Andreoni - extended the BBV model to include utility directly from giving, not just from the total contribution to the public good; BBV: ui ( xi

Transcription

1 Pvt Pvs Pulc Gds wth Imu Altusm Ad. "Gvg wth Imu Altusm: Alcts t Chty d Rcd Equvlc." Jul Pltcl Ecmy. Vl. 97, N. 6, Dc 989, Ad - xtdd th BBV mdl t clud utlty dctly m gvg, t ust m th ttl ctut t th ulc gd; BBV: u x, Y vs. Ad: u x, Y, g Rsult - wll t hv th utlty sult th BBV mdl Mdl - - Utlty - u x, Y, g s csg, dtl d qus-ccv,,, u - Evy Gvs - lm... guts t slut g 0 g 0 - Vluty Dts - dvdul gvs g Ttl vluty gvg: G g ; gvg y vy xct dvdul : G 7 g - Fcd Dts - τ s xgus lum sum tx dvdul ; ll th tx vu s usd t suly th ulc gd.., cd dt Ttl cd gvg txs: T τ ; txs m vy xct dvdul : T τ - Ttl Gvg - dvdul ctuts y g τ ; ttl m vy xct : Y y - Ttl Pulc Gd - cms vluty d cd ctuts: Y g τ G T - Tw Ects - utlty css m cts: Altusm Ect - g Y u x, Y, g Egsm Ect - g u x, Y, g... ls clld "wm glw" L clls t "h's studt ct" ly d cmmuty svc cgt - Edwd Wlth - sm s - Evythg Dlls - ulc gd s msud uts th um x whch s msud dlls Nsh Equlum - wt t xm ts ctuts d mld vt csumt tkg tx lcy s gv.., gv τ, τ,, τ Exstc & Uquss - ssumd th ds't v thm lk BBV dd Csum Plm - mx u x, Y, g x, Y, g s.t. x g τ Csum ut cstt Y T G g Pulc gd cstt Nt: g 0 t qud y th vy gvs ssumt Slv g : g Y T G Su T T τ d Y T G : g Y T τ G Y Y τ Su ths sult t : x Y Y τ τ x Y Y

2 Su g d x t th cstt t gt ucstd mxmzt lm: mx u Y Y, Y, Y Y τ Y u x u u FOC: 0 x u u u 0 x x Y Y Y Y τ Bcus & Dmd - thtclly slvg th FOC Y ylds th dmd ulc gd: Y Y, Y τ Dmd dds u, x & g wh th tw Ectv Icm Egsm Ect gumts stsy th cstts th Altusm Ect csum lm v Su Y y Y : y Y, Y τ Y Eq 3 Su y g τ : g Y, Y τ Y τ Cll ths 3'... t, ut t'll cm hdy "Whv yu h 'clly,' yu wt t suscus." Rm Pts - ssumg x, Y, g ll ml gds Ntt - Y Y, Y τ, ltusm & gsm cts / d / Fct - g d τ t s y csum s ct susttuts s thy u ltusm mdl whch gvs utlty sult P: ttlly dtt 3' g & τ vy; & Y cstt: Nutlty qus.g., txs cs y $0 s gvg dcss y $0... ths qus 0, whch sults th s gsm ct Fct - > 0 ml qus ml gd ssumt; w w't d t Itutv P: gsm ct, th ct, th mut s tx d Fct - > 0 utlty; th s gsm <.., csum w't cut ck vt ctut y sm, w must hv > 0 P: m 3': > 0... cus x s ml gd d I ct, x & g th ml, th 0 < < Fct v - 0 < gds ml scl wlth Y dy P: dtt Y wt Y : > 0... cus Y s ml gd dy Sum s < vythg s ml gd Budy Css 7

3 Fct v - u ltusm: 0 BBV mdl wth u x, Y ; gt ct cw ut utlty sult; s txs y mut lss th g d gvg dcls y sm mut Fct v - u gsm: u x, g ; s ds't c ut ttl mut ulc gd, ust hw much h ctuts t t; gt cw ut; Kt: "thk t s tw vt gds" P: t cg ut ttl ms th l's gvg ds't ct wht gvs; tht ms / 0 dy Us 3' hl τ xd t clcult / dy 0 Msu Altusm - ltv t gsm; msus t wht xtt scl wlth Y d w wlth susttuts v th ulc gd Y ;.., w dcs y hw much must w cs Y t gt th sm Y Pu Altust - Y & Nt Pu Altust - y $, ct susttuts utlty; y $ Y y $ y $ Y y lss th $ cus csum ds't duc hs ctuts y $ sc h gts utlty m dy Dt - α d Y cstt g dctly Sc w wt t hld Y cstt, st dy 0 Ttlly dtt Y Y, Y τ wth τ xd dy d dy dy 0 Rg tms t gt dy / d / Budy Css Pu Altusm - 0 s α Pu Egsm - s α... α Alyss Nutlty d Cw Out Ics Tx - sus w cs tx csum d τ > 0, 0 ; csum 's wlth ds't chg cstt, ut th csums c chg th ctuts sd csum 's ctut s T vs Ttlly dtt 3: dy dy dy dy Su dy dy dy dy dy dy : dy dy dy dy dy dy dy Multly ut th tms: dy dy dy dy dy dy dy Cm tms: dy dy Slv dy : dy dy 3 7

4 Tck: α α Su tht : dy dy α Eq 4 Oth csums cll 0 : dy dy,,3, Eq 5 Add ths u t gt ttl ct th mut th ulc gd: dy dy dy dy α α Slv dy : dy dy Smly y lttg c dy dy c α Eq 6 Sgg - m th cts., w kw: > 0, > 0, 0 whch ms 0 tht 0 < c ; w ls kw < α < s lg s csum s t u ltust, w hv lss th ct cw ut.., dy > 0 Nt ls tht c α < s w d u wth th -utlty sult: 0 < dy < Eglsh - ttl s th ulc gd css, ut t y s much s th tx csum ; csum w u ltust, th th s ct cw ut dy 0 Wlth Rdstut - csd tsg wlth m csum t csum ds t vs; w c vw ths s chgg th txs s d τ > 0 d dy dy Rsult - us vus sult chgg txs t cmut dy dy c α c α Su : dy c α α Css - Nutlty - α α th dy 0.., csums qully ltustc, th's chg th ulc gd m chgg wlth... NOTE: w' t tlkg ut csumt th vt gd whch wll NOT utl G Y - α > α th dy > 0.., w ts wlth t m ltustc s, th's cs th ulc gd; ths sults cus th lss ltustc s m whm th my s tk wll ctu t ctut s sd t u ltustc s whs ctut wll dcs y th sm mut th tx Ls Y - α < α th dy < 0... st g cs 4 7

5 u N-Gvs -. w ssumd lm s vy gvs; w d ths g 0 ssumt d tx -gv, w stll gt th -utlty sult m dstut; th csum ws't gvg, csg hs txs wll t chg hs ctuts, ut gvg tht my t ctut wll cs gvg vll utlty ms dy 0 Ovll - ths mdl hs lt m gvs th BBV's u ltusm mdl, whch mks ths m lstc u ltusm ds't hld mclly; s.8 BBV ts Imlcts Rcd Equvlc Rcd Equvlc - tmg txt t c ulc gd s lvt.., gvmt dt cg ds't mtt Imlct - th gl RE sult qus u ltusm th quvlc t css gts Sml Mdl - mhsz "sml" - gts wth ds gt ch d: ts d sg - Pulc Gd - S t d ulc xdtu d th dul ulc gd th s ulc gd vstmt d t txs t t ulc gd d y t d dt cg t ulc gd d y sg - Pt Utlty- u C, G βu C, G wh u t's utlty m w csumt; u sg's w utlty C t's vt csumt, G t's csumt ulc gd C sg's vt csumt, G sg's csumt ulc gd β sl dscut ct hw much t vlus sg's wllg Nt: G G S wth G ' 0 ; G G S wth G ' 0 > - Bqust - t tss mut t sg, ut sd utlty, t ds't c ut th mut qust, ust th sg's ttl utlty - Blcd But Rqumt - tx sg must y th dt d th tst ccumultd t xgus tst t : t d > - Exgus Icm - ch gt ys cm I d I, sctvly - Psl But Blc - C I t C I t - Gvmt Plcy - cmltly dd y S, d... m tht w c d t, t, C, C, G, G, th tms: β,, I, I gv Pt's Plm tkg gvmt lcy s gvg: t t mx u I t, Gt β u I t, Gt FOC: u β u 0 ssumg > 0 5 7

6 Rcd Equvlc - gv S lvl ulc gd, csumt vct d utlts vt t mthd cg.., t,t P: g wth tl lcy S, t, t, d qulum... s FOC s stsd Nw chg t,t whl hl S cstt t dt Rcll S t ds dt 0 cus S s cstt Sus ts ct y d dt, th: dc dt d 0 t vt csumt uchgd dt d dt dt dc 0 llws m ds 0 Sc vt csumt d lvl ulc gd d't chg, utlts ls uchgd Add Wm Glw - dd xt utlty tm wh t gs utlty dctly m mut quthd t sg: u wth u ' > 0 d u '' < 0 mx u I t t, Gt u β I t, G t t u EXERCISE - shw Rcd Equvlc ds't hld.., shw w duc t's tx ud dt 0, th t s tt d sg s ws < U Otml Nutl FOC: u β u u ' 0 Sus tl tx systm t,t systm s qulum s FOC hlds Nw chg tx kg S lvl ulc gd xd s dt < 0 dt ds 0 dt dt dt Assum utlty s d dt dc dt d 0 t vt csumt uchgd dt d dt dt dc 0 ls uchgd Bcus u '' < 0, sc d dt 0, w must hv du < 0.., > t th tx chg But lk t th FOC: u & u uchgd m dc dc 0 d smll s w ctully hv u β u u ' 0 < u s smll u s th utl qust s t lg; tml qust s smll d* < dt Kys Ect - ct lwg tx ths cs css cut t csumt dc dt d 0 dc > dt d dt d < 0 6 7

7 Fm : "Pts muly ltustc wth sct t th gts t hs, whl th hs c thught s 'uly ltustc' wth sct t th w csumt. As such, dstuts m chld m ltustc t ts lss ltustc wll duc th vt suly th ulc gd th csumt th h. Pts wll uwllg t ctly susttut qusts dt; hc thy wll k sm th w 'wlth' thmslvs."