Equlbra Exst and Trade S effcent proportionally

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1 On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané February 27, 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 conference partcpants at the 7 th ENSAI Economc Day and at the Unversté Pars-Dauphne Workshop n Honor of Rose-Anne Dana for many useful dscussons. 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 types 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 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 q k, k t k (q k ) always has a soluton, we requre the tarffs t k 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. (1) 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, when ther value s fnte, 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 (13) (14), 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 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 ). 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 2 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 14 (15)

16 () 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: D 1 (p) =... = D I 1 (p) = 0 < D I (p). (16) 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. 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 2 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 8 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 (15) (16). 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 2, 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 (13) 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, 15

17 defne { } V 0 (t 1,..., t K ) sup m [t k (q ) c (q )] k (17) 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 V 0 (t 1,..., t K ) k v k. (18) But the program (17) defnng V 0 (t 1,..., t K ) can be smplfed nto { } nf m c (q ) k 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 (13) and (18) 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. 16

18 Appendx Proof of Lemma 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 utlty-maxmzaton problem (1) reduces to mnmzng total payment for Q : { mn t k (q k ) : q k R + for all k and k k q k = Q }. Ths s a convex program, so that by the Kuhn Tucker theorem one can assocate to any soluton (q 1,..., q K ) a Lagrange multpler p such that p t k (q k ) for all k. If there are two dfferent solutons (q 1,..., q K ) and (q 1,..., q K ) wth dfferent multplers p < p, then because each tarff s convex one obtans q k q k for all k, and because both solutons sum to the same Q they must be dentcal, a contradcton. Ths shows that two dfferent solutons must share the same p. Consequently 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. Moreover, by the same argument as above, p s nondecreasng n. For each and each k, one can thus buld the nonempty set {q : p t k (q)}. Let s k be ts mnmum element, and let s k be ts maxmum. Both s k and s k are nondecreasng n. The nterval [s k, s k ] s n fact the set of quanttes that are provded by seller k at a margnal prce equal to p. If ths nterval s nontrval, then t k s affne over t, wth slope p. Consequently solutons (q 1,..., q K ) to type payment mnmzaton problem must verfy q k = Q and s k q k s k for all k, (19) k and these condtons are n fact suffcent, as all tarffs have the same slope p for quanttes n these ntervals. Our problem thus reduces to fnd a famly of nondecreasng quanttes verfyng (19). We n fact prove a stronger result, whch wll be useful for future reference. Choose a famly of strctly convex functons (f 1,..., f I ), and consder the followng famly of problems, ndexed by : { } mn f (q k ) k subject to (19). By strct convexty of the functons f, each 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 (19), and shows the lemma. 17

19 To do so, proceed by contradcton and suppose that a famly of solutons has q k > q k +1, for some k and < I. Under (19), ths mples s k s k +1 q k +1 < q k s k s k +1. (20) 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 = s k +1 and s k = s k +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 = s k +1 q k < q k +1 s k = s k +1. (21) Gven (20) (21), one can slghtly reduce q k and ncrease q k by the same amount, so that (19) 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 thus 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, < I, 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), 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. Suppose that (q, t) < (q, t ) (22) and let >. Because Q q s an admssble canddate n the problem that defnes (q, t), we must have u (Q, T(Q )) (q, t) < (q, t ) = u (Q, T (Q )). Suppose frst that Q < Q. Usng Assumpton 1, we get (q, t) = u (Q, T(Q )) < u (Q, T (Q )) (q, t ), 18

20 where the last nequalty stems from the fact that Q q s an admssble canddate n the problem that defnes (q, t ). Ths shows (5) n ths case. Otherwse we have Q Q q. Then Q s an admssble canddate n the problem that defnes (q, t ), and we get If T (Q ) < T(Q ), we obtan whch shows (5) n ths case. u (Q, T (Q )) (q, t ). (q, t) = u (Q, T(Q )) < u (Q, T (Q )) (q, t ), The only remanng case s when Q Q q and T (Q ) T(Q ). Note that because q < q and T k s convex, T (Q) T(Q) s nonncreasng n Q for Q q. Because Q Q q, we get T (Q ) T(Q ) and thus, as Q q > q s an admssble canddate n the problem that defnes (q, t), (q, t) u (Q, T(Q )) u (Q, T (Q )) = (q, t ), n contradcton wth (22). Hence we have shown (5). The proof of (4) follows by contnuty. Indeed, assume that thus (q, t) = (q, t ). Then, for each ε > 0, (q, t + ε) < (q, t) from (5). Because goes to zero to obtan (4). The result follows. Proof of Lemma 3. The proof conssts of two steps. (q, t + ε) < (q, t) and s contnuous, one can take lmts as ε Step 1 Pck a menu µ = {(0, 0),..., (q, t ),...} that satsfes the ncentve-compatblty and ndvdual-ratonalty constrants (7) (8) for all and, and that has nondecreasng quanttes q +1 q for all < I. We buld a new menu µ = {(0, 0),..., (q, t ),...} by applyng the followng algorthm. At each step n 0 of the algorthm, let µ (n) = { ( (n) (0, 0),..., q, t (n) ) },... be the current menu, wth µ (0) µ by conventon. If there exsts < I such that q (n) both bnd: < q (n) +1 and the followng local ncentve-compatblty constrants +1 ( (n) q, t (n) ) ( = z k (n) q ( (n) ) ( q = z k (n) q +1, t(n) , +1) t(n), then take the smallest such, (n), and pool types (n) and (n) + 1 on the same trade ( (n+1) q, t (n+1) ) ( (n+1) (n) = q, ) ( (n) (n) +1 t(n+1) (n) (n) +1 equal to ether q, t (n) ) ( (n) (n) or q, (n) (n) +1 t(n) +1) accordng to (n) 19, t (n) ),

21 the maxmum value t gves to the proft on the ( (n), (n) + 1) par [ [ m (n) t (n+1) c (n) (n)q ]+ (n+1) m (n) (n) +1 t (n+1) c (n) +1q ]. (n+1) (n) (n) Otherwse, the algorthm stops, and µ µ (n). Note that the algorthm stops n a fnte number of steps as there are fntely many types. Moreover, applyng the algorthm only affects the way tes are broken. Therefore, the menu µ remans ncentve compatble and ndvdually ratonal. Moreover, by constructon, t has nondecreasng quanttes q +1 q for all < I. Fnally, at each step of the algorthm, seller k s proft cannot be decreased. Indeed, the algorthm s actve at step n 0 only f q (n) < q (n). In that case, ether (n) (n) +1 t (n) c (n) q (n) < t (n) (n) (n) (n) (n) +1 c(n) q (n) (n) (n) +1 and then seller k s proft s ncreased by poolng (n) and (n) + 1 on ( q (n) (n) +1, t(n) (n) +1), or t (n) c (n) q (n) t (n) (n) (n) (n) (n) +1 c(n) q (n), (n) (n) +1 so that, as c (n) c (n) +1 by Assumpton 3 and q (n) < q (n) by constructon, seller k s proft (n) (n) +1 cannot be decreased by poolng (n) and (n) + 1 on ( q (n), t (n) ) (n). As a result, (n) I I m (t c q ) m (t c q ), (23) =1 that s, seller k s proft under µ s as least as large as under µ. Step 2 We may now proceed to the second step of the proof. =1 Let ε > 0 be gven. We are gong to modfy transfers (t 1,..., t I ) nto transfers (t 1,..., t I ) such that the menu µ = {(0, 0),..., (q, t ),...} satsfes the followng ncentve-compatblty and ndvdualratonalty constrants for all and : (q, t ) (q, t ), (24) (q, t ) (0, 0), (25) where now these nequaltes are strct as soon as, respectvely, q q and q 0. Moreover, we wll perform ths modfcaton n such a way that transfers reman almost the same: t t ε. (26) Suppose ths modfcaton performed. Then for each ε > 0 seller k could devate to the menu µ. Because of the above propertes, each type must then choose to trade (q, t ) wth seller k. Hence by playng so seller k can secure a proft I =1 m (t c q ) I m (t c q ) ε =1 20 I m (t c q ) ε, =1

22 where the frst and second nequaltes follow from (26) and (23). As ε can be made arbtrarly small, ths shows that seller k s equlbrum proft s at least (9), and the result follows. To conclude the proof, there only remans to modfy the transfers as announced above. We now turn to ths task. Because the quanttes (q 1,..., q I ) are gven, and because the functons are contnuous and strctly decreasng n transfers, we can defne two famles of (extended) real-valued functons γ k and δ k for < I such that, for each t, (q, t) = (q +1, γ k (t)) and +1 (q, t) = +1 (q +1, δ k (t)). (27) Here γ k (t) = or δ k (t) = by conventon f there exsts no soluton to the relevant equaton, whch may occur for t below some threshold. Both γ k and δ k are contnuous and strctly ncreasng where they are fnte. If q = q +1, then clearly γ k (t) = δ k (t) = t. If q < q +1, then, accordng to (4) along wth the fact that the functons are strctly decreasng n transfers, γ k (t) δ k (t) for all t 0. Fnally, by constructon of the menu µ, f q < q +1, then γ k (t ) t +1 δ k (t ) wth at least one strct nequalty. (One may have γ k (t ) =, but δ k (t ) s necessarly fnte.) Gven ε > 0, we can recursvely construct a famly of strctly postve real numbers (ε 1,..., ε I ) as follows. Let ε I ε. Then, for each < I, consder ε +1 > 0 as gven. If q = q +1, choose ε such that 0 < ε < ε +1, whch s clearly feasble. If q < q +1, choose ε such that 0 < ε < ε +1 /2 and such that γ k (t) < δ k (t) and γ k (t) ε +1 2 < t +1 < δ k (t) + ε +1 2 (28) for all t that satsfy t t < ε. Ths s feasble because f ε +1 > 0, all these propertes hold for t = t, and because the functons γ k and δ k are contnuous at t. Observe that the famly (ε 1,..., ε I ) s strctly ncreasng. We now recursvely construct a famly of transfers (t 1,..., t I ) such that t t < ε for all. Set t 1 t 1 f q 1 = 0, and set t 1 t 1 ε 1 /2 otherwse. Note that t 1 t 1 < ε 1. Suppose next that t t < ε for some < I, and defne t +1 as follows: () If q +1 = q, set t +1 t. Note that because t +1 = t n ths case, we then have t +1 t +1 = t t < ε < ε +1, as requred. () If q < q +1, then, as t t < ε by assumpton, we know from the frst part of (28) that γ k (t ) < δ k (t ). Choose any ˆε such that 0 < ˆε < mn{ε +1 /2, δ k (t ) γ k (t )}, and consder the followng three subcases. If t +1 δ k (t ), set t +1 δ k (t ) ˆε. If t +1 γ k (t ), set t +1 γ k (t ) + ˆε. Otherwse, set t +1 t +1. The second part of 21

23 (28) ensures that n each of these three subcases t +1 t +1 < ˆε + ε +1 /2 < ε +1, as requred. By constructon, we have γ k (t ) < t +1 < δ k (t ) f q < q +1. Ths shows that the local ncentve-compatblty constrants (q, t ) (q +1, t +1), +1 (q +1, t +1) +1 (q, t ) are satsfed, wth strct nequaltes f q < q +1. Smlarly, our choce of t 1 ensures that the ndvdual ratonalty constrant for = 1, 1 (q 1, t 1) 1 (0, 0), s also a strct nequalty f q 1 0. Gven the sngle-crossng property (4) (5), a standard argument can be used to establsh that ths set of local constrants mples that the menu µ satsfes the ncentve-compatblty and ndvdual-ratonalty constrants (24) (25) for all and, wth strct nequaltes as soon as, respectvely, q q we have t t < ε < ε I = ε, whch yelds (26). The result follows. and q 0. Fnally, for each Proof of Lemma 4. Fx a seller k, and suppose by way of contradcton that j (q k j, t k (q k j )) > j (qj 1, k t k (qj 1)) k and j (qj k, t k (qj k )) > j (0, 0) (29) for some j 2. Because the quanttes (q1, k..., qi k ) are gven, we can defne a famly of (extended) real-valued functons δ k for < I as n (27). Usng ths notaton, the frst nequalty n (29) equvalently says that t k (q k j ) < δ k j 1(t k (q k j 1)). Now, choose ε such that 0 < ε < δj 1(t k k (qj 1)) k t k (qj k ) and defne a famly of transfers (t 1,..., t I ) as follows: for < j, set t t k (q k ); for = j, set t j t k (q k j ) + ε; for > j, defne recursvely t max{t k (q k ), δ k 1(t 1)}. The menu {(0, 0),..., (q k, t ),...} has three notceable features: () It has the same nondecreasng quanttes and hgher sometmes strctly hgher transfers as the equlbrum allocaton. () It satsfes the ncentve-compatblty constrants (7). Ths s obvous for types < j, because ther transfers are unchanged, whereas the transfers of types j are (weakly) ncreased. As for type j, observe that she cannot be better off mmckng type j 1 because t j = t k (q k j ) + ε, t j 1 = t k (q k j 1), and ε has been chosen so that 22

24 j (qj k, t k (qj k ) + ε) > j (qj 1, k t k (qj 1)). k Usng the fact that the quanttes (q1, k..., qj k ) are nondecreasng along wth the sngle-crossng property (4), t follows that type j cannot be better off mmckng any type < j 1 ether. Fnally, type j cannot be better off mmckng any type > j. that > j s the frst type such that j Indeed, suppose by way of contradcton (qj k, t j) < j (q k, t ). Then j (q 1, k t 1) = j (qj k, t j) < j (q k, t ) so that, from q 1 k q k along wth the sngle-crossng property (5), we have (q k 1, t 1) < (q k, t ). But (q k, t ) (q k, δ 1(t k 1)), so that (q 1, k t 1) < (q k, δ 1(t k 1)), n contradcton wth the defnton (27) of δ 1. k The clam follows. The proof that no type > j can be better off mmckng any type s smlar, and s therefore omtted. () It satsfes the ndvdual-ratonalty constrants (8). Ths s obvous for types < j, because ther trade s unchanged. Thus n partcular j 1 (qk j 1, t j 1) (0, 0). The sngle-crossng property (4) then mples that (qj 1, k t j 1) (0, 0) for all j, from whch the clam follows as the menu {(0, 0),..., (q k, t ),...} s ncentve compatble. Gven () (), we can apply Lemma 3 to conclude that ths menu must gve seller k at most hs equlbrum proft. Ths however contradcts () above. The contradcton establshes that for each j 2, at least one nequalty n (29) must bnd. The proof that the ndvdual ratonalty constrant (8) bnds at = 1 s smlar, and s therefore omtted. Proof of Lemma 5. As shown n Lemma 1, there exsts a nondecreasng sequence of prces (p 1,..., p I ) such that p t k (q k ) for all k and all best responses (q 1,..., q K ) of type. In fact, those best responses are exactly the quanttes verfyng (19) gven the equlbrum tarffs (t 1,..., t K ). As a prelmnary result, let us show that there s no type such that Q > 0 and q k = s k for all k. Indeed, these condtons mply that type has a unque best response that exhausts all avalable supply at the margnal prce p. Moreover, because Q > 0 there exsts at least one seller k such that q k > 0. For any such k, the ndvdual-ratonalty constrant (8) of type s slack n equlbrum, because type has a unque best response. Ths mples from Lemma 4 that 2, and that the ncentve-compatblty constrant constrant (7) for = 1 bnds n equlbrum. As type has a unque best response, t must thus be that q k = q 1. k Ths equalty also holds for those k such that q k = 0, as ndvdual quanttes are nondecreasng by assumpton. Overall we have shown that 2, and that type 1 also exhausts supply at prce p, so that p 1 = p, q 1 k = s k 1 = s k for all k, and Q 1 = Q > 0. 23