Environmental taxes in an economy with distorting taxes and a heterogeneous population Michael Hoel

Transcription

1 Envronmental taxes n an economy wth dstortng taxes and a heterogeneous populaton Mchael Hoel

2 Ths seres s publshed by the Unversty of Oslo Department of Economcs P. O.Box 1095 Blndern N-0317 OSLO Norway Telephone: Fax: Internet: e-mal: econdep@econ.uo.no In co-operaton wth The Frsch Centre for Economc Research Gaustadalleén 21 N-0371 OSLO Norway Telephone: Fax: Internet: e-mal: frsch@frsch.uo.no Last 10 Memoranda No 3/08 No 2/08 No 1/08 No 23/07 No 22/07 No 21/07 No 20/07 No 19/07 No 18/07 No 17/07 Fedor Iskhakov Dynamc programmng model of health and retrement Atle Seerstad and Sgve D. Stabrun Dscontnuous control systems Kjell Arne Brekke, Rolf Golombek and Sverre A. C. Kttelsen Is electrcty more mportant than natural gas? Partal lberalzaton of the Western European energy markets Ger B. Ashem and Taoyuan We Sectoral Income Ger B. Ashem and Seung Han Yoo Coordnatng under ncomplete nformaton Jo Thor Lnd and Halvor Mehlum Wth or Wthout U? The approprate test for a U shaped relatonshp Chng-to Albert Ma A Journey for Your Beautful Mnd: Economcs Graduate Study and Research Smen Markussen Trade-offs between health and absenteesm n welfare states: strkng the balance Torbjørn Hægeland, Oddbjørn Raaum and Kjell Gunnar Salvanes Pennes from heaven - Usng exogenous tax varaton to dentfy effects of school resources on pupl achevement B. Bratsberg, T. Erksson, M. Jäntt, R. Naylor, E. Österbacka, O. Raaum and K. Røed Martal Sortng, Household Labor Supply, and Intergeneratonal Earnngs Moblty across Countres A complete lst of ths memo-seres s avalable n a PDF format at:

3 Envronmental taxes n an economy wth dstortng taxes and a heterogeneous populaton Mchael Hoel y 15 January 2008 Abstract Durng the last couple of decades, there has been a large lterature dscussng how the propertes of emsson taxes are a ected by the exstence of dstortonary taxes. Most of ths lterature gnores dstrbutonal aspects of envronmental taxes and other types of envronmental polcy nstruments.the present paper consders a very smple model wth heterogeneous households, d erng n ncome earnng ablty. The tax system s not necessarly fully optmal. Instead, a tax functon s assumed to be exogenously gven, but the parameters of ths tax functon are optmally chosen. The rule for the second-best optmal envronmental tax s derved and compared wth the Pgovan rule. The results derved n the present paper are related to the results from the lterature on publc goods provson under dstortonary taxes. Keywords: envronmental taxes, publc goods, dstortonay taxaton JEL class caton: H23, H41, Q58 I am grateful for many useful comments and suggestons from partcpants at the conference "Skatteforum 2007" (at Refsnes gods, Norway, June 2007), and n partcular from Vdar Chrstansen. The support of the Research Councl of Norway, under the Tax Research Programme, s also gratefully acknowledged. y Department of Economcs, Unversty of Oslo. P.O Box 1095 Blndern, N-0317 Oslo. Emal: mhoel@econ.uo.no 1

4 1 Introducton The dea that there s a so-called "double dvdend" of envronmental taxes goes back to the late 1980s. 1 The background for the double dvdend hypothess s the observaton that there are dstortng taxes n the economy. By ntroducng an envronmental tax one therefore not only ncreases the envronmental qualty (the rst "dvdend") but the envronmental tax also rases revenue that can be used to reduce other dstortng taxes so that the performance of the economy s mproved (the second "dvdend"). Whle ths mght seem plausble at rst sght, t s obvously not generally correct. If the tax system before any consderatons of the envronment s optmally desgned, socal welfare (gnorng envronmental aspects) s by de nton maxmzed subject to whatever constrants there are on what taxes and tax rates one may use. Changng the ntal tax system by ntroducng a new tax and adjustng other taxes wll n such a stuaton necessarly reduce socal welfare (gnorng envronmental aspects). The second dvdend s thus not present. Introducng an envronmental tax may of course nevertheless be welfare mprovng; ths wll be the case f the mprovement n envronmental qualty s consdered larger than the loss n "non-envronmental socal welfare" (.e. socal welfare gnorng envronmental aspects). If the ntal tax system s not optmally desgned, there wll by de nton exst changes n the tax system that mprove socal welfare. In partcular, among the set of such possble welfare mprovng tax changes there may be changes that nvolve ntroducng a tax that may be labeled "envronmental" and adjustng some other taxes. If ths s the case, non-envronmental socal welfare wll ncrease, n addton to the envronmental bene t the envronmental tax gves. In ths case we thus have a double dvdend. There was a consderable lterature n the early 1990s dentfyng cases were the ntal tax system was non-optmal and where the ntroducton of an envronmental tax n combnaton wth a reducton of other tax rates gave a double dvdend. However, one can queston how much nsght ths lterature gave, and n partcular what mplcatons t had for envronmental polcy. The "second dvdend" (.e. the ncrease n non-envronmental socal welfare) n all these cases had nothng to do wth the envronmental tax as such, t was smply a "dvdend" from changng a non-optmal tax system 1 The rst use of the concept "double dvdend" seems to have been by Pearce (1991), but the general dea goes back to at least Terkla (1984). 2

5 to some tax system that was better (measured by non-envronmental socal welfare). Such a tax reform therefore made economc sense even f there was no concern for the envronment. The fact that such a tax reform also mproved the envronment was of course nce, but rrelevant. A related, and n my opnon more nterestng, ssue that has been dscussed durng the last couple of decades, s how the propertes of emsson taxes and other envronmental polcy nstruments are a ected by the exstence of dstortonary taxes. An mportant nsght from ths lterature s that polcy nstruments that rase revenue (such as an emsson tax or auctoned quotas) become more favorable compared wth non-revenue rasng nstruments (such as drect regulaton or non-auctoned quotas) than n a stuaton wthout dstortng taxes, see e.g. Parry (1997) and Goulder et al. (1999). A second mportant queston that has been rased s how the optmal rate of an emsson tax wll be a ected by the exstence of dstortng taxes. From the earler double dvdend lterature dscussed above t mght be argued that even f there s no double dvdend, at least some of the costs of ntroducng an envronmental tax are o set by the reducton n other dstortng taxes. Snce ths reduces the cost of envronmental polcy, one could furthermore argue that the optmal envronmental tax should be hgher than the Pgovan level (.e. the sum of everyone s wllngness to pay for mprovng the envronment). Lee and Msolek (1986) gve a formal analyss wth ths concluson: The rate of the envronmental tax should be hgher than f the tax was chosen only to control the amount of polluton, provded the revenue from the envronmental tax s ncreasng n the tax rate at the optmum. However, studes by e.g. Bovenberg and van der Ploeg (1994), Bovenberg and de Mooj (1994), Goulder (1995), Parry (1995), Bovenberg and Goulder (1996), Bovenberg (1999), and Metcalf (2003) have shown that t s not so obvous whether or not the optmal envronmental tax should exceed the Pgovan level. Most of these studes consder the case of a proportonal ncome tax. An mportant result derved n several of these papers s that f preferences are separable between polluton and other goods, and the uncompensated labour supply elastcty s postve, the second-best optmal emsson tax s lower than the Pgovan level. Somewhat surprsngly, there s relatvely lttle n ths lterature concernng dstrbutonal aspects of envronmental taxes and other types of envronmental polcy nstruments. The reason why ths s surprsng s that dstrbutonal consderatons are the reason one has dstortng taxes. Wthout any concern for ncome/consumpton dstrbuton, the necessary tax revenue 3

6 could smply be rased by an equal tax per person, and there would be no nteracton (except ncome e ects) between such taxaton and the use of envronmental taxes. There s a small lterature that dscusses envronmental taxaton n models that explctly allow for varous types of heterogenety n the populaton. Perhaps the earlest contrbuton was Sandmo (1975), who adressed ths ssue wth a relatvely restrcted set of possble taxes. More recently, Cremer et al. (1998), Cremer and Gahvar (2001), and Prttla and Tuomala (1997) consder a heterogeneous populaton, and ncome taxaton s n these studes assumed to be optmally desgned. The present paper starts by ntroducng a smple one person economy that s used to explan the result that the second-best optmal emsson tax s lower than the Pgovan level (secton 2). In secton 3 the model s extended to a heterogeneous populaton and the rules for an optmal emsson tax are derved n a settng wth a smple two-parameter lnear ncome tax. The specal case of a homogeneous populaton s bre y dscussed agan n secton 3.1, whle secton 3.2 gves results for the case of a heterogeneous populaton. In Secton 4 t s shown that there s a strong smlarty between the ssue of optmal emsson taxaton and the optmal provson of publc goods. Whle ths smlarty s partcularly transparent for the model used n the present analyss, the smlarty s a general feature: Optmal emsson taxaton s smply a queston of the optmal amount of a publc bad, whch obvously s almost equvalent to the queston of the optmal amount of a publc good. There s a large lterature on the optmal provson of publc goods n the presence of dstortng taxaton, gong back to Pgou (1947), and wth mportant early contrbutons ncludng Damond and Mrrlees (1971), Stgltz and Dasgupta (1971) and Atknson and Stern (1974). Contrbutons such as Feldsten (1997) have argued that the presence of dstortng taxaton mples that publc goods should be suppled by a lower amount than what s mpled by the Samuelson rule (.e. margnal costs of producng a publc good should be equal to the sum of wllngness to pay for the good). The formal analyses of e.g. Stgltz and Dasgupta (1971) and Atknson and Stern (1974) mply that when a publc good s optmally suppled, the margnal costs of producng t should be lower than the sum of wllngness to pay for the good, provded the publc good s separable from consumpton and lesure n preferences and that the labour supply curve s upward slopng. Ths corresponds completely to the clam that the optmal emsson tax should be lower than the Pgovan level. Unlke the envronmental lterature 4

7 however, the lterature on publc goods has for a long tme explctly taken populaton heterogenety nto consderaton. In addton to Damond and Mrrlees (1971), early contrbutons nclude Hylland and Zeckhauser (1979) and Chrstansen (1981). The latter contrbuton shows that f (a) ncome taxes are optmal and (b) labour s separable from other goods n preferences, the optmal provson of publc goods s gven by the Samuelson Rule. Ths result s dscussed further by Boadway and Keen (1993), and t s extended by Kaplow (1996). Kaplow shows that even f the tax system s not ntally optmal, t s possble to acheve a Pareto mprovement f the Samuelson Rule does not hold, assumng that labour s separable from other goods and that a partcular type of tax adjustment s possble smultaneously wth a change n the supply of the publc good. In the lterature on envronmental taxes, there has been remarkably lttle reference to the large lterature on the optmal supply of publc goods. None of the contrbutons on envronmental taxes referred to above refer to the mportant results of Chrstansen, Boadway and Keen, and Kaplow. It s partcularly surprsng that the most recent of the above contrbutons do not refer to Kaplow s result, snce Kaplow explctly shows that hs result s of drect relevance to the queston of how large envronmental taxes should be. 2 In Secton 4, the results derved n Secton 3 are related to the results from the lterature on publc goods provson. Secton 5 concludes. 2 Envronmental taxes n the presence of a tax wedge n the labour market Consder a consumer wth a utlty functon u(c; L; E) where C s consumpton, L s labour supply and E s aggregate emssons of some pollutant. Ths utlty functon s ncreasng n C and decreasng n L and E. The economy s resource constrant s C = F (L; E) G, where G s an exogenously gven publc expendture and F (L; E) s a reduced form (net) producton functon. If e.g. the pollutant s greenhouse gases, the nterpretaton of E could be the use of fossl fuels n gross producton (L; E). If these fuels have an 2 Prttla and Tuomala refer bre y to Boadway and Keen (1993) and Cremer and Gahvar refer bre y to Kaplov (1996), but nether refer explctly to the results n these papers and ther relevance for optmal envronmental taxaton. 5

8 nternatonal prce equal to p, the country s net producton (.e. GDP) s F (L; E) = (L; E) pe. Wthout any concern for the envronment, the country would maxmze ts welfare by choosng E so that E = p, gvng F E = 0. If the use of fossl fuels s reduced below ths level, we get F E = E p > 0, and F E may be nterpreted as a tax on the use of fossl fuels n producton. The optmal choce of E s the choce that maxmzes u(c; L; E) = u(f (L; E) G; L; E), and the rst-order condton of ths maxmzaton problem s or u C F E + u E + (u C F L + u L ) dl de = 0 F E = u E u C F L u L u C dl de As explaned above, F E s equal to the emsson tax. Moreover, the gross wage s equal to F L : Wthout any tax wedge n the labour market, u the consumer chooses hs labour supply so that L u C s equal to the gross wage rate. In ths case the second term on the r.h.s. of (1) s therefore zero, and we get the standard expresson for the Pgou tax: The tax should be equal to the margnal wllngness to pay for reduced emssons. However, wth a postve tax wedge n the labour market, the term n brackets on the r.h.s. of (1) s postve. The optmal envronmental tax s therefore lower or hgher than the Pgovan level dependng on whether dl s postve or negatve. The de sgn of dl generally ambguous. An mportant specal case s characterzed de by preferences beng weakly separable n emssons and the two other goods (consumpton and lesure),.e. that u(c; L; E) = ~u(w(c; L); E). In ths case the sze of E wll have no drect e ect on L. In ths case the sgn of dl wll de depend only on the sgn of the uncompensated supply elastcty for labour: As E s ncreased,.e. envronmental polcy s relaxed, the after-tax wage rate goes up (otherwse there would have been no cost of reducng emssons). If the labour supply responds postvely to ths ncrease n the after-tax wage, t follows that dl s postve, mplyng that the optmal emsson tax s below de the Pgovan level u E u C. (1) 6

9 3 Envronmental taxes wth an optmal lnear ncome tax. We now turn to the more nterestng case of a heterogeneous populaton. There are I persons n the economy, and persons s labour supply n e cency unts s ` when hs or her labour supply measured n hours s `: The parameters re ect d erences n productvty across the populaton. The producton functon s as before F (L; E), where aggregate labour nput n e cency unts now s equal to L = P `. The gross ncome of person s F L `; and the tax s tf L ` s; where t s postve and s may be postve or negatve. The net (after tax) margnal wage rate (n e cency unts) s denoted n, and s n = (1 t)f L (2) snce the margnal productvty of labour F L s equal to the gross wage rate. The dsposable ncome of person s thus n ` + s. It s useful to derve the ndrect utlty functon of each person. Ths s de ned by v = v(n ; s; E) = max u(c ; `; E) s.t. C = n ` + s (3) The soluton of ths maxmzaton problem gves demand and supply functons C(n ; s; E) and `(n ; s; E), and the aggregate labour supply s L(n; s; E) = `(n ; s; E) (4) The government s optmzaton problem can be formulated as max s.t. F (L(n; s; E); E) v(n ; s; E) (5) C(n ; s; E) G 0 (6) The Lagrangan to ths problem s 7

10 = v(n ; s; E) + " F (L(n; s; E); E) C(n ; s; E) G # (7) and the rst order condtons determnng E, s and n are " # ve + F E + F L L E = 0 (8) " vs + F L L s " vn + F L L n C s C E # = 0 (9) # Cn = 0 (10) Our concern here s the optmal emsson tax,.e. F E. In Appendx A t s shown that (8) mples (usng well-known propertes of the ndrect utlty functon gven by (3)) P F E = ( u E ) tf L L E (11) Before dscussng ths expresson further for the general case, t s useful to bre y consder the specal case of a homogeneous populaton, whch s dentcal to the one person case. 3.1 A homogeneous populaton In the Appendx t s shown that when all I persons have the same value of, (10) gves 1 = 1 t 1 u C 1 t " (12) where " s the average labour supply elastcty ( = L n n=l) Insertng (12) nto (11) gves 8

11 F E = 1 t 1 t " I ue u c tf L L E (13) If preferences are weakly separable n emssons and the two other goods as explaned n Secton 2, L E = 0. If the term n square brackets was 1, we would have the Pgou rule for the I-person case,.e. F E = I ue u c. When the tax rate t s postve, the term n square brackets wll be smaller than or larger than one dependng on whether the labour supply elastcty s postve or negatve. Ths con rms the result we derved somewhat less formally n Secton 2 (and whch s well known from prevous lterature, see e.g. Bovenberg and de Mooj, 1994). Even f the labour supply s ndependent of the net wage rate (" = 0), the second-best-optmal emsson tax wll d er from the Pgovan level f preferences are not separable. The tax s hgher than the Pgovan level f the labour supply s ncreasng n emssons (e.g. f hgher emssons make tme ntensve lesure actvtes less attractve) whle the opposte s true f the labour supply s decreasng n emssons (e.g. f hgher emssons have a negatve e ect on health and thus on labour supply). Equaton (13) s however not partcularly nterestng. If the populaton s homogeneous, t well known from elementary economc theory that there s no reason to have a dstortonary tax f a unform head tax s feasble. And f t = 0, (13) smply gves us the standard Pgou tax. The reason we n practce see dstortonary taxes used, s that the populaton s not homogeneous, and that the government has some dstrbutonal preferences. We therefore now return to the case of a heterogeneous populaton. 3.2 Envronmental taxes wth a heterogeneous populaton. Consder agan equaton (11). When there s separablty n preferences between emssons and the other two goods, the second term n (11) s zero. In ths case (11) s very smlar to the standard Pgou tax. The d erence s that the Pgou tax n the I person case s P ( u E ), whle t n the present u P C case s ( u E ). Cremer and Gahvar (2001) conclude from ths that n the case of separablty between emssons and other goods, the optmal emsson tax s equal to the Pgovan level. As we shall see below, ths seems to be a rather msleadng use of the term "Pgovan level". 9

12 It s shown n Appendx A that (9) may be rewrtten as 1 = 1 1 tf LLs u C where the two averages u C and L s are de ned by P u C = u C I and L s = P `s I (14) (15) (16) Insertng (14) nto (11) gves or F E = 1 tf L Ls P ( u E ) u c F E = 1 tf L Ls u c u c u E u c tf L L E tf L L E (17) As before, the last term n (17) s zero f preferences are weakly separable n emssons and the two other goods. The term P u c u E u c s a weghted u c sum of the wllngness to pay for reduced emssons for all persons. The weghts are margnal utltes of consumpton. Wth realstc assumptons about the utlty functon, consumpton wll be hgher and margnal utlty of consumpton lower the hgher s the productvty (and thus also wage per hour). In other words, hgh ncome persons wll have a lower weght n the term P u c u E u c than low ncome persons. A natural phrase for ths term u c s therefore the equty weghted Pgovan level of the emsson tax. Notce that f lump-sum taxaton was possble,.e. d erent values of s for d erent persons, u c would be dentcal for all persons, so there would be no d erence between a weghted and unweghted sum of the wllngness to pay. The term 1 tf LLs s larger than one provded that Ls < 0,.e. provded that lesure s a normal good (on average). For the reasonable case that 10

13 lesure s a normal good, the optmal emsson tax s thus hgher than the equty weghted Pgovan level (n the case of weakly separable preferences). Returnng to the expresson (17), an obvous queston s why the optmal emsson tax should be hgher than the equty weghted Pgovan level for the case when preferences are separable from the two other goods. The nterpretaton s as follows. Startng wth a tax at the weghted Pgovan level, a small ncrease n the emsson tax wll have a small cost for households, through a lower transfer s to all persons. Dsregardng nduced e ects on labour supply, ths change n the emsson tax has an mpact of measure zero on welfare. However, snce s goes down, labour supply ncreases (when lesure s a normal good). Due to the postve margnal ncome tax, labour supply s lower than what s optmal. The ncrease n labour supply thus gves an ncrease n the welfare level of measure one, thus domnatng the drect welfare loss of devatng from the weghted Pgovan level. An obvous queston s how the expresson for the second-best tax gven by (17) d ers from the Pgovan level (.e. the unweghted sum of the wllngness to pay for reduced emssons for all persons). It s generally not possble to say n what drecton ths d erence goes. The fact that the term n square brackets s larger than one (when lesure s a normal good) draws n the drecton of the second-best tax beng hgher than the Pgovan level. On the other hand, f the wllngness to pay for reduced emssons s hgher for hgh-ncome than for low-ncome persons, the weghng of the wllngness to pay terms n (17) draws n the drecton of the second-best tax beng lower than the Pgovan level. Although we have restrcted the analyss n ths secton to the case of a lnear ncome tax, we have assumed that the two parameters n ths tax functon (t and s) are optmally chosen. To understand the mportance of the tax rate beng optmally chosen, consder a startng pont wth gven values of E, t and s. Assume that for the gven value of E, both t and s are hgher than ther optmal values. Moreover, assume that a small reducton n E wll lead to a surplus n the government s budget, whch can be used to ether reduce t or ncrease s. Snce t and s by assumpton are both hgher than ther optmal values, t s clear that a small reducton n E s more favorable f t s accompaned by a reducton n t than f t s accompaned by an ncrease n s. It may even be the case that f the tax rate t s smultaneously adjusted, welfare ncreases f E s reduced, whle f s s smultaneously adjusted welfare ncreases f E s ncreased. The optmal envronmental tax wll thus n ths case depend on whch tax rate s assumed to be adjusted f the envronmental 11

14 tax s changed. 4 Envronmental taxes and the optmal provson of publc goods The model we have used may easly be re-nterpreted as that of a model of publc goods provson. Rede ne E to be the provson of a partcular publc good. The sgns of u E and F E must be reversed compared wth our analyss so far, so that u E > 0 and F E < 0 ( F E s thus the margnal cost of producng the publc good). The rst-best optmum for the supply of ths good s gven by the Samuelson Rule: F E = u E u c (18) The nterpretaton of ths equaton s that the margnal cost of producng the publc good should be equal the sum of the wllngness to pay for good. In the early lterature on the optmal provson of publc goods t has often been argued that one should provde less of the good than (18) suggests, due to the dstorton caused be rasng taxes to nance the good. 3 Ths argument s very much n the sprt of the argument leadng to (13), suggestng that the optmal emsson tax should be lower than the Pgovan level (n the case of weakly separable preferences). The analyss n Secton 3 remans vald wth the renterpretaton of E, so that the results (13) and (17) remans vald. Rewrtng these we get and F E = 1 t 1 t " I ue u c + tf L L E (19) 3 For nstance, Wlson (1991) ctes Stgltz (1988, p. 180): "Snce t becomes more costly to obtan publc goods when taxaton mposes dstortons, normally ths wll mply that the e cent level of publc goods s smaller than t would have been wth non-dstortonary taxaton". 12

15 F E = 1 tf L Ls u c u c u E + tf L L E (20) Expressons smlar to (19) for the case of a homogeneous populaton - but wth somewhat d erent notaton - were derved by e.g. Stgltz and Dasgupta (1971) and Atknson and Stern (1974). In the absence of lump-sum taxes the publc good should be suppled at a level makng the margnal cost of producng ths good lower than the margnal wllngness to pay, provded labour supply s ncreasng n the net wage rate and preferences are weakly separable n the publc good and the two prvate goods (consumpton and lesure). For the case of a heterogeneous populaton and the possblty of a unform lump-sum tax t s from (20) not obvous whether the supply of the publc good should be lower or hgher than the level mpled by the Samuelson Rule. The fact that we n (20) have an equty weghted sum of the wllngness to pay for all persons wll tend to make the r.h.s. of (20) lower than the r.h.s. of (18), as long as the publc good s a normal good. On the other hand, f lesure s a normal good the term n square brackets n (20) tends to make the r.h.s. of (20) hgher than the r.h.s. of (18). Fnally, f the supply of the publc good has a drect e ect on labour supply, the second term n (20) wll be a thrd source of d erence between (20) and (18). Unlke the lterature on emsson taxes, there s a large lterature on the optmal provson of publc goods where dstrbutonal consderatons are explctly taken nto account. An expresson smlar to (20) - but wth somewhat d erent notaton and assumptons - can for nstance be found n Atknson and Stgltz (1980, Sec. 16.3). An mportant early contrbuton to ths lterature s Chrstansen (1981), who shows that f the ncome tax s optmally desgned, and f consumers preferences have separablty between labour and other goods, then the optmal supply of the publc good s characterzed by the Samuelson Rule (18). Notce that the separablty assumpton used s d erent from the separablty assumpton requred for the second tern n (20) to become zero. The latter condton was that u(c; L; E) = ~u(w(c; L); E), whle the assumpton used by Chrstansen s that u(c; L; E) = ~u((c; E); L). Kaplow (1996) has generalzed the result of Chrstansen to cases where the ntal taxes are not necessarly second best optmal. 4 Wth the same sep- 4 A related result was derved already n 1979 by Hylland and Zeckhauser. u c 13

16 arablty assumpton as Chrstansen, Kaplow shows that unless (18) holds, t s possble to change the provson of publc goods and smultaneously adjust taxes so that a Pareto mprovement s possble. A Pareto optmal outcome must thus be characterzed by the Samuelson Rule (18). Kaplow also shows that hs result apples to the ssue of an optmal envronmental tax, thus argung that the Pgovan tax s optmal also n a second-best settng. It s mportant to note that Kaplow s result assumes the possblty of adjustng taxes so that a Pareto mprovement s possble for any change n the supply of the publc good f (18) does not hold ntally. Ths requres qute a rch tax system (see also Chrstansen, 2007). Wth the smple lnear tax functon used n secton 3 ths wll generally not be possble, so that we wth ths tax functon cannot expect the Samuelson Rule for publc goods to hold. Smlarly, we cannot expect the optmal emsson tax to be equal to the Pgovan level for the smple lnear tax functon used n Secton 3. 5 An obvous queston s whether the strong smlarty between the optmal emsson tax and the optmal provson of a publc good s smply a partcular property of the spec c model used, or whether t holds more generally. To see that t s a qute general property, consder the followng generalzaton of our model: Let the economy s resource constrant be H(C;G; L; E) 0 n stead of our prevous resource constrant C = F (L; E) G. H s ncreasng n L and E (whch as before represent labour nput and emssons) and decreasng n the other varables: C = (C 1 ; :::; C K ) s a vector of prvate consumpton goods, and G s now the supply of publc goods (generalzaton to several publc goods s trval). Let C 1 be the numerare good. The margnal producton cost of prvate good k s n terms of the numerare good gven by (n obvous notaton) H k H 1, and the margnal producton cost of the publc good s (n terms of the numerare good) H G H 1. Notce that both of these margnal producton costs nclude envronmental costs through a shadow prce of E, snce E s held constant. The margnal abatement cost n terms of the numerare good (.e. the emsson tax) s equal to H E H1, and the margnal productvty of labour (.e. the gross wage rate) s H L H1 : In much of the lterature on envronmental taxes t s assumed that emssons are lnked to the use of "drty" consumer goods,.e. as E = P k kc k where k s postve for all "drty" goods. Ths case can be approxmated by the general descrpton of the producton possbltes gven above. A smple 5 An example of a utlty functon mplyng that tha Kaplov result holds wth the lnear tax functon assumed n Secton 3 s gven n Appendx B 14

17 case s where producer prces (excludng envronmental costs) p k and gross wages w are exogenous. Let H(C;G; L; E) = k p k C k p k C k + wl k C k E! 2 k where s "large". Moreover, let H 1 = p 1 = 1; and let the numerare good be "clean", so that 1 = 0: Then the margnal shadow prce of emssons s H E = 2 ( P k kc k E) 2 1 and the margnal cost of consumer good k, ncludng P envronmental costs, s H k = p k + k H E. Fnally, the optmum k kc k can be made arbtrarly close to E by choosng a su cently large value of : The case often consdered n the envronmental lterature s thus a specal case of the model used n the present secton. When there are several consumer goods, C must be replaced by C n the P utlty functon, and C n the budget equaton (3) must be replaced by k q kck, where q k s the consumer prce of good k. The commodty tax of H good k s thus k = q k k H 1. For the numerare good we have q 1 = 1 and 1 = 0. Wthout loss of generalty the numerare good can be chosen so that all commodty taxes are non-negatve. Proceedng as we dd n Secton 3, t s straghtforward to derve the followng formulas for the optmal emsson tax and the optmal provson of the publc good (for the case of a lnear ncome tax): H E H 1 = " 1 t H L H 1 Ls + k k Cks # u 1 u 1 u E u 1 t H L H 1 L E + k k C ke (21) H G H 1 = " 1 t H L H 1 Ls + k u 1 u 1 u G + t H L H 1 L G k C kg # kcks u 1 k (22) where the margnal utlty of ncome s now equal to the margnal utlty of the numerare good, and where 15

18 C ke and C kg = C ks = 1 k Equatons (21) and (22) are practcally dentcal, and correspond completely to the prevous expressons (17) and (20) except for the addtonal terms nvolvng how the non-numerare consumpton goods are e ected by changes n E and s. Notce that f all goods are normal and all commodty taxes are non-negatve, the terms contanng C ks n the square brackets are postve, so that the terms n square brackets as before are larger than one. Also, f E and G are separable form the prvate goods n preferences, the second and thrd terms n (21) and (22) wll be zero. The results above hold for any arbtrarly gven commodty taxes. Wthout any restrctons on preferences, the optmal commodty taxes wll typcally be postve for some goods. However, although the choce both of E and G typcally wll a ect producer prces (.e. margnal producton costs) of the prvate consumpton goods, these choces wll have no consequence for the rule for determnng optmal commodty taxes. Ths pont was rst made by Sandmo (1975). 5 Concludng remarks The prevous sectons have demonstrated that there s no smple answer to the queston of how large the second-best optmal emsson tax should be compared to the Pgovan level. In partcular, the answer to ths queston does not depend on smple propertes such as whether the tax revenue form the envronmental tax s (locally) ncreasng n the tax rate, or the sgn of the labour supply elastcty. Even wth spec c assumptons about separablty 16

19 n preferences we do not know the answer to ths queston. If e.g. preferences are weakly separable n emssons and the two other goods (consumpton and lesure) the last term n (17) s zero. However, as argued after (17), even f ths case we do not generally know whether the optmal emsson tax s hgher or lower than the Pgovan level. 6. From the lterature on the provson of publc goods we know that f consumers preferences have separablty between labour and other goods, and the ncome tax system s su cently "rch" so that Kaplow s result holds, then the optmal envronmental tax rate s equal to the Pgovan tax rate. In practce, these two condtons wll seldom be ful lled. However, an nterestng queston would be to analyze what the welfare loss would be from (wrongly) settng the tax rate equal to the Pgovan rate for realstc devatons from the two assumptons above. Obvously, ths s a very d cult queston to answer. If I were to guess, my guess would be that ths welfare loss would be much lower than the loss we may get from an naccurate measurement of peoples wllngness to pay for reduced emssons, gven the large and wellknown d cultes n makng such measurements. Appendx A: Dervatons of the second-best optmal emsson tax The demand and supply functons correspondng to (3) satsfy C n (n ; s; E) = ` + n `n (n ; s; E) C s (n ; s; E) = 1 + n `s(n ; s; E) (23) C E (n ; s; E) = n `E(n ; s; E) Moreover, the envelope theorem appled to (3) gves 6 In an earler verson of ths paper I have shown that a smlar concluson s true for a more general tax functon than the smple tax functon consderred n Sectons 3-5. Ths ssue s also dscussed by Wlson (1991) for qute general assumptons about what taxes are feasble. 17

20 v s (n ; s; E) = u C (C ; `; E) v n (n ; s; E) = u C (C ; `; E)`(n ; s; E) (24) v E (n ; s; E) = u E (C ; `; E) Usng (2), (23) and (24) we can rewrte (8) as Rearrangng gves (11). u E + [F E + (n + tf L )L E nl E ] = 0 Usng (2), (23) and (24) we can rewrte (9) as u C + [(n + tf L )L s I nl s ] = 0 (25) Together wth (15) and (16) ths gves (14). The optmal tax can alternatvely be derved by usng (10) together wth (2), (4), (23) and (24). Ths gves or " `u C + (n + tf L ) ( ) 2`n `u C + [tf L L n L] = 0 # (` + n `n) = 0 Ths can be rewrtten as 1 = 1 1 ~u C t 1 t " where " s the average labour supply elastcty ( = L n n=l) and ~u C followng weghted average of u C : (26) s the 18

21 ~u C = P `u C P ` (27) Ths weghted average of margnal utltes d ers from the unweghted average u C de ned n (15). Snce ` s hgher the hgher s (at least f consumpton s a normal good), low values of the margnal utlty of consumpton are gven more weght n ~u C than n u C : We therefore have ~u C < u C. Insertng (27) nto (11) gves t F E = 1 1 t " uc u c ~u C u c u E u c tf L L E (28) For the specal case of a homogeneous populaton, u C = ~u C = u C, so that ths equaton may be rewrtten as (13). For ths case t also follows from from (13) and (17) that t = 0. Ths con rms the well-known result that dstortonary taxes are not needed n an economy wth a completely homogeneous populaton. Appendx B: The Kaplow result Assume that preferences are gven by the separable utlty functon u(c ; `; E) = ~u((c ; E); `) (29) Consder a change n E combned wth a change n taxes so that for each person the utlty level s unchanged for any potental choce of labour supply. (We shall return to the queston of whether such a tax change s feasble.) Gven the utlty functon (29), no one wll change hs or her choce of labour supply, snce the e ect of changng ` on (C ; E) wll be the same as before the change n E and taxes. For person the change n consumpton that compensates the change n E, denoted E; s C = u E u C E so that the change n total consumpton must be 19

22 C = u E u C E Snce labour supply s unchanged, the change n total output (F (L; E)), denoted F, s F = F E E The surplus (producton mnus consumpton) from the combned change n E and taxes, whch wll be a surplus n the government budget snce households consume all of ther ncome, s thus " # u F C = E E (30) F E If the emsson tax (= F E ) s not at the pgovan level,.e. f the term n square brackets n (30) s not zero, the government wll get a surplus by a sutable choce of E: (E wll be postve (negatve) f the ntal tax s above (below) the Pgovan level.) Ths surplus can be used to make everyone better o by an approprate change n tax rates, for nstance by gvng a unform tax credt. Snce a Pareto mprovement s possble f the Pgou condton F E = u E (31) u C does not hold, ths condton must hold at any Pareto optmum. The reasonng above assumed that t was possble to supplement a change n emssons wth tax changes so that for each person the utlty level was unchanged for any potental choce of labour supply. Gahvar (2006) has shown that f the only restrcton on the tax functon s the nformaton constrant (.e. that the government knows the dstrbuton of productvtes, but not each ndvdual s productvty), then such a compensatng tax change s possble. In ths case we can therefore conclude that the optmal emsson tax rate s the Pgovan rate (provded the assumptons about preferences gven ntally hold). For a smple lnear tax as gven n secton 3 t s generally not possble to adjust taxes n response to a change n E as descrbed above. Such a tax adjustment s however possble for a partcular spec caton of the utlty 20 u C

23 functon: In the utlty functon (29), let be a Cobb Douglas functon n consumpton and "envronmental qualty" Q E;where Q s a postve parameter (representng envronmental qualty n the absence of any emssons): (C ; E) = (C ) a (Q E) b (32) where a and b are postve parameters. Wth ths spec caton u E u C = E C = C Q E.e. the margnal wllngness to pay for mproved envronmental qualty s for any gven E proportonal to the person s consumpton of prvate goods. Insertng the households budget equatons nto (32) gves (C ; E) = (1 t)f L ` + s a (Q E) b (33) The tax adjustment descrbed n the begnnng of ths Appendx requred that as E s changed, (C ; E) must be left unchanged for all and all `. From (33) t s clear that a change n E must be accompaned by a change n the term n square brackets that s proportonal to the ntal value of ths term. Ths can be acheved by adjustng s and t so s and (1 t)f L are changed n the same proporton, and wth a change just su cent to make (C ; E) unchanged n spte of E beng changed. After such a combned change n E and the tax rates t and s t follows from the reasonng above that f (31) does not hold t s possble to change E, t and s such that all utlty levels reman unchanged whle (6) wll hold wth a strct nequalty. From ths poston t s possble to acheve a Pareto mprovement by ncreasng s and adjustng t so (1 t)f L s left unchanged, see (2) and (3). For the preference functon gven by (29) and (32) we can thus conclude that the optmal emsson tax s equal to the Pgovan level. References Atknson, A.B. and Stern, N.H. (1974), "Pgou, taxaton and publc goods", Revew of Economc Studes 41, Atknson, A.B. and Stgltz, J.E. (1980), Lectures on Publc Economcs. McGraw Hll. 21

24 Bodway, R. and Keen, M. (1993), "Publc goods, self-selecton and optmal ncome taxaton", Internatonal Economc Revew 34, Bovenberg, A.L., and van der Ploeg, F. (1994): "Envronmental polcy, publc nance and the labour market n a second-best world", Journal of Publc Economcs 55, Bovenberg, A.L., de Mooj, R.A. (1994): "Envronmental Leves and Dstortonary Taxaton". Amercan Economc Revew 84, Bovenberg, A.L., Goulder, L.H. (1996): "Optmal envronmental taxaton n the presence of other taxes: General-equlbrum analyses". Amercan Economc Revew 86, Bovenberg, A.L. (1999), "Green Tax Reforms and the Double Dvdend: An Updated Reader s Gude". Internatonal-Tax-and-Publc-Fnance. August 1999; 6(3): Chrstansen, V. (1981), "Evaluaton of publc projects under optmal taxaton", Revew of Economc Studes 48, Chrstansen, V. (2007), "Two Approaches to Determne Publc Good Provson under Dstortonary Taxaton," Natonal Tax Journal 60, Cremer, H., F. Gahvar and N. Ladoux (1998), Externaltes and optmal taxaton, Journal of Publc Economcs 70, Cremer, H. and F. Gahvar (2001), Second-best taxaton of emssons and pollutng goods, Journal of Publc Economcs 80, Damond, P.A. and Mrrlees, J.A. (1971), "Optmal taxaton and publc producton" I: "Producton e cency", and II: "Tax rules", Amercan Economc Revew 61, 8-27 and Feldsten, M. (1997): How Bg Should Government Be?, Natonal Tax Journal 50 No. 2, Gahvar, F. (2006), "On the margnal cost of publc funds and the optmal provson of publc goods", Journal of Publc Economcs 90,

25 Goulder, L.H. (1995), "Envronmental Taxaton and the Double Dvdend: A Reader s Gude", Internatonal Tax and Publc Fnance 2(2): Goulder, L.H., Parry, I.W.H., Wllams, R.C., Burtraw, D. (1999), "The cost-e ectveness of alternatve nstruments for envronmental protecton n a second-best settng", Journal of Publc Economcs, 72 (3): Hylland, Aa. and Zeckhauser, R. (1979): Dstrbutonal Objectves Should A ect Taxes but Not Program Choce or Desgn, Scandnavan Journal of Economcs 81 No. 2, Kaplow, L. (1996), "The optmal supply of publc goods and the dstortonary cost of taxaton", Natonal Tax Journal 48, Lee, D.R. and Msolek, W.S. (1986), "Substtutng polluton taxaton for general taxaton: Some mplcatons for e cency n polluton taxaton", Journal of Envronmental Economcs and Management, 13, Metcalf, G.E. (2003): Envronmental Leves and Dstortonary Taxaton: Pgou, Taxaton and Polluton, Journal of Publc Economcs 87, Parry, I.W.H. (1995), "Polluton taxes and revenue recyclng", Journal of Envronmental Economcs and Management, 29 (3): S64-S77 Part 2 Suppl. S. Parry, I.W.H. (1997), "Envronmental Taxes and Quotas n the Presence of Dstortng Taxes n Factor Markets", Resource and Energy Economcs 19(3): Pearce, D. (1991), "The role of carbon taxes n adjustng to global warmng", Economc Journal 101, Pgou, A.C. (1947), A Study n Publc Fnance, 3rd edton, Maxmllan, London. Prttla, J. and M. Tuomala (1997), Income tax, commodty tax and envronmental polcy, Internatonal Tax and Publc Fnance 4, Sandmo, A. (1975), "Optmal taxaton n the presence of externaltes", Swedsh Journal of Economcs 77,

26 Stgltz, J.E. and Dasgupta, P.S. (1971), "D erental taxaton, publc goods and economc e cency", Revew of Economc Studes 38, Stgltz, J.E. (1988), Economcs of the Publc Sector, New York: Norton. Terkla, D. (1984), "The e cency value of e uent tax revenues", Journal of Envronmental Economcs and Management, 11, Wlson, J.D. (1991), "Optmal publc good provson wth lmted lumpsum taxaton", Amercan Economc Revew 81,