Buyers and sellers cartels on markets with indivisible goods

Transcription

1 Rev. Econ. Design 5, (2000) c Springer-Verlag 2000 Buyers and sellers cartels on markets with indivisible goods Francis Bloch 1, Sayantan Ghosal 2 1 IRES, Department of Economics, Université Catholique de Louvain, Belgium 2 Department of Economics, University of Warwick, Coventry, UK Abstract. This paper analyzes the formation of cartels of buyers and sellers in a simple model of trade inspired by Rubinstein and Wolinsky s (1990) bargaining model. When cartels are formed only on one side of the market, there is at most one stable cartel size. When cartels are formed sequentially on the two sides of the market, there is also at most one stable cartel configuration. Under bilateral collusion, buyers and sellers form cartels of equal sizes, and the cartels formed are smaller than under unilateral collusion. Both the buyers and sellers cartels choose to exclude only one trader from the market. This result suggests that there are limits to bilateral collusion, and that the threat of collusion on one side of the market does not lead to increased collusion on the other side. JEL classification: C78, D43 Key words: Bilateral collusion, buyers and sellers cartels, collusion in bargaining, countervailing power 1 Introduction Recent theoretical models of collusion only consider the formation of cartels on one side of the market. Studies of bidding rings in auctions focus on collusion on the part of buyers in models with a unique seller. On oligopolistic markets, the This research was started while both authors were at CORE. The first author gratefully acknowledges the financial assistance of the European Commission (Human Capital Mobility Fellowship) which made his visit to CORE possible, and the second author a CORE doctoral fellowship. Discussions with Ulrich Hege, Heracles Polemarchakis and Asher Wolinsky helped us formalize the ideas in this paper. We also benefitted from comments by participants at seminars in Namur and Bilbao, at the Hakone Conference in Social Choice, the Valencia Workshop on Game Theory, the Bangalore and Saint Petersburg Conferences in Game Theory and Applications and the Econometric Society European Meeting in Istanbul. This paper is an extended version of a paper formerly entitled Bilateral Bargaining and Stable Cartels.

2 130 F. Bloch, S. Ghosal formation of cartels of producers is analyzed under the assumption that demand is atomistic, so that buyers react in a competitive fashion to the choices of sellers. While these models are well-suited to analyze collusion on some markets (simple auctions, markets for consumer goods), their conclusions can hardly be extended to other markets, such as markets for primary commodities, where a small number of buyers and sellers interact repeatedly. However, some of the best known examples of cartels are actually found on thin markets with a small number of traders. The commodity cartels grouping producer countries (OPEC, the Uranium, Coffee, Copper and Bauxite cartels) face a very small number of buyers of primary commodities. Similarly, the famous shipping conferences, legal cartels grouping all shipping companies operating on the same route, interact repeatedly with the same shippers. On markets with a small number of strategic buyers and sellers, the study of collusion on one side of the market must take into account the reaction of traders on the other side. The formation of a cartel by traders on one side of the market may induce collusion on the other side. In fact, it is often argued that commodity cartels were formed in the 1970 s as a response to the increasing concentration of buyers on the market (see the case studies by Sampson (1975) on the oil market, by Holloway (1988) on the aluminium market and by Taylor and Yokell (1979) on the uranium market). In oceanliner shipping, the monopoly power of shipping companies has led to the emergence of cartels of buyers, the shippers councils which negotiate directly with the shipping conferences (Sletmo and Williams 1980). Our purpose in this paper is to analyze the formation of buyers and sellers cartels on markets with a small number of strategic buyers and sellers. In particular, we study how collusion on one side of the market may induce collusion on the other side. When do cartels emerge on the two sides of the market? What are the sizes of those cartels? Does the formation of cartels on the two sides of the market lead to a higher restriction in trade than in the case of unilateral collusion? Does bilateral collusion induce a balance in the market power of buyers and sellers, as suggested by Galbraith (1952)? In order to answer these questions, we study a sequential model of interaction between an equal number of buyers and sellers of an indivisible good. In the first stage, buyers decide to form a cartel and restrict the number of traders they put on the market. In the second stage, sellers form a cartel and restrict trade in the same way. In the third stage, once the number of buyers and sellers excluded from the market is determined, the remaining traders trade on the market. The model of trade for an indivisible commodity that we use is inspired by Rubinstein and Wolinsky (1990) s model of matching and bargaining among a small number of traders. Buyers and sellers bargain over the surplus generated by the indivisible good, which is normalized to one. At each point in time, buyers and sellers are randomly matched and make decentralized offers. However, to guarantee the existence of a unique price at which trade occurs, we add to the model a centralization mechanism: trade only occurs when all offers are accepted in the same round. It is easy to see that, as the discount factor converges to 1,

3 Buyers and sellers cartels on markets with indivisible goods 131 the outcome of this model of trade converges to the competitive outcome, giving all of the surplus to traders on the short side of the market. On the other hand, when the discount factor converges to 0, the trading mechanism approaches a simple bargaining model with take-it-or-leave-it offers. As the price of the good traded depends on the numbers of buyers and sellers on the markets, traders have an incentive to restrict the quantities of the good they buy or sell on the market. However, given the indivisibility of the good traded, the only way to restrict offer or demand on the market is to exclude some agents from trade. Hence, we assume that cartels are formed in order to exclude some traders from the market and to compensate them for withdrawal. 1 We model the formation of the cartel as a simple, noncooperative game, where traders simultaneously decide on their participation to the cartel. This participation game implies that a cartel is stable when (i) no trader has an incentive to join the cartel and (ii) no trader has an incentive to leave the cartel. In a first step, we analyze the formation of a stable cartel on one side of the market and show that there exists at most one stable cartel size. This is the unique cartel size for which, upon departure of a member, the cartel collapses entirely. 2 If there are originally more sellers than buyers on the market, sellers form a cartel in order to equalize the number of active buyers and sellers. If there are more buyers than sellers, there does not exist any stable cartel of sellers. Finally, if originally buyers and sellers are on equal number on the market, sellers form a cartel and exclude one trader from the market. Next, we analyze the response of one side of the market to collusion on the other side when buyers form a cartel, they anticipate that sellers will respond by colluding themselves. We suppose that buyers and sellers are originally in equal number on the market. Using our earlier characterization of stable cartels on one side of the market, we show that in the sequential game of bilateral cartel formation, there exists a unique stable cartel configuration, where both buyers and sellers form cartels, the cartels are of equal size, and both cartels exclude one trader from the market. It thus appears that the formation of cartels on the two sides of the market leads to the same restriction in trade as in the case of unilateral collusion. Furthermore, the size of the cartels formed under bilateral collusion is smaller than the size of the cartel formed under unilateral collusion. We interpret these results by noting that there exist limits to bilateral collusion. The threat of collusion on one side of the market does not lead to a higher level of collusion among traders on the other side. In order to gain some insights about these results, it is instructive to consider the limiting case of a competitive market, where traders on the short side of the market almost obtain the entire surplus. 3 Suppose that originally buyers form a 1 Alternatively, we could assume that cartels are formed for traders to coordinate their actions at the bargaining stage. This is a much more complex issue that we prefer to leave for further research. 2 This characterization of the stable cartel emphasizes the role played by the indivisibility of the good traded. On markets with divisible goods, the formation of a cartel is prevented by the traders incentives to leave the cartel and free ride on the cartel s trading restriction. 3 It has long been noted that traders have an incentive to collude on these competitive markets for indivisible commodities. See Shapley and Shubik (1969), fn. 10 p. 344.

4 132 F. Bloch, S. Ghosal cartel and exclude one buyer from the market. What will the seller s response be? Clearly, by forming a cartel which excludes two sellers from the market they could capture a surplus of 1 ɛ per unit traded, whereas by excluding one trader they obtain a surplus of 1 2. Hence, at first glance, it seems that sellers should form a cartel which excludes two traders. However, we argue that this cartel cannot be stable, and that the only stable cartel is a cartel of size three which excludes one seller from the market. To see this, note that the minimal cartel size for which two sellers are excluded is four. Each cartel member then receives a payoff of 2 2ɛ 4 = 1 2 ɛ 2. By leaving the cartel, a member would obtain a higher payoff of 1 2, so that the cartel is unstable. By the same free-riding argument, no cartel of size greater than four can be stable. Hence, the only stable cartel is the cartel of size three, showing that free-riding prevents the formation of a cartel in which sellers could capture the entire trading surplus. While our analysis departs from recent studies of collusion in auctions and competitve markets, its roots can be traced back to the debate surrounding Galbraith s (1952) book on countervailing power. In this famous book, Galbraith (1952) argues that the concentration of market power on the side of buyers is the only check to the exercise of market power on the part of sellers. (see Scherer and Ross (1990), Ch. 14, for a survey of recent contributions to the theory of countervailing power ). As was already noted by Stigler (1954) in his discussion of the book, Galbraith s (1952) assertions are not easily supported by formal economic arguments. In fact, we show that the existence of countervailing power may balance the market power of buyers and sellers, but does not help to reduce the inefficiencies linked to the existence of market power. In the 1970 s, the formation of stable cartels has been studied in general equilibrium models, using various solution concepts (see, for example the survey by Gabszewicz and Shitovitz 1992 for the core, Legros 1987 for the nucleolus, Hart 1974 for the stable sets and Okuno et al for a strategic market game). While these models provide some general existence and characterization results on stable cartels, these results cannot be easily compared to the results we obtain here. Finally, our analysis relies strongly on the study of stable cartels on oligopolistic markets initiated by d Aspremont et al. (1983), Donsimoni (1985) and Donsimoni et al. (1986). The stability concept we use is due to d Aspremont et al. (1983). In spite of differences in the models of trade, our results bear some resemblance to the characterization of stable cartels in Donsimoni et al. (1986). As in their analysis, we find that free-riding greatly limits the size of stable cartels, thereby reducing collusion on the market. The rest of the paper is organized as follows. We present and analyze the model of trade and describe the cartel s optimal choice in Sect. 2. In Sect. 3, we define the game of cartel formation and characterize stable cartels, both when cartels are formed only on one side of the market and when cartels are formed sequentially by buyers and sellers. Finally, Sect. 4 contains our conclusions and directions for future research.

5 Buyers and sellers cartels on markets with indivisible goods Trade and collusion on the market In this section, we present the basic model of trade and analyze the behavior of cartels formed on the market. We consider a market for an indivisible good with a finite set B of identical buyers and a finite set S of identical sellers and let b and s denote the cardinality of the sets B and S respectively. Each buyer i in B wants to purchase one unit of the indivisible good traded on the market, and each seller j in S owns one unit of the good. Without loss of generality, we normalize the gains from trade to 1. The interaction between participants on the market is modeled as a threestage process. In the first stage, a cartel is formed; in the second stage, members of the cartel choose the number of active traders they put on the market. Finally, in the third stage of the game, buyers and sellers trade on the market. Since the model is solved by backward induction, we start our formal description of the game by the final stage of the game and proceed backwards to the first stage. 2.1 A model of matching and trade After cartels are formed, and the number of active traders on the market is determined, agents engage in trade. The trading mechanism we analyze combines elements of bilateral bargaining as in Rubinstein and Wolinsky (1990) and a centralization mechanism. We suppose that, at each period of time, traders are matched randomly and engage in a bilateral bargaining process. However, in order for trade to be concluded, we require that all agents unanimously agree on the offer they receive. Formally, we let t =1, 2,... denote discrete time periods. At each period t, the traders remaining on the market are matched randomly. If s t and b t denote the numbers of sellers remaining on the market in period t, and if b t < s t, this implies that each buyer i is matched with a seller j, whereas a seller j is matched with a buyer i only with probability bt s t. Each match (i, j ) is equally likely. Once a match (i, j ) is formed, one of the traders is chosen with probability 1 2 to make an offer. The other trader then responds to the offer. If, at some period t, all offers are accepted, the transactions are concluded and traders leave the market. If, on the other hand, one offer is rejected, all traders remain on the market and enter the next matching stage. If a transaction is concluded at period T for a price of p, the seller obtains a utility equal to T p whereas the buyer obtains T (1 p). While our model shares some formal resemblance to models of bilateral bargaining, it differs sharply from traditional models of decentralized trade since transactions are concluded only when all traders unanimously accept the offers. While this coordination device is clearly unnatural in a model of decentralized trade, we need it to guarantee that the bargaining environment is stationary, so that classical methods of characterization of stationary perfect equilibria can be used (see for example, Rubinstein and Wolinsky 1985). Without this coordination

6 134 F. Bloch, S. Ghosal device, utilities obtained in any subgame following the rejection of an offer would depend on the number of pairs of buyers and sellers who have concluded trade at this stage. Since bargaining occurs simultaneously in all matched pairs of buyers and sellers, traders cannot observe the outcome of bargaining among other traders. Hence, this is a game of incomplete information, and in order to compute utilities obtained after the rejection of offer, we need to specify the players beliefs about the behavior of the other traders in the game. As in any extensive form game with incomplete information, there is some leeway in defining those beliefs off the equilibrium path, and there is no natural way to select a sequential equilibrium in this game. 4 It is well known that the sequential game we consider may have many equilibria. In order to restrict the set of equilibria, we assume that traders use stationary strategies, which only depend on the number of active traders in the game, and on the current proposal. Formally, a stationary perfect equilibrium of the trading game is a strategy profile σ such that For any player i, σ i only depends on the number of active traders and the current proposal. At any period t at which player i is active, the strategy σ i is a best response to the strategy choices σ i of the other traders. Proposition 1. In the model of trade, there exists a unique stationary perfect equilibrium. The equilibrium is symmetric and all offers are accepted immediately. If b s, sellers propose a price p s = (1 )(2s b) (2 )s b and buyers propose a price p b = (1 )b (2 )s b.ifs b, sellers propose a price p s = (b s)(2 ) b(2 ) s and buyers propose a price p b = (b s) b(2 ) s. Proof. Without loss of generality, suppose b s. First observe that, since unanimous agreement is needed for the conclusion of trade, in a stationary perfect equilibrium, all players must make acceptable offers. If, on the other hand, one trader makes an offer which is rejected, the game moves to the next stage and, by stationarity, the bargaining process continues indefinitely. Hence, given a fixed set of buyers indexed by i =1, 2,...b and a fixed set of sellers indexed by j =1, 2,...,s, the strategy profile σ must be characterized by price offers (p b (i, j ), p s (i, j )) where p b (i, j ) represents the price offered by buyer i in the match (i, j ) and p s (i, j ) the price offered by seller j in the match (i, j ). In a subgame perfect equilibrium, the price p b (i, j ) is the minimum price accepted by seller j and must satisfy p b (i, j )= 1 s i p b (i, j )+p s (i, j ). 2 4 Rubinstein and Wolinsky (1990) suggest to specify the beliefs in such a way that, after observing an offer off the equilibrium path, each agent believes that the other agents stick to their equilibrium behavior. Under this restriction on beliefs, we are able to characterize a symmetric stationary sequential equilibrium in the game without the coordination device. Unfortunately, we have been unable to characterize the formation of cartels of buyers and sellers in that setting.

7 Buyers and sellers cartels on markets with indivisible goods 135 Similarly, p s (i, j ) is the maximum price accepted by buyer j and must satisfy 1 p s (i, j )= 1 1 p b (i, j )+1 p s (i, j ). s 2 j By the two preceding equations, the minimum price accepted by seller j is independent of the identity of the buyer i and the maximum price accepted by buyer i is independent of the identity of the seller j. Hence p b (i, j )=p b (j ) and p s (i, j )=p s (i). Replacing in the two equations, we get p b (j ) = 1 s t 1 p s (i) = 1 s j p b (j )+p s (i) 2 1 p b (j )+1 p s (i). 2 Hence, the strategies (p b, p s ) are independent of the identity of the buyers and sellers, and can be found as the unique solution to the system of equations p b = b 2s (p s + p b ) (1 p s ) = 1 2 (1 p s +1 p b ). It remains to check that the strategies (p b, p s ) form indeed a subgame perfect equilibrium. First note that no seller has an incentive to lower the price it offers and no buyer can benefit from increasing the price. Suppose next that a seller deviates and proposes a price p > p s. If the buyer rejects the offer, no trade is concluded at this period, and the buyer obtains her continuation value 1 2 (1 p s +1 p b )=1 p s > 1 p. Hence, the offer p will be rejected. Similarly, if a buyer offers a price p < p b, her offer will be rejected by the seller. Hence the strategy (p b, p s ) forms a subgame perfect equilibrium of the game. To complete the proof, it suffices to use the same arguments to obtain the unique stationary perfect equilibrium in the case s b. Proposition 1 allows us to compute the expected payoff of a seller, U (b, s) as a function of the number of buyers and sellers on the market. 5 U (b, s) = b(1 ) s(2 ) b if b s U (b, s) = b s b(2 ) s if s b. Figure 1 graphs the utility of a seller as a function of the number of sellers s on the market for various values of when b = 50. Notice that for all values of, this function has an inverted-s shape. The marginal effect of a reduction in the number of sellers on the price of the good is highest around the point 5 The expected payoff of a buyer is simply given by V (b, s) =1 U (b, s).

8 136 F. Bloch, S. Ghosal Fig. 1. where the market is balanced (b = s) and lowest when the numbers of buyers and sellers are very different. As converges to 1, the outcome of the bargaining game approaches the symmetric competitive solution, with a price of 1 if s < b, 0ifs > b and 1 2 if s = b. On the other hand, as converges to 0, the trading mechanism converges to a model where each agent makes a take-it-or-leave-itoffer, and the seller s expected payoff converges to 1 2 if b s and b 2s if b s.

9 Buyers and sellers cartels on markets with indivisible goods Collusion and cartel behavior In the model of trade we consider, agents benefit from the exclusion of traders on the same side of the market. We assume that cartels are formed precisely to withdraw some traders from the market and compensate them for their exclusion. These collusive arrangements, whereby some traders agree not to participate on the market, have commonly been observed among bidders in auctions. The wellknown phase of the moon mechanism, used by builders of electrical equipment in the 50 s, specified exactly which of the companies was supposed to participate in an auction at any period of time (see Mac Afee and Mac Millan 1992 and the references therein). Clearly, agreements to exclude traders from the market face two types of enforcement issues. First, the excluded traders could decide to renege on the agreement and reenter the market after receiving compensation. We assume that the market for the indivisible good opens repeatedly, so that this deviation can be countered by an appropriate dynamic punishment strategy. Second, members of the cartel could find it in their interest to organize different rounds of trade, and sell (or buy) the goods of the excluded traders in a second trading round. However, in equilibrium, this behavior will be perfectly anticipated by traders on the other side of the market. As the time between trading rounds goes to zero, the Coase conjecture indicates that trade should then occur as if no good was ever excluded from the market (see Gul et al. 1986). In order to avoid this problem, we assume that cartel members have access to a technology which allows them to credibly commit not to sell (or buy) the goods of excluded traders. For example, we could assume that buyers and sellers can ostensibly destroy their endowments in money and goods. Formally, we analyze in this section the formation of a cartel of sellers on the market. If a cartel K of size k forms on the market, and decides to withdraw r sellers, the total number of active sellers is given by s r since independent sellers always participate in trade. Hence the total surplus obtained by cartel members is given by U K (b, s) =(k r)u (b, s r). Members of the cartel K thus select the number r of traders excluded from the market, 0 r k, in order to maximize U K (b, s) = b(1 ) (k r) (s r)(2 ) b = b (s r) (k r) b(2 ) (s r) if r s b if r s b. This optimization problem is a simple integer programming problem, and it admits generically a unique solution. For a fixed value b and s of the total number of traders on the market, we let ρ(k) denote the optimal choice of traders excluded from the market by the cartel K.

10 138 F. Bloch, S. Ghosal Proposition 2. The optimal choice of a cartel K of sellers, ρ(k) is given by the following expressions. ρ(k) = 0 if s b 2 k, (3 2)b ρ(k) = max{0, s b} if s +2 k s ρ(k) = r (3 2)b if k s +2, where r is the first integer following the root of the equation b 2, (b (s r + 1))(b(2 ) (s r)) + b(k r)(1 ) =0. (1) Proof. In order to characterize the optimal choice of a cartel, we consider the incremental value of excluding one additional seller from the market f (r) =(k (r + 1))U (b, s (r + 1)) (k r)u (b, s r). Assume first that r s b 1, then sign f (r) = sign b (s k)(2 ). It thus appears that the sign of the incremental value of an additional exclusion is independent of r. Ifk s b b 2, f (r) 0 and, if k s 2, f (r) 0. Next suppose that r s b. Then sign f (r) = sign(1 )b(k r) (b (s r 1))(b(2 ) (s r)). Notice that the function f ( ) is a quadratic function in r and has at most two roots. We show that it has at most one root on the domain [s b, k]. First, notice that f (k) < 0. Next, note that, at r = s b(2 ) < s b, f (r) > 0. This implies that f ( ) has at most one root in the relevant domain. Furthermore, note that f (s b) < 0 if and only if (3 2)b k < s +2. Otherwise, as long as k s +2 (3 2)b, there is a unique root to the equation f (r) =f (r + 1) in the relevant domain. 3 2 Finally, note that < 2 so that we can summarize our findings as follows. If k s b (3 2)b 2, f (r) < 0 so that ρ(k) =0.Ifs+2 > k s b 2, f (r) > 0 for r s b and negative afterwards, so that ρ(k) = max{0, s b}. Finally, if k s +2 (3 2)b, there exists a unique integer r such that f (r) > 0 for all integers r < r and f (r) < 0 for all integers r r. Proposition 2 shows that three different situations may arise depending on the size k of the cartel and the numbers b and s of traders on the market. If k is small, the cartel has no incentive to restrict the number of traders on the market. If, on the other hand, k is large, the optimal choice of the cartel is to exclude

11 Buyers and sellers cartels on markets with indivisible goods 139 enough sellers from the market so that the number of active sellers becomes smaller than the number of active buyers. Finally, there exists an intermediate situation where the cartel chooses to restrict the number of sellers in order to match the number of buyers on the market. Notice that, if the discount parameter is too low, ( < 2 3 ), the cartel never chooses to restrict the number of sellers below the number of buyers. In fact, when is low, the increase in per unit surplus obtained by excluding sellers is too small to outweigh the decrease in total surplus due to the restriction in trade. We now derive some properties of the function ρ(k) assigning to each cartel of size k the number of traders withdrawn from the cartel. Lemma 1. For any two cartels k and k with k > k, ρ(k ) ρ(k). Proof. To prove that the function ρ( ) is weakly increasing, it suffices to check that it is monotonic for k s +2 (3 2)b. In that case, ρ(k) is defined by the unique root of Eq. (1). It is easy to see that this root is strictly increasing in k, so that the function ρ(k) is weakly increasing. Lemma 1 shows that the number of sellers withdrawn from the market is a weakly increasing function of the size of the cartel. Hence, larger cartels choose to exclude more sellers from the market. Lemma 2. For any k such that ρ(k) s b, ρ(k +1) ρ(k)+1. Proof. Suppose by contradiction ρ(k + 1) ρ(k) + 2. By definition of ρ(k + 1), we have (b (s ρ(k +1) 1))(b(2 ) (s ρ(k + 1)) k +1 ρ(k +1). b(1 ) Observe that the right hand side of the inequality is increasing in r. Since ρ(k +1) ρ(k) + 1, we thus have (b (s ρ(k)))(b(2 ) (s ρ(k) 1)) k +1 ρ(k +1). b(1 ) Next note that, since ρ(k +1) ρ(k)+2,k +1 ρ(k +1) k (ρ(k) + 1). Hence we obtain (b (s (ρ(k)+1) 1))(b(2 ) (s ρ(k) + 1)) k (ρ(k)+1), b(1 ) contradicting the fact that ρ(k) is the first integer following the root of Eq. (1). Lemma 2 shows that the total number of active sellers in the cartel (k ρ(k)) is an increasing function of the size of the cartel, as long as ρ(k) > s b. Hence, larger cartels put more active traders on the market. Alternatively, Lemma 2 shows that, for any value r of excluded members, where s b r ρ(n), there exists a unique cartel size κ(r) such that for any k <κ(r), ρ(k) < r, ρ(κ(r)) = r and, for any k >κ(r), ρ(k) r. The function κ(r) thus assigns to each number r of excluded traders, the minimum size of a cartel excluding r traders. We conclude this section by establishing the following Lemma.

12 140 F. Bloch, S. Ghosal Lemma 3. For any r, ρ(n) r max{0, s b}, κ(r +1)>κ(r)+1. Proof. Notice first, that, by Lemma 1, the function κ(r) is weakly increasing. Hence, to prove the Lemma, it suffices to show κ(r +1)/= κ(r) + 1. Suppose by contradiction that κ(r +1)=κ(r) + 1. Let k = κ(r). We then have ρ(k 1) < r and ρ(k +1)=k + 1. Hence we obtain (b (s r + 1))(b(2 ) (s r)) k 1 < r + b(1 ) (b (s r))(b(2 ) (s r 1)) k +1. b(1 ) After some manipulations, this system can be rewritten as b(2 ) (3 )(s r) k < r b(2 ) (3 )(s r) k r implying (s r) > b, + (s r)(s r +1) b(1 ) 1 (s r)(s r 1) + 1 b(1 ) which contradicts the fact that r s b. Lemma 3 shows that, for any fixed value r, there are at least two cartel sizes k and k such that ρ(k) =ρ(k )=r. Summarizing the findings of the three preceding lemmas, Fig. 2 depicts a typical function ρ(k) in the case where s > b. Fig. 2.

13 Buyers and sellers cartels on markets with indivisible goods Stable cartels In this section, we analyze the formation of cartels by buyers and sellers on the market. We first describe the noncooperative game of coalition formation, and then consider both the situation where a cartel is formed only on one side of the market and where cartels are formed on the two sides of the market. 3.1 Cartel formation The formation of the cartel is modeled as a noncooperative participation game. All traders on one side of the market simultaneously announce whether they want to participate in the cartel (C) or not (N). The cartel is formed of all the traders who have announced (C). A cartel K of size k > 1isstable if it is a Nash equilibrium outcome of the game of cartel formation. This simple game of cartel formation embodies the notions of internal and external stability proposed by d Aspremont et al. (1983). In a stable cartel, no insider wants to leave the cartel, and no outsider wants to join. It should be noted that this model imposes several crucial restrictions on the formation of cartels. First, it is assumed that only one cartel is formed. The issue of the formation of several cartels and the possible competition between cartels is ignored in the analysis. Second, our focus on the Nash equilibria of the game, excluding coordination between cartel members at the participation stage, implies that whenever a member leaves the cartel, all other cartel members remain in the cartel. Given that outsiders benefit from the formation of a cartel, this assumption makes deviations easy and thus leads to a more stringent concept of cartel stability than if we allowed cartel members to respond to a defection. On the other hand, by focussing on Nash equilibria, we ignore the possibility that traders deviate together. Hence, we make deviations harder than in games where coalitions of traders can cooperate 6. Once a cartel is formed, we assume that sellers share equally the profits of the cartel. Since all sellers are identical, this assumption can be made without loss of generality in the following way. If a stable cartel with unequal sharing exists, it must be that the cartel is also stable under equal sharing. Hence, the stable cartels under equal sharing are the only candidates for stable cartels under arbitrary sharing rules. On the other hand, note that a cartel may be stable under equal sharing but unstable under an alternative sharing rule. 7 6 In games of cartel formation, where the formation of a coalition induce externalities on the other traders, there is no simple way to define a stable cartel structure. The concept of stability depends on the assumptions made on the reaction of external players to a deviation. See Bloch (1997) for a more complete discussion of the alternative concepts of stability in games with externalities across coalitions. 7 In a different model of coalition formation, where traders make sequential offers consisting both of a coalition and the distribution of gains within the coalition, Ray and Vohra (1999) show that, when traders are symmetric, the assumption of equal sharing can also be made without loss of generality.

14 142 F. Bloch, S. Ghosal 3.2 Cartel formation on one side of the market We first analyze the formation of a cartel on one side of the market. As before, we assume that only sellers can organize on the market and characterize the stable cartels of sellers. Proposition 3. There exists at most one stable cartel sixe. If s < b, no cartel is stable. If s = b 2 3 2, no cartel is stable. If s = b 2 3 2, the unique stable cartel size is the first integer k following 2+ 2b(1 ) and ρ(k )=1.Ifs> b, the unique stable cartel size is the first integer k following s b 2 and ρ(k )=s b. Proof. Pick a cartel K of size k. First observe that, if there exists a number r such that κ(r) < k <κ(r + 1), the cartel K cannot be stable, since, following the departure of a member, the cartel of size k 1 still selects to exclude the same number r of sellers. Hence, the only candidates for stable cartels are the cartels of size k = κ(r) for some r s b. Suppose now that the cartel K of size k = κ(r) is indeed stable. Since no member wants to leave the cartel, k r U (b, s r) > U (b, s r +1). (2) k Furthermore, by Lemma 2, any cartel of size k 1 chooses ρ(k 1) = r 1, k r 1 k 1 U (b, s r) < k r U (b, s r +1). (3) k 1 Inequalities 2 and 3 imply k r k > k r 1. k r Rearranging, we obtain r 2 r 1 > k. Next observe that, using Eq. (1), we derive the following inequality k r > 2(r (s b)). By the two previous inequalities, we obtain O > r +2(r (s b))(r 1)), a condition which can only be satisfied by r =1ifs = b and r = s b is s > b. Hence, the only candidates for stable cartels are k = κ(1) if s = b and k = κ(s b) ifs > b. It is clear that no insider wants to leave the cartel. Furthermore, by Lemma 3, no outsider wants to join the cartel, since the addition of a new member does not change the cartel s optimal choice. Hence, the two cartels we have found are indeed stable.

15 Buyers and sellers cartels on markets with indivisible goods 143 Proposition 3 characterizes the unique stable cartel formed by sellers on the market. It appears that the formation of cartels cannot lead to a large imbalance in the number of buyers and sellers on the market. If sellers are initially on the long side of the market, the cartel they form leads to an equal number of active buyers and sellers on the market. If initially, buyers and sellers are present in equal numbers on the market, the cartel formed only leads to the exclusion of one seller. 8 As opposed to the classical model of Cournot competition with divisible goods studied by Salant et al. (1983), we show that, when the good traded is indivisible, there exists a stable cartel size. To understand the differences between the two models, it is useful to recall the free-rider problem associated to the formation of a cartel. Since outsiders obtain a higher payoff than insiders, most cartels are not sustainable, because members have an incentive to leave the cartel. As a result, the only sustainable cartels are those which break down entirely upon the departure of a member. These cartels can be characterized when the good traded is indivisible, but do not exist in models with divisible commodities. The characterization of Proposition 3 allows us to compute the stable cartel sizes when the market outcome converges to the competitive outcome. As approaches 1, the stable cartel sizes converge to 3 if s = b and to s b + 1 when s > b. It is easy to see that these are the minimal cartel sizes for which it is profitable to exclude some traders from the market. While the arguments underlying Proposition 3 rely on computations using the specific trading mechanism based on Rubinstein and Wolinsky s (1990) bargaining model, we believe that some of the conclusions are robust to changes in the model of trade. In fact, the characterization of the unique stable cartel relies on the fact that gains from excluding a trader are highest when the market is balanced. Hence, the trader s incentives to join a cartel are largest when the cartel forms to match the number of buyers and sellers or to exclude one trader when the market is originally balanced. The characterization of the unique stable cartel thus seems to rely primarily on the inverted S-shape of the function relating the utility of a trader to the number of traders on his side of the market. The extension of our results to other trading mechanisms is thus related to the shape of this function. The Shapley value of this market, analyzed by Shapley and Shubik (1969), also yields an inverted S-shape relation. Similarly, small perturbations around the competitive equilibrium solution also result in an inverted S shape relation. On the other hand, the distribution of the trading surplus obtained by Rubinstein and Wolinsky (1985) at the steady-state of a matching and bargaining market with entry yields a completely different picture. In that case, the utility of a seller is b 2(1 )b+s+b if s b and b 2(1 )s+s+b given by if s b. These functions are not inverted S-shape functions of the number s of sellers, and our conclusions do not extend to this trading mechanism. In fact, with this specification of gains 8 Note that, for a stable cartel to exist when s = b, the following condition must be satisfied: s = b If this condition is violated, the stable cartel size k is larger than s.

16 144 F. Bloch, S. Ghosal from trade, it turns out that no cartel is stable since it is never optimal to exclude any trader from the market. 3.3 Cartel formation on the two sides of the market In this section, we characterize the stable cartels formed on the two sides of the market. In order to analyze the response of agents on one side of the market to collusion on the other side, we adopt a sequential framework where buyers form a cartel first, and sellers organize in response to the formation of the buyers cartel. Hence, the sequence of stages in the game is as follows. First, buyers engage in the game of cartel formation; second, the buyers cartel selects the number of active traders; third, sellers engage in the game of cartel formation; fourth, the sellers cartel chooses the number of active traders, and finally buyers and sellers meet and trade on the market. In order to focus on the endogenous response of sellers to collusion on the part of buyers, we assume that the market is originally balanced and let n denote the initial number of buyers and sellers on the market. In order to analyze the behavior of the buyers, we note that, for any choice b of active buyers, there exists a unique stable cartel size on the side of sellers as given by Proposition 3. The expected utility obtained by a cartel of buyers K of size k can then be computed as U K = k if r = 0 and n 2 U K = ku (n 1, n) if r = 0 and n U K = k r for all k r > 0. 2 This definition of the cartel s surplus takes into account the reaction of sellers to the formation of the cartel of buyers. If buyers do not collude on the market, sellers either respond by forming a cartel which withdraws one trader from the market (if n ) or else do not collude. If buyers withdraw some traders from the market (r > 0), a cartel of sellers is formed in order to match the number of active buyers on the market. The definition of the cartel s surplus can be used to determine the optimal behavior of the cartel. First observe that, as long as n, for any size k of the cartel, the optimal restriction is ρ(k) = 0. When n 2 3 2, the cartel must choose between a restriction r = 0, yielding a surplus of ku (n 1, n) = k(n 1)(1 ) 2n(1 )+ and a restriction r = 1 yielding a payoff of k 1 2. Simple computations show that the optimal choice is given by 2n(1 )+ ρ(k) = 0 if k 2 2n(1 )+ ρ(k) = 1 if k. 2

17 Buyers and sellers cartels on markets with indivisible goods 145 Using the same arguments as in the proof of Proposition 3, one can easily show that the only stable cartel size is the first integer k following 2n(1 )+ 2. We then obtain the following Proposition. Proposition 4. When cartels are formed sequentially on the two sides of the market, there is at most one stable cartel configuration. If n 2 3 2, no cartel is stable. If n 2 3 2, buyers and sellers form cartels of size k where k is the first integer following 2n(1 )+ 2. Both the buyers and sellers cartels choose to withdraw one trader from the market. Proof. The determination of the buyers stable cartel follows from the preceding arguments. The determination of the sellers stable cartel is obtained by applying Proposition 3 to the case b = n 1, s = n. Proposition 4 shows that, when the numbers of buyers and sellers are initially equal, the stable cartels formed on the two sides of the market have the same size. Furthermore, both cartels select to withdraw one trader from the market so that bilateral collusion leads to a balanced market where the number of active buyers and sellers are identical. The requirement that the cartels formed be stable greatly limits the scope of collusion and the sizes of the cartels. For example, as converges to 1, the stable cartel sizes converge to 2: in equilibrium, the cartels formed by buyers and sellers only group two traders on both sides. In order to understand how the threat of collusion on one side of the market affects the incentives to form cartels on the other side, it is instructive to compare the sizes of the cartels obtained under bilateral collusion with the size of the cartel formed under unilateral collusion with b = s = n. It appears that the cartel formed under unilateral collusion is always larger than the cartels formed under bilateral collusion. Furthermore, the total numbers of trades obtained under unilateral and bilateral collusion are equal. We interpret this result by noting that there are limits to bilateral collusion. The threat of collusion on one side of the market does not lead to a higher level of collusion on the other side. In fact, there is no escalation process by which buyers would choose to form large cartels and to restrict trade by a large amount, anticipating the reaction of sellers on the market. 4 Conclusion This paper analyzes the formation of cartels of buyers and sellers in a simple model of trade inspired by Rubinstein and Wolinsky s (1990) bargaining model. We show that, when cartels are formed only on one side of the market, there is at most one stable cartel size. When cartels are formed sequentially on the two sides of the market, there is also at most one stable cartel configuration. It appears that, under bilateral collusion, buyers and sellers form cartels of equal sizes, and that the cartels formed are smaller than under unilateral collusion. Both cartels choose to exclude only one trader from the market. This result suggests that there are limits to bilateral collusion, and that the threat of collusion on one side of the market does not lead to increased collusion on the other side.

18 146 F. Bloch, S. Ghosal Our results thus show that the formation of a cartel of buyers induces the formation of a cartel of sellers yielding a balance in market power on the two sides of the market. Clearly, our model is much too schematic to account for the emergence of cartels of producers of primary commodities. However, we believe that our model gives credence to the view that these cartels were formed partly as a response to increasing concentration on the part of sellers. Furthermore, our results indicate that the cartels formed would only group a fraction of the active traders on the market, in accordance with the actual evidence at the time of the formation of OPEC, the Copper and Uranium cartels. While our analysis provides a first step into the study of bilateral collusion on markets with a small number of buyers and sellers, we are well aware of the limitations of our model. The process of trade we postulate, the specific model of cartel formation we analyze, allow us to derive some sharp characterization results, but clearly restrict the scope of our analysis. Furthermore, we assume in this paper that collusion results in the exclusion of traders from the market. In reality, the cartel can choose to enforce different collusive mechanisms. It could for example specify a common strategy to be played by its members at the trading stage or delegate one of the cartel members to trade on behalf of the other members. The analysis of these forms of collusion is a difficult new area of investigation in bargaining theory, and we plan to tackle this issue in future research. References d Aspremont, C., Jacquemin, A., Gabszewicz, J.J., Weymark, J. (1983) The stability of collusive price leadership. Canadian Journal of Economics 16: Bloch, F. (1997) Noncooperative models of coalition formation in games with spillovers. In: Carraro, C., Siniscalco, D. (eds.) New Directions in the Economic Theory of the Environment. Cambridge University Press, Cambridge Donsimoni, M.P. (1985) Stable heterogeneous cartels. International Journal of Industrial Organization 3: Donsimoni, M.P., Economides, N.S., Polemarchakis, H.M. (1986) Stable cartels. International Economic Review 27: Gabszewicz, J.J., Shitovitz, B. (1992) The core in imperfectly competitive economies. In: Aumann, R.J., Hart, S. (eds) Handbook of Game Theory with Economic Applications. Chap. 15, Elsevier Science, Amsterdam Galbraith, J.K. (1952) American Capitalism: The Concept of Countervailing Power. Houghton Mifflin, Boston Gul, F., Sonnenschein, H., Wilson, R. (1986) Foundations of dynamic monopoly and the coase conjecture. Journal of Economic Theory 39: Halloway, S.K. (1988) The Aluminium Multinationals and the Bauxite Cartel. Saint Martin s Press, New York Hart, S. (1974) Formation of cartels in large markets. Journal of Economic Theory 7: Legros, P. (1987) Disadavantageous syndicates and stable cartels. Journal of Economic Theory 42: Mac Afee, P., Mac Millan, J. (1992) Bidding rings. American Economic Review 82: Okuno, M., Postlewaite, A., Roberts, J. (1980) Oligopoly and Competition in Large Markets. American Economic Review 70: Ray, D., Vohra, R. (1999) A theory of endogenous coalition structures. Games and Economic Behavior 26:

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