The Package Assignment Model

Transcription

1 1 Introducton The Package Agnment Model y Suhl Bkhchandan and Joeph M. Otroy May 1998 We tudy agnment prolem nvolvng package, where a package cont of one or more unt of homogeneou or heterogeneou oject (Kelo and Crawford (1982), Gul and Stachett (1995), Bkhchandan and Mamer (1997) ). Such prolem are an extenon of the tandard agnment model n whch the permle package are retrcted to at mot one oject (Koopman and Beckman (1957), Gale (1960), Shapley and Shuk (1972) ). We demontrate the extent to whch the reult of the tandard prolem can(not) e generalzed to the package veron. We alo how that a pecal cae of the package agnment model n whch there a ngle eller ued to model the auctonng of package enjoy derale properte not alway hared y the model wth everal uyer and eller. The package agnment prolem may alo e regarded a a general equlrum model wth the mplfcaton that ndvdual preference are qualnear and the complcaton that except for the money commodty, all package are ndvle and peronalzed n the ene that whle each uyer can exchange wth many eller and each eller wth many uyer, any uyer/eller par can exchange only one package. Makowk (1979) ha hown that for a general equlrum model wth peronalzed tradng Walraan equlrum can e uefully extended to nclude non-lnear prce. A key element of our analy an enlargement of the commodty pace o that lnear prce wth repect to package can repreent non-lnear prce wth repect to oject. The tandard agnment model rule out complementarty and ututalty over oject. Even f a uyer utlty were non-addtve over oject, the contrant that he can chooe at mot one preclude potental non-addtvte from playng any role. In the package veron, the non-addtvty of uyer utlte over oject an eental ngredent. The tandard agnment prolem alo aume that each eller own one oject. In the package model, f a eller were to own only one oject that would elmnate h packagng opportunte. When th contrant relaxed, eller face the followng prolem: Should they offer the oject they own for ale eparately or n package, and how hould the package e aemled? We hall aume that eller have zero reervaton value for what they own. Conequently, n a model contng of a gven collecton of uyer and a gven collecton of oject, the maxmum gan from trade wll e ndependent of the numer of eller and the 1

2 dtruton of ther ownerhp of oject. The dtruton of thoe gan wll, of coure, depend on the dtruton of ownerhp. Neverthele, when attenton confned to prce that are lnear (addtve) over oject, the dvon of the total gan etween all the uyer and all the eller doe not depend on the dtruton of ownerhp nce a lt of prce (p k ) per unt of each oject k wll e a Walraan equlrum for one dtruton of ownerhp only f t an equlrum for any other. Wth non-lnear prce, however, the dvon of the gan etween uyer and eller doe depend on the dtruton of ownerhp. The more concentrated the ownerhp of oject, the larger the et of pole dvon of the total gan from trade that can e prced and converely, the more dpered the ownerhp the maller the et of dvon of the gan that are content wth prce equlrum. The tandard agnment model an nteger programmng prolem that well-known to e amenale to analy y lnear programmng. Moreover, () the dual of the lnear programmng prolem can e ued to determne the payoff to uyer and eller and the prce of oject, () the dual oluton form a lattce and () the core can e prced,.e., the payoff to uyer and eller n Walraan equlra concde wth the core. We follow th mode of analy n our extenon to package. In partcular, we demontrate that jut a n the tandard agnment model, Walraan equlrum n the package model ext f and only f the aocated prmal lnear programmng prolem ha an nteger-valued oluton, n whch cae the dual oluton form the et of Walraan prce. But wthout further retrcton, n the package agnment A dual oluton compatle wth an nteger-valued oluton to the prmal (Walraan equlrum) need not ext; ndeed, the core may e empty. When Walraan equlra do ext: The prce of package may exht econd degree prce dcrmnaton,.e., the prce of package are the ame for all (anonymou) ut they may e non-lnear (non-addtve) wth repect to the prce of the oject they contan. We alo examne veron of the model wth Walraan equlra exhtng oth econd and thrd degree prce dcrmnaton where the prce of package are oth non-lnear and non-anonymou. There are package agnment model for whch non-lnear prce equlra, and only non-lnear prce equlra, ext. The et of non-anonymou, non-lnear prce equlra contaned n the core, ut the core not necearly prced. Further, thee prce need not form a lattce. The tandard agnment model exht the derale property that an effcent oluton can e acheved va a ealed-d aucton that non-manpulale y the uyer (Demange, 2

3 Gale, and Sotomayor (1986)). The extence of th aucton depend on the fact that (a) the core can e prced and that () all uyer can multaneouly acheve ther margnal product, where an ndvdual margnal product an upper ound on h pole payoff n the core. A nether of thee properte can e guaranteed for the package agnment model, t would appear that there lttle hope for a mlar domnant trategy aucton n th ettng. Neverthele, f there only one eller n the package agnment model then Prce (Walraan) equlra poly requrng non-lnear non-anonymou prcng alway ext. The core can e prced. Even though the core prced n the ngle eller package model, uyer need not acheve ther margnal product n any core payoff (therefore at any prce equlra); hence all uch prce equlra may e manpulale. Wth an aumpton that we call decreang margnal product for uyer, there wll ext prce equlra compatle wth uyer recevng ther margnal product. Therefore, wth th qualfcaton, The core ha a lattce property. There ext non-manpulale ealed-d aucton whoe outcome are prce equlra for the ngle eller package agnment model. The RTC and FCC aucton are example of mult-oject aucton n whch uyer valuaton may e non-addtve n oject,.e., a ngle eller package agnment model. Our reult ugget the pole advantage of allowng for uyer d on package rather than on oject prced ndvdually. We decre the package agnment model n Secton 2 and defne four veron of Walraan equlrum wth repect to t n Secton 3. In Secton 4 we etalh an equvalence etween the extence of equlrum for each of the four veron of Walraan equlrum and the extence of optmal oluton to four correpondng lnear programmng prolem. Secton 5 ntroduce an orderng on the pattern of ownerhp among eller and we how that the extence of Walraan equlra and the core n an economy wth le concentrated ownerhp mple the ame property n the more concentrated veron of that model. In Secton 6 we etalh everal properte of the model wth the mot concentrated pattern of ownerhp, namely the ngle eller model. In Secton 7 we provde uffcent condton for the ngle eller model to exht truth-tellng prce equlrum. Example are gven throughout to llutrate our reult or to how why they cannot e extended. Concludng remark are contaned n Secton 8. Proof are gven n an Appendx. 3

4 2 The model Buyer = 1; 2; :::; B are ntereted n purchang package of ndvle oject k = 1;:::;Kfrom eller = 1; 2; :::; S. LetZ+ = f0; 1; 2; 3; :::g denote the et of nonnegatve nteger. The ntal endowment of ndvle oject owned y eller The aggregate endowment n the economy Defne! =(! 1 ;! 2 ; :::;! k ; :::;! K ) 2Z K + : S! =! : =1 C fw 2Z K + j w! g; = 1; 2; :::; S; C fw 2Z K + j w!g: Buyer have utlty over package z 2 C and (dvle) money, m 2 <. Buyer utlty functon W (z; m) = û (z) +m; where û (z) uyer reervaton value for z. Aume û (0) =0andû () weakly ncreang. Buyer ha no endowment of the non-money commodte ut a large enough quantty of the money commodty (> û (!)) to purchae any z!. It convenent to aume that the uyer ntal endowment of the money commodty normalzed to e zero and that the uyer can upply any (negatve) quantty requred. Hence the utlty of no trade for the uyer W (0; 0) =û (0) +0 = 0. Seller utlty functon over the undle (y; m) 2 C < W (y; m) = m; where y the amount of the K oject old. The eller ha no endowment of the money commodty and therefore the utlty of no trade for the eller W (0; 0) =0. Rather than eparately qualfyng the doman on whch W (;m) and W (;m) are defned, we adopt the followng conventon o that oth are defned on Z+ K : u (z) û (z) f z 2 C,1 otherwe W (z; m) = u (z) +m 0 f y 2 C u (y),1 otherwe W (y; m) = u (y) +m 4

5 Let th economy e E(u ; 8; u ; 8). Each uyer uy S package, one from each eller. (Snce null package are ncluded, th not a retrcton.) Let z 2Z+ K denote a package that uyer requet from eller and let Z =(z 1 ;:::;z S ) denote a collecton of uch package. Each eller rng B package to the market, one for each uyer. Let y 2Z+ K denote a package that eller earmark for uyer and let Y =(y 1 ;:::;y B ) denote a collecton of uch package. Let Z =(Z ) denote the uyer collecton of package requeted from eller and let Y =(Y ) denote the eller collecton of package earmarked for uyer. A neceary condton for the par (Z; Y ) to e feale that each eller can upply h package from h endowment,.e., for each, Y =(y ) atfe y! : Our pont of departure to conder more than one way to decre the remanng condton for a feale agnment, namely that demand for package equal to upply. Thnk of the demand and upple for package a arrvng at a central clearnghoue. The dfferent defnton elow can e nterpreted a dfferent role that the clearnghoue mght play n etalhng fealty. The par (Z; Y ) a frt order agnment f 1 z w f:z =wg 0 w f:y =wg P f:z=wg z mply another way to wrte P y 1 A : P z ; mlarly, P P w f:y=wg y Note that P w the ame a P P y. For a frt order agnment, the clearng houe take an actve role undlng and unundlng the oject n the package uppled to meet the demand. Therefore, the avalale upply of package need only e uffcent n term of the total numer of oject P P y they contan o that they can e repackaged, f neceary, to atfy the total requrement for oject P P z contaned n the demand for package y uyer. The par (Z; Y ) a econd order agnment f for every w f:z =wg z f:y =wg Wth a econd order agnment, the clearnghoue take a le actve role. Package uppled y eller cannot e unundled to atfy the demand of uyer, a they can wth a frt order agnment. But package can e reagned n the ene that a package earmarked y eller for uyer n Y can e reagned a a package from 0 to 0 to atfy the demand n Z. Thepar (Z; Y ) a thrd order agnment f for every and every w f:z =wg z 5 f:y =wg y : y :

6 Th lke a econd order agnment n that package uppled y eller cannot e unundled to atfy the demand of uyer. But a package earmarked y eller for uyer n Y may e reagned a a package from eller 0 to the ame uyer to atfy the demand n Z. Thu, a thrd order agnment anonymou n the eller. Thepar (Z; Y ) a fourth order agnment f for every and z = y or 0: No reaemly or reagnment of package permtted wth th defnton of an agnment. Remark 1: One other type of agnment lyng etween econd and fourth order agnment : for every and every w f:z =wg z f:y =wg Th uyer anonymou agnment allow the clearnghoue to reagn package of type w uppled y a eller among the uyer a long a the total numer of uch package uppled y at leat a large a the numer demanded from. For our purpoe th agnment, whch a counter-part of thrd order agnment, doe not add much to the four veron condered aove. Clearly, the more the clearnghoue ntervene, the eaer t to atfy thee nequalte,.e., they are neted. Therefore, we have Lemma 1: If (Z; Y ) j order agnment, t j, 1 order agnment, j = 4; 3; 2. It wll e convenent to extend the defnton of utlte for package to utlte for agnment y lettng U (Z )=u ( z ) and U (Y )=u ( The utlty of an agnment for a uyer aed on the total package the uyer otan after collectng the package purchaed from each of the eller. For example, f the uyer purchae half of the ook n a matched collecton from one eller and the other half from another eller, the utlty the ame a f the entre collecton were purchaed from a ngle eller. The utlty of an agnment for the eller ether zero or,1; although agnment that yeld,1 utlty are unattanale y a eller, we do not explctly exclude thee from the model. The agnment (Z; Y ) and (Z 0 ;Y 0 ) are utlty equvalent f U (Z )=U (Z 0 ) for all and U (Y )=U (Y 0 ) for all. 6 y ): y :

7 The par (Z; Y ) a j(= 1; 2; 3; 4) order feale agnment f t a j order agnment and P y!, 8. Thu, f (Z; Y ) j order feale then U (Y ) = 0, 8, and U (Z ) >,1, 8. Even though the hgher the order, the maller the et of feale agnment, the added contrant ued to defne econd, thrd, and fourth order fealty are not ndng n term of ther utlty equvalent. Lemma 2: If (Z; Y ) j order feale, there a utlty equvalent agnment (Z 0 ;Y 0 ) that j + 1 order feale, j = 1; 2; 3. An effcent agnment of the non-money commodte n a model wth qua-lnear preference otaned y maxmzng the um total of utlte for the non-money commodte. In the preent cae, effcency acheved y v j (E) =max U (Z )+ U (Y ); uject to the contrant that (Z; Y ) a j order agnment. Note that the maxmum otaned only when the agnment feale for each eller,.e., for all, U (Y )=0. From Lemma 1, what conttute a feale agnment change wth the dfferent order of fealty, ut Lemma 2 ay that the defnton of j order effcency are neverthele utlty equvalent. Therefore, v(e) =v j (E); j = 1; 2; 3; 4 the maxmum gan from an effcent allocaton from the package model E no matter whch defnton of fealty employed. 3 Varete of Walraan equlra Let p (w) 0 e the amount of money that uyer pay eller for the package w (the amount of money eller receve from uyer for w). 1 Thu, n the mot general cae, prce are non-lnear and depend on the dentte of the eller and the uyer. The cot to uyer of the agnment Z =(z ), P (Z ), and the revenue to eller of the agnment Y =(y ), P (Y ),: P (Z )= P (Y )= p (z ); p (z ): 1 Although t uffce to conder only w 2 C, we do not make th explct. The defnton of u n Secton 2 enure that n equlrum any eller wll not ell a package w 62 C. 7

8 Ordnarly, prce of package would e aed on prce of the oject p k, k = 1; 2;:::;K n the vector p =(p k ) uch that p (w) = p w; 8; ; w; (1) P (Z ) = p 2S z ; P (Y ) = p 2B y : That, the prcng functon lnear. We call th a frt order prcng functon. Makowk (1979) ha demontrated that the defnton of Walraan equlrum can e extended to more general prcng functon. A econd order prcng functon defned y p (w) = p(w); 8; ; w; (2) P (Z )= 2S p(z ); P (Y )= 2B p(y ); whch permt non-lnearty n prcng. That, we may have p(w + w 0 ) 6= p(w) +p(w 0 ). Wth econd order prcng all cot and revenue are aed on prce p(w) for all w 2Z+ K. A thrd order prcng functon of the form p (w) = p (w); 8; ; w; (3) P (Z )= 2S p (z ); P (Y )= 2B p (y ): The prmtve prce are p (w), forallw 2Z+ K,forall. The prce pad (receved) for a package non-lnear and depend on the dentty of the uyer. A fourth order prcng functon of the form p (w) = p (w); 8; ; w; (4) P (Z ) = 2S p (z ); P (Y ) = 2B p (y ): The prmtve prce are p (w), forallw 2Z+ K,forall;. Th veron of prcng non-lnear and completely non-anonymou n that prce depend on the dentte of the uyer and the eller etween whom a tranacton occur. Thu, only a frt order prcng functon lnear and only frt and econd order prcng functon are anonymou. It clear from (1) (4) that a j(= 1; 2; 3) order prcng functon a pecal cae of j + 1 order prcng functon. 2 To ratonalze non-lnear prcng, non-anonymou prcng we aume that: B1. Although each uyer can deal wth many eller and each eller wth many uyer, a uyer/eller par can exchange at mot one package. 2 Another prcng functon, analogou to thrd order prcng, one where prce are non-lnear and depend on the dentty of the eller (ut not the uyer). Th prcng functon doe not add much to the four veron condered aove. 8

9 B2. Two uyer may not form a purchang cartel. B3. Buyer may not reell package to each other. B1 mple that prce can e uperaddtve. That, p(w) +p(w 0 ) <p(w + w 0 ),for ome (w + w 0 ) 2 C pole. 3 B2 mple that prce can e uaddtve. That, p(w) +p(w 0 ) >p(w + w 0 ) for ome (w + w 0 ) 2 C pole. 4 B3 mple that t pole to utan p (z) 6= p 0 0(z) for (; ) 6= (0 ; 0 ). The defnton mply that f (Z; Y ) a j order agnment and (P ;P ) a j order prcng functon then, a p () 0, P (Z ) P (Y ): (5) A Walraan equlrum an agnment and prce uch that conumer maxmze utlty, producer maxmze proft and market clear. The package agnment model an exchange economy (there no producton), ut the uyer and eller reemle houehold and frm, repectvely. A houehold, uyer maxmze utlty uject to the udget contrant P (Z )+m = 0. Wth qua-lnear preference, maxmzaton of utlty uject to the udget contrant equvalent to the ojectve max U (Z ), P (Z ): A frm, eller chooe package from among ther feale et to maxmze proft. Gven our conventon wth repect to U, proft maxmzaton equvalent to max U (Y )+P (Y ): A Walraan equlrum of order j(= 1; 2; 3; 4) for the package agnment prolem E(u ; 8;! ; 8) a [(Z ;Y ); (P ;P )] uch that (P ;P ) a j order prcng functon, (Z ;Y ) a j order feale agnment, uyer maxmze utlte: for all, U (Z ), P (Z ) U (Z 0 ), P (Z0 ),allz0 eller maxmze proft: for all, U (Y )+P (Y ) U (Y 0 )+P (Y 0 ),ally 0 P P (Z )=P P (Y ) 3 If B1 doe not hold and p(w)+p(w ) 0 <p(w + w 0 ),for(w + w ) 0 2 C then a uyer whng to uy w + w 0 wll uy the two package w, w 0 eparately from eller. 4 Suppoe that for a gven et of prce uyer and 0 wh to uy package w and w 0 from eller. IfB2 doe not hold and p(w)+p(w ) 0 >p(w + w 0 ), (w + w ) 0 2 C, then uyer 1 and 2 wll form a cartel to jontly purchae w + w 0. 9

10 The lat condton that total money payment equal total recept mple that the prce of package n exce upply mut e zero. Lemma 1 and the fact that a j(= 1; 2; 3) order prcng functon a j + 1 order prcng functon mple that a j(= 1; 2; 3) order Walraan prce a j + 1 order Walraan prce. Kelo and Crawford (1982), Gul and Stacchett (1995), Ma (1996), and Bkhchandan and Mamer (1997) condered frt order Walraan equlrum for the package agnment model. The dfferent order of Walraan equlra wll mply dfferent dtruton of the total gan from trade, ut they do not dffer wth repect to overall effcency of ther agnment of non-money commodte. Theorem 1: If [(Z ;Y ); (P ;P )] Walraan equlrum of order j(= 1; 2; 3; 4) for E, then U (Z )+ U (Y )=v(e): Snce a Walraan equlrum of order j a pecal cae of order j +1, Theorem 1 along wth the qualnearty hypothe mply that the Walraan equlra of order four permt the wdet range of pole money payment wth repect to a gven agnment of nonmoney commodte. Conequently, the dtruton of the gan from trade whch can e prced enlarged. Th wll lead, elow, to the mplcaton that a larger, and ometme complete, et of allocaton n the core can e prced. Alo, when the narrower et of pole dtruton of the gan requred y lower order prcng equlra wth ther more trngent retrcton cannot e prced, a dtruton of the gan compatle wth hgher order prcng mght ext. Th alo true n Makowk (1979) model peronalzed tradng. 4 LP characterzaton of Walraan equlra In th ecton we how that each of the aove varete of Walraan equlra [(Z ;Y ); (P ;P )] equvalent to an nteger-valued oluton to a correpondng lnear programmng prolem, where (Z ;Y ) a oluton to the prmal and (P ;P ) a oluton to the dual. In th repect, therefore, the package agnment model mlar to the tandard agnment model. 5 Unlke the tandard agnment model, however, an nteger-valued oluton to the correpondng lnear programmng prolem doe not alway ext. Regard each Z and Y a an actvty operated at unt level. If thee actvte are operated at level x (Z ) 0andx (Y ) 0, ther payoff are U (Z )x (Z ) and 5 An LP characterzaton of Walraan equlra wth qua-lnear preference alo hold when commodte are dvle (Makowk and Otroy (1996)). 10

11 U (Y )x (Y ). Recallng our conventon, for each and each there are only a fnte numer of (ndvdually feale) actvte yeldng U or U greater than,1. Therefore, throughout the followng, attenton can e confned to them. Let IP j (E) denote the nteger programmng prolem of order j(= 1; 2; 3; 4) defned y package model E,.e., v(ip j (E)) = max Z U (Z )x (Z )+ Y U (Y )x (Y ); uject to the contrant, Z x (Y ) 1; 8 x (Z ) 1; 8 Y A j x 0 x (Z );x (Y ) 2f0; 1g; where A j the matrx of coeffcent of the lnear nequalte dtnguhng IP j and x =(x (Z );x (Y )). Let F (w) =fz : z = wg denote thoe Z n whch uyer requet package w from eller. Smlarly, G (w) =fy : y = wg are thoe Y n whch eller earmark package w for uyer. The lnear nequalte for A 1 are w Z 2F (w) x (Z ), 1 x (Y ) A w 0: Y 2G (w) The lnear nequalte for A 2 are Z 2F (w) x (Z ), The lnear nequalte for A 3 are Z 2F (w) x (Z ), Y 2G (w)g Y 2G (w)g x (Y ) 0; 8w: x (Y ) 0; 8; w: The lnear nequalte for A 4 are x (Z ), x (Y ) 0; 8; ; w: Z 2F (w) Y 2G (w) P When x () 2f0; 1g, P Z 2F (w) x (Z ) count the numer of package of type w requeted y uyer. Smlarly, when x () 2f0; 1g, P P Y 2G (w) x (Y ) count the numer of package of type w earmarked for uyer. Snce each w 2Z+ K defned y the 11

12 numer and knd of oject n t, the nequalte for A 1 retate the condton for a frt order agnment. For example, w x (Z ) Z 2F (w) 1 A w = w z = f:z =wg the total demand for oject contaned n the package demanded n Z. The nequalte for A 2 requre that the total numer of package of type w uppled y eller are at leat a large a the total numer requeted y uyer; hence, t equvalent to a econd order agnment. The nequalte for A 3, whch are equvalent to thrd order agnment, requre that the total numer of package of type w uppled y eller to a uyer are at leat a large a the total numer requeted y from eller. The defnton of F (w) and G (w) mply that when x ();x () 2f0; 1g, thenequalte defnng A 4 are equvalent to z = y or 0, 8;, the condton for fourth order agnment. Let LP j (E) denote the lnear programmng prolem of order j defned y changng the nteger contrant x ();x () 2f0; 1g n IP j (E) to x ();x () 0. The frt two contrant n each LP j (E) are the ame. Snce thee two contrant are the only one wth non-zero rght-hand de term, they determne the ojectve functon for ther dual. Hence, the ojectve functon for each DLP j (E) the ame, namely, v(dlp j (E)) = mn + A the remanng contrant for the LP j (E) dffer, o do the dual contrant. The contrant for DLP 1 (E) are : z ; + p, p z U (Z ); 8Z =(z ); 8; y U (Y ); 8Y =(y ); 8; The contrant for DLP 2 (E) are + ; ;p k 0; 8; ; k: The contrant for DLP 3 (E) are p(z ) U (Z ); 8Z =(z ); 8;, p(y ) U (Y ); 8Y =(y ); 8; ; ;p(w) 0; 8; ; w: + p (z ) U (Z ); 8Z =(z ); 8; 12

13 , p (y ) U (Y ); 8Y =(y ); 8; The contrant for DLP 4 (E) are + ; ;p (w) 0; 8; ; w: p (z ) U (Z ); 8Z =(z ); 8; (6), p (y ) U (Y ); 8Y =(y ); 8; (7) ; ;p (w) 0; 8; ; w: Although the prce of package n the contrant do not appear n the ojectve functon (ecaue the aocated rght-hand de contrant have zero value), t helpful to emphaze that the mnmum n the dual taken over the prce of package a well. Indeed, mnmzaton wth repect to the prce of package determne the mnmum value of and n the ojectve functon of the dual. Let fu (P ) maxfu (Z ), P (Z )g; Z fu (P ) maxfu (Y )+P (Y )g; Y e the ndrect utlty (or conjugate) functon, where the form of the ndrect utlty functon depend upon whch of the four form of prcng functon employed. Inpecton of the dual contrant reveal that the are feale f and only f U f (P ) and U f (P ). Snce the ojectve of the dual to mnmze the value, at ther optmal value they wll e equated to the value of the ndrect utlty functon for ome prce (P ;P ). Therefore, although the prce of package only appear n the dual contrant, they determne the prce of uyer and eller n the ojectve functon. To ummarze, the prolem for DLP j (E) to fnd prcng functon of order j whch mnmze the total gan to the uyer and eller, whle the prolem for LP j (E) to fnd agnment of order j whch maxmze thoe gan. Theorem 2: [(Z ;Y ); (P ;P )] a Walraan equlrum of order j(= 1; 2; 3; 4) for the package agnment model E f and only f t an optmal oluton to LP j (E) and DLP j (E),.e., v(lp j (E)) = U (Z )+ U (Y + = v(dlp j(e)); where )=v(e) = 8; = f U (P ) and 8; = f U (P ): The LP characterzaton of Walraan equlra mply the followng. Corollary 1: Suppoe a Walraan equlrum of order j(= 1; 2; 3; 4) ext. 13

14 () If [(Z ;Y ); (P ;P )] and [(Ẑ;Ŷ ); ( ˆP ; ˆP )] are Walraan equlra of order j, o are [(Z ;Y ); ( ˆP ; ˆP )] and [(Ẑ;Ŷ ); (P ;P )]. () If (P ;P ) a j order Walraan prcng functon (Z ;Y ) a j order effcent agnment, then [(Z ;Y ); (P ;P )] a j order Walraan equlrum. () The et of Walraan prce a cloed, ounded, convex et. If [(Z ;Y ); (P ;P )] a Walraan equlrum of order j,then(z ;Y ) a feale oluton to IP j (E). Moreover, y Theorem 1 U (Z )+ U (Y )=v(e) =v(ip j(e)): Theorem 2 ay that a Walraan equlrum of order j ext f and only f t a oluton to the aocated LP j. Conequently, the crtcal condton for extence of Walraan equlrum of order j the equalty Becaue of the evdent nequalty v(e) =v(ip j (E)) = v(lp j (E)): v(e) =v(ip j (E)) v(lp j (E)); (8) extence n general prolematc. The followng explan why appeal to more general (.e., hgher order) prcng functon can enlarge the doman of extence. Note that a j ncreae the contrant n LP j (E) ecome ncreangly retrctve (a j order agnment a j, 1 order agnment) and the contrant n DLP j (E) ecome decreangly retrctve (frt order prcng a pecal cae of econd order prcng, etc.). Therefore, v(lp j+1(e)) = v(dlp j+1(e)) v(lp j (E)) = v(dlp j (E)); j = 1; 2; 3; where the equalte are guaranteed y the ac Theorem of Lnear Programmng. If v(lp 1 (E)) > v(e) [ = v(ip j (E))], frt order Walraan equlra cannot ext. Neverthele, f the retrcton mpoed y j>1order equlra are uffcently ndng that v(lp j (E)) = v(e) for j = 2; 3, or 4, then j order equlra wll ext. The followng example from Kelo and Crawford (1982) (who ue t to how nonextence of frt order Walraan prce) how that Walraan equlrum of order 2 may not ext; ut n th example Walraan equlra of order 3 do ext. Example 1: There one eller endowed wth one unt each of three commodte. The reervaton value of the two uyer, 1 and 2, are n Tale 1. 14

15 TABLE 1 z (1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,0,1) (1,1,1) u 1 (z) u 2 (z) Let Y 0 =((1; 1; 0); (0; 0; 1)) denote a collecton of package earmarked y the eller for uyer 1 and 2 repectvely and let Y 00 = ((0; 1; 0); (1; 0; 1)) denote another uch collecton. Let Z1 0 = (1; 1; 0) e a package requeted y uyer 1 from the eller and let Z1 00 = (0; 0; 1) e another uch requet y uyer 1. Smlarly, Z0 2 = (0; 1; 0) and Z00 2 = (1; 0; 1) are two package requeted y uyer 2. The unque optmal oluton to LP 2 (E) () x (Y 0 )=x (Y 00 )=0:5, and x (Y )=0; 8Y 6= Y 0 ;Y 00, () x 1 (Z1 0 )=x 1 (Z1 00 )=0:5, and x 1 (Z 1 ) =0, 8Z 1 6= Z1 0 ;Z00 1, and () x 2 (Z2 0 ) =x 2 (Z2 00 ) =0:5, and x 2 (Z 2 ) =0, 8Z 2 6= Z2 0 ;Z00 2, At th optmal oluton, v(lp 2 (E)) = 11:75. An effcent agnment Z = Y = ((0; 1; 0); (1; 0; 1)) mplyng that v(ip(e)) = 11:5. A v(lp 2 (E)) >v(e), Theorem 2 mple nonextence of Walraan prce of order 2. 2 However, there are Walraan prce of order 3 n th example. TABLE 2 z (1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,0,1) (1,1,1) p 1 (z) p 2 (z) Tale 2 how order 3 Walraan prce, p (), = 1; 2. The only agnment that thee prce upport are the two effcent agnment. Another example how that Walraan equlrum of order 4 need not ext. Example 2: There are three uyer, ;, and, and three eller, 1, 2, and 3. There are three dtnct commodte and eller, = 1; 2; 3, endowed wth one unt of commodty. The three uyer reervaton value functon are hown n Tale 3. TABLE 3 z (1; 0; 0) (0; 1; 0) (0; 0; 1) (1; 1; 0) (1; 0; 1) (0; 1; 1) (1; 1; 1) u (z) u (z) u (z) Each eller ha four feale ale, dependng on whether ell h unt to uyer or to uyer or to uyer or to no one (uyer ). Let Y () e the feale ale n whch eller, = 1; 2; 3, agn h commodty to uyer, = ;;;. A there a one-to-one correpondence etween Z = (z ) and P z n th example, we wll denote Z y 15

16 P z. Conder the followng optmal feale oluton to LP 4 (E): () x 1 (Y 1 ()) = x 1 (Y 1 ()) = 0:5, x 1 (Y 1 ()) = x 1 (Y 1 ()) = 0. () x 2 (Y 2 ()) = x 2 (Y 2 ()) = 0:5, x 2 (Y 2 ()) = x 2 (Y 2 ()) = 0. () x 3 (Y 3 ()) = x 3 (Y 3 ()) = 0:5, x 3 (Y 3 ()) = x 3 (Y 3 ()) = 0. (v) x (1; 1; 0) =0:5, x (Z ) =0forallZ 6= (1; 1; 0). That, the agnment n whch uyer uy a unt of commodty 1 from eller 1, a unt of commodty 2 from eller 2, and nothng from eller 3 agned weght 0.5 and every other agnment agned a weght of zero. (v) x (1; 0; 1) =0:5, x (Z )=0forallZ 6=(1; 0; 1). (v) x (0; 1; 1) =0:5, x (Z )=0forallZ 6=(0; 1; 1). The ojectve functon value of LP 4 (E) at th optmal oluton v(lp 4 (E)) = 4:5. In any effcent agnment, all three commodte are agned to the ame uyer; th yeld a um of reervaton value equal to 4. A 4 = v(e) <v(lp 4 (E)) = 4:5, Theorem 2 mple that order 4 Walraan prce do not ext. 2 Remark 2: (The role of numer) Let E r e an r-replca of E. Then rv(lp 1 (E)) = v(lp 1 (E r )) v(ip j (E r )) rv(ip j (E));.e., whle the LP 1 extenon of the model cale nvarant, the underlyng model not. If a frt order Walraan equlrum ext for E, hence v(lp 1 (E)) = v(ip j (E)), then that allocaton can e replcated wth prce remanng the ame to provde a frt order Walraan equlrum n E r. The aence of cale nvarance n E another way to ee why frt order Walraan equlra wll not generally ext n the package agnment model. Contrat th wth the LP j extenon of the model, j = 2; 3; 4 whch doe not exht cale nvarance,.e., v(lp j (E r )) rv(lp j (E)): Thu, j order prce equlra, j 2, more nearly reflect the properte of E. Let E 1 denote a contnuum economy wth ma one of each type of uyer and eller n E. It readly demontrated that v(lp 1 (E)) = v(lp j (E 1 )) = v(ip j (E 1 )): In other word, n the contnuum there no dtncton etween the IP j and LP j veron of the model and oth acheve the ame maxmum gan a the LP 1 extenon of the orgnal model E. Conequently, all prce equlra n E 1 are utlty equvalent to frt order Walraan equlra. Moreover, uch equlra alway ext n E 1. 16

17 The extence concluon n E can e refned: There ext an nteger r uch that for each n = 1; 2; 3;:::, nrv(lp 1 (E)) = nv(ip j (E r )): To ee th, oerve that all the contrant n LP j (E) have nteger coeffcent and therefore all extreme pont of the feale et have ratonal co-ordnate. Hence, there ext an optmal oluton n ratonal numer. Let r e an nteger whch when multpled wth th ratonal optmal oluton reult n an nteger. For all nteger multple of the replca economy E r, a frt order Walraan equlrum ext. 5 Prcng, the pattern of ownerhp, and the core The frt order defnton of Walraan equlrum nentve to the pattern of ownerhp y eller. To llutrate, uppoe E and E 0 are two package agnment prolem dfferng n the numer and dtruton of ownerhp of the endowment of eller, ut havng the ame uyer and the ame total!. Suppoe [(Z; Y ); (P ;P )] a frt order equlrum for E, wth frt order prcng functon P and P defned y the vector of prce p =(p k ) for the oject. If Y proft-maxmzng at prce P defned y p, thenp (Y )=p!, for any!. Hence, no matter what the dtruton of ntal endowment n E 0, the total upply n term of total oject uppled!. Snce the uyer reman unchanged n E 0, ther demand at p would alo reman the ame. Becaue frt order fealty (market clearance) aed on market total of oject, p reman an equlrum prce vector for E 0. Converely, the pattern of ownerhp can retrct the varety of prcng equlra. Note that B1 mple that prce are addtve acro eller. That, f a uyer uy package z from eller and package z 0 from eller 0, then the cot necearly addtve. Suppoe each eller endowed wth a ngle unt of a ngle oject. Then, f prcng anonymou, t necearly frt order. Th the approach taken n Kelo and Crawford (1982). To derve anonymty, t uffce to allow uyer to reell to each other. In general, however, the pattern of ownerhp n the package agnment model mportant determnng the extence and order of equlrum. We decre ome of thee mplcaton elow. A (fnte) numer of exchange econome may e formed around a et of uyer and an aggregate ntal endowment. Defne a trct partal order on th et of exchange econome a follow. Let E(u ; 8 = 1; 2; :::; B;! ; 8 = 1; 2; :::; S) and E 0 (u ; 8 = 1; 2; :::; B;! 0 0; 80 = 1; 2; :::; S 0 ) e two exchange econome uch that P P S =1! S = 0 0 =1!0 =!. Suppoe that S <S0. Let (T 0 1 ;T 2 ; :::; T S ), T 6= ; 8 = 1; 2; :::; S, e a partton of the et of eller f1; 2; :::; S 0 g n E 0. Ownerhp n E more P concentrated than n economy E 0, denoted E c E 0, f and only f S < S 0 and! = 0 2T! 0 0; 8 = 1; 2; :::; S. If ome of the eller n an economy E 0 form cartel whch reult n a new economy 17

18 E then E c E 0. Gven a et of uyer and an aggregate endowment, the economy n whch there one eller per unt of a commodty the leat concentrated economy and the economy wth only one eller the mot concentrated economy. Let E c E 0. Suppoe [(Z 0 ;Y 0 ); (P 0 ;P0 )] a Walraan equlrum of order j for E 0. Let Y =(y ) where y = 0 2T y 0 0 e a collecton of package that the eller n the more concentrated economy could upply from h endowment of oject; and let P (Y )= P 0 0(Y 0 0) 0 2T e the aocated revenue derved from P 0 0. Note that th reformulaton of prce to acknowledge the more concentrated et of eller leave unchanged the prce that uyer face for package; hence, ther demand would reman unchanged. From the hypothe that for each eller 0 2 S P 0, Y 0 proft-maxmzng at P, 0 t follow that for each 2 S, Y =(y ),wherey = 0 2T y 0 proft-maxmzng at P. Th lead mmedately to 0 the followng concluon. Theorem 3: Suppoe that ownerhp more concentrated n E than n E 0. Then, f a Walraan equlrum of order j = 1; 2; 3 ext n E 0 then a Walraan equlrum of the ame order ext n E. The followng example how that the convere to Theorem 3 fale. Example 3: There are two uyer, 1 and 2, and the aggregate endowment three unt of one commodty. Thu C = f1; 2; 3g. The two uyer have dentcal reervaton value functon gven y u(1) =0, u(2) =3, and u(3) =4. There are no lnear Walraan prce n th example (ee Tale 1, Bkhchandan and Mamer (1997)). It may alo e verfed that there no Walraan equlrum of order 2 and 3 n the leat concentrated economy. However, prce p(1) =0, p(2) =3, and p(3) =4 are order 2 Walraan prce n the mot concentrated veron of th economy The core To decre the core, addtonal notaton requred. The et of player N f1; 2; :::; Bg[ f1; 2; :::; Sg. For any uet T N, lett S T \f1; 2; :::; Sg e the et of eller n T and let T B T \f1; 2; :::; Bg e the et of uyer n T. 6 There are other order 2 Walraan prce n the mot colluve economy. Prce whch atfy p(1)+p(2) 4, p(2) 3, and p(3)=4 are Walraan. 18

19 Recall that v(e) the maxmum gan avalale n the economy E. RegardENand let V (N )=v(e). Smlarly, let V (T ) e the maxmum gan n the economy contng of thoe uyer and eller n T. Let (Z(T );Y(T )) e any j order agnment uch that z = 0forall 2 T B and 62 T S,andy = 0forall 62 T B and 2 T S. That, at (Z(T );Y(T )) uyer and eller n T trade only among themelve. Lemma 2 aure u that regardle of j, V (T ) max Z(T);Y (T) f U (Z (T )) + U (Y (T ))g: (9) 2T B 2T S Note that: () If T B = or T S = then V (T )=0. () If T 1 T 2 then V (T 1 ) V (T 2 ). () V uperaddtve,.e., f T 1 \ T 2 = then V (T 1 )+V (T 2 ) V (T 1 [ T 2 ). Let Π B =( ) 2< B + and ΠS =( ) 2< S +.Then (ΠB ; Π S ) n the core of the game defned y E, denoted (Π B ; Π S ) 2 core(e), f 2T B + B + =1 =1 2T S V (T ); 8T N S = V (N ): Theorem 4 etalhe that a ownerhp ecome more concentrated t ecome more lkely that the core nonempty. Th analagou to Theorem 3, n whch a mlar reult wa etalhed for Walraan equlrum. Theorem 4: Let E and E 0 e uch that E c E 0.Ifcore(E 0 ) 6=, then core(e) 6=. The next theorem how that the et of j(=1 4) Walraan payoff n the core. Th, together wth Theorem 1, whch etalhed the effcency of all four type of Walraan equlrum, mply that thee tandard properte of Walraan equlrum do not depend on lnear, anonymou prce. Theorem 5: Let = f U (P ) and = f U (P ),where(p ;P ) are Walraan equlrum prce of order j(= 1; 2; 3; 4) for the economy E. Then(Π B ; Π S ) 2 core(e). In the tandard agnment model every allocaton n the core can e prced,.e., every vector of utlty payoff n the core concde wth the ndrect utlte aocated wth ome 19

20 Walraan equlrum prce. But an example how that n the package model the convere of Theorem 5 not true. Example 4: Let the uyer and the aggregate endowment e a n Example 1 of Secton 4. There are two eller, and. Seller ntal endowment (1; 0; 0) and eller ntal endowment (0; 1; 1). Conder the followng optmal oluton to LP 4 (E). Seller agnment: Let Y =((1; 0; 0); (0; 0; 0)), denote a collecton of package earmarked y eller for uyer 1 and 2 repectvely and let Y 0 =((0; 0; 0); (1; 0; 0)) denote another uch collecton. Smlarly, Y =((0; 1; 0); (0; 0; 1)) and Y 0 =((0; 0; 1); (0; 1; 0)) are two collecton of package earmarked y eller. For = ; ; let ( 0:5; f Y 2fY ;Y 0 x (Y g, )= 0; otherwe. Buyer agnment: Let Z 1 = ((1; 0; 0); (0; 1; 0)) denote a collecton of package requeted y uyer 1 from eller and repectvely, and let Z1 0 =((0; 0; 0); (0; 0; 1)) e another uch collecton. Smlarly, let Z 2 =((1; 0; 0); (0; 0; 1)), Z2 0 =((0; 0; 0); (0; 1; 0)) denote two collecton of package requeted y uyer 2. For = 1; 2, let ( 0:5; f Z 2fZ ;Z x (Z g, 0 )= 0; otherwe. The ojectve functon value of v(lp 4 (E)) at th optmal oluton A v(ip(e)) = 11:5, Theorem 2 mple that order 4 Walraan equlrum doe not ext. Hence order 1 or 2 or 3 Walraan equlra do not ext ether. The charactertc functon of th exchange economy V (f; 1g) = V (f; 2g) = V (f; 1; 2g) = 4 V (f; 1g) = V (f; 2g) = 7 V (f; 1; 2g) = 8:5 V (f; ; 1g) = V (f; ; 2g) = 9 V (f; ; 1; 2g) = 11:5; wth V (T )=0 for all other coalton. The pont ( 1 ; 2 ; ; )=(2:5; 2:5; 2; 4:5) n the core. However, there no order j(= 1; 2; 3; 4) Walraan equlrum whch yeld th (or any other) core pont The ngle eller model Denote the package agnment model wth a ngle eller, the mot concentrated pattern of ownerhp, y E[1]. We how that Walraan prce of order 3 ext. 8 Under an aumpton 7 The core cont of the convex hull of pont (2.5, 2.5, 2, 4.5), (1, 1, 3, 6.5), (1.5, 1.5, 3, 5.5), and (2.5, 2.5, 1.5, 5). 8 In th model, there no dfference etween the Walraan equlra of order 3 and 4. 20

21 that we call decreang margnal product for uyer, the core ha a lattce property and t can e prced. In E[1], each Z = z a calar. Thu, the nteger programmng prolem IP 3 (E[1]) mplfe to uject to v(e[1]) = max z u (z )x (z )+ z x (Y ) 1; 8 x (z ) 1; 8 Y Y U (Y )x (Y ); x (w), x (Y ) 0; 8; w (10) Y 2G (w) x ();x () 2f0; 1g: Theorem 6: In E[1], v(e[1]) = v(lp 3 (E[1])): Therefore, a Walraan equlrum of order 3 ext n E[1]. An mmedate conequence of Theorem 5 and 6 that core(e[1]) non-empty. Example 1 of Secton 4 mple that Theorem 6 not true for Walraan equlra of order le than 3. The next reult etalhe core equvalence for the ngle eller model. Theorem 7: If (Π B ; ) 2 core(e[1]), then there are Walraan equlrum prce (P ;P ) of order 3 uch that = f U (P ) and for all, = f U (P ). For any two pont (Π B ; ); (Π B0 ; 0 ) 2 core(e[1]) we defne two addtonal pont, one of whch preferred y the uyer and the other y the eller: maxf ; 0 g; 8; V (N ), mnf ; 0 g; 8; V (N ), : (11) The core ha a lattce property f (Π B ; ); (Π B ; ) 2 core(e[1]). The followng example how that, n general, core(e[1]) doe not have the lattce property. Example 5: There are three dentcal uyer, laeled = 1; 2; 3, and one eller,, endowed wth three dentcal unt of an oject. The reervaton value functon of the uyer : 21

22 u(1) =7, u(2) =8, and u(3) =10. Thu, for any T f; 1; 2; 3g, V (T ) = 8 >< >: 10; f 2 T; jt j = 2 15; f 2 T; jt j = 3 21; f 2 T; jt j = 4 0; otherwe: The core cont of pont ( ; = 1; 2; 3; ) 0uchthat 6, , = 21. Thu, the pont (6; 5; 5; 5) and (5; 6; 5; 5) le n the core ut the pont (6; 6; 5; 4) doe not. 2 The followng condton enure that the core ha the lattce property: Buyer have decreang margnal product n E[1] f for all T 1 ;T 2 ;T 3 f1; 2; :::; Bg uch that T \T j = for all 6= j,wehave V (fg[t 1 [ T 3 ), V (fg[t 1 ) V (fg[t 1 [ T 2 [ T 3 ), V (fg[t 1 [ T 2 ): (12) That, the margnal product of addtonal uyer (the et T 3 ) jonng a coalton whch nclude the eller, decreae a the coalton (that, the et T 3 jon) ecome larger. Theorem 8: Suppoe that uyer have decreang margnal product. Then core(e[1]) ha the lattce property. Theorem 8 ha the followng corollary. Corollary 2: Suppoe that uyer have decreang margnal product n E[1]. Then there ext pont (Π B ; ); (Π B ; ) 2 core(e[1]) uch that for any (Π B ; ; ) 2 core(e[1]), ; 8 : Theorem 7 aure u that each of the core pont, (Π B ; ), whch mot preferred y all uyer and leat preferred y the eller, and (Π B ; ), whch leat preferred y all uyer and mot preferred y the eller, can e prced y thrd order Walraan prce. 7 A Domnant Strategy Aucton E[1] the ettng for a mult-oject aucton. Buyer may umt d on any z 2 C. Ina Vckrey aucton, truth-tellng a domnant trategy, o n th dcuon we aume that uyer umt ther reervaton value functon a d for all the oject eng auctoned. The auctoneer chooe a (Z ;Y ) uch that u (z )+U (Y )=v(e[1]): 22

23 Let E [1] denote the economy wthout uyer. Buyer margnal product MP v(e[1]), v(e [1]): A well-known condton from Vckrey-Clarke-Grove mechanm theory that f an effcent mechanm (aucton) to guarantee truth-tellng for uyer, then n addton to the utlty u (z ) that the uyer receve from h part of the effcent agnment (Z ;Y ), the money payment m mut e uch that the uyer total utlty equal to h margnal product u (z ), m = v(e[1]), v(e [1]); plu poly a lump um. Moreover, when total utlty receved y the uyer mut alo atfy an ndvdual ratonalty or partcpaton contrant condton, the lump um mut e zero (Makowk and Otroy (1987)). Th condton (that each uyer receve h margnal product) alo neceary for any effcent, domnant trategy aucton (Holmtrom (1979)). 9 The aove equaton mple that m = v(e [1]), u 0(z 0) 0; 0 6= The nequalty follow ecaue (z1 ;z 2 ; :::; z,1 ;z +1 ; :::; z B ) a feale allocaton n E [1]. Thu, the money payment pad y uyer hould jut compenate the ret of the economy for any change n ther total welfare a a conequence of the preence of. Call th charge the Vckrey payment,.e., the amount of money that allow uyer to receve h margnal product. Let R e the amount that the eller receve when all the uyer receve ther margnal product. Note that R v(e[1]), = = = = m 0: MP v(e [1]), (B, 1)v(E[1]) v(e [1]), (B, 1) h v(e [1]), 0 6= u 0(z 0) u (z ) Call the pont (MP 1 ; MP 2 ; :::; MP ; :::; MP B ;R ) the uyer margnal product pont (BMP). It an ovou conequence of the defnton of the core that an ndvdual margnal product an upper ound of the et of pole payoff an ndvdual can receve. In the 9 Therefore all effcent domnant trategy aucton on the doman of qualnear preference are revenue equvalent. 23

24 preent cae, f (Π B ; ) 2 core(e[1]), then MP ; 8 (13) =) = v(e[1]), MP R : By Theorem 7, f (Π B ; ) 2 core(e[1]), then(π B ; ) can e prced. Hence, f BMP2 core(e[1]) then () th pont can e prced and () (13) mple that the BMP the core pont mot preferred y all uyer and leat preferred y the eller that wa dentfed n Corollary 2. Further, that would mean that the money payment etalhed y the prce whch yeld the BMP would concde wth Vckrey payment. Suppoe that BMP2 core(e[1]). Conder the followng aucton mechanm: Buyer umt d on every uet. The eller compute thrd order Walraan prce whch yeld the BMP for the tated d. Buyer report ther demand at thee prce to the eller, who then agn each uyer a uet that he demand. The aove dcuon mple that truth-tellng a domnant trategy for uyer and th aucton reult n an effcent agnment. Example 5 of Secton 6 how, however, that BMP may le outde the core. In th example, the BMP (6,6,6,3): 3 to the eller and 6 to each of the uyer. Th pont not n the core a <V(f; 1g) =10. The condton of decreang margnal product for uyer uffcent to enure that BMP le n the core. Theorem 9: Aume uyer have decreang margnal product n E[1]. Then BMP 2 core(e[1]) and there ext a thrd order Walraan prcng functon, p (), whch gve uyer ther margnal product. That f (Z ;Y ) an effcent agnment for E[1],then u (z ), p (z ) = MP ; p (y ) = R : The example elow how that under the hypothe of Theorem 9, Walraan prcng of order le than 3 may not prce the BMP and therefore may not reult n a domnant trategy aucton. Example 6: One eller,, own four dentcal unt of an oject. There are two uyer, a and, for the oject wth reervaton value n Tale 4. TABLE 4 z u a (z) u (z) The charactertc functon of th economy V (f; ag) =9, V (f; g) =10, V (f; a; g) = 14, and V (T )=0 for all other T f; a; g. TheBMP(MP a = 4; MP = 5;R = 5). 24

25 A uyer have decreang margnal product, Theorem 9 mple that the BMP n the core and that a thrd order Walraan prcng functon yeld th pont. At th Walraan equlrum, each uyer otan two unt ut uyer a pay 3, and uyer pay 2. Therefore, although Walraan equlrum of order 2 ext n th example, there no Walraan equlra of order le than 3 whch yeld the BMP. 2 8 Concludng Remark It well-known that many commonly ued aucton for ellng multple ndvle oject are neffcent (ee Auuel and Cramton (1995), Engelrecht-Wggan and Kahn (1995), and Bkhchandan (1996)). The mult-oject veron of the Vckrey aucton effcent when uyer value are prvately known. Moreover, t a domnant trategy for uyer to truthfully report ther value. However, a the Vckrey aucton requre d on every concevale undle of oject, t mpractcal to ue for ellng more than a handful of oject. An effcent and domnant trategy aucton ext n a ettng where everal, poly heterogenou, oject are old to uyer who have zero utlty for a econd oject,.e., under the unt demand aumpton (ee Leonard (1983), Demange and Gale (1985), and Demange, Gale, and Sotomayor (1986)). In th aucton, the oject are old at the lowet Walraan prce. The extence of th domnant trategy and effcent aucton depend on two properte: A. Walraan equlrum ext. B. All uyer can multaneouly acheve ther margnal product at a Walraan equlrum. In our model, property A atfed n the ngle eller cae. If uyer have decreang margnal product then property B atfed, and there ext an effcent mult-oject ealed-d aucton n whch truth-tellng a domnant trategy for uyer. The ealed-d aucton proceed a follow: () dder umt d on every package of oject, () the eller ue thee d to compute the lowet Walraan prce for the package, and () each uyer chooe a package whch maxmze h utlty at thee prce. Th aucton not any mpler than the mult-oject Vckrey aucton. However, n our framework t pole to nvetgate when mpler Walraan prce and therefore a mpler ealed-d aucton ext. In partcular, when properte A and B are atfed wth ether (I) Walraan prce of order 2 n the ngle eller economy or (II) wth Walraan prce of order 3 n a le concentrated economy then the ealed-d aucton can e mplfed. We plan to nvetgate condton on uyer preference under whch ether I or II otan. 25

26 9 Appendx Proof of Lemma 2: In vew of Lemma 1, t uffcent to how that for any frt order feale agnment there a utlty equvalent fourth order feale agnment. Let (Z; Y ) e a frt order feale agnment. Thu, U (Y )=0, 8,and y! z ; y ; 8: (14) For any two K vector, V 1 =(v 1 1 ;v1 2 ; :::; v1 K ) and V 2 =(v 2 1 ;v2 2 ; :::; v2 K ),let[v 1,V 2 ] + (maxfv 1 k,v 2 k ; 0g) k=1;2;:::;k and [V 1,V 2 ], [V 2,V 1 ] +.LetZ 0 1 =(z0 11 ;z0 12 ; :::; z0 1 ; :::; z0 1S ) e uch that z 1, z 0 1 = 0; (15) z 0 1! ; 8: (16) If for every Z1 0 atfyng (16) we have P [ z 1, P z 0 1 ]+ > 0then P z 1 6 P!, contradctng (14). Thu, we can fnd a Z1 0 uch that P [ z 1, P z 0 1 ]+ = 0 and, y reducng ome of the z 0 f neceary, [P 1 z 1, P z 0 1 ], = 0. Hence, there ext a Z1 0 whch atfe (15) and (16). Equaton (15) mple that U 1 (Z 1 )=U 1 (Z1 0 ). The nducton hypothe that there ext Z 0 1 ;Z0 2 ; :::; Z0 uch that z`, `=1 z 0` = 0; ` = 1; 2; :::; ; (17) z 0`! ; 8: (18) We have hown (17) and (18) to e true for = 1. Equaton (17) mple that U`(Z`) = U`(Z 0`), 8` = 1; 2; :::;. Suppoe that Z1 0 ;Z0 2 ; :::; Z0 atfe (17) and (17). Chooe a Z+1 0 uch that z +1;, z 0 +1; = 0; (19) +1 `=1 z 0`! ; 8: (20) If for every Z 0 +1 atfyng (20) we have [P z +1;, P z 0 +1; ]+ > 0then P P +1 P!, contradctng (14). Thu t pole to chooe Z+1 0 atfyng (19) and (20). Hence, there ext Z 0 uch that U (Z 0 ) =U (Z ), 8, and P z 0!, 8. Defne y 0 z 0 and Y 0 (y 0 1 ;y0 2 ; :::y0 ).AP P B z 0 = y 0!,wehave8, U (Y 0 )=0, 26 `=1 z` 6

27 8. Therefore, (Z 0 ;Y 0 ) a fourth order feale agnment that utlty equvalent to (Z; Y ). Proof of Theorem 1: Buyer and eller maxmzaton at a j(= 1; 2; 3; 4) order Walraan equlrum [(Z ;Y ); (P ;P )] mple that for any j order agnment (Z; Y ) [U (Z ), P (Z)] + [U (Y )+P (Y )] [U (Z ), P (Z )] + [U (Y )+P (Y )]: The defnton of Walraan equlrum mple that P P (Z)=P P (Y ).Further,a (P ;P ) a j order prcng functon and (Z; Y ) a j order agnment, we have from (5) that P P (Z ) P P (Y ). Therefore, U (Z )+ U (Y ) U (Z )+ U (Z )+ U (Y )+ U (Y ): P (Y ), P (Z ) Proof of Theorem 2: We gve a proof for fourth order Walraan equlrum; the proof for j order Walraan equlrum, j = 1; 2; 3, mlar. Suffcency. Suppoe that LP 4 (E) ha an nteger optmal oluton, (Z ;Y ).Let( ; ;P ;P ) e an optmal oluton to DLP 4 (E). The complementary lackne condton atfed y thee optmal oluton are: h 1, x (Z ) = 0; 8 (21) Z h 1, x (Y ) = 0; 8 (22) Y h x (Z ), x (Y ) p (w) = 0; 8; 8; 8w (23) Z 2F (w) Y 2G (w) h + p (z ), U (Z ) x (Z ) = 0; 8Z =(z ); 8 (24) h, U (Y ), p (y ) x (Y ) = 0; 8Y =(y ); 8 (25) where x (Z ) 1 f Z = Z and 0 otherwe. Smlarly, x (Y ) 1 f Y = Y and 0 otherwe. Thu (6), the uyer maxmzaton contrant n DLP 4 (E), and (24) mply U (Z ), p (z )= U (Z ), p (z ); 8Z =(z ); 8: 27

28 Smlarly (7), the eller maxmzaton contrant n DLP 4 (E), and (25) mply U (Y )+ p (y )= U (Y )+ p (y ); 8Y =(y ); 8: A (Z ;Y ) LP 4 (E) feale t a fourth order agnment. A (P ;P ) DLP 4(E) feale, t a fourth order prcng functon. Further, (23) mple that f a package w agned y eller to uyer n Y ut not n Z (.e., y = w ut z = 0) then p (w) =0. Therefore, P P (Z)=P P (Y ). Hence, [(Z ;Y ); (P ;P )] a Walraan equlrum of order Necety. Let [(Z ;Y ); (P ;P )] e a Walraan equlrum of order 4. Therefore, U (Z ), P (Z ) U (Z ), P (Z ); 8Z ; 8 U (Y )+P (Y ) U (Y )+P (Y ); 8Y ; 8: Let x (Z ) = x (Y ) = 1 f Z = Z 0 otherwe 1 f Y = Y 0 otherwe A (Z ;Y ) a fourth order agnment, x ();x () a feale oluton to LP 4(E). Smlarly, DLP 4 (E) fealty of ( ; ;P ;P ) mpled y the fact that (P ;P ) a fourth order prcng functon. Therefore, v(lp 4 (E)) = v(dlp 4 (E)) + = = [U (Z ), P (Z )] + = v(e) U (Z )+ v(lp 4 (E)) U (Y ) [U (Y )+P (Y )] where the frt nequalty follow from DLP 4 (E) fealty of ( ; ) and the econd nequalty from (8), the frt equalty follow from the defnton of and aove, the econd nequalty from the fact that P P (Z)=P P (Y ) at a Walraan equlrum, and the thrd equalty from Theorem The complementary lackne condton (21) [ (22) ] tate the ovou fact that a uyer [eller] who doe not uy [ell] anythng make zero proft. 28