Exclusive Contracts and Market Dominance

We develop a theory of exclusive dealing that rehabilitates pre-Chicago-school analyses. Our theory rests on two realistic assumptions: that firms are imperfectly informed about demand, and that a dominant firm has a competitive advantage over its rivals. Under those assumptions, exclusive contracts tend to be pro-competitive when the dominant firm's competitive advantage is small, but are anti-competitive when it is more pronounced. In this latter case, the dominant firm uses exclusivity clauses as a means to increase its market share and profit, without necessarily driving its rivals out of the market, or impeding their entry. We discuss the implications of these results for competition policy.


Introduction
Exclusive dealing has long been a controversial practice. In the past, the primary basis for condemnation was a naive theory of a …rm that competes with rivals supplying substitute goods. The theory contended that by requiring buyers not to purchase from its competitors, the …rm can boost the demand for its product. This raises its market share and pro…t, but harms its customers and rivals. In the last decades, however, this theory has fallen into disrepute as a result of heavy attacks by the Chicago school, 1 and it has been replaced as a reason for prohibiting exclusive contracts by entry deterrence arguments (discussed below).
The main purpose of this paper is to show that under realistic assumptions the old theory is substantially correct. To understand what assumptions are needed, consider the reformulation of the Chicago-school critique proposed by O'Brien and Sha¤er (1997) and Bernheim and Whinston (1998). When dealing with the …rm, a buyer has a reservation payo¤ equal to the bene…t that he can obtain by trading exclusively with the …rm's competitors. If the …rm can price discriminate perfectly, it will then extract all the surplus in excess of said reservation payo¤. Thus, it will o¤er a contract that maximizes the buyer's surplus -a property referred to as bilateral e¢ ciency. Generally speaking, exclusivity clauses cannot be optimal as they would violate bilateral e¢ ciency if buyers have a preference for variety, reducing the buyer's surplus and hence the …rm's pro…t.
In reality, however, full extraction of surplus is almost impossible to achieve. One major obstacle is incomplete information about demand. 2 This is, therefore, the …rst key ingredient of our model. 3 Suppose, for example, that the buyer's demand can be either high or low, but the …rm does not know which state is realized. The …rm will then be unable to extract all the surplus when demand is high. Furthermore, in order to extract more surplus in the high-demand state, it must distort the contract that applies in the low-demand one.
What is the optimal distortion? If the …rm is restricted to non-linear pricing, all it can do is to reduce its own quantity below the e¢ cient level. With exclusive contracts, by contrast, the …rm can set its rivals' volume to zero. This latter strategy will often be more pro…table than the former, since the cost of creating the distortion is shifted onto the rivals.
These arguments imply that bilateral e¢ ciency, and hence the non-optimality of exclusive dealing, holds only when demand is highest. In all other cases, the contract o¤ered to the buyer must be distorted, and the optimal distortion may well involve exclusivity clauses. This simple observation is the starting point of our analysis.
To proceed, observe that when …rms are symmetric, a …rm's attempt to use exclusive contracts to implement the optimal distortion is destined to fail. The reason for this is that when all …rms o¤er exclusive contracts because doing so is unilaterally pro…table, they end up competing for exclusives. Now, competition for exclusives is competition in utility space, where …rms'products are e¤ectively homogeneous. As a result, competition with exclusive contracts becomes very intense and reduces both prices and pro…ts. Symmetric …rms are therefore caught in a prisoner's dilemma, as we show in a related article (Calzolari and Denicolò, 2013).
However, antitrust authorities and the courts are rarely concerned with equally sized competitors that compete vigorously; more often, they are concerned with dominant …rms facing smaller rivals. The second key assumption of this paper is, therefore, asymmetry of …rms: we assume that one …rm has a competitive advantage over its rivals, in terms of lower cost, higher demand, or a combination of the two.
The competition-enhancing e¤ect of exclusive contracts will still prevail if the dominant …rm's competitive advantage is small. When it is more pronounced, however, the dominant …rm can win the competition for exclusives easily. The competition-enhancing e¤ect weakens, and it may even vanish altogether if the competitive advantage gets large enough. The reason for this is that the buyer obtains an information rent even if the dominant …rm acts as a monopolist. If that rent is larger than the rent that the buyer can obtain by trading exclusively with the dominant …rm's competitors, exclusive contracts e¤ectively shelter the dominant …rm from competition. As a result, there is no longer any prisoner's dilemma, and the dominant …rm can simply increase its market share and pro…t, exactly as argued by the old theory.
Buyers are harmed by exclusive contracts, both in terms of higher prices and reduced variety. Rivals are harmed as well, but they need not be driven (or kept) out of the market for the strategy to be pro…table. This marks a key di¤erence with other post-Chicago theories that regard exclusive contracts as anti-competitive.
Why do buyers sign exclusive contracts if they get harmed by doing so? Because those are the best contracts that are actually o¤ered to them. Buyers are harmed with respect to the hypothetical non-linear pricing equilibrium that would arise if exclusive contracts were prohibited. However, when exclusive contracts are permitted …rms change their entire pricing strategies, and so non-exclusive contracts are no longer available at the same conditions. Why do not the dominant …rm's rivals resist being foreclosed? They actually do, both by o¤ering exclusive contracts in their turn, and by reducing their non-exclusive prices as compared to the non-linear pricing equilibrium. However, the dominant …rm may exploit its competitive advantage to o¤er exclusive deals so attractive that they cannot be matched by rivals. 4 Furthermore, the decrease in its competitors'non-exclusive prices limits, but does not eliminate, the pro…tability of exclusionary tactics.
Remarkably, our model can easily reproduce the key stylized facts of many real world antitrust cases. 5 These often involve two main actors: (i ) a dominant …rm that controls a substantial share of the market and has entered into some kind of exclusive arrangement with its customers, and (ii ) a smaller competitor (or group of competitors) that has been active in the industry for some time and in principle could have used exclusive contracts, too, but apparently has not. 6 The equilibria generated by our model are consistent with this pattern. In particular, even if the dominant …rm's competitors can o¤er exclusive contracts, only those o¤ered by the dominant …rm are accepted in equilibrium by some buyers.
We close the introduction by contrasting the theory proposed here with other post-Chicago theories that regard exclusive contracts as anti-competitive. All these theories rely on the absence from the contracting game of some of the agents that are a¤ected by exclusivity clauses. Therefore, their practical relevance depends on the realism of this crucial assumption. For example, in Aghion and Bolton (1987) and Rasmusen, Ramseyer and Wiley (1991) the missing agent is the dominant …rm's competitor -in those models, this is a potential entrant that initially cannot contract with the buyers. Thus, these theories cannot apply to the many antitrust cases in which the dominant …rm's competitors are already active. In Bernheim and Whinston (1998, sect. IV), the missing agents are future buyers whose demand is essential for the dominant …rm's rival to achieve economies of scale. Their explanation can therefore apply only to markets where a substantial increase in demand is expected in the not too distant future.
Furthermore, theories that rely on the absence of a¤ected agents from the contracting stage explicitly or implicitly assume that signing a contract is a form of commitment. 7 In fact, should the missing agents materialize, the parties would have an incentive to renegotiate -a point that has been forcefully made by Spier and Whinston (1995). In some cases, for exclusion to be prevented it may su¢ ce that exclusivity clauses can be breached upon payment of reasonable damages (Simpson and Wickelgren, 2007).
Finally, theories that view exclusive dealing as a means to deprive a rival of economies of scale are faced with the di¢ culty that exclusivity clauses may not be necessary for that purpose. The same outcome can sometimes be achieved by simple non-linear pricing, e.g. via quantity forcing or quantity discounts. If this is so, then a prohibition of exclusive contracts might be easily overcome and could even be welfare reducing.
While these di¢ culties may not be insurmountable, they limit the applicability of existing anti-competitive theories in antitrust practice. 8 The pre-Chicago theory rehabilitated in this paper does not su¤er from any of these drawbacks. It is consistent with the stylized facts mentioned above, including the fact that all involved parties can often participate in the contracting game. It produces an equilibrium in which exclusive contracts are (trivially) renegotiation proof and need not be long-term to be e¤ective, as they play no commitment role. It does not rely on the dominant …rm's rival being driven out of the market -so no proof of eviction or recoupment is needed. Finally, it uses as a benchmark the non-linear pricing equilibrium, which already allows for quantity discounts (including quantity forcing).
In sum, the theory developed in this paper seems in several ways more robust than other anti-competitive theories. Having said this, it must be stressed that the anti-competitive e¤ects of exclusive contracts must be weighted not only against the well-known pro-e¢ ciency rationales already discussed in the literature, 9 but also against the pro-competitive e¤ects that arise in our model when …rms are not too asymmetric.
The rest of the paper is structured as follows. Section 2 sets out the model. In section 3, we discuss how the possibility of o¤ering exclusive contracts changes the formulation of the …rm's pricing problem. Section 4 analyzes the case in which the dominant …rm faces a competitive fringe. Section 5 studies the case of duopoly, where the dominant …rm's competitor, too, may have some market power. Section 6 summarizes the paper's results and discusses their implications for competition policy. Proofs are in an Appendix.
We consider a one-period model of price competition. There are two substitute goods, A and B. Good A is supplied by …rm A, whereas good B may be supplied either by …rm B (the duopoly model ) or by a competitive fringe (the competitive fringe model ). A buyer who buys q A units of good A and q B units of good B obtains a bene…t, measured in monetary terms, of u (q A ; q B ; ) : We may think of buyers as downstream …rms, and of u as their gross pro…ts, 10 or as …nal consumers, with u as their utility function. The function u is symmetric and strictly concave in q A and q B , and it satis…es the single-crossing condition u qi (q A ; q B ; ) 0. The one-dimensional parameter is the buyer's private information; it is distributed over an interval [ min ; max ] according to a distribution function F ( ) with density f ( ): Notice that since heterogeneity is one-dimensional, the demand for the two products is correlated.
In the symmetric case, marginal production costs are normalized to zero. In the asymmetric case, …rm A (the dominant …rm) has a competitive advantage in terms of lower cost, higher demand, or both. With cost asymmetry, the unit production cost of product B is c > 0, whereas that of product A is still normalized to zero. With demand asymmetry, the utility function becomes u (q A ; q B ; ) cq B . The parameter c can now be interpreted as an index of vertical product di¤erentiation, with product A being of better quality, and hence having a larger demand, than product B. These formulations are clearly equivalent; however, they entail di¤erent equilibrium prices. To …x ideas, in what follows we shall stick to the cost interpretation.
Buyers have no bargaining power, but are large enough so that …rms can monitor whether they purchase from their competitors. Their reservation payo¤, u (0; 0; ), is normalized to zero. We abstract from …xed costs, and hence from economies of scale. 11 Firms are expected pro…t maximizers. They compete by simultaneously and independently o¤ering a menu of contracts. We distinguish two di¤erent modes of competition according to the type of contract that the …rms may o¤er. With simple non-linear pricing, the payment to each …rm depends only on its own quantity. A strategy for …rm i then is a function P i (q i ) in which q i is the quantity …rm i is willing to supply and P i (q i ) is the corresponding total payment it asks. With exclusive contracts, by contrast, a strategy for …rm i comprises two price schedules, P E i (q i ) and P N E i (q i ): The former applies to exclusive contracts (q i = 0), the latter to non exclusive ones (q i > 0). 12 1 0 In this interpretation, downstream …rms must operate in separate markets and must not interact strategically with each other. This prevents the emergence of contractual externalities that would complicate the analysis: see, for instance, Fumagalli and Motta (2006), Simpson and Wickelgren (2007) and Wright (2009). 1 1 As long as all …rms remain active, this is with no loss of generality. Furthermore, in the competitive fringe model one can interpret c as the minimum average cost of a number of identical …rms, thus allowing for economies of scale at the …rm level. 1 2 To guarantee that the buyer's maximization problem has a solution, we assume that each price schedule P i must be non decreasing in q i (a free disposal assumption which also implies that price schedules must be di¤erentiable almost everywhere), that it satis…es P i (0) = 0, and Buyer observes the …rms'o¤ers and then chooses the quantities fq A ( ); q B ( )g that maximize his net payo¤. Notice that the buyer may choose to purchase only one product even if he is faced with non-exclusive contracts. We shall call "common representation" the outcome in which the buyer buys positive quantities of both products (which of course can only arise when the buyer does not choose an exclusive contract). We denote 13 The full information, …rst-best quantities are n To make the analysis interesting, we assume that q f b B ( max ) > 0; if this condition is violated, good B should not (and would not) be produced in equilibrium.

The pricing problem
Before comparing the equilibrium with and without exclusive contracts, it may be instructive to discuss how the possibility of using exclusive contracts changes the formulation of the …rms' optimal pricing problem. To …x ideas, we shall focus on the dominant …rm, but the same approach applies, in the duopoly model, to its rival.

Non-linear pricing
When exclusive contracts are prohibited, the dominant …rm must solve a fairly standard problem of non-linear pricing. The only twist is that buyers can also trade with the …rm's rivals, so they e¤ectively have an indirect utility function v(q A ; ) = max In other words, v(q A ; ) is the maximum rent that buyer can obtain by purchasing q A and then trading optimally with the dominant …rm's rivals. 14 Let us brie ‡y review the solution technique with non-linear pricing. The …rm maximizes its pro…t R max By invoking the Revelation Principle, we can reformulate the problem as if the …rm could control q A ( ) directly, i.e. as a direct mechanism. Using the change of variables U ( ) = v(q A ( ); ) P A (q A ( )), the …rm's problem becomes that it is upper semi-continuous. 1 3 Here P i (q i ; q i ) is used as a general notation that covers both exclusive and non-exclusive contracts. 1 4 The notion of indirect utility function has been introduced by Martimort and Stole (2009). Its role is similar to that of residual demand in oligopoly models of linear pricing.
We can then replace the constraint that q A ( ) = arg max q A 0 [v(q A ; ) P A (q A )] with the familiar incentive compatibility and participation constraints. Provided that the indirect utility function satis…es the single-crossing condition v q A (q A ; ) 0, the incentive compatibility constraint is equivalent to the requirements that U 0 ( ) = v (q A ; ) and that q A ( ) is non-decreasing. The participation constraint is U ( ) 0. Since U 0 ( ) 0, we can optimally set U ( min ) = 0. The program then becomes U ( min ) = 0 and q A ( ) non-decreasing. This is an optimal control program with q A ( ) as the control variable and U ( ) as the state variable. Once the optimal quantity has been found, one can then recover the tari¤ that supports it.

Exclusive contracts
When exclusive contracts are permitted, the dominant …rm can control not only q A ( ), but also whether q B ( ) may be positive or must be nil. When q B ( ) may be positive, the …rm's problem is similar to (2), except that the …rm must guarantee to the buyer at least the type-dependent reservation utility which the buyer could obtain by trading with the dominant …rm's rivals only. 15 The dominant …rm's problem then becomes and q A ( ) non-decreasing, where~ is the lowest demand buyer served by the …rm. We denote by q N E A ( ) the solution to this problem.
, however, it must be dealt with separately. For an extensive treatment of type-dependent participation constraints in monopolistic screening problems see Jullien (2000).
If q B ( ) is set to zero, however, the …rm's problem is (following the same steps as above) and q A ( ) non-decreasing. Compared with problem (4), the indirect utility function v(q A ( ); ) is now replaced by u(q A ( ); 0; ). In other words, the dominant …rm can behave as a monopolist (except for the type-dependent reservation utility). Let us denote by q E A ( ) the solution to problem (5). Since the …rm can choose whether to impose an exclusivity clause or not, 16 it is faced with a hybrid optimal control problem involving two di¤erent control systems, (4) and (5), and the possibility of switching from one system to the other. 17 To solve this problem, one needs to choose a sequence of control systems, the switching points, and the control function q A ( ) for each system that maximize the …rm's pro…t. Clearly, the possibility of switching from one system to the other is generally valuable to the dominant …rm.
Let us start from the choice of the optimal control function q A ( ). A basic insight in the theory of hybrid optimal control problems is the following: if one particular control system is optimal over an interval of types, then the optimal control function over that interval must maximize the objective function for that system and that interval, given appropriate boundary conditions. 18 In general, the solution depends on the boundary conditions. For the particular non-linear pricing problem at hand, however, it does not: boundary conditions do not a¤ect the optimal quantity q A ( ), and hence the marginal prices that support it; they can only a¤ect the …xed fee, or subsidy, of the optimal tari¤.
To prove this, consider the choice of the optimal control function on a generic sub-interval [ 1 ; 2 ] of the support of the distribution of types. One boundary condition is the reservation payo¤ for type 1 , which depends on the contracts o¤ered to lower types, via the incentive compatibility constraints. However, this is optimally accommodated through a …xed fee or subsidy, which shifts the function U ( ) up or down in a parallel fashion. As for the upper bound of the interval, ours are free boundary problems. Furthermore, the value of the upper bound itself, i.e. 2 , is irrelevant, since the optimal quantity for any type 2 [ 1 ; 2 ] depends only on how many higher types there are, both inside and outside the sub-interval. In other words, all that matters is 1 F ( ). 1 6 Notice that the constraint U ( ) U R A ( ) guarantees that buyers would accept the exclusive contracts o¤ered by the …rm. 1 7 Generally speaking, a hybrid optimal control problem is a problem involving both continuous and discrete control variables. Our problem is a special case of a hybrid system, called a switched control system, in which the discrete variable is a dummy that describes which control system actually applies: see Sun and Ge (2005) for an introduction. 1 8 See, for instance, Sussmann (1999).
These properties imply that the optimal quantity can be found by pointwise maximization of a "virtual surplus function," 19 which does not depend on the extremes of the interval. This greatly simpli…es the analysis. Taken together, the two properties above imply that the optimal control function q A ( ) for the complete problem is formed by appropriately joining the control functions q N E A ( ) and q E A ( ) that are optimal for problems (4) and (5) separately. When the solution does not involve any exclusivity clause, so that the optimal quantity q A ( ) results from the control system (4), it must therefore coincide with q N E A ( ). When instead the solution involves an exclusivity clause, the optimal quantity q A ( ) results from the control system (5), so it will coincide with q E A ( ) (and of course q B ( ) must be nil).
To proceed, we must determine the optimal sequence and the optimal switching points. Generally speaking, this is the most di¢ cult part of a hybrid optimal control problem. At this point, it is therefore convenient to abandon the direct mechanism approach and address the optimal pricing problem straightaway. Let us denote by P N E A (q A ) and P E A (q A ) the tari¤s that implement the control functions q N E A ( ) and q E A ( ), respectively. At this point, all that remains to be found are the constant terms of these schedules.
Consider the optimal sequence …rst. In all the problems that we shall analyze, the equilibrium rent function U ( ) is steeper under non-exclusivity than under exclusivity. 20 Since …xed fees or subsidies can only shift the rent functions up or down in a parallel fashion, the optimal sequence is necessarily from exclusivity (for low-demand types) to non-exclusivity (for high-demand ones). This makes intuitive sense: the no-distortion-at-the-top property implies that the solution for high-demand types must be nearly e¢ cient. As argued by O'Brien and Sha¤er (1997) and Bernheim and Whinston (1998), this rules out exclusivity clauses. But exclusivity clauses can be optimal for low-demand buyers, whose quantities are distorted more heavily.
The fact that the exclusive solution must apply, if ever, to low-demand types, and so also to the marginal buyer, implies that the constant term of the corresponding tari¤ must be nil. 21 We are therefore left with only one degree of freedom, i.e. the …xed fee for the non-exclusive price schedule P N E A (q A ), A . This …xed fee is pinned down by the optimal choice of the switching point 1 9 This is the Hamiltonian of problem (4) or (5). 2 0 By the envelope theorem, the slope of U ( ) is u (q E A ( ); 0; ) under exclusive dealing, and u (q N E A ( ); q N E B ( ); ) under common representation. Since the goods are substitutes, q E A ( ) is generally lower than q N E A ( ) + q N E B ( ). Given the single-crossing condition, this by itself implies that U ( ) tends to be steeper under non-exclusivity than under exclusivity. To guarantee the conclusion, however, some regularity conditions are generally needed. In the uniform-quadratic model that we shall focus on later, for instance, the conclusion always holds. 2 1 The marginal buyer is the lowest type who purchases a positive amount of a good. When the market is uncovered, as we shall assume later, the marginal buyer purchases a negligible quantity of the goods. This implies that the price schedules which apply to the marginal buyer cannot involve any …xed fee or subsidy. This property was …rst noted by Wilson (1994) for the case of monopoly, and Martimort and Stole (2009) for the case of duopoly.
^ . This must maximize the dominant …rm's pro…t under the constraint that at the switching point^ the buyer must be indi¤erent between exclusive and non-exclusive contracts: Implicit di¤erentiation of the constraint (6) yields The …rst-order condition for the optimal switching point is therefore Taken together, (6) and (7) determine the optimal switching point^ and the corresponding constant term of the non-exclusive price schedule, A (which turns out to be positive, i.e. a …xed fee). This completes the solution to the …rm's optimal pricing problem with exclusive contracts. 22

Competitive fringe
In this section, we focus on the case in which the dominant …rm A faces a competitive fringe. This assumption serves two main purposes. First, it sim-pli…es the analysis, as the competitive fringe will always price at cost (i.e., P B (q B ) = cq B ), without imposing any exclusivity clause. 23 Given the passive behavior of the competitive fringe, …nding the model's equilibrium is tantamount to …nding the dominant …rm's optimal pricing strategy. Secondly, the dominant …rm has no way to eliminate the competitive pressure from the fringe, which will always stand ready to supply product B at a unit price of c. This highlights the di¤erence between our theory and other post-Chicago theories in which the role of exclusive contracts is to deter entry, or deprive a rival of economies of scale so as to drive it out of the market. If exclusive contracts are ever optimal in the competitive fringe model, their role must evidently be di¤erent.
The downside of the competitive fringe assumption is that …rms which just break even cannot really be harmed. Thus, if exclusive contracts are anticompetitive, they will harm only the buyers. However, in the next section we shall see that the main insights extend to the duopoly model, where exclusive contracts always decrease the dominant …rm's rival's pro…t.
The uniform-quadratic model. In order to get explicit solutions, we shall henceforth focus on a uniform-quadratic speci…cation of the model. The parameter is assumed to be uniformly distributed over the interval [0; 1], and the function u is taken to be: The parameter captures the degree of substitutability among the products: it ranges from 1 2 (perfect substitutes) to 0 (independent goods). The factor 1 2 in the middle term in (8) prevents changes in from a¤ecting the size of the market. 24 The strict concavity of u implies that buyers have a preference for variety. For example, the extra bene…t from buying q 2 units of both goods rather than q units of one is: u( q 2 ; q 2 ; ) u(q; 0; ) = 1 4 q 2 > 0. The uniform-quadratic model has two properties that will be used repeatedly in what follows. First, the market is uncovered; in other words, in equilibrium the marginal buyer's demand is negligible, and so the price schedules which apply to the marginal buyer cannot involve any …xed fee or subsidy. Second, the envelope theorem directly implies that Thus, the equilibrium payo¤ of any buyer is fully determined by the quantities obtained in equilibrium by all lower demand buyers. Finally, notice that in the uniform-quadratic speci…cation, the condition

Equilibrium
Under the uniform-quadratic speci…cation, the equilibrium of the competitive fringe model can be calculated explicitly by using the methods discussed in section 3. It involves several di¤erent sub-cases, depending on the size of the dominant …rm's competitive advantage, c.
In particular, the size of c, together with the demand parameter , determines the form of the optimal pricing strategy for the dominant …rm when it is restricted to simple non-linear pricing. When c is large, low-demand buyers are e¤ectively captive, so the dominant …rm can engage in monopoly pricing in the low-demand segment of the market. We denote by the monopoly price schedule, and by the corresponding quantity. 25 As the size of the competitive advantage decreases and/or demand increases, however, the buyer's temptation to purchase also product B increases. But if a buyer purchased a positive amount of product B, his demand for product A would decrease, as the products are substitutes. To prevent this, the dominant …rm therefore engages in limit pricing, raising the sales of product A just up to the point where the buyer's marginal willingness to pay for product B falls below the competitive fringe's cost c. Let us denote by this limit pricing price schedule, and by the corresponding quantity. 26 Finally, when the buyer's demand gets still higher, foreclosing the competitive fringe becomes unpro…table. The dominant …rm therefore accommodates, so that in equilibrium high-demand buyers purchase both goods. We show in the Appendix that the price schedule that supports such common representation outcome is: The corresponding quantities are Whether any buyers will be faced with monopoly or limit pricing depends on the size of the dominant …rm's competitive advantage in the following way: 2 5 The calculation of these functions is a standard exercise in optimal non-linear pricing. 2 6 The function q lim A ( ) is implicitly de…ned by the condition uq B (q lim A ( ); 0; ) = c: It is then easy to verify that this quantity is implemented by the price schedule (13).

Proposition 1
In the competitive fringe model, there is a unique non-linear pricing equilibrium where P B = cq B and: when 0 c 1 2 2 3 , …rm A o¤ ers the price schedule P A (q) = P cr A (q); where B is implicitly de…ned by the condition q cr B ( B ) = 0 and the constant guarantees the continuity of the price schedule; when c 1 2 , …rm A o¤ ers the price schedule and the constants guarantee the continuity of the price schedule.
Proof. See the Appendix.
In equilibrium, the highest type, i.e. = 1, purchases both goods under common representation. In particular, it can be easily veri…ed that he obtains the e¢ cient quantities: q cr i (1) = q f b i (1). As we shall see, this no-distortion-atthe-top property holds in all equilibria, with and without exclusive contracts. 27 It implies that in the absence of economies of scale the dominant …rm's rivals would never be driven out of the market.
When exclusive contracts are permitted, the dominant …rm can still o¤er non-exclusive contracts similar to those described above. In addition, it can also o¤er exclusive contracts. In this case, it can either undercut the competitive fringe, pricing just below c and selling an amount of its product, or engage in monopoly pricing. Obviously, the latter strategy is always more pro…table, but it may not be feasible because of the pressure from the competitive fringe. While the equilibrium outcome is still unique, it can be supported by di¤erent price schedules. The reason for this is that when the dominant …rm o¤ers both exclusive and non-exclusive contracts, some contracts are destined not to be accepted and may therefore be speci…ed arbitrarily, at least to some extent. Accordingly, the following proposition speci…es only the relevant parts of the equilibrium price schedules.
when c c, …rm A o¤ ers the price schedules: where^ , A and c are de…ned in the Appendix; when c c 1 2 , …rm A o¤ ers the price schedules: where = 1 c is the solution to q e c ( ) = q m A ( ); and the constant guarantees the continuity of the price schedule, and where^ and A are de…ned in the Appendix; when c 1 2 , …rm A o¤ ers the price schedules: where^ and A are as in the previous case.
Proof. See the Appendix. Figure 1 shows, in the parameter space ( ; c), the various regions where di¤erent equilibrium patterns arise.

Discussion
We start our discussion from the extremes cases, in which the dominant …rm's competitive advantage is either quite small or quite large. The role of exclusive contracts turns out to be quite di¤erent in these two cases. Intermediate cases are just a combination of the extreme ones and will be brie ‡y considered later.

Small competitive advantage (c c)
When the dominant …rm's competitive advantage is small, in the non-linear pricing equilibrium the marginal buyer purchases product B only. The intuition for this is simple. On the one hand, product A is less costly to produce, or of higher quality. On the other hand, product B is supplied competitively whereas the dominant …rm exercises its market power in the market for product A. This latter e¤ect prevails when the competitive advantage is small (the precise condition for this is, in fact, c 1 2 2 3 ). In equilibrium, therefore, while low-demand types ( c) do not purchase any good, intermediate demand types (c A ) purchase an amount q e c ( ) of good B only. High demand types ( A 1) purchase both goods, in which case the equilibrium quantities are the common representation quantities q cr A ( ) and q cr B ( ). The threshold A is the solution to q cr A ( ) = 0. The non-linear pricing equilibrium quantities are depicted as the solid lines in Figure 2.
Starting from this equilibrium, consider what role exclusive contracts may play. The fact that some buyers purchase only product B is disappointing from the point of view of the dominant …rm (as well as being ine¢ cient from the social viewpoint). If only the dominant …rm could replace the competitive fringe in the low-demand segment of the market, it would save the production cost cq B and increase its pro…ts by the same amount. However, this would require undercutting the competitive fringe's price c. With simple non-linear pricing, this would give high-demand buyers the opportunity of purchasing a certain amount of product A at a unit price slightly below c, in addition to that of purchasing product B at a unit price of c. This would be so attractive an option for them that the dominant …rm's pro…ts would actually decrease.
With exclusive contracts, however, the dominant …rm can undercut the competitive fringe conditioning on exclusivity. This leaves the high-demand buyers'outside option substantially unchanged, as they can already purchase any amount of just one product, namely B, at a price only nominally higher than that charged by …rm A. It follows that by imposing an exclusivity clause the dominant …rm can replace the competitive fringe in the low-demand segment of the market, without losing any pro…t on the high-demand segment.
Notice that the dominant …rm would ideally want to restrict its supply to low-demand buyers to the monopoly quantity. However, buyers still have the option of refusing the exclusivity clause and trading with the competitive fringe only. Therefore, under exclusivity the dominant …rm must o¤er a quantity of at least q e c ( ). When c is su¢ ciently low, as we assume here, this is greater than the monopoly quantity, so the constraint is binding. 28 Under the exclusivity clause, then, the dominant …rm sells q e c ( ) units of product A and makes a pro…t of c per unit of output. This is greater than the pro…t it would make, without imposing exclusive dealing, not only on types A , but also on buyers of type just above A , who would purchase a small amount of good A. Therefore, the dominant …rm will impose exclusivity also on those buyers. As a result, their quantities will be distorted more heavily than in the non-linear pricing equilibrium, and they will also su¤er a loss in terms of reduced variety. Only for su¢ ciently high will the dominant …rm stop imposing the exclusivity clause. At that point, the equilibrium quantities must coincide with those of the non-linear pricing equilibrium. 29 The optimal switching point,^ , will therefore exceed A . Notice that since q cr , the denominator of left hand side of (7) is strictly positive. This implies , so the dominant …rm extracts more rents, at the margin, from buyers who accept non-exclusive contracts than from those who accept exclusive ones. The equilibrium quantities with exclusive contracts are depicted as the dotted lines in Figure 2.
Let us now compare the equilibrium with and without exclusive contracts. By revealed preferences, it is clear that the dominant …rm's pro…t must increase. It is also clear that the competitive fringe will just break even anyway. The interesting question then is how exclusive contracts impact on buyers. To answer this question, remember that a buyer's equilibrium rent must be equal to the sum of the equilibrium quantities of all lower-demand buyers (equation (9)). The equilibrium aggregate quantities are depicted in panel c of Figure 2.
Clearly, low-demand buyers (i.e., types A ) are una¤ected by exclusive contracts. However, all higher types are harmed. Some intermediate demand buyers (i.e., types A ^ ) su¤er from both lower volumes and reduced variety. Higher demand buyers obtain the same quantities as in the non-linear pricing equilibrium. However, the dominant …rm can now extract more rents from them, thanks to the fact that they have less attractive alternatives (since the quantities for lower types are distorted more heavily) than in the non-linear pricing equilibrium. The additional rents are extracted by adding a positive …xed fee to the non-linear pricing equilibrium tari¤. 30 31 The e¤ect of exclusive contracts on social welfare is ambiguous, though. On the one hand, aggregate equilibrium quantities, which are already ine¢ ciently low, are further reduced. This is bad for e¢ ciency. A countervailing e¤ect, however, is the replacement of the competitive fringe with the more e¢ cient dominant …rm in the low-demand segment of the market. This reduces total production costs, with a positive production-e¢ ciency e¤ect. In general, either 2 9 This follows from the fact that since 3 0 This follows from the fact that marginal prices cannot di¤er from those prevailing under non-linear pricing, as they must support the same quantities.

Large competitive advantage (c 1
2 ) Next, consider the case in which the competitive advantage of the dominant …rm is so large that (i ) in the non-linear pricing equilibrium the marginal buyer purchases product A only and (ii ) under exclusivity, the competitive fringe does not exert any competitive pressure on the dominant …rm, which can therefore behave as an unconstrained monopolist. In the uniform-quadratic model, the exact condition for this is c 1 2 . In this case, in the non-linear pricing equilibrium the dominant …rm can sell the monopoly quantity in the low-demand segment of the market (i.e. 1 2 m ), must sell the limit-pricing quantity to intermediate-demand buyers, and eventually accommodates, selling the common representation quantity, in the high-demand segment. The high demand segment comprises types B 1, where B is the highest such that q cr B ( ) = 0. The non-linear pricing equilibrium quantities are depicted as the solid lines in Figure 3.
Here, the problem for the dominant …rm is that the only way to foreclose its rivals is to engage in limit pricing. This is costly, as prices must be reduced relative to the monopoly solution. By imposing an exclusivity clause, however, the dominant …rm can keep selling the pro…t-maximizing monopoly quantity q m A ( ) to a larger set of types, without having to resort to limit pricing. In other words, exclusive contracts now allow the dominant …rm to foreclose its competitors more e¢ ciently.
Of course, buyers have the option of refusing the exclusivity clause and trading with the competitive fringe only. When c 1 2 , however, the dominant …rm's competitive advantage is so large that this option does not really constrain its pricing strategy. This is so because the buyer obtains an information rent even under exclusive dealing. When c 1 2 , this is greater than the rent that he could obtain by trading with the competitive fringe only, so exclusive dealing e¤ectively shelters the dominant …rm from competition. 32 Since the foreclosure strategy via exclusivity clauses is more pro…table than limit pricing, the dominant …rm will want to use it more extensively. Therefore, we must have^ > B . In other words, the marginal buyer who switches to a non-exclusive contract will be a higher type than the one who switches to the common representation quantities q cr A ( ) and q cr B ( ) under non-linear pricing. Eventually, however, no exclusivity clause will be imposed. The reason for this is that the distortion due to information asymmetry must optimally vanish at the top of the distribution of types. Intuitively, it is pro…table for the dominant …rm to allow high-demand types to purchase quantities that are nearly e¢ cient and then extract the surplus by adding a …xed fee A . This …xed fee can be interpreted as a "tax" that the dominant …rm levies to allow buyers to bene…t from product variety. Thanks to this …xed fee, the dominant …rm can extract more pro…ts under common representation than under exclusivity, even if in the latter case it charges monopoly prices.
When no exclusivity clause is imposed, buyers will purchase exactly the same quantities as in the non-linear pricing equilibrium, i.e. q cr A ( ) and q cr B ( ). The equilibrium quantities with exclusive contracts are depicted as the dotted lines in Figure 3.
Comparing the equilibrium with and without exclusive contracts, it is clear that buyers are once again harmed. For some intermediate-demand buyers (i.e. m B ), equilibrium quantities are reduced. Hence, so are the equilibrium rents for them and also all higher types. The latter obtain the same quantities as in the non-linear pricing equilibrium, but are left with lower rents as they now have less attractive alternatives. As in the previous case, the dominant …rm extracts the extra rents by adding a positive …xed fee to the non-linear pricing equilibrium tari¤.
Unlike the previous case, however, the impact of exclusive contracts on social welfare is now unambiguously negative. This follows immediately from the fact that equilibrium quantities decrease and there is no countervailing cost-saving e¤ect, since the dominant …rm no longer replaces the competitive fringe's output.

Intermediate cases
We have seen that in the competitive fringe model exclusive contracts can serve two purposes: when c is small, they allow the dominant …rm to replace the competitive fringe in the low-demand segment of the market; when c is large, they allow the dominant …rm to foreclose its rivals more e¢ ciently. When c < c < 1 2 , exclusive contracts serve both purposes. For low-demand buyers, the dominant …rm replaces the competitive fringe, pricing just below c and selling q e c ( ) units of product A. For higher demand buyers, the dominant …rm uses exclusive contracts as a substitute for limit pricing.
In any case, buyers are always harmed by exclusive contracts. The welfare e¤ects, however, depend on whether the q e c ( ) units of product A replace an equal amount of product B (which happens when c < c < 1 2 2 3 ), or the limit quantity q lim A ( ) (which happens when 1 2 2 3 < c < 1 2 ). In the former case, there is a positive cost-saving e¤ect that may make the total welfare e¤ect ambiguous. In the latter case, exclusive contracts unambiguously decrease social welfare, for the same reasons as in the large competitive advantage case.

Duopoly
In this section, we turn to the case in which there is only one supplier of product B (…rm B ), which will then have some market power. Unlike the competitive fringe, …rm B can therefore respond actively to the dominant …rm's attempt at foreclosing it. In particular, it may o¤er exclusive contracts in its turn, lower its non-exclusive prices, or both.
There are also two other important di¤erences with the competitive fringe model. Firstly, since …rm B may reap positive pro…ts in equilibrium, it can be hurt by the dominant …rm's exclusionary strategy. Secondly, the duopoly model allows to endogenize the choice of who o¤ers exclusive contracts, and whose exclusive contracts are accepted in equilibrium.

Non-linear pricing equilibrium
The analysis of the duopoly model is more complex than that of the competitive fringe model, as the solution to a …rm's pricing problem (discussed in section 3) does not yield directly the equilibrium, but only its best response to its rival's strategy. The non-linear pricing equilibrium must be found by adapting to the asymmetric case the solution procedure proposed by Martimort and Stole (2009) for the symmetric case (i.e. c = 0). 33 This is a non-trivial exercise; although in this paper it serves mainly to provide a benchmark for comparison, it may also be of some interest in its own right.
Like in the competitive fringe model, the dominant …rm can engage in monopoly or limit pricing, or it can accommodate its rival. The monopoly schedules are exactly the same as in the competitive fringe model. The limit pricing schedules are also similar, except that now the unit cost c is replaced by the marginal price that …rm B charges for the …rst unit it o¤ers, P 0cr B (0). The new price and quantity schedules under common representation are 34 0 is a decreasing function of that vanishes when = 1 2 . In the limiting case of perfect substitutes ( = 1 2 ), the solution therefore converges to the standard limit pricing equilibrium in which the dominant …rm undercuts its rival pricing just below its unit cost c. As the degree of substitutability decreases, competition becomes less intense, prices increase, and quantities decrease. Equilibrium quantities are distorted downward; as it turns out, the size of the distortion is the same for both goods and is independent of c. 35 With independent goods ( = 0), one obtains the monopoly solution. When c = 0, the Martimort and Stole solution returns.
In the non-linear pricing equilibrium, when demand is high buyers always obtain the common representation quantities. However, some low-demand buyers (i.e. types B ; where B is implicitly de…ned by q cr B ( B ) = 0) must purchase product A only. 36 Whether these buyers are contested (in which case limit pricing applies) or captive (which would allow the dominant …rm to engage in monopoly pricing) depends on the size of the competitive advantage c. The equilibrium is described in the following: Proposition 3 In the duopoly model, the following is a non-linear pricing equilibrium. Firm B o¤ ers the price schedule and: conform to the guess, if there are any. However, this is not a serious problem for our purposes.
If there were multiple non-linear pricing equilibria, for each there would exist a corresponding equilibrium with exclusive contracts, with the same comparative statics properties. 3 4 Notice that the guess and check procedure is used only to …nd the common representation price schedules. The monopoly and limit pricing schedules are pinned down uniquely. 3 5 This property, however, does not play any special role in what follows. 3 6 Unlike the competitive fringe model, the case in which the marginal buyer purchases product B only can no longer arise, as both …rms now have market power. (1 ) 2 , …rm A o¤ ers the price schedule where B is implicitly de…ned by the condition q cr B ( B ) = 0 and the constant guarantees the continuity of the price schedule; when c c, …rm A o¤ ers the price schedule where m is implicitly de…ned by the condition q m A ( m ) = q lim A ( m ) and the constants guarantee the continuity of the price schedule.
Proof. See the Appendix.

Exclusive contracts
With exclusive contracts, the nature of the equilibrium depends on the size of the dominant …rm's competitive advantage even more profoundly than in previous cases. In particular, when c is small the equilibrium outcome is no longer unique, as the …rms are faced with coordination problems that may have multiple solutions. In all equilibria, however, the e¤ect of exclusive contracts is to reduce prices and pro…ts. When the dominant …rm's competitive advantage is large, by contrast, there is a unique equilibrium outcome, and exclusive contracts tend to be anti-competitive.
Because of these di¤erences, it is convenient to deal with the two cases separately. Figure 4 shows the parameter values for which the di¤erent equilibrium patterns may arise.

Small asymmetry (c c)
When the dominant …rm's competitive advantage is relatively small, …rm B can compete for exclusives e¤ectively. Competition for exclusives, though, is competition in utility space, where the …rms' products e¤ectively become homogeneous. The familiar Bertrand undercutting process then drives exclusive prices towards marginal costs. 37 In particular, there is always an equilibrium in which (i ) exclusive prices are equal to marginal cost, with …rm A just undercutting its rival, and (ii ) non-exclusive prices are exorbitantly high, so that no buyer purchases both products. In this equilibrium, …rm B 's pro…t vanishes, and …rm A's is reduced to a minimum. However, this equilibrium arises only in the case of very serious failures of coordination between the …rms. Both …rms can obtain larger pro…ts, and buyers larger surpluses, but this requires some coordination among the …rms. Speci…cally, they must lower their non-exclusive prices, and raise their exclusive prices, in coordinated fashion. The …rst move induces some buyers to purchase both products, allowing …rms to extract the buyers'preference for variety; the second reduces the intensity of competition.
The degree to which …rms can coordinate their strategies may vary, though. When …rms lower their non-exclusive prices in order to extract the buyers' preference for variety, they may do so in an asymmetric fashion. If one …rm reduces its prices excessively, however, buyers will purchase a disproportionate amount of its product, and so the bene…t from variety that can be extracted is reduced. This is the …rst source of multiplicity of equilibria. Secondly, if buyers are purchasing both products, …rms have no de…nite incentive to undercut one another's exclusive prices. Thus, exclusive prices may be raised above marginal cost. This, too, can be done to a variable extent, providing another source of multiple equilibria.
Since our aim is to show that when c is small exclusive contracts increase the intensity of competition, we focus on the equilibrium in which …rms coordinate their strategies as best as they can. This is the "most cooperative"equilibrium, in which pro…ts are largest. 38 This equilibrium can be obtained as follows. Let U E ( ) be the (typedependent) reservation utility that buyers could obtain by choosing their most preferred exclusive contract. To extract the buyers'preference for variety, …rms must introduce non-exclusive price schedules implicitly de…ned by the condition: with a small tie-breaking discount if necessary. However, high-demand buyers will actually obtain more than U E ( ) simply thanks to competition with non-exclusive contracts. Therefore, each of the non-exclusive price schedules comprises two branches: a lower branch that is intended for buyers who obtain exactly the reservation utility U E ( ); and an upper branch that is intended for high-demand buyers who obtain strictly more. The latter must coincide with the equilibrium price schedules under non-linear pricing, except possibly for constant terms that serve to guarantee the continuity of equilibrium prices. 39 Since we look for the equilibrium in which …rms'pro…ts are largest, we posit that …rms maximize the rents that they extract from low-demand buyers. This requires maximization of the total surplus u(q A ; q B ; ) cq B , subject to the constraint that these buyers must obtain U E ( ). Using the envelope theorem, the constraint can be rewritten as where q E ( ) is the optimal quantity under exclusivity (to be determined). Generally speaking, the more e¢ cient …rm must produce more than the less e¢ cient one. In particular, the problem of total-surplus maximization may have a corner solution in which some low-demand types must buy good A only. In this case, exclusive contracts must be accepted in equilibrium by those types, and so Bertrand competition in utility space implies that exclusive prices must fall to marginal costs. Therefore, q A ( ) must coincide with q e c ( ) as de…ned by condition (17), and q B ( ) must vanish.
When instead the total-surplus maximization problem has an interior solution, which is buyers purchase both products. In this case exclusive contracts are not accepted, and hence there may be room for coordinating the exclusive prices. Thus, q E ( ) depends on what exclusive prices are sustainable in the most cooperative equilibrium. 40 Let us denote by an upper bar the highest exclusive prices that are consistent with playing the game non-cooperatively (i.e. the exclusive prices in the "most cooperative" equilibrium). To …nd them, we can assume, with no loss of generality, that both …rms o¤er the same exclusive price schedule P E (q). 41 By construction, low-type buyers must be just indi¤erent between an exclusive contract and a non-exclusive one (equation (20)). Thus, any arbitrarily small discount would trigger a switch to an exclusive contract. In equilibrium, no such deviation can be pro…table. This implies the following no undercutting conditions: which in the most cooperative equilibrium must hold as equalities. Denote by q E ( ) the optimal quantity associated with the exclusive prices P E (q), and by q cr i ( ) the values of q i ( ) given by (22) when q E ( ) = q E ( ). Rewrite (20) as and use the no-undercutting conditions (23) to get From (21), the term inside the …rst square brackets on right-hand side can be interpreted as the preference for variety, while the term inside the second square bracket is the cost saving. Using (22), we …nally get The corresponding quantity is The non-exclusive prices that generate the same net utility and maximize the preference for variety that the …rms can extract, thus supporting the quantities q cr A ( ) and q cr B ( ), are P cr A (q A ) = cq + (1 2 )q 2 + cq e c (^ ); P cr B (q B ) = 2cq + (1 2 )q 2 ; (26) 4 0 Notice that while exclusive contracts are not accepted, they do a¤ect the equilibrium outcome. The less aggressively …rms bid for exclusivity, the greater the payments …rms can obtain for non-exclusive contracts. 4 1 The proof that this does not entail any loss of generality is by contradiction. Suppose to the contrary that one …rm o¤ered more attractive exclusive contracts than its rival. Since these contracts are not accepted in equilibrium, the …rm could increase its exclusive prices without losing any pro…ts on its exclusive contracts. In fact, the buyers' reservation utility would decrease, allowing both …rms to increase their pro…ts from non-exclusive contracts.
where^ is now the solution to q e c (^ ) = q cr A (^ ) and the constant term in P cr A (q A ) guarantees a smooth-pasting condition from exclusive to non-exclusive contracts.
We are now ready to provide the characterization of the most cooperative equilibrium.
Proposition 4 Suppose that c c 2(1 2 ) Then, in the duopoly model the most cooperative equilibrium with exclusive contracts is as follows. 42 Both …rms o¤ er the exclusive price schedules with …rm A slightly undercutting …rm B, though. Furthermore: where^ is the solution to q e c (^ ) = q cr A (^ ) and the solution to q cr A ( ) = q cr A ( ) (and to q cr B ( ) = q cr B ( )), and the constants guarantee the continuity of the price schedules.
Notice that the only exclusive contracts that are accepted in equilibrium are those o¤ered by the dominant …rm. While …rm B o¤ers exclusive contracts as well, these contracts are not accepted by any buyer. The equilibrium is thus broadly consistent with the observation that only the dominant …rm uses exclusive contracts: for one thing, contracts that are o¤ered but not accepted are di¢ cult to observe; for another thing, to sustain the equilibrium it su¢ ces, in practice, that …rm B just stands ready to o¤er its prescribed exclusive contracts.
We can now compare the equilibrium with and without exclusive contracts. In the non-linear pricing equilibrium, low-demand buyers purchase only product A, while high-demand buyers buy both products. The non-linear pricing equilibrium quantities are depicted as the solid lines in Figure 5.
In the most cooperative equilibrium with exclusive contracts, buyers are divided into four groups. For c; buyers do not buy any product; for c < ^ , buyers purchase q e c ( ) units of product A only, under an exclusivity clause and at a price just below c; for^ < , buyers purchase q cr A ( ) units of good A and q cr B ( ) units of good B, and so obtain the same net surplus as if they accepted an exclusive contract; …nally, for < 1, buyers buy q cr A ( ) units of good A and q cr B ( ) units of good B, and strictly prefer their non-exclusive contract to any exclusive one. These equilibrium quantities are depicted as the dotted lines in Figure 5. Clearly, exclusive contracts are unambiguously pro-competitive. To see why, notice that equilibrium quantities are larger than under non-linear pricing, and are everywhere closer to the …rst best. Since the social surplus (i.e., the sum of buyers' surplus and …rms' pro…ts) is concave in q A and q B , it is clear that exclusive contracts increase social welfare.
In particular, buyers now bene…t from exclusive contracts. Low-demand buyers ( A < < ) increase their purchases. High-demand buyers ( ) purchase the same quantities as in the non-linear pricing equilibrium, but they too are better o¤ as they now have more attractive alternatives. The bene…t now is obtained via …xed subsidies that in the equilibrium with exclusive contracts are added to the non-linear pricing equilibrium price schedules.
As for …rms, prices are lower than in the non-linear pricing equilibrium (to support higher quantities). Since prices are already lower than under monopoly, the fact that they are further reduced means that exclusive contracts decrease …rms' pro…ts. Thus, …rms are caught in a prisoner's dilemma: each has a unilateral incentive to o¤er exclusive contracts if it can, but both would gain if exclusive contracts were prohibited.
All of these conclusions agree with those obtained in Calzolari and Denicolò (2013) for the symmetric case. This is natural, as the asymmetry is small.

Large asymmetry (c > c)
When the dominant …rm's competitive advantage is relatively large, things are di¤erent. The dominant …rm can engage in monopoly pricing on a segment of the market. If c is not too large, however, …rm B can still compete for exclusives, and in any case it can resist being foreclosed by reducing its non-exclusive prices.
The competition for exclusives has now a unique outcome. When c > c, there is no longer any room for extracting the buyers'preference for variety. 43 The exclusive contracts o¤ered by the dominant …rm are therefore accepted. This implies that there is no scope for coordinating the exclusive prices. As a result, …rm B will always price at cost, whereas …rm A will either undercut …rm B or engage in monopoly pricing -whichever leads to lower prices.
Next, let us focus on the non-exclusive prices. Even though it cannot win the competition for exclusives, …rm B can at least try to induce high-demand buyers, who value product variety more, to reject the exclusive contracts o¤ered by …rm A and buy both products. To get buyers to purchase both products, …rm B must be willing to lower its non-exclusive prices, as compared to the nonlinear pricing equilibrium. Firm A, by contrast, would prefer to serve buyers under an exclusivity clause. However, it can now raise its non-exclusive prices -again, taking the non-linear pricing equilibrium as a benchmark -which directly increases its pro…t and also makes the exclusive deals more appealing, comparatively speaking.
In other words, both …rms try to a¤ect the critical buyer^ who is just indi¤erent between accepting exclusive and non-exclusive contracts: …rm A tries to push^ up, …rm B to pull it down. In order to do so, …rms modify the constant terms of their non-exclusive price schedules. 44 Consider, then, the optimal choice of these constant terms, which we denote by A and B . Each …rm maximizes its pro…t. This is The critical buyer^ is the one who is just indi¤erent between exclusive 4 3 Intuitively, the dominant …rm's competitive advantage is so large that if B 's output was positive, the bene…ts from greater product variety would be o¤set by the increase in total production costs. 4 4 The reason why they use only those constant terms as their strategic weapons at this stage is that marginal prices are pinned down by the fact that once buyers start purchasing both goods, the equilibrium quantities must be given by (19). 4 5 This formula accounts for the fact that under an exclusivity clause the dominant …rm now enforces the monopoly solution. and non-exclusive contracts: (27) It follows that the equilibrium conditions are: The …rst condition implies that the dominant …rm charges a …xed fee: A > 0. The proof is very simple: since the right-hand side and the denominator of the left-hand side are both positive, it must be P N E A (q N E A (^ )) > P m A (q m A (^ )). However, we know that q m A (^ ) > q N E A (^ ) as the goods are substitutes, and on the other hand in the absence of a …xed fee we would have P N E A (q) < P m A (q) as competition reduces the equilibrium prices. Therefore, the inequality can hold only if the non-exclusive price schedule involves also a …xed fee.
Likewise, the second equilibrium condition implies that less e¢ cient …rm must charge a …xed subsidy (the argument is similar). Intuitively, …rm B trades o¤ market share and pro…tability.
Given that c > c, we still must distinguish between two sub-cases. If c < 1 2 , …rm B exerts a competitive pressure on …rm A's exclusive prices. This implies that for small volumes, …rm A must just undercut …rm B ; only for higher volumes (and hence higher demand buyers) can it engage in monopoly pricing. When c 1 2 , by contrast, the information rent under exclusive dealing and monopoly pricing is greater than the rent that the buyer could obtain by trading with …rm B only. Thus, exclusive dealing e¤ectively shelters the dominant …rm from competition.

Proposition 5 Suppose that c > c.
When c < c < 1 2 the two …rms o¤ er the following exclusive price schedules where m is such that q e c ( m ) = q m A ( m ) and the constant guarantees the continuity of the price schedule, and the following non-exclusive price schedules where^ , A and B are the solution to the system (27)-(29) When c 1 2 the two …rms o¤ er the following price schedules where^ , A and B are de…ned as in the previous case.
The role of exclusive contracts in this case is similar to the competitive fringe model: by imposing an exclusivity clause, the dominant …rm can keep selling the pro…t-maximizing monopoly quantity q m A ( ) without having to resort to limit pricing. Thus, exclusive contracts allow the dominant …rm to foreclose its competitor more e¢ ciently. Like in the small asymmetry case, however, the only exclusive contracts that are accepted in equilibrium are those o¤ered by the dominant …rm.
Let us now compare the equilibrium with and without exclusive contracts. We start from the case c 1 2 : In this case, with simple non-linear pricing the dominant …rm serves some low-demand buyers under monopoly but then must engage in limit pricing (the equilibrium quantities are the solid lines in Figure  6). With exclusive contracts, by contrast, the dominant …rm can engage in monopoly pricing straightaway. Since the foreclosure strategy via exclusivity clauses is more pro…table than limit pricing, the dominant …rm will also want to use it more extensively. Therefore, we must have^ > B . In other words, the marginal buyer who switches to a non-exclusive contract will be a higher type than the one who switches to the common representation quantities q cr A ( ) and q cr B ( ) under non-linear pricing. The equilibrium quantities under exclusive contracts are depicted as the dotted lines in Figure 6.
Comparing the equilibrium with and without exclusive contracts, one …nds that exclusive contracts are now unambiguously anti-competitive, exactly as in the corresponding case of the competitive fringe model. Buyers are harmed, as the equilibrium quantities for some intermediate demand buyers are decreased. High-demand buyers obtain the same quantities as in the non-linear pricing equilibrium, but they are left with lower rents as they now have less attractive alternatives.
The main di¤erence with the competitive fringe model is that now the less e¢ cient …rm is also harmed, both in terms of market share and pro…t margins. The dominant …rm gains, as exclusive contracts have no competition-enhancing e¤ect. Remarkably, however, the dominant …rm's gain does not rest on …rm B being driven out of the market.
Social welfare decreases, since equilibrium quantities, which are already distorted downward in the non-linear pricing equilibrium, further decrease when exclusive contracts are permitted. Thus, the dominant …rm's gain from exclusive contracts does not compensate for the losses that it in ‡icts to its customers and competitors. When c < c < 1 2 , the equilibrium is similar to the case c 1 2 , except for the fact that now low-demand types are faced with exclusive prices just below c and so buy q e c ( ). This is higher than the limit pricing quantity that they would have purchased under non-linear pricing; furthermore, less buyers are excluded under exclusive contracts. Therefore, low demand buyers bene…t from exclusive contracts. High-demand buyers, by contrast, are harmed, for the same reasons as above. The dominant …rm gains and its rival loses, but the total e¤ect on social welfare is now ambiguous. When c is close to c, the pro-competitive e¤ect on the low-demand segment of the market must prevail. When c is close to 1 2 , by contrast, what prevails is the anti-competitive e¤ect on the high-demand segment.

Conclusion
In this paper, we have developed a modern, consistent version of the pre-Chicago theory of exclusive dealing. The theory maintains that a dominant …rm may …nd it pro…table to use exclusive contracts just to increase its market share, without necessarily driving its rivals out of the market or impeding entry. This theory is valid under two assumptions. First, …rms are imperfectly informed about demand. Second, the dominant …rm has a competitive advantage over its rivals, in terms of lower cost, higher demand, or a combination of the two.
Not only are these assumptions realistic, but the model's predictions are also consistent with the facts of many antitrust cases. In addition to a dominant …rm that controls a substantial share of the market and has entered into some kind of exclusive arrangement with its customers, these often involve one or more smaller competitors, which have been active in the industry for some time and in principle could themselves use exclusive contracts, but apparently have not. Existing theories have found it di¢ cult to explain this recurrent situation without making ad hoc assumptions. Ours, by contrast, can reproduce these stylized facts naturally.
Our theory o¤ers new insights for competition policy. Since exclusive contracts may be either pro or anti-competitive, the theory does not call for a radical change in the current policy, which is based on the rule of reason. However, it may suggest that di¤erent factors should be considered for the purposes of antitrust evaluation.
In our model, the key factor is the size of the dominant …rm's competitive advantage. This determines whether the dominant …rm's rival can compete for exclusives e¤ectively or not. If it can, exclusive contracts tend to be procompetitive, reducing prices and pro…ts and bene…ting buyers. If it cannot, exclusive contracts are anti-competitive. The dominant …rm gains, but both its rival and buyers are harmed, and social welfare goes down.
While the size of the dominant …rm's competitive advantage can not be observed precisely, it is correlated with two variables that often can. One is the dominant …rm's market share, the other is the fraction of the market foreclosed. When both these two variables are large, an anticompetitive e¤ect is more likely.
Factors other than the dominant …rm's competitive advantage, which are often emphasized by antitrust authorities and the courts, turn out to be less important in our analysis. For example, the length of exclusive contracts is irrelevant, since contracts are not used for commitment purposes. Another factor that should be reconsidered is economies of scale. Their existence would be crucial if a negative impact of exclusive contracts on competition could only arise because rivals are driven, or kept, out of the market. However, our analysis clari…es that exclusive contracts may have anti-competitive e¤ects even if the dominant …rm's rivals are and stay active. This is so because exclusive contracts may be pro…table even in the short run; in other words, without entailing a sacri…ce for the dominant …rm. This implies that no proof of eviction and recoupment may be needed.
If, however, economies of scale do matter, anti-competitive e¤ects may arise under even broader circumstances than when they do not. In our duopoly model, exclusive contracts always reduce the less e¢ cient …rm's pro…t. Thus, they have the potential to foreclose a rival, or deter its entry, even if the dominant …rm's competitive advantage is small. In this case, the fraction of the market that is foreclosed may be small as well. However, the e¤ects of a small segment of the market being foreclosed may reverberate throughout the entire market thus having a signi…cant impact on the less e¢ cient …rm's pro…ts. 46 4 6 This observation may cast doubts on policies that provide a safe harbour when exclusive dealing arrangements foreclose 30% of the market or less, such as those adopted in the US Department of Justice's guidelines. Such an high threshold re ‡ects the view that exclusive contracts can only be anti-competitive if they deprive a rival of economies of scale. From that perspective, it may seem reasonable that if 70% of the market remains contestable, it should su¢ ce for a rival to prosper. But we have shown that the anti-competitive e¤ects of exclusive contracts may extend well beyond the segment of the market that is actually foreclosed.

Appendix
This Appendix contains the proofs of Propositions 1-5.
Proof of Proposition 1. Because the proof is based on the use of direct mechanisms, it is convenient to report the equilibrium quantities …rst. They are: when c 1 2 , where the thresholds B , m and A are calculated explicitly below. The marginal buyer is = c when c 1 2 , and = 1 2 when c 1 2 . Obviously, the competitive fringe will always price at cost: P B (q B ) = cq B . To prove the proposition, it then su¢ ces to show that the dominant …rm's equilibrium pricing strategy is indeed optimal. To do so, we shall focus on direct mechanisms and hence …nd the optimal quantity q A ( ), showing that it coincides with the equilibrium quantity reported above. It is then straightforward to conclude that the price schedules that support these quantities, which are the equilibrium price schedules, are indeed optimal.
To begin with, we calculate the indirect utility function v(q A ; ) when P B (q B ) = cq B . This is piecewise quadratic, with two branches corresponding to the cases in which the quantitỹ is 0 or is strictly positive, and a kink between the two branches. That is: On both branches, the coe¢ cients of the quadratic terms are negative. Furthermore, so the function v is globally concave. It can also be easily checked that the single-crossing condition v q A (q A ; ) 0 is satis…ed since: The single-crossing condition guarantees that the participation constraint binds only for the marginal buyer, whom we indicate here as~ , so that …rm A's optimization program (2) becomes By a standard integration by parts, the problem reduces to …nding the function q A ( ) that pointwise maximizes the indirect virtual surplus: Like the indirect utility function, the indirect virtual surplus is a piecewise quadratic function, with two branches and a kink at q A = q lim A ( ). Since the additional term (1 )v (q A ; ) is linear in q A and v(q A ; ) is globally concave, s(q A ; ) is also globally concave in q A .
Generally speaking, for any the maximum can occur in either one of the two quadratic branches, or at the kink. Let and recall that the kink q lim A ( ) is implicitly de…ned by the condition c q lim A = 0. It is easy to verify that q m ( ), q lim A ( ) and q cr A ( ) are given precisely by the expressions (12), (14) and (16), respectively. We can then conclude that q A ( ) = q m A ( ) if the maximum is achieved on the upper branch, q A ( ) = q cr A ( ) if the maximum is achieved on the lower branch, and q A ( ) = q lim A ( ) if the maximum is achieved at the kink.
Global concavity of s(q A ; ) implies that if q m A ( ) > q lim A ( ), then s(q A ; ) is increasing at the kink and the maximum is achieved at q m A ( ). If instead q m A ( ) < q lim A ( ), then s(q A ; ) is decreasing to the right of the kink, and one must further distinguish between two cases. If q cr A ( ) > q lim A ( ), then s(q A ; ) is increasing to the left of the kink and so the maximum is achieved at the kink q lim A ( ). If instead q cr A ( ) < q lim A ( ), the maximum is achieved to the left of the kink.
It remains to …nd out when each type of solution applies. Provided that c; B i , to whom the limit pricing solution applies. Again, the corresponding equilibrium quantity of good B must be nil.
Finally, consider the case in which B , so that q cr A ( ) q lim A ( ) and the maximum is achieved on the lower branch of the virtual surplus function. Here, we must distinguish between two sub-cases, depending on whether the solution is interior, or is a corner solution at q A ( ) = 0. Clearly, the solution is interior, and is q cr A ( ), when A . In this case, the corresponding equilibrium quantity of good B is q cr B ( ) =q B (q cr A ( ); ): Now, notice that 1 2 2 3 is also the critical threshold for c such that when c < 1 2 2 3 we have B < A , whereas the inequality is reversed when c 1 2 2 3 . This means that if c 1 2 2 3 and the maximum is achieved in the lower branch, it must necessarily be an interior solution. However, when c < 1 2 2 3 we have B < A . In this case, for c A , we have a corner solution for q A , and the corresponding equilibrium quantity of good B is q e c ( ); for A , the solution is again interior. This completes the derivation of the optimal quantities in all possible cases. It is then easy to check that they coincide with the equilibrium quantities reported above, and that they are implemented by the equilibrium price schedules. This completes the proof of the Proposition.
Notice that since equilibrium quantities are everywhere continuos, the equilibrium price schedules must be continuous. The constant terms that guarantee continuity are all negative, i.e. …xed subsidies. In fact, it can be veri…ed that the equilibrium price schedules are also everywhere smooth.
Proof of Proposition 2. Since the proof will use direct mechanisms, we start again by reporting the equilibrium quantities. They are: when c c, where^ is de…ned in the text of the Proposition. Obviously, the competitive fringe will always price at cost, i.e. P E B (q B ) = P N E B (q B ) = cq B . To prove the proposition, it then su¢ ces to show that the dominant …rm's equilibrium pricing strategy is indeed optimal. We shall focus on direct mechanisms and hence look for the optimal quantity q A ( ), showing that it coincides with the equilibrium quantity reported above. It is then straightforward to conclude that the price schedules that support these quantities, which are the equilibrium price schedules, are indeed optimal. From the arguments in section 3, we know that the solution to the dominant …rm's problem is formed by appropriately joining the solution to the maximization program (5), which applies to low-demand buyers ( <^ ), and that to the maximization program (4), which applies to high-demand buyer ( >^ ). The solution to problem (4) has been characterized in the proof of Proposition 1. (Notice that since P E B (q B ) = P N E B (q B ), the constraint U ( ) U R ( ) is already subsumed into the indirect utility function.) We therefore start by focusing on problem (5).
Problem (5) is a standard monopolistic non-linear pricing problem with a utility function except that buyers now have a type-dependent reservation utility Thus, the problem becomes Its solution is given in the following When instead c 1 2 , the solution is Proof. Consider …rst the unconstrained problem. The solution is q A ( ) = max[0; q m A ( )], and the corresponding utility is When c 1 2 , we have U m ( ) U R A ( ) for all , so the unconstrained solution applies. Now suppose that c < 1 2 , so that the type-dependent participation constraint must bind for a non-empty set of types. To deal with this constraint, we use the results of Jullien (2000), and in particular his Proposition 3. To apply that proposition, we must show that our problem satis…es the conditions of Weak Convexity, Potential Separation, Homogeneity, and Full Participation. Following Jullien (2000), de…ne the virtual surplus function where the "weight" g 2 [0; 1] accounts for the possibility that the participation constraint may bind over any subset of the support of the distribution of types. Pointwise maximization of the virtual surplus function yields The condition of Weak Convexity requires that whereĝ( ) is implicitly de…ned by q e c ( ) =`E(ĝ( ); ): Straightforward calculations show that this inequality always holds. Potential Separation is also satis…ed, since it requires simply that`E(g; ) is nondecreasing in , which is obviously true. Homogeneity is obvious, as it requires that U R A ( ) can be implemented by a continuous and non decreasing quantity. which in our case is, by construction, q e c ( ). Finally, the condition of Full Participation requires that in equilibrium all types > c obtain positive quantities, which is obvious given that their reservation utility is strictly positive. Proposition 3 in Jullien (2000) then implies that the solution to problem (5) is and obviously q A ( ) = 0 for c.
Next, we proceed to the characterization of the optimal switching point,^ . To begin with, we show that the equilibrium rent function U ( ) is steeper under non-exclusivity than under exclusivity. By (9), the slope of U ( ) is the sum of the equilibrium quantities. It is easy to verify that the sum of equilibrium quantities in the non-linear pricing equilibrium is always at least as large as in the solution to problem (A.1): with a strict inequality whenever q N E B ( ) > 0. This implies that the solution to the hybrid optimal control problem involves a unique switch from problem (A.1) (which applies to low-demand types) to problem (4) (which applies to high-demand types).
The next lemma says that the switch must be from exclusive dealing to a common representation equilibrium. In other words, at the switching point the solution to problem (4) is given by the common representation quantities q cr A ( ); q cr B ( ) > 0. This rules out the possibility that the switch occurs for types who obtain the monopoly or limit pricing quantity of product A.
Lemma 7 When >^ , both q A ( ) and q B ( ) are strictly positive.
Proof. In the uniform-quadratic model, condition (7) becomes , so the dominant …rm extracts more rents, at the margin, from buyers who accept non-exclusive contracts than from those who accept exclusive ones. From this, it follows immediately that that q N E A (^ ) > 0 (otherwise, P N E A (q N E A (^ )) must be nil). The proof that also q N E B (^ ) > 0 is equally simple. If the solution to problem (4) entails q B ( ) = 0, it must be either max[q m A ( ); q e c ( )] or q lim A ( ). In the former case, the dominant …rm would obtain the same rent from buyers who accept non-exclusive contracts as from those who accept the exclusive one; in the latter, it would actually obtain less. Since we have just shown that it must obtain more, these two cases are not possible. Lemma 7 implies that at the switching point the total quantity q A ( ) + q B ( ) is discontinuous, and in particular that it jumps upward (thus preserving monotonicity). To the right of^ we have q A ( ) = q cr A ( ) and q B ( ) = q cr B ( ). When c 1 2 , to the left of^ we must necessarily have q A ( ) = q m A ( ). Therefore, conditions (6) and (7) become where the indirect utility function v(q A ( ); ) is that derived in the proof of Proposition 1. The explicit solutions for^ and A are complicated and are reported in a Mathematica …le that is available upon request from the authors. When instead c < 1 2 , to the left of^ we can have either q A ( ) = q m A ( ) or q A ( ) = q e c ( ). The former case applies when^ > 1 c; the latter when^ < 1 c. In the former case, the switching point^ is determined by the same conditions as above. In the latter case, conditions (6) and (7)  Once again, the explicit solutions for^ and A are reported in a Mathematica …le that is available upon request from the authors. The threshold c is implicitly de…ned as the solution to^ = 1 c. The explicit expression for c is reported in the Mathematica …le mentioned above. In any case, it can be shown that c < 1 2 2 3 1 2 and that^ > max[ A ; B ]. This completes the derivation of the equilibrium quantities in all possible cases. It is then easy to check that these equilibrium quantities are implemented by the price schedules reported in the statement of the Proposition.
Although the explicit expressions for A are complicated, it can be shown that in any case we have A > 0. In other words, while high-demand buyers are faced with the same marginal non-exclusive prices as in the non-linear pricing equilibrium, they now pay a …xed fee instead of receiving a …xed subsidy.
Proof of Proposition 3. As usual, we start by reporting the equilibrium quantities, which are when c c, when c >c, To prove the proposition, we must show that the equilibrium price schedules satisfy the best response property. Given its rival's price schedule, a …rm is faced with an optimal non-linear pricing problem that can be solved by invoking the Revelation Principle and thus focusing on direct mechanisms. The strategy of the proof is to show that for each …rm i = A; B the optimal quantities q i ( ), given P i (q i ), coincide with the equilibrium quantities reported above. It is then straightforward to conclude that the price schedules that support these quantities must be equilibrium price schedules.
Given P i (q i ), …rm i faces a monopolistic screening problem where type has an indirect utility function v i (q i ; ) = max which accounts for any bene…t he can obtain by optimally trading with its rival. Since u is quadratic and P i (q i ) is piecewise quadratic, v i is also piecewise quadratic. It may have kinks, but we shall show that any such kink preserve concavity, so the indirect utility function is globally concave.
Provided that the single-crossing condition holds, …rm i's problem reduces to …nding a function that pointwise maximizes the "indirect virtual surplus" where c i is zero for i = A and c for i = B. It is easy to verify ex post that the maximizer q i ( ) satis…es the monotonicity condition. Consider, then, …rm A's best response to the equilibrium price schedule of …rm B, P B (q B ). The indirect utility function is piecewise quadratic, with two branches corresponding to the case in which arg max q B 0 [u(q A ; q B ; ) P B (q B )] is 0 or is strictly positive, and a kink between the two branches: The coe¢ cients A 0 , A 1 and A 2 can be calculated as On both branches of the indirect utility function, the coe¢ cients of the quadratic terms are negative. In addition, it can be checked that so the function v A (q A ; ) is globally concave in q A . It can also be checked that the sorting condition @ 2 v A @ @q A > 0 is satis…ed as We can therefore obtain A's best response by pointwise maximizing the virtual surplus function s A (q A ; ). Like the indirect utility function, the virtual surplus function is piecewise quadratic with a kink. The maximum can occur in either one of the two quadratic branches, or at the kink. To be precise: But these are precisely the monopoly, limit-pricing and common representation quantities de…ned in the main text. Note also that the case in which < 1 3 and 1 2 m cannot arise if c <c. In this case, the optimum is never achieved on the upper branch of the indirect utility function; in other words, …rm A's best response never involves setting the quantity at the monopoly level. It is therefore apparent that …rm A's best response is to o¤er precisely the equilibrium quantities. This can be achieved by o¤ering the equilibrium price schedules. This veri…es that …rm A's equilibrium price schedule satis…es the best response property.
Consider now …rm B: The procedure is the same as for …rm A, but now we must distinguish between two cases, depending on whether A's price schedule comprises the lowest monopoly branch or not.
Consider …rst the case in which there is no monopoly branch in A's price schedule. The indirect utility function of a buyer when trading with …rm B then is The …rst branch corresponds to …rm B acting as a monopolist. Along the second branch, …rm B competes with …rm A's limit-pricing price schedule. Clearly, neither case can occur in equilibrium, but these may be out of equilibrium outcomes which therefore must be considered. Finally, the third branch corresponds to the case in which …rm A accommodates. The coe¢ cients of the lower branches of the indirect utility functions arê All branches are concave, and global concavity can be checked by comparing the left and right derivatives of v B (q B ; ) at the kinks. The sorting condition can also be checked as for …rm A. We can therefore …nd B's best response by pointwise maximization of the virtual surplus function.
It is easy to verify that there is never an interior maximum on the upper or intermediate branch of the virtual surplus function. This is equivalent to saying that …rm B is active only when …rm A supplies the common representation quantity q cr A ( ). Pointwise maximization of the relevant branch of virtual surplus function then leads to This coincides with q cr B ( ), thereby con…rming that the equilibrium price schedule P B (q B ) is indeed its best response to …rm A's strategy.
The case where …rm A's price schedule comprises also the monopoly branch is similar. The indirect utility function v B (q B ; ), and hence the virtual surplus s B (q B ; ), now comprise four branches (all quadratic). The equation of the fourth branch, which corresponds to However, it turns out that the optimum still lies on the same branch as before and that it therefore entails a quantity equal to q cr B ( ). This observation completes the proof of the Proposition.
Proof of Proposition 4. The equilibrium quantities are: where^ and , which are de…ned in the text of the Proposition, are given bŷ The claim that this is the most cooperative equilibrium is justi…ed in the text above the Proposition. Here, we just verify that this is indeed an equilibrium of the game. The logic of the proof is the same as for Proposition 3. We must show that for each …rm the equilibrium price schedules satisfy the best response property. When calculating the best response, we take fP E i (q); P N E i (q)g as given and hence can invoke the Revelation Principle and focus on direct mechanisms. We must therefore show that for each …rm i = A; B the optimal quantities q i ( ) coincide with the equilibrium quantities reported above. It is then straightforward to conclude that the price schedules P E i (q); P N E i (q) that support these quantities must be equilibrium price schedules.
Given its rival's exclusive and non exclusive price schedules, a …rm must solve a monopolistic screening problem in which the buyer has an indirect utility function v i (q i ; ) = max and a reservation utility Since …rm i can impose exclusivity clauses, it must solve a hybrid optimal control problem in which the two control systems are It is then easy to verify that^ is now lower than 1 c, so the only relevant part of the solution is q e c ( ). Consider now problem (A.2). Several properties of this problem must hold for both …rms. By construction, the indirect utility functions v i (q i ; ) are almost everywhere di¤erentiable. At any point where the derivatives exist, by the envelope theorem we have Generally speaking, the indirect utility functions v i (q i ; ) have two branches, according to whetherq i (q i ; ) q i ( ) orq i (q i ; ) q i ( ) respectively. Whenq i (q i ; ) q i ( ), we have P N E i (q i ) = P cr i (q i ). Whenq i (q i ; ) q i ( ), we have P N E i (q i ) = P cr i (q i ) (plus a constant). The indirect utility functions v i (q i ; ) are continuous, almost everywhere di¤erentiable, and satisfy v i qi (q i ; ) > 0. Continuity and a.e. di¤erentiability follows directly from the de…nition of v i (q i ; ). To prove the sorting condition, observe that v i qi (q i ; ) = 1 @q i (q i ; ) @q i 0: Consider the two branches of the indirect utility function in turn. Whenq i (q i ; ) q i ( ), v i qi (q i ; ) = 1 + @q i (q i ; ) @ ( ) 2 = 3 6 3 5 > 0: When insteadq i (q i ; ) q i ( ) the sorting condition is immediately veri…ed since v qi (q i ; ) = 1 2 1 0: Now consider problem (A.2). Because of the type-dependent participation constraint, following Jullien (2000) we de…ne the virtual surplus function: where the "weight" g 2 [0; 1] accounts for the possibility that the participation constraint may bind for a whole set of types. Let i (g; ) = arg max qi 0 i (g; q i ; ) 49 be the maximizer of the virtual surplus function. This solution is still in implicit form, as it depends on the value of g, which is still to be determined. This can be done by exploiting Proposition 5.5 of Jullien (2000).
To apply that Proposition, we …rst prove the following lemma.
Lemma 8 Problem (A.2) satis…es the conditions of Potential Separation, Homogeneity and Weak Convexity.
Proof. Potential Separation requires that`i(g; ) is non-decreasing in . This follows from the fact that the virtual surplus function has increasing di¤erences. To show this, consider each branch of the indirect utility function separately. First, whenq i (q i ; ) q i ( ) we have The …rst term is positive, as we have just shown. The second term is positive because d d (g ) < 0 and 1 + @q i @q i = 1 2 3 5 > 0: Second, whenq i (q i ; ) q i ( ) the indirect utility function coincides, modulo a constant, with the one arising in the equilibrium with non-linear pricing. In this case, it is immediate to show that i qi (q i ; ) > 0. This completes the proof that problem (A.2) satis…es the condition of Potential Separation.
Homogeneity requires that U R i ( ) can be implemented by a continuous and non decreasing quantity. This is obvious, since U R i ( ) is implemented by q E ( ), where q E ( ) is the optimal quantity given the exclusive price schedule P E i (q): To prove Weak Convexity, we …rst show that`i(0; )+q i (`i(0; ); ) q E ( ) for all 2 [0; 1]. By de…nition, Thus,`i(0; ) is implicitly de…ned by the …rst order condition v i qi (q i ; ) + v i qi (q i ; ) = 0: Since v i qi (q i ; ) > 0, this implies that v i qi (q i ; ) < 0, or u qi (q i ;q i (q i ; ); ) < 0. In other words,`i(0; ) exceeds the satiation consumption u qi (q i ;q i (q i ; ); ) = 0. The quantity q E ( ), on the contrary, is lower than the satiation consumption. It follows that`i(0; ) +q i (`i(0; ); ) q E ( ).
In addition, Weak Convexity requires that the curve q E ( ) cuts the curvè i (1; ) +q i (`i(1; ); ) = q cr A ( ) + q cr B ( ) from above. Noting that`i(1; ) = q cr i ( ), the fact that q E ( ) can only cut the curve q cr A ( ) + q cr B ( ) from above as irrespective of whether q E ( ) is q e c ( ) or q E ( ). This …nally proves Weak Convexity and hence the lemma.
With these preliminary results at hand, let us now consider the dominant …rm's problem. The solution when q B ( ) = 0 has been already characterized. If q B ( ) > 0, Proposition 5.5 in Jullien (2000) guarantees that generally speaking the solution partitions the set of types into three sets: buyers who are excluded, buyers who obtain their reservation utility U R A ( ), and buyers whose payo¤ is strictly greater than U R A ( ). Clearly, the …rst set is always empty: if q B ( ) > 0, we always have q A ( ) > 0.
Next consider the second group of buyers. When the participation constraint binds, …rm A can guarantee to each low type consumer his reservation utility U R A ( ) in two ways. First, it can o¤er an exclusive price schedule that just undercuts that of …rm B. Alternatively, it can implement via non-exclusive prices the quantities that satisfy the condition q cr A ( ) + q cr B ( ) = q E ( ); which by the envelope theorem guarantees that the participation constraint is met as an equality. The maximum payment that …rm A can requested for q cr A ( ) is P cr A ( q cr A ( )) = c q cr A ( ) + (1 2 ) [ q cr A ( )] 2 + cq e c (^ ): The second strategy is at least as pro…table as the …rst one if P cr A ( q cr A ( )) P E ( q E ( )); which is precisely the no-undercutting condition (23) which holds by construction. This shows that o¤ering P cr A (q A ) is indeed a best response for …rm A when the participation constraint is binding.
Finally, when the participation constraint does not bind, the solution to …rm A's program is obtained simply by setting g = 1. Assume that`A(1; ) q cr A ( ) when >^ (this will be proven shortly). Since the virtual surplus function A (1; q A ; ) is exactly the same as in the non-linear pricing equilibrium, modulo a constant, the maximizers of the virtual surplus functions must coincide and the optimal quantity is`A (1; ) = q cr A ( ): Finally, the cuto¤ is implicitly given by the condition This also establishes that`A(1; ) q cr A ( ) when >^ . To complete the veri…cation of the best response property for …rm A, it remains to consider the switch from exclusive to non-exclusive contracts. By the no-deviation condition (23), which in the most cooperative equilibrium holds as an equality, …rm A is just indi¤erent between imposing an exclusivity clause or not for all . Exclusive dealing arises just when q cr B ( ) 0; which is equivalent to ^ . Because …rm A is indi¤erent between the exclusive and non-exclusive regimes, at the switching point a smooth-pasting condition must now hold, which implies that aggregate quantities must be continuous, and hence that P N E A ( q cr A (^ )) = P E A ( q E A (^ )). The problem faced by …rm B is similar, except that …rm B can never make a pro…t by selling under an exclusivity clause. Thus, we can focus on problem (A.2). Proceeding as for …rm A, one can show that the optimal quantity is q cr B ( ) when the participation constraint U ( ) U R A ( ) is binding, and q cr B ( ) when it is not.
These arguments complete the proof that the solution to the problem of …rm i coincides with q i ( ) as shown in the text of the proposition. By construction, this solution can be implemented by …rm i using the equilibrium price schedules The strategy of the proof is the same as for Proposition 4. Many of the arguments are indeed the same as in previous proofs and so need not be repeated here. In particular, notice that: …rst, when c > c, there is no longer any scope for coordinating exclusive prices (this was shown in the proof of Proposition 4). Hence, …rm B always sets exclusive prices at the competitive level P E B (q B ) = cq B . This implies that when …rm A imposes an exclusivity clause, the buyers' reservation utility is exactly the same as in the competitive fringe model. It follows that the solution to problem (5) is still given by Lemma 6; second, without exclusivity the problems that are faced by the …rms are exactly the same as in the proof of Proposition 4 when the participation constraint does not bind.
These remarks imply that Proposition 5 can be proved simply by combining arguments already presented in the proofs of Proposition 2 and Proposition 4. The only di¤erence is that now the switch from the exclusive to the non-exclusive regime is the result of the interaction between the pricing choices of …rm A and …rm B. This point, however, has already been discussed in the main text, which shows that the equilibrium switching point must satisfy conditions (27)-(29). The explicit expressions for A and B are complicated and are reported in a Mathematica …le that is available upon request from the authors. In any case, it can be veri…ed that A > 0 and B < 0; as argued in the main text.