The Unique Informational Effciency of the Lindahl Allocation Process in Economies with Public Goods

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1 MPRA Munch Personal RePEc Archve The Unque Informatonal Effcency of the Lndahl Allocaton Process n Economes wth Publc Goods Guoqang Tan 2001 Onlne at MPRA Paper No , posted 12 September :54 UTC

2 The Unque Informatonal Effcency of the Lndahl Allocaton Process n Economes wth Publc Goods Guoqang TIAN Department of Economcs Texas A&M Unversty College Staton, Texas (gtan@tamu.edu) May, 2003/revsed October, 2005 Abstract Ths paper nvestgates the nformatonal requrements of resource allocaton processes n publc goods economes wth any number of frms and commodtes. We show that the Lndahl mechansm s nformatonally effcent n the sense that t uses the smallest message space among smooth resource allocaton processes that are nformatonally decentralzed and realze Pareto optmal allocatons over the class of publc goods economes where Lndahl equlbra exst. Furthermore, we show that the Lndahl mechansm s the unque nformatonally effcent decentralzed mechansm that realzes Pareto effcent and ndvdually ratonal allocatons n publc goods economes wth Cobb-Douglas utlty functons and quadratc producton functons. Journal of Economc Lterature Classfcaton Number: D5, D61, D71, D83, P51. 1 Introducton Snce the poneerng work of Hurwcz (1960) and Mount and Reter (1974), there has been a lot of work on studyng the nformatonal requrements of decentralzed resource allocaton mechansms. The focus n ths lterature has partcularly been on the dmenson of the message space beng used for communcaton among agents. These nformatonal requrements depend upon two basc components: the class and types of economc envronments over whch a mechansm s supposed to operate and the partcular outcomes that a mechansm s requred to realze. Ths paper wll study the nformatonal requrements of resource allocaton mechansms that select Pareto optmal allocatons for publc goods economes wth general convex producton technologes and any number of producers and goods. Fnancal support from the Texas Advanced Research Program as well as from the Prvate Enterprse Research Center, and the Lews Faculty Fellowshp at Texas A&M Unversty s gratefully acknowledged. 1

3 The nterest n studyng the nformatonal requrements and the desgn of resource allocaton mechansms was greatly stmulated by the socalst controversy the debate over the feasblty of central plannng between Mses-Hayek and Lange-Lerner. In lne wth the prevalng tradton, nterest n ths area was focused on Pareto-optmalty and nformatonally decentralzed decson makng. Allocatve effcency (Pareto optmalty) and nformatonal effcency are two hghly desred propertes for an economc mechansm to have. Pareto optmalty requres resources be allocated effcently whle nformatonal effcency requres an economc system have the mnmal nformatonal cost of operaton. The noton of an allocaton mechansm was frst formalzed by Hurwcz (1960). Such a mechansm can be vewed as an abstract plannng procedure; t conssts of a message space n whch communcaton takes place, rules by whch the agents form messages, and an outcome functon that translates messages nto outcomes (allocatons of resources). Mechansms are magned to operate teratvely. Attenton, however, may be focused on mechansms that have statonary or equlbrum messages for each possble economc envronment. A mechansm realzes a prespecfed welfare crteron (called socal choce rule, or socal choce correspondence) f the outcomes gven by the outcome functon agree wth the welfare crteron at the statonary messages. The realzaton theory studes the queston of how much communcaton must be provded to realze a gven performance, or more precsely, studes the mnmal nformatonal cost of operatng a gven performance n terms of the sze of the message space and determnes whch economc system or socal choce rule s nformatonally the most effcent n the sense that a mnmal nformatonal cost s used to operate the system. Such studes can be found n Hurwcz (1972, 1977), Mount and Reter (1974), Calsamgla (1977), Walker (1977), Sato (1981), Hurwcz, Reter, and Saar (1985), Calsamgla and Krman (1993), Tan (1990, 1994, 2004, 2006) among others. One of the well-known results n ths lterature establshes the mnmalty of the compettve mechansm n usng nformaton for pure exchange economes. Hurwcz (1972), Mount and Reter (1974), Walker (1977), Hurwcz (1986b) among others proved that, for pure exchange prvate goods economes, the Walrasan allocaton process s the nformatonally effcent process n the sense that any smooth nformatonlly decentralzed allocaton mechansm that acheves Pareto optmal allocatons must use nformaton as least as large as the compettve mechansm,.e., the compettve allocaton process has a message space of mnmal dmenson among smooth resource allocaton processes that are prvacy preservng (nformatonally decentralzed) and non-wasteful (.e., yeldng Pareto effcent allocatons). 1 For brevty, these results have been referred to as the Effcency Theorem. Jordan (1982) and Calsamgla and Krman (1993) further provded the Unqueness Theorem for prvate goods pure exchange economes. Jordan (1982) 1 The term smoothness used here s not referred as the usual dfferentablty of a functon. Instead, the smoothness of a mechansm referees that the statonary message correspondence s ether locally threaded or f the nverse of the statonary message correspondence has a Lpschzan-contnuous selecton n the subset. Ths termnology was used by Hurwcz (1999). We wll gve the defnton of the local threadedness below. 2

4 proved that the compettve allocaton process s unquely nformatonally effcent. Calsamgla and Krman (1993) proved the equal ncome Walrasan mechansm s unquely nformatonally effcent among all allocatons mechansms that realze far allocatons. Recently, Tan (2004) nvestgates the nformatonal requrements of resource allocaton processes n pure exchange economes wth consumpton externaltes. It s shown that the dstrbutve Lndahl mechansm s a unquely nformatonally effcent allocaton process that s nformatonally decentralzed and realzes Pareto effcent allocatons over the class of economes that nclude non-malevolent economes. Tan (2006) further proved the unque nformatonal effcency of the compettve market mechansm for prvate ownershp producton economes. These effcency and unqueness results are of fundamental mportance from the pont of vew of poltcal economy. They show the unqueness of the compettve market mechansm n terms of allocatve effcency and nformatonal effcency for prvate goods economes. The concept of Lndahl equlbrum n economes wth publc goods s, n many ways, a natural generalzaton of the Walrasan equlbrum noton n prvate goods economes, wth attenton to the well-known dualty that reverses the role of prces and quanttes between prvate and publc goods, and between Walrasan and Lndahl allocatons. In the Walrasan case, prces must be equalzed whle quanttes are ndvdualzed; n the Lndahl case the quanttes of the publc good must be the same for everyone, whle prces charged for publc goods are ndvdualzed. In addton, the concepts of Walrasan and Lndahl equlbra are both relevant to prvate-ownershp economes. Furthermore, they are characterzed by purely prce-takng behavor on the part of agents. It s essentally ths property that one can also explot to defne the Lndahl process as an nformatonally decentralzed process. For the class of publc goods economes, Sato (1981) obtaned a smlar result showng that the Lndahl allocaton process has a message space of mnmal dmenson among a certan class of resource allocaton processes that are prvacy preservng and non-wasteful. However, Sato (1981) only dealt wth the class of publc goods economes wth ust a sngle producer, and n such a case, a specal class of economes s constructed usng a class of lnear producton sets for the producer. Qute clearly, more complex producton sets must be devsed when the number of frms ncreases. So one of the purpose of the paper s to fll ths gap, although our man purpose n the paper s to establsh the unque nformatonal effcency of the Lndahl mechansm. In ths paper we establsh the nformatonal optmalty and unqueness of the Lndahl mechansm for publc goods economes wth any number of producers. The task of ths paper s three-fold. Frst, we establsh the lower bound of nformaton, as measured by the Fréchet topologcal sze of the message space, that s requred to guarantee an nformatonally decentralzed mechansm to realze Pareto effcent allocatons over the class of publc goods economes. Theorem 1 shows that any smooth nformatonally decentralzed mechansm that realzes Pareto effcent allocatons on a class of publc goods economes that ncludes a test famly of Cobb- Douglas utlty functons and quadratc producton functons as a subclass has a message space of dmenson no smaller than (L + K 1)I + (L + K)J, where I s the number of consumers, J 3

5 s the number of frms, L s the number of prvate goods, and K s the number of publc goods. Second, we establsh the nformatonal optmalty of the Lndahl mechansm. Theorem 2 shows that the lower bound s exactly the sze of the message space of the Lndahl mechansm, and thus any smooth nformatonally decentralzed mechansm that realzes Pareto effcent allocatons over the class of publc goods economes n whch Lndahl equlbra exst has a message space of dmenson no smaller than the one for the Lndahl allocaton mechansm, and thus the Lndahl mechansm s nformatonally the most effcent process among smooth prvacy preservng and non-wasteful resource allocaton mechansms. Thrd, we show that the Lndahl mechansm s actually the unque nformatonally effcent process that realzes Pareto effcent and ndvdually ratonal allocatons over the class of publc goods economes wth Cobb-Douglas utlty functons and quadratc producton functons. Theorem 3 shows that any nformatonally decentralzed, ndvdually ratonal, and non-wasteful mechansm wth the (L+K 1)I +(L+L)J-dmensonal message space and a contnuous snglevalued statonary message functon s essentally the Lndahl mechansm on the test famly wth Cobb-Douglas utlty functons and quadratc producton functons. Thus, any other economc nsttuton that acheves Pareto effcent and ndvdually ratonal allocatons for publc goods economes must use a message space whose nformatonal sze s bgger than that of the Lndahl mechansm. In an unpublshed paper, Nayak (1982) attempted to establsh the nformatonal effcency of the Lndahl mechansm for publc goods economes. However, he consdered an unusual class of producton technology sets that results n postve outputs wth zero nputs. Nevertheless, to the author s knowledge, there s no Unqueness Theorem on the Lndahl mechansm for publc goods economes n the lterature. It may also be worthwhle to menton that the proof of Lemma 6 reles on the local homology of manfolds. However, no knowledge of algebrac topology s requred to understand the statements of the other lemmas and theorems. The remander of ths paper s as follows. In Secton 2, we provde a formal descrpton of the model. We specfy publc goods economc envronments wth any number of goods and frms, and gve notaton and defntons on resource allocaton, performance correspondence, outcome functon, allocaton mechansm, etc. Secton 3 establshes a lower bound of the sze of the message space that s requred to guarantee that a smooth nformatonally decentralzed mechansm that realzes Pareto effcent allocatons on the class of publc goods economes. Secton 4 gves an Effcency Theorem on the allocatve effcency and nformatonal effcency of the Lndahl mechansm for the class of publc goods economes where Lndahl equlbra exst. Secton 5 gves a Unqueness Theorem that shows that only the Lndahl mechansm s nformatonally effcent over the class of publc goods economes wth Cobb-Douglas utlty functons and quadratc producton functons. Concludng remarks are presented n secton 6. 4

6 2 Model In ths secton we wll gve notaton, defntons, and a framework that wll be used n the paper. 2.1 Publc Goods Economc Envronments Consder publc goods economes wth L prvate goods, K publc goods, I consumers (characterzed by ther consumpton sets, preferences, and endowments), and J frms (characterzed by ther producton sets). It wll often be convenent to dstngush a vector representng a prvate commodty bundle wth an ndex ρ, a vector of publc goods wth an ndex σ. Throughout ths paper, subscrpts are used to ndex consumers or frms, and superscrpts are used to ndex goods unless otherwse stated. By an agent, we wll mean ether a consumer or a producer, thus there are N := I + J 2 agents. 2 For the th consumer, hs characterstc s denoted by e = (X, w, R ), where X R L+K s hs consumpton set, w R L s hs ntal endowments of the prvate goods, and R s a preference orderng on X whch s assumed to be strctly monotoncally ncreasng, convex 3, and contnuous. Let P be the strct preference (asymmetrc part) of R. We assume that there are no ntal endowments of publc goods, and the publc goods wll be produced from prvate goods. For producer, hs characterstc s denoted by e = (Y ) where Y R L+K s hs producton possblty set. We assume that, for = I + 1,..., N, Y s nonempty, closed, convex, and 0 Y. Denote by E the set of the -th agent s characterstcs. An economy s the full vector e = (e 1,..., e I, e I+1,..., e N ) and the set of all such producton economes s denoted by E that s endowed wth the product topology. 2.2 Allocatons Let x = (x ρ, xσ ) denote a consumpton bundle of commodtes by consumer, where xρ s the net exchange of prvate goods and x σ s a consumpton vector of the publc goods by consumer. Denote by x = (x 1,..., x I ) a (net) consumpton dstrbuton. A consumpton dstrbuton x s sad to be ndvdually feasble f (x ρ + w, x σ ) X for all = 1,..., I. Smlarly, let y = (y ρ, yσ ) denote producer s (net) output vector that has postve components for outputs and negatve ones for nputs. Here y ρ s a producton vector of the prvate goods and yσ s a producton vector of the publc goods by producer. Note that, by the assumpton of no publc goods nputs, y σ 0. Denote by y = (y I+1,..., y N ) a producton plan. A producton plan y s sad to be ndvdually feasble f y Y for all = I + 1,..., I + J. An allocaton of the economy e s a vector z := (x, y) R N(L+K) wth (x ρ + w, x σ ) X for = 1,..., I and y Y for = I = 1,..., I + J. An allocaton z = (x, y) s sad to be consstent 2 As usual, vector nequaltes are defned as follows: Let a, b R m. Then a b means a s b s for all s = 1,..., m; a b means a b but a b; a > b means a s > b s for all s = 1,..., m. 3 R s convex f for bundles a, b, c wth 0 < λ 1 and c = λa + (1 λ)b, the relaton a P b mples c P b. Note that the term convex s defned as n Debreu (1959), not as n some recent textbooks. 5

7 f and x σ = I x ρ = =1 N =I+1 N =I+1 y ρ, (1) y σ, = 1,..., I, (2).e., the total net demand for the prvate goods by consumers s equal to the total net supply of prvate goods by producers, and the consumpton of the publc goods by each consumer s equal to the total producton of publc goods by producers, and thus all consumers consume the same amounts of publc goods. An allocaton z = (x, y) s sad to be feasble f t s consstent and ndvdually feasble for all ndvduals. An allocaton z = (x, y) s sad to be Pareto effcent f t s feasble and there does not exst another feasble allocaton z = (x, y ) such that (x ρ + w, x σ )R (x ρ + w, x σ ) for all = 1,..., I and (x ρ + w, x σ )P (x ρ + w, x σ ) for some = 1,..., I. Denote by P (e) the set of all such allocatons. An mportant characterzaton of a Pareto optmal allocaton s assocated wth the followng concept. A prce system (p, q) = (p, q 1,..., q I ) R L+IK + s called a vector of effcency prces for a Pareto optmal allocaton (x, y) f (a) p x ρ +q x σ p x ρ +q x σ for all = 1,..., I and all x such that (x ρ +w, x σ ) X fo and (x ρ + w, x σ )R (x ρ + w, x σ ); (b) p y ρ + ˆq yσ p yρ + ˆq yσ for all y Y, = I + 1,..., N. Here ˆq = I =1 q. Smlar to Debreu (1959, p. 93), we may call (x, y) an equlbrum relatve to the prce system p. It s well known that under certan regularty condtons such as convexty, contnuty, etc, as we assumed n the paper, every Pareto optmal allocaton (x, y) has an effcency prce assocated wth t (see Foley (1970) and Mlleron (1972)). Note that by the strct monotoncty of preferences, we must have (p, q) R L+IK ++ It s perhaps not obvous what the approprate generalzaton of the ndvdual ratonalty concept should be for publc goods economes n the presence of decreasng returns to scale. It s natural to seek a dstrbuton (called the reference dstrbuton) that would play a role analogous to that played by the ntal endowment n the case of constant returns. The reference dstrbuton then should depend on the envronment as well as how much was produced, by whom, and other factors. Thus, smlar to Hurwcz (1979), we ntroduce the followng defnton of ndvdual ratonalty of an allocaton for publc goods economes, whch ncludes the usual ndvdual ratonalty for publc goods economes wth constant returns as a specal case. An allocaton z = (x, y) s sad to be ndvdually ratonal wth respect to the fxed share guarantee structure γ (e; θ) f (x ρ + w, x σ )R (γ (e; θ) + w, 0) for all = 1,..., I. Here, γ (e; θ) 6

8 s gven by γ (e; θ) = N =I+1 θ [p y ρ + ˆq yσ ] p w w, = 1,..., I, (3) where (p; q) s an effcency prce system for e and the θ are non-negatve fractons such that n =1 θ = 1 for = I + 1,..., N. Denote by I θ (e) the set of all such allocatons. Now we defne the Lndahl equlbra of a prvate ownershp economy wth publc goods n whch the -th consumer owns the share θ of the -th producer, and s, consequently enttled to the correspondng fracton of ts profts. Thus, the ownershp structure can be denoted by the matrx θ = (θ ). Denote by Θ the set of all such ownershp structures. An allocaton z = (x, y) = (x 1, x 2,..., x I, y I+1, y I+2,..., y N ) R I(L+K) ++ Y s a θ-lndahl allocaton for an economy e f t s feasble and there s a prce system (p, q) = (p, q 1,..., q I ) wth the prce vector p R L ++ and personalzed prce vectors q R K ++, one for each, such that: (1) p x ρ + q x σ = N =I+1 θ [p y ρ + ˆq yσ ] for all = 1,..., I; (2) (x ρ + w, x σ ) P (x ρ + w, x σ for all = 1,..., I; (3) p y ρ + ˆq yσ p yρ ) mples p xρ + q x σ > N =I+1 θ [p y ρ + ˆq yσ ] + ˆq yσ for all y Y and = I + 1,..., N. Here ˆq = I =1 q. Denote by L θ (e) the set of all such allocatons, and by L θ (e) the set of all such prce-allocaton trple (p, q, z). It may be remarked that, every θ-lndahl allocaton s clearly ndvdually ratonal wth respect to γ (e; θ), and also, by the strct monotoncty of preferences, t s Pareto effcent. Thus we have L θ (e) I θ (e) P (e) for all e E. 2.3 Allocaton Mechansms Let Z = {(x, y) R (L+K)(I+J) : I =1 xρ = N =I+1 yρ & x σ = N =I+1 yσ ( = 1,..., I)} and let F be a socal choce rule,.e., a correspondence from E to Z. Followng Mount and Reter (1974), a message process s a par M, µ, where M s a set of abstract messages and called message space, and µ : E M s a statonary or equlbrum message correspondence that assgns to every economy e the set of statonary (equlbrum) messages. An allocaton mechansm (process) s a trple M, µ, h defned on E, where h : M Z s the outcome functon that assgns every equlbrum message m µ(e) to the correspondng trade z Z. An allocaton mechansm M, µ, h, defned on E, realzes the socal choce rule F, f for all e E, µ(e) and h(m) F (e) for all m µ(e). In ths paper, nformatonal propertes wll be nvestgated for a class of mechansms that realze Pareto effcent outcomes. Let P(e) be a set of Pareto effcent allocatons for e E. An allocaton mechansm M, µ, h s sad to be non-wasteful on E wth respect to P f for all e E, µ(e) and h(m) P(e) for all m µ(e). If an allocaton mechansm M, µ, h s non-wasteful on E wth respect to P, the set of all Pareto effcent outcomes, then t s sad smply to be non-wasteful on E. 7

9 An allocaton mechansm M, µ, h s sad to be prvacy-preservng or nformatonally decentralzed on E f there exsts a correspondence µ : E M for each such that µ(e) = n =1 µ (e ) for all e E. Thus, when a mechansm s prvacy-preservng, each ndvdual s messages are dependent on the envronments only through the characterstcs of the ndvdual and the ndvdual does not need to know the characterstcs of the other ndvduals. Remark 1 Ths mportant feature of the communcaton process mples that the so called crossng condton has to be satsfed. Mount and Reter (Lemma 5, 1974) showed that an allocaton mechansm M, µ, h s prvacy-preservng on E f and only f for every and every e and e n E, µ(e) µ(e ) = µ(e, e ) µ(e, e ), where (e, e ) = (e 1,..., e 1, e, e +1,..., e N ),.e., the th element of e s replaced by e. Thus, f two economes have the same equlbrum message, then any crossed economy n whch one agent from one of the two ntal economes s swtched wth the agent from the other must have the same equlbrum message. Hence, for a gven mechansm, f two economes have the same equlbrum message m, the mechansm leads to the same outcome for both, and further, ths outcome must also be the outcome of the mechansm for any of the crossed economes because of the crossng condton. Let M, µ, h be an allocaton mechansm on E. The statonary message correspondence µ s sad to be locally threaded at e E f t has a locally contnuous sngle-valued selecton at e. That s, there s a neghborhood N(e) E and a contnuous functon f : N(e) M such that f(e ) µ(e ) for all e N(e). The statonary message correspondence µ s sad to be locally threaded on E f t s locally threaded at every e E. 2.4 The Lndahl Process We now gve a prvacy-preservng process that realzes the Lndahl correspondence L θ, and n whch messages consst of prces and trades of all agents. To do so, we restrct ourselves to the subset, denoted by E L, of producton economes on whch L(e) for all e E L. For convenence, n ths secton, we normalze the prce system by makng the frst prvate goods the numerare so that p 1 = 1. by Defne the demand correspondence of consumer ( = 1,..., I) D : R L ++ R IK ++ Θ R J + E D (p, q, θ, π I+1,..., π N, e ) = (4) N {x : (x ρ + w, x σ ) X, p x ρ + q x σ = θ π =I+1 (x ρ + w, x σ ) P (x ρ + w, x σ ) mples p x ρ + q x σ > where π s the proft of frm ( = I + 1,..., N). N =I+1 θ π, } (5) 8

10 Defne the supply correspondence of producer ( = I + 1,..., N) S : R L ++ R IK ++ E by S (p, q, e ) = {y : y Y, p y ρ + ˆq yσ p y ρ + ˆq yσ y Y }. (6) Note that (p, q, x, y) s a θ-lndahl equlbrum for economy e wth the prvate ownershp structure θ f p R L ++, q R IK ++, x D (p, q, θ, p y ρ I+1 + ˆq yσ I+1,..., p yρ N + ˆq yσ N, ) for = 1,..., I, y S (p, q, e ) for = I + 1,..., N, and the allocaton (x, y) s consstent. The Lndahl process M L, µ L, h L s defned as follows. Defne M L = R L ++ R IK ++ Z. Defne µ L : E M L by µ L (e) = where µ L : E M L s defned as follows: N µ L (e ), (7) =1 (1) For = 1,..., I, µ L (e ) = {(p, q, x, y) : p R L ++, q R IK ++, x D (p, q, θ, p y ρ I+1 +ˆq yσ I+1,..., p yρ N +ˆq yσ N, e ), I =1 xρ = N =I+1 yρ, and xσ = N =I+1 yσ, = 1,..., I}. (2) For = I + 1,..., N, µ L (e ) = {(p, q, x, y) : p R L ++, q R IK ++, y S (p, q, e ), I =1 xρ = N =I+1 yρ, and xσ = N =I+1 yσ, = 1,..., I}. Thus, we have µ L (e) = L θ (e) for all e E. Fnally, the Lndahl outcome functon h L : M L Z s defned by whch s an element n L θ (e). h L (p, q, x, y) = (x, y), (8) The Lndahl process can be vewed as a formalzaton of resource allocaton, whch s nonwasteful and ndvdually ratonal wth respect to the fxed share guarantee structure γ (e; θ). The Lndahl message process s prvacy-preservng by the constructon of the Lndahl process. Remark 2 For a gven prvate ownershp structure matrx θ, snce an element, m = (p, q, x 1,..., x I, y I+1,..., y N ) R L ++ R IK ++ R N(L+K), of the Lndahl message space M L satsfes the condtons p 1 = 1, I =1 xρ = N =I+1 yρ, xσ = N =I+1 yσ, p xρ + q x σ = N =I+1 θ [p y ρ + ˆq yσ ] for = 1,..., I, and one of these equatons s not ndependent, any Lndahl message s contaned wthn a Eucldean space of dmenson (L+IK +IL+IK +JL+JK) (1+L++IK +I)+1 = (L + K 1)I + (L + K)J and thus, an upper bound on the Eucldean dmenson of M L s (L + K 1)I + (L + K)J. 2.5 Informatonal Sze of Message Spaces The noton of nformatonal sze can be consdered as a concept that characterzes the relatve szes of topologcal spaces that are used to convey nformaton n the resource allocaton process. 9

11 It would be natural to consder that a space, say S, has more nformaton than the other space T whenever S s topologcally larger than T. Ths suggests the followng defnton, whch was ntroduced by Walker (1977). Let S and T be two topologcal spaces. The space S s sad to have as much nformaton as the space T by the Fréchet orderng, denoted by S F T, f T can be embedded homeomorphcally n S,.e., f there s a subspace of S of S whch s homeomorphc to T. Let S and T be two topologcal spaces and let ψ : T S be a correspondence. The correspondence ψ s sad to be nectve f ψ(t) ψ(t ) mples t = t for any t, t T. That s, the nverse, (ψ) 1, of ψ s a sngle-valued functon. A topologcal space M s an n-dmensonal manfold f t s locally homeomorphc to R n. 2.6 Cobb-Douglas-Quadratc Economes To establsh the nformatonal effcency of the Lndahl mechansm, we wll adopt a standard approach that s wdely used n the realzaton lterature: For a set of admssble economes and a smooth nformatonally decentralzed mechansm realzng a socal choce correspondence, f one can fnd a (parametrzed) subset (test famly) of the set such that the subset s of dmenson n, and the statonary message correspondence s nectve, that s, f the nverse of the statonary message correspondence s sngle-valued, then the dmenson of the message space requred for an nformatonally decentralzed mechansm to realze the socal choce correspondence cannot be lower than n on the subset. Thus, t cannot be lower than n for any superset of the subset, and n partcular, for the entre class of economes. It s ths result that was used by Hurwcz (1977), Mount and Reter (1974), Walker (1977), Sato (1981), Calsamgla and Krman (1993) among others to show the mnmal dmenson and thus nformatonal effcency of the compettve mechansm, Lndahl mechansm, and the equal-ncome Walrasan mechansm over the varous classes of economc envronments. It s also ths result that was used by Calsamgla (1977) and Hurwcz (1999) to show the non-exstence of a smooth fnte-dmensonal message space mechansm that realze Pareto effcent allocatons n certan economes wth ncreasng returns and economes wth producton externaltes that result n non-convex producton sets. It s the same result that wll be used n the present paper to establsh the lower bound of the sze of the message space requred for an nformatonally decentralzed and non-wasteful smooth mechansm on the test famly that we wll specfy below, and consequently over the entre class of publc goods economes wth general convex preferences and producton sets. The test famly, denoted by E cq = N =1 Ecq, are a specal class of publc goods economes, where preference orderngs are characterzed by Cobb-Douglas utlty functons, and effcent producton technology are characterzed by quadratc functons. For = 1,..., I, consumer s admssble economc characterstcs n E cq are gven by the set of all e = (X, w, R ) such that X = R L+K +, w > 0, and R s represented by a Cobb- Douglas utlty functon u(, a, c ) wth a R L 1 ++ and c R K ++ such that u(x ρ +w, x σ, a, c ) = 10

12 (x ρ1 + w 1)[ L l=2 (xρl + w l )al ][ K k=1 (xσk ) ck ]. For = I + 1,..., N, producer s admssble economc characterstcs are gven by the set of all e = Y(b, d ) such that Y(b, d ) = {y R L : b 1 y y ρl 1 b l L l=2 [y ρl for l = 2,..., L + bl 2 (yρl )2 ] + K [y σk k=1 + dk 2 (yσk ) 2 ] 0 0 y σk 1 d k for k = 1,..., K}, (9) where b = (b 1,..., bl ) wth bl > 1 for l = 1,..., L, and d = (d 1,..., dk ) wth dd > 1 for k = 1,..., K. It s clear that any economy n E cq s fully specfed by the parameters a = (a 2,..., a I ), b = (b I+1,..., b N ), c = (c 1,..., c I ), and d = (d I+1,..., d N ). Furthermore, producton sets are nonempty, closed, and convex by notng that 0 Y(b, d ) and ther effcent ponts are represented by quadratc producton functons n whch y ρ1 components of y are outputs. s an nput, and all other Gven an ntal endowment w R LI ++, wth w 1 2(L + K 1)J, defne a subset Ēcq of E cq by Ēcq = {e E cd : w = w = 1,..., I}. That s, endowments are constant over Ēcq. A topology s ntroduced to the class Ēcq as follows. Let be the usual Eucldean norm on R L+K. For each consumer, ( = 1,..., I), defne a metrc δ on Ēcq by δ[u (, a, c ), u(, ā, c )] = a ā + c c. Note that, snce endowments are fxed over Ēcq, ths defnes a topology on Ēcq. Smlarly, for each producer, ( = I + 1,..., N), defne a metrc δ on Ēcq by δ[y(b, d ), Y( b, d )] = b b + d d. We may endow Ēcq wth the product topology of the Ē cq ( = 1,..., N) and we call ths the parameter topology, whch wll be denoted by T p. Then t s clear that the topologcal space (Ēcq, T p ) s homeomorphc to the (L + K 1)I + (L + K)J dmensonal Eucldean space R (L+K 1)I+(L+K)J. 3 The Lower Bound of Informatonal Requrements of Mechansms In ths secton we establsh a lower bound (the mnmal amount) of nformaton, as measured by the Fréchet nformaton sze of the message space, that s requred to guarantee that an nformatonally decentralzed mechansm realzes Pareto effcent allocatons on, E, the class of publc goods economes. As usual, to establsh the effcency results, we need to mpose the nterorty assumpton that Pareto effcent allocatons are nteror. A suffcent condton that guarantees nteror outcomes s that a mechansm s ndvdually ratonal. In fact, a mechansm that gves everythng to a sngle ndvdual yelds Pareto effcent outcomes and no nformaton about prces s needed. Thus, gven a class E of economes that ncludes E cq, we defne an optmalty correspondence 11

13 P : E Z such that the restrcton P E cq assocates wth e E cq the set of P(e) of all the Pareto effcent allocatons that assgn strctly postve consumpton to every consumer. The followng lemma, whch s based on the test famly of Cob-Douglas Quadratc economes Ē cq specfed n the above secton, s central n fndng the lower bound of nformatonal requrements of resource allocaton processes by the Fréchet orderng. Lemma 1 Suppose M, µ, h s an allocaton mechansm on the specal class of economes Ēcq such that: () t s nformatonally decentralzed; () t s non-wasteful wth respect to P. Then, the statonary message correspondence µ s nectve on Ēcq. That s, ts nverse s a sngle-valued mappng on µ(ēcq ). Proof. Suppose that there s a message m µ(e) µ(ē) for e, ē Ēcq. It wll be proved that e = ē. Snce µ s a prvacy-preservng correspondence, for all = 1,..., N by Remark 1, and hence, n partcular, µ(e) µ(ē) = µ(ē, e ) µ(e, ē ) (10) m µ(e) µ(ē, e ) (11) for all = 1,..., N. Let z = (x, y) = h(m). Snce the process M, µ, h s non-wasteful wth respect to P, z = h(m) and (11 ) mply that z P(e) P(ē, e ). Snce Cobb-Douglas utlty functons u (x) are strctly quas-concave and producton functons defned by effcent ponts of L l=2 (yρl K k=1 (yσl producton sets, y ρ1 = 1 + bl b 1 2 (yρl )2 ) dk b 1 2 (yσk by the usual Lagrangan method of constraned maxmzaton, z P(e) mples ) 2 ), are strctly convex, a l (xρ1 + w 1) (x ρl + w l) = 1 + b l yρl b 1 l = 2,..., L, = 1,..., I, = I + 1,..., N, (12) and I =1 b 1 y 1 = c k (xρ1 + w 1 ) x σk L l=2 [y ρl = 1 + dk yσk b 1 + bl 2 (yρl )2 ] K [y σk k=1 k = 1,..., K, = I + 1,..., N, (13) + dk 2 (yσk ) 2 ] = I + 1,..., N. (14) (12) and (13) are well-known condtons for Pareto effcency for economes wth publc goods. At Pareto optmalty, (12) means the margnal rate of substtuton between two prvates goods for each consumer should be equal to the margnal rate of techncal substtuton between the two goods for all producers, and (13) means the sum of margnal rate of substtutons between 12

14 a publc good and a prvates good for all consumers should be equal to the margnal rate of techncal substtuton between these two goods for all producers. Smlarly, z P(ē, e ) mples ā l (xρ1 + w 1) (x ρl + w l) = 1 + b l yρl b 1 l = 2,..., L, = 1,..., I, = I + 1,..., N, (15) c k (xρ1 + w 1) x σk + I s c k s(x ρ1 s + w 1 s) x σk s From equatons (12) and (15), we have = 1 + dk yσk b 1 k = 1,..., K, = I + 1,..., N. (16) and from equatons (13) and (16), we have a l = ā l l = 2,..., L, = 1,..., I, (17) and thus we have c k (xρ1 + w 1 ) x σk = ck (xρ1 + w 1 ) x σk (18) c k = c k k = 1,..., K, = 1,..., I, (19).e., a = ā and c = c. and As for producers, z P(ē, e ) mples a l (xρ1 + w 1) (x ρl + w l) = I =1 b1 y 1 = c k (xρ1 + w 1 ) x σk L l=2 [y ρl 1 + b l yρl b1 l = 2,..., L, = 1,..., I, = I + 1,..., N, (20) = 1 + d k yσk k = 1,..., K, = I + 1,..., N, (21) b1 + bl 2 (yρl )2 ] From equatons (12) and (20), we derve K [y σk + k=1 d k 2 (yσk ) 2 ] = I + 1,..., N. (22) b 1 = 1 + bl yρl b1 1 + b l yρl l = 2,..., L, ; = I + 1,..., N. (23) Also, from equatons (13) and (21), we derve b 1 = 1 + dk yσk b1 1 + d l yσk k = 1,..., K, = I + 1,..., N. (24) From equatons (14) and (22), we derve b 1 L l=2 = (1 + bl b1 2 yρl )yρl + K k=1 (1 + dk L l=2 (1 + b l 2 yρl )yρl + K k=1 (1 + d k 2 yσk 2 yσk )y σk )y σk = I + 1,..., N. (25) 13

15 Thus, from equatons (23) and (25), we have 1 + b l yρl L l=2 (1 + bl 2 yρl )yρl + K k=1 (1 + dk 2 yσk )y σk = 1 + b l yρl L l=2 (1 + b l 2 yρl )yρl + K k=1 (1 + d k for l = 2,..., L, = I + 1,..., N, and from equatons (24) and (25), we have 2 yσk )y σk (26) 1 + d k yσk L l=2 (1 + bk 2 yρl )yρl + K k=1 (1 + dk for l = 2,..., L, = I + 1,..., N. 2 yσk )y σk = 1 + d k yσk L l=2 (1 + b l 2 yρl )yρl + K k=1 (1 + d k 2 yσk Multplyng y ρl on the both sdes of equaton (26) and y σk on the both sdes of equaton (27), and then makng summatons oven these two equatons, we have )y σk (27) L l=2 (1 + bl yρl )yρl + K k=1 (1 + dk yσk L l=2 (1 + bl 2 yρl )yρl + K k=1 (1 + dk 2 yσk )y σk )y σk for = I + 1,..., N. Smplfyng equaton (28), we have = L l=2 (1 + b l yρl )yρl + K k=1 (1 + d k yσk L l=2 (1 + b l 2 yρl )yρl + K k=1 (1 + d k 2 yσk )y σk )y σk (28) L l=2 b l (y ρl )2 + K k=1 d k (y σk ) 2 = L l=2 bl (y ρl )2 + Multplyng 1/2 and addng L l=2 yρl + K k=1 yσk applyng equatons (14) and (22), we have b 1 y ρ1 K k=1 d k (y σk ) 2 = I + 1,..., N. (29) on the both sdes of equaton (29), and then = b 1 y ρ1 = I + 1,..., N, (30) whch mples b 1 = b 1 = I + 1,..., N. (31) Fnally, from equatons (23), (24) and (31), we have b l = b l l = 2,..., L, = I + 1,..., N. (32) and Thus, we have proved and d k = d k k = 1,..., K, = I + 1,..., N. (33) b = b = I + 1,..., N, (34) d = d = I + 1,..., N, (35) whch means b = b and d = d. Thus, equatons (17), (19), (34), and (35) mean that e = ē. Consequently, the nverse of the statonary message correspondence, (µ) 1 s a sngle-valued mappng from µ(ēcq ) to Ēcq. Q.E.D. The followng theorem establshes a lower bound of the Fréchet orderng nformatonal sze of messages spaces of any smooth allocaton mechansm that s nformatonally decentralzed and non-wasteful over the class of economes E. 14

16 Theorem 1 (Informatonal Boundedness Theorem) Suppose that M, µ, h s an allocaton mechansm on the class of publc goods economes E such that: () t s nformatonally decentralzed; () t s non-wasteful wth respect to P; () M s a Hausdorff topologcal space; (v) µ s locally threaded at some pont e Ēcq. Then, the sze of the message space M s at least as large as R (L+K 1)I+(L+K)J, that s, M F R (L+K 1)I+(L+K)J. Proof. As was noted above, Ē cq s homeomorphc to R (L+K 1)I+(L+K)J. Hence, t suffces to show M F Ē cq. By the nectveness of Lemma 1, we know that the restrcton µ Ē cq of the statonary message correspondence µ to Ēcq s an nectve correspondence. Snce µ s locally threaded at e Ēcq, there exsts a neghborhood N(e) of e and a contnuous functon f : N(e) M such that f(e ) µ(e ) for all e N(e). Then f s a contnuous necton from N(e) nto M. Snce µ s an nectve correspondence from Ēcq nto M, thus f s a contnuous one-to-one functon on N(e). Snce Ēcq s homeomorphc to R (L+K 1)I+(L+K)J, there exsts a compact set N(e) N(e) wth nonempty nteror pont. Also, snce f s a contnuous one-to-one functon on N(e), f s a contnuous one-to-one functon from the compact space N(e) onto a Hausdorff topologcal space f( N(e)). Hence, t follows that the restrcton f N(e) s a homeomorphc mbeddng on N(e) by Theorem 5.8 n Kelley (1955, p. 141). Choose an open ball N(e) N(e). Then N(e) and f( N(e)) are homeomorphc by a homeomorphsm f N(e) : N(e) f( N(e)). Ths, together wth the fact that Ēcq s homeomorphc to ts open ball N(e), mples that Ēcq s homeomorphc to f( N(e)) M, mplyng that µ L (E L ) = F Ē cd can be homeomorphcally mbedded n µ(e L ). Hence, t follows that M F Ē cd = F R (L+K 1)I+(L+K)J. Q.E.D. 4 Informatonal Effcency of Lndahl Mechansm In the prevous secton, we found that the lower bound of the Fréchet nformatonal sze of message spaces for smooth allocaton mechansms that are prvacy-preservng and non-wasteful over the class E of publc goods economes that ncludes Ēcq s the (L + K 1)I + (L + K)J-dmensonal Eucldean space R (L+K 1)I+(L+K)J. In ths secton we assert that the lower bound s exactly the sze of the message space of the Lndahl mechansm, and thus the Lndahl mechansm s nformatonally effcent among all smooth resource allocaton mechansms that are nformatonally decentralzed and non-wasteful over the set E L of producton economes on whch L(e) for all e E L. From Remark 2, we know that the upper bound dmenson of the message space of the Lndahl mechansm s also (L + K 1)I + (L + K)J. As a result, f we can show that ths 15

17 upper bound can be reached on the restrcton of the message space of the Lndahl mechansm to the test famly Ēcq of Cobb-Douglas Quadratc economes,.e., f we can show that µ L Ē cq s homeomorphc to the (L + K 1)I + (L + K)J-dmensonal Eucldean space R (L+K 1)I+(L+K)J, then we know that the Fréchet nformatonal sze of the message space of the Lndahl mechansm s R (L+K 1)I+(L+K)J and thus the Lndahl mechansm s nformatonally effcent among all resource allocaton mechansms that are nformatonally decentralzed and non-wasteful over the class of economes n whch Lndahl equlbra exst. Hence, to show the nformatonal effcency of the Lndahl mechansm, t suffces for us to show that ths upper bound can be actually reached on the test famly of economes for the Lndahl mechansm. We wll frst state the followng lemmas that shows that the Lndahl mechansm s snglevalued and contnuous so that t s locally threaded on the test famly set Ēcq of Cobb-Douglas Quadratc economes. Lemma 2 For any gven prvate ownershp structure matrx θ, every economy n Ēcq has a unque θ-lndahl equlbrum,.e., L θ (e) s a sngle-valued mappng from Ēcq to Z. Proof. To show the exstence and unqueness of θ-lndahl equlbrum, we want to frst derve the supply and demand functons of agents. Produce ( = I + 1,..., N) chooses hs producton plan so as to maxmze proft wthn Y(b, d ). Thus, he solves the followng proft maxmzng problem: max[p y ρ + ˆq yσ ] subect to b 1 y ρ1 + L l=2 [y ρl + bl 2 (yρl )2 ] + K [y σk k=1 + dk 2 (yσk ) 2 ] = 0, (36) and 0 y ρl 0 y σk 1 b l 1 d k for all l = 2,..., L, for all k = 1,..., K. An nteror soluton y must satsfy the followng frst-order condtons: p 1 = λ b 1 (37) p l = λ (1 + b l y l ), l = 2,..., L, (38) where λ s a Lagrange multpler. From (37) and (38), and the restrctons 0 y ρl l = 2,..., L and 0 y σk 1 for k = 1,..., K, we can obtan the supply functons d k y ρl (p, q) = 1 b l b 1 pl 1 b l p1 b l f pl p 1 f 1 b 1 0 f pl p b 1 < pl p 1 1 b 1 < 2 b 1 1 b l for

18 for l = 2,..., L, y σk (p, q) = 1 d k b 1 qk 1 d k p1 d k f qk p 1 f 1 b 1 0 f qk p 1 for k = 1,..., K and = 1,..., I, and thus, by (36), 2 b 1 < qk p 1 1 b 1 < 2 b 1 y ρ1 (p, q) = 1 b 1 L l=2 [y ρl (p, q) + bl 2 (yρl (p, q))2 ] 1 b 1 K [y σk k=1 (p, q) + dk 2 (yσk (p, q)) 2 ]. (39) It may be remarked that 3(L + K 1)/2 y 1 l = 2,..., L, f 1 < pl < 2, then y ρl b 1 p 1 b 1 (p, q) = b1 pl 1 b l p1 b l by notng that b l y ρl (p, q) + bl 2 (yρl (p, q))2 = = > 1 and b1 pl p 1 < 2. If b l 2 (y ρl (p, q))2 = < 3/2. If pl b l 2b l p 1 2,..., L, 0 y ρl (p, q) + bl 2 (y ρl 0 y σk [ 1 + bl [ 1 + bl 2 = 1 2b l = 1 2b l 1 b 1 p l p 1 2 b 1 (p, q) 0 for all (p, q) RL+K ++. Indeed, for 2 yρl < 1 b l ] (p, q) ( b 1 p l whch means b1 pl p 1 y ρl (p, q) )] [ b 1 p l < 2, and thus b l 1 p1 b l b l 1 p1 b l ) ( (1 + b1 pl b 1 p l ) p 1 p 1 1 [ (b 1 p l ) 2 1] < 3/2 (40) p 1, then y ρl (p, q) = 1 b l ] and thus y ρl (p, q) +, then y ρl (p, q) = 0 by (39). Thus, for each l = (p, q))2 < 3/2 for all (p, q) R++ L+K. Smlarly, we can show. Therefore by (39), we have (p, q) < 3/2 for k = 1,..., K and for all (p, q) R L+K ++ (p, q) 0 for all (p, q) R L+K 3(L + K 1)/2 < y ρ1 ++. Consumer (for = 1,..., I) chooses hs consumpton so as to maxmze hs utlty subect to hs budget constrant. Snce all utlty functons are Cobb-Douglas, the net demand functons for prvate goods are gven by x ρ1 (p, q) = and x ρl a (p, q) = 1 p 1 (1 + L l=2 al + L k=1 ck ) l p l (1 + L l=2 al + L k=1 ck ) p w + p w + N =I+1 N =I+1 θ [p y ρ (p, q) + ˆq yσ (p, q)] w 1, (41) θ [p y ρ (p, q) + ˆq yσ (p, q)] w l (42) for l = 2,..., L. The demand functons for publc goods are gven by x σk c k N (p, q) = q k(1 + L l=2 al + p L k=1 ck ) w + θ [p y ρ (p, q) + ˆq yσ (p, q)]. (43) =I+1 17

19 Defne the aggregate net excess demand functon for prvate goods by ẑ ρ (p, q) = I =1 x ρ (p, q) N =I+1 y ρ (p, q). (44) Defne the -the consumer s excess demand functon for publc goods by for = 1,..., I. ẑ σ (p, q) = x σ (p, q) N =I+1 y σ (p, q) (45) To show the exstence and unqueness of Lndahl equlbrum, we defne a transformed economy e wth only prvate commodtes such that there s one-to-one correspondence between the transformed prvate goods economy e and the orgnal publc goods economy e. Ths approach s standard and has been adopted by Foley (1970) and Mlleron (1972) to show the exstence of Lndahl equlbrum. Extend the commodty space by consderng each consumer s bundle of publc goods as a separate group of commodtes. In ths KI + L space, extend the sets X by wrtng zeros for all publc good components not correspondng to the -th consumer. The transformed producton sets are defned by Y (b, d ) = {(y ρ, yσ 1 ),..., (yρ, yσ I ) : yσ 1 =... = y σ I = yσ & (y ρ, yσ ) Y(b, d )} for = I + 1,..., I + J. Notce that, snce every consumer s budget constrant holds wth equalty, and the demand and supply functons are clearly contnuous, the aggregate excess demand functon ẑ(p, q) = (ẑ ρ, ẑ σ 1,..., ẑσ I ) s contnuous and satsfes Walras s Law,.e., p ẑρ (p, q) + q ẑ σ (p, q) = 0 for all p R L ++. Thus, by the exstence theorem on Walrasan equlbrum (cf. Varan (1992)), there exsts some (p, q) R L ++ such that ẑ(p, q) 0, whch means (p, q, ẑ(p, q)) s a θ-walrasan equlbrum for the transformed economy e and hence (p, q, ẑ(p, q)) s a θ-lndahl equlbrum for the orgnal publc goods economy e Ēcq. 4 Now we show that every economy e Ēcq has a unque θ-lndahl equlbrum, or, t s equvalent to show that the correspondng transformed economy e has a unque θ-walrasan equlbrum. For ths, t suffces to show that all goods for the economy e are gross substtutes at any prce (p, q) R L+IK ++,.e., an ncrease n prce, s, brngs about an ncrease n the excess demand for good t. When ẑ s dfferentable, the gross substtutes condton becomes ẑt (p,q) p > 0 s for t s. 4 Another way to show the exstence of a θ-lndahl equlbrum s to apply the exstence theorem of Mlleron (1972) drectly by notng that Y s closed and convex, 0 Y, ( R L +) Y and Y ( Y ) {0}. 18

20 For each = I + 1,..., N, from (39), f 1 b 1 < pl p 1 < 2 b 1, we have y ρl (p, q) p l = b1 b l > 0 l = 2,..., L, (46) p1 y ρl (p, q) p s = 0 s l, l 1, (47) y ρl (p, q) p 1 = b1 pl b l < 0 l = 2,..., L, (48) (p1 ) 2 y ρl (p, q) q k = 0 = 1,..., I, k = 1,..., K, l = 2,..., L. (49) From (39), f 1 b 1 < ql p 1 < 2 b 1, we have y σk (p, q) q k y σk (p, q) q t = b1 d k > 0 = 1,..., I, k = 1,..., K, (50) p1 = 0 t k, k = 1,..., K, = 1,..., I, (51) y σk (p, q) p 1 = b1 qk d k < 0 k = 1,..., K, = 1,..., I, (52) (p1 ) 2 y σk (p, q) p s = 0 s = 1,..., L, k = 1,..., K. (53) As a result, from (39), (48), and (52), we have When pl p 1 y ρ1 (p, q) p 1 = 1 b 1 L [1 + b l y ρl ] yρl p 1 1 b 1 l=2 K [1 + d k y σk k=1 ] yσk > 0, (54) p1 y ρ1 (p, q) p l = 1 b 1 [1 + b l y ρl ] yρl p l < 0, l = 2,..., L, (55) y ρ1 (p, q) q k 1 b 1 (resp. = 1 b 1 q k p 1 [1 + d k y σk 1 ) or pl b 1 p 1 ] yσk q k 2 b 1 < 0 k = 1,..., K, = 1,..., I. (56) (resp. q k p 1 2 b 1 ), y ρl (p, q) (resp. yσk (p, q)) are constant functons for l = 2,..., L (resp. for k = 1,..., K and = 1,..., I). Thus, y ρl (p, q) s a nonncreasng functon n p s and q t for any l s, t = 1,..., K, and any (p, q) RL+IK ++, and y σk (p, q) s a nonncreasng functon n q t and p l for any k t, l = 1,..., L, and any (p, q) R L+IK ++. Note that, by Hotellng s Lemma (cf. Varan (1992, p. 43), [p yρ (p,q)+ˆq yσ (p,q)] p s and [p yρ (p,q)+ˆq yσ (p,q)] q t = y σt (p, q). Also, note that yρl 0 for l = 2,..., L, y σk = y ρs (p, q), 0 for k = 1,..., K, and 3(L + K 1)/2 < y ρ1 0 for all = 1,..., I. Then, for each = 1,..., I, from (41), we have x ρl (p, q) p s = a l p l (1 + L l=2 al + L k=1 ck ) 19 w s + N =I+1 θ y ρs (p, q) > 0 (57)

21 for l s, s 1 by notng that y ρs (p, q) 0 for all s = 2,..., L, x ρl (p, q) q t = a l p l (1 + L l=2 al + L k=1 ck ) w s + N =I+1 θ y σt (p, q) > 0 (58) for t = 1,..., K, = 1,..., I by notng that y σk (p, q) 0 for all k = 1,..., K, and x ρl (p, q) p 1 = > a 1 p l (1 + L l=2 al + L k=1 ck ) a l p l (1 + L l=2 al + L k=1 ck ) w 1 + w 1 + N =I+1 N =I+1 θ y ρ1 (p, q) θ ( 3(L + K 1)/2) > a l p l (1 + L l=2 al + [2(L + K 1)J 3(L + K 1)J/2] > 0 (59) L k=1 ck ) for l 1. Smlarly, we can show that xσk and xσk (p,q) q t v (p,q) > 0 for all v = 1,..., I, t k, and k = 1,..., K, p > 0 for all s = 1,..., L and k = 1,..., K. Thus, the net demand functon x s s (p, q) s an ncreasng functon and the supply functon y s (p, q) s nonncreasng functon n prce t s and for every (p, q) R L+IK ++. Therefore, an ncrease n prce, t, brngs about an ncrease n the excess demand for good s, and thus all goods are gross substtutes. Hence, the θ-walrasan equlbrum must be unque for the transformed economy e (cf. Varan (1992)), and consequently θ-lndahl equlbrum s unque for the orgnal economy e. Q.E.D. Lemma 3 Let µ cq be the Lndahl equlbrum message correspondence on Ēcq. The µ cq s a contnuous functon. Proof. By Lemma 2, we know µ cq = (p, x, y) s a (sngle-valued) functon. Also, from (39), (39), (39), (41), (42), and (43), we know that the demand functon x(p, q; ξ) and supply functon y(p, q; ξ) are contnuous n (p, q) and ξ := (a, b, c, d). So we only need to show the prce system (p, q) s a contnuous functon on Ēcq. Snce the demand functon x(p, q; ξ) and supply functon y(p, q; ξ) are homogenous of degree zero n (p, q), we can normalze the prce system as an element n the compact smplex set L+IK 1 = {(p, q) R L+IK + : L l=1 p + I =1 K k=1 qk = 1}. Let {e(k)} be a sequence n Ēcq and e(k) e Ēcq. Snce any economy n Ēcq s fully specfed by the parameter vector ξ, e(k) e mples ξ(k) ξ. Let µ cq = (p, q, x(p, q; ξ), y(p, q; ξ)) and µ cq (k) = (p(k), q(k), x(p(k), q(k); ξ(k)), y(p(k), q(k); ξ(k))). 20

22 Then we have, ẑ(p, q; ξ) = 0 and ẑ(p(k); ξ(k)) = 0, e.., I N x ρ (p, q; ξ) = =1 x σ (p, q; ξ) = =I+1 N =I+1 I N x ρ (p(k); ξ(k)) = =1 x σ (p(k); ξ(k)) = =I+1 N =I+1 y ρ (p, q; ξ), (60) y σ (p, q; ξ) = 1,..., I, (61) y ρ (p(k); ξ(k)), (62) y σ (p(k); ξ(k)) = 1,..., I. (63) Snce the sequence {p(k)} s contaned n the compact set L+IK 1, there exsts a convergent subsequence {p(k t ), q(k t )} whch converges to, say, ( p, q) L+IK 1 and ẑ(p(k t ), q(k t ); ξ(k t )) = 0. Snce x (p(k), q(k); ξ(k)) and y (p(k), q(k); ξ(k)) are contnuous n ξ, ẑ(p, q; ξ) s contnuous n ξ and thus we have ẑ(p(k t ), q(k t ); ξ(k t )) ẑ( p, q, ξ) as k t and ξ(k t ) ξ. However, snce every e Ēcq has the unque θ-lndahl equlbrum prce system (p, q) whch s completely determned by ẑ(p, q; ξ) = 0, so we must have p = p and q = q. Q.E.D. Lemma 4 Let µ L be the Lndahl equlbrum message correspondence on E L, where E L E so that L(e) for all e E L. Then µ L (E L ) s homeomorphc to Ēcq. Proof. Let µ cq be the restrcton of µ L to Ēcq. By Lemmas 2 and 3, we know that µ cq s a contnuous functon. We want to show that the nverse of µ cq, (µ cq ) 1 s also a functon. Let m µ cq (Ēcq ) and let e, e (µ cq ) 1 (m). Then m µ cq (e) µ cq (e ) = µ c (e) µ c (e ) = µ c (e, e ) µ c (e, e ) for all = 1,..., N by Remark 1. Let z = h cd L(Ēcq ) be the Lndahl outcome functon. Snce u s monotoncally ncreasng, we know z s Pareto effcent by the Frst Theorem of Welfare Economcs. Then, the allocaton process M L, µ cq, h cq s prvacy-preservng and non-wasteful over Ēcq wth respect to P. Then, by Lemma 1, e = e and thus (µ cq ) 1 s a functon. Therefore, µ cq s a contnuous one-to-one functon on Ēcq. Snce every e s fully characterzed by (a, b, c, d) R (L+K 1)I+(L+K)J ++, Ē cq s homeomorphc to the fnte-dmensonal Eucldean space R (L+K 1)I+(L+K)J. Thus, t must be homeomorphc to any open ball centered on any of ts ponts, and also locally compact. It follows that for any e Ēcq, we can fnd a neghborhood N(e) of e and a compact set N(e) N(e) wth a nonempty nteror pont. Snce µ cd s a contnuous one-to-one functon on N(e), µ cd s a contnuous one-to-one functon from the compact space N(e) onto an Eucldean (and hence Hausdorff topologcal) space µ cd ( N(e)). Hence, t follows that the restrcton µ cd restrcted to N(e) s a homeomorphc mbeddng on N(e) by Theorem 5.8 n Kelley (1955, p. 141). Choose an open ball N(e) N(e). Then N(e) and µ cd ( N(e)) are homeomorphc by a homeomorphsm µ cd N(e) : N(e) µcd ( N(e)). Ths, together wth the fact that Ēcq s homeomorphc to ts 21

23 open ball N(e), mples that Ēcq s homeomorphc to µ cd ( N(e)) M L, mplyng that µ cd (E L ) can be homeomorphcally mbedded n µ L (E L ). Fnally, by Remark 2, the Lndahl message space M L s contaned wthn an Eucldean space of dmenson (L + K 1)I + (L + K)J. Ths necessarly mples that M L and thus µ L (Ēcq ) s homeomorphc to R (L+K 1)I+(L+K)J because ths restrcton µ cq (Ēcq ) s homeomorphc to R (L+K 1)I+(L+K)J, and consequently, µ L (E L ) s homeomorphc to Ēcq. Q.E.D. From the above lemmas and Theorem 1, we have the followng theorem that establsh the nformatonal effcency of the Lndahl mechansm wthn the class of all smooth resource allocaton mechansms that are nformatonally decentralzed and non-wasteful over the class of producton economes E L on whch L(e) for all e E L. Theorem 2 (Informatonal Effcency Theorem) Suppose that M, µ, h s an allocaton mechansm on the class E L of publc goods economes E L such that: () t s nformatonally decentralzed; () t s non-wasteful wth respect to P; () M s a Hausdorff topologcal space; (v) µ s locally threaded at some pont e Ēcq. Then, the sze of the message space M s at least as large as that of the Lndahl mechansm, that s, M F M L = F R (L+K 1)I+(L+K)J. Proof. Let M L, µ L, h L be the Lndahl mechansm and let Ēcq be the specal class of publc goods economes defned n the prevous secton. Snce L(e) for all e E L, the Lndahl mechansm s well defned. Furthermore, snce u s locally non-satated on E by assumpton, we know (x, y) s Pareto effcent by the Frst Theorem of Welfare Economcs. Then, the Lndahl process M L, µ L, h L s prvacy-preservng and non-wasteful over E L. Also, was noted above, Ē cq s homeomorphc to R (L+K 1)I+(L+K)J. Also by Lemma 4, µ L (Ēcq ) and M L are homeomorphc to Ēcq. Thus, by Theorem 1, we have M F M L = F R (L+K 1)I+(L+K)J. Q.E.D. 5 The Unqueness Theorem In ths secton we establsh that the Lndahl allocaton process s the only nformatonally effcent decentralzed mechansm for publc goods economes among smooth mechansms that acheve Pareto optmal and ndvdually ratonal allocatons. We frst show the followng lemmas. Lemma 5 Suppose that M, µ, h s an allocaton mechansm on the class of publc goods economes E cq such that: 22