Fundamental impossibility theorems on voluntary participation in the provision of non-excludable public goods

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1 Rev. Econ. Desgn 00 4:5 73 DOI 0.007/s ORIGINAL PAPER Fundamental mpossblty theorems on voluntary partcpaton n the provson of non-excludable publc goods Tatsuyosh Sajo Takehko Yamato Receved: 9 October 008 / Accepted: November 009 / Publshed onlne: December 009 The Authors 009. Ths artcle s publshed wth open access at Sprngerlnk.com Abstract Groves and Ledyard Econometrca 45: , 977 constructed a mechansm attanng Pareto effcent allocatons n the presence of publc goods. After ths path-breakng paper, many mechansms have been proposed to attan desrable allocatons wth publc goods. Thus, economsts have thought that the free-rder problem s solved, n theory. Our vew to ths problem s not so optmstc. Rather, we propose fundamental mpossblty theorems wth publc goods. In the prevous mechansm desgn, t was mplctly assumed that every agent must partcpate n the mechansm that the desgner provdes. Ths approach neglects one of the basc features of publc goods: non-excludablty. We explctly ncorporate non-excludablty and then show that t s mpossble to construct a mechansm n whch every agent has an ncentve to partcpate. We thank an anonymous referee and Takesh Suzuk for useful comments. Research was partally supported by the Grant n Ad for Scentfc Research of the Mnstry of Educaton, Culture, Sports, Scence and Technology n Japan. Yamato thanks the Dvson of the Humantes and Socal Scences at the Calforna Insttute of Technology for ther hosptalty durng the perod when ths draft was wrtten. T. Sajo B Insttute of Socal and Economc Research, Osaka Unversty, Ibarak, Osaka , Japan e-mal: sajo@ser.osaka-u.ac.jp T. Sajo CASSEL, UCLA, Los Angeles, CA , USA T. Yamato Department of Socal Engneerng, Graduate School of Decson Scence and Technology, Tokyo Insttute of Technology, -- Ookayama, Meguro-ku, Tokyo 5-855, Japan e-mal: yamato@valdes.ttech.ac.jp

2 5 T. Sajo, T. Yamato Keywords Impossblty theorems Voluntary partcpaton Non-excludable publc goods Lndahl equlbrum Voluntary contrbuton mechansm Olson s conjecture JEL Classfcaton C7 D7 D78 H4 Introducton Hurwcz 97, n hs path-breakng paper, showed that Walrasan mechansm has an ncentve problem although many researchers at that tme consdered that t solves agents ncentve problem. That s, some agents have ncentve not to reveal true excess demand functons. Later, Ledyard and Roberts 974 showed the same problem n publc good economes. In other words, t s mpossble to desgn a mechansm that satsfes ncentve compatblty where each agent reveals her true utlty functon or excess demand functon as her domnant strategy n prvate or publc good economes. On the other hand, Groves and Ledyard 977 desgned a Nash mplementable mechansm to acheve Pareto effcency n the presence of publc goods. Rght after ths dscovery, Hurwcz 979a and Walker 98 desgned Nash mplementable mechansms for Lndahl allocatons. Hereafter, many mechansms havng nce features have been proposed. Thus, economsts have thought that the free-rder problem s solved, n theory. Our vew to ths problem s not so optmstc. Rather, we propose fundamental mpossblty theorems wth publc goods. In the prevous mechansm desgn, t was mplctly assumed that every agent must partcpate n the mechansm that the desgner provdes. Ths approach neglects one of the basc features of publc goods: non-excludablty. In Sajo and Yamato 999, we explctly ncorporated non-excludablty n mechansm desgn by examnng a two-stage game on voluntary partcpaton n a mechansm for provdng a non-excludable publc good: n the frst stage, each agent smultaneously decdes whether or not to partcpate n the mechansm; and n the second stage, after knowng the other agents partcpaton decsons, the agents who chose partcpaton n the frst stage play the mechansm. We fully characterzed the equlbrum set of partcpants n the two-stage game for any second-stage mechansm satsfyng symmetry, feasblty, and Pareto effcency only for partcpants n symmetrc Cobb-Douglas economes. In partcular, we found that there exst economes for whch full partcpaton of all agents s not an equlbrum, mplyng that t s mpossble to desgn reasonable mechansms n whch all agents always have partcpaton ncentves. The same negatve result on voluntary partcpaton holds for the voluntary contrbuton mechansm. In Sajo and Yamato 999, however, we made a restrctve assumpton that each agent has the same Cobb-Douglas utlty functon as well as the same endowment. Our soluton concept for the two-stage game assumes sequental ratonalty, although any equlbrum concept wth complete nformaton s allowed for the second stage. For example, f Nash equlbrum s the equlbrum concept for the second stage, then we examne subgame perfect Nash equlbra of the two-stage game.

3 Fundamental mpossblty theorems on voluntary partcpaton 53 In ths paper, we show the above negatve results on partcpaton ncentves are robust n the sense that they occur n more general envronments. We formulate full partcpaton of all agents as an axom on a mechansm called the voluntary partcpaton condton: each agent always prefers partcpaton to non-partcpaton n the mechansm when all other agents partcpate n t. Frst, we consder any mechansm mplementng the Lndahl correspondence, called Lndhal mechansm. We show that any Lndahl mechansm fals to satsfy the voluntary partcpaton condton n asymmetrc Cobb-Douglas utlty as well as quas-lnear utlty economes n whch agents may have dfferent utlty functons and endowments. Moreover, we dentfy the classes of Cobb-Douglas and quas-lnear utlty economes for whch the voluntary partcpaton condton s satsfed. These classes become smaller and eventually vansh as the number of agents become larger, whch can be nterpreted as a support for Olson 965 conjecture: a publc good s less lkely provded as the sze of a group grows large. The Lndahl correspondence satsfes ndvdual ratonalty and Pareto effcency, and moreover t s the only Nash mplementable socal choce correspondence that s Pareto effcent and ndvdually ratonal under sutable condtons Hurwcz 979b. However, nether Pareto effcency nor ndvdual ratonalty s necessary to obtan a negatve result on voluntary partcpaton. We demonstrate that the voluntary contrbuton mechansm, whch does not satsfy Pareto effcency under Nash equlbrum, fals to meet the voluntary partcpaton condton n asymmetrc Cobb-Douglas and quas-lnear utlty economes, even though the name of the mechansm contans the term voluntary. Moreover, we nvestgate a large class of mechansms that are necessarly nether ndvdually ratonal nor Pareto effcent. We establsh mpossblty results on voluntary partcpaton n mechansms meetng mld condtons. For the case of two agents, there s no feasble mechansm satsfyng the voluntary partcpaton condton and the Robnson Crusoe condton, whch requres that f only one agent partcpates n the mechansm, then she choose an outcome that s best for her, on a doman of economes that nclude a Cobb-Douglas utlty economy or a quas-lnear utlty economy satsfyng a certan condton. Furthermore, for the case of more than two agents, there s no feasble mechansm satsfyng the voluntary partcpaton condton and contrbuton monotoncty, whch means that f n agents chose partcpaton and a new agent becomes a partcpant addtonally, then the sum of contrbutons to the publc good by the prevous n partcpants n the mechansm does not rse, on a suffcently large doman of economes. Palfrey and Rosenthal 984, Mouln 986, and Dxt and Olson 000 studed the partcpaton ncentve problem n the provson of a publc good. In those papers, however, the publc good s dscrete, whle t s contnuous n our model. Moreover, the mechansms studed there are dfferent from ours. Palfrey and Rosenthal 984 examned voluntary contrbuton or provson pont mechansms wth and wthout a refund to decde whether to produce a dscrete publc project or not. Contrbu- More specfcally, Hurwcz 979b proved that f all Nash equlbrum allocatons are Pareto effcent and ndvdually ratonal, then every Lndahl allocaton s a Nash allocaton and every nteror Nash allocaton s a Lndahl allocaton under approprate assumptons on envronments and mechansms.

4 54 T. Sajo, T. Yamato tons are bnary and makng a fxed contrbuton can be nterpreted as partcpaton n a mechansm. They dentfed mxed strategy Nash equlbra of the mechansms. Mouln 986 usedtheno free rde axom, requrng each agent have a partcpaton ncentve n a mechansm, to characterze the pvotal mechansm n economes wth a dscrete publc good and quas-lnear preferences. Dxt and Olson 000 ndependently consdered a two-stage partcpaton game, smlar to that n Sajo and Yamato 999, from the vewpont of the Coase theorem rather than mechansm desgn: n the frst stage, each agent smultaneously decdes whether to partcpate or not, and n the second stage, those who selected partcpaton play a cooperatve game of Coasean barganng wth no costless enforcement of contracts. They examned a bnary publc good model lke Palfrey and Rosenthal 984. In partcular, they found that the effcent equlbrum outcome of the partcpaton game s not robust when ntroducng even very small transacton costs. Ths casts doubt on the valdty of Coasean clams of unversal effcency, whch s smlar to our negatve vew n the desgn of effcent resource allocaton mechansms wth partcpaton decsons. The paper s organzed as follows. In Sect., we explan examples llustratng our basc dea. In Sect. 3, we ntroduce notaton and defntons. We establsh mpossblty results on voluntary partcpaton n Lndahl mechansms n Sect. 4. In Sect. 5, we characterze a condton for whch each agent loses a partcpaton ncentve n any Lndahl mechansm n a replca of a Cobb-Douglas or quas-lnear utlty economy. In Sect. 6, we consder the partcpaton problem on the voluntary contrbuton mechansm and a class of mechansms satsfyng mld condtons. In the fnal secton, we make concludng remarks. Examples Let us consder the followng two-agent economes wth one prvate good x and one pure publc good y. Agent s consumpton bundle s denoted by x, y R + where x R + s the level of prvate good she consumes on her own, and y R + s the level of publc good. Each agent has a Cobb-Douglas utlty functon: u α x, y = α ln x + α ln y, where α 0, and =,. Agent s ntal endowment s gven by ω, 0 for =,, that s, there s no publc good ntally. However, the publc good can be produced from the prvate good by means of a constant return to scale technology, and let y = ω x be the producton functon of the publc good. Consder any mechansm mplementng the Lndahl correspondence for example, see Hurwcz 979a; Walker 98; Hurwcz et al. 984, and Tan 990 for Nash mplementaton, 3 and Moore and Repullo 988 and Varan 994 for subgame perfect mplementaton. Suppose that each agent s able to choose whether she partcpates n the mechansm. Then n order to acheve the desred Lndahl equlbrum allocaton by usng the mechansm, every agent must choose partcpaton. Therefore, we ask a 3 The Lndahl correspondence s not Nash mplementable due to the boundary problem, but the constraned Lndahl correspondence s Nash mplementable.

5 Fundamental mpossblty theorems on voluntary partcpaton 55 crucal queston of whether each agent always has an ncentve to partcpate n the mechansm. Unfortunately, our answer to ths queston s negatve. To see why, let T {, } be the set of agents who partcpate n the mechansm. An equlbrum allocaton of the mechansm when the agents n T partcpate n t s denoted by x T T, yt. 4 If two agents decde to partcpate n the mechansm, then x {,}, x {,}, y {,} should be a Lndahl allocaton of the economy consstng of two agents, snce the mechansm mplements the Lndahl correspondence. 5 It s straghtforward to check that there exsts a unque Lndahl allocaton gven by x {,}, x {,}, y {,} = α ω,α ω, =, α ω. Now suppose that some agent does not partcpate n the mechansm, whle the other agent j = does,.e., T ={j}. Then x { j} j, y { j} s a unque Lndahl allocaton of the economy consstng of only one agent j. It s easy to see that x { j} j, y { j} = α j ω j, α j ω j. Notce that non-partcpant can enjoy her ntal endowment, ω, as well as the non-excludable publc good produced by agent j =, y { j}.onthe other hand, she s no longer able to affect the decson on the provson of the publc good. Because of ths trade-off, t s not obvous whether or not each agent has an ncentve to partcpate n the mechansm. The followng condton should be satsfed f each agent has such a partcpaton ncentve: u α x {,}, y {,} u α ω, y { j} for, j =,, j =, where u α s any Cobb-Douglas utlty functon. We call condton the voluntary partcpaton condton. 6 We show that no mechansm mplementng the Lndahl correspondence satsfes ths condton. Ths fact can be llustrated by usng Kolm s trangle. See Fg. n whch α,α = 0.5, 0.7 and ω,ω = 0, 0. In ths economy, agent s valuaton of the publc good s hgher than agent s, but agent s poorer than agent. We wll see that nether agent has a partcpaton ncentve. Pont A n Fg. denotes the Lndahl equlbrum allocaton when both agents partcpate n the mechansm: A = x {,}, x {,}, y {,} = 5, 4,. Pont B represents the allocaton when agent does not partcpate n the mechansm, but agent does: B = ω, x {}, y{} = 0, 4, 6. Snce u α x {,}, y {,}.004 < u α ω, y {}.047 for α = 0.5, agent prefers Pont B to Pont A and she does not partcpate n the 4 Here we consder a general defnton of a mechansm whch specfes a strategy set of each partcpant n T and an outcome functon for each T {, }. 5 A mechansm s sad to mplement the Lndahl correspondence f for each set of partcpants T {, } and each economy consstng of the partcpants n T, every equlbrum allocaton s a Lndahl allocaton and every Lndahl allocaton s an equlbrum allocaton. 6 The voluntary partcpaton condton s dfferent from the ndvdually ratonal condton whch requres that u α x {,}, y {,} u α ω, 0 for =,. Snce u α ω, y { j} u αω, 0, the voluntary partcpaton condton s stronger than the ndvdually ratonal condton.

6 56 T. Sajo, T. Yamato Fg. No Lndahl mechansm satsfes the voluntary partcpaton condton when preferences are Cobb-Douglas A 4 5 C B ω 0 ω=0,0,0, A=5,4,, B=0,4,6, C=5, 0, 5 mechansm when agent does. The same thng holds for agent. In Fg., the allocaton when agent does not partcpate n the mechansm, but agent does s represented by Pont C = x {},ω, y {} = 5, 0, 5. Agent prefers Pont C to Pont A: u α x {,}, y {,}.567 < u α ω, y {}.580 for α = 0.7. A smlar negatve result on voluntary partcpaton n any Lndahl mechansm holds wth quas-lnear preferences. Suppose that each agent has a quas-lnear utlty functon: u β x, y = x + β ln y, where β 0,ω =,. It s easy to check that a unque Lndahl allocaton when both agents partcpate n the mechansm s gven by x {,}, x {,}, y {,} = ω β,ω β, =, β and a unque Lndahl allo- caton when only one agent j partcpates n t s x { j} j, y { j} = ω j β j,β j.the followng voluntary partcpaton condton should be satsfed f each agent has a partcpaton ncentve: u β x {,}, y {,} u β ω, y { j} for, j =,, j =, where u β s any quas-lnear utlty functon. We wll see that no Lndahl mechansm satsfes ths condton. Suppose that β,β =, 3 and ω,ω = 3, 4. Then nether agent has a partcpaton ncentve. Pont A n Fg. represents the Lndahl equlbrum allocaton when both agents partcpate: A = x {,}, x {,}, y {,} =,, 5. Pont B stands for the allocaton when agent does not partcpate, but agent does: B = ω, x {}, y{} = 3,, 3. Snce u β x {,}, y {,} 4.9 < u β ω, y {} 5.97 for β =, agent prefers Pont B to Pont A, n other words, she has no partcpaton ncentve when agent partcpates. The same thng holds for agent. In Fg., the allocaton when agent does not partcpate, but agent does s denoted by Pont C = x {},ω, y {} =, 4,. Agent prefers Pont C to Pont A: u β x {,}, y {,} 5.88 < u β ω, y {} for β = 3.

7 Fundamental mpossblty theorems on voluntary partcpaton 57 Fg. No Lndahl mechansm satsfes the voluntary partcpaton condton when preferences are quas-lnear 3 Notaton and defntons In the prevous secton, we see that any Lndahl mechansm fals to satsfy the voluntary partcpaton condton n economes wth two agents by lookng at certan values of Cobb-Douglas and quas-lnear preference and endowment parameters. We wll show smlar negatve results hold for any number of agents. Also, we wll dentfy classes of preference and endowment parameters for whch agents lose partcpaton ncentves. In partcular, these classes become larger as the number of agents ncreases. Frst of all, we ntroduce notaton and defntons. As n Sect., there are one prvate good x and one publc good y wth a constant return to scale technology. Let N ={,,...,n} be the set of agents, wth generc element. Each agent s preference relaton admts a numercal representaton u :R + Rwhch s contnuously dfferentable, strctly quas-concave, and strctly monotonc. Let U be the class of utlty functons admssble for agent and U N U. Agent s ntal endowment s denoted by ω, 0. There s no publc good ntally. Let be the class of prvate good endowments admssble for agent and N. An economy s a lst of utlty functons and endowments of all agents, e = u,ω= u N,ω N and the class of admssble economes s denoted by E = U. Let an economy e = u,ω E be gven. Also, let PN be the collecton of all no-empty subsets of N. GvenT PN, e T = u T,ω T = u T,ω T s a sub-economy consstng of agents n T.Afeasble allocaton for e T s a lst x T, y #T + x T, y R + such that T ω x = y. The set of feasble allocatons for e T s denoted by Ae T. A mechansm s a functon Ɣ that assocates wth each T PN a par ƔT = S T, g T, where S T = T S T and g T : S T R #T +.HereS T s the strategy space of agent T and g T s the outcome functon when the agents n T play the mechansm. Gveng T s = x T, y, letg T s x, y for T and gy T s = y. The allocaton for the entre economy when the agents n T PN partcpate n the mechansm s provded by x T,ω N T, y, that s, every non-partcpant enjoys her endowment ω and the publc good y produced by the partcpants. Notce that

8 58 T. Sajo, T. Yamato we assume nether ndvdual feasblty g T #T + s R + for all s S T nor balancedness g T s Ae T for all s S T. Our negatve results hold wthout requrng these condtons. An equlbrum correspondence s a correspondence µ whch assocates wth each mechansm Ɣ, each economy e E, and each set of agents T PN, asetof strategy profles µ Ɣ e T S T, where S T, g T = ƔT. Thesetofµ-equlbrum allocatons of Ɣ for e T s denoted by g T µ Ɣ e T {x T, y R #T + there exsts s S T such that s µ Ɣ e T and g T s = x T, y}, where S T, g T = ƔT. In ths paper, we consder an arbtrary equlbrum correspondence wth complete nformaton among agents. Examples of equlbrum correspondences nclude the Nash equlbrum correspondence, the strong Nash equlbrum correspondence, and any refnement of the Nash equlbrum correspondence such as the perfect and proper equlbrum correspondences. Gven an economy e = u,ω E and a set of agents T PN, a feasble allocaton x T, y Ae T s a Lndahl allocaton for e T f there s a prce vector p R #T + such that for each agent T, x + p y = ω and u x, y u x, y for any x, y R + such that x + p y ω.letle T be the set of Lndahl allocatons for e T. Let an equlbrum correspondence µ be gven. A Lndahl mechansm under µ s a mechansm such that for each economy e = u,ω E and each set of agents T PN, g T µ Ɣ e T = Le T. A Lndahl mechansm under µ s a mechansm mplementng the Lndahl correspondence n µ-equlbrum, that s, for each set of partcpants T PN and each economy consstng of the partcpants n T, every µ-equlbrum allocaton s a Lndahl allocaton and every Lndahl allocaton s a µ-equlbrum allocaton. The above defnton of a mechansm mplementng the Lndahl correspondence s a generalzaton of the usual one, n whch all agents are supposed to partcpate, to the case n whch voluntary partcpaton s allowed. 4 Impossblty results on voluntary partcpaton n Lndahl mechansms We ntroduce the followng condton on voluntary partcpaton n mechansms. Let an equlbrum correspondence µ be gven. Defnton The mechansm Ɣ satsfes voluntary partcpaton for aneconomy e = u,ωunderµ f for all x N, y N g N µ Ɣ e N and all N, u x N, y N u ω, y N {} mn, where y N {} mn arg mn y N {} g N {} y µ Ɣe N {} u ω, y N {}. Also, the mechansm Ɣ satsfes voluntary partcpaton on the class of economes E under µ f t satsfes voluntary partcpaton for all economes e = u,ω E under µ.

9 Fundamental mpossblty theorems on voluntary partcpaton 59 Snce there s one publc good and preferences satsfy monotoncty, y N {} mn s the mnmum equlbrum level of publc good when all agents except partcpate n the mechansm. Consder an agent who decdes not to partcpate n the mechansm. Then she can enjoy the non-excludable publc good produced by the other agents wthout provdng any prvate good, whle she cannot affect the decson on the provson of the publc good. Voluntary partcpaton requres that no agent can beneft from such a free-rdng acton. Note that when an agent chooses non-partcpaton, she has a pessmstc vew on the outcome of her acton: an equlbrum outcome that s most unfavorable for her wll occur. Mouln 986 proposed a smlar condton, called the No Free Rde axom, when publc goods are dscrete and costless, and preferences are quas-lnear. We wll show any Lndahl mechansm fals to satsfy the voluntary partcpaton condton under mld condtons. Frst of all, consder the class of Cobb-Douglas utlty economes: E CD {u N,ω N N, u x, y = u α x, y = α ln x + α ln y,α 0,, ω R ++ }. Such an economy s specfed by a lst of Cobb-Douglas preference parementers and endowments of n agents, α, ω α,...,α n, ω,...,ω n such that α 0, and ω 0, ω] for all. Here ω s the upper bound of each endowment. Wthout loss of generalty, we assume that ω =. Hence, the set of economes s represented by the product of ntervals E CD = 0, n 0, ] n endowed wth Lebesgue measure λ. Let αn 0, be a unque value satsfyng α ln α/α = lnn/n.we have the followng negatve result on voluntary partcpaton regardng Cobb-Douglas utlty economes. Theorem Let α, ω E CD be any Cobb-Douglas utlty economy n whch α > αn for agent such that α ω α j ω j for all j =. Then any Lndahl mechansm fals to satsfy voluntary partcpaton for α, ω under µ. Proof Fx any α, ω E CD. For each T PN, t s easy to check there exsts a unque Lndahl equlbrum allocaton for α,ω T, whch concdes wth a unque µ-equlbrum allocaton of the mechansm when agents n T partcpate n t, gven by x T, y T = α ω, j T α jω j. Therefore, the dfference between agent s utlty level when all agents partcpate n the mechansm and that when all agents except partcpate n t s gven by u α, ω u α x N, y N u α ω, y N {} ln α ω + α j ω j j = = α ln α + α ln α j ω j. 4. j = Take N such that α ω α j ω j for all j =. Wthout loss of generalty, let =. We prove that u α, ω < 0fα >αn, so that the voluntary

10 60 T. Sajo, T. Yamato partcpaton condton s volated. Snce n α ω j = α jω j, u α, ω α ln α + α ln /n + α j ω j j = ln α j ω j j = = α [lnn/n α ln α /α ]. Let hα α ln α/α. Frst, we show that hα s strctly ncreasng n α 0,. Note that dhα/dα = α ln α/α. Snce α > 0, t remans to prove that Bα α ln α>0. It s easy to check that dbα/dα <0 f α 0,, dbα/dα = 0fα =, and B = 0. Therefore, Bα > 0for α 0,. Second, by L Hôptal s rule, lm α 0 hα = lm α 0 [ln α/{ /α}] = lm α 0 [α] =0 and lm α hα = lm α [α] =. Snce dhα/dα > 0 and lnn/n < forn, t follows that there exsts a unque αn 0, such that hαn = lnn/n and hα > lnn/n for α>αn, mplyng that u α, ω < 0fα >αn. Each agent contrbutes α ω of the prvate good to the producton of the publc good n a Lndahl allocaton for a Cobb-Doulas utlty economy see the proof of Theorem. Consder agent such that α ω α j ω j for all j =, that s, agent s contrbuton to the publc good s the mnmum and hence the reducton n the publc good provson level by agent s non-partcpaton s the lowest among agents. Theorem says that f ths agent s value of the prvate good relatve to the publc good s large enough α >αn, then she has a non-partcpaton ncentve and the voluntary partcpaton condton s volated. By Theorem, f the class of admssble economes E contans a Cobb-Douglas utlty economy satsfyng the above condton, then any Lndahl mechansm fals to satsfy voluntary partcpaton on E. Table llustrates how the value of αn depends on the number of agents, n. Snce αn s strctly decreasng n n and lm n αn = 0, the measure of the set of Cobb- Douglas utlty economes for whch any Lndahl mechansm satsfes the voluntary partcpaton condton becomes smaller and converges to zero, as the number of agents grows large. 7 As Olson 965 asserted, a publc good would be less lkely provded as the sze of an economy becomes larger. The above result verfes ths conjecture from the perspectve of a partcpaton ncentve for any Lndahl mechansm n asymmetrc Cobb-Douglas economes. { A smlar negatve result holds for quas-lnear utlty economes. Let E QL } u N,ω N N, u x, y = u β x, y = x + β ln y,β 0,ω be 7 The condton n Theorem for whch the voluntary partcpaton condton s volated becomes dentcal to that dscussed n Sajo and Yamato 999 f each agent has the same Cobb-Douglas utlty functon and endowment.

11 Fundamental mpossblty theorems on voluntary partcpaton 6 Table The value of αn n αn the class of quas-lnear utlty economes. Here we assume that ω >β for each to ensure an nteror soluton. Such an economy s specfed by a lst of quas-lnear preference parementers and endowments of n agents, β, ω β,...,β n, ω,...,ω n such that β 0,ω and ω 0, ] for all. In ths case, the set of economes s gven by E QL {β, ω 0, n 0, ] n : β <ω, }. Theorem Let β, ω E QL be an arbtrary quas-lnear utlty economy. Any Lndahl mechansm fals to satsfy voluntary partcpaton for β, ω under µ. Proof Fx any β, ω E QL. For each T PN, t s easy to check there exsts a unque Lndahl equlbrum allocaton at β,ω T, whch concdes wth a unque µ-equlbrum allocaton of the mechansm when agents n T partcpate n t, gven by x T, y T = ω β, j T β j. Therefore, the dfference between agent s utlty level when all agents partcpate n the mechansm and that when all agents except partcpate n t s gven by u β u β x N, y N u β ω, y N {} = β + ln β + β j ln β j. 4. j = j = We wll show that there exst some such that u β < 0. Take N such that β β j for all j =. Wthout loss of generalty, let =. Snce n β j = β j, u β β + ln /n + β j ln β j j = j = = β { + ln n lnn }. Snce the functon ln n lnn s decreasng n n and ln ln 0.693<, t follows from the above nequalty that u β < 0forn. Therefore, the voluntary partcpaton condton s volated. By Theorem, f the class of admssble economes E contans a quas-lnear utlty economy β, ω E QL, then any Lndahl mechansm fals to satsfy voluntary partcpaton on E. Notce that Theorem holds wthout makng any condton on parameters of quas-lnear utlty functons, on endowments, nor on the number of agents, except that the endowment of each agent s large enough to guarantee an nteror soluton.e., ω >β. In ths sense, the result for quas-lnear utlty economes s stronger than Theorem for Cobb-Douglas utlty economes.

12 6 T. Sajo, T. Yamato 5 A partcpaton ncentve of each agent n a replca economy In the prevous secton, we show that there exst Cobb-Douglas and quas-lnear utlty economes for whch some agent fals to have a partcpaton ncentve n any Lndahl mechansm, so that the mechansm does not satsfy the voluntary partcpaton condton. In ths secton, we check whether or not each agent has a partcpaton ncentve n any Lndahl mechansm n a replca of any gven Cobb-Douglas or quas-lnear utlty economy. We wll fnd that n a suffcently large replca of any economy of n-type agents, every type of agent has no partcpaton ncentve. Let µ be any equlbrum correspondence, Ɣ be any Lndahl mechansm under µ, and α, ω E CD be any Cobb-Douglas utlty economy of n-type agents. Consder the k-replca of ths economy n whch there are k agents of type α,ω for each N. Denote the set of all kn agents n the k-replca economy by kn.let x kn, y kn be the consumpton bundle each agent of type α,ω receves at the unque Lndahl allocaton for kn and y kn {} be the publc good level at the unque Lndahl allocaton for kn {}. Also, let u α, ω, k u α x kn, y kn u α ω, y kn {} be the dfference between the utlty level of each agent of type α,ω when all agents partcpate n the mechansm Ɣ and that when all agents except her, that s, k agents of type α,ω as well as kn agents of other types partcpate n Ɣ n the k-replca economy. If the mechansm Ɣ satsfes the voluntary partcpaton condton for the k-replca economy, then for any α, ω E CD and for any N,we must have u α, ω, k 0. However, we have the followng negatve result. Gven any n, any α, ω E CD, and any N, letk α, ω be the largest nteger less than or equal to e hα α ω / {e hα } n j= α j ω j, where e s the base of the natural logarthm and hα α ln α /α. Theorem 3 Consder an arbtrary Lndahl mechansm Ɣ under µ. Gven any n, any α, ω E CD, and any N, u α, ω, k 0 for any postve nteger k k α, ω; and u α, ω, k <0 for any postve nteger k > k α, ω. Proof It s not hard to check that x kn, y kn = α ω, k j N α jω j and y kn {} = k α ω + k j = α jω j. Therefore, u α, ω, k = α ln α + α ln k α ω + k α j ω j j = ln k α ω + k α j ω j. j =

13 Fundamental mpossblty theorems on voluntary partcpaton Type does not partcpate. u 0 Nether partcpates Type does not partcpate. Nether partcpates. Type does not partcpate Nether partcpates u 0 Both partcpate. u 0 Type does not partcpate. u Type does not partcpate a The k = -replca Both partcpate. b The k = -replca Both Type does not partcpate. partcpate. c The k = 5-replca. Fg. 3 Partcpaton ncentves n two-agent Cobb-Dogulas utlty economes wth symmetrc endowments Let k α, ω be a value satsfyng the equaton u α, ω, k α, ω = 0. Ths equaton can rewrtten as [ k ln α, ω α ω + k α, ω j = α ] jω j k α, ω α ω + k α, ω j = α jω j } = hα = α ln α /α. { e Thus, k α, ω = ehα α ω / hα } n j= α j ω j. Notce that u α, ω, k s strctly decreasng n k: k u α ω α, ω, k = k [k α ω + k ] < 0. j = α jω j Therefore, u α, ω, k < = > 0 f and only f k > = < k α, ω. Ths mples the desred result. Theorem 3 mples that for any Cobb-Douglas economy α, ω E CD, no agent has a partcpaton ncentve n a suffcently large replca of the economy. Fgure 3 llustrates the result n Theorem 3 when there are n = agents, each agent has the same endowment, ω = ω, and the number of replcaton s k =,, and 5. As k ncreases, the regon of preference parameters α,α for whch u < 0 and u < 0 becomes larger and t converges to the entre space 0, 0,. In other words, the measure of the set of Cobb-Dogulas utlty economes for whch at least one of two agents has a partcpaton ncentve vanshes as the replcaton sze grows large. That s, the partcpaton ncentve dsappears n a large economy. Ths could be nterpreted as another support for Olson 965 conjecture that a publc good would be less lkely provded as the number of agents ncreases. A smlar negatve result holds for quas-lnear preferences. Gven a quas-lnear utlty economy β, ω E QL consstng of n-type agents, we consder the k-replca of ths economy n whch there are k agents of type β,ω for each N. For each agent of type β,ω,let

14 64 T. Sajo, T. Yamato u β, ω, k u β x kn, y kn u β ω, y kn {} where x kn, y kn s the consumpton bundle each agent of type β,ω receves at the unque Lndahl allocaton when all agents partcpate n a Lndahl mechansm and y kn {} s the publc good level produced at the unque Lndahl allocaton when all agents except one of agents of type β,ω partcpate n the mechansm n the k-replca economy. We have the followng result: Theorem 4 Consder an arbtrary Lndahl mechansm Γ under µ. Gven any n, any β, ω E QL, and any N, u β, ω, = > > 0 f and only f β =< e < j = β j, where e s the base of the natural logarthm; and u β, ω, k <0 for any postve nteger k. Proof It s not dffcult to see that x kn, y kn = ω β, k j N β j and y kn {} = k β + k j = β j. Hence, u β, ω, k = β + ln kβ + k j = β j ln k β + k j = β j. Let k β, ω be a value satsfyng the equaton u β,ω,k β, ω [{ = 0. Ths equaton can be rewrtten as ln k β, ωβ + k β, ω } j = β j / { k β, ω β + k β, ω }] j = β j =. Therefore, k β, ω = eβ /{e n j= β j }. Also, note that u β, ω, k s strctly decreasng n k: k u β, ω, k = [ k k β + k ] < 0. j = β j β Hence, u β, ω, k > = < 0 f and only f k < = > k β, ω.letk =. Then u β, ω, > = < 0 > f and only f β =< e j = β j. On the other hand, f k, then k > k β, ω = eβ / {e } n j= β j, so that u β, ω, k <0. Fgure 4 llustrates the result n Theorem 4 for the case of two agents, n = and no replcaton, k =. In ths case, t follows from Theorem 4 that u β, ω, > = < 0 < f and only f β => β /e and u β, ω, = > > 0 f and only f β =< e β. < Notce there s no possblty for whch u β, ω, >0 holds for all {, }, that s, both agents have partcpaton ncentves. In other words, the measure of the set of economes for whch the voluntary partcpaton condton s satsfed s zero. No replcaton of an economy s necessary to obtan ths negatve result. Moreover, for the just k = -replca of any economy β, ω E QL, u β, ω, <0 holds for any {, }, that s, no agent has a partcpaton ncentve. Ths negatve concluson

15 Fundamental mpossblty theorems on voluntary partcpaton 65 Fg. 4 Partcpaton ncentves n two-agent quas-lnear economes: the k = -replca case 0.8 Type partcpates, but not Type. u 0 Nether partcpates. 0.6 u 0 u 0 u u 0 u 0 0. Type partcpates, but not Type holds for at least two-replca of an arbtrary quas-lnear utlty economy. In ths sense, the result for quas-lnear utlty economes s stronger than that for Cobb-Douglas utlty economes. 6 Impossblty results on voluntary partcpaton n general mechansms So far we have lmted our attentons to Lndahl mechansms, whch satsfy ndvdual ratonalty and Pareto effcency, n Cobb-Douglas and quas-lnear utlty economes. However, nether Pareto effcency nor ndvdual ratonalty s necessary to obtan a negatve result on voluntary partcpaton. In ths secton, we consder a large class of mechansms that are necessarly nether ndvdually ratonal nor Pareto effcent, ncludng the voluntary contrbuton mechansm. We establsh mpossblty results that no mechansms satsfy the voluntary partcpaton condton on a suffcently large doman of economes. 6. The voluntary contrbuton mechansm Frst, let us study the voluntary contrbuton mechansm that does not satsfy Pareto effcency when the equlbrum concept s Nash equlbrum. We wll fnd that ths mechansm fals to meet the voluntary partcpaton condton, even though the name of the mechansm contans the term voluntary. Defnton The voluntary contrbuton mechansm s a mechansm such that for all T PN and T, S T =[0,ω ] and g T s = ω s, T s for s S T. The above defnton of the voluntary contrbuton mechansm s a generalzaton of the usual one, n whch all agents are supposed to partcpate, to the case n whch voluntary partcpaton s allowed. When the equlbrum concept s Nash equlbrum,

16 66 T. Sajo, T. Yamato Fg. 5 The proof of Lemma y y N A N-{} y mn B u 0 N x x each agent selects her contrbuton out of her endowment to the provson of the publc good, s, to maxmze her utlty u ω s, j T s j, gven contrbutons of the other agents n T,s j j T {} n the voluntary contrbuton mechansm. We begn by provng the followng result that s smple, but useful below. Let an equlbrum correspondence µ be gven. Lemma Suppose that a mechansm Ɣ and an economy u,ωsatsfy the followng condton: for some x N, y N g N µ Ɣ e N and some N, y N y N {} mn ω x N MRS ω, y N {} mn, 6. where M RS ω, y N {} mn u ω,y N {} mn x / u ω,y N {} mn y s agent s margnal rate of substtuton at ω, y N {} mn. Then the mechansm Ɣ fals to satsfy voluntary partcpaton for the economy u,ωunder µ. Proof The basc dea behnd the proof s llustrated n Fg. 5 n whch the horzontal axs denotes agent s consumpton level of prvate good x, and the vertcal axs stands for the publc good level, y. The above nequalty 6. says that the slope of the lne gong through Pont A = ω, y N {} mn and Pont B = ω, y N {} mn s smaller than or equal to the slope of the tangent to agent s ndfference curve, that s, the margnal rate of substtuton at B = ω, y N {} mn. By strct quas-concavty of u, agent strctly prefers Pont B to Pont A, so that she fals to have the partcpaton ncentve n ths economy. The usefulness of Lemma s that t s applcable to any mechansm and any utlty functon form. Frs of all, we apply Lemma to the voluntary contrbuton mechansm and any Cobb-Douglas utlty economy. In what follows, we assume that for all N,

17 Fundamental mpossblty theorems on voluntary partcpaton 67 ω > α j = ω j { α + } 6. j = [α j/ α j ] to ensure an nteror equlbrum. 8 We have the followng negatve result: Theorem 5 Let α, ω E CD be any Cobb-Douglas utlty economy n whch α /n for agent such that ω ω j for all j =.Then the voluntary contrbuton mechansm fals to satsfy voluntary partcpaton for α, ω under the Nash equlbrum correspondence. The proof of Theorem 5 s n the appendx. Theorem 5 says that f the agent whose endowment s the smallest has a suffcently large value of the prvate good relatve to the publc good.e., α /n, then she loses her partcpaton ncentve n the voluntary contrbuton mechansm. Notce that the regon of α for whch the voluntary partcpaton condton s volated that s, α /n expands as the number of agents n grows larger. Ths result s nterpretable as another support of Olson s conjecture that a publc good would be less lkely provded as the sze of an economy becomes larger. A smlar negatve result holds for any quas-lnear utlty economy. Gven a quaslnear utlty economy β, ω E QL and T PN, let β T max max T β be the maxmal value of β among the agents n T and M T { T β = β T max} be the set of agents who have the maxmal value β T max.if#m T, then there are multple Nash equlbrum allocatons n whch the publc good level and the sum of contrbutons by the agents belongng to M T are equal to β T max. Theorem 6 Let β, ω E QL be any quas-lnear utlty economy n whch ether #M N, or M N ={} and β < e βmax N {}, where e s the base of the natural logarthm. Then the voluntary contrbuton mechansm fals to satsfy the voluntary partcpaton condton for β, ω under the Nash equlbrum correspondence. The proof of Theorem 6 s n the appendx. Theorem 6 means the followng. Frst, when #M N, at least one agent n #M N, say agent, makes a postve contrbuton at each of Nash equlbra. If agent chooses non-partcpaton, then her contrbuton becomes zero, whle the publc good level s unchanged and provded by the other agents n M N, so that agent becomes better off. 9 Second, when M N ={}, there s a unque equlbrum allocaton n whch the publc good level s β and only agent contrbutes β to the publc good. By decdng not to partcpate n the mechansm, agent does not have to make any contrbuton, whereas the publc good level becomes 8 It s easy to check that each agent s contrbuton to the publc good n the voluntary contrbuton mechansm for a Cobb-Doulas utlty economy α, ω E CD when all agents partcpate s provded by ω x N = + j = [α j / α j ] + α j = n ω l= [α l / α l ] ω j α {+ j = [α, N, j / α j ]} whch s postve by 6., ncreasng n ω, and decreasng n α. 9 Notce that the mechansm does not satsfy the voluntary partcpaton condton f n at least one of equlbrum allocatons, some agent loses a partcpaton ncentve. However, we have a stronger result: at each of the equlbrum allocatons, there s some agent who wll not partcpate n the mechansm. See the proof of Theorem 6.

18 68 T. Sajo, T. Yamato β N {} max. If ths new publc good level s suffcently large.e., β < e βmax N {}, then agent has a non-partcpaton ncentve. By Theorem 5 resp. 6, f the class of admssble economes E contans a Cobb- Douglas resp. quas-lnear utlty economy satsfyng the condton n the theorem, then the voluntary contrbuton mechansm fals to satsfy the voluntary partcpaton condton on E. 6. General mechansms In ths secton we wll consder a large class of mechansms and show mpossblty results on voluntary partcpaton n mechansms n the class. Frst, we nvestgate the case of two agents. We mpose the followng mld condtons on a mechansm. Defnton 3 The mechansm Ɣ satsfes non-emptness for an economy e under µ f for all T PN, g T µ Ɣ e T = Ø. Defnton 4 The mechansm Ɣ satsfes feasblty for an economy e under µ f for all T PN, g T µ Ɣ e T A T. Non-emptness says that there should always exst an equlbrum. Feasblty demands that every equlbrum allocaton of the mechansm be feasble. Note that we requre feasblty only at equlbrum, but not out of equlbrum. Moreover, a feasble mechansm does not necessarly satsfy ndvdual feasblty.e., for all T PN and all s S T, g T #T + s R + nor balancedness.e., for all T PN and all s S T, g T s A T. Defnton 5 The mechansm Ɣ satsfes the Robnson Crusoe condton for an economy e = u,w under µ f for all N, f x {}, y {} g {} µ Ɣ e, then x {}, y {} Arg max u x, y. x,y A {} The Robnson Crusoe condton means that f only one agent partcpates n the mechansm, then she chooses an outcome that s best for her. Clearly, any Lndahl mechansm and the voluntary contrbuton mechansm satsfy ths condton. We have the followng two-agent mpossblty theorems on voluntary partcpaton: Theorem 7 Let n = and α, ω E CD be any Cobb-Douglas utlty economy n whch α > 0.5, =,, and α ω = α ω. If a mechansm satsfes nonemptness, feasblty, and the Robnson Crusoe condton for α, ω under µ, then t fals to satsfy the voluntary partcpaton condton for α, ω under µ. Theorem 8 Let n = and β, ω E QL be any quas-lnear utlty economy n whch β < e / β j for agent such that β >β j, j =, where e s the base of the natural logarthm. If a mechansm satsfes non-emptness, feasblty, and the Robnson Crusoe condton for β, ω under µ, then t fals to satsfy the voluntary partcpaton condton for β, ω under µ.

19 Fundamental mpossblty theorems on voluntary partcpaton 69 The proofs of Theorems 7 and 8 are n the appendx. Both Theorems 7 and 8 can be appled to any mechansm, whereas the condton n Theorem 7 resp. 8 for whch the voluntary partcpaton condton s volated s stronger than that n Theorem resp. focusng on any Lndahl mechansm as well as that n Theorem 5 resp. 6 for the voluntary contrbuton mechansm n the case of two agents wth Cobb-Douglas resp. quas-lnear utlty economes. Fnally, we examne the general case of at least two agents. We ntroduce the followng condton on a mechansm: Defnton 6 The mechansm Ɣ satsfes contrbuton monotoncty for an economy e = u,ω under µ f for all x N, y N g N µ Ɣ e N,all N, and all x N {}, y N {} g N {} µ Ɣ en {}, j = ω j x N j j = ω j x N {} j. Contrbuton monotoncty means the followng. Suppose that n agents except ntally partcpate n the mechansm, and each of those n partcpants contrbutes ω j x N {} j of the prvate good to provde the publc good at equlbrum. Now magne that the non-partcpant also partcpates, so that the equlbrum contrbuton by each of the prevous n partcpants becomes ω j x N j. Then contrbuton monotoncty requres that the sum of equlbrum contrbutons by the n partcpants should not ncrease, that s, j = ω j x N j j = ω j x N {} j. Roughly speakng, contrbuton monotoncty means that the burdens by partcpants n the mechansm do not rse as the number of agents who choose partcpaton become larger. It s not hard to check that any Lndahl mechansm and the voluntary partcpaton mechansm satsfy contrbuton monotoncty for any Cobb-Douglas or quas-lnear utlty economy. By applyng Lemma, we have the followng mpossblty result: Theorem 9 Suppose that a mechansm Ɣ satsfes non-emptness, feasblty, and contrbuton monotoncty for an economy u,ωunder µ; and u ω,y N {} mn u ω,y N {} mn y under µ. x / for some. Then Ɣ fals to satsfy voluntary partcpaton for u,ω The proof of Theorem 9 s n the appendx. By Theorem 9, there s no mechansm satsfyng non-emptness, feasblty, contrbuton monotoncty, and voluntary partcpaton on a doman of economes E f E contans an economy u,ω satsfyng the condton n Theorem 9. Theorem 9 s applcable to any mechansm and any utlty functon, but the condton for whch the voluntary condton s not satsfed s stronger than that n Theorem for Lndahl mechansms and Cobb-Douglas utlty economes. 0 Ths s because Theorem s derved by comparng the utlty level of partcpaton wth that of non-partcpaton drectly, whereas Theorem 9 s obtaned by 0 For any Lndahl mechansm and any Cobb-Dogulass economy, t s straghtforward to check that the condton n Theorem 9 holds f α /n for agent such that α ω α j ω j for all j =, whch s stronger than the condton α >αn n Theorem.

20 70 T. Sajo, T. Yamato usng only local nformaton on a utlty functon,.e., the margnal rate substtuton see Fg. 5 n Lemma. 7 Concludng remarks We see that the solutons to the free-rder problem, whch have been proposed n mechansm desgn theory, are not necessary solutons to the free-rder problem when partcpaton n mechansms s voluntary. It s qute dffcult or mpossble to desgn mechansms wth voluntary partcpaton: any reasonable mechansm, ncludng any Lndhal mechansm and the voluntary contrbuton mechansm, fals to satsfy the voluntary partcpaton condton on a suffcently large doman of economes ncludng Cobb-Douglas or quas-lnear economes. There are several open questons to examne. Frst, we assumed that preferences and endowments are mutually known among agents, although the equlbrum noton s arbtrary as long as t s consstent wth the complete nformaton assumpton. It remans to nvestgate the voluntary partcpaton problem when the agents do not know the preferences of the others by usng an equlbrum concept wth ncomplete nformaton such as Bayesan Nash equlbrum. Second, we assumed that all agents can access the technology to the producton of the publc good freely. Instead, f only the mechansm desgner s able to use the producton technology and cancel a mechansm that s, produce no publc good unless all agents partcpate, then a mechansm satsfyng the standard ndvdual ratonalty condton should work well. However, ths s possble only f the desgner have enough power to force all agents not to produce the publc good. It s an open queston to examne ths ssue. Fnally, Cason et al. 00 and Cason et al. 004 observed that cooperaton has emerged though spteful behavor n ther experments on the voluntary contrbuton mechansm wth voluntary partcpaton. Our theory n ths paper suggests that no cooperaton wll emerge. Reconclng theoretcal results to expermental results s an open area of our future research. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Noncommercal Lcense whch permts any noncommercal use, dstrbuton, and reproducton n any medum, provded the orgnal authors and source are credted. On the other hand, for any Lndahl mechansm and any quas-lnear utlty economy, t s easy to see that the condton n Theorem 9 always holds regardless of the values of preference parameters, because MRS ω, y N {} mn = j = β j /β n for agent such that β β j for all j =. The result s the same as that n Theorem, andtheorem can be obtaned as a corollary of Theorem 9, although the proofs are qute dfferent. Nevertheless, even when the utlty functons are prvately known, a Nash equlbrum can be nterpreted as a rest pont of the dynamc learnng process Hurwcz 97. In fact, Nash equlbrum outcomes have been observed n economc experments even n ncomplete nformaton envronments Cason et al. 006.

21 Fundamental mpossblty theorems on voluntary partcpaton 7 Appendx Proof of Theorem 5 Fx any α, ω E CD. For each T PN, t s easy to check that there s a unque Nash equlbrum allocaton of the voluntary contrbuton mechansm for α,ω T when agents n T partcpate n t, gven by x T, y T = α l T ω l α { + l T [α l/ α l ] }, l T ω l { + l T [α l/ α l ] } for T. Pck agent such that ω ω j for all j =. Snce n ω j = ω j, MRS ω, y N {} = mn ω α α j = ω j { + } j = [α j/ α j ] α n { α + }. j = [α j/ α j ] Also, y N y N {} mn = ω x N { + }. Therefore, y N y N {} mn j = [α j / α j ] ω x N f α /n. By Lemma, we have the desred result. MRS ω, y N {} mn Proof of Theorem 6 Fx any β, ω E QL. For each T PN, t s easy to check that the set of Nash equlbrum allocatons of the voluntary contrbuton mechansm for β,ω T when agents n T partcpate n t, g T µ Ɣ e T, s gven by { #T + x T, y R + y = ω x = βmax T, x T } M T, x j = ω j, j / M T. [ ] ω βmax T,ω, There are multple Nash equlbra unless #M T =, but the equlbrum level of the publc good s always unquely determned. Let x N N, y N g N µ Ɣ e N be gven. Frst, suppose that #M N. Notce that there s some agent M N such that x N <ω and y N = y N {} = βmax N. Snce MRS ω, y N {} mn = βmax N /β = > 0 = y N y N {} mn, t follows from Lemma that ω x N the voluntary partcpaton condton s not satsfed. Second, suppose that M N ={} and β < e βmax N {}. Then x N = ω βmax N = ω β, y N = βmax N = β, and y N {} = βmax N {}. Snce β < e βmax N {}, u β = u β x N, y N u β ω, y N {} { [ ]} = β ln β /βmax N {} < 0. Thus we have the desred result.

22 7 T. Sajo, T. Yamato Proof of Theorem 7 Suppose by way of contradcton that the mechansm satsfes the voluntary partcpaton condton. It s easy to check that by the Robnson Crusoe condton, a unque equlbrum allocaton of the mechansm for one agent Cobb- Douglas utlty economy s gven by x {}, y {} = α ω, α ω, =,. Let V ω, y { j}, u α { x, y R + u α x, y u α ω, y { j}} be agent s weak upper contour set at ω, y { j} for u α, where ω, y { j} = ω, α j ω j and j =. Pckany x {,}, x {,}, y {,} g {,} µ Ɣ u α, uα. By the voluntary partcpaton condton,, y {,} V ω, y { j}, u α, j =, ; j =. 6.3 x {,} We clam that x, y V x {,} ω, y { j}, u α, x + y > ω, j =, ; j =. 6.4 Suppose that 6.4 does not hold. Then for some and some x, ȳ R +, uα x, ȳ u α ω, y { j} and x +ȳ ω.let x, y be a maxmzer of the utlty functon u α x, y = α ln x + α ln y subject to the constrant x + y ω.itseasyto see that x, y = α ω, α ω and u α x, y u α ω, y { j} = α ln α + α ln α ω / α j ω j = α ln α + α ln < 0 snce α ω = α j ω j and α > 0.5. Thus, u α x, ȳ u α ω, y { j} >u α x, y, whch contradcts the fact that x, y s the maxmzer of u α x, y subject to x + y ω. However, by 6.3 and 6.4, x {,} + x {,} + y {,} >ω + ω. Ths contradcts the feasblty condton on the mechansm. Proof of Theorem 8 Suppose by way of contradcton that the mechansm satsfes the voluntary partcpaton condton. It s easy to see that by the Robnson Crusoe condton, a unque equlbrum allocaton of the mechansm for one agent quaslnear utlty economy s gven by x {}, y {} = ω β,β, =,. Let ω V, y { j} {, u β x, y R β + u x, y u β ω, y { j}} be agent s weak upper contour set at ω, y { j} for u β, where ω, y { j} = ω,β j and j =. Pck any x {,}, x {,}, y {,} g {,} µ Ɣ u β, uβ. By the voluntary partcpaton condton,, y {,} V ω, y { j}, u β, j =, ; j =. 6.5 We clam that x, y V ω, y { j}, u β, x + y > ω, j =, ; j =. 6.6 Suppose that 6.6 does not hold. Then for some and some x, ȳ R +, uβ x, ȳ ω, y { j} and x +ȳ ω.let x, y be a maxmzer of the utlty functon u β