- if lump-sum taxes are infeasible, level of public good provision must take account of the method of finance

Transcription

1 Fiace of Public oods - if lump-sum taxes are ifeasible, level of public good provisio must take accout of the method of fiace - gai i welfare due to provisio of public good must be offset by ay distortios due to method of fiace - focus o the case of a sigle-cosumer ecoomy where oly commodity taxes are available - each cosumer maximizes utility fuctio U(x,), subject to a budget costrait qx = 0, where q is vector of post-tax prices ad x is a vector of et demads - govermet reveue must equal expediture o the public good: x i (28) - reveue costrait ca be used iterchageably with the productio costrait because of market clearig

2 - productio costrais used, where F(X,) = F( x,) = 0, ad assumed that: F 1 F 1 X (29) 1 good 1 is chose as the umeraire, q 1 = p 1 = 1, ad pre-tax prices are chose so that F k = p k - the Lagragea is: V( q, ) F ( X (q, ), ) (30) - first-order coditio for choice of is: V F i F 0 (31) - usig defiitio of pre-tax prices, (31) becomes: V p k F F k q k p i (32) *see the appedix - where is margial utility of icome for each household, ad from each households first-order coditio, U/x k =, ece,

3 V U U (33) x k The LS of (32) is the sum of the margial rates of substitutio betwee public good ad private good k (equals the ratio of utility derivatives) - evaluatig (32) for k=1, so that q 1 =1, it ca be rearraged: F F 1 U U x 1 [q i ] (34) - cosumer budget costraits imply: q i 0

4 - re-writig (34): MRT,1 h1 MRS h,1 (35) - (35) is the Samuelso rule i the presece of distortig commodity taxes, which is differet to (7) i two ways: (i) the sum of the margial rates of substitutio is multiplied by / which may ot be =1 (ii) a additioal term o right-had side that measures effect of public good provisio o tax reveue due to substitutability/complemetarity i demad betwee private goods ad public good - latter effecmplies thaf provisio of public good icreases tax reveue, i.e., if is a complemet to private goods, this reduces cost of providig the public good - this teds to icrease provisio above Samuelso level; coverse is true if provisio of public good reduces tax reveue

5 - to isolate the first effect, assume the secod term o right-had side is zero, i.e., public good is reveue eutral, so / determies departure from first-best - from the Lagragea, tax rate for good k is: V F i p i (36) - usig Roy s idetity, ad fact that: p i q i 0 (37) - (36) is re-writte: X k (38) * see the hadout

6 - usig the Slutsky equatio: 1 I S ik X k (39a) * see the hadout the divergece of / from 1 is separated ito two parts, the first a reveue effect, the secod a distortioary effect, i.e.: I ad S ik X k - the reveue effect caot be uambiguously siged, uless all goods are ormal, ad the effect is positive, so that < - as the Slutsky matrix is egative semi-defiite, it follows that the distortioary effecs egative, so teds to be reduced below - implies that the true beefit of the public good is less tha the sum of the margial rates of substitutio

7 Level of Provisio i Presece of Taxes - ca the aalysis outlied aswer questio: will more or less of public good be provided i presece of distortioary taxatio? - cosider a ecoomy where fiace is possible through a lump-sum tax T or a tax t levied upo a sigle factor of productio L, the Lagragea beig: V ( t,t, ) [ T t L ] (39b) * see the hadout with first-order coditios: V t L t L t o (40) V 0 (41) T t L 0 (42) where T ca vary from 0 to

8 -i Figure 8, (41) ca be draw as the demad curve for public goods, =V t L =/(1 + ) L t (43) where s a fuctio of from (42) - i the case of lump-sum taxes, = -i Figure 8, for CT < LS, requires that CT > LS (= LS ); Stiglitz ad Dasgupta have show that CT > CT, if the supply curve of labor is upward slopig, but this is ot sufficiet to show CT < LS, has to be show CT > LS -Atkiso ad Ster have show provisio is lower with commodity taxatio with utility fuctio: U( X, L,) = alog X + [1 - a]log[1 - L] + f ( ) (44) * See the appedix

9 Figure 8: Lump-sum vs. Commodity Taxes =/[1+[t/L]L/t] {T=0} CT = {T=} LS =V CT LS