Dynamics of social ties and local public good provision

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1 Jounal of Publc Economcs 64 (1997) Dynamcs of socal ts and local publc ood povson * Fans van Dk, Fans van Wndn CREED, Faculty of Economcs and Economtcs, Unvsty of Amstdam, 1018 WB Amstdam T Ntlands Rcvd 1 Dcmb 1993; vsd 1 Jun 1996; accptd 1 Jun 1996 Abstact A modl s psntd n wc socal ts btwn ndvduals and pvat contbutons to a local publc ood a ntlatd. Ts a fomalzd by mans of utlty ntdpndnc, and dpnd on t stoy of socal ntacton, n ts cas t ont povson of t publc ood. T sultn dynamc modl nats qulbum valus of t ntnsty of ts and t pvat povson lvl. T mpact of publc povson on ts vaabls s analyzd. Ou sults a vy dffnt fom tos obtand wt t standad modl, w ndvduals a only ntstd n t utlty fom own consumpton. Kywods: Local publc oods; Socal ts; Voluntay povson JEL classfcaton: A13; D64; H41; H70 1. Intoducton In many conomc ssus socal ts btwn ndvduals a mpotant. Ts as bn aud by socal-psycolosts and socolosts ov t yas and, mo cntly, by a numb of conomsts, amon wc s Bck (1974). Postv socal ts povd ndvduals wt addtonal soucs n tms of nd, lp to contol xtnalts, fom a bass fo a coopatv outcom of a pson s dlmma, and nsu fa busnss dalns (Colman, 1990; Ganovtt, 1985). Natv ts may bn about t vs. W ts ts a lvant, t * Cospondn auto / 97/ $ Elsv Scnc S.A. All ts svd PII S (96)

2 324 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) factos dtmnn t fomaton bcom of ntst. In ts atcl w nvstat t dvlopmnt and mpact of suc ts n t contxt of t pvat povson of publc oods n small communts w ty a xpctd to play a 1 snfcant ol. Studs of t pvat povson of a publc ood usually assum tat ndvduals dtmn t contbutons non-coopatvly n a on-sot am. Contbutons of ot ndvduals a takn as vn, and only t utlty dvd fom own consumpton s consdd. W f to ts st-up as t standad modl (fo a oous tatmnt, s Bstom t al. (1986)). Equlbum contbutons fall blow t socal optmum n ts modl, and ts sotfall ncass wt t numb of plays. Dsadn con solutons, t total avalablty of t publc ood s nvaant to t ncom dstbuton as wll as publc povson. Wn t ntacton s patd a known fnt numb of tms, t outcom s t sam as tat of t on-sot am. Wt an nfnt o unctan tmspan any outcom can b sustand (s,.., Fudnb and Tol (1991)). Howv, f ndvduals a myopc, t sults of t standad modl a stll vald. W wll us ts modl as a bnc-mak. Many vaants of t standad modl and altnatv modls av bn dvlopd (.., Sudn (1984); Cons and Sandl (1985); Andon (1989)), but also ts modls do not xplctly allow fo socal ts. Intstnly, xpmntal sults sow tat oup dntty affcts contbutons, ndcatn tat t s mo to t povson of publc oods tan s cuntly captud by conomc modls (s Daws and Tal (1988)). Ou modl focuss on t affctv componnt of socal ts, ts dvlopmnt, and ts mpact on pvat povson. Ts a fomalzd by mans of wtd 2 ntdpndnt utlty functons. Wts attacd to t utlty of ot ndvduals can b postv, zo o natv. An mpotant dffnc wt t xstn ltatu s tat ts wts a not constant ov tm, but dpnd on t stoy of ntacton btwn t ndvduals. T fomaton of ts s dtmnd by t contbutons of ndvduals, sultn n a smultanous, dynamc modl of socal ts and publc ood contbutons. W also nvstat t mpact of publc povson and taxaton. It sould b mpaszd mmdatly, owv, tat w do not clam to fully captu t vy complx socal pocss tat s at stak. T ncopoaton of t wts n t modl, and t us mad of psycolocal notons and fndns s stll at ad oc. In t absnc of a fundamntal toy of ntpsonal latons tat duly taks account of t 1 Fo convnnc, w f to ts oods as local publc oods, n contast wt t publc fnanc ltatu, w t tm nally fs to publc oods povdd by muncpalts. 2 In t conomc ltatu utlty ntdpndnc s fquntly assumd (ad oc) to tackl pnomna tat a ad to xplan by pu ndvdualsm, suc as bavo wtn famls (ncludn ntnatonal tansfs), voluntay dstbuton, caty and cm (.., Bao (1974); Bck (1981); Hocman and Ntzan (1985)). In ts modls utlty ntdpndnc s usd, but not xpland. An xcpton s Guttman t al. (1992) w a unfom lvl of altusm s dtmnd by ducaton.

3 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) affctv componnt (sntmnts and motons), ts s unavodabl. Hopfully, t analyss psntd blow wll motvat suc fundamntal sac. Ou man sults a n stak contast wt tos of t standad modl. Fst, n many cass pvat povson s. It may vn appoac t povson lvl tat follows fom t maxmzaton of a socal wlfa functon. A low povson lvl can also occu, owv. T outcom s dtmnd by t combnd ffct of t spad of ncoms and pfncs, and t so-calld tolanc lvl of ndvduals, wc fs to t appcaton by an ndvdual of t contbutons of ots. Wn any contbuton s valud postvly, and ncoms a qual, a povson lvl s acd. If popl av a natv appcaton of any contbuton small tan t own, povson lvls cannot b attand. Wn n addton ncoms dff t povson lvl falls blow tat of t standad modl. Tus, n ou modl pvat povson s not nvaant to t ncom dstbuton. Scond, publc povson lads to a dcas of total povson, by mpdn t dvlopmnt of socal ts. Aan nvaanc s not found. Total povson acs a mnmum wn, n t two-pson cas psntd n dtal, at last on of t pvat contbutons as ust bn ducd to nl. It s only, wn publc povson s fut asd tat total povson stats to ncas. Futmo, as t dvlopmnt of ts taks tm, a ducton n publc povson s not mmdatly takn ov by pvat ntatv, f at all. Consquntly, publc poams a mo dffcult to nd tan to ntat. T oanzaton of t atcl s as follows. Socal ts a fomalzd n Scton 2. In addton, t mpact of ts on publc ood povson s bfly xamnd. Scton 3 os nto t dynamcs of socal ts. In Scton 4 t complt modl of pvat povson n t absnc of ovnmnt ntvnton s analyzd, wl publc povson s addd n Scton 5. Scton 6 concluds. 2. Socal ts and t mpact on publc ood povson Amon t vaabls usd by socolosts and socal psycolosts to caactz socal ts motonal ntnsty, cpocal svcs, mutual confdnc and ots t affctv componnt s consdd to b t ky lmnt of a t; s Ganovtt (1973) and t vw of t mpcal ltatu on psonal attacmnts by Baumst and Lay (1995). Ts nabls a on-dmnsonal concptualzaton as a fst appoac, and t povds a natual way of dfnn postv and natv ts, as wll as t ntnsty of ts (wc may b dffnt fo t subcts nvolvd; s Wllman (1988)). A socal t btwn two ndvduals and s assumd to consst of s ( s) sntmnts about (), w sntmnts o flns a dfnd as t xtnt to wc ( ) cas about s ( s) wlfa and dvs satsfacton fom t. Ts s fomalzd by usn t standad concpt of utlty ntdpndnc. A t btwn two ndvduals and s dscbd by t pa (a, a ) n t follown xpsson:

4 326 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) a k V 5U U (,k5,,±k) (1) k w V stands fo t total utlty of ndvdual, and U fo t utlty dvs 3 fom s own consumpton.. If ndvduals mantan ts wt ot psons as wll, t wtd utlty nts Eq. (1) n t sam way. Altnatvly, on could substtut total utlty Vk fo U k, as n Bnm and Stak (1988). In t cas of two ndvduals, tat wll b focusd on blow, sults a not affctd by t coc of spcfcaton. T doman of ak s assumd to b stctd to t ntval (21, 1), wc mpls tat popl nt lov no at ots mo tan ty lov tmslvs. Ts sms to b a asonabl stcton fo wc substantal xpmntal vdnc xsts (.., Sawy (1966); Lband (1984)). W now tun to t mpact of socal ts on publc ood povson. To bn out t ntstn aspcts of t poblm, t suffcs to focus on a stylzd local communty wt two nabtants. T sults can asly b nalzd to la communts (s Scton 6). Lt U 5U (x,) (5, ), w x dnots t consumpton of a pvat ood, and t consumpton of a publc ood. Pfncs a assumd to b stctly convx, and t pcs of bot oods a takn to b on. Wt yand, spctvly, dnotn s ncom and contbuton to t publc ood, y 5x 1 and 5 1. Indvduals tak ac ot s contbuton to t publc ood as vn. Wn bot a k (, k5,, ±k) a zo, t standad modl s obtand. T fst-od condton fo t maxmzaton of Eq. (1) s vn by: U Uk U U ]] 1ak]] 5 ]]. (2) U x k Eq. (2) psupposs tat t lft-and sd s postv, wc s not ncssaly so. Wn non-postv, s zo and x quals y. Ts cas s covd by t qumnt tat contbutons a non-natv. It follows fo s acton functon tat 5max ( y, y k, k, a k), 0. In nal, / a k.0. Assumn tat x and a nomal oods, 0, / y,1 and 21, / k,0. Fnally, / y k$,0, as t dcton of can sultn fom an ncas n yk (x k, vn k) s not dtmnd. It can b sown tat t pfncs psntd by t xtndd utlty functons a stctly convx, as lon as t ndvduals dv postv utlty fom t publc ood (t poblm s tval f ts s not t cas). As a sult t stuctu of t poblm s t sam as tat of t standad modl, and a unqu qulbum xsts (s Bstom t al. (1986)). Assumn an nto soluton, 3 T sultn total (o xtndd) utlty functons a fomally dntcal to t ndvdual socal wlfa functons usd by Aow (1981). Howv, t undlyn concpts a compltly dffnt. In Aow s study a socal wlfa functon flcts an ndvdual s socal conscnc concnn t dstbuton of ncom (a nomatv poblm), was total utlty V psnts t actual concn fo and satsfacton dvd fom t wlfa of spcfc ots.

5 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) ( y, y, a, a )(,k5,,±k) (3) k k k It follows fom t popts of t acton functons tat / a.0, / k k 4 a,0 and / a.0. Tus, t mpact of an ncas n a s unambuous.. k k k T spcal cass wt ad to t a s a of ntst. Fst, t standad modl s obtand wn a5a50. A scond ntstn cas s a5a 1, w V 5V, wc mans tat and maxmz t sam functon und spaat budt constants. Contbutons ac a maxmum ( ), and t can b asly sown tat ts lads to t sam sult as t maxmzaton of a socal wlfa functon S5VV, o S5UU, und an aat budt constant and t constant tat t consumpton of t pvat ood by an ndvdual dos not xcd o s ncom. Consquntly, full coopaton s acvd. A td spcal cas olds wn a 5a 21. In ts cas and bcom maxmally uncoopatv and acs a mnmum ( ). Wn pfncs and ncoms of mn and a qual, s zo. Otws, o s zo. T standad modl s an ntmdat cas, w t pvat povson lvl falls sot of t maxmum, but s abov t potntal mnmum. W conclud tat t sn and ntnsty of socal ts amon t nabtants of a communty nflunc pvat povson. T dvlopmnt of ts ts s dscussd n t nxt scton. In concludn ts scton w consd t consquncs of ovnmnt ntvnton, assumn tat publc povson s fnancd tou (local) taxaton. T budt constant of ndvdual (5, ) now ads y5x11t, w dnots t quantty of t publc ood povdd by t ovnmnt, and t s s tax sa. Futmo, In cas of an nto soluton, t qulbum s vn by: mn 5 ( y, y, a, a ) 2 t (, k 5,, ± k). (4) k k k Ts s t famla nvaanc sult obtand by Wa (1982) fo t standad cas. Fo vn socal ts, publc povson cowds out pvat contbutons on to on. max 3. Fomaton of socal ts Evdntly, a unfom lvl of sntmnts towads ot ndvduals dos not xst. Sntmnts not only dff btwn ndvduals, ty also vay ov tm. Wt som bas lvl of affcton o concn occus amon ndvduals wo av nv ntactd bfo, s opn to dbat, and w abstact fom t. Radn t dynamcs of socal ts, w tak nto account tat sntmnts 4 Du to t sconday, undtmnd coss-ffct of y on k n t acton functon, t sns of t dvatvs wt spct to y a not as cla-cut. In nal, / y.0, k/ y,0 and / y.0, and ts wll b assumd touout.

6 328 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) dvlop tou polond ntacton (Baumst and Lay, 1995). Futmo, as aud by Colman (1990), ts a subct to dcay ov tm, wn not actvly mantand. As to t dvlopmnt of ts w us t ypotss tat postvly valud ntacton s lkly to nat postv sntmnts, was t vs olds fo natvly valud ntacton. Ts cospondnc btwn ntpsonal xpncs and sntmnts as bn aud and substantatd by many socal scntsts ov t yas (s Homans (1950); Smon (1952); Faao (1989); Fda (1986)). Wt n ou cas t ntacton wll b valud postvly o natvly dpnds on t contbutons. Two aspcts a lkly to play an mpotant ol. Fst, as any contbuton by s lpful n poducn t publc ood wc s postvly valud by, ts s xpctd to lad to a postv valuaton by of t ntacton (wt smla asonn applyn to ). Scond, t stands to ason tat t valuaton of t ntacton wll b postvly (natvly) affctd f t contbuton by t ot xcds (falls sot of) s o s own contbuton. Ts compason of contbutons may b lctd by a noton of 5 fanss. Wn t on aspct aanst t ot lads to t follown cton fo postv (natv) sntmnts: G k;k2..(,) 0(,k5,,±k), w t wt (0# #1) flcts t ndvdual s tolanc towads small contbutons by t ot ndvdual. T follown dffntal quatons dscb t dvlopmnt of t socal t (a, a ) ov tm: da /dt5f (G, a )(,k5,,±k). (5) k k k T can of sntmnts dpnds on t wtd latv contbutons (G k) and t stat of t sntmnts (a k). A dmnsn manal (postv) ffct of Gk s assumd. T ffct of ak s twofold. Fst, ak s takn to nflunc t mpact of Gk suc tat sntmnts man wtn t boundas (21, 1). Scond, sntmnts dcay ov tm. Consquntly, f s takn to b an ncasn, S-sapd, functon of G (t mpuls ), suc tat f 12a (212a ) wn k k k G ` (2`). Wn G 50, f 50 f ts do not dcay ov tm, was f,0 f k k dcay occus and a k±0. T functon f s contnuous and twc dffntabl. In t numcal xampl of t appndx (Appndx A) a functon xbtn ts popts s psntd. F. 1 vs t pas-daam of Eq. (5) fo.0. Fo clafcaton, suppos tat ts would not dcay ov tm. In tat cas t statonay ponts, w da k /dt5f50, a psntd by t bokn ln b. Wn t mpuls G s zo, statonaty s obtand fo any a. Fo all postv k valus of G k, ak ows to ts uppbound 1, and fo all natv valus to ts low bound 21. Fo any a k±0 to qualfy as a statonay pont t countffct of t dcay of t t qus G ±0. Ts s sown by t fat ln n t fu. k k 5 T lvanc of fanss as bn sown patculaly n t contxt of ultmatum-baann ams (Tal, 1988). Its ol n publc ood xpmnts s stll subct of sac (Ldyad, 1993). S also Rabn (1993) on fanss n nal.

7 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) F. 1. Pas-daam of Eq. (5) (.0) (a) f 50 wt attton; (b) f 50 wtout attton. In tat cas ak only appoacs ts bounday 1(21) fo nfnt postv(natv) valus of t mpuls G k. Tus, n t psnc of attton t xsts an upwad slopn dffntabl functon tou t on ( f * ) dscbn t statonay ponts of Eq. (5): ak 5 f * (G k). (6) 4. Dynamcs of socal ts and pvat povson 4.1. Modl and qulbum outcoms Combnn t sults of Scton 2 Scton 3, w can now xamn t dvlopmnt of t socal t (a, a ) and t cospondn contbutons (, ). Follown Colman (1990) w assum tat socal ts a nally t unconscous bypoduct of socal ntacton. Indvduals a supposd to b myopc, wc mpls tat popl do not tak nto account t potntal mpact of t 6 contbutons on t own flns and t flns of ots towads tm. T dcson stuctu can tn b modld as a squnc of contbuton dcsons 6 Statc bavo s dscussd n van Dk and van Wndn (1995).

8 330 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) psntd by Eq. (3), wc a only connctd by t dvlopmnt of t t ov tm. Fnally, w no t possblty tat socal ts a affctd by motalty o maton (s Scton 6). Usn Eqs. (3) and (5) t dvlopmnt of t socal t s dscbd by t follown dffntal quatons: da /dt5f (G (a, a ), a )(,k5,,±k) (7) k k k k k w t ncom vaabls av bn dltd, fo convnnc. T statonay ponts of ts quatons, dtmnd by f50, can b mappd n (a, a )-spac. Nlctn xtm stuatons w o s zo, t cuvs psntn ts statonay ponts a upwad slopn. Usn Eq. (6), t slops a vn by: S f * k ]] ]] 2 ]] dak Gk ak ak ]] 5 ]]]]]]]].0. da k f* k 12]] S]] 2 ]] G a ad S D D k k k As ads t poston of t cuvs n t (a, a )-plan, t stuaton w ndvduals a dntcal s consdd fst. F. 2 vs t pas-daam fo t nal cas w 0, 5 5,1. Fo natv valus of a, t dcay of t k t ov tm causs an ncas of a k. Fo statonaty ( f50), ts ncas nds to b offst by a natv valu of G, and s contbuton as to b la tan k k s. Ts mpls tat ak would av to b mo natv tan a k. Fo ak50, statonaty qus tat G 50, and tus 5, wc aan mpls tat a k k k would av to b small tan a k, and natv. Wn ak bcoms postv, Gk must also b postv to compnsat t dcln of sntmnts du to attton. Consquntly, a ncass and swtcs sn. Gvn t popts of f, t k F. 2. Pas daam of Eq. (7) (dntcal ndvduals) : f50, : f50.

9 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) manal mpact of Gk tn dcass wt t ncass of ak and G k. But t at of dcln of ak du to dcay dos not smlaly dcas. Tus, as ak bcoms mo and mo postv, ak as to ncas at an v fast at, and at som pont ak would av to bcom ncssaly la tan a k. Snc ndvduals a assumd to b dntcal, t cuvs fo and a symmtc, and ntsct w a5a. Dffncs btwn and may xst wt ad to ncom, pfncs fo t pvat and publc oods, t tolanc lvl, and t spd at wc ts dvlop and dcay. Fo xpostonal asons, only dffncs n ncom o pfncs a consdd. Wt ad to ncom w assum tat y,y, and as to pfncs x x z tat. and, fo all x 5x and, w dnots t lastcty of utlty of wt ad to z (z5x, and 5, ). Ts mpls tat ( U / )/ ( U /x ),( U / )/( U / x ). In bot cass f 50 and f 50 a no lon symmtc. Symmty qus G fo (a, a )5(a, b) to b qual to G fo (a, a )5(b, a), was n t cass consdd G.G. Bot cuvs sft to t t, and at t pont of ntscton a must b la tan a. Wn ncoms o pfncs dff suffcntly, f50 sfts to suc an xtnt tat a s postv fo t wol doman of a, wl f50 may sft so fa tat a s natv fo all a. T follown poposton psnts t qulbum outcoms. Poposton 1. (a) A unqu socal-t qulbum, dnotd by (a, a ), xsts; (b) wt qual pfncs and ncoms, t t s symmtc and postv n qulbum (a 5a 5a.0); (c) wt dffn pfncs o ncom lvls, t t s asymmtc n qulbum (n t xamnd cas a.a ), and t sntmnt wt t low ntnsty may b natv (a $,0); (d) t qulbum s locally stabl. Pats (a) and (b) follow dctly fom t omty of F. 2. Not tat t functons cannot ntsct n t natv otant, snc f 50 tn qus tat. (G,0) and f 50 tat, (G,0). Pat (c) s basd on t mpact of ts dffncs on t pas-daam, as dscussd abov. Pat (d) can smply b vfd by xamnn t Jacoban of t systm at ts qulbum. Ts sults a consstnt wt socolocal nsts mpaszn t dpndnc of fndsp ts (symmtc postv latons) on smlaty of atttuds, attbuts and socal postons (Fld, 1981). It s notd tat n t psnt modl a stuaton w bot ndvduals av natv flns towads ac ot cannot occu. In cas of sval publc oods and dffn pfncs, suc a stuaton may as, owv. A fw maks about t xtm cass 50 and 51 a n od fo lat fnc. Wn 50, G 5 $0. Consquntly, f cannot b statonay fo k k natv valus of a. Bot cuvs ntsct onc n t postv otant, k spctv of dffncs n pfncs o ncom. T qulbum a s a la

10 332 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) tan n t nal cas. Wn 51, Gk5k2. Wt qual pfncs and ncoms, bot f50 and f50 pass tou t on. Hnc, (0,0) s t qulbum. Wt dffn pfncs o ncoms, bot cuvs sft to t t. In tat cas, t qulbum a s a of oppost sn, but qual n absolut valu (snc G52G ). Moov, ty a small tan n t nal cas Povson lvls: compason wt t standad modl As a bncmak w us t standad modl, wt a5a50. Compan t b bncmak povson lvl ( ) wt t qulbum lvls natd by ou modl ( ), t follown poposton olds fo t nal cas, w 0,,1: Poposton 2. (a) Wt qual pfncs and ncoms:. ;(b) wt dffnt b pfncs and qual ncoms:. ; (c) wt dffnt ncoms and qual b pfncs: $,. b Pat (a) of t poposton s vdnt fom poposton 1(b) and t popts of Eq. (3). To poof pats (b) and (c), w pocd by xamnn t xtm cass b 50 and 51 fst. If 50,., bcaus t qulbum a s a always postv spctv of dffncs n pfncs o ncom. Now, consd 51. Wn pfncs and ncoms, and, tfo, contbutons a qual, t a s a b zo n qulbum, and 5. Allown fo dffncs n pfncs o ncom vs n ts cas: (I) wt dffnt pfncs adn x and, but qual b ncom:. ; and (II) wt dffncs n ncom, but qual pfncs: b,. If (I) olds, ndvdual wo attacs a low wt to t publc ood ncass o s contbuton as a sult of t ntacton, was t contbuton of dcass, but to a lss xtnt. T ntuton s tat ndvdual taks t utlty of wt a latvly valuaton of t publc ood postvly nto account, wl taks to t sam xtnt t utlty of, wt a latvly low valuaton, natvly nto account. Howv, wn ncoms dff (II), t ncas of t contbuton by t poo pson as a sult of ntacton dos not compnsat fo t dcas of t c ndvdual s contbuton. A poof s vn n Appndx B. As t nal cas, 0,,1, ls btwn ts xtm cass, t poposton follows. To summaz, takn t dynamcs of socal ts nto account wdns t an of possbl outcoms. Dpndn on t omonty of communts and tolanc wt spct to contbutons, povson may b abov o blow t bncmak lvl of t standad modl. In addton, t s mpotant to not tat t povson lvl ncass wt tolanc and t qualty of t ncom dstbuton (ts follows fom t poof n Appndx B). In t xtm cas of 50 and max qual ncoms, t maxmum povson lvl s appoacd. Appndx A povds a numcal xampl.

11 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) Pvat povson and publc povson W now xamn t mpact of publc povson. Tatn publc povson ( ) as xonous and usn Eq. (4), t dvlopmnt of a s dscbd by: da /dt5f ( (a, a )2 (a, a )2(t 2 t),a ) (8) k k k k k k k k 7 8 (, k5,, ±k). Assumn tat t tax budn s qually splt (t 50.5), and concntatn aan on t nal cas w 0,,1, t mpact of publc povson on t poston of t pas-lns n F. 2 s adly sn. Publc povson causs a dcln of Gk f bot pvat contbutons a postv: k(a k, a )2 (a, a )20.5(12 ), (a, a )2 (a, a ). Wn s nk k k k k k k k casd fom zo, f50 sfts to t lft, and f50 to t t. Wt dffn pfncs o ncoms t pas-daam s aan asymmtcal. T follown two popostons psnt t qulbum sults fo t socal t and t total povson lvl of t publc ood. k Poposton 3. If ncass fom zo, tn: (a) wt qual pfncs and ncom, t qulbum a s dcln unfomly untl ty bcom zo; (b) wt dffn pfncs o ncoms, t qulbum a s dcln, and t low (a ) bcoms natv, o, wn alady natv, bcoms mo natv; wn ncass fut and t low contbuton ( ) bcoms zo, bot qulbum a s o to zo,.., t postv on (a ) dcass fut, wl t natv on (a ) ncass. Poposton 4. If ncass fom zo, tn: (a) t total povson lvl of t publc ood (wc ncluds ) dcass untl, n cas of dntcal n- dvduals, bot contbutons a zo o, ls, t low contbuton ( ) s zo; wn ncass fut, ncass; (b) wn ncoms dff, total povson stats to ncas at a low valu of tan n cas of dntcal ndvduals (kpn total ncom constant); wt qudstant dffncs n pfncs 9 nally t sam sult s found. 7 Not tat w focus on obsvabl ffot,.., t voluntay contbutons to t publc ood. W assum tat contbutn to t publc ood by payn taxs dos not nat affctv ts. Poltcal pocsss sm to consttut a wak mcansm fo t fomaton of socal ts tan voluntay povson. In cas of votn n lctons t lnk btwn dcsons and contbutons s muc lss dct. Also, t sccy typcal fo votn sults n lmtd nfomaton about t wllnnss of ndvduals to contbut. 8 If t ±0.5 t tax at can b splt n an ava tax at and an qually szd dstbutv tax and subsdy at. In tat cas t analyss of t ffct of dffncs n ncom of Scton 4.2 appls. 9 Equdstant mans tat t manal at of substtuton dx/d fo s (12 ) and fo (11 ) tms tat at n cas of unfom pfncs. As to t nalty of t sult, an xcpton occus n t xtm cas w s clos to 1 and t attton appoacs zo fo bot ndvduals (s Appndx C).

12 334 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) Pat (a) of poposton 3 follows dctly fom t dscussd nwad sft of f 50 and f50. As to pat (b), t s asly sn tat ncasn n a cas of dffncs n pfncs o ncoms causs on qulbum a to bcom natv. In t cas w attacs low mpotanc to t publc ood, o as a low ncom tan, s contbuton bcoms zo bfo s. In tat vnt t mpuls on a bcoms natv, snc G52 [ (a, a )20.5 ],0. It follows tat a must b natv fo f50. T ntscton of f50 and f50 occus at a natv a and a postv a. A fut ncas of sults n a dcln of t natv mpuls, wc causs a to bcom lss natv. Evntually bot qulbum a s a ducd to zo. Pat (a) of poposton 4 follows fom t dcln of t qulbum a s. T ncas of s ovcompnsatd by t dcas of t contbutons of and. Total povson stats to ncas only wn at last on of t voluntay contbutons as bn ducd to zo. Pat (b) flcts tat wt dffncs n ncom t low contbuton s always, and wt dffncs n pfncs n naly all cass, small n qulbum tan t contbuton n cas of qual pfncs and ncoms. T poof of poposton 4 s statfowad, but tdous (s Appndx C). T numcal xampl of t appndx (Appndx A) llustats t sults. Tot, t popostons lad to t follown conclusons. Publc povson mpds t dvlopmnt of postv and natv sntmnts, w t latt occus f t povson lvl s suffcntly. Popl bcom mo nutal/ ndffnt towads ac ot. Fo dntcal ndvduals, ts and, tfo, t total povson lvl dcln unfomly as publc povson ncass. Wn pfncs o ncoms dff, t mpact s mo complx. At low lvls of publc povson, sntmnts bcom vn (mo) natv, and aan total povson dcass. Tus, n nal, ncasn publc povson fom zo fst causs total povson to dcln. Ts consttuts a mao dpatu fom t standad modl. It s only wn at last on of t contbutons s ducd to zo tat total povson stats to ncas. T la t dffncs btwn ndvduals t low t lvl of publc povson at wc ts vsal occus, and t small t dcln of total povson. T consquncs of dcasn publc povson, to cut back ovnmnt xpndtu fo nstanc, can also b xamnd. Suppos tat fo a lon tm publc povson compltly cowdd out pvat contbutons, and s now totally abolsd. At fst, total povson dclns saply to t lvl (0,0)1 (0,0). Wn tolanc s low and la dffncs n ncom xst, pvat povson subsquntly dclns fut. If tolanc s suffcntly, pvat povson stats to ncas and acs a substantal lvl, but only aft som tm. T communty would fac a dffcult tanston pod. Bcaus t lctoat may not accpt t tmpoay dcln of povson, t ovnmnt mt fl ud to mantan xpndtus. Also not tat, fom a lon tm, socal wlfa pspctv lmtd xpndtu ductons only mak sns as lon as t ducd publc povson lvl mans abov t lvl tat would b pvatly attand n t

13 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) absnc of publc povson. Otws, total povson s ducd blow t lvl tat can b pvatly acvd. T modl susts tat t s a cla coc to b mad: t ly on pvat o on publc povson. 6. Concludn maks Ou analyss ponts at condtons und wc ndvduals succd n povdn 10 local publc oods wtout ovnmnt ntvnton. Ty wll fal to do so n cass of low tolanc wt spct to t contbutons of ots, n patcula wn combnd wt an unqual ncom dstbuton. In tat cas t s a atonal fo ovnmnt ntvnton. Moov, cowdn out pvat contbutons would also fostall natv sntmnts. Fom a wd pspctv, ts s mpotant n tslf as suc sntmnts may caus socal tnsons tat ndan publc od and sult n costs to socty. Anot atonal fo ovnmnt ntvnton sustd by t modl concns oapcal moblty. In contast wt t standad modl, maton affcts t povson lvl of t publc ood n ou modl, bcaus t dsupts xstn socal ts and ncsstats t buld-up of nw ts. van Dk (1997) sows tat nally t ava ntnsty of socal ts dcass and tat t povson lvl ts clos to t pdcton of t standad modl. Communty sz could povd a td atonal fo ovnmnt ntvnton. In t standad modl t povson lvl, latv to t lvl tat s found maxmzn a socal wlfa functon, bcoms apdly nconsquntal wn t numb of nabtants ncass. Extndn ou modl to la communts, t can b sown tat t latv dcln of t povson lvl pocds muc mo adually as lon as t nabtants a nfomd about t contbutons of a substantal numb of t fllow-nabtants. If ts condton s not mt, t sults appoac tos of t standad modl. In ot nstancs pvat povson can ac a lvl, and publc povson s unncssay. Wn t dos not cowd out pvat povson compltly, publc povson s countpoductv, as t ducs t total povson lvl. At any at, t amps t dvlopmnt of postv socal ts tat would dvlop tou t pvat povson pocss, and may tby nduc fut publc ntvnton. A cas n pont s voluntay assstanc to popl n nd. Follown ou modl, t s no nal wllnnss to lnd suc suppot; only ndvduals wo av postv sntmnts towads a pson n nd would b wlln to lp. Wn t dvlopmnt of suc sntmnts s blockd, t nd fo costly wlfa and socal scuty aanmnts ncass. Wl stayn clos to standad conomc toy, ou modl undscos t 10 It may also sd a nw lt on t occunc of coopaton n publc ood xpmnts. T modl susts tat account sould b takn of t possblty tat t motvaton of subcts cans und t socal ntacton tat taks plac wtn t xpmnt.

14 336 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) mpotant ol playd by socal ts n t povson of publc oods. Inon t ffcts of publc polcy on ts ts, may, on t on and, unwttnly sult n t dcln of socal condtons tat stmulat ndvduals to tak ca of t own nds, and, on t ot and, n socal tnsons and nsuffcnt publc amnts. In t fst cas, ovnmnt ntvnton s too ston, n t scond too wak. In bot cass publc polcy s not attund to t socal nvonmnt. Acknowldmnts W tank Gay Bck, Fdk Calsn, Ec Dssn, Hnz Holland, Smul Ntzan, Tomas Rom and, n patcula, an anonymous f fo stmulatn commnts on al vsons of ts pap. W a also atful to t Ntlands Oanzaton fo Scntfc Rsac fo fnancal suppot. Appndx A Numcal xampl To llustat t socal-ts mcansm, Tabl 1 psnts som numcal xampls. T (own consumpton) utlty functons a spcfd as: U 5 b 12b x (5, ). Fo Eq. (5) a dsct-tm vson of t follown spcfcaton s usd: s ( k2. ) k k k s ( k2. ) k k da 2(1 2d a )/(11da ) ]] 5]]]]]]]]]] 2a k. dt 1(1 2d a )/(11da ) Tabl 1 Povson of a publc ood and socal ts y 5y 5100 y 570, y 5130 b b a a a a Not: b50.5, d50.8 and s fo 5,. T mnmum povson lvl s acd at 550 fo y570, y5130, and at 567 fo y5y5100. In cas of t maxmzaton of a socal wlfa functon S5VV fo a5a51, o S5UU, t optmum s 100.

15 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) T paamt s flcts t spd wt wc ts dvlop, wl d s 1 mnus t lna at of dcay ov tm. Tabl 1 vs t socal-ts qulba and t cospondn povson lvls fo qual as wll as unqual ncoms. In t fst t ows publc povson s zo, and t nflunc of dffnt valus of s xamnd. In t low pat of t tabl sults a vn fo dffnt lvls of publc povson. Rsults a compad wt tos of t standad modl. Appndx B Poofs of pats (b) and (c) of poposton 2 T conclusons as to 51 man to b povn: wn pfncs o ncoms b b dff suc tat as a low dmand fo t publc ood, tn:,, and b b b a52a.0,.,,, and,. W wll xamn wt $,. In t bncmak cas a and a a bot zo. Can of t a s s always symmtcal. Tus, t must b povn tat: / a 2 / a$,0 fo a5a50 and fo a5d and a52d,0,d,d, 1 w d1 follows fom t qumnt.. It s assumd tat an nto soluton xsts. Wn a5a50, n qulbum ndvduals and coos ponts dnotd by (1) and (2) n t (x, )-plan, suc tat: [( U / )/( U / x )] 5[( U / )/( U / x )]. W fst (1) (2) consd dffncs n pfncs. Of cous, s qual fo bot ndvduals and x.x. By sttn a 5a 5a, w a s manally la tan zo, w can xamn t can of t slops of and s ndffnc cuvs tou t ponts (1) and (2), wn a and a ncas fom zo. Fo postv a t slops a wt ad to, spctvly, and n absolut valu: s 5 [( U / )/( U / x )] 1 a[( U / )/U ] /[( U / x )/U ], (1) (2) (1) s 5[( U / )/( U / x )] 1 a[( U / )/U ] /[( U / x )/U ]. (2) (1) (2) Fo a wd an of utlty functons (.., CES functons wt lastcty of substtuton #21, and tus also a Cobb Doulas functon): [( U / )/U ]/ x # 0. Ts mpls: [( U k/ )/U k ](1)#[( U k/ )/U k ](2) fo k5,. And tus: s $s (1) 5 [( U / )/( U / x )] 1 a[( U / )/U ] /[( U / x )/U ], (1) (1) (1) s #s (2) 5 [( U / )/( U / x )] 1 a[( U / )/U ] /[( U / x )/U ]. (2) (2) (2) As dfnd n Scton 4.1, wt dffncs n pfncs: ( U / )/U,( U / )/U fo all x 5x and. Tus: s (1). (1 1 a)[( U / )/( U / x )] (1)

16 338 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) and s (2), (1 1 a)[( U / )/( U / x )]. (2) As t..s. of bot nqualts a qual, s (1).s (2) and tus also s.s. Consquntly, t slop dx/d of s nw ndffnc cuv tou (1) s stp tan t slop of s nw ndffnc cuv tou (2). As also s ffctv ncom y1 s la tan s ffctv ncom y1, t must ncssaly old tat / a. /a. And, usn t sults of Scton 2, / a. /a,ast can b sown tat t dffnc btwn / and / woks n t sam dcton. Fo a5d and a52d wt d manally dffnt fom zo, w can pat ts analyss. Ts s lft to t ad (s, owv, blow fo t cas of dffncs n ncom). Wt dffncs n ncoms ( y,y ), t ndffnc cuvs of and a dntcal, wn a 5a 50 and also wn a 5a 5a. In t cas a 5a 50, n qulbum x5x and tus y15y1. Ts mpls tat at a 50 / a 5 / a. Wn a 5d and a 52d, and coos ponts n t (x, )-plan dnotd aan as (1) and (2), suc tat: s95s 9, w: s9 5[( U / )/( U / x )](1) 1 d [( U / )/U ](2)/[( U / x )/U ](1), s9 5[( U / )/( U / x )](2) 2 d [( U / )/U ](1)/[( U / x )/U ](2). Now, n qulbum x,x. It may also b notd tat: [( U / )/( U / x )](1), [( U / )/( U / x )](2).Incasn a and a manally wt, t slops of t ndffnc cuvs bcom: s5 s9 1 [( U / )/U ](2)/[( U / x )/U ](1), s5s9 1 [( U / )/U ](1)/[( U / x )/U ](2). As x, x and (dct) pfncs a qual, [( U / )/U ] /[( U / x )/U ] #[( U / )/U ] /[( U / x )/U ] (2) (1) (1) (1) 5[( U / )/U ] /[( U / x )/U ], (1) (1) [( U / )/U ] /[( U / x )/U ] $[( U / )/U ] /[( U / x )/U ] (1) (2) (2) (2) 5[( U / )/U ] /[( U / x )/U ]. (2) (2) And tus: s,s, as t xtm..s. fo s small tan tat fo. Consquntly, dx /d otats lss tan dx /d, and, as x,x mpls tat also y1,y1, / a, /a. Aan t follows tat also / a, /a. Ts pocdu can aan b patd.

17 Appndx C F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) Poof of poposton 4 Pat (a): w dfn s qulbum pvat contbuton n t absnc of publc povson ( 50) as: 5 (a, a ); and wt publc povson as: * 5 (a *, a * )20.5 $0. W fst xamn *.0 and *.0. As follows fom poposton 3(a), a *,a and a *,a and tus (a *, a * ), (a, a ). Consquntly, *, A smla sult olds fo *, and tus: * 1 *, 1 2. Tus: * 1,. Tus w av povn tat n t nw qulbum wt publc povson, wn pvat contbutons a postv, t total povson lvl s low tan bfo. W wll now nvstat wat appns, wn on o bot contbutons a ducd to zo. Wn ndvduals a dntcal wt spct to pfncs and ncom, * and * a ducd to zo by t sam. Wn s fut ncasd, pvat contbutons man of cous zo, and total povson ncass. Tus, ncasn fom zo, fst sults n a dcln of total povson. Only, wn bot pvat contbutons a ducd to zo, total povson stats to ncas. Total povson s at a mnmum, wn bot contbutons a ust ducd to zo. Wn ndvduals dff (assumn aan ( U / )/( U / x ),( U / )/( U / x )oy,y), w know tat,. Ts s also ncssaly t cas wn t s publc povson and t tax budn s sad qually (t 50.5): *, *. Tus, as s ncasd, s ducd to zo bfo. Usn Eq. (4), w fnd (suppssn y and y ): * 5 (t, a * )2(12t). Total povson s: * 5 (t, a * )1t, and: */ 5t / t1t. Not tat 21, / t,0. Tus, an ncas of ncass t total povson lvl. Consquntly, wn s ncasd fom zo, total povson dops untl s zo. Wn s fut ncasd, total povson ncass. Pat (b): Usn poposton 1(c) wl n cas of dffncs n ncom oldn total ncom constant and n cas of dffncs n pfncs consdn qudstant pfncs (s footnot 9), a (qual),a (unqual) and a (qual). a (unqual). And,. T nqualty adn a mpls: (12 ) (qual), (unqual)2 (unqual), and t nqualty adn a :(12 ) (qual). (unqual)2 (unqual). W wll fst xamn t xtm cass wt ad to. Wn 50, t nqualts duc to: (qual), (unqual) and (qual). (unqual), wc mpls tat (unqual) s ducd to zo bfo (qual) wn ncasn publc povson. Ts sult olds fo dffncs n ncom as wll as pfncs. Wn 51, bot nqualts duc to: (unqual). (unqual), wc dos b not allow dct conclusons. W know, owv, tat: (qual)5 (qual), and b b n cas of dffncs n ncom: (qual)5 (unqual) and fom t poof n b Appndx B: (unqual), (unqual). Tus: (qual). (unqual). And, as

18 340 F. van Dk, F. van Wndn / Jounal of Publc Economcs 64 (1997) (unqual). (unqual) and of cous (qual)5 (qual), (unqual), (qual). Ts mpls tat n cas of dffncs n ncom (unqual) s aan ducd to zo bfo (qual) wn ncasn publc povson. Ts sult now obvously olds also fo t nal cas 0,,1. In cas of qudstant pfncs as dfnd al (stll fo 51), t can b b b b asly sown tat (qual). (unqual). As (qual)5 (qual) and fom t t poof n Appndx B (unqual). (unqual), w can not ac mmdat conclusons as to (qual)$, (unqual). By consdn aan an xtm cas, sults can b acd. T small t waknn of latons, t a and a. In t xtm cas, wn latons do not wakn at all ov tm, (unqual)5 (unqual). Ts mpls tat and maxmz t sam xtndd utlty functon. By substtutn t dfntons of qudstant pfncs n t optmalty condtons of and and consdn xtm cass adn t allocaton of t dffnc btwn t manal ats of substtuton of and ov ( U / x )/U and ( U / )/U fo 5,, t can b sown tat: b -n all stuatons wt ad to t allocaton ov x and, s qulbum contbuton n cas of dffnt pfncs s qual o clos to s qulbum contbuton n cas of qual pfncs; -wn bot a not qual, (unqual) may b la o small tan (qual), dpndn on t allocaton ov x and. To summaz t sults fo dffn pfncs, only wn s clos to 1 and t at of dcay s clos to zo, s qulbum contbuton n cas of dffnt pfncs may b (somwat) la tan (o qual to) s qulbum contbu ton n cas of qual pfncs. In all ot cass (unqual) s small tan (qual), and, consquntly, (unqual) s aan ducd to zo bfo (qual), wn publc povson s ncasd. Rfncs Andon, J., 1989, Gvn wt mpu altusm: applcatons to caty and Rcadan quvalnc. Jounal of Poltcal Economy 97, Aow, K.J., 1981, Optmal and voluntay ncom dstbuton, n: S. Rosnfld, Economc wlfa and t conomcs of Sovt Socalsm: Essays n ono of Abam Bson (Cambd Unvsty Pss, Cambd). Bao, R.J., 1974, A ovnmnt bonds nt walt? Jounal of Poltcal Economy 82, Bck, G.S., 1974, A toy of socal ntactons. Jounal of Poltcal Economy 82, Bck, G.S., 1981, A tats on t famly (Havad Unvsty Pss, Cambd). Bstom, T., L. Blum and H. Vaan, 1986, On t pvat povson of publc oods. Jounal of Publc Economcs 29,

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