Distribution, Distortionary Taxation, and the Evaluation of Public Goods

Transcription

1 Dstrbuton, Dstortonary Taxaton, and the Evaluaton of Publc Goods Roberton C. Wllams III Unversty of Texas at Austn and NBER December 2005 Key Words: second-best, tax dstortons, polluton tax, envronmental tax, commodty tax, publc good, dstrbuton, equty, representatve-agent, multple-agent JEL Classfcaton Nos.: H23, H21, Q58, D60, D62, D63 Please address correspondence to Roberton Wllams, Department of Economcs, Unversty of Texas, 1 Unversty Staton #C3100, Austn, TX , E-mal: robwllams@mal.utexas.edu. For ther helpful comments and suggestons, I thank Don Fullerton, Larry Goulder, Gb Metcalf, and semnar partcpants at Stanford Unversty and the ASSA meetngs.

2 Dstrbuton, Dstortonary Taxaton, and the Evaluaton of Publc Goods Abstract Ths paper compares the results of representatve-agent and multple-agent models for three dfferent polcy problems: optmal commodty taxes, optmal provson of a publc good, and the optmal polluton tax. It shows that representatve-agent models yeld fundamentally the same results as multple-agent models f the dstrbuton of the net beneft of the polcy s proportonal to ncome. Other studes have found very dfferent results for the two types of models, but these turn out to be the result of dfferences n assumptons and defntons, rather than of fundamental dfferences between the models. Ths suggests that even though multpleagent models are far more realstc, usng a smpler representatve-agent model wll produce reasonable results as long as the dstrbuton of the net benefts of the polcy n queston s roughly proportonal to ncome, but that f t s sgnfcantly regressve or progressve, representatve-agent models can be msleadng.

3 Ths paper uses a sngle ntegrated model to compare the results of representatve-agent (RA) and multple-agent (MA) models for three dfferent polcy problems: optmal commodty taxes, optmal provson of a publc good, and the optmal polluton tax. In each of these cases, pror studes usng RA models have found substantvely dfferent results from those usng MA models. For example, RA models of optmal publc good provson typcally fnd that because rasng tax revenue entals a dstortonary cost, the optmal quantty of the publc good s less than t would be n a frst-best economy, but MA models suggest that ths should have no effect, and that the optmal quantty of the publc good should smply equate the margnal beneft of the publc good to ts margnal cost. Smlarly, RA models of second-best optmal envronmental taxaton suggest that n the presence of pre-exstng tax dstortons, the optmal polluton tax should be somewhat less than margnal damages, whereas MA models suggest that the optmal tax s equal to margnal damages. But other dfferences n the models complcate comparsons across studes. By usng the same assumptons, basc model structure, and approach to solvng for the optmum n both the RA and MA versons of the model, the comparson n ths paper dstngushes between dfferences n results that arse because of dstrbutonal effects (whch are the most obvous dfference between RA and MA models, gven that the former are nherently unable to consder such effects) and those that stem from dfferences n other assumptons. RA models are attractve because they are substantally smpler both for authors to construct and for readers to understand than are MA models. Gven that even the RA models of these problems can be qute complex, any addtonal complexty comes at a substantal cost. 1

4 On the other hand, RA models are clearly unrealstc, and are ncapable of lookng at dstrbutonal ssues. To the extent that such ssues have an mportant effect on the model s results, t may be necessary to use the more complex MA models. Ths paper shows, however, that most of the dfferences n results are not caused by dstrbutonal effects at all. Instead, they are caused by other dfferences n defntons and assumptons across studes. For example, the MA models n Kaplow (1996 and 2004a) assume that lesure s weakly separable from all other terms n utlty, whereas RA models of publc good provson typcally assume that the quantty of the publc good s separable. Ths dfference n assumptons accounts for the dvergent results; f both types of models use the same separablty assumpton, then they yeld the same central-case results. Ths suggests that the central-case results from RA models are perfectly consstent wth MA models. Ths s also true for devatons from the central case along some dmensons. In such cases, RA models provde a relable gude for polcy, and are substantally smpler than MA models. However, for polces wth sgnfcantly progressve or regressve dstrbutonal effects, RA and MA models can yeld very dfferent results. Ths suggests that MA models are necessary n such cases, or at least that the results of RA models for such cases should be nterpreted wth substantal cauton. The next secton of ths paper presents a unfed model of commodty taxaton, publc good provson, and envronmental taxaton wth both RA and MA versons. It then descrbes the novel approach used to solve for the optmum n ths model, and uses t to compare the results of the RA and MA models for each of the problems. The fnal secton concludes. 2

5 I. A Theoretcal Model Ths secton presents two closely related versons of a smple analytcal general equlbrum model, and uses them to analyze the optmal set of dfferentated commodty taxes, optmal provson of a publc good, and optmal envronmental tax. The frst verson of the model has only a sngle representatve agent and an ncome tax that s restrcted to be lnear wth a zero ntercept assumptons smlar to those n the smpler models used n pror work on these problems. The second verson of the model has a contnuum of agents and a nonlnear ncome tax. Equatons that apply only to the RA verson of the model wll have an equaton number followed by an R and those that apply only to the MA verson wll have an equaton number followed by an M. Equaton numbers wthout a letter ndcate an equaton that apples to both versons of the model. A. Model Structure Utlty for ndvdual s gven by (1) U! U ( l,x,g,q), where l s the quantty of lesure (non-market tme) consumed by the ndvdual, the elements of the vector X are the quanttes consumed of each of a set of consumpton goods, G s the quantty of a government-provded publc good, and Q s the level of envronmental qualty. Ths functon s assumed to be quas-concave and twce dfferentable. The terms l and X (wthout superscrpts) wll denote total consumpton. In the RA verson of the model, there s only one ndvdual, so total consumpton of each good s smply equal to the consumpton of that representatve ndvdual. The MA verson of the model has a contnuum of agents, so total consumpton of each good s gven by an ntegral over that contnuum. All ndvduals have the 3

6 same utlty functon. The quantty of the publc good G and envronmental qualty Q are also constant across households, whch mples that both are nonexcludable. Each ndvdual faces the tme constrant (2) L = L + l where L s a fxed tme endowment and L s labor. The ndvdual budget constrant s (3)! p j X j " Y # Y $ T Y j ( ), where p j s the consumer prce of good j and X j s the consumpton of good j by ndvdual (an element of the vector X ). The total amount spent on consumpton goods must be less than or equal to after-tax ncome (Y ), whch s n turn equal to pre-tax ncome (Y ) mnus ncome taxes pad (T). 1 The government can observe an ndvdual s ncome, but not hs wage or amount of labor suppled, and thus the ncome tax s a functon of ncome. Pre-tax ncome s gven by (4) Y! w L, where w s the pre-tax wage. In the RA model, unts of labor are normalzed such that w equals one, whch entals no loss of generalty. In the RA model, the ncome tax s restrcted to be lnear wth a zero ntercept. Ths mples that (5R) T ( Y ) = T! Y = " L Y, where T! s the margnal tax rate per unt of ncome, and! L s the margnal tax rate per unt of labor. In the RA model,! L = T ", because w = 1, but that s not true n the MA verson of the 1 Throughout the paper, a varable wth an overbar wll represent a pre-tax amount, as n ths case. 4

7 model, so defnng both terms separately s necessary to keep the notaton constant across models. In the MA verson of the model, the ncome tax functon s assumed to be contnuous and twce dfferentable, but s otherwse unrestrcted. 2 The margnal tax rate per unt of labor n ths model s gven by (5M)! L = w T ", where T! s the margnal tax rate per unt of ncome for ndvdual, equal to the frst dervatve of the functon T, evaluated at Y. In both models, the after-tax wage follows (6) w = w! " L. The consumer goods and the publc good are produced under constant returns to scale, wth labor as the only nput. Markets for those goods and for labor are assumed to be perfectly compettve, so producer prces equal margnal cost, and pre-tax wages equal margnal product. Quantty unts are normalzed such that the producer prce equals one for any of these goods. Fnally, n the MA model, labor by any ndvdual s perfectly substtutable n producton for labor by any other ndvdual. Together, these assumptons mply that pre-tax wages and producer prces are exogenously fxed, and that the aggregate producton constrant s! (7) G + " X j = " w L. j The government also mposes a commodty tax of! j per unt of consumpton good j. The consumer prce for good j s equal to that tax plus the producer prce, whch was normalzed to one for all goods. Thus, 2 Even these assumptons are not strctly necessary, but they greatly smplfy the analyss. 5

8 (8) p j = 1+! j. Government revenue from the ncome tax and commodty taxes must equal or exceed spendng on the publc good (whch s smply equal to the quantty of the publc good, because of the quantty normalzaton). Ths gves the government budget constrant $ ' (9) R! T ( Y * & ) + #" j X j ) + G, 0, % j ( where R denotes government revenue mnus expendture on the publc good. Fnally, there s a polluton externalty assocated wth the producton and/or consumpton of one of the consumer goods, whch wll be denoted wth a subscrpt P (for pollutng ). Unts of envronmental qualty are normalzed such that one unt of the pollutng good reduces envronmental qualty by one unt. Ths gves (10) Q = Q! X P, where Q s the exogenous baselne level of envronmental qualty. Each ndvdual maxmzes utlty (1) by choosng l and X, subject to the tme constrant (2) and budget constrant (3 and 4), takng as gven all of the other varables (taxes, prces, the after-tax wage, the quantty of the publc good, and the level of envronmental qualty). 3 The consumer frst-order condtons are gven by (11) U X j =! p j and U l =! w, where! s the margnal utlty of ncome for ndvdual, and the subscrpts on U denote partal dervatves. It s also useful to defne the correspondng expendture functon 3 If the ncome tax schedule s not lnear, then after-tax wages are not fxed, even for a gven ncome tax schedule, because the margnal tax rate vares along the tax schedule. For margnal changes, assumng that households take after-tax wages as gven s equvalent to takng a local frst-order approxmaton to the tax schedule. 6

9 ( )! p j X j * (12) E! E U,p,w,G,Q " # w L *. j Ths s the mnmum level of lump-sum ncome expendtures on goods mnus after-tax labor ncome (gnorng the nonlnearty of the tax schedule, f any) necessary to acheve a utlty of U for a gven vector of after-tax prces, a gven wage, and gven levels of the publc good and envronmental qualty. In ths expresson, X j * and l * are the quanttes of the consumpton goods and lesure, respectvely, that mnmze expendture subject to the constrant that utlty has to be at least U. Partal dervatves of U wth respect to p and w yeld compensated demand and labor supply functons (by Shephard s Lemma): (13) X j ( U,p,w,G,Q) =!E!p j and L ( U,p,w,G,Q) =! "E "w. B. Solvng for Optmal Government Polcy Ths paper consders three dstnct polcy ssues: commodty taxes, provson of a publc good, and a polluton tax. The fundamental approach used to solve for the optmal polcy s the same for each of these polces, and for both the RA and MA versons of the model. In each case, I analyze the combnaton of a margnal ncrease n the polcy nstrument n queston (the commodty tax rate, quantty of the publc good, or polluton tax rate) and a compensatng change n the ncome tax. For the RA model, ths s a change n the ncome tax rate that wll exactly offset the effect of the polcy n queston on the utlty of the representatve agent (e.g., f the polcy makes the representatve agent better off, then the compensatng ncome tax change would be an ncrease n the ncome tax rate). The compensatng ncome tax change for the MA model s smlar, but s a change n the whole ncome tax schedule that exactly offsets the effect 7

10 of the polcy n queston on the utlty of the ndvduals at each pont on the tax schedule. 4 Thus, n ether model, the combnaton of the margnal polcy change and the compensatng ncome tax change holds utlty constant for all ndvduals. Usng the expendture functon (12) and the envelope theorem shows that the compensatng ncome tax change follows (14)!E!P dp + dt Y ( ) = 0 " dt ( Y ) dp = #!E!P, where P s the polcy nstrument n queston (the commodty tax rate, quantty of the publc good, or polluton tax rate). The ntuton behnd ths expresson s very smple: the frst term s the change n expendture necessary to hold utlty constant, and the compensatng ncome tax change must exactly offset that change n expendture. 5 To obtan an expresson for the change n the margnal tax per unt of labor n the RA model, substtute together (14) and a dervatve of (5R) to get (15R) d! L dp = 1 L dt ( Y ) = " 1 #E dp L #P. In ths case, the ncome tax s just a multple of the quantty of labor suppled, so the change n the margnal tax rate on labor equals the compensatng ncome tax change dvded by the quantty of labor suppled. To get the analogous expresson for the MA model, take the dervatve of (14) wth respect to Y and substtute t nto (5M). Ths gves 4 If there s any poolng n the equlbrum of the MA model that s, f the tax schedule s such that ndvduals of dfferent abltes choose to earn the same pre-tax ncome then the compensaton necessary mght dffer for ndvduals wth the same pre-tax ncome. Ths would make t mpossble to hold utlty constant for all households. Wllams (2004) deals wth ths ssue by settng the compensatng tax change for any ncome equal to the average compensaton necessary for ndvduals wth that ncome. But, for smplcty, I assume here that there wll be no poolng, and thus that ths case wll not occur. 5 Note that the dt term here s the compensatng change n ncome tax for a constant pre-tax ncome equal to Y : t does not nclude the effect of any change n Y as a result of the combnaton of the polcy change and the compensatng ncome tax change. Thus, ths term wll not generally equal the change n ncome tax pad by. 8

11 (15M) d! L dp = w! " # $ dt Y ( ) dp % & = 'w "Y! " "E "P ( ) "Y, where the arrow over the! denotes a cross-sectonal dervatve. In other words, the dervatve wth respect to Y s not the effect of a margnal ncrease n ncome for ndvdual, but rather the dfference across ndvduals for a margnal move along the ncome dstrbuton startng from Y. Ths equaton shows that f the compensatng ncome tax change s rsng wth ncome, then t wll cause an ncrease n the margnal tax rate, whereas f t s fallng wth ncome, t wll cause a decrease n the margnal tax rate. If ths combnaton yelds a net ncrease n government revenue net of spendng (R), then that margnal ncrease could yeld a pure Pareto mprovement; the combnaton of the polcy change and compensatng ncome tax change holds utlty constant for everyone, and the addtonal revenue can be used to make some (or all) agents better off. If the combnaton yelds a net loss of revenue, then the same argument works n reverse; a margnal decrease n the polcy nstrument n queston plus a compensatng ncome tax change could acheve a pure Pareto mprovement. The optmum, then, s the pont at whch the change n revenue from the margnal polcy change exactly offsets the change n revenue from the compensatng ncome tax change, so that the combnaton yelds no net change n revenue. Thus, the government frst-order condton s dr dp = 0. Ths approach s based on that used by Wllams (2004), whch s n turn a generalzaton of the approach used n Kaplow (2004a and 2004b). Kaplow used smplfyng assumptons that ensure that labor supply does not change n response to a polcy shft, whereas Wllams (2004) relaxed those assumptons and derved expressons for effects of the resultng changes n labor supply. The approach used here for the MA model s essentally the same as that n Wllams 9

12 (2004), but that paper dd not nclude a RA model, nor dd t even recognze that ths approach could be used n a RA model. Prevous RA models have typcally derved expressons for revenue-neutral polcy changes, and solved for an optmum by settng the dervatve of utlty for a margnal polcy change equal to zero. The approach used here takes the opposte approach: t holds utlty constant, and solves for an optmum by settng the dervatve of net revenue equal to zero. Even though these approaches are qute dfferent, notce that under ether approach, the dervatves of net revenue and of utlty for a margnal polcy change are both equal to zero at the optmum as defned under ether approach. Thus, ths approach wll yeld the same results as the approach used n prevous RA models. As noted above, the approach used here s an extenson of that used n Kaplow (2004a and 2004b). For the cases consdered by Kaplow, both approaches wll produce the same results. The results from ths approach must also match those that would result from settng government polcy to maxmze socal welfare, whch s a more common way to solve for the optmum n an MA model. Suppose ths weren t true: that the pont that maxmzes socal welfare s not the optmum under ths approach. But, as dscussed above, at any pont other than the optmum under ths approach, t s possble to use a combnaton of a polcy change and a compensatng ncome tax change to produce a pure Pareto mprovement. And f t s possble to produce a pure Pareto mprovement startng from a partcular pont, that pont obvously can t be the pont that maxmzes socal welfare. Thus, whle the approach used n ths paper dffers from that used n most prevous MA models and from the approaches used n all prevous RA models, t wll nonetheless produce the same results as those other approaches. C. The Optmal Commodty Tax Problem 10

13 Ths subsecton looks at the optmal commodty tax rates on the consumpton goods. To focus on ths problem, the analyss n ths secton assumes that the quantty of the publc good remans constant, and that there s no polluton externalty (.e.,!u!q = 0 ). Consder the effect of a margnal change n the tax rate on good k, holdng all other commodty tax rates constant. To obtan an expresson for the change n government revenue n ths case, take total dervatves of the government budget constrant (9) and the defnton of pretax ncome (4) and substtute them together wth the defntons of the margnal tax rate per unt of labor (5R and 5M) to get (16) ( ) dr dp = dr = X k + dt Y # dx j dl + " &! j d! k ) % +! L (. $ % d! k j d! k d! k '( The frst term on the rght-hand sde of ths equaton s the change n revenue that would result from the commodty tax ncrease f ndvdual consumpton and labor supply decsons ddn t change at all. The second term s the analogous change n revenue from the compensatng ncome tax change. The thrd and fourth terms are the effects of changes n consumpton and labor supply on the revenue from the commodty taxes and the ncome tax, respectvely. To obtan an expresson for the compensatng ncome tax change n ths case, take the dervatve of the equaton for the prce of good k (8) wth respect to! k and substtute t together wth the equatons for the compensated demand functon for k (13) and for the compensatng ncome tax change (14) to get (17) dt ( Y ) = "X k. d! k Ths expresson says that to hold utlty constant for a margnal ncrease n the tax on good k, at any level of ncome, the ncome tax pad must fall by an amount equal to the consumpton of 11

14 good k by ndvduals at that ncome level. Substtutng (17) nto (16), cancelng terms, expandng the terms for the changes n labor supply and commodty demands, and rearrangng gve (18) dr dp = dr "X = # j! j + d! k j "p 1 + "L "L % "X j ( d!! L $! L + #! L. - k "p ' k "w j & "w * 0,- j ) d! k / 0. Note that f the changes dd not affect ndvduals consumpton and labor supply decsons, then the extra revenue from the commodty tax ncrease would be exactly offset by the cost of the compensatng ncome tax cut; the frst two terms on the rght-hand-sde of (16) cancel out. To obtan the frst-order condton for the optmal tax rate n the RA model, substtute (17 ) nto the equaton for the change n the margnal tax rate on labor (15R), then substtute the result nto (18), set equal to zero, and rearrange terms to get, #X (19R)! *RA k = "! j #L % j +! L + X & k. j $ k #p k #p k L ( -. '! L #L #w +! #X j ) / % j #w + 1 j * 01 #X k #p k To get an expresson that s smlar to those from the pror lterature on optmal commodty taxes, t s useful to defne two terms. The frst s the margnal cost of publc funds (MCPF), gven by (20R)! = L $ U L L " # L $w " % # j j $ U X j $w (where a superscrpt U denotes an uncompensated dervatve). The MCPF s the cost to the representatve ndvdual of rasng a margnal dollar of government revenue va the ncome tax. Thus, the numerator n (20R) s the cost to the ndvdual of a margnal ncrease n the ncome tax rate, and the denomnator s the margnal revenue from that ncrease. The second useful term s the socal margnal utlty of ncome, gven by 12

15 ' %L (21R)! = " + #" ) $ L %I + & $ j ( j %X j %I *, +, where I s lump-sum ncome. Ths s the socal shadow value of one addtonal unt of after-tax ncome, equal to the prvate margnal utlty of ncome plus the value of the change n government revenue caused by ncome effects on labor supply and commodty demands. 6 Usng these two defntons and the Slutsky equaton, one can rewrte the frst-order condton n (19R) as (22R) "X j "L %& $ ' #! j +! L = $X k j "p k "p k %&. Ths s the standard Ramsey tax result from the pror lterature on optmal commodty taxes. 7 To get a frst-order condton for the MA model that s analogous to (19R), follow the same set of steps, but use (15M) for the change n the margnal tax rate on labor nstead of (15R). Ths gves 57 #X (19M)! *MA k = "! j. #L % j +! L +! 6 & j $ k #p k 4 0 #p Xk 'Y 87 / 0 k X k L ( * )! L #L #X +! j % #w j #w - 3: j, 23 ;7 #X k #p k, where! Xk "Y s the cross-sectonal elastcty of X k wth respect to Y. The ntuton n ths case s the same as that n the RA model, but here the compensatng tax change s no longer necessarly proportonal to ncome. Consequently, the optmal tax rate depends on the ncdence of the tax. If the taxed good s consumed prmarly by hgh-ncome people (.e., f! Xk "Y s large), then the compensatng tax change wll lower ncome taxes more for hgher-ncome people than for lower- 6 Ths defnton mplctly assumes that the labor tax s set optmally, whch mples that the shadow value of government revenue equals the MCPF tmes the prvate margnal utlty of ncome. 7 For example, ths equaton s equvalent to equaton (3.9) n Auerbach and Hnes (2002), as long as the labor tax s set optmally. 13

16 ncome people, and thus wll result n lower margnal tax rates, reducng deadweght loss. Consequently, the hgher s! Xk "Y, the hgher wll be the optmal tax rate on k. Most pror work on optmal commodty taxes wth a nonlnear ncome tax mposed the smplfyng assumpton that lesure s weakly separable n the utlty functon. Ths mples that (20M)!L = " X k! #!p k L Xk $Y!L!w (see appendx for dervaton). Under ths assumpton, then, (19M) reduces to / 1 #X (22M)! *MA k = "! j 0% j + 21 j $ k #p k. (! * ) & Xk 'Y X k L % j! j #X j #w , 51 #X k #p k, whch s satsfed f the tax rate s zero on all goods. Ths s the same zero-commodty-tax result from the pror lterature (e.g., Atknson and Stgltz 1976 and Kaplow 2004b). Here, the effect of nteractons wth the ncome tax exactly offsets the effect of dstrbuton. As dscussed above, the more progressve the tax s, the hgher the optmal tax rate wll be. However, (20M) shows that when lesure s separable n utlty, the more progressve the tax s, the more t wll dscourage labor supply, whch wll lead to a lower optmal tax rate. Note that the expressons for the optmal tax n the RA model (19R) and MA model (19 M) are very smlar, though the expresson for the MA model dffers n two respects. Frst, t obvously must cover a contnuum of agents, rather than a sngle representatve agent. So the second term n the numerator of (19M) s an ntegral over that contnuum, whereas the correspondng term n (19R) s just the value for one ndvdual. Ths could cause the result from the MA model to dffer from the result from the RA model, even f the assumed elastcty and tax rates for the representatve agent are equal to the correspondng averages over all ndvduals n the MA model. In ths case, the results of the two models would stll dffer f the elastctes of 14

17 labor supply wth respect to vary across the ncome dstrbuton, and that varaton s pk and w correlated wth varaton n the margnal tax rate. For the RA model s result to match the MA model s result exactly, one would need to use a weghted average for the tax rate and or labor supply elastcty, rather than a smple average. But ths seems unlkely to cause a qualtatve dfference n the models results, though t certanly could be quanttatvely mportant. The second dfference between (19R) and (19M) could lead to a more fundamental dfference n results. There s no concern for dstrbuton n the RA model, whereas dstrbuton matters n the MA model. As a result, the second term n the numerator n (19M) ncludes! Xk "Y whle the analogous term n (19R) does not. In the MA model, the more progressve s a tax on good k (.e., the hgher s! Xk "Y ), the hgher the optmal tax rate wll be, all else the same. The, more regressve s the tax (the lower s! Xk "Y effect occurs n the RA model. ), the lower the optmal tax rate wll be. No such If consumpton of good k s proportonal to ncome (! Xk "Y = 1), then the two models wll yeld fundamentally the same result for the optmal tax on good k. Ths should not be especally surprsng. The fundamental dfference between the two models s that the MA model takes dstrbuton nto account, whle the RA model does not. But f consumpton of the taxed good s proportonal to ncome, then that dfference doesn t matter, because the ncdence f the tax s smply proportonal to ncome. The approach used n ths paper to derve the optmal tax rates provdes an even smpler ntutve explanaton: f consumpton of the taxed good s proportonal 15

18 to ncome, then the compensatng ncome tax change n the MA model wll be proportonal to ncome, and thus wll be the same as n the RA model, and thus wll lead to the same results. 8 D. The Optmal Publc Good Provson Problem Ths subsecton looks at the problem of fndng the optmal quantty of the publc good, gven that t must be fnanced usng the dstortonary ncome tax. Thus, the analyss n ths secton assumes that all of the commodty tax rates equal zero, and that there s no polluton externalty (.e.,!u!q = 0 ), and focuses on the effect of a margnal change n the quantty of the publc good (G). To obtan an expresson for the change n government revenue n ths case, take total dervatves of the government budget constrant (9) and the defnton of pre-tax ncome (4) and substtute them together wth the defntons of the margnal tax rate per unt of labor (5R and 5M) to get (23) dr dp = dr dg = ( ) # dt Y!1 + dg + " dl & ) % L (. $ % dg '( Just as wth the analogous equaton for the commodty tax problem, the frst term n ths expresson s the cost of provdng one more unt of the publc good (whch has been normalzed to 1), the second term s the drect effect on revenue of the compensatng ncome tax change, and the remanng term represents the effects on revenue of polcy-nduced changes n labor supply. Settng the expresson n (23) equal to zero, expandng the term for the change n labor supply, substtutng n the equatons for the compensatng ncome tax change (14) and the correspondng change n the margnal tax on labor n the RA model (15R), and rearrangng gve (24R)! "E "G = 1! # L $ % & "L "G + 1 "L L "w "E ' "G ( ). 8 More generally, the two models wll yeld the same results as long at the ncdence of the commodty tax n 16

19 The left-hand sde of ths expresson s the representatve agent s margnal wllngness to pay for another unt of the publc good. The rght-hand sde s the margnal cost per unt (equal to 1) mnus any change n government revenue resultng from the ndvdual s labor supply response to the polcy shft. To obtan an expresson that more closely resembles those n the pror lterature, rearrange (24R), usng the Slutsky equaton and the defnton of the MCPF (20R), whch wll gve (25R)! "E "G = # + - 1! $ L, % & ' "L "G! "L "I "E (. "G ) * 0 / = # % 1! $ L & ' where a superscrpt U denotes an uncompensated dervatve. Expresson (25R) s the modfed " U L "G ( ) *, Samuelson rule for publc good provson. 9 If the quantty of the publc good does not affect labor supply (as wll be the case f G s weakly separable n utlty), then the second term on the rght-hand sde wll equal zero. In ths case, the optmal quantty of the publc good wll be where the margnal beneft (wllngness to pay) for the publc good s equal to the margnal cost (whch has been normalzed to one) tmes the MCPF. Ths equates the socal margnal cost whch s not just the drect cost to produce a margnal unt, but also the addtonal deadweght loss that results from rasng the tax rate to fnance that cost wth the socal margnal beneft. If the quantty of the publc good does affect labor supply, then the second term wll not be zero. The ncome tax causes the margnal socal beneft of a unt of labor (ts margnal product) and ts margnal socal cost (the value of forgone lesure). Therefore, f an ncrease n G leads to an ncrease n labor supply, ths wll produce a general-equlbrum welfare gan, and queston s the same as the ncdence of the ncome tax. 9 Ths expresson s equvalent to equaton (5.4) from Auerbach and Hnes (2002). See also Ballard and Fullerton (1992). 17

20 thus the optmal level of G wll be hgher. If t leads to a decrease n labor supply, then the optmal level of G wll be lower. To obtan a frst-order condton for the optmal quantty of G n the MA model, follow the same set of steps that led to the analogous frst-order condton n the RA model (24R), but use the equaton for the margnal tax on labor from the MA model (15M) nstead of (15R). Ths yelds (24M)! #! "E & "L ) # $ % "G ' ( = 1! * L "G + + WTPG,Y "L "E & ) $ % L "w "G ' (, where! WTPG"Y s the cross-sectonal elastcty of wllngness-to-pay for the publc good (!E!G ) wth respect to Y. Here, just as n the optmal commodty tax problem, the results of the RA and MA models are fundamentally qute smlar, but not dentcal. And the expressons dffer n the same two respects as dd the expressons for the optmal commodty tax rates. Frst, f the labor supply elastctes (n ths case, wth respect to the wage and the quantty of the publc good) vary across the ncome dstrbuton, and that varaton s correlated wth varaton n the margnal tax rate, then the value of the ntegral on the rght-hand sde of (24M) wll dffer from that of the analogous term n (24R). Just as n the commodty tax case, though, ths seems unlkely to have a qualtatve effect on the result. The second and more notable dfference s that n the MA model, the optmal quantty of G depends on the dstrbuton of wllngness to pay, whereas n the RA model, t does not. In the MA model, the more progressve s the dstrbuton of benefts from the publc good (.e., the lower s! WTPG"Y good. ), the hgher s the optmal quantty of the publc 18

21 ! WTPG"Y If wllngness to pay for the publc good s smply proportonal to ncome (.e., f = 1), then the results of the MA model and RA model wll be qualtatvely the same. 10 Ths parallels the result that the optmal commodty tax s qualtatvely the same across the two models as long as consumpton of the taxed good s proportonal to ncome, and the ntuton s the same. Ths smlarty n results mght seem to conflct wth pror work. As dscussed above, the standard result n pror studes that use RA models s the modfed Samuelson rule: the sum of wllngness to pay should equal margnal cost tmes the MCPF. In contrast, papers such as Hylland and Zeckhauser (1979), Chrstansen (1981), and Kaplow (1996 and 2004a) show that n a MA model, f lesure s weakly separable n utlty, the optmal quantty of a publc good s gven by the standard Samuelson rule: the sum of wllngness to pay equals margnal cost. But the results are n fact perfectly consstent wth each other. Weak separablty mples that (25M)!L!G = "! # WTPG$Y!L L!w!E!G (see appendx for dervaton). In ths case, the two terms n the ntegral on the rght-hand sde of (24M) cancel out, so that expresson reduces to the smple Samuelson rule. And f! WTPG"Y = 1, so the results of the MA and RA models are equal, then the second and thrd terms on the rghthand sde of (24R) wll also cancel out, and the result n the RA case wll also reduce to the smple Samuelson rule. But ths s true only when lesure s separable. If the utlty functon s nonseparable, or f t has a dfferent knd of separablty (for example, many papers assume that G s weakly separable) then ths smple rule does not hold. 10 The two models wll yeld exactly the same result f, n addton, any varaton n elastctes across the ncome dstrbuton s uncorrelated wth varaton n the tax rate. 19

22 E. The Optmal Polluton Tax Problem Ths subsecton looks at the optmal polluton tax n a settng wth a dstortonary ncome tax. Thus, the analyss here assumes that the commodty tax rates equal zero for all goods except the pollutng good and that the level of the publc good s fxed, and looks at a margnal ncrease n the tax rate on the pollutng good. The results for ths case wll resemble those for the publc good provson problem, because envronmental qualty enters utlty n the same way that the publc good does. The results wll also resemble those for the commodty tax problem, because the effects of the polluton tax (other than ts effect on envronmental qualty) are the same as those of a commodty tax. Thus, the polluton tax problem combnes elements of the two problems consdered thus far. Just as n the prevous two cases, takng total dervatves of the government budget constrant (9) and the defnton of pre-tax ncome (4) and substtutng them together wth the defntons of the margnal tax rate per unt of labor (5R and 5M) gves an expresson for the net change n revenue from a margnal polluton tax ncrease and the compensatng ncome tax change. Ths gves (26) ( ) dr dp = dr " = X P + dt Y dl % +! L d! P ( $ '. # $ d! P d! P &' The compensatng ncome tax change n ths case can be obtaned by takng the dervatve of the equaton for the prce of good k (8) wth respect to! k and substtutng t together wth the equatons for the compensated demand functon for k (13) and for the compensatng ncome tax change (14). Ths gves (27) dt ( Y ) = "X P + #E d! P #Q dx P d! P. 20

23 Each ndvdual s made worse off by the hgher tax rate on the pollutng good, but s also made better off by the reducton n the externalty. The compensatng tax change s thus a tax reducton equal to the quantty of the taxed good consumed, plus a tax ncrease equal to margnal wllngness to pay for envronmental qualty tmes the decrease n polluton resultng from the tax change. Substtutng (27) nto (26), cancelng terms, and expandng the term for the change n labor supply gve (28) dr dp = dr dx =! P P + d! P d! P "E dx P "L +! L # "L dx P # "L * $ d! L ' - 0, "Q d! P % & "p P "Q d! P "w d! P ( ) /. +. To obtan the frst-order condton for the optmal polluton tax rate n the RA model, substtute (27) nto the equaton for the change n the margnal tax rate on labor (15R), then substtute the result nto (28), set equal to zero, and rearrange terms to get (29R)! *RA P = " #E $ #Q 1 "! L % & L #L ' #L #w ( ) +! L #Q "! $ L % & #L + #L X P #p P #w L ' ( ) dx P d! P. Just as n the publc good problem, ths can be rewrtten usng the Slutsky equaton and the defnton of the MCPF (20R) as (30R)! *RA P = 1 % " # $E ( & ' $Q ) * +! L $ U L $Q #! L % & ' $L + $L X P $p P $w L ( ) * dx P d! P. Ths expresson s smlar to those derved n many of the recent papers on second-best optmal envronmental taxes n RA models. Such papers frequently make the smplfyng assumptons that envronmental qualty s separable n utlty (whch mples that! U L!Q = 0, thus elmnatng the second term on the rght-hand sde of (30R)) and that the pollutng good s an average substtute for lesure (whch mples that!l +!L X P, and thus elmnates the thrd!p P!w L term on the rght-hand sde). Under these assumptons, the optmal polluton tax equals margnal 21

24 polluton damage (MED) dvded by the MCPF. Ths s equvalent to the central-case result from many pror studes. 11 Ths result s fundamentally smlar to the result from the optmal publc good provson problem, n whch the optmal level of the publc good s where the margnal cost (whch n ths case s the polluton tax rate) equals the MCPF tmes the margnal beneft. And relaxng the assumpton that Q s separable n utlty has the same effect as n the publc good provson problem: f mproved envronmental qualty ncreases labor supply, then that ncreases the optmal polluton tax rate. 12 Fnally, f the pollutng good s not an average substtute for lesure, then the thrd term on the rght-hand sde of (30R) wll not equal zero; f the good s a relatve complement to lesure, then ths term wll lower the optmal tax rate, and f t s a relatve substtute, ths term wll rase the optmal tax. 13 To obtan a frst-order condton for the optmal polluton tax n the MA model that s analogous to (29R), follow the same set of steps used to derve that equaton, but use (15M) for the change n the margnal tax rate on labor nstead of (15R). Ths gves #E (29M)! *MA P = " #Q 1"!, &! $ L #L ) ' ( WTPQ%Y L #w * + "! #L / & #L 2. L 1 "! L +! - #Q 2 $ 0 #p XP %Y ' ( P #L X P ) #w L * + dx P d! P, where! XP "Y and! WTPQ"Y are the cross-sectonal ncome elastctes of demand for X P and of wllngness-to-pay for envronmental qualty, respectvely. These parallel the elastctes! Xk "Y 11 See for example Bovenberg and de Mooj (1994), Parry (1995), Goulder (1995), Goulder et al. (1999) or Parry et al. (1999). 12 In ths case, the second term on the rght-hand sde of (30R) wll not equal zero. The result n ths case s equvalent to results from Bovenberg and van der Ploeg (1994), Schwartz and Repetto (2000), and Wllams (2002 and 2003). 13 Ths s equvalent to results from Km (2002) and West and Wllams (2004). 22

25 and! WTPG"Y that appeared n the expressons for the optmal commodty tax rate (19M) and optmal quantty of the publc good (24M). Just as n the commodty tax and publc good provson problems, the expressons for the optmal polluton tax n the RA model (29R) and MA model (29M) are smlar, and the two sources of dfferences between these expressons are the same as those n the earler problems. Frst, f the labor supply elastctes vary across the ncome dstrbuton, then the ntegrals n (29 M) may not equal the correspondng terms n (29R). Second, and more mportant, dstrbutonal effects nfluence the optmal polluton tax rate n the MA model, but not n the RA model. The dstrbutonal effects are more complcated n ths case, however, because the dstrbutonal effects of the polluton tax depend both on how the consumpton of the pollutng good s dstrbuted and on the dstrbuton of wllngness to pay for envronmental qualty. Just as n the commodty tax case, the hgher s the cross-sectonal ncome elastcty of demand for the taxed good (! XP "Y ), the more progressve wll be the dstrbuton of costs, and thus the hgher wll be the optmal tax rate, all else the same. And just as n the publc good provson case, the lower the cross-sectonal ncome elastcty of wllngness to pay for envronmental qualty, the more progressve wll be the dstrbuton of benefts, and thus the hgher wll be the optmal tax rate. The ntuton for these effects s the same as n those earler problems. And, just as n the earler problems, f the dstrbutonal effects of the polcy are smply proportonal to ncome, then the RA and MA models wll yeld qualtatvely the same result. Ths wll be true f both the consumpton of the pollutng good and wllngness to pay for envronmental qualty are proportonal to ncome, but that sn t necessary. The net effect of the polcy can be proportonal to ncome even f nether the costs nor the benefts are. For example, 23

26 f the costs are more than proportonal to ncome, but so are the benefts, then those dfferences could offset. A suffcent condton for the effects to be proportonal to ncome s (30M) (! # 1) $E WTPQ"Y $Q ( ) X P dx P =! d% XP # 1 "Y P. If ths condton holds, then the results of the RA and MA models wll be qualtatvely equal. Despte ths smlarty n results between the RA and MA models, pror studes usng these two dfferent types of models have reached very dfferent conclusons about the optmal polluton tax. Studes usng MA models typcally fnd that the optmal polluton tax should smply equal MED, whereas those usng RA models typcally fnd that the polluton tax should equal MED dvded by the MCPF. 14 But ths dfference s caused by dfferences n assumptons and defntons between these two strands of the lterature, not by fundamental dfferences between the two types of models. Most RA models assume that envronmental qualty s separable n utlty, whch mples that the second term n (29R) equals zero. Makng the same assumpton n a MA model would cause the analogous term n (29M) to equal zero, so both models would stll generate the same results (as long as (30M) holds). But most of the MA models do not make ths assumpton. Instead, Prttlä and Tuomala (1997) and Kaplow (1996 and 2004a) assume that lesure s separable n utlty. Just as n the commodty tax case, ths causes the terms n the second ntegral n (29M) to cancel out. And, parallelng the result n the publc good case, t causes the frst ntegral to reduce to!e ", so the!q 14 These results are sometmes referred to as the Pgouvan rule and modfed Pgouvan rule, respectvely. Footnote 11 lsts examples of papers usng RA models that fnd that the optmal tax equals MED/MCPF. Papers usng MA models that fnd that the optmal tax equals MED nclude Cremer et al. (1998 and 2003, as well as several other papers) and Prttlä and Tuomala (1997). Kaplow (1996 and 2004a) does not explctly model the polluton tax case, but explans that t parallels the publc good case, and thus the optmal tax should equal MED. 24

27 optmal tax rate smply equals MED. Makng the same assumpton n a RA model would produce the same result, though, as long as (30M) holds, because the analogous terms n (29R) would also cancel. Thus, the dfference n results between these MA models and papers usng RA models stems from a dfference n separablty assumptons, not from a fundamental dfference n the models. Cremer et al (1998 and 2003), however, make the same separablty assumpton as n the RA models that Q s separable. The result n those papers dffers because they use a dfferent defnton of margnal damages. Rather than defnng MED as the sum over all ndvduals of margnal wllngness to pay, these papers defne MED as the sum of margnal wllngness to pay tmes the weght (relatve to government revenue) put on ncome to that ndvdual. In other words, f socety would lke to redstrbute ncome to a partcular ndvdual, then that ndvdual s wllngness to pay gets a hgher weght n MED. Applyng the analogous defnton n a RA model would mply that MED equals wllngness to pay dvded by the MCPF. Thus, the dfference n results stems from the dfferent defnton: a tax equal to MED as defned n Cremer et al. s the same as a tax equal to MED dvded by MCPF under the defnton used n the RA papers. Thus, many of the apparent dfferences between the results of RA and MA models are n fact caused by dfferences n assumptons and defntons, not by fundamental dfferences n the models. Ths s not true, however, for cases n whch the dstrbutonal effects of polcy are not proportonal to ncome. For example, Ballard et al. (2005) shows that f lesure s separable from consumpton goods n utlty and the utlty functon s not homothetc, then the polluton tax should be hgher f the pollutng good s a necessty, and lower f t s a luxury. Smlarly, Parry and Small (2005) fnd that the optmal gasolne tax exceeds MED (under the assumpton that 25

28 lesure s separable), because gasolne s a necessty. 15 Both papers use RA models, and would get very dfferent results n a MA model. If lesure s separable, then n both MA and RA models, the hgher the ncome elastcty for a partcular good, the more a tax on that good wll dscourage labor supply (as can be seen n (20M)), and thus the lower the optmal tax wll be. In MA models, though, ths effect s exactly offset by the nfluence of dstrbutonal consderatons, whch mply hgher tax rates on goods wth hgher ncome elastctes. Thus, whle a tax on a necessty good s very effcent, and thus s an attractve polcy opton n a RA model, t s also regressve, makng t a less attractve polcy opton n a MA model an effect that s gnored by the RA model. II. Conclusons These results ndcate that many of the dfferences n results between studes usng RA models and those usng MA models are actually caused by dfferences n other assumptons and defntons, and are not the result of any fundamental dfference between RA and MA models. Indeed, the two types of models yeld smlar results n many cases, whch should be welcome news, gven the substantal complexty that a MA model entals. However, n other cases notably those n whch the dstrbuton of the net beneft of the polcy s hghly progressve or hghly regressve RA and MA models can yeld very dfferent results. RA models should be used wth cauton n such cases, snce they can be substantally msleadng. These results also show that the partcular type of separablty assumed can substantally affect the results of the model. Gven that there s no obvous a pror reason to prefer one 15 The same effect s partly responsble for West and Wllams s (2004) fndng that the optmal gas tax exceeds MED, though that paper does note that dstrbutonal consderatons would act n the opposte drecton. 26

29 separablty assumpton over another, ths suggests that further emprcal work to test these assumptons could be qute valuable. 27

30 References Atknson, Anthony and Joseph Stgltz, The Desgn of Tax Structure: Drect Versus Indrect Taxaton. Journal of Publc Economcs 6: Auerbach, Alan and James Hnes, Taxaton and Economc Effcency. In A. Auerbach and M. Feldsten (eds.) Handbook of Publc Economcs, vol. 3, North-Holland. Ballard, Charles, and Don Fullerton, Dstortonary Taxes and the Provson of Publc Goods. Journal of Economc Perspectves 6: Ballard, Charles, John Goddeers and Sang-Kyum Km Non Homothetc Preferences and the Non-Envronmental Effects of Envronmental Taxes. Internatonal Tax and Publc Fnance, 12: Bovenberg, A. Lans and Ruud A. de Mooj, Envronmental Leves and Dstortonary Taxaton. Amercan Economc Revew 85: Bovenberg, A. Lans and Frederck van der Ploeg, Green polces and publc fnance n a small open economy. Scandnavan Journal of Economcs 96: Chrstansen, Vdar, Evaluaton of Publc Projects under Optmal Taxaton. Revew of Economc Studes 48: Cremer, Helmuth, Frouz Gahvar, and Norbert Ladoux, 1998, Externaltes and Optmal Taxaton, Journal of Publc Economcs 70: Cremer, Helmuth, Frouz Gahvar, and Norbert Ladoux, Envronmental Taxes wth Heterogeneous Consumers: An Applcaton to Energy Consumpton n France. Journal of Publc Economcs 87: Goulder, Lawrence, Envronmental Taxaton and the Double Dvdend: A Reader s Gude. Internatonal Tax and Publc Fnance 2: Goulder, Lawrence H., Ian W. H. Parry, Roberton C. Wllams III, and Dallas Burtraw, The Cost-Effectveness of Alternatve Instruments for Envronmental Protecton n a Second-Best Settng. Journal of Publc Economcs, 72: Hylland, Aanund, and Rchard Zeckhauser, Dstrbutonal Objectves Should Affect Taxes but not Program Choce or Desgn. Scandnavan Journal of Economcs 81: Kaplow, Lous, 1996, The Optmal Supply of Publc Goods and the Dstortonary Cost of Taxaton, Natonal Tax Journal 49: Kaplow, Lous, 2004a. On the (Ir)relevance of Dstrbuton and Labor Supply Dstorton to Government Polcy. Journal of Economc Perspectves 18(4):

31 Kaplow, Lous, 2004b. On the Undesrablty of Commodty Taxaton Even When Income Taxaton s Not Optmal. NBER workng paper no Km, Seung-Rae, "Optmal Envronmental Regulaton n the Presence of Other Taxes: The Role of Non-separable Preferences and Technology." Contrbutons to Economc Analyss & Polcy: Vol. 1: No. 1, Artcle 4. Parry, Ian, Polluton Taxes and Revenue Recyclng. Journal of Envronmental Economcs and Management 29:S64-S77. Parry, Ian W. H., Roberton C. Wllams III, and Lawrence H. Goulder, When Can Carbon Abatement Polces Increase Welfare? The Fundamental Role of Dstorted Factor Markets. Journal of Envronmental Economcs and Management, 37: Parry, Ian, and Kenneth Small, Does Brtan or Amerca Have the Rght Gasolne Tax? Amercan Economc Revew, forthcomng Pgou, A.C., The Economcs of Welfare (4 th ed.). London: Wedenfeld and Ncolson. Prttlä, J. and M. Tuomala, 1997, Income Tax, Commodty Tax and Envronmental Polcy, Internatonal Tax and Publc Fnance, 4: Saez, Emmanuel, Usng Elastctes to Derve Optmal Tax Rates. Revew of Economc Studes 65: Saez, Emmanuel, The Desrablty of Commodty Taxaton Under Non-Lnear Income Taxaton and Heterogeneous Tastes Journal of Publc Economcs, 83: Schwartz, Jesse and Robert Repetto, Nonseparable Utlty and the Double Dvdend Debate: Reconsderng the Tax-Interacton Effect Envronmental and Resource Economcs, 15: West, Sarah and Roberton C. Wllams III, Emprcal Estmates for Envronmental Polcy Makng n a Second-Best Settng. NBER workng paper no Wllams, Roberton C., III, Envronmental Tax Interactons When Polluton Affects Health or Productvty. Journal of Envronmental Economcs and Management, 44: Wllams, Roberton C., III, Health Effects and Optmal Envronmental Taxes. Journal of Publc Economcs, 87: Wllams, Roberton C., III, An Estmate of the Optmal Second-Best Gasolne Tax Consderng Both Effcency and Equty. Workng paper, Unversty of Texas at Austn. 29