WORKING PAPER SERIES

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1 DEPARTMET OF ECOOMCS UVERSTY OF MLA - BCOCCA WORKG PAPER SERES Euilibrium Conribuions and Locally Enoyd Public Goods Luca Corazzini o ovmbr 004 Diparimno di Economia Poliica Univrsià dgli Sudi di Milano - Bicocca hp://dipco.conomia.unimib.i

2 EQULBRUM COTRBUTOS AD LOCALLY EJOYED PUBLC GOODS Luca CORAZZ * Absrac Th main rsuls of h radiional hory of priva provision of public goods in h cas of idnical individuals ar: hr xiss a uniu ash uilibrium parn of conribuions in which vrybody conribus h sam amoun Brgsrom al. [986]; his parn is locally sabl Corns [980]. Undr homohic prfrncs, show ha hs rsuls gnrally no longr hold in h conx of locally noyd public goods. n paricular, whn h symmric ash uilibrium is no h uniu uilibrium parn, i is locally unsabl and hr xiss a las a locally sabl asymmric ash uilibrium. Journal of Economic Liraur Classificaion umbrs: C6, C7, H4. Ky words: Local nracion, Public Goods, ash Euilibria. * School of Economics and Social Sudis, Univrsiy of Eas Anglia, orwich, R4 7TJ, Unid Kingdom. siuo di Economia Poliica Eor Bocconi, Univrsià L. Bocconi, via Gobbi,, 036, Milan, aly. Tlphon numbr: [ o l.corazzini@ua.ac.u]. am graful o profssor Robr Sugdn for his coninuing suprvision. han Luigino Bruni, Giuspp Capplli, Claudio Esaico, Ugo Gianazza, Bndo Gui, Bruno annazzo and Pr Morovic for hlpful suggsions.

3 . TRODUCTO: PUBLC GOODS, ASH EQULBRA AD LOCAL TERACTO is widly accpd ha Brgsrom al. [986] hncforh BBV rprsns on of h mos imporan horical wors on h priva provision of public goods. Thy rfr o a siuaion in which anonymous individuals hav o dcid how o shar hir prsonal incom bwn consumpion and conribuion owards a public good which affcs h wll bing of all h conribuors in h sociy. Assuming ha a individuals ar uiliy maximisrs, b individuals considr ohrs conribuions as indpndn of hir own, c boh public good and priva consumpion ar normal goods, hy prov ha, for any siz of h populaion, hr xiss a uniu ash uilibrium parn of conribuions. Morovr, by using a simpl adusmn procss, Corns [980] hncforh CO provs ha h uniu ash uilibrium parn is also locally sabl. f w assum ha individuals ar idnical in rms of boh prfrncs and incoms, h conclusion is ha h uniu, locally sabl ash uilibrium is rprsnd by a parn of idnical conribuions. L us dfin his parn symmric ash uilibrium hncforh SE. Howvr, in h ral world, volunary conribuion rgims clarly do gnra unual conribuions from popl who ar ual in obciv circumsancs such as prfrncs and incoms i.. som popl fr rid and som do no. How can his mpirical puzzl b solvd? On possibl soluion is o inroduc an assumpion on h alruisic aiud of individuals and allow i o vary across individuals. 3 An alrnaiv soluion which savs h hypohsis of idnical prfrncs and incoms across individuals could consis of modifying h spaial srucur assumd by BBV. n paricular, i appars naural o as whhr h rsuls of uniunss and sabiliy implid by h BBV modl coninu holding in a conx in which agns who ar idnical in rms Th liraur on h priva provision of public goods couns many conribuions. ncluding Samulson s sminal wors Samulson [94], [9], h radr can also rfr o Warr [983], Androni [988], Corns al. [984]. Corns [980] provs local sabiliy undr a simpl coninuous adusmn procss for a mor gnral xrnaliis modl. Th radr can also rfr o Sandmo [980]. 3 An xampl is h hory of warm-glow giving. S for xampl Androni [989], [990], Diamond [003].

4 of boh prfrncs and incoms prsn h following characrisics: a hy diffr from ach ohr for h nighbourhood hy blong o, and b hy noy h lvl of public good collcd by h mmbrs of his nighbourhood. 4 Alhough hr ar many horis which mov in his dircion, non of hm can b considrd as an xnsion of h BBV modl o h conx of locally noyd public goods. On of h firs amps o a ino accoun h spaial localisaion of public goods is Tibou s modl of local xpndiurs Tibou [96]. n his modl individuals can frly mov across communiis which ar ndowd wih a fixd uaniy of som public good and which ar characrisd by an opimal communiy siz. Communiis blow h opimal siz ry o arac nw rsidns o lowr avrag coss whil communiis abov h opimal siz do h opposi. Givn his srucur, h conomy would convrg auonomously owards an uilibrium sinc [ ] xcp whn h sysm is in uilibrium, hr will b a subs of consumr-vors who ar disconnd wih h parns of hir communiy. Anohr s will b saisfid. pag 40. is clar ha h modl rfrs o a oally diffrn problm wih rspc h on considrd by BBV. ndd, insad of inroducing h local srucur o xplain how much individuals conribu, Tibou uss i o analys whr individuals dcid o mov o. Mor rlvan for h opic of his papr is h modl buil by Eshl al. [998] hncforh ESS. Thy adap Ellison s modl of local inracion Ellison [993] o by public good gams. Thr ar individuals spacd around a circl. Each individual has o dcid how o conribu o a public good which affcs h wll bing of hos who blong o his nighbourhood. ndividual i s nighbourhood is composd by h firs prson on his righ, h firs prson on his lf and himslf. Each individual can bhav ihr alruisically or goisically. An alruis conribus a uaniy of public good which supplis on uni of uiliy o ach mmbr of his nighbourhood. Th n cos o h alruis for providing ha uaniy of public good is xprssd by a loss of uiliy ual o c > 0. An gois dos no conribu a all and dos no bar any cos. H simply noys h oal amoun of 4 For insanc, h nvironmn is a valid xampl of locally noyd public good. 3

5 public good supplid by h alruiss. A h nd of ach priod, ach individual dcids how o bhav on h basis of a spcific larning procss. n paricular, wih probabiliy µ wih > µ > 0 h imias h bhaviour of hos in his nighbourhood who arnd h highs payoffs whil wih probabiliy µ h rains his sragy. On h basis of hs assumpions, h auhors prov ha alruiss can surviv if hy ar groupd oghr, so ha h bnfis of alruism ar noyd primarily by ohr alruiss, who hn arn rlaivly high payoffs and ar imiad. Morovr alruiss manag o surviv also in h prsnc of muaions ha coninually inroduc goiss ino h populaion. ESS modl significanly diffrs from BBV modl for wp rasons. Firsly, ESS obain h rsul of coxisnc bwn high conribuors and low conribuors assuming ha popl do no prfcly bhav as raional agns. Scondly, individuals can only dcid ihr o conribu fixd uaniy of public good which incrass h uiliy of h nighbours of on uni or o conribu nohing. am going o prsn a vrsion of h BBV modl basd on h sam spaial srucur assumd by ESS. Thr ar individuals who ar disribud around a circl. Thy ar idnical in rms of boh prfrncs and incoms. ndividual s nighbourhood is dfind as h firs individuals on his righ, h firs individuals on his lf and himslf. Th oal conribuion collcd wihin his nighbourhood rprsns h lvl of public good noyd by individual. Finally, individuals ar assumd o b radiional uiliy maximisrs. This simply mans ha, aing nighbours bhaviour has givn, ach individual dcids how o shar his incom bwn conribuion and consumpion in ordr o maximis his uiliy funcion. Th main conclusion of my modl is ha h inroducion of h assumpion of locally noyd public goods dramaically affcs boh h rsuls of uniunss and sabiliy of h SE implid by BBV modl in a conx characrisd by idnical individuals. n paricular, boh hs rsuls coninu bing ru in siuaions in which h prfrncs for h public good ar unralisically Th xisnc of an voluionary uilibrium characrisd by coxisnc of alruisic and goisic conribuors is also implid by Brgsrom al. [993]. 4

6 srong. Bu no ohrwis, whn uniunss dos no hold, all h uilibria bu h SE ar asymmric ash uilibrium parns hncforh AE in which high conribuors coxis wih low conribuors. Coxisnc is a naural rsul in h sns ha, rahr han bing causd by ad hoc assumpions ihr on h bhaviour of agns or on hir prsonal characrisics, i appars as a dirc consunc of h spaial srucur assumd in my modl. Th papr is organisd as follows. n scion, prsn h assumpions of h modl. n scion, prov an xisnc horm for a gnric siz of h nighbourhood. n scion V, sudy h condiions undr which h SE coninus bing uniu and locally sabl whn individual s nighbourhood is dfind as h firs individual on his lf, h firs individuals on his righ and himslf. n scion V, discuss a simpl xampl which highlighs h diffrn implicaions of my modl wih rspc hos of h BBV modl. Finally, in scion V, gnralis h rsuls obaind in scion V o conxs characrisd by largr nighbourhoods.. THE STRUCTURE OF THE MODEL: ASSUMPTOS AD DEFTOS Suppos ha hr ar individuals disribud around a circl. ndividuals ar idnical in rms of boh prfrncs and incoms. Each individual blongs o a nighbourhood. n paricular, individual s nighbourhood is dfind as h firs individuals on his righ, h firs individuals on his lf and himslf, so ha {,,...,,,,...,, } mod. Th prfrncs of individual ar dscribd by a homohic uiliy funcion, U U c, Q, which is sricly incrasing in boh h lvl of priva consumpion, c, and h lvl of public good collcd in his nighbourhood, Q. o ha h hypohss on h uiliy funcion imply h on of normaliy of boh public good and priva consumpion usd by BBV in hir modl. L us assum ha Q is an addiiv funcion of h conribuions of h mmbrs of s nighbourhood, so ha h lvl of public good noyd by individual can b wrin as follows:

7 Q [] s s whr s is h conribuion of individual s. ndividual has o dcid how o shar his incom,, bwn consumpion and conribuion by aing h conribuion of his nighbours as givn and by subcing his dcision o boh a budg consrain and a non ngaiviy consrain. Formally, if w assum ha boh h pric for a uni of priva consumpion and h pric for a uni of public good ar ual o on, h dcisional problm facd by individual can b wrin as follows: max c, s.. U U c 0 c, Q [] Following BBV l us dfin f W h dmand funcion of conribuor whn h is ndowd wih oal walh W Q whr Q is h oal amoun conribud by all h mmbrs of bu. simply rprsns h valu of public good ha individual would choos for diffrn valus of W if h could ignor h non ngaiviy consrain. 6 Givn h assumpion of homohiciy, f W is linar. Thn, i can b rwrin as follows: f W Q [3] Th variabl, wih 0 < <, rprsns h proporion of incom an individual would li o conribu o h public good if nobody conribus and i basically xprsss h imporanc of h public good in h uiliy funcion. For ypical cass ha modl applis o, rasonabl valus of ar vry low. For insanc, in a Cobb-Douglas uiliy funcion of h yp U c Q, w could xpc o b lowr han 0. maning ha h priva consumpion is mor imporan han h public good. 6 n ohr words, f W rprsns h Engl curv of individual. 6

8 Taing ino accoun h non ngaiviy consrain, w obain h following xprssion which rprsns individual s bs rspons o Q : BR max[ Q Q ;0] [4] L m inroduc som simpl dfiniions which will b widly usd hroughou h papr. DEF.. For givn,,, an uilibrium is a vcor of conribuions,..., such ha,. BR n paricular, if, hn,, and h spaial srucur of h conomy coincids wih h on assumd by BBV. Thrfor, h SE is h uniu uilibrium parn of h modl and i is globally sabl. DEF.. A plain of siz m is a s of m conscuiv individuals {,,..., m } who conribu h sam shar of incom. mod mod DEF.3. For a givn, a BBV communiy of siz z is a s of z conscuiv individuals {,,..., z } such ha a z, b individuals mod mod {,...,, z,..., z } ar null conribuors i.. individuals who mod mod mod mod conribu nohing. Th prvious dfiniion implis ha if C is a BBV communiy of siz z for a givn and C, hn a vryon ls in C is s nighbour, b if d is a nighbour of bu d C, hn h is a null conribuor and finally c h lvl of public good noyd by individual is Q. s s C 7

9 n ohr words, for a givn, a BBV communiy of siz z is an isolad subgroup of individuals who noy h sam lvl of public good. This mans ha h conclusions of h BBV modl can b xndd o ach BBV communiy. ow, l us urn o h dynamic srucur of h modl which will b usd o sudy h sabiliy propris of h uilibrium parns. L us suppos ha individuals adus hir conribuion ovr im according o h sam coninuous adusmn procss adopd by CO:, BR &, µ [,, ], µ > 0 [] Whr µ rprsns h spd of adusmn which is assumd o b sam across individuals. n ohr words, individual aduss his conribuion o h uilibrium lvl gradually ovr im. 7. EXSTECE OF A EQULBRUM Suppos ha individual s nighbourhood is composd by h firs individuals on his righ, h firs individuals on his lf and himslf wih. Givn h dfiniion of h nighbourhood, xprssion [4] bcoms: BR max[ s s mod s mod s mod s mod ;0] s, [6] Rgarding h xisnc of an uilibrium, horm prsnd in BBV pag 33 can b innocuously xndd o his conx. PROP.. For any and for any, a ash uilibrium xiss. [Proof in appndix] Th prvious proposiion dos no say anyhing abou h numbr and h sabiliy propris of h uilibria. n h nx scions, shall show ha boh h propris of uniunss and sabiliy 7 Economiss xplain his adusmn mchanism in rms of larning procss. For furhr informaion on larning procsss in gam hory s Fudnbrg al [999]. 8

10 significanly chang whn rmov h assumpion of globally noyd public goods in favour of local noymn. n paricular, shall bgin wih h cas of and, afr having prsnd a simpl xampl, will gnralis h rsuls o any >. V. O UQUEESS AD STABLTY OF THE EQULBRA WHE Whn, h bs rspons funcion of individual bcoms: max[, [7] BR mod mod mod mod ;0] Givn boh h adusmn procss followd by ach individual xprssion [] and xprssion [7], w can spcify h following sysm which dscribs h dynamics of h modl: &, M &, M &, µ {max[ µ {max[, µ {max[,,,,,,,,, ;0],, ;0] ;0],,, } } } [8] Sysm [8] is composd by non linar diffrnial uaions of h firs ordr. Considr h sysm wihou h non ngaiviy consrain. n his cas, h dynamic sysm bcoms linar and i can b rwrin in marix noaion as follows: & µ µ A [9] Whr &,, R, R A µ, and finally: 3 rms [,,0,..., 0, ] R xr circulan [0] Th circulan naur 8 of A is imporan for wo rasons. Firsly, sinc A is symmric, is ignvalus ar all ral. Scondly, sinc A is circulan, i can b diagonalisd hrough h following normalizd Fourir marix F * R xr : F * F [] 8 For furhr rfrncs on circulan marics, s Davis [994]. 9

11 Th gnric lmn of h marix F R x R siuad on h h -h row -h column is: F h, π i h h w [] πi whr i is h imaginary numbr, and w is h -h of h roos of h cycloomic uaion x. Ths roos somims calld h D Moivr umbrs rprsn h coordinas in h complx plan of h vrics of a rgular polygon wih sids and uni radius. For xampl, whn 6, w hav: i w w w 3 r w 6 w 0 r w w 4 Fig.. h coordinas in h complx plan of h vrics of a rgular polygon wih 6 sids and uni radius. h From h graph w hav ha Fh, [ w ] w h mod. Th h -h ignvalu of a circulan marix is obaind by muliplying h firs row of h column of A : A by h h - F. By applying his procdur, w obain a gnral xprssion for h ignvalus of λ h h πi h πi w h w h, h,..., [3] Morovr, hans o h propris of h D Moivr umbrs, w hav ha w h is h complx conuga of w h. 9 Thrfor, xprssion [3] can b rwrin as follows: λ h h wh wh wh con w, h,..., [4] 9 f w h is ral, hn is conuga is islf. 0

12 By using h propris of h D Moivr umbrs, w can prov h following lmma: LEMMA.. L Ο b h s of valus of includd bwn 0 and such ha a las on ignvalu of A is null. 0 L vn and b h dimnsion of h populaion whn h vn numbr of individuals is vn and whn i is rspcivly, wih 4 and. W hav ha: a min / Ο vn ; b Thr xiss a sricly dcrasing succssion of valus { a, a,..., } ha min Ο and lim / [Proof in appndix] a a. 7 a such f Ο and w do no considr h non ngaiviy consrain, hn h SE in which vrybody conribus 3, [] is h uniu uilibrium of sysm [9]. On h ohr hands, if w inroduc h non ngaiviy consrain and w allows for varying in 0,, h SE migh los h propry of uniunss. ndd, alhough i is clar ha, for any and for any, sysm [8] can hav a mos on and only on SE parn, hr ar paricular condiions on ha imply if saisfid h xisnc of ohr AE parns. Bfor xamining hs condiions, l m inroduc hr furhr dfiniions which will b widly usd in his scion. 0 Th cardinaliy of his s dpnds on h siz of h populaion.

13 DEF.4. An up and down uilibrium parn on vn individuals is an uilibrium parn such ha vn vn... vn a if and only if mod 4 mod mod vn vn vn... vn b wih a b. mod 3 mod mod mod DEF.. A pa and hill uilibrium parn on individuals wih an upwards spi is an uilibrium parn such ha if h smalls local minima bar in posiions mod and mod wih mod mod, hn h scond smalls local minima ar in posiions 3 mod and 3 mod wih 3 mod 3 mod, h hird smalls local minima ar in posiions mod and mod wih and so on unil posiions mod and mod mod mod which consiu a plain of siz. DEF.6. A pa and hill uilibrium parn on individuals wih a downwards spi is an uilibrium parn such ha if h smalls local minimum is in posiion i, hn h scond smalls local minima ar in posiions mod and mod wih, h hird smalls local minima ar in posiions 4 mod mod mod and 4 mod wih, and so on unil posiions 4 mod 4 mod mod and mod which consiu a plain of siz. ow, can sa and prov undr which condiions h SE coninus bing h uniu uilibrium parn.

14 PROP.. For and for any 4 conribuions if and only if ihr < / cas. [Proof in appndix], h SE is h uniu uilibrium parn of in h vn cas or < a in h Th prvious proposiion sas ha if and only if ihr / in h vn cas or a in h cas, w can find a las an AE parn in addiion o h SE on. This rsul is ui counrinuiiv sinc i suggss h xisnc of asymmric uilibrium parns of conribuion in a world populad by individuals who ar idnical in rms of prfrncs and incoms. Surprisingly, vn h sabiliy propris of h uilibrium parns dpnd on h valu of. n paricular, h following proposiion sas ha h condiions on such ha h SE is h uniu uilibrium parn coincid wih hos which imply is local sabiliy. PROP.3. For and for any 4, h SE parn of conribuions is sabl if and only if ihr < / in h vn cas or < a in h cas. Morovr, hr xiss a las on AE parn of conribuions which is locally sabl if and only if Ο and ihr > / in h vn cas or > [Proof in appndix] a in h cas. Combining h main implicaions of PROP.. and PROP.3, w can sa h following corollary. 3

15 COR.. For and for any 4, if and only if / in h vn cas or a in h cas, h SE is no h uniu uilibrium parn of conribuions and, morovr, i is locally unsabl. Morovr, if and only if > / in h vn cas or > a in h cas and AE parn which is locally sabl. Ο, hn i is always possibl o find a las an Th prvious corollary sas ha for rasonabl valus of h imporanc of h public good in h uiliy funcion, a h main rsuls on h SE parn of conribuions obaind by BBV and CO no longr hold in a conx characrisd by locally noyd public goods wih, b i is always possibl o find a las on parn of asymmric conribuions which rprsns a locally sabl ash uilibrium. Finally, h following proposiion sas ha, whn h prfrncs for h priva consumpion ar sufficinly srong, i is always possibl o idnify a paricular class of locally sabl AE parns of conribuions which ar obaind by combining of BBV communiis of siz and siz. PROP.4. For and for any 4, if and Ο, hn h gnric combinaion of BBV communiis of siz wih BBV communiis of siz which xhauss and such ha i dos no prsn wo or mor conscuiv BBV communiis of siz rprsns a locally sabl AE parn. [Proof in Appndix] For insanc, if 9 and 0. 7, hn h following parn of conribuions obaind by combining hr BBV communiis of siz wih on BBV communiy of siz is a locally sabl AE: 4

16 * * * * 3 7 * * * * * Fig.. A locally sabl AE whn 9 and V. A LLUSTRATVE EXAMPLE: THE BBV MODEL VS THE LOCALLY EJOYED PUBLC GOODS MODEL WHE Considr a sociy of fiv individuals. L us sar assuming ha so ha,, and w hav h BBV modl. Givn h xprssion of BR, w can spcify h following dynamic sysm composd by fiv non linar diffrnial uaions of h firs ordr:, , 4, 3 3 4, 3, 4 4 3,, ,, , ;0] max[ ;0] max[ ;0] max[ ;0] max[ ;0] max[ µ µ µ µ µ µ µ µ µ µ & & & & & [6] Considr h sysm wihou h non ngaiviy consrain. n his cas, w hav a dynamic sysm composd by linar diffrnial uaions which can b rwrin in marix noaion as follows:, 4, 3,,,, 4, 3,,, µ µ & & & & & [7] L us indica wih BBV A h suard marix of siz which appars in [7]. By imposing h uilibrium condiion 0 &, w obain:

17 ,, 3, 4,, [8] Th ignvalus of ar λ 4, λ λ3 λ4 λ. Thrfor, givn BBV A 0 < <, h drminan of BBV A is diffrn from zro. Th prvious linar sysm has a uniu soluion in which vrybody conribus h sam amoun i.. h SE:, [9] 4 3 Th inroducion of h non ngaiviy consrain dos no add any furhr uilibrium parn. Morovr, by valuaing h Jacobian marix in h SE parn, w obain ha for 0 < < all h ignvalus of BBV A which ar h ignvalus of ngaiv. Thrfor, h SE is locally sabl. BBV A wih opposi sign ar sricly ow, l us assum ha individual s nighbourhood is dfind as h firs individual on his righ, h firs individual on his lf and himslf. n his cas, sysm [8] bcoms: & & & & &,, 3, 4,, µ max[ µ max[ µ max[ µ max[ µ max[ ;0] 3 ;0] 4 ;0] ;0] ;0] 4 µ µ µ µ µ,, 3, 4,, [0] Again, by imposing h uilibrium condiion wihou considring h non ngaiviy consrain, w obain h following linar sysm: ,, 3, 4,, [] 6

18 f and only if h drminan of A is diffrn from zro, h unconsraind sysm has a uniu soluion rprsnd by h SE in which vrybody conribus h sam amoun, [] 3 By using uaion [4], i is possibl o show ha h ignvalus of ar λ 3, A λ λ , λ λ.68a and ha h ignvalus of A ar h ignvalus of A 3 4 wih opposi sign. Clarly, Ο { a }. f and only if a, w hav an infini numbr of pa and hill uilibrium parns in addicion o h SE. Wihou considring h plain, h pa and hill uilibrium parn wih an upwards spi prsns h highs numbr of local minima. Among h infini uilibrium parns of his yp l us considr h on in which, l us say, 0. By solving h linar sysm in which w 4 impos hs wo consrains, w obain ha 3 and. Givn h prvious xprssions, h bs rspons funcions of individuals and 4 imply ha 0 if and only if 4 a. Thrfor, if and only if < a, h SE is h uniu uilibrium parn. On h conrary, if and only if a, h SE is no uniu and i is always possibl o find a las an AE parn. For insanc, h following parn of conribuions rprsns an uilibrium: * * * * 3 4 * Wihou considring h plain, a pa and hill uilibrium parn wih an upwards spi prsns local minima whil a pa and hill uilibrium parn wih a downwards spi prsns local minimum. 7

19 Fig.3. Pa and hill uilibrium parn wih an upwards spi in 3 and smalls local minima ual o zro. ow, l us urn o h sabiliy propris of h uilibria. Th Jacobian marix in h SE coincids wih A. Thrfor, h SE is locally sabl whn all h ignvalus of A ar sricly ngaiv. This holds if and only if 0 < < a. f and only if a, hr is an infini numbr of locally unsabl pa and hill uilibrium parns. Finally, h Jacobian marix in h pa and hill uilibrium parn wih an upwards spi in and smalls local minima ual o zro is: J µ 0 0 [3] f and only if a <, all h ignvalus of J ar sricly ngaiv and, hrfor, < h pa and hill uilibrium parn wih an upwards spi and smalls local minima ual o zro is locally sabl. V. EGHBOURHOODS COMPOSED BY MORE THA THREE DVDUALS > Wha happns whn h siz of h nighbourhood incrass? Suppos ha h nighbourhood of individual is composd by h firs individuals on his righ, h firs individuals on his lf and himslf, wih > and <. n his cas, h bs rspons funcion of individual is xprssd by xprssion [6]. Givn boh h adusmn procss sad in h assumpions and xprssion [6], w can spcify h following dynamic sysm: 8

20 &, M &, M &, µ {max[ µ {max[ µ {max[ s s s s mod, s mod, s mod, s mod, s s mod, s mod, s mod, s mod, s s mod, s mod, s mod, s mod, s ;0] ;0] ;0],,, } } } [4] f w do no considr h non ngaiviy consrain, sysm [4] bcoms a linar dynamic sysm of h yp: & µ µ A [] Whr &,, R, R µ, and A rms [,,...,,0,..., 0,,..., ] R xr circulan [6] rms rms By using h propris of h circulan marix, h xprssion of h ignvalus of A can b wrin as follows: sπi λ R, 0,..., [7] s whr R [ d] is h ral par of h complx numbr d. Spcularly o wha said in scion, if consrain, hn h SE in which vrybody conribus Ο and w do no considr h non ngaiviy, [8] is h uniu uilibrium of sysm [4]. n h ohr cass, h SE rmains h uniu uilibrium parn if and only if spcific condiions on ar saisfid. Morovr, as in h cas of, hs condiions coincid wih hos which allow h SE for bing locally sabl. n paricular, h following proposiion gnraliss h rsuls of uniunss and sabiliy of h SE sad in COR.. o largr nighbourhoods: 9

21 PROP.. L Ο b h s of valus of includd bwn 0 and such ha a las on ignvalu of A is null. L a b h lows valu in Ο. For any > and for any, h SE is h uniu uilibrium parn and i is locally sabl if and only if < a. Morovr, if and only if o find a AE which is locally sabl. [Proof in appndix] and > Ο a, i is always possibl Can w say anyhing ls on h valu of rlaion bwn a? Unforunaly, whn > h monoonic a and h numbr of individuals highlighd in LEMMA.. no longr holds. Howvr, h following propry sas, for any and for any, h xisnc of a lowr bound such ha if is grar han or ual o such a bound, hn h SE in no uniu and unsabl. PROP.6. For any > and for any, if /, hn h SE is no h uniu uilibrium parn and, morovr, i is unsabl. [Proof in appndix] Finally, in ordr o idnify a paricular class of AE parns which ar obaind by combining BBV communiis of siz wih BBV communiis of siz, w can xnd PROP.4. o any >. PROP.7. For any > and for any, if and 0 Ο, hn h gnric combinaion of BBV communiis of siz wih BBV communiis of siz which xhauss rprsns a locally sabl AE parn. [Proof in appndix]

22 V. COCLUSOS AD SUGGESTOS FOR FURTHER RESEARCHES Th samn idnical individuals conribu h sam amoun of rsourcs is h mos naural and rasonabl rsul of h radiional hory of priva provision of public goods. Unforunaly, h ral world inducs public conomiss o b vry scpical wih his implicaion. Volunary conribuion rgims clarly do gnra unual conribuions from popl who ar ual in obciv circumsancs such as prfrncs and incoms. Could w us h framwor of h radiional modls o g rid of his mbarrassing puzzl? Rahr han inroducing hrogniy in prfrncs and incoms, hav shown ha symmry, uniunss and sabiliy which characris h uilibrium parn of conribuions in h radiional modls dramaically dpnd on h implici hypohsis of globally noyd public goods. n paricular, by inroducing h Ellison s local inracion srucur Ellison [993] in h BBV modl Brgsrom al. [986] wih agns who hav idnical homohic prfrncs and ar ndowd wih h sam incom, hav provd ha: a h SE is h uniu uilibrium parn and i is locally sabl if and only if h prfrncs ar unralisically biasd owards h public good; b all h ohr uilibrium parns bu h SE ar characrisd by asymmric conribuions; c if and only if h SE is locally unsabl, i is always possibl o find a locally sabl AE parn; d h prvious rsuls hold for any siz of h nighbourhood such ha i dos no includ h nir populaion. is possibl o idnify a las wo aspcs of my modl which ruir furhr dvlopmns. Firsly, i could b inrsing o analys h rsuls of uniunss and sabiliy of h SE undr diffrn spcificaions of h uiliy funcion which saisfy h minimal ruirmns highlighd by BBV. Scondly, on could ry o rmov h assumpion of idnical individuals assuming ihr hrognous incoms or hrognous prfrncs. Luca Corazzini

23 APPEDX L BR Proof of PROP.. R b h vcor which conains h bs rsponss funcions of h conribuors in h conomy. dfins a coninuous funcion from h compac and convx s { x R : 0 x,,..., } BR Ι o islf. Thrfor, by h Brouwr s Fixd Poin Thorm hr mus xis a fixd poin,...,, which is a ash uilibrium vcor of conribuions. Proof of LEMMA.. Whn is vn, h coordinas of h vrx w ar 0, h gras ignvalu and ha / vn. Thn, xprssion [4] implis boh ha h valu of such ha λ 0 vn λ is vn is h smalls lmn in Ο vn. Whn is, h vrxs w and w whr and ar h minor and h maor ingr of rspcivly ar on h complx conuga of h ohr and, morovr, hy hav h gras ral par in absolu valu. Thrfor, λ and λ ar h gras ignvalus and h valu of such ha λ λ 0 is h smalls lmn in nds o infiniiv, h vrxs succssion of a w and is monoon dcrasing in Ο. L us call his valu a w g closr and closr o h vrx, wih lim /. Morovr, as 0. Thrfor, h a. Proof of PROP.. Th uilibria of h unconsraind sysm [9] ar obaind by imposing is LHS ual o zro, so ha: µ µa 0 [A], marix Ο SE., marix Ο A is no full ran. Thrfor, [A] has an infini numbr of soluions which includs h A is full ran. Thrfor [A.] has a uniu soluion which is ual o h SE:

24 wih A [A] 3 R h gnric lmn of which is. ow, l us idnify AE parns implid by h non ngaiviy consrain. Suppos ha h dimnsion of h populaion is vn. Considr h parn in which vn... vn 0 mod mod, and vn vn... vn. Givn mod 3 mod mod vn mod and vn mod, h bs rply funcion of individual implis ha 0 if and only if /. Thrfor, if and only if < /, h non ngaiviy consrain dos no bind and h SE is h uniu uilibrium parn. Suppos ha h dimnsion of h populaion is o u u ϑ wih and ϑ ; 3 a. Saring from h SE, l o chang from 3. By solving h sysm composd by h bs rspons funcions of h unconsraind conribuors and h condiion u ϑ, w find h xprssion of h uilibrium conribuions s wih s. Thn, givn mod ha ϑ if and only if a and uilibrium parn wih an upwards spi. Sinc mod, h bs rspons funcion of individual implis. Th s of uilibrium conribuions idnifis a pa and hill u ϑ ; 3, hr xiss an infini numbr of uilibrium parns of his yp. Among hm, considr h on such ha h smalls local minima ar rprsnd by wo null conribuors. Bcaus of h non ngaiviy consrain, h condiion on such ha his parn is an d a. By rpaing h prvious procdur wih ϑ ; 3 AE bcoms 0, w nd up wih a pa and hill uilibrium parn wih a downwards spi rprsnd by a null conribuor. Th condiion on such ha i is an uilibrium parn is h sam of h on obaind in h prvious cas. Among h wo AE w hav idnifid, l us considr h on which prsns h highs numbr of local minima wihou considring h, hn ' plaau. By inducion w hav ha, if and only if a > a ' > wih a such ha i is possibl o find an uilibrium parn associad wih ' which prsns a las on furhr null conribuor. ndd, h scond smalls local minima ar ual o zro if and only if a, h hird smalls local minima ar ual o zro if and only if a so on unil a. Whn a, hn for any, and, hr always xiss an uilibrium For xampl, if, a pa and hill uilibrium parn wih an upwards spi prsns local minima wihou considring h plaau whil a pa and hill uilibrium parn wih a downwards spi prsns local minimum wihou considring h plaau. 3

25 parn which is composd by a combinaion of a uniu BBV communiy of siz and siz. 3 BBV communiy of f and only if Ο Proof of PROP.3., a las on ignvalu of marix A is null. Thrfor w hav an infini numbr of unsabl uilibrium parns. f and only if and ihr < / Ο 4 in h vn cas or < a in h cas, h SE is h uniu uilibrium parn. n ordr o sudy h sabiliy of h SE, w dircly sudy h sign of h ignvalus of A. LEMMA.. implis ha hy ar sricly ngaiv. Thrfor h SE is locally sabl. n ordr o sudy h sabiliy propris of h uilibria whn > / in h vn cas or > Ο and ihr a in h cas, w us h linarizaion procdur. ndd, l us rwri h gnric diffrnial uaion of h dynamic sysm [] as follows: { m [ } & ] [A3] µ mod mod whr m is ual o if and only if and i is null if and only if <. mod mod mod mod Th gnric uilibrium parn,..., is locally sabl if and only if all h ignvalus of h Jacobian marix valuad in i ar sricly ngaiv. n paricular, h Jacobian marix J is a suard marix of siz such ha is -h row is obaind by muliplying all h rms ual o which appar in h -h row of m. L us considr h SE. n his cas h Jacobian marix coincids wih wih ihr > / in h vn cas or > A. Thrfor, for A by Ο a in h cas, h SE uilibrium is locally unsabl. ow, suppos ha h dimnsion of h populaion is. L us assum ha is such ha h configuraion which prsns h highs numbr of local minima wihou considring h plaau is a pa and hill uilibrium parn wih an upwards spi. As alrady said, if and only if < a a, w can idnify ohr wo uilibria: a pa and hill uilibrium parn wih a downwards spi rprsnd by a null conribuor and a pa and hill uilibrium parn wih an upwards spi surroundd by wo null conribuors. Th condiion such ha all h ignvalus of h Jacobian marix valuad in h firs uilibrium parn ar sricly ngaiv is < of h Jacobian marix valuad in h scond uilibrium parn ar sricly ngaiv is < a Thrfor, if and only if and < < a which clarly rprsns a conradicion. Th condiion such ha all h ignvalus. Ο a a, a pa and hill uilibrium parn wih an upwards spi such ha h firs smalls local minima ar rprsnd by null conribuors is locally sabl. ow, l a <, w can idnify a pa and o incras. f and only if a 4 hill uilibrium parn wih an upwards spi such ha h firs and h scond smalls local minima ar null. By

26 using h Jacobian marix valuad in his uilibrium parn w hav ha h condiion such ha all h ignvalus ar sricly ngaiv is < a 4 < a 4. Thrfor, if and only if Ο and a, a pa and hill uilibrium parn wih an upwards spi such ha h firs and h scond smalls local minima ar rprsnd by null conribuors is locally sabl. Th rs of h proof consiss in rpaing h sam procdur ling a o incras unil a. f and only if Ο and, h pa and hill parn wih an upwards spi such ha all h local minima ar ual o zro is locally sabl. Suppos ha h dimnsion of h populaion is vn. f and only if / w can idnify an vn up and down uilibrium parn composd by h combinaion of / BBV communiy of siz. By applying h linarisaion procdur w can asily prov ha his uilibrium parn is locally sabl if and only if and /. > vn Ο Proof of PROP.4. Firsly, l us prov ha an uilibrium parn canno conain wo conscuiv BBV communiis of siz. L C b a BBV communiy of siz. By dfiniion, i is sraighforward o prov ha, if C is a BBV communiy of siz z hn, in uilibrium:, C [A4] z z Morovr, i has o b a null conribuor bwn wo BBV communiis of siz. L us call his individual x. Givn x mod x mod, w hav ha h bs rspons funcion of individual x implis x 0 and only if. Th prvious condiion is saisfid whn 0 Scondly, l us prov ha if, hn, such ha which is impossibl. Ο and 4 if, h gnric combinaion of BBV communiis of siz wih BBV communiis of siz which xhauss and such ha i dos no prsn wo or mor conscuiv BBV communiis of siz rprsns a locally sabl AE parn. Whn h siz of h populaion is such ha i can b dcomposd ino BBV communiis of siz xclusivly, hn h nighbourhood of h gnric null conribuor b is composd by wo individuals who blong o a BBV communiy of siz. By using h bs rspons funcion of individualb, w obain ha 0 if and only if 0.. Whn h siz of h b populaion is such ha i canno b dcomposd ino BBV communiis of siz xclusivly, w can always find a combinaion of BBV communiis of siz wih BBV communiis of siz which xhauss and such ha i dos no prsn wo or mor conscuiv BBV communiis of siz. Any of hs combinaions is characrisd by h prsnc of a las on individual x such ha: a h is a null conribuor; b his nighbourhood is composd by himslf, an individual who blongs o a BBV communiy of siz and, finally, an individual who blongs o BBV communiy of siz. o ha, among h null conribuors who spara a BBV communiy of siz form a BBV

27 communiy of siz, w hav ha Q x Q, x wih x. By using h bs rspons funcion of individual x, w obain ha 0 if and only if Finally, by using h linarisaion procdur x prsnd in h proof of PROP.3., w hav ha, such ha Ο, any combinaion of BBV communiis of siz wih BBV communiis of siz which xhauss and which dos no prsn wo or mor conscuiv BBV communiis of siz is locally sabl. Ο Proof of PROP.., w hav an infini numbr of AE parns in addiion o h SE. n his cas, hr is no any locally sabl uilibrium parn. Ο, h SE is h uniu soluion of sysm [4]. n ordr o find ohr uilibria parns implid by h non ngaiviy consrain, l us rplica h procdur prsnd in PROP.. Saring from h SE associad wih a, l o chang from u ϑ ; unconsraind individuals and h condiion o u ϑ wih. By solving h sysm composd by h bs rspons funcions of h u ϑ, w find h xprssion of h uilibrium conribuions s wih s. Thn, by using h uilibrium conribuions of h individuals who blong o s nighbourhood, xprssion [6] implis ha u ϑ if and only if a. Thrfor, w hav idnifid a paricular parn of conribuions h shap of which dpnds h siz of h populaion and h valu of. u Sinc ϑ ;, w hav an infini numbr of uilibrium parns of his yp. L ϑ u incras unil h smalls local minimum of h parn of conribuions gs valu zro. L us call his parn l u. Bcaus of h non ngaiviy consrain, h condiion on such ha u l is an AE bcoms d a. By rpaing h sam procdur wih ϑ 0 ;, w find anohr AE wih null conribuors as smalls local minima, linarisaion procdur, w hav ha if and only if u l and d l, which is implid by h condiion and > Ο a. By applying h a, on and only on bwn d l is locally sabl sinc all h ignvalus of h Jacobian marix valuad in i ar sricly ngaiv. n paricular, h Jacobian marix J is a suard marix of siz such ha is -h row is obaind by muliplying all h rms ual o which appar in h -h row of A by m. Saring from h locally sabl uilibrium parn, by inducion w hav ha if and only if a > a, hn 6

28 ' wih a ' > such ha h uilibrium parn associad wih ' las on furhr null conribuor, whr a is h highs valu of such ha > a numbr of null conribuors associad wih is h sam of h on associad wih prsns a h. By applying h linarisaion procdur, w hav ha h uilibrium parns idnifid by inducion ar locally sabl if and only if and h condiions on Ο is locally sabl if and only if h ignvalus of marix a such ha hy ar uilibria ar saisfid. Finally, h SE uilibrium A ar sricly ngaiv which holds if and only if < a. Proof of PROP.6. By dfining π sπi β, w can rwri h rm R s of xprssion [7] as follows: R s iβ iβs iβs R R [A] iβ s 0 By using boh h Eulr s formula iβ R i β cos R s iβ s β s and h rigonomric idniis, w obain: cos β i sin β R cos β i sin β sin β β sin Thrfor, by combining uaion [A6] wih h xprssion of h gnric ignvalu λ, w obain: [A6] β sin λ g β [A7] β sin Suppos for h sa of simpliciy ha h paramr is a ral variabl. is clar ha, in ordr o maximis λ wih rspc, w hav o minimis h xprssion g β. Apar from a pur numrical facor, h funcion g β is a Dirichl rnl of ordr and is minimum valu is aaind a is firs local minimum. Using sandard calculus chnius, w hav o solv h following uaion: g β β β β β 0 cos sin cos sin 0 β β β an an L us dfin h minimum poin β m. Thn: [A8] 7

29 β m an β m an Using h rigonomric rlaions, w hav: βm sin βm an β βm β m m an cos an [A0] βm βm βm βm an an cos sin and coming bac o g β, w obain: ming β g β W conclud ha: m βm an βm an βm 4 4cos βm βm cos sin βm βm cos an β m max λ 4 4cos [A] Thans o h propris of boh g β and h an funcion, w hav ha h firs non rivial roo diffrn from zro of [A8] is includd in h s [ π,3π / ]. Thrfor, givn h priodiciy of h angn funcion w obain: π β 3 m π By combining h prvious condiion wih [A], w obain boh an uppr bond and a lowr bond for h gras ignvalu: [A9] [A] [A3] max λ [A4] From h LHS of [A4] w hav ha max 0 / λ if and only if /. Thrfor, if and only if, h SE is boh no uniu and unsabl. Proof of PROP.7. Th proof of PROP.7. is vry similar o ha of PROP.4. Th only diffrnc consiss of h fac ha, for >, an AE can prsn wo conscuiv BBV communiis of siz. ndd, considr a combinaion which prsns wo conscuiv BBV communiis of siz and a las on BBV communiy of siz. Thn, i has o b characrisd by h prsnc of a las on individual x and a las individuals s such ha: a hy ar boh null conribuors; b x s nighbourhood is 8

30 composd by himslf, an individual who blongs o a BBV communiy of siz, an individual who blongs o a BBV communiy of siz and null conribuors whil s s nighbourhood is composd by himslf, 3 individuals who blong o BBV communiis of siz and 3 null conribuors. From h prvious considraion, i follows ha Q x Q s. Thrfor, among h x s null conribuors who spara a BBV communiy of siz form a BBV communiy of siz, w hav ha Q x x Q, wih x. Th rs of h proof is idnical o h prvious cas. REFERECES Androni, J. [988]. Privaly Providd Public Goods in a Larg Economy: h Limis of Alruism. Journal of Public Economics, vol. 3 p Androni, J. [989]. Giving wih mpur Alruism: Applicaions o Chariy and Ricardian Euivalnc. Journal of Poliical Economy, vol. 97, p Androni, J. [990]. mpur Alruism and Donaions o Public Goods: a Thory of Warm Glow Giving. Economic Journal, vol. 00, p Brgsrom, T. Blum, L. Varian, H. [986]. On h Priva Provision of Public Goods. Journal of Public Economics, vol. 9, p Brgsrom, T. Sar, O. [993]. How Alruism Can Prvail in an Evoluionary Environmn. Amrican Economic Rviw, vol. 83, p Corns, R. [980]. Exrnal Effcs: an Alrnaiv Formulaion. Europan Economic Rviw, vol. 4, p Corns, R. Sandlr, T. [984]. Easy Ridr, Join Producion, and Public Goods. Economic Journal, vol. 94, p Davis, P. J. [994]. Circulan Marics: nd Ediion. AMS Chlsa Publishing Company. Diamond. P. [003]. Opimal Tax Tramn of Priva Conribuions for Public Goods wih and wihou Warm Glow Prfrncs. Manuscrip, MT. 9

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