Pigouvian Taxation in a Ramsey World

Transcription

1 gouvan Taxaon n a Ramsey orld Robn Boaday a* and Jean-Franços Tremblay b a Queen s Unversy anada b Unversy of Oaa anada Absrac Ths paper sudes he opmal gouvan ax for correcng polluon hen he governmen also uses dsoronary axes o rase revenues. hen preferences are uaslnear n lesure and addve he gouvan ax can be separaed from he Ramsey revenue-rasng ax. e characerze he relaonshp beeen he gouvan ax and margnal socal damages n a varey of crcumsances. n a seng h homogeneous households he gouvan ax exceeds margnal damages f goods have nelasc demands and vce versa. hen households are heerogeneous so axes can be redsrbuve he gouvan ax gves more egh o damages suffered by loncome persons. The analyss s exended o allo for cosly abaemen. n general correcve axes have o be appled o boh emssons and oupu of he pollung good. Keyords: gouvan Tax opmal axes polluon ax JL lassfcaon: H H3. nroducon Ths paper explores he ssue of he opmal ax for correcng exernales from polluon hen he governmen s rasng revenues usng dsoronary axes. n parcular ll he gouvan componen of he ax on a pollung good eual he margnal damages o households (as does n a frs-bes orld) and ho ll he gouvan ax change as revenue reuremens rse? The general analyss of opmal axes n he presence of exernales has been avalable n he leraure a leas snce Sandmo s (975) semnal paper. remer e al (998) and remer and Gahvar (00) have refned he analyss o ake accoun of nonlnear ncome axaon and he exsence of an abaemen echnology. Hoever her resuls are someha formal and he precse applcably of hem o envronmenal axaon remans unclear. arly hs s because opmal revenue-rasng axes and gouvan axes nerac n a complcaed ay n opmal ax characerzaons snce closed-form soluons are generally no possble: formulas for opmal ax raes are generally only n mplc erms. Ths lack of clary * orrespondng auhor: Robn Boaday eparmen of conomcs Queen s Unversy Kngson O K7L 36 anada. mal: boadayr@econ.ueensu.ca Fax number:

2 84 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) has been sressed n he recen leraure on he double dvdend from envronmenal axes hch has cas doub on heher here s n fac a full double dvdend: one dvdend from correcng exernales he oher from he revenue rased. Bovenberg and de Mooj (994) for example sugges ha envronmenal axes mgh exacerbae he ax dsorons ha already exs. Ths ould sugges ha he envronmenal (gouvan) ax should be less han margnal damages o households. Ohers such as Jaeger (00) have argued he oppose. ar of he problem as noed by remer e al (00) les h denfyng precsely ha s he gouvan componen of he ax on he pollung good. n an opmal ax orld he ax on a pollung good ll comprse boh a Ramsey (revenue-rasng) effec and a gouvan (correcve) effec and dsenanglng hem ll generally no be possble (snce he mposon of a correcve ax ll affec opmal revenue-rasng axes and vce versa). Moreover as Auerbach and Hnes (00) have sressed n her recen survey of opmal axaon he nerpreaon of gouvan commody axes becomes confounded dependng on heher commody axes or ncome axes are used for revenue rasng. n parcular hs affecs he margnal uly of ncome for he consumer and herefore he value of he numerare for measurng margnal damages. To address hese ssues e adop a smple formulaon for he household uly funcon ha allos for a naural separaon beeen Ramsey and gouvan axes smlar n spr o ha used by remer e al (00). They characerze he opmal ax on a pollung good n a Ramsey seng h many non-pollung goods and sho ha h separable preferences all nonpollung goods are axed a he same rae. As a resul hey defne he correcve ax as he ax dfferenal beeen pollung and nonpollung goods. remer e al (00) also examne he mplcaons for he opmal ax on a pollung good of dfferen preference separably assumpons hen agens are heerogeneous and an opmal nonlnear ncome ax sysem s used. n order o derve explc soluons o opmal ax raes e assume he uly funcon o be uaslnear n lesure so ha he demand for goods depends only on on prces relave o he age rae and no on eher ncome or oher goods prces. e explore he sze of he gouvan ax relave o he sze of margnal damages o households n a varey of sengs. These nclude he basc case here only commody axes are used he case here a age ncome ax s used he case here here s an abaemen echnology and he case here households have dfferen age raes and an opmal ncome ax eher lnear or nonlnear s used.. The Basc Ramsey Opmal ommody Tax Model The model e use for our benchmark analyss s he smples and mos ransparen one for our purposes. consss of a populaon of dencal households h uaslnear preferences n labor/lesure from hom he governmen mus exrac a gven amoun of ax revenues a he leas cos usng dsorng commody axes. dffers from he sandard opmal commody ax model n ha he consumpon of one good ems some harmful envronmenal exernaly. The choce of uaslnear preferences mples ha goods demands depend only on on prces and no on he prces of oher goods or

3 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) ncome effecvely leadng o a uas-paral eulbrum seng. Ths formulaon s chosen parly because he dsaggregaon of axes no Ramsey (revenue-rasng) and gouvan (correcve) componens can be made as clearly as possble and also because closed-form soluons for opmal axes can be derved. More formally he economy consss of households each of hose uly funcon akes he uaslnear form U () U () L here U () and U () are ncreasng and srcly concave funcons. Usng mnemonc noaon good s a clean good good s a dry good s he ualy of he envronmen and L s labor suppled. One can hnk of labor supply as beng L T H here T s me avalable and H s lesure. The ualy of he envronmen s gven by here s s ualy n he absence of polluon and s he margnal damage o he envronmen per un of consumpon of he dry good h assumed for smplcy o be consan. Thus each person s consumpon of affecs he ualy of he envronmen enjoyed by all households. e suppose ha s large enough ha each person reas he ualy of he envronmen as gven and hus ndependen of her on consumpon of. roducon s lnear n hs economy and e normalze he producer prces of he o goods o uny. The age rae s aken o be alhough oo could be normalzed as desred. s useful for exposonal purposes no o se o uny. The governmen reures an amoun of resources valued a R so he economy s aggregae producon consran s: R L () Follong he Ramsey opmal ax approach (e.g. Aknson and Sglz 980) he governmen canno levy lump-sum axes on consumers so mus mpose axes on ransacons. n hs economy h hree goods and lesure or labor he governmen could use hree axes. Hoever s ell-knon ha one ax rae s alays redundan (e.g. a ax on labor s euvalen o proporonal axes on goods). n our basc model e assume ha he governmen dspenses h axes on labor and uses only axes on he o goods denoed and. onsumer prces are hen gven by and. The ax on akes accoun of he exernaly assocaed h s consumpon. Gven ha households have no nal endomens her ndvdual budge consran can be ren: ( ) ( ) L () ombnng he aggregae of () for all households h () e mmedaely oban he governmen revenue consran: R (3) Thus only o of he economy s hree consrans () () and (3) need o be aken accoun of explcly and he oher s mpled. More generally our analyss apples hen here are many clean goods as n remer e al (00). Snce lle nsgh s ganed by havng more han one clean good e aggregae hem all no one compose good.

4 86 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) To solve he governmen s opmal ax problem consder frs household behavor. Households maxmze uly akng as gven subjec o her budge consran (). The Lagrangan s: ( L) ( L ) U U L ( ) ( ) Ω The frs-order condons on and L are: U ' ( ) 0 ' ( ) 0 U 0 : From hese e oban he demands for and and he margnal uly of ncome here '(. ) < 0 and '(. ) < 0. The fac ha s consan s a conseuence of he uaslnear uly funcon and s useful for purposes of nerpreaon. oce ha because of he addve uly funcon no only do cross-prce and ncome effecs dsappear bu also goods demands are ndependen of envronmenal ualy. The ndrec uly funcon can be ren: V here by he envelope heorem e have: V.. V.. and V.. (.) (. V ) Gven he behavor of he households he governmen s problem s o choose ax raes o maxmze he sum of ndrec ules subjec o s revenue consran (3) and reang he ualy of he envronmen as endogenous. The governmen s Lagrangan s: ( V.. ) (.) (.) ( ) ( R) The frs-order condons on and usng he envelope condons (4) are: ' 0 (5) (4) ' ' 0 (6) Gven ha demands for and depend only on on relave prces hch depend on on ax raes (5) and (6) mplcly deermne ax raes as a funcon of he shado

5 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) prce of governmen revenue and belo. Frs noe ha (5) and (6) can be rearranged as follos usng. e shall explo hese relaonshps / : ' (7) ' here and are elasces of demand: ' ' < 0 < 0 (9) n he absence of envronmenal exernales 0 (7) and (8) are he famlar nverse elascy rules for opmal axaon: here he ax raes are expressed as ad valorem axes based on consumer prces. oe ha > hen he governmen s usng dsoronary axes. f he governmen had access o non-dsoronary axes so / he ax on good by (8) ould be he frs-bes gouvan ax /. Tha s ould be se eual o margnal damages measured n erms of household ncome. hen axes are dsoronary so > (8) corresponds o he opmal ax expresson obaned by remer e al (998) albe here n a much smpler conex h homogeneous households. (Ther analyss nvolves heerogeneous households and nonlnear ncome axaon as ell as commody axes.) The opmal ax on consss of o separae componens one seemng o nvolve a Ramsey componen and he oher a gouvan componen. Hoever he decomposon s ncomplee and ambguous for a couple of reasons. For one he oal ax rae hch ncludes boh he Ramsey and he gouvan componen also appears on he rgh-hand sde of (8) ncludng as an argumen of. As ell he second erm nvolvng he polluon exernaly s no exacly he margnal damages suffered by consumers snce s dscouned by he shado prce of funds o he governmen. n order o explore hs more carefully e pu slghly more srucure on he problem n he follong secon by assumng elasces of demand are consan. (8) 3. gouvan Taxes n he Basc Model: A Specal ase To faclae he nerpreaon of gouvan axes n he above Ramsey opmal commody ax model le us make he follong o specal assumpons. Assume ha he elasces of demand for he o goods are consan and assume ha hey are eual o one anoher. Thus here s a consan. These assumpons are

6 88 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) pedagogcally useful snce hey allo us o solve for he properes of correcve axes explcly. Moreover n hs conex he meanng of he Ramsey componen of he ax srucure s apparen. f he elasces of demand are eual he Ramsey componen of he commody ax srucure ould be unform. e can herefore nerpre he dfference n commody ax raes as he gouvan componen of he ax on he dry good denoed. Gven hese assumpons (8) becomes: Usng (7) h hs can be solved for : (0) ( ) Thus our assumpons have alloed us o oban an explc soluon for hch e can use o oban o relevan properes of he gouvan ax. Frs noe ha f (0) reduces o /. Tha s he gouvan ax s eual o margnal socal damages measured n erms of household uly. Furhermore dfferenang (0) h respec o e oban: ( ) ) > 0 () Ths mples ha ncreases as decreases mplyng ha > / for < and vce versa. e summarze hs n he follong proposon. roposon : n he basc model h elasces of demand consan for boh goods he gouvan componen of he ax on good ll be greaer (less) han margnal socal damages o he households f goods are nelasc (elasc) n demand. The nuon for hs seems o be as follos. mposng a gouvan ax on good reures devang from eual axes on boh goods hch nvolves an effcency cos. The opmal ax mus rade off hs effcency cos agans he benefs of decreasng envronmenal damages. Snce devang from eual axes on boh goods nvolves a greaer effcency cos hen demands are more elasc he gouvan ax ll be smaller n hs case. ex recall ha n hs case h uaslnear preferences he frs order condons (5) and (6) or euvalenly (7) and (8) mplcly deermne he ax raes and as a funcon of respecvely ha s he sze of revenue reuremens. n he specal case here he elasces of demand are consan and eual e can nvesgae ho he Our analyss also apples n he case here and are consan bu dfferen n sze bu he analyss s smpler hen hey are eual. h dfferen elasces he Ramsey axes ould be proporonal o he rao of elasces raher han beng dencal o one anoher.

7 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) gouvan componen of he ax on vares h revenue reuremens. To do so dfferenae (0) h respec o o oban: ( ) ( ) ) > < 0 as > < () Thus ll ncrease h revenue reuremens f he demand for goods s nelasc and decrease f demand s elasc. Ths leads o he follong proposon: roposon : n he basc model h elasces of demand consan for boh goods he gouvan componen of he ax on good ll dverge more from margnal socal damages (posvely n he case of nelasc demands negavely n he case of elasc demands) as revenue reuremens ncrease. The resuls of hs secon apply hen he governmen uses only commody axes. n he specal case suded here here demand elasces for boh goods are consan and eual he governmen could apply a proporonal ax on he consumer purchases of boh goods such as a unform VAT and accompany h a gouvan ax on he consumpon of good. The gouvan ax ould hen be greaer or less han margnal damages o he consumer accordng o heher he demands are nelasc or elasc. ex e nvesgae he case here he unform ax s appled o ncome (ages) raher han consumpon. 4. gouvan Taxes ombned h a age Tax e connue o assume ha preferences are uaslnear and ha demand elasces are eual and consan. Hoever nsead of mposng axes on he o goods e use a ax on labor ncome along h a ax on good. Le be he ax rae on age ncome and be he ax on he dry good hch n hs case e can hnk of as he gouvan ax. Then he household budge consran s ( ) ( )L and he Lagrangan funcon for he household problem becomes: ( L ) U U L ( ( ) ( ) L) Ω The frs-order condons on and L are no: U ' 0 U ' ( ) 0 ( ) 0 hch yeld he demands for and (and mplcly he supply of labor) and he margnal uly of ncome : ( ) ( ) ( )

8 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) The ndrec uly funcon s: V and he envelope heorem gves: Turnng o he governmen s problem e frs oban an expresson for he governmen s budge consran. To do so e can use he resource consran for he economy hch as above s L R. ombnng hs h he aggregae budge consran of he consumers e oban R L. From he household budge consran labor ncome L s gven by: L nserng hs no he governmen budge consran e oban: () () R.. (3) Gven he governmen s budge consran (3) he Lagrangan funcon for he governmen s opmal ax problem can be ren: () () () ().... R V The frs-order condons on and p are: L ' ' 0 ' 0 ' ' Usng he second euaon o elmnae erms from he frs euaon he value of L from he household budge hese o euaons smplfy o: 0 ' 0 ' ' (4) () () V and () () L V L.... (). ().

9 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) Analogously o he basc model e can use (4) o ge expressons for he gouvan ax p. Frs noe ha usng he relave prces n hs case he common elascy of demand η can be ren as follos: ' ' ( ) ( ) Usng hese expressons and he fac ha / ( ) reren: ( ) ( ) 0 he euaons n (4) can be ( ) 0 (5) ombnng hese o expressons and solvng for p e oban: ( ) ( ) ( ) hch s he analog of (0) n hs case. roceedng as before e see ha hen / so he gouvan ax euals margnal socal damages o he household. Moreover dfferenang (6) by η e fnd ha: ( ( ) ) ( ) ( ) ( ) > 0 Ths s he analog of () so roposon apples. Smlarly dfferenang (6) h respec o λ e oban: ( )( ) ( ) ( ) > < 0 as > < Ths s euvalen o () mplyng ha roposon also apples. o surprsngly our resuls are ndependen of heher he governmen uses a se of dfferenal commody axes or a age ax combned h a ax on good. 3 (6) 5. gouvan Taxes h olluon Abaemen So far e have assumed ha he exernaly s proporonal o he oupu of he dry good. Le us no suppose ha polluon emssons arsng from he use of good can be reduced by an abaemen echnology as n remer and Gahvar (00). olluon abaemen s modeled n he smples ay conssen h makng he pon. Le A be 3 The analyss ould be slghly more complcaed f he elasces of demand ere dfferen for he o goods snce hen he Ramsey axes ould no be eual o one anoher.

10 9 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) oal abaemen such ha A are oal emssons of polluon by he dry good ndusry. The ualy of he envronmen hen becomes ( A) follong he above noaon. The oal cos of abaemen borne by producers s gven by Z(A) here Z '( A) Z'' ( A) > 0. e rever o he commody ax model of secon 3 bu n addon o commody axes c and mposed on he sales of goods and here s also a per un ax mposed on emssons by producers. Gven he level of abaemen A he ax apples on ncremenal producon of oupu. Assumng A < he producer prce of s herefore. roducer coss nclude npus no he producon of hch gven un coss are smply as ell as axes on emssons and he cos of abaemen. Gven he producer prce profs of he producers n ndusry are herefore: ( ) ( A) Z ( A) A Z( A) (7) roducers ll choose A o maxmze her profs hch leads o he frs-order condon Z '( A) hose soluon s A ( ) h A '( ) > 0. e assume for smplcy ha he governmen axes hese profs fully. onsumer prces for and are and and as before he age rae s fxed a. The consumer budge consran s ( ) ( ) L. onsumers maxmze uly gven subjec o her budge consran. The Lagrangan s: ( L ) U U L ( ) ( ) Ω ( L) roceedng as earler he soluon gves he demand funcons and ndrec uly funcon: h and V /. Applyng he envelope heorem yelds: V and V V (8) The resource consran for he economy s Z ( A) R L here sasfes (7). Togeher h he aggregae household budge consran hs leads o he governmen revenue consran ( A) R. Gven hs he Lagrangan expresson for he problem of he governmen usng (7) s: A( )) ( ( ) Z( A( )) R) ( V.... The frs-order condons for and decson of producers are: usng Z '( A) from he abaemen ' 0 (9)

11 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) ' ' ( ) A' ( ) ' 0 ( ) ' A '( ) 0 Assume ha he elasces of demand for and are dencal and consan so ' Then usng ' / e can rere condons (9) and (0) respecvely as: 93 (0) () () (3) hch are analogous o (7) and (8) n our basc case. n hs case he oal ax on good s and follong he earler logc he gouvan ax can be defned as. By (3) e have: hch reduces usng () o: (4) ( ) Ths gouvan componen of he oal ax on good s he same as n he basc case hou abaemen gven by (0). The same analyss as before leads o he analogs of roposons and n hs case gven he defnon of he gouvan ax used here. n hs case he gouvan componen s more complcaed han earler snce no ncludes he ax on emssons as ell as he dfference n commody axes. e can use he frs-order condon () on o gve furher nsgh. Subsung (0) no () e oban mmedaely: (5) Ths componen of he gouvan ax s loer han margnal damages / >. The oher componen s gven by combnng (4) and (5) o yeld: snce

12 94 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) Thus > f < 0 (snce he numeraor s negave). n fac hs condon s sasfed as can be seen from () hch can be ren: Therefore assumng > 0 hs mples ha < hch n urn mples ha >. Ths means ha he gouvan ax ncorporaes no jus a ax on emssons bu also a dfferenal ax on good. oe furher ha f he governmen could levy a non-dsorng lump-sum ax hen so here ould be no dfferenal ax on ( ) and he ax on emssons ould eual margnal damages o he consumers / /. e can summarze hese resuls n he follong proposon. roposon 3: hen producers can reduce emssons of polluon by cosly abaemen. he gouvan componen of he second-bes commody ax sysem ncludes boh a ax on emssons and a dfferenal ax on sales of he dry good. he second-bes ax on emssons s less han s frs-bes level hle he dfferenal ax on exceeds s frs-bes level hch s zero and 3. he compose gouvan ax sasfes roposons and. A fnal apparen mplcaon of hs analyss s ha f he governmen uses a age ax raher han commody axes on and can only acheve he second-bes opmum by mposng a o-componen gouvan correcve ax. One componen s he ax on emssons hle he oher s a ax on sales of good. 6. gouvan Taxes and Lnear rogressve Taxaon Suppose no ha households are heerogeneous. Follong he opmal ncome ax leraure e assume ha hey dffer only n her age raes bu oherse he model used above apples. suffces o resrc our aenon o o age-ypes and h > here and are he numbers of he o ypes. The governmen s assumed o be able o observe ncome L bu neher he age rae nor labor supply L. e begn h he case here he governmen uses a lnear progressve ncome ax as ell as a gouvan ax on he pollung good. The ncome ax consss of a consan margnal ax rae combned h an eual per capa subsdy s. The budge consran for a household h age rae becomes ( ) ( ) L s. The household maxmzes uly subjec o hs budge consran and he soluon gves analogous demand funcons and margnal uly of ncome o he age ax case: ( ) ( ) ( )

13 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) The ndrec uly funcon for he ype- household s no: V s ( ) ( ) here s no (. The envelope heorem gves: V ( ) ( ) ( ( ) V L (.) ( ). L. V s Usng he aggregae household budge consran and he economy s resource consran as before he governmen revenue consran can be ren: ( ) ( ) R s The governmen problem s reaed as a areo maxmzng one and e can ake he objecve funcon o be ( V (. ) ) ( V (. ) ). The socal eghs are arbrary alhough s useful o suppose ha hey are chosen such ha he governmen ans o redsrbue from he hgh-age o he lo-age ypes. Tha mples ha he margnal socal uly of ncome of he laer exceeds he margnal socal uly of ncome of he former or > n he opmum. f he governmen could use lumpsum axaon ould euae hese margnal socal ules n a frs-bes opmum. The laer serves as a useful benchmark belo. The governmen maxmzes s objecve funcon subjec o s revenue consran. Usng he frs-order condon on s he frs-order condons h respec o and reduce o he follong afer some smplfcaon: ' ' 0 (6) ' ' ' ' 0 (7) here. These are he analogs of (4) above n he age ax case. Le us agan assume a consan elascy of demand so: ' ' ( ) ( ) ( ) ( ) ' Usng hese defnons (6) and (7) can be ren: ( ) ( ) 0 ' ( ) 0 (8)

14 96 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) here: (9) These are eghed averages of he margnal socal ules of ncome of he o age ypes eghed by her shares of consumpon of he o goods. 4 Gven our assumpon abou uly funcons preferences are homohec n and and separable from lesure hch mples ha / / so ha. Then he euaons n (8) are he same as hose n (5) n he homogeneous-consumer case hen replaces and replaces so he same dervaon yelds he analog of (6): ( ) ( ) ( ) The analogs of roposons and hen apply. n parcular hen / hle s ncreasng n. As ell / > < 0 as > <. To nerpre hs consder he frs-bes oucome h lump-sum redsrbuve axes and a gouvan ax. The lump-sum axes are used o euae margnal socal ules of ncome o he shado prce of governmen revenue hle he gouvan ax s se eual o he sum of margnal damages o he households: (3) h lnear progressve axes margnal socal ules of ncome canno be euaed. n hese crcumsances e can use / obaned hen as a benchmark and nerpre by rerng n he follong ay usng : / / Usng (9) for and recallng ha > e have: (30) < > The mplcaon s ha compared h he frs bes he gouvan ax pus more egh on margnal damages o he lo-age persons han o he hgh-age persons hus akng on some redsbuve role o complemen he lnear ncome ax sysem. Ths s remnscen of he resul of Sandmo (006) ha n he absence of full nernaonal redsrbuve ransfers lo-ncome counres should have loer polluon axes han hgh-ncome counres. e can summarze hese resuls n he follong proposon hch s relaed o roposons and. 4 The expressons and are analogous o he dsrbuve eghs for opmal commody axes (or publc secor prces) defned by Feldsen (97).

15 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) roposon 4: Suppose households dffer n age raes bu have he same uaslnear-n-lesure preferences and he governmen uses a lnear progressve ncome ax.. hen he elascy of demand for goods s uny / / hch dffers from socal margnal damages by pung relavely more egh on damages o lo-age persons and less egh on damages o hgh-age persons.. The gouvan ax falls h he absolue value of he elascy of demand. 3. The gouvan ax ncreases h revenue reuremens f < and vce versa. 7. gouvan Taxes and a onlnear ncome Tax Suppose no ha he governmen leves a nonlnear ax on ncome. Le he beforeax ncome of a household h age rae be L. Gven ha he governmen can observe s useful o ransform he uly funcon no one nvolvng ncome raher han labor supply as follos: V ( ) U U / (3) here V s ype-specfc. hle he governmen can observe ncome canno observe ndvdual purchases of and. can herefore levy a non-lnear ncome ax on bu only mpersonal ndrec axes on commody purchases. From an analycal perspecve suffces o consder only a ax on : proporonae axes on and can be subsumed no he ncome ax funcon so e can normalze one commody ax rae o be zero. Le be he per un ax rae on commody so he consumer prce s. Suppose ha T ( ) s a nonlnear ax funcon appled o pre-ax ncome. e can hen defne as dsposable ncome (oal consumpon expendures) here T ( ). Alhough he governmen canno observe ho s dvded beeen and does kno he consumer uly funcon. f kne a person s ype could nfer a household s labor supply from s ncome and ould herefore kno ho each household spends s ncome. Hoever households of one ype can mmc one anoher s ( ) bundle hch precludes he governmen from nferrng L and consrans governmen polcy. All hs s ell knon from opmal ncome ax heory (e.g. Sglz 987). As above e frs characerze household behavor and hen urn o he governmen. A household of ype maxmzes V ( ) subjec o a budge consran T ( ). urns ou o be useful for he purposes of analyss o dsaggregae household behavor no o successve sages. n he frs sage hey choose her labor supply and herefore he combnaon of and. n he second sage hey allocae beeen and. The analyss of household behavor s consdered n reverse order. n sage and have been deermned and are aken as gven. From he household s budge consran. Therefore he problem of a represenave household of a gven ype can be ren (h superscrps suppressed): max { } V ( ).

16 98 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) The frs-order condon s V V 0 hch yelds he demand for h he follong properes (here s separable so does no affec demand): ( ): < 0 > 0 0 The sgn of / depends on he subsue/complemen relaons beeen and L. Snce preferences are separable labor supply and hus ncome do no affec preferences for and. So gven a change n ll no affec he demand for. Therefore e can re he demand for as smply ( ). The maxmum value funcon for he consumer he ndrec uly funcon s. Applyng he envelope heorem o hs problem e oban: denoed V U ' V U ' V (33) For a gven value of consumer ndfference curves n and can be dran. They have a slope of / / ( ' U ) and sasfy he sngle crossng propery: / > /. As menoned households of a gven ype may mmc he consumpon-ncome () bundles of he oher ype. f e assume ha he governmen ll redsrbue from he hgh- o he lo-age ypes sarng n he lassez fare he case of neres for us s here he hgh-age ypes are emped o mmc he lo-age ypes. Le V ˆ ( ) denoe he uly of he ype s hen hey are mmckng he consumpon-ncome bundle of he ype s. The problem for he hgh-age mmckers n sage s hen: max {} Vˆ ( ). Ths yelds he demand funcon ˆ ( hch has he same properes as for he non-mmckers. The ndrec uly funcon for he mmckers s denoed ˆ ( ). The relaonshp beeen and ˆ s mporan n ha follos. Boh ype s and mmckng ype s ll have he same consumpon expendures and ncomes hle labor supply s loer for he mmcker (snce he same ncome can be earned h loer L). Therefore snce consumpon goods are separable from lesure n he uly funcon ˆ. f had been more complemenary han h lesure < ˆ and vce versa. Turn no o sage. n hs sage households choose her labor supply or euvalenly her ncome. n effec hey choose he mos preferred bundle () from he budge consran T ( ). Gven hs behavor he governmen chooses s ax polces boh T() and. Follong he opmal ncome ax approach nsead of analyzng he governmen choce of T() e use he drec approach and le he governmen offer he bundles ( ) ( ) for he o household ypes. Then households faced h he bundles offered choose her mos preferred. As before e characerze a areo opmal ax polcy focusng on hose areo opmal oucomes for hch he ncenve consran on he ype s s bndng. (Ths ll be conssen h a socal elfare funcon ha exhbs averson o neualy.) The opmal polcy problem for he case here he ncenve consran apples only o he hgh-ably ypes s as follos here h no loss of generaly: (

17 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) { } Max ) ( ) ( subjec o: ˆ R ) ( ) ( here recall and ) ( ) ( hle and refer o he Lagrange mulplers n he respecve consrans. The frs-order condons on and are: 0 ˆ (34) ˆ 0 (35) 0 (36) 0 (37) ˆ 0 (38) here as before. These condons gve he srucure of he nonlnear ncome ax sysem as ell as he opmal ax on. Gven he assumpons abou he form of he uly funcon he so-called Aknson and Sglz (976) Theorem apples. Tha s n he absence of envronmenal exernales here ould be no dfferenal ax on goods. Tha mples ha e can nerpre as he gouvan ax. To deermne he value of combne (34) (36) and (38) usng from (33) and ˆ : 0 Snce he erms n he large brackes are non-zero hs mples ha he gouvan ax sasfes: (39)

18 00 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) n erms of he margnal value of governmen revenue. e reurn o he nerpreaon of hs belo. Frs consder he margnal ax rae on he hgh-age households. ondon (36) can be ren: n ords he gouvan ax s he margnal damages ( ) ( ) 0 Usng (39) hs becomes ( ) 0. ombnng hs h (37) e oban / mplyng ha he margnal ax rae a he op s zero. Ths famlar condon hus connues o apply hen polluon axes are se opmally. Smlar reasonng shos ha he margnal ax rae on he lo-age ypes has he sandard form: s posve bu less han 00 percen. onsder no he nerpreaon of he gouvan ax gven by (39). As a benchmark s useful o recall he frs-bes case here axes are non-dsoronary. n hs conex he ncenve consran s no bndng so 0. Then from he frs-order condons (35) and (37) and usng he envelope condons (33) e oban: V V Thus he governmen euaes socal uly of ncome for boh ypes. ondon (39) can herefore be ren: (40) The opmal gouvan ax euals he sum of margnal damages measured n erms of he ncome of each consumer. hen he ncome ax s dsorng so he ncenve consran bnds he governmen can no longer euae margnal socal ules of ncome because 0. ombnng (35) and (37) no yelds: ( ˆ ) ( ) By (33) ˆ / and /. Therefore e oban: / / Thus he shado value of publc funds s a eghed average of margnal socal ules of ncome. To compare he second-bes gouvan ax h he frs bes rere (36) as follos: / /

19 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) Usng he above expresson for and nong ha n he second bes margnal socal uly of ncome ll be hgher for he lo-age ypes > e oban: < > The mplcaon s ha compared h he frs bes he gouvan ax pus more egh on margnal damages o he lo-age persons han o he hgh-age persons hus akng on some redsrbuve role o complemen he nonlnear ncome ax sysem. Ths mgh be conrased h he resuls of he basc model hereby he gouvan ax devaes sysemacally from margnal damages dependng on he elascy of demand. h more general preferences he ax on good ould devae from zero accordng o he subsue-complemen relaonshp beeen he demand for and lesure as analyzed by dards e al (994) and ava e al (996). n hs case he formula for ould nclude no jus a gouvan componen bu also a componen reflecng he role of he ndrec ax sysem as a par of he governmen s redsrbuon polcy as n remer e al (998). The ssue of dsaggregang he gouvan ax componen and he redsrbuve componen ould rase ssues smlar o hose n earler secons. Fnally suppose e nroduce he possbly of cosly abaemen no he nonlnear ax model usng he same abaemen echnology as n secon 5. As before he producer prce n ndusry s and aggregae profs are A Z ( A) hch are axed aay by he governmen. The consumer prce for s. The oucome of sage of he consumers uly maxmzaon s exacly as before yeldng consumer demands ( ) for demand ˆ ( ) for he mmcker ndrec ules ( ) and ˆ ( ) and he envelope resuls (33). The governmen polcy problem s he same as before h replacng and profs added o he governmen budge consran. The Lagrangan expresson becomes: ( ) ( ) ( ( ) ( )) A( ) ( ˆ ( )) ( ( ) ( ( ) ( )) Z( A( )) R) The frs-order condons on and are he same as (34) (38) h replacng here. The frs-order condon on s: ( ) ˆ A' ( ) (4) ( ) ( ) Z '( A) A' ( ) 0

20 0 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) Subsung he frs-order condon for (he analog of (38)) no (4) and usng Z' A e oban: (4) ex proceed exacly as n he prevous secon by combnng he frs-order condons on and o oban he analog of (39): ( ) (43) Then snce / by (4) e oban 0. Thus unlke n he lnear ax case he gouvan ax consss only of a ax on emssons: no ax on s reured. Moreover he nerpreaon of he gouvan ax s he same as above. More egh s pu on margnal damages o he lo-age han he hgh-age persons. e can summarze hese resuls of gouvan axaon n he presence of nonlnear ncome axaon as follos. roposon 5: hen households dffer n age raes bu have he same uaslnearn-lesure preferences and he governmen can use a nonlnear ncome ax:. he gouvan componen of he second-bes commody ax sysem s a ax on good hch dffers from socal margnal damages by pung relavely more egh on damages o lo-age persons and less egh on damages o hgh-age persons. he srucure of he opmal nonlnear ncome ax s he same as n he absence of polluon and 3. hen a cosly abaemen echnology s avalable he gouvan ax apples only o emssons and no o he oupu of he dry good and dffers from socal margnal damages n he same ay as. above. 8. oncludng Remarks Our purpose n hs paper has been o sudy ho gouvan axes dffer from margnal damages o households hen he axes are leved alongsde dsoronary axes. Because axes used for correcve purposes and axes used for revenue-rasng hch e refer o as Ramsey axes are nerdependen s generally no possble o dsenangle one from he oher. e have specfed a model n hch gouvan axes can readly be dsaggregaed from Ramsey axes and used ha o deermne ho he former devaes from margnal socal damages and ho ha devaon vares h he amoun of revenue ha mus be rased. The formulaon e use nvolves household preferences ha are uaslnear n lesure so ha demands for goods depend only on on prces. n hs seng ualavely clear resuls emerge. hen he governmen reles only on commody axes he gouvan componen of he ax s he devaon from unformy of commody axes on he clean and dry goods. The gouvan componen ll exceed margnal damages hen goods demands are nelasc and vce versa. o surprsngly he same resul apples hen nsead of

21 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) commody axes a labor ncome ax s used along h a ax on he dry good. hen an abaemen echnology s nroduced no he basc seng he gouvan componen of axaon follos he same paern as n he basc model. Hoever hle damage s no a resul of emssons he gouvan componen ncludes boh a ax on emssons and a ax on he sales of he dry good even hough n a frs-bes orld only he former ould be used. Fnally hen households are heerogeneous and he ax sysem s used no jus for revenue-rasng bu also for redsrbuon maers change. n he case here he governmen uses all he nsrumens a s dsposal and leves a nonlnear ncome ax here s no longer a presumpon ha he gouvan ax ll be eher hgher or loer han margnal socal damages. nsead he ax s se eual o a sum of damages o he lo- and hgh-age persons n he economy bu h relavely more egh pu on margnal damages o he former and less o he laer compared h socal damages. The addon of abaemen possbles changes lle n hs case. Unlke h lnear axaon only a ax on emssons should be used and no a ax on oal use of he dry good. Our analyss reles on some mporan smplfyng assumpons parcularly he uaslneary of preferences. As menoned hese preferences ere used o faclae he clean separaon of correcve from revenue-rasng componens of axaon. n a more complcaed seng analycal soluons ould generally be hard o come by so one may have o resor o compuaonal mehods. References Aknson A.B. and J.. Sglz 976 The esgn of Tax Srucure: rec vs. ndrec Taxaon Journal of ublc conomcs Aknson A.B. and J.. Sglz 980 Lecures on ublc conomcs (e ork: McGra-Hll). Auerbach A.J. and J.R. Hnes Jr. 00 Taxaon and conomc ffcency n: A.J. Auerbach and M.S. Feldsen eds. Handbook of ublc conomcs Vol. 3 (Amserdam: lsever) haper. Bovenberg A.L. and R.A. de Mooj 994 nvronmenal Leves and soronary Taxaon Amercan conomc Reve remer H. and F. Gahvar 00 Second-Bes Taxaon of mssons and ollung Goods Journal of ublc conomcs remer H. F. Gahvar and. Ladoux 998 xernales and Opmal Taxaon Journal of ublc conomcs remer H. F. Gahvar and. Ladoux 00 Second-Bes olluon Taxes and he Srucure of references Souhern conomc Journal dards J. M. Keen and M. Tuomala 994 ncome Tax ommody Taxes and ublc Good rovson: A Bref Gude Fnanzarchv Feldsen M.S. 97 srbuonal uy and he Opmal Srucure of ublc rces Amercan conomc Reve Jaeger.K. 00 arbon Taxaon hen lmae Affecs roducvy Land conomcs

22 04 Robn Boaday and Jean-Franços Tremblay Asa-acfc Journal of Accounng & conomcs 5 (008) ava M. F. Schroyen and M. Marchand 996 Opmal Fscal and ublc xpendure olcy n a To-lass conomy Journal of ublc conomcs Sandmo A. 975 Opmal Taxaon n he resence of xernales Sedsh Journal of conomcs Sandmo A. 006 Global ublc conomcs: ublc Goods and xernales oregan School of conomcs and Busness Admnsraon mmeo. hp://nhh.no/sam/res-publ/006/3.pdf Sglz J The Theory of areo-ffcen and Opmal Redsrbuve Taxaon n: A.J. Auerbach and M.S. Feldsen eds. Handbook of ublc conomcs Vol. (Amserdam: lsever) haper 5.