BY JOHN O. LEDYARD AND THOMAS R. PALFREY 1

Transcription

1 Econometrca, Vol 67, No 2 March, 1999, A CARACTERIZATION OF INTERIM EFFICIENCY WIT PUBLIC GOODS BY JON O LEDYARD AND TOMAS R PALFREY 1 1 INTRODUCTION IN TIS PAPER, WE CONSIDER the followng classcal publc goods problem A group of ndvduals must decde on a level of a publc good that s produced accordng to constant returns to scale up to some capacty constrant In addton to decdng the level of publc good, the group must decde how to tax the ndvduals n the group n order to cover the cost The dstrbuton of the burden of taxaton s mportant because dfferent ndvduals have dfferent margnal rates of substtuton between the prvate good taxes and the publc good, and may have dfferent ncomes as well These ndvdual margnal rates of substtuton are prvate nformaton; that s, each ndvdual knows hs or her own margnal rate of substtuton, but not those of the other members of the group Adoptng a Bayesan mechansm desgn framework, we assume that the dstrbuton of margnal rates of substtuton s common knowledge We are nterested n characterzng effcency n ths envronment and are also nterested n characterzng those mechansms that one mght expect to actually arse n practce Ths suggests two approaches, one from normatve consderatons and one from postve consderatons On the normatve sde, we ask: What should an actve planner a mechansm desgner do? A well-known specal case of ths problem has been solved for one partcular socal welfare functon eg, d Aspremont and Gerard-Varet 1979 that s nsenstve to the dstrbuton of cost shares What dstngushes our work here s that we consder a planner who s maxmzng a welfare functon that s senstve to the allocaton of cost shares over the dfferent valuaton-types Smply put, the planner may care who pays Ths s represented formally by type-contngent welfare weghts Why mght the consderaton of such dstrbutonal goals be relevant? What ratonale can be gven for nonconstant welfare weghts? Perhaps the smplest example to answer these questons corresponds to publc decsons wth zero producton costs Such cases are well-approxmated n the real world by socal legslaton such as blue laws, smokng and drnkng prohbtons, clothng requrements at beaches, and so forth Suppose one s consderng the mplementaton of one such socal regulaton Many would argue that f mplementaton takes place, then the losers e those wth a negatve valuaton to the 1 We are grateful for the support of the Natonal Scence Foundaton Grant No SBR , and of the New Mllennum Program of the Jet Propulson Laboratory of NASA The second author s grateful for the hosptalty and research support at Laboratore d Econome Industrelle and at Centre d Ensegnement et de Recherche en Analyse Soco-Economque An early draft of ths artcle was prepared for the 1996 Francqu Prze Colloquum We are grateful to ts organzer, Claude d Aspremont, and to the partcpants, for useful comments The artcle has also benefted from the comments of semnar partcpants at Unverste de la Medterranee, Northwestern Unversty, arvard Unversty, Unverste de Cergy-Pontose, the Conference on Effcency n Economcs wth Publc Goods and Prvate Informaton at the Unversty of Vence, the Roy Semnar at Ecole des Ponts et Chaussees, and from dscussons wth Lous-Andre Gerard-Varet and Jean-Charles Rochet Three referees and a coedtor provded addtonal helpful suggestons whch have mproved the artcle The usual dsclamer apples 435

2 436 J O LEDYARD AND T R PALFREY 2 proposed regulaton should be compensated But one runs nto the ncentve compatblty problem that f you navely say you are gong to compensate all losers, then everyone wll clam to be a loser, possbly leadng to producton never occurrng A planner mght want, therefore, to gve some weght to the losers but not to the excluson of all others Obvously, n order to compensate the losers n such decsons, ncentve taxes need to be carefully constructed that wll acheve such type-contngent redstrbuton, at least to the extent lmted by ncentve compatblty constrants As we wll show below, there s a drect and ntutve lnk between the desred degree of such compensaton and the correspondng dstortons away from the Lndahl-Samuelson optmum In ths partcular example, sgnfcant compensaton of losers would necesstate a correspondng degree of underproducton relatve to the classc soluton Other weghtng schemes would correspond to other type-dstrbutonal goals, and could lead to ether under- or overproducton A second reason to consder nonconstant welfare weghts arses f one concedes that ths partal equlbrum model s embedded n a rcher general equlbrum structure, where ncome or wealth dstrbuton s a goal of the planner If preferences for the publc good are correlated wth ncome or wealth n a systematc way, then the publc good mechansm can be used as an nstrument for redstrbuton, and unequal welfare weghts would be a reflecton of the planner s redstrbutve goals A thrd ratonale for unequal weghts s more drect For reasons that may have to do only remotely wth ssues of compensatng losers or wealth redstrbuton, certan knds of type-dependent cost-sharng may be deemed desrable on ther own merts A classc example of ths s the class of proportonal cost-sharng rules, whereby ndvduals valung the publc good more should bear a proportonally larger share of the costs eg, Jackson and Mouln 1992 Such normatve goals would correspond to a system of welfare weghts that decrease n valuatons n a partcular way For the postve approach to the mechansm desgn problem, we ask: What would we expect to see n practce? ere we are lookng for a concept of effcency or stablty because one would expect neffcent or unstable mechansms to be replaced by others Under complete nformaton, these concepts correspond to Pareto optmalty and the core, respectvely Under asymmetrc nformaton the problem s a bt more subtle, and there remans no true consensus on the approprate equvalent concepts 3 We therefore take a mnmalst approach, and look at a natural extenson of Pareto optmalty to asymmetrc nformaton In the analyss below we assume that all decsons, ncludng whether to change the mechansm, are made at the nterm stagethat s, when each agent knows hs or her type, but not anyone else s type If there s no communcaton, then the set of nterm ncentve effcent mechansms conssts of those ncentve compatble mechansms for whch t cannot be common knowledge that there s another mechansm whch generates a unanmous mprovement We would expect therefore that survvng nsttutons would be, mnmally, nterm ncentve effcent Luckly we do not have to choose between normatve and postve approaches to ths problem As ponted out n olmstrom and Myerson 1983, a mechansm s nterm effcent f and only f there exst type-dependent socal welfare weghts for whch that mechansm solves the planner s optmzaton problem subject to feasblty and ncentve compatblty constrants Thus, by varyng the welfare weghts n our planner s problem, 2 Sometmes, ths requrement s mplctly mposed as a voluntary partcpaton constrant 3 See olmstrom and Myerson 1983, Cramton and Palfrey 1995, and Crawford 1985 for good dscussons of the dffcultes of extendng these concepts to asymmetrc nformaton

3 PUBLIC GOODS 437 we map out the entre set of mechansms that are nterm ncentve effcent Thus, a complete soluton to ths problem, posed ether from a normatve or postve standpont, s equvalent to fully characterzng the set of nterm effcent mechansms for the producton of publc goods n ths framework A complete characterzaton of nterm effcency has been done for the specal case where the types are dentcally dstrbuted and can only take on two values Ledyard and Palfrey 1994 There t was shown that optmal producton always takes a specal form n whch the publc good s provded f and only f the number of hgh valuaton types exceeds a threshold number that depends on the welfare weghts and the dstrbuton of types The greater the welfare weght on hgh valuaton types, the lower the optmal threshold Wth more than two types as n ths paper the optmal mechansm generally depends on the exact profle of types n a more complcated way In ths paper, we fully characterze nterm effcent mechansms and obtan some comparatve statcs about how the optmal mechansm changes wth the underlyng dstrbuton of types and wth the welfare weghts of the welfare functon Secton 2 presents the basc model Secton 3 presents the characterzaton and ts proof Secton 4 explans the ntuton behnd the characterzaton In Secton 5, we summarze these fndngs and offer some concludng remarks about some possble future drectons of research 2 TE MODEL There are N people who must decde on the quantty, q, ofapublc good that s produced accordng to constant returns to scale 4 and has a maxmum level Y1 The cost of producng q0, 1 s equal to Kq Inaddton, they must decde how to dstrbute the producton costs Because of the lnear producton technology, the optmal level of the publc good wll always be ether 0 or 1, so ths s equvalent to a problem of decdng on whether or not to produce a dscrete publc good We let a denote ndvduals s share of the cost, n unts of the consumpton of the prvate good, and assume t can take any real value Therefore the set of feasble levels of producton and cost shares are gven by 1 N N a,,a, q 0,1 such that N Ý 1 Kq a Indvdual preferences are assumed to be rsk-neutral and quaslnear n the level of publc good producton and the taxes cost shares, so the utlty to type of agent for an allocaton q, a s gven by V qa Thus, represents the margnal rate of substtuton between the publc and prvate good We refer to as player s value We assume that each ndvdual knows hs own value,, and does not know the values of the other ndvduals We assume that the 4 We explan n the next secton how the model and the results are easly extended to arbtrary producton technologes

4 438 J O LEDYARD AND T R PALFREY ndvdual values are ndependently dstrbuted, wth the common knowledge cdf of s value denoted F and the support of F s V,, where KN We assume F has a contnuous postve densty on V Note that 0sallowed Clearly under these assumptons, our choce of normalzaton of the utlty functon s arbtrary up to an affne transformaton In partcular, t s equvalent n terms of ndvdual decson theory to the models of asymmetrc nformaton about contrbuton 5 costs a, where utltes are normalzed so that the margnal utlty of the publc good equals 1, so that u q1 a owever, the class of ex-ante ncentve effcent mechansms n the sense of olmstrom and Myerson 1983 wll be dfferent under the two normalzatons 6 So, below, we wll focus on the set of nterm-ncentve effcent mechansms That set s ndependent of whatever type dependent normalzaton one chooses A mechansm conssts of a message space for each agent and an outcome functon mappng message profles nto probablty dstrbutons over the set of feasble allocatons By the revelaton prncple, the propertes n terms of allocatons of any optmal mechansm can be duplcated by an ncentve compatble, drect mechansm n whch the message space for agent s smply the set of possble types values n the support of F A strategy for s a mappng : V V, that s, a decson rule that specfes a reported type for each possble type We refer to the dentty mappng as the truthful strategy By the lnearty of the ndvdual utlty functons, there s also no loss of generalty n restrctng attenton to determnstc mechansms Thus, we denote a feasble drect mechansm smply as a functon ½ Ý 5 N N 1 N N : V a,,a, q R 0,1 a Kq We denote the publc good allocaton component of at type profle by q, and the prvate good tax for by a Besdes feasblty, the man restrcton on s that t be ncente compatble, whch means that t s a Bayesan equlbrum of for all agents to adopt a strategy of truthfully reportng ther type Gven a strategy profle : V V and a mechansm,, let the nterm utlty of type of agent, assumng all others truthfully report ther type, be denoted by: 1 V uˆ,, q, a, df Let u, uˆ,, I where I denotes the truthful strategy I Then s ncentve compatble f and only f u, uˆ,, for all, The set of nterm ncentve effcent allocaton rules 7 can be represented as the solutons to a set of maxmzaton problems Let 0 beasystem of welfare weghts, a profle of measurable functons mappng types nto the postve real lne such that df 1, where represents the welfare weght assgned to type of 5 Ths normalzaton can be made as long as 0 6 The fact that ex ante effcency s senstve to utlty normalzatons s dscussed n Ledyard and Palfrey 1994, p For the remander, we smply refer to such allocatons as nterm effcent

5 PUBLIC GOODS 439 agent Then s nterm effcent f and only f there s a such that maxmzes 8 Ý u, df over the set of feasble and ncentve compatble mechansms We now proceed to characterze ths set 3 TE CARACTERIZATION As ndcated above, we represent nterm effcent rules as a soluton to a constraned maxmzaton problem Frst we need to dentfy ncentve compatble mechansms n a useful way For smooth mechansms, when preferences are lnear, the characterzaton of ncentve compatblty n terms of dervatves s well-known There are bascally two features of such mechansms Frst, an envelope condton s satsfed, namely that the total dervatve of the nterm utlty for wth respect to type when players adopt truthful strateges s equal to the partal dervatve wth respect to type e, fxng the reports of all agents Second, the nterm utlty to under truthful reportng s convex n s type Ths s stated formally below, wthout proof LEMMA Rochet 1987 : If uˆ s lnear n and s twce contnuously dfferentable, then s ncente compatble f and only f u, uˆ,, I, u, s conex n For our problem u, Q V q df Sou s convex n f and only f Q 0 Usng these facts we can see that a mechansm q, a s nterm effcent f and only f there s a such that q, a solves max Ý q V a df subject to 0q 1, Q 0,, u uˆ, I,, and Ý a Kq Usng the approach of Mrrlees 1971 and Wlson 1993 we construct the Lagrangan equvalent problem Ý, max mn u, df Ý ˆ u, u,, I d Ý a Kq d subject to: V 0q 1 V, Q 0, V, where and are multplers for frst order ncentve compatblty and feasblty, respectvely Applyng Green s Theorem and substtutng the dentty u, 8 See olmstrom and Myerson 1983

6 440 J O LEDYARD AND T R PALFREY uˆ,, I converts the maxmzaton problem to: Ý ˆ, max mn u,, I f ˆ ž Ý / 4 u,, I d a Kq d Ý V uˆ,, I d subject to: 0q 1 V, Q 0, V, where V denotes the boundary of V and ponts outward at We are now n a poston to gve a complete characterzaton of the class of nterm effcent mechansms TEOREM 1: q, a s an nterm effcent mechansm f and only f 0 wth df 1, such that: a, q maxmzes Ý w K4 subject to: 0q 1 V, Q 0, V, where 1F t df t w ; f f and b a tdq t where Ý Ý Kq tdq t and df 0, PROOF: A sketch s gven For further detals see Ledyard and Palfrey 1996 V Notce that the restrcton of to df 1 s wthout loss of generalty Snce utltes are lnear n the transfers, for some welfare weghts total welfare can be made arbtrarly large smply by makng ex ante transfers from one ndvdual to another ndvdual That s, f, for two agents and j, twere the case that j j j j j j df df, then total welfare could be made arbtrarly large by makng ex ante transfers of the prvate good from to j Thus, a soluton to the maxmzaton problem only exsts when the welfare weghts are, n expectaton, the same for all agents Thus, wthout loss of generalty, we restrct the welfare weghts to satsfy s df s 1

7 We can wrte as ž / PUBLIC GOODS 441 max mn Ý q a q df f f, V ž / Ý Ý a Kq d uˆ uˆ V From the frst order condtons wth respect to a,, and we obtan, for, 1 f f 0, 2 a Kq 0, f 0, Ý ½ 0 f 0, V 3 a df A Q, where f From 1 t follows that s constant n Integraton of 1 gves F C where s the expected value of condtonal on s valuaton beng less than or equal to 9 Part b of the theorem follows from 2 and 3 Fnally, the contnuty of along wth the frst order condtons for a at and mply that 0 So C0 and df 1 Substtutng all of ths nto mples that we must fnd q to solve Ý ž f / max K q df subject to: q 0q 1 V, Q 0, V QED REMARK: The technque above apples equally to the case of general producton functons Let Cq be the cost of producng a publc good level equal to q Frst observe that the ncentve compatblty constrants do not depend on CqAs aconsequence, substtuton of the ncentve constrants results n the program: Ý ž f / max q C q df subject to: q Q 0, V 9 The exstence of such an for any gven q was frst shown by d Aspremont and Gerard-Varet 1979 One that satsfes b s K 1 1 j j q Q Q V sdq s Ý j Ý j j j N N1 N1 j j j

8 442 J O LEDYARD AND T R PALFREY The only regularty assumptons needed to guarantee a soluton are that C s nondecreasng n q and that Ý ž f / q C q has a soluton for every valuaton profle, The correspondng taxes are then constructed n a manner smlar to part b of the theorem Therefore, ths general approach can be appled to problems wth ncreasng or decreasng returns to scale, U-shaped average cost functons, and even lumpy publc goods 10 wth fxed costs and jumps n the cost functon Call 4 INTERPRETING TE CARACTERIZATION 41 Vrtual Cost-Beneft Crteron F w 1, f 11 type of agent s rtual aluaton `alamyerson Suppose w 0, Then, snce Q probý w j Kw t wll be true that Q j j 0 s never bndng So for, F such that w 0,, nterm effcent q satsfy Ý q 1 f w K, 0 otherwse Ths s a rtual cost-beneft crteron 12 The vrtual utlty has a famlar nterpretaton see, for example, Myerson 1981 It equals the true publc good valuaton of the -type nflated 13 by a factor that depends on the dstrbuton of types and on the welfare weghts The benchmark case s the one where 1 for all and Inths case the frst best optmal level of publc good s 1 or 0 dependng only on whether or not Ý KN 0 That s, produce f and only f the sum of the margnal rates of substtuton exceeds the margnal producton cost Ths s the Lndahl-Samuelson soluton, precsely the soluton nvestgated n most prevous papers on the optmal provson of publc good See d Aspremont and Gerard-Varet 1979 Ths smplfcaton arses because the allocaton of the prvate good e, the ncdence of the taxes on dfferent types does not affect socal welfare For ths reason, ncentve compatblty does not reduce socal welfare relatve to the frst best soluton owever, t must be emphaszed 10 These are sometmes referred to as threshold or step-level publc goods The smplest knd s just a bnary publc good, whch s mathematcally equvalent to the standard model presented n Secton 2: constant returns wth a maxmum capacty 11 Ths s the so-called regular case, where the second order condton s never bndng 12 Notce that smlar ex post vrtual cost-beneft condtons characterze the second-best optmum n the case of nonlnear producton technologes, provded the second order condton s satsfed 13 Ths could be deflated f 1

9 PUBLIC GOODS 443 that ths s a very specal case It s n fact the only system of welfare weghts where ncentve compatblty does not cause dstortons relatve to the frst best soluton 14 To better understand the ntuton behnd the vrtual valuatons, one can thnk of the mechansm operatng n the followng way Each agent truthfully reports a valuaton If the publc good s produced, then each agent pays the ncentve tax, whch equals a constant plus that agent s valuaton mnus hs nformatonal rent, 1F f Recall from standard ncentve theory that ths s the amount that can be extracted from an agent, gven ncentve constrants Of course, n ths publc good problem, the objectve of the mechansm s not to extract rent from agents, so any excess ncentve tax wll be dstrbuted lump sum back to the agents, by adjustng the ncentve tax by a constant Thus, f the good s provded, the government spends K to produce the publc good and makes a lump-sum refund, whch s formally captured by the constant e ndependent of that s added to each agent s ncentve tax The porton of ths refund that comes from type of agent equals ž / 1F K f N There are two other terms that complete the socal costbeneft pcture, as t concerns type of agent One s smply that producng the publc good, produces a drect beneft of to agent, whch s valued socally as Last, but not least, s the fact that the ncentve tax before refund s a socal cost, and ths socal cost equals t df t f Collectng all these terms, gves us type of agent s contrbuton to the margnal net socal value of producng the publc good Denotng ths by w, gves us t df t 1F K w N f f 1F t df t K f f N K w, N whch s the cost adjusted vrtual valuaton of type of agent Notce that n the specal case of neutral dstrbutonal weghts, t df t 1 F,sothat t df t 1F f f 14 Actually, ths s the only system of welfare weghts for whch a frst best soluton exsts For any other weghts, welfare can be arbtrarly ncreased by shftng the allocaton of the prvate good to one partcular type of some ndvdual Snce we mpose no feasblty bounds on the allocaton of the prvate good, ths means that the frst best soluton does not exst Of course, wth ncentve compatblty constrants, the second-best problem s well defned

10 444 J O LEDYARD AND T R PALFREY and as a result there are no welfare costs assocated wth chargng the ncentve taxes n a type-dependent way and then redstrbutng them back n a lump sum fashon Otherwse there s a cost to dong ths The form of vrtual utltes also makes t easy to see how dstortons away from the classc optmum are related to the welfare weghts For example, f s decreasng n type then generally the nterm effcent soluton calls for underproducton relatve to the Lndahl-Samuelson soluton, snce s postve for all types That s, the vrtual valuatons are always less than true valuatons, so the sum of the true valuatons must more than exceed the producton cost n order for producton to be optmal Conversely, f s ncreasng n type, then there should be oerproducton relatve to the Lndahl Samuelson soluton 42 Second Order Condtons The dscusson above assumes monotone vrtual utltes, whch ensures that maxmza ton of the relaxed program, wthout the Q 0 constrant, automatcally satsfes that constrant It s straghtforward to see what s requred for vrtual utltes to be monotone n type, and ths provdes a nce ntuton for how our results dffer from standard ncentve problems of ths type eg Guesnere and Laffont 1984 From above, 1F t df t f f w The frst term,, s clearly ncreasng n The second term, 1F f, the nformatonal rent, s typcally assumed to be monotone n n adverse selecton models n prvate goods envronments, by requrng the dstrbuton to satsfy a monotone hazard rate condton Snce the ncdence of ncentve taxes can have welfare effects, there s a thrd term to worry about, ndcatng that one may need more or sometmes less! than the standard monotone hazard rate condton to guarantee that Q 0 sautomatcally satsfed when one smply plugs n vrtual utltes and maxmzes subject only to producton feasblty These addtonal condtons wll mply restrctons on the dstrbuton of welfare weghts, as we llustrate n the example below 43 Example Let be dstrbuted unformly on 0, 1 for all, so F and f 1 Then w 2 tdtand w 2 0 Therefore, the second order condton s globally satsfed for unform dstrbutons of valuatons f the maxmum welfare weght s less than or equal to 2 Thus, f 2 ab 2 ab, where a0 and 2 ab0, then we are always n the regular case where vrtual valuatons are monotonc n type and the second order condtons are satsfed If b0 hgh valuaton types receve more weght then producton wll occur more often than n the Lndahl-Samuelson soluton, whle f b0, the reverse s true owever, there are such that vrtual valuatons are 2 decreasng, even for the unform dstrbuton For example, f 3, then vrtual valuatons are decreasng for 23 The optmal soluton n ths case wll nvolve '

11 PUBLIC GOODS 445 equal treatment of all types wth valuatons above some 1; Q 1 for all such types 15 If K1, the soluton above s straghtforward, because the Q obtaned from the relaxed problem s n fact weakly ncreasng, even though the vrtual valuatons are decreasng n some regon 16 The reason why Q obtaned from the relaxed problem s nondecreasng s that vrtual valuatons are decreasng only for very hgh values of, where w 1 So, when K1, the relaxed soluton sets Q 1 whenever w 1 for at least one agent, and all these hgh types are treated the same But n general, for hgher values of K, the relaxed soluton may produce volatons of the second order condtone Q decreasng n some regon In such cases, one apples a procedure called ronng Rochet and Chone 1998 The prncple behnd ths procedure s to flatten out Q n the decreasng regon and for some adjacent types as well The geometry s llustrated clearly n a seres of fgures n Guesnere and Laffont 1984 for the sngle agent case It s also nstructve to use ths example to llustrate the range of publc good provson rules or cost-beneft crtera that are nterm effcent Suppose N2, K1 The Lndahl-Samuelson effcent outcome s to produce f and only f the average valuaton exceeds 12, so the publc good wll be provded half the tme Next suppose one shfts welfare weght to the low valuaton types, to the pont where 2 for all 12 and 0 for all 12 Ths satsfes monotoncty of vrtual valuatons 17 and t s easy to see that the optmal mechansm s to produce f and only f the sum of valuatons exceeds 32 In other words, ths weghtng scheme effectvely nflates the cost of the publc good by 50 percent, so t should be produced only f the actual beneftcost rato exceeds 15 At frst blush t seems as though ths mechansm could be mproved, snce there are some states where both agents are hgh types e 12 for both of them, and the publc good s not provded Snce all hgh types receve the same welfare weght, and snce low types do not bear any of the cost of producton n these states, t would seem to lead to an mprovement n welfare Why doesn t ths lead to an mprovement? The answer s that the mechansm s desgned to acheve redstrbutve goals n addton to decdng on publc good producton In ths case, the welfare weghts ndcate that there should be a transfer from hgh valuaton to low valuaton types ence n the optmal mechansm there are some states where there s one low type and one hgh type, and the publc good s not produced, but a prvate good transfer takes place between the low and hgh types The extent of such transfers would be hndered by greater publc good producton due to ncentve compatblty problems We conjecture that ths choce of welfare weghts corresponds to the lowest possble expected output Q125 of all nterm effcent mechansms for the unform case wth N2 and K1 15 Smlarly, f the optmal soluton wll nvolve equal treatment of all types wth valuatons below some 0; Q0 for all such types 16 In other words, ths s an example demonstratng why monotoncty of w s not a necessary condton for the second order condtons to be satsfed 17 If one shfts the welfare weghts even further downward, so that A2 for all 1A and 0 for all 1A, then the vrtual valuatons are nonmonotonc and ronng must be done Nevertheless, from the characterzaton n Theorem 1, t s easy to verfy that standard ronng procedures can be used and wll generate an optmal mechansm wth the same property: produce f and only f the sum of valuatons exceeds 32

12 446 J O LEDYARD AND T R PALFREY At the other extreme, suppose the welfare weghts are shfted n the opposte drecton, wth 2 for all 12 and 0 for all 12 In ths case the optmal mechansm s to produce f and only f the sum of valuatons exceeds 12 In other words, the cost of the publc good s effectvely deflated by 50 percent, so that t should be produced f the actual beneftcost rato s at least 5 Agan t would seem that effcency would dctate that when both types are low types, the good should never be produced owever, redstrbutve goals mpled by ths welfare weghtng scheme requre the low types to subsdze the cost of the publc good The most effcent way to perform ths subsdzaton requres some overproducton of the publc good We conjecture that ths choce of welfare weghts corresponds to the hghest possble expected output Q875 of all nterm effcent mechansms for the unform case wth N2 and K1 5 CONCLUSIONS In ths paper, we have characterzed the nterm effcent publc good allocaton rules n a smple Bayesan publc good envronment We fnd that the optmal mechansm nvolves ether more or less producton of the publc good dependng on whether the welfare weghts are shfted n the drecton of types wth hgher or lower valuatons for the publc good Thus, compared to the classcal optmal level of publc good provson the Lndahl-Samuelson soluton, there should generally be some dstorton The reason for ths dstorton s that unless welfare weghts are perfectly neutral, effcent allocatons wll depend n general on both the level of publc good and the ncdence of taxes to fnance the publc good Because of ncentve compatblty, the effcent way to reduce the tax burden on low-valuaton resp: hgh-valuaton consumers s to reduce resp: ncrease the level of provson of the publc good In the borderlne case, the frst-best soluton s attanable only because the welfare functon s ndependent of dstrbuton of the prvate good There are several drectons worth pursung One drecton s to explore the use of smple mechansms The publc good mechansms proposed here nvolve complcated transfer schemes that can necesstate the use of very large taxes and subsdes In a companon paper Ledyard and Palfrey 1998 we explore smple mechansms n large populatons, and show that for any nterm effcent allocaton rule there exsts a smple domnant-strategy referendum mechansm that perfectly approxmates the effcency of that allocaton rule In a referendum, ndvduals smply submt a bnary message a vote ether for or aganst producton of the publc good If a suffcently large fracton of the ndvduals vote n favor, then the publc good s provded and the costs are dstrbuted equally n the populaton Otherwse, the publc good s not produced Ths provdes an approxmate welfare theorem for publc goods: effcent allocaton rules can be approxmately decentralzed by an approprately chosen votng rule Moreover, f there s a common value component to the dstrbuton of preferences, then the optmal referendum s unque We prove ths by approxmatng the soluton to the optmal mechansm where the second order condton s gnored Thus a by-product s the result that the second order condtons are nconsequental n large economes There are several other drectons Partcpaton constrants were not mposed n our soluton for the optmum It s farly easy to show that when these constrants are bndng, ths mples a reducton n the level of the publc good, snce these constrants are necessarly bndng on the low valuaton types Ledyard and Palfrey 1994 It s also

13 PUBLIC GOODS 447 true that, except n unnterestng cases, these constrants wll mply QN0 nlarge populatons Ledyard and Palfrey 1994, Malath and Postlewate 1990 But for the case of large N, twould usually seem more realstc to assume that partcpaton s generally oblgatory to all members of the group under consderaton, as we have assumed here Related to the general ssue of partcpaton s the applcaton of the general approach presented here to excludable publc goods In that case, partcpaton constrants can be relaxed by the no-cost excluson of low valuaton types More nvolved extensons, such as relaxng the assumpton of ndependent types, consderaton of utlty functons where the valuaton parameter enters nonlnearly, or ntroducng multdmensonal types, appear to be more dffcult open questons Fnally, as we remarked at the end of Secton 3, the analyss s easly extended to accommodate arbtrary producton technologes D ofumantes and Socal Scences, Calforna Insttute of Technology, SS , 1200 E Calforna Bld, Pasadena, CA 91125, USA; jledyard@hsscaltechedu and D ofumantes and Poltcal Scence, Calforna Insttute of Technology, SS , 1200 E Calforna Bld, Pasadena, CA 91125, USA; trp@hsscaltechedu Manuscrpt receed August, 1996; fnal reson receed Aprl, 1998 REFERENCES D ASPREMONT, C, AND L-A GERARD-VARET 1979 : Incentves and Incomplete Informaton, Journal of Publc Economcs, 11, 2545 CRAMTON, P, AND T PALFREY 1995 : Ratfable Mechansms: Learnng from Dsagreement, Games and Economc Behaor, 10, CRAWFORD, V 1985 : Effcent and Durable Decson Rules: A Reformulaton, Econometrca, 53, GUESNERIE, R, AND J-J LAFFONT 1984 : A Complete Soluton to a Class of Prncpal-Agent Problems wth an Applcaton to the Control of a Self-Managed Frm, Journal of Publc Economcs, 25, OLMSTROM, B, AND R MYERSON 1983 : Effcent and Durable Decson Rules wth Incomplete Informaton, Econometrca, 51, JACKSON, M, AND MOULIN 1992 : Implementng a Publc Project and Dstrbutng Its Cost, Journal of Economc Theory, 57, LEDYARD, JON O, AND TOMAS PALFREY 1994 : Votng and Lottery Drafts as Effcent Publc Goods Mechansms, Reew of Economc Studes, 61, : Votng s Approxmately Optmal, Workng Paper, Calforna Insttute of Technology 1999 : Interm Effcency n a Publc Goods Problem, n Socal Organzaton and Mechansm Desgn, Proceedngs of the 1996 Francqu Prze Colloquum, ed by Claude d Aspremont Belgum: De Boeck Unversty, MAILAT, GEORGE, AND ANDREW POSTLEWAITE 1990 : Asymmetrc Informaton Barganng Problems wth Many Agents, Reew of Economc Studes, 57, MIRRLEES, JA 1971 : An Exploraton n the Theory of Optmum Income Taxaton, Reew of Economc Studes, 38, MYERSON, R 1981 : Optmal Aucton Desgn, Mathematcs of Operatons Research, 6,5873 ROCET, JEAN-CARLES 1987 : A Necessary and Suffcent Condton for Ratonalzablty n a Quas-Lnear Context, Journal of Mathematcal Economcs, 16,

14 448 J O LEDYARD AND T R PALFREY ROCET, JEAN-CARLES, AND PILIPPE CONE 1998 : Ironng, Sweepng and Multdmensonal Screenng, Econometrca, 66, WILSON, ROBERT 1993 : Desgn of Effcent Tradng Procedures, n The Double Aucton Market Insttutons, Theores, and Edence, ed by D Fredman and J Rust Proceedngs of the Workshop on Double Aucton Markets eld June, 1991 n Santa Fe, New Mexco Proceedngs Volume XIV Readng, MA: Addson-Wesley