- contrast so-called first-best outcome of Lindahl equilibrium with case of private provision through voluntary contributions of households

Transcription

1 Prvate Provson - contrast so-called frst-best outcome of Lndahl equlbrum wth case of prvate provson through voluntary contrbutons of households - need to make an assumpton about how each household expects ther contrbuton to affect contrbutons of others - standard Nash assumpton made - n plannng contrbuton to publc good provson, each household takes contrbuton of other households as gven - smlar set of basc assumptons used as wth the Lndahl equlbrum - utlty functon s: U h U h x h,, h 1,...,H 21 x h s prvate good, = g h and g h s contrbuton of household h - contrbuton of households other than h, h, s defned:

2 h g h 22 - usng the budget constrant n 16, utlty can be wrtten as: U h x h, U h h p g h,g h h V h g h, h,p 23 - household h maxmzes 23 gven h and subect to: g h 0, h p - ndfference curves of V h can be drawn n g h, h space; ncreasng h leads to a hgher V for gven g h, the latter lmted by the budget constrant - gven the Nash assumpton, optmal choce of g h for a gven h occurs at a tangency of an ndfference curve to lne h * as shown n Fgure 3 - locus of these ponts traces out the Nash reacton functon, whch slopes down

3 Fgure 3: Indrect Preferences over Prvate Provson h Budget Constrant h * g h

4 - for all g h on the reacton functon: g h argmax U h h p g h,g h h U h x p U h varatons n g h and h satsfyng the frst-order condton 24 are: dg h d h U h xx p 2 U h x p U h 2U h x p U h s always negatve when U h x > 0. - equlbrum of prvate provson economy s where all reacton functons are smultaneously satsfed - for a two-household economy, prvate provson equlbrum s shown n Fgure 4 - a property of Nash equlbra s that they are typcally not Pareto-effcent; n Fgure 5, Paretoeffcent allocatons are ponts of tangency between households ndfference curves - prvate provson generates under-supply

5 Fgure 4: Prvate Provson Equlbrum g 2 g 1 = 1 g 2 g 2 * g 2 = 2 g 1 g 1* g 1

6 Fgure 5: Non-Optmalty of Prvate Provson g 2 g 1 = 1 g 2 Set of Pareto Improvements Locus of Pareto Optma Prvate Provson g 2 = 2 g 1 g 1

7 Varatons n Provson Quantty - result that prvate provson equlbrum s domnated by other allocatons does not necessarly mean prvate provson leads to undersupply relatve to socal optmum - ths follows from noton that a socal optmum s not necessarly on that part of locus that Pareto domnates the prvate provson equlbrum - n Fgure 6, prvate provson equlbrum s at N, and set of Pareto effcent allocatons gven by locus of tangences between ndfference curves - CC s an aggregate level of publc good supply equal to that at N, so f locus of Pareto optma cuts CC, and socal optmum s at P, there wll be a reducton n provson of publc good - Damond and Mrlees 1973 show such anomales can only be ruled out f restrctons are placed on the second dervatves of households utlty functons

8 Fgure 6: Prvate Provson and Over-Supply g 2 C P N Locus of Pareto Optma C g 1

9 Number of Households - mght be expected that an ncrease n number of households would lead to greater dvergence between prvate provson and optmal provson - assume all households are dentcal n terms of preferences and endowments, so that wth H households, the symmetrc equlbrum s: g 26 H 1 where g s common contrbuton of a household; n g, space, an allocaton satsfyng 26 must le on a ray from the orgn wth slope of H-1 - n Fgure 7, for a level of H, prvate provson where ray cuts reacton functon, and welfare optma are where ray s tangent to an ndfference curve - quantty of publc good at prvate provson vares n H as functon of the gradent of the reacton functon,.e., f gradent s < 1 >1, provson Hg, s an ncreasng decreasng functon of H * See the appendx

10 Fgure 7: Equlbra and Optma for H ˆ H=3 H=2 Pareto Optma g= g

11 Alternatve Formulatons - noton that an ncrease n contrbuton of one household leads to a reducton n that of others has been crtczed - behavoral assumptons other than Nash have been nvestgated, such that n 24, household s consder the choces of others as beng dependent on ther decson,.e., a conectural varaton: -frst-order condton becomes: U h x p U h H 1 1,h g g h 0 27 * see the appendx - Cornes and Sadler 1984 show that f conectures are postve, equlbrum has greater prvate provson of the publc good - f conectures are consstent,.e., they agree wth actual responses of households, Sugden 1985 has shown the only consstent conectures are negatve, so that zero provson of the publc good s possble - conectural varatons are ultmately arbtrary

12 Mechansm Desgn -analyss of the Lndahl equlbrum assumed households were honest n revealng ther reactons to cost shares for a publc good - a household can play strategcally and msrepresent ther preferences, assumng other households do not do lkewse - n Fgure 9, EL s the Lndahl equlbrum f households play honestly - suppose, household 2 knows household 1 s preferences, and clams ts preferences to be L 2 2, rather than L 2 2, equlbrum can be pushed to E, where household 2 maxmzes utlty subect to household 1 s reacton functon -household 1 s reacton functon can be thought of as nverse supply curve for publc good facng household 2 measures per unt or average cost to household 2

13 - m s margnal cost to household 2 of publc good, so household 2 chooses level of where margnal value of publc good from true reacton functon s ust equal to margnal cost, and reports L 2 2, gvng and - household 2 tax bll s reduced by ab*, whch offsets ther reducton n consumpton of publc good ac - cost savng on publc good s ed*, losses to household 1 s cdee and ac* for household 2, so effcency loss s ace -due to ths type of result, there has been a focus desgn of preference revelaton mechansms -these are games where each household chooses a strategy to maxmze ther payoff, the game often beng one of ncomplete nformaton, households only havng knowledge of ther own payoff functon -a number of equlbrum concepts have been employed n the lterature ncludng domnant strateges, Nash, and more recently Bayesan

14 Fgure 9: Manpulatng the Lndahl Equlbrum 1 L 1 1 L d e m b a c E L E' L 2 2 * I 2 ' 1 * ' 2

15 Clarke-roves-Vckrey CV Mechansm Board asks ndvduals to report ther margnal valuatons B' of dfferent amounts of publc good, and supply s gven by equatng summed margnal valuatons to margnal cost of the publc good: B' = c 1 Frst part of CV tax s a per unt tax prce t wth t = c whch s fxed and cannot be altered by ndvdual s announcement Second part of tax on s determned by calculatng - the amount of the publc good whch would be suppled f announced a constant margnal valuaton equal to ther tax prce: B' + t c 2 = Wth s announcement of B ' good changes to, satsfyng:, supply of the publc B ' = c 3 Second part of tax on s change n reported benefts to all other ndvduals, less ther payments of per unt taxes:

16 = t B B T ] [ ] [ 4 Ths calculaton s possble because B B s the area under ndvdual s reported demand curve or margnal valuaton schedule between - and Under the CV tax, ndvdual has utlty of: T t x B + 5 They choose to maxmze 5, so satsfes: Thus, chooses a level of publc good whch s effcent gven announced valuatons of other ndvduals,.e., CV mechansm s a domnant strategy mechansm As all are motvated to reveal true margnal valuatons, there s an effcent supply of the publc good + = + = c B' B' t B' t B' d dt t B'

17 Fgure 10: The CV Mechansm 1 L 1 1 L d e c E L b a E' L 2 2 * I 2 ' * ' 1 2

18 CV mechansm s llustrated n fgure 10 Suppose tax prce s t =', -2 = ', where B' 1 + τ ' = c If supply s ncreased to *, ndvdual 1 wll have a gross beneft equal to acde, and pays extra taxes of abde, therefore, T 2 * = abc In order to ncrease to *, ndvdual 2 has to pay addtonal taxes of ac*'= '*-'+ T 2 * ac*' s also the socal cost of the ncrease n,.e., the dfference n the resource cost '*de, and ts gross value to ndvdual 1 of acde The CV mechansm confronts ndvdual 2 wth a margnal cost of the publc good equal to the heght of ndvdual 1 s demand curve, so that they choose *, where ther true margnal valuaton cuts the margnal cost curve under the mechansm There are two key features of the process: - per unt taxes ' do not change wth ndvdual s announcement, so they can be set arbtrarly, as long as they add up to c

19 - as per unt taxes cover cost of publc good, there s surplus of T, whch should be wasted f there s correct revelaton, although ths becomes arbtrarly small as the populaton of ndvduals becomes very large