Cost sharing: efficiency and implementation

Transcription

1 Journal of Mathematcal Economcs Cost sharng: effcency and mlementaton Todd R. Kalan, Davd Wettsten ) Deartment of Economcs, Ben-Guron UnÕersty of the egeõ, P.O.B. 653, Beer-SheÕa 84105, Israel Receved 7 July 1997; receved n revsed form 7 Setember 1998; acceted 15 December 1998 Abstract We study envronments where a roducton rocess s jontly shared by a fnte grou of agents. The socal decson nvolves the determnaton of nut contrbuton and outut dstrbuton. We defne a comettve soluton when there s decreasng-returns-to-scale whch leads to a Pareto otmal outcome. Snce there s a fnte number of agents, the comettve soluton s rone to manulaton. We construct a mechansm for whch the set of ash equlbra concdes wth the set of comettve soluton outcomes. We defne a margnal-cost-rcng equlbrum MCPE. soluton for envronments wth ncreasng returns to scale. These solutons are Pareto otmal under certan condtons. We construct another mechansm that realzes the MCPE. q 1999 Elsever Scence S.A. All rghts reserved. JEL classfcaton: D51; D61; D78 Keywords: Cost sharng; Margnal-cost-rcng equlbrum; Increasng returns to scale 1. Introducton The sharng of costs s revalent n many facets of economc actvty. Large enterrses allocate overhead costs among varous deartments. Members of a unversty share the cost of a software ste lcense. The artes watchng a ay-er-vew boxng match share the fee. ) Corresondng author. E-mal: wettstn@bgumal.bgu.ac.l r99r$ - see front matter q 1999 Elsever Scence S.A. All rghts reserved. PII: S

2 Issues of cost sharng surface exlctly as well as mlctly. The roblem arses exlctly whenever a grou of ndvduals jontly uses a common resource or undertakes a jont roject for an excellent survey, see Young, A cost-sharng method arses mlctly n any rvate-ownersh comettve economy. In such an economy, an ndvdual s share of the roducton costs of a frm s the amount he ays for goods urchased mnus the rofts he earns from shares owned n that frm. The roertes of the cost-sharng method are of major concern. Does t lead to effcent outcomes? Is the outcome unque? Can t be manulated by the ndvduals nvolved? Mouln and Shenker rovde the seral cost-sharng method and have demonstrated ts aeal as far as manulablty and unqueness are concerned. Mouln and Shenker and Mouln and Watts analyze more tradtonal methods such as average cost sharng. Both seral and average cost sharng do not guarantee Pareto otmalty. Our contrbuton s to resent two mechansms that wll generate Pareto otmal outcomes for several classes of envronments. We start by observng that n a neoclasscal economy, the mlct cost sharng mentoned above mles effcency. A cost-sharng method that attemts to relcate a neoclasscal economy by creatng a fcttous frm would have two shortcomngs: manulaton may exst wth a fnte number of ndvduals and convexty of the technology s requred. Both manulaton and non-convexty may lead to an undesrable outcome, whle non-convexty may also lead to nonexstence of equlbra. We suggest two cost-allocaton mechansms that obtan effcent outcomes by mtatng mlct cost sharng and addressng both roblems. The manulaton ssue s resolved n art by elmnatng the market ower ossessed by the ndvduals. To address the nonexstence ssue, we resort to margnal-cost-rcng equlbra that exst for a large class of non-convex envronments see Brown, The aer roceeds as follows. In Secton 2, we resent the comettve soluton. In Secton 3, the comettve soluton s mlemented and the resultng mechansm s comared to other cost-sharng methods. In Secton 4, we address the roblems created by ncreasng returns to scale. Fnally, n Secton 5, we conclude the aer and menton further drectons of research. 2. Allocaton of costs the comettve soluton The large varety of cost-allocaton roblems makes t ntractable to resent a general method to allocate costs effcently. In ths secton, we defne the class of envronments that we analyze n ths aer. Then, we ntroduce the comettve soluton concet for ths class. We show that for a subclass the comettve soluton yelds effcent allocatons only when gnorng ossble manulaton by ndvduals.

3 We consder a class of cost allocaton roblems 1 where there s a fnte number of ndvduals, greater than 2, that consume two goods, x and y, and X X have access to technology c: R R, where c y. q q s the cost of roducng y unts of good y. The references of each ndvdual can be reresented by a utlty functon u x, y. where utlty s strctly ncreasng, dfferentable,. concave and satsfes the Inada condtons lm u x, y s lm u x, y. x 0 1 y 0 2 s `, lm... u x, y s lm x ` 1 y `u2 x, y s 0. The ndvduals are endowed wth strctly ostve amounts w of good x and none of good y. An allocaton s gven by a 2 q 2-tule x, y., x, y. s1 where the frst comonents denote the ndvduals consumton levels and the last two comonents the roducton levels. The allocaton s feasble f: Ýx qx F Ý s1 s1 Ý y Fy s1 c y. Fx. w A feasble allocaton s Pareto otmal f there does not exst a feasble allocaton whch makes no ndvdual worse off and at least one ndvdual strctly better off. We wll say the cost-allocaton roblem belongs to class D I. f the cost functon s dfferentable and convex concave., wth c 0. s0. In the frst case, we are n the decreasng-returns-to-scale scenaro, whereas n the second case, roducton s characterzed by ncreasng returns to scale. In order to defne a comettve soluton, we need to create a frm that owns the technology and endow ndvduals wth strctly ostve ownersh shares a.a X X. X X X comettve soluton s a feasble allocaton x, y, x, y. s1 and a rce X for good y n terms of good x. such that gven the rce, the frm s maxmzng ts rofts and the ndvduals are maxmzng ther utlty subject to ther budget constrants: x X, y X. solves max X y yx x, y s.t. c y. Fx 1 These roblems encomass both tradtonal cost sharng when ndvduals demand oututs and the mechansm determnes nuts. and surlus sharng when ndvduals suly nuts and the mechansm determnes oututs.. Our mechansm can be nterreted as a hybrd constructon snce t determnes both nuts and oututs.

4 X x, y X. solves max u x, y x, y s.t. x q X y Fw qa where denotes the rofts of the frm. When the comettve soluton exsts ths s guaranteed only n class D., the standard arguments leadng to the Frst Welfare Theorem show that the comettve soluton yelds a Pareto otmal allocaton. Prooston 1. All comettõe solutons for a gõen cost allocaton roblem are Pareto otmal. Restrctng attenton to the class D of cost-allocaton roblems, the comettve soluton exsts and generates an allocaton wth an mlct sharng of costs. The cost of roducton mlctly mosed on ndvdual s the dfference between s exendtures on y and s share n the roft. The secfc allocaton realzed deends uon the ownersh structure and may not ossess an axomatc characterzaton lke several cost allocaton methods ut forward n the lterature. It s, however, Pareto otmal. Ths may seem to be a vable method to reach an effcent cost allocaton, but strategc behavor by the ndvduals may undermne the effcency. An ndvdual, by msreresentng hs references, may be able to secure an outcome referable to the comettve cost allocaton acheved wth hs true reference. Several aers have addressed the ncentves roblem nherent n the Walrasan aradgm for ure exchange economes Hurwcz, 1979; Schmedler, 1980; Postlewate and Wettsten, and for roducton economes Hong, In Secton 3, we offer a contnuous and feasble mechansm that would realze the comettve soluton to the cost allocaton roblem. All the ash equlbra of ths mechansm yeld Pareto otmal outcomes. ( ) 3. Realzaton of the comettve soluton mechansm A Mechansm A conssts of an n-tule of strategy sets and an outcome functon mang strateges nto allocatons. The strategy sace of ndvdual s S s 2 R =R =R=R wth a generc element denoted by, c, t, r. qq qq. The frst comonent s a rce for commodty y submtted by ndvdual, the second s a net consumton bundle, the thrd s an nut level nto the roducton rocess, and the fourth s a number used n averagng out the ossbly conflctng demands of all the ndvduals. We wll now outlne the way mechansm A oerates nformally, before we formally descrbe t. The mechansm constructs an average rce based on the announced rces and an average roducton lan based on the announced roducton lans. The requred amount of nut, secfed n the roducton lan, s collected from the ndvduals and used n roducton. Indvdual budget sets are

5 constructed based on the average rce and the rofts generated from the roducton lan. The consumton bundles requested are rojected onto these budget sets. The resultng bundles may not be feasble n the aggregate, but aggregate feasblty s reached by scalng down the bundles. We assume the ndvduals are comletely nformed as regards the technology, references and endowments. We also assume the desgner knows the ndvduals ntal endowments, but we do not assume that the desgner knows the technology. 2 otce that the frm s a fcttous entty, whch s created n order to defne the outcome functon. It s thus controlled by the desgner and has no strategc role. We show that mechansm A has ash equlbra that are all comettve solutons, thereby, yeldng an effcent soluton to the roblem of cost allocaton. In order to resent more clearly the formal descrton of the mechansm, we wll roceed n several stes even though the mechansm tself s a one-stage game. Ste 1: An average rce s constructed as follows. Defne: X t t 2 Ý Ý a s < y < ; as a X t,t / a b s a)0 a 1 s s Ý b s1 as0 s1 The constructon of mles that f all ndvduals other than ndvdual announce the same rce q, the average rce constructed wll be q and furthermore, ndvdual s announcement wll have no effect on the rce reached. Ste 2: The roducton lan x, y. used by the mechansm s determned by: y1 ž ½ 55 ž ½ ½ 55/ / s1 s1 s1 s1. Ý ½ Ý Ý Ý x, y s mn w,max 0, t,c mn w,max 0, t The aearance of the cost functon n the roducton lan does not mly that the desgner needs to know the technology. The oeraton of the mechansm 2 Knowledge of ntal endowments can be relaxed at the cost of a more comlcated mechansm that would handle destructon and wthholdng of ntal endowments as n the work of Hong Furthermore, note that each ndvdual needs only to know the set of references and endowments that exst n the entre socety and not the secfc reference or endowment for any artcular other. ndvdual.

6 reveals the value of c at a sngle ont the roducton ont yelded by the choces of the ndvduals. Ste 3: ndvdual budget sets are constructed elements are net trades., based on the average rce and the roducton lan: z qz Fa y yx. 1 2 ~ Ý 1 s1 B s z, z gr z qw F w yx ; z qw G0 z Fy ; z G0 ß 2 2 Let Õ be the closest ont n B to c. In order to nsure the fnal allocaton s feasble, the followng set J s constructed: rpr F1 for s1..., ~ Ý Ý Ý 1 2 ß Js rgr qq r r Õ qw F w ; r r Õ Fy s1 s1 s1 Let rsmax ˆ r g J r. The bundles allocated to the ndvduals by the mechansm are: g srpr ˆ Õ qw ; g srpr ˆ Õ for s1,..., The mechansm n addton to the ndvduals utlty functons consttutes a well defned game. We analyze the ash equlbra of games resultng from our mechansm A. Several other soluton concets lke subgame erfect equlbra Moore and Reullo, 1988; Abreu and Sen, 1990 and more recently Varan, 1994., equlbra n undomnated strateges Palfrey and Srvastava, and vrtual equlbra Matsushma, 1988 and Abreu and Sen, have been used to analyze mechansms n the lterature. Mechansms relyng on these soluton concets may requre more strngent nformatonal assumtons or larger strategy saces. ext, we wll show that all ash equlbra generated by our mechansm gve rse to Pareto otmal allocatons. Prooston 2. For any cost allocaton roblem n D, the ash equlbra of the mechansm constructed aboõe yeld a comettõe soluton that s Pareto otmal. X X X X X X X Proof. Denote, x, y, r, rˆ and x, y s1 as the values and allocatons generated at the ash equlbrum ont. We show ths s a comettve soluton va the followng lemmata. X Lemma 1. IndÕdual can get arbtrarly close to any ont u n B.

7 Proof. Announcng the net trade leadng to u as c and a large enough r wll generate an outcome arbtrarly close to u. The large r nullfes the effect of all the other terms n the constructon, and the calculaton of the fnal bundles allocated to the ndvduals wll leave ndvdual arbtrarly close to u.b X Lemma 2. IndÕdual can generate a B ( ) that contans net trades leadng to strctly ostõe consumton bundles for hmself gõen any choce of strateges by the other ndõduals. X Proof. Snce w )0 and )0 ndvdual can, by sendng n an arorate t, force a roducton lan that has x and y strctly ostve and yelds a ostve ncome level for consumer even f rofts are always negatve, t s ossble to choose a small enough roducton level to guarantee ostve ncome.. Ths X mles that B. contans net trades that lead to strctly ostve consumton bundles for ndvdual.b X Lemma 3. The equlbrum allocaton must be strctly nteror x, y X. s1 g 2 R. qq. Proof. Assume by way of contradcton, there exsts an ndvdual for whom X X. 2 x, y f R qq. By Lemma 2, ndvdual can, by sendng n a ossbly X dfferent t, obtan a B. that contans net trades leadng to strctly ostve consumton bundles. Any one of those consumton bundles s strctly referred X to x, y X.. By Lemma 1 and contnuty of references, there exsts an obtanable consumton that s referred to the equlbrum consumton. Ths contradcts that ndvdual was layng a ash equlbrum strategy. Hence, the equlbrum outcome entals a strctly nteror allocaton.b Lemma 4. The roducton lan ( x, y ) maxmzes rofts under rce. X X X Proof. Assume, by way of contradcton, there s a roducton lan x, y. that X X yelds hgher rofts. Any ont of the form l x q 1yl. x, l y q 1yl. y. wth 0-l-1 yelds hgher rofts by convexty of the cost functon. By Lemma 3, there exsts a l close enough to 1, where such a ont s feasble. By Lemma 2, any ndvdual could obtan ths ont by alterng the t message. Thus, ndvdual X exands the B. set, and by Lemma 1 and contnuty of references can obtan a referred outcome, n contradcton to the orgnal outcome beng an equlbrum outcome.b Lemma 5. The consumton lan ( x X, y X ) maxmzes ndõdual s utlty subject X X X to the budget constrant wth rce and roducton lan ( x, y ).

8 Proof. Assume, by way of contradcton, there s a consumton lan x, y. that X satsfes ndvdual s budget constrant and s strctly better than x, y X..By X X X Lemma 3, only the frst constrant n B. can be bndng at the ont x, y.. X By ths fact, there exsts a l close enough to 1 such that l x q 1 y l x, X X l y q 1yl. y. belongs to B.. By convexty of references, ths ont s X referred to x, y X.. By Lemma 1 and contnuty of references, ths contradcts that ndvdual s layng a ash equlbrum strategy.b By Lemmata 4 and 5, any equlbrum s a comettve soluton and by Prooston 1, ths soluton yelds a Pareto otmal allocaton.b The result of Prooston 2 may be vacuously satsfed f the mechansm suggested does not ossess any ash equlbra. We show that ths s not the case. Gven our assumtons, a comettve soluton always exsts. Prooston 3 shows that any comettve soluton s a ash equlbrum. Hence, the mechansm ossesses a ash equlbrum. Prooston 3. For any cost-allocaton roblem n D, the set of comettõe solutons s contaned n the set of ash equlbra outcomes of mechansm A. X X X. X X X Proof. Let A s x, y, x, y. s1 and rce consttute a comettve X X X. X soluton. A set of strateges realzng t s: s ; c s x yw, y ; t sx r; r s1 for all. Ths -tule of strateges yelds the average rce X and the consumton roducton allocaton A X. We now show that these strateges form a ash equlbrum, snce they are best resonses. Frst, we note that an ndvdual s unable to change the rce constructed, X. Second, the roducton lan n A X maxmzes rofts gven X ; thereby, an ndvdual s choce of t gves hm the largest budget set. Fnally, the choce of c and r leads to the most referred consumton bundle n the budget set. Therefore, changes n c, t or r wll not X mrove uon the x, y X. outcome for ndvdual.b The man features dstngushng mechansm A from other cost-allocaton methods s the Pareto otmalty of the outcome reached and the relaxaton of nformatonal assumtons. In contrast to other cost-sharng methods, our mechansm by vrtue of concdng wth comettve solutons yelds Pareto otmal levels of y. Furthermore, ts oeraton does not requre the desgner to know the technology, as assumed wth seral cost sharng. Mechansm A, on the other hand, s not mmune to coaltonal devatons lke the seral cost-sharng method and s more comlex than the revous methods suggested. The mechansm s otmalty of outcomes and exstence of a soluton crtcally deend uon the assumton of convexty of the technology. Ths henomena s arallel to the one encountered n a comettve economy. Secton 4 secfes soluton concets arorate for envronments wth ncreasng returns non-con- vextes. and dscusses ther mlementaton.

9 4. Increasng returns to scale In ths secton, we consder the allocaton of costs n envronments wth ncreasng returns to scale. We use the analogy of these cost allocaton roblems to economes wth ncreasng-returns-to-scale roducton to suggest a soluton. A common construct for such roducton economes s a margnal-cost-rcng equlbrum MCPE.. Ths conssts of dctatng the roducton lan of the frm and allowng the ndvduals to urchase goods at margnal cost after ayng for ther share of the frm s losses. Exstence of such equlbra under certan condtons has been shown n a seres of aers Mantel, 1979; Beato, 1982; Kamya, 1988a; Bonnsseau and Cornet, Also, the otmalty of these equlbra s guaranteed wth strngent enough condtons on the curvature of the ndfference curves and roducton ossblty fronters Derker, 1986; Qunz, Further results can be found n the work of Cornet Hence, a devce that leads to margnal-cost-rcng equlbra would be an nterestng soluton to cost-sharng roblems. As before, the standard constructon gnores the ossblty of manulaton by the ndvduals. To address ths ssue, we rovde a contnuous, feasble and fnte-dmensonal mechansm that realzes the MCPE soluton. Calsamgla demonstrates that n the resence of ncreasng returns to scale t s mossble to obtan Pareto otmal outcomes va a fnte-dmensonal mechansm. Our mechansm s comatble wth ths result, snce the MCPE that t yelds s not always Pareto otmal. In order to defne an MCPE soluton, we create a fcttous frm and endow w 3 ndvduals wth strctly ostve ownersh shares gven by a s. Alterna- j Ý js 1w tvely, we can choose a desred share structure a and redstrbute endowments as w sa Ý js1w j. An MCPE soluton s a feasble allocaton and a rce of y where the rce equals the margnal cost of roducton, the frm carres out the rescrbed roducton lan and the ndvduals maxmze utlty subject to ther budget constrants. These constrants ncororate both the rce and ther share of the X negatve rofts. Formally, the MCPE soluton s a feasble allocaton x, X. X X X X. X X. X y, x, y and rce where s c y and x, y X. s1 solves: max u x, y. x, y where s.t. x q X y Fw qa denotes the rofts of the frm. 3 The creaton of shares n ths manner revents ndvduals from gong bankrut when held resonsble for the frm s losses. Ths s a verson of the survval assumton, whch aears n the exstence roofs for MCPE Mantel, 1979; Beato, 1982; Kamya, 1988a; Bonnsseau and Cornet, A counter-examle for nonexstence when the survval assumton does not hold s rovded by Kamya 1988b..

10 The cost of roducton mosed on ndvdual by ths soluton s the sum of s exendture on y and s share n the losses. We cannot acheve these outcomes n a straghtforward manner due to ossble msreresentaton of references by the ndvduals. In order to revent these roblems, we offer a contnuous and feasble mechansm mlementng MCPE solutons Mechansm B Excet for the constructon of the budget sets, mechansm B s defned just lke the mechansm mlementng the comettve soluton. The frst constrant n the X y y constructon of B n Secton 2 s relaced wth z qz F2 a y yx. 1 2 ya y yx., where: ž y y j ž s1 j/1 ½ / 5 Ý Ý x, y s mn w, max 0, t, ½Ý ž Ý / 5 ž // y1 j c mn w, max 0, t s1 j/1 The RHS s twce the rofts of the frm at the roducton lan determned by the announcements of all ndvduals other than mnus the rofts of the frm at the roducton lan determned by the announcements of all ndvduals, wth all terms adjusted for feasblty. In equlbrum, all ndvduals announce the same roducton lan. When ths occurs, the desgner can construct the budget sets wth only the knowledge ganed from roducng the announced lan and does not need to know the whole technology as before, the desgner should be able to measure the outut that s eventually roduced.. Outsde of equlbrum, ndvduals may not announce the same roducton lan. In ths case, the desgner needs to know the technology at several onts, that s, he should be able to dscover the outut that would be roduced for several dfferent levels of nuts n order to construct the outcome functon. For some fnte cost he should be able to obtan ths nformaton, for nstance, by ether rerunnng the technology for several onts or stong the technology at several levels of nut. Dong so for the entre curve may ental nfnte costs, whch would not be a credble oton even outsde of equlbrum. These requrements both nsde and outsde equlbrum. are notceably weaker than havng to know the entre roducton technology. 4 4 An alternate route to revealng the entre technology can be acheved by aendng to the mechansm a game smlar to the Hurwcz Maskn Postlewate Hurwcz et al., 1995 construct.

11 The ash equlbra resultng from mechansm B are shown to be MCPE solutons n Prooston 4. In Prooston 5, we show that any MCPE soluton for the cost allocaton roblem can be realzed as a ash equlbrum of the mechansm. Prooston 4. For any cost-allocaton roblem n I, the ash equlbra of mechansm B yeld an MCPE soluton. Proof. The roof concdes wth the roof of Prooston 2 excet for Lemmata 2 and 4. X Lemma 2. B ( ) contans net trades leadng to strctly ostõe consumton bundles for ndõdual. Proof. We consder three dstnct cases:. y Case 1: 0-x -Ýs1w Indvdual can, by adjustng the t announcement, set ysy y and x sx y, hence both x and y are strctly ostve. Lettng z1syw and z2s0 turns the X w X frst nequalty n the defnton of B. nto yw F yyx. or 0 Ýs 1w w X s1 s1 Ýs 1w X 1 2 F Ý w yx q y. The RHS s strctly ostve snce x -Ý w. Hence, Bcontans net trades where z ) yw and z ) 0. These net trades lead to strctly ostve consumton bundles for ndvdual. y Case 2: x s Ýs1w Once more, ndvdual can by adjustng the t announcement set ysy y and x sx y. Indvdual would then have an ndvdual budget constrant for X B. mentoned above. that would allow for ostve consumton of both X goods. However, the aggregate constrants n B. would restrct the ndvdual to receve zero consumton of the x good. By submttng n a smaller t such that x -x y, the ndvdual can relax the aggregate constrants whle stll keeng the ndvdual budget constrant not bndng. Ths would allow strctly ostve consumton of both goods. y Case 3: x s 0 The frst term on the RHS s zero leavng ya yyx. Snce w )0 and X )0 ndvdual can, just as before, choose a strctly ostve roducton lan X that leaves hm wth strctly ostve ncome ncludng w.. Thus, B. contans net trades leadng to strctly ostve consumton bundles.b

12 Lemma 4. The roducton lan ( x, y ) s such that c ( y ) s. X X X X X Proof. The negatve roft of the frm s n the ndvdual budget constrant for X B.. By the same argument as n Lemma 4 of Secton 3, the ndvdual would be able to exand hs budget set f losses were not maxmzed. Maxmzaton of X X X losses mles the condton c y. s.b Ths demonstrates that the ash equlbra outcomes of mechansm B consttute MCPE solutons.b Prooston 5. For any cost-allocaton roblem n I, the set of MCPE solutons s contaned n the set of ash equlbra outcomes of mechansm B. Proof. Smlar to revous roofs.b 5. Conclusons In ths aer, we study the allocaton of costs for envronments wth both decreasng and ncreasng returns to scale. In the decreasng-returns-to-scale case, we construct a mechansm that leads to Pareto otmal outcomes, correctly recognzng the ncentves of ndvduals. In the ncreasng-returns-to-scale case, Pareto otmalty s harder to acheve. We construct a mechansm that leads to margnal-cost-rcng equlbra that generate Pareto otmal outcomes under certan condtons. The exstence of such a constructon further justfes the MCPE concet. The outcomes of revously suggested mechansms are not guaranteed to be Pareto otmal even n the decreasng-returns-to-scale case. Furthermore, n contrast to revous mechansms, our cost-sharng mechansms do not requre the desgner to know ether the technology or ndvdual references. Whether or not there exsts a mechansm that s sueror to ours for envronments where MCPE outcomes fal to be Pareto otmal s a queston of nterest. One also may be nterested n the equty roertes of our mechansm. Such ssues can be used to determne share ownersh wth decreasng-returns-to-scale. Ths s not an oton n our mechansm for ncreasng-returns-to-scale where share ownersh s roortonal to endowments; however, snce rofts are negatve, ndvduals wth hgher endowments bear a larger cost. The constructon technque we use s not lmted to the secfed envronment. Extendng the mechansm to envronments wth more than two goods or multle technologes s straghtforward. Creatng a smlar mechansm to allocate resources when externaltes are resent s also ossble. Modfyng the mechansm to handle asymmetrcally nformed ndvduals nvolves major changes as well as movng to the Bayes ash equlbrum concet and remans a toc of further research.

13 Acknowledgements We wsh to thank Andrew Postlewate and a referee for helful comments. We are grateful to the artcants of the Stony Brook Internatonal Conference on Game Theory w1996 x. Also, we gratefully acknowledge the suort from the Kretman Foundaton and the Monaster Center for Economc Research. References Abreu, D., Sen, A., Subgame erfect mlementaton: a necessary and almost suffcent condton. Journal of Economc Theory 50, Abreu, D., Sen, A., Vrtual mlementaton n ash equlbrum. Econometrca 59, Beato, P., The exstence of margnal cost rcng equlbra wth ncreasng returns. The Quarterly Journal of Economcs 89, Bonnsseau, J.-M., Cornet, B., Exstence of margnal cost rcng equlbra n economes wth several nonconvex frms. Econometrca 58, Brown, D., Equlbrum analyss wth non-convex technologes. In: Hldenbrand, W., Sonnenschen, H. Eds.., Handbook of Mathematcal Economcs, Vol. 4, Cha. 36. orth-holland, Amsterdam, Calsamgla, X., Decentralzed resource allocaton and ncreasng returns. Journal of Economc Theory 14, Cornet, B., Margnal cost rcng and areto otmalty. In: Essays n honor of Edmond Malnvaud, Vol. 1. MIT Press, Cambrdge, MA. Derker, E., When does Margnal Cost Prcng lead to Pareto-effcency? Zetschrft fur atonalokonome 5, Hong, L., ash mlementaton n roducton economes. Economc Theory 5, Hurwcz, L., Outcome functons yeldng Walrasan and Lndahl allocatons at ash equlbrum onts. Revew of Economc Studes 46, Hurwcz, L., Maskn, E., Postlewate, A., Feasble ash mlementaton of socal choce rules when the desgner does not know endowments or roducton sets. In: Ledyard, J.O. Ed.., The Economcs of Informaton Decentralzaton: Comlexty, Effcency, and Stablty. Kluwer Academc, Boston, Kamya, K., 1988a. Exstence and unqueness of equlbra wth ncreasng returns. Journal of Mathematcal Economcs 17, Kamya, K., 1988b. On the survval assumton n margnal cost. rcng. Journal of Mathematcal Economcs 17, Mantel, R., Equlbro con rendmento crecentes a escala. Anales de la Asocaton Argentne de Economa Poltca 1, Matsushma, H., A new aroach to the mlementaton roblem. Journal of Economc Theory 45, Moore, J., Reullo, R., Subgame erfect mlementaton. Econometrca 56, Mouln, H., Shenker, S., Seral cost sharng. Econometrca 60, Mouln, H., Shenker, S., Average cost rcng versus seral cost sharng: an axomatc aroach. Journal of Economc Theory 64, Mouln, H., Watts, A., Two versons of the tragedy of the commons. Economc Desgn 2, Palfrey, T., Srvastava, S., ash mlementaton usng undomnated strateges. Econometrca 59,

14 Postlewate, A., Wettsten, D., Contnuous and feasble mlementaton. Revew of Economc Studes 56, Qunz, M., Effcency of margnal cost rcng equlbra. In: Majundar, M. Ed.., Equlbrum and Dynamcs: Essays n Honour of Davd Gale, Cha. 14. St. Martn s Press, ew York, Schmedler, D., Walrasan analyss va strategc outcome functons. Econometrca 48, Varan, H., A soluton to the roblem of externaltes when agents are well nformed. Amercan Economc Revew 84, Young, H.P., Cost allocaton. In: Aumann, R.J., Hart, S. Eds.., Handbook of Game Theory, Vol. 2, Cha. 36. orth-holland, Amsterdam,