A Theory of Buyer Fragmentation: Divide and Conquer Intensifies Competition

Transcription

1 TSE 543 November 2014 A Theory of Buyer Fragmentation: Divide and Conquer Intensifies Competition Doh Shin Jeon and Domenico Menicucci

2 A Theory of Buyer Fragmentation: Divide-and-Conquer Intensifies Competition Doh-Shin Jeon and Domenico Menicucci November 26, 2014 Abstract It is well-known from Innes and Sexton (1993, 1994) that divide-and-conquer contracts allow an incumbent facing a potential entrant to extract more surplus from buyers and hence buyers suffer from the strategy. In this paper, we show that when sellers compete by offering personalized non-linear tariffs, divide-and-conquer strategies intensify competition among sellers in the presence of indirect contracting externalities. Therefore, buyers prefer remaining fragmented to forming a buyer group. When buyer group formation is decided before the entry of an entrant, our result implies that buyers may deliberately induce a socially suboptimal entry by remaining fragmented in order to benefit from more intense competition upon the entry. Keywords: Divde-and-Conquer, Buyer Group, Indirect Contracting Externalities, Common Agency, Competition in Non-linear Tariffs, Multimarket Contact. JEL numbers: D4, K21, L41, L82 We thank Zhiun Chen, Jay Pil Choi, Vincenzo Denicolo and the audience who attended our presentations at EARIE, EEA-ESEM, Postal Economics Conference on "E-commerce, Digital Economy and Delivery Services" (Toulouse, 2014), Yonsei University. Toulouse School of Economics (GREMAQ, IDEI) and CEPR. dohshin.eon@gmail.com Università degli Studi di Firenze, Italy. domenico.menicucci@dmd.unifi.it

3 1 Introduction In many sectors, buyers purchasing multiple products can form a buyer group. Examples of buyer groups abound in retailing, health care, agriculture, academic ournals, etc. In addition, existing buyer groups can expand their size by merging with other groups. Understanding how buyer group affects buyer power is very important given the increasing policy makers concerns about buyer power (European Commission, 1999 and OECD, 2008). 1 In this paper, we study how the formation of a buyer group affects the total payoff of the buyers and each seller s payoff. We investigate this question in a situation when multiple sellers compete by offering personalized non-linear tariffs and each buyer (and buyer group) is interested in buying a positive quantity from each seller. More precisely, the case of buyer group is modelled as a common agency à la Bernheim and Whinston (1986, 1998). For the case of no buyer group, we consider a natural extension of the common agency to multiple buyers in which each buyer operates in a separate market. We show that buyers prefer to remain fragmented since then each seller can use divide-andconquer strategies, which intensify competition among sellers in the presence of indirect contracting externalities among buyers. 2 We consider a model of two sellers and two buyers. In the absence of the buyer group, the indirect contracting externalities between the buyers exist if the modification of the vertical contract between seller and buyer i creates an incentive for seller to modify its contract offer to buyer h ( i). For instance, when the sellers costs are convex, the contracting externalities exist. In addition, without buyer group, seller can deviate with a divide-and-conquer strategy, which means that he induces one buyer to buy exclusively from seller, but induces the other buyer to buy from both seller and seller k ( ). 3 When seller k s offer is given, sometimes a divide-and-conquer strategy allows seller to obtain a strictly higher profit than the profit it can realize otherwise. This captures the conventional wisdom that a monopolist facing buyers with exogenously given outside offers can realize a higher profit with the divide-and-conquer strategy (Innes and Sexton, 1993, 1994). However, when sellers compete, seller k s offer cannot be an equilibrium offer if seller s best response consists in divide-and-conquer. 4 1 In US, the Federal Trade Commission organized a workshop on slotting allowances in 2000, a maor buyer power issue in grocery retailing. See Chen (2007) for a survey of the literature on buyer power and antitrust policy implications. 2 Note that in our model, there is no direct contracting externalities among buyers since each buyer operates in a separate market. 3 We use "he" for a seller and "she" for a buyer. 4 We assume that social welfare maximization requires each buyer to buy a positive quantity from each seller and show that all equilibria are effi cient regardless of whether or not the buyers form a group. 1

4 Hence in equilibrium seller k provides a more attractive offer which can resist the divideand-conquer strategy of seller, and the same reasoning applies to seller s offer with respect to the divide-and-conquer strategy of seller k. Therefore, divide-and-conquer strategies intensify competition between the sellers. In order to benefit from this intense competition, the buyers have an incentive to remain fragmented. However, if there are no indirect contracting externalities, which occurs if the marginal costs are constant, there is no link between buyer i s market and buyer h( i) s market as long as the sellers sell substitutes. Therefore, in this case, divide-and-conquer strategies have no bite and buyer group affects no player s payoff with respect to no buyer group. Examples in which buyers (or buyer group) buy multiple products from multiple sellers and sellers compete by offering personalized non-linear tariffs abound. Hospitals (or groups of hospitals) buy different drugs or vaccines from multiple pharmaceutical firms, and the latter often offer different non-linear tariffs to different hospitals (or cooperatives). 5 Similarly, in the case of the market for academic ournals, each library (or library consortium) buys subscriptions to distinct academic ournals from multiple publishers, and publishers provide personalized prices to different libraries (or library consortia). 6 As long as each buyer participating in a buyer group is a firm, it is very natural to consider that each buyer buys multiple inputs from multiple sellers, and that these sellers compete by offering personalized non-linear tariffs. However, the existing literature on buyer group studies only situations in which each buyer or buyer group buys exclusively from one among competing sellers: see in particular Inderst and Shaffer (2007) and Dana (2012). 7 Although exclusive purchase is often practiced, non-exclusive purchase should be the norm when firms buy diverse inputs which are not very close substitutes. We add some novel insights to the literature on buyer group by analyzing the situation of non-exclusive purchases. For the most part of the paper, we assume that the two products offered by the sellers are strict substitutes (and we consider independent products as a benchmark and in Section 6.3 allow for complements). We consider tariffs such that what buyer i pays to seller depends only on the quantity that the buyer purchases from seller, not on the quantity it purchases from seller k ( ). In addition, we assume that each buyer s 5 See for instance the French antitrust case against GlaxoSmithKline France (Autorité de Concurrence, 2007) in which GlaxoSmithKline offered different non-linear prices to different hospitals or groups of hospitals. In the U.S., Tyco Health Care has been accused of entering into contracts requiring miniumshare with group-purchasing organizations (Masimo v. Tyco Health Care, Case No. CV , Lexis (CD Cal. Mar 2006). 6 According to Derk Haank (2001), CEO of Elsevier Science What we are basically doing is to say that you pay depending on how useful the publication is for you - estimated by how often you use it. 7 An exception is our paper on the library consortium (Jeon and Menicucci, 2014): see the literature review in Section 1.1 for more details. 2

5 utility function is strictly concave and each seller cost function is either linear or strictly convex. In the case of buyer group, we start with the sell-out equilibrium, which achieves the effi cient allocation (Bernheim and Whinston, 1986, 1998). We also show that there exists an effi cient two-part tariff equilibrium and, furthermore that any convex combination of the sell-out strategy and the two-part tariff constitutes an equilibrium tariff. We remind the result from Bernheim and Whinston (1998) that in no equilibrium a seller can obtain a payoff strictly higher than the payoff he obtains in the sell-out equilibrium. We also show that under our assumption of concave utility and convex cost, all equilibria are effi cient. 8 In the case of no buyer group, we first show that all equilibria are effi cient and that in no equilibrium a seller can obtain a payoff strictly higher than the payoff he obtains in the sell-out equilibrium under buyer group. These results imply also that in no equilibrium the buyers can obtain a total payoff strictly lower than the payoff they obtain in the sell-out equilibrium under buyer group. Given these first results on the case of no buyer group, we first show two different conditions for buyer group neutrality (i.e., buyer group affects neither any seller s payoff nor the total payoff of the buyers). First, when the products have independent values, each seller is a monopolist. Then each seller can realize the monopoly profit regardless of whether or not the buyers form a group. Second, when costs are linear, there is no indirect contracting externality in the sense that when seller and buyer i modify the quantity they trade, it has no impact on the marginal cost of seller and hence seller has no incentive to modify its contract offer to buyer h ( i). Therefore each buyer s market can be studied in isolation. The main case of interest is when the products are strict substitutes and the sellers costs are strictly convex. In this case, there exist indirect contracting externalities. We first show that the sell-out equilibrium does not exist because of the divide-and-conquer strategy which exploits the indirect contracting externalities. However, an effi cient twopart tariff equilibrium always exists. As the main result, we show that each seller s highest equilibrium payoff without buyer group is strictly lower than the payoff in the sell-out equilibrium under buyer group, which implies that the lowest total equilibrium payoff of the buyers is strictly larger than their payoff in the sell-out equilibrium under buyer group. To gain further insight, we consider the setting of symmetric buyers and show that 8 This is because in any equilibrium, given the output that the buyer group buys from seller k( ), seller chooses its output to maximize the oint surplus from the trade with the group. Therefore, the first-order conditions for each seller s profit maximization problem are identical to those of a benevolent social planner, and they are suffi cient for maximization of social welfare given our assumptions of concave utility functions and convex cost functions. 3

6 starting from any equilibrium tariffs under buyer group, we can find tariffs that implement the same outcome as an equilibrium without buyer group if divide-and-conquer strategies are prohibited. This shows that it is essentially the divide-and-conquer strategy that creates the difference between the case with buyer group and the case without group. In addition, in the symmetric setting (i.e., buyers and sellers are symmetric), when we consider the convex combination of sell-out strategy and effi cient two-part tariff, we find that there is a cut-off weight (strictly smaller than one) on the sell-out strategy such that the convex combination constitutes an equilibrium under no buyer group only if the weight is smaller than the cut-off. Given that the sellers extract more surplus from the buyers as the weight increases, this result shows that in equilibrium a seller cannot extract too high surplus under no buyer group since otherwise the other seller could profitably deviate by using the divide-and-conquer strategy. In Section 6.3, we consider the case in which the two products can be substitutes or complements while the cost functions are linear (hence there is no indirect contracting externalities). Then, buyer group has no effect on any player s payoff if the products sold by the sellers are complements to both buyers (or substitutes to both buyers). But if the products are strict substitutes for buyer 1, say, and strict complements for buyer 2, then buyer group reduces the buyers total payoff. 9 This is because the sellers have some residual market power with respect to buyer, as each seller charges less than the incremental value of its product because charging the incremental value leads to a strictly negative payoff for buyer 2. Buyer group allows the sellers to transfer the residual market power to buyer 1 in the same way as multi-market contact facilitates collusion by transferring residual collusive power from one market to another (Bernheim and Whinston, 1990). 10 Finally, we apply our insight to the situation in which one seller s entry is endogenous and the buyers decide whether or not to form a group before the entry decision. The entrant has to incur a fixed cost of entry, which is a random draw from a (commonly known) distribution as in Aghion and Bolton (1987), Chen and Shaffer (2014) and Innes and Sexton (1994). When the buyers decide to form a group or not, they do not know the fixed cost of entry. In this situation, the buyers face a clear trade-off. Formation of the buyer group increases the probability of entry but reduces the total payoff of the buyers conditional on entry. In particular, if we assume that the sellers play the Pareto-superior equilibrium, then the private decision of entry coincides with the socially optimal one under buyer group since the entrant captures the social incremental value of his product in the sell-out equilibrium. This suggests that whenever the buyers decide 9 As an example, a car and an airplane can be substitutes for a traveller of short distance but complements for a traveller of long distance. 10 Inderst and Shaffer (2007) and Dana (2012) do not find this result because they do not allow for complementary products. 4

7 to remain fragmented, there is a socially suboptimal entry. The existing literature on exclusion has explained suboptimal entry either by focusing on the incumbent s extracting surplus from the entrant (Aghion and Bolton, 1987) or the incumbent s taking advantage of coordination failure among buyers (Rasmusen, Ramseyer and Wiley 1991, Segal and Whinston, 2000, Chen and Shaffer, 2014). We provide an alternative explanation for the insuffi cient entry from the self-interest of buyers. The paper is organized as follows. Section 1.1 presents a review of the related literature. Section 2 introduces our model and Section 3 illustrates our main results through a simple quadratic setting. Section 4 studies the case of buyer group, Section 5 studies the case of no buyer group and Section 6 compares the two cases. Section 7 analyzes endogenous entry and Section 8 provides concluding remarks. 1.1 Literature review Divide-and-conquer contracts were first studied by Innes and Sexton (1993, 1994). They analyze the situation in which a monopolist seller faces identical buyers who may form a coalition to introduce a rival seller by sharing a fixed cost. They show that when buyers make acceptance decisions sequentially, the monopolist finds it optimal to discriminate the buyers by using the divide-and-conquer strategy: it makes a higher compensation to early buyers than to late buyers since once the former accepted the contract, the latter can share the fixed cost among a smaller number of buyers and hence has a weaker outside option. Therefore, the divide-and-conquer strategy hurts the buyers since it allows the monopolist to extract more surplus. 11 By contrast, we show that when sellers compete, buyers can benefit from divide-and-conquer strategy by remaining fragmented since divide-and-conquer strategy intensifies competition among sellers. To the best of our knowledge, we are the first to analyze the effect of buyer group in a framework that extends the common agency (Bernheim and Whinston, 1986, 1998) to multiple buyers. Prat and Rustichini (2003) analyze a more general framework than ours that has multiple sellers and buyers and prove the existence of an effi cient equilibrium, under the assumptions that buyers utility functions are concave and sellers cost functions are convex, by allowing for more complex tariffs than ours. Precisely, Prat and Rustichini (2003) allow seller to make buyer i s payment depend on the whole vector of i s purchases such that the payment depends on q i and also on qk i for each seller k different from. However, they do not specify the equilibrium strategies and do not compare the case of buyer group with the case of no group. In our two-buyer-two-seller setting, we consider tariff such that the payment of each buyer i to each seller only depends on q, i the 11 The divide-and-conquer idea has been subsequently used in the literature on two-sided markets (Caillaud and Jullien, 2001, 2003 and Jullien 2011). 5

8 quantity buyer i buys from seller. Under the assumption of substitute goods and convex cost functions, we show that two-part tariffs constitute an effi cient equilibrium. We show additional results by considering the class of tariffs composed of a convex combination of sell-out strategy and two-part tariff. Jeon and Menicucci (2012) extends the common agency to competition between portfolios in the presence of the buyer s slot constraint and provide conditions to make the "sell-out equilibrium" the unique equilibrium. 12 Our paper is related to the papers that study buyer group when sellers compete: Inderst and Shaffer (2007), Marvel and Yang (2008), Dana (2012), Chen and Li (2013). 13 Among those paper, our paper is more closely related to Inderst and Shaffer (2007) and Dana (2012) since they assume that sellers have complete information on buyers preferences and hence can offer personalized tariffs, regardless of whether or not buyers form a group. 14 Although the two papers differ in the way they generalize their results, 15 the main insight can be obtained by considering a two-seller-two-buyer setting in which each buyer buys one unit from only one of the two sellers. They assume that the buyer group makes exclusive purchase commitment and that sellers make a take-it-or-leave-it offer. In this setup, a buyer group never decreases its members total payoffs: it strictly increases the members total payoffs unless they have identical preferences; it has no impact on the members payoffs in case of identical preferences. There are three main differences between our paper and Inderst and Shaffer (2007) and Dana (2012). First, in our paper, 12 Contrary to what happens under slot constraint, Jeon and Menicucci (2006) show that in the presence of the buyer s budget constraint, the well-known result in the common agency literature that competition between sellers achieves the outcome that maximizes all parties oint payoff fails to hold. 13 Some papers study buyer coalition in a monopoly setting. Chipty and Snyder (1999) and Inderst and Wey (2007) consider some specific bargaining models and find that a convex cost function for the seller helps to make profitable the formation of a buyer group. This occurs because the incremental costs to serve the group is lower than the sum of the incremental cost to serve each member, and buyers are assumed to have some bargaining power. Since we consider take-it-leave-it offers from sellers, this effect is absent in our setting. Alger (1999) studies a monopolist s optimal menu of price-quantity pairs when (a continuum of) consumers can purchase multiple times and/or ointly in a two-type setting. While the previous papers consider buyer coalition formation under complete information, Jeon and Menicucci (2005) study a monopolist s optimal menu of price-quantity pairs when buyers form a coalition under asymmetric information between themselves. 14 For instance, in Marvel and Yang (2008) and Chen and Li (2013), buyers are located on the Hotelling line and each seller makes the same price offer to all buyers in the absence of buyer group. However, Marvel and Yang (2008) and our paper are similar in one aspect: when buyers form a group, sellers propose non-linear tariffs and the group can buy a positive quantity from both sellers. 15 Inderst and Shaffer (2007) consider competition in non-linear tariff between the two sellers and extend their result to a bargaining setup. They also make each seller s choice of product characteristics endogenous. Dana (2012) considers n sellers, a continuum of buyers, and allows that different buyer groups are formed. He proves that the grand coalition is a coalition-proof subgame perfect equilibrium when there are two sellers. 6

9 the buyer group does not make exclusive purchase commitment and each buyer and buyer group can buy multiple units from both sellers. Second, in their models, there is no indirect contracting externalities when buyers are not integrated. Last, contrary to what happens in their papers, in our paper, buyers never gain by forming a buyer group. Our paper is closely related to our companion paper (Jeon and Menicucci, 2014) which studies library consortium in the market for academic ournals. Each seller (i.e., publisher) is a monopolist of its ournals and sells a bundle of its electronic academic ournals at personalized price(s). However, there are no indirect contracting externalities such that in the absence of buyer group (i.e., library consortium), each library s market can be studied in isolation. Competition among sellers occurs because of the budget constraint of each buyer. We find that depending on the degree of correlation between each member library s preferences, building a library consortium can increase or reduce the total payoffs of the buyers. 16 Contracting externalities have been studied in several papers such as Hart and Tirole (1990), Bernheim and Whinston (1998), Segal (1999) etc. However, all these papers deal with direct contracting externalities that arise for instance when buyers compete in product market or sellers compete in input market. Since we assume away any such direct interactions between buyers in the product market or between sellers in the input market, there is no direct contracting externalities in our paper. Instead, the existence of indirect contracting externalities is a necessary condition for the divide-and-conquer strategy to create a difference between the case of buyer group and the case of no group. 2 The Model There are two sellers, A and B, and two buyers, 1 and 2. Let q i 0 represent the quantity of the product that buyer i (= 1, 2) buys from seller (= A, B). For each buyer i, the gross utility from consuming (qa i, qi B ) is given by U i (qa i, qi B ), with U i strictly increasing and strictly concave in (qa i, qi B ); precisely, we suppose that U i / q i 0 as q i +, and that the Hessian matrix of U i is negative definite at any (qa i, qi B ). Moreover, we assume that the two goods are substitutes for each buyer i: 2 U i / qa i qi B 0 at any point.17 For each seller, the cost of serving the buyers is C (q 1 + q 2 ), with C convex and strictly increasing. Each seller offers a non-linear tariffto each buyer (or to the buyer group). In particular, we consider tariffs such that what buyer i pays to seller depends only on the quantity that the buyer purchases from seller, not on the quantity she purchases from the other 16 Therefore, in terms of the results, the companion paper is situationed between Inderst and Shaffer (2007) and Dana (2012), on the one hand, and the current paper, on the other hand. 17 We consider a case with complementary goods in Proposition 8. 7

10 seller. When there is no buyer group, we allow each seller to offer personalized non-linear tariffs: the tariff offered by seller to buyer i is denoted by T i (q) (with T i (0) = 0), and can be different from the tariff seller offers to buyer h ( i). After seeing the tariffs, buyer i chooses (qa i, qi B ) in order to maximize U i (qa i, qi B ) T A i (qi A ) T B i (qi B ). When the buyers form a buyer group, the sellers compete for serving the group. Let G denote the buyer group, q G the quantity G buys from seller, and T G (q) the non-linear tariff that seller offers to G (with T G (0) = 0). We define U G (qa G, qg B ) as follows: U G (q G A, q G B) max U 1 (qa, 1 qb) 1 + U 2 (qa, 2 qb) 2 (1) qa 1,q1 B,q2 A,q2 B subect to q 1 A + q 2 A = q G A, q 1 B + q 2 B = q G B. (2) Thus U G (qa G, qg B ) is the group s gross utility from buying ( qa G, B) qg, as it results from the optimal allocation of ( qa G, B) qg between the two buyers. Hence, after seeing the tariffs, G chooses ( qa G, B) qg in order to maximize U G (qa G, qg B ) T A G(qG A ) T B G(qG B ). For expositional facility, we will consider sometimes two special settings: the setting of symmetric buyers, and the symmetric setting. The setting of symmetric buyers is such that U 1 ( ) = U 2 ( ) = U( ). The symmetric setting is such that U 1 ( ) = U 2 ( ) = U( ), U(q A, q B ) = U(q B, q A ), and C A ( ) = C B ( ) C( ). Define q (qa 1, q1 B, q2 A, q2 B ) as the unique allocation vector that maximizes social welfare, 18 and VAB G as the social welfare in the first-best allocation q : q arg max (q 1 A,q1 B,q2 A,q2 B) U 1 (q 1 A, q 1 B) + U 2 (q 2 A, q 2 B) C A (q 1 A + q 2 A) C B (q 1 B + q 2 B) (3) V G AB U 1 (q 1 A, q 1 B ) + U 2 (q 2 A, q 2 B ) C A (q 1 A + q 2 A ) C B (q 1 B + q 2 B ). (4) We say that an equilibrium is effi cient if the equilibrium allocation is q. From Proposition 4 to Proposition 9, we assume that each buyer buys a positive quantity from each seller in the first best allocation: q i > 0 for i = 1, 2 and = A.B. We consider a game with the following timing: Stage 0: The buyers decide whether they will form a group or not. Stage 1: When there is no buyer group, each seller (= A, B) simultaneously chooses T i (q) for i = 1, 2. When the buyer group is formed, each seller (= A, B) simultaneously chooses T G (q). Stage 2: Each buyer i, or the buyer group, makes purchase decisions. At stage 0, the buyers form a group if and only if this leads to a higher oint payoff than under no group. 18 Our assumptions imply that for each maximization problem below, a maximizer exists and is unique, since the obective function is strictly concave. 8

11 3 A simple illustrative example Before describing our analysis, we here illustrate our main results through a simple quadratic example. Consider a specific symmetric setting described by U i (q i A, q i B) = q i A + q i B 1 2 (qi A) (qi B) qi Aq i B for i = 1, 2; (5) C (q) = 1 2 q2 for = A, B, (6) where 1/2 in 1 2 qi A qi B is a parameter of substitution. In the effi cient allocation, each buyer buys quantity q 2 from each seller and the marginal cost at the effi cient quantity is 7 C (2q ) = 4. We below present a number of results with explanations. 7 Result 1: All equilibria are effi cient regardless of whether the buyers form a group or not. This result comes from the concavity of the utility functions and the convexity of the cost functions. Result 2: Under buyer group, for any β [0, 1], there exists an equilibrium with TA G(q) = T B G(q) = F + 4(1 β)q + β 7 2 q2 where F = (4β+1)(4β+3). The buyer group s 49(2β+1) payoff decreases (and each seller s payoff increases) with β. β = 1 corresponds to the sell-out equilibrium (Bernheim and Whinston 1986, 1998): see the next section for the details. β = 0 corresponds to a two-part tariff equilibrium. Hence, according to Result 2, any convex combination of the sell-out strategy and the two-part tariff constitutes an equilibrium under buyer group. In equilibrium, the buyer group is indifferent between trading with both sellers and trading with only one seller. When the group trades with only seller, the payoff it obtains strictly decreases with β as the marginal price increases with β for any q G > 2q. Therefore, the buyer group s payoff decreases (and each seller s payoff increases) with β. When there is no buyer group, given seller k s offer, we can distinguish three deviations of seller ( k) depending on the number of buyers who are induced to keep buying from seller k. This number can vary from zero to two. In particular, the deviation inducing only one buyer to keep buying from the rival (while the other buyer buys uniquely from seller ) can be made only when there is no buyer group. We define this deviation as the divide-and-conquer strategy, as it essentially treats the two buyers differently. Result 3: When we neglect the divide-and-conquer strategy, the case of buyer group is equivalent to the case of no group: starting from any equilibrium under 9

12 buyer group TA G, T B G, we can construct an equilibrium without buyer group in which T i (q) = 1T G 2 (2q) for each i = 1, 2 and = A, B, which generates the same payoff for each player. Hence, this result shows that essentially what creates a difference between the two cases is the divide-and-conquer strategy. Result 4: Fix seller B s offer such that he offers TB G(q) = F + 4(1 β)q + β 7 2 q2 with F = (4β+1)(4β+3) under buyer group, and T i 49(2β+1) B (q) = 1T G 2 B (2q) without group. Then, seller A s payoff under no group is the same as under buyer group for β [0, 0.226], but the former is strictly higher than the latter for β (0.226, 1]. This result captures the conventional wisdom that a monopolist facing buyers who have a fixed outside option does better when buyers are divided than when they are integrated. 19 In particular, A s deviation using the divide-and-conquer strategy is profitable only when the buyers outside option is relatively weak (i.e., for β > 0.226). This in turn implies that for β > 0.226, B s offer cannot constitute an equilibrium when B can choose its offer. Therefore, we have: Result 5: Without buyer group, T i (q) = 1T G 2 (2q) (where T G (q) = F + 4(1 7 β)q + β 2 q2 with F = (4β+1)(4β+3) ) constitutes an equilibrium only for β [0, 0.226]. 49(2β+1) Therefore, when we consider the class of tariffs composed by a convex combination of the sell-out strategy and the two-part tariff, the lowest payoff for each buyer without buyer group is strictly larger than the lowest payoff under buyer group and the highest payoff for each seller without buyer group is strictly lower than the highest payoff under buyer group. Our main result (Proposition 7) shows that the above payoff comparison extends very generally to asymmetric setting (and beyond the case of quadratic utilities and costs) in which sellers can use any non-linear tariffs, as long as the cost functions are strictly convex and the products are strict substitutes. 4 Buyer group: Common agency Since the setting with the buyer group is essentially an environment with a unique buyer, our model is a model of common agency (Bernheim and Whinston 1986, 1998). Hence, it is natural to start the analysis with the so-called sell-out equilibrium (Bernheim and 19 However, the buyers are not hurt by not forming a group, since their payoff is determined by the outside option which is the same regardless of they form a group or not since T i B (q) = 1 2 T G B (2q). 10

13 Whinston 1998). In order to determine the equilibrium strategies, we need to study the case in which G trades only with one seller, for instance only with seller : ) 1 ( q, q 2 arg max U 1 (q 1, 0) + U 2 (q 2, 0) C (q 1 + q 2 ) (7) q 1,q2 V G U 1 ( q 1, 0) + U 2 ( q 2, 0) C ( q 1 + q 2 ) (8) Hence V G is the maximal social welfare when G trades with seller, but not with the other seller. A sell-out strategy for seller is defined as T G (q) = F G + C (q) for any q > 0. Given this strategy, G is required to pay the fixed fee F G for the right to buy any quantity from seller at cost. When each seller uses a sell-out strategy, the first-best allocation is achieved (provided that FA G and F B G are not too high) since G will buy qg q 1 +q 2 from each seller and allocate ( ) q 1, q 2 to the buyers. This is because after paying the fixed fees, G becomes the residual claimant over social welfare. In equilibrium it is necessary that 20 V G AB F G A F G B = V G A F G A = V G B F G B. (9) The sell-out equilibrium is given by the sell-out strategies with the fixed fees obtained from (9): FA G = VAB G VB G, FB G = VAB G VA G. The group s payoff is u G VA G +V B G V AB G, and ug 0 since goods are substitutes for each buyer. The profit π G of seller is given by F G, which is the marginal contribution to social welfare of seller, that is the first best social welfare minus the social welfare that is generated when G trades only with the other seller k( ). The sell-out equilibrium is not the unique equilibrium in the game of competition for G. For instance, a two-part tariff equilibrium exists in which each seller proposes a tariff given by T G (q) = F G + α q for any q > 0, with α = C (q G ). (10) This tariff is such that the buyer group is required to pay the fixed fee F G for the right to buy any quantity from seller at the constant marginal price α, which is the marginal cost of seller at the first best allocation q. When both sellers use tariffs of this kind, the group buys ( qa G, ) qg B and the allocation q is achieved. In fact, the next proposition shows that any convex combination of the sell-out strategy and the two-part tariff is an equilibrium tariff, and implements the allocation q. In addition, the proposition highlights two properties that any equilibrium satisfies. 20 For instance, if VAB G F A G F B G > V A G F A G, then seller B can increase his own profit by slightly increasing FB G as G will still buy from seller B. 11

14 Proposition 1 (buyer group) Suppose that the buyers have formed a group. (i) Any convex combination of the sell-out strategy and the two-part tariff (with seller s marginal price equal to α ) constitutes an equilibrium tariff: for any β [0, 1], there exists an equilibrium with tariffs T G (q) = F G + βc (q) + (1 β)α q, and a suitable F G, for = A, B. (ii) In no equilibrium, seller (=A,B) obtains a strictly higher payoff than in the sell-out equilibrium (Bernheim and Whinston 1998). (iii) Any equilibrium is effi cient. Proof. The proof of (i) is in the Appendix. Below we provide the proof of (ii) and a part of the proof of (iii) the remaining part is in the Appendix. About Proposition 1(i), notice that if C is strictly convex and β [0, 1), then the marginal price in T G for q > q G is smaller than in the sell-out strategy of seller. This matters when products are substitutes, as it increases the payoff G obtains by trading only with seller, and forces seller k to leave a higher payoff to G than in the sell-out equilibrium. This reduces the profit of seller k, and a similar argument applies to seller. In fact, each seller s profit is increasing in β. Proposition 1(ii), proved in Bernheim and Whinston (1998), applies to any equilibrium (not only to those described in Proposition 1(i)) and is based on a simple remark which we now describe. In any equilibrium, the sum of the buyer group s payoff and the profit of seller cannot be smaller than V G, otherwise seller may deviate by offering to G a suitable trade, in which G buys only from : see (7) and (8). This implies that in any equilibrium the profit of seller k( ) is at most VAB G V G, i.e., it is weakly smaller than ( ) π G k = F G k. Proposition 1(iii) is a consequence of Proposition 1 in O Brien and Shaffer (1997), which implies that in any equilibrium under buyer group, the equilibrium quantities, denoted by ( qa Ge, ) qge B, are such that q Ge A ( arg max U G (qa, G q Ge qa G B ) C A (qa) ) G, q Ge B ( arg max U G (q Ge qb G A, qb) G C B (qb) ) G. This means that qa Ge maximizes the sum of G s payoff and firm A s profit, given qg B = qge and an analogous property holds for qb Ge. Therefore { UA G(qGe A, qge B ) C A (qge A ), with equality if qge A > 0; UB G(qGe A, qge B ) C B (qge B ), with equality if qge B > 0. (11) Since U 1 and U 2 are concave, it follows that U G is concave and therefore social welfare U G (q G A, qg B ) C A(q G A ) C B(q G B ) is a concave function of ( q G A, qg B). The conditions in (11) are the necessary first order conditions for maximization of social welfare, and moreover they B, 12

15 are suffi cient since social welfare is concave. Hence ( qa Ge, ) qge B maximizes social welfare, that is q Ge A = q1 A + q2 A, qge B = q1 B + q2 B.21 5 No buyer group In this section, we study what happens when there is no buyer group. 5.1 Preliminaries When there is no buyer group, we restrict attention to non-linear tariffs such that what buyer i pays to seller depends only on what i buys from, but not on what i buys from the other seller, k. In this setting we find a result analogous to Proposition 1(ii) and (iii): Proposition 2 Suppose that there is no buyer group. (i) Any equilibrium is effi cient. (ii) In any equilibrium, seller (=A,B) obtains a weakly lower payoff than in the sell-out equilibrium under buyer group. (iii) In any equilibrium, the buyers obtain a weakly higher total payoff than in the sell-out equilibrium under buyer group. The intuition for Proposition 2(i) is similar to the intuition for Proposition 1(iii): We can prove that in any equilibrium, the equilibrium quantities (qa 1e, q1e B, q2e A, q2e B ) are such that { (qa 1e, q2e A ) arg max qa 1 (U 1 (q 1,q2 A A, q1e B ) + U 2 (qa 2, q2e B ) C A(qA 1 + q2 A )) ; (qb 1e, q2e B ) arg max qb 1 (U 1 (q 1e,q2 B A, q1 B ) + U 2 (qa 2e, q2 B ) C B(qB 1 + q2 B )). (12) This means that (qa 1e, q2e A ) maximizes the sum of the buyers total payoff and firm A s profit, given (qb 1, q2 B ) = (q1e B, q2e B ), and an analogous property holds for (q1e B, q2e B ). Since social welfare U 1 (qa 1, q1 B )+U2 (qa 2, q2 B ) C A(qA 1 +q2 A ) C B(qB 1 +q2 B ) is a concave function, it follows from (12) that (qa 1e, q1e B, q2e A, q2e B ) maximizes social welfare, that is (q1e A, q1e B, q2e A, q2e B ) = (qa 1, q1 B, q2 A, q2 B ). The intuition for Proposition 2(ii) is similar to the intuition for Proposition 1(i). Regardless of whether or not the buyers form a group, the sum of seller s payoff and the total payoff of the buyers cannot be strictly smaller than V G ; otherwise, seller can profitably deviate by proposing all buyers to exclusively deal with him. Hence, seller k s profit is weakly lower than π G k. 21 O Brien and Shaffer (1997) exhibit a setting in which an ineffi cient equilibrium exists, but that setting has discontinuous cost functions for sellers (because of a fixed cost each seller bears for each positive quantity produced). Therefore the social welfare function is not concave. 13

16 Proposition 2(iii) is a consequence of Proposition 2(i) and (ii). This result is interesting since in Inderst and Shaffer (2007) and Dana (2012), buyers never lose from building a group. This contrast in the results arises since in their papers, the buyer group commits to purchasing from a single seller, whereas in our paper there is no such commitment. 5.2 Divide-and-conquer In this subsection, we introduce the definitions of the core concepts for our analysis. We first define divide-and-conquer strategies: Definition 1 Divide-and-conquer: Suppose that there is no buyer group. A seller uses a divide-and-conquer strategy if he induces one buyer to buy exclusively from him and induces the other buyer to buy from both sellers. Definition 2 A simple divide-and-conquer strategy is a divide-and-conquer strategy such that the deviating seller deviates only in one market by inducing the buyer of the market to buy exclusively from him. We now define indirect contracting externalities: Definition 3 Indirect contracting externalities: Given (qa h, qh B ) with h {1, 2}, suppose that (q i A, q i B) with i {1, 2} and i h maximizes the surplus from the multilateral trade among seller A, seller B and buyer i. We say that there are no indirect contracting externalities if (q i A, q i B) is the same for any possible (qa h, qh B ) R2 +. We say that there are indirect contracting externalities if a change in q h for = A, B induces a change in q i. As will be clear in the next subsection, the divide-and-conquer strategy has no bite if there are no indirect contracting externalities. Direct contracting externalities (Segal, 1999) exist when the utility of buyer i directly depends on (qa h, qh B ) with h i. This can occur if the buyers compete in the same market as in Hart and Tirole (1990). In our model, there is no direct contracting externalities between the two buyers since each buyer s utility function depends only on the pair of quantities she buys, and the tariff she pays to a seller depends only on the quantity she buys from the seller. However, there can be indirect contracting externalities in the sense that a change in (qa h, qh B ) indirectly affects the payoff of buyer i ( h). This can occur if the change modifies a seller s marginal cost such that the seller in turn modifies his offer to buyer i. This is the case if sellers have strictly convex cost functions, but not if the cost functions are linear. Some analogy can be made to the distinction between direct network externalities and indirect network externalities. Direct network externalities exist if a consumer s consumption utility directly depends on the number of other consumers using the same 14

17 good (say, hardware). Even if direct network externalities do not exist, indirect network externalities exist if an increase in the number of consumers using the same hardware modifies the number of applications that can be used with the hardware. 5.3 Buyer group neutrality Proposition 2 reveals that the buyers never gain strictly by forming a group. We now show two different conditions for buyer group neutrality, which means that buyer group affects neither the buyers total payoff nor any seller s payoff. We say that products have independent values if there exist functions U i A, U i B such that U i (q i A, qi B ) = U i A (qi A )+U i B (qi B ). Proposition 3 (buyer group neutrality) Buyer group affects neither the buyers total payoff nor any seller s payoff if either of the following conditions is satisfied: (i) the products have independent values; (ii) the sellers costs are linear and the products are substitutes. When the products have independent values as in Proposition 3(i), each seller behaves like a perfectly discriminating monopolist because each buyer s valuation for seller s products does not depend on the quantity the buyer buys from the other seller. Therefore, seller can fully extract the buyers surplus regardless of whether or not they form a group. When the sellers cost functions are linear, there are no indirect contracting externalities since the effi cient allocation in market i does not depend on the allocation implemented in market h. More precisely, (q i ) is defined as follows: A, qi B (q i A, q i B ) arg max (q i A,qi B) U i (q i A, q i B) c A (q i A + q h A) c B (q i B + q h B), where c is the constant marginal cost of seller. Therefore, (qa i, qi B ) does not depend on ( qa h, B) qh. As there exist no links between competition for buyer 1 and competition for buyer 2, we can study each market in isolation and the divide-and-conquer strategy has no bite. There exists a sell-out equilibrium in each market such that without buyer group, each seller obtains the same payoff as in the sell-out equilibrium under buyer group and the buyers obtain the same total payoff as in the sell-out equilibrium under buyer group. 5.4 Indirect contracting externalities and non-existence of the sell-out equilibrium In this subsection, we consider the case in which the sellers cost functions are strictly convex and the products are strict substitutes. Therefore, indirect contracting externalities exist when there is no buyer group: an increase in qa h, for instance, increases A s 15

18 marginal cost and hence creates an incentive for A to modify the terms of its offer to buyer i ( h). We show that no sell-out equilibrium exists when there is no buyer group and q i > 0 for i = 1, 2 and = A, B. 22 As a generalization of the sell-out equilibrium to the case of two buyers, it seems natural to consider tariffs such that the tariff offered by seller to buyer i is given by a fixed fee plus the incremental cost for seller of serving buyer i given that it serves the effi cient quantity to buyer h (recall that any equilibrium is effi cient from Proposition 2(i)). More precisely, we consider the following tariffs: T i (q) = F i +C (q+q h ) C (q h ) for any q > 0, for each i = 1, 2 and = A, B. (13) The term C (q + q h ) C (q h ) is the incremental cost for seller to produce q for buyer i, given that already accepted to produce q h for buyer h.( i) The tariff in (13) is such that buyer i is required to pay the fixed fee F i for the right to buy any quantity from seller at the incremental cost incurred by. If the other seller uses a tariff analogous to (13), then each buyer i buys (qa i, qi B ) and the first-best allocation q is realized. Since it is important to understand how divide-and-conquer strategies in the presence of indirect contracting externalities make this equilibrium not exist, we below provide an explanation for the symmetric setting. 23 Suppose U 1 ( ) = U 2 ( ) U( ) with U( ) symmetric and C A ( ) = C B ( ) C( ). This implies qa 1 = q2 A = q1 B = q2 B q. Let U U(q, q ) and C C(2q ). We consider a symmetric strategy profile (i.e., each seller offers the same tariff T (q) to each buyer) that implements the first best allocation, as it occurs in any equilibrium. Let T TA 1(q ) = TA 2(q ) = TB 1(q ) = TB 2(q ). Therefore, the profit of each seller is π = 2T C. The payoff of each buyer is U 2T, which is equal to the payoff u each buyer can obtain from trading with a single seller, u max q [U(q, 0) T (q)] = U 2T. Suppose now that seller A uses a simple divide-and-conquer strategy. Namely, in market 1, A makes a take-it-or-leave-it offer (q, t) to induce buyer 1 to buy only from A (i.e., (q, t) satisfying U(q, 0) t u ) but A does not deviate in market 2. Since A still has revenue T from buyer 2, his profit from the deviation is U(q, 0) u +T C(q +q ), and therefore the deviation is profitable for A if and only if max [U(q, 0) u + T C(q + q )] > 2T C. (14) q This inequality can be simply interpreted as the condition that the deviation increases the sum of seller A and buyer 1 s payoffs 24, and is equivalent to max q [U(q, 0) (C(q + q ) C )] > max [U(q, 0) (T (q) T )]. (15) q 22 For instance, if qa 1 > 0, q1 B > 0 and q2 A = q2 B = 0 then Proposition 4 below does not hold. 23 In the Appendix, we provide a general proof: see the proof of Proposition For this interpretation, write (14) as max q [U(q, 0) + T C(q + q )] > u + 2T C. 16

19 In the case of the sell-out contract, we have T (q) = F + C(q + q ) C(q ) as in (13) and we find that the left hand side in (15) is equal to the right hand side. Hence, A is indifferent between not deviating at all and deviating only in market 1. However, after the deviation, A supplies buyer 1 some quantity larger than qa 1 because the products are strict substitutes. This increases A s marginal cost and creates an opportunity to profitably deviate in market 2 by selling qa 2 (< q2 A ) to buyer 2. Hence, indirect contracting externalities imply that A can find a suitable divide-and-conquer strategy to profitably deviate in both markets. This implies that no sell-out equilibrium exists. Proposition 4 (non-existence of the sell-out equilibrium) Suppose that there is no buyer group, that the cost functions are strictly convex, and that the products are strict substitutes. Then no sell-out equilibrium exists. Since products are strict substitutes, what really matters for the result is the comparison between T (q) T and C(q + q ) C for q > q in (15). In order to have an equilibrium, it is necessary that the left hand side in (15) is smaller than the right hand side, and this is achieved if T (q) T < C(q + q ) C for q > q. The result in the next subsection relies on this insight. Remark: In Subsection 6.1 we consider the symmetric setting in which each seller (= A, B) offers T i (q) = F + 1 C(2q) for i = 1, 2 and find that (15) is satisfied as 2 T (q) T = 1C(2q) C is greater than C(q + q ) C for q > q. Hence, no equilibrium exists such that each seller offers the tariff T (q) = F + 1 C(2q) to each buyer Two-part tariff equilibrium In this subsection we show that a two-part tariff equilibrium exists. Precisely, the tariff seller offers to buyer i is: T i (q) = F i +α q for any q > 0, with α = C (q i +q h ), for any i = 1, 2 and = A, B. (16) This tariff is analogous to the two-part tariff we have introduced in (10) under buyer group. In order to determine the equilibrium fees, we define VAB i, qi, V i as follows: q i arg max q i VAB i = U i (qa i, qb i ) α A qa i α B qb i ; (17) ( U i (q, i 0) α q) i and V i U i ( q i, 0) α q i. (18) Thus VAB i is the payoff of buyer i (gross of the fixed fees) if she buys (qi A, qi B ); V i is the payoff of buyer i (gross of the fixed fee) if she trades only with seller, and q i is the 17

20 payoff-maximizing quantity in that case. As in the case of buyer group, in equilibrium it is necessary that VAB i F A i F B i = V A i F A i = V B i F B i, and from these equalities we obtain the equilibrium fees: F i A = V i AB V i B, F i B = V i AB V i A, for i = 1, 2. (19) Proposition 5 (equilibrium with two-part tariffs) Suppose that there is no buyer group and that the cost functions are strictly convex and the products are strict substitutes. The tariffs described by (16) and (19) constitute an equilibrium. The intuition for why the two-part tariffs constitute an equilibrium can be given by considering simple divide-and-conquer strategies. For the symmetric setting, we have proved that for seller A, a simple divide-and-conquer strategy is profitable if and only if (15) is satisfied. When T (q) = F + αq, with α = C (2q ) as in (16), we find that (15) is violated because T (q) T is smaller (i.e., grows less quickly) than C(q + q ) C for q > q. Therefore, no simple hybrid deviation is profitable. Although indirect contracting externalities suggest that more powerful deviations involving both markets exist, the proof of Proposition 5 establishes that no profitable deviation exists. 6 Comparison between buyer group and no group In this section, we show that buyer group reduces the buyers payoffs. In the first two subsections, we consider the case in which the cost functions are strictly convex and the products are strict substitutes, and show that divide-and-conquer intensifies competition between sellers, to the benefit of buyers. In the last subsection, we consider a setting without indirect contracting externalities in which we relax the assumption of substitute goods. We show a mechanism, different from divide-and-conquer, which makes buyers lose from building a group. 6.1 Symmetric buyers We first consider the setting of symmetric buyers. Starting from tariffs TA G, T B G which are an equilibrium under buyer group, we consider the following tariffs under no group: T 1 A(q) = T 2 A(q) = 1 2 T G A (2q), T 1 B(q) = T 2 B(q) = 1 2 T G B (2q). (20) These tariffs help us to understand how competition changes when moving from buyer group to no buyer group. Precisely, if sellers offer the tariffs in (20) under no buyer group, then we can prove that buyers purchases are the same as under buyer group, and each seller and the buyers have the same payoff as under buyer group. Moreover, the payoff 18