sets), (iii) differentiable - Nash equilibrium in gifts (i.e., contributions to public good) from consumer i is g i + G = amount of public good

Transcription

1 Prvate Provso o Publc oods Bergstrom, Blume, & Vara. "O the Prvate Provso o Publc oods." Joural o Publc Ecoomcs. Vol. 9, 986, BBV - most mportat paper o ths subect; also apples to gvg to ay charty, poltcal party, the arts, etc.; we'll ocus o pure publc good Pure Altrusm Model - cosumers oly care about total amout o cotrbutos, ot ther dvdual cotrbuto: u, vs. u,, g rom Adreo's Impure Altrusm or Warm low model whch we'll study later - Assumptos o u, : strctly creasg, quascocave.e., cove better-tha sets, deretable - Nash equlbrum gts.e., cotrbutos to publc good rom cosumer s g such that, g solves: ma u, g, g s.t. g g Note: takg everyoe else's equlbrum gt as gve Eectve Wealth - other people's cotrbutos eectvely crease cosumer 's wealth although wth a trucated budget set; re-wrte the problem: ma u,, Eectve wealth s.t. kds o cosumers: those who cotrbute: g > those who do't: g Neutralty Result - Theorem, p.9 Assume that cosumers have cove preereces ad that cotrbutos are orgally a Nash equlbrum. Cosder a redstrbuto o come amog cotrbutg cosumers such that o cosumer loses more come tha hs orgal cotrbuto. Ater the redstrbuto there s a ew Nash equlbrum whch every cosumer chages the amout o hs gt by precsely the chage hs come. I ths ew equlbrum, each cosumer cosumes the same amout o the publc good ad the prvate good that he dd beore the redstrbuto Proo: let deote tal equlbrum ad s the chage cosumer 's wealth theorem requres g... ca't take more tha perso s curretly gvg Assume all cosumers The,,, where publc good ot coutg 's cotrbuto g g amout o publc good chage ther cotrbuto by amout o prvate good purchased by cosumer Trucated budget set cosumer 's edowed wealth rom the assumpto ad because o 9 g, g Cotrbutes Does't cotrbute It's a redstrbuto

2 raphcally, cosumer 's budget costrat s trucated < or legtheed >, but ot eough to chage the result; sce cosumer does't chage, g whe others do't chage ther decsos, ths s stll a Nash equlbrum Cosumer pays egatve wealth traser Cosumer gets pad postve wealth traser Cosumer who does't cotrbute o chage g, g g, g, g g No Cotrbutors - ot addressed theorem, but they do't chage ther decso ether Aother Vew - usg demad curves: Stadard utlty mamzato problem: ma, y s.t. u, y, p y Soluto yelds demad ucto: y Apply to equlbrum problem or cotrbutor: Smlar to stadard utlty problem wth: - p p everythg $ - Eectve come socal wealth: - Igore o 9 p y y usg p Soluto yelds suppress the prce argumet sce t's costat For cotrbutors > the amout o publc good demaded at p s the amout provded For o-cotrbutors the amout o publc good demaded at p s less tha the amout provded Neutralty Result - sce <, eectve wealth does't chage; that meas demad curve does't chage soluto to utlty mamzato problem s the same Icome Eects - everythg ths model operates through come eects Demad - combg results or cotrbutors ad o-cotrbutors: ma, { } Optmal Respose - ust subtract g ma rom both sdes o the demad equato:, { } $ / P ma u,, cosumer cotrbutes s.t. $ / P < do't cotrbute -

3 Estece Assumptos - u, s quascocave ',... rom the paper: "there s a sgle-valued demad ucto or the publc good,, whch s a deretable ucto o wealth. The margal propesty to cosume the publc good s greater tha zero ad less tha so that < ' < or all,,."... ths holds as log as both the publc ad prvate goods are ormal goods or all cosumers Estece - Theorem, p.33 A Nash equlbrum ests Proo: stadard applcato o Brower's Fed Pot Theorem: W R :,, s a doma { } compact ad cove set; g s a cotuous ucto rom W to tsel Note: the ucto does't map to tsel or t's ot cotuous there may ot be a ed pot. For more paul detals o ed pot theorems, see ECO 745 otes "Sgle Play Nash Equlbrum" ad "Weake Assumptos o Nash Theorem" ed pot pure strat. Nash eq. I does't map to there may ot be a ed pot Cotrbutors Set - C s set o cosumers who cotrbute equlbrum; the umber o cosumers s c Fact - A cogurato o gts s a Nash equlbrum ad oly the ollowg codtos are satsed: or C or ot C... we covered why o the prevous page 45 I ot cotuous there may ot be a ed pot Fact - There ests a real valued ucto F,, deretable ad creasg, such that a Nash equlbrum: F, the sum o the wealth o all cotrbutors C poor otato: usg a set C as a argumet or a ucto Proo: s a strctly creasg ucto by assumpto, so t has a verse φ Apply the verse to the rst equato rom act : φ C 3 o 9 ths s lke a verse demad ucto; t gves the eectve wealth ecessary to get cosumer to demad amout o the publc good Add these equatos: φ C The last term ca be smpled: Put that back : C C C C g g C C C c φ c φ c F, C Ths part s "less tha obvous; that's why I have a ob" Romao F φ ' c C C C

4 Recall dervatve o a verse ucto: y ' Sce ' <, φ ' > φ ' c < C so φ ' ' 4 o 9 > ' F F Put that back ad we see that φ ' c > c c or > so C F, s strctly creasg Uqueess - Theorem 3, p.34 There s a uque Nash equlbrum wth a uque quatty o publc good ad a uque set o cotrbutg cosumers. Proo: uses F, ; paper skps a step Fact 3 - Let g g,, g ad g g ',, g ' be Nash equlbra gve the wealth,, ' ' ',, dstrbutos ad ' Note: these are deret dstrbutos o the same aggregate level o wealth; o ew wealth s created, let C ad C ' be the correspodg sets o cotrbutg cosumers, ad let the ucto F, be as deed act, the: Proo: rom act, F C F C ',, ' C ' ' ' C there's o guaratee that C s also C' Apply φ : ' ' φ ' C Sum these up: C ' C As beore, we ca smply ' φ ' C C ' ' g ' ' g C C C ' c' ' ths last step s because the set o cotrbutors may be deret; members rom the orgal set are ot C ', the g ' so eect, t's a subset o the cotrbutos o the ew cotrbutors Put that back : ' φ ' c ' From act : From act : C C F ', φ ' c ' F ', ' F, C C Subtract ths rom the rst oe: C F C F C ',, ' C Comparatve Statcs - Theorem 4, p.35 I a Nash equlbrum: Ay chage the wealth dstrbuto that leaves uchaged the aggregate wealth o curret cotrbutors wll ether crease or leave uchaged the equlbrum supply o publc good. Math - ay redstrbuto that keeps ' C C C results Proo: From act 3, F ', F, ' uchaged agg. wealth From act, F, s creasg so F ', F, ' '

5 Note : ths s't the same as eutralty result because we dd't restrct wealth redstrbuto to the cotrbutos g Note : ot sayg aythg about prvate cosumpto, but that chages too Eample - g $, g $, g 3 $ so $ 3 Step - take $ rom perso ad gve t to perso ; eutralty result says publc good s uchaged ow g ad g Step - take $.5 rom perso ad gve t to perso ; perso 's wealth has rse so we'll by more publc good ' > Ay chage the wealth dstrbuto that creases the aggregate wealth o curret cotrbutors wll ecessarly crease the equlbrum supply o the publc good. Math - ay redstrbuto that has ' > results ' > C C ', F, ' > C Proo: From act 3, F From act, F, s creasg so F ', F, > ' > I a redstrbuto o come amog curret cotrbutors creases the equlbrum supply o the publc good, the the set o cotrbutg cosumers ater the redstrbuto must be a proper subset o the orgal set o cotrbutors. Math - redstrbuto amog C ad ' >, the C' C.e., ewer cotrbutors ew equlbrum Proo: From act, ' >, people who dd't cotrbute beore, wo't cotrbute ow so C' C I C ' C, the the chage rom the come dstrbuto leaves the total come o cotrbutors C ' uchaged ad part o ths theorem says ', but we kow ' > so we ca't have C ' C C' C v Ay smple traser o come rom a cosumer to a curretly cotrbutg cosumer wll ether crease or leave costat the equlbrum supply o the publc good. Proo: ollows drectly rom ad overmet Provso - assume govermet uses lump sum taes o cosumers.e., ot codtoed o aythg to provde some o the publc good ad cosumers are allowed to volutarly cotrbute to purchase more o the publc good Neutralty - more geeral orm o Theorem 6, part, p.4; theorem starts wth absece o govermet supply, but ths result does't make that assumpto I govermet places a lump sum ta o cotrbutors that s less tha ther cotrbutos, there s a % crowdg out.e., cotrbutors reduce doatos by the amout o the ta ad there s o chage the total amout spet o the publc good Proo: Dee t lump sum ta o cotrbutor g t... ta less tha cotrbuto C Dee g govermet provso o publc good t For ay cotrbutor g g Budget costrat o cosumer wll be 5 o 9 g t or Follow reasog rom Facts & ad we see Nash equlbrum we have t g Suppose govermet chages ta o cosumer such that g t t

6 Suppose all have amout o ther respectve taes C g t.e., other cotrbutors reduce doatos by the g t ad t g t Put these to cosumer 's demad or publc good: t g t g t g There s o chage the cosumer's demad holds or all other cosumers sce we pcked a geerc Theorem 6 p.4 - Suppose that startg rom a tal posto where cosumers supply a publc good volutarly, the govermet supples some amout o the publc good whch t pays or rom taes. The: I the taes collected rom each dvdual do ot eceed hs volutary cotrbuto to the publc good the absece o govermet supply, the the govermet's cotrbuto results a equal reducto the amout o prvate cotrbutos. Math - t ad t g... slghtly deret tha eutralty result we ust proved I the govermet collets some o the taes that pay or ts cotrbuto rom ocotrbutors, the, although prvate cotrbutos may decrease, the equlbrum total supply o the publc good must crease. Proo: tag o-cotrbutors s equvalet to a traser that creases the eectve wealth o the set o cotrbutors... by part o Theorem 4, that meas equlbrum supply o the publc good creases I the govermet collects some o the taes that pay or ts cotrbuto by tag ay cotrbutor by more tha the amout o ths cotrbuto, the equlbrum total supply o the publc good must crease. Proo: rst take the cotrbutor the etre amout he's cotrbutg; part says amout o publc good does't chage ad cotrbutor reduces hs cotrbuto to zero; ow he's a o-cotrbutor so govermet keeps takg hm, wealth s traserred as part so amout o publc good creases Berhem - "O the Volutary ad Ivolutary Provso o Publc oods" 989; geeralzes to other orms o taato; eutralty holds geeral case... dcult to prove emprcally Epple & Romao - do't have ttle IER 3; looks more at dual provso o publc good Idetcal Tastes - make model more specc to get more specc results; look at case where dvduals are detcal ecept or edowmet.e., u, s same so we ca drop the rom u ; oly derece betwee cosumers s edowmet... we'll use ths model or the remader o the dscusso o the paper Assumptos - as beore u, s strctly creasg, quascocave, deretable; also have ', Subscrpts - also drop subscrpts rom ad φ 6 o 9 Equalzg Redstrbuto - redstrbuto o wealth s sad to be equalzg t s equvalet to a seres o blateral trasers whch the absolute value o the wealth derece betwee the two partes to the traser s reduced e.g., $6K ad $5K; traser o more tha $K; results less dsperse dstrbuto o wealth

7 Fact 4 - I all cosumers have detcal preereces ad s a equlbrum supply o the publc good, the there s a crtcal wealth level φ such that all cosumers wth wealth cotrbute othg ad every cosumer wth come > cotrbutes the amouts g to the supply o the publc good Math - equlbrum supply o publc good φ... cut o level o come or cotrbutors > cotrbute g do't cotrbute g Proo: From Fact, equlbrum a cotrbutor has ad a o-cotrbutor has wth equalty g > Take verse o both sdes: φ wth equalty g > throughout Recall g : g φ g φ Let φ perso cotrbutes, g g perso does't cotrbute, g ad g Equlbrum Propertes or Idetcal Tastes - Theorem 5, p.38 I preereces are detcal, the a Nash equlbrum: All cotrbutors wll have greater wealth tha all o-cotrbutors. Math - cotrbutes ad does't, > Proo: mmedate rom Fact 4 Romao: "I you thk about t, these thgs ollow mmedately, whch meas you do't have to thk about t' All cotrbutors wll cosume the same amout o the prvate good as well as o the publc good. Math - all cotrbutors cosume o the prvate good cotrbute g to publc good; all cosume o the publc good Proo: mmedate rom Fact 4 A equalzg wealth redstrbuto wll ever crease the volutary equlbrum supply o the publc good. Proo: ollows rom v ad v v Neutralty - Equalzg wealth redstrbutos amog curret o-cotrbutors or amog curret cotrbutors wll leave the equlbrum supply uchaged. Proo: "obvous" rom Fact 4; or o-cotrbutors, redstrbuto keeps comes so they stll do't cotrbute; or cotrbutors, redstrbuto keeps comes > so they stll cotrbute... appeal to part o Theorem 4 v Equalzg come redstrbutos that volve ay trasers rom cotrbutors to ocotrbutors wll decrease the equlbrum supply o the publc good. Proo: outle: cotrbutor's wealth so he cotrbutes less; o-cotrbutors wealth, but t's below he stll does't cotrbute so ot a vgorous proo because s also chagg 7 o 9

8 Idetcal & Homothetc Tastes - ow add homothetc tastes.e., come cosumpto curve s a straght le; dvdual cosumes goods same proporto; MRS g /g Demad - αwealth... proportoal to eectve wealth Recall, ', so we must have α, Sort - label dvduals such that lower umber s hgher wealth: Whe Oly Rchest Type Cotrbutes - Assume oly perso cotrbutes: α.e., Make sure perso does't: α α... oly does so Sub α : α α α α α depeds o both α ad the derece wealth Cosder α that meas the rchest guy has to have more tha twce the wealth o the secod guy order or oly the rchest guy to cotrbute Wder Dstrbuto o Wealth More Volutary Provso - ust applyg part v o Theorem 5 F total wealth the ecoomy at W Wdest wealth dstrbuto s W > 3 I ths case oly the rchest type cotrbutes whch we ust solved: α αw W Splt wealth equally betwee top : > 3 Equlbrum s symmetrc: g g / α α α W/ / α / αw/ W α So we have < W 3 Splt wealth equally betwee top k : k > k k Equlbrum s symmetrc: g g g k W k α 3 α α 3 W k k 3 α k α typo paper Note: 3 < ad s decreasg k 3 g Subscrpt or case ICC rowg Ecoomy - wat to show large ecoomy, very ew types wll cotrbute; ths s the key law the BBV model does't match the real world where lots o types cotrbute How to row - replcate cloe cosumers so we crease the umber o members o each type; deote k {,,} as umber o each type o dvdual Assumptos - model we ust used: detcal & homothetc tastes; keep dvduals ordered as beore but assume strct equaltes: > > > Oly Rchest Cotrbutes - d codto where oly type cotrbutes lke we ust dd at the top o the page: k α Cotrbutor - α k α α No-Cotrbutor - α 8 o 9

9 α α Sub rom above: α α α α α α Do a lttle algebra: α α α α α α α α α α α k α Large α meas publc good s hghly valued.e., less lkely to have sgle cotrbutor Cosder α. 5 :., the k meas oly rchest cotrbutes I., the k meas oly rchest cotrbutes Result - these are ot very bg ecoomes do't eed a bg ecoomy beore oly the rchest type s the oly cotrbutor eeral Case - ths result holds geeral or the Pure Altrusm Model.e., u, Adreo uses a cotuum o types Romao, Fres, & oldg IER, 986? use dscrete types 9 o 9