Optimal Control of Externalities in the Presence of Income Taxation

Transcription

1 Optmal Control of Externaltes n the Presence of Income Taxaton Lous Kaplow * Abstract A substantal lterature examnes second-best envronmental polcy, focusng partcularly on how the Pgouvan drectve that margnal taxes should equal margnal external harms needs to be modfed n lght of the preexstng dstorton due to labor ncome taxaton. Addtonal lterature s motvated by the possblty that dstrbutve concerns should amend the nternalzaton prescrpton. It s demonstrated, however, that smple frst-best rules unmodfed for labor supply dstorton or dstrbuton are correct n a natural, basc formulaton of the problem. Specfcally, settng all commodty taxes equal to margnal harms (and subsdes equal to margnal benefts) can generate a Pareto mprovement. Lkewse, a margnal reform n the drecton of the frst-best can yeld a Pareto mprovement. Qualfcatons and explanatons for dfferences from prevous work are also presented. JEL Classes D61, D62, D63, H21, H23, K32 * Harvard Unversty and Natonal Bureau of Economc Research. I am grateful to Steven Shavell, Roberton Wllams, and workshop partcpants at Harvard Unversty and the NBER for comments and to the John M. Oln Center for Law, Economcs, and Busness at Harvard Unversty for fnancal support. Ths artcle s related to the larger project, Kaplow (2008).

2 1. Introducton The control of externaltes s a complex second-best problem. The frst-best prescrpton, followng Pgou (1920), s to set margnal taxes and subsdes equal to margnal external harms and benefts. Because of the preexstng dstorton due to labor ncome taxaton, however, researchers have explored how the Pgouvan rule needs to be modfed on account of the nteracton between envronmental regulaton and the ncome tax, partcularly concernng the need to rase revenue and the ndrect effects of envronmental polcy on labor supply. Work began n the 1970s wth Sandmo (1975) and others, followng the growng nterest n optmal taxaton more generally. Subsequently, researchers became nterested n the possblty of a double dvdend that correctve taxaton mght both enhance welfare by nternalzng externaltes and also rase revenue that would allow a reducton n dstortonary ncome taxaton. See, for example, Ballard and Medema (1993), Cordes, Ncholson, and Sammartno (1990), and Pearce (1991). Ths work, n turn, led to an extensve modern lterature on envronmental regulaton, much of whch suggests that there s no double dvdend and, nstead, that optmal envronmental control may well fall short of the Pgouvan frst best. For a survey and a collecton of lterature, see respectvely Bovenberg and Goulder (2002) and Goulder (2002). An addtonal second-best problem nvolves dstrbuton. See, for example, Casler and Rafqu (1993) and West (2004). Because many proposed correctves, such as heghtened taxaton of gasolne, are beleved to be regressve and, moreover, envronmental benefts may have values that rse (perhaps dsproportonately) wth ncome, dstrbutve concerns seem potentally mportant. Indeed, Pgou (1928) hmself argued that dstrbutve effects needed to be taken nto account n assessng correctve taxes and subsdes. Ths artcle seeks to advance our understandng of the regulaton of externaltes by examnng the problem n what seems to be a natural and foundatonal settng n lght of work on taxaton more generally: namely, n a model n whch there can be taxes or subsdes on each commodty and a nonlnear ncome tax, and n whch ndvduals dffer n ther earnng abltes. Specfcally, the queston addressed s, n a world n whch there may be externaltes, what reforms of commodty taxes and subsdes can generate Pareto mprovements, begnnng wth any arbtrary ntal system. By ncludng a preexstng ncome tax wth no restrcton on how t s set ntally nteractons nvolvng tax revenue and labor supply are addressed. And by focusng on Pareto mprovements, concerns about dstrbuton are taken nto account. The prncple result s that, n a standard, smple settng (specfcally, n whch utlty s weakly separable n labor), t s possble to move from any set of commodty taxes and subsdes to frst-best Pgouvan taxes and subsdes wheren each tax or subsdy equals margnal harm or beneft n a manner that generates a Pareto mprovement. In partcular, ths result arses when the ncome tax s adjusted such that the reform as a whole s dstrbuton neutral. An extenson shows that, subject to some addtonal assumptons, any margnal change n commodty taxes and subsdes that s proportonally n the drecton of the frst best produces a Pareto mprovement f a dstrbuton-neutral ncome tax adjustment s employed. In addton, a necessary and suffcent condton s offered that ndcates whch other commodty tax reforms - 1 -

3 make possble Pareto mprovements. The artcle also addresses how relaxng dstrbuton neutralty and weak separablty affects the results and dscusses how these assumptons explan dfferences from pror work. Before proceedng, t s useful to relate the present analyss and results to two addtonal strands of lterature. The frst pertans to the method of proof employed here, whch as mentoned nvolves an adjustment to the preexstng ncome tax such that the reform as a whole s dstrbuton neutral. Ths approach follows Hylland and Zeckhauser (1979), Kaplow (1996), and some subsequent work that focuses on publc goods. Kaplow (1996, 2004, 2008) has dscussed nformally the applcaton of ths approach to the regulaton of externaltes; the contrbuton of the present artcle s to analyze the problem formally, offer extensons, and relate the results to exstng lterature on envronmental regulaton. When dstrbuton-neutral ncome tax adjustments are utlzed, dstrbutve concerns are obvously taken nto account, and t becomes possble to characterze Pareto mprovements. Furthermore, as wll be explaned, when reforms are mplemented n ths dstrbuton-neutral manner, labor supply s unaffected n a baselne case; hence, concerns about second-best nteractons wth the preexstng ncome tax also become moot. Accordngly, there s an mportant sense n whch the concerns about labor supply and dstrbuton that have occuped much of the second-best lterature on externaltes are ndependent of the queston of how best to control externaltes. The present nvestgaton can also be related to the lterature on optmal commodty taxaton n the presence of an ncome tax. The semnal analyss of ths model for the case n whch there are no externaltes s provded by Atknson and Stgltz (1976), who examne optmal commodty taxaton when the ncome tax s also set optmally. They fnd that, wth weak labor separablty, unform commodty taxaton (equvalent to no commodty taxaton) s optmal. A way to understand ther result s to realze that, wth the separablty assumpton, dfferental commodty taxaton cannot help to reduce the labor-lesure dstorton due to labor ncome taxaton; hence, ts only effect s to dstort consumpton allocatons. Accordngly, t s optmal for commodty prce ratos to equal the correspondng ratos of socal resource costs, whch n ther model consst solely of producton costs. In the present settng, socal resource costs nclude externaltes, so the natural extenson of ther fndng s that prce ratos should equal the ratos of the sum of producton costs and external effects, whch s precsely what obtans f one follows Pgou s (1920) prescrpton by settng commodty taxes and subsdes equal to margnal external harms and benefts. The artcle formalzes ths ntuton, thereby extendng Atknson and Stgltz s (1976) mportant result to settngs wth externaltes. Furthermore, the analyss here s not restrcted to cases n whch the ncome tax s optmal and s not lmted to commodty tax reforms n the neghborhood of ther optmum. 1 In these two respects, the present results are most analogous to 1 For prelmnary analyses of the case wth externaltes, see Prttlä and Tuomala (1997) and Cremer, Gahvar, and Ladoux (1998), each of whch examnes formula for optmal commodty taxes (not permttng analyss of reforms away from the optmum) for the case n whch the ncome tax s assumed to be optmal. Specfcally, Cremer, Gahvar, and Ladoux (1998) fnd that, when the only effect of an externalty regardng a sngle good s addtvely separable, a term nvolvng the externalty appears only n the frst-order condton nvolvng that good, and Prttlä and Tuomala (1997) - 2 -

4 those n Kaplow s (2006) recent artcle on commodty taxaton. See also Laroque (2005) and Kaplow (2008). Nevertheless, the bulk of the analyss here s dfferent precsely because of the presence of externaltes. Furthermore, t s dffcult to vew the present results however ntutvely plausble they may seem on reflecton as obvous when most exstng lterature on externaltes appears to advance qualtatvely dfferent prescrptons. Formal analyss s necessary to understand when and why the conclusons from varous models dverge and to assess the mplcatons of the dfference n approaches. 2. Optmalty of Frst-Best Pgouvan Taxes and Subsdes 2.1. Model Indvduals earn ncome y, whch s equal to wl, the product of ther wage w and ther labor effort l. Wages (types) are dstrbuted accordng to the densty functon f(w). The government mposes a (nonlnear) ncome tax T(y). As s standard, the government observes ncome but does not separately observe ndvduals wages or choces of labor effort. Indvduals spend ther after-ncome-tax ncome, y! T(y), on n commodtes, x 1,..., x n. Commodty prces for the x (whch equal producton costs) are denoted p and are assumed to be postve constants. There are also commodty taxes on each x of τ (whch may be subsdes, n whch case they are negatve). Indvduals thus face net prces of p + τ. An ndvdual of type w s choce of labor l s equvalent to a choce of ncome y that wll be denoted y(w). For the chosen level of ncome, and takng the tax system (and the level of externaltes, see below) as gven, the ndvdual chooses how much to expend on each commodty, x (y(w)). The ndvdual s budget constrant can thus be wrtten as ( 1) ( p + τ ) x ( y( w)) = y( w) T( y( w)), where summatons throughout are from equals 1 to n. Each commodty x s assocated wth an externalty e, whch denotes the total consumpton by all ndvduals of that commodty: ( 2) e = x( y( w)) f ( w) dw. When ndvduals choose y(w) and the x (y(w)) s, they are assumed to take the e as gven (whch s approprate snce there s a contnuum of ndvduals, each of measure zero). Indvduals utlty functons are u(v(x 1,..., x n, e 1,..., e n ), l), where v s a subutlty fnd that the commodty tax formula for the drty good n ther model has an addtonal term whch, under a smlar separablty assumpton, nvolves only the externalty

5 functon. Ths utlty functon s assumed to be contnuously dfferentable, strctly concave, ncreasng n commodtes, and decreasng n labor effort. Utlty may have any relatonshp to the levels of the e s; that s, the external effect due to each of the commodtes may be postve, negatve, or nonexstent. Ths form of the utlty functon, wth the subutlty functon v, entals what s referred to as weak separablty of labor (or lesure): For a gven level of after-ncometax ncome, ndvduals wll allocate ther dsposable ncome among commodtes n the same manner regardless of the level of labor effort requred to earn that level of ncome; furthermore, n choosng labor effort, t does not matter what levels of consumpton and externaltes combne to produce a gven level of subutlty v. In the analyss that follows, use wll be made of the ndrect subutlty functon V(τ, T, y), whch s the value of v(x 1,..., x n, e 1,..., e n ), maxmzed over the x s, where the commodty tax vector τ, the ncome tax schedule T, and before-tax ncome y (as well as the e s) are taken as gven. Observe that snce v depends only on the x s (and the e s), and snce the budget constrant (1) depends only on the x s, τ, T, and y and not on w or l ndependently the ndrect subutlty functon V s the same for all ndvduals, regardless of ther type w. It wll also be useful to use ths ndrect subutlty functon to defne U(τ, T, w, l) = u(v(τ, T, y), l). The government s budget constrant s [ τ ] ( 3) T( y( w)) + x ( y( w)) f ( w) dw = R, where R s a gven revenue requrement. Fnally, the margnal external harm assocated wth any commodty s ( 4) h uv( w) ve ( w) = λ ( w) f ( w) dw, where subscrpts ndcate partal dervatves, the notaton (w) ndcates evaluaton for an ndvdual of type w, and λ(w) refers to the Lagrange multpler (for ndvduals of type w), whch sgnfes the margnal utlty of dsposable ncome. The fracton n the ntegrand, therefore, s the margnal effect of the externalty on utlty dvded by the margnal utlty of dsposable ncome, whch denotes the margnal external effect for a gven type w, measured n dollars. Note that for postve externaltes, h < 0. Now we can defne: Frst-best Pgouvan taxes and subsdes. A commodty tax vector {τ 1,..., τ n } conssts of frst-best Pgouvan taxes and subsdes f and only f (p +τ )/(p j +τ j ) = (p +h )/(p j +h j ), for all, j. Notce that the defnton does not requre that τ = h, for all. The reason has to do wth normalzaton: If all commodty taxes are rased or lowered n such a manner as to leave all consumer prce ratos unchanged, ndvduals behavor wll be unaffected f the level of the ncome tax s also adjusted to produce the same effectve dsposable ncome. (Ths pont s - 4 -

6 analogous to the famlar dea that unform commodty taxes are equvalent to a proportonal adjustment to the ncome tax.) 2.2. Analyss The approach s to begn wth any regme that does not consst of frst-best Pgouvan taxes and subsdes and then to construct a regme wth frst-best Pgouvan taxes and subsdes that makes everyone better off. As an ntermedate step n analyzng any reform (not necessarly a move to the socal optmum), say, from tax regme {τ, T} to regme {τ*, T*}, t s useful to examne the ntermedate ncome tax schedule T that has the property that each ndvdual s utlty under {τ*, T } s the same as that under {τ, T} f the ndvdual contnues to choose the same level of labor effort l. (Whether ndvduals wll choose the same labor effort under ths ntermedate regme s the subject of Lemma 1, below.) To be more precse, defne T (y) such that V(τ, T, y) = V(τ*, T, y) for all y. In general, a reform from τ to τ* wll, for a gven y, change the value of subutlty V. For each y, the tax schedule T (y) s set at the level that restores the orgnal level of subutlty (by changng aftertax ncome for the stpulated level of before-tax ncome). Ths tax adjustment, T (y)!t(y), s smply the schedule of utlty-compensatng changes n dsposable ncome. Gven how the ntermedate ncome tax schedule s constructed, t s possble to establsh the followng result. Lemma 1: Every type of ndvdual w chooses the same level of labor effort under {τ*, T } as under {τ, T}. Proof: It s straghtforward to establsh that U(τ, T, w, l) = u(v(τ, T, y), l) = u(v(τ*, T, y), l) = U(τ*, T, w, l), for all w, l. The frst equalty follows by the defnton of U. The second equalty follows because T (y) s constructed such that V(τ, T, y) = V(τ*, T, y) for all y. And the thrd equalty also follows from the defnton of U. Therefore, U(τ, T, w, l) = U(τ*, T, w, l), for all w, l. Because, for any type w, ths equalty holds for all l, the level of utlty an ndvdual acheves for each possble choce of l s the same n each of the two regmes. Therefore, for each type w, whatever l maxmzes U n the ntal regme {τ, T} must be the l that maxmzes U n the ntermedate regme {τ*, T }. The ntuton s as follows: The frst part of the lemma merely shows the obvous pont that, f the subutlty V s unchanged for every y and thus for every choce of l (whch s true by constructon), then ndeed U s unchanged for every choce of l. The second part of the lemma smply observes that snce U as a functon of l s unchanged when movng from {τ, T} to {τ*, T }, the choce of l that maxmzes U s lkewse unchanged. Ths result that labor supply s unaffected greatly smplfes the analyss of any tax reform, ncludng those of partcular nterest here. To begn the constructon of a Pareto-mprovng tax reform, start wth an ntal regme {τ, T} that s not a Pgouvan frst best (as defned above). For smplcty, choose from among the multtude of equvalent frst-best Pgouvan tax systems the one for whch τ * = h, for all

7 For purposes of the argument to follow, t wll be assumed that the h s are constant and thus ndependent of the x s and the e s. (The more general case wll be commented on below.) Movng to ths new commodty tax vector wll tend to change ndvduals utltes on account of three effects: changes n the payment and recept of commodty taxes and subsdes, changes n consumpton due to the new relatve prce vector, and changes n externaltes on account of others changes n consumpton. Whatever s the net effect on utlty for any type w and gven labor effort for that type, we wll consder the ntermedate ncome tax schedule T (y), defned above, that offsets the net effect on utlty at each ncome level. We know that from Lemma 1 that movng to ths ntermedate regme leaves labor supply unchanged. The next queston to consder s how revenue compares between the ntal regme and the ntermedate regme. Lemma 2: If the h s are constant, then regme {τ*, T } (wth frst-best Pgouvan taxes and subsdes τ * = h, for all ) rases more revenue than does regme {τ, T} (wthout frst-best Pgouvan taxes and subsdes). Proof: To begn, we can see from expresson (3) that the change n revenue s gven by [( ) + + ( )] ( 5) T ( y( w)) T( y( w)) ( τ * τ ) x ( y( w)) τ * x ( y( w)) x ( y( w)) f ( w) dw. The frst term n the ntegrand s the change n ncome tax revenue for an ndvdual of type w, who earns y(w), from movng to the ntermedate regme. The next two terms are the change n commodty tax revenue for the ndvdual, whch s decomposed for convenence: The second term s the revenue change from movng to the frst-best Pgouvan commodty tax vector, holdng constant the orgnal level of consumpton, and the thrd term s the revenue change from the ndvdual s adjustment n consumpton, evaluated usng the new commodty tax vector, x. These three terms are ntegrated over the populaton to yeld the total change n revenue. In analyzng expresson (5), t s useful to decompose the frst term, the change n ncome tax revenue, by magnng that the shft to the ntermedate regme occurs n three stages: 1. The commodty tax reform s ntroduced but the x s are magned to be held constant, and the T(y) schedule s adjusted to hold ndvduals at the same utlty levels takng nto account that commodty tax payments are changed the second term n expresson (5). Let T 1 (y) denote the resultng ncome tax schedule. 2. Indvduals adjust ther x s but the e s are magned to be held constant, and the T(y) schedule s further modfed, to T 2 (y), for the effect of these adjustments on ndvduals own levels of utlty. 3. The e s are allowed to reflect the new level of the x s (whch n turn may be further adjusted because of the change n the levels of the e s, ultmately yeldng the x s), and the T(y) schedule s further modfed, now to T (y). Usng ths notaton, expresson (5) can be rewrtten as - 6 -

8 [( ) + ( ) + ( )] ( ) T ( yw ( )) T ( yw ( )) T ( yw ( )) T ( yw ( )) T ( yw ( )) Tyw ( ( )) f( wdw ) [ ( )] + ( τ * τ ) x ( y( w)) + τ * x ( y( w)) x ( y( w)) f ( w) dw. Stage 1: Begnnng wth the frst stage, n whch the x s are held constant, the ncome tax schedule T(y) for each type w s rased (lowered), to the schedule T 1 (y), by an amount equal to the negatve of the fourth term (the frst term n lne 2) n expresson (6), whch denotes the ncrease (or reducton) n expendtures on commodtes necessary to fnance a type-w ndvdual s orgnal consumpton bundle. Note that, after ths adjustment to the ncome tax, each type of ndvdual wll just be able to afford the orgnal level of consumpton. Because the ncrease (decrease) n ncome tax revenue for each type of ndvdual, T 1 (y)! T(y), precsely equals the decrease (ncrease) n commodty tax and subsdy revenue for each, t follows that the thrd and fourth terms n (6) are precsely offsettng at each y and thus also when each s ntegrated over the populaton. Therefore, the net revenue effect wth regard to ths frst stage s nl. Stage 2: Now we move to the second stage and allow ndvduals to adjust ther x s, but we magne that the e s are stll held constant. It wll now be demonstrated that there s a postve revenue effect on account of the change n T(y) necessary to offset the drect effect on ndvduals utlty from ther own changes n consumpton. That s, T 2 (y)! T 1 (y) > 0 for all y. Ths arses because, n ths hypothetcal settng, all ndvduals would wsh to change ther consumpton, whch they would do only f they stood to gan. To show ths result, begn wth the fact that an ndvdual s optmal consumpton vector s determned by standard frst-order condtons, whch n the ntal regme are (Mv/Mx )/(Mv/Mx j ) = (p +τ )/(p j +τ j ), for all, j. Gven that the commodty taxes n the orgnal regme are not frst-best Pgouvan taxes and subsdes, there exsts, j such that (p +τ )/(p j +τ j ) (p +h )/(p j +h j ). However, the latter term equals (p +τ *)/(p j +τ j *). Because the frst-order condtons are all satsfed n the ntal regme, at least one must therefore be volated after the commodty tax reform f the consumpton bundle s unchanged. As a consequence, the consumpton vector that was optmal under the orgnal regme cannot be optmal under the hypotheszed ntermedate regme. Now, for ndvduals thus adjustng ther consumpton on account of the change n prce ratos, t follows that ther utlty must be hgher under the ntermedate regme unless the ncome tax schedule s adjusted upwards to reduce ther dsposable ncome. Of course, precsely ths adjustment must be done n the ntermedate regme because T 2 (y) s, by defnton, constructed so as to hold everyone s utlty constant at the end of stage 2. Accordngly, wth regard to ths second stage, t s ndeed true that T 2 (y)! T 1 (y) > 0 for all y; that s, the second term n (6) s postve. The effect of changes n consumpton on commodty tax revenue s consdered n combnaton wth stage 3. Stage 3: Fnally, n the thrd stage, we allow the e s to reflect the new level of the x s whch n turn may be further adjusted on account of the change n the levels of the e s, - 7 -

9 ultmately yeldng the x s and we examne what further change n the T(y) schedule, from T 2 (y) to T (y), s necessary to compensate for ths change n the levels of externaltes. Furthermore, we can compare ths ncome tax revenue effect to the change n commodty tax and subsdy revenue due to the shfts n consumpton, the fnal term n expresson (6). It wll now be demonstrated that these two effects must offset because the commodty taxes are beng used as correctve taxes (and subsdes), set at the level that nternalzes any externaltes. The compensaton for all changes n the level of externaltes s, as stated, accomplshed by movng the ncome tax schedule from T 2 (y) to T (y). The aggregate effect on ncome tax revenues s: ( ) ( ) T ( y( w)) T ( y( w)) f ( w) dw 7 2 ( ( x y w x y w ) f w dw) f w dw = h( e e) uv( w) ve ( w) = ( ( )) ( ( )) ( ) ( ). λ( w) The left sde of the second lne of expresson (7) s the aggregate effect on ndvduals utlty (measured n dollars) due to changes n the level of the externalty. The ntegrand of the nternal ntegral s the change n the levels of consumpton; ntegrated over all ndvduals, ths gves the change n the levels of the externaltes, as recorded on the rght sde. For a unt change n the level of any partcular externalty e, the ntegral of the effect on utlty, measured n dollars, over the populaton equals the negatve of the margnal external harm, h, as stated n expresson (4). Let us compare ths revenue effect due to the compensatory adjustment n the ncome tax schedule to the fnal term n expresson (6), the change n revenue through the commodty taxes and subsdes on account of ndvduals changes n consumpton. [ ( )] = ( ) [ ] ( 8) τ * x ( y( w)) x ( y( w)) f ( w) dw h x ( y( w)) x ( y( w)) f ( w) dw ( e ) = h e The left sde of (8) shows that the change n total commodty tax (and subsdy) collectons on account of consumpton adjustments equals the ntegral over the populaton of each ndvdual s change n commodty tax payments due to the shft n consumpton n movng from the orgnal regme to the ntermedate regme. On the rght sde of (8) n the top lne, h s substtuted for τ * because the τ * s are defned to be frst-best Pgouvan taxes and subsdes (normalzed to equal external harm), and the orders of the summaton and ntegraton are reversed for convenence. Fnally, the move to the second lne of (8) reflects, as wth expresson (7), that the levels of the externaltes are defned as the ntegral of consumpton over the populaton as a whole

10 Comparng expressons (7) and (8), t s apparent that the amount of revenue lost (ganed) through ths thrd stage n the ncome tax adjustment the compensaton for externaltes precsely equals the revenue gan (loss) through commodty taxes whch are now set at frstbest Pgouvan levels on account of ndvduals adjustng ther levels of consumpton. Therefore, the frst and last terms n expresson (6) offset. Summarzng, from the thrd stage, the frst and last terms n (6) offset; from the frst stage, the thrd and fourth terms n (6) offset; and, from the second stage, the second term n (6) s postve. Therefore, t has been demonstrated that expresson (6) s postve. The ntuton behnd ths proof s as follows. For stage 1, the only effects are on taxes (behavor s held constant); hence, to keep each ndvdual at the same level of utlty, t must be that the frst component of the ntermedate ncome tax adjustment just offsets the effect of the commodty tax reform, at each level of ncome. (For example, f the reform makes ndvduals at a gven ncome y spend more because taxes on commodtes causng negatve externaltes are hgher, t must be that ncome taxes at ncome y are reduced accordngly so that ndvduals can afford the same consumpton bundle.) Ths stage obvously s revenue neutral. In stage 2, ndvduals are adjustng ther x s and everythng else s held constant, except for the second component of the ncome tax adjustment. Because ndvduals necessarly ncrease ther utlty when they voluntarly choose to change ther consumpton, the ncome tax adjustment that holds utlty constant must reduce ther dsposable ncome and therefore rase revenue. Fnally, the stage 3 ncome tax adjustment that compensates for externaltes just offsets the change n commodty tax revenue due to changes n consumpton. The basc reason s that commodty taxes now equal margnal external effects, so they collect (expend) an amount equal to margnal harm (beneft) from consumpton adjustments, whch n turn equals the amount that ncome tax collectons must fall (rse) to offset the effect of the change n externaltes on ndvduals utltes. To complete the argument that a Pareto mprovement s possble n movng to frst-best Pgouvan taxes and subsdes, construct T*(y) from T (y) by performng two operatons smultaneously: Gradually rebate the surplus pro rata, and, as ndvduals adjust ther levels of consumpton, adjust the ncome tax schedule n order to offset the effects on utlty of changes n the levels of externaltes, as descrbed n the proof of Lemma 2 regardng stage 3. As wth stage 3, ths latter operaton s revenue neutral when one combnes the revenue effects from the ncome tax adjustment wth those from consumpton adjustments changng the level of commodty tax recepts. The former operaton (rasng the rebate) wll ultmately exhaust the surplus. Moreover and of central mportance to the present argument the effect from ths operaton s that all ndvduals utlty wll ncrease. 2 At ths pont, we have constructed a new ncome tax schedule, to accompany the move to frst-best Pgouvan taxes and subsdes, such that every type of ndvdual s strctly better off, 2 Note that, as the ncome tax schedule s reduced, ndvduals may reduce labor supply and thus tax revenue may fall, but as long as there ntally s a surplus and aggregate behavor s contnuous, some net reducton n everyone s ncome tax payment through the former operaton wll be possble

11 thereby establshng the followng result: Proposton 1: If the h s are constant, then for any tax system {τ, T} for whch commodty taxes and subsdes are not at frst-best Pgouvan levels, there exsts a tax system {τ*, T*} wth frst-best Pgouvan taxes and subsdes that s strctly Pareto superor.e., U(τ*, T*, w, l(w)) > U(τ, T, w, l(w)), for all w. The role of the assumpton that the h s are constant and thus ndependent of the x s and the e s n the proof pertans to stage 3 n Lemma 2 and to the rebate process. These arguments rely on the fact that wth τ s that have been set equal to the h s changes n commodty tax and subsdy payments due to changes n consumpton are necessarly equal to changes n the level of external harm and beneft due to those same changes n consumpton. If one allows the h s to be endogenous, one would need to modfy the argument by employng a contnuous adjustment process, roughly as follows. Intally, reform the τ s to equal the h s at the exstng level of the x s. Then, (hypothetcally) allow the x s to adjust gradually. One can then perform all of the pertnent operatons used n the proof of Lemma 2 and n the rebate process, but n each case wth the level of the τ s beng contnuously adjusted as the x s change so that, at every pont, the τ s equal the h s. For each ncrement of these margnal changes, the precedng analyss apples. Hence, once the adjustment consdered n Lemma 2 s complete, the net effect on revenue wll be the ntegral of postve margnal changes due to the stage 2 ncome tax adjustments all other effects nettng to zero as before so revenue wll necessarly ncrease. Lkewse, the rebate process can be mplemented as before, subject to further adjustment of the τ s. Also note that, because n all phases of the constructon the τ s are set equal to the h s, when the constructon s complete the system {τ*, T*} wll ndeed be one that has frst-best Pgouvan taxes and subsdes. (Fnally, observe that the endogenety of the h s would be moot f we were consderng only local changes near the Pgouvan frst best.) It s useful to remark on the ntuton underlyng Proposton 1. Any drect effects of the move to frst-best Pgouvan taxes and subsdes on ndvduals budget constrants can be offset by an ncome tax adjustment, after whch ndvduals can contnue to afford ther orgnal consumpton bundle. Gven that prce ratos have changed, ndvduals wll want to change ther levels of consumpton, and the drect effect of such adjustments on the ndvduals who make them s utlty ncreasng (for otherwse they would not have chosen to alter ther consumpton). External effects of these changes n consumpton are fully compensated through further adjustments to the ncome tax schedule, and ths compensaton s fully funded by changes n recepts from the commodty taxes and subsdes, each of whch equals the pertnent margnal external effect. Hence, just as n a world n whch there s fxed labor supply or smply endowments of goods and no labor decson and no redstrbutve, dstortonary ncome tax, all ndvduals can be made better off by the complete nternalzaton of externaltes. Dstrbuton and labor supply effects do not nterfere wth the argument because the ncome tax schedule s adjusted to keep them fxed. The ntermedate tax schedule (before the rebate, whch makes everyone better off) keeps everyone s utlty constant; hence, there are no dstrbutve effects to consder. Moreover, as explaned n provng Lemma 1, when ths s done, the return to any choce of labor effort, for ndvduals of any type w, s the same, so ndvduals

12 wll be nduced to keep ther labor supply unchanged. Thus, n the present constructon, the ncome tax problem nvolvng the tradeoff of dstrbuton and labor supply dstorton s ndependent of the problem of controllng externaltes, so the soluton to the latter problem has ts famlar frst-best character. 3. Extensons 3.1. Reforms other than to Frst-Best Pgouvan Taxes and Subsdes When reformed commodty taxes and subsdes do not all equal the level of margnal external harm or beneft, the change n commodty tax revenue due to shfts n consumpton need not be suffcent to fnance the compensaton for the change n the levels of the externaltes provded by the stage 3 ncome tax adjustment. Consder frst reforms that move proportonately toward the frst-best commodty tax regme. Specfcally, begnnng wth an ncome tax T(y) and an ntal set of τ s that are not frst-best Pgouvan taxes and subsdes, such a reform can be descrbed formally as a move to τ (α) = αh + (1!α)τ, for all, where α 0 [0, 1]. Furthermore, because the h are themselves endogenous, beng nfluenced by changes n α, consder the margnal reform dτ (α)/dα, for all, evaluated at α = 0. For the ncome tax adjustment, we can defne the ncome tax schedule T(y,α) that changes as a functon of α so as to keep utlty constant at every level of ncome and thus for every type. Because ncreasng α moves n the drecton of the frst best, changes n consumpton tend to be desrable. Indvduals wll tend to consume less of commodtes that are underprced (for example, those wth undercorrected negatve externaltes) because those prces wll rse, and ndvduals wll tend to consume more of overprced commodtes (overcorrected negatve externaltes and undercorrected postve externaltes) because those prces wll fall. There s, however, a qualfcaton due to possble cross-effects. To llustrate, suppose that ntally all commodty taxes are zero and only two goods nvolve externaltes. Flowers produce a postve externalty (to those drvng by), and drvng, correspondng, say, to the commodty gasolne, produces a negatve externalty (polluton). The reform would provde a small subsdy to flowers and mpose a small tax on gasolne. Suppose, however, that drvng and the postve externalty produced by flowers are hghly complementary. Then the ncrease n flowers mght nduce ndvduals to ncrease drvng that s, the effect due to the complementarty mght exceed the contrary effect due to the tax on gasolne. Addtonally, f ths ncrease s suffcently great, t s possble that the reform wll be neffcent. A suffcent condton that rules out ths possblty s M 2 v/me j Mx = 0, for all, j. One further complcaton s that the pro rata rebate of the revenue surplus could actually make ndvduals worse off rather than better off when there are uncorrected negatve externaltes. (In such a stuaton, t would tend to be optmal for the government to mpose a tax, such as a unform lump-sum tax, and destroy the proceeds.) To set ths problem asde, assume that more s better for everyone, specfcally, that f the government fnds tself wth a surplus, there exsts some manner of rebatng t to the populaton (not necessarly pro rata, gven that externaltes may affect ndvduals earnng dfferent levels of ncome dfferently) that produces a Pareto mprovement

13 Although the analyss of ths case turns out to be more nvolved than that whch establshes Proposton 1, t s possble to demonstrate the followng result (whch s proved n the Appendx): Proposton 2: If M 2 v/me j Mx = 0, for all, j, and f more s better for everyone, then for any tax system {τ, T} for whch commodty taxes and subsdes are not at frst-best Pgouvan levels, for a margnal ncrease n α, evaluated at α = 0, where τ (α) = αh + (1!α)τ, for all, there exsts T(y,α) such that the margnal reform s strctly Pareto mprovng.e., du(τ(α), T(y(w),α), w, l(w))/dα* α=0 > 0, for all w. It s more dffcult to characterze reforms that do not move proportonally toward the frst best. Takng a smple reform, t s famlar that movng a sngle tax or subsdy n the drecton of the frst-best need not be effcent. For example, rasng a tax that was prevously too low may cause substtuton to another good whose tax was also too low, perhaps to an even greater extent. Nevertheless, t turns out that there s a conceptually smple, f not always easy to apply, test of when a Pareto mprovement s possble. Ths test nvolves a narrow, tradtonal concept of effcency. Effcency-ncreasng commodty tax reform. For any tax system {τ, T}, a commodty tax reform τ* s effcency ncreasng f, when combned wth the ntermedate ncome tax schedule T (y) as defned prevously, ( 9) p x ( yw ( )) f( wdw ) < p x( yw ( )) f( wdw ). Expresson (9) states that the total real resource cost of everyone s consumpton vectors n the ntermedate regme s less than the total real resource cost n the ntal regme. Because everyone s utlty s the same n these two regmes, ths condton ndcates that the ntermedate regme s more effcent wth regard to consumpton choces n a narrow, conventonal sense.e., when concerns wth the labor-lesure dstorton and dstrbuton are gnored. Note that ths condton makes no explct reference to externaltes; externaltes are relevant, however, because the extent of postve and negatve externaltes wll affect the level of consumpton necessary to brng ndvduals to the same level of utlty as n the ntal regme. It s straghtforward to demonstrate the followng (whch s proved n the Appendx): Proposton 3: If more s better for everyone, then begnnng wth any tax system {τ, T}, for any effcency-ncreasng commodty tax reform τ*, there exsts T*(y) such that the reform regme s strctly Pareto superor.e., U(τ*, T*, w, l(w)) > U(τ, T, w, l(w)), for all w. In essence, Proposton 3 states that, f a commodty tax reform ncreases effcency n a tradtonal sense that s, f t ncreases surplus n a world n whch labor supply s constant, tantamount to a world wth fxed labor supply or smply one n whch ntal wealth endowments are gven then the reform wll be desrable, ndeed strctly Pareto mprovng, when combned

14 wth an approprate ncome tax adjustment, even n a world n whch labor supply s not constant and there exsts a dstortonary labor ncome tax. Ths fnal proposton renforces the sense n whch the envronmental tax problem and the ncome tax problem can be vewed as ndependent (gven the assumpton of weak separablty) Non-Dstrbuton-Neutral Income Tax Adjustment The foregong results assume that the ntermedate ncome tax adjustment holds utlty fxed, whch necessarly entals that t s dstrbuton neutral (n terms of utltes). Ths subsecton consders the mplcatons of relaxng ths assumpton and also comments on how devatons from dstrbuton neutralty explan some (but not all) of the dfference between the present results and those n the pertnent lterature. Dstrbutve effects may have three sources. The frst s the ncdence of commodty tax changes. The drect effect n terms of commodty taxes pad and subsdes receved may depend on ncome (and thus type and utlty) due to dfferng aggregate consumpton and dfferng relatve consumpton; lkewse, the utlty effect from adjustments to prvate consumpton may vary wth ncome. Second, the benefts of controllng externaltes may vary wth ncome. (As elaborated further below, note that even f these benefts are addtvely separable n utlty, the benefts when measured n dollars wll rse wth ncome and, n general, nonlnearly, dependng on the rate of change n the margnal utlty of consumpton.) Thrd, the ncome tax adjustment can n prncple take any form. Ths artcle constructs the thrd to just offset the frst two n terms of effects on utlty. When ths s not done, the overall result wll not be dstrbuton neutral. 3 Fortunately, the analyss of non-dstrbuton-neutral ncome tax adjustments s conceptually straghtforward, whch can be seen by the followng decomposton: (1) Implement the commodty tax reform usng the dstrbuton-neutral ncome tax adjustment specfed prevously. (2) Instantly thereafter, mplement a further reform of the ncome tax, from the dstrbuton-neutral adjustment of step 1 to whatever (non-dstrbuton-neutral) ncome tax adjustment one wshes to consder. The analyss of step 1 s that presented prevously. To complete the analyss of the nondstrbuton-neutral reform, one needs to examne step 2. Observe that ths second step s a purely redstrbutve adjustment to the ncome tax, and accordngly t would be analyzed as such. For example, f the ntal regme s close to the redstrbutve optmum and the reform s small, then t s approxmately true that no further adjustment to the welfare analyss should be requred because at the optmum the dstrbuton-dstorton tradeoff s a wash. If nstead the reform, say, ncreased redstrbuton and the status quo redstrbuted too lttle, then there would be a further 3 Absent some specfc publc choce argument, the most natural conjecture, f one were requred, seems to be that such reforms wll, on average, tend to leave the preexstng poltcal equlbrum regardng the extent of redstrbuton unaltered. In that case, dstrbuton-neutral analyss would not merely be a useful conceptual benchmark; t would also be a plausble approxmaton of poltcal realty

15 welfare gan, whereas f the status quo redstrbuted too much, there would be an assocated welfare loss. Note that n these latter cases, there would stll be partal offsets: For example, f redstrbuton becomes even more excessve, the ncremental dstorton would overstate the welfare loss because there would stll be some welfare gan from the addtonal redstrbuton. Ths dscusson llumnates some of the exstng lterature. The explanaton may not be mmedately apparent because much work uses representatve-ndvdual models, n whch dstrbutve effects are lterally moot. Yet a concern for dstrbuton s what motvates the common stpulaton that proportonal ncome tax adjustments rather than adjustments to a unform grant are the margnal source of fnance. Accordngly, t s natural to ask what the dstrbutve effects would be f one supermposed the reform and stpulated ncome tax adjustment on a model lke that examned here n whch ndvduals do vary n ncome-earnng ablty. In that case, dstrbutve effects may well be present. Furthermore, the second step of the decomposton makes apparent that, f the contemplated reforms mplctly ncrease (decrease) redstrbuton, they would be assocated wth addtonal (reduced) dstorton, whch would make the seemngly optmal commodty tax rate below (above) the Pgouvan level. Upon examnaton, some of the pror lterature does consder reforms that mplctly ncrease redstrbuton and thus should be expected to nvolve dstorton due to that effect. For example, work on polluton quotas (surveyed n Bovenberg and Goulder, 2002) compares envronmental regmes that themselves have dfferent effects on revenue, wth any gap made up by changng margnal ncome tax rates. The regmes that rase more revenue are thus less dstortonary, but the dstorton reducton s a consequence of reduced use of redstrbutve ncome taxaton. In those models, the other revenue dfferences have a lump-sum character; hence, regmes wth lower revenue are equvalent to modfcatons of a lnear ncome tax that ncrease the unform grant fnanced by an ncrease n the margnal tax rate, a combnaton that obvously rases redstrbuton as well as dstorton. Ths characterzaton s renforced by Fullerton s (1997, p. 248 n.8) explanaton of the second-best lterature s weak double-dvdend clam: f an uncorrected externalty s subjected to ntal taxaton, then welfare s hgher f the revenue s used to reduce other dstortng taxes than f t s returned to consumers lump sum. To take a dfferent type of example, Goulder et al. (1999) use a tax adjustment that holds constant the real purchasng power of the ncome transfer component of the tax-transfer system. Ths property mples that the full cost of mproved envronmental protecton s borne by rcher ndvduals, so ncreased envronmental protecton s mplctly lnked to greater redstrbuton, whch n turn produces labor supply dstorton. A more broadly applcable yet subtle effect s due to the fact that the curvature of (sub)utlty as a functon of consumpton s smultaneously related to the slope of the labor supply curve (see Chetty 2006) and also to the ncdence of commodty tax reforms as well as that of the accompanyng benefts from controllng externaltes. Although ths connecton s not wdely apprecated, the mplcaton s that explct assumptons about labor supply typcally, that the labor supply curve s upward slopng ental mplct assumptons about the dstrbutve ncdence of reforms

16 To llustrate ths elusve pont, consder the followng addtvely separable utlty functon: u(x c, x d, e, l) = (x c.5 x d.5 ) 1!ρ /(1!ρ)! b(e)! z(l), where x c s a clean good (no externalty), x d s a drty good (negatve externalty), and e s total consumpton of x d. (Ths utlty functon exhbts constant relatve rsk averson n consumpton of ρ; n the lmt as ρ approaches 1, t s approprate to substtute the log of the value n the frst parentheses.) Suppose further that there s a sngle commodty tax τ d on x d and that there s a lnear ncome tax wth rate t and a zero ntercept (.e., dsposable ncome s (1!t)wl). The polcy experment s to rase τ d whle smultaneously reducng t to mantan budget balance. It s possble to show that ρ < 1 (ρ > 1) s assocated wth all three of the followng: labor supply s upward (downward) slopng; the experment reduces (ncreases) labor supply; and the beneft from controllng the externalty, although equal for all types n utls, rses less (more) than proportonately wth ncome when measured n dollars, whch mples that the reform s overall effect s to ncrease (reduce) redstrbuton. 4 If ρ = 1, labor supply s ndependent of the wage, the experment has no effect on labor supply, and the reform s dstrbuton neutral. Although merely a smple, specal case, ths model s close to common ones employed n the lterature. Furthermore, t brngs to lght a central pont of ths artcle that s not mmedately obvous, namely, that dstrbutve effects from a reform as a whole can arse n subtle ways even when the beneft of controllng externaltes s addtvely separable and that net dstrbutve effects are mplctly related to effects on labor supply n precsely the manner suggested by step 2 of the decomposton. That s, addtonal labor supply dstorton goes hand n hand wth greater redstrbuton. Indeed, wth weak separablty of labor, Lemma 1 demonstrates that ths s a general, hard-wred relatonshp Nonseparablty The model also assumes that utlty s weakly separable n labor, wth the mplcaton that nether the commodty mx nor the level of partcular externaltes affects labor effort only the total subutlty due to commodtes and externaltes matters. As s famlar from Corlett and Hague (1953), f some commodtes are substtutes for or complements wth labor (lesure), dfferental taxes or subsdes tend to be optmal. In the present settng, the benchmark s not unformty but rather Pgouvan dfferentals. To llustrate the practcal sgnfcance of nonseparablty, West and Wllams (2007) provde evdence that gasolne consumpton s a lesure complement; accordngly, t s optmal to tax gasolne at a rate above the Pgouvan frstbest level. (Observe that, even f gasolne consumpton nvolved no negatve externaltes, t would stll be optmal to tax gasolne above the Pgouvan level (zero) that s, to mpose a postve dfferental tax on gasolne because of ts complementarty wth lesure.) It s also possble that externaltes themselves nteract wth labor effort. For example, reducng some negatve externaltes may make outdoor recreaton more attractve, n whch case t would be optmal to stop short of the Pgouvan frst best. If, nstead, central cty ambence s 4 The demonstraton s omtted because t s somewhat tedous. 5 Further corroboraton of the often hdden but nevertheless real relatonshp between dstrbuton and dstorton s provded by Allgood and Snow (1998), who show that much of the dsparty n estmates of the margnal cost of funds for publc projects and of redstrbuton s due to mplct dfferences n the extent of redstrbuton that s assumed to take place

17 mproved, whch n turn makes work more appealng, then gong beyond full nternalzaton would rase welfare. Much pror lterature on optmal nternalzaton of externaltes, lke the present paper, assumes at least weak separablty wth regard to envronmental benefts but, n contrast, does not assume that prvate consumpton s weakly separable. Accordngly, some dfferences n results are due to ths dstncton. As explaned, relaxng separablty could change the optmum n ether drecton, dependng on whether, say, a negatve-externalty-causng good was a lesure complement (as n West and Wllams s study of gasolne consumpton) or a lesure substtute. Snce pror work more often suggests that optmal taxes are below Pgouvan levels, ths dfference could be due n part to an mplct assumpton that drty goods are lesure substtutes Concluson Ths artcle examnes the problem of controllng externaltes n a model wth commodty taxes and subsdes as the polcy nstrument, an ncome tax (ntally set arbtrarly), and a contnuum of ndvduals whose earnng abltes dffer. It consders consumpton tax reforms that nvolve adjustments to the ncome tax schedule that n aggregate are dstrbuton neutral (takng nto account both the drect effects of reforms, ncludng on commodty tax revenue and externaltes, and the effects of the ncome tax adjustments). Such reforms allow one to set asde dstrbutve concerns and focus on Pareto mprovements. Moreover, as demonstrated, there s also no effect on labor supply n the baselne case wth separable labor. Whch reforms are optmal, therefore, depends on what mght be vewed as narrow, tradtonal consderatons of effcency. The man result s that settng commodty taxes and subsdes equal to margnal harms and benefts s optmal. Dstrbuton-neutral mplementaton results n a Pareto mprovement. An extenson shows that, wth some addtonal assumptons, margnal reforms n the drecton of the nave frst-best also result n Pareto mprovements (wth dstrbuton-neutral mplementaton). These results suggest that basc frst-best prncples of nternalzaton provde a useful focal pont that clarfes thnkng and asssts n the examnaton of reforms. The analyss here employs a number of dfferent types of assumptons. The addtonal assumptons for partal reforms pertan to second-best problems that can arse when externaltes are not fully nternalzed. (Thus, they were unnecessary n demonstratng Proposton 1.) These qualfcatons are orthogonal to the concerns about dstrbuton and about labor supply dstorton due to the ncome tax that have been the focus of most second-best work on the control of externaltes: In a world wth ndvdualzed lump-sum taxes that acheve the desred dstrbuton wthout any dstorton, the pertnent qualfcatons ndcate that movng commodty taxes n the drecton of (but not all the way to) the frst best mght not rase welfare. The assumpton of dstrbuton-neutral mplementaton was examned at length. It was 6 On the possble nteracton between consumpton as a whole and labor supply, see Chetty (2006)