domain: shs qfin

text:
J Ér Ôme Pouyet

Thomas Tr

Égou Ët

Özlem Bedre-Defolie

Bruno Jullien

Daniel O'brien

Markus Reisinger

Paul Belleflamme

Alexandre De Corniére

Jacques Crémer

Vincent Lefrere

David Martimort

Patrick Rey

Nicolas Schutz

The Competitive Effects of Vertical Integration in Platform Markets *

Keywords: Vertical integration, platform markets, network effects, foreclosure. JEL Code: L40, L10, D43

We analyze vertical integration in a setting where, first, platforms compete to provide an operating system to manufacturers of devices and attract developers of applications, and, second, there are indirect network effects between buyers of devices and developers of applications. Vertical integration creates market power over developers, and over non-integrated manufacturers but only under certain circumstances. That market power enables to coordinate pricing decisions across both sides of the market, which leads to a better internalization of network effects. Vertical integration does not systematically lead to foreclosure and can benefit all parties, even in the absence of efficiency gains. Its competitive impact depends on the strength and the structure of indirect network effects.

Introduction

Motivation. Software platform industries have recently witnessed many changes in the nature of the relationship between software and hardware producers. Traditional suppliers of operating systems have ventured into the hardware market and prominent hardware manufacturers have developed their own operating systems. In the smartphone market, while Apple further intensified its hallmarked integration between hardware and software, Google launched in 2016 the Pixel, its first device conceptualized and engineered inhouse. It also acquired a major handset manufacturer, HTC, in 2018 and recently started designing mobile processors, the Google Tensors. 1 Samsung and Huawei, subjugated to Google for the use of its Android platform while delivering Google substantial money through services installed on their phones, have started to equip some of their devices with their own operating systems. 2 In the online retail sector, Amazon sells devices powered by FireOS, an operating system built on Android's technology but stripped from Google's applications. Microsoft, once praised for its software-only model, has ventured in the electronic devices market with the Surface brand. Other industries are witnessing a similar momentum of integration along the value chain. 3 These changes raise several questions, and whether the usual competitive assessment of integration could readily be applied to platform markets remains an on-going debate in the antitrust arena. 4 In this article, we address the following question: what are the competitive effects of vertical integration between platforms offering operating systems and device manufacturers? We show that indirect network effects, which are prevalent in digital markets, substantially impact the competitive assessment of vertical integration. Although vertical integration still creates some market power, as the common wisdom has it, the sources of such market power are different than in one-sided markets. Perhaps more importantly, the exercise of such market power does not necessarily harm either consumers or non-integrated competitors. For instance, when indirect network effects are sufficiently strong and asymmetric (in a sense to be defined properly later on), vertical integration can benefit all parties, even in the absence of efficiency gains. A recurrent intuition of our analysis is that vertical integration enables to coordinate several pricing decisions and such coordination sometimes allows a better internalization of indirect network effects.

The Model. Several platforms compete to license their operating systems to two manufacturers of devices. Manufacturers equip their devices with an operating system, pay some fee (whose precise nature is detailed later on) to the corresponding platforms, and then compete to sell devices to buyers. Developers pay fees to platforms to publish their applications on the operating systems. Therefore, a device gives its buyers access to applications developed for the operating system it is equipped with. This interaction between buyers of devices and developers of applications is the source of indirect network effects in our analysis.

Our benchmark is the situation in which none of the platforms are integrated with a manufacturer. There, competition between equally-efficient platforms leads to a Bertrandlike outcome in which neither the developers nor the manufacturers pay anything to the platforms. Competition prevents platforms from exerting any market power, either on manufacturers or on developers.

Vertical Integration. We then consider vertical integration between a platform and a manufacturer. Because it faces competition from equally-efficient platforms, the vertically-integrated platform cannot exert any market power on the non-integrated manufacturer; a result that is standard from the literature on strategic vertical integration (see, e.g., [START_REF] Salop | Raising Rivals' Costs[END_REF][START_REF] Ordover | Equilibrium Vertical Foreclosure[END_REF][START_REF] Chen | On Vertical Mergers and Their Competitive Effects[END_REF]). 5Vertical integration creates, however, market power over developers because the integrated platform has monopoly power over the access to the buyers of its device. This is a new source of market power, which comes from the two-sided nature of our model. The next step of our analysis consists in assessing how the integrated firm exercises such market power. To do so, observe that the integrated platform has two pricing instruments: the fee paid by developers to publish their applications on its operating system and the price paid by buyers for its device. The integrated platform's prices are guided by two forces: a one-sided logic, according to which increasing prices (above their pre-merger levels) allows to extract more profit from developers and from buyers; a two-sided logic, according to which setting an asymmetric pricing structure allows to better internalize network effects between buyers and developers. Which logic prevails overall depends on the strength and the structure of indirect network effects. For instance, and in the spirit of the literature on two-sided markets [START_REF] Armstrong | Competition in Two-Sided Markets[END_REF][START_REF] Rochet | Two-Sided Markets: A Progress Report[END_REF][START_REF] Caillaud | Chicken & Egg: Competition among Intermediation Service Providers[END_REF], when buyers value strongly applications, the integrated platform finds it optimal to decrease the developer fee and increase the price for its device.

Next, we analyze the consequences of that market power.

Foreclosure. Because vertical integration does not create market power on the nonintegrated manufacturer, foreclosure cannot be the result of a 'raise the rival's cost' effect. Foreclosure may arise, or not, because the integrated firm has some market power over developers, which ultimately impacts the non-integrated manufacturer's profit. For instance, when buyers value strongly applications, the integrated platform subsidizes developers and increases the price of its device, which boosts the non-integrated manufacturer's demand. As a result, the vertical merger benefits the non-integrated manufacturer. A reverse conclusion obtains when developers value more the participation of buyers. Summarizing, foreclosure of the non-integrated manufacturer is neither systematic nor the result of a 'raise the rival's cost' effect. It is, rather, the mere collateral damage of the integrated firm's market power over developers that, sometimes, depending on the strength and the structure of network effects, leads to an asymmetric pricing structure.

Welfare. Assuming a particular specification of our model, we fully characterize the impact of vertical integration on buyer and developer surpluses. In a nutshell, when indirect network effects are strong and sufficiently asymmetric, a situation that may characterize more infant platform markets, large social gains can be generated by implementing an asymmetric pricing structure that internalizes these effects. This is precisely what the integrated platform does, and vertical integration benefits buyers and developers. Otherwise, when network effects are balanced or weak, a situation that may characterize more mature platform markets, the pricing structure chosen by the integrated platform aims more to directly extract surpluses from buyers and developers through price increases. In these cases, developers and buyers tend to be harmed by the vertical merger.

Importantly, there is no obvious correlation between buyer/developer harm and foreclosure of the non-integrated manufacturer. For instance, when network effects are much stronger on the developers side than on the buyers side of the market, buyers and developers may benefit from the merger; internalization of indirect network effects by the integrated platform may require to increase the developer fee, which hurts the non-integrated manufacturer.

Efficiency Gains. Next, we consider that vertical integration creates synergies. In a traditional one-sided framework, synergies have two facets. They are pro-competitive because they are passed through partly to buyers in the form of a lower price for the integrated platform's device. They are anti-competitive because they create some market power that allows the integrated firm to command some payment from the non-integrated manufacturer, thereby softening competition on the buyers' market through a 'raise the rival's cost' effect.

The analysis becomes more complex in our two-sided framework. Although a more efficient integrated platform is able to command a higher fee from the non-integrated manufacturer, it is not always willing to do so. This holds because, again, the integrated platform uses its pricing instruments to extract surplus (from buyers, developers and the non-integrated manufacturer) but also to internalize network effects across both sides of the market. Such internalization requires, sometimes, to lower the fee paid by the nonintegrated manufacturer below its pre-merger level. To illustrate, when network effects are stronger on the developers' side than on the buyers' side, subsidizing buyers can be done by setting a low price for the integrated platform's device and charging a low fee to the non-integrated manufacturer. Whether the fees charged by the integrated platform increase or decrease following the vertical merger depends, again, on the structure and the strength of indirect network effects.

We then study whether vertical integration leads to foreclosure and harms buyers or developers. Overall, and in line with the situation without efficiency gains, vertical integration tends again to be beneficial (respectively, detrimental) to welfare when indirect network effects are strong and asymmetric (respectively balanced or weak).

Coordination Motives and Porting Costs. Finally, we discuss the impact of vertical integration when platform users gain if manufacturers adopt the same operating system (perhaps because of direct network effects between users), or when developers have a cost to port their applications on different platforms. In these situations, there are motives of coordination between manufacturers. Much as in the case of efficiency gains, coordination motives create market power over the non-integrated manufacturer, because vertical integration somewhat forces the coordination of manufacturers on the integrated firm's operating system. However, such market power is not necessarily detrimental to welfare; this depends on the strength and the structure of network effects.

Related literature. From the literature on two-sided markets, we borrow the general insight that indirect network effects are key to understanding platform pricing and competition [START_REF] Caillaud | Chicken & Egg: Competition among Intermediation Service Providers[END_REF][START_REF] Armstrong | Competition in Two-Sided Markets[END_REF][START_REF] Rochet | Two-Sided Markets: A Progress Report[END_REF][START_REF] Weyl | A Price Theory of Multi-sided Platforms[END_REF][START_REF] Jullien | Chapter 7 -Two-sided markets, pricing, and network effects[END_REF]. That literature has considered the effect of exclusive dealing between a platform and content providers (that is, developers in our model): [START_REF] Evans | Economics of Vertical Restraints for Multi-Sided Platforms[END_REF] discusses the antitrust of such vertical relations in platform industries; [START_REF] Doganoglu | Exclusive Dealing with Network Effects[END_REF] and [START_REF] Hagiu | Exclusivity and Control[END_REF] provide a rationale for why platforms sign exclusive contracts with content providers; [START_REF] Church | Systems Competition, Vertical Merger, and Foreclosure[END_REF] describe the incentives of a manufacturer that is integrated with a developer to make its applications compatible with the hardware of a rival manufacturer; [START_REF] Hagiu | First-Party Content and Coordination in Two-Sided Markets[END_REF] show that investment in first-party content (that is, vertical integration with one side of the market) depends on whether a platform faces a 'chicken-and-egg' coordination problem; in the video game industry, [START_REF] Lee | Vertical Integration and Exclusivity in Platform and Two-Sided Markets[END_REF] finds that exclusivity tends to be pro-competitive, in that it benefits an entrant platform more than an incumbent platform. While we share with these papers the focus on the competitive impact of vertical restraints in twosided markets, our work also differs, for we are interested in the interactions between platforms/operating systems and manufacturers when devices are an essential link to connect buyers and developers.

Several papers have recently analyzed how a dominant platform may foreclose a fringe upstream producers when those producers can reach the downstream market either using the platform's marketplace or through alternative channels. Considering different models of the downstream product market (horizontal or vertical differentiation, endogenous entry/variety or fixed number of sellers), [START_REF] Hagiu | Should Platforms Be Allowed to Sell on Their Own Marketplaces?[END_REF], [START_REF] Etro | Product Selection in Online Marketplaces[END_REF] and [START_REF] Anderson | Hybrid Platform Model[END_REF] study various modes of organization for the platform, the platform's incentives to steer consumers away from the competitors' product and favor its own products, and the possible regulatory responses. [START_REF] Madsen | Insider Imitation[END_REF] examine how platforms may use information about downstream demand to decide whether or not to launch their own version of the same product. [START_REF] Kang | Contracting and Vertical Control by a Dominant Platform[END_REF] allow for a rich contracting space to model the relationship between the dominant platform and the privately-informed upstream producers. We share with these papers the broad concern about platform-driven foreclosure of downstream competitors. In our model, however, several platforms compete to provide an essential input to manufacturers and manufacturers have market power downstream. Moreover, indirect network effects enabled by platforms are central to our analysis (as in [START_REF] Anderson | Hybrid Platform Model[END_REF].

Our framework is much closer to those belonging to the so-called strategic approach of vertical integration initiated by [START_REF] Ordover | Equilibrium Vertical Foreclosure[END_REF]. 6 A message conveyed by that literature is that vertical integration can lead to input foreclosure and be detrimental to consumer surplus. Analyses that feature trade-offs between the pro-and the anticompetitive effects of vertical integration include the following: [START_REF] Ordover | Equilibrium Vertical Foreclosure[END_REF] and [START_REF] Reiffen | Equilibrium Vertical Foreclosure: Comment[END_REF], in which integration generates an extra commitment power; [START_REF] Chen | On Vertical Mergers and Their Competitive Effects[END_REF], in which manufacturers have switching costs; [START_REF] Choi | Vertical Foreclosure with the Choice of Input Specifications[END_REF], in which upstream suppliers can choose the specification of their inputs; [START_REF] Chen | Vertical Integration, Exclusive Dealing, and Expost Cartelization[END_REF], in which exclusive dealing can be used in combination with integration; [START_REF] Nocke | Do Vertical Mergers Facilitate Upstream Collusion?[END_REF] and [START_REF] Normann | Vertical Integration, Raising Rivals' Costs and Upstream Collusion[END_REF], in which upstream suppliers tacitly collude; [START_REF] Hombert | Anticompetitive Vertical Merger Waves[END_REF], in which there are more manufacturers than upstream suppliers; and [START_REF] Hunold | Passive Vertical Integration and Strategic Delegation[END_REF], in which integration can be either controlling or passive. 7 None of these papers address multi-sided markets with indirect network effects, and our analysis provides sev-6 Riordan (2008) considers a dominant firm facing a fringe of competitors and shows that vertical integration is always anticompetitive. [START_REF] Loertscher | Market Structure and the Competitive Effects of Vertical Integration[END_REF] extend [START_REF] Riordan | Competitive Effects of Vertical Integration[END_REF]'s framework to allow for the exercise of market power upstream by several firms and show that vertical integration can be procompetitive.

7 For empirical analyses, see, e.g., [START_REF] Lafontaine | Vertical Integration and Firm Boundaries: The Evidence[END_REF] and [START_REF] Crawford | The Welfare Effects of Vertical Integration in Multichannel Television Markets[END_REF] and the references therein.

eral new insights. For instance, when indirect network effects are sufficiently strong and asymmetric, vertical integration does not systematically lead to foreclosure and may benefit both consumers and non-integrated manufacturers, even in the absence of efficiency gains. By contrast, when network effects are sufficiently balanced or weak, our findings echo those of this literature.

Organization of the Paper. Section 2 considers a situation with only one platform and only one manufacturer. That simpler situation proves to be useful to understand the impact of vertical integration when several platforms compete to supply several manufacturers competing on a product market, as studied in Section 3. Section 4 analyzes the impact of vertical integration with efficiency gains. Section 5 discusses the role of coordination motives and porting costs. Section 6 concludes. All proofs are relegated to an Appendix.

Monopoly Platform and Monopoly Manufacturer

We begin with a simplified version of our model, with only one platform I and one manufacturer of devices M . Buyers of devices and developers of applications may interact. These interactions require: buyers to purchase devices from the manufacturer; developers to decide how much applications to develop for the platform; the manufacturer to set the price for its device; the platform to set the licensing condition imposed on the manufacturer and the fee charged to developers to publish applications on its operating system.

Model. The contractual relationship between a manufacturer and a platform typically specifies which party owns user-generated data and, accordingly, who can monetize these data through targeted advertising for instance. Let r be the per-user benefit generated by a buyer of M 's device equipped with I's operating system. The platform decides how this benefit is shared with the manufacturer: β ∈ [0, 1] (resp., 1 -β) denotes the share of r kept by the manufacturer (resp., the platform). 8

Platform I also charges a fee to application developers. 9 Denote by a the fee charged by I to allow a developer to make its application available on its operating system. That fee can be either positive or negative.

When manufacturer M adopts platform I's operating system, I's profit can be expressed as follows

Π I = (1 -β) r Q B + a Q S ,
8 As a broader illustration, in the recent Google and Alphabet v Commission case, the European Commission found that Google has infringed Article 102 TFEU by entering into several agreements with Original Equipment Manufacturers and Mobile Network Operators (Commission Decision C(2018) 4761 final of 18 July 2018, available at https://eur-lex.europa.eu/legal-content/GA/TXT/?uri=CELEX: 52019XC1128(02)). Of particular relevance were: (i) the mobile application distribution agreements, which required OEMs and MNOs wishing to pre-install Google Play on their devices to also pre-install other Google applications; (ii) the portfolio-based revenue sharing agreements, according to which Google provided payments to OEMs and MNOs in return for having the Google Search application exclusively pre-installed on a given portfolio of smart mobile devices.

9 Software platforms often charge developers on participation (Google charges developers $25 for each application published on the Play Store) or on transaction each time an application is sold on the platform (both Apple and Google charge a roughly 30% royalty on each transaction on their respective applications stores).

where Q B is the number of buyers of device and Q S is the number of applications available on platform I's operating system.

Manufacturer M produces at a constant marginal cost normalized without loss of generality to 0. The number of buyers of devices depends on the price charged to buyers, denoted by p, and on the number of applications running on the devices, denoted by n S . Hence, it may be written as Q B (p, n S ), with ∂Q B /∂p < 0 and ∂Q B /∂n S > 0. 10 M 's profit when it chooses I's operating system is thus given by 11

π M = (p + β r)Q B .
Last, to compute the buyer surplus, we consider that there exists a representative buyer with utility function

U B (q B , n S ) such that Q B is solution of max q B ≥0 U B (q B , n S ) - pq B . Let V B (p, n S ) denote the corresponding indirect utility.
We assume that there is a representative developer which bears a strictly increasing convex cost C S (q S ) to develop q S applications (with C S (0) = C ′ S (0) = 0). Once applications are developed, the developer can make these applications available on I's operating system at no further cost.12 Let n B be the number of buyers using a device running platform I's operating system. When the developer creates q S applications, its profit is given by (u S n B -a)q S -C S (q S ).

In words, q S applications published on platform I with a number of users n B yields a gross benefit U S (q S , n B ) = u S n B q S -C S (q S ) for developers. Parameter u S relates to the strength of indirect network effects from the developer side of the market. Let Q S (n B , a) = arg max q S ≥0 U S (q S , n B )-aq S and V S (n B , a) be the corresponding developer's (net) profit.

We can restrict the analysis to cases where platform I sets a developer fee a such that a ≤ u S n B , for otherwise there would be no applications developed and I would be strictly better off setting a = 0 as soon as n B > 0. Therefore, Q S (n B , a) = (C ′ S ) -1 (u S n B -a). Last, the timing is as follows. In stage 1, platform I sets the share of the per-user benefit left to manufacturer M for the use of its operating system and the fee charged to developers. In stage 2, manufacturer M chooses the operating system for its devices and the price of its devices. Last, in stage 3, buyers decide whether to buy a device, and, simultaneously, developers decide how many applications to develop. All decisions are public and we look for the subgame-perfect equilibrium of the game. Figure 1 summarizes the structure of the simplified model. Proofs for the results stated in this section are gathered in Appendix A.1.

Running Example. We sometimes use the following specification of the model, in particular to compute the welfare impact of vertical integration.

-Buyers. Demand for the device is given by Q B (p, n S ) = v -p + u B n S , where u B relates to the strength of indirect network effects from the buyer side, and

U B (q B , n S ) = (v + u B n S )q B -1 2 q 2 B .
10 Q B (•) is often called the 'quasi-demand function' of buyers. The number of applications available on the device can thus be viewed as an endogenous quality attribute.

11 The sharing parameter β acts thus like a negative perceived marginal cost for the manufacturer.

Indirect network effects

Integration

Platform I M p β a q S q B Buyers q B devices Developers q S applications
Figure 1: The model with a monopoly platform and a monopoly manufacturer.

-

Developers. The development cost is C S (q S ) = 1 2 q 2 S . Hence, Q S (n B , a) = u S n B -a.
Buyers' and Developers' Participation Decisions. At the last stage of the game, given a price of device p and a fee a paid by developers, the number of buyers of devices and the number of applications developed must be consistent with each other and solve

(2.1)

n B = Q B (p, n S ), n S = Q S (n B , a).
The solution of (2.1) is unique and interior provided that indirect network effects are not too strong. 13 That solution defines the buyers' demand for the device, denoted by D B (p, a), and the number of applications developed (also called the developers' demand), denoted by D S (p, a). The following intuitive properties hold: both demands are decreasing in the price of the device and in the developer fee (∂D S /∂p < 0 and ∂D S /∂a < 0 for the developers' demand; ∂D B /∂p < 0 and ∂D B /∂a < 0 for the buyers' demand).

Social

p * + r = - ∂U S ∂n B < 0, a * = - ∂U B ∂n S < 0.
(2.2) 13 The condition is that ∂Q B ∂n S ∂Q S ∂n B < 1 in the relevant range of (p, a). 14 Throughout the paper, we will omit some arguments in order to ease the exposition.

Because of indirect network effects, welfare-maximizing prices are below the corresponding marginal costs. Decreasing the developer fee (resp. the price of devices) boosts the number of applications (resp. the number of buyers), which increases in turn the developer's gain from interacting with buyers (resp. the buyers' benefit from having more applications).

Ramsey Pricing. Socially optimal prices defined by (2.2) lead the industry to a loss since both the margin on buyers and that on developers are negative. Ramsey prices (p R , a R ) maximize welfare while ensuring that the industry makes a non-negative profit. Denoting by λ the Lagrange multiplier associated to the break-even constraint Π(p, a) ≥ 0, Ramsey prices satisfy

p R + r + 1 1 + λ ∂U S ∂n B = λ 1 + λ 1 η B - ∂Q S ∂n B 1 η S , a R + 1 1 + λ ∂U B ∂n S = λ 1 + λ 1 η S - ∂Q B ∂n S 1 η B , (2.3)
where

η B = -1 n B ∂Q B ∂p and η S = -1 n S ∂Q S
∂a . At the Ramsey optimum, the externality created by one side of the market on the other side (∂U i /∂n j ) is now weighted by the shadow cost of the budget constraint (1 + λ). The industry makes no profit and one side is taxed and faces a positive margin, whereas the other side is subsidized and enjoys a negative margin; which case occurs depends on the relative strength of indirect network effects.

Proposition 1. Ramsey prices (p R , a R ) that maximize welfare subject to the industry break-even constraint are such that buyers of devices are taxed (p R +r ≥ 0) and developers of applications are subsidized (a R ≤ 0) if and only if

(2.4) 1 η B ∂U B ∂n S + ∂U S ∂n B ∂Q B ∂n S ≥ 1 η S ∂U S ∂n B + ∂U B ∂n S ∂Q S ∂n B .
In the running example, (2.4) amounts to u B ≥ u 3 S .

There are three terms in the right-hand side of (2.4). The first term (1/η S ) is the inverse of the semi-elasticity of the developers' demand. The second term ( ∂U S ∂n B ) measures the extent to which developers benefit directly from an increase in the participation of buyers. The third term ( ∂U B ∂n S ∂Q S ∂n B ) measures how much buyers benefit from an increase in their own participation through a feedback effect: an increase in the number of buyers boosts the number of applications, which ultimately benefits buyers. Therefore, Condition (2.4) shows that developers are subsidized (and buyers are thus taxed) when they are the 'high-elasticity group of users,' that is, when their demand is more price elastic and when they benefit less from the participation of buyers and more from their own participation.

Industry Profit Maximizing Prices. Suppose now that platform I and manufacturer M are vertically integrated and maximize their joint profit Π(p, a). In an interior optimum, the integrated monopoly prices (p I , a I ) satisfy

p I + r = 1 η B - ∂Q S ∂n B 1 η S , a I = 1 η S - ∂Q B ∂n S 1 η B , (2.5)
Monopoly prices defined by (2.5) differ from Ramsey prices given in (2.3) in two ways.

Since the integrated monopoly only cares about its profit and not about welfare, the margins allocated on both sides of the market are excessive; a standard market power effect. Second, the integrated platform internalizes the effect on the higher interaction benefits of the marginal users on the other side, and not the inframarginal ones; a distortion in the spirit of [START_REF] Spence | Monopoly, Quality, and Regulation[END_REF]. [START_REF] Tan | Pricing Distortions in Multi-Sided Platforms[END_REF] provide a general framework to compare socially optimal prices with monopoly ones in multi-sided platforms, and show that the comparison is a priori ambiguous. Taking advantage of our more structured setting, we observe that the integrated monopoly may still be willing to subsidize one side of the market when this sufficiently boosts revenues earned from the other side.

Proposition 2. Monopoly prices (p I , a I ) that maximize industry profit are such that:

-Buyers of devices are taxed and developers of applications are subsidized (p

I + r ≥ 0 and a I ≤ 0) iff η B ≤ η S ∂Q B ∂n S ;
-Buyers of devices are subsidized and developers of applications are taxed ( p I +r ≤ 0 and a

I ≥ 0) iff η S ≤ η B ∂Q S
∂n B ; -Both sides of the marked are taxed otherwise.

In the running example, a

I ≥ 0 iff u B ≤ u S and p I + r ≥ 0 iff u S (u B + u S ) ≤ 2.
Vertically-Separated Monopolies. Suppose platform I and manufacturer M are now separated. At stage 2, M chooses the price of devices p so as to maximize its profit (p + βr)D B (p, a). Assuming an interior optimum, the optimal price for the manufacturer ∂p (p, a)) -1 is the manufacturer's profit. Denote by p S and a S the device price and the developer fee that emerge under separation.

P (β,
With respect to integration, the separation outcome involves a first type of inefficiency in the form of a double marginalization. In the case of successive monopolies with a single-product downstream firm, it is well-known that eliminating double marginalization through vertical integration leads to a lower price and improves consumer surplus [START_REF] Spengler | Vertical Integration and Antitrust Policy[END_REF]. However, with a downstream monopoly producing several substitutable products, the elimination of one double marginalization has a priori ambiguous consequences on prices, and thus on consumer surplus, as shown by [START_REF] Salinger | Vertical Mergers in Multi-Product Industries and Edgeworth's Paradox of Taxation[END_REF]. Indeed, when it becomes more efficient on one of its products, the firm has incentives to boost the sales of that product by increasing the price of the other substitutable good. 15 In our framework, devices and applications are (imperfect) complements rather than substitutes, but a similar ambiguity emerges and there is no hope to obtain a general result on the impact of vertical integration on prices or surpluses.

Second, pricing decisions are not fully internalized under separation. The manufacturer does not internalize the impact of the price of devices on the participation of application developers, and the platform does not fully internalize the impact of the developer fee on the demand for devices. In particular, the platform now uses its prices also to limit the manufacturer's rent φ(p, a).

Third, the sharing parameter β is a crude instrument to control the price of devices p set by the manufacturer. 16 This limits the ability of the platform to operate crosssubsidies between buyers of devices and developers of applications.

In the running example, we can fully compare the outcomes under separation and under integration. As can be observed from Figure 2, the developer fee a as well as the margin on devices p + r are always positive under separation; no cross-subsidization ever occurs since the downstream manufacturer always chooses a price such that its margin p + βr remains positive . By contrast, the integrated firm is able and willing to implement some cross-subsidization between buyers of devices and application developers, most notably when indirect network effects are strong and asymmetric (that is, either u B ≫ u S or u S ≫ u B ); in those cases with sufficiently strong and skewed network effects, the price structure under integration somehow looks like the one under Ramsey pricing. We now turn on to the welfare impact of vertical integration. A simple revealed preferences argument shows that integration always leads to a higher industry profit than separation. From the perspective of the surpluses of buyers and developers, the comparison is a priori ambiguous. Nevertheless, using brute force computations we can show the following result.

Proposition 3. Consider the running example with v = r = 1. Buyers of devices and developers of applications always benefit from vertical integration between the platform and the manufacturer. Hence, welfare increases following vertical integration.

Vertical Integration with Competing Platforms and Competing Manufacturers

We now analyze vertical integration but in the presence of competition both between platforms and between manufacturers.

Competition between Platforms, Competition between Manufacturers

Platforms. There are N + 1 (with N ≥ 2) symmetric platforms, denoted by I, E 1 , ..., E N . Let P denote the set of platforms. We sometimes refer to platforms E 1 , ..., E n as the fringe of platforms. There are two manufacturers, denoted by M 1 and M 2 . In the following, platform I will be the one contemplating a merger with manufacturer M 1 .

Let β k

i ∈ [0, 1] (respectively, 1 -β k i ) be the share of r kept by manufacturer M k (respectively, platform i) if it equips its devices with platform i's operating system. Denote by a i the fee charged by platform i to allow a developer to publish applications on its operating system. The profit of a platform i can thus be expressed as follows

Π i = k=1,2 1 {M k adopts i} (1 -β k i )rQ k B + a i q i ,
where (i) 1 {M k adopts i} is the indicator function equal to 1 when M k chooses platform i's

operating system and 0 otherwise, (ii) Q k B is the number of buyers of device k, and (iii) q i is the number of applications available on platform i's operating system.

Manufacturers and Buyers of Devices. Manufacturers are symmetric and produce at a zero marginal cost. The number of buyers of M k 's device depends on the prices charged by manufacturers to buyers, denoted by p k and p ℓ , with k ̸ = ℓ ∈ {1, 2}, and on the number of applications running on the devices, denoted by n k S and n ℓ S . Hence, it may be written as Q k B (p k , p ℓ , n k S , n ℓ S ). Assume that these so-called 'quasi-demand functions' are symmetric:

Q k B (p k , p ℓ , n k S , n ℓ S ) = Q ℓ B (p ℓ , p k , n ℓ S , n k S ).
The profit of M k when it chooses platform i's operating system can thus be written as follows

π k = (p k + β k i r)Q k B .
From the buyer side, assume that given some numbers of applications running on the manufacturers' devices (n 1 S , n 2 S ): devices are demand substitutes for buyers, or ∂Q k B /∂p k < 0 < ∂Q k B /∂p ℓ ; the direct price effect is stronger than the indirect one, or ∂Q k B /∂p k + ∂Q k B /∂p ℓ < 0; buyers of device k value positively the number of applications available on their devices, or ∂Q k B /∂n k S > 0, but negatively the number of applications available on the other device, or ∂Q k B /∂n ℓ S < 0. Last, to compute the buyer surplus, we consider that there exists a representative buyer with utility function U B (q 1 , q 2 , n 1 S , n 2 S ) such that Q 1 B and Q 2 B are solutions of max (q 1 ≥0,q 2 ≥0) U B (q 1 , q 2 , n 1 S , n 2 S )-p 1 q 1 -p 2 q 2 . Let V B (p 1 , p 2 , n 1 S , n 2 S ) denote the corresponding indirect utility.

Application Developers. There is a representative developer which develops q S applications at cost C S (q S ). We assume that there are no platform-specific costs to port applications on the various platforms; an assumption relaxed in Section 5 where we show that porting costs create a specific source of market power. Let n i B be the number of buyers using a device running platform i's operating system. Since there are no porting costs, all applications are published on platform i as soon as u S n i B -a i ≥ 0, which we assume from now on. 17 The developer's gross profit is thus given by U S (q S , (n i B ) i∈P ) = q S i∈P u S n i B -C S (q S ) and its net profit writes as follows

(3.1) q S i∈P (u S n i B -a i ) -C S (q S ).
Let Q S ((n i B , a i ) i∈P ) be the number of applications q S that maximizes (3.1) and denote by V S ((n i B , a i ) i∈P ) the corresponding developer profit. Figure 3 summarizes the structure of the model. Timing. In stage 1, platforms set the shares of the per-user benefit left to manufacturers in exchange of using their operating systems and the fees charged to developers. In stage 2, manufacturers choose the operating system for their devices. Once operating systems have been chosen, manufacturers set the prices of their devices in stage 3. Last, in stage 4, buyers decide whether to buy a device, and, simultaneously, developers decide how 17 As in Section 2, this is without loss of generality since, otherwise, there would be no applications developed for platform i and that platform would be strictly better off setting a i = 0 as soon as n i B > 0.

Indirect network effects

Integration

Platform I Platforms E i M 1 M 2 p 1 p 2 (β 1 I , β 2 I ) (β 1 Ei , β 2 Ei ) a I a Ei q S q S q 1 q 2 Buyers Developers q S applications
much applications to develop and on which platforms to publish. All decisions are public and we look for the subgame-perfect equilibrium of the game.

Running Example under Competition.

-Buyers. The demand for device k is given by

(3.2) Q k B (p k , p ℓ , n k S , n ℓ S ) = v -p k -γ p k - p k + p ℓ 2 + u B n k S + γ 2 u B (n k S -n ℓ S ) .
Terms in the first parenthesis in Equation (3.2) correspond to a standard productmarket interaction with imperfectly substitutable products. Terms in the second parenthesis illustrate how indirect network effects between users of devices and application developers impact the demand for the manufacturers' products.

The utility function of the representative buyer is given by18

U B (q 1 , q 2 , n 1 S , n 2 S ) = q 0 + k=1,2 (vq k + u B n k S q k ) - 1 2 1 2(1 + γ) 2 k=1,2 q 2 k + γ( k=1,2 q k ) 2 ,
where q 0 is the numéraire and q k is the quantity of device k bought.

-

Developers. C S (q S ) = 1 2 q 2 S . Hence, Q S ((n i B , a i ) i∈P ) = i∈P (u S n i B -a i ).

Participation Decisions

Consider now the last stage of the game. At that stage, platforms have set their fees (β i 1 , β i 2 , a i ) i∈P and manufacturers have chosen their operating system and the prices (p 1 , p 2 ) for the devices.

There are two consequences to the fact that the developer is willing to publish all its applications on all the platforms. First, whatever the choices of operating systems by the manufacturers, the developer is able to interact with all the buyers of devices, or i∈P n i B = n 1 B + n 2 B . The developer's profit can thus be rewritten more simply as (u S (n 1 B + n 2 B ) -a)q S -C S (q S ) where a ≡ i∈P a i denotes the 'total developer fee.' The number of applications n S that maximizes this profit is given by n S = Q S (u S (n 1 B +n 2 B )-a), where Q S = (C ′ S ) -1 . Second, whatever their choices of operating systems, manufacturers benefit from the same number of applications running on their devices, or n 1 S = n 2 S ≡ n S . The demand for device k may now be written more simply as

n k B = Q k B (p k , p ℓ , n S )
. Buyers' and Developers' Participation Decisions. Given the prices of devices p 1 and p 2 and a total fee a paid by developers, the number of buyers of each device and the number of applications must be consistent with each other and solve

(3.3)    n 1 B = Q 1 B (p 1 , p 2 , n S ), n 2 B = Q 2 B (p 2 , p 1 , n S ), n S = Q S (u S (n 1 B + n 2 B ) -a).
Assume that the solution of (3.3) is unique and interior for the relevant range of prices. 19 That solution defines, as functions of the prices of the devices and the developer fee, the buyers' demands for devices, denoted by D k (p k , p ℓ , a) with k ̸ = ℓ ∈ {1, 2}, and the number of applications developed (also called the developers' demand), denoted by D S (p 1 , p 2 , a).

The following intuitive properties hold: the developers' demand is decreasing in the prices of devices and in the developer fee (∂D S /∂p k < 0 and ∂D S /∂a < 0); the demand for a device is decreasing in its own price and in the developer fee (∂D k /∂p k < 0 and ∂D k /∂a < 0). We further impose that the demand for a device is more responsive to its own price than to the price of the other device (∂D k /∂p k + ∂D k /∂p ℓ < 0). In the running example, all these properties hold provided that 2u B u S < 1.

3.3. Indirect Networks Effects impact Product Market Interactions.

Perhaps more surprising is the fact that indirect network effects impact the nature of the interaction between manufacturers on the product market. The demand faced by a manufacturer may, indeed, either increase or decrease with the price of the rival manufacturer, depending on the strength of indirect network effects relative to the degree of product market competition. Formally, using (3.3), it follows that

∂D k ∂p ℓ = ∂Q k B ∂p ℓ + ∂Q k B ∂n S ∂D S ∂p ℓ ,
which can be positive or negative. The intuition is as follows. If p ℓ increases, then some buyers are diverted from M ℓ , and M k 's demand increases by ∂Q k B /∂p ℓ . This is a standard rivalry effect created by product market competition between manufacturers. The increase in p ℓ has, moreover, a negative impact on the total number of buyers, since the direct price effect on buyers of device ℓ is stronger than the indirect price effect on buyers of device k (∂Q ℓ B /∂p ℓ + ∂Q k B /∂p ℓ < 0). Since there are less buyers overall, there are fewer applications too, for developers find it less attractive to develop (∂D S /∂p ℓ < 0). Because buyers value applications, this negatively affects M k 's demand by ∂Q k B /∂n S . We therefore expect that when indirect network effects are small (that is, when (∂Q k B /∂n S )(∂D S /∂p ℓ ) ≈ 0), the rivalry effect created by product market competition dominates and ∂D k /∂p ℓ ≥ 0, that is, devices are demand substitutes. By contrast, when product market competition is weak (that is, when ∂Q k B /∂p ℓ ≈ 0), then the interaction created by indirect network effects dominates and ∂D k /∂p ℓ ≤ 0, that is, devices are demand complements. In our running example, devices are demand substitutes when γ -2u B u S (1 + γ) > 0, and demand complements otherwise.

In the sequel, we shall focus on the case studied by the bulk of the literature on strategic vertical integration, namely the case where manufacturers' products are demand substitutes:

Assumption 1. Indirect network effects are not too strong relative to product market competition so that manufacturers' products are demand substitutes: for k ̸ = ℓ, for all (p k , p ℓ , a)

∂D k ∂p ℓ (p k , p ℓ , a) ≥ 0.

In the running example, this amounts to σ ≡ γ -2u B u S (1 + γ) ≥ 0.

In stage 3, manufacturers compete on the product market. Given a share β k of the per-user benefit that M k receives from the platform it has chosen and a total fee a paid by developers, let π k (β k , p k , p ℓ , a) = (p k + β k r)D k (p k , p ℓ , a) denote M k 's profit. We now make some assumptions that ensure the price competition subgame is 'well-behaved.' M k 's best response, denoted by R k (β k , p ℓ , a), is uniquely characterized by the firstorder condition ∂π k ∂p k (β k , R k , p ℓ , a) = 0. Moreover, 0 ≤ ∂R k /∂p ℓ < 1 for all (β k , p k , p ℓ , a), so that prices of devices are strategic complements and best responses satisfy the usual stability assumption. 20 Last, M k 's best response decreases with a, that is, ∂R k /∂a ≤ 0 for all (β k , p k , p ℓ , a). 21 This assumption seems reasonable since an increase in the developer fee negatively impacts the demand for device k. Together, these assumptions ensure that there exists a unique pair of prices ( p 1 (β 1 ,β 2 ,a),p2 (β 2 ,β 1 ,a)) that form the Nash equilibrium of stage 3 of the game, and that the equilibrium price of a manufacturer is decreasing in its share of the per-user benefit and in the developer fee, or ∂ pk /∂β k < 0 and ∂ pk /∂a ≤ 0. We further impose that |∂ pk /∂β k | < r. 22,23 Let πk (

β k , β ℓ , a) = π k (β k , pk (β k , β ℓ , a), pℓ (β ℓ , β k , a), a) denote M k 's
profit at the equilibrium of the subgame starting at stage 3. From the assumptions made above, we obtain the following: (i) ∂ πk ∂a (β k , β ℓ , a) ≤ 0 for all (β k , β ℓ , a) because an increase in the developer fee reduces the number of applications and acts thus as a negative shock on the demands faced by manufacturers; (ii) a manufacturer's profit increases with the share of the per-user benefit it receives, or ∂ πk ∂β k (β k , β ℓ , a) > 0 for all (β k , β ℓ , a). Roughly speaking, the assumptions on the manufacturers subgame ensure that the direct shift in their profit functions (associated to a change in the sharing parameter or in the developer fee) is stronger than the indirect shift in the marginal profit, which in turn changes the equilibrium between manufacturers. 24

Separation

The situation of 'separation,' in which none of the manufacturers are integrated with either platforms, serves as our benchmark to assess the impact of vertical integration. 25 Because several operating systems are available to manufacturers, there may exist several Nash equilibria in the subgame of choice of operating systems by the manufacturers (stage 2 of our game). We want to avoid situations where a platform obtains some unduly market power thanks to the mere lack of coordination between manufacturers. To that end, we impose the following selection on the equilibrium set: If there exists several Nash equilibria in the subgame starting at stage 2, we select the one that maximizes the manufacturers' joint profit. This is a mild yet meaningful restriction commonly found in 20 This holds when 0 [START_REF] Seade | The Stability of Cournot Revisited[END_REF] and [START_REF] Dixit | Comparative Statics for Oligopoly[END_REF]. 22 This is the equivalent of the usual assumption that cost pass-throughs are smaller than one. It can be linked to the log-curvature of demand functions as discussed in [START_REF] Weyl | Pass-Through as an Economic Tool: Principles of Incidence under Imperfect Competition[END_REF] and [START_REF] Ritz | The Simple Economics of Asymmetric Cost Pass-Through[END_REF] for instance.

< ∂ 2 π k ∂p k ∂p ℓ < -∂ 2 π k ∂p 2 k for all (β k , p k , p ℓ , a). See
21 Observe that ∂R k ∂a < 0 amounts to ∂ 2 π k ∂p k ∂a = ∂D k ∂a + D k ∂ 2 D k ∂p k ∂a (-∂D k ∂p k ) -1 < 0. Hence, R k decreases with a if either ∂ 2 D k ∂p k ∂a < 0 or if ∂ 2 D k ∂p k ∂a > 0 but small enough.
23 All these assumptions are satisfied in our running example. 24 Although intuitive, these properties may not always hold. In a Cournot oligopoly, Seade (1985), [START_REF] Kimmel | Effects of Cost Changes on Oligopolists' Profits[END_REF] and [START_REF] Linnemer | Backward Integration by a Dominant Firm[END_REF], among others, find conditions under which an increase in the marginal cost of several firms increases or decreases equilibrium profits; see [START_REF] Février | Idiosyncratic Shocks in an Asymmetric Cournot Oligopoly[END_REF] for a unifying framework. [START_REF] Cowan | Demand Shifts and Imperfect Competition[END_REF] extends the analysis to demand shocks. See also [START_REF] Dixit | Comparative Statics for Oligopoly[END_REF] and [START_REF] Leahy | Public Policy Towards R&D in Oligopolistic Industries[END_REF] for the case of Bertrand oligopolies. 25 The social optimum and Ramsey benchmarks are qualitatively similar to those derived in Section 2 and available in Appendix A.2. the literature. 26 Then, we obtain the following result.

Proposition 4. In equilibrium with competitive platforms, developers pay no fee (a S = 0), manufacturers obtain all the per-user benefit (β S = 1), platforms make no profits and manufacturers are indifferent between any of the platforms' operating systems.

Proof. See Appendix A.3.

To provide some intuition, suppose all platforms set β 1 = β 2 = 1 and a nil developer fee. Platforms make no profit, developers publish their applications on all platforms and manufacturers are indifferent between all the operating systems. Consider now a deviation by, say, platform I, which sets a I < 0 and β I < 1 such that its profit remains nil. Even though such deviation could be profitable for manufacturers, each manufacturer individually has the incentives to choose a platform from the fringe. Indeed, the developer can costlessly port its applications on platforms from the fringe while still enjoying the subsidy offered by platform I; even if a manufacturer chooses a platform from the fringe, it will still benefit from the same number of applications. And platforms from the fringe offer a higher share of the per-user benefit, which attracts each manufacturer individually. Put differently, a platform can freeride on the subsidy offered to the developer by another platform.

Vertical Integration

Consider now that platform I is integrated with manufacturer M 1 . To streamline the exposition, we directly consider that competition between non-integrated platforms E 1 , ..., E N leads them to set a nil developer fee and a sharing parameter equal to 1, a result that follows from the logic of Proposition 4. Our analysis proceeds as follows. First, we show that vertical integration creates market power over developers. Second, we study how such market power is exercised and its consequences on welfare.

Price Competition and Choice of Operating System. At stage 3 of the game, the integrated platform and the non-integrated manufacturer compete in prices to sell their devices to buyers. The outcome of that price competition subgame depends on whether the non-integrated manufacturer chooses one the non-integrated platform's operating system (in the following, we will refer to this case by saying that M 2 chooses 'platform E') or that of the integrated platform: (E). When M 2 chooses E's operating system, its profit is (p 2 + r)D 2 and the integrated platform's profit is

(3.4) (p 1 + r)D 1 + a I D S
because developers are willing to reach buyers of the integrated manufacturer's device.

26 To see its purpose in our setting, suppose that platform I sets (β I = 1, a I > 0) and platforms from the fringe set (β E = 1, a E = 0). There are several Nash equilibria in the subgame starting at stage 2, one in which both manufacturers choose I's operating system (because if a manufacturer deviates, the developer still pays a I to reach buyers of the other manufacturer's device), another one in which each manufacturer adopts the operating system of a platform from the fringe. Clearly, the former is Pareto-dominated (for manufacturers) by any of the latter.

(I). When M 2 chooses the integrated platform's operating system, its profit writes now as (p 2 + β I r)D 2 and that of the integrated platform is given by

(3.5) (p 1 + r)D 1 + a I D S + (1 -β I )rD 2
because it perceives some per-user benefit from the non-integrated manufacturer on top of the revenues earned from developers.

In case (E) (resp. case (I)), price competition on the product market then leads to equilibrium prices denoted by p E 1 (1, 1, a I ) andp E 2 (1, 1, a

I ) (resp. p I 1 (1, β I , a I ) and p I 2 (β I , 1, a I )). 27 Let π E 1 (1, 1, a I ) = (p E 1 + r)D 1 (p E 1 , p E 2 , a I ) + a I D S (p E 1 , p E 2 , a I ) and π E 2 (1, 1, a I ) = (p E 2 + r)D 2 (p E 2 , p E 1 , a I ) (resp. π I 1 (1, β I , a I ) = (p I 1 + r)D 1 (p I 1 , p I 2 , a I ) + (1 -β I )rD 2 (p I 2 , p I 1 , a I ) + a I D S (p I 1 , p I 2 , a I ) and π I 2 (β I , 1, a I ) = (p I 2 + β I r)D 2 (p I 2 , p I 1 , a I )) denote profits corresponding to case (E) (resp. (I)).
Pricing incentives of the integrated firm depend on whether or not it licenses its operating system to the non-integrated manufacturer. Indeed, when this is the case, increasing the price of device 1 increases the demand for device 2 since manufacturers' products are demand substitutes, and, therefore, increases the revenues (1 -β I )rD 2 generated by the licensing of the operating system.28 This is the so-called 'accommodation effect' found in the literature on strategic vertical integration. This effect suggests that the non-integrated manufacturer may be willing to accept a sharing parameter smaller than 1 to make the integrated manufacturer a softer competitor. Equivalently, the integrated platform has some market power over the non-integrated manufacturer.

That market power is, however, constrained by the competitive pressure exerted by platforms from the fringe. Indeed, the non-integrated manufacturer always has the option to choose the fringe's operating system. Therefore, the non-integrated manufacturer adopts the integrated platform's operating system if

(3.6) π I 2 (β I , 1, a I ) ≥ π E 2 (1, 1, a I ).
The Integrated Platform's Pricing Policy. At the first stage of the game, the integrated platform's profit writes as

π I 1 (1, β I , a I ) = (p I 1 + r)D 1 (p I 1 , p I 2 , a I ) + (1 - β I )rD 2 (p I 2 , p I 1 , a I ) + a I D S (p I 1 , p I 2 , a I ). Formally, β I and a I are solution of 29 (3.7) max (β I ,a I ) π I 1 (1, β I , a I ) s.t. π I 2 (β I , 1, a I ) ≥ π E 2 (1, 1, a I ), 0 ≤ β I ≤ 1.
To study this problem, we proceed in two steps. First, we consider a relaxed problem in which none of the constraints are taken into account. This allows to understand the logic underlying the exercise of market power by the integrated platform. Second, we solve for the constrained problem. 30Maximizing the integrated firm's profit requires to extract revenues from developers and from the non-integrated manufacturer. This calls for increasing a I and decreasing β I , while keeping an eye on how this impacts competition on the buyer's market. But it also requires to take advantage of network effects across both sides of the market. When, for instance, buyers value strongly the applications offered by developers, the integrated platform wants to boost the number of applications available on its device with a low developer fee and extract some of the buyer surplus through a low sharing parameter asked to the non-integrated manufacturer.

Let us assume that the solution of this relaxed problem is interior and denote by (β * I , a * I ) the solution of the system formed by the two first-order conditions ∂π I 1 /∂β I = 0 and ∂π I 1 /∂a I = 0. In Appendix A.4, we characterize this solution for the running example. Figure 4 below represents the two curves β * I = 1 and a * I = 0 in the (u B , u S )-space to allow a comparison with the separation benchmark.

Three broad cases must be considered:

-'Strong and buyer-skewed network effects.' This corresponds to the green region in Figure 4, in which buyers' valuation for number of applications is larger than developers' valuation for the number of buyers (that is, u B > u S ). There, the solution of the relaxed problem calls for setting a negative developer fee (that is, a * I < 0) to boost the participation of developers and extracting the buyer surplus thereby created with a low sharing parameter (that is, β * I < 1).

-'Strong and developer-skewed network effects.' This corresponds to the red region in Figure 4, in which u S ≫ u B . There, the solution of the relaxed problem leads to a high sharing parameter (that is, β * I > 1) to boost the participation of buyers and a high developer fee to extract the developer surplus (that is, a * I > 0).

-In the blue region, network effects are rather balanced and weak across both sides of the market. There, the integrated platform sets a positive developer fee (that is, a * I > 0) and a low sharing parameter (that is, β * I < 1).

The comparison between the outcome under separation (a S = 0, β S = 1) and the optimal pricing policy obtained in the relaxed problem illustrates a central feature of our model. The way that the integrated firm exercises its market power on developers and on the non-integrated manufacturer depends on the strength and the structure of network effects across both sides of the market. In a nutshell, when network effects are strong and asymmetric, a two-sided market logic is at work: the integrated firm implements an asymmetric pricing structure, subsidizing one side and taxing the other, to harness those network effects. When network effects are weak and balanced, a one-sided market logic is at work: the integrated platform exercises its market power by raising the developer fee and decreasing the sharing parameter.

Let us now come back to the platform's constrained problem as defined in (3.7) and denote by (β * * I , a * * I ) its solution. With respect to the relaxed problem, the first constraint is that the sharing parameter must be positive and smaller than 1. The second constraint is the participation constraint of the non-integrated manufacturer, namely (3.6). Observe that the non-integrated manufacturer's profit if it buys the fringe's operating system, namely π E 2 (1, 1, a I ), coincides with its profit if it chooses the integrated platform's operating system for a sharing parameter β I = 1: this holds because there is no accommodation effect when the integrated platform gives up all the per-user benefit. Therefore,

u B 0 u S u B = u S σ = 0 β * I = 1 a * I = 0 β * I >1 a * I >0 β * I <1 a * I >0 β * I <1 a * I <0
π E 2 (1, 1, a I ) = π I 2 (1, 1, a I ), which implies that 31 (3.8) π I 2 (β I , 1, a I ) ≥ π E 2 (1, 1, a I ) ⇔ β I ≥ 1.
In words, the competitive pressure exerted by the fringe prevents the integrated platform from exerting any market power on the non-integrated manufacturer, or β * * I = 1. This illustrates another standard result from the literature on strategic integration: absent efficiency gains, vertical integration does not create market power over non-integrated manufacturers.

The integrated platform still has some market power over the developer who wants to access the buyers of its device. The optimal developer fee maximizes 32 π I 1 (a

I ) = (p I 1 (a I ) + r)D 1 (p I 1 (a I ), p I 2 (a I ), a I ) + a I D S (p I 1 (a I ), p I 2 (a I ), a I )
, and is given by the first-order condition dπ I 1 (a * * I )/da I = 0, or (3.9)

D S + a I ∂D S ∂a I + ∂D S ∂p 2 dp I 2 da I + (p I 1 + r) ∂D 1 ∂a I + ∂D 1 ∂p 2 dp I 2 da I a I =a * * I = 0.
Increasing the developer fee has both a direct impact on the number of applications and the demand for the integrated manufacturer's device, and an indirect impact through the strategic effect on the non-integrated manufacturer's price. Remember that, under our assumptions, dp I k /da I ≤ 0. Hence, Equation (3.9) shows that increasing the developer fee allows to capture revenues from developers (first bracketed term) but depreciates the profit earned from buyers (second bracketed term). The first effect calls for increasing the developer fee above the marginal cost, but the second one calls for decreasing it. The integrated platform may still be willing to subsidize the developer if this boosts sufficiently the demand for its own device.

Proposition 5. The integrated platform's optimal pricing policy is as follows:

-All the per-user benefits is left to the non-integrated manufacturer:

β * * I = 1;
-The developer fee is the solution a * * I of (3.9).

In the running example, a * * I > 0 if and only if u B < h(u S ) (with h(u S ) > u S for all u S > 0).

Proof. See Appendix A.5.

Figure 5 represents graphically Proposition 5. The integrated platform boosts the number of applications with a negative developer fee only when network effects are sufficiently strong and buyer-skewed. Otherwise, it sets a positive developer fee. Indeed, with respect to the relaxed problem, the competitive pressure exerted by platforms from the fringe forces the integrated platform to give up all the per-user benefits to the nonintegrated manufacturer. It is therefore less profitable to subsidize developers because such a loss cannot be recouped with benefits earned from the licensing of its operating system.

u B 0 u S u B = u S σ = 0 a * * I >0 β * * I =1 a * * I <0 β * * I =1 uB = h ( uS )
Figure 5: The integrated platform's optimal pricing policy (β * * I , a * * I ) in the running example.

Competitive Impact of Vertical Integration. We now assess the welfare impact of vertical integration. Since the integration outcome coincides with that under separation when a I = 0, we only need to study how the non-integrated manufacturer's profit π I 2 (a I ) = (p I 2 (a I )+r)D 2 (p I 2 (a I ), p I 1 (a I ), a I ), the buyer surplus V B (p I 1 (a I ), p I 2 (a I ), D I S (p I 1 (a I ), p I 2 (a I ), a I )), and the developer surplus V S (a I , D 1 (p I 1 (a I ), p I 2 (a I ), a I ) + D 2 (p I 2 (a I ), p I 1 (a I ), a I )) vary with the developer fee a I .

To study whether vertical integration leads to foreclosure of the non-integrated manufacturer, we can differentiate the non-integrated manufacturer's profit with respect to the developer fee to get

dπ I 2 da I (a I ) = (p I 2 + r) ∂D 2 ∂a I + ∂D 2 ∂p 1 dp I 1 da I < 0.
Since prices of devices decrease with the developer fee under our assumptions, we obtain immediately the next proposition.

Proposition 6. Vertical integration creates foreclosure if and only if the developer fee increases above the pre-merger level. Therefore, in the running example, foreclosure arises if and only if u S > h(u B ).

Proof. Immediate from the text.

This foreclosure effect is different from the one found in the literature for two reasons.

First, it does not stem from a 'raise the rival's cost' effect. In our setting, foreclosure is a collateral damage of the integrated platform's market power on developers, but not the result of its desire to soften competition on the buyer's market. Second, foreclosure is not systematic. When network effects are strong and buyerskewed, Proposition 5 has shown that the integrated firm lowers the developer fee (with respect to the pre-merger level), which increases the non-integrated manufacturer's profit. A reverse result obtains when the integrated firm raises the developer fee above the premerger level.

Consider now the impact of vertical integration on buyer and developer surpluses. The analysis is slightly more involved since these surpluses are intertwined through indirect network effects. We consider a small variation of the developer fee around its value under separation (that is, 0). Simple manipulations lead to (omitting some arguments)

dV B da I a I =0 = -Q B dp I 1 da I + dp I 2 da I + ∂U B ∂n S dD S da I , (3.10) dD S da I a I =0 = 1 1 -2 ∂Q B ∂n S ∂Q S ∂n B ∂Q S ∂n B ∂Q B ∂p k + ∂Q B ∂p ℓ dp I 1 da I + dp I 2 da I + ∂Q S ∂a . (3.11)
Equation (3.11) describes how the number of applications varies when the developer fee charged by the integrated platform increases. First, since publishing applications becomes more costly, developers are less willing to participate; this corresponds to the term ∂Q S /∂a < 0. Second, the prices paid by buyers decrease, so that there are more buyers overall, which benefits developers through indirect network effects; this corresponds to the term (∂Q B /∂p k + ∂Q B /∂p ℓ )(dp I 1 /da I + dp I 2 /da I ) > 0. Equation (3.10) describes how the surplus of buyers varies when the developer fee increases. There are two effects again. First, the prices paid by buyers decrease, which boosts the demand from those buyers (this corresponds to the first term in the righthand side). Second, fewer or more applications are developed, which impacts negatively or positively buyers through indirect network effects. This suggests that the impact of vertical integration on buyers and on developers is a priori ambiguous. For instance, taxing developers with a positive fee may actually improve the surpluses of buyers and developers, if doing so sufficiently reduces the prices of devices and boosts the number of buyers. Next proposition provides a complete characterization of the impact of vertical integration on buyer and developer surpluses in our running example.

Proposition 7. In the running example, vertical integration

-increases buyer surplus if either u B ≥ h(u S ) or u S ≥ h B (u B ); -increases developer surplus if and only if either u B ≥ h(u S ) or u S ≥ h S (u B ). 33
Proof. See Appendix A.6.

u B 0 u S u B = u S σ = 0 h(u S ) h B (u B ) h S (u B ) π 2 ↗ V B ↗ V S ↗ π 2 ↘ V B ↗ V S ↘ π 2 ↘ V B ↘ V S ↘ π 2 ↘ V B ↗ V S ↗
Figure 6: Impact of vertical integration on the non-integrated manufacturer's profit (π 2 ), buyer surplus (V B ), and developer surplus (V S ) in the running example.

Several comments are worth making. First, and quite remarkably, vertical integration can improve buyer and developer surpluses simultaneously even in the absence of efficiency gains. The intuition is that, when indirect network effects are sufficiently strong and asymmetric, the integrated firm's market power over developers leads to an asymmetric pricing structure that better internalizes network effects. The price structure is thus closer to the one that would be socially desirable, that is, prices under vertical integration are closer to their Ramsey counterparts than under separation. 34More precisely, with buyer-skewed network effects (u B > h(u S )), the integrated platform subsidizes developers (see Proposition 5). Although the prices of devices increase, buyers benefit from an increase in the number of applications, which explains that their surplus increases following the merger. By contrast, with developer-skewed network effects (u S > h B (u S )), the integrated platform taxes developers, which tends to reduce the number of applications. However, the prices of devices decrease and buyer surplus increases following the merger. In these two cases of strongly-skewed indirect network effects, which correspond to the green and orange regions in Figure 6, the surpluses of buyers and developers increase because the integrated firm implements an asymmetric pricing structure that is more in line with the Ramsey optimum.

Second, the impact of vertical integration on foreclosure is now disconnected from its impact on buyers or on developers. We already know from Proposition 6 that foreclosure is related to the developer fee chosen by the integrated platform. Proposition 7 shows that what matters for buyer and developer surpluses is the asymmetry between network effects. Figure 6 illustrates that buyers and developers gain from the vertical merger when network effects are either strongly buyer-skewed (u B > h(u S )), a region where the developer fee decreases, or strongly developer-skewed (u S > h B (u S ), a region where the developer fee increases.

Third, when network effects become smaller or more symmetric, there is less value to create through an asymmetric pricing structure and the integrated firm's market power is more likely to be detrimental. In Appendix A.6, we show that, when u B < h(u S ) and u S < h B (u S ), the buyer surplus may either increase or decrease. 35Our analysis provides therefore a new efficiency defense for vertical integration in platform markets. Vertical integration creates market power on the developer side of the market. That market power is used by the integrated firm to extract more profit from developers and from the non-integrated manufacturer; a potentially harmful effect for buyers. In a two-sided market, however, that market power is also used to internalize indirect network effects between the two sides of the market; a potentially beneficial effect for buyers.

Although consumer surplus seems to be the standard pursued by antitrust authorities, it is also interesting to briefly look at total welfare. Figure 7 computes total welfare for a particular specification of our running example. 36 It shows that welfare also improves following vertical integration when network effects are sufficiently strong and asymmetric.

These simulations can be used to study the role of parameter γ that describes the degree of substitutability between manufacturers' products on the downstream market (Figure 8). Intuitively, as γ increases, products become more substitutes and price competition between manufacturers intensifies. As a result, prices become more rigid and cost-based. For the integrated platform, this implies that internalizing network effects through an asymmetric price structure becomes less interesting: there is no point in subsidizing developers if the price charged to buyers cannot be raised. Therefore, as the simulations reported in Figure 8 suggest, vertical integration becomes more harmful to buyers when manufacturers' products are more demand substitutes. Both network effects and the intensity of competition between manufacturers matter to determine whether vertical integration benefits or hurts buyers.

Efficiency Gains

We now study vertical mergers that bring efficiency gains. Efficiency gains are modeled as follows: following the merger, the per-buyer benefit associated to the integrated platform's operating system becomes r 0 > r. Put differently, the integrated platform I is better able to create value from user-generated data obtained from the manufacturers that use its operating system. 37 The separation benchmark is unchanged (see Section 3.4).

From now on, we consider that platform I is integrated with manufacturer M 1 . Under vertical integration, we can still apply the logic of Lemma 4 to show that, in equilibrium, platforms from the fringe set β E = 1 and a E = 0.

As in Section 3.5, pricing incentives at stage 3 of the game depend on the choice of operating system by the non-integrated firm. To track changes in the per-user benefit, we adopt now the following notations. When the non-integrated manufacturer adopts the fringe's operating system, M 2 obtains a net per-user benefit of 1 • r. Let (p E 1 (r 0 , r, a I ), p E 2 (r, r 0 , a I )) and (π E 1 (r 0 , r, a I ), π E 2 (r, r 0 , a I )) be the prices of devices and the profits in that case. When the non-integrated manufacturer adopts the integrated firm's operating system, M 2 obtains a net per-user benefit of β I •r 0 . Let (p I 1 (r 0 , β I r 0 , a I ), p I 2 (β I r 0 , r 0 , a I )) and (π I 1 (r 0 , β I r 0 , a I ), π I 2 (β I r 0 , r 0 , a I )) be the prices of devices and the profits in that case. For future reference, we have in particular

π I 1 (r 0 , β I r 0 , a I ) = (p I 1 + r 0 )D 1 (p I 1 , p I 2 , a I ) + (1 -β I )r 0 D 2 (p I 2 , p I 1 , a I ) + a I D S (p I 1 , p I 2 , a I ), π I 2 (β I r 0 , r 0 , a I ) = (p I 2 + β I r 0 )D 2 (p I 2 , p I 1 , a I ) and π E 2 (r, r 0 , a I ) = (p E 2 + r)D 2 (p E 2 , p E 1 , a I
). The roadmap of our analysis is as follows. First, we show that efficiency gains provide the integrated firm with some market power over the non-integrated manufacturer (Section 4.1). Second, we study the way such market market power is exercised (Section 4.2) and show that it is not necessarily detrimental either to buyers and developers or to the non-integrated manufacturer (Sections 4.3 and 4.4).

Efficiency Gains Create Market Power

Efficiency gains create market power vis-à-vis the non-integrated manufacturer. To understand why, observe that, at stage 3 of the game, the integrated platform's profit when it supplies the non-integrated manufacturer writes now as

(4.1) (p 1 + r 0 )D 1 + (1 -β I )r 0 D 2 + a I D S .
Comparing the integrated platform's profit with efficiency gain (Equation (4.1)) and without (Equation (3.5)) shows that, with efficiency gains, even when the integrated platform provides the same value as the fringe in terms of per-user benefits, that is, when β I r 0 = r, it earns some strictly positive profit from licensing its operating system, namely (1 -β I )r 0 D 2 = (r 0 -r)D 2 > 0. Hence, even when both operating systems are licensed on the same terms, the non-integrated manufacturer now strictly prefers adopting the integrated firm's operating system because this makes that firm more accommodating on the buyers market: π I 2 (β I r 0 = r, r 0 , a I ) > π E 2 (r, r 0 , a I ). This implies that the integrated platform is now able to license its operating system 37 We could have assumed that synergies between platform I and manufacturer M 1 decrease the (marginal) cost of I's operating system. Those two formulations are equivalent in our model. against a sharing parameter strictly smaller than the fringe's. Put differently, efficiency gains create market power over the non-integrated manufacturer. The integrated firm's market power remains constrained by the fringe's behavior, though, and the following participation constraint must be satisfied

(4.2) π I 2 (β I r 0 , r 0 , a I ) ≥ π E 2 (r, r 0 , a I ).
To relate with the case of no efficiency gains, the participation constraint (4.2) rewrites as follows in the running example

(4.3) β I ≥ β I (r 0 , r) ≡ 1 - r 0 -r r 8 + σ(8 + σ) 8(1 + σ) .
Comparing with (3.8) shows that as soon as the merger creates efficiency gains (r 0 > r), it also empowers the integrated platform with some market power over the non-integrated manufacturer (that is, β I (r 0 , r) < 1). The next step consists in analyzing how the integrated platform uses its market power over the developer and the non-integrated manufacturer.

The Integrated Firm's Pricing Policy

The integrated platform's problem can be written as follows

(4.4) max (β I ,a I ) π I 1 (r 0 , β I r 0 , a I ) s.t. π I 2 (β I r 0 , r 0 , a I ) ≥ π E 2 (r, r 0 , a I ), 0 ≤ β I ≤ 1.
It is again useful to introduce the solution of the relaxed problem (that is, when none of the constraints in the above problem are taken into account), which we denote by (β * I (r 0 ), a * I (r 0 )). Up to the fact that the integrated firm's sharing parameter is now r 0 , the outcome of the relaxed problem can be represented in a similar way as in Figure 4.

Consider now the constrained problem (4.4) in our running example. In Appendix A.7, we show that the main features of the solution to that problem, denoted by (β * * I (r 0 ), a * * I (r 0 )), can be summarized with Figure 9. There are three regions of interest depending on which constraints are binding.

Suppose that network effects are buyer-skewed (green region below the 45 • -line in Figure 9). In that case, absent any constraints the integrated platform would like to set a low sharing parameter and a negative developer fee (β * I (r 0 ) < 1 and a * I (r 0 ) < 0). With respect to Section 3.5, setting a sharing parameter below the pre-merger level is now feasible thanks to the efficiency gains that create upstream market power. Hence, with buyer-skewed network effects, we expect that the non-integrated manufacturer's participation constraint becomes binding and that the integrated platform charges a sharing parameter below the pre-merger level (that is, β * * I (r 0 ) = β I (r 0 , r) < 1). The integrated platform may also subsidize developers if network effects are sufficiently asymmetric.

Suppose now that indirect network effects are sufficiently developer-skewed (red region in Figure 9). The integrated firm then wants to subsidize buyers with a high sharing parameter and tax developers with a positive fee. Hence, we expect that the integrated firm gives up all the per-user benefit to the non-integrated manufacturer and taxes developers

u B 0 u S u B = u S σ = 0 β * I = 1 β * I r 0 = r β * I = β I β * * I =1 a * * I >0 β * * I =β * I <1 a * * I =a * I >0 β * * I =β I <1 a * * I >0 β * * I =β I <1 a * * I <0
Figure 9: The integrated platform's optimal pricing policy (β * * I (r 0 ), a * * I (r 0 )) in the running example with efficiency gains.

(that is, β * * I (r 0 ) > 1 and a * * I (r 0 ) > 0). Last, it is also possible that none of the constraints are binding (blue region in Figure 9), in which case the solution of the integrated platform's problem is actually the solution of the relaxed problem. This arises with moderately developer-skewed network effects. In that case, with respect to the pre-merger outcome, the integrated firm moderately decreases the sharing parameter but taxes developers.

Competitive Impact of Vertical Integration with Efficiency Gains: Polar Cases

With efficiency gains, the impact of vertical integration on buyers, developers and the non-integrated manufacturer is less straightforward to assess because both the developer fee and the sharing parameter change with respect to the separation benchmark. For instance, when the developer fee increases beyond its pre-merger level, this does not necessarily imply foreclosure, for the sharing parameter could be lowered leading to a net gain for the non-integrated manufacturer. We explore this intuition by analyzing two polar cases.

Consider first that developers do not value the participation of buyers (that is, u S = 0 in our running example). Without efficiency gains, buyers and developers, as well as the non-integrated manufacturer, benefit from integration. With efficiency gains, the integrated platform decreases the sharing parameter below the pre-merger level (β * * I (r 0 ) < 1) and sets a negative fee for developers (a * * I (r 0 ) < 0). Since developers do not value the participation of buyers, they are better off following integration because their participation is subsidized. Things are more complicated for the buyers and the non-integrated manufacturer. On the one hand, both benefit from the fact that the participation of developers is subsidized. On the other hand, the integrated platform exploits its com-petitive advantage and extract more from the non-integrated manufacturer. Intuitively, this latter effect prevails when network effects are weak overall, that is, when buyers also value weakly the participation of developers. Next proposition formalizes this intuition.

Proposition 8. Consider the running example and assume u S = 0. Following integration, -there is foreclosure if and only if u B is small enough; -buyers are better off if and only if u B is large enough; -developers are always better off.

Proof. See the Online Appendix.

Let us now consider another polar case in which buyers do not value the participation of developers (that is, u B = 0 in our running example). The integrated platform sets a developer fee above the pre-merger level (a * * I (r 0 ) > 0), as in the case without efficiency gains. The difference is that efficiency gains provide incentives to increase the sharing parameter. When u S is small (resp. large), the integrated platform sets a sharing parameter below (resp. beyond) the pre-merger level. Therefore, intuitively, not only buyers and developers but also the non-integrated manufacturer may benefit from integration when developers value strongly the participation of buyers. Next proposition formalizes this intuition.

Proposition 9. Consider the running example and assume u B = 0. Following integration, -there is foreclosure if and only if u S is either large enough or small enough; -buyers are better off if and only if u S is large enough; -developers are better off if and only if u S is large enough.

Proof. See the Online Appendix.

Competitive Impact of Vertical Integration with Efficiency Gains: Numerical Simulations

In this Section, we extend Propositions 8 and 9 using numerical simulations. 38 The results are depicted in Figures 10, 11 and 12, which represent the range of parameters u B and u S for which integration harms or benefits buyers, developers and the non-integrated manufacturer respectively.

Overall, the results of the simulations are in line with those of Section 3.5. First, there is no foreclosure when network effects are skewed toward buyers (see Figure 10). Second, buyers and developers benefit from integration when network effects are sufficiently asymmetric (see Figures 11 and12 respectively). Third, when network effects are rather balanced, buyers, developers and the non-integrated manufacturer all lose from the vertical merger (see Figures 10,11 Consider first that network effects are developer-skewed. As efficiency gains increase, Figure 10 suggests that foreclosure becomes less of an issue. This arises because the integrated platform sets a sharing parameter close or equal to 1. Put differently, efficiency gains are passed through almost entirely to the non-integrated manufacturer. This effect can be strong enough to compensate for the increase in the developer fee. This happens when efficiency gains are sufficiently large and when manufacturers' products are weak substitutes, that is when γ is small (Figures 10d, 10e and10f). In this case, competition on the downstream market is weak and the non-integrated manufacturer benefits fully from the high sharing parameter.

Second, the non-integrated manufacturer is foreclosed when network effects are balanced and weak, that is in the neighborhood of (u B , u S ) = (0, 0) (see Figure 10). This is reminiscent of the literature on the strategic effects of vertical integration: when vertical integration creates efficiency gains, the non-integrated rival is (partially) foreclosed (see, e.g., [START_REF] Chen | On Vertical Mergers and Their Competitive Effects[END_REF].

Third, buyers and developers are better off following integration when network effects are strong and asymmetric, that is when either u S is large and u B is small or the opposite (see Figures 11 and12). These are situations where a platform would like to internalize network effects through an asymmetric price structure. When it creates efficiency gains, vertical integration allows the platform to implement an even more asymmetric price structure because it now has some control over its sharing parameter. Buyers and developers benefit as well from such a more asymmetric price structure. The results from the simulations reinforce the main message of our analysis. A vertically integrated platform is empowered with some market power on users on both sides of the market and on the non-integrated manufacturer. However, because vertical integration allows for a better internalization of network effects, the exercise of this market power is not necessarily at the expense of the non-integrated firm. Buyers and developers may also benefit from integration.

Coordination Motives and Porting Costs

Coordination Motives. Although a quintessential feature of platform markets is the presence of indirect network effects linking various groups of agents, these markets also frequently exhibit direct network effects of paramount importance. For instance, operating systems often feature applications or software programs that aim to take advantage of direct network effects between buyers. 39 Hence, when more manufacturers adopt the same operating system, this creates extra benefits for buyers of devices using the same operating system.40 Ultimately, part of these benefits may end up being pocketed by manufacturers, which creates coordination motives between manufacturers. Our goal is to analyze how such coordination motives impact on our assessment of vertical integration.

To do so, we consider our model without efficiency gains (in which vertical integration does not create any market power over the non-integrated manufacturer) and modify it as follows. If manufacturers choose different operating systems, then quasi-demands are given as before by (3.2) and the analysis is the same as in Section 3.5. If manufacturers choose the same operating system, then quasi-demands are now given by41

(5.1)

   Q1 B = Q 1 B + α B , Q2 B = Q 2 B + α B , QS = Q S .
Parameter α B is positive and used as a shortcut to capture the magnitude of the extra gains for buyers when manufacturers adopt the same operating systems. Demands ( D1 , D2 , DS ) that solve (5.1) are clearly increasing in α B . Let us further assume that the profits of manufacturers associated with these demands, denoted by (π i 1 , πi 2 ) with i ∈ P, are also increasing in α B . 42The analysis under separation is immediate. In any equilibrium, manufacturers coordinate on one platform i ∈ P, obtain the whole per-user benefit, and the developer is charged a nil fee by platforms. Essentially, the addition of the extra benefit α B simply leads manufacturers to choose the same platform because there is now a coordination motive.

Consider now that platform I is vertically-integrated with manufacturer M 1 . If M 2 chooses platform E from the fringe, manufacturers equip their devices with different operating systems and there are no extra benefits. M 2 's profit is then given by π E 2 (1, 1, a I ). If, instead, M 2 chooses I's operating system, there are extra benefits and M 2 's profit is denoted by πI 2 (β I , 1, a I ). 43 We have πI 2 (1, 1, a I ) > π E 2 (1, 1, a I ) because, if platform I sets the same sharing parameter as the fringe, M 2 now strictly prefers I's operating system to take advantage of the extra benefits. Therefore, the integrated firm can now offer β I < 1 while still ensuring that M 2 adopts I's operating system. To illustrate, in the running example the participation constraint (3.8) becomes with coordination motives (5.2)

β I ≥ βI (α B ) ≡ 1 - α B r 4 + 3σ 4(1 + σ) .
Much as in the case of efficiency gains (see in particular Equation (4.3)), coordination motives create market power over the non-integrated manufacturer. That market power arises even though platforms are symmetric (i.e., there are no efficiency gains) because vertical integration somewhat forces the coordination of the manufacturers on the integrated firm's operating system.

The integrated platform's problem can be written as follows:

(5.3) max

(β I ,a I ) πI 1 (1, β I , a I ) s.t. πI 2 (β I , 1, a I ) ≥ π E 2 (1, 1, a I ), 0 ≤ β I ≤ 1.
This problem is quite similar to the one analyzed in Section 4 and its solution will feature the same main properties. For the running example, the complete resolution is provided in Appendix A.8. One noticeable difference is the following. Since vertical integration does not create any efficiency gains, the non-integrated manufacturer M 2 and, indirectly, the developers, do not benefit from a more advantageous sharing parameter when the integrated platform decides to subsidize buyers. This happens when network effects are developer-skewed and the integrated platform prefers to tax developers, which harms both M 2 and the developers. We therefore expect, contrary to Section 4, that when network effects are developer skewed, there is always foreclosure and developers are worse off.

We can confirm these two intuitions by studying two polar cases (Propositions 10 and 11) and by performing numerical simulations (Figures 13,14 and 15). 44Proposition 10. Consider the running example with coordination motives and assume u S = 0. Following integration, -there is foreclosure if and only if u B is small enough; -buyers are better off if and only if u B is large enough; -developers are always better off.

Proof. See the Online Appendix.

Proposition 11. Consider the running example with coordination motives and assume u B = 0. Following integration, -there is always foreclosure; -buyers are better off if and only if u S is large enough; -developers are always worse off.

Proof. See the Online Appendix.

Porting Costs. Throughout our analysis, we have maintained the assumption that there are no platform-specific cost for the developer. In practice, the programming languages used in Android and iOS strongly differ45 and it appears that Android application development is usually longer and more costly because of the fragmentation issue that impacts this operating system. We now discuss informally how platform-specific costs to port applications on operating systems impact on our analysis. Roughly speaking, we show that porting costs create a coordination motive for manufacturers. Figure 14: Impact of vertical integration on developer surplus (V S ) for different levels of the gain for buyers when manufacturers adopt the same operating system (α B ): V S decreases (resp. increases) following integration in the red area (resp. the blue area).

Let us assume now that the developer bears a unit cost c i > 0 to port each application on platform i's operating system. To streamline the analysis, let us further assume that c i = c for all i ∈ P. At the last stage of the game, the developer decides to publish its applications on platform i if u S n i B ≥ a i + c. Therefore, the number of applications developed is now given by Q

S ( i∈P (u S n i B -(a i + c))1 {u S n i B -(a i +c)≥0}
). Last, assume that c is sufficiently small so that u S n i B > c when n i B > 0; this ensures that all applications are published on any platform i that has attracted at least one manufacturer and that sets a nil developer fee. 46 Consider that I and M 1 are integrated. If M 2 chooses E's operating system, the number of applications is given by (5.4)

Q E S u S (n 1 B + n 2 B ) - i=I,E (a i + c)1 {u S n i B -(a i +c)≥0} .
46 When c becomes sufficiently large, and when manufacturers have chosen different operating systems, the developer might find it profitable to publish on only one operating system, i.e., to single-home rather than to multi-home. for different levels of the gain for buyers when manufacturers adopt the same operating system (α B ): π 2 decreases (resp. increases) following integration in the red area (resp. the blue area).

If M 2 chooses instead I's operating system, the number of applications is then given by (5.5)

Q I S u S (n 1 B + n 2 B ) -(a I + c)1 {u S n I B -(a I +c)≥0} .
Comparing (5.4) and (5.5) immediately leads to Q I S > Q E S : With porting costs, choosing the integrated platform's operating system leads to more applications because it saves on the developer's cost. This therefore leads to higher demands for both devices. Hence, porting costs create a coordination motive for the non-integrated manufacturer. Put differently, and using notations that should be familiar by now, we have now π I 2 (1, 1, a I ) > π E

2 (1, 1, a I ) and the integrated platform is thus empowered with some market power over the non-integrated manufacturer. To illustrate further, considering the running example and assuming that platforms from the fringe offer (β E = 1, a E = 0), the participation constraint (3.8) becomes with porting costs

β I ≥ 1 - c u B r 4 + 3σ 4(1 + σ) .
Given the analysis above (see in particular equations (4.3) and (5.2)), we expect that such market power may not always be detrimental to welfare. 47

Conclusion

We develop a model of a platform market, in which platforms interact with manufacturers of devices and there are indirect network effects between buyers of devices and 47 Porting costs also introduce a novelty, which is best seen by considering the case of separation. If non-integrated manufacturers choose the same platform, say I, setting a nil developer fee (i.e., a E = 0) no longer ensures that the developer publishes on E's operating system because of the porting costs. This means that platform I can now set any developer fee such a I + c ≥ 0 without being threaten that a rival platform attracts the developer with a lower developer fee. Although platforms still compete fiercely and should make no profits in equilibrium, some partial cross-subsidization between both sides of the market becomes possible, even under separation. A complete characterization of the equilibrium outcome under separation and integration with porting costs is left for future research.

developers of applications. We study the consequences of vertical integration between one of the platforms and one of the manufacturers.

The sources of upstream market power, and their consequences on foreclosure or on consumer surplus, are different from those unveiled in the extant literature. Even absent any efficiency gains, vertical integration creates market power over developers who want to access the buyers of the integrated manufacturer's device. With efficiency gains, coordination motives or porting costs, vertical integration creates additionally some market power over the non-integrated manufacturer. However, what is key is how the integrated firm exploits these sources of market power. Our analysis unveils that this depends both on the strength and on the structure of indirect network effects. When network effects are strong in level but also sufficiently asymmetric in structure, the integrated firm implements an asymmetric pricing structure, which may well improve buyer and developer surpluses as well as the non-integrated manufacturer's profit. Our analysis therefore warns policy-makers against a blind application of the standard view on foreclosure when dealing with platform markets.

As in standard markets, antitrust authorities may want to limit the anti-competitive effects of vertical integration by constraining the pricing instruments between the integrated platform and non-integrated manufacturers. In the context of platform markets, such remedy raises some issues. For instance, constraining the integrated firm's sharing parameter is likely to impact the pricing on the developer side of the market, thereby dampening the internalization of network effects.

Our setting could be extended in various directions. First, throughout the analysis, we have maintained the assumption that manufacturers' products remain demand substitutes. With (imperfect) demand complements, the integrated platform should have less incentives to foreclose the non-integrated manufacturer. Second, and relatedly, analyzing the incentives of both manufacturers and platforms to differentiate their products, and the consequences on the assessment of vertical integration, would be worth investigating. Third, there is always multi-homing on the developer side and single-homing on the buyer side of the market in our analysis. Different patterns may be more relevant depending on the platform markets under consideration and this is likely to impact on the assessment of vertical integration. These extensions are left for future research.

A. Appendix

A.1. Monopoly Platform and Monopoly Manufacturer

This Appendix is devoted to the analysis of the case with a monopoly platform and a monopoly manufacturer.

Demand Functions. At the last stage of the game, given prices (p, a), the number of buyers and the number of developers solve

(A.1) n B = Q B (p, n S ), n S = Q S (u S n B -a).
In the following, with a slight abuse of notations, let ∂Q S ∂a (a, n B ) = -Q ′ S (u S n B -a) and

∂Q S ∂n B (a, n B ) = u S Q ′ S (u S n B -a).
To avoid 'cornered-market' solutions, in which all buyers or all developers participate in equilibrium, we assume that indirect network effects are not too strong so that, in the relevant range, each manufacturer faces a demand that is locally elastic with respect to prices.

Assumption A.1. In the relevant range of (p, a), (n B , n S ) satisfy ∂Q S ∂n B (a, n B ) ∂Q B ∂n S (p, n S ) < 1.

Assumption A.1 is maintained throughout this section. We can then show the next result.

Lemma A.1. System (A.1) has a unique interior solution.

Proof. From system (A.12), we have

(A.2) D B (p, a) = Q B (p, Q S (u S D B (p, a) -a)).
For a given (p, a), D B (p, a) is thus a fixed point of ψ We can then show the following result.

(x) = Q B (p, Q S (u S x -a)). Notice then that ψ ′ (x) = ∂Q S ∂x (a, x) ∂Q B ∂n S (p, Q S (u S x -a)). Assumption A.1 implies that |ψ ′ (•)| < 1, so that ψ(•)
Lemma A.2. ∂D B ∂p (p, a) < 0, ∂D S ∂p (p, a) < 0, ∂D B ∂a (p, a) < 0 and ∂D S ∂a (p, a) < 0.

Proof. By the implicit function theorem, D B (p, a) is continuously differentiable. Differentiating Equation (A.2) with respect to p and rearranging terms, we find (omitting some arguments)

∂D B ∂p 1 - ∂Q S ∂n B (a, D B ) ∂Q B ∂n S (p, D B ) = ∂Q B ∂p (p, D B ).
By Assumption A.1, the term in squared brackets is positive. Therefore, ∂D B /∂p is negative.

Since ∂Q S /∂n B > 0, D S (p, a) = Q S (u S D B (p, a) -a) is decreasing in p.
From Equation (A.1), the developer demand solves At the social optimum, (n B , n S ) solve max (n B ,n S ) W (P (n B , n S ), A(n B , n S )) = U B (n B , n S )-P (n B , n S )n B +U S (n B , n S )-A(n B , n S )n S +(P (n B , n S )+r)n B +A(n B , n S )n S . Assume an interior optimum. Using the fact that ∂U B ∂q = P and ∂U S ∂q = A, the first-order conditions wrt n B and n S yield respectively P + r + ∂U S ∂n B = 0 and A + ∂U B ∂n S = 0. These two equations define the socially optimal prices (p * , a * ).

At the Ramsey optimum, (n B , n S ) solve max (n B ,n S ) W (P (n B , n S ), A(n B , n S )) subject to Π(P (n B , n S ), A(n B , n S )) = (P (n B , n S ) + r)n B + A(n B , n S )n S ≥ 0. Denote by λ the Lagrange multiplier associated to the break-even constraint. Observe that λ must be strictly positive at the Ramsey optimum since the break-even constraint is strictly violated at the unconstrained optimum. Assume an interior optimum. Using the fact that ∂P ∂n B = 1 ∂Q B /∂p and ∂P ∂n S = -∂Q B /∂n S ∂Q B /∂p , ∂A ∂n B = -∂Q S /∂n B ∂Q S /∂a and ∂A ∂n S = 1 ∂Q S /∂a , the first-order conditions on the Lagrangean W (P (n B , n S ), A(n B , n S )) + λΠ(P (n B , n S ), A(n B , n S )) yield (A.4) where

P + r + 1 1 + λ ∂U S ∂n B = λ 1 + λ 1 η B - 1 η S ∂Q S ∂n B , A + 1 1 + λ ∂U B ∂n S = λ 1 + λ 1 η S - 1 η B ∂Q B ∂n S ,
η B = -1 n B ∂Q B ∂p and η S = -1 n S ∂Q S
∂a . These two conditions define the Ramsey prices (p R , a R ) as functions of the multiplier λ.

Since the break-even constraint must bind at the optimum, we have

(A.5) (P + r)n B + An S = 0.
This shows in particular that either P + r < 0 and A > 0, or P + r > 0 and A < 0, or P + r = A = 0 at the optimum. Combining (A.4) and (A.5), we obtain

(A.6) λ = - n S ∂U B ∂n S + n B ∂U S ∂n B n 2 B ∂P ∂n B + n 2 S ∂A ∂n S + n B n S ∂P ∂n S + ∂A
f (n B , n S ) = 1 η B ∂U B ∂n S + ∂U S ∂n B ∂Q B ∂n S - 1 η S ∂U S ∂n B + ∂U B ∂n S ∂Q S ∂n B ,
where

η B = -1 n B ∂Q B ∂p (p, n S ) and η S = -1 n S ∂Q S ∂a (a, n B ). At the monopoly optimum, (n B , n S ) solve max (n B ,n S ) Π(P (n B , n S ), A(n B , n S )) = (P (n B , n S )+ r)n B + A(n B , n S )n S .
Assume an interior optimum. The first-order conditions write as follows Running Example. Next, we study the same problems but in the context of our running example. This allows, first, to determine the conditions under which the solutions are interior, and, second, to obtain a neat characterization of which side is taxed/which side is subsidized as function of network effects. Assumption A.1 amounts to u B u S < 1 and is maintained throughout this section.

P + r + ∂P ∂n B n B + ∂A ∂n B n S = 0, A + ∂A ∂n S n S + ∂P ∂n S n B = 0.
Social Optimum. Consider the problem max (p,a) W . We have

∂ 2 W/∂p 2 = -(1 -u S (2u B + u S ))/(1 -u B u S ) 2 , ∂ 2 W/∂a 2 = -(1 -u B (u B + 2u S ))/(1 -u B u S ) 2 and (∂ 2 W/∂p 2 )(∂ 2 W/∂a 2 ) - (∂ 2 W/∂a∂p) 2 = (1 -(u B + u S ) 2 )/(1 -u B u S ) 2
. Therefore, a necessary and sufficient condition for the Hessian to be negative definite is

u B + u S < 1. Notice that u B + u S < 1 ⇒ u B u S < 1.
Ramsey Pricing. Consider the problem max (p,a) W subject to Π(p, a) ≥ 0. Π(p, a) is strictly concave in (p, a) under the assumption u B + u S < 2. We have indeed

∂ 2 Π/∂p 2 = -2/(1 - u B u S ) < 0, ∂ 2 Π/∂a 2 = -2/(1 -u B u S ) < 0 and (∂ 2 Π/∂p 2 )(∂ 2 Π/∂a 2 ) -(∂ 2 Π/∂a∂p) 2 = (4 - (u B + u S ) 2 )/(1 -u B u S ) 2 > 0. The Lagrangian L(p, a) = W (p, a) + λΠ(p, a)
, with λ ≥ 0, is thus strictly concave as the sum of two strictly concave functions when u B + u S < 1. The optimum is then characterized by the first-order conditions ∂L/∂p = 0 and ∂L/∂a = 0, and the complementary slackness condition λ∂L/∂λ = 0. The constraint must bind at the optimum since the unconstrained outcome violates the break even constraint (except in the degenerate case u B = u S = 0). Therefore, λ > 0 at the optimum. Using the first-order conditions ∂L/∂p = 0 and ∂L/∂a = 0, we can express the optimal price p R and developer fee a R as functions of the multiplier λ

p R + r = (r + v) -λ(2λ + 1) + (λ + 1) 2 u B u S + (λ + 1) 2 u 2 S (λ(u B + u S -2) + u B + u S -1)(λ(u B + u S + 2) + u B + u S + 1) , (A.9) a R = (λ + 1)(r + v)(λu B + u B -λu S ) (λ(u B + u S -2) + u B + u S -1)(λ(u B + u S + 2) + u B + u S + 1) . (A.10)
Replacing these values in the break-even constraint Π(p R , a R ) = 0, the multiplier λ must satisfy

(A.11) (u B + u S ) 2 = λ(2λ + 1) 2 (λ + 1) 3 .
The right-hand side in (A.11) is strictly increasing and takes values in [0, 4) for λ ∈ [0, +∞). Therefore, (A.11) has a unique strictly positive solution in λ when (u B + u S ) 2 < 4, which is ensured by our assumption u B + u S < 1. Denote that solution by λ R . We can use (A.11) to replace (u B + u S ) 2 as a function of λ R in the denominator of (A.10) (which is the same as the denominator of (A.9)) to show that this denominator is equal to -4λ R -1/(λ R + 1) < 0. As a consequence, and using again (A.10), we obtain

Sign(a R ) = -Sign(p R + r) = -Sign(λ R u B + u B -λ R u S ).
Observe now that a R = 0 (or equivalently p R + r = 0) amounts to λ R = u B /(u S -u B ) with λ R the unique positive solution of (A.11). Plugging this expression in (A.11), we obtain that the following condition between u B and u S must hold to have a R = 0 (or equivalently p R + r = 0): u B = u 3 S . Integration. Consider the problem max (p,a) Π(p, a). We have already established that Π(p, a) is strictly concave in (p, a) if (u B + u S ) 2 < 4. Monopoly prices are given by

p I + r = (r + v)(2 -u S (u B + u S )) 4 -(u B + u S ) 2 and a I = (r + v)(u S -u B ) 4 -(u B + u S ) 2 .
Therefore, a I ≥ 0 ⇔ u B ≤ u S and p I + r ≥ 0 ⇔ u S (u B + u S ) ≤ 2. From this, we

A.2. Competing Platforms and Competing Manufacturers: Generalities

Demand Functions with Indirect Network Effects. In the last stage of the game, given prices (p 1 , p 2 , a), the number of buyers and the number of developers solve (A.12)

   n 1 B = Q 1 B (p 1 , p 2 , n S ), n 2 B = Q 2 B (p 2 , p 1 , n S ), n S = Q S (u S (n 1 B + n 2 B ) -a).
In the following, with a slight abuse of notations, let ∂Q S /∂a(a, n

B ) = -Q ′ S (u S n B -a) and ∂Q S /∂n B (a, n B ) = u S Q ′ S (u S n B -a).
To avoid 'cornered-market' solutions, in which all buyers or all developers participate in equilibrium, we assume that indirect network effects are not too strong so that, in the relevant range, each manufacturer faces a demand that is locally elastic with respect to prices.

Assumption A.2. In the relevant range of (p 1 , p 2 , a), n 1 B + n 2 B and n S satisfy

∂Q S ∂n B (a, n 1 B + n 2 B ) ∂Q 1 B ∂n S (p 1 , p 2 , n S ) + ∂Q 2 B ∂n S (p 2 , p 1 , n S ) < 1.
This assumption is maintained throughout the analysis.

Lemma A.3. System (A.12) has a unique interior solution.

Proof.

Let D B (p 1 , p 2 , a) = D 1 (p 1 , p 2 , a) + D 2 (p 2 , p 1 , a).
From system (A.12), we have (omitting some notations)

(A.13) D B = Q 1 B (p 1 , p 2 , Q S (u S D B -a)) + Q 2 B (p 2 , p 1 , Q S (u S D B -a)).
For a given (p 1 , p 2 , a), D B (p 1 , p 2 , a) is thus a fixed point of ψ

(x) = Q 1 B (p 1 , p 2 , Q S (u S x -a)) + Q 2 B (p 2 , p 1 , Q S (u S x -a)). Notice then that ψ ′ (x) = ∂Q S ∂n B (a, x) ∂Q 1 B ∂n S (p 1 , p 2 , Q S (u S x -a)) + ∂Q 2 B ∂n S (p 2 , p 1 , Q S (u S x -a)) .
Assumption A.2 implies that |ψ ′ (•)| < 1, so that ψ(•) is a contraction mapping and Equation (A.13) has a unique solution. It follows that D B (p 1 , p 2 , a) is uniquely defined; hence,

D S (p 1 , p 2 , a) = Q S (u S D B (p 1 , p 2 , a) -a)
is also uniquely defined, as well as

D k (p k , p ℓ , a) = Q k B (p k , p ℓ , D S (p 1 , p 2 , a)).
Lemma A.4. The following properties hold:

∂D k ∂p k (p k , p ℓ , a) < 0, ∂D S ∂p k (p k , p ℓ , a) < 0, | ∂D k ∂p k (p, p, a)| > | ∂D k ∂p ℓ (p, p, a)|, ∂D k ∂a (p k , p ℓ , a) < 0 and ∂D S ∂a (p k , p ℓ , a) < 0. Proof. By the implicit function theorem, D B (p 1 , p 2 , a) = D 1 (p 1 , p 2 , a) + D 2 (p 2 , p 1 , a
) is continuously differentiable. Differentiating Equation (A.13) with respect to p 1 and rearranging terms, we find (omitting some arguments)

∂D B ∂p 1 1 - ∂Q S ∂n B ∂Q 1 B ∂n S + ∂Q 2 B ∂n S = ∂Q 1 B ∂p 1 + ∂Q 2 B ∂p 1 .
By Assumption A.2, the term in squared brackets is positive. Therefore, ∂D B /∂p 1 is negative.

Similarly,

∂D B /∂p 2 < 0. Since ∂Q S /∂n B > 0, D S (p 1 , p 2 , a) = Q S (u S D B (p 1 , p 2 , a) -a) is de- creasing in both p 1 and p 2 . Then, ∂D 1 /∂p 1 = ∂Q 1 B /∂p 1 + (∂D B /∂p 1 )(∂Q 1 B /∂n S )(∂Q S /∂n B ), which shows that ∂D 1 /∂p 1 < 0. Similarly, ∂D 1 /∂p 2 = ∂Q 1 B /∂p 2 +(∂D B /∂p 2 )(∂Q 1 B /∂n S )(∂Q S /∂n B ). For symmetric prices, ∂D B /∂p 1 = ∂D B /∂p 2 , and therefore, |∂D 1 /∂p 1 |-|∂D 1 /∂p 2 | = |∂Q 1 B /∂p 1 |- |∂Q 1
B /∂p 2 | < 0, which is negative under our assumptions. Let λ ≥ 0 be the Lagrange multiplier associated to the constraint. Assume that this problem is well-behaved so that its solution can be characterized through first-order conditions.

W = U B (1/2 n B , 1/2 n B , n S ) -P (n B , n S )n B + U S (n S , n B ) -A(n B , n S )n S +(P (n B , n S ) + r)n B + A(n B , n S )n S s.t. (P (n B , n S ) + r)n B + A(n B , n S )n S = 0.
The first-order conditions on n B and n S can be written as follows (omitting notations) (A.18)

1 2 ∂U B ∂q 1 + ∂U B ∂q 2 + r + ∂U S ∂n B + λ ∂P ∂n B n B + P + r + ∂A ∂n B n S = 0, ∂U B ∂n S + ∂U S ∂q S + λ ∂P ∂n S n B + ∂A ∂n S n S + A = 0.
Then, noticing that the maximization problems of the representative buyer and developer give With no break-even constraint, the welfare-maximizing prices are obtained by setting λ = 0 in (A.19): P + r = -∂U S ∂n B < 0 and A = -∂U B ∂n S < 0. These conditions violate the break-even constraint. Therefore, λ > 0 and the break-even constraint is binding at the Ramsey optimum (A.20) (P + r)n B + An S = 0, which shows in particular that either P + r ≤ 0 and A ≥ 0, or P + r > 0 and A < 0 at the optimum. ∂a (a, n B ). Next, we study the same problem but in the context of our running example. This allows, first, to determine the conditions under which the constrained-maximization problem is concave, and, second, to obtain a neat characterization of which side is taxed/which side is subsidized as function of network effects.

Ramsey Prices in the Running Example. Consider the unconstrained problem max (p,a) W . Assume 2(u B +u S ) 2 < 1, which ensures that the Hessian is negative definite so that W is strictly concave. We have indeeed

∂ 2 W/∂p 2 = -(2 -4u S (2u B + u S ))/(1 -2u B u S ) 2 < 0, ∂ 2 W/∂a 2 = -(1 -2u B (u B + 2u S ))/(1 -2u B u S ) 2 < 0 and (∂ 2 W/∂p 2 )(∂ 2 W/∂a 2 ) -(∂ 2 W/∂a∂p) 2 = (2 - 4(u B +u S ) 2 )/(1-2u B u S ) 2 > 0.
Solving for the first-order conditions, we obtain p = -(2u

S (u B + u S )v + r(1 -2u B (u B + u S )))/(1 -2(u B + u S ) 2 ) and a = -(2u B (v + r))/(1 -2(u B + u S )
2 ), which yields p + r ≤ 0 and a ≤ 0.

Consider now problem (A.17). Π is strictly concave under the assumption 2(u

B + u S ) 2 < 1. We have indeed ∂ 2 Π/∂p 2 = -4/(1 -2u B u S ) < 0, ∂ 2 Π/∂a 2 = -2/(1 -2u B u S ) < 0 and (∂ 2 Π/∂p 2 )(∂ 2 Π/∂a 2 ) -(∂ 2 Π/∂a∂p) 2 = (8 -4(u B + u S ) 2 )/(1 -2u B u S ) 2 > 0.
The Lagrangian L = W + λΠ, with λ ≥ 0, is thus strictly concave as the sum of two strictly concave functions. The optimum is then characterized by the first-order conditions ∂L/∂p = 0 and ∂L/∂a = 0, and the complementary slackness condition λ∂L/∂λ = 0. The constraint must bind at the optimum since the unconstrained outcome violates the break even constraint (except in the degenerate case u B = u S = 0). Therefore, λ > 0 at the optimum.

Using the first-order conditions ∂L/∂p = 0 and ∂L/∂a = 0, we can express the optimal price p R and developer fee a R as functions of the multiplier .24) Replacing in the constraint Π = 0, the multiplier λ satisfies

λ a R = 2(λ + 1)(v + r)(λu B + u B -λu S ) -4λ + 2 (λ 2 ((u B + u S ) 2 -2) + 2λ(u B + u S ) 2 + (u B + u S ) 2 ) -1 , (A.23) p R + r = (v + r) -λ(2λ + 1) + 2(λ + 1) 2 u B u S + 2(λ + 1) 2 u 2 S -4λ + 2 (λ 2 ((u B + u S ) 2 -2) + 2λ(u B + u S ) 2 + (u B + u S ) 2 ) -1 . (A
(A.25) (u B + u S ) 2 = λ(2λ + 1) 2 2(λ + 1) 3 .
The right-hand side in (A.25) is strictly increasing and takes values in [0, 2) for λ ∈ [0, +∞). Therefore, (A.25) has a unique strictly positive solution in λ when (u B + u S ) 2 < 2, which is ensured by our assumption 2(u B + u S ) 2 < 1. Denote that solution by λ R . We can use (A.25) to replace (u B + u S ) 2 as a function of λ R in the denominator of (A.23) (which is the same as the denominator of (A.24)) to show that this denominator is equal to -4λ R -1/(λ R + 1) < 0. As a consequence, and using again (A.25), we obtain

Sign(a R ) = -Sign(p R + r) = -Sign(λ R u B + u B -λ R u S ).
Observe now that a R = 0 (or equivalently p R + r = 0) amounts to λ R = u B /(u S -u B ) with λ R the unique positive solution of (A.25). Plugging this expression in (A.25), we obtain that the following condition between u B and u S must hold to have a R = 0 (or equivalently p R + r = 0):

u B = 2u 3 S .

A.3. Proof of Proposition 4 (Separation Outcome)

Existence. Suppose all platforms offer (β = 1, a = 0) and thus make no profit. The developer publishes all its applications on all the operating systems and manufacturers are indifferent between any of the operating systems. There are a priori three possible configurations to consider: (i) both manufacturers choose I; (ii) M 1 chooses I and M 2 chooses E; (iii) both manufacturers choose E. However, in all these configurations, all platforms make a nil profit and manufacturers' profits are given by π1 (1, 1, 0) = π2 (1, 1, 0). These configurations are thus essentially equivalent from the platforms' perspective.

Consider now the possible deviations by platform I.

(i). First, consider a deviation (β I < 1, a I < 0). Because a I < 0, the developer publishes its applications on I even if no manufacturers choose I's operating system.

If both manufacturers choose I's operating system, M 1 's profit is given by π1 (β I , β I , a I ). If M 1 chooses E and M 2 chooses I, M 1 's profit is given by π1 (1, β I , a I ). Since π1 (1, β I , a I ) > π1 (β I , β I , a I ), M 1 chooses the fringe's operating system rather than I's. Hence, following I's deviation, it is not possible that both manufacturers choose I. A configuration in which M 1 chooses I and M 2 chooses E is not an equilibrium either because π1 (β I , 1, a I ) < π1 (1, 1, a I ). Both manufacturers choosing E is the only continuation following I's deviation because π1 (1, 1, a I ) > π1 (β I , 1, a I ).

Hence, the deviation leads to both manufacturers choosing E and to a strictly negative profit for I (no manufacturers and a I < 0). It is thus not profitable for I.

(ii). Second, consider a deviation (β I < 1, a I ≥ 0). Consider that the developer publishes on I when at least one manufacturer chooses I's operating system, or equivalently that a I is not too large (this is the most favorable scenario for I's deviation). If both manufacturers choose I's operating system, M 1 's profit is given by π1 (β I , β I , a I ). If M 1 chooses E and M 2 chooses I, profits are given by π1 (1, β I , a I ) and π2 (β I , 1, a I ). Therefore, both manufacturers choosing I is not part of the continuation following I's deviation. Again, it is immediate to show that, following I's deviation, both manufacturers choose E and the developer does not publish on I. The deviation is thus not profitable.

(iii). Third, consider a deviation (β I = 1, a I ≥ 0). Consider that the developer publishes on I when at least one manufacturer chooses I's operating system, or equivalently that a I is not too large (again, this is the most favorable scenario for I's deviation). If both manufacturers choose I, M 1 's gain is π1 (1, 1, a I ). If M 1 chooses E and M 2 chooses I, M 1 's gain is again π1 (1, 1, a I ). Hence, given that the other manufacturer chooses I, each manufacturer is indifferent between I and E. Therefore, both manufacturers choosing I is a part of the continuation equilibrium. It is immediate to show that both manufacturers choosing E is also an equilibrium since π1 (1, 1, 0) > π1 (1, 1, a I ).

Observe, though, that manufacturers collectively gain if they both choose E rather than I, for their total profit would be π1 (1, 1, 0) + π2 (1, 1, 0) > π1 (1, 1, a I ) + π2 (1, 1, a I ) when a I > 0. Our criterion selects both manufacturers choosing E as the continuation following the deviation by I. Hence, the deviation is not profitable for I.

Uniqueness. It remains to show that (β = 1, a = 0) for all platforms is the unique equilibrium.

(i). Consider a situation where I sets (β I < 1, a I ) and attracts both manufacturers. Assume that a I is such that the developer is not discouraged from publishing on I. Then, E can set β E = β I + ε, ε > 0 but small, and a E = 0. Both manufacturers choose E since π1 (β I , β I , a I ) < π1 (β E , β I , a I + 0) (both manufacturers choosing I is not a Nash equilibrium) and π1 (β E , β E , 0 +

1 {a I ≤0} ) ≥ π1 (β I , β E , a I + 0) (both manufacturers choosing E is a Nash equilibrium).
(ii). Consider now a situation where I sets (β I < 1, a I ) and attracts M 1 only. Consider that M 2 chooses E that offers (β E < 1, a E ). Assume that a I and a E are such that that the developer is not discouraged from publishing on I and on E respectively. Then, E ′ can set

β E ′ = max(β I , β E ) + ε, ε > 0 but small, and a E ′ = 0. Both manufacturers choose E ′ since π1 (β I , β E , a I + a E ) < π1 (β E ′ , β E , 1 {a I ≤0} + a E + 0) (both manufacturers choosing I is not a Nash equilibrium) and π1 (β E ′ , β E ′ , 1 {a I ≤0} + 1 {a E ≤0} + 0) ≥ π1 (β I , β E , a I + 1 {a E ≤0} + 0) (both manufacturers choosing E ′ is a Nash equilibrium).
A consequence of (i) and (ii) is that a strategy (β I < 1, a I ) leads at best to a nil profit in equilibrium. That strategy is thus strictly dominated by (β I = 1, a I = 0).

(iii). A strategy (β I = 1, a I < 0) is never used at equilibrium because it leads at best to a nil profit. It is also strictly dominated by (β I = 1, a I = 0).

(iv). It remains to study the strategy (β I = 1, a I > 0). If both manufacturers choose I, then E can offer (β E = 1, a E = a I -ε), with ε > 0 but small. Since π1 (1, 1, a I ) > π1 (1, 1, a I + a E ) and π1 (1, 1, a E ) > π1 (1, 1, a I + a E ), there are two Nash equilibria in the subgame starting at stage 2: both manufacturers choose I; and both manufacturers choose E. According to our selection criterion, both manufacturers choose E, which leads to a nil profit for I. Similarly, if M 1 only chooses I, and

M 2 chooses E that offers (β E , a E ), E ′ can offer (β E ′ = 1, a E ′ = a I -ε), ε > 0 but small, and ensures that M 1 choose E ′ . The strategy (β I = 1, a I > 0) is dominated by (β I = 1, a I = 0). A consequence of (i)-(ii)-(iii)-(iv)
is that there is no other equilibrium than the one in which all platforms offer (β = 1, a = 0).

A.4. Unconstrained Outcome with No Efficiency Gains in the Running Example

We study the unconstrained outcome and derive formally the curves drawn in Figure 4. Let

π I 1 (β I , a I ) = (p I 1 + r)D 1 (p I 1 , p I 2 , a I ) + (1 -β I )rD 2 (p I 2 , p I 1 , a I ) + a I D S (p I 1 , p I 2 , a I )
, where prices p I 1 and p I 2 are given in Appendix A.2. Conditions for Concavity. We find conditions that ensure the concavity of the maximization problem max (β I ,a I ) π I 1 (β I , a I ). One can show that a sufficient condition is 2(u B + u S ) 2 < 1. This is, however, an overly restrictive condition that prevents from studying situations with quite asymmetric network effects. In the sequel, we establish a set of necessary and sufficient conditions that ensure the concavity of the previous problem. Computations show that:

(i) ∂ 2 π I 1 /∂β 2 I = -[4(σ + 1)(σ + 2)(σ(9σ + 32) + 32)]/[(σ + 4) 2 (3σ + 4) 2 (1 -2u B u S )], which is strictly negative. (ii) ∂ 2 π I 1 /∂a 2 I = -[2(σ +4) 2 (3σ +4) 2 -4(σ +2)(3σ +4) 2 u 2 B -4(3σ +4)(σ(7σ +32)+32)u B u S - 4(σ(σ(7σ + 40) + 64) + 32)u 2 S ]/[(σ + 4) 2 (3σ + 4) 2 (1 -2u B u S )]. (iii) (∂ 2 π I 1 /∂β 2 I )(∂ 2 π I 1 /∂a 2 I ) -(∂ 2 π I 1 /∂a I ∂β I ) 2 = -4r 2 (σ + 1)[5σ 2 (9u 2 B + 22u B u S + 9u 2 S -20) + 16σ(5u 2 B +14u B u S +5u 2 S -12)+9σ 3 ((u B +u S ) 2 -2)+16((3u B +u S )(u B +3u S )-8)]/((σ + 4) 2 (3σ + 4) 2 (1 -2u B u S ) 2 ).
Simplifying these expressions further, the Hessian is negative definite if and only if

-2(σ + 4) 2 (3σ+4) 2 +4(σ+2)(3σ+4) 2 u 2 B +4(3σ+4)(σ(7σ+32)+32)u B u S +4(σ(σ(7σ+40)+64)+32)u 2 S < 0 and 5σ 2 (9u 2 B + 22u B u S + 9u 2 S -20) + 16σ(5u 2 B + 14u B u S + 5u 2 S -12) + 9σ 3 ((u B + u S ) 2 -2) + 16((3u B + u S )(u B + 3u S ) -8) < 0.
Instead of working with (u B , u S ), it turns out to be easier to work with (σ, u S ) with σ = γ -2(1 + γ)u B u S (which is possible since such transformation is a C 1 -diffeomorphism). Remind that σ ≥ 0 by assumption and σ ≤ γ by definition. Equipped with this change of variables, we have that

∂ 2 π I 1 /∂a 2 I < 0 and (∂ 2 π I 1 /∂β 2 I )(∂ 2 π I 1 /∂a 2 I ) -(∂ 2 π I 1 /∂a I ∂β I ) 2 > 0 are equivalent to (A.26) f (x) = 1 x(1 + γ)(1 + σ)(4 + σ) 2 (4 + 3σ) 2 4x 2 (1 + γ) 2 (32 + σ(64 + σ(40 + 7σ))) -2x(1 + γ)(4 + 3σ)(2(2 + σ)(16 + 5σ(4 + σ)) + γ(32 + 3σ(16 + σ(7 + σ)))) +(γ -σ) 2 (2 + σ)(4 + 3σ) 2 < 0 and (A.27) g(x) = 1 x(1 + σ)(4 + σ) 2 (4 + 3σ) 2 4x 2 (1 + γ) 2 (4 + 3σ)(12 + σ(11 + 3σ)) -4x(1 + γ)(128 + γ(4 + 3σ)(12 + σ(11 + 3σ)) + σ(272 + σ(212 + σ(73 + 9σ)))) +(γ -σ) 2 (4 + 3σ)(12 + σ(11 + 3σ)) < 0, where x = u 2 S .
Notice that the denominators in (A.26) and (A.27) are identical and positive. Let N 1 and N 2 denote the numerators in (A.26) and (A.27) respectively. We have

N 2 -N 1 = (γ -σ) 2 (4 + σ)(4 + 3σ) + 4x 2 (1 + γ) 2 (16 + σ(16 + σ(5 + 2σ))) + 2x(1 + γ)(γ(4 + 3σ)(8 + σ(26 + 3σ(5 + σ))) + 2σ(48 + σ(76 + σ(37 + 6σ)))) > 0.
Put differently, Condition (A.27) is more demanding than Condition (A.26), that is N 2 < 0 is a necessary and sufficient condition for the concavity of the maximization problem. Then, simple computations show that condition N 2 < 0 amounts to

(A.28) (u B + u S ) 2 -2 (u S -u B ) 2 < 16 + σ(16 + 5σ) (2 + σ)(32 + σ(32 + 9σ))
.

Since the right-hand side in (A.28) is positive, a sufficient condition for the concavity of the maximization problem is (u .28) is more likely to be satisfied if (u S -u B ) 2 large, that is, roughly speaking when network effects are sufficiently asymmetric.

B + u S ) 2 < 2. When (u B + u S ) 2 > 2, condition (A
Let us now describe the set of (u B , u S ) such that (A.28) is satisfied, which as we have seen amounts to g(x) < 0. The numerator in g(x) is a polynomial of degree 2 in x, whose discriminant is equal to 64(1 + γ) 2 (1 + σ)(2 + σ)(32 + σ(32 + 9σ))(γ(4 + 3σ)(12 + σ(11 + 3σ)) + 2(32 + σ(56 + σ(33 + 7σ)))) > 0. Therefore, it has two distinct real roots x(σ) and x(σ). Since the numerator in g(x) is positive when x = 0, the smallest root is positive: x(σ) > 0. It follows that, for a given γ ≥ 0, g(x) < 0 amounts to x(σ) < u 2 S < x(σ). Define u σ) . By construction, the set of (u B , u S ) such that (A.28) is satisfied is the set of (u B , u S ) whose frontiers are given by the two parametric curves C = (u B (σ), u S (σ)) and C = (u B (σ), u S (σ)) for all σ ∈ [0, γ]. Curves C and C are represented in Figure 16.

S (σ) = x(σ), u B (σ) = γ-σ 2(1+γ)u S (σ) , u S (σ) = x(σ) and u B (σ) = γ-σ 2(1+γ)u S (
Simple computations show that the slope of the parametric curves (given by uS (σ)/ uB (σ)) is equal to

(A.29) - 2(1 + γ)x(σ) (γ -σ) + 2 x(σ) x ′ (σ)
, where x(σ) = x(σ) for the curve C and x(σ) = x(σ) for the curve C. Computations show that x ′ (σ) < 0 < x ′ (σ). Plugging this in Equation (A.29) shows that the curve C is downward sloping in the plane (u B , u S ). Computations then show that (γ -σ) + 2 x(σ) x ′ (σ) > 0, which proves that the curve C is downward sloping in the plane (u B , u S ).

Figure 16 represents the sufficient condition (area below the red curve) and the necessary and sufficient condition (area below the blue curves) for the concavity of the maximization problem (given that we focus on σ ≥ 0). by the two first-order conditions ∂π I 1 /∂β I = 0 and ∂π I 1 /∂a I = 0. Simple computations lead to

Unconstrained

r(1 -β * I ) = (v + r)(3σ + 4) -3σ(σ + 4) + (σ(3σ + 13) + 20)u B u S + (σ(3σ + 11) + 12)u 2 S - 16 Den , (A.30)
a * I = (v + r)(3σ + 4)(σ(3σ + 11) + 12)(u B -u S ) Den , (A.31) with Den < 0 since it is proportional to -(∂ 2 π I 1 /∂β 2 I )(∂ 2 π I 1 /∂a 2 I ) + (∂ 2 π I 1 /∂a I ∂β I ) 2 < 0. First, a * I = 0 is equivalent to u B = u S . Second, β * I = 1 amounts to -3σ(σ + 4) + (σ(3σ + 13) + 20)u B u S + (σ(3σ + 11) + 12)u 2 S - 16 = 0 with σ = γ -2(1 + γ)u B u S .
Expressing u B as a function of σ, β * I = 1 amounts to (σ(3σ + 13) + 20)(γ -σ))/(2(γ + 1)) -3σ(σ + 4) + (σ(3σ + 11) + 12)u 2 S -16 = 0, second degree polynomial in u S with a positive root equal to

ûβ * I =1 S (σ) = γ + 8σ 9σ 2 +33σ+36 + σ + 8 3 2(γ + 1) . Define ûβ * I =1 B (σ) = (γ -σ)/(2(γ + 1)û S (σ)).
Then, the set of (u B , u S ) such that β * I = 1 is described by the curve associated to the parametric equations (u

B = ûβ * I =1 B (σ), u S = ûβ * I =1 S (σ)) for all σ ∈ [0, γ].

Straightforward computations show that (û

β * I =1 S (σ)) 2 > (û β * I =1 B (σ)
) 2 for all σ, so that the set of (u B , u S ) such that β * I = 1 lies strictly above the 45 • -degree line. We have (û

β * I =1 B (γ), ûβ * I =1 S (γ)) =
(0, (3γ(γ + 4) + 16)/(γ(3γ + 11) + 12)). Simple computations then show that (û

β * I =1 S ) 2 (γ) ∈ [x(γ), x (γ)
] and (û

β * I =1 S ) 2 (0) ∈ [x(0), x(0)].
Last, we also have

(A.32) d dσ ûβ * I =1 S (σ) d dσ ûβ * I =1 B (σ) = - C D
with C = ( σ σ 9σ 2 + 66σ + 185 + 264 + 176 (γ(σ(3σ+11)+12)+(3σ+4)(σ(σ+5)+8))) > 0 and D = (σ(3σ + 11) + 12)(γ(σ(σ(9σ(3σ + 22) + 571) + 792) + 464) + σ(σ(σ(3σ(3σ + 38) + 569) + 1352)+1584)+768) > 0; this derivative is thus strictly negative for all σ. Hence, the parametric curve (û

β * I =1 B (σ), ûβ * I =1 S (σ)) for σ ∈ [0, γ] is strictly downward-sloping in the (u B , u S )-plane.
Simple computations also show that: (i) (û (γ)) 2 = (γ + 4)/(γ(3γ + 11) + 12) + 1 < x(γ) = 2 3 (γ/(γ(3γ + 11) + 12) + 4/(3γ + 4) + 3). To summarize, β * I = 1 describes a curve in the (u B , u S )-space that is always above the 45 • -degree line and is strictly downward-sloping, starts and ends within the sets of admissible values, as depicted in Figure 4.

β * I =1 S ( 0

A.5. Proof of Proposition 5

We consider here the constrained outcome with no efficiency gains in the running example.

Let π I 1 (a I ) = (p I 1 + r)D 1 (p I 1 , p I 2 , a I ) + a I D S (p I 1 , p I 2 , a I )
, where the prices p I 1 and p I 2 are given in Appendix A.2. From the analysis of Appendix A.4, this is a strictly concave function of a I provided that (A.28) holds. Therefore, as shown in Appendix A.4, the objective is concave in

a I iff u 2 S ∈ [x(σ), x(σ)] for σ ∈ [0, γ].
The first-order condition leads to (σ) ∈ [x(σ), x(σ)] (so that the curve associated to the parametric equations always lies within the set of admissible values) using brute force computations that are similar to those performed in Appendix A.4 and are not reported here.

a * * I = - (3σ + 4)(v + r)(2(σ + 2)(3σ + 4)u B -(3σ(σ(σ + 7) + 16) + 32)u S ) H with H = (σ + 4) 2 (3σ + 4) 2 -2(σ + 2)(3σ + 4) 2 u 2 B -2(3σ + 4)(σ(7σ + 32) + 32)u B u S -2(σ(σ(7σ + 40) + 64) + 32)u 2 S > 0 when u 2 S ∈ [x(σ), x (σ)
We have: lim σ→γ (û

a * * I =0 B (γ), ûa * * I =0 S (γ)) = (0, 0), (û a * * I =0 B (0), ûa * * I =0 S (0)) = ( γ/(γ + 1), (1/2) γ/(γ + 1)).
Simple computations also show that (û (σ) σ=γ = 2(γ + 2)(3γ + 4) 3γ(γ(γ + 7) + 16) + 32 .

a * * I =0 S (σ)) 2 ≤ (û a * * I =0 B (σ)) 2 for all σ ∈ [0, γ],
To summarize, a * * I = 0 describes a curve in the (u B , u S )-space such that: it goes through (0, 0), it is always below the 45 • -degree line, and it is strictly increasing if γ < 4 or increasing then decreasing if γ ≥ 4. Therefore, there exists a uniquely defined function h : u S → h(u S ) such that, for all (u B , u S ), a * * I (u B , u S ) > 0 if and only if u B < h(u S ).

A.6. Proof of Proposition 7

Impact on Buyers. Let V I B (a I ) be the buyer surplus under vertical integration when β I = 1 and the developer fee is set at some value a I . Let V S B be the buyer surplus under separation. Let ∆V B (a

I ) = V I B (a I ) -V S B . Computations show that ∆V B (a I ) = a I a I - 2(σ + 2)(3σ + 4) 2 (v + r)((σ + 2)u B -u S ) K K (σ + 4) 2 (3σ + 4) 2 (1 -2u B u S ) 2
where K = (σ+2) 2 (3σ+4) 2 u 2 B + (σ + 2)(σ(σ + 16) + 16) -2u B u S (σ + 1)(σ + 4) 2 u 2 S -2u B u S (σ+ 2)(3σ + 4) 2 . We show first that K > 0. Since 2u B u S < 1, the term in brackets in K is strictly greater than (σ + 2)(σ(σ + 16) + 16) -(σ + 1)(σ + 4) 2 = (3σ + 4) 2 . Plugging this in K then gives K > (3σ + 4) 2 (u S -u B (2 + σ)) 2 ≥ 0. This implies that the sign of ∆V B (a I ) is given by the sign of a I (a

I -2(σ + 2)(3σ + 4) 2 (v + r)((σ + 2)u B -u S )/K).
First Sufficient Condition. Consider that a * * I < 0, which amounts to u B > h(u S ), with h defined in Appendix A.5 and such that h(u S ) > u S . Consequently, a * * I < 0 implies (σ + 2)u Bu S > 0, so that ∆V B (a * * I ) > 0. Therefore, we have established a first sufficient condition a * * I < 0 ⇒ ∆V B (a * * I ) > 0.

Second Sufficient Condition. Consider that a * * I ≥ 0. If (σ + 2)u B -u S ≤ 0, then ∆V B (a * * I ) ≥ 0. Notice that (σ + 2)u B -u S ≤ 0 is equivalent to

u S ≥ h B (u B ) ≡ (γ + 2)u B 2(γ + 1)u 2 B + 1 .
h B is first increasing then decreasing, reaching a maximum at u B = 1/ 2(γ + 1) and leading to u S = (γ + 2)/(2 2(γ + 1)). These values of (u B , u S ) satisfy σ ≥ 0 iff γ ≥ 2. h B can also be represented with the parametric equations (u B = (γ -σ)/(2(γ + 1)(σ + 2)), u S = ûS (σ) ≡ (σ + 2)(γ -σ)/(2(γ + 1))) for σ ∈ [0, γ]. This rewriting allows to get immediately that h B is strictly above the 45 • -degree line, and goes through (0, 0) (for σ = γ) and ( γ/(4(γ + 1)), γ/(γ + 1)) (for σ = 0). This also shows that the conditions a * * I ≥ 0 (i.e., u B ≤ h(u S )) and (σ + 2)u B -u S ≤ 0 (i.e., u S ≥ h B (u B )) define a non-empty set. Last, straightforward manipulations show that û2 S (σ) belongs to [x 1 (σ), x 1 (σ)] for all σ, so that it always belongs to the admissible set.

Intermediate Region. It remains to study the sign of ∆V B (a * * I ) when a * * I ≥ 0 and (σ+2)u Bu S ≥ 0. Observe that (σ + 2)u B -u S ≥ 0 amounts to u 2 S ≤ x a (σ) ≡ (σ + 2)(γ -σ)/(2(γ + 1)). Similarly, a * * I ≥ 0 amounts to u 2 S ≥ x a (σ) ≡ (σ+2)(3σ+4)(γ-σ)/((γ+1)(3σ(σ(σ+7)+16)+32)) (see Appendix A.5). Simple computations show that x a (σ) > x a (σ) for all σ in [0, γ), so the interval in non empty.

On this interval, the sign of ∆V B (a * * I ) is given by the sign of

a * * I - 2(σ + 2)(3σ + 4) 2 (v + r)((σ + 2)u B -u S ) K .
Replacing u B by (γ -σ)/(2(γ + 1)u S ) and up to some positive multiplicative terms (namely, v + r, H and K), the previous expression has the same sign as h 1 (x) = (σ + 2) 3 (3σ + 4) 3 (γσ) 3 + 4(γ + 1)(σ + 2)(3σ + 4)x 2 [γ 2 (σ(σ(σ(23σ + 183) + 507) + 584) + 240) + γ(σ(σ(σ(σ(49 -5σ) + 562)+1618) + 1840) + 736) + σ(σ(σ(σ(37 -4σ) + 422) + 1184) + 1312) + 512] -4(γ + 1) 2 x 3 (γ(3σ + 4)(σ(σ + 4)(σ(19σ + 65) + 88) + 128) -σ(σ(σ + 3)(σ + 8) + 16)(3σ(σ(σ + 6) + 10) + 16)) -(γ + 1)(σ + 2) 2 (3σ + 4) 2 x(γ -σ)(3γ(3σ(σ(σ + 7) + 16) + 32) + σ(σ(σ(3σ + 61) + 288) + 480) + 256) with x = u 2 S . One can then show that h 1 (x a ) < 0 < h 1 (x a ) and also h

′ 1 (x a ) < 0 < h ′ 1 (x a ). Since h ′ 1 (x) is a polynomial of degree 2
, it has at most two real roots and only one of these roots belongs to [x a , x a ]. Since h ′ 1 (x a ) < 0 < h ′ 1 (x a ), this implies that there exists a unique x ∈ (x a , x a ) such that h 1 decreases for x ∈ [x a , x] and increases for x ∈ [x, x a ]. This finally implies that there exists a unique

x B 0 (σ) ∈ (x a , x a ) such that h 1 (x B 0 (σ)) = 0. Let û∆V B (a * * I )=0 S (σ) = x B 0 (σ) and û∆V B (a * * I )=0 B = (γ -σ)/(2(1 + γ)û ∆V B (a * * I )=0 S (σ)). By construction, the parametric curve C = (û ∆V B (a * * I )=0 B , û∆V B (a * * I )=0 S ), σ ∈ [0, γ],
is the frontier of the set of (u B , u S ) such that ∆V B (a * * I ) ≤ 0 when a * * I ≥ 0 and (σ +2)u B -u S ≥ 0. Since x B 0 (σ) is continuous and differentiable with respect to σ and the function σ → (û

∆V B (a * * I )=0 B , û∆V B (a * * I )=0 S
) is injective, the curve C divides the (u B , u S )-space into two connected subsets, one in which the buyer surplus increases, the other in which it decreases.

Impact on Developers. Let V I S (a I ) be the developer surplus under vertical integration when β I = 1 and the developer fee is set at some value a I . Let V S S be the developer surplus under separation. Let ∆V S (a

I ) = V I S (a I ) -V S S . Computations show that ∆V S (a I ) = (σ -2u S (2u B + u S ) + 4) 2 2(σ + 4) 2 (1 -2u B u S ) 2 a I a I - 4(σ + 2)u S (v + r) σ -2u S (2u B + u S ) + 4
.

Using σ = γ-2(1+γ)u B u S , we rewrite σ-2u S (2u B +u S )+4 = 0 as 4+γ-2u 2 S -2u S u B (γ+3) = 0 and denote the unique positive solution of this second degree polynomial equation in u S by

h S (u B ) = 1 2 2(γ + 4) + (γ + 3) 2 u 2 B -(γ + 3)u B .
We thus have σ -2u S (2u B + u S ) + 4 > 0 ⇔ u S < h S (u B ). The following facts are easily established: (i) h S is strictly decreasing and strictly convex;

(ii) h S (0) = 2 + γ/2 > 0; (iii) h S (u B ) = u B ⇔ u B = u S = 1/ √ 2 ⇔ σ = -1.
This implies that, in the (u B , u S )-space, and for σ ≥ 0, h S is strictly decreasing and above the 45 • -degree line.

First Sufficient Condition. Therefore, if a * * I < 0 (which amounts to u B > h(u S ), with h below the 45 • -degree line), then we also have σ -2u S (2u B + u S ) + 4 > 0 (which amounts to u S < h S (u B ), with h S above the 45 • -degree line). As a consequence, we obtain a first sufficient condition a * * I < 0 ⇒ ∆V S (a * * I ) > 0.

Second Sufficient Condition. Next, we want to show the existence of another sufficient condition σ -2u S (2u B + u S ) + 4 < 0 ⇒ ∆V S (a * * I ) > 0. Notice that σ -2u S (2u B + u S ) + 4 < 0 (which amounts to u S > h S (u B ) and thus implies u S > u B ) implies a * * I > 0 (because a * * I < 0 amounts to h(u S ) < u B and thus implies u S < u B ). Therefore, σ -2u S (2u B + u S ) + 4 < 0 implies ∆V S (a * * I ) > 0.

It remains to show that the condition σ -2u S (2u B + u S ) + 4 < 0 is compatible with the conditions for concavity. The condition u S > h S (u B ) can be equivalently expressed in terms of (u S , σ) as u S > ũS (σ) ≡ (γ(σ + 2) + 3σ + 4)/(2(γ + 1)) with σ ∈ [0, γ]. Simple computations show then that ũ2 S (σ) ≥ x(σ) and ũ2 S (σ) ≤ x(σ) for all σ. Intermediate Region. Last, we establish that when a * * I > 0 and σ -2u S (2u B + u S ) + 4 > 0, ∆V S (a * * I ) is strictly negative. From the computations made to analyze the buyer surplus, a * * I > 0 is equivalent to u 2 S > x a (σ). Simple computations show that σ -2u S (2u B +u S )+4 > 0 amounts to u 2 S < xa (σ) ≡ (γ(σ + 2) + 3σ + 4)/(2(γ + 1)). Computations show that ∆V S (a * * I ) = 0 amounts to h 2 (x) ≡ (σ + 2)(3σ + 4) 2 (γ -σ)(γ(σ + 2) -σ(2σ + 7) -4) + 2(γ + 1) 2 (σ(σ + 4)(σ(19σ + 65) + 88) + 128)x 2 -(γ + 1)(3σ + 4)x(γ(σ + 2)(3σ(3σ(σ + 7) + 46) + 88) + σ(σ(σ(31σ + 251) + 720) + 880) + 384) = 0, with x = u 2 S and x ∈ [x a , xa ]. The previous expression is a strictly convex second degree polynomial in x with a strictly positive discriminant. Hence, it admits two reals roots. Computations show that h 2 (x a ) < 0 and h 2 (x a ) < 0, which proves that h 2 (x) < 0 for all x ∈ [x a , xa ]. Hence, if a * * I > 0 and σ -2u S (2u B + u S ) + 4 > 0, then ∆V S (a * * I ) < 0.

A.7. Efficiency Gains

We consider here the constrained outcome with efficiency gains in the running example. Let ∆ = r 0 -r > 0 denote the efficiency gain. The integrated platform's problem writes as follows max

(β I ,a I ) π I 1 (r 0 , β I r 0 , a I ) s.t. π I 2 (β I r 0 , r 0 , a I ) ≥ π E 2 (r, r 0 , a I )
In the running example, the constraint π I 2 (β I r 0 , r 0 , a I ) ≥ π E 2 (r, r 0 , a I ) can be rewritten as β I ≥ β I with

β I = 1 -1 - r r 0 8 + σ(8 + σ) 8(1 + σ) .
Let us then define: a I = arg max a I π I 1 (r 0 , β I r 0 , a I ); the unconstrained outcome (β * I , a * I ) = arg max (β I ,a I ) π I 1 (r 0 , β I r 0 , a I ); a I (r 0 ) = arg max a I π I 1 (r 0 , r 0 , a I ).

Condition for Concavity. Up to the fact that the integrated firm's sharing parameter is now r 0 , the maximization problem max (β I ,a I ) π I 1 (r 0 , β I r 0 , a I ) is the same as the one described in Section 3.5. Hence, the Hessian is negative definite under the same conditions as those stated in Appendix A.4.

Curve β *

I = 1. Up to the fact that the integrated firm's sharing parameter is now r 0 , the solution of the relaxed problem is the same as in Section 3.5. The curve describing the set of parameters (u B , u S ) such that β * I = 1 is therefore described by the parametric curve (û

β * I =1 B (σ), ûβ * I =1 S (σ)) for σ ∈ [0, γ] (see Appendix A.4).
Curve β * I = β I . Computations show that β * I -β I has the sign of r(16(-8

+ (3u B + u S )(u B + 3u S )) + 16(-12 + 5u 2 B + 14u B u S + 5u 2 S )σ + 5(-20 + 9u 2 B + 22u B u S + 9u 2 S )σ 2 + 9(-2 + (u B + u S ) 2 )σ 3 )(8+σ(8+σ))-8v(1+σ)(4+3σ)(-16-3σ(4+σ)+u 2 S (12+σ(11+3σ))+u B u S (20+σ(13+ 3σ)))+r 0 (-512+u 2 B (4+3σ)(8+σ(8+σ))(12+σ(11+3σ))+σ(-1280+u 2 S σ(4+3σ)(12+σ(11+ 3σ
)) -2σ(656 + σ(340 + σ(86 + 9σ)))) + 2u B u S (320 + σ(768 + σ(764 + σ(384 + σ(91 + 9σ)))))). Expressing u B as a function of σ, computations show that the previous expression has the sign of -16vx(1 +σ(1600+σ(2080+ σ(1444+σ(556+σ(109+9σ))))))), where x = u 2 S . The previous expression is a polynomial of degree 2 in x. Computations show that, if r 0 -r is not too large, this polynomial has two distinct real roots, one of which is positive and the other negative. To find out which one is positive, we compute their values when r 0 = r. Indeed, in this case, β I = 1 and thus the curves β * I = β I and β * I = 1 are the same. The values of the two roots when r 0 = r must thus be those found when studying the β * I = 1 curve. The computations are not reported here for the sake of brevity. (σ)) when r 0 = r is given by -A/B where A = (((4 + 3σ)(8 + σ(5 + σ)) + γ(12 + σ(11 + 3σ)))(176 + σ(264 + σ(185 + 66σ + 9σ 2 ))) and B = (12 + σ(11 + 3σ))(768 + γ(464 + σ(792 + σ(571 + 9σ(22 + 3σ)))) + σ(1584 + σ(1352 + σ(569 + 3σ(38 + 3σ)))))). Since both A and B are positive, the slope of the parametric curve is negative when r 0 = r. By continuity, this shows that, if r 0 -r is not too large, the parametric curve (û

+ γ)(1 + σ)(4 + 3σ)(-((4 + 3σ)(8 + σ(5 + σ))) -γ(12 + σ(11 + 3σ)) + 2x(1 + γ)(12 + σ(11 + 3σ))) + r(8 + σ(8 + σ))(4x 2 (1 + γ) 2 (4 + 3σ)(12 + σ(11 + 3σ)) + (γ -σ) 2 (4 + 3σ)(12 + σ(11 + 3σ)) -4x(1 + γ)(128 + γ(4 + 3σ)(12 + σ(11 + 3σ)) + σ(272 + σ(212 + σ (73 + 9σ))))) + r 0 (4x 2 (1 + γ) 2 σ 2 (4 + 3σ)(12 + σ(11 + 3σ)) + (γ -σ) 2 (4 + 3σ)(8 + σ(8 + σ))(12 + σ(11 + 3σ)) -4x(1 + γ)(512 + γ(2+σ) 2 (4+3σ)(12+σ(11+3σ))
β * I =β I B (σ), ûβ * I =β I S (σ)
) is downward sloping in the (u B , u S )-space. Simple computations also show that: (û

β * I =β I B (0)) 2 = (3(r + v)γ 2 )/((1 + γ)(2r(8 + 3γ) -(r 0 -v)(8 + 3γ) + (64(2r - r 0 + v) 2 + 48(2r -r 0 + v) 2 γ + 9(r 0 + v) 2 γ 2 ) 1/2 ); (û β * I =β I S (0)) 2 = (2r(8 + 3γ) -(r 0 -v)(8 + 3γ) + (64(2r -r 0 + v) 2 + 48(2r -r 0 + v) 2 γ + 9(r 0 + v) 2 γ 2 ) 1/2 )/(12(r + v)(1 + γ))); (û β * I =β I B (γ)) 2 = 0;
and (û

β * I =β I S (γ)) 2 = ((8(r + v)(1 + γ)(4 + 3γ)(16 + 3γ(4 + γ)) -2(256 + γ(640 + γ(656 + γ(340 + γ(86 + 9γ)))))∆))/((4 + 3γ)(12 + γ(11 + 3γ))(8(r + v)(1 + γ) -γ 2 ∆))), where ∆ = r 0 -r.
Binding Constraints. Let us first notice that the two parametric curves (û

β * I =1 B (σ), ûβ * I =1 S (σ))
and (û Curve a I = 0. Straightforward but tedious computations show that a I ≥ 0 amounts to 8r(4 + 3σ)(-2u B (2 + σ)(4 + 3σ) + u S (32 + 3σ(16 + σ(7 + σ)))) + u S (-8v(4 + 3σ)(32 + 3σ( 16

β * I =β I B (σ), ûβ * I =β I S (
+ σ(7 + σ))) + ∆ (-512 + σ(4 + σ)(-256 + σ(4 + σ)(-20 + 9σ)))) + u B (4 + 3σ)(16v(2 + σ)(4 + 3σ) + ∆(256 + σ(384 + σ (184 + 3σ(12 + σ))))) ≥ 0. Let us now work with (σ, u S ) by using u B = (γ -σ)/(2u S (1 + γ)). Introducing also x = u 2 S , a I ≥ 0 amounts to 16r(4 + 3σ)(-((γ - σ)(2+σ)(4+3σ))+x(1+γ)(32+3σ(16+σ(7+σ))))-16v(4+3σ)(-((γ -σ)(2+σ)(4+3σ))+x(1+ γ)(32 + 3σ(16 + σ(7 + σ)))) + ∆ (2x(1 + γ)(-512 + σ(4 + σ)(-256 + σ (4 + σ)(-20 + 9σ))) + (γ - σ)(4 + 3σ)(256 + σ(384 + σ(184 + 3σ(12 + σ)))
)) ≥ 0. The left-hand side is linear and decreasing in x. It is therefore positive when

x < x(σ) = (((γ -σ)(4 + 3σ)(-16r(2 + σ)(4 + 3σ) + 16v(2 + σ)(4 + 3σ) + ∆(256 + σ(384 + σ(184 + 3σ(12 + σ))))))/(-2(1 + γ)(8 r(4 + 3σ)(32 + 3σ(16 + σ(7 + σ))) + 8v(4 + 3σ)(32 + 3σ(16 + σ(7 + σ))) -∆(-512 + σ(4 + σ)(-256 + σ(4 + σ)(-20 + 9σ)))))). Let ûa I =0 S (σ) = x(σ) and ûa I =0 B (σ) = (γ -σ)/(2(1 + γ) x(σ))
. By construction, a I > 0 above the parametric curve (û a I =0 B (σ), ûa I =0 S (σ)), σ ∈ [0, γ], in the (u B , u S )-space. Computations then show that the slope of the parametric curve is positive when ∆ goes to 0 and that the parametric curve is below the 45 • -degree line. By continuity, the slope will remain strictly positive and the parametric curve below the 45 • -degree line for all σ when ∆ is sufficiently close to 0.

Curve β *

I r 0 = r. Following the same reasoning, one can show that there exists a downward slopping parametric curve (u

β * I r 0 =r B (σ), u β * I r 0 =r S (σ)), σ ∈ [0, γ],
such that, in the (u B , u S )-space, β * I r 0 > r above this curve and β * I r 0 < r below this curve.

A.8. Coordination Motives

We consider the situation with coordination motives studied in Section 5. First, it is straightforward to show that the Hessian matrix associated to the unconstrained problem does not depend on α B . Hence, the Hessian is negative definite under the same conditions as those stated in Appendix A.4.

The optimal pricing policy associated to the unconstrained problem is as follows: β

(u B + u S ) -1) + α B (-3u 2 B -2u B u S + u 2 S + 2)) + σ 2 (4(r + v)(2u S (8u B + 7u S ) -15) + α B (-45u 2 B - 46u B u S + 11u 2 S + 40)) + 4σ((r + v)(33u B u S + 23u 2 S -28) + α B (-20u 2 B -23u B u S + 3u 2 S + 20)) + 16(r + v)(5u B u S + 3u 2 S -4) -16α B (3u 2 B + 5u B u S -4) ≥ 0.
Let us now work with (σ, u S ) by using u B = (γ -σ)/(2u S (1 + γ)). Introducing also x = u 2 S , β * I ≥ βI amounts to 8(γ + 1)r(σ + 1)x(-γ(σ(3σ + 11) + 12) -((3σ + 4)(σ(σ + 5) + 8)) + 2(γ + 1)(σ(3σ + 11) + 12)x) + 8(γ + 1)(σ + 1)vx(-γ(σ(3σ + 11) + 12) -((3σ + 4)(σ(σ + 5) + 8)) + 2(γ + 1)(σ(3σ + 11) + 12)x) + α B (-(3σ + 4)(σ(3σ + 11) + 12)(γ -σ) 2 + 4(γ + 1) 2 σ(σ(3σ + 11) + 12)x 2 + 4(γ + 1)x(γ(σ + 2)(σ(3σ + 11) + 12) + σ(σ + 6)(σ(3σ + 11) + 20) + 64)) ≥ 0. The left-hand side is a strictly convex second order polynomial in x, which takes a strictly negative value at x = 0. That polynomial admits two roots x(σ) and x(σ) with x(σ) < 0 < x(σ). Therefore, β * I ≥ βI if and only if x ≥ x(σ). Using α B = α(v + r) to simplify further, we have x

(σ) = A/B with A = σ 3 (-α(3γ+29)+6γ+44)+σ 2 (-17αγ-86α+28γ+126)+2σ(-α(17γ+60)+23γ+76)+8(1- α)(3γ + 8) + 3(2 -α)σ 4 + [α(3σ + 4)(σ(3σ + 11) + 12) 2 ((α + 4)σ + 4)(γ -σ) 2 + (α(γ(σ + 2)(σ(3σ + 11)+12)+σ(σ+6)(σ(3σ+11)+20)+64)-2(σ+1)(γ(σ(3σ+11)+12)+(3σ+4)(σ(σ+5)+8))) 2 ] 1/2
and B = 2(γ + 1)(σ(3σ + 11) + 12)((α + 4)σ + 4).

Consider the parametric curve defined as follows: (û

β * I = βI B = (γ-σ)/(2(1+γ) x(σ)), ûβ * I = βI S = x(σ)) for σ ∈ [0, γ]. Computations show that ûβ * I = βI B (γ) = 0 < ûβ * I = βI S (γ) and that 0 < ûβ * I = βI B (0) < ûβ * I = βI S (0) ⇔ α ≤ 1.
The slope of the parametric curve for α = 0 is given by (A.32). By continuity, it remains strictly negative for all σ when α is sufficiently close to 0.

Last, straightforward but tedious computations show that ãI ≥ 0 amounts to (8ru

B (2 + σ)(4 + 3σ) + u B (4 + 3σ)(8v(2 + σ) + α B (4 + σ)(8 + 3σ)) -4ru S (32 + 3σ(16 + σ(7 + σ))) + u S (-α B (4 + σ)(16 + σ(20 + 3σ)) -4v(32 + 3σ(16 + σ(7 + σ))))) ≤ 0. Let us now work with (σ, u S ) by using u B = (γ -σ)/(2u S (1 + γ)). Introducing also x = u 2 S , ãI ≥ 0 amounts to α B (4+σ)(-((γ -σ)(4+3σ)(8+3σ))+2x(1+γ)(16+σ(20+3σ)))+8r(-((γ -σ)(2+σ)(4+3σ))+ x(1+γ)(32+3σ(16+σ(7+σ))))+8v(-((γ-σ)(2+σ)(4+3σ))+x(1+γ)(32+3σ(16+σ(7+σ)))) ≤ 0.
The left-hand side is linear and increasing in x. It is therefore positive when

x > x(σ) = ((γ -σ)(4 + 3σ)(8r(2 + σ) + 8v(2 + σ) + α B (4 + σ)(8 + 3σ)))/(2(1 + γ)(64(2(r + v) + α B ) + 96(2(r + v) + α B )σ + 4(21(r + v) + 8α B )σ 2 + 3(4(r + v) + α B )σ 3 )). Let ûã I =0 S (σ) = x(σ) and ûã I =0 B (σ) = (γ -σ)/(2(1 + γ) x(σ))
. By construction, ãI > 0 above the parametric curve (û ãI =0 B (σ), ûã I =0 S (σ)), σ ∈ [0, γ], in the (u B , u S )-space. Computa-tions then show that the slope of the parametric curve when α B goes to 0 is (2(2 + σ)(4 + 3σ) (256 + σ(640 + 3σ(200 + σ(80 + 11σ))) + γ(64 + 3σ(48 + σ(46 + σ(20 + 3σ))))))/((32 + 3σ(16 + σ(7 + σ)))((8 + 3σ(4 + σ))(32 + 3σ(16 + σ(9 + 2σ))) -γ(64 + 3σ(48 + σ(46 + σ(20 + 3σ)))))), which is positive. Moreover, when α B goes to 0, we have ûã

I =0 B (σ) 2 -ûã I =0 S (σ) 2 = ((γ -σ)(1 + σ)(16 + 3σ(4 + σ))(48 + σ(68 + 3σ(9 + σ))))/(4(1 + γ)(2 + σ)(4 + 3σ)(32 + 3σ(16 + σ(7 + σ))
)), which is nonnegative since γ ≥ σ. This thus shows that the parametric curve is below the 45 • -degree line when α B = 0. By continuity, the slope will remain strictly positive and the parametric curve below the 45 • -degree line for all σ when α B is sufficiently close to 0.

Figure 17 summarizes the main features of the optimal pricing policy with coordination motives.

u B 0 u S u B = u S σ = 0 β * I = 1 β * I = βI β * * I =1 a * * I >0 β * * I =β * I <1 a * * I =a * I >0 β * * I = βI <1 a * * I >0 β * * I = βI <1 a * * I <0
Figure 17: The integrated platform's optimal pricing policy (β * * I , a * * I ) in the running example with coordination motives.

B. Online Appendix (Not For Publication)

B.1. Impact of Vertical Integration with Efficiency Gains: Polar Cases

Let ∆ = r 0 -r > 0 denote the efficiency gain. Remind that the integrated platform's problem writes as follows max

(β I ,a I ) π I 1 (r 0 , β I r 0 , a I ) s.t. π I 2 (β I r 0 , r 0 , a I ) ≥ π E 2 (r, r 0 , a I ) 0 ≤ β I ≤ 1.
In the running example, the constraint π I 2 (β I r 0 , r 0 , a I ) ≥ π E 2 (r, r 0 , a I ) can be rewritten as β I ≥ β I with

β I = 1 -1 - r r 0 8 + σ(8 + σ) 8(1 + σ) .
Let us then define: a I = arg max a I π I 1 (r 0 , β I r 0 , a I ); (β * I , a * I ) = arg max (β I ,a I ) π I 1 (r 0 , β I r 0 , a I ); a I (r 0 ) = arg max a I π I 1 (r 0 , r 0 , a I ). Proof of Proposition 8 (Buyer-skewed Network Effects, u S = 0). For later reference, note that when u S = 0 the integrated platform sets β I = β I and a I = a I . Simple computations show that

a I = - u B (16(v + r)(2 + γ)(4 + 3γ) + (16 + 3γ(4 + γ))(8 + γ(8 + γ))∆) 8(4 + 3γ)((4 + γ) 2 -2u 2 B (2 + γ))
.

Moreover, when u S = 0, a necessary condition to ensure that the maximization problem max (β I ,a I ) π I 1 (r 0 , β I r 0 , a I ) is concave is u B ≤ ūB = 2(2+γ)(32+γ(32+9γ)) (4+3γ)(12+γ(11+3γ)) . Impact on Manufacturer M 2 . Let π I 2 (β I r 0 , r 0 , a I ) be M 2 's profit under vertical integration when the sharing parameter is equal to β I and the developer fee is set at some value a I . Let π S 2 be M 2 's profit under separation. Let ∆π 2 (β I , a I ) = π I 2 (β I r 0 , r 0 , a I ) -π S 2 . Computations show that

(B.1) ∆π 2 (β I , a I ) = 2(2 + γ)[v(4 + 3γ) -r 0 (γ -4β I (1 + γ)) -a I u B (4 + 3γ)] 2 (4 + γ) 2 (4 + 3γ) 2 - 2(2 + γ)(v + r) 2 (4 + γ) 2 .
One can show that the term in brackets in the right-hand side of (B.1) is nonnegative, for otherwise M 2 's markup is negative under integration. Therefore, ∆π 2 (β I , a I ) has the same sign as

f (β I , a I ) = (v(4 + 3γ) -r 0 (γ -4β I (1 + γ)) -a I u B (4 + 3γ)) (4 + 3γ) -(v + r).
Then, computations show that

(B.2) f (β I , a I ) = 16(v + r)u 2 B (2 + γ)(4 + 3γ) -(4 + γ) 4γ(2 + γ)(4 + γ) -u 2 B (8 + 3γ)(8 + γ(8 + γ)) ∆ 8(4 + 3γ)((4 + γ) 2 -2(2 + γ)u 2 B )
.

For all u B ≤ ūB , computations show that the denominator in (B.2) is positive. Therefore f (β I , a I ) has the same sign as its numerator. Then, simple computations show that the numerator of f (β I , a I ) is increasing in u B , is negative when u B = 0 and positive when u B = ūB . It follows that there exists a unique ûB ∈ (0, u B ) such that f (β I , a I ), and thus ∆π 2 (β I , a I ) as well, is negative if and only if u B < ûB .

Impact on Buyers. Let V I B (β I r 0 , a I ) be the buyer surplus under vertical integration when the sharing parameter is equal to β I and the developer fee is set at some value a I . Let V S B be the buyer surplus under separation. Let ∆V B (β I r 0 , a I ) = V I B (β I r 0 , a I ) -V S B . Our first step is to show that V I B (β I r 0 , a I ) is increasing in u B . Omitting notations, we have

(B.3) dV I B du B (β I r 0 , a I ) = ∂V B ∂u B + ∂V B ∂β I dβ I du B + ∂V B ∂a I da I du B .
Computations show that the first term in Equation (B.3) is positive, the second term is nil since dβ I /du B = 0 and the third term is positive. This shows that V I B (β I r 0 , a I ) is increasing in u B , and so is ∆V B (β I r 0 , a I ) consequently.

Then, we notice that ∆V B (β I r 0 , a I ) is positive when u B = u B . Therefore, there exists ûB ≤ 0 such that ∆V B (β I r 0 , a I ) is nonnegative if and only if u B ≥ ûB . 48 Impact on Developers. When u S = 0, the developers surplus is simply given by V I S (β I r 0 , a I ) = a 2 I /2 and therefore ∆V S (β I r 0 , a I ) ≥ 0.

Proof of Proposition 9 (Developer-skewed Network Effects, u B = 0). When u B = 0, a necessary condition to ensure that the maximization problem max .

(β I ,a I ) π I 1 (r 0 , β I r 0 , a I ) is concave is u S ≤ ūS = 2(2+γ)(32+γ(32+9γ)) (4+3γ)(12+γ( 11+3γ
Impact on Manufacturer M 2 . Let π I 2 (β I r 0 , r 0 , a I ) be M 2 's profit under vertical integration when the sharing parameter is equal to β I and the developer fee is set at some value a I . Let π S 2 be M 2 's profit under separation. Let ∆π 2 (β I , a I ) = π I 2 (β I r 0 , r 0 , a I ) -π S 2 . We have π I 2 (β I r 0 , r 0 , a I ) = 2(2 + γ)(v(4 + 3γ) -a I u S γ + r 0 (4(1 + γ)β I -γ)) 2 (4 + γ) 2 (4 + 3γ) 2

48 One can additionally find conditions under which ûB is positive. If this is the case, ∆V B (β I r 0 , a I ) is negative in the neighborhood of u B = 0. The buyer surplus under separation is given by

V S B = (v + r) 2 (2 + γ) 2 (4 + γ) 2 .
Tedious calculations then show that: (i) ∆V I B (β I r 0 , a I ) is increasing in u S , (ii) ∆V I B (β I r 0 , a I ) is positive when u S = ūS , and (iii) that the sign of ∆V I B (β I r 0 , a I ) depends on the values of γ and ∆ when u S = 0. Indeed, when u S = 0, we have The previous equation shows that, for instance, when ∆ is small ∆V I B (β I r 0 , a I ) is positive. From the three observations above, we conclude that there exists a cutoff ũB S such that ∆V B (β I I r 0 , a I I ) is nonnegative if and only if u S ≥ ũB S . Impact on Developers. When u B = 0, calculations show that the developer surplus under integration is given by V I S (β I r 0 , a I ) = (2u S (v(2 + γ) + r 0 (1 + β I (1 + γ))) -a I (4 + γ -2u 2 S )) 2 2(4 + γ) 2 and the developer surplus under separation is given by

V S S = 2u 2 S (2 + γ) 2 (v + r) 2 (4 + γ) 2 .
As is standard in models with linear demands, V I S (β I r 0 , a I ) is proportional to the square of the developers demand. It follows that ∆V S (β I r 0 , a I ) = V I S (β I r 0 , a I ) -V S S has the sign of (2u S (v(2 + γ) + r 0 (1 + β I (1 + γ))) -a I (4 + γ -2u 2 S )) -2u S (2 + γ)(v + r). . Moreover, when u B = 0, a necessary condition to ensure that the maximization problem of platform I is concave is u 2 S ≤ ū2 S = 2(2 + γ)(32 + γ(32 + 9γ))/(4 + 3γ)(12 + γ(11 + 3γ)).

Impact on Manufacturer M 2 . Computations show that ∆π 2 (β I , a I ) has the sign of 3(v + α B )a I u S γ + 3(v + α B )γ + r(-1 + 4β I )(1 + γ), which is increasing in β I and decreasing in a I . Since, following integration and when u B = 0, β I (weakly) decreases and a I increases, we have that ∆π Figure 22: Impact of vertical integration on buyer surplus (V B ) for different degrees of substitutability (γ) and levels of the gain for buyers when manufacturers adopts the same operating system (α B ): V B decreases (resp. increases) following integration in the red area (resp. in the blue area). Figure 24: Impact of vertical integration on the non-integrated manufacturer's profit (π 2 ) for different degrees of substitutability (γ) and levels of the gain for buyers when manufacturers adopts the same operating system (α B ): V B decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 25: Impact of vertical integration on total welfare (W ) for different degrees of substitutability (γ) and levels of the gain for buyers when manufacturers adopts the same operating system (α B ): V B decreases (resp. increases) following integration in the red area (resp. in the blue area).

  Optimum. Let Π(p, a) = (p + r)D B (p, a) + aD S (p, a) denote the industry profit and W (p, a) = V B (p, D S (p, a)) + V S (a, D B (p, a)) + Π(p, a) denote welfare. Prices (p * , a * ) that maximize welfare are given by 14

  a) is characterized by D B (P (β, a), a) + (P (β, a) + βr) ∂D B ∂p (P (β, a), a) = 0. At stage 1, I chooses the share of per-user benefit β and the developer fee a so as to maximize its own profit, (1-β)rD B (p, a)+aD S (p, a), anticipating the price of devices chosen at stage 2 by M . That problem may be written as follows: max (β∈[0,1],a) Π(P (β, a), a)φ(P (β, a), a), where Π(p, a) is the industry profit and φ(p, a) = -(D B ) 2 (p, a)( ∂D B

  I +r p I +r>p S +r>0 p S +r>p I +r>0 (a) Margin on buyers p + r.

Figure 2 :

 2 Figure 2: Comparison of margins under Integration and under Separation with a monopoly platform and a monopoly manufacturer in the running example (with r = v = 1). Note: the grey area corresponds to non admissible values of the parameters (u B + u S < 2).

Figure 3 :

 3 Figure 3: The model with competing platforms and competing manufacturers.

Figure 4 :

 4 Figure 4: The solution of the relaxed problem (β * I , a * I ) in the running example (with v = r = 1).

Figure 6

 6 Figure 6 represents graphically Proposition 6 and Proposition 7.

Figure 7 :

 7 Figure7: Impact of vertical integration on the buyers surplus (V B , left panel) and on total welfare (W , right panel): V B and W decrease (resp. increase) following integration in the red area (resp. the blue area).

Figure 8 :

 8 Figure8: Impact of vertical integration on buyers surplus (V B ) for different degrees of substitutability between manufacturers (γ): V B decreases (resp. increases) following integration in the red area (resp. the blue area).

  and 12 respectively).

Figure 10 :

 10 Figure10: Impact of vertical integration on the non-integrated manufacturer's profit (π 2 ) for different levels of the efficiency gains (r 0 -r): π 2 decreases (resp. increases) following integration in the red area (resp. the blue area).

Figure 11 :

 11 Figure11: Impact of vertical integration on buyer surplus (V B ) for different levels of efficiency gains (r 0 -r): V B decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 12 :

 12 Figure12: Impact of vertical integration on developer surplus (V S ) for different levels of efficiency gains (r 0 -r): V S decreases (resp. increases) following integration in the red area (resp. the blue area).

Figure 13 :

 13 Figure13: Impact of vertical integration on buyer surplus (V B ) for different levels of the gain for buyers when manufacturers adopt the same operating system (α B ): V B decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 15 :

 15 Figure15: Impact of vertical integration on the non-integrated manufacturer's profit (π 2 ) for different levels of the gain for buyers when manufacturers adopt the same operating system (α B ): π 2 decreases (resp. increases) following integration in the red area (resp. the blue area).

  is a contraction mapping and Equation (A.2) has a unique solution. It follows that D B (p, a) = Q B (p, D S (p, a)) is uniquely defined. D S (p, a) = Q S (u S D B (p, a)-a) is thus uniquely defined as well.

(.

  A.3) D S (p, a) = Q S (u S (Q B (p, D S (p, a)). By the implicit function theorem, D S (p, a) is continuously differentiable. Differentiating Equation (A.3) with respect to a and rearranging terms, we obtain (omitting some arguments) By Assumption A.1, the denominator is positive, and therefore, ∂D S /∂a has the sign of ∂Q S /∂a, which is negative. Since D B (p, a) = Q B (p, D S (p, a)) and ∂Q B /∂n S > 0, ∂D B /∂a is also negative. Benchmarks: Social Optimum, Ramsey Pricing and Monopoly Pricing. Remind that Π(p, a) = (p + r)D B (p, a) + aD S (p, a) and W (p, a) = V B (p, D S (p, a)) + V S (a, D B (p, a)) + Π(p, a). Consider the following change of variables: φ : (p, a) → (n B , n S ) = (D B (p, a), D S (p, a)). It is a C 1 -diffeomorphism since, under Assumption A.1, the system of equations n B = Q B (p, n S ) and n S = Q S (u S n B -a) has a unique solution, namely (n B , n S ) = (D B (p, a), D S (p, a)), in the relevant range of parameters. Therefore, instead of optimizing wrt (p, a) we can optimize wrt (n B , n A ). Let (P (n B , n S ), A(n B , n S )) = φ -1 (n B , n S ).

Let(.

  Den denote the denominator in(A.6). Since λ > 0 and the numerator in (A.6) is positive, we must have Den < 0. Then, by combining Equations (A.4) and (A.5), we obtain the following expression for the margin P + r Since the denominator is negative, Equation (A.7) shows that P + r has the sign of f (n B , n S ) = ∂U S ∂n B (n B ∂P ∂n S + n S ∂A ∂n S ) -∂U B ∂n S (n B ∂P ∂n B + n S ∂A ∂n B ). Then, using the fact that ∂P ∂n B = 1 ∂Q B /∂p , ∂P ∂n S = -∂Q B /∂n S ∂Q B /∂p , ∂A ∂n B = -∂Q S /∂n B ∂Q S /∂a and ∂A ∂n S = 1 ∂Q S /∂a , we have after rearranging terms

(A. 8 )

 8 We can use again the fact that ∂P ∂n B =1∂Q B /∂p, ∂P ∂n S = -∂Q B /∂n S ∂Q B /∂p , ∂A ∂n B = -∂Q S /∂n B ∂Q S /∂a and ∂A ∂n S = 1 ∂Q S /∂a , as well as the notations η B = -1 n B ∂Q B ∂p (p, n S ) and η S = -1 n S ∂Q S∂a (a, n B ), to obtain the characterization of the optimal monopoly prices (p m , a m ) given in Section 3.

V(

  B (p, p, D S (p, a)) + V S (a, D B (p, a)) + Π(p, a). Ramsey prices solve the following problem

  Consider the following change of variables: φ : (p, a) → (n B , n S ) = (D B (p, a), D S (p, a)). It is a C 1 -diffeomorphism since, under Assumption A.2, the system of equations n B = Q 1 B (p, p, n S ) + Q 2 B (p, p, n S ) and n S = Q S (u S n B -a) has a unique solution, namely (n B , n S ) = (D B (p, a), D S (p, a)), in the relevant range of parameters. Let (P (n B , n S ), A(n B , n S )) = φ -1 (n B , n S ). Problem (A.17) then rewrites as follows max (n B ,n S )

Figure 16 :

 16 Figure 16: Necessary and sufficient conditions for the concavity of the maximization problem in the running example.

  ]. The curve a * * I = 0 is given by -(2(σ + 2)(3σ + 4)u B -(3σ(σ(σ + 7) + 16) + 32)u S ) = 0 and can be expressed as a function of (u S , σ) as (3σ(σ(σ + 7) + 16) + 32)u S -(σ + 2)(3σ + 4)(γσ)/((γ + 1)u S ) = 0, which has a unique positive root ûa ). Then, the set of (u B , u S ) such that a * * I = 0 is characterized by the parametric equations (u B = ûa * *I =0 B (σ), u S = ûa * * I =0 S (σ)) for all σ in the relevant range. We can check that ûa * * I =0 S

  which shows that the curve a * * I = 0 lies below the 45 • -degree line in the (u B , u S )-space. Its derivative is given by σ + 2)(3σ + 4)(γ(3σ(σ(σ(3σ + 20) + 46) + 48) + 64) + σ(3σ(σ(11σ + 80) + 200) + 640) + 256) > 0 and J = (3σ(σ(σ + 7) + 16) + 32)((3σ(σ + 4) + 8)(3σ(σ(2σ + 9) + 16) + 32) -γ(3σ(σ(σ(3σ + 20) + 46) + 48) + 64)). The denominator J is equal to zero when γ = [(3σ(σ + 4) + 8)(3σ(σ(2σ + 9) + 16) + 32)]/[3σ(σ(σ(3σ + 20) + 46) + 48) + 64]. The right-hand side is strictly increasing in σ and is equal to 4 for σ = 0. Hence, if γ < 4, the slope is always positive. If γ ≥ 4, there is a vertical asymptote and the slope is first positive and then negative.

  set of (u B , u S ) such that β * I = β I is described by the parametric curve associated to the parametric equations (u B = ûβ *

  σ)), for σ ∈ [0, γ], cannot cross each other when r 0 > r, for otherwise we would have β I = 1, which is impossible. Computations then show that ûβ * in the (u B , u S )-space. To conclude, we have the following in the (u B , u S )-space: the constraint β I ≤ 1 is binding at optimum above the parametric curve (û for σ ∈ [0, γ]; none of the constraints β I ≤ 1 and β I ≥ β I are binding at the optimum between the two parametric curves; the constraint β I ≥ β I is binding at the optimum below the parametric curve (û for σ ∈ [0, γ].

  )) . We know from Section 4.2 that, first, M 2 's participation constraint is binding when u S ≤ûβ * I =β I S (γ) and, second, that the constraint β I ≤ 1 is binding when u S ≥ ûβ * γ) -γ 2 ∆ (8(1 + γ) (r + ∆) , a I = 8u S (r + v)(4 + 3γ)(32 + 3γ(16 + γ(7 + γ))) -u S (-512 + γ(4 + γ)(-256 + γ(4 + γ) (-20 + 9γ)))∆ 8(4 + γ) 2 (4 + 3γ) 2 -16u 2 S (32 + γ(64 + γ(40 + 7γ))) 3γ)(-16 -3γ(4 + γ) + u 2 S (12 + γ(11 + 3γ))) + r(64 + γ(96 + γ(52 + 9γ))) + (64 + γ(96 + γ(52 + 9γ)))∆ 2(2 + γ)(32 + γ(32 + 9γ)) -u 2 S (4 + 3γ)(12 + γ(11 + 3γ)))(r + ∆) 3γ)(12 + γ(11 + 3γ))(r + v + ∆) 2(2 + γ)(32 + γ(32 + 9γ)) -u 2 S (4 + 3γ)(12 + γ(11 + 3γ)) , a I (r 0 ) = u S (4 + 3γ)(32 + 3γ(16 + γ(7 + γ)))(r + v + ∆) (4 + γ) 2 (4 + 3γ) 2 -2u 2S (32 + γ(64 + γ(40 + 7γ)))

  Tedious calculations then show that: (i) π I 2 (β I I r 0 , r 0 , a I I ) is decreasing in u S when u S ≤ ûβ * ; (iii) π I 2 (β I I r 0 , r 0 , a I I ) is decreasing in u S when u S > ûβ * I =1 S (γ); (iv) ∆π 2 (β I I , a I I ) < 0 when u S = 0. Together, since ∆π 2 (β I I , a I I ) is continuous in u S , these observations show that ∆π 2 (β I I , a I I ) is positive for some parameters values only if ∆π 2 (β I I , a I I ) > 0 when u S = ûβ * I =1 S (γ). Put differently, there exists an interval [u M 2 S , ūM 2 S ] such that: (i) if ∆π 2 (β I I , a I I ) > 0 when u S = ûβ * Impact on Buyers.When u B = 0, calculations show that the buyer surplus under integration is given by(B.4) V I B (β I r 0 , a I ) = 1 (4 + γ) 2 (4 + 3γ) 2 (2 + γ)(2a I u S v(4 + 3γ) 2 + v 2 (2 + γ)(4 + 3γ) 2 + a 2 I u 2 S (16 + γ(16 + γ)) + r 2 0 (16 + 16β I γ(1 + γ) + γ(16 + γ) + β 2 I (1 + γ)(16 + γ(16+9γ))) + 2r 0 (v(4 + 3γ) 2 (1 + β I + β I γ) + a I u S (16 + γ(16 + γ + 8β I (1 + γ))))).

  ∆V I B (β I r 0 , a I ) = 2 + γ (64(1 + γ)(4 + γ) 2 (4 + 3γ) 2 ) (64(r + v) 2 (1 + γ)(2 + γ)(4 + 3γ) 2 -16(r + v)(1 + γ)(4 + 3γ) 2 (-8 + γ 2 )∆ + (1024 + γ(2048 + γ(1088 + γ(-64 + γ(-112 + γ(16 + 9γ))))))∆ 2 )).

.

  Tedious calculations then show that: (i) ∆V S (β I I r 0 , a I I ) is negative when u S ≤ ûβ * I =β I S (γ), (ii)∆V S (β I I r 0 , a I I ) is increasing in u S when u S ≥ ûβ * I =1 S (γ), and (iii) that the sign of ∆V I S (r 0 , a I (r 0 )) depends on the values of γ and ∆ when u S = u S . Indeed, when u S = u S , simple calculations show that ∆V I S (r 0 , a I (r 0 )) has the sign of (B.5) -(r + v)(4 + 3γ)(32 + 3γ(16 + γ(7 + γ)))(64 -γ(16 -3γ(20 + 3γ(5 + γ))))+ (4 + γ)(6144 + γ(22016 + γ(33728 + γ(29008 + γ(15316 + γ(5021 + 9γ(106 + 9γ)))))))∆, which can be positive or negative depending on the values of γ and the efficiency gain. From the observations above, we conclude that there exists a cutoff ũS S such that ∆V S (β I I r 0 , a I I ) is nonnegative if and only if u S ≥ ũS S .B.2. Impact of Vertical Integration with Coordination Motives: Polar CasesProof of Proposition 10. As shown in Appendix A.8, when u S = 0, β * I = βI and a * I = ãI . Moreover, when u S = 0, a necessary condition to ensure that the maximization problem of platformI is concave is u 2 B ≤ ū2 B = 2/3(3 + 4/(4 + 3γ) + γ/(12 + γ(11 + 3γ))). Impact on Manufacturer M 2 . Let ∆π 2 (β I , a I ) = πI 2 (β I , 1, a I ) -π S 2 . Simple computations show that ∆π 2 (β I , a I ) has the sign of 3(v +α B )(1+γ)+r(-1+ 4β I )(1+γ)-a I u B (4+ 3γ).Evaluating this expression at a I = ãI and β I = βI and differentiating with respect to u B , we find that d∆π 2 ( βI , ãI )/du B has the sign of (uB (4 + γ) 2 (4 + 3γ)(8r(2 + γ) + 8v(2 + γ) + α B (4 + γ)(8 + 3γ)))/(2((4 + γ) 2 ) 2 -2u 2 B (2 + γ)), which is positive. Put differently, ∆π 2 ( βI , ãI ) is increasing in u B . Then, computations show that, when α B > 0, ∆π 2 ( βI , ãI ) evaluated at u B = 0 and at u B = ūB is negative and positive respectively. Therefore, by continuity, there exists û∆π 2 =0 B > 0 such that ∆π 2 ( βI , ãI ) is negative if and only u B < û∆π 2 =0 B Impact on Buyers. Let V S B and V I B ( βI , ãI ) denote respectively the buyers surplus under separation and integration. Let ∆V I B ( βI, ãI ) = V I B ( βI , ãI )-V S B . Computations show that d∆V I B ( βI , ãI )/du B is equal to (u B (2 + γ)(4 + γ)(8r(2 + γ) + 8v(2 + γ) + α B (4 + γ)(8 + 3γ))(4r(2 + γ)(4 + γ) + 4v(2 + γ)(4 + γ) + α B (3u 2 B (2 + γ) 2 + (4 + γ) 2 )))/(4(-2u 2 B (2 + γ) + (4 + γ) 2 ) 3 ).Computations then show that the denominator in the previous expression is positive when u B ≤ ūB . Therefore, d∆V I B ( βI , ãI )/du B ≥ 0 and ∆V I B ( βI , ãI ) is increasing in u B . Then, computations show that, when α B > 0, ∆V I B ( βI , ãI ) evaluated at u B = 0 and u B = ūB is negative and positive respectively. Therefore, by continuity, there exists û∆V I B =0 B > 0 such that ∆V I B ( βI , ãI ) is negative if and only u B < û∆V I B =0 B . Impact on Developers. When u S = 0, the developers surplus is simply given by V I S (β I , a I ) = a 2 I /2 and therefore ∆V S ( βI , ãI ) ≥ 0. Proof of Proposition 11. As shown in Appendix A.8, when u B = 0, there exists thresholds ûβ * solution of platform I's maximization problem under integration is: β * * I = βI and a * * I = ãI when u S ≤ ûβ *

  2 (β * * I , a * * I ) ≤ 0. Impact on Buyers. In the following, we are going to show that ∆V I B (β * * I , a * * I ) is increasing in u S . Differentiating with respect to u S , we have d∆V I B (β * * I , a * * I )/du S = ∂V I B /∂u S + (∂V I B /∂β I )(dβ * * I /du S ) + (∂V I B /∂a I )(da * * I /du S ), omitting arguments for the sake of conciseness. Then, computations show:(i) ∂V I B /∂u S = (1/((4 + γ) 2 (4 + 3γ) 2 ))2a I (2 + γ)((v + α B )(4 + 3γ) 2 + a I u S (16 + γ(16 + γ)) + r(16 + γ(16 + γ + 8β I (1 + γ)))), which is positive since a I = a * * I ≥ 0 when u B = 0; (ii) ∂V I B /∂β I = (1/((4 + γ) 2 (4 + 3γ) 2 ))2r(1 + γ)(2 + γ)(16(v + α B + rβ I ) + 8(r + a I u S + 3(v + α B ) + 2rβ I )γ + 9(v + α B + rβ I )γ 2 ), which is positive; (iii) ∂V I B /∂a I = (1/((4 + γ) 2 (4 + 3γ) 2 ))2u S (2+γ)((v +α B )(4+3γ) 2 +a I u S (16+γ(16+γ))+r(16+γ(16+γ +8β I (1+γ)))),which is positive. Then, computations show that dβ * I /du S = (2u S (r + v + α B )(4 + 3γ)(12 + γ(11 + 3γ))(64 + γ(96 + γ(52 + 9γ))))/(r(u 2 S (4 + 3γ)(12 + γ(11 + 3γ)) -2(2 + γ)(32 + γ(32 + 9γ))) 2 ) > 0 and, since dβ * * I /du S = 0 when β * * I ̸ = β * I , we thus have dβ * * I /du S ≥ 0 for all u S . Similarly, computations show that da * * I /du S ≥ 0 for all u S . Together, this shows that ∆V I B (β * * I , a * * I ) is increasing in u S . Then, computations show that, when α B > 0, ∆V I B (β * * I , a * * I ) is negative when u S = 0 and ∆V I B (β * * I , a * * I ) is positive when u S = ūS . Therefore, by continuity, there exists û∆V I B =0 S > 0 such that ∆V I B (β * * I , a * * I ) < 0 if and only if u S < û∆V I B =0 S . Impact on Developers. Computations show that ∆V I S (β I , a I ) has the sign of a I (-4 + 2u 2 Sγ) + 2ru S (-1 + β I )(1 + γ) which is increasing in β I and decreasing in a I if 2u 2 S < 4 + γ. Computations show that the last inequality holds when u S < ûβ * I =1 S . Since β I decreases and a I increases following integration when u B = 0, we therefore have that ∆V I S (β * * I , a * * I ) ≤ 0 when u S < ûβ * I =1 S . When u S ≥ ûβ * I =1 S , ∆V I S (β * * I , a * * I ) has the sign of -((u S (r + v + α B )(-4 + 2u 2 Sγ)(4 + 3γ)(32 + 3γ(16 + γ(7 + γ))))/(-(4 + γ) 2 (4 + 3γ) 2 + 2u 2 S (32 + γ(64 + γ(40 + 7γ))))). Simple calculations then show that this last expression is increasing in u S and negative in u S = ūS . Therefore, ∆V I S (β * * I , a * * I ) < 0 when u S ≥ ûβ * I =1 S .

Figure 18 :

 18 Figure18: Impact of vertical integration on buyer surplus (V B ) for different degrees of substitutability (γ) and levels of efficiency gains (r 0 -r): V B decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 19 :

 19 Figure19: Impact of vertical integration on developer surplus (V S ) for different degrees of substitutability (γ) and levels of efficiency gains (r 0 -r): V S decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 20 :

 20 Figure20: Impact of vertical integration on the non-integrated manufacturer's profit (π 2 ) for different degrees of substitutability (γ) and levels of efficiency gains (r 0 -r): π 2 decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 21 :

 21 Figure21: Impact of vertical integration on total welfare (W ) for different degrees of substitutability (γ) and levels of efficiency gains (r 0 -r): W decreases (resp. increases) following integration in the red area (resp. in the blue area).

Figure 23 :

 23 Figure23: Impact of vertical integration on developers surplus (V S ) for different degrees of substitutability (γ) and levels of the gain for buyers when manufacturers adopts the same operating system (α B ): V B decreases (resp. increases) following integration in the red area (resp. in the blue area).

  (a) γ = 1, α B = 0 (b) γ = 1, α B = 0.2 (c) γ = 1, α B = 0.4 (d) γ = 4, α B = 0 (e) γ = 4, α B = 0.2 (f) γ = 4, α B = 0.4 (g) γ = 8, α B = 0 (h) γ = 8, α B = 0.2 (i) γ = 8, α B = 0.4

.

  Since the denominator is negative, Equation(A.22) shows that P +r has the sign of f (n B , n S ) = ∂Q B /∂n S ∂Q B /∂p (where, with a slight abuse of notations, ∂Q B /∂p = ∂Q B /∂p 1 + ∂Q B /∂p 2 ), ∂A ∂n B = -∂Q S /∂n B ∂Q S /∂a and ∂A ∂n S =

	Then, combining Equations (A.19) and (A.20), we obtain λ = -n S ∂U B ∂n S + n B ∂U S ∂n B n 2 B ∂P ∂n B + n 2 S ∂A ∂n S + n B n S ∂P ∂n S + ∂A ∂n B Let Den denote the denominator in (A.21). Since λ > 0 and the numerator in (A.21) is positive, (A.21) . we have Den < 0. Then, by combining Equations (A.19) and (A.20), we obtain the following expression for the margin P + r (A.22) P + r = n S ∂U B ∂n S (n B ∂P ∂n B + n S ∂A ∂n B ) -∂U S ∂n B (n B ∂P ∂n S + n S ∂A ∂n S ) Den -n S ∂U B ∂n S -n B ∂U S ∂n B ∂U S ∂n B (n B ∂P ∂n S + n S ∂A ∂Q S /∂a , we have after rearranging terms f (n B , n S ) = 1 η B ∂U B ∂n S + ∂U S ∂n B ∂Q B ∂n S -1 η S ∂U S ∂n B + ∂U B ∂n S ∂Q S ∂n B , ∂n 1 where η B = -1 n B ∂Q B n S ∂p (p, n S ) and η S = -1 ∂Q S

S ) -∂U B ∂n S (n B ∂P ∂n B + n S ∂A ∂n B ). Then, noticing that ∂P ∂n B = 1 ∂Q B /∂p and ∂P ∂n S = -

We focus on the literature that determines under which circumstances vertical integration creates some market power and leads to harmful foreclosure of non-integrated competitors. Another strand, following[START_REF] Hart | Vertical Integration and Market Foreclosure[END_REF], shows that vertical integration may be used as a mean to restore the upstream market power that was eroded by a lack of commitment; see[START_REF] Rey | A Primer on Foreclosure[END_REF] and[START_REF] Riordan | Competitive Effects of Vertical Integration[END_REF] for surveys.

This assumption will be relaxed in Section 5 where we show that porting costs create a specific source of market power.

This was originally discussed by[START_REF] Edgeworth | The Theory of Pure Monopoly[END_REF] in the context of excise taxes on a subset of the goods sold by a multiproduct firm.[START_REF] Luco | The Competitive Impact of Vertical Integration by Multiproduct Firms[END_REF] measure the causal effects of vertical integration on prices caused by vertical mergers in the carbonated-beverage industry. They find that vertical integration caused a decrease in the prices of products with eliminated double margins, but a price increase on average.

The constraints β ≥ 0 and β ≤ 1 may be binding at the optimum under separation. See Appendix A.1 for the detailed analysis in the case of the running example.

This is[START_REF] Shubik | Market Structure and Behavior[END_REF]'s linear demands system, to which we append indirect network effects additively.

For both cases, we adopt implicitly the same assumptions on best responses as those made in Section 3.3 to ensure that the price competition subgame is 'well-behaved.'

Formally, the integrated firm's best response in price changes from D 1 + (p 1 + r)∂D 1 /∂p 1 + a I ∂D S /∂p 1 = 0 when M 2 buys from the fringe, to D 1 + (p 1 + r)∂D 1 /∂p 1 + a I ∂D S /∂p 1 + (1β I )r∂D 2 /∂p 1 = 0 when M 2 chooses the integrated firm's operating system. Since devices are demand substitutes, ∂D 2 /∂p 1 ≥ 0 and the integrated firm's best response in price shifts upward when it licenses its operating system.

 29 In the main text (but not in the numerical simulations), we neglect the constraint β I ≥ 0. When it binds, it simply means the integrated firm wants to extract as much as possible of the per-user benefit associated to licensing its operating system to the non-integrated manufacturer.

Some conditions are required to ensure that the maximization problems are quasi-concave. In the running example, these conditions are fully characterized in Appendix A.4.

Remind that under our assumption the profit of a manufacturer is increasing in its sharing parameter.

In Appendix A.6, we show that h B (u B ) = ((γ + 2)u B )/(2(γ + 1)u 2 B + 1) and h S (u B ) = (1/2)( 2(γ + 4) + (γ + 3) 2 u 2 B -(γ + 3)u B ).

In Appendix A.2, we characterize the Ramsey optimum with competing manufacturers. For the running example, the Ramsey optimum is such that p 1 + r = p 2 + r ≥ 0 and a ≤ 0 iff u B ≥ 2u 3 S .

In Proposition 7, the conditions for the buyer surplus to increase are sufficient only. In Appendix A.6, we show that the intermediate region {(u B , u S ) : u B ≤ h(u S ) and u S ≤ h B (u B )} can be divided into two subsets, one in which the buyer surplus increases and the other in which it decreases. The frontier between these two subsets is represented by the red dashed curve in Figure6.

The Python code of the simulations is available on the authors' webpages, as well as other simulations. If otherwise not specified, we use the following set of parameters values: γ = 4, v = 2, r = 0.5.

The Python code of the simulations is available on the authors' webpages, as well as other simulations. Unless specified in the figure caption, we use the following set of parameters values: γ = 4, v = 2, r = 0.5.

iMessage, FaceTime and Airdrop are prime examples for iOS.

One could argue similarly that developers care about the community of programmers using a particular operating system because this may help lowering development costs.

 41 This formulation where only buyers received a fixed benefit when manufacturers choose the same operating system is chosen for its

tractability.42 Although intuitive, this property does not always hold; see the discussion

in footnote24. 43 This profit is defined as before (see Section 3.5) except that demands (D 1 , D 2 , D S ) are replaced by ( D1 , D2 , DS ).

The Python code of the simulations is available on the authors' webpages, as well as other simulations. As in Section 4.4, we use the following set of parameters values: γ = 4, v = 2, r = 0.5.

Mainly Java, C, C++ and Kotlin for the former and Objective C or Swift for the latter.

The Python code of the simulations is available on the authors' webpages. We use the following value for parameter v: v = 2.

This work was supported by a grant overseen by the French National Research Agency (ANR-12-BSH1-0009), by the Cepremap (Paris) and by the Labex MME-DII (ANR11-LBX-023-01). All remaining errors are ours.

can compute profit and surpluses: Π

(2-(u B +u S )) 2 (2+(u B +u S )) 2 , V I S =

(r+v) 2 (u B +u S ) 2 2(2-(u B +u S )) 2 (2+u B +u S ) 2 . Separation. Consider last the case of separation. At stage 2 of the game, M chooses its price p to maximize its profit π M (p) = (p + βr)D B (p, a). This profit is strictly concave in p when u B u S < 1. Hence, M 's best response is uniquely characterized by the first-order condition, which yields P (β, a) = 1 2 (v -βr -au B ). At stage 1 of the game, I chooses (β, a) to maximize its profit π I (β, a) = (1 -β)rD B (P (β, a), a) + aD S (P (β, a), a). π I (β, a) is strictly concave in (β, a) iff 8 -u 2 B -6u B u S -u 2 S = 4(2 -u B u S ) -(u B + u S ) 2 > 0, a condition which is implied by the concavity condition under integration (u B + u S ) 2 < 4; hence, we assume from now on that (u B + u S ) < 2 in order to carry meaningful comparisons between integration and separation.

Solving for the first-order conditions with respect to β and a leads to

,

S > 0 by assumption. Notice that three cases have to be considered:

The previous optimization did not take into account the constraint β ∈ [0, 1]. We have that

. I's optimal price under separation is thus characterized as follows

where (1) corresponds to parameters values such that vf (u S , u B ) ≥ rf (u B , u S ); (2) corresponds to f (u S , u B ) ≥ 0 and vf (u S , u B ) ≤ rf (u B , u S ); and (3) corresponds to f (u S , u B ) ≤ 0 . M 's equilibrium price is thus given by p S = P (β S , a S ), or

We can then compute profit and surpluses under separation:

Running Example: Comparison between Integration and Separation. We can now compare prices under integration (p I , a I ) with those under separation (p S , a S ). We assume that u B + u S < 2, which ensures that the optimization problem is strictly concave both under integration and under separation. This also implies that f

To streamline the exposition, we limit ourselves to the case v = r = 1; the condition vf (u S , u B ) ≶ rf (u B , u S ) then boils down to u B ≶ u S . Price comparisons.

-Consider Case (1) or u B ≥ u S . Then, p I -p S = -

. This expression can be either positive or negative for the relevant values of parameters.

, which is negative under our assumptions.

, which is positive under our assumptions. Profit comparison. As far as the industry profit is concerned, a simple revealed preference argument shows that this profit is always larger under integration than under separation since the integrated structure can replicate any (p, a) implemented by the separated entities. Comparisons of the surplus of buyers.

-Case (1).

2 , which is positive under our assumptions.

-Case (3).

In that case, the assumption is that 4 -3u B u S ≤ u 2 S (which implies u S ≥ u B ) and u B u S < 1. Hence,

. Then,

2 ≥ 0, which always holds under our assumptions.

-Case (2).

, which holds under our assumptions.

-Case (3).

1-u B u S , which always holds under our assumptions.

From Equation (A.12), the developer demand solves

By the implicit function theorem, D S (a, p 1 , p 2 ) is continuously differentiable. Differentiating Equation (A.14) with respect to a and rearranging terms, we find (omitting some arguments)

.

By Assumption A.2, the denominator is positive, and therefore, ∂D S /∂a has the sign of ∂Q S /∂a,

Assumptions in the Running Example. Assumption A.2 writes as 2u B u S < 1. Solving for n k B , k = 1, 2, and n S in (3.3), we obtain .16) where σ = γ -2u B u S (1 + γ), which is positive by assumption. From (A.15), and since σ ≥ 0 and 0 ≤ 2u B u S < 1, D k is strictly decreasing in p k and a. Manufacturers' products are demand substitutes since ∂D k /∂p ℓ = σ/(2(1 -2u B u S )) ≥ 0. From (A.16), we have that D S is decreasing in p 1 , p 2 and a. Finally, the direct price effect is stronger than the indirect one since -∂D k /∂p k -∂D k /∂p ℓ = 1/(1 -2u B u S ) > 0.

We now check that the price competition subgame is 'well-behaved' in the running example. Consider the separation benchmark. Developers pay a total fee a, and M k earns a profit equal to

< 0, so that π k is strictly concave in p k and M k 's best response is uniquely characterized by the first-order condition ∂π k /∂p k = 0. The best response is given by R

We then have 0 < ∂R k /∂p ℓ < 1 and ∂R k /∂a ≤ 0. Equilibrium prices are given by pk = (-2au B (4 + 3σ) + v(8 + 6σ) + (2 + σ)(-β ℓ rσ -2β k r(2 + σ)))/((4 + σ)(4 + 3σ)). We obtain then that: ∂ pk /∂β k = -r(2(2 + σ) 2 )/((4 + σ)(4 + 3σ)), which belongs to (-2r/3, r/2] for σ ≥ 0; ∂ pk /∂β ℓ = -r(σ(2 + σ))/((4 + σ)(4 + 3σ)), which belongs to (-1/3, 0] for σ ≥ 0; and ∂ pk /∂a = -2u B (4+σ) ≤ 0. Last, we check that these assumptions are satisfied when I and M 1 are integrated and the integrated platform sets a developer fee a I and a sharing parameter β I for M 2 . I's profit writes as (p

, which is increasing with a slope smaller than 1 in p 2 and decreasing in a I . M 2 's profit writes as (p 2 + β I r)D 2 and its best response is given by R 2 (p 1 , β I , a I ) = (2(v -β I r) + σ(p 1 -β I r) -2u B a I )/(2(2 + σ)). Those best responses satisfy our assumptions. Additionally, equilibrium prices are given by p I 1 = -β I r + (-r -a I (2u B + u S ) + 2v + 3β I r)/(4 + σ) + (-r -a I u S + β I r)/(4 + 3σ) and p I 2 = -β I r + (-r -a I (2u B + u S ) + 2v + 3β I r)/(4 + σ) + (r + a I u S -β I r)/(4 + 3σ). These prices are decreasing in β I at a rate smaller than r and decreasing in a I .

Ramsey Prices in the General Case. Note that we can assume that p 1 = p 2 ≡ p without loss of generality. To ease the exposition, consider the following notations: