On Competitive Nonlinear Pricing

Transcription

1 On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané July 4, 2013 Abstract A buyer of a dvsble good faces several dentcal sellers. The buyer s preferences are her prvate nformaton, and they may drectly affect the sellers profts (common values). Sellers compete by postng menus of nonexclusve contracts, so that the buyer can smultaneously and prvately trade wth several sellers. We focus on the fntetype case, and we provde a full characterzaton of pure-strategy equlbra n whch sellers post convex tarffs. All equlbra nvolve lnear prcng. When the sellers cost functons are lnear and do not depend on the buyer s type (prvate values), equlbra exst and trade s effcent. Under common values, or when the sellers costs are strctly convex, there s a severe form of market breakdown as at most one type of the buyer may actvely trade. Moreover equlbra exst only under restrctve condtons. Keywords: Adverse Selecton, Competng Mechansms, Nonexclusvty. JEL Classfcaton: D43, D82, D86. We thank Bruno Bas, Alessandro Pavan, Jean-Charles Rochet, and Mchael Whnston for very valuable feedback. We also thank semnar audences at Northwestern Unversty and Toulouse School of Economcs, as well as conference partcpants at the 7 th ENSAI Economc Day, the 9 th Cowles Foundaton Annual Conference on General Equlbrum and ts Applcatons, the LUISS Workshop on Macroeconomcs and Fnancal Frctons, and the Unversté Pars-Dauphne Workshop n Honor of Rose-Anne Dana for many useful dscussons. Elena Panova provded excellent research assstance. Fnancal support from the Chare Marchés des Rsques et Créaton de Valeur and the European Research Councl (Startng Grant ACAP) s gratefully acknowledged. Toulouse School of Economcs (IDEI, PWRI) and Unverstà degl Stud d Roma Tor Vergata. Toulouse School of Economcs (CNRS, GREMAQ, IDEI). Toulouse School of Economcs (INRA, LERNA, IDEI).

2 1 Introducton Many markets for goods and servces do not restrct n any way the ablty of each trader to sgn secret, blateral contracts wth dfferent partners. Ths prevents outsde partes from montorng the whole of a trader s actvtes. As a consequence, the formaton of prces on such nonexclusve markets s by nature a decentralzed process, unlke on dealzed markets ruled by a Walrasan auctoneer. Blateral contracts are necessarly ncomplete as they only bear on a fracton of each trader s actvty. Moreover, blateral negotatons allow to talor contracts at wll, at odds wth contracts that are normalzed for quotaton. In partcular, contracts may be dscrmnatory, and the balance between supply and demand may be ensured not by a sngle prce, but by nonlnear tarffs. These tarff offers n turn are formulated n a strategc envronment n whch sellers take nto account both the reacton of buyers and the other sellers offers. The am of ths paper s to understand the formaton of prces on nonexclusve markets. In the case of fnancal markets, our results shed lght on the robustness of organzed exchanges such as lmt-order books to trades that take place n the dark, outsde vsble order books. As we wll see, the nonexclusve nature of such transactons s a major obstacle to the effcent functonng of these markets. We study these ssues n the context of the followng model of trade under uncertanty. There are two commodtes, money and a physcal good. Trade takes place between a buyer and a fnte number of sellers offerng ths good. The sellers frst post possbly nonlnear tarffs expressng how much they ask for any quantty of the good. The buyer then learns her preferences and she decdes whch quantty to purchase from each seller. There s an arbtrary fnte number of states of nature. In each state, the buyer has strctly convex preferences. These preferences are ordered across states accordng to how much she s wllng to trade at the margn, reflectng a strct sngle-crossng property. As for the sellers, they weakly prefer to sell lower quanttes when the buyer s more eager to trade, reflectng a reverse weak sngle-crossng property. Our model thus encompasses prvate-value and adverse-selecton envronments as specal cases. In addton, sellers may have constant or ncreasng margnal costs of servng the buyer n each state of nature. In ths context, we provde a complete characterzaton of pure-strategy equlbra n whch sellers post convex tarffs. Such tarffs can be nterpreted as sequences of lmt orders, and are natural canddates to consder n nonexclusve models of trade wth adverse selecton (Bas, Martmort, and Rochet (2000, 2013), Back and Baruch (2013)) or ncreasng margnal costs (Bas, Foucault, and Salané (1998)). Importantly, we allow sellers to devate by postng arbtrary nonconvex tarffs, so as to ft our defnton of a nonexclusve market. Our 1

3 man result s that all equlbra must nvolve lnear prcng. Hence competton n our model s powerful enough to make a sngle equlbrum prce emerge. Sellers then cannot beneft from usng nonlnear tarffs. When sellers have constant and state-ndependent margnal costs, one ends up wth a unque equlbrum outcome whch s effcent n the strongest sense, as t concdes wth the equlbrum outcome of a perfectly compettve market. 1 When there s adverse selecton or sellers have ncreasng margnal costs, lnear-prce equlbra are such that the buyer trades n at most one state of nature, and does not trade at all n any other state. Hence the market breaks down n a very strong sense. Moreover, n such cases necessary condtons for the exstence of an equlbrum are severe. An mplcaton of our analyss s that organzed exchanges such as lmt-order books can be destablzed by decentralzed exchanges such as over-the-counter markets. Standard analyses of nonexclusve markets take lnear prcng as a defnng feature of such markets. The opportunty to trade small quanttes from several sellers, the argument goes, allows buyers to arbtrage away any nonlneartes n the sellers tarffs. In lne wth ths ntuton, Pauly (1974) analyzed a nonexclusve nsurance market n whch nsurance companes are restrcted to post lnear tarffs, and showed that equlbra then nvolve crosssubsdes between sellers profts across states. 2 Our analyss suggests that these outcomes do not survve when strategc nteractons between sellers are explctly taken nto account. The ntuton s that due to adverse selecton or ncreasng margnal costs, the sellers face a hgh demand from the buyer precsely n those states n whch the cost of servng her s hgh. To hedge aganst ths rsk, each seller has an ncentve to devate by proposng a lmt order specfyng the maxmal quantty of the good he s ready to trade at the standng prce. In these crcumstances, lnear prcng can be reconcled wth nonexclusve competton only f the buyer trades a postve quantty n at most one state. In contrast wth ths result, Attar, Marott, and Salané (2011) showed that the restrcton to lnear prces s wthout loss of generalty n a lemons market where an nformed seller can trade up to a capacty and all market partcpants have lnear preferences. Cross-subsdes between states can then resst lmt-order devatons because, at any gven unt prce, and dependng on the state, the seller s ether ready to trade up to the maxmum quantty demanded at ths prce (as long as t does not exceed her capacty) or prefers not to trade at all. By contrast, the nformed 1 The exstence of an effcent equlbrum n ths Bertrand-lke envronment wth prvate values s qute straghtforward. Stll we could not fnd any prevous work showng that no other equlbra wth convex tarffs can exst. A smlar effcency result appears n Pouyet, Salané, and Salané (2008), albet n the case of an exclusve market n whch the buyer can trade wth at most one seller. 2 The same restrcton to lnear prcng s postulated n recent analyses of the annuty market, whch s nonexclusve n many countres (Rothschld (2007), Sheshnsk (2008), and Hossen (2010)). 2

4 buyer n our model has strctly convex preferences and faces no capacty constrant. Ths mples that, at any gven unt prce, the buyer typcally has dfferent aggregate demands n dfferent states. Ths n turn gves lmt-order devatons ther bte and destablzes lnearprce canddate equlbra n whch trade takes place n more than one state. One may then turn to equlbra wth nonlnear tarffs, n the hope that they yeld more tradng under adverse selecton or ncreasng margnal costs. Glosten (1994) proposed a natural canddate n a framework n whch the buyer faces an exogenously gven tarff and sellers have lnear producton costs. Specfcally, he showed that there s a unque convex tarff that ressts entry. Ths tarff can be nterpreted as a generalzaton of Akerlof (1970) prcng, for margnal quanttes. It specfes that each addtonal quantty above any quantty q s sold at a prce equal to the expected cost of servng t, condtonal on the fact that the buyer buys at least q. Under sngle crossng, ths amounts to compute an upper-tal expectaton, namely, the expectaton of the cost gven that the buyer s ready to purchase at least q. In each state, the buyer then trades exactly her demand at the tal prce. An addtonal nce property s that by constructon such a tarff yelds zero proft to the sellers. In our settng, the queston becomes whether we can fnd convex tarffs for the sellers that once aggregated yeld the Glosten (1994) tarff, and such that no seller can proftably devate by postng another tarff. 3 Suppose that n equlbrum two dfferent types of the buyer, correspondng to two dfferent states of nature, end up tradng at two dfferent tal prces. Then there must exst a seller that sells more to the type tradng at the hghest prce than to the other type. Note that when facng ths seller, the former type does not want to devate and choose the quantty traded by the latter type because, when tarffs are convex, optmalty condtons mply that all quanttes traded by a gven type wth the sellers are traded at the same prce. On the other hand, the seller desgns hs tarff so as to maxmze hs expected proft, under ncentve-compatblty constrants. Gven convex tarffs and sngle crossng, we show that downward local ncentve-compatblty constrants must be bndng at the soluton of such a problem. But ths contradcts the fact that the hghest type does not want to mmc the lowest type. Hence n a Glosten-lke equlbrum all trades must take place at the same prce. Moreover, we show that the above logc also apples to any convex tarff. Therefore, the only equlbra are lnear-prce equlbra. We are then back to the concluson that at most one type may trade n equlbrum under adverse selecton or ncreasng margnal costs. Our results confrm those obtaned by Attar, Marott, and Salané (2013). That paper 3 Ths study was not performed n Glosten (1994), see the dscusson n Glosten (1998). 3

5 examnes the case wth two states of nature, adverse selecton, and constant margnal costs n each state. A complete characterzaton of aggregate equlbrum allocatons s provded, wth no restrcton on equlbrum tarffs. It turns out that all equlbrum allocatons can be supported by lnear tarffs, wth at most one type tradng. Focusng on equlbra wth convex tarffs, ths paper shows that, strkngly, the result that the buyer may trade n at most one state extends to an arbtrary fnte number of states. We thus exhbt a new form of market falure, characterzed by a dramatc market breakdown that exceeds by far the one frst characterzed by Akerlof (1970). On the other hand, our results stand n stark contrast wth those obtaned n Bas, Martmort, and Rochet (2000), who consder a parametrc verson of our model wth a quaslnear, quadratc utlty functon for the buyer, and constant margnal costs wth adverse selecton for the sellers. The man dfference s that the set of states s assumed to be contnuous, nstead of fnte as n ths paper. Ths allows Bas, Martmort, and Rochet (2000) to focus on equlbra wth strctly convex tarffs. 4 They show that such an equlbrum exsts, s unque n ths class, and s symmetrc across sellers. Moreover the buyer trades n a nontrval set of states n equlbrum, at a tarff between the perfectly compettve tarff that would obtan under complete nformaton and the monopoly tarff under ncomplete nformaton. We thus exhbt n ths paper a remarkable dscontnuty between the fntestate case and the contnuous-state case: the equlbrum characterzed n the latter case s not a lmt of equlbra n the former case as the number of states grows large. The paper s organzed as follows. Secton 2 descrbes the model. Secton 3 states and dscusses our central result, the proof of whch s outlned n Secton 4. Secton 5 dscusses varous extensons of our analyss. Secton 6 concludes. 2 The Model Our model features a buyer who can purchase nonnegatve amounts of a dvsble good from several sellers. The good s homogeneous, so the buyer only cares about aggregate trade. The possblty of adverse selecton plays an mportant role, as n well-known models of nsurance provson, labor supply, or more generally compettve screenng. 2.1 The Buyer 4 Equlbra wth strctly convex tarffs do not exst when there are fntely many states. The reason s that otherwse, each type of the buyer would have a unque best response, and no ncentve-compatblty constrant would bnd, n contradcton wth one of our key fndngs. 4

6 The buyer s prvately nformed of her preferences. Her type may take a fnte number of values n the set {1,..., I}, wth postve probabltes m such that m = 1. Each type of the buyer only cares about the aggregate quantty Q 0 she purchases from the sellers and the aggregate transfer T she makes n return. Type s preferences over aggregate quanttytransfer bundles (Q, T ) are represented by a utlty functon u defned over R + R. For each, u s assumed to be contnuous and strctly quasconcave n (Q, T ), and strctly decreasng n T. The followng strct sngle-crossng assumpton s the man determnant of the buyer s behavor n our model, and s also used throughout the related lterature. Assumpton 1 For all <, Q < Q, T, and T, u (Q, T ) u (Q, T ) mples u (Q, T ) < u (Q, T ). In words, hgher types are more eager to ncrease ther purchases than lower types are. At the end of our analyss, we shall also use an addtonal property that we now ntroduce. For each p R, let D (p) be type s demand at prce p, that s, the unque soluton to max {u (Q, pq)}. Q R + { } The contnuty and strct quasconcavty of u mply that D (p) s unquely defned and contnuous n p. Assumpton 1 mples that for each p, D (p) s nondecreasng n the buyer s type. We strengthen ths monotoncty property as follows. Assumpton 2 For all < and p R, 0 < D (p) < mples D (p) < D (p). A suffcent condton for both Assumptons 1 and 2 to hold s that the margnal rate of substtuton MRS (Q, T ) of the good for money be well defned and strctly ncreasng n for all (Q, T ). 2.2 The Sellers There are K 2 dentcal sellers. There are no drect externaltes between them: each seller only cares about the quantty q 0 he provdes the buyer wth and the transfer t he receves n return. Such par (q, t) we call a trade. The seller s profts from a trade may depend on the buyer s type. Our key assumpton here s a reverse sngle-crossng property: we mpose that each seller weakly prefers to sell lower quanttes to hgher types. Ths assumpton 5

7 ntroduces adverse selecton n our model: a hgher type s wllng to buy more, but faces sellers that are more reluctant to sell. To allow comparsons wth the lterature, we represent each seller s preferences over trades (q, t) by a lnear proft functon: f a seller provdes type wth a quantty q and receves a transfer t n return, he earns a proft t c q, where c s the cost of servng type. Our reverse sngle-crossng property can thus be wrtten as follows. Assumpton 3 For all <, c c. Assumpton 3 s consstent wth prvate-value envronments, n whch the sellers cost s ndependent of the buyer s type, and common-value envronments, n whch the sellers cost strctly ncreases wth the buyer s type. Secton 5.2 provdes a general defnton of the reverse sngle-crossng property and hghlghts ts role n our analyss. 2.3 Strateges and Equlbrum The game unfolds as follows: 1. Sellers smultaneously post tarffs, whch are mappngs t k : R + R { } such that t k (0) = 0. We let t k (q) f seller k does not offer the quantty q. 2. After prvately learnng her type, the buyer purchases a nonnegatve quantty q k from each seller k, for whch she pays n total k tk (q k ). A pure strategy for type s a functon s that maps any tarff profle (t 1,..., t K ) nto a quantty profle (q 1,..., q K ). We let s = (s 1,..., s I ) be the buyer s strategy. To ensure that type s problem max (q 1,...,q K ) R K + { u ( k always has a soluton, we requre the tarffs t k q k, k t k (q k ) )} (1) to be lower semcontnuous, and the sets {q R + : t k (q) < } to be compact. Ths defnton s general enough to allow sellers to offer menus contanng a fnte number of trades, ncludng the (0, 0) trade. It also allows us to use perfect Bayesan equlbrum as our equlbrum concept. In lne wth Bas, Martmort, and Rochet (2000, 2013) and Back and Baruch (2013), we focus on pure-strategy equlbra (t 1,..., t K, s) n whch sellers post convex tarffs t k that one can nterpret as sequences of lmt orders. 5 5 By conventon, all functons n Gothc letters refer to equlbrum objects. Two elementary mplcatons of ths 6

8 restrcton are worth mentonng at ths stage. Frst, because the utlty functons u are strctly quasconcave, any type has unquely determned aggregate equlbrum demand Q and transfer T, whch addtonally are nondecreasng n under Assumpton 1. Second, convexty of equlbrum tarffs s preserved under aggregaton. In partcular, suppose that the buyer wshes to trade an aggregate quantty Q k wth the sellers other than k. Then the mnmum transfer she has to make n return s { T k (Q k ) mn t k (q k ) : q k R + for all k k and q k = Q }. k (2) k k k k The aggregate tarff T k s the nfmal convoluton of the ndvdual tarffs t k posted by the sellers other than k, and s convex f each of them s convex (Rockafellar (1970)). 3 The Man Result Our central result s the followng theorem. Theorem 1 Suppose that Assumptons 1 3 are satsfed, and let (t 1,..., t K, s) be an equlbrum wth convex tarffs. If some trade takes place n equlbrum, then () All trades take place at unt prce c I and each type purchases D (c I ) n the aggregate. () If D (c I ) > 0, then c = c I. Thus each seller earns zero proft on each trade. The frst nsght of Theorem 1 s that nonexclusve competton leads to lnear prcng, at least when attenton s restrcted to equlbra wth convex tarffs. Ths shows the dscplnng role of competton n our model: although sellers are allowed to propose arbtrary tarffs, they end up tradng at the same prce. From the standard Bertrand undercuttng argument, ths prce cannot be strctly above the hghest possble cost c I. In an equlbrum t cannot le below nether. If t dd, then sellers would want to lmt the quanttes they sell to the hghest types, whch they can do by postng a lmt order at the equlbrum prce wth a well-chosen maxmum quantty. We then have a tenson between zero profts n the aggregate, and the hgh equlbrum prce c I. In the pure prvate-value case n whch the cost c s ndependent of the buyer s type, ths tenson s easly relaxed, and we obtan the usual Bertrand result, leadng to an effcent outcome. By contrast, n the pure common-value case n whch the cost c s strctly ncreasng wth the buyer s type, our result mples that only the hghest type I may actvely trade n equlbrum, whereas all types < I must be excluded from trade. 7

9 Ths market falure s much more dramatc than n Akerlof (1970) or Rothschld and Stgltz (1976), as only a sngle type may actvely trade n equlbrum. Addtonally condtons for the exstence of an equlbrum are very restrctve: from Theorem 1() one must have D (c I ) = 0 for all < I f an equlbrum s to exst at all. Hence the hghest type must have preferences dfferent enough from those of other types. 4 Proof Outlne Throughout ths secton, we suppose the exstence of an equlbrum (t 1,..., t K, s) wth convex tarffs, and we nvestgate ts propertes. Recall that from the vewpont of seller k the aggregate tarff T k of the sellers other than k can be computed from the tarffs t k as n (2). In turn T k determnes how type evaluates any bundle (q, t) she may trade wth seller k through the followng ndrect utlty functon (q, t) max Q k R + {u (q + Q k, t + T k (Q k ))}. (3) Observe that the maxmum n (3) s always attaned and that the ndrect utlty functons are strctly decreasng n t and contnuous n (q, t). 6 Two types of arguments are used n the proof. Some rely only on the convexty of tarffs and preferences. Because we only assume weak convexty, gven a convex functon f : R + R we use the notaton f(x), f(x), and + f(x) to denote respectvely the subdfferental of f at x, the mnmum element of f(x), and the maxmum element of f(x). Hence f(x) = [ f(x), + f(x)]. Other arguments rely on sngle-crossng propertes, n partcular when t comes to examnng the buyer s best response to a devaton. Most often the devatons we consder correspond to fnte menus, ncludng as many optons as there are types. We denote such a menu by {(0, 0),..., (q, t ),...}. Fnally, we say that ndvdual quanttes are nondecreasng f, gven a famly of tarffs, the quanttes q k one has q k q k +1. traded by each type wth each seller k are such that for any k and < I 4.1 The Buyer s Behavor Consder frst the buyer s choce problem when she faces an arbtrary famly of convex tarffs. When these tarffs are strctly convex, the buyer clearly has a unque best response, wth ndvdual quanttes that are nondecreasng n her type. On the other hand, when some 6 The last statement follows from Berge s maxmum theorem (Alprants and Border (2006, Theorem 17.31)). 8

10 tarffs are affne wth the same slope on some ntervals of quanttes, then the buyer may have multple best responses. Stll we can show the followng result. Lemma 1 Let (t 1,..., t k ) be a famly of convex tarffs. Then the buyer has a best response to (t 1,..., t k ) wth nondecreasng ndvdual quanttes. The proof of Lemma 1 ntroduces some notatons and addtonal results that wll be used later on. It only reles on convexty, by showng the exstence of a best response wth ndvdual quanttes that are comonotonc wth aggregate quanttes. Consder next the choce problem faced by the buyer n her relatonshp wth any seller k, fxng the equlbrum tarffs t k of the sellers other than k. From these tarffs one can buld T k as n (2), and mples that the ndrect utlty functons the prmtve utlty functons u. as n (3). The convexty of the aggregate tarff T k crucally nhert a weak sngle-crossng property from Lemma 2 For all k, <, q q, t, and t, (q, t) (q, t) < (q, t ) mples (q, t ) mples (q, t) (q, t ), (4) (q, t) < (q, t ). (5) In words, hgher types are more eager to buy hgher quanttes from a gven seller. As an applcaton, suppose that seller k devates and posts an arbtrary tarff t k. From the vewpont of seller k, type s maxmzaton problem amounts to max { q k (q k, t k (q k ))}. (6) R + Gven Lemma 2, t follows from standard monotone-comparatve-statcs consderatons that there exsts for each a soluton to (6) that s nondecreasng n. Lemma 2 therefore complements Lemma 1: f all tarffs but the k th one are convex, then there exsts a best response of the buyer such that the quanttes traded wth seller k are nondecreasng n her type. Ths property, whch plays a central role n our analyss, suggests that the restrcton to convex equlbra allows one to make use of standard screenng technques. 4.2 How the Sellers Can Break Tes We now consder the behavor of a sngle seller k, n a stuaton n whch all other sellers post ther equlbrum tarffs t k. Suppose that seller k devates to a menu {(0, 0),..., (q, t ),...}. 9

11 For each type of the buyer to select the trade (q, t ) n ths menu, t must be that the followng ncentve-compatblty and ndvdual-ratonalty constrants hold for all and : (q, t ) (q, t ), (7) (q, t ) (0, 0). (8) These constrants are not suffcent to ensure that each type wll choose to trade (q, t ) after the devaton. Indeed, a gven type may be ndfferent between two trades, thus creatng some tes. The followng result shows that, as long as he stcks to nondecreasng quanttes, seller k can secure the proft he would obtan f he could break tes n hs favor. Defne { I } V k (t k ) sup m (t c q ) =1 over all menus {(0, 0),..., (q, t ),...} that satsfy (7) (8) for all and, and that have nondecreasng quanttes q +1 q for all < I. Lemma 3 In an equlbrum (t 1,..., t K, s) wth convex tarffs, seller k s proft s no less than V k (t k ). Any seller k can thus control the quanttes he trades wth the buyer f, gven the other sellers tarffs, he devates to an ncentve-compatble menu that dsplays nondecreasng quanttes. Ths last requrement s not a drect consequence of (7) (8), gven that the buyer s preferences only satsfy the weak sngle-crossng property characterzed n Lemma 2. Indeed, ths requrement s lkely to be costly because, gven Assumpton 3, any seller would prefer to sell less to hgher types. However, t cannot be dspensed wth as the buyer always has a best response wth nondecreasng quanttes. Therefore, V k (t k ) s the hghest payoff that seller k may expect by devatng, f he faces a buyer who systematcally selects a best response wth nondecreasng quanttes. The proof for Lemma 3 goes as follows. (9) Consder a menu of trades that verfes the constrants n the Lemma, and suppose that two consecutve types and + 1 are both ndfferent between ther trade and the other type s trade. Then seller k can modfy hs menu by poolng both types on the same trade. Under Assumpton 3, because q q +1 ths can be done wthout reducng the profts on the rght-hand sde of (9). Ths frst step s key to the proof, as t shows that between two neghborng types only one ncentve-compatble constrant can be bndng. The proof then shows that seller k can slghtly perturb the transfers n the menus so as to make all the relevant ncentve-compatblty constrants slack. Hence the buyer has a unque best response, whch guarantees that seller k gets the proft on the rght-hand sde of (9). 10

12 4.3 Equlbra wth Nondecreasng Quanttes The above results suggest that we frst focus on equlbra wth nondecreasng ndvdual quanttes, that s q k q+1 k for all k and < I. In ths secton, we characterze these equlbra. We then show n Secton 4.4 that the latter restrcton on the buyer s behavor actually s nconsequental. So suppose that such an equlbrum (t 1,..., t K, s) exsts. The equlbrum trades of seller k then verfy all the constrants n program (9). An mmedate consequence of Lemma 3 s thus that these trades must be soluton to ths program, and that the equlbrum proft of seller k s equal to V k (t k ). Consderng program (9), t s clear that for each type at least one constrant must bnd, for, otherwse, one could slghtly ncrease t. Our next result reles on Lemma 2 to determne whch constrants are bndng. Lemma 4 Let (t 1,..., t K, s) be an equlbrum wth convex tarffs and nondecreasng ndvdual quanttes. Then, for any seller k, f for some the equlbrum trades of type are such that the ndvdual-ratonalty constrant (8) s slack, one has 2 and the ncentve-compatblty constrant (7) for = 1 bnds. Therefore, from the perspectve of each seller, the ndvdual-ratonalty constrant bnds at the bottom, or more generally for all types below a threshold, and the downward local ncentve-compatblty constrants bnd for all other types. Ths result s remnscent of those obtaned under monopolstc screenng, wth the dfference that they are formulated n terms of the ndrect utlty functons nstead of the prmtve utlty functons u. Under monopolstc screenng, the am s to characterze Pareto-optmal allocatons, whch mples that tes are broken n the most favorable way to the monopolst. 7 In our compettve settng, Lemma 3 offers a condton under whch the seller can break tes as desred, namely, that quanttes are nondecreasng. Ths allows us to proceed wthout ntroducng further restrctons on the buyer s behavor. Our next result bulds on Lemma 4 to show that equlbra wth convex tarffs and nondecreasng quanttes actually feature lnear prcng f trade takes place at all. Lemma 5 Let (t 1,..., t K, s) be an equlbrum wth convex tarffs and nondecreasng ndvdual quanttes such that some trade takes place n equlbrum. Then there exsts p R such that all trades take place at unt prce p, and each type purchases D (p) n the aggregate. 7 See Hellwg (2010) for a complete treatment of the monopolstc case under weak sngle-crossng and prvate values, and Chade and Schlee (2012) for a smpler approach to the common values case. 11

13 The proof of Lemma 5 goes as follows. When sellers offer convex tarffs, every best response of each type s such that she buys the last unt of the good at some prce p, ndependently of the sellers she trades wth. Because the correspondng aggregate quantty s nondecreasng n the type, t s easly shown that one must have p p 1. Consder now an equlbrum, and suppose that type trades at a prce p > p 1. Clearly, t s not optmal for type to mmc type 1 and trade the quantty q k 1 wth seller k, as ths would mply tradng at a margnal prce dfferent from p. Hence the downward local ncentve constrant from type to type 1 cannot bnd. A fortor, t s not optmal for type to trade a zero quantty wth seller k. Hence the ndvdual ratonalty constrant of type cannot bnd. But these results contradct Lemma 4. We now show that each equlbrum trade must yeld zero proft to the seller who makes t. The ntuton s smple. Under lnear prcng, sellers collectvely have to share a rsky demand D (p). Under Assumpton 2, we know that D I (p) > D I 1 (p) f some trade takes place at all, so the prce p must be hgh enough to convnce some of the sellers to provde addtonal quanttes to the hghest type. In fact, one must have p c I, otherwse a seller could devate by postng a lmt order wth unt prce p and maxmum quantty qi 1 k. On the other hand, aggregate profts cannot be postve, by a standard Bertrand argument. Because c I s the hghest possble cost, we get the followng result. Lemma 6 Let (t 1,..., t K, s) be an equlbrum wth convex tarffs and nondecreasng ndvdual quanttes such that some trade takes place n equlbrum. Then, for p defned as n Lemma 5, we have p = c = c I for any type who actvely trades. The proof of ths result, unlke that of Lemmas 1 to 5, reles on Assumpton 2. If we relax t, we can stll prove that n equlbrum the types who trade are exactly those above a threshold 0, and that the equlbrum prce s E[c 0 ]. Moreover, these types must demand exactly the same aggregate quantty at that prce, mplyng n most setups that there s only one such type 0 = I, leavng Theorem 1 unaffected. 4.4 Other Equlbrum Outcomes It follows from Lemmas 5 and 6 that the conclusons of Theorem 1 hold n the case of equlbra wth nondecreasng ndvdual quanttes. To complete the proof of Theorem 1, we now show how to turn any equlbrum wth convex tarffs nto an equlbrum wth the same tarffs, but now wth nondecreasng quanttes. So let (t 1,..., t K, s) be an equlbrum wth convex tarffs. Let v k be the equlbrum profts of seller k. Lemma 3 offered a lower bound V k (t k ) for ths proft. We can buld 12

14 another lower bound by mposng n program (9) the addtonal constrant that the transfers t must be computed usng the equlbrum schedule t k. So defne { I } V k (t 1,..., t K ) sup m [t k (q ) c q ] over all (q 1,..., q I ) R I + that satsfy =1 (10) (q, t k (q )) (q, t k (q )), (11) (q, t k (q )) (0, 0), (12) and such that q +1 q for all < I. By Lemma 3, we therefore have v k V k (t k ) V k (t 1,..., t K ) (13) for all k. Now, recall from Lemma 1 that the buyer has at least one best response wth nondecreasng ndvdual quanttes. Choose one such best response, and let v k be the resultng proft for seller k. Because the correspondng trades for seller k verfy the constrants n the above program, one must have V k (t 1,..., t K ) v k (14) for all k. Fnally, gven the convexty of the tarffs (t 1,..., t K ), the aggregate quanttes Q and the aggregate transfers T are the same for any best response of the buyer. Due to the lnearty of the sellers profts, we get k vk = m (T c Q ) = k v k. Usng the nequaltes (17) (18), we fnally obtan v k = V k (t k ) = V k (t 1,..., t K ) = v k for all k. Ths proves n partcular that, n any equlbrum, each seller k earns V k (t k ). Therefore, no seller can get more than the proft he could secure by stckng to nondecreasng quanttes. If we now specfy that the buyer s strategy must select nondecreasng quanttes whenever possble, t s easly understood that wth ths new strategy we have bult an equlbrum wth nondecreasng ndvdual quanttes. Ths last result s proven more formally n the Appendx. Lemma 7 If (t 1,..., t K, s) s an equlbrum wth convex tarffs, then there exsts a strategy ŝ for the buyer such that (t 1,..., t K, ŝ) s an equlbrum wth nondecreasng ndvdual quanttes that yelds the same proft to each seller. Note that the aggregate equlbrum quanttes Q and the ndrect utlty functons are the same n the ntal and the fnal equlbrum. Combnng Lemmas 5, 6, and 7 then shows that Theorem 1 apples to all equlbra wth convex tarffs. 13

15 5 Extensons In ths secton, we consder some extensons of the basc settng to test the robustness of our man result. 5.1 Competton n Convex Tarffs: The Lmt-Order Book Many fnancal markets are centralzed and heavly regulated, and are seemngly at odds wth non-exclusve markets for whch transactons are hdden and freely barganed upon. In ths part, we show how to generalze our results to an mportant group of markets, those that use a prcng mechansm known as the dscrmnatory (or open) lmt order book. Ths (electronc) book records all offers made by market-makers. These offers are lmt orders, that specfy a prce and a maxmum quantty to be sold (or bought) at ths prce. When a new order arrves, f t can be executed aganst orders n the book then trade occurs. The key feature s that the trade s settled usng the market makers prces, whch may dffer across market makers. 8 So n ths part we study the same game as n Secton?? except that sellers are now restrcted to use convex tarffs. Our treatment reles a lot on the followng specfcaton for the buyer s payoff: u (Q, t) = v (Q) T where v s dfferentable and strctly concave. We requre that for any Q the dervatve v (Q) s ncreasng wth ; ths requrement s strong enough to mply both Assumpton 1 and Assumpton 2. The quaslnearty of the buyer s preferences s not wthout loss of generalty, as t excludes wealth effects and all nsurance consderatons. Stll t encompasses the specfcatons n BMR and BB for the case of fnancal markets. In the followng, we refer to ths game as the convex game. Focusng on convex tarffs has two man advantages. Frstly t allows us to rely on smple tools such as supply functons and frst-order condtons, whose propertes are well-known under convexty assumptons; nstead of usng arbtrary menus, wth ther cohort of ncentve-compatblty constrants. Ths makes for more ntutve proofs; some of our arguments are n fact qute drect when consderng graphs. Secondly, compared to the orgnal model we reduce the avalablty of devatons. Ths change can only enlarge the set of equlbra. 8 By contrast, n a unform lmt order book trades are settled usng an equlbrum prce that equates global demand to global supply; ths prce then apples to all transactons. Ths system s often studed usng supply functons equlbra (Klemperer-Meyer, Vves, XXXX), whle our model fts better the case of the dscrmnatory order book. 14

16 Stll we obtan essentally the same results as above: Theorem 2 Let (τ 1,..., τ K, σ) be an equlbrum of the convex game. If at least one type actvely trades n equlbrum, then () All trades take place at unt prce c I and each type purchases D (c I ) n the aggregate. () If D (c I ) > 0, then c I = c. Thus each seller earns zero proft on each trade. As already explaned, exstence condtons are also very severe. Overall ths result casts a doubt on whether even well-organzed markets, n whch partcpants are requred to post lmt orders, may functon satsfactorly. The proof of ths result follows smlar steps as the proof of Theorem 1. Frstly notce that Lemmas 1 and 2 stll hold. Then we have to check how sellers can break tes; the case of equlbra wth nondecreasng quanttes; and fnally other equlbrum outcomes How the Sellers Can Break Tes We reformulate here Lemma 3. Consder the behavor of a sngle seller k, n a stuaton n whch all other sellers post ther equlbrum tarffs t k. Suppose that seller k devates to a convex tarff t. For each type of the buyer to select the trade q n ths tarff, t must be that q s a best-response to the tarffs (t, t k ), or equvalently q arg max q (q, t(q)) (15) Ths constrant s not suffcent to ensure that each type wll choose to trade (q, t(q )) after the devaton. Indeed, a gven type may be ndfferent between two trades, thus creatng some tes. The followng result shows that, as long as he stcks to nondecreasng quanttes, seller k can secure the proft he would obtan f he could break tes n hs favor. Defne { I } V k (t k ) sup m (t(q ) c q ) (16) =1 over all convex tarffs t and nondecreasng quanttes (q ) that satsfy (15) for all. Lemma 8 In an equlbrum (t 1,..., t K, s) wth convex tarffs, seller k s proft s no less than V k (t k ). 15

17 Under quas-lnearty of buyer s preferences, only the slope of the schedule t k matters for q to be a best-response of type-. As llustrated n Fgure XXXXX, one can therefore replace t k by a pecewse lnear tarff nducng the same best-reply for the buyer and yeldng seller k a proft at least equal to that of t k. Moreover consder a segment of ths pecewse lnear tarff wth slope p, and the set of types that trade on ths segment. If there exsts a quantty ˆq on ths segment such that all types that trade above ˆq have p < c, then seller k would rase profts by truncatng ths segment at ˆq, as llustrated on Fgure XXXXX. Indeed ths would reduce the quanttes traded by those, wth transfers that are as least as hgh. Fnally seller k can reduce each slope slghtly. Ths ensures that all types buy the maxmum quantty at prce p, and thus seller k can secure the announced proft Equlbra wth nondecreasng ndvdual quanttes A key mplcaton of Lemma 9 s that n equlbrum, sellers offer pecewse lnear tarffs that can be nterpreted as fnte sequences of lmt orders. The followng result parallels Lemma 5. Lemma 9 Let (t 1,..., t K, s) be an equlbrum wth convex tarffs and nondecreasng ndvdual quanttes such that some trade takes place n equlbrum. Then there exsts p R such that all trades take place at unt prce p. The followng result parallels Lemma 6. Lemma 10 Let (t 1,..., t K, s) be an equlbrum wth convex tarffs and nondecreasng ndvdual quanttes such that some trade takes place n equlbrum. Then, for p defned as n Lemma 9, we have p = c I and only type I actvely trades, an amount equal D I (c I ) n the aggregate Other Equlbrum Outcomes It follows from Lemmas XXXXX that the conclusons of Theorem 3 hold n the case of equlbra wth nondecreasng ndvdual quanttes. To complete the proof, we now show how to turn any equlbrum of the convex game nto an equlbrum wth the same tarffs, but now wth nondecreasng ndvdual quanttes. The proof strctly parallels the proof gven n Secton XXXXX. [So why gve t here? We could stop. I only dd elementary changes below.) So let (t 1,..., t K, s) be an equlbrum of the convex game. Let v k be the equlbrum profts of seller k. Lemma 8 offered a lower bound V k (t k ) for ths proft. We can buld 16

18 another lower bound by mposng n program (??) the addtonal constrant that the transfers t must be computed usng the equlbrum schedule t k ; ths defnes V k (t 1,..., t K ). By Lemma 3, we therefore have v k V k (t k ) V k (t 1,..., t K ) (17) for all k. Now, recall from Lemma 1 that the buyer has at least one best response wth nondecreasng ndvdual quanttes. Choose one such best response, and let v k be the resultng proft for seller k. Because the correspondng trades for seller k verfy the constrants n V k (t 1,..., t K ), one must have V k (t 1,..., t K ) v k (18) for all k. Fnally, gven the convexty of the tarffs (t 1,..., t K ), the aggregate quanttes Q and the aggregate transfers T are the same for any best response of the buyer. Due to the lnearty of the sellers profts, we get k vk = m [T c Q ] = k v k. Usng the nequaltes above, we fnally obtan v k = V k (t k ) = V k (t 1,..., t K ) = v k for all k. Ths proves n partcular that, n any equlbrum, each seller k earns V k (t k ). Therefore, no seller can get more than the proft he could secure by stckng to nondecreasng quanttes. If we now specfy that the buyer s strategy must select nondecreasng quanttes whenever possble, t s easly understood that wth ths new strategy we have bult an equlbrum wth nondecreasng ndvdual quanttes. Ths last result s the analogue of Lemma 7. Note that the aggregate equlbrum quanttes Q and the ndrect utlty functons are the same n the ntal and the fnal equlbrum. Combnng our results then shows that Theorem 3 apples to all equlbra of the convex game. 5.2 Convex Costs So far, we have assumed that sellers have constant and possbly type-dependent margnal costs. An examnaton of the proof of Lemmas 1 5 reveals that we can handle much more general cases. We now endow each seller k wth a proft functon v k (q, t), whch we take to be contnuous and strctly ncreasng n t, and such that the followng generalzed reverse sngle-crossng assumpton holds. Assumpton 4 For all k, <, q < q, t, and t, v k (q, t) v k (q, t ) mples v k (q, t) vk (q, t ). 17

19 Each seller k therefore weakly prefers to sell lower quanttes to hgher types. followng result holds. Then the Corollary 1 Under Assumptons 1 2 and 4, any equlbrum wth convex tarffs and nondecreasng ndvdual quanttes such that some trade takes place n equlbrum dsplays lnear prcng: there exsts p R such that all trades take place at unt prce p, and each type purchases D (p) n the aggregate. Extendng ths result to equlbra wth quanttes that may be decreasng requres some addtonal structure. Assume that each seller s cost of provdng type wth a quantty q s c (q), where c : R + R + s now a strctly convex cost functon, wth c (0) = 0. In ths settng, the analogue of Assumpton 4 can be stated n terms of the one-sded dervatves of these cost functons. Assumpton 5 For all < and q < q, c (q ) + c (q). Assumpton 5 s consstent wth prvate-value and common-value envronments. Theorem 1 generalzes as follows. Theorem 3 Suppose that Assumptons 1 2 and 5 are satsfed, and let (t 1,..., t K, s) be an equlbrum wth convex tarffs. If some trade takes place n equlbrum, then there exsts p R soluton to and such that: ( ) DI (p) p c I K () All trades take place at unt prce p and each type purchases D (p) n the aggregate, and D (p)/k from each seller. () Only type I actvely trades n equlbrum: (19) D 1 (p) =... = D I 1 (p) = 0 < D I (p). (20) When there s a sngle type I, ths result states that any equlbrum s compettve n the sense that the equlbrum prce equalzes type I s demand and the sum of the sellers supples. Equlbrum outcomes are hence frst-best effcent, as n the case of lnear costs. 18

20 The ntroducton of multple types does not affect ths property, the only change beng that all types below I must demand a zero quantty at the equlbrum prce. The structure of the proof of Theorem 3 s smlar to that of Theorem 1. Frst, gven Corollary 1, one has to show that the result holds for all equlbra wth convex tarffs and nondecreasng ndvdual quanttes. Lemma 11 Let (t 1,..., t K, s) be an equlbrum wth convex tarffs and nondecreasng ndvdual quanttes such that some trade takes place at prce p n equlbrum. satsfes (19) (20). Then p The result that no trade may take place except perhaps at the top of the buyer s type dstrbuton now holds whether or not the envronment features common values. As n the lnear cost case, sellers collectvely have to share a rsky demand D (p), but under convex costs the precse sharng now matters. Under Assumpton 2, we know that D I (p) > D I 1 (p), so the prce p must be hgh enough to convnce some of the sellers to provde addtonal quanttes to the hghest type. In fact, one must have p c I (qi k ) for all k, otherwse seller k could devate by postng a lmt order wth a unt prce p and a maxmum quantty slghtly below qi k. But, at such a hgh prce, sellers are wllng to sell hgh quanttes to lower types, whch s consstent wth equlbrum only f all these types demand a zero quantty. To complete the proof of Theorem 3, there thus only remans to show that the restrcton to equlbra wth nondecreasng ndvdual quanttes s nnocuous. To ths end, consder an equlbrum (t 1,..., t K, s) wth convex tarffs. Denote by v k the equlbrum profts of seller k. Replacng the lnear cost functons n the defntons (9) and (10) of V k (t k ) and V k (t 1,..., t K ) by the now convex cost functons, we formally get the lower bound (17) for the profts v k. On the other hand, the sum of these profts cannot exceed the value they would reach f the buyer were to break tes n favor of the coalton of sellers. Formally, defne { } V 0 (t 1,..., t K ) sup m [t k (q ) c (q )] k over all (q 1,..., q I ) R I + that satsfy (11) (12). Note that we do not mpose the constrant that quanttes be nondecreasng. We thus have (21) V 0 (t 1,..., t K ) k v k. (22) But the program (21) defnng V 0 (t 1,..., t K ) can be smplfed nto { } nf m c (q ) k 19

21 under the same constrants, as the aggregate transfer chosen by the buyer s unquely defned gven the tarffs. The proof of Lemma 1 shows that such a rsk-sharng problem admts a soluton wth nondecreasng ndvdual quanttes: ths s the effcent manner to share rsk. Let v k be the assocated proft for seller k; note that k v k = V 0 (t 1,..., t K ). Moreover, n such a soluton, each seller k trades a famly of quanttes that are nondecreasng, and thus hs assocated proft v k must be no more than V k (t 1,..., t K ). Summarzng, we get from (17) and (22) that V k (t 1,..., t K ) v k = V 0 (t 1,..., t K ) v k V k (t k ) V k (t 1,..., t K ), k k k k k and thus these nequaltes are n fact equaltes. In partcular, ths mples for every k that v k = V k (t k ). We can then apply Lemma 7 wthout changes. 20

22 Appendx Proof of Lemma 1. As a prelmnary remark, observe that when seller k selects a convex tarff t k, hs supply correspondence s the nverse of the subdfferental of t k (Bas, Martmort, and Rochet (2000, Defnton 2)): for each p R, the supply of seller k at the margnal prce p s the set {q : p t k (q)}. Ths set s a nonempty compact nterval, wth lower and upper bounds s k (p) and s k (p) that are nondecreasng n p. When ths nterval s nontrval, t k s affne over t, wth slope p. Note also for future reference that s k s rght-contnuous. The proof conssts of two steps. Step 1 Recall that gven a famly (t 1,..., t K ) of convex tarffs, the aggregate equlbrum demand Q of type s unquely defned and nondecreasng n. Gven Q, type s utltymaxmzaton problem (1) reduces to mnmzng her total payment for Q : { mn t k (q k ) : q k R + for all k and k k q k = Q }. (23) Ths s a convex problem, so that by the Kuhn Tucker theorem one can assocate to any of ts solutons (q 1,..., q K ) a Lagrange multpler p such that p t k (q k ) for all k. If there were two dfferent solutons (q 1,..., q K ) and (q 1,..., q K ) wth dfferent multplers p < p, then because each tarff s convex one would obtan q k q k for all k, and because both solutons must sum to the same Q they would be dentcal, a contradcton. shows that two dfferent solutons must share the same p. Thus one can assocate to each type a prce p such that whatever the soluton (q 1,..., q K ) to type s problem, one has p t k (q k ) for all k. Note that we can wthout loss of generalty adopt the conventon that p s nondecreasng n. Indeed, f p > p +1 for some < I, then, because p t k (q k ) and p +1 t k (q k +1) for all k, one has q k Ths q k +1 for all k. Because these quanttes sum respectvely to Q and Q +1, and because Q Q +1, t actually follows that q k = q k +1 for all k. Hence p t k (q k +1) for all k and one may replace p +1 by p. Gven ths conventon, s k (p ) and s k (p ) are nondecreasng n for all k. Step 2 Accordng to Step 1 and our prelmnary remark, solutons (q 1,..., q K ) to type payment-mnmzaton problem must verfy q k = Q and s k (p ) q k s k (p ) for all k, (24) k and these condtons are n fact suffcent, as all tarffs have the same slope p for quanttes n these ntervals. Thereby, our problem reduces to fndng a famly of nondecreasng quanttes 21

23 verfyng (24). For future reference, we shall prove a stronger result. Choose a famly of strctly convex functons (f 1,..., f I ), and consder the followng famly of mnmzaton problems ndexed by : { } mn k f (q k ) (25) subject to (24). By strct convexty of the functons f, any such problem admts a unque soluton. We show below that the famly of these solutons must dsplay nondecreasng ndvdual quanttes. Ths naturally mples the exstence of a famly wth nondecreasng ndvdual quanttes verfyng (24), and shows the lemma. To do so, let us proceed by contradcton and suppose that a famly of solutons to the problems (25) has q k > q+1 k for some k and < I. Under (24), ths mples s k (p ) s k (p +1 ) q k +1 < q k s k (p ) s k (p +1 ). (26) Because the ntervals for and + 1 have a nontrval ntersecton, t must be that p = p +1. Therefore, for any seller k we have s k (p ) = s k (p +1 ) and s k (p ) = s k (p +1 ). Moreover, because q k > q k +1 and Q Q +1, we know that there exsts k k such that q k < q k +1. Usng the equaltes we have just shown, ths mples s k (p ) = s k (p +1 ) q k < q k +1 s k (p ) = s k (p +1 ). (27) Gven (26) (27), one can slghtly reduce q k and ncrease q k by the same amount, so that (24) s stll verfed. Because (q 1,..., q K ) s assumed to mnmze k f (q k ), t must be that at the margn f (q k ) + + f (q k ) 0. Because f s strctly convex, ths mples that q k q k. Alternatvely, one could slghtly ncrease q k +1, and reduce q k +1 by the same amount. Once more, t must be that at the margn + f +1 (q k +1) f +1 (q k +1) 0. Because f +1 s strctly convex, ths mples that q k +1 q k +1. Overall, we have shown that q k q k < q k +1 q k +1, n contradcton wth our assumpton that q k > q k +1. Ths concludes the proof. Proof of Lemma 2. Fx some k, q < q, t, and t. Let T (Q) t + T k (Q q), defned for Q q. Smlarly, let T (Q) t + T k (Q q ), defned for Q q. Accordng to (3), for each, computng (q, t) amounts to maxmze u (Q, T (Q)) wth respect to Q q. Let Q q be the soluton to ths problem; t s unque as u s strctly quasconcave and strctly decreasng n aggregate transfers, and T (Q) s convex n Q. Smlarly, computng (q, t ) amounts to maxmze u (Q, T (Q)) wth respect to Q q. Let Q q be the unque soluton to ths problem. The proof conssts of two steps. 22