Nonexclusive competition under adverse selection

Mariotti, Thomas; Salanié, François; Attar, Andrea Article Nonexclusive competition under adverse selection Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Mariotti, Thomas; Salanié, François; Attar, Andrea (2014) : Nonexclusive competition under adverse selection, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 9, Iss. 1, pp. 1-40, https://doi.org/10.3982/TE1126 This Version is available at: https://hdl.handle.net/10419/150213 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Theoretical Economics 9 (2014), 1–40 1555-7561/20140001 Nonexclusive competition under adverse selection Andrea Attar Toulouse School of Economics and Facoltà di Economia, Università degli Studi di Roma “Tor Vergata” Thomas Mariotti Toulouse School of Economics François Salanié Toulouse School of Economics A seller of a divisible good faces several identical buyers. The quality of the good may be low or high, and is the seller’s private information. The seller has strictly convex preferences that satisfy a single-crossing property. Buyers compete by posting menus of nonexclusive contracts, so that the seller can simultaneously and privately trade with several buyers. We provide a necessary and sufficient condition for the existence of a pure-strategy equilibrium. Aggregate equilibrium trades are unique. Any traded contract must yield zero profit. If a quality is actually traded, then it is efficiently traded. Depending on parameters, both qualities may be traded, or only one of them, or the market may break down to a no-trade equilibrium. Keywords. Adverse selection, competing mechanisms, nonexclusivity. JEL classification. D43, D82, D86. Andrea Attar: [email protected] Thomas Mariotti: [email protected] François Salanié: [email protected] We thank a co-editor, Johannes Hörner, and two anonymous referees for very thoughtful and detailed comments. We also thank David Bardey, Bruno Biais, Antoine Bommier, Catherine Casamatta, Pradeep Dubey, John Geanakoplos, Piero Gottardi, Martin Hellwig, David Martimort, Enrico Minelli, Alessandro Pavan, Nicola Pavoni, David Pérez-Castrillo, Gwenaël Piaser, Jean-Charles Rochet, Bernard Salanié, Larry Samuelson, Dimitri Vayanos, David Webb, and Robert Wilson for very valuable feedback. Finally, we thank seminar audiences at Collegio Carlo Alberto, Columbia University, ETH Zürich, European University Institute, Helsinki Center of Economic Research, London School of Economics and Political Science, LudwigMaximilians-Universität München, New York University, Princeton University, Rijksuniversiteit Groningen, Toulouse School of Economics, Universitat Pompeu Fabra, Université de Cergy-Pontoise, Université de Franche-Comté, and Universiteit van Tilburg, as well as conference participants at the 2011 CESifo Area Conference on Applied Microeconomics, the 2011 Conference of the Society for the Advancement of Economic Theory, the 2011 European Summer Symposium in Economic Theory, the 2011 IDEI/CSIO Workshop on Industrial Organization, the 2011 Toulouse Workshop of the Paul Woolley Research Initiative on Capital Market Dysfunctionalities, and the 2011 Universität Zürich-ETH Zürich Workshop in Honor of Ivar Ekeland for many useful discussions. Financial support from the Agence Nationale de la Recherche (Grant ANR09-BLAN-0358-01), the Chaire Marchés des Risques et Création de Valeur, the European Research Council (Starting Grant 203929-ACAP), and the Europlace Institute of Finance is gratefully acknowledged. Copyright ©2014 Andrea Attar, Thomas Mariotti, and François Salanié. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at http://econtheory.org. DOI: 10.3982/TE1126 2Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) 1. Introduction The recent financial crisis has spectacularly recalled that the liquidity of financial markets cannot be taken for granted, even for markets that usually attract many traders and on which exchanged volumes tend to be very high. For instance, Adrian and Shin (2010) document that the issuance of asset-backed securities declined from over 300 billion dollars in 2007 to only a few billion in 2009. Similarly, Brunnermeier (2009) emphasizes the severe liquidity dry-up of the interbank market over the 2007–2009 period, when many banks chose to keep their liquidity idle instead of lending it even at short maturities. It is tempting to associate these difficulties with asymmetries in the allocation of information among traders. Indeed, during the crisis, one of the banks’ main concerns was the unknown exposure to risk of their counterparties.1Moreover, structured financial products such as mortgage-backed securities, collateralized debt obligations, and credit default swaps often involve many different underlying assets, and their designers are likely to hold private information about their quality; this creates an adverse selection problem that reduces liquidity provision.2Finally, most of these securities are traded outside of organized exchanges on over-the-counter markets, with poor information on the trading volumes or on the net positions of traders. Hence agents are able to interact secretly with multiple partners, at the expense of information release. These two features, adverse selection and nonexclusivity, are at the heart of the present paper. Theoretical studies of adverse selection in competitive environments have mainly been developed in the context of two alternative paradigms. Akerlof (1970)studiesan economy where privately informed sellers and uninformed buyers act as price takers. All trades are assumed to take place at the same price. Competitive equilibria typically exist, but feature a form of market failure: because the market-clearing price must be equal to the average quality of the goods offered by the sellers, the highest qualities are generally not traded in equilibrium. It seems, therefore, natural to investigate whether such a drastic outcome can be avoided by allowing buyers to screen goods of different qualities. In this spirit, Rothschild and Stiglitz (1976) consider a strategic model in which buyers offer to trade different quantities at different unit prices, thereby allowing sellers to credibly communicate their private information. They show that low-quality sellers trade efficiently, while high-quality sellers end up trading a suboptimal, but nonzero quantity. For instance, on insurance markets, high-risk agents are fully insured, while low-risk agents obtain only partial coverage; no pure-strategy equilibrium exists if the proportion of low-risk agents is too high. The present paper revisits these classical approaches by relaxing the assumption of exclusive competition, which states that each seller is allowed to trade with at most one buyer. This assumption plays a central role in Rothschild and Stiglitz’s (1976) model, and is also satisfied in the simplest versions of Akerlof’s (1970) model, in which sellers can trade only one or zero unit of an indivisible good. However, situations where sellers can simultaneously and secretly trade with several buyers naturally arise on many 1See, among others, Taylor and Williams (2009) and Philippon and Skreta (2012). 2See Gorton (2009). There is also some evidence that lending standards and the intensity of screening have been progressively deteriorating with the expansion of the securitization industry in the pre-2007 years. See, for instance, Keys et al. (2010) and Demyanyk and Van Hemert (2011). Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 3 markets—one may even say that nonexclusivity is the rule rather than the exception. In addition to the contexts we have already mentioned, well known examples include the European banking industry, the U.S. credit card market, and the life insurance and annuity markets of several Organization for Economic Cooperation and Development (OECD) countries.3 Our aim is to study the impact of adverse selection in markets with such nonexclusive trading relationships. To do so, we allow for nonexclusive trading in a generalized version of Rothschild and Stiglitz’s (1976) model. This exercise is interesting per se: as we shall see, the reasonings that lead to the characterization of equilibria are quite different from those put forward by these authors. The results are also different: the equilibria we construct typically feature linear pricing, possibly with a bid–ask spread, and trading is efficient whenever it takes place. Alternatively, pure-strategy equilibria may fail to exist, as in Rothschild and Stiglitz (1976), and some types may be excluded from trade, as in Akerlof (1970). It might even be that the only equilibrium involves no trade. Our analysis builds on the following simple model of trade. There is a finite number of buyers, who compete for a divisible good offered by a seller.4The seller is privately informed of the quality of the good, which may be low or high. The seller’s preferences are strictly convex, but otherwise arbitrary, provided they satisfy a single-crossing property. Buyers compete by simultaneously posting menus of contracts, where a contract specifies both a quantity and a transfer. After observing the menus offered and taking into account her private information, or type, the seller chooses which contracts to trade. Our model encompasses pure-trade, insurance, and credit environments as special cases.5 In this context, we fully characterize the seller’s aggregate trades in any pure-strategy equilibrium. The contribution of this paper is twofold. First, we provide a necessary and sufficient condition for such an equilibrium to exist. This condition can be stated as follows: Let vbe the average quality of the good. Then a pure-strategy equilibrium exists if and only if, at the no-trade point, the low-quality type would be willing to sell a small quantity of the good at price v, whereas the high-quality type would be willing to buy a small quantity of the good at price v. Second, we show that there exists a unique aggregate equilibrium allocation. Each buyer earns zero profit in equilibrium. If the willingness to trade at the no-trade point varies enough across types, equilibria are firstbest efficient: the low-quality type sells the efficient quantity, while the high-quality type 3Detragiache et al. (2000) and Ongena and Smith (2000) document that multiple banking relationships have become very widespread in Europe. Rysman (2007) provides recent evidence of multi-homing in the U.S. credit card industry. Cawley and Philipson (1999) and Finkelstein and Poterba (2004) report similar findings for the U.S. life insurance market and the U.K. annuity market. The structure of annuity markets is of particular interest because some legislations explicitly rule out the possibility of designing exclusive contracts: for instance, on September 1, 2002, the U.K. Financial Services Authority ruled in favor of the consumers’ right to purchase annuities from suppliers other than their current pension provider (Open Market Option). 4We argue in Section 5 that our results extend to the case of multiple sellers, provided contracting is bilateral and private. 5The labels seller and buyers are only used for expositional purposes. On financial markets, one may sell as well as buy assets. This translates in our model to allowing for negative as well as positive quantities. We argue in Section 5 that our results extend to the case where only nonnegative quantities can be traded. 4Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) buys the efficient quantity. By contrast, if the two types have similar willingness to trade at the no-trade point, there is no trade in equilibrium. Finally, in intermediate cases, one type of seller trades efficiently, while the other type does not trade at all. These results suggest that under nonexclusivity, the seller may only signal her type through the sign of the quantity she proposes to trade with a buyer. This is, however, a very rough signalling device, which is only effective when one type acts as a seller, while the other type acts as a buyer. As a consequence, there is no equilibrium in which both types trade nonzero quantities on the same side of the market. In the context of insurance markets, for instance, this rules out situations in which both the low-risk and the high-risk agents purchase a basic policy at a medium price, with the high-risk agent purchasing on top of this a supplementary policy at a higher price. The general message is thus that nonexclusive competition exacerbates the adverse selection problem: if the first-best outcome cannot be achieved, a nonzero level of trade for one type can be sustained in equilibrium only if the other type is left out of the market. In particular, no cross-subsidization between types takes place in equilibrium. That is, each buyer earns zero profit on any contract he trades in equilibrium. To establish this result, we exhibit a class of deviations that make it possible for at least one buyer to keep trading with the type with which he would hypothetically make a profit, while minimizing the loss he would make with the other type by exploiting the equilibrium offers of his rivals. Overall, our analysis shows that a partial or complete market breakdown may arise under nonexclusive competition when buyers compete in arbitrary menu offers, with very few restrictions on the set of instruments available to them. Related literature The implications of nonexclusive competition have been extensively studied in moralhazard contexts. Following the seminal contributions of Hellwig (1983)andArnott and Stiglitz (1993), many recent works emphasize that in financial markets where agents can make noncontractible effort decisions, the impossibility of enforcing exclusive contracts can induce positive profits for financial intermediaries and a reduction in trades. Positive profits arise in equilibrium because none of the intermediaries can profitably deviate without inducing the agents to trade several contracts and select inefficient levels of effort.6The present paper rules out moral-hazard effects and argues that nonexclusive competition under adverse selection drives intermediaries’ profits to zero. Pauly (1974), Jaynes (1978), and Hellwig (1988) pioneered the analysis of nonexclusive competition under adverse selection. Pauly (1974) suggests that Akerlof-like outcomes can be supported in equilibrium when buyers are restricted to offer linear price schedules. Jaynes (1978) points out that the separating equilibrium characterized by Rothschild and Stiglitz (1976)isvulnerabletoentrybyaninsurancecompanyproposing additional trades that can be concealed from its competitors. He further argues that the nonexistence problem identified by Rothschild and Stiglitz (1976)canbeovercome if insurance companies can share the information they have about the agents’ trades. 6See, for instance, Parlour and Rajan (2001), Bisin and Guaitoli (2004), and Attar and Chassagnon (2009) for applications to credit and insurance markets. Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 5 Hellwig (1988) discusses the relevant extensive form for the interfirm communication game. Biais et al. (2000) study a model of nonexclusive competition among uninformed market-makers who supply liquidity to an informed insider whose preferences are quasilinear, and quadratic in the quantities she trades. Although our model encompasses this specification of preferences, we develop our analysis in a two-type framework, whereas Biais et al. (2000) consider a continuum of types. Despite the similarities between the two setups, their results stand in stark contrast to ours. Indeed, restricting attention to equilibria where market-makers post convex menus of contracts, they argue that nonexclusivity leads to a Cournot-like equilibrium outcome, in which each market-maker earns a positive profit. This is very different from our Bertrand-like equilibrium outcome, in which each traded contract yields zero profit. We postpone until Section 5.3 a more detailed comparison between these contrasting sets of results. Attar et al. (2011) consider a situation where a seller is endowed with one unit of a good, the quality of which she privately knows. The good is divisible, so that the seller may trade any quantity of it with any of the buyers, as long as she does not trade more than her endowment in the aggregate. Both the buyers’ and the seller’s preferences are linear in quantities and transfers. It is shown that pure-strategy equilibria always exist and that the corresponding aggregate allocations are generically unique. Depending on whether quality is low or high, and on the probability with which quality is high, the seller may either trade her whole endowment or abstain from trading altogether. Buyers earn zero profit in any equilibrium. These results offer a fully strategic foundation for Akerlof’s (1970) classic study of the market for lemons, based on nonexclusive competition. Besides equilibrium existence, a key difference with our setting is that equilibria in Attar et al. (2011) may exhibit nontrivial pooling and, hence, cross-subsidies across types. This reflects the notion that trades are subject to an aggregate capacity constraint. By contrast, the present paper considers a situation where the seller’s trades are unrestricted, as in a financial market where agents can take arbitrary positions. Another feature of our model is that we consider general preferences for the seller, provided that they are strictly convex and satisfy a single-crossing property. Thus the range of applications of the present paper is different than in Attar et al. (2011). In contemporaneous work, Ales and Maziero (2011) study nonexclusive competition in an insurance context similar to that analyzed by Rothschild and Stiglitz (1976). Relying on free-entry arguments, they argue that only the high-risk agent can obtain a positive coverage in equilibrium. This is consistent with the results derived in the present paper; however, a distinguishing feature of our analysis is that it is fully strategic and avoids free-entry arguments. Our results are also more general in that we do not rely on a particular parametric representation of the seller’s preferences, which allows us to uncover the common logical structure of a broad class of models.7 This paper also contributes to the common-agency literature that analyzes situations where several principals compete through mechanisms to influence the decisions 7For instance, a special feature of the insurance model is that efficiency requires that both types of agents be fully insured, whereas our analysis covers situations where efficiency requires that different types of sellers trade different quantities. 6Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) of a common agent. In our bilateral-contracting setting, the trades between the seller and the buyers are not public, and the seller may choose to trade with any subset of buyers. Moreover, in line with our focus on competitive environments, the profit of each buyer depends only on the trade he makes with the seller, not on the other trades his competitors may make with her. In the terminology of common agency, our model is thus a private and delegated common-agency game with no direct externalities between principals.8In contrast to most of the common-agency literature, our analysis yields a unique prediction for aggregate equilibrium trades and equilibrium payoffs. In our view, this uniqueness result is tied to three key ingredients of our model. First, there are no direct externalities between principals.9Second, each buyer’s profit is linear in the contract he trades; whereas if some convexity were introduced in the buyers’ preferences, then multiple equilibrium outcomes would arise even in a complete-information version of our model.10 Finally, each type of the seller cares only about the aggregate quantity she sells to the buyers and the aggregate transfer she receives in return, whereas if the buyers’ offers were not perfectly substitutable from the seller’s viewpoint, then one would again expect multiple equilibrium outcomes to arise even under complete information.11 Observe that these three assumptions are natural in a broad range of situations, including financial and insurance markets. Finally, it should be stressed that our uniqueness result obtains despite the fact that very few restrictions are imposed on the set of instruments available to the buyers, who are basically free to propose arbitrary menus of contracts. In this respect, our results contrast with the literature on supply-function equilibria, which considers oligopolistic industries where firms compete in supply schedules instead of simple price or quantity offers. Wilson (1979)andGrossman (1981) are the first to observe that this additional degree of freedom may significantly expand the set of equilibrium outcomes. Klemperer and Meyer (1989)andKyle (1989) suggest that the introduction of some uncertainty, either in the form of imperfect information over market demand or in the form of noise traders, may limit the multiplicity of equilibria. Vives (2011) develops these intuitions in a general setting where rational traders interact in the presence of idiosyncratic shocks; 8The distinction between delegated common-agency games, in which the agent can trade with any subset of principals, and intrinsic common-agency games, in which the agent must either trade with all principals or with none of them, was introduced by Bernheim and Whinston (1986). Martimort (2006) formulates the distinction between public-agency settings, in which each principal’s transfer can be made contingent on all the agent’s decisions, and private-agency settings, in which the transfer made by each principal is only contingent on the trades that the agent makes with him. Finally, the role of direct externalities between principals has been emphasized by Martimort and Stole (2003) and Peters (2003). 9Direct externalities between principals typically lead to multiple equilibrium outcomes even in complete-information environments, as shown by Martimort and Stole (2003) and Segal and Whinston (2003). 10This setting is analyzed by Chiesa and Denicolò (2009), who show that although the aggregate quantity traded in equilibrium always coincides with the first-best quantity, equilibrium transfers and payoffs are not uniquely determined. 11Examples in this direction are provided by d’Aspremont and Dos Santos Ferreira (2010),whoprovide a strategic analysis of competition between firms selling differentiated goods to a representative consumer under complete information, both in the cases of intrinsic and delegated agency. Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 7 he shows that there exists a unique symmetric equilibrium in which supply functions are linear. The paper is organized as follows. Section 2 describes the model. Section 3 characterizes pure-strategy equilibria. Section 4 derives necessary and sufficient conditions under which such equilibria exist. Section 5 discusses extensions of our analysis, imposing nonnegative trades or allowing for multiple sellers and more than two types. Section 6 concludes. 2. The model Our model features a seller who can simultaneously trade with several identical buyers. To simplify the general description and the analysis of the model, in most of the paper, and unless otherwise mentioned, we impose no restriction on the sign of the quantities traded by the seller or, for that matter, on the sign of the transfers she receives in return. The labels seller and buyers, although useful, are, therefore, to a large extent conventional. In some of the applications presented in Section 2.4, however, it is more natural to impose that quantities traded be nonnegative. As explained in Section 5.1, our analysis and results extend to these cases as well, with minor modifications. Which assumption is more appropriate should be clear from the context. 2.1 The seller The seller is privately informed of her preferences. She may be of two types, Lor H,with positive probabilities mLand mHsuch that mL+mH=1.Subscriptsiand jare used to index these types, with the convention that i= j. Each type cares only about the aggregate quantity Qshe sells to the buyers and the aggregate transfer Tshe receives in return. Type i’s preferences over aggregate quantity-transfer bundles (QT) are represented by a utility function uidefined over R2.Foreachi, we assume that uiis continuously differentiable, with ∂ui/∂T > 0,andthatuiis strictly quasiconcave. Hence, type i’s marginal rate of substitution of the good for money τi≡−∂ui/∂Q ∂ui/∂T is everywhere well defined and strictly increasing along her indifference curves. Note that τi(Q T ) can be interpreted as type i’s marginal cost of supplying a higher quantity, given that she already trades (Q T ). We impose no restriction on the sign of τi(QT). The following assumption is key to our results. Assumption SC. For each (Q T ),τH(QT) > τL(Q T ). Assumption SC expresses a strict single-crossing property: type His less eager to sell a higher quantity than type Lis. As a result, in the (QT) plane, a type-Hindifference curve crosses a type-Lindifference curve only once, from below. 8Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) 2.2 The buyers There are n≥2identical buyers. There are no direct externalities between them: each buyer cares only about the quantity qhe purchases from the seller and the transfer t he makes in return. Each buyer’s preferences over individual quantity-transfer bundles (q t) are represented by a linear profit function: if a buyer receives from type ia quantity qand makes a transfer tin return, he earns a profit viq−t. We impose no restriction on the sign of vi. The following assumption will be maintained throughout the analysis. Assumption CV. We have vH>v L. We let v≡mLvL+mHvHbe the average quality of the good, so that vH>v>v L.Assumption CV reflects common values: the seller’s type has a direct impact on the buyers’ profits. Together with Assumption SC,Assumption CV captures a fundamental trade-off of our model: type Hprovides a more valuable good to the buyers than type L,butata higher marginal cost. These assumptions are natural if we interpret the seller’s type as the quality of the good she offers. Together, they create a tension that will be exploited later on: Assumption SC leads type Hto offer less of the good, but Assumption CV induces buyers to demand more of the good offered by type H, if only they could observe quality. 2.3 The nonexclusive trading game Trading is nonexclusive in that no buyer can control and a fortiori contract on the trades that the seller makes with other buyers. The timing of events is as follows. First, buyers compete in menus of contracts for the good offered by the seller.12 Next, the seller can simultaneously trade with several buyers. Accordingly, the extensive form is as follows. 1. Each buyer kproposes a menu of contracts, that is, a set Ck⊂R2of quantitytransfer bundles that contains at least the no-trade contract (00).13 2. After privately learning her type, the seller selects one contract from each of the menus Ckoffered by the buyers. A pure strategy for type iis a function that maps each menu profile (C1Cn) into a vector of contracts ((q1t1)(qntn)) ∈C1×···×Cn.Toensurethattypei’s utility-maximization problem maxui k qk k tk:(qktk)∈Ckfor each k always has a solution, we require the buyers’ menus Ckto be compact sets. This allows us to use perfect Bayesian equilibrium as our equilibrium concept. Throughout the paper, we focus on pure-strategy equilibria. 12As shown by Peters (2001) and Martimort and Stole (2002), there is no need to consider more general mechanisms in this multiple-principal single-agent setting. 13This requirement allows one to deal with participation in a simple way. It reflects the fact that the seller cannot be forced to trade with any particular buyer. Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 15 3.2 The zero-profit result In any Bertrand-like setting, the usual argument consists in making buyers compete for any profit that may result from serving the whole demand. This also applies to our setting, although the logic is different. Specifically, the following zero-profit result obtains. Proposition 2. In any equilibrium, B=0,sothatbk=0for each k. Proof. Denote type-by-type aggregate profits by Bi≡kbk iand recall that the expected aggregate profit is denoted by B. We first prove that for each jand k, Bj>b k jimplies B−bk≤miSi(5) Indeed, if Bj>b k j,buyerkcan deviate by proposing a menu that consists of the no-trade contract, and of the contracts ck i=(qk itk i+εi)and ck j=(QjTj+εj), for some positive numbers εiand εj.BecauseUj≥z−k j(qk itk i)and the function z−k jis continuous, it is possible, given the value of εj, to choose εismall enough so that type jtrades ck jfollowing buyer k’s deviation. Turning now to type i, observe that she must trade either ck ior ck jfollowing buyer k’s deviation: indeed, because εiis positive, type istrictly prefers ck i to any contract she could have traded with buyer kbefore the deviation. If type iselects ck i,thenbuyerk’s profit from this deviation is mi(bk i−εi)+mj(Bj−εj),which,because Bj>b k jby assumption, is strictly higher than bkwhen εiand εjare small enough, a contradiction. Therefore, type imust select ck jfollowing buyer k’s deviation and for this deviation not to be profitable, one must have vQj−Tj−εj≤bk. In line with (4), this can be rewritten as B−miSi−εj≤bk,fromwhich(5) follows by letting εjgo to zero. Now, if B>0,thenB>b kfor some k.BecauseSi≤0and Sj≤0by Proposition 1,it follows from (5) that Bi≤bk iand Bj≤bk jfor each k. Averaging over types yields B≤bk for each k, a contradiction. Hence the result.  The intuition for Proposition 2 can easily be understood in the context of a freeentry equilibrium. Indeed, suppose, for instance, that the aggregate profit from trading with type jis positive, Bj>0.ThenanentrantcouldproposetobuyQjin exchange for a transfer slightly above Tj. This contract would certainly attract type j,whichwould benefit the entrant; in equilibrium, it must, therefore, be that this trade also attracts type iand that vQj−Tj≤0. Now recall that the aggregate profit can be written as B= vQj−Tj+miSi. Our first result in Proposition 1 is that Si≤0and we just argued that vQj−Tj≤0when Bj>0. Hence the aggregate profit must be zero. Proposition 2 shows that the same result holds when the number of buyers is fixed, which is not a priori obvious. In line with the proof of Proposition 1, the proof of Proposition 2 amounts to showing that if Bis positive, then at least one buyer must have a profitable deviation. Remark. An inspection of the proofs of Propositions 1and 2reveals that these results require only weak assumptions on feasible trades, namely that if the quantities qand q are tradable, then so are the quantities q+qand q−q. Hence we allow for negative and 16 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) positive trades, but we may, for instance, have integer constraints on quantities. Finally, we did use in Lemma 1 the fact that the functions uiand, thus, the functions z−k iare continuous with respect to transfers, but, for instance, we did not use the fact that the seller’s preferences are convex. 3.3 Pooling versus separating equilibria We say that an equilibrium is pooling if both types of the seller trade the same aggregate quantity, QL=QH, and that it is separating if they trade different aggregate quantities, QL>Q H. We now investigate the basic price structure of these two kinds of candidate equilibria. Lemma 2. The following statements hold. •In any pooling equilibrium, TL=vQL=TH=vQH. •In any separating equilibrium, the following cases occur. (i) If QL>0>Q H,thenTL=vLQLand TH=vHQH. (ii) If QL>Q H≥0,thenTH=vQHand TL−TH=vL(QL−QH). (iii) If 0≥QL>Q H,thenTL=vQLand TH−TL=vH(QH−QL). The first statement of Lemma 2 is an immediate consequence of the zero-profit result. Otherwise, the equilibrium is separating and the three cases may, in principle, arise. In case (i), type Lsells a positive quantity QL, while type Hbuys a positive quantity |QH|. There are no cross-subsidies in equilibrium, as each type itrades at the fair price vi.In case (ii), everything happens as if, in the aggregate, both types were selling a quantity QHat the fair price v,withtypeLselling an additional quantity QL−QHat the fair price vL. Two scenarios are conceivable. If QH>0, there are cross-subsidies in equilibrium, with BL<0<B H. In that case, the structure of aggregate equilibrium trades is similar to that obtained by Jaynes (1978)andHellwig (1988) in a nonexclusive version of Rothschild and Stiglitz’s (1976) model, where insurance companies can share information about their clients. It is also reminiscent of the equilibrium of the limit-order book analyzed by Glosten (1994). Further results in Section 3.4 rule out this scenario and, more generally, any equilibrium in which both types trade nonzero quantities on the same side of the market. Alternatively, if QH=0, the structure of aggregate equilibrium trades is similar to that which prevails in a two-type version of Akerlof’s (1970) model when adverse selection is severe. Finally, case (iii) is the mirror image of case (ii). 3.4 The no-cross-subsidization result In this section, we prove that our nonexclusive competition game has no equilibrium with cross-subsidies. We first establish that the aggregate profit earned on each type must be zero in equilibrium. As discussed below, this drastically reduces the set of candidate equilibria. We then refine this result by showing that any traded contract must actually yield zero profit in equilibrium. Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 17 The first step of the analysis consists of showing that if buyers make profits in the aggregate when trading with type j,thentypejmust trade inefficiently in equilibrium. Specifically, her marginal rate of substitution at her aggregate equilibrium trade is not equal to the quality of the good she sells, but rather to the average quality of the good. Lemma 3. If in equilibrium Bj>0,thenτj(QjTj)=v. The intuition for Lemma 3 is as follows. If τj(QjTj)were different from v,thenany buyer could propose a contract in the neighborhood of (QjTj)that would attract type j, thereby generating a positive profit close to Bj, and that would generate a small positive profit even if it were traded by both types. This, however, is impossible according to the zero-profit result. Remark. Consider a candidate equilibrium in which Bj>0and let kbe such that bk j>0. Then, for any such buyer k,typeicould get her equilibrium utility Ui= z−k i(qk itk i)by trading (qk jtk j)instead of (qk itk i)with buyer k.Thatis, Ui=z−k i(qk itk i)=z−k i(qk jtk j) (6) which can be interpreted as a binding incentive compatibility constraint, taking into account the nonexclusivity of trades. (Note the difference with the exclusive competition case, in which incentive constraints bear only on aggregate quantities, Ui=ui(QiTi)≥ ui(QjTj).) The argument goes as follows. Suppose that z−k i(qk itk i)>z −k i(qk jtk j).Then buyer kcould deviate by proposing a menu that consists of the no-trade contract and of the contracts ck i=(qk itk i−εi)and ck j=(qk jtk j+εj)for some positive numbers εiand εjsuch that miεi>m jεj. Clearly, type jselects ck jfollowing buyer k’s deviation. Turning now to type i, observe that because z−k i(qk itk i)>z −k i(qk jtk j)and the function z−k iis continuous, she is better off selecting ck irather than ck jfollowing buyer k’s deviation as long as εiand εjare small enough. If she decides to trade ck i,thenbuyerkmakes a positive profit miεi−mjεj.Thustypeimust not trade with buyer kfollowing his deviation and for this deviation not to be profitable, one must have vjqk j−tk j−εj≤0. Letting εj go to zero, we get bk j≤0. By contraposition, (6) must hold as soon as bk j>0. The second step of the analysis consists of showing that if buyers make profits in the aggregate when trading with type j, then the aggregate trade made by type jin equilibrium must remain available if any buyer withdraws his menu offer. In our oligopsony model, this rules out Cournot-like outcomes in which the buyers share the market in such a way that each of them needs to provide type jwith her aggregate equilibrium trade, as is the case in the equilibrium described in Biais et al. (2000).Thisismoreinthe spirit of Bertrand competition, where cross-subsidies are harder to sustain. Lemma 4. In equilibrium, if Bj>0,thenforeachk, the seller can trade (QjTj)with buyers other than k. 18 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) The proof of Lemma 4 proceeds as follows. First, we show that if Bjis positive, then the equilibrium utility of type jmust remain available following any buyer’s deviation; the reason for this is that otherwise, a buyer could deviate and reap the aggregate profit on type j.Asaresult,foranybuyerk, there exists an aggregate trade (Q−kT−k)with buyers other than kthat allows buyer jto achieve the same level of utility as in equilibrium, uj(Q−kT−k)=Uj. From the strict quasiconcavity of uiand Lemma 3,weobtain that if Q−k= Qj,thenT−k>vQ −k. We finally show that this would allow buyer kto profitably deviate by pivoting on (Q−kT−k). We are now ready to state and prove the main result of this section. Proposition 3. In any equilibrium, Bj=0for each j. Proof. Suppose,tothecontrary,thatBj>0for some j. Then any buyer ksuch that bk j>0can deviate by proposing a menu that consists of the no-trade contract and of the contracts ck i=(Qi−Qj+δivi(Qi−Qj)+εi)and ck j=(qk jtk j+εj)for some numbers δi,εi,andεj. Choose δiand εisuch that τi(QiTi)δi<ε i. This ensures that when δi and εiare small enough, type ican strictly increase her utility by trading ck iwith buyer kand trading (QjTj)with buyers other than k, thereby trading (Qi+δiTi+εi)in the aggregate; according to Lemma 4, this is feasible as Bj>0.BecauseUi≥z−k i(qk jtk j)and the function z−k iis continuous, it is possible, given the values of δiand εi, to choose εj positive and small enough so that type itrades ck ifollowing buyer k’s deviation. Turning now to type j, observe that she must trade either ck ior ck jfollowing buyer k’s deviation: indeed, because εjis positive, type jstrictly prefers ck jto any contract she could have traded with buyer kbefore the deviation. If type jselects ck j, then buyer k’s profit from this deviation is mi(viδi−εi)+mj(vjqk j−tk j−εj),which,becausevjqk j−tk j=bk j>0 by assumption, is positive when δi,εi,andεjare small enough, in contradiction to the zero-profit result. Therefore, type jmust select ck ifollowing buyer k’s deviation and for this deviation not to be profitable, one must have v(Qi−Qj+δi)−vi(Qi−Qj)−εi≤0(7) Now, recall that as a consequence of Assumption SC,(v −vi)(Qi−Qj)≥0. Therefore, letting δiand εigo to zero in (7), we get Qi=Qjand, hence, the equilibrium must be pooling. Using the equality Qi=Qjto simplify (7), we obtain that for any small enough δiand εisuch that τi(QiTi)δi<ε i,onehasvδi≤εi.Asδican be positive or negative, it follows that τi(QiTi)=v. However, according to Lemma 3,onealsohasτj(QjTj)=vas Bj>0.Because(QiTi)=(QjTj), this contradicts Assumption SC. Hence the result.  Along with Lemma 2, this no-cross-subsidization result leads to the conclusion that one must have QH≤0≤QLin any equilibrium. This excludes two types of equilibrium outcomes that have been emphasized in the literature: first, pooling outcomes such as the one described in Attar et al. (2011), in which both types trade the same nonzero quantity at a price equal to the average quality of the good; second, separating outcomes such as the one described by Jaynes (1978), Hellwig (1988), and Glosten Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 19 Figure 1. Depiction of a candidate Jaynes–Hellwig–Glosten equilibrium with QL>Q H>0. (1994), and illustrated in Figure 1. If one leaves aside the case in which both types trade nonzero quantities on opposite sides of the market, the remaining possibilities for equilibrium outcomes are either that there is no trade in the aggregate or that only one type actively trades at a fair price in the aggregate. To illustrate the logic of the no-cross-subsidization result, consider a candidate separating equilibrium with positive quantities QL>Q H>0, as illustrated in Figure 1.The basic price structure of such an equilibrium is delineated in Lemma 2(ii). Let kbeabuyerwhoseprofitbk Hfrom trading with type His positive. According to Lemma 4, the aggregate trade (QHTH)remains available if buyer kremoves his menu offer. He can thus attempt to pivot on (QHTH)to attract type L,whichamountsto offering a contract ck L=(QL−QHTL−TH+εL)for some positive number εL.WhenεL is small enough, the loss for buyer kfrom trading ck Lwith type Lis negligible, as the slope 20 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) of the line segment that connects (QHTH)and (QLTL)is the fair price vL.Forbuyer k’s deviation to be profitable, he must make a profit when trading with type H.Todo so, he can offer an additional contract ck H=(qk Htk H+εH)for some positive number εH. Because (qk Htk H)was available for trade in equilibrium, ck Lis more attractive than ck Hfor type Las long as εLis large enough relative to εH.Now,iftypeHtrades ck H, the deviation is profitable, because when εHis small enough, ck Hyields a profit close to bk H>0when traded by type H, whereas the loss from trading ck Lwith type Lis negligible. If type Htrades ck Linstead, the deviation is still profitable, because ck Lyields a positive profit when traded by both types. This shows that there exists no separating equilibrium with positive quantities. The reasoning for a pooling equilibrium is slightly more involved, but reaches the same conclusion. Remark. The proof of Proposition 3 shows that cross-subsidies are not sustainable in equilibrium because it would otherwise be possible for some buyer to neutralize the type on which he makes a loss by proposing that she mimic the behavior of the other type when facing the other buyers. A key feature of this deviation is that it is performed by a buyer who is actively and profitably trading with one type in equilibrium.17 Moreover, it is crucial for the argument that this buyer deviates to a menu that includes two nontrivial contracts targeted at the two types of sellers. Observe that this class of deviations was not considered in the early contributions of Jaynes (1978)and Glosten (1994). Jaynes (1978), who studies strategic competition between insurance providers under nonexclusivity, indeed restricts firms to the use of simple insurance policies. That is, each firm can propose at most one contract that is different from the no-trade contract.18 As a consequence, an incumbent firm cannot profitably deviate by simultaneously making a loss when trading with the high-risk agent and compensating for this loss when trading with the low-risk agent. Glosten (1994) characterizes an aggregate price–quantity schedule that is robust to entry. In our setting, this schedule would be as depicted in Figure 1. By contrast, we do not take the aggregate price– quantity schedule as given, but we derive it from the individual menus offered by the buyers. So far, we have focused on the aggregate equilibrium implications of our model. We now briefly sketch a few implications for individual equilibrium trades. The following result shows that each traded contract yields zero profit and that aggregate and individual equilibrium trades have the same sign. 17It is unclear that an entrant would be able to upset the above candidate equilibrium. One might think that an entrant could successfully attempt to nearly reap the aggregate profit on type H, say by proposing a contract of the form (QHTH+εH), while making limited losses on type Lby proposing a contract of the form (QL−QHTL−TH+εL)as above. Yet this would overlook the fact that by proposing such a contract to type H, the entrant would globally modify the structure of available trades, unlike the local deviation (qk Htk H+εH)we used in the proof of Proposition 3. As a result, type Lmight well be attracted by the contract (QHTH+εH)because she may find it profitable to trade this contract along with some contracts offered by the incumbents, thereby upsetting the attempt at a successful entry. 18This assumption is maintained in the reformulation of Jaynes (1978)proposedbyHellwig (1988). Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 21 Proposition 4. In any equilibrium, bk j=0and qk L≥0≥qk Hfor all jand k. Proposition 4 reinforces the basic insight of our model, according to which, in equilibrium, the seller can signal her type only through the sign of the quantities she trades. It follows that if a type does not trade in the aggregate, then she does not trade at all. Hence a pooling equilibrium, when it exists, is actually a no-trade equilibrium. 3.5 Aggregate equilibrium trades In this section, we fully characterize the candidate aggregate equilibrium trades and we provide necessary conditions for the existence of an equilibrium. Given the price structure of equilibria delineated in Section 3.3 and the no-cross-subsidization result established in Section 3.4, all that remains to be done is to give restrictions on each type’s equilibrium marginal rate of substitution. Two cases need to be distinguished, according to whether a type’s aggregate trade is zero in equilibrium. Ourfirstresultisthatiftypejdoes not trade in the aggregate, then her equilibrium marginal rate of substitution must lie between vand vj. This is why an equilibrium may fail to exist for some parameter values. Lemma 5. In equilibrium, if Qj=0,thenvj−τj(00)and τj(00)−vhave the same sign. The intuition for Lemma 5 is as follows. Suppose, for instance, that QH=0.If vH>τ H(00), then any buyer could attract type Hby proposing a contract that offers to buy a small positive quantity at a unit price lower than vH. For this deviation not to be profitable, type Lmust also trade this contract, and one must have τH(00)≥v,so that the deviator makes a loss when both types trade this contract. The same reasoning applies if vH<τ H(00), by considering a contract that offers to sell a small positive quantity at a unit price higher than vH.ThecaseQL=0canbehandledinasymmetric way. Our second result is that if type itrades a nonzero quantity in the aggregate, then she must trade efficiently in equilibrium. Lemma 6. In equilibrium, if Qi= 0,thenτi(QiTi)=vi. The intuition for Lemma 6 is as follows. Suppose, for instance, that QL>0. As crosssubsidization cannot occur in equilibrium, TL=vLQL.IftypeLwere trading inefficiently in equilibrium, that is, if τL(QLTL)= vL, then there would exist a contract that offers to buy a positive quantity at a unit price lower than vL, and that would give type La strictly higher utility than (QLTL). Any of the buyers could profitably attract type L by proposing this contract, which would be even more profitable for the deviating buyer if traded by type H.HencetypeLmust trade efficiently in equilibrium. The case QH<0 can be handled in a symmetric way. To state our characterization result, it is necessary to define first-best quantities. The following assumption ensures that these quantities are well defined. 22 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) Assumption FB. For each i, there exists Q∗ isuch that τi(Q∗ iviQ∗ i)=vi. Assumption FB states that Q∗ iis the efficient quantity for type ito trade at a unit price vithat gives an aggregate zero profit for the buyers. An important consequence of the strict quasiconcavity of uiis that Q∗ i≥0if and only if τi(00)≤vi,andthatQ∗ i=0 if and only if τi(00)=vi. In the pure-trade model, Q∗ iis defined by c i(Q∗ i)=vi.Inthe insurance model, because of the agent’s risk aversion, efficiency requires full insurance for each agent i,sothatQ∗ i=WG−WB. The credit model is special in that the constraint that quantities must remain nonnegative may be binding. Efficiency requires that the net present value of the project, πifi(T ) −T, be maximized; if this leads to a positive and finite investment, the promised repayment Q∗ ithat makes the investors just break even satisfies πif i(πiQ∗ i)=1. In contrast, if πif i(0)≤1,borroweri’s investment project has a nonpositive net present value and it is efficient not to invest in her project. We can now state our main characterization result. Theorem 1. If an equilibrium exists, then τL(00)≤v≤τH(00). Moreover, the following statements hold. •If vL≤τL(00)≤v≤τH(00)≤vH, all equilibria are pooling, with QL=QH=0. •Otherwise, all equilibria are separating and the following cases occur. (i) If τL(00)<v L<v<v H<τ H(00),thenQL=Q∗ L>0and QH=Q∗ H<0. (ii) If τL(00)<v L<v≤τH(00)≤vH,thenQL=Q∗ L>0and QH=0. (iii) If vL≤τL(00)≤v<v H<τ H(00),thenQL=0and QH=Q∗ H<0. The first message of Theorem 1 is negative: the nonexclusive competition game need not have an equilibrium. A necessary condition for an equilibrium to exist is that at a price equal to the average quality v,typeLwould like to sell some of the good, whereas type Hwould like to buy some of it. In the pure-trade model, no equilibrium exists if c L(0)>vor c H(0)<v, that is, if the low-cost seller Lis not eager enough to sell or if the high-cost seller His too eager to sell. In the insurance model, no equilibrium exists if [πH/(1−πH)]u(WG)/u(WB)<π/(1−π),whereπ≡mLπL+mHπH,thatis,if the low-risk agent His too eager to buy insurance.19 In the credit model, no equilibrium exists if πf  H(0)>1, where again π≡mLπL+mHπH, that is, if the low-defaultrisk borrower His too eager to invest.20 Overall, Theorem 1 reinforces the insight of the no-cross-subsidization result: an equilibrium exists only if the adverse selection problem is severe enough so that both types’ incentives to trade are not too closely aligned. On a more positive note, we show in Theorem 2 that the necessary condition τL(00)≤v≤τH(00)also turns out to be sufficient for the existence of an equilibrium. 19This result is also obtained in Ales and Maziero (2011), assuming free entry. The second existence condition τL(00)≤vor, equivalently, [πL/(1−πL)]u(WG)/u(WB)≤π/(1−π), is automatically satisfied in the insurance model as π>π Land u(WB)>u (WG). 20The second existence condition τL(00)≤vor, equivalently, πf  L(0)≥1, is irrelevant in the credit model because the borrower cannot raise negative amounts of capital; see Section 5.1 and the Appendix. Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 23 Figure 2. Depiction of the structure of equilibrium aggregate trades as a function of τL(00) and τH(00)>τ L(00)for fixed parameters vL,vH,andv. Thus Theorem 1 provides a complete description of the structure of possible aggregate equilibrium outcomes, which is summarized in Figure 2. The second message of Theorem 1 is that pooling additionally requires vL≤τL(00) and vH≥τH(00); by the no-cross-subsidization result, we already know that a pooling equilibrium involves no trade for both types. The conditions vL≤τL(00)and vH≥τH(00)together imply that Q∗ L≤0≤Q∗ H. When one of these inequalities is strict, the first-best quantities are not implementable. Thus pooling requires a strong form of nonresponsiveness: namely, in the first-best scenario, type Lwould like to buy and type Hwould like to sell. This cannot arise in the insurance model, for in that case Q∗ L=Q∗ H=WG−WB. Therefore, the insurance model admits no pooling equilibrium. In 24 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) the pure-trade model, a pooling equilibrium requires that c L(0)≥vLand c H(0)≤vH.21 In the credit model, a pooling equilibrium requires that πLf L(0)≤1or, equivalently, that the investment project of the high-default-risk borrower Lhas nonpositive net present value.22 The third message of Theorem 1 is that in a separating equilibrium, at least one of the types trades efficiently. In case (i), the preferences of types Land Hare sufficiently far apart from each other, in the sense that Q∗ L>0>Q ∗ H: in the first-best scenario, type Lwould like to sell and type Hwould like to buy—a strong form of responsiveness. In that case, both types end up trading their first-best quantities in equilibrium. Clearly, neither the insurance model nor the credit model admits an equilibrium of this kind. In the pure-trade model, a first-best equilibrium exists if c L(0)<v Land c H(0)>v H.In case (ii), both Q∗ Land Q∗ Hare nonnegative: in the first-best scenario, both types would like to sell. The seller’s preferences may or may not satisfy responsiveness. The unique candidate equilibrium outcome is then that seller Ltrades efficiently, while seller H does not trade at all. This is the situation that prevails in the insurance model when an equilibrium exists: in that case, the high-risk agent Lobtains full insurance at an actuarially fair price, while the low-risk agent Hpurchases no insurance. In the puretrade model, this type of equilibrium exists only if c L(0)<v Land c H(0)≤vH.Inthe credit model, this type of equilibrium exists only if πLf L(0)>1, that is, if the investment project of the high-default-risk borrower Lhas positive net present value.23 Finally, case (iii) is symmetric to case (ii), exchanging the roles of types Land H. Note that in any separating equilibrium, each type strictly prefers her aggregate equilibrium trade to that of the other type. This contrasts with the predictions of models of exclusive competition under adverse selection, such as Rothschild and Stiglitz’s model (1976), in which the high-risk agent Lis indifferent between her equilibrium contract and that of the lowrisk agent H. Remark. It is interesting to compare the conclusions of Theorem 1 with those reached by Attar et al. (2011) in a nonexclusive version of Akerlof’s (1970) market for lemons. Compared to the present setup, the two distinguishing features of their model is that the seller has linear preferences, ui(Q T ) =T−θiQ, and makes choices under an aggregate capacity constraint, Q≤1. Observe that in this context, type i’s marginal rate of substitution is constant and equal to θiup to capacity. In a two-type version of their model in which there are potential gains from trade for each type, that is, vL>θ Land vH>θ H, Attar et al. (2011) show that the nonexclusive competition game always admits an equilibrium, that the buyers earn zero profits, and that the aggregate equilibrium allocation is generically unique. If θH>v, the equilibrium is similar to the separating equilibrium found in case (ii) of Theorem 1:typeLtrades efficiently, QL=1and TL=vL, while type Hdoes not trade at all, QH=TH=0. In contrast, if θH<v, the situation is markedly 21This is, for instance, the case in the Biais et al. (2000) setting if θL≥vLand θH≤vH. It should, however, be noted that they explicitly rule out this parameter configuration. 22The second pooling condition τH(00)≤vHor, equivalently, πHf H(0)≥1, is irrelevant in the credit model because the borrower cannot raise negative amounts of capital; see Section 5.1 and the Appendix. 23Again, the condition τH(00)≤vHis irrelevant in the credit model. Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 31 However, from the assumption that viq−t>b k i, this is strictly higher than bkwhen εi and εjare small enough, a contradiction. Hence type jmust select ck ifollowing buyer k’s deviation. In equilibrium, this deviation cannot be profitable, so that vq −t−εi≤bk. Letting εigo to zero yields the desired implication. The result follows.  Proof of Lemma 2. In the case of a pooling equilibrium, the conclusion follows immediately from the zero-profit result. Consider next a separating equilibrium and let us start with case (ii): QL>Q H≥0. We know from Proposition 1 that SL≤0. Suppose that SL<0.From(5) and the zero-profit result, we get BH≤bk Hfor each k, which implies that BH≤0. Now, notice from (4) that B=vQH−TH+mLSL=BH+mL[SL−(vH−vL)QH] Because BH≤0,SL<0and QH≥0, we obtain that B<0, a contradiction. Therefore, it must be that SL=0,sothatTL−TH=vL(QL−QH). This implies that B=vQH−TH, so that TH=vQHas B=0. The result follows. Case (iii) follows in a similar manner, exchanging the roles of Land H. Finally, consider case (i): QL>0>Q H.Asabove, B=BH+mL[SL−(vH−vL)QH]=0. Suppose that BH>0and thus BH>b k Hfor some k. Again, from (5), this implies that SL=0and thus that BH−mL(vH−vL)QH=B=0. Because vH>v Land BH>0,onemusthaveQH>0, a contradiction. Hence BH≤0. Symmetrically, using that B=BL+mH[SH−(vL−vH)QL]=0,wegetBL≤0.Thus BL=BH=0as B=0, and hence TL=vLQLand TH=vHQH. The result follows.  Proof of Lemma 3.IfBj>0, then one must have Tj=vQjby Lemma 2.Anybuyer kcan deviate by proposing a menu consisting of the no-trade contract and of the contract ck j=(Qj+δjTj+εj)for some numbers δjand εj. Suppose, to the contrary, that τj(QjTj)= v. Then one can choose δjand εjsuch that τj(QjTj)δj<ε j<vδ j.Whenδj and εjare small enough, the first inequality guarantees that type jcan strictly increase her utility by trading ck jwith buyer k.Iftypeitrades ck j,thenbuyerk’s profit from this deviation is v(Qj+δj)−(Tj+εj)=vδj−εj>0, in contradiction to the zero-profit result. Therefore, type imust not trade with buyer kand for this deviation not to be profitable, one must have mj[vj(Qj+δj)−(Tj+εj)]=mj(Bj+vjδj−εj)≤0. Letting δjand εjgo to zero yields Bj≤0, a contradiction. The result follows.  Proof of Lemma 4. Suppose first that Uj>z −k j(00)for some k.Thenbuyerkcan deviate by proposing a menu consisting of the no-trade contract and of the contract ck j=(QjTj−εj)for some positive number εj.Whenεjis small enough, one has uj(QjTj−εj)>z −k j(00),sothattypejtrades the contract ck jfollowing buyer k’s deviation. If type idoes not trade the contract ck j,buyerk’s profit from this deviation is mj(vjQj−Tj+εj)=mj(Bj+εj)>0, in contradiction to the zero-profit result. If type itrades the contract ck j, then, because Tj=vQjby Lemma 2,buyerk’s profit from this deviation is vQj−Tj+εj=εj>0, again in contradiction to the zero-profit result. As in any case Uj≥z−k j(00),itmustbethatUj=z−k j(00)for each k. It follows that for any buyer k, there exists an aggregate trade (Q−kT−k)with buyers other than ksuch that uj(Q−kT−k)=Uj. 32 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) Suppose now that Q−k= Qj. Then, from the strict quasiconcavity of uiand Lemma 3,onemusthaveT−k>vQ −k. We now examine two deviations for buyer k that pivot on (Q−kT−k). First, define (q1t1)such that (q1t1)+(Q−kT−k)=(QjTj). Then the seller can trade (Qj−q1Tj−t1)with buyers other than k. Moreover, using the fact that Tj=vQjby Lemma 2 and that T−k>vQ −k,weget vq1−t1=v(Qj−Q−k)−(Tj−T−k)=T−k−vQ−k>0 Therefore, by Lemma 1,onemusthavevjq1−t1≤bk j, that is, again using Tj=vQj, T−k−vjQ−k+(vj−v)Qj≤bk j.AsT−k>vQ −k, this implies that (vj−v)(Qj−Q−k)<b k j(8) Second, define (q2t2)such that (q2t2)+(Q−kT−k)=(QiTi). Then the seller can trade (Qi−q2Ti−t2)with buyers other than k. Moreover, using the fact that Si=0and Tj=vQjby Lemma 2,thatT−k>vQ −k,andthat(v−vi)(Qi−Qj)≥0by Assumption SC, we get vq2−t2=v(Qi−Q−k)−(Ti−T−k) =T−k−vQ−k+vQi−[Tj+vi(Qi−Qj)−Si] =T−k−vQ−k+(v −vi)(Qi−Qj) >0 Therefore, by Lemma 1,onemusthaveviq2−t2≤bk i, that is, using again Si=0and Tj=vQj,T−k−viQ−k+(vi−v)Qj≤bk i.AsT−k>vQ −k, this implies that (vi−v)(Qj−Q−k)<b k i(9) Because v=mivi+mjvjand mibk i+mjbk j=0by the zero-profit result, averaging (8)and (9) yields 0<0, a contradiction. Therefore, one must have Q−k=Qjand, thus, T−k=Tj as uj(Q−kT−k)=Uj=uj(QjTj). The result follows.  Proof of Proposition 4.Wefirstprovethatbk j=0for all jand k. Suppose, to the contrary, that bk j>0for some jand k. We first show that Si=Sj=0.ToprovethatSi=0, observe that by the no-cross-subsidization result, one has bl j<0=Bjfor some l= k. From (5), this implies that miSi≥B−bl.BecauseB−bl=0by the zero-profit result and because Si≤0by Proposition 1, it follows that Si=0.ToprovethatSj=0,observe that if bk j>0, then bk i<0=Biby the zero-profit result and the no cross-subsidization result. Arguing as for Si, it follows that Sj=0.HenceSi=Sj=0, as claimed. As Si+Sj=(vi−vj)(Qi−Qj),onemusthaveQi=Qj, and the equilibrium is pooling, with (QiTi)=(QjTj)=(00).Now,becausebk j>0and because (QjTj)=(00)can obviously be traded with buyers other than k, one can show as in the proof of Proposition 3 that τi(00)=v. Finally, consider buyer las above. As bl j<0,onehasbl i>0 by the zero-profit result. Because (QiTi)=(00)can obviously be traded with buyers Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 33 other than l, it follows along the same lines that τj(00)=vas well, which contradicts Assumption SC. Hence the result. We next prove that qk L≥0≥qk Hfor each k.BecausevH>v Land sk i=vi(qk i−qk j)−(tk i−tk j)=bk i−bk j−(vi−vj)qk j=(vj−vi)qk j as bk i=bk j=0, we need to show only that sk i≤0for all iand k. Choose i,k,andl= k, and set q≡qk i+ql i−qk jand t≡tk i+tl i−tk j. Then the seller can trade (Qi−qTi−t) = (qk j+m=kl qm itk j+m=kl tm i)with buyers other than l. We can thus apply Lemma 1. One has viq−t−bl i=vi(qk i+ql i−qk j)−(tk i+tl i−tk j)−bl i=sk i and vjq−t−bl j=vj(qk i+ql i−qk j)−(tk i+tl i−tk j)−bl j=−(sk j+sl j) Therefore, according to (1), sk i>0implies misk i≤mj(sk j+sl j) (10) Now, suppose, to the contrary, that sk i>0for some iand k.Then,by(10), misk i≤mj(sk j+sl j)(11) for each l= k. Summing on l= kyields (n −1)misk i≤mj[Sj+(n −2)sk j] From Proposition 1, we know that Sj≤0.Hence,ifsk i>0, one must also have sk j>0. Exchanging the roles of iand jin (10) yields mjsk j≤mi(sk i+sl i)(12) for each l= k. Combining (11)and(12)leadstomisk i≤mjsl j+mi(sk i+sl i)or, equivalently, misl i+mjsl j≥0for each l= k. Note that we also have misk i+mjsk j>0, as both sk iand sk j are positive. Summing all these inequalities yields miSi+mjSj>0, in contradiction to Proposition 1. Hence the result.  Proof of Lemma 5. Suppose that Qj=0.Ifτj(00)=vj, the result is immediate. Suppose then that τj(00)= vj.Anybuyerkcan deviate by proposing a menu consisting of the no-trade contract and of the contract ck j=(δjεj)for some numbers δjand εj. Choose δjand εjsuch that τj(00)δj<ε j<v jδj. This ensures that when δjand εjare small enough, type jcan strictly increase her utility by trading ck jwith buyer kand that buyer kthereby makes a positive profit with type j. Therefore, type imust also trade ck j following buyer k’s deviation and for this deviation not to be profitable, one must have 34 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) εj≥vδj. Thus we have shown that for any small enough δjand εj,τj(00)δj<ε j<v jδj implies that εj≥vδj, which is equivalent to the statement of the lemma. The result follows.  Proof of Lemma 6. By the no-cross-subsidization result, if Qi= 0, the equilibrium must be separating. Moreover, from Lemma 2,onemusthaveTi=viQi. Suppose, to the contrary, that τi(QiTi)= vi. Then any buyer kcan deviate by proposing a menu consisting of the no-trade contract and of the contract ck i=(qiti)for some numbers qiand ti.Asτi(QiTi)= vi, it follows from the strict quasiconcavity of uithat one can choose (qiti)close to (QiTi)such that Ui<u i(qiti)and ti<v iqi,whereqiis positive if i=Land negative if i=H. The first inequality guarantees that type itrades ck ifollowing buyer k’s deviation. As viqi>t i,typejmust also trade ck ifollowing buyer k’s deviation and one must have ti≥vqi, for, otherwise, this deviation would be profitable. Overall, we have shown that viqi>vq i.Becauseqiis positive if i=Land negative if i=H,and because vH>v>v L, we obtain a contradiction in both cases. The result follows.  Proof of Theorem 1. Suppose first that a pooling equilibrium exists. Then, according to the no-cross-subsidization result, QL=QH=0.Lemma 5 then implies that vL≤τL(00)≤v≤τH(00)≤vH(13) Suppose next that a separating equilibrium exists. Then, according again to the nocross-subsidization result, only three scenarios are possible. (i) In the first case, QH<0<Q L.Then,byLemma 2,TL=vLQLand TH=vHQH. Moreover, by Lemma 6,τL(QLTL)=vLand τL(QHTH)=vH.Asaresult, QL=Q∗ Land QH=Q∗ H,sothatQ∗ H<0<Q ∗ L. The strict quasiconcavity of ui then implies that τL(00)<v Land τH(00)>v H(14) (ii) In the second case, QH=0<Q L.Then,byLemma 5,v≤τH(00)≤vH.Moreover, by Lemma 2,TL=vLQL. Finally, by Lemma 6,τL(QLTL)=vL.Asaresult, QL=Q∗ L,sothatQ∗ L>0. The strict quasiconcavity of uithen implies that τL(00)<v Land v≤τH(00)≤vH(15) (iii) In the third case, QH<0=QL.Then,byLemma 5,vL≤τL(00)≤v.Moreover, by Lemma 2,TH=vHQH. Finally, by Lemma 6,τH(QHTH)=vH.Asaresult, QH=Q∗ H,sothatQ∗ H<0. The strict quasiconcavity of uithen implies that vL≤τL(00)≤vand τH(00)>v H(16) To conclude the proof, observe that from (13)–(16), an equilibrium exists only if τL(00)≤v≤τH(00). As conditions (13)–(16) are mutually exclusive, the characterization of the candidate aggregate equilibrium trades is complete. Hence the result.  Proof of Theorem 2. Choose an integer m,2≤m≤n,andfixQand Qsuch that Q < min{0QH}/(m −1)and Q>max{0QL}/(m −1). Suppose that mbuyers post the tariff Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 35 tdefined as in the theorem, while the other buyers stay inactive and propose only the no-trade contract. Consider any buyer. In the aggregate, his competitors post the tariff T−(Q−)≡min{vLQ−vHQ−}Q 1≤Q−≤Q1 where Q−refers to the aggregate quantity they trade. Here Q1is either mQ or (m −1)Q, and thus is no greater than QH,andsimilarlyforQ1, which cannot be smaller than QL. Note also that if the efficient trade Q∗ His negative, then Q1≤Q∗ H≤Q1; the symmetrical statement applies for Q∗ L. Suppose that our buyer deviates and ends up trading (qLtL)with type Land (qHtH)with type H. For his deviation to be profitable, he must make a positive profitwithatleastonetype,saytypeH(the proof for type Lis symmetrical). Hence vHqH>t H.DefineQ− i∈[Q1 Q1]as the quantity traded by type iwith the deviator’s competitors following his deviation. Also define ˆ Qias the total quantity traded by type i,sothat ˆ Qi=qi+Q− i,anddefine ˆ Tias the total transfer obtained by type i,sothat ˆ Ti=ti+T−(Q− i).ThetariffT−is such that the deviator’s competitors cannot make losses following the deviation. Therefore, as vHqH>t H,onemusthavevHˆ QH>ˆ TH. Because the no-trade contract is available, we get uH(ˆ QHvHˆ QH)>u H(ˆ QHˆ TH)≥uH(00) (17) If ˆ QH<0,then(17)impliesthatτH(00)>v H,sothatQ∗ H<0.Byconstructionofthe tariff T−,typeHcanthentrade(Q∗ H,vHQ∗ H)with the deviator’s competitors, thereby getting utility uH(Q∗ HvHQ∗ H)=maxQ{uH(Q vHQ)}>u H(ˆ QHˆ TH)by (17), a contradiction. As the case ˆ QH=0is trivially ruled out by (17), it must be that ˆ QH>0.From(17), we now get τH(00)<v H, for, otherwise, type Hwould be strictly better off not trading at all than trading (ˆ QHvHˆ QH).28 Because τH(00)≥vby assumption, from (17)again we get ˆ TH≥vˆ QH, for, otherwise, type Hwould be strictly better off not trading at all than trading (ˆ QHˆ TH). Finally, notice that T−(Q−)≤vQ−for all Q−∈[Q1 Q1].Thus vˆ QH=vqH+vQ− H≤ˆ TH=tH+T−(Q− H)<v HqH+vQ− Hand hence qH>0. Type Lmay also choose to trade (qHtH)with the deviator. He would then have to choose some Q−to maximize uL(qH+Q−tH+T−(Q−)),subjecttoQ1≤Q−≤Q1. Notice first from the definition of QLthat the constraint Q−≤Q1does not play any role: indeed qH>0, so that when Q−reaches its upper bound Q1, the total quantity traded qH+Q−is higher than QLand, therefore, type L’s marginal rate of substitution is higher than vLby Assumption T. We can thus eliminate the constraint Q−≤Q1, taking care of extending the tariff T−beyond Q1by setting T−(Q−)≡vLQ−for all Q−> Q1. Now ˆ QL−qHsatisfies the remaining constraint Q1≤Q−; indeed, thanks to Assumption SC,wehave ˆ QL≥ˆ QH,sothat ˆ QL−qH≥Q− H≥Q1. We thus have shown that type Lcan get at least utility uL(ˆ QLtH+T−(ˆ QL−qH)). Observe that the transfer in this 28Note that the condition τH(00)<v Hexcludes the efficient case (i) of Theorem 1.Inthatsimplecase, an inspection of the above lines reveals that we have used only the fact that (Q∗ HvHQ∗ H)is offered by the deviator’s competitors. We have thus shown that in the efficient case (i) of Theorem 1, any equilibrium can be sustained by having at least two buyers posting the two trades (Q∗ HvHQ∗ H)and (Q∗ LvLQ∗ L). 36 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) expression can be rewritten as ˆ TH+T−(ˆ QL−qH)−T−(Q− H), which is no less than ˆ TH+ vL(ˆ QL−ˆ QH)by concavity of T−.BecausetypeLis supposed to end up with utility uL(ˆ QLˆ TL)following the deviation, it follows that ˆ TL≥ˆ TH+vL(ˆ QL−ˆ QH).Moreover, as shown above, ˆ TH≥vˆ QH. Therefore, the aggregate profit, which may as usual be written as vˆ QH−ˆ TH+mL[vL(ˆ QL−ˆ QH)−(ˆ TL−ˆ TH)],isatmostzero. Becausethetariff T−is such that the deviator’s competitors cannot make losses, the deviation cannot be profitable. Hence the result.  When only nonnegative quantities can be sold A careful rereading of the proofs leads to the following changes. Lemma 1 is still valid, but only when qis nonnegative. Proposition 1 now allows only to conclude that SL≤0.BecauseSL=BL−BH+(vH−vL)QHand QH≥0by assumption, a useful consequence of Proposition 1 is that BH≥BL. Proposition 2 still holds. Indeed (5) still holds, as its derivation involves only nonnegative trades. Once (5) is proven, one has to include the following argument. Suppose B>0. From the above remark that BH≥BL,itmustbethatBH>0and there exists ksuch that BH>b k H.From(5) applied to kat (i j) =(L H),wegetmLSL≥B−bk= l=kbl≥0.AsSL≤0by Proposition 1,wegetSL=0. But by the single-crossing property, SL+SH=(vL−vH)(QL−QH)≤0, so we now know that both SLand SHare nonpositive. One can then use the last lines of the proof of Proposition 2 to conclude that profits must be zero. Lemma 2 still holds. Only the pooling case and case (ii) remain. Lemma 3 and its proof are unchanged. Indeed, notice that if Bj>0,thenonemust have Qj>0, so that deviations that involve a small change in this quantity are feasible. Lemma 4 still holds, but the proof has to be adapted somewhat. Suppose Bj>0. From the zero-profit result and the above remark that BH≥BL,itmustbethatj=H.As BH>0,thereexistsksuch that BH>b k Hand, thus, we can apply (5)tokat (i j) =(L H) to get SL=0. The first step of the proof shows without changes that there exists an aggregate trade (Q−kT−k)with buyers other than ksuch that uH(Q−kT−k)=UH.If Q−k<Q H, then the two deviations used in the rest of the proof are feasible, as both q1 and q2are positive. If QH<Q −k<Q L, then only deviation (q2t2)is feasible (recall that j=H), so that (9) holds. Therefore, bk L>0.BecauseBL≤0, this implies that there exists l= ksuch that BL>b l L. We can then apply (5)tolat (i j) =(H L) to get SH=0. Because SL=SH=0and SL+SH=(vL−vH)(QL−QH),wegetQL=QH, in contradiction to our assumption that QH<Q −k<Q L. Finally, it cannot be that Q−k≥QL, for, otherwise, type Lwould strictly prefer (Q−kT−k)to (QLTL)by the single-crossing property. Proposition 3 still holds. Recall that the assumption Bj>0implies that j=Hand, thus, the proof of Proposition 3 needs no change. At this point, we know from Lemma 2 and Proposition 3 that QH=TH=0in any equilibrium, so that necessarily qk H=tk H=0for each kand that TL=vLQL. Deriving the other results is then easy and requires only minor adaptations to the proofs. The Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 37 only important difference concerns Lemma 5 and, as a result, the statement of the necessary condition for the existence of an equilibrium in Theorem 1. Indeed, although the reasoning in Lemma 5 remains correct, we must take into account the restriction δj≥0.Whenj=L,wegetthatτL(00)δL<ε L<v LδLimplies that εL≥vδLand, thus, vLδL>ε L≥vδL, which is impossible if δL≥0. By contraposition, we can thus conclude only that τL(00)≥vLif QL=0.Asaresult,τL(00)≤vis no longer a necessary condition for the existence of an equilibrium. Indeed, one may have QL=0and yet τL(00)>v>v L.Whenj=H, in contrast, the conclusion of Lemma 5 remains correct, from which it follows that τH(00)≥vis a necessary and indeed sufficient condition for the existence of an equilibrium. References Adrian, Tobias and Hyun Song Shin (2010), “The changing nature of financial intermediation and the financial crisis of 2007–2009.” Annual Review of Economics, 2, 603–618. [2] Akerlof, George A. (1970), “The market for ‘lemons’: Quality uncertainty and the market mechanism.” Quarterly Journal of Economics, 84, 488–500. [2,3,5,16,24] Ales, Laurence and Pricila Maziero (2011), “Adverse selection and non-exclusive contracts.” Unpublished paper. [5,10,22] Arnott, Richard J. and Joseph E. Stiglitz (1993), “Equilibrium in competitive insurance markets with moral hazard.” Unpublished paper. [4] Attar, Andrea and Arnold Chassagnon (2009), “On moral hazard and nonexclusive contracts.” Journal of Mathematical Economics, 45, 511–525. [4] Attar, Andrea, Thomas Mariotti, and François Salanié (2011), “Nonexclusive competition in the market for lemons.” Econometrica, 79, 1869–1918. [5,11,18,24,25,27,29] Attar, Andrea, Thomas Mariotti, and François Salanié (2013), “On competitive nonlinear pricing.” Unpublished paper. [28,29] Bernheim, B. Douglas and Michael D. Whinston (1986), “Common agency.” Econometrica, 54, 923–942. [6] Biais, Bruno, David Martimort, and Jean-Charles Rochet (2000), “Competing mechanisms in a common value environment.” Econometrica, 68, 799–837. [5,9,12,17,24, 28,29] Bisin, Alberto and Danilo Guaitoli (2004), “Moral hazard and nonexclusive contracts.” RAND Journal of Economics, 35, 306–328. [4] Brunnermeier, Markus K. (2009), “Deciphering the liquidity and credit crunch 2007– 2008.” Journal of Economic Perspectives, 23, 77–100. [2] Caillaud, Bernard, Roger Guesnerie, Patrick Rey, and Jean Tirole (1988), “Government intervention in production and incentives theory: A review of recent contributions.” RAND Journal of Economics, 19, 1–26. [9] 38 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) Calzolari, Giacomo (2004), “Incentive regulation of multinational enterprises.” International Economic Review, 45, 257–282. [12] Cawley, John and Tomas J. Philipson (1999), “An empirical examination of information barriers to trade in insurance.” American Economic Review, 89, 827–846. [3] Chiappori, Pierre-André (2000), “Econometric models of insurance under asymmetric information.” In Handbook of Insurance (Georges Dionne, ed.), 365–394, Kluwer, Boston. [14] Chiesa, Gabriella and Vincenzo Denicolò (2009), “Trading with a common agent under complete information: A characterization of Nash equilibria.” Journal of Economic Theory, 144, 296–311. [6,11] Cook, Philip J. and Daniel A. Graham (1977), “The demand for insurance and protection: The case of irreplaceable commodities.” Quarterly Journal of Economics, 91, 143–156. [10] d’Aspremont, Claude and Rodolphe Dos Santos Ferreira (2010), “Oligopolistic competition as a common agency game.” Games and Economic Behavior, 70, 21–33. [6, 11] Demyanyk, Yuliya and Otto Van Hemert (2011), “Understanding the subprime mortgage crisis.” Review of Financial Studies, 24, 1848–1880. [2] Detragiache, Enrica, Paolo Garella, and Luigi Guiso (2000), “Multiple versus single banking relationships: Theory and evidence.” Journal of Finance, 55, 1133–1161. [3] Finkelstein, Amy and James Poterba (2004), “Adverse selection in insurance markets: Policyholder evidence from the U.K. annuity market.” Journal of Political Economy, 112, 183–208. [3] Glosten, Lawrence R. (1994), “Is the electronic open limit order book inevitable?” JournalofFinance, 49, 1127–1161. [16,18,19,20] Gorton, Gary (2009), “Information, liquidity, and the (ongoing) panic of 2007.” American Economic Review Papers and Proceedings, 99, 567–572. [2] Grossman, Sanford J. (1981), “Nash equilibrium and the industrial organization of markets with large fixed costs.” Econometrica, 49, 1149–1172. [6,11] Han, Seungjin (2006), “Menu theorems for bilateral contracting.” Journal of Economic Theory, 131, 157–178. [27] Hellwig, Martin F. (1983), “On moral hazard and non-price equilibria in competitive insurance markets.” Discussion Paper 109, Institut für Gesellschaftsund Wirtschaftswissenschaften, Sonderforschungsbereich 21, Universität Bonn. [4] Hellwig, Martin F. (1988), “A note on the specification of interfirm communication in insurance markets with adverse selection.” Journal of Economic Theory, 46, 154–163. [4, 5,16,18,20] Theoretical Economics 9 (2014) Nonexclusive competition under adverse selection 39 Jaynes, Gerald D. (1978), “Equilibria in monopolistically competitive insurance markets.” Journal of Economic Theory, 19, 394–422. [4,16,18,20] Keys, Benjamin J., Tanmoy Mukherjee, Amit Seru, and Vikrant Vig (2010), “Did securitization lead to lax screening? Evidence from subprime loans.” Quarterly Journal of Economics, 125, 307–362. [2] Khalil, Fahad, David Martimort, and Bruno Parigi (2007), “Monitoring a common agent: Implications for financial contracting.” Journal of Economic Theory, 135, 35–67. [12] Klemperer, Paul D. and Margaret A. Meyer (1989), “Supply function equilibria in oligopoly under uncertainty.” Econometrica, 57, 1243–1277. [6,11] Kyle, Albert S. (1989), “Informed speculation with imperfect competition.” Review of Economic Studies, 56, 317–355. [6,11] Laffont, Jean-Jacques and Jérôme Pouyet (2004), “The subsidiarity bias in regulation.” Journal of Public Economics, 88, 255–283. [12] Martimort, David (2006), “Multi-contracting mechanism design.” In Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Volume 1 (Richard Blundell, Whitney K. Newey, and Torsten Persson, eds.), 57–101, Cambridge University Press, Cambridge. [6] Martimort, David and Lars Stole (2002), “The revelation and delegation principles in common agency games.” Econometrica, 70, 1659–1673. [8] Martimort, David and Lars Stole (2003), “Contractual externalities and common agency equilibria.” Advances in Theoretical Economics,3,Article4.[6,12] Martimort, David and Lars Stole (2009), “Market participation under delegated and intrinsic common-agency games.” RAND Journal of Economics, 40, 78–102. [12] Ongena, Steven and David C. Smith (2000), “What determines the number of bank relationships? Cross-country evidence.” Journal of Financial Intermediation, 9, 26–56. [3] Parlour, Christine A. and Uday Rajan (2001), “Competition in loan contracts.” American Economic Review, 91, 1311–1328. [4] Pauly, Mark V. (1974), “Overinsurance and public provision of insurance: The roles of moral hazard and adverse selection.” Quarterly Journal of Economics, 88, 44–62. [4,14] Peters, Michael (2001), “Common agency and the revelation principle.” Econometrica, 69, 1349–1372. [8] Peters, Michael (2003), “Negotiation and take it or leave it in common agency.” Journal of Economic Theory, 111, 88–109. [6] Philippon, Thomas and Vasiliki Skreta (2012), “Optimal interventions in markets with adverse selection.” American Economic Review, 102, 1–28. [2] Quiggin, John (1982), “A theory of anticipated utility.” Journal of Economic Behavior and Organization, 3, 323–343. [10] 40 Attar, Mariotti, and Salanié Theoretical Economics 9 (2014) Rothschild, Michael and Joseph E. Stiglitz (1976), “Equilibrium in competitive insurance markets: An essay on the economics of imperfect information.” Quarterly Journal of Economics, 90, 629–649. [2,3,4,5,9,10,13,14,16,24] Rysman, Marc (2007), “An empirical analysis of payment card usage.” Journal of Industrial Economics, 55, 1–36. [3] Segal, Ilya and Michael D. Whinston (2003), “Robust predictions for bilateral contracting with externalities.” Econometrica, 71, 757–791. [6] Sheshinski, Eytan (2008), The Economic Theory of Annuities. Princeton University Press, Princeton. [14] Stiglitz, Joseph E. and Andrew Weiss (1981), “Credit rationing in markets with imperfect information.” American Economic Review, 71, 393–410. [10] Taylor, John B. and John C. Williams (2009), “A black swan in the money market.” American Economic Journal: Macroeconomics, 1, 58–83. [2] Vives, Xavier (2011), “Strategic supply function competition with private information.” Econometrica, 79, 1919–1966. [6,11] Wilson, Robert (1979), “Auctions of shares.” Quarterly Journal of Economics, 93, 675–689. [6,11] Yaari, Menahem E. (1965), “Uncertain lifetime, life insurance, and the theory of the consumer.” Review of Economic Studies, 32, 137–150. [30] Submitted 2011-12-1. Final version accepted 2012-10-30. Available online 2012-10-30.