We study the dynamics of competition in a model with network effects, an incumbent and entry. We propose a new way of representing the strategic advantages of incumbency in a static model and embed it in a dynamic framework with heterogeneous consumers. We completely identify the conditions under which inefficient equilibria with two platforms emerge at equilibrium; explore the reasons why these inefficient equilibria arise; compute the profits of the incumbent and demonstrate that the incumbency advantage does not improve much, if at all, when going from a static to a dynamic framework.

Introduction

"A standard concern is that if a platform becomes dominant, there may be dynamic inefficiencies because users are coordinated and locked-in to a single platform. It may be difficult for an innovative new platform to gain market share, even if its underlying attributes and technology are better. This concern has helped to motivate antitrust actions in industries such as operating systems and payment cards. . . Indeed, even in industries such as social networking, where one might expect positive feedback effects to generate agglomeration, it is easy to point to examples of successful entry (Twitter) or rapid decline (MySpace)." [START_REF] Levin | The economics of internet markets[END_REF] Although the importance of competition for as opposed to in the market has been stressed by economists studying the new information technologies, relatively little work has been done to explore the strategies used by platforms to maintain their dominant positions in the presence of network externalities, and, when they are able to do so, to determine the value of these dominant positions. To help remedy this gap, we study a dynamic market with network effects and free entry (comparisons with the extant literature can be found in Section 8). We are, in effect, analyzing the following conversation between competition authorities and a dominant platform:

-Because of the network externalities in your industry, you yield enormous market power and can extract large profits from your clients.

-You forget that if I try to use this market power, entrants will be able to convince my consumers to join their platform.

-This is your standard argument, but according to your reasoning, these entrants will be scared of entry in future periods, and therefore will not be aggressive.

In order to appraise the argumentation of the two parties in this dialogue, we construct a model in which at the outset a single platform controls the market. There are positive network externalities so that consumers prefer to be on the same platform as other consumers. We study dynamic competition, assuming that there are (at least two) potential entrants in each of an infinite sequence of periods, 1 both in models where consumers are all similar to each other, and in models where consumers are heterogenous, with two types of consumers. We derive bounds on the intertemporal profit of the incumbent and compare it to the profit which would be computed using a static one period model. When consumers are heterogenous, we characterize the conditions under which there is one platform or (inefficiently) several platforms in equilibrium.

This analysis yields rich insights about the strategies used by platforms to defend their incumbency advantage. In particular, we are able to pinpoint the roles of consumers whose willingness to pay is low, but who are valuable as their presence increases what other consumers are willing to pay.

We are also able to throw light on some of the difficulties of competition policy when network externalities are present. Competition has a beneficial consequence: it lowers prices which benefits consumers. It potentially also has a harmful consequence: it can lead to the choice of different platforms by different groups of consumers, which is inefficient in the presence of positive network externalities. Our full characterization of the circumstances under which two platforms can co-exist at equilibrium in a dynamic model (Proposition 3) provides some useful lessons. First, static models have a tendency to exaggerate the extent of inefficiencies: they predict the separation of consumers on different platforms more often than do dynamic models. Thus, it is important to think through the long run consequences of competition. Second, in our framework, the presence of entrants generally increases social welfare. Not only does it decrease prices and ensures that more consumers join a platform, it also makes it more likely that they join the same platform. This seems to indicate that entry deterring strategies by an incumbent typically cannot be justified on the grounds that they help consumers coordinate on the same platform.

Our analysis also allows us to compare the value of incumbency in static and dynamic settings. As in switching cost models, we find that if consumers are all identical to each other, or if their preferences are not too different, then the profits which would be computed for the incumbent in a one period model are exactly the same as those obtained from a fully dynamic model. In the first period, entrants price low enough in order to try to attract clients that competition "eats up" all the incumbent's future profits. On the other hand, when consumers are heterogenous, the one period model underestimates the profits obtained through a fully dynamic model. However, this difference is relatively limited and in any case the dynamic profits are always strictly less than the value of a flow of one period profit. The value of incumbency is more limited than what a naive analysis would predict. This should give policy makers pause before they react too aggressively in markets with network externalities.

For the purposes of conducting this analysis, we develop a new and, we believe, more convenient way to represent the reluctance of consumers to migrate from one platform to the other. In policy discussions economists often argue that network effects make consumers reluctant to migrate and hence provide a strong advantage to incumbents: Levin's statement that "It may be difficult for an innovative new platform to gain market share, even if its underlying attributes and technology are better"provides a typical example. However, formal models of competition between platforms do not naturally lead to this conclusion. Absent switching costs, there is no reason for all the members of an incumbent platform not to purchase from a new entrant who would offer better conditions. If all consumers are willing to pay a premium of more than 10e to belong to the same platform as the other consumers, and an incumbent platform charges 25e while an entrant offers its service at a price of 15e, it is an equilibrium for the consumers to all purchase from the incumbent, but it is also a (Pareto better from their point of view) equilibrium for all of them to purchase from the entrant. The characterization of network externalities as "social switching costs" is due to our intuition that in many such cases the most likely equilibrium is for the consumers to purchase from the incumbent.

Without formalizing this intuition it is impossible to study the constraints that potential entry puts on the strategies of incumbents, but, as we discuss in our literature review in section 8, much of the small amount of work which has been conducted on this issue tackles the problem by modelling the belief of consumers. This approach has been used in a dynamic model by [START_REF] Ha Laburda | Dynamic competition with network externalities: Why history matters[END_REF] with homogenous consumers, but cannot readily be extended to the case of heterogenous consumers.

In section 2, we propose to represent the coordination of consumers through what we call Attached Consumers (AC) Equilibria. We do so through a very simple model where we essentially assume that consumers only change platforms when it is individually rational for them to do so -they are very bad at coordinating their moves even when it would be Pareto efficient for them to do so. This enables us to select an equilibrium of the game played by the consumers when they choose which platform to join; this equilibrium depends on the prices charged by the platforms and also on the initial allocation of consumers among the various platforms; our equilibrium is tractable even when there are several types of consumers. We show that AC Equilibria always exist and that they are often unique. This equilibrium concept gives a great deal of power to the incumbent(s) and can be viewed as choosing the best equilibrium from the their point of view. This makes our results that an incumbent's profit are limited in the dynamic model more striking. In order to show that the concept of AC equilibrium is reasonable, in section 3 we illustrate its application to simple one period games with one incumbent.

Starting in section 4, we turn to the core of the paper, the study of an infinite horizon model with free entry by presenting the model which we will be using. In section 5, we present simple versions of our dynamic model, among them one where consumers all have identical preferences and one where some consumers are insensitive to network externalities. The former allows us to identify situations where the incumbent has the same profit in the one period and the infinite number of period models. while the latter shows that the presence of consumers who derive little utility from network externalities can increase the profit of the incumbent, despite the fact that they never join its platform.

We provide a complete characterization of the circumstances where there are inefficiently several platforms in equilibrium in section 6, while in section 7 we characterize the circumstances where there is, efficiently, only one platform in equilibrium. We discuss the literature in section 8. Section 9 concludes by presenting some open research questions.

Modeling incumbency

The main focus of the paper is on dynamic multi-period models. However, in this section we study a one period model and propose a new way to represent incumbency advantage. Starting in section 4, we will embed it in dynamic models.

We use the following strategy. We first define unattached consumers (uc) equilibria, in which there are no incumbency advantages (because consumers move easily from platform to platform) -these are the standard Nash equilibria of the games played by the consumers. We then define attached consumers (ac) equilibria, which are the outcome of a migration process between the incumbent(s) and the entrants, and which are essentially unique. We finally show that ac equilibria are uc equilibria. Using the concept of ac equilibrium yields in the static model predictions which are not very different from those of [START_REF] Caillaud | Chicken and Egg: Competition Among Intermediation Service Providers[END_REF], but is much easier to manipulate in the dynamic games that we will consider starting in section 4.

In order to avoid introducing unnecessary complications, we assume that there are two types of consumers, but the results of this section extend to any number of types. We also assume that the only choice that the consumers face is which platform to join, but there is no difficulty extending the definitions to situations where one of the choices is to join no platform (and indeed we do so in section 3).

There is a mass α h of high network effects (hne) consumers and a mass α of low network effects (lne) consumers.2 We will refer to h or as the "types" of the consumers.

Using, as we will throughout the paper, θ to denote "the other type," different from θ, a consumer of type θ derives utility ψ θ (γ iθ , γ iθ ) from belonging to platform i when γ iθ consumers of type θ and γ iθ consumers of type θ also belong. The functions ψ θ are strictly increasing in both arguments and satisfy ψ θ (0, 0) = 0.3

Even though consumers like to have more consumers of both types on the platform to which they belong, they prefer consumers of their own type:

∂ψ θ (γ iθ , γ iθ )/∂γ iθ > ∂ψ θ (γ iθ , γ iθ )/∂γ iθ ≥ 0.
(1)

We make no concavity or convexity assumptions on these utilities. An allocation γ of consumers among m platforms is a 2 × m vector of nonnegative real numbers {γ ih , γ i } i=1,...,m with i γ iθ = α θ and i γ i = α where γ ih and γ i are the measures of hne and lne consumers on platform i.

Let p i be the price charged by platform i. An allocation γ is an unattached consumers (uc) equilibrium (that is an equilibrium in which incumbency plays no role) if and only if for all i and θ

γ iθ > 0 =⇒ ψ θ (γ iθ , γ iθ ) -p i = max j ψ θ (γ jθ , γ jθ ) -p j .
The definition of uc equilibrium treats all platforms in the same way and is the standard definition of equilibrium in the economics of network externalities: there is no incumbency advantage. We now build to the definition of an ac equilibrium, presented in Definition 2, which depends on the initial allocation of consumers, and show in Lemma 1 that it selects a uc equilibrium. This definition is illustrated on Figure 1. It is the result of a migration process, formally defined in Definition 1, where at each step a group of consumers myopically move from one platform to another if, given the allocation of the other consumers, they find it profitable to do so. Notice that we are not modelling the equilibrium of a well specified migration game,4 although, as discussed after Lemma 1, the migration process is indeed one such equilibrium.

Definition 1 (migration path). Let β be an initial allocation of consumers among the platforms. The allocation γ is on a migration path from β if there Start from the initial allocation β of consumers between platforms.

Would some consumers gain from moving?

A mass η t iθ -η t-1 iθ of the consumers with highest gains from moving move.

The allocation is an ac equilibrium. exists an integer T ≥ 0 and sequences {η t } t=0,1...,T of allocations of consumers to platforms ( i η t iθ = α θ for all t = 0, 1, . . . T and all θ) which lead from β to γ: η 0 iθ = β iθ and η T iθ = γ iθ , for all i and θ, and which satisfy the following four properties: a) for all t = 1, 2, . . . , T -1 there exists a type transferred θ(t), a source platform s(t), and a destination platform d(t) such that a strictly positive mass of consumers of type θ(t) migrates from platform s(t) to platform d(t):

η t d(t)θ(t) -η t-1 d(t)θ(t) = η t-1 s(t),θ(t) -η t s(t)θ(t) > 0;
b) the other consumers do not change platform:

η t iθ = η t-
ψ θ η t-1 iθ , η t-1 i,θ -p i -ψ θ η t-1 jθ , η t-1 j,θ -p j . (3)
The allocation γ is a final allocation if there is no migration path leading from γ to another allocation.

Definition 2 (ac equilibrium). An allocation γ is an ac equilibrium if it is on a migration path from the original allocation and is a final allocation.

If an initial allocation is a final allocation, then there can be no other allocation that can be reached by a migration path of length 1. It is straightforward to see that this implies that the allocation is a uc equilibrium and therefore proves the following lemma.

Lemma 1. All ac equilibria are also uc equilibria. An initial allocation is an ac equilibrium if and only if it is a uc equilibrium. Furthermore, if an initial allocation is a uc allocation, it is the only ac equilibrium.

Lemma 1 has two consequences for the interpretation of ac equilibria. First, trivially from a technical viewpoint (this is a consequence of the fact that there is a continuum of consumers) but importantly for interpretation, the sequence of moves from an initial allocation to a uc equilibrium is a (perfect Nash) equilibrium of the dynamic game played by the consumers. 7 Second, we have presented the migration of consumers between platforms as happening in "real time". Thanks to Lemma 1, it can also be interpreted in terms of fictitious play, where all consumers move at the same time. In that interpretation, consumers think about how their fellow consumers will react to the price offerings that they all face in a subgame. They would all correctly predict how the consumers will choose which platform to belong to. 6 An alternative assumption would have any group of consumers with a strictly positive gain from moving move, i.e, we would dispense with (3) and (2) would be the only constraint on the utilities of the migrating consumers. This is not sufficient to prove our results. Indeed, we have built an example using this relaxed assumption where in the initial allocation the hne consumers are on one platform and the lne consumers on another. The migration leads to an uc equilibrium where all the hne consumers and some of the lne consumers migrate to one platform and the rest of the lne consumers to another one. Lemma 3 does not hold.

7 A precise statement of this fact would require a formal description of the dynamic game, which is beyond the scope of this paper. Of course, there would be many equilibria of this game.

Of course, no consumer will have an incentive to deviate from their platform choice. The description of moves in Figure 1 then represents the way in which consumers think about the choices of the other consumers.

We will call a migration path a migration path through large steps if at every step all the consumers of type θ(t) on platform s(t) migrate to platform d(t): η t s(t)θ(t) = 0 for all t. The following lemma makes easier both the identification and proof of existence of ac equilibria -its proof, as well as the proofs of most of the other results of the paper are presented in the Appendix .

Lemma 2. The set of ac equilibria is not changed if we impose the restriction that the migration path is a migration path through large steps.

This lemma holds because once some consumers have begun migrating from a platform to another a 'snowballing' effect arises. The idea is very simple: if in any migration process not all the consumers of type θ(t) migrate from platform s(t) to platform d(t), then we must have θ(t + 1) = θ(t), s(t + 1) = s(t) and d(t + 1) = d(t). Indeed, for consumers of type θ(t), the utility of being of platform d(t) has strictly increased while those of being on platform s(t) has decreased; there is no other move between two platforms which would yield a greater increase the utility of a consumer of type θ(t).

Condition 1 shows that this also holds for consumers of type θ (t), but for a smaller gain than for type θ(t).

This implies the following lemma, which we will use extensively in the sequel.

Lemma 3. If all the consumers of type θ belong to the same platform in the initial allocation (i.e., if β iθ = α θ for some i), then they also belong to the same platform in any ac equilibrium.

The following lemma shows that we can also interpret our migration paths as a sequence of "individual moves". Lemma 4. The set of ac equilibria is not changed if we add the restriction that η

t d(t)θ(t) -η t-1 d(t)θ(t) = η t-1 s(t),θ(t) -η t s(t)θ(t)
must be smaller than some > 0 for all t.

Lemma 4 is proved by "cutting" each step of a large step algorithm into smaller steps with the same source and destination platforms and the same migrating type. It shows that we can think of migration paths as approximating a process in which the consumers move "one by one" from one platform to the other; in each stage it is the consumer with the greatest gain from moving who moves.

It is easy to show that large step migrations must eventually stop at an ac equilibrium, which proves the following lemma.

Lemma 5. Whatever the initial allocation {β ih , β i } i=1,...,m and prices p i charged by the platforms, there exists an ac equilibrium.

3 One Period Games

Notation and assumptions

In the models which we study in the rest of this paper, there is at the start an Incumbent platform from which all consumers purchased in the past. By Lemma 3, this implies that in any subsequent period consumers of the same type will all join the same platform, and the following shorthand notation will prove useful:

u θ = ψ θ (α θ , 0), v θ = ψ θ (0, α θ ), w θ = ψ θ (α θ , α θ ).
We assume that the consumers prefer to be with consumers of the same type:

w θ > u θ > v θ for θ ∈ {h, }. (4)
Condition (1) implies w θ > v θ and w θ > u θ . It also implies

u θ > v θ if α h = α .
If, for instance, α were much larger than α h , hne consumers might rather belong to the same platform as lne consumers than belong to the same platform as other hne consumers. The right most inequality in (4) assumes this away. We are representing the fact that the hne consumers value network effects more than lne consumers by the following conditions:

w h > w and u h > u .
(5)

Finally, unless we explicitly state the opposite, we assume

w < u h -v h . (SmallCE)
The right hand side is the amount an hne consumer would be willing to pay to move from a platform to which all the lne and no hne consumers belong to another platform to which all the hne and no lne consumers belong. The left hand side is what lne consumers are willing to pay to be on the same platform as all other consumers. Written v h < u h -w , it puts an upper bound on v h , hence its name which stands for "Small Cross Effects". When it does not hold there is only one platform in equilibrium whether in a one period or in an infinite horizon model (see section 5.2).

Equilibrium in one period models

In the rest of this section, we study ac equilibria in one period models. This will both prepare the study of dynamic games and demonstrate that ac equilibria are intuitive. First, as a benchmark, suppose that there is no entry: the Incumbent announces a price and the consumers decide whether or not to stay on its platform. They all stay if the Incumbent charges w , and its profit is then (α h + α )w . If the Incumbent charges u h , which is greater than w by (SmallCE), its only clients are the hne consumers and its profit is α h u h . All other prices are dominated by one of these two, and we have therefore proved the following lemma. Lemma 6. In a one period model with no entrants, the incumbent sells only to hne consumers if

u h > α h + α α h w , (6)
and to all consumers if α h u h < (α h + α )w . It charges u h and its profit is α h u h in the first case; in the second, it charges w and its profit is (α h + α )w .

Let us now add entry and assume that there are least two entrants. Although it is easy to see that Nash timing would give exactly the same results, for simplicity we assume Stackelberg timing where the Incumbent first chooses its price p I followed by the entrants; afterwards, the consumers decide which platforms to join. Entrants will never charge less than 0, and competition among them implies that any entrant who attracts consumers will do so at a price of 0. The incumbent either charges w and keeps all the consumers or u h -v h and keeps only the hne consumers. This implies the following lemma.

Lemma 7. In the one period model with entry, if

u h -v h > α h + α α h w (7)
the incumbent sells only to the hne customers at price u h -v h and its profit is

α h (u h -v h ). If u h -v h < (α h + α )w /α h it
sells to all consumers at price w and its profit is (α h + α )w .

Lemmas 6 and 7 together imply the following corollaries, which follow immediately from ( 6) and ( 7). The first is stated for future reference (see Lemma 9).

Corollary 1. In a one period model with or without entry, if (SmallCE) does not hold, the Incumbent sells to all consumers.

The second corollary is important from an economic viewpoint.

Corollary 2. Entry makes the separation of lne and hne consumers less likely and therefore improves efficiency.8 When the incumbent sells to both types of consumers the price it charges and its profit are the same with or without entry.

Note the reason why entry improves efficiency: the incumbent finds it more costly to let the lne consumers "go" to an entrant, since it will be more costly to keep the hne consumers. This result has ramifications for policy, since it says, somewhat counter-intuitively, that entry keeps more consumers on an incumbent's platform despite network effects. This is a gain in efficiency in two senses. One, it has all consumers on the same platform. Second, without entry, instead of joining another platform lne consumers opt out of the market. We will review the role of entry and find more ambiguous results for intermediate values of the discount factor in Section 6.2 -see Corollary 5. Fudenberg and Tirole (2000, p. 375) also discuss the role of entry in a dynamic model and find that "in equilibrium, the welfare effects of inducing additional entry are ambiguous".

The dynamic model 4.1 Equilibria in the dynamic game with entry

The dynamic model which we will use is represented in Figure 2. At the beginning of period 1 there is one incumbent, which, as in Section 3, we will denote the "Incumbent". In each subsequent period, there will be one or more incumbents: the firms that sold to a strictly positive measure of consumers in the previous period. There will also be n E ≥ 2 entrants in each period.

For simplicity, we assume Stackelberg timing where all the incumbents first set prices simultaneously and then the entrants, having seen these prices, choose their own prices. 9 Afterwards, the consumers choose their platforms, Period 1 starts with one Incumbent Each period t > 1 starts with one or several incumbents Incumbent(s) set prices Entrants set prices (free entry)

The consumers play the "within period" dynamic game of choosing their platforms and the game moves to period t + 1. We assume that firms with no consumer at the end of a period "drop out"of the game. 10 Note that there are two dynamics in the game which we are describing: the "large scale" dynamics from period to period and the "small scale" or "within period" dynamics, when consumers choose which platform to join according to the process described in section 2. We assume there is no discounting within a period, as in the one period model, but there is a common discount factor, δ < 1, between periods. This formalizes the assumption that consumers move very quickly between platforms, and then have a full period to consume the benefits of belonging to a platform.

We focus on measurable equilibria. This implies that along the equilibrium path, in any period t the price charged by the incumbents i depends only on the β t jθ s; the prices charged by the entrants depend only on the β t jθ s and the prices charged by the incumbents; the equilibrium of the game played between the consumers depend only of β t jθ s and on the prices charged by the platforms.

When the horizon is infinite, we will restrict ourselves to Markov equilib- [START_REF] Biglaiser | The value of switching costs[END_REF] and Biglaiser, Crémer, and Dobos (2016a) for discussion of similar issues in a model of switching costs). Along the equilibrium path, we would observe more switching between platforms by consumers than under Stackelberg timing.

(see
10 Formally, this would be done by assuming that in any period τ > t, their strategy set is a singleton, and that purchasing from these firms is not in the consumers strategy set.

ria, which we define more precisely in 4.3. The results of 4.2 do not depend on stationarity.

The myopia principle

We now turn to the extension of the notion of ac equilibrium appropriate for dynamic games. In each period t, there is a set of incumbents {1, 2, . . . , n t I } (in equilibrium, n t I will actually be equal to either 1 or 2), and a set of entrants {1, 2, . . . , n E }. In period 1, the initial allocation allocates all consumers to the Incumbent. In future periods, the initial allocation is the allocation of consumers at the end of the previous period. For incumbent i in period t, we call β t ih and β t i the mass of hne and lne consumers in his initial clientele; because it is an incumbent, we must have β t ih + β t i > 0. The purchasing decisions of the consumers depend on the β t jθ s, on the prices charged by the firms, and on their expectations of the decisions of other consumers. We call W iθ,t+1 β t+1 the expected discounted utility measured at the beginning of period t + 1, before incumbents have chosen their prices of a consumer of type θ who has purchased from platform i in period t.

Because consumers are "small" and do not affect the market through their individual choices and there are no switching costs, W iθ,t+1 β t+1 does not depend on i, and can therefore be written W θ,t+1 β t+1 . If the equilibrium allocation of consumers in period t has γ t jh hne consumers and γ t j lne consumers in platform j, the utility of a consumer of type θ who purchases from platform i which charges p t i will be

ψ θ (γ t iθ , γ t iθ ) -p t i + δW θ,t+1 {γ jh , γ j } j∈I(t+1) ,
where I(t + 1) is the set of incumbents at stage t.

We can apply the same reasoning as in section 2 to define migration paths within period t. At each step τ , the consumers who change platforms are those consumers of type θ(τ ) such that there exists source and destination platforms, s(τ ) and d(τ ), which are solution of max θ ,i,i

ψ θ (η τ -1 iθ , η τ -1 i,-θ ) + W θ ,t η τ -1 jθ , η τ -1 j,-θ j∈I(t) -p t i -ψ θ (η τ -1 i θ , η τ -1 i ,-θ ) + W θ ,t η τ -1 jθ , η τ -1 j,-θ j∈I(t) -p t i , (8)
as long as the value of this solution is strictly positive. The same W term appears in both terms of this expression and therefore solving (8) is equivalent to solving (2). We obtain the following "myopia principle", which also holds in the belief based analysis of Ha laburda et al. ( 2016) and plays an important role in the sequel.

Lemma 8 (Myopia principle). Given the prices chosen by the firms, the set of equilibria of the game played by the consumers in any period t of a dynamic game is the same as if the game were a one period game.

The myopia principle does not imply that the prices charged by the platforms will be the same in a multi-period game as in a one period game -it is only the consumers who act as if they are "myopic", not the firms.

The myopia principle does not hold in switching cost models unless future switching costs are uncorrelated with current switching costs: as shown in Biglaiser et al. (2016a) high switching cost consumers try to "hide among" low switching cost consumers who induce firms to charge low prices -the high switching cost consumers are willing to incur higher costs in the current period in order to do so. The fact that there are no switching cost and, as in the one period model, that there is a continuum of consumers, and strategies and continuation values are measurable are necessary for lemma 8 to hold.

Markov equilibria

We focus our attention on Markov equilibria, defined as follows. Along the equilibrium path, in any period t the prices charged by the platforms depend on the initial allocation, as described in 4.1, Furthermore, we assume anonymity: all that matters is the β t jθ s and not the name of the platforms (this is one of the important differences between our model and the dynamic model of Ha laburda et al. ( 2016)). 11By Lemmas 3 and 8, consumers of the same type will all "stay together"; therefore, along the equilibrium path, there will be only either one or two incumbents in every period and we need only distinguish the following equilibrium prices and profits for the incumbents: p h , Π h : the price charged by and the total discounted profit of a firm whose clients in the previous period were (only) the hne consumers;

p , Π : the price charged by and the total discounted profit of a firm whose clients in the previous period were (only) the lne consumers;

p 2 , Π 2 : the price charged by and the total discounted profit of a firm who sold to both types of clients in the previous period.

The main focus of the paper will be on two platform equilibria where along the equilibrium path, after the first period, consumers purchase from two different platforms (this implies that in the first period at least one type of consumers purchase from an entrant). By stationarity, if after a deviation there is only one incumbent platform, consumers will reallocate themselves among two different platforms. We study these equilibria in section 6. In section 7, we study the equilibria with only one platform.

Before studying our main model, section 5 presents some simple versions of our dynamic model that will illustrate the use of our equilibrium concept and develop intuition.

Simple infinite horizon models with free entry

In this section, we examine a series of simple dynamic models with free entry. They both illustrate how our solution concept works in dynamic models and show that the value of incumbency can be quite limited.

Identical Consumers

We begin by examining the equilibrium when there is only one type of consumer -for definitiveness only hne consumers. Our main result will be that the profits in the infinite horizon are exactly the same as in the static model. Let Π denote the equilibrium profits of the incumbent when all consumers are on the same platform. Entrants are willing to "price down to" -δΠ/α h and no further, as any lower price would yield negative profits.12 The Incumbent chooses the largest price p I which enables it to keep all the consumers: it satisfies u h -p I = δΠ/α h . Because Π = α h p I /(1 -δ), this proves the following proposition.

Proposition 1. If all consumers are of the same type, then the essentially13 unique equilibrium has a single platform. The Incumbent keeps all the consumers and charges p I = (1 -δ)u h in each period. Its profit is α h u h , the same as in a static model. [START_REF] Biglaiser | The value of switching costs[END_REF] establish the same result of equality of static and dynamic profit with homogenous consumers in the case of switching costs. Competition from the entrants prevents the incumbent from enjoying the rents of incumbency more than once: it can take only one bite from the apple. The result would also hold in a model with a finite number of periods. In the infinite horizon case, it requires the stationarity assumption [START_REF] Biglaiser | Equilibria in an infinite horizon game with an incumbent, entry and switching costs[END_REF].

Equilibrium when condition (SmallCE) does not hold

The aim of this subsection is to show the following proposition.

Proposition 2. If Condition (SmallCE) does not hold, there exists a unique equilibrium. It is a single platform equilibrium. where the Incumbent charges (1 -δ)w in every period and its discounted profit is (α h + α )w , the same as in the one period model.

We begin by proving the following lemma, which is stronger than we need to prove Proposition 2 and which is of independent interest. It mirrors Lemma 1 in the one period case. In the proof (and in the rest of the paper), we will use the notation p E for the lowest price charged by an entrant.

Lemma 9. Condition (SmallCE) is necessary for the existence of a two platform equilibrium.

Proof. Because w < w h , a first period entrant cannot attract the hne consumers without also attracting the lne consumers. 14 In the first period of a two platform equilibrium, we have w -p 2 < -p E because the lne consumers purchase from the entrant and u h -p 2 ≥ v h -p E because the hne consumers purchase from the Incumbent. These two conditions imply (SmallCE).

Trivially, Lemma 9 implies that when (SmallCE) does not hold, any equilibrium is a single platform equilibrium. In such an equilibrium, the Incumbent sells to all the consumers in all periods at price p 2 . If -p E ≤ wp 2 < w h -p 2 , both lne and hne consumers purchase from the Incumbent. On the other hand, by the myopia principle, if -p E > w -p 2 , the lne consumers will migrate to one of the entrants who charges p E . Hne consumers will follow the lne consumers, as u h -p 2 < w + v h -p 2 < v h -p 2 . Therefore, the Incumbent keeps all the consumers if p 2 -p E ≤ w and loses them all otherwise.

The lowest price that entrants are willing to price to attract all consumers is -δΠ 2 /(α + α h ), where Π 2 is the discounted intertemporal profit of the incumbent starting from any period. Therefore the profit maximizing p 2 satisfies p 2 = -δΠ 2 /(α h + α ) + w . Along with Π 2 = (α h + α )p 2 /(1 -δ), this proves Proposition 2.

Equilibria when LNE consumers do not derive any

utility from belonging to a platform15

We now turn to a case where the lne consumers derive no utility from belonging to a platform: w = u = v = 0. Hne consumers only derive utility from the presence of other hne consumers and do not care about the presence of lne consumers: v h = 0 and w h = u h > 0 (but see footnotes 18 and 19). We will show that, even under these circumstances, lne consumers affect the equilibrium by dampening the aggressiveness of entrants. As proved formally in section 6, there are two16 platforms at equilibrium: this is obvious as the Incumbent must charge 0 in order to keep the lne consumers while he can make a strictly positive profit by selling only to the hne consumers.

Let Π be the discounted profits of the Incumbent measured from the start of a period (in section 6, we show that this profit is the same whether all the consumers or only the hne consumers were its clients in the last period). If an entrant attracts17 the hne consumers, it will also attract the lne consumers and the lowest price that it is willing to offer is -δΠ/(α h + α ). To keep the hne consumers, the Incumbent chooses a price p I that makes them just indifferent between staying on its platform and purchasing from the entrant at that price: we must have

u h -p I = δΠ/(α h + α ) =⇒ Π = p I 1 -δ = (α h + α )α h u h α h + (1 -δ)α . ( 9
)
The profit of the Incumbent is increasing in the number of lne consumers: because they accept the offers designed to attract hne consumers but do not contribute to profits, they make the entrants less aggressive. This is especially striking when δ converges to 1, as Π converges to (α h + α )u h : adding an lne consumer, who never purchases from the Incumbent, contributes as much to the profit of the Incumbent as adding an hne consumer! 18,19,20 This result is similar to the results presented by [START_REF] Biglaiser | The value of switching costs[END_REF]Biglaiser et al. ( , 2016a) ) in the framework of a switching cost model, where, with δ close to 1, a consumer with zero switching cost is worth as much to the Incumbent as a consumer with strictly positive switching cost.

It is worthwhile noting that in a one period model, the Incumbent would charge u h and its profit would be α h u h . The "equivalence" between lne and hne consumers do not hold.

6 Two platform equilibria

Main results

In this section, we study the conditions under which two platforms coexist at equilibrium. Given our Markov assumption and the results of section 2, which show that consumers of the same type always purchase from the same platform if they are initially together, such an equilibrium must look as follows. In the first period, the Incumbent charges p 2 . After the first period, there will be two platforms on the equilibrium path. This can only happen if in the first period an entrant charges p E and attracts one type of consumer.

18 This reasoning assumes that u h is not affected by changes in α h . If we make explicit the dependence of the utility of the hne consumers on the size of the platforms, lim δ→1 Π is equal to (α h + α )ψ h (α h , 0). When α = 0, the derivative of this profit with respect to α is ψ h (α h , 0) as ∂ψ h /∂α = 0. The derivative with respect to α h is equal to ψ h (α h , 0) + α h ∂ψ h (α h , 0)/∂α h . The second term, α h ∂ψ h (α h , 0)/∂α h , is the increase in the value of the platform for the other hne consumers. A lne consumer is worth as much as the "direct" effect of a hne consumer. If ψ h is concave, then α h ∂ψ h (α h , 0)/∂α h < ψ h (α h , 0) and therefore a lne consumer is worth more than half a hne consumer.

19 Very similar results hold when hne consumers do care about the presence of lne consumers; i.e., if v h > 0. Entrants will still be willing to price down to -δΠ/(α h + α ), but the first part of (9) becomes u h -p I = v h + δΠ/(α h + α ): the presence of the lne consumers increase the attractiveness of the entrant. Then Π is given by ( 13). An increase in v h decreases profits: it makes it easier for entrants to attract the hne consumers by first attracting the lne consumers.

It is also be worthwhile noticing that the separation of the consumers in two different platforms is now inefficient.

Because w < w h , this implies

-p E -(w -p 2 ) > -p E -(w h -p 2 ),
lne consumers gain the most from purchasing from the entrant. Therefore, the entrant will attract the lne consumers and the Incumbent will sell to the hne consumers. In subsequent periods, along the equilibrium path, there will be two incumbents: the Incumbent, who sells to the hne consumers, and the successful first period entrant, who sells to the lne consumers. This implies that, using the notation introduced on page 14, we have Π

h = α h p h /(1 -δ), Π = α p /(1 -δ) and Π 2 = α h p 2 + δΠ h .
If, off equilibrium, in some period all the consumers belong to one platform, by the Markov hypothesis in the subsequent period they would again split among two platforms as described in the previous paragraph.

The following proposition summarizes our results, which we prove in the rest of this section.

Proposition 3. There exists a two platform equilibrium if and only if

u h -v h ≥ (1 -δ)α + α h (1 -δ)α h (w -δu ). ( 2NetCond
)
In this equilibrium L incumbents charge the same price and have the same profit as if there were only lne consumers:

p = u (1 -δ) (10)
and

Π = α u . (11)
H incumbents and firms which, after a deviation, have sold to every consumers in the preceding period charge the same price,

p 2 = p h = (1 -δ)(α h + α )(u h -v h ) (1 -δ)α + α h , (12)
and have the same profit,

Π 2 = Π h = α h (α h + α )(u h -v h ) (1 -δ)α + α h . ( 13
)
This profit, which is also the profit of the Incumbent, 1. is greater than the profit of the Incumbent of the two platform equilibrium in the one period model, α h (u h -v h ), and smaller than the value of a flow of this one period profit,

α h (u h -v h )/(1 -δ); 2. is less than (α + α h )(u h -v h );
3. is increasing in u h , decreasing in v h and independent of w h , w , u and v ;

4. is increasing in α h and in α .

The difficult part of the proof is proving that (2NetCond) is a necessary and sufficient condition and that equations ( 11) to ( 13) hold -this is done in 6.3. The rest of the proposition is an immediate consequence of (13).

As explained below, the binding deviation for the existence of a two platform equilibrium is the attempt by the Incumbent to keep all the consumers. By (11) the lowest price that entrants are willing to charge is -δu . Because the lne consumers are the most eager to change platforms, the Incumbent has to charge at most w -δu if it wants to keep all the consumers. The profits resulting from repeating this strategy forever are

Π D = (α + α h )(w -δu ) 1 -δ . (14) Condition (2NetCond) is equivalent to Π D ≤ Π 2 .
Point 3 shows that the profits of the Incumbent are independent of the preferences of the lne consumers. These preferences do play a role in the existence of a two platform equilibrium. Once such an equilibrium exists, it is only the strength of attraction that lne consumers hold for hne consumers that affect the Incumbent's profit.

Point 4 states that, as in section 5.3, lne consumers have value for the Incumbent even though they never join its platform, since they get in the way of an entrant who would try to attract high value consumers. Points 1 and 2 put bounds on its profit. In particular, the profit of the Incumbent is increasing in δ, and, as stated in point 2, is always smaller than (α h + α )(u h -v h ). By Lemma 6, the Incumbent charges u h -v h when there are two platforms in the one period model. Therefore in the dynamic infinite horizon model, its profit is always inferior to what it would be in the one period model if all the consumers were hne consumers. Section 7 will show that this result also holds true when there is only one platform in equilibrium.

In 6.2, we examine in greater details the positive and normative consequences of Proposition 3.

Welfare implications

We now turn to a detailed discussion of the existence of two platform equilibria; this is of policy importance since, in our setup with positive network externalities, it is always more efficient to have one rather than two platforms. We will show that two platform equilibria are, in general, more likely in the static than in the dynamic setting: more precisely, under most parameter values and especially when δ is large, if a two platform equilibrium exists in the dynamic model, then a two platform equilibrium also exist in the static model, while the reverse is not necessarily true. Thus, a policymaker who uses the static model in order to predict the inefficiency of market outcomes, will in general reach overly pessimistic conclusions.

For most of the analysis we focus on the case when w > u and then turn to the special case when w = u , i.e., when lne consumers get no benefit from belonging to the same platform as the hne consumers.

It is useful to rewrite (2NetCond) as

u h -v h ≥ g(δ) with g(δ) def = (1 -δ)α + α h (1 -δ)α h (w -δu ).
Condition (7), which holds for the one period model, is equivalent to u h -v h > g(0). The function g(δ) is strictly convex 21 and satisfies lim δ→1 g(δ) = +∞. Defining δ as the largest δ < 1 such that u h -v h = g(δ), we have the following important corollary.

Corollary 3. There exists no two platform equilibrium when δ ∈ δ, 1 .

On the other hand, as stated in Corollary 6, there always exist a one network equilibrium for large enough δ. Thus, a competition authority who thinks that an inefficient equilibrium will exist in a static setting, should be less concerned that this is also true in the dynamic case, especially for high discount factors.

To go further, we need to focus on the sign of

g (0) = w - α + α h α h u .
For all u h -v h ≥ g(0), there is a two platform equilibrium in the static model. Figure 3 illustrates the case where g (0) < 0, which is equivalent to w < (α h +α )u /α h . The function g reaches a minimum V min for some δ min < 21 Indeed,

g (δ) = 1 (1-δ) 2 (w -δu ) -α α h + 1 1-δ u and g (δ) = 2(w -u ) (1-δ) 3 > 0. u h -v h 0 δ α α h g(0) g(δ) V min δ min = .5 m 1 δ 1 δ m 2
There exists a 2 platform equilibrium in this region 2NetCond) is then equal to (2 -δ)(1 -.8δ)/(1 -δ), which is equal to 2 when δ = 0. Its minimum, V min is equal to 1.8 and is obtained for

δ min = .5. For u h -v h = m 1 ∈ [1.8, 2],
there exists an interval of δs such that there exists a two platform equilibrium. For u h -v h = m 2 > 2, there exists a two platform equilibrium if δ is small enough.

1. The set of parameters such that there exist a two network equilibrium is represented on the figure. Note that when u h -v h < g(0) there exists a two platform equilibrium only for intermediate values of δ.

On the other hand, if g (0) > 0, then by the convexity of g, a two platform equilibrium exists if and only if δ ≤ δ. This proves the following corollary.

Corollary 4. Assume u < w . 1. If u h -v h ≥ (α h + α )w /α h so
, δ ] with 0 < δ 1 < δ < 1.
There are two complimentary ways to think about this result. First, in industrial organization economics, the discount factor is thought of as being influenced both by the interest rate and the probability of the "end of the world", which for our model would be interpreted as the appearance of a new disruptive technology. Our results indicate that efficiency, under the form of the existence of a single platform, is more likely in a more stable world. Second, one can view δ as a proxy for the frequency with which consumers ponder whether to change platforms. From this angle, the more often consumers check, the more intense the competition between platforms. Corollary 4 shows that this increased competition generally, but not always, leads to higher efficiency. Corollary 5 presents a similar result when the intensity of competition is linked to entry.

We can see the influence of cross effects by looking at case when w = u , when the lne consumers obtain no additional utility being with the hne consumers. Under this assumption, if there exists a two platform equilibrium in the static model, then there is a two platform equilibrium in the dynamic model, but the converse is not always the case. Why is there a difference when u = w ? From ( 14), Π D is increasing in δ when u < w and independent of δ when u = w . Combined with the fact that condition (2NetCond) is equivalent to Π D ≤ Π 2 , this explains the sharp contrast between the two cases.

Finally, it is interesting to see whether entry enhances welfare in the dynamic model as was the case in the static model. We have the following corollary, which is similar to Corollary 2, but in the dynamic case.

Corollary 5. For δ close to 0 or 1, entry makes the separation of lne and hne consumers less likely in the dynamic model and therefore entry improves efficiency.

In the static model, entry improves welfare as it increases the cost to the Incumbent of letting an entrant capture the lne consumers who are valuable to the hne consumers (see Corollary 2). The situation is more complicated in the dynamic model. Without entry, Condition (6) must be satisfied both in the static and dynamic model for the Incumbent to keep only hne consumers while with entry, Condition (2NetCond) is necessary and sufficient for a two platform equilibrium to exist in the dynamic model. When δ is close to either 0 or 1, then entry makes it more likely that all consumers remain on the Incumbent network, just as in the static model. For intermediate values of δ, however, for some values of the parameters there exists a two platform equilibrium in the dynamic model without entry but not in the dynamic model with entry. On the one hand, the Incumbent finds it more costly to keep the lne consumers in the dynamic model where entrants are willing to price down to -δu while in the static model they are not willing to offer a negative price. On the other hand, without entry the profit in the two platform dynamic model is equal to 1/(1 -δ) times the profit in the static model, but with entry when a two platform equilibrium exists in the dynamic model, the profit is smaller than 1/(1 -δ) times the static profit. This leads to the corollary.

The rest of this section is devoted to the analysis of the strategies of the platforms and to the proof of Proposition 3. The reader who is interested mostly in the results can skip forward to Section 7 for discussion of one platform equilibria.

Condition (2NetCond) is necessary for existence of a two platform equilibrium

As one would expect, in any equilibrium, the net surplus of the hne consumers is larger than the net surplus of the lne consumers:

u -p ≤ u h -p h . (15)
This implies that, along the equilibrium path, an entrant cannot attract the hne consumers without having first attracted the lne consumers. 22 In the

p E p E = -(u h -p h ) + v h p E = -(u -p )
All consumers purchase from their respective incumbent.

The lowest price entrant attracts all consumers; its profits are (α h + α )p E + δΠ 2 .

The lowest price entrant attracts the lne consumers; its profits are α p E + δΠ .

Figure 4: The response of consumers to entry when (16) holds.

main text, we assume that (15) holds and show in section D of the appendix that this must indeed be the case whenever a two platform equilibrium exists.

We first show that we can strengthen (15). The proof of all the claims in this subsection can be found in section C of the appendix.

Claim 1. If (15) holds, then in any two platform equilibrium

v h + u -p < u h -p h . (16)
Condition ( 16) obviously implies v h -p < u h -p h : along the equilibrium path, hne consumers strictly prefer to purchase from the H incumbent than from the L incumbent.

Condition ( 16) implies that the continuation equilibria in the consumers' game as a function of p E are as represented on Figure 4.

From the definition of the cutoff prices p E and p E in Figure 4, the following conditions are necessary to ensure that there is no profitable entry:

-an entrant cannot profitably attract only the lne consumers:

α p E + δΠ ≤ 0; (17)
-an entrant cannot profitably attract all consumers:

(α h + α )p E + δΠ 2 ≤ 0. ( 18
)
It is relatively intuitive, and proved in claim C-1 of the appendix, that ( 17) is binding at equilibrium: otherwise, the L incumbent could raise its price and keep its consumers. Because Π = α p /(1 -δ), equations ( 10) and ( 11) hold. Thus, once the two groups are separated, the L incumbent behaves in the same way and obtain the same profit as if it where the Incumbent with only the lne consumers present (see Proposition 1). Similarly, the fact that ( 18) is binding follows; if not, the H incumbent could raise its price and increase its profit.

The constraint that the L incumbent does not try to attract the hne consumers is not binding. Indeed the L incumbent finds it less attractive to attract the hne consumers than do the entrants, as its opportunity cost to do so is greater because it obtains a positive profit from the lne consumers.

The reasoning which precedes show the "necessity" part of the following lemma. The sufficiency part, which is quite straightforward, is proved in the appendix -the proof relies on the non profitability of deviations in period 2 and subsequent periods, when the consumers have already split between the two platforms.

Lemma 10. Equations (10) and (12) are sufficient and necessary for the fact that once lne and hne consumers have purchased from different platforms they will continue to do so in the continuation equilibrium.

We now turn to the study of the first period and on the incentives of the agents to create two platforms out of one. First, we must have

-(α h + α ) [p 2 -(u h -v h )] + δΠ 2 ≤ 0. (19)
Otherwise, in the first period an entrant could attract all the consumers by charging a price "slightly below" p 2 -(u h -v h ) and make a strictly positive profit. Claim C-3, presented in the appendix, shows that in equilibrium this constraint must be binding: otherwise the Incumbent could profitably increase its price in period 1. Along with (18) this implies p 2 = p h and therefore Π 2 = Π h . Then, Π h = α h p h /(1 -δ) implies ( 12) and ( 13).

Summarizing the discussion so far, we have proved that if there is an equilibrium satisfying (16), then the prices must satisfy equations ( 10) and ( 12). Furthermore, if the prices satisfy these equations, then there is no profitable entry by ( 17) and ( 18).

In the first period, the entrant who attracts the lne consumers must charge less than p 2 -w and its profit will be less than -α (w -p 2 ) + δΠ = α (p 2 -(w -δu )). Thus, no entrant will be willing to attract the lne consumers unless p 2 > w -δu , which is therefore a necessary condition for the existence of a two platform equilibrium.

If (α h + α )(w -δu ) > α h p 2 , the Incumbent finds it profitable to charge w -δu + ε and keep all the consumers. Hence, (α h + α )(w -δu ) ≤ α h p 2 , which is equivalent to (2NetCond) and also p 2 > w -δu .

We have used ( 15) and (SmallCE) to prove that (2NetCond) is necessary for the existence of a two platform equilibrium. It is easy to show (see claim C-4 in the appendix) that both of these conditions hold whenever (2NetCond) holds, we have therefore proved (2NetCond) to be a necessary conditions for the existence of a two platform equilibrium.

6.4 Condition (2NetCond) is a sufficient condition for a two platform equilibrium.

We now show that (2NetCond) is sufficient for the existence of a two platform equilibrium. Much of the construction of the equilibrium in the preceding subsection can be used in this proof. We will proceed by going through the possible deviations showing that they are not profitable.

First period and any subsequent period where, off equilibrium, there is only one incumbent Incumbent: By the reasoning leading to (19), if the Incumbent increased its price an entrant would find it profitable to attract all the consumers. Decreasing the price to p 2 ≥ w -δu would not change the demand facing the Incumbent and hence would lower its profit. Decreasing the price below w -δu would enable the Incumbent to keep all the consumers, but, by (2NetCond), at the cost of lower profits.

Entrants: Competition between the entrants will lead them to charge a price equal to -δΠ /α = -δu . At that price, lne consumers find it profitable to purchase from the entrants. The proof of Proposition 3 shows that no entrant will find it profitable to attract all the consumers.

Subsequent periods with two incumbents

Incumbents The same reasoning as for period 1 shows that the H incumbent has no incentive to deviate. The incentives of the L incumbent are the same as in 5.1 as far as competing for the lne consumers. In order to attract the hne consumers it would have to choose a price smaller to p 2 -(u h -v h ), which is unprofitable for the same reason that it would be unprofitable for an entrant to attract all the consumers in the first period.

Entrants For the same reasons as in 5.1 they cannot profitably attract the lne consumers. For the same reason as in the first period, they cannot attract profitably all the consumers.

Analysis of equilibria with one platform

In this section, we present our main results on the existence and the properties of single platform equilibria paying particular attention to δ close to 1 -a full analysis is presented in section E of the Appendix.

For simplicity, we take it as granted that Condition (15), u -p ≤ u h -p h , holds -which would be the case whenever δ > u /u h . 23 The main result of this section is the following corollary:24 Corollary 6. There exists a δ such that for any δ ≥ δ: a) there exists at least one single platform equilibrium; b) the Incumbent's profit in all single platform equilibria is

(α h +α )(u h - v h );
c) the profit of the Incumbent is larger than in the static model, but smaller than the value of a flow of one period profit.

Corollary 6 implies for large δ there exist only a single platform equilibria, since, as Corollary 4 and the comment that follows demonstrates, there exists no two platform equilibrium for large δ.

From item b), the Incumbent's profit is equal to the product of the total number of consumers, α h +α , and the price that it would charge to maximize its profit while selling only to hne consumers in the static model, u h -v h .

Finally, we obtain the same result as in the dynamic switching cost model of [START_REF] Biglaiser | The value of switching costs[END_REF]: for large discount factors lne consumers are as valuable to the Incumbent as hne consumers: the derivative of the profit with respect to α is the same as the derivative with respect to α h (the same caveat discussed in footnote 18 also applies here). This is for the same reason: the lne consumers will accept aggressive offers from entrants but also be very footloose in future periods. Thus, entrants will be reluctant to offer low prices.

For small δ the profits of the incumbent can be smaller, but never larger, than in Corollary 6. The ratio of long run profits to profits in the one period model is quite small. Corollary 7. The Incumbent's equilibrium profit in the dynamic model never exceeds

(α h + α ) min u h -v h , w 1 -δ .
Notice that this upper bound on the profits holds both for one and two platform equilibria.

For a small set of parameters there exist both a single platform and a two platform equilibrium. 25 As the following corollary states, the Incumbent always prefers the single platform equilibrium, which is also welfare maximizing. 26 Corollary 8. For parameter values such that both a single platform equilibrium and a two platform equilibrium exist, the profit of the incumbent is larger in the single platform equilibrium.

We now turn to a discussion of the structure of one platform equilibria, which is complicated by the fact that they differ along two dimensions. The first dimension describes what happens off the equilibrium path if the consumers ever get "separated" in two different platforms: consumers can either stay separated in subsequent periods -the S (for Separated) equilibria, or they can all purchase from the same platform in the period after they have split so that two platforms coexist for only one period -the T (for Together) equilibria.

The second dimension is the the entry constraint which binds on the Incumbent along the equilibrium path when it sells to both types of consumers: either preventing profitable entry which would attract only the lne consumers or preventing profitable entry which would attract all consumers.

As a consequence, there are four types of single platform equilibria which are represented in Figure 5. For large δ the binding constraint is preventing entry which attracts both lne and hne consumers: equilibria are either of type S2 or T2. To attract both types of consumers, an entrant must charge a price p E which satisfies v h -p E > u h -p 2 . 27 This is unprofitable only if (α h + α )p E + δΠ 2 < 0, which is equivalent to p E + δp 2 /(1 -δ) < 0 because 25 More precisely, for a small set of parameters there exist both a two platform equilibrium and a "T type" equilibrium (see definition below). However, whatever δ, there never exist both a two platform equilibrium and an "S type" one platform equilibrium.

26 Corollary 8 is a direct consequence of Proposition 3 and Lemma E-4 (and Lemma E-2 for the set of measure zero in which there exist both a two platform equilibrium and a S2 one platform equilibrium).

27 By (SmallCE), this condition is sufficient for the entrant to attract first the lne consumers and then the hne consumers. Π 2 = (α h + α )p 2 /(1 -δ). Therefore, to prevent this type of entry p 2 must satisfy

p 2 ≤ (1 -δ)(u h -v h ). ( 20
)
Along the equilibrium path, constraint ( 20) is binding and this yields the profits of Corollary 6.

The study of the existence of T equilibria raises some difficulties. In Section 2, we assumed that consumers left their current platform only if this strictly increased their utility (this is the strict inequality in ( 2)). If we maintain this assumption, no T equilibrium exist: after, out of equilibrium, some consumers have purchased from an entrant, the Incumbent would have to choose the highest possible price that makes them strictly prefer to come back to its platform, but no such highest price exists. 28 To finesse this issue, only in the analysis of T equilibria, we assume that, when the consumers are separated and indifferent between changing platform and not, they change whenever the platform they are purchasing from would generate negative profits if it lowered its price, whereas the destination platform would still make positive profits if it decreased its price by a small enough amount.29

Literature

We have built the model used in this paper to focus on the role of the incumbency advantage in competition for the market in the presence of different subgroups of consumers with different tastes for network externalities. This has enabled us to present clean conditions under which competition will lead, inefficiently, to the presence of two platforms and also to crisp comparisons of the difference of the incumbent's profits in a dynamic and a static model.

The relevant literature is very vast -the interested reader will find a very complete survey of the literature up to about 2005 in [START_REF] Farrell | Coordination and Lock-In: Competition with Switching Costs and Network Effects[END_REF] and newer references in [START_REF] Cabral | Dynamic price competition with network effects[END_REF]; we will selectively discuss some papers to highlight what we believe is new in our approach.

Dynamic competition between platforms

First, our paper is related to the recent literature on dynamic competition between platforms. This literature has focused on oligopoly models where platforms are in somewhat stable competition with each other. [START_REF] Cabral | Dynamic price competition with network effects[END_REF] provides an interesting and representative example. In every period, a new consumer chooses to join one of two differentiated platforms; once a consumer has joined a platform he stays with this platform until "death"this is equivalent to assuming infinite switching costs. The dynamics of the model are driven by the interplay of two forces: platforms would like to price low to attract consumers, which increases both their current profits and their future attractiveness. On the other hand, as they become larger, they have incentives to increase their prices in order to reap high profits. The analysis focuses on the dynamics of dominance and stresses that convergence to monopoly is unlikely despite the fact that platforms will be of unequal size. Because in each period only one consumer chooses a platform, the issue of coordination between consumers does not arise. Some authors have used similar models to study compatibility between platforms (for instance, [START_REF] Chen | Avoiding market dominance: product compatibility in markets with network effects[END_REF] or introduce some simple version of two sided competition [START_REF] Zhu | Entry into platform-based markets[END_REF]Laussel and Resende, 2014a,b). In this tradition, several platforms compete over an infinite horizon. To ensure that they both survive, they must attract consumers even when they are smaller than their rivals. To obtain this result it is typically assumed that horizontal differentiation is "stronger" than platform effects.

Contrary to this strand of literature, we are interested in the dynamics of competition for the market rather than competition in the market. We study the persistence of market power and endogenize the number of firms at equilibrium. Consumers can switch platforms and the platforms can attract each other's customers.

In the earlier literature, [START_REF] Katz | Technology adoption in the presence of network externalities[END_REF] built a two period, two platforms model which tackled similar issues. Consumers, who all have the same utility function, are unattached at the start; some choose a platform in the first period, some in the second -the platforms are identical except for their costs. They study the circumstances under which first and second period consumers join the same platform. They also show that firms compete aggressively in the first period, to benefit of the incumbency advantage in the second period -in the switching cost context [START_REF] Klemperer | Competition When Consumers Have Switching Costs[END_REF] named this strategy, also discussed in [START_REF] Beggs | Multiperiod Competition and Switching Costs[END_REF], "invest and harvest". [START_REF] Fudenberg | Pricing a Network Good To Deter Entry[END_REF] present an infinite horizon model which builds on Katz and Shapiro's. Consumers live for two periods and the new consumers, who are all identical, coordinate on the best platform from their viewpoint. In some periods all the consumers purchase from the incumbent. In others, a low cost entrant will price low enough to attract the new consumers. However, despite the fact that the incumbent has an installed based advantage, the entrant technology is assumed to be better than the incumbent in such a way that, at equal platform effects benefits, it provides more utility to the consumers. This technological gap caused by entry is larger than the platform effects. As a consequence, the focus of the paper is more on the way in which the incumbent uses platform effects to defend against entry of a technically superior entrant than on the way in which incumbency advantage can be exploited. 30 In particular they show that despite the fact that in a static model the incumbent would choose to price high enough to exclude consumers with a low stand alone valuation for platform services, in a dynamic model, it charges a lower price to make its services more attractive to the new consumers.

Before turning to the discussion of the modeling of incumbency advantage, we should mention that some authors have been studied the incentives for a monopolist to allocate its customers among several platforms that it owns (see [START_REF] Board | Monopolistic group design with peer effects[END_REF][START_REF] Veiga | Platforms exclude but do not segregate[END_REF]: several platforms opens the possibility of price discrimination but lessens the benefits of platform externalities.

Modeling incumbency advantage

Policy makers have been quite concerned about the long lasting market power of platforms as consumers find it difficult to coordinate on joining other platform; yet, the literature proposes very few theoretically exploration of the sources of incumbency advantage. In their early survey of platform effects, [START_REF] Katz | Systems competition and network effects[END_REF] say "Asymmetries involving reputation, product differentiation, and installed base 31 are especially likely when one of the firms is an entrant and the other an incumbent." In this view, it is not incumbency by itself that induces the competitive advantage. We believe that most economists would feel that the incumbent would have a competitive advantage even if the entrants were subsidiaries of large firms with established reputation, even if products were similar and even if there were no switching costs so that there were no installed base competitive advantage. It is this view which we have taken in this paper, despite the fact that we do agree that the points raised by Katz and Shapiro are often relevant.

Most papers which try to model incumbency advantage have done so by modelling the beliefs of the consumers. The pioneering work in this strand of literature is [START_REF] Caillaud | Chicken and Egg: Competition Among Intermediation Service Providers[END_REF] who assume, in a two sided framework, that the agents on both sides of the market coordinate on the equilibrium which is the less favorable to the entrant. Ha laburda and Yehezkel (2013) follow the same strategy while [START_REF] Ambrus | Asymmetric networks in two-sided markets[END_REF] use the concept of coalitional rationizability to solve the consumer coordination problem. [START_REF] Argenziano | Differentiated networks: Equilibrium and efficiency[END_REF] adopts the global game approached introduced in the economic literature by [START_REF] Carlsson | Global games and equilibrium selection[END_REF] to study competition between symmetric platforms; Gunay [START_REF] Bendas | Competition in markets with network externalities[END_REF] follow her lead but assumes that consumers have better information about the incumbent platform than about the entrant.

Ha laburda et al. ( 2016) build a dynamic infinite horizon duopoly model with heterogenous firms on the basis of this "belief approach" to incumbency. They assume that last period's incumbent is "focal" in the beliefs of the consumers, who all have the same preferences. They show that a firm can stay dominant even with lower quality. Because of their belief bases approach, there can be several equilibria when the horizon is infinite.

Other authors use strategies closer to those of the present paper, modelling either the way in which consumers would migrate from an incumbent to an entrant or the way in which they would choose to join a platform where none existed previously. For instance, [START_REF] Ochs | Overcoming the coordination problem: Dynamic formation of networks[END_REF] assume that consumers are uncertain about the tastes of other consumers and go through several rounds of choosing whether or not to join a firm. [START_REF] Farrell | Standardization, Compatibility and Innovation[END_REF], 1986[START_REF] Farrell | Coordination through committees and markets[END_REF] study games where consumers choose one after the other whether or not to join a platform. [START_REF] Biglaiser | Migration between platforms, free riding and and incumbency advantage[END_REF] studies the way in which consumers would attempt to free ride on each other while attempting not to be the first to abandon the incumbent (see also [START_REF] Ostrovsky | Adoption of standards under uncertainty[END_REF] for a similar approach).

Finally, we note that two companion papers, [START_REF] Biglaiser | The value of switching costs[END_REF] and Biglaiser et al. (2016a) examine models with free entry, an incumbent, and consumer switching costs. As in the current paper, the Incumbent's profit does not grow very much when expanding the time horizon from one to an infinite number of periods.

Network externalities have often been called "social switching costs", however there are subtle but important differences in their consequences for the strategies of firms. These differences are analyzed in [START_REF] Crémer | Switching costs and network effects in competition policy[END_REF].

Conclusion

In this paper, we studied the long run incumbency value in a market with consumer platform externalities and free entry. Competition for the market greatly limits the additional profits in a dynamic model relative to the static market outcome. Consumer heterogeneity can have great strategic value and even consumers who never join the incumbent's platform enhance the incumbent's profits. In order to study the value of incumbency we define a criterion for equilibrium selection which is based on a model of migration between platforms.

To identify the main economic forces at play, we have purposefully used a very sparse model. In the rest of this conclusion, we discuss two dimensions in which it could be fruitfully expanded: two sided markets and the addition of switching costs.

Many markets with platform externalities are two-sided markets. Using our selection approach in two sided settings introduces some interesting possibilities. The 'snowballing' effect discussed in Section 2 when examining migration through large steps would not necessarily arise. Also, many two sided platforms offer multiple functionalities on at least one side. They may therefore compete on some dimensions and not others (for instance, eBay and Amazon compete for the sales of some goods, but not on the ebook market). Combined with the fact that consumers often multi-home, this opens up a very rich area for investigation which has not been sufficiently explored.

There are many ways that the model can be extended. First, we have assumed that the platforms offered by the Incumbent and entrants have the same stand alone value to consumers. One question that should be investigated is how much better would an entrant's platform have to be to completely displace an incumbent's platform. It would also be useful to have a better understanding of the incentives to invest in quality. [START_REF] Levin | The economics of internet markets[END_REF] states "In traditional industries with platform effects, high switching costs are often an important compounding factor" but argues that for Internet platforms switching costs are small. However, we know very little about the interaction between switching costs and platform externalities and more theoretical and empirical work needs to be done on the topic. Until then, it will be impossible, for instance, to know whether relatively small switching costs can have large effects. In [START_REF] Crémer | Switching costs and network effects in competition policy[END_REF], we present an example that shows that switching costs and platform externalities do not necessarily have additive effects. We plan to investigate further this

Figure 1 :

 1 Figure 1: This figure represents in algorithmic form the definition of ac equilibria.

Figure 2 :

 2 Figure 2: The dynamic model.

Figure 3 :

 3 Figure 3: This figure illustrates part 2 of Corollary 4 for α = α h , w = 1 and u = .8. The right hand side of (2NetCond) is then equal to(2 -δ)(1 -.8δ)/(1 -δ), which is equal to 2 when δ = 0. Its minimum, V min is equal to 1.8 and is obtained for δ min = .5. For u h -v h = m 1 ∈ [1.8, 2], there exists an interval of δs such that there exists a two platform equilibrium. For u h -v h = m 2 > 2, there exists a two platform equilibrium if δ is small enough.

Figure 5 :

 5 Figure 5: The type of equilibria in the one platform case.

  that there exists a two platform equilibrium in the static model, there exists a two platform equilibrium in the dynamic model if and only if δ ≤ δ.2. If w <(α h + α )u /α h and u h -v h is less than (α h + α )w /α h butgreater than V min , then there exists no two platform equilibrium in the static model and there exists one in the dynamic model if and only if δ belongs to an interval [δ 1

We make this assumption for expositional reasons. All our results hold with two infinitely lived entrants at the start of the game.

For the purpose of this section, the fact that some consumers derive more utility than the others from the presence of other consumers play no role.

All our results still hold true if ψ h (0, 0) = ψ (0, 0) > 0. On the other hand we would have to change our analysis, but in non-essential ways, if hne and lne consumers had different stand alone utilities for the platforms, i.e., if ψ h (0, 0) = ψ (0, 0).

See Biglaiser, Crémer, and Veiga (2016b) for such an attempt.

This assumption considerably simplifies the reasoning below. In an ac equilibrium, consumers will stay on the incumbent platform when they are indifferent between doing so and joining an entrant. We could do without the assumption, but this would require that when studying competition between platforms, we use the type of limit pricing arguments standard in, for instance, the study of Bertrand competition with different marginal costs. In our framework, this would make the proofs much more complicated without changing the equilibrium payoffs. We relax this condition in subsection E.2 for off the equilibrium path events to guarantee existence for a certain class of equilibria.

There are some parameter values for which separation occurs with entry but not without entry, and none for which the opposite is true.

We can also obtain the same basic results regarding profits using a Nash timing, where firms simultaneously set prices. There would only exist mixed strategy equilibria, but the equilibrium profits of the platforms would be the same as with Stackelberg timing

These assumptions are sufficient to exclude "collusive" equilibria when we assume a finite number of infinitely lived entrants. See[START_REF] Biglaiser | Equilibria in an infinite horizon game with an incumbent, entry and switching costs[END_REF] for discussion of such outcomes in a switching cost framework.

We allow entrants to offer negative prices. They can be thought of as a discount below the cost of providing service or as the value of goods that the entrants give away in addition to access to the platform.

The "essentiality" refers to the fact that the prices charged by the entrants, who acquire no consumers, are not uniquely determined. In the sequel, essentiality will be implicit whenever we speak about uniqueness.

The Incumbent will have some sales in the first period. If in period 1 it sells only to the lne consumers, the price p I charged by the Incumbent and p E satisfy w h -p 2 ≤ -p E (hne consumers purchase from the entrant), and u -p I ≥ v -p E (lne consumers purchase from the incumbent). This implies w h ≤ p I -p E ≤ u -v , which contradicts (4) and (5).

Formally, the results of this subsection are a special case of those of section 6; our aim here is to bring out some of the economics of competition between the incumbents and the entrants which might not be as transparent in the analysis of the general case.

We are assuming that the lne consumers join a platform. There could be an equilibrium where the lne consumers do not join a platform.

 17 This assumes that the lne consumers all coordinate on the same entrant. They need not do so if there are indifferent to network effects. This coordination must either be assumed or the results which we present here can be considered as limit results when lne consumers are close to indifferent to network effects.

This result will also arise in the more general model below. See equation (13).

If (15) holds as an equality, then an entrant charging -p E = v h -p , will obtain no consumers. If -p E > v h -p , the entrant will attract all consumers. It will not matter which type of consumers moves first in the migration to the entrant.

If u h -p h < u -p , then by the same argument as in the proof of claim C-1,p h = u h (1 -δ) and therefore u h -p h = δu h . Since p ≥ 0, u h -p h < u -p is possible only if u > δu h .Assuming that condition (15) holds is only relevant in the analysis of "S equilibria" (definition below). From claim C-1, when (15) holds the profits of an entrant who attracts the lne consumers are α u . From Appendix D, when (15) fails, they would be greater. We conjecture that this will make entrants more aggressive along the equilibrium path and lead to lower profits for the Incumbent.

This corollary is an easy consequence of lemmas E-2, E-3, E-5, and E-6: for δ close to 1, only type S2 and T2 equilibria exist and in all these equilibria the Incumbent's profit is (α h + α )(u h -v h ).

The set of prices that make consumers prefer to purchase from the entrant than from the incumbent has a supremum, but no maximum.

A more fundamentalist approach would conduct a full Bertrand game analysis, including the continuation game played by the consumers.

In current policy discussions, it is generally assumed that it is the incumbent which has a technological advantage, due to the possession of data.

[START_REF] Crémer | Connectivity in the Commercial Internet[END_REF] this strategy and assume that there is a mass of "trapped" consumers who cannot leave the Incumbent.

It is sufficient to note that the proof of Claim 1 depends on the relative sizes of the platform effect only through (15).

interaction in future work. The loss of the myopia principle will make the study of dynamic models quite challenging.

Appendix A Proofs of lemmas in section 2

Proof of lemma 2. Assume that a migration path which leads to an ac equilibrium is not a migration path through large steps. Then, there exists a t such that η t+1 θ(t)s(t) > 0 (at the beginning of step t + 1 there remains some consumers of type θ(t) on platform s(t)). It is easy to see that {θ(t + 1), d(t + 1), s(t + 1)} = {θ(t), d(t), s(t)} as at step t + 1, the migration will involve the same type of consumers moving from the same platform to the same platform as at step t. Indeed, in all platforms the utility of agents of type θ(t) is the same at the end and at the beginning of step t, except for the fact that it is strictly higher in d(t) and strictly smaller in s(t). For agents of type θ (t), the "other type", the same property holds true; however by (1), the increase in the utility they derive from d(t) and the the decrease in the utility they derive from s(t) are smaller than for agents of type θ(t). Hence, condition (2) holds true when the superscript t -1 is replaced by t. We can therefore construct a new migration path, which will lead to the same final allocation by replacing steps t and t + 1 by one "larger" step with the same θ, d and s. Iterating on this procedure will lead to a migration path through large steps which leads to the same allocation as the original path.

Proof of lemma 4. It is sufficient to show that a migration path through large steps can be replaced by a migration path with η t d(t)θ(t) -η t-1 d(t)θ(t) < ε for all t. Let θ(t), d(t), s(t) t=1,T define a large step migration path. We construct a new migration path in the following way. Let θ(1) = θ(1), d(1) = d(1), s(1) = s(1), and η such that

At the end of step 1 on the new migration path, by the same reasoning as in the proof of lemma 2,

Appendix -1 for all ( θ, i, j) = (θ(1), d(1), s(1)). Therefore

and by an easy recurrence it is possible to build a new migration path which after a finite number t * 1 of steps will rejoin the original migration path:

θj for all θ and j. We can then take

. By the same reasoning as in the previous paragraph there will exist t * 2 such that after t * 2 steps the new migration path will have the same allocation as the original migration path at t = 2. The result is proved by noticing that we can repeat the process until convergence to the final allocation along the original path.

Proof of lemma 5. We have defined migration paths by the fact that they lead from one initial allocation to a final allocation. To show that there exists a final allocation define the following procedures, inspired by large steps migration paths, but without guarantee that they lead to a final allocation. At every step, check whether there exist a {θ(t), d(t), s(t)} satisfying (2). If there is move all the consumers of type θ(t) from s(t) to d(t). If there is not, we have identified an ac equilibrium. To finish the proof, we only need to show that any such procedure will eventually find itself at a stage when this happens. At every step, either the destination platform already has clients or it charges a strictly lower price that the source platform, or both. To each platform which has a strictly positive mass of consumers, associate an index equal to the number of platforms which charge strictly lower prices multiplied by either 1 if it has a positive mass of only one type of consumers and 2 if it has a positive mass of both types of consumers. The sum of these platform indexes decreases by at least one at each stage of the migration. Given that this sum cannot be smaller than 1, the result is proved. 2NetCond) is satisfied for δ = 0 and b) by (SmallCE) and footnote 21 g (0) > 0. By convexity of g this implies point 1 of the corollary.

B Proof of Corollary 4

From footnote 21, g (δ)×α h (1-δ) 2 is equal to -α u (1-δ) 2 +(w -u )α h . This implies point 2 of the corollary with

C Proof of Proposition 3

In this appendix, we provide the formal proofs of the lemmas and claims of section 6.3 of the main text.

Proof of Claim 1. Assume that the H incumbent charges a price p h which satisfies (15) and

If an entrant charges p E ≥ -(u -p ), it attracts no consumer as, by ( 15), we have

On the other hand if the entrant charges p E < -(u -p ), it attracts all the consumers: the lne consumers as -p E > u -p , and, once it has attracted the lne consumers, the hne consumers as v h -p E > v h +u -p ≥ u h -p h . In equilibrium, this must not be profitable; a sufficient and necessary condition for this is -(α h + α )(u -p ) + δΠ 2 ≤ 0. This condition does not depend on p h . Therefore, the H incumbent could increase p h without affecting the demand for its services and therefore increase its profits.

Proof. From ( 17), it is sufficient to prove that we cannot have -α (up ) + δΠ < 0. Assume this were the case. In any period after the first, the L incumbent could increase its profit by charging p ∈ (p , u -δΠ /α ). Indeed, in order to attract the lne consumers an entrant would have to charge at most p E = -(u -p ) < -δΠ /α and would therefore make negative profits, α p E + δΠ .

Proof. By ( 16) and ( 18), if the claim does not hold, there exists p h > p h which satisfies both

We will show that a deviation by the H incumbent to such a p h would be profitable.

The H and L incumbents announce their prices simultaneously; therefore the deviation by the H incumbent would not affect p . By (C-2), after such a deviation the lne consumers would respond by purchasing either from the lowest price entrant or from the L incumbent, as in Figure 4 (replacing, of course, p h by p h ). Therefore, the deviation would be unprofitable for the H incumbent only if an entrant could profitably attract all the consumers.

Appendix -3

It could do this only by charging a price p E which satisfies

The profits of the entrant, (α h + α )p E + δΠ 2 , would be strictly negative, which proves the result.

Proof of Lemma 10. Only the sufficiency part is left to prove. From Figure 4 an entrant could try either to a) attract only the lne consumers by charging a price strictly smaller than -(u -p ), but, by claim C-1, this is not profitable as α (-(u -p )) + δΠ = α u (-1 + (1 -δ) + δ) = 0, or b) attract all consumers by charging a price strictly smaller that -(u h -p h ) + v h , but this is not profitable by ( 19).

Claim C-3. If (15) holds, then

would be profitable. At the original p 2 , there was profitable entry by attracting only the lne consumers; a fortiori, it will also be profitable to attract the lne consumers when the price is p 2 . Therefore, the deviation by the period 1 incumbent is unprofitable only if an entrant could profitably attract all the consumers when the price is p 2 . By (C-4), in order to attract the hne consumers as well as the lne consumers, a entrant needs to charge a price p E which satisfies

The profits of the entrant, (α h + α )p E + δΠ 2 , would be strictly negative, which proves the result.

Claim C-4. If (2NetCond) holds, then a) (SmallCE) holds and b) the prices defined by (10) and (12) satisfy condition (15).

Appendix -4 D Proof that (15) holds

In the main text, we have assumed that condition (15) holds. In this part of the appendix, we show that this must indeed be the case whenever a two platform equilibrium exists. We proceed by contradiction. If (15) did not hold, we would have u hp h < u -p . We first show that the results which we obtained in 6.3 based on the fact that consumers do not change platforms from period 2 onwards hold with h and inverted. Then, we show that these results are incompatible with the separation of the consumers in two different platforms in the first period.

The proof of claim 1 can be reproduced with h and inverted 32 and therefore

Similarly, adapting the reasoning which leads to claim C-2 we obtain:

Along with (D-1) and (D-2), this implies

To compute Π 2 , we eliminate p 2 from the system composed of the two equations a) (C-3), which still holds as Claim C-3, whose proof is based on period 1 deviations which attract all consumers, is still valid as it stands, and b) Π 2 = α h p 2 + δα h u h . Substituting into (D-4), we obtain

which establishes the contradiction.

E One platform equilibria

In this appendix, we provide a full analysis of one platform equilibria starting with S equilibria in E.1 and turning to T equilibria in E.2.

Appendix -5 E.1 S equilibria: consumers stay separated after they split

In S equilibria, if, off the equilibrium path, hne and lne consumers join different platforms in some period, then they stay on these platforms in subsequent periods.

As proven in claim C-1, when condition (15) holds Π = α u and p = u (1 -δ). Along the equilibrium path, in order to attract only the lne consumers, an entrant must charge a price p E which satisfies -p E > w -p 2 as well as v h -p E ≤ u h -p 2 ; such a p E exists by (SmallCE). This is profitable if α p E + δΠ > 0, which is equivalent to -p E < δΠ /α = δu .

To make this type of entry impossible the incumbent must ensure that if p E = -δu the lne consumers choose not to purchase from the entrant. Therefore, it must choose p 2 such that p 2 ≤ w -δu .

(E-1)

To attract both types of consumers, an entrant must charge a price p E which satisfies v h -p E > u h -p 2 . 33 This is unprofitable only if (α h + α )p E + δΠ 2 < 0, which is equivalent to p E + δp 2 /(1 -δ) < 0 because Π 2 = (α h + α )p 2 /(1 -δ). Therefore, to prevent this type of entry p 2 must satisfy

Along the equilibrium path, both constraints (E-1) and (E-2) must be met, and at least one of them must be binding. When (E-1) is binding, we have an S equilibrium; when (E-2) is binding we have an S2 equilibrium. This implies the following lemma.

Lemma E-1. In S type single platform equilibria the profit of the Incumbent is

A full characterization of the conditions under which these equilibria exist is rather complicated. However, it is a direct consequence of Lemmas E-2 and E-3 that a) S equilibria exist only for "small" u h -v h and b) except on a set of parameters of measure zero 34 no S equilibrium exists when (2NetCond) holds, i.e., when a two platform equilibrium exists. Furthermore, the set of parameters for which there exist either a two platform equilibrium or an 33 By (SmallCE), this condition is sufficient for the entrant to attract first the lne consumers and then the hne consumers.

34 This arises when (2NetCond) and the right hand side of (E-3) are both binding.

Appendix -6 S equilibrium is quite large. S2 equilibria are described in Lemma E-2, while S equilibria are described in Lemma E-3.

Lemma E-2. If δα -(1 -δ)α h > 0, then an S2 equilibrium exists if and only if

If δα -(1 -δ)α h < 0, then an S2 equilibrium exists if and only if

In both cases, the profit of the Incumbent is

proves the last sentence of the lemma. Along with (E-1) this implies u hv h ≤ (w -δu )/(1 -δ). By the discussion on page 26, after lne and hne consumers are separated Claim C-2 holds. This implies p h = p 2 and Π h = α h (u h -v h ). In order to attract the lne consumers the H incumbent would have to charge a price p h which satisfies v -

, which proves the lemma.

Lemma E-3. An S equilibrium exists if and only if

The profit of the Incumbent is then

Proof. The fact that (E-1) is binding immediately yields p 2 = (w -δu )/(1δ) and Π 2 . The lowest price that an entrant is willing to charge in order to attract all the consumers is -δΠ 2 /(α h + α ), and therefore the incumbent can price up to (u h -v h ) -δΠ 2 /(α h + α ) and sell only to the hne consumers. In subsequent periods, it would set the same price by Claim C-2. Therefore, this deviation is unprofitable, only if

which is equivalent to the right most inequality of (E-3).

Off the equilibrium path, consumers stay separated. In order to attract the lne consumers away from the L incumbent, the H incumbent would have to announce a price not larger than v -u + p = v -δu . This is unprofitable only if Π h ≥ (α h + α )(v -δu ) + δΠ 2 , where Π h , the profit of the H incumbent, is equal to the right hand side of (E-4). This inequality is equivalent to the left most inequality in (E-3). 35 Out of equilibrium, once the consumers are separated, the profit of either the L incumbent or of an entrant which would charge p and attract all the consumers would be (α h + α )p + δΠ 2 . For such a strategy to be profitable, we must have p ≥ -δΠ 2 /(α h + α ). It is only feasible if p + (u h -v h ) < 0 -otherwise the H incumbent can profitably ensure the fidelity of the hne consumers. By (E-2), these two bounds on p cannot hold simultaneously and therefore no such deviation is possible.

E.2 T equilibria: consumers come back together after they split

In a T equilibrium, we must have

Otherwise, by charging, for instance, (p 2 -w )/2 > 0, an entrant would attract the lne consumers and make positive profits even if it "lost" all these consumers in the following period. An entrant must also find it unprofitable to attract all the consumers. This occurs if and only if

When (E-5) is binding, we have a T equilibrium; when (E-6) is binding we have a T2 equilibrium. Because at least one of these two constraints is always binding, we have the following lemma.

Lemma E-4. In a T type single platform equilibria the profit of the incumbent is

35 There are other possible deviations. It is possible to show that they are not profitable.

Appendix -8

Contrary to what happens for S equilibria, there can exist both a two platform equilibrium and a T equilibrium: this is true, for instance, when δ = 1/2, α = 3α h /2 and u = v = w = 8(u h -v h )/15. Then, (SmallCE), (2NetCond) and the conditions of lemma E-5 below are satisfied.

The following fact is both economically interesting and technically important for the characterization of T equilibria: if, off the equilibrium path, the consumers are separated in two platforms, in the next period they will all purchase from the H incumbent, not from the L incumbent. The L incumbent profitably attracts all the consumers only if p ≥ -δΠ 2 . It will attract the hne consumers if for all non negative prices the consumers prefer to "leave" the H incumbent (otherwise, the H incumbent could deviate and profitably keep its consumers). Thus, we must have p < -(u h -v h ). But, from Lemma E-4, -(α h + α )(u h -v h ) + δΠ 2 < 0, and these two bounds on p cannot be satisfied simultaneously. A similar argument shows that no entrant can attract all consumers to its platform.

The following two lemmas provide the conditions for existence and the profits for each of the types of T equilibria.

Lemma E-5. A T2 equilibrium exists if and only if

The equilibrium profit of the Incumbent is

Proof. In a T2 equilibrium, the Incumbent cannot profitably raise its price and sells only to the hne consumers at price p

Thus, it immediately follows that Π 2 = (α h + α )(u h -v h ) and this gives us the right hand side of the necessary and sufficient condition.

Off the equilibrium path, we need to find conditions for the Incumbent to be willing and able to attract the lne consumers if the consumers are ever separated. The Incumbent attracts the lne consumers, by charging a p h smaller than or equal to v -u , since the L incumbent is willing to charge any positive price to keep the lne consumers. 36,37 Furthermore, the Incumbent must choose a price that will induce the hne consumers to stay on its platform instead of joining an entrant platform. Entrants are willing to price down to -δ(u h -v h ) in order to attract all the consumers and the incumbent must 36 We assume that firms do not use weakly dominated strategies and it is clear that if the L firm charged a positive price and lost consumers that it could profitably deviate and lower its price.

37 Recall that we are using the weak inequality definition of AC equilibria for this class of equilibria.

Appendix -9 therefore charge a price p h smaller than or equal to (u

The most profitable deviation which allows the H to keep only the hne consumers is to charge (1 -δ)(u h -v h ). Thus, for the H incumbent to prefer to attract the lne consumers to deviating and keeping only the hne consumers we must have

which is equivalent to the left hand side of the necessary and sufficient condition.

Lemma E-6. A T equilibrium exists if and only if:

The Incumbent's equilibrium profit is (α + α h )w /(1 -δ).

Proof. The binding pricing constraint when the incumbent has all the consumers is (E-5) and the left hand side of condition (E-7) reflects this. It follows immediately that p 2 = w /(1 -δ) and Π 2 =(α + α h )w /(1 -δ).

Off the equilibrium path, in order to attract back the lne consumers, the Incumbent must offer a price less than or equal to v -u , since the L incumbent will price at 0. 38 The lowest price an entrant is willing to offer to attract all the consumers is -δw /(1 -δ). Thus, the Incumbent's price must not exceed u h -v h -δw /(1 -δ). Since v -u < 0 < u h -v h -δw /(1 -δ), it is the first constraint which is binding. The most profitable deviation which would allow the Incumbent to keep only the hne consumers is to charge u h -v h -δw /(1 -δ). Thus, for the Incumbent to prefer to bring all consumers onto its platform we must have

which is the first term of the right hand side of (E-7).

On the equilibrium path, the Incumbent must prefer to keep all consumers to just keeping the hne consumers and then bringing them back the lne onto its platform the following period at a price of v -u . Hence, we must have

Appendix -10 where the left hand side is the equilibrium profit, and the right hand side is the sum of profits in the defection period plus the discounted left hand side of expression (E-8). This can be rewritten

which is the second term of the right hand side of (E-7).

Appendix -11