Journal of Mathematical Economics

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1 Journal of Mathematcal Economcs 45 (2009) 3 23 Contents lsts avalable at ScenceDrect Journal of Mathematcal Economcs journal homepage: Implementaton of Pareto effcent allocatons Guoqang Tan a,b, a Department of Economcs, Texas A&M Unversty, College Staton, TX 77843, USA b School of Economcs and Insttute for Advanced Research, Shangha Unversty of Fnance and Economcs, Shangha , Chna. artcle nfo abstract Artcle hstory: Receved 0 October 2005 Receved n revsed form 7 July 2008 Accepted 22 July 2008 Avalable onlne 5 August 2008 JEL classfcaton: C72 D6 D7 D82 Ths paper consders Nash mplementaton and double mplementaton of Pareto effcent allocatons for producton economes. We allow producton sets and preferences are unknown to the planner. We present a well-behaved mechansm that fully mplements Pareto effcent allocatons n Nash equlbrum. The mechansm then s modfed to fully doubly mplement Pareto effcent allocatons n Nash and strong Nash equlbra. The mechansms constructed n the paper have many nce propertes such as feasblty and contnuty. In addton, they use fnte-dmensonal message spaces. Furthermore, the mechansm works not only for three or more agents, but also for two-agent economes Elsever B.V. All rghts reserved. Keywords: Incentve mechansm desgn Implementaton Pareto effcency Prce equlbrum wth transfer. Introducton.. Motvaton Ths paper consders mplementaton of Pareto effcent allocatons for producton economes by presentng well-behaved and smple mechansms that are contnuous, feasble, and use fnte-dmensonal spaces. Pareto optmalty s a hghly desrable property n desgnng ncentve compatble mechansms. The mportance of ths property s attrbuted to what may be regarded as mnmal welfare property. Pareto optmalty requres that resources be allocated effcently. If an allocaton s not effcent, there s a waste n allocatng resources and therefore there s room for mprovement so that at least one agent s better off wthout makng the others worse off under gven resources. However, due to techncal dffcultes, study on mplementaton of Pareto effcent allocatons n Nash equlbrum so far has been largely devoted to mplementng a subset, but not the whole doman, of Pareto effcent allocatons. The most commonly used equlbrum prncples that result n Pareto optmal allocatons are the Walrasan equlbrum, proportonal equlbrum and Lndahl dstrbutve equlbrum solutons for prvate goods economes, Lndahl equlbrum, rato equlbrum and cost share equlbrum solutons for publc goods economes. Many specfc mechansms have been provded n the lterature that mplement theseequlbrum prncples such as those n Hurwcz (979); Schmedler (980); Hurwcz et Correspondng author at: School of Economcs and Insttute for Advanced Research, Shangha Unversty of Fnance and Economcs, Shangha , Chna. E-mal address: gtan@tamu.edu /$ see front matter 2008 Elsever B.V. All rghts reserved. do:0.06/j.jmateco

2 4 G. Tan / Journal of Mathematcal Economcs 45 (2009) 3 23 al. (995); Postlewate and Wettsten (989); Tan (989, 994, 996, 999, 2003); Hong (995); Peleg (996a, b); Suh (995, 997); Yoshhara (999); Duggan (2003) among others. The mplementaton lterature rarely dscusses mplementaton of the whole set of Pareto effcent allocatons under prvate ownershp or more generally under other ownershp structures. From a postve pont of vew, Pareto outcomes are often taken as a benchmark-outcomes that expect to arse even though nformaton s ncomplete or prvate. A fundamental queston studed n the mechansm desgn lterature s whether or not a socal choce correspondence s (fully) mplementable on a rch doman of economc envronments, and n partcular a socal choce correspondence s the Pareto correspondence. Although many mechansms, such as the Groves mechansm (that does not take care of ndvdual ratonalty or equty) and the market type mechansms, have been proposed to mplement socal/pareto optmal allocatons, an open queston s f the socal choce rule that s gven by the Pareto effcent correspondence can be (fully) mplemented by an ncentve compatble mechansm. From a normatve perspectve, an often consdered prce equlbrum allocaton (such as Walrasan or Lndahl equlbrum allocaton) that may result n Pareto effcency s only concerned wth effcency of allocatons and has nothng to say about dstrbuton of welfare and equty of allocatons. Even f we agree wth Pareto optmalty, we stll do not know whether or not a Pareto allocaton that satsfes a desrable property such as equty can be mplemented by an ncentve compatble mechansm. However, by the well-known Second Theorem of Welfare Economcs, such an equtable allocaton can be obtaned by a prce equlbrum allocaton wth transfers. Indeed, under some regularty condtons such as contnuty, convexty, and monotoncty, the Second Theorem of Welfare Economcs shows that every Pareto effcent allocaton can be supported as a prce equlbrum wth transfers. As such, one may want to know f every Pareto effcent allocaton s mplementable. If so, one may further desre that all Pareto effcent allocatons can be mplemented by a sngle ncentve mechansm. In other words, one may desre to have a mechansm that mplements the full set of Pareto effcent allocatons..2. Related lterature Groves and Ledyard (977) were the frst to consder Nash mplementaton of Parato effcent allocatons for publc goods economes. However, ther mechansm only (weakly) mplements, but does not fully mplement, the Pareto correspondence, that s, t only mplements a subset of Pareto effcent allocatons. Furthermore, the mechansm s not ndvdually feasble. Osann (997) dscussed Nash mplementaton of weak Pareto effcent allocatons for decomposable (externalty-free) economc envronments. Agan, hs mechansm only mplements a subset of weak Pareto effcent allocatons although the set s much bgger than the set of Lndahl allocatons. Besdes, the mechansm s not ndvdually feasble and contnuous. Anderln and Sconole (2004) consder mplementaton of Pareto effcent and ndvdually ratonal allocatons for publc goods economes. However, ther mechansms are not contnuous on the boundary of the feasble set ether, and further they consder publc goods economes wth only one prvate and one publc good. It seems that ther equlbrum noton and approach to mplementaton of Pareto effcent allocatons may be hard to extend to a more general case of publc goods economes wth arbtrary numbers of prvate and publc goods. Thus, an mportant unanswered queston s whether one can desgn an ncentve compatble mechansm that mplements the whole doman, but not just a subset, of Pareto effcent allocatons. If so, the mechansm should be well-behaved n the sense that t s contnuous, feasble, realstc, has a fnte-dmensonal dmenson message space, and works for any number of agents, etc. Some of these propertes have been studed systematcally n Dutta et al. (995); Sajo et al. (996). We wll answer ths queston by usng the approach of the Second Theorem of Welfare Economcs, whch tells us, under some condtons, ncludng essental condton of convexty of preferences and producton sets, that any desred Pareto optmal allocaton can be acheved as a market-based equlbrum wth transfers..3. Results of the paper In ths paper, we deal wth the problem of desgnng an ncentve mechansm that mplements Pareto effcent allocatons for general neoclasscal producton economes. We allow producton sets and preferences to be unknown to the planner, and present a mechansm that fully mplements Pareto effcent allocatons n Nash equlbrum n the sense that the set of Nash equlbrum allocatons of a mechansm concdes wth the set of Pareto optmal allocatons over the class of general convex producton economes. To do so, we frst ntroduce the noton of constraned prce equlbrum wth transfers whch s equvalent to the conventonal prce equlbrum wth transfer when each agent s consumpton bundle at equlbrum s non-zero. We then reduce the mplementaton problem of Pareto effcent allocatons to the problem of mplementng constraned prce equlbrum allocatons wth transfers. Ths approach s based on the Second Theorem of Welfare Economcs. It s a converse of the Frst Theorem of Welfare Economcs. It s ths general noton of prce equlbrum wth transfers that permts an arbtrary dstrbuton of wealth among consumers. The mechansm s then slghtly modfed to doubly mplement Pareto effcent allocatons n Nash and strong Nash equlbra. That s, by the mechansm, not only Nash equlbrum allocatons, but also strong Nash equlbrum allocatons concde wth Pareto optmal allocatons. By double mplementaton, the soluton can cover the stuaton where agents n coaltons may cooperate and n some other coaltons may not. Thus, the desgner does not need to know whch coaltons are permssble and, consequently, t allows the possblty for agents to manpulate coalton patterns.

3 G. Tan / Journal of Mathematcal Economcs 45 (2009) The mechansms presented n the paper are well-behaved and elementary mechansms that have many desred propertes such as feasblty and contnuty. In addton, they are prce-quantty market type mechansms, and use fnte-dmensonal message spaces. Furthermore, our mechansms work not only for three or more agents, but also for two-agent economes, and thus they are unfed mechansms that are rrespectve of the number of agents. It may be remarked that, because of transfer payments, the mplementaton of all Pareto effcent allocatons s harder than those for the conventonal solutons mentoned above and the mechansms we wll construct n the paper are qute dfferent from the exstng ones for Walrasan or Lndahl solutons. For nstance, we need to gve the managers of frms specal ncentves to maxmze profts at equlbrum, but at the same tme the proft cannot affect ndvduals wealth. Thus, some of the technques used n mplementng other market-lke-equlbrum solutons such as Walrasan equlbrum for prvate goods economes or Lndahl equlbrum for publc goods economes may not be applcable, and some new technques must be developed. Tan (n press) used smlar technques to consder the problem of ncentve mechansm desgn n non-convex producton economes when producton sets and preferences both are unknown to the desgner. The remander of ths paper s as follows. Secton 2 sets up a general model, ntroduces the noton of constraned prce equlbrum wth transfers, and gves the defnton of mplementaton by a mechansm. Secton 3 presents a well-behaved mechansm that fully mplements constraned prce equlbrum allocatons wth transfers, and consequently fully mplements Pareto effcent allocatons n Nash equlbrum. Secton 4 modfes the mechansm so that the modfed mechansm doubly mplements Pareto effcent allocatons n Nash and strong Nash equlbra. These mechansms have the desred propertes mentoned above. Concludng remarks are presented n Secton The setup 2.. Economc envronments We consder producton economes wth L commodtes, n 2 consumers and J frms. Let N ={, 2,...,n} denote the set of consumers. Each agent s characterstc s denoted by e = (C,R ), where C = R L + s the consumpton set, and R s the preference orderng defned on R L +.LetP denote the asymmetrc part of R (.e., ap b f and only f ar b, but not br a). We assume that R s contnuous, convex, and strctly monotoncally ncreasng on R L +.2 Let ŵ be the total ntal endowment vector of commodtes. Producton technologes of frms are denoted by Y,...,Y j,...,y J. We assume that, for j =,...,J, Y j s closed, convex, contans 0 (possblty of nacton), and R L + Y j (free-dsposal). We assume that there are no externaltes or publc goods. An economy s the full vector e = (e,...,e n, ŵ, Y,...,Y J ) and the set of all such economes s denoted by E whch s assumed to be endowed wth the product topology Pareto effcency and constraned prcng equlbrum wth transfers An allocaton of the economy e s a vector (x,...,x n,y,...,y J ) R L(n+J) such that: () x := (x,...,x n ) R nl +, and (2) y j Y j for j =,...,J. Denote by y = (y,...,y J ) the profle of producton plans of frms. An allocaton (x, y) sfeasble f n n J x ŵ + y j. () = = j= Denote the aggregate consumpton and producton by ˆx = n = x and ŷ = J j= y j, respectvely. 3 Then the feasblty condton can be wrtten as ˆx ŵ + ŷ. An allocaton (x, y) spareto-optmal wth respect to the preference profle R = (R,...,R n ) f t s feasble and there s no other feasble allocaton (x,y ) such that x R x for all N and x P x for some N. Denote by P(e) the set of all such Pareto optmal allocatons. From Debreu (952); Mas-Colell et al. (995), one knows that Pareto effcent allocatons can be characterzed by prce equlbrum wth transfers under the condtons mposed n the paper. 4 Ths equvalence result s referred as the well-known Second Theorem of Welfare Economcs. We wll use ths characterzaton result to study mplementaton of Pareto effcent allocatons. The noton of prce equlbrum wth transfers s the generalzaton of Walrasan (compettve) equlbrum As usual, vector nequaltes,,, and, are defned as follows: Let a, b R m. Then a b means a s b s for all s =,...,m; a b means a b but a /= b; a>bmeans a s >b s for all s =,...,m. 2 R s convex f, for bundles a and b, ap b mples a + ( )bp b for all 0 <. Note that the term convex s defned as n Debreu (959), not as n some recent textbooks. 3 For notatonal convenence, â wll be used throughout the paper to denote the sum of vectors a,.e., â := a l. 4 A prce equlbrum wth transfers s also called an equlbrum relatve to a prce system n Debreu (952).

4 6 G. Tan / Journal of Mathematcal Economcs 45 (2009) 3 23 allocatons. Whle the Walrasan equlbrum concept alles to the case of prvate ownershp economy, the more general noton of a prce equlbrum wth transfers allows nstead for an arbtrary dstrbuton of wealth among consumers. A prce equlbrum wth transfers for an economy e s a lst of consumpton plans (x ), a lst of producton plans (y ), and j a prce vector p such that (a) every consumer maxmzes hs preferences subject to hs budget constrant {x R L + : p x p x }, (b) every frm maxmzes ts proft on Y j for all j =,...,J, and (c) the excess demand over supply s zero. Formerly, we have the followng defnton: An allocaton z = (x,y ) = (x,x 2,...,x n,y,y 2,...,y J ) RnL + Y s a prce equlbrum allocaton wth transfers for an economy e f there s a prce vector p R L + such that () for every N, x P x mples p x >p x,.e., x s a greatest element for R n the budget set {x R L + : p x I p x }, (2) for j =,...,J, p y p y j j for all y j Y j, n (3) = x J = ŵ + j= y j. Denote by PE(e) the set of all such prce equlbrum allocatons. Debreu (952) proved that, under the condtons mposed n the paper, an allocaton (x,y ) s Pareto effcent f and only f t s a prce equlbrum wth transfers (also see Propostons 6.C., 6.D. and 6.D.3. n Mas-Colell et al., 995). 5 Thus, f we can construct a mechansm that mplements prce equlbrum allocatons wth transfers, the mechansm then mplements Pareto effcent allocatons wth transfers. Therefore, the problem of mplementng Pareto effcent allocatons can bereduced to the problem of mplementng prce equlbrum allocatons wth transfers. However, due to the dffculty of constructng a feasble mechansm that mplements prce equlbrum allocatons wth transfers, we would ntroduce the noton of constraned prce equlbrum wth transfers whch resembles the one where the constraned Walrasan prncple s used nstead of the Walrasan prncple when one consders mplementaton of compettve equlbrum allocatons (cf. Hurwcz et al., 995). As we show below, whle the set of constraned Walrasan allocatons n general s bgger than the set of Walrasan allocatons, the set of constraned prce equlbrum allocatons wth transfers concdes wth prce equlbrum allocatons wth transfers for non-zero consumpton bundles. Thus, we can reduce the problem of mplementng Pareto effcent allocatons to the problem of mplementng constraned prce equlbrum allocatons wth transfers. An allocaton z = (x,y ) = (x,x 2,...,x n,y,y 2,...,y J ) RnL + Y s a constraned prce equlbrum allocaton wth transfers for an economy e f there s a prce vector p R L + such that () for every N, x P x and x J ŵ + j= y mples that p x j I >p x, (2) for j =,...,J, p y p y j j for all y j Y j, n (3) = x J = ŵ + j= y j. Denote by PE c (e) of the set of all such prce equlbrum allocatons. Remark. A constraned prce equlbrum allocaton wth transfers dffers from a prce equlbrum allocaton wth transfers, only n the way that each consumer maxmzers hs preferences not only subject to hs budget constrant but also subject to the total supply. Thus, t s clear that every prce equlbrum allocaton wth transfers s a constraned prce equlbrum allocaton wth transfers. What about the converse? We know that a constraned Walrasan allocaton may not be a Walrasan allocaton on boundary ponts. Contrary to the conventonal Walrasan equlbrum, the set of constraned prce equlbrum allocatons wth transfers actually concdes wth the set of prce equlbrum allocatons wth transfers when each consumer s consumpton bundle at equlbrum s non-zero. To show ths, we frst show that, under local non-sataton of preferences, every constraned prce equlbrum allocaton wth transfers s Pareto effcent,.e., PE c (e) P(e). Lemma. Suppose (p, x, y) s a constraned prce equlbrum wth transfers. If each consumer s locally non-satated at x, then (x, y) s Pareto effcent. Proof. Suppose, by way of contradcton, that (x, y) s not Pareto effcent. Then there s another feasble allocaton (x,y ) such that x R x for all N and x P x for some N. Snce x s a constraned prce equlbrum wth transfers, x P x and x ŵ + J j= y j mply that p x >p x for some, and, by local non-sataton of preferences, x R x and x ŵ + J j= y j mply that p x p x for all N. Therefore, f (x,y n ) s Pareto superor to (x, y), we have = p x > n = p x by local J non-sataton. However, because y j s a proft maxmzng producton plan for frm j at prce p,wehave j= p y J j j= p y j, 5 The condton they used are actually weaker. For nstance, only local non-sataton s requred. Also a consumer s consumpton space s not necessarly gven by R L +.

5 G. Tan / Journal of Mathematcal Economcs 45 (2009) n and thus = p x > n = p x n = = p ŵ + J j= p y n j = p ŵ + J j= p y j. Ths contradcts the fact that (x,y )s a feasble allocaton. Q.E.D. We now show the equvalence on the two notons of constraned prce equlbrum wth transfers and prce equlbrum wth transfers. Lemma 2. Suppose for each N, preference orderngs R are contnuous on R L + and strctly ncreasng on RL ++, and producton sets Y j are non-empty, convex, and closed. Then, an allocaton (x, y) wth x /= 0 s Pareto effcent f and only f t s a constraned prce equlbrum wth transfers. Proof. By Propostons 6.C., 6.D. and 6.D.2. n Mas-Colell et al. (995), we know that the set of prce equlbrum allocatons wth transfers concdes wth the set of Pareto effcent allocatons wth non-zero consumptons for all ndvduals,.e, PE(e) = P(e) wth x /= 0 under the assumptons mposed. Also, by Lemma, we have PE c (e) P(e). Thus, combnng these facts, we have P(e) = PE(e) PE c (e) P(e), and thus PE c (e) = P(e) for all e E. Q.E.D. Thus, n ths paper, we consder mplementaton of Pareto effcent allocatons n whch every consumer s consumpton x s non-zero. The set of such Pareto effcent allocaton s stll denoted by P(e). We wll present a feasble and contnuous mechansm whch fully mplements Pareto effcent allocatons and has fnte-dmensons of message spaces. To do so, we frst gve some basc concepts on economc mechansm Mechansm In ths subsecton, we gve some basc concepts, notaton and defntons used n the mechansm desgn lterature. Let F : E R L(n+J) + be a socal correspondence to be mplemented. Let M denote the -th agent s message doman. n Its elements are wrtten as m and are called messages. Let M = = M denote the message space whch s assumed to be endowed wth the product topology. Denote by h : M R L(n+J) + the outcome functon, or more explctly, h(m) = (X (m),...,x n (m),y (m),...,y J (m)). Then a mechansm, whch s defned on E, conssts of a message space M and an outcome functon. It s denoted by M, h. A message m = (m,...,m n ) M s sad to be a Nash equlbrum of the mechansm M, h for an economy e f, for all N and m M, X (m )R X (m,m ), (2) where (m,m ) = (m,...,m,m,m +,...,m n ). The outcome h(m ) s then called a Nash (equlbrum) allocaton of the mechansm for the economy e. Denote by V M,h (e) the set of all such Nash equlbra and by N M,h (e) the set of all such Nash equlbrum allocatons. A mechansm M, h s sad to Nash-mplement a socal choce correspondence F on E, f, for all e E, N M,h (e) F(e). It s sad to fully Nash-mplement a socal choce correspondence F on E, f, for all e E, N M,h (e) = F(e). A coalton C s a non-empty subset of N. A message m = (m,...,m n ) M s sad to be a strong Nash equlbrum of the mechansm M, h for an economy e f there does not exst any coalton C and m C C M such that for all C, X (m C,m C )P X (m ). (3) The outcome h(m ) s then called a Nash (equlbrum) allocaton of the mechansm for the economy e. Denote by SV M,h (e) the set of all such strong Nash equlbra and by SN M,h (e) the set of all such strong Nash equlbrum allocatons. A mechansm M, h s sad to (fully) doubly mplement a socal choce correspondence n Nash and strong Nash equlbra F on E, f, for all e E, SN M,h (e) = N M,h (e) F(e) (SN M,h (e) = N M,h (e) = F(e)). A mechansm M, h s sad to be feasble, f, for all m M, () X(m) R nl +, (2) Y n j(m) Y j for j =,...,J, and (3) = X (m) J ŵ + j= Y j(m). A mechansm M, h s sad to be contnuous, f the outcome functon h s contnuous on M. 3. Implementaton of pareto effcent allocatons 3.. The mechansm for pareto effcent prncple In the proposed mechansm below, the desgner does not need to know frms true producton sets. However, t s well known by now that to have feasble mplementaton, the desgner has to know some nformaton about producton sets. That s, we have to requre that the mechansm desgner can dentfy one of the consumers as a frm s manager, who s asked to announce a producton plan from the producton set for each frm. Although ths requrement s a strong assumpton,

6 8 G. Tan / Journal of Mathematcal Economcs 45 (2009) 3 23 t s necessary to guarantee feasblty at all ponts, ncludng dsequlbrum ponts. 6 Thus, t s assumed that the manager of a frm knows the frm s producton possblty set whle others may or may not have ths nformaton. Although we can reasonably assume that one ndvdual s at most a manager for one frm, to smplfy the exposton, wthout loss of generalty, consumer s assumed to be the manager of all frms. The message space of the mechansm s defned as follows. For each N, let the message doman of agent be of the form L ++ Ỹ Y Z R ++ (0, ] f = M = L ++ RJL Z R ++ (0, ] f = 2 Z R ++ (0, ] f 3 where { ++ L = p R L ++ : Ỹ = y Y : ŵ + } L p l =, l= J y j 0, j= (4) (5) and Z ={(z,...,z n ) R nl + : z,+ /= 0} (6) for =,...,n, where n + sreadtobe. Note that, unlke the exstng mechansms, we do not requre all consumers, except for frst two consumers, to announce prces and producton profles of producton plans. Thus, the dmenson of the message space s lower than that of some of the exstng mechansms that mplement Walrasan allocatons for convex producton economes wth more than two agents. A generc element of M s m = (p,y,t,z,, ), m 2 = (p 2,y 2,z 2, 2, 2 ), and m = (z,, )for = 3,...,n, whose components have the followng nterpretatons. The component p s the prce vector proposed by agent and s used as a prce vector by the other agents k /=. The components y = (y,...,y J ) and t = (t,...,t J ) are producton profles announced by agent who works as the manager of frms, and wll be used to nduce the feasble producton plans and proft maxmzng rule, respectvely. The component y 2 = (y 2,...,y 2J ) s a producton profle announced by agent 2 who may be nterpreted as the owner or a group of nvestors of frms, and wll determne the proft maxmzng producton profle. Whle y and t are requred to be producton profles n Y, 7 the producton profle y 2 s not necessarly n Y snce agent 2 s not assumed to know producton sets. Note that n order for a mechansm to have a fnte-dmensonal message space, unlke the mechansm constructed n Hurwcz et al. (995), whch requres agents to report producton sets that result n an nfnte-dmensonal message space, we only requre agents to report producton plans for frms, but not the producton sets. The component z = (z...,z n ) s an allocaton proposed by agent, where z k s the consumpton bundle of agent k proposed by agent. The component s a shrnkng ndex of agent used to shrnk the consumpton of other agents n order to have a feasble outcome. The component s the penalty ndex mposed to agent f she proposes consumpton bundles for the others whch are dfferent from those announced by agent +. Before we formally defne the outcome functon of the mechansm, we gve a bref descrpton and explan why the mechansm works. For each announced message m M, the prce vector facng consumer s determned by p 2 announced by agent 2, and the other s prce vector s determned by p announced by agent. Thus each ndvdual takes prces as gven and cannot change them by changng hs own messages. The feasble producton outcome Y(m) Ỹ s then determned by the feasble producton profle y proposed by agent, who s the manger of the frms. Also to gve the manager of frms an ncentve to produce at a proft maxmzng outcome at equlbrum, a specal compensaton s provded to the manager of frm j n the way that she wll receve a postve amount of compensaton f she proposes a more proftable producton plan t than one announced by agent 2. Her prelmnary consumpton outcome s determned by the product of her consumpton n B (m) that s closest to the consumpton bundle z announced by the consumer and the penalty dscount factor [/( + ( z 2 x 2 (m) + z, z 2, + (m))) + (m)] f she announced a dfferent consumpton profle z, from z 2, and a dfferent producton profle t from y 2, where z j, = (z j,,...,z j,,z j,+,...,z j,n ) whch denotes consumpton bundles proposed by agent j for all agents except for agent. To gve ncentves for agent 2, regarded as the owner of frms, to match the prce vector and producton profle announced by the manager of frms and match the consumpton bundles announced by agent 3, agent 2 s prelmnary consumpton 6 If one does not lke ths assumpton and s wllng to gve up the requrement of feasblty of a mechansm lke most mechansms that mplement Walrasan or Lndahl allocatons n the economcs lterature, t s much easer to construct a mechansm that mplements Pareto effcent allocatons. 7 Actually, we requre y be a feasble producton profle so that non-negatve aggregate consumptons for consumers.

7 G. Tan / Journal of Mathematcal Economcs 45 (2009) outcome s determned by the product of the consumpton n B 2 (m) that s closest to the consumpton bundle z 2 announced by the consumer 2 and the penalty dscount factor, /( + p p + y y ( z 23 x 3 (m) + z 2, 2 z 3, 2 )). For agent 3, the prelmnary consumpton outcome x (m) s determned by the product of her consumpton n B (m) that s closest to the consumpton bundle z announced by the consumer 3 and the penalty dscount factor /( + ( z,+ x + (m) + z, z +, )) f he announces a dfferent consumpton profle z, from z +,. To obtan the feasble outcome consumpton X(m), we need to shrnk the prelmnary outcome consumpton x (m) n the way specfed below. We wll show that the mechansm constructed n such a way have the propertes we desre, and fully mplements Pareto effcent allocatons over the class of producton economes under consderaton. Now we formally present the outcome functon of the mechansm. In order for each agent to take prces as gven, defne agent s proposed prce vector p : M L ++ by { p (m) = p 2 f = p otherwse Defne the feasble producton outcome functon Y : M Ỹ by Y(m) = y. (7) To gve the manager ncentves to follow the proft maxmzng rule at Nash equlbrum, some specal compensaton wll be provded to her accordng to the followng formula. Defne : M R + by J (m) = j (m) (8) where j= j (m) = max{0,p (m) (t j y 2j )}. (9) The compensaton formula means that agent wll receve a postve amount of compensaton j (m) f the manager can announce a more proftable producton plan y j than the one proposed by agent 2, or she wll receve a zero amount of compensaton. In other words, j (m) > 0 f and only f p (m) t j >p (m) y 2j. Remark 2. The reason the rewardng system works s the followng. In determnng the proft maxmzng producton outcome Y(m), f the producton plan y 2j, announced by agent 2, does not maxmze p (m) y j over producton set Y j, then there s a test producton plan y j Y j such that p (m) y j >p (m) y 2j, and thus agent can gan from usng ths test producton plan. The feasble consumpton set B (m) s then defned by { {x R L + : p 2 x = p 2 z n, &x ŵ + ŷ } f = B (m) = {x R L + : p x p z, &x ŵ + ŷ } otherwse, whch are contnuous correspondences wth non-empty, compact, and convex values. Defne x : M B by x (m) ={x : mn x B (m) x z }, (0) whch s the closest to z. Defne agent s prelmnary consumpton outcome functon x : M R L + by [ + ( z 2 x 2 (m) + z, z 2, + (m)) + (m)]x (m) f = x (m) = + p p 2 + y y ( z 23 x 3 (m) + z 2, 2 z 3, 2 ) x 2 (m) f = 2 + ( z,+ x + (m) + z, z +, ) x (m) f 3 where z, = (z,,...,z,,z,+,z,n ). Defne the -correspondence A: M 2 R+ by A(m) ={ R + : N& n x (m) ŵ + = Let (m) be the largest element of A(m),.e., (m) A(m) and (m) for all A(m). J Y j (m)} () j=,

8 20 G. Tan / Journal of Mathematcal Economcs 45 (2009) 3 23 Fnally, defne agent s outcome functon for consumpton goods X : M R L + by X (m) = (m) x (m), (2) whch s agent s consumpton resultng from the strategc confguraton m. Thus the outcome functon (X(m),Y(m)) s contnuous and feasble on M snce, by the constructon of the mechansm, (X(m),Y(m)) R nl + Y, and ˆX(m) ŵ + Ŷ(m) (3) for all m M. Remark 3. The above mechansm works not only for three or more agents, but also for a two-agent world The result The remander of ths secton s devoted to provng the followng theorem. Theorem. For the class of producton economc envronments E specfed n Secton 2, f the followng assumptons are satsfed: () ŵ>0; (2) For each N, preference orderngs, R, are contnuous, and strctly ncreasng on R L + ; (3) The producton sets Y j are non-empty, convex, closed, 0 Y j, and R L + Y j. then the mechansm defned n the above subsecton, whch s contnuous, feasble, and uses a fnte-dmensonal message space, fully mplements constraned prce equlbrum allocatons wth transfers and consequently fully mplements Pareto effcent allocatons wth non-zero consumpton bundles n Nash equlbrum on E. Proof. The proof of Theorem conssts of the followng two propostons whch show the equvalence between Nash equlbrum allocatons and constraned prce equlbrum allocatons wth transfers. Proposton below proves that every Nash equlbrum allocaton s a constraned prce equlbrum allocaton wth transfers. Proposton 2 below proves that every constraned prce equlbrum allocaton wth transfers s a Nash equlbrum allocaton. Therefore, we show that the mechansm constructed n the prevous secton fully mplements constraned prce equlbrum allocatons. Then, by Lemma 2, we show that the mechansm fully mplements Pareto effcent allocatons. To show these propostons, we frst prove the followng lemmas. Lemma 3. Suppose x (m)p x. Then agent can choose a very large such that X (m)p x. Proof. If agent declares a large enough, then (m) becomes very small (snce (m) ) and thus almost nullfes the n effect of other agents n (m) = J x (m) ŵ + j= Y j(m). Thus, X (m) = (m) x (m) can arbtrarly approach as close to x (m) asagent wshes. From x (m)p x and contnuty of preferences, we have X (m)p x f agent chooses a very large. Q.E.D. Lemma 4. If m V M,h (e), then X (m ) /= 0, x (m ) /= 0 and x (m ) /= 0 for N. Proof. We argue by contradcton. Suppose X (m J ) = 0 for some N. Snce ŵ + j= Y j(m ) > 0, p (m ) > 0, and z /= 0by, constructon, we have p (m ) z, > 0. Thus there s some x R L ++ such that p (m ) x p (m ) z,, x J ŵ + j= Y j(m ), and x P X (m ) by monotoncty of preferences. Now suppose that agent chooses z = x, >, and keeps the other components of the message unchanged. Then, z B (m,m ), and thus x (m,m ) = z. Snce x (m,m ) s proportonal to x (m,m ), x (m,m ) > 0, we have x (m,m )P X (m ) by monotoncty of preferences. Therefore, by Lemma 3, agent can choose a very large such that X (m,m )P X (m ). Ths contradcts m V M,h (e) and thus we must have X (m ) /= 0 for all N. Snce X (m ) s proportonal to x (m ) and x (m ), x (m ) /= 0 and x (m ) /= 0for N.Q.E.D. Lemma 5. If m s a Nash equlbrum, then p = p 2, y = y 2, (m ) = 0, z,+ = x +(m ), and z, = z for all N. +, Consequently, p(m ) p (m ) = p = p 2 for N, Y(m ) = y = y 2, x (m ) = x (m ) for all N, z = z 2 =...= z n = x (m ) = x(m ). Proof. We frst show that p = p 2 and y = y 2. Suppose, by way of contradcton, that p /= p 2,ory /= y 2. Snce x (m ) /= 0 for all agent by Lemma 4, agent 2 can choose p 2 = p,ory 2 = y so that her consumpton becomes larger and she would be better off by monotoncty of preferences. Hence, m s not a Nash equlbrum strategy f p /= p 2,ory /= y. Thus, we must 2 have p = p 2 and y = y at Nash equlbrum. 2 We now show that (m ) = 0, z,+ = x +(m ), and z, = z for all N. Suppose not. Then agent can choose a +, smaller < n (0, ] so that her consumpton becomes larger and she would be better off by monotoncty of preferences. Hence, no choce of could consttute part of the Nash equlbrum strategy when (m ) /= 0, z,+ = x +(m ), or z /= z, +,. Thus, we must have (m ) = 0, z,+ = x +(m ), and z, = z for all N. +,

9 G. Tan / Journal of Mathematcal Economcs 45 (2009) Consequently, by the constructon of the mechansm, p(m ) p (m ) = p = p 2 for all N, Y(m ) = y = y 2, x(m ) = x (m ) = z = z 2 =...= z n. Q.E.D. Lemma 6. If (X(m ),Y(m )) N M,h (e), then (m ) = for all N and thus X(m ) = x(m ) = x (m ). Proof. Suppose, by way of contradcton, that (m ) < for some N. Then X (m ) = (m ) x (m ) <x (m ), and thus x (m )P X (m ) by monotoncty of preferences. Therefore, by Lemma 3, agent can choose a very large such that X (m,m )P X (m ). Ths contradcts m V M,h (e). Thus we must have (m ) =, and therefore X(m ) = x(m ) = x (m ). Q.E.D. Lemma 7. If m s a Nash equlbrum, then Y(m ) maxmzes profts on Y j for j =,...,J,.e., p(m ) Y j (m ) p(m ) y j for all y j Y j. Consequently, j (m,m ) = j(m ) = 0 for all m G (M). Proof. Suppose, to the contrary, that for some frm j, there exsts a producton plan y j Y j such that p(m ) Y j (m ) < p(m ) y j. Then, f agent chooses t j = y j, =, and >, and keeps other components of the message unchanged, we have (m,m ) = p(m ) y j p(m ) Y j (m ) > 0, and thus x (m,m ) = [ (m,m ) + /( + (m,m ))]x (m ) > x (m ) = X (m ) by notng (m ) = 0 and the functon f () = + /( + ) s strctly ncreasng n. Then, x (m,m )P X (m ) by monotoncty of preferences, and thus, by Lemma 3, agent can choose a very large such that X (m,m )P X (m ), whch contradcts the fact m s a Nash equlbrum. Hence Y j (m ) must a proft maxmzng producton plan for frm j. Fnally, snce there s no producton plan y j Y j such that p(m ) y j >p(m ) Y j (m ), we must have j (m,m ) = j(m ) = 0 for all m I G (M). Q.E.D. Proposton. If the mechansm defned above has a Nash equlbrum m, then the Nash equlbrum allocaton (X(m ),Y(m ),p(m )) s a constraned prce equlbrum wth transfers,.e., N M,h (e) PE(e). Proof. Let m be a Nash equlbrum. We need to prove that (X(m ),Y(m ),p(m )) s a constraned prce equlbrum allocaton wth transfers. Note that the feasblty condton s also satsfed by the constructon of the mechansm, all ndvduals take p(m ) as gven by the constructon of the mechansm, and Y(m ) maxmzes the profts of frms over Y j for all j by Lemma 7. So we only need to show that X (m ) s the greatest element for R n the budget set {x R L + : p(m ) x p(m ) X (m )} for all N. Suppose, by way of contradcton, that there s some x R L + and x J ŵ + j= Y j(m ) such that x P X (m ) and p(m ) x p(m ) X (m ). Then, f agent chooses z = x, >, and keeps other components of the message unchanged. Then, z B (m,m ), and thus x (m,m ) = z P X (m ). Therefore, by Lemma 3, agent can choose a very large such that X (m,m )P X (m ). Ths contradcts (X(m ),Y(m )) N M,h (e). Thus, (X(m ),Y(m )) s a constraned prce equlbrum allocaton wth transfers. Proposton 2. If (x,y,p ) s a constraned prce equlbrum wth transfers and x /= 0 for all N, then there s a Nash equlbrum m of the above mechansm such that Y(m ) = y, p(m ) = p, and X (m ) = x for all N,.e., PE(e) N M,h (e). Proof. We frst note that by the strct monotoncty of preference orderngs, the normalzed prce vector p must be n L ++. We need to show that there s a message m such that (x,y ) s a Nash equlbrum allocaton. For each N, defne m by p = p, y = t = y (for =, 2), z = x, =, and =. Then, t can be easly verfed that Y(m ) = y = y 2, p (m ) = p, and X (m ) = x for all N. Then, p (m,m ) = p (m ), p(m ) Y j (m ) p(m ) y j for all y j Y j, and thus (m,m ) = (m ) = 0 for all m M. Hence p(m ) X (m,m ) p(m ) X (m ) (4) for all m M. Thus, X (m,m ) satsfes the budget constrant for all m M. Thus, we must have X (m )R X (m,m ), or else t contradcts the fact that (X(m ),Y(m )) s a constraned prce equlbrum allocaton wth transfers. So (X(m ),Y(m )) must be a Nash equlbrum allocaton. Q.E.D. Thus, by Propostons and 2, we know that N M,h (e) = PE(e), whch means the mechansm fully mplements constraned prce equlbrum allocatons wth transfers n Nash equlbrum, and thus by Lemma 2, the mechansm fully mplements Pareto effcent allocatons n Nash equlbrum. The proof of Theorem s completed. Q.E.D. 4. Double mplementaton of pareto effcent allocatons The above mechansm only fully mplements Pareto effcent allocatons n Nash equlbrum, but does not fully doubly mplement Pareto effcent allocatons n Nash and strong Nash equlbra. When agents and 2 form a coalton, they may be better off. However, f one s wllng to ncrease the dmensons of the message by askng every consumer to report a prce vector and producton profle, the above mechansm can be slghtly modfed so the modfed mechansm fully doubly mplementaton Pareto effcent allocatons. In ths secton, we brefly dscuss such a modfed mechansm. The message doman of agent s gven by { L ++ Ỹ Y Z R ++ (0, ] f = M = L ++ RJL Z R ++ (0, ] otherwse

10 22 G. Tan / Journal of Mathematcal Economcs 45 (2009) 3 23 where Ỹ and Z are defned by (6). A generc element of M s m = (p,y,t,z,, )for = and m = (p,y,z,, )for = 2,...,n. Note that, whle agent, the manager of frms, s requred to report producton profles y = (y,...,y J ) and t = (t,...,t J ) n the producton sets, the others are not requred to announce producton profles n the producton sets. Agent s prce vector p : M L + s defned by p (m) = p + (5) and the feasble producton outcome functon Y : M Ỹ by Y(m) = y. (6) Agents s compensaton functon for nducng frms to produce at the proft maxmzng producton plans s specfed as the same as before. The feasble consumpton set B (m) and x for agent are also defned as the same as before. Agent s prelmnary consumpton outcome functon x : M R L + s modfed to [ + x (m) = ( z 2 x 2 (m) + z, z 2, +(m)) + (m)]x (m) f = + p p + + y y + + ( z,+ x + (m) + z, z +, ) x (m) otherwse Fnally, agent s outcome functon for consumpton goods X : M R L + s gven by X (m) = (m) x (m), (7) whch s the same as before. Then, the outcome functon (X(m),Y(m)) s contnuous and feasble on M, and the mechansm clearly satsfes the best response property. Smlarly, we have the followng theorem. Theorem 2. For the class of producton economc envronments E specfed n Secton 2, f the followng assumptons are satsfed: () ŵ>0; (2) For each N, preference orderngs, R, are contnuous on R L +, and strctly ncreasng on RL + ; (3) The producton sets Y j are non-empty, convex, closed, 0 Y j, and R L + Y j. then the mechansm defned n the above subsecton, whch s contnuous, feasble, and uses a fnte-dmensonal message space, fully doubly mplements constraned prce equlbrum allocatons wth transfers and consequently fully doubly mplements Pareto effcent allocatons wth non-zero consumpton bundles n Nash and strong Nash equlbra on E. Proof. The proof of the equvalence between Nash equlbrum allocatons and constraned prce equlbrum allocatons wth transfers s the same as the one of Theorem. We only need to prove the followng proposton that shows that every Nash equlbrum s a strong Nash equlbrum. As a result, we show that the mechansm constructed n the prevous secton doubly mplements constraned prce equlbrum allocatons wth transfers. Consequently, by Lemma 2, the mechansm constructed n the prevous secton doubly mplements Pareto effcent allocatons. Proposton 3. SN M,h (e). Every Nash equlbrum m of the mechansm defned above s a strong Nash equlbrum, that s, N M,h (e) Proof. Let m be a Nash equlbrum. Snce the mechansm fully mplements constraned prce equlbrum allocatons wth transfers, we know that (X(m ),Y(m )) s a constraned prce equlbrum allocaton wth transfers. Then (X(m ),Y(m )) s Pareto optmal by Lemma 2, and thus the coalton N cannot be mproved upon by any m M. Now for any coalton C wth /= C /= N, choose C such that + / C. Then no strategy played by C can change p(m) and Y(m) snce they are determned by m +. Furthermore, because (X(m ),Y(m )) P(e) and p(m ) X (m C,m C ) p(m ) X (m ), (8) X (m ) s the maxmal consumpton n the budget set of, and thus S cannot mprove upon (X(m ),Y(m )). Q.E.D. Thus, we have showed that N M,h (e) = SN M,h (e) = PE(e) for all e E. The proof of Theorem 2 s completed. Q.E.D. 5. Concludng remarks In ths paper we consdered mplementaton and double mplementaton of Pareto optmal allocatons for general producton economes. We presented a specfc mechansm that uses a fnte-dmenson of message space and fully Nash mplements Pareto optmal allocatons when preferences and productons sets are unknown to the desgner. The mechansm then s slghtly modfed to doubly mplement Pareto correspondence n Nash and strong Nash equlbra. That s, by the mechansm, not only Nash equlbrum allocatons, but also strong Nash equlbrum allocatons concde wth constraned prcng equlbrum allocatons wth transfers. By double mplementaton, the soluton can cover the stuaton where agents n some

11 G. Tan / Journal of Mathematcal Economcs 45 (2009) coaltons may cooperate and n some other coaltons may not. Thus, the desgner does not need to know whch coaltons are permssble and, consequently, t allows for the possblty of agents manpulatng coalton patterns. The mechansms constructed n the paper are well-behaved and have several desred propertes: () they use fntedmensonal message spaces. (2) They are stable n the sense that they are contnuous. A slght change of strateges does not result n a drastc change of the outcome. (3) They are credble n the sense that they are feasble. Every partcpant receve a consumpton bundle n her consumpton set. Every producton plan s n the producton set, and aggregate consumpton do not excess aggregate supply even at non-equlbrum. A mechansm would not be credble f t was not feasble. (4) They are market type mechansms. The prce and quantty are components of the message spaces. (5) They are more realstc and relatvely more nformatonally effcent. The mechansm that mplements Pareto effcent allocatons n Nash equlbrum only requres two agents, one as the manager and the other as the owners of frms, to announce prce vectors and producton plans of frms. Thus, ths mechansm not only uses smaller message spaces, but also they are more realstc snce the others that are not nvolved n or related to producton are not requred to announce producton plans. (6) They work not only for three or more agents, but also for two-agent economes. Thus they are unfed mechansms that are rrespectve of the number of agents. Although ths paper only consders mplementaton of Pareto effcent allocatons for prvate goods economes, the technques developed n the paper can be used and the mechansms can be modfed to consder mplementaton of Pareto effcent allocatons for publc goods economes. Acknowledgements I wsh to thank an anonymous referee for helpful comments and suggestons. Fnancal support from the Natonal Natural Scence Foundaton of Chna (NSFC ) and the Prvate Enterprse Research Center at Texas A&M Unversty as well as from Cheung Kong Scholars Program at the Mnstry of Educaton of Chna s gratefully acknowledged. References Anderln, L., Sconole, P., Effcent provson of publc goods wth endogenous redstrbuton. Revew of Economc Desgn 8, Debreu, G., 952. A socal equlbrum exstence theorem. In: Proceedngs of the Natonal Academy of Scences of the USA, 38, pp Duggan, J., Nash Implementaton wth a Prvate Good. Economc Theory 2, 7 3. Dutta, B., Sen, A., Vohra, R., 995. Nash mplementaton through elementary mechansms n economc envronments. Economc Desgn, Groves, T., Ledyard, J., 977. Optmal allocaton of publc goods: a soluton to the free rder problem. Econometrca 45 (4), Hong, L., 995. Nash mplementaton n producton economy. Economc Theory 5, Hurwcz, L., 979. Outcome functon yeldng Walrasan and Lndahl allocatons at Nash equlbrum pont. Revew of Economc Studes 46, Hurwcz, L., Maskn, E., Postlewate, A., 995. Feasble Nash mplementaton of socal choce rules when the desgner does not know endowments or producton sets. In: Ledyard, J.O. (Ed.), The Economcs of Informatonal Decentralzaton: Complexty, Effcency, and Stablty (Essays n Honor of Stanley Reter). Kluwer Academc Publshers. Mas-Colell, A., Whnston, M., Green, J., 995. Mcroeconomc Theory. Oxford Unversty Press. Osann, H., 997/998. Nash-mplementaton of the weak Pareto choce rule for ndecompposable envronments. Revew of Economc Desgn 3, Peleg, B., 996a. A contnuous double mplementaton of the constraned Walrasan equlbrum. Economc Desgn 2, Peleg, B., 996b. Double mplementaton of the Lndahl equlbrum by a contnuous mechansm. Economc Desgn 2, Postlewate, A., Wettsten, D., 989. Contnuous and feasble mplementaton. Revew of Economc Studes 56, Schmedler, D., 980. Walrasan analyss va strategc outcome functons. Econometrca 48, Sajo, T., Tatamtan, Y., Yamato, T., 996. Toward natural mplementaton. Internatonal Economc Revew 37, Suh, S., 995. A mechansm mplementng the proportonal soluton. Economc Desgn, Suh, S., 997. Double mplementaton n Nash and strong Nash equlbra. Socal Choce and Welfare 4, Tan, G., 989. Implementaton of the Lndahl correspondence by a sngle-valued, feasble, and contnuous mechansm. Revew of Economc Studes 56, Tan, G., 994. Implementaton of lnear cost share equlbrum allocatons. Journal of Economc Theory 64, Tan, G., 996. Contnuous and feasble mplementaton of ratonal expectaton Lndahl allocatons. Games and Economc Behavor 6, Tan, G., 999. Double mplementaton n economes wth producton technologes unknown to the desgner. Economc Theory 3, Tan, G., A soluton to the problem of consumpton externaltes. Journal of Mathematcal Economcs 39, Tan, G., Implementaton n economes wth non-convex producton technologes unknown to the desgner, Games and Economc Behavor, n press. Yoshhara, N., 999. Natural and double mplementaton of publc ownershp solutons n dfferentable producton economes. Revew of Economc Desgn 4, 27 5.