David Spector
email: spector@pse.ens.fr

Severin Borenstein

Bernard Caillaud

Jeff Ely

Joseph Farrell

Bruno Jullien

Zvika Neeman

Debraj Ray

Patrick Rey

Daniel Rubinfeld

Jean Tirole

Thibaud Vergé

Exclusive contracts and demand foreclosure

Keywords: L12, L13, L14, L41, L42 exclusive dealing, foreclosure, exclusionary strategies

A Þrm may decide to have some of its customers sign exclusive contracts in order to deprive a rival of the minimum viable size, exclude it from the market, and enjoy increased market power. If contracts are required to be simple enough, this strategy may induce inefficient exclusion even if the excluded Þrm is present at the contracting stage. Exclusive contracts may thus cause inefficient eviction, not only entry-deterrence, even though the former is less likely than the latter. However, complex enough contracts, if feasible, would allow agents to reach a Pareto-optimum, without inefficient exclusion.

Introduction

This paper aims to clarify the circumstances under which a Þrm may sign exclusive contracts with some of its customers in order to exclude a rival, even though exclusion reduces social welfare. This clariÞcation is important both for its own sake and from the viewpoint of antitrust policy. Indeed, the legal treatment of exclusive dealing appears to be responsive to the twists and turns of economic theory. For example, the so-called Chicago critique, which purported to prove that the reasonings underpinning the traditional hostility toward exclusive dealing were ßawed, seems to have induced U.S.

Courts to progressively soften their handling of exclusivity clauses. 1From a theoretical viewpoint, the need for clariÞcation stems from the fact that, while the recent body of literature rigorously described several scenarios of welfare-reducing exclusion through the use of exclusive contracts, it still leaves readers wondering which conditions exactly are necessary for these anticompetitive outcomes to occur in equilibrium. The main contribution of this paper is to show that, although the existing theories of socially harmful foreclosure through exclusive contracts assume that some of the adversely affected parties are not present during the contracting stage, this assumption is unnecessary. Such an outcome can also occur when all affected parties are present, as long as contracts are required to be simple enough. This theoretical result is relevant for antitrust policy, because most of the relevant case law involves situations in which the allegedly excluded Þrms were initially in the market and thus perfectly able to respond to the disputed contracts with counteroffers of their own.

Relation to the literature

The Þrst analysis of exclusive contracts emanated from the "Chicago school" and dismissed the view that these contracts could be used by a Þrm in order to exclude a rival and increase its market power. The Chicago school argument is simply that if such exclusion is socially inefficient, the payment which the excluding Þrm has to grant consumers in order to "bribe" them into agreeing to exclusivity would exceed the incumbent Þrm's gain from deterring entry or inducing exit 2 Exclusive dealing must them have other, probably procompetitive motives. 3 The "post-Chicago" literature has identiÞed several circumstances under which socially harmful exclusive contracts may arise. Matthewson and Winter (1987), for example, showed that a manufacturer may proÞtably use impose exclusivity to a local retailer in order to foreclose a rival in a local market, and that this outcome may be (but need not be) socially harmful. 4 But this result breaks down if nonlinear pricing is feasible 5 .

2 See Posner, (1976, p. 212) and Bork (1978, p. 309). For a generalized and rigorous version of the Chicago critique, see Bernheim and Whinston (1998, hereinafter "BW"), especially Section III; and O' Brien and Shaffer (1997).

3 See, e.g., [START_REF] Marvel | Exclusive Dealing[END_REF] and Segal and Whinson (2000b). 4 See also [START_REF] Comanor | The Competitive Effect of Vertical Agreements: Comment[END_REF] for a related analysis. 5 See the discussion in BW and O'Brien and Shaffer (1997). However, [START_REF] Spector | Adverse Selection and Exclusive Supply Contracts[END_REF] shows that [START_REF] Mathewson | The Competitive Effect of Vertical Agreements: Comment[END_REF] result may still hold under nonlinear pricing 3 On the other hand, various papers have shown that exclusivity clauses may facilitate proÞtable entry deterrence or competitors' eviction. Their common theme is that inefficient exclusion may occur when some adversely affected parties (a potential entrant, or future consumers) are absent at the contracting stage, and thus cannot make the payments to other parties which are necessary for Coasian bargaining to take place and lead to efficiency.

The seminal paper in this branch of the literature is Rasmusen et al. (1991, henceforth, RRW), complemented by Segal and Whinston (2000a, henceforth SW) -these papers are the ones most closely related to ours.

They show that, if increasing returns make a minimum scale of operation necessary for proÞtable entry, an incumbent can achieve full exclusion cheaply by exploiting the lack of buyers' coordination, or by discriminating between buyers. The idea is that even if buyers as a whole lose when entry is deterred, entry deterrence can be proÞtable because the excluding Þrm does not need to bribe all its potential customers into signing an exclusivity agreement. It only needs to have some of them sign exclusive contracts, just enough to deprive the potential entrant from the minimum viable scale. The incumbent can then fully exploit its market power vis-à-vis all the potential customers, including those who did not sign an exclusive contract and whose consent was not bought. The entrant's need for a minimum scale of operation thus generates a contracting externality across customers which the incumbent if the seller does not know each customer's demand function. This is simply a consequence of the more general fact that nonlinear pricing together with asymmetric information resembles linear pricing. can exploit 6 . In this case, the Chicago critique breaks down because the excluding Þrm does not need to compensate the loss suffered by all buyers, but only that suffered by some of them. A coordination failure among buyers may lead to the same result. 7 Variants of RRW consider the possibility of buyers forming coalitions 8 or the impact of competition among buyers 9 . But all of them stick to the assumption that the potential entrant does not take part to the contracting game.

A related idea can be found in models assuming that some adversely affected parties, other than the excluded Þrm, cannot take part in the contracting game. These parties can be consumers in a future market 10 , or agents whom transaction costs deter from participating to the contracting game 11 .

In yet one more category of papers, socially harmful exclusion may result from contracts signed between wholesalers and retailers because Þnal consumers are left aside 12 .

All these models assume that some adversely affected parties are absent from the contracting game. This raises two issues. The Þrst one is theoret-6 For a systematic treatment of contracting with externalities, see [START_REF] Segal | Contracting with Externalities[END_REF][START_REF] Segal | Coordination and Discrimination in Contracting with Externalities: Divide and Conquer?[END_REF] and [START_REF] Segal | Robust Predictions for Bilateral Contracting with Externalities[END_REF].

7 SW stresses that the equilibria in which exclusion is achieved by exploiting the lack of coordination among buyers are not perfectly coalition-proof.

8 [START_REF] Innes | Strategic Buyers and Exclusionary Contracts[END_REF]. 9 Motta and [START_REF] Fumagalli | Exclusive dealing and entry, when buyers compete[END_REF], and Simpson andWickelgren (2004, 2005). 10 BW, section IV 11 [START_REF] Gans | Exclusionary Contracts and competition for large buyers[END_REF]. 12 [START_REF] Hart | Vertical Integration and Market Foreclosure[END_REF]; [START_REF] Lin | The Dampening-of-Competition Effect of Exclusive Dealing[END_REF]; [START_REF] O'brien | On the Dampening-of-Competition Effect of Exclusive Dealing[END_REF]. [START_REF] Simpson | Naked Exclusion, Efficient Breach, and Downstream Competition[END_REF] belongs to this set of papers (since the presence of downstream consumers exacerbates the inefficiency) while at the same time being a variant of RRW (since the excluded Þrm in their model is a potential entrant unable to participate to the contracting game).

ical. In most of the aforementioned models, contracts are restricted to take a rather simple form. For example, RRW and SW only consider simple exclusive contracts, ruling out both breach penalty provisions and conditional contracts. 13 A natural question emerges then: is the possibility of socially harmful eviction driven by the absence of some adversely affected parties during the contracting game, or by the restrictions imposed on the nature of feasible contracts? Clearly, if these two assumptions were lifted, i.e., if all affected parties could enter into very complex contracts, unhindered Coasian bargaining would induce an efficient outcome. But what is the respective importance of each of these two assumptions?

This question brings us to the second issue. The aforementioned literature is at odds with the facts of much of the relevant case law, which deals with settings where the excluded Þrm(s), or other potentially harmed parties, could in principle have responded to the alleged exclusionary strategy. This point has been made by Whinston (2001, pp. 68-69) in his discussion of the US v Microsoft case, and by [START_REF] Rasmusen | Naked Exclusion: Theory and Law[END_REF], who noticed that their theory failed to apply to the landmark Lorain Journal case, because the alleged victim was already present in the market when the disputed exclusivity clauses were offered. The same remark applies to several recent nonlinear pricing cases in which the disputed schemes were close to exclusivity requirements, like the European Michelin II case.14

This paper's contribution

This paper attempts to precisely identify the factors which may cause inefficient exclusion to occur, and in particular to disentangle between two possible factors, namely limits on the nature of contracts and limits on which agents may offer contracts. Section 2 presents a very general, reduced-form model, of which RRW, SW, and BW are subcases. It is Þrst applied to the familiar incumbent-entrant case (Section 3), with the following results.

• Inefficient entry deterrence may occur even when discriminatory offers are prohibited, even without any coordination failure among buyers.

A necessary condition for this is that Þrms' proÞts in a no-exclusion equilibrium be strictly positive (Proposition 1).

• Inefficient entry deterrence is even more likely to occur when discriminatory offers are possible. This is proved in Proposition 2, of which the results in RRW and SW are special cases.

• The possibility to include breach penalty provisions in exclusive contracts decreases the likelihood of inefficient entry deterrence, but does not eliminate it (Proposition 3).

• The combination of breach penalty provisions and conditional offers is enough to make inefficient entry deterrence impossible (Proposition 4).

The paper then considers the two-incumbent case, in which the potential excluder and the potentially excluded Þrm are on an equal footing as regards the possibility of making offers (Section 4). The results are as follows.

• There always exists a perfect coalition-proof equilibrium without exclusion, and there exists no equilibrium with exclusion if discrimination is prohibited (Proposition 5), or if exclusive contracts can include breach penalty provisions (Proposition 7).

• A perfect coalition-proof equilibrium with inefficient eviction may exist alongside a no-exclusion one if simple, discriminatory exclusive contracts can be offered, without breach penalty provisions. But the existence of an exclusionary equilibrium is in general less likely than in the entrant-incumbent case (Proposition 6). We view this result as the most important one of this paper, because it means that exclusive contracts can induce inefficient eviction and not only entry deterrence.

Even more strikingly, one of the implications of Proposition 6 is that under the special assumptions of RRW and SW, exclusion is as likely to occur in the two-incumbent case as in the incumbent-entrant case.

These results are summarized in Table 2, in Section 5.

The model

The assumptions of the model are kept as simple as possible for the sake of tractability. There are two Þrms, labeled Firm 1 and Firm 2, and two consumers (a and b). Firms' cost structures are different, while consumers have identical preferences. 15In order to conduct the analysis at a general enough level, we specify reduced forms rather than detailed preferences and technologies. More speciÞcally, we assume that the institutional context is as follows.

During the Þrst phase (which may be subdivided into different periods), consumers and Þrms can enter into contracts, which may involve lump-sum transfers, a commitment by a Þrm to serve a consumer, possibly an exclusivity requirement, as well as, possibly, other characteristics. Which Þrms can offer contracts (depending on whether the situation considered is an incumbententrant one, or a two-incumbent one), which type of contract can be offered, whether discrimination is allowed, the timing of moves, is described in greater detail below. Different assumptions correspond to different "contractual environments", which this paper compares. At the end of this Þrst period, some lump-sum transfers have been made, some Þrms are committed to deal with some consumers, and some consumers are committed not to deal with some Þrms.

During the second phase, subject to the constraints induced by the contracts which have been signed (i.e. an obligation to deal or a prohibition from dealing), Þrms and consumers deal with each other, which results into payoffs for the various parties. How these payoffs are determined is not the point of this paper. They may result from bargaining, or from Þrms, or consumers, making take-it-or-leave-it offers, involving linear or nonlinear pricing, etc.

For our purposes, there is no need to enter into these details, because the only relevant variables are, for each possible situation, the proÞt accruing to each Þrm, and each consumer's surplus.

Finally, the list of Þrm-consumer deals actually consummated (as per second phase decisions, themselves constrained by Þrst phase outcomes) and the Þrst phase lump-sum transfers together determine each Þrm's proÞt and each consumer's surplus level.

Consumer preferences

Consumers are assumed to have identical preferences, given by Table 1 The following assumptions about parameters are made throughout the paper for i=1 and i = 2:

π i ≥ 0 (1) V > U i (2) V + π 1 + π 2 > U i + b π i (3) s 2 < b s 2 < 0 (4)
These assumptions have the following interpretation.

(1) implies that competition leads to nonnegative proÞts for both Þrms if each has unrestricted access to both consumers. ( 2) and (3) imply that consumer as well as aggregate welfare is greater under competition than under monopoly. Finally, (4) means that because of economies of scale, Firm 2 is better off serving no consumer at all than serving only one of them, even as a monopolistic supplier. This assumption is central to all our results: it implies that if Firm 1 succeeds in signing an exclusive contract with one consumer, then Firm 2 is better off not serving any of the two consumers, because serving only one of them would not allow it to recover its Þxed costs. This assumption is indeed central to the results in RRW, SW, and BW.

All the assumptions above are satisÞed in RRW. 17 , but our model is far more general. For example, it allows for the possibility that both Þrms sell differentiated goods as well as for nonlinear pricing.

3 Foreclosing an entrant through exclusive contracts

In this section, we deal with the situation analyzed in RRW and SW, i.e. that of an incumbent able to offer exclusive contracts. We start by assuming that the incumbent is only able to offer simple exclusive contracts (i.e. without any breach possibility), and we examine two alternative cases, depending on the incumbent's ability to discriminate across consumers. Our results in the case where discrimination is allowed coincide with those of SW. We then turn to more complex contracts.

Simple exclusive contracts without discrimination

In accordance with RRW and the related literature, we start by considering the following simple game.

makes losses when serving a single consumer is equivalent to the identity N * =1 in RRW.

There is however a difference between RRW and this paper. In RRW, both Þrms have identical costs, but the assumption that the incumbent would make losses if it served only a small number of customers is irrelevant because of its Þrst-mover advantage. In this paper, because Þrms are on equal terms regarding the timing of offers, we need to depart from the assumption of identical costs for simplicity, in order to limit the number of equilibria. In this sense, Þrms are not on equal terms as regards costs. But the point of this paper is simply to investigate the consequences of them being in equal terms as regards contracts, not costs.

Stage 1. Firm 1 may offer each consumer a contract specifying that (i) the consumer commits not to purchase from Firm 2; and (ii) a lump-sum transfer from Firm 1 to the consumer. Equilibrium multiplicity is pervasive when exclusive contracts are possible 18 . Following the existing literature, and in order to rule out situations where exclusive contracts result only from of a lack of coordination between consumers or Þrms (a possibility arising for example in RRW), we restrict our attention to perfect coalition-proof Nash equilibria (PCPNE), as deÞned in [START_REF] Bernheim | Coalition-Proof Nash Equilibria. I. Concepts[END_REF].

Proposition 1 In any PCPNE, inefficient entry deterrence occurs if and only if it maximizes the joint surplus of the incumbent and the two consumers,

i.e. if U 1 + b π 1 > V + π 1 .
Notice that for exclusion to happen in Proposition 1, it must be the case

that π 2 > 0. The reason is that, if π 2 = 0, Assumption (3) is equivalent to U 1 + b π 1 < V + π 1 .
Consequently, under the assumptions of RRW and SW, exclusion cannot occur in a PCPNE is discrimination is prohibited.

Simple exclusive contracts with discrimination

We assume now that the incumbent can discriminate among consumers. In this case, exclusion is more likely because in order to exclude, the incumbent only needs to convince one consumer to sign an exclusive contract: if it does, the entrant will decide to serve no consumer at all rather than only one, and the incumbent will exert its market power vis-à-vis both consumers while only having to compensate one of them.

Proposition 2 In any PCPNE, inefficient entry deterrence occurs if and

only if U 1 + 2b π 1 > V + 2π 1 , i.e.
if the joint surplus of Firm 1 and one consumer is greater under exclusion than under no-exclusion.

Proposition 2 coincides with Proposition 3 in SW if the special assumptions of RRW are made. The proof of this result is straightforward. Coalitionproofness implies that for Firm 1 to induce a consumer to enter into an exclusive contract, it must guarantee this consumer a welfare level equal to at least V . Indeed, if in equilibrium both consumers earn less than V , they could form a coalition and jointly decide not to sign contracts. This would yield each consumer a welfare level of V (because Firm 2's entry would not be deterred) and not signing would be an equilibrium (since signing an exclusive contract when the other consumer does not would cause the signing consumer's welfare level to fall below V ). Therefore, Firm 1 can sign an exclusive contract only against a lump-sum transfer equal to at least V -U 1 , which is the minimum amount needed to keep the signing consumer's welfare level equal to V while deterring entry. Firm 1's maximum proÞt when offering such an exclusive contract is thus 2b π 1 + V -U i (because, as a consequence of assumption (4), Firm 2 will decide not to enter if it observes that a consumer signed an exclusive contract), and such a contract is offered only if this is greater than the proÞt which Firm 1 could earn by offering no contract at all, i.e. 2π 1 .

Exclusive contracts with discrimination and breach penalties

We now increase contract complexity by allowing Firm 1 to offer contracts including a breach penalty clause. 19 The game is changed accordingly, as follows.

Stage 1. Firm 1 may offer each consumer a contract specifying that (i) the consumer commits not to purchase from Firm 2; (ii) a lump-sum transfer from The following result, proved in the appendix, shows that if exclusive contracts can include breach penalty provisions, then Firm 2's exclusion occurs only if the joint surplus of Firm 1, Firm 2, and one consumer, is greater under exclusion than under competition.

Proposition 3 If breach penalties are possible, then inefficient entry deterrence occurs if and only if the joint surplus of both Þrms and one consumer is greater under exclusion than under no exclusion: (i

) if U 1 + 2b π 1 > V +2π 1 +2π 2 , then Firm 2 is excluded in any PCPNE; and (ii) if U 1 +2b π 1 < V + 2π 1 + 2π 2 ,
then both Þrms serve both consumers in any PCPNE.

Remarks.

1. If the assumptions of RRW are made, then allowing for breach penalties has no impact on the likelihood of exclusion. This is because, following the logic of [START_REF] Aghion | Contracts as a Barrier to Entry[END_REF], the role of breach penalties is to induce the entrant to transfer part of its rent to the pair comprising the incumbent and one consumer. But in RRW, the entrant has no rent, because both Þrms earn zero proÞts under competition. In a more general setting however, Þrms may earn positive proÞts under competition. The possibility of introducing breach penalties in exclusive contracts in that case reduces the likelihood of exclusion, without eliminating it. What makes inefficient exclusion still possible is that the condition stated in Proposition 3 only takes into account one consumer (in addition to both Þrms).

2. The fact that breach penalties make exclusion less likely should come as no surprise, because they facilitate the transfer of an entrant's rent to the incumbent, and thus makes entry deterrence less attractive. This may seem to counter Aghion and Bolton's famous (1987) result that breach penalty provisions may cause inefficient exclusion. But there is in fact no contradiction, because that result is entirely driven by informational asymmetries, which sometimes cause the incumbent to exclude by mistake, having set too high a penalty. In [START_REF] Aghion | Contracts as a Barrier to Entry[END_REF] just like in the present paper, the purpose of exclusive contracts with breach penalty provisions is not to exclude, but rather to be breached and generate a penalty payment.

Exclusive contracts with breach penalties and conditional offers

We now further complexify the institutional setup by considering the possibility for contracts to be conditional on the acceptance of other contracts by other consumers.

Foreclosing an already present competitor

We assume now that both Þrms are present at the time when contracts can be offered to consumers and that both are on an equal footing as regards the ability to offer contracts. This section assesses whether exclusion may occur in a PCPNE in different contractual environments.

Simple exclusive contracts without discrimination

We assume in this subsection that (i) each Þrm must offer the same contracts to both consumers, and (ii) only simple exclusive or non-exclusive contracts are allowed. A contract mentions a lump-sum transfer and, possibly, an exclusivity clause. The timing of the game is as follows.

Stage 1. Both Þrms offer as many contracts as they wish (possibly none)

to each consumer. In this setting, as Proposition 5, proved in the appendix, establishes, inefficient exclusion cannot occur in equilibrium. The reason is simple: the prohibition of discrimination implies that when offering contracts, the potentially excluding Þrm (Firm 1) takes into account the utility level of both consumers. But the fact that both Þrms make simultaneous offers implies that Firm 2 is able to transfer its proÞts to consumers (through non-exclusive contracts together with lump-sum payments) in order to deter them from signing exclusive contracts with Firm 1. This ability for Firm 2 to make lump-sum payments to consumers implies that Firm 2's proÞts are taken into account in the relationship between Firm 1 and consumers. Therefore, all parties are taken into account, and the equilibrium outcome must be efficient.

Proposition 5 Inefficient exclusion cannot occur in a PCPNE.

Simple exclusive contracts with discrimination

In this subsection, the structure of the game is unchanged and we still assume that only simple exclusive or non-exclusive contracts are allowed, but we relax the no-discrimination rule. As Proposition 6 shows, the possibility to discriminate causes inefficient exclusion to occur in equilibrium for some parameter values. The reason is that, if discrimination is allowed, Firm 1 needs only one consumer to sign an exclusive contract in order to evict Firm 2. The parties whose surplus is taken into account when contracts are offered and taken up are thus Firm 1 (which has the choice whether to offer an exclusive contract)20 , one consumer (the one to whom a hypothetical exclusive contract is offered), and Firm 2 (which can transfer its expected proÞts by offering non-exclusive contracts). This leaves out one of the two consumers. Exclusion thus occurs if it maximizes the joint surplus of Þrms and one consumer -which is possible even if exclusion is socially inefficient.

Proposition 6 Under the assumptions made above, there exists a PCPNE in which no Þrm is excluded, and there exists no PCPNE in which Firm 1 is excluded. If, and only if 2b π 1 +U 1 > 2π 1 +2π 2 +V , there also exists a PCPNE in which the inefficient exclusion of Firm 2 occurs.

Corollary. 1. Inefficient eviction of an already present competitor is possible. In particular, in the RRW model, if cost and preference parameters are such that exclusion occurs when Firm 2 is a potential entrant, it also occurs (in one of the two PCPNE) when Firm 2 is already present and Þrms can only offer simple exclusive contracts.

2. With simple exclusive contracts, efficient eviction is however less likely than inefficient entry deterrence.

Proof of the corollary. Claim 1 results from the fact that under the assumptions of RRW, π 1 = π 2 = 0, so that the condition for discrimination to occur in Proposition 6 coincides with that in Proposition 2. Claim 2 results from the assumption that π 2 ≥ 0.

Proposition 6 provides additional support to the idea that in the presence of increasing returns, exclusive contracts can be used to foreclose consumers by discriminating among them so as to exclude rivals. This strategy may deter entry, as stated in RRW or SW, but it may also cause the inefficient eviction of a rival already present in the market.

Exclusive contracts with breach penalties

We now increase contract complexity by allowing Þrms to offer contracts including a breach penalty clause. In order to make this possibility meaningful, the time structure of the game is modiÞed as follows: The following Proposition, proved in the appendix, shows that if both

Þrms make offers simultaneously and contracts may include provisions for breach penalties, then exclusion cannot occur in a PCPNE.

Proposition 7 If both Þrms can offer contracts simultaneously and exclusive contracts can include provisions for breach penalties, then inefficient exclusion cannot occur in a PCPNE. In addition, both Þrms earn zero proÞts in any PCPNE.

The reason why both Þrms earn zero proÞts if breach penalty provisions are allowed is that these clauses allow Firm i to induce Firm j to transfer its proÞts to the pair formed by Firm i and any consumer. When both Þrms can do that, competition results into each Þrm "forcing" the other one to transfer its entire proÞts to consumers. As is proved in the appendix, this logic implies that inefficient exclusion cannot occur in equilibrium.

Proposition 7 means that when both Þrms can simultaneously make offers, the degree of contract complexity needed for inefficient exclusion to be ruled out is less than in the incumbent-entrant case, as there is no need to supplement breach penalty provisions with the possibility of making conditional offers. One can however check that conditional offers are redundant if both Þrms can offer contracts and breach penalty provisions are possible, in that if they were allowed, inefficient exclusion still could not occur in a PCPNE.

Conclusion [INSERT TABLE 2]

The main result of the paper is that as soon as too complex contracts are ruled out, exclusive contracts may cause socially inefficient eviction, and not only entry deterrence, even though the former is less likely than the latter.

In terms of real-world applicability, this theory of foreclosure (as well related ones such as in RRW, SW and BW) does not require an outcome as extreme as full exit by the excluded Þrm. "Exit" should be considered as a continuous variable: rather than fully exiting, a Þrm may scale down investment (in R&D, production facilities, or marketing). The analysis is exactly the same as long as the investment variable involves some increasing returns, in the sense that for a given level of investment, the induced increase in demand is reduced if the Þrm is barred from serving a given set of consumers (for example, a Þxed investment allowing a Þrm to lower its variable costs could Þt into this theory). In such a setting, partial consumer foreclosure induces a Þrm to scale down investment, which reduces the competitive constraint it is able to exert vis-à-vis those consumers who did not sign an exclusive contract -thus enhancing the market power enjoyed by the Þrm offering exclusive contracts vis-à-vis all consumers.

While this paper extends the existing theory and shows that anticompetitive exclusive contracts may arise in more general settings than had been established so far, further work is probably required in order to clarify the "complexity metric". In our quest for an institutional setting rich enough to permit Coasian bargaining, we did not proceed according to a predetermined metric over contractual environments. For example, while it seems straightforward to consider that contracts conditional on many variables are more complex than contracts conditional on fewer variables, there is much less clarity as to whether a game with many moves and countermoves is more or less complex, and more or less plausible, than a game with simultaneous moves. In other words, our claim that "simple" contractual environments allow inefficient exclusion to take place, while "rich" contractual environments do not, is not founded on a rigorous theory of what a simple or a rich environment is. Progress towards a complexity metric could pave the way for more general results about how much complexity is required for Coasian bargaining to take place, both in the "incumbent-entrant" case and in the "two-incumbent" case. the same utility level). It must be the case that V * ≥ V . Assume indeed that this is not the case and that V * < V . In this case, there necessarily exists an equilibrium in which no consumer accepts the exclusive contract offered by Firm 1 and both consumers enjoy a utility level of V . Indeed, if consumer a rejects the contract, then consumer b's utility from rejecting it as well is V (since rejection by both induces Firm 2 to enter and allows each consumer to be served by both Þrms). If consumer a takes up the contract, then he will be served by Firm 1 alone and be paid the lump-sum transfer provided for in the exclusive contract offered by Firm 1. He will get the utility level V * . If V * < V , rejection by both consumers is thus a Pareto-superior equilibrium of the continuation game than acceptance. This implies that in any PCPNE involving exclusion, V * ≥ V . Since the joint surplus of Firm 1 and consumers is 2(b π 1 +U 1 ), Firm 1's proÞt is at most equal to 2(b π 1 +U 1 -V ).

No exclusion

(V-U 1 )/2 (V-U1)/2+π 2 V-U 1 V-U 1 +π 2
For exclusion to be an equilibrium outcome, it must then be the case that

2(b π 1 + U 1 -V ) ≥ 2π 1 , or equivalently b π 1 + U 1 ≥ π 1 + V Otherwise, Firm 1
could increase its proÞt to 2π 1 by offering no contract at all.

We show now that the converse is true: if b π 1 +U 1 > π 1 +V , then exclusion must occur in equilibrium. Under no exclusion, Firm 1's proÞt is π 1 . But Firm 1 could earn greater proÞts by offering the following contract: offer each consumer an exclusive contract against a lump-sum transfer equal to

V -U 1 + ε, with ε > 0.
Clearly, accepting such a contract increases each consumer's utility level by ε (if acceptance is pivotal in deterring Firm 2's entry) or by V -U 1 + ε (if acceptance is not pivotal). Accepting this contract is thus a dominant strategy, and this contract is accepted by both consumers in equilibrium. Firm 1's proÞt is equal to b π 1 -(V -U 1 )ε, which is strictly greater than π 1 if ε is small enough. Offering an exclusive contract which will be accepted by consumers is thus a dominant strategy for Firm 1. QED.

Proof of Proposition 2

First step: if U 1 + 2b π 1 > V + 2π 1 , then exclusion occurs in all PCPNE. We assume that U 1 + 2b π 1 > V + 2π 1 . In a hypothetical no-exclusion equilibrium, Firm 1's proÞt would be 2π 1 . Now consider the following strategy for Firm 1:

offer consumer a an exclusive contract against a payment equal to V -U 1 +ε, where ε is strictly positive and small, and no contract to consumer b. If consumer a accepts this offer, Firm 2 is deterred from entering and consumer a's

surplus is thus U 1 + (V -U 1 + ε) = V + ε,
which is greater than the utility level V it would earn if it rejected the offer, triggering Firm 2's entry. But then, Firm 1's proÞt will be 2b π 1 -(V -U 1 + ε), which is strictly greater than 2π 1 if ε is small enough. Therefore, offering an exclusive contract which is taken up by consumer a allows Firm 1 to increase its proÞt with respect to a hypothetical equilibrium in which Firm 2 is not excluded. This implies that exclusion occurs in equilibrium. More precisely, one can check that the only PCPNE of this game is such that Firm 1 offers one consumer an exclusive contract against a payment equal to V -U 1 , which that consumer accepts, thus deterring Firm 2 from entering.

Second step: if U 1 + 2b π 1 < V + 2π 1 , then exclusion does not occur in any PCPNE.

We assume that U 1 + 2b π 1 < V + 2π 1 . By offering no exclusive contract, Firm 1 can obtain a proÞt equal to 2π 1 . We consider now a hypothetical PCPNE involving Firm 2's exclusion. First, we claim that in any such PCPNE, at least one consumer's utility level is no smaller than V , and that each consumer's utility level is above U 1 . By signing no contract at all, any consumer is certain to be served at least by Firm 1, which yields a surplus equal to U 1 . This implies that each consumer's surplus is greater than or equal to U 1 . Then, assume that both consumers' equilibrium utility levels are below

V . This means that each exclusive contract taken up in equilibrium involves a lump-sum payment strictly smaller than V -U 1 . But this implies the existence of an equilibrium of the subgame starting in Period 2 (when consumers decide whether they will sign contracts) in which no consumer signs a contract, yielding each consumer a utility level of V (because Firm 2 enters if no consumer signs an exclusive contract with Firm 1). This equilibrium of the continuation subgame starting in Period 2 yields both consumers a strictly greater utility than the hypothetical equilibrium, which is therefore not a PCPNE. Second, the fact that in any PCPNE involving exclusion, one con-sumer's utility level is greater than or equal to V , while the other consumer's utility level is greater than or equal to U 1 implies that in such an equilibrium, Firm 1's proÞt is no greater than 2(U 1 + b π 1 ) -V -U 1 , which is smaller than 2π 1 by assumption. Therefore, Firm 1 could increase its proÞt by offering no contract at all and earning 2π 1 , so that exclusion does not occur in a PCPNE. QED.

Proof of Proposition 3. First, notice that in any equilibrium with exclusion, each consumer gets a utility level of at least U 1 and at least one of them gets a utility level greater than or equal to V (the proof of this claim is the same as in the proof of Proposition 2). This implies that the maximum proÞt which Firm 1 can earn by excluding Firm 2 is 2b π 1 -(V -U 1 ). We now show that the maximum proÞt which Firm 1 can earn in an equilibrium in which Firm 2 is not excluded is equal to 2π 1 + 2π 2 . In a no-exclusion equilibrium, no consumer signs a contract, so that each consumer's utility level is equal to V . Since Firm 2's equilibrium proÞt must be nonnegative (otherwise it could do better by deciding to serve no consumer), this implies that Firm 1's proÞt is smaller than or equal to 2π 1 + 2π 2 . What remains to be shown is that Firm 1 can offer a contract such that the only PCPNE of the continuation game yields it a proÞt arbitrarily close to 2π 1 + 2π 2 while inducing no exclusion. Consider the following strategy: Firm 1 offers consumer a an exclusive contract, together with a lump-sum payment of V -U 1 and a breach penalty equal to 2π 2 + V -U 1ε (with ε very small).

If consumer a accepts such a contract, Firm 2 will be ready to offer him to breach it against a payment comprised between 2π 2ε and 2π 2 , and consumer a will be better off accepting such an offer. His utility will then be greater than

V + (V -U 1 ) -(2π 2 + V -U 1 -ε) + (2π 2 -ε) = V ,
so that consumer a is better off accepting Firm 1's offer than rejecting it.

This proves that Firm 1 can offer a contract inducing Firm 2 to enter, and yielding itself a proÞt arbitrarily close to 2π 1 + 2π 2 . As a consequence, if

2π 1 + 2π 2 > 2b π 1 -(V -U 1 )
, then Firm 1's optimal strategy is to offer an exclusive contract together with a breach penalty clause which will induce Firm 2 to enter. In this case, one can easily check that the only PCPNE is such that Firm 1 offers consumer a (or b) an exclusive contract together with a lump-sum payment of V -U 1 and a breach penalty equal to 2π 2 + V -U 1 .

Firm 2 then enters and offers this consumer to breach the exclusive contract against a payment of 2π 2 . Conversely, if

2π 1 + 2π 2 < 2b π 1 -(V -U 1 ), then
Firm 1's optimal strategy is to offer an exclusive contract with no provision for breach (or with prohibitively high breach penalties). QED.

Proof of Proposition 4. Consider the following offer by Firm 1 in Stage 1: Firm 1 offers consumer a an exclusive contract together with a lump-sum payment of V -U 1 +ε and a breach penalty equal to 2π 2 +2(V -U 1 )-ε (with ε very small). Consumer a obviously takes up this contract in Stage 2, since it guarantees him a utility level equal to at least V + ε (in case he decides not to breach it). Then, consider Stage 3. Firm 2 cannot induce consumer a to breach the contract by making an unconditional offer. Indeed, Firm 2 is willing to give up at most the entirety of its foreseeable proÞt, or 2π 2 . But if consumer a breaches the exclusive contract against a payment of 2π 2 , this results into (i) a gain of V -U 1 due to the shift from being served by Firm 1 alone to being served by both Þrms, (ii) a gain of 2π 2 (Firm 2's bribe for breaching the contract), and (iii) the breach penalty 2π 2 + 2(V -U 1 )ε, leading to a loss of (V -U 1 )ε. Firm 2 can induce consumer a to breach the exclusive contract by simultaneously (i) offering consumer a to breach the contract against a payment

2π 2 + (V -U 1 ) -η (with 0<η < ε); (ii)
requiring consumer b to pay V -U 1ε 0 with ε 0 < η; and (iii) stating that the offer to consumer a is conditional on consumer b accepting to make this payment. Clearly, in any PCPNE of the continuation game starting in Stage 4, both consumers decide to accept these offers. Consumer a risks nothing: if consumer b does not accept, then accepting has no consequence for consumer a, who remains bound by his exclusive contract with Firm 1. On the other hand, if consumer b accepts, consumer a ends up with a utility level equal to

V +ε + (εη), which is greater than the level V +ε which it would achieve if it did not accept Firm 2's offer. Regarding consumer b's incentive to accept Firm 2's requirement for a payment of V -U 1ε 0 , notice that if consumer a takes up Firm 2's conditional offer, then (i) if consumer b rejects Firm 2's requirement, consumer a will remain bound by the exclusive contract signed with Firm 1 (by virtue of the conditional nature of Firm 2's offer), so that Firm 2 will decide to serve no consumer in Stage 6, and consumer b will be subjected to Firm 1's monopoly power and have a utility level equal to U 1 . On the other hand, if consumer b accepts Firm 2's requirement, consumer a will be released from the exclusive contract signed with Firm 1, Firm 2 will thus decide to serve both consumers, and consumer b will enjoy a utility level equal to V -(V -U 1ε 0 ) = U 1 + ε 0 . Thus, both consumers have an interest in accepting Firm 2's offer, inducing Firm 2 to serve both consumers. Firm 2's proÞt is in turn equal to the proÞt it would normally earn when serving both consumers, minus its "bribe" to release consumer a from his exclusive contract, plus the payment made by consumer b, i.e.

2π 2 -[2π 2 + (V -U 1 ) -η] + [V -U 1 -ε 0 ] = η -ε 0 > 0.
Thus, faced with the aforementioned contract offered by Firm 1 in Stage 1, consumer a has an incentive to take it up, and Firm 2 can make a positive proÞt by offering to bribe consumer a into breaching the exclusive contract conditionally on consumer b making a payment to Firm 2. Therefore, in any PCPNE of the continuation game starting in Stage 2, the breach penalty is paid to Firm 1, which thus earns its "duopoly proÞt" 2π 1 minus the lump-sum transfer (V -U 1 + ε) paid to consumer a upon signing the exclusive contract, plus the breach penalty 2π 2 + 2(V -U 1 )ε, or in total 2π 1 + 2π 2 + (V -U 1 ) -2ε.

Therefore, Firm 1 can earn a proÞt arbitrarily close to

[2π 1 + 2π 2 + (V -U 1 )]
while inducing Firm 2 to serve both consumers.

In contrast, the maximum proÞt which Firm 1 can earn while excluding Firm 2 is 2b π 1 + U 1 -V , because in any equilibrium with exclusion, each consumer gets a utility level of at least U 1 and at least one of them gets a utility level greater than or equal to V (the proof of this claim is the same as in the proof of Proposition 2). Assumption (3), stating that exclusion is socially inefficient, is equivalent to

2π 1 + 2π 2 + (V -U 1 )>2b π 1 + V -U 1 , implying that
Firm 1's proÞt-maximizing strategy in Stage 1 involves offering a contract such as the one described above, leading to no exclusion. Therefore, no PCPNE involves exclusion in equilibrium. Finally, one can easily check that the strategies described above, with η = ε = ε 0 = 0, deÞne a PCPNE. QED.

Proof of Proposition 5.

Step 1. There exists a subgame perfect equilibrium such that (i) the equilibrium of the continuation subgame starting in period 2 is coalition-proof, (ii) Firm i (for i = 1 and i = 2) offers both consumers an exclusive contract with a lump-sum transfer

t ei = b π i -π i + Max[0,(b π j + U j ) -(π j + V )] and a non-exclusive contract with a lump-sum transfer of t ni = Max[0,(b π j + U j ) - (π j + V )]
(with the notation {i; j} = {1; 2}); and (iii) consumers choose to accept both Þrms' non-exclusive offers.

Proof. First, these transfers are such that each consumer is better off accepting both non-exclusive offers (which yields a payoff V + t n1 + t n2 ) than accepting Firm i's exclusive offer (which yields a payoff U i + t ei ), while being indifferent between these two options if t nj > 0. Accepting both Þrms' non-exclusive offers is thus a weakly dominant strategy and thus deÞnes a PCPNE of the continuation game. Second, consider Þrms' actions in period 1. If the contracts offered by Firm j are as described above, the cheapest way for Firm i to induce a consumer to be served by both Þrms in equilibrium involves offering a non-exclusive contract with the smallest positive lump-sum transfer required to make the consumer indifferent between accepting both Þrms' non-exclusive contracts and accepting Firm j's exclusive contract.

This minimal lump-sum transfer is equal to Max [0;

U j + t ej -V -t nj ] = Max[0,(b π j + U j ) -(π j + V )] = t ni .
Similarly, the cheapest way for Firm i to induce a consumer to purchase from Firm i only in equilibrium involves offering an exclusive contract with the smallest positive lump-sum transfer required to make the consumer indifferent between accepting Firm i's exclusive contract, and either of the two contracts offered by Firm j. This minimal lump-sum transfer is equal to Max [0;

U j + t ej -U i ; V + t nj -U i ] = Max[0; b π j -π j + Max[0, (b π i + U i ) -(π i + V )];(b π j + U j ) -(π j + V )] ≥ t ei .
But, since b π it ei = π it ni , Firm i's maximal proÞt, given Firm j's offers, can be obtained by offering a non-exclusive contract with a lump-sum payment equal to t ni . Firm i's best response to Firm j's actions is thus such that Firm i offers a non-exclusive contract with a lump-sum payment of t ni , and an exclusive contract with a lump-sum payment smaller than or equal to Max [0;

U j + t ej -U i ; V + t nj -U i ].
Offering a non-exclusive contract with a lump-sum payment of t ni , and an exclusive contract with a lump-sum payment of t ei is thus a best response to Firm j's actions. This proves that the actions described above deÞne a subgame-perfect equilibrium. This also proves that there exists a PCPNE involving no exclusion, and that in this PCPNE, Firm i's proÞt is greater than or equal to

2 (Min[π i , (π 1 + π 2 + V ) -(b π j + U j )]
). Indeed, in any hypothetical exclusionary subgame perfect equilibrium, one of the Þrms earns zero, and is thus worse off than in the non-exclusionary equilibrium described above, which implies that there exists at least one non-exclusionary PCPNE.

Step 2. We show that the equilibrium described above is a PCPNE.

Proof. This is equivalent to showing that there exists no subgame-perfect equilibrium such that the continuation subgame starting in period 2 is a PCPNE, and such that (i) each consumer is served by both Þrms and (ii)

for one Þrm at least, say Þrm i, the equilibrium proÞt is strictly greater

than 2 (Min[π i , (π 1 + π 2 + V ) -(b π j + U j )]
). Assume Þrst that (b π j + U j ) ≤ (π j + V ) and that Firm i's equilibrium proÞt is greater than 2π i . This means that one of the consumers signs a non-exclusive contract with Firm i involving a strictly positive payment to Firm i, which is impossible since this strategy would be dominated by not signing any contract with Firm i.

Assume now that (b π j + U j ) > (π j + V ) and that Firm i's equilibrium proÞt is strictly greater than 2[(π

1 + π 2 + V ) -(b π j + U j )].
This means that at least one consumer receives a lump-sum payment from Firm 1 which is strictly lower than (b π j + U j ) -(π j + V ). But if this is the case, then faced with Firm i's equilibrium non-exclusive offer and a hypothetical exclusive offer proposed by Firm j together with a lump-sum transfer equal to b π jπ jε for small enough ε, this consumer would choose the latter. This implies that, by offering such an exclusive contract, Firm j could increase its proÞt by at least ε (the reason for this is that in equilibrium the non-exclusive contracts picked by consumers necessarily involve nonnegative transfers from Þrms to consumers). This contradicts the assumption that both consumers are served by both Þrms in equilibrium.

Step 3. We assume now that there exists a PCPNE in which one of the consumers (at least) is served by only one Þrm. Let t ei and t ni denote respectively Firm i's lump-sum transfer associated with the best (from the consumers' viewpoint) exclusive and a non-exclusive contract it offers (offering no contract can be interpreted as offering a transfer equal to minus inÞnity, which will be rejected by consumers in any subgame perfect equilibrium of the continuation game). Assume that consumer a is served by one Þrm only, say Firm i. Since both consumers are offered the same contracts, they both get the same utility level U * . But U * ≥ Max(V + π j , U j + b π j ), because otherwise Firm j would have an interest in offering to each consumer it does not serve in the hypothetical equilibrium a non-exclusive contract together with a lump-sum payment equal to π jε, as well as an exclusive contract together with a lump-sum payment equal to b π j -ε, with ε > 0 being very small. Such a contract would be picked by any consumer being offered it (because it would yield a payoff greater than U * ), and this would increase Firm j's proÞt. Since both consumers are offered the same contracts, Firm i's proÞt is the same as the one it would earn if both consumers were picking an exclusive contract it offers (otherwise, this means that Firm i could increase its proÞt by offering only an exclusive contract to both consumers, or only a non-exclusive contract to both consumers). Firm i's equilibrium proÞt is

thus equal to 2(b π i + U i -U * ) ≤ 2Min(b π i + U i -V -π j , b π i + U i -U j -b π j ) ≤ 2Min(π i , (π 1 + π 2 + V ) -(b π j + U j ))
, which is smaller than or equal to Firm i's proÞt in some non-exclusionary PCPNE, as shown in Step 1. This implies that the hypothetical exclusive PCPNE is Pareto-dominated, from the point of view of Þrms, by some non-exclusive PCPNE. This implies that all PCPNE are non-exclusive.

Proof of Proposition 6.

First, it can easily be checked that the PCPNE described in Step 1 of the proof of Proposition 5 is still a PCPNE when discriminatory offers are allowed.

Second, we show that in any PCPNE, Firm 1 serves both consumers.

Assume that there exists an equilibrium in which one consumer at least, say consumer a, is not served by Firm 1. This means that in equilibrium consumer a signs an exclusive contract with Firm 2, against some lump-sum

payment t e2 . But the inequality U 2 + t a e2 ≥ Max[b π 1 + U 1 ; π 1 + V ] must
≥ Max[b π 1 + U 1 ; π 1 +V ] implies that Firm 2's equilibrium proÞt is smaller than π 2b +b π 2 +U 2 -Max[b π 1 +U 1 ; π 1 +V ]. But,
as proved above, there also exists an equilibrium which is a PCPNE of the subgame starting in Period 2, such that in period 1 Firm i (for i = 1 and i = 2) offers consumer a an exclusive contract with a lump-sum transfer U 2 ). This means that the equilibrium originally considered is not a PCPNE.

t 0 ei = b π i -π i + Max[0,(b π j + U j ) -(π j + V )]
Third, in order to prove that there exists a PCPNE in which Firm 2 is excluded if and only if 2b π 1 + U 1 > 2π 1 + 2π 2 + V , we distinguish two cases.

First case:

π 2 + V > b π 2 + U 2 . Step 1. Assume that 2b π 1 + U 1 > 2π 1 + 2π 2 + V .
We show that there exists an equilibrium in which Firm 1 offers consumer a an exclusive contract together with a lump-sum payment equal to 2π 2 + V -U 1 , Firm 2 offers consumer a a non-exclusive contract together with a lump-sum payment equal to 2π 2 , and consumer a chooses to accept Firm 1's exclusive contract. Clearly, consumer a is indifferent between both contracts and accepting either makes him better off than accepting none.

Firm 1 cannot offer less in an exclusive contract (this would cause consumer a to accept Firm 2's non-exclusive contract and cause Firm 1's proÞt to fall from 2b π 1 + U 1 -2π 2 -V to 2π 1 ), and Firm 2 cannot avoid being excluded (offering a more generous non-exclusive contract would cause losses, and the inequality b π 2 + U 2 < π 2 + V implies that providing consumer a with a given utility level is more expensive for Firm 2 using an exclusive than a nonexclusive contract). This proves that the aforementioned actions deÞne an equilibrium, which is also a PCPNE of the subgame starting in period 2 (this is because consumer a is the only agent making a decision in period 2). Finally, Firm 1's proÞt in this equilibrium, at 2b π

1 + U 1 -2π 2 -V , is
greater than its level in the only non-exclusive PCPNE (where it is equal to

2π 1 if b π 2 + U 2 < π 2 + V ).
Also, this equilibrium is a PCPNE because Firm 2 would earn zero proÞt in the out-of-equilibrium event in which consumer a would choose to accept its contract rather than Firm 1's exclusive one.

Step 2. We now prove the converse and consider a PCPNE in which Firm 2 is excluded.

-V + U 1 >2π 1 , or equivalently 2b π 1 + U 1 > 2π 1 + 2π 2 + V . Second case: b π 2 +U 2 > π 2 +V. Step 1. Assume that 2b π 1 +U 1 > 2π 1 +2π 2 +
V . We show that there exists a PCPNE in which Firm 1 offers consumer a an exclusive contract together with a lump-sum payment equal to 2(b π 2 + U 2 ) -(V + U 1 ), Firm 2 offers consumer a an exclusive contract together with a lump-sum payment equal to 2b π 2 + U 2 -V , and in equilibrium consumer a chooses to accept Firm 1's exclusive contract. In order to prove that Firm 2's strategy is a best response, notice that in order to avoid exclusion at lowest possible cost, Firm 2 has to offer consumer a an exclusive contract (this is because the inequality b π 2 + U 2 > π 2 + V implies that exclusion is jointly optimal for the Firm 2 -consumer a pair) together with a transfer yielding consumer a at least the same utility as it would get by accepting Firm 1's offer, i.e., a transfer equal to 2b π 2 + U 2 -V . But in any subgame-perfect equilibrium such that consumer a signs a contract with Firm 2, consumer b knows that it can get a utility of at least V simply by signing no contract.

Thus, the maximum proÞt Firm 2 can derive from serving consumer b is b π 2 + U 2 -V , obtained by offering consumer b an exclusive contract against a transfer of V -U 2 , leaving consumer b with a utility level of at least

V . Thus, given Firm 1's offer, Firm 2 cannot obtain a proÞt greater than

2b π 2 -(2b π 2 + U 2 -V ) + U 2 -V =0
. Firm 2's strategy is thus a best response.

Conversely, Firm 1's strategy is a best response to Firm 2's, because the transfer offered to consumer a is the smallest leading consumer a not to choose to accept Firm 2's offer, and Firm 1's ensuing proÞt is greater than the proÞt level π 1 which it would earn if it offered no contract at all (or no contract which consumer a would prefer over Firm 2's offer). Finally, in order to prove that these strategies form a PCPNE, it is enough to check that Firm 1's proÞt is greater than that it would earn in the only non-exclusive PCPNE, i.e. that 2b π 1 -[2(b π 2 + U 2 ) -(V + U 1 )] > 2π 1 -2[(b π 2 + U 2 ) -(π 2 + V )], which is equivalent to the assumed inequality, i.e. 2b π 1 + U 1 > 2π 1 + 2π 2 + V .

Step 2. Conversely, we prove that if Firm 2 is excluded in a PCPNE, then The fact that the equilibrium considered is a PCPNE implies that Firm 1's equilibrium proÞt is strictly greater than Firm 1's proÞt in the only nonexclusionary PCPNE, i.e. greater than 2π 1 -2[(b π 2 + U 2 ) -(π 2 + V )], which implies that 2b π 1 + U 1 > 2π 1 + 2π 2 + V .

2b π 1 + U 1 > 2π 1 + 2π 2 + V .

Proof of Proposition 7

Step 1. We prove that in any equilibrium, both Þrms earn zero proÞts.

The reason is that if Firm i earned positive proÞts in a hypothetical equilibrium, the other Þrm could increase its proÞt by offering in stage 1 an exclusive contract together with a breach penalty clause having the effect of inducing Firm i to relinquish some of its proÞt in Stage 3, to its own proÞt.

Proof.

We start by showing that it is impossible in equilibrium for a consumer, say consumer a, to make a strictly positive payment to Firm i even though it is not served by Firm i. This could happen only if consumer a signed an exclusive contract with Firm i together with a transfer t and a breach penalty t 0 > t, before breaching it and paying the penalty . But such an outcome cannot be an equilibrium, because in that case Firm j could proÞtably deviate in Stage 1 by offering consumer a an exclusive contract (with no breach possibility), together with a transfer U i + t + (t 0t)/2.

Assume now that Firm i earns a strictly positive proÞt in equilibrium.

This means that there exists a consumer, say consumer a, such that in equilibrium (with t denoting the transfer from Firm i to consumer a) either (i)

Firm i is the only Þrm serving consumer a,and t < b π i , or (ii) both Þrms serve consumer a,and t < π i . Let U * a denote consumer a's equilibrium surplus level. Firm j could increase its proÞt by offering consumer a, in Stage 1, an exclusive contract together with a transfer of U * a -U j + ε and a penalty for breach equal to t + V -U j + 2ε (case (i)), or t + U i -U j + 2ε (case (ii)), with ε > 0 small enough. Such a contract would be accepted by consumer a in Stage 2 (because it would provide consumer a with a surplus greater than in the original equilibrium, by ε, and thus greater than that induced by any other contract offered to consumer a after the abovedescribed deviation).

Following this action by Firm j, and consumer a's subsequent acceptance decision, Firm i would be better off making making an offer which consumer itself at least zero proÞts by deciding to offer no contracts and not to deal with any consumer. This implies that in any PCPNE, each Þrm serves both consumers.

Firm 1

 1 to the consumer; and (iii) a penalty which the consumer must pay to Firm 1 if it breaches the exclusivity requirement. Firm 1 can discriminate across consumers. Stage 2. Consumers simultaneously decide whether to sign the contracts possibly offered to them in Stage 1. Stage 3. Observing the outcome of Stage 2, Firm 2 may decide to offer some consumers a lump-sum payment in exchange for these consumers breaching the exclusive contract signed with Firm 1. Stage 4. Consumers who signed an exclusive contract in Stage 2 and were offered by Firm 2 to breach it in Stage 3 decide whether to accept Firm 2's offer. A consumer breaching an exclusive contract signed in Stage 2 receives the lump-sum payment proposed by Firm 2 in Stage 3, and pays Firm 1 the breach penalty provided for as per the exclusive contract it breaches. Stage 5. Firm 2 decides which consumers it wants to serve. It cannot decide to serve a consumer who signed an exclusive contract in Stage 2 and did not breach it in Stage 4. Each consumer's welfare level and each Þrm's proÞt is then determined according to Firm 2's choices in Stage 5. Lump-sum payments or breach penalties provided in the various contracts are added to or subtracted from Þrms' proÞts and consumers' welfare level.

Stage 1 .

 1 Both Þrms offer contracts to each consumer. A contract may be exclusive or non-exclusive. An exclusive contract between Firm i and a consumer may include a breach penalty clause specifying (i) a payment to be made to Firm i should the consumer breach the exclusivity requirement while still dealing with Firm i; and (ii) a payment to be made to Firm i should the consumer breach the exclusivity requirement by not dealing at all with Firm i. 21 Discrimination across consumers is allowed. Stage 2. Consumers simultaneously decide whether to sign the contracts possibly offered to them in Stage 1. Stage 3. Each Þrm may offer contracts to consumers who signed in Stage 2 an exclusive contract with the other Þrm. 22 Stage 4. Consumers simultaneously decide whether to sign the contracts possibly offered to them in Stage 3, taking into account the obligations imposed upon them by the contracts signed in Stage 2. Stage 5. Observing the outcome of Stages 2 and 4, both Þrms simultaneously decide which consumers they want to serve. A Þrm is obliged to serve a consumer who signed (and did not breach) a contract it proposed, and it cannot serve a consumer who signed an exclusive contract with the other Þrm in Stage 2 unless this contract has been breached. It is free to serve or not to serve consumers with whom it is not bound by a contract, and who are not bound by an exclusive contract signed with the other Þrm.

  and a non-exclusive contract with a lump-sum transfer of t 0 ni = Max[0,(b π j + U j ) -(π j + V )] (with the notation {i; j} = {1; 2}); and (iii) consumer a chooses to accept both Þrms' nonexclusive offers. If both Þrms changed their offers to consumer a accordingly (while leaving unchanged their offers to consumer b), the outcome would still be an equilibrium, Firm 1's proÞt would increase by Min[π 1 , (π 1 + π 2 + V ) -(b π 2 + U 2 )] and Firm 2's proÞt would increase by at least (π 1 + π 2 + V ) -(b π 2 +

  Two cases can arise, depending on whether one or two consumers sign an exclusive contract with Firm 1 in equilibrium. If two consumers do, let t k e1 (k = a, b) denote the equilibrium lump-sum transfer paid by Firm 1 to consumer k. If t a e1 + t b e1 < 2b π 2 + 2U 2 -2U 1 , then Firm 2 can proÞtably avoid exclusion by offering consumer k an exclusive contract together with a lump-sum transfer equal to t k e1-U 2 + U 1 + ε, with ε small enough. Thus t a e1 + t b e1 ≥ 2π 2 + 2U 2 -2U 1 , implying that Firm 1's proÞt is smaller than or equal to 2 (b π 1 -b π 2 -U 2 + U 1 ), which is strictly smaller than 2[π 1 -(b π 2 + U 2 ) + (π 2 + V )] by (3). Since the right-hand side of this inequality is equal to Firm 1's proÞt in the non-exclusive PCPNE, it follows that there is no exclusive PCPNE in which both consumers sign an exclusive contract. If only one consumer, say consumer a, signs an exclusive contract with Firm 1 in equilibrium, against a lump-sum transfer t a e1 , then necessarilyt a e1 ≥ 2(b π 2 + U 2 ) -(V + U 1 ), because otherwise Firm 2 could proÞtably avoid exclusion by offering consumer a an exclusive contract together with a lumpsum transfer equal to t k e1 -U 2 + U 1 + ε, and consumer b an exclusive contract together with a lump-sum transfer equal to V -U 2 + ε, with ε small enough. Firm 1's proÞt is thus smaller than or equal to 2b π 1 -[2(b π 2 + U 2 ) -(V + U 1 )].

be- low. 16Table 1 :

 1 consumer preferences16 The magnitudes V and U i in Table1are indirect utility levels. They are equal to the utility levels achieved by a consumer dealing with either Þrm (or with both), given the nature of the strategic interaction between Þrms and consumers, left outside of the model.

	Situation	Consumer's utility
	A consumer is served by both Þrms V
	A consumer is served by Firm i only U i
	2.2 Firm proÞts
	Firm 1's proÞt is equal to λπ 1 + µb π 1 where λ and µ are, respectively, the
	number of consumers served by both Þrms and the number of consumers
	served by Firm 1 only. It is assumed that b π 1 > π 1 in order to capture
	the idea that Firm 1's proÞt is greater in monopoly than in duopoly. The
	assumption that Firm 1's per customer proÞt only depends on the degree of
	competition for that customer is consistent with the assumption that Firm
	1's technology displays constant returns to scale.
	Firm 2's technology is assumed to be characterized by economies of scale
	in the following sense. If Firm 2 serves both consumers, then its proÞt is
	equal to λπ 2 + µb π 2 where λ and µ are, respectively, the number of consumers
	served by both Þrms and the number of consumers served by Firm 2 only
	(with, obviously, λ+µ = 2, and b π 2 > π 2 ). However, if Firm 2 serves a single
	consumer, its proÞt is equal to b s 2 (if it is alone in serving that consumer)
	or s 2 (if that consumer is also served by Firm 1), with b s 2 > s 2 (in order
	to account for the fact that Firm 2's proÞt is greater under monopoly than
	under duopoly).

  The previous game is modiÞed as follows. In Stage 3, having observed the outcome of Stage 2, Firm 2 can make the following offers

	The following result, proved in the appendix, shows that this contrac-
	tual environment is rich enough to allow Coasian bargaining to take place,
	implying that inefficient exclusion cannot occur in a PCPNE.
	Proposition 4 Under the above assumptions, inefficient entry deterrence
	does not occur in a PCPNE.

to consumers. To consumers who signed an exclusive contract with Firm 1, it may offer a lump-sum payment in exchange for breaching it. From consumers who did not sign an exclusive contract with Firm 1, it may ask for a lump-sum transfer (from the consumer to itself). Finally, if Firm 2 makes offers to both consumers (counting as an offer also the demand for a lumpsum transfer from a consumer), it may state that a given offer to a consumer is valid only if some other offer is accepted by another consumer in Stage 4.

Then, in Stage 4, consumers who were made offers by Firm 2 in Stage 3 decide whether to take them up or not. If these offers were made conditionally on acceptance by both consumers, and only one of them accepts the offer made by Firm 2, then the acceptance decision is not taken into account. In particular, if a consumer (say, consumer a) signed an exclusive contract in Stage 2, agreed to breach it in Stage 3, but Firm 2's offer to breach it was conditional on consumer b making a lump-sum payment to Firm 2, and this last offer was not taken up, then consumer a remains bound by the exclusive contract signed with Firm 1.

  They cannot discriminate across consumers.

	to the contracts which end up being signed are made.
	Stage 3. Observing the outcome of Stage 2, both Þrms simultaneously
	decide which consumers they want to serve. A Þrm cannot serve a consumer
	who signed an exclusive contract with the other Þrm in Stage 2 (assuming
	such a contract has been offered and accepted). However a Þrm can serve a
	consumer with whom it did not sign a contract at the previous stage, and it
	is obliged to serve consumers with whom it is bound by a contract signed in
	Stage 2.
	Stage 2. Consumers simultaneously decide whether to sign the contracts
	possibly offered to them in Stage 1. The lump-sum payments corresponding

Table 2 .

 2 Summary of the results

		Entrant-		No exclusion	Exclusion
		Incumbent		(Proposition 1)	(Proposition 1)
	No discrimination
		Two		No exclusion
		Incumbents		(Prop. 5)
			No
		Entrant-	exclusion	Exclusion
		Incumbent	(Prop. 2)	(Prop. 2, generalizing RRW and SW)
	Discrimination
		Two Incumbents	No exclusion (Prop. 6)	Existence of PCPNE with and without exclusion (Prop. 6)
		Entrant-Incumbent	No exclusion (Prop. 3)	Exclusion (Prop. 3)
	Discrimination +
	breach penalties
		Two		No exclusion
		Incumbents		(Prop. 7)
		Entrant-		No exclusion
	Discrimination +	Incumbent		(Prop. 4)
	breach penalties +
	conditional offers
		Two
		Incumbents

  Wiley, John. 1998. "Exclusionary Agreements", in The New Palgrave Dictionary of Economics and the Law, ed. Peter Newman, 110-116. Vol. 2.

	6 Appendix
	Proof of Proposition 1
	London: MacMillan.
	Consider a hypothetical equilibrium in which exclusion occurs, and let
	V * denote each consumer's utility level in this hypothetical equilibrium (the
	no-discrimination clause implies that in equilibrium both consumers enjoy

  which would be chosen by consumer a while yielding Firm 1 a strictly positive proÞt if ε is small enough. In equilibrium, Firm 2 necessarily serves consumer b (this is because Firm 2 loses money when serving one consumer only, so that if it serves consumer a it must also serve consumer b). Let us deÞneπ 2b =π 2t 2 b ifin equilibrium consumer b is served by both Þrms and Firm 2 pays consumer b a lump-sum transfer t 2 b , and π 2b =b π 2t 2 b if in equilibrium consumer b is served by Firm 2 only and Firm 2 pays consumer b a lump-sum transfer t 2 b . The inequality U 2 +t a e2

hold, because otherwise Firm 1 could increase its proÞt by offering consumer a an exclusive contract against a lump-sum transfer equal to U 2 +t a e2 -U 1 +ε and a non-exclusive contract against a lump-sum transfer equal to U 2 + t a e2 -V + ε,

  Two cases can arise, depending on whether one or two consumers sign an exclusive contract with Firm 1 in equilibrium. If two consumers do, + U 1 + ε, with ε small enough. Firm 1's proÞt is thus smaller than or equal to 2b π 1 -2π 2 -V + U 1 . The fact that the equilibrium considered is a PCPNE implies that Firm 1's equilibrium proÞt is strictly greater than Firm 1's proÞt in the only non-exclusionary PCPNE, i.e. greater than 2π 1 , which implies that 2b π 1 -2π 2

	than or equal to 2b π 1 -2π 2 -2V + 2U 1 , which is strictly smaller than 2π 1
	by (3). If only one consumer, say consumer a, signs an exclusive contract
	with Firm 1 in equilibrium, against a lump-sum transfer t a e1 , then necessarily
	t a e1 ≥ 2π 2 +V -U 1 , because otherwise Firm 2 could proÞtably avoid exclusion
	by offering consumer a a non-exclusive contract together with a lump-sum
	transfer equal to t k e1 -V
	let t k e1 (k = a, b) denote the equilibrium lump-sum transfer paid by Firm 1
	to consumer k. If t a e1 + t b e1 < 2π 2 + 2V -2U 1 , then Firm 2 can proÞtably
	avoid exclusion by offering consumer k a non-exclusive contract together
	with a lump-sum transfer equal to t k e1 -V + U 1 + ε, with ε small enough.
	Thus t a e1 + t b e1 ≥ 2π 2 + 2V -2U 1 , implying that Firm 1's proÞt is smaller

SeeWiley (1998) and[START_REF] Gilbert | Exclusive Dealing, Preferential Dealing, and Dynamic Efficiency[END_REF] for a survey of the legal treatment of exclusive contracts and an economic discussion.

[START_REF] Aghion | Contracts as a Barrier to Entry[END_REF] is an exception. They show that inefficient exclusion may occur if the incumbent can offer exclusive contract together with a breach penalty provision. But this result is driven by uncertainty about the potential entrant's costs, which causes the incumbent to exclude a more efficient entrant "by mistake", as a result of poorly calculated breach penalties. Also, a section of SW considers breach penalties, but under very speciÞc assumptions.

European Commission Decision 2002/405/EC, Michelin, 2002 O.J. (L 143).

The assumption that consumers have identical preferences is made in RRW as well. RRW also assumes that Þrms are identical (except for the fact that one is incumbent), but SW relaxes this assumption when discussing breach penalties (in Section IV).

The correspondence is as follows.In RRW, π 1 = π 2 = 0 (since both Þrms' duopoly proÞts are zero), V =CS(c),U 1 = U 2 =CS(p m ),and bπ 1 =b π 2 = (p m -c)q(p m ). RRW's assumption stated as "π < x * " is equivalent to (3), since it means that aggregate welfare is greatest when both Þrms serve consumers. Finally, the assumption that the entrant

See, e.g., BW, SW, O'Brien and Shaffer, (1997).

Unlike SW, we choose to address the question of breach penalties without modifying the assumptions about the Þrms' cost structure. This will allow us to assess what exactly the impact of allowing for breach penalty provisions is, leaving everything else unchanged.

Firm 2 also has the option of offering an exclusive contract, but this cannot be an equilibrium strategy because Firm 1 faces no increasing returns and there are thus no cross-customer externalities regarding the acceptance of exclusive contracts hypothetically offered by Firm 2.

For the sake of tractability, we do not allow for breach penalties in non-exclusive contracts, i.e. for clauses stating that a consumer must compensate a Þrm for deciding not to deal with it (for example by signing an exclusive contract with a competitor) after having signed a non-exclusive contract. Adding this possibility would only complicate the resolution of the game without changing the results.

The assumption that a Þrm may offer a contract in Stage 3 only to consumers who previously signed an exclusive contract with the other Þrm is made for tractability. The results would not change under more general assumptions (e.g., if both Þrms were allowed to offer contracts to both consumers in Stage 3, irrespective of the actions taken in previous stages).

a wants to accept than not making such an offer, because it could for example increase its proÞt (relative to the situation in which it would not offer a contract inducing consumer a to breach Firm 2's abovementioned exclusive contract) by offering consumer a in Period 3 a non-exclusive (case (i)) or an exclusive (case (ii)) contract together with a transfer t + 2ε. The payment of the breach penalty by consumer a would increase Firm j's proÞt by ε, making Firm j's deviation from the postulated equilibrium proÞtable.

Step 2. The following actions always form a PCPNE in which no exclusion takes place. Both Þrms offer in Stage 1 the same type of contract: Firm i offers each consumer an exclusive contract, together with (i) a transfer

(ii) a clause specifying a payment for breach equal to π j + V -U i (i 6 = j), owed to Firm i if the consumer, having signed the contract, later decides to drop the exclusivity requirement while still dealing with Firm i; and (iii) a clause specifying an inÞnite payment to Firm i if the consumer later decides not to deal with it. In Stage 2, each consumer signs one of the two such contracts offered to him (being indifferent between both). In Stage 3, if Firm i's Stage 1 contract was picked by a consumer, then Firm j offers this consumer a non-exclusive contract together with a transfer equal to π j , which is accepted in Stage 4, causing the breach penalty to be paid to Firm i, as per the contract signed in Stage 2.

Step 3. In any PCPNE, each consumer's equilibrium utility level is at least π 1 + V . Step 4. In any PCPNE,

Assume that this were not the case. Then Firm 2 could increase its proÞt by making the following offers in Stage 1: offer each consumer an exclusive contract together with a transfer U * c -U 2 +ε (with c = a and c = b), together with a clause specifying a penalty π 1 +V -U 2 -ε for breaching the exclusivity requirement, and an inÞnite penalty for not dealing with Firm 2 at all. Each consumer would be induced to choose this contract offered by Firm 2. The reason is that, by not doing so, a consumer ends up with a utility level no greater than U * c (it may earn U * c by signing Firm 1's equilibrium offer, or at most V < U * c by signing no contract at all). Firm 1's optimal action in Stage 3 is thus to offer a non-exclusive contract inducing both consumers to breach Firm 2's contract, because this can be done by offering a transfer equal to (for example) π 1ε/2, leaving Firm 1 with a proÞt ε/2 per consumer. Firm 2's proÞt in any continuation subgame following its deviation away from equilibrium would thus be equal to

Step 5. Assumption (3) implies that total surplus (i.e. the sum of Þrms'

proÞts and the surplus of both consumers) is maximized when each Þrm serves both consumers; and that it is then equal to 2(π 1 + π 2 + V ). The inequality U * a + U * b ≥ 2(π 1 + π 2 + V ), proved in Step 4, implies that if either

Þrm served zero or one consumer, then total surplus would be strictly less than consumer surplus, so that at least one Þrm's proÞt would be strictly negative, which is impossible in equilibrium, since each Þrm can guarantee