac {\partial \Pi_ {B} (\delta)}{\partial V _ {B}}\right)}{\partial \delta} = - \frac {1}{9 (t - \eta)} \frac {\partial (\pi (\delta) + v (\delta))}{\partial \delta} > 0.\]

To analyze the marginal incentive of the weak firm to invest in the fully compatible application we observe that

\[\frac {\partial \Pi_ {B} (\delta)}{\partial W _ {2}} = \frac {(t - \frac {1}{3} (\Delta + w (\delta) + \pi (\delta)) - \eta)}{(t - \eta)} \left(\frac {(2 - \frac {1}{b}) (1 - \delta^ {2}) W _ {2}}{6 (1 - \sigma^ {2}) b}\right),\]

so that, diferentiating expression above and rearranging gives

\[\frac {\partial \left(\frac {\partial \Pi_ {B} (\delta)}{\partial W _ {2}}\right)}{\partial \delta} = \frac {2 (1 - \sigma^ {2}) (2 b - 1) \delta W _ {2}}{9 (t - \eta) (\sigma + 1) ^ {2} (1 - \sigma) ^ {2} b ^ {2}} \left(w (\delta) + \pi (\delta) - \frac {1}{2} (3 (t - \eta) - \Delta)\right).\]

Note that

\[\frac {\partial \left(\frac {\partial \Pi_ {B} (\delta)}{\partial W _ {2}}\right)}{\partial \delta} \lessgtr 0\]

as long as

\[w (\delta) + \pi (\delta) - \frac {1}{2} (3 (t - \eta) - \Delta) \lessgtr 0,\]

or equivalently if

\[\frac {1}{3} \left(2 \left(w (\delta) + \pi (\delta)\right) + \Delta\right) + \eta \leqslant t.\]

Then, and the weak firm is type I, if the asymmetries are very strong, i.e., if either or w are large enough. Otherwise this firm is type II.

\[\begin{array}{l} \text {Finally, note that} \frac {\partial \left(\frac {\partial \Pi_ {A} (\delta)}{\partial V _ {A}}\right)}{\partial \delta} = - \frac {\partial \left(\frac {\partial \Pi_ {B} (\delta)}{\partial V _ {B}}\right)}{\partial \delta} \text {and that} \frac {\partial \left(\frac {\partial \left(\Pi_ {A} (\delta)\right)}{\partial W _ {1}}\right)}{\partial \delta} + \frac {\partial \left(\frac {\partial \Pi_ {B} (\delta)}{\partial W _ {2}}\right)}{\partial \delta} = \\ \frac {G}{3 (t - \eta)} \frac {\partial (w (\delta) + \pi (\delta))}{\partial \delta} (1 - \delta^ {2}) (W _ {1} - W _ {2}) \\ - \frac {2 \delta G}{(t - \eta)} (t (W _ {1} + W _ {2}) + \left(\frac {1}{3} (\Delta + w (\delta) + \pi (\delta)) - \eta\right) (W _ {1} - W _ {2})) \end{array}\]

where . Since , and by Assumption 1, the last statement of the proposition follows.

Proof of Proposition 4

The changes in the marginal incentives to invest in the stand-alone value of the dominant platform are given by

\[\frac {\partial \left(\frac {\partial (\Pi_ {A} (\delta , \beta))}{\partial V _ {A}}\right)}{\partial \beta} = - \frac {\eta}{9 (t - \eta (1 - \beta)) ^ {2}} (\pi (\delta) + w (\delta) + \Delta) < 0,\]

and the changes in the marginal incentives to invest in its fully-compatible application are given by

\[\frac {\partial \left(\frac {\partial (\Pi_ {A} (\delta , \beta))}{\partial W _ {1}}\right)}{\partial \beta} = - \frac {2 \eta W _ {1}}{9 (t - \eta (1 - \beta)) ^ {2}} H \left(H (W _ {1} ^ {2} - W _ {2} ^ {2}) + \Delta + \pi (\delta)\right) < 0,\]

where , so that i) follows. Regarding the weak firm, the corresponding expressions are given by

\[\frac {\partial \left(\frac {\partial (\Pi_ {B} (\delta , \beta))}{\partial V _ {B}}\right)}{\partial \beta} = \frac {\eta}{9 (t - \eta (1 - \beta)) ^ {2}} (\pi (\delta) + w (\delta) + \Delta) > 0,\]

and

\[\frac {\partial \left(\frac {\partial (\Pi_ {B} (\delta , \beta))}{\partial W _ {2}}\right)}{\partial \beta} = \frac {2 \eta W _ {2}}{9 (t - \eta (1 - \beta)) ^ {2}} H \left(H \left(W _ {1} ^ {2} - W _ {2} ^ {2}\right) + \Delta + \pi (\delta)\right) > 0,\]

consequently ii) follows.

\[\begin{array}{r l} & {\mathrm{Finally,since} \frac {\partial \left(\frac {\partial (\Pi_ {A} (\delta , \beta))}{\partial V _ {A}}\right)}{\partial \beta} = - \frac {\partial \left(\frac {\partial (\Pi_ {B} (\delta , \beta))}{\partial V _ {B}}\right)}{\partial \beta} \mathrm{andthetotalincentive} \frac {\partial \left(\frac {\partial (\Pi_ {A} (\delta , \beta))}{\partial W _ {1}}\right)}{\partial \beta} +} \\ & {\frac {\partial \left(\frac {\partial (\Pi_ {B} (\delta , \beta))}{\partial W _ {2}}\right)}{\partial \beta} \mathrm{becomes}} \\ & {- (W _ {1} - W _ {2}) \frac {2 \eta H (H (W _ {1} ^ {2} - W _ {2} ^ {2}) + \Delta + \pi (\delta))}{9 (t - \eta (1 - \beta)) ^ {2}} < 0,} \end{array}\]

part iii) follows.

Appendix B

We first introduce an assumption to ensure that system 2 is active, namely,

Assumption 2: for all

From the utility (6) in the text, the demand function for system i becomes

\[q _ {i} = \frac {\left(\alpha_ {i} - \sigma \alpha_ {j}\right) - p _ {i} + \sigma p _ {j}}{b \left(1 - \sigma^ {2}\right)}.\]

Assuming marginal costs are null firm i obtains profits of

\[\Pi_ {i} = p _ {i} \left(\frac {\left(\alpha_ {i} - \sigma \alpha_ {j}\right) - p _ {i} + \sigma p _ {j}}{b \left(1 - \sigma^ {2}\right)}\right).\]

Next lemma formalizes the discussion in the main text.

Lemma B1

If the systems are complementary or independent the dominant platform in the market ofapplications benefits from an increase in the degree ofcompatibility. If the systems are substitutes, it benefits if the reducing diferentiation efect induced by compatibility is not too strong. The weak platform always benefits from compatibility an increase in the degree of compatibility.

Proof: From the expressions for the equilibrium prices given in (8), equilibrium quantities and profits are respectively given by

\[\begin{array}{r c l} q _ {1} & = & \frac {(2 - \sigma^ {2}) \alpha_ {1} - \sigma \alpha_ {2}}{b (1 - \sigma^ {2}) (4 - \sigma^ {2})} \\ q _ {2} & = & \frac {(2 - \sigma^ {2}) \alpha_ {2} - \sigma \alpha_ {1}}{b (1 - \sigma^ {2}) (4 - \sigma^ {2})} \\ \Pi_ {1} & = & \frac {1}{b (1 - \sigma^ {2})} \left(\frac {(2 - \sigma^ {2}) \alpha_ {1} - \sigma \alpha_ {2}}{4 - \sigma^ {2}}\right) ^ {2} \\ \Pi_ {2} & = & \frac {1}{b (1 - \sigma^ {2})} \left(\frac {(2 - \sigma^ {2}) \alpha_ {2} - \sigma \alpha_ {1}}{4 - \sigma^ {2}}\right) ^ {2}. \end{array}\]

σ ≤ 0. σ ≤ 0.

^{2 9} The assumption is trivially satisfied if systems are independent or complementary, i.e.,

We now show that if then for all i, whereas if then 2 but if and only if . To see this note that

\[\begin{array}{r c l} \frac {\partial \Pi_ {1}}{\partial \delta} & = & \frac {2}{b (1 - \sigma^ {2}) (4 - \sigma^ {2}) ^ {2}} ((2 - \sigma^ {2}) \alpha_ {1} - \sigma \alpha_ {2}) ((2 - \sigma^ {2}) \frac {\partial \alpha_ {1}}{\partial \delta} - \sigma \frac {\partial \alpha_ {2}}{\partial \delta}) \\ & = & \frac {2}{b (1 - \sigma^ {2}) (4 - \sigma^ {2}) ^ {2}} ((2 - \sigma^ {2}) \alpha_ {1} - \sigma \alpha_ {2}) ((2 - \sigma^ {2}) W _ {2} - \sigma W _ {1}), \text {and} \\ \frac {\partial \Pi_ {2}}{\partial \delta} & = & \frac {2}{b (1 - \sigma^ {2}) (4 - \sigma^ {2}) ^ {2}} ((2 - \sigma^ {2}) \alpha_ {2} - \sigma \alpha_ {1}) ((2 - \sigma^ {2}) W _ {1} - \sigma W _ {2}). \end{array}\]

Since because of Assumption 2 and 0 (recall that and then , and sign . Consequently, the dominant platform in the market of applications benefits with compatibility if , an inequality that always holds if

We next study how changes in the degree of compatibility as measured by afect consumers surplus and total welfare.

Proposition B1

both consumer surplus and total welfare are increasing in otherwise they may be increasing or decreasing.

Proof:Consumer surplus and total welfare are is given by

\[\begin{array}{r c l} C S & = & \left(\alpha_ {1} - p _ {1}\right) q _ {1} + \left(\alpha_ {2} - p _ {2}\right) q _ {2} - \frac {1}{2} b \left(q _ {1} ^ {2} + 2 \sigma q _ {1} q _ {2} + q _ {2} ^ {2}\right), \text {and} \\ W & : & \alpha_ {1} q _ {1} + \alpha_ {2} q _ {2} - \frac {1}{2} b \left(q _ {1} ^ {2} + 2 \sigma q _ {1} q _ {2} + q _ {2} ^ {2}\right). \end{array}\]

Substituting prices and quantities for their equilibrium prices expressions above can be rewritten as

\[\begin{array}{r l r} {C S} & {:} & {B \left(\left(4 - 3 \sigma^ {2}\right) \left(\alpha_ {1} ^ {2} + \alpha_ {2} ^ {2}\right) - 2 \sigma^ {3} \alpha_ {1} \alpha_ {2}\right), \mathrm{and}} \\ {W} & {=} & {B \left(\left(1 2 + 2 \sigma^ {4} - 9 \sigma^ {2}\right) \left(\alpha_ {1} ^ {2} + \alpha_ {2} ^ {2}\right) - 2 \sigma \left(8 - 3 \sigma^ {2}\right) \alpha_ {1} \alpha_ {2}\right),} \end{array}\]

where . Thus,

\[\begin{array}{r c l} \frac {\partial C S}{\partial \delta} & = & B \left(\left(4 - 3 \sigma^ {2}\right) \frac {\partial \left(\alpha_ {1} ^ {2} + \alpha_ {2} ^ {2}\right)}{\partial \delta} - 2 \sigma^ {3} \frac {\partial \left(\alpha_ {1} \alpha_ {2}\right)}{\partial \delta}\right), \\ \frac {\partial W}{\partial \delta} & = & B \left(\left(1 2 + 2 \sigma^ {4} - 9 \sigma^ {2}\right) \frac {\partial \left(\alpha_ {1} ^ {2} + \alpha_ {2} ^ {2}\right)}{\partial \delta} - 2 \sigma \left(8 - 3 \sigma^ {2}\right) \frac {\partial \left(\alpha_ {1} \alpha_ {2}\right)}{\partial \delta}\right) \end{array}\]

where

\[\frac {\partial \left(\alpha_ {1} ^ {2} + \alpha_ {2} ^ {2}\right)}{\partial \delta} = 2 \delta \left(W _ {1} ^ {2} + W _ {2} ^ {2}\right) + 2 \left(2 W _ {1} W _ {2} + V _ {L} W _ {1} + V _ {H} W _ {2}\right) > 0\]

and

\[\frac {\partial \left(\alpha_ {1} \alpha_ {2}\right)}{\partial \delta} = \left(W _ {1} ^ {2} + W _ {2} ^ {2}\right) + 2 \delta W _ {1} W _ {2} + V _ {H} W _ {1} + V _ {L} W _ {2} > 0.\]

If it follows that and , consumer surplus and welfare can decrease or increase as the following example shows. Let and . Figure A1 depicts (solid line) and (dotted line) as a function of σ.

Figure A1

Note that there are values of σ for which both and can be either negative or positive, which shows our claim. The picture suggests that high values of σ make and negative.

We next analyze the incentives to invest

Proposition B2

(i) There exist , such that

if dominant firm is type I

if the dominant firm is type II

if the dominant firm is type IV

(ii) The weak firm is either type I or type II. It is more likely type I if asymmetries in either stand-alone values or in application values are large.

Proof.

Consider first the dominant system. The efect of δ on the marginal profit from a higher stand-alone value is given by

\[\frac {\partial \left(\frac {\partial \Pi_ {1} (\delta)}{\partial V _ {H}}\right)}{\partial \delta} = \frac {2 (2 - \sigma^ {2})}{b (1 - \sigma^ {2}) (4 - \sigma^ {2}) ^ {2}} ((2 - \sigma^ {2}) W _ {2} - \sigma W _ {1}),\]

which is increasing if the inequality

\[\left(2 - \sigma^ {2}\right) W _ {2} - \sigma W _ {1} > 0\]

holds. Let . Then if . Note that as it is strictly decreasing in w.

Regarding its marginal incentive to invest in the application, we have

\[\frac {\partial \Pi_ {1} (\delta)}{\partial W _ {1}} = 4 B (2 - \sigma (\delta + \sigma)) ((2 - \sigma^ {2}) (V _ {H} + W _ {1} + \delta W _ {2}) - \sigma (V _ {L} + W _ {2} + \delta W _ {1}))\]

and

\[\frac {\partial \left(\frac {\partial \Pi_ {1} (\delta)}{\partial W _ {1}}\right)}{\partial \delta} = 4 B \binom{\left(4 (1 - \sigma \delta) - \sigma^ {2} (3 - \sigma (\sigma + 2 \delta))\right) W _ {2} - 2 \sigma (2 - \sigma (\delta + \sigma)) W _ {1}}{- \sigma ((2 - \sigma^ {2}) V _ {H} - \sigma V _ {L})}\]

Depending on the parameters, this expression can be negative or positive. The first term evaluated at . Since we only care about its sign we divide it by and we find that it is negative as it equals for all . It is positive at , while negative and increasing at . Consequently, there is such that the first term equals zero at . Since the second term is always negative we can conclude that for all . Furthermore since at the second term equals zero, there is such that for any . The dominant firm type does hence depend on it is type I. If it is type II. And if it is type then (i) follows.

Consider now the weak system. The efect of on the marginal profit from a higher stand-alone value is given by

\[\frac {\partial \left(\frac {\partial \Pi_ {2} (\delta)}{\partial V _ {L}}\right)}{\partial \delta} = \frac {2 (2 - \sigma^ {2})}{b (1 - \sigma^ {2}) (4 - \sigma^ {2}) ^ {2}} ((2 - \sigma^ {2}) W _ {1} - \sigma W _ {2})\]

which is always positive. When analyzing the incentives to invest in the application value we have that

\[\frac {\partial \Pi_ {2} (\delta)}{\partial W _ {2}} = 4 B (2 - \sigma (\delta + \sigma)) \binom{2 (V _ {L} + W _ {2} + \delta W _ {1}) -}{\sigma (W _ {1} + V _ {H} + \delta W _ {2}) - \sigma^ {2} (W _ {2} + V _ {L} + \delta W _ {1})}\]

and

\[\frac {\partial \left(\frac {\partial \Pi_ {2} (\delta)}{\partial W _ {2}}\right)}{\partial \delta} = 4 B \binom{(4 (1 - \sigma \delta) - \sigma^ {2} (3 - \sigma (\sigma + 2 \delta))) W _ {1} - 2 \sigma (2 - \sigma \delta - \sigma^ {2}) W _ {2}}{+ \sigma (\sigma V _ {H} - (2 - \sigma^ {2}) V _ {L})}.\]

Depending on the parameters expression above can be negative or positive. It is strictly positive at both and . Furthermore, it is strictly increasing in both and and strictly decreasing in and . Similarly, it is increasing in both w and v and decreasing in δ. To analyze the impact of asymmetries on the sign of , consider first a full symmetric scenario with . In this case evaluated at is positive if , and . If we introduce an asymmetry in the stand alone value, for instance by setting , while keeping , then is positive if where both and increase with W with and . Finally, if we introduce an asymmetry in the applications value by setting , while keeping then is positive if where . Thus, asymmetries lower values of δ increase the probability of being positive. The weak system is hence either type I or type II. It is more likely type I if asymmetries either in stand-alone values or in application values are large, and statement in (ii) follows.

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