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1 ECONOMIC GROWTH CENTER YALE UNIVERSITY P.O. Box New Haven, CT htt:// CENTER DISCUSSION PAPER NO. 844 COMPETITION IN OR FOR THE FIELD: WHICH IS BETTER? Eduardo Engel Yale University Ronald Fischer University of Chile and Alexander Galetovic University of Chile March 2002 Note: Center Discussion Paers are reliminary materials circulated to stimulate discussions and critical comments. This aer can be downloaded without charge from the Social Science Research Network electronic library at: htt://aers.ssrn.com/abstract= An index to aers in the Economic Growth Center Discussion Paer Series is located at: htt://

2 COMPETITION IN OR FOR THE FIELD: WHICH IS BETTER? BY EDUARDO ENGEL, RONALD FISCHER AND ALEXANDER GALETOVIC 1 March, 2002 Abstract In many circumstances, a rincial, who wants rices to be as low as ossible, must contract with agents who would like to charge the monooly rice. This aer comares a Demsetz auction, which awards an exclusive contract to the agent bidding the lowest rice (cometition for the field) with having two agents rovide the good under (imerfectly) cometitive conditions (cometition in the field). We obtain a simle sufficient condition showing unambiguously which otion is best. The condition deends only on the shaes of the surlus function of the rincial and the rofit function of agents, and is indeendent of the articular duooly game layed ex ost. We aly this condition to three canonical examles rocurement, royalty contracts and dealershis and find that whenever marginal revenue for the final good is decreasing in the quantity sold, a Demsetz auction is best. Moreover, a lanner who wants to maximize social surlus also refers a Demsetz auction. Key words: Demsetz auction, double marginalization, franchising, joint vs. searate auctions, monooly, rocurement, dealershis, royalty contracts JEL classification: D44, L12, L92 1 Engel: Deartment of Economics, Yale University, 28 Hillhouse Ave., New Haven, CT Fischer and Galetovic: Center for Alied Economics (CEA), Deartment of Industrial Engineering, University of Chile, Av. Reública 701, Santiago, Chile. s: eduardo.engel@yale.edu, rfischer@dii.uchile.cl, agaleto@dii.uchile.cl. Financial suort from FONDE-CYT (Grant ) is gratefully acknowledged.

3 1 Introduction Consider a rincial that needs to buy a good or service and has two rocurement alternatives. On the one hand, it can award an exclusive contract to the agent offering the lowest rice, as in a standard Demsetz (1968) auction. In this case, there will be intense ex ante rice cometition ( cometition for the field ), but in the aftermath, the agent will always charge the maximum rice allowed by the contract. The other otion is to award two indeendent suly contracts and rely on ex ost cometition ( cometition in the field ). Then agents will consider the robabilities of different ex ost duooly games when negotiating their contracts. If they anticiate that with high robability there will be intense rice cometition, they will articiate only if they obtain high rofits in the rare cases in which they collude. Thus, the rosect of intense cometition in the field softens cometition for the field. 2 This motivates the question we address in this aer: Should the rincial contract with one or two agents? The answer does not seem straightforward. A basic difficulty is that the aroriate secification of the ex ost duooly game deends on articular asects of the situation. 3 Moreover, many duooly games have multile equilibria and there are no a riori comelling reasons to choose one over another. Nevertheless, in this aer we obtain a simle sufficient condition that shows unambiguously which otion is better. This condition deends only on the shaes of the surlus function of the rincial and the rofit function of each agent, and is indeendent of the duooly game layed ex ost. We then aly this condition to study three canonical examles rocurement, royalty contracts and dealershis. We find that whenever marginal revenue is decreasing in the quantities, a Demsetz auction is unambiguously better. Moreover, a lanner who wants to maximize social surlus also refers a Demsetz auction. Our oint of dearture is a standard setting where a rincial wants the final rice of the good or service to be as low as ossible, but agents refer the monooly rice. The rincial can either run a Demsetz auction for an exclusive contract, or auction two searate contracts to different agents who then roduce erfect substitutes and comete. With a Demsetz auction, the ex ost equilibrium rice equals the winning bid. By contrast, when there are two contracts, the rice deends on the outcome of ex ost cometition. We do not secify the second stage game, but summarize its outcome as follows: ex ante the equilibrium rice is a random variable with an arbitrary distribution F whose suort is bounded above by the maximum rice allowed by the contract. In some states of nature agents will succeed in colluding and rices will be close to the winning bid; in other states agents will comete intensely and rices will be much lower. 4 We assume that the rincial and the agents are risk neutral. Nevertheless, the main result of the aer exloits the fact that a Demsetz auction eliminates variability in the equilibrium rice. To get the intuition 2 This terminology is due to Chadwick (1859). 3 In this aer we abstract from comlications due to incomlete contracting and asymmetric information. 4 There are several interretations for F. In one of them, it describes agents uncertainty about ex ost market conditions and otential collusive rices. In another, firms always collude at the rice that maximizes joint rofits, but there is a ositive robability of a successful antitrust case against them, leading to a rice equal to marginal costs. Similarly, in the secific case of dealershis, the ustream firm may try to revent the double marginalization associated with collusion by enalizing those agreements between franchisors that are detected. 2

4 in a simle setting, assume that the agents rofits, π, are increasing and concave in the equilibrium rice, and that the rincial s surlus function, S, is decreasing and linear in the equilibrium rice. 5 Consider next what haens if the rincial substitutes two indeendent contracts by a Demsetz auction. Clearly, this eliminates the rice variability described by F. And since π is concave in rice and the articiation constraint always binds, the rice that results from a Demsetz auction must be lower than the average with two agents and ex ost cometition, i.e., cometition for the field leads to lower rices than cometition in the field. Thus, a Demsetz auction is better for the rincial when her surlus function is linear. It is easily seen, as well, that if the surlus function S is sufficiently convex, searate contracts may be better, because then the rincial likes rice variability. Our main result generalizes this intuition and shows that a Demsetz auction is unambiguously better when the comosition of the rincial s surlus function and the inverse rofit function, S π 1, is strictly concave. Conversely, when this comosite function is strictly convex two searate contracts are unambiguously better. As in the theory of exected utility, we find that this general result is equivalent to a simle condition that comares the curvatures of the surlus and rofit functions. This condition is quite similar to the necessary and sufficient condition for a utility function to be more risk averse than the other (Pratt s [1964] theorem) and makes it easier to comare cometition in the field with cometition for the field. The condition amounts to checking a relation that involves only the first and second derivatives of S and π. We illustrate the usefulness of this condition in the alications section, showing that a Demsetz auction is referred by the rincial in all cases considered rocurement, royalty contracts, dealershis whenever marginal rofits are decreasing in quantities. Our aer is related to the literature of monooly regulation via franchising which was ioneered by Chadwick (1859) and Demsetz (1968) (see also Stigler [1968], Posner [1972], Williamson [1975], Riordan and Saington [1987], Sulber [1989, ch. 9], Laffont and Tirole [1993, chs. 7 and 8], Harstad and Crew [1999] and Engel, Fischer and Galetovic [2001 a, b]). We extend this literature by studying Demsetz auctions in contexts where imerfect cometition in the field is feasible and is an alternative to a standard Demsetz auction. The alications we study suggest that our aer is also related to the literature on the double marginalization roblem in monooly ricing (see Sengler [1950] for the seminal contribution and Tirole [1988, ch. 4] for a review of the literature). Our result imlies that when marginal revenue is decreasing in the quantity sold and downstream cometition is imerfect, auctioning an exclusive contract is better than relying on ex ost imerfect cometition. The rest of the aer roceeds as follows. In Section 2 we describe the general model and rove the main result of the aer. In section 3 we aly this general result to study four alications. Section 4 concludes 5 While we assume that rofit and surlus functions are linear in money (that is, they are risk neutral in money terms), neither the agents rofit function nor the rincial s surlus function need, in general, be linear in the equilibrium rice, i.e., they are risk averse (or loving) in rices. For examle, rofit functions are tyically quasiconcave in rice. By contrast, when the agent is a lanner who wants to maximize consumer surlus, the rincial s objective function is convex in rices. 3

5 and is followed by a brief aendix. 2 General model and main result A risk neutral rincial wants to contract the roduction of a good at two lants or locations. 6 Outut from one lant is a erfect substitute for the outut of the other. If the equilibrium rice is, an agent roducing at one lant makes rofits π(), with π () > 0 for [, m ), where π() = 0 and m = argmax π(). Furthermore, π ( m ) = 0 and π ( m ) < 0. On the other hand, the rincial s surlus is S() if agents charge, with S () < 0. Hence there is a conflict of interest: while agents would like to increase rices u to m, the rincial wants the rice to be as low as ossible. The rincial may award both lants jointly (J), so that they are run by one agent; or searately (S), so that two agents run one lant each and comete. The rincial auctions both contracts. When both lants are awarded jointly, the winning bid is denoted by J and er-lant rofits for the agent are equal to π( J ). 7 On the other hand, when lants are awarded to different agents, the minimum winning bid, common across lants, is denoted by S. In this case agents are uncertain both about whether they will be able to collude, 8 and, if they do, about the rice above at which they will collude. 9 We assume that each agent serves half the demand at a common equilibrium rice, and denote by F() the cdf with suort [, S ] that describes their common beliefs about the realization of this rice. 10 We make the essential assumtion that ex-ante exected gross rofits er lant under a joint or a searate auction are the same, that is Condition 1 S π()df() = π( J ) = u + I, where u is the agent s reservation utility and I stands for any sunk setu cost. There exist many agents that could roduce the good, all of them with the same value of (u + I). Condition 1 imlies that benefits for agents are indeendent of whether the rincial auctions roduction at both lants jointly or searately. Or, in the standard guise of rincial-agent theory, Condition 1 is the articiation constraint that the rincial must obey. Note also that if F() is degenerate, Condition 1 imlies that under searate auctions the rice will be J, so that joint and searate auctions are identical. We rule out this ossibility by assumtion in what follows. 6 All that follows extends trivially to the case of n locations. 7 We assume J m. 8 Caillaud and Tirole (2001) consider this ossibility in the context of essential facilities. 9 That is, we assume that rices are such that agents do not lose money ex ost, since π() = Cometition in ractice is generally neither static nor symmetric. We avoid comlications by concentrating on stationary equilibria and we use symmetry due to the lack of consensus on how to model collusion in asymmetric games. 4

6 When lants are awarded jointly, the rincial s benefit is denoted W J S( J ). On the other hand, when when they are awarded searately, the rincial s exected benefit deends on the distribution of collusive rices F and equals S W S S()dF(). From the assumtions we made on π, it follows that π 1 : [π(),π( m )] [, m ] is well defined, increasing and convex. We then have the following central result of the aer: Proosition 1 If S π 1 is strictly concave, then W J > W S. If S π 1 is strictly convex, then W S > W J. Proof: We consider the case where S π 1 is concave. The case where it is convex is analogous. We have: W S = S S S()dF() < S π 1 [ S S π 1 [π()]df() = S π 1 [π( J )] = S( J ) W J, ] π()df() where the inequality follows from Jensen s inequality and our assumtion that F is not degenerate, and the identity after the inequality from Condition 1. A surrising feature of this result is that we have not imosed any condition on the distribution of ossible collusive outcomes F. Hence, in order to comare joint and searate auctions, it is sufficient to examine the rimitive functions π, and S, and one can ignore the exact secification of the ex ost game between the agents. This result deends crucially on Condition 1, which ensures that softer cometition when the articiation constraint becomes more demanding (that is, ū + I increases). In the case of joint roduction, this means softer cometition for the franchise, while in the case of searate roduction it means less cometition between both agents after they begin roducing. The following result rovides a simle characterization for the concavity of S π 1. Proosition 2 A necessary and sufficient condition for S π 1 be concave is that (1) S S > π π 5

7 for all [, m ). Since, by assumtion, π > 0 and S < 0 in the relevant range, equation (1) is equivalent to (2) S π < S π. Moreover, the converse of condition (1) is necessary and sufficient for S π 1 to be convex. Proof: See the Aendix. Corollary 1 If π is strictly concave, then the concavity of S is sufficient for a joint contract to be better than two searate contracts. We can use Proosition 2 and Figures 1 and 2 to examine the intuition underlying our main result. Suose that S is linear, π strictly concave, two searate contracts are auctioned, and in equilibrium can take only two values, and m, with equal robability. In this case each agent makes exected rofits equal to 1 2 π() π( m) = 1 2 π( m) and the rincial s surlus equals 1 2 S() S( m) (see Figure 1). Since S is linear and π concave, Proosition 2 holds, and a joint auction is better than a searate auction. Why? Condition 1 imlies that 1 2 π( m) = π( J ). As is straightforward from Figure 1a, concavity of π imlies that J < m. Hence the rincial obtains a lower average rice with a joint auction. 11 Because in this examle S is linear, 1 2 S()+ 1 2 S( m) = S( m) < S( J ) (see Figure 1b). Note that the same reasoning alies to any robability distribution F with suort in the interval [, m ]. It can now easily be seen why strict concavity of S is sufficient for a joint auction to be better when π is concave. Eliminating variability in is an added bonus for the rincial, since ES() < S(E) for all distributions F. Conversely, when S is convex, a searate auction may (but need not) be better. Figure 2 deicts exactly the same case as Figure 1, excet that S is convex, so that now the rincial likes rice variability. For the articular distribution deicted in this figure, the rincial is indifferent between a joint and a searate auction. Essentially, the gain of a lower exected rice attained with a joint auction is exactly offset by the fall in the exected surlus due to lower rice variability. With S sufficiently convex and for a given π, the gains from a lower exected rice are outweighed by the utility loss which stems from losing high surluses. 3 Alications In this section we use Proosition 2 to study three canonical alications: rocurement (the rincial buys the roduction of the lants), dealershis (agents buy an inut from the rincial and incur some costs to transform and resell it) and royalties (the rincial receives a fixed fee er unit sold by the agent without 11 This can be ut in the more standard terms of rincial agent theory. From Condition 1 it follows that the agent s articiation constraint is 1 2 π( m) = π( J ). Since π is concave, the average rice that the agent requires in order to articiate is lower with a joint contract, which eliminates risk. 6

8 π π( m ) π( J ) = 1 2 π( m) (a) J ( + m )/2 m S S( m ) S( J ) (b) S() J ( + m )/2 m Figure 1: Intuition underlying the main result: When S is linear a joint auction is always better. 7

9 π π( m ) π( J ) = 1 2 π( m) (a) J ( + m )/2 m S S( m ) S( J ) (b) S() J ( + m )/2 m Figure 2: Intuition underlying the main result: When S is convex, anything goes. 8

10 engaging in roduction). In these cases functions S and π can be derived from standard demand and cost functions. In all the cases that follow we assume that the value of the marginal unit at q is P(q), with P < 0. We also assume that the inverse function of P, P 1 () D() is well defined in the aroriate range. Obviously D < 0. Agent i incurs in total cost c(q i ) when roducing q i units of outut at a given lant, with c > 0 and c We find that in all three cases a sufficient condition for a joint contract to be better for the rincial is that marginal revenue be decreasing in quantities. In addition, we show that whenever this holds, a social lanner also refers a joint auction. The following two lemmas, which are roven in the aendix, will be useful when establishing this result: Lemma 1 (i) P (q) = 1 D (P(q)) ; (ii) P (q) = D (P(q)) {D (P(q))} 3 ; (iii) D () = 1 P (D()) ; (iv) D () = P (D()) {P (D())} 3. Lemma 2 2P (q) + qp (q) < 0 if and only if DD 2(D ) 2 < Procurement We first consider fixed-rice rocurement. 13 The rincial wants to buy an inut as chealy as ossible, and can choose between one or two suliers. Clearly, the rincial cares (directly) only about the rice aid er unit, and not about roduction costs c (of course, as in any rincial-agent roblem, the rincial cares about the agents costs indirectly through the articiation constraint). Hence S() D(s)ds is the ( ) rincial s surlus, and π() = 1 2 D() c D() 2 is the surlus of each agent with a searate auction; with { ( )} a joint auction the agent s surlus is 2π(). In this case m = argmax 1 2 D() c D() 2 and is such that 1 2 D() c ( D() 2 ) = 0. Therefore S = D < 0, S = D > 0 (i.e. S is convex and Corollary 1 does not aly). Also, π = 1 2 [D + ( c )D ], 12 This condition imlies no loss of generality. If c < 0, marginal and average costs are decreasing and auctioning jointly is clearly better. 13 In fixed-rice rocurement contracts, the buyer and the seller agree on a rice, and the seller assumes all cost risk. At the other extreme, in a cost-lus contract the buyer reimburses the seller s cost. As argued in Laffont and Tirole (1993,. 662), only fixed-rice contracts are relevant when it is too costly for the buyer to audit the subcost of the sulier. 9

11 which we assume strictly ositive for [, m ), and π = 1 2 [( c )D + 2D 12 (D ) 2 c ]. Alying Proositions 1 and 2 to this case, the following result follows: Proosition 3 A sufficient condition for W J > W S is that (3) 2P + qp < 0, that is, that marginal revenue be decreasing in q. Proof: Since in the relevant range we have S < 0 and π > 0, it follows from Proosition 2 that it suffices to show that (3) ensures that (4) S π < S π for all [, m ). Some straightforward calculations show that (4) is equivalent to: (5) ( c )[D D 2(D ) 2 ] < 2π D (D ) 2 Dc. Lemma 2 imlies that the left hand side of (5) is negative for all [, m ] if marginal revenue is decreasing in q. On the other hand, the right hand side of (5) is ositive, because c 0 and π > 0 for all [, m ). 3.2 Royalties Consider a licensing agreement where the licensee ays the rincial a royalty t er unit roduced and sold, but no fixed fee. 14 For examle, this is the case when a atent holder licences the right to manufacture the good, but does not articiate in the roduct market. 15 In this case the rincial is worried about downstream double marginalization, and, given t, would like the licensee to sell as much as ossible. The rincial s surlus is S() = td() (where now D is market demand for the good), and π() = 1 2 ( t)d() c(d()/2) is the surlus of each licensee. We then have that Proosition 3 also alies, i.e. 2P + qp < 0 is sufficient for W J > W S. We ostone the roof until the next subsection. As is well known, a disadvantage of licensing through royalties is that any market ower exercised downstream reduces industry rofits (this is the double marginalization roblem). One solution is to auction 14 Calvert (1964) and Taylor and Silberston (1973) observe that about 50% of all licensing contracts secify royalties only. Also, Lafontaine and Shaw (1999) show that, on average, franchise fees amount to no more than 8% of actual ayouts from franchise holders to franchisees. 15 See Tirole (1988, ch. 10.8) for a review of the literature on licensing. 10

12 off the monooly for a fixed fee and charge no royalty. 16 In that case, the rincial does not want cometition downstream. When fixed lum-sum fees are not feasible, the atent holder must make her income through royalties. But then the double marginalization roblem bites, and controlling market ower downstream becomes an issue. Our result imlies that decreasing marginal revenue is sufficient for a Demsetz auction (ex ante cometition) to be better than ex ost cometition when ex ost market structure is uncertain. One could argue that an exclusive contract with a two art tariff is enough to revent the double marginalization roblem. Note, however, that to choose the right fixed fee the rincial must know the demand curve. Our analysis imlies that when marginal revenue is decreasing in quantities a joint franchise solves the roblem. 3.3 Dealershis Dealershis are similar to licensing, excet for the fact that the rincial s cost increases with the number of units sold. For examle, consider the case of car dealershis. Cars are rovided by the manufacturer at a fixed rice and the dealers are free (within limits set by list rice of the manufacturer) to bargain their marku with clients. The question for the manufacturer then is whether to have dealershis that are, say, satially close and thus comete with each other, or to have one dealer with a ca on the resale rice. 17 Assume that c(q) is the rincial s cost function, with c, c > 0. Then S() = td() c(d()), and, as with licensing, π() = 1 2 ( t)d() c(d()/2). Then:18 and S = (t c )D, S = (t c )D (D ) 2 c ; π = 1 2 [D + ( t c )D ], π = 1 2 [D + (1 c D ) D + ( t c )D ]. 2 We now show that 2P + qp < 0 is again sufficient for W J > W S. As before, since π > 0 for all [, m ), it follows, after some straightforward but tedious algebra, that S π < S π for all [, m ) is equivalent to (6) DD 2(D ) 2 < 2π (D ) 2 c (t c ) 1 2 (D ) 3 c, which holds because the right hand side is (obviously) ositive, while the left hand side is negative because 16 See Gallini (1984) and Katz and Shairo (1985). 17 We abstract from other imortant considerations in these contracts, such as service quality. 18 Note that royalties corresonds to the case where c 0. 11

13 of Lemma The social lanner Next, we consider the case of a social lanner who wants to contract for the rovision of a service. As an examle, consider highway franchises. In many develoing countries roads are being franchised to rivate firms. In exchange for toll revenue, the franchise holder finances, builds, oerates and maintains the road. 19 In some cases roads are natural monoolies, and are awarded to the firm offering to charge the lowest toll. Nevertheless, when there is more than one way to get from one oint to another, as is often the case in large cities, different roads could be awarded to different franchise holders. 20 Should the regulator award all franchises to the same firm or award several highway franchises and let them comete? Another examle is the auction of the rights to rovide local telehony in rural areas, where the auction is based on the rice of a standard local call. Is it better to have a single comany rovide the service or would it be better to allocate two comanies to the area? In this case the rincial cares about social surlus (i.e. consumer and roducer surlus) so that S() = ( ) [ ] D(s)ds + D() 2c D() 2 ; and π() = 1 2 D() c D() 2. Then S = ( c )D, S = ( c )D + D 1 2 (D ) 2 c, and π, π are the same as in the case of dealershis (with t = 0). Condition (1) now leads to which, after some algebra yields ( c )D + D 1 2 (D ) 2 c ( c )D > ( c )D + 2D 1 2 (D ) 2 c D + ( c )D, (7) ( c ) [ DD 2(D ) 2] < 2π D (D ) 2 Dc, which is the same condition as in the case of rocurement. Hence, again decreasing marginal revenue is sufficient for W J > W S. Finally, we resent a concrete examle which suggests that welfare gains may be imortant when using a Demsetz auction instead of searate auctions. Examle We use the notation and definitions from the receding subsection and assume D() = 1, 19 See Engel, Fischer and Galetovic (2001b) for a discussion of highway franchising. 20 One examle is the La Dormida roject in Chile, which would comete with Route 68, the highway that currently joins Chile s caital, Santiago, with the ort of Valaraíso. The Ministry of Public Works weighed the benefits of joint and searate auctions. 12

14 c(q) = c 0. We also assume that the searate and joint auction of both franchises dissiate all rents. 21 Since S () = D() = 1 and π() = (1 ) c 0, we have that π /π < S /S if and only if 2/(1 2) < 1/(1 ) for all < m = 1/2, which indeed holds. It follows that the sufficient condition for Proosition 2 is satisfied. Assume that the ex-ante distribution of market structure outcomes in the case of searate franchises F is a oint distribution with weights of 1/2 on collusion at the bid rice and 1/2 on rice cometition with rice equal to marginal cost (zero). Then J is the smallest solution to (1 ) = c 0 while S is the smallest solution to 1 2 (1 ) = c 0. The existence of a solution in both cases requires c 0 < 1/8. Then: W J W S = 1 2 (1 J) { (1 S ) } = 1 8 [2 1 4c c 0 ] > 0, c 0 > 0, Furthermore, in this examle a joint auction can lead to a welfare increase of as much as 17%. 4 Conclusion We have shown a simle condition for a rincial to refer to contract the rovision of a good from a single agent via a Demsetz auction, rather than by having multile agents rovide the good under (imerfectly) cometitive conditions. In the canonical cases of rocurement, royalty contracts and dealershis, decreasing marginal revenue ensures that a Demsetz auction (ex ante cometition) is better for the rincial than ex ost cometition. This result is surrising, because it is indeendent of the exected intensity of ex ost cometition. The results in this aer do not necessarily imly an endorsement of monoolies, since many relevant factors were left out of our analysis. First, we assumed a single service quality, which can be verified at no cost, even though in most cases quality will be worse in the absence of cometition. Second, we ignored olitical economy and asymmetric information considerations, which may be worse when a regulator deals with a monooly. Third, we rule out incomlete contracting and the hold-u roblem. For examle, a manufacturer might refer to have cometing dealers in order to avoid a bilateral monooly. Finally, we have not considered the ossibility of technical change in the delivery of franchise services, a factor that if resent makes cometition more desirable if it accelerates the introduction of new technologies. On the other hand, there are some asects we left out which strengthen the case for a joint auction. First, if agents are risk averse, the reference for joint auctions increases. More imortantly, we have assumed that a joint contract does not lead to any cost savings; or, conversely, that ex ost cometition, which imlies more than one agent, does not lead to (fixed) cost dulication. A common concern when formerly monoolistic markets are liberalized is that cometition may lead to inefficient cost dulication through excessive entry. 22 With a Demsetz auction, however, cost dulication is no longer an issue. Our result 21 Thus the common value of both exressions in Condition 1 is zero. 22 See, for examle, Armstrong (2000). 13

15 indicates that an exclusive contract may be referable even when it does not revent fixed-cost dulication. 14

16 Aendix A Proof of Proosition 2 Differentiating both sides of π π 1 () = leads to: (8) (π 1 ) () = 1 π (π 1 ()). Hence: (9) (S π 1 ) () = S (π 1 ())(π 1 ) () = S (π 1 ()) π (π 1 ()). Differentiating both sides of (8) with resect to and using (9) leads to: (S π 1 ) () = S π S π (π ) 3, where all terms on the r.h.s. are evaluated at π 1 (). Since π > 0 this imlies that S π 1 is concave if and only if S π < S π. And since S < 0 and π > 0 we conclude that π /π < S /S is necessary and sufficient for concavity of S π 1. The result now follows. B Proof of Lemma 2 To rove (i) and (ii) totally differentiate the identity q D[P(q)] with resect to q to obtain 1 = D P, from which (i) follows. Next, totally differentiate P (q) 1 D [P(q)] with resect to q to obtain P = D (D ) 2 P, from which (ii) follows by substituting 1 D for P. The roof of (iii) and (iv) is analogous and we omit it. C Proof of Lemma 3 Proof: Sufficiency: use (i) and (ii) in Lemma 1 to substitute for P and P. Necessity: use (iii) and (iv) to substitute for D, D and D. 15

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