When redistribution makes personalized pricing of externalities useless

Fleu baey, Ma c; Ko nek, Ul ike A icle — Published Ve sion When edis ibu ion makes pe sonalized p icing o ex e nali ies useless Jou nal o Public Economic Theo y P o ided in Coope a ion wi h: John Wiley & Sons Sugges ed Ci a ion: Fleu baey, Ma c; Ko nek, Ul ike (2021) : When edis ibu ion makes pe sonalized p icing o ex e nali ies useless, Jou nal o Public Economic Theo y, ISSN 1467-9779, Wiley, Hoboken, NJ, Vol. 23, Iss. 2, pp. 363-375, h ps://doi.o g/10.1111/jpe .12505 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/241857 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h p://c ea i ecommons.o g/licenses/by/4.0/ Recei ed: 17 Feb ua y 2020 | Re ised: 15 Sep embe 2020 | Accep ed: 12 Janua y 2021 DOI: 10.1111/jpe .12505 ORIGINAL ARTICLE When edis ibu ion makes pe sonalized p icing o ex e nali ies useless Ma c Fleu baey 1 |Ul ike Ko nek 2,3,4 1 CNRS and Pa is School o Economics, Pa is, F ance 2 Me ca o Resea ch Ins i u e on Global Commons and Clima e Change, Be lin, Ge many 3 Depa men o Economics, Ch is ian Alb ech s Uni e si y in Kiel, Kiel, Ge many 4 Resea ch Depa men III (T ans o ma ion Pa hways), Po sdam Ins i u e o Clima e Impac Resea ch, Po sdam, Ge many Co espondence Ul ike Ko nek, Depa men o Economics, Ch is ian Alb ech s Uni e si y in Kiel, Wilhelm‐Seelig‐Pla z 1, 24118 Kiel, Ge many. Email: [email p o ec ed], [email p o ec ed] and [email p o ec ed] Funding in o ma ion Ho izon 2020 F amewo k P og amme, G an /Awa d Numbe : 776608; Bundesminis e ium ü Bildung und Fo schung, G an /Awa d Numbe : 01LS1904A‐B Abs ac We conside a s anda d op imal axa ion amewo k in which consume s' p e e ences a e sepa able in consump ion and labo and iden ical o e con- sump ion, bu a e a ec ed by consump ion ex- e nali ies. Fo e e y nonlinea , income‐dependen p icing o goods he e is a linea p icing scheme, combined wi h an adjus ed income ax schedule, ha lea es all consume s equally well‐o and weakly inc eases he go e nmen 's budge . The e- sul depends on whe he a linea p icing scheme exis s ha keeps he agg ega e amoun o con- sump ion a i s ini ial le el obse ed unde non- linea p icing. We p o ide su icien condi ions o he assump ion o hold. I adjus ing he income ax a e is no a ailable, pe sonalized p ices o an ex- e nali y can enhance social wel a e i hey a e e- dis ibu i e, ha is, a o consume s wi h a la ge ma ginal social alue o income. 1|INTRODUCTION In a amous esul , A kinson and S igli z (1976) showed ha , a he second‐bes op imal alloca ion, commodi y axa ion is supe luous when income axa ion is p esen . La oque (2005)ex ended his esul o subop imal alloca ions, showing ha i he income ax can be adjus ed, e e y alloca ion wi h commodi y axa ion is weakly domina ed by an alloca ion wi hou commodi y axa ion. J Public Econ Theo y. 2021;23:363–375. wileyonlinelib a y.com/jou nal/jpe | 363 This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. © 2021 The Au ho s. Jou nal o Public Economic Theo y Published by Wiley Pe iodicals LLC Recen wo ks modi y he A kinson and S igli z (1976) model o include consump ion ex- e nali ies. They show condi ions unde which op imal p ices o ex e nali ies ollow i s ‐bes ules and a e unique and linea , ha is, do no depend on he p e ax income o agen s o he quan i y o ex e nali y‐gene a ing goods ha is consumed (Gau hie & La oque, 2009; Jacobs & de Mooij, 2015; Kaplow, 2012). We ex end his li e a u e o subop imal p icing o ex e nali ies. Ou main inding is ha o e e y nonlinea , income‐dependen p icing scheme he e is a linea p icing scheme o he ex e nali ies, combined wi h an adjus ed income ax schedule, ha lea es all consume s equally well‐o and weakly inc eases he go e nmen 's budge . Pe sonalized p ices a e useless when dis ibu ional conce ns a e ully handled ia income axa ion. Ou main inding holds unde he ollowing assump ions. Common assump ions include (i) cons an e u ns o scale, (ii) sepa abili y o consume s' u ili y in consump ion and labo , and (iii) as e homogenei y in consump ion. Ex ending he model, we allow o mul iple ex e nali ies ha a ec p oduc i i y and he e o e p ices and wages, and o consume p e e ences o be he e ogeneous wi h espec o labo and he ex e nal e ec s. Ou esul holds ue i i is possible o ind a linea p icing scheme o he ex e nali ies so ha he agg ega e amoun o consump ion emains a i s le el unde he nonlinea , income‐dependen p icing scheme, while keeping con- sume s' consump ion subu ili ies a hei ini ial le el. Apa om ha , we only equi e duali y be ween u ili y maximiza ion and expendi u e minimiza ion. Fo he case o only one good causing an ex e nali y, we p o ide su icien condi ions o he exis ence o a linea p ice ha keeps he agg ega e amoun o consump ion, and consume s' subu ili ies, a he o iginal le el. The same condi ions ensu e duali y be ween u ili y maximiza ion and expendi u e minimiza ion. We ex end ou esul o he se ing o clima e change and he C O 2 ‐emission ex e nali y. In his case, he agg ega e consump ion o mul iple goods causes he same ex e nali y. We show ha a linea p ice, equal in all sec o s o he economy, is also weakly p e e able o a nonlinea , income‐dependen and sec o ‐speci ic p ice. Fu he mo e, we conside ex e nali ies a ising om he income o consume s, and show ha pe sonalized p ices a e again supe luous. Ou gene al se ing ex ends p e ious li e a u e in se e al dimensions. Gau hie and La oque (2009) and Jacobs and de Mooij (2015) analyze he op imal p ice o a single good causing an ex e nali y. We allow o mul iple ex e nali ies, whose le el may be op imal o nonop imal. Gau hie and La oque (2009) u he assume ha he nonlinea p ice depends on he income, bu no consump ion, o consume s; in Jacobs and de Mooij (2015) he nonlinea p icing is con inuous and di e en iable in indi idual consump ion o he good causing an ex e nali y. Ou se ing in- cludes any unc ional o m o he nonlinea p ice scheme in income and consump ion. Policies o command and con ol o making excep ions o ce ain g oups o consume s based on hei income o consump ion a e accoun ed o , which a e pe asi e in policy‐making. In addi ion, we le he ex e nali ies no only a ec u ili ies, bu also p oduce p ices and labo p oduc i i ies. Kaplow (2012) conside s ax e o ms in case o mul iple ex e nali ies, and p o ides su icien condi ions ensu ing ha mo ing o, o in he di ec ion o , i s ‐bes Pigou ian axa ion is Pa e o‐supe io . Kaplow es ic s commodi y p icing o be linea , whe eas we allow o nonlinea , income‐ dependen p icing. In addi ion, u ili y o consume s may no be di e en iable in consump ion o ex e nali ies in ou model, an assump ion ound in Kaplow (2012) and Jacobs and de Mooij (2015). Ou main esul shows a possibili y: gi en a nonlinea , income‐dependen p icing sys em o ex e nali ies, and gi en ou assump ions, a linea p icing sys em wi h an app op ia ely adjus ed income ax schedule pe o ms (weakly) be e . Redis ibu ion wa an s linea p icing o ex e nali ies. Res ibu ion he e means ha he income ax schedule can be adjus ed eely. Fo policy make s who wan o implemen , o example, an en i onmen al ax e o m i migh be in easible o 364 | FLEURBAEY AND KORNEK edis ibu e eely ia he income ax schedule. Di e en ial p icing could be supe io o add ess dis ibu ional conce ns, as ecen ly highligh ed by S igli z (2019). We hus las ly conside he case when income ax a es a e ixed, and show ha social wel a e can be imp o ed wi h a nonlinea p ice o an ex e nali y ha again keeps he ex e nali y a i s o iginal le el. Mo e p ecisely, we show ha such a nonlinea p ice a o s consume s wi h a highe ma ginal social alue o income. In e es ingly, his esul holds i espec i e o whe he he s a us quo income ax edis ibu es oo much o oo li le. E en i consume s' u ili ies a e oo close oge he compa ed o a second‐bes o hi d‐bes (e.g., linea income ax) alloca ion, a nonlinea p ice has o be edis ibu i e o enhance social wel a e. In case he social wel a e unc ion is u ili a ian and consume s' ma ginal u ili y o income dec eases, he nonlinea p ice impac has o a o he poo . Ou las esul echoes he impo an esul o Chichilnisky and Heal (1994), who showed ha ma ginal cos s o public good p o ision a e no necessa ily equalized a he op imum when edis ibu ion is es ic ed. Sheshinski (2004) also showed ha op imal axes o an ex e nali y should be highe o iche consume s i he income ax sys em does no su icien ly edis ibu e. We ex end hese esul s by conside ing subop imal commodi y axa ion and in oducing labo choices. S igli z (2019) also discusses cases in which he bene i s o alle ia ing dis ibu ional conce ns by imposing g ea e en i onmen al axes on ich consume s can ou weigh he e iciency cos s om di e en ia ed p icing, gi en ha he income ax sys em canno be adjus ed eely, albei wi hou in oking a ull op imal axa ion model. The es o he pape is s uc u ed as ollows. Sec ion 2in oduces he model and no a ions, ollowed by Sec ion 3in which he esul s a e s a ed and p o ed. Sec ion 4discusses he policy implica ions o ou esul s. 2|MODEL Conside an economy wi h a con inuum o consume s ∈ i [0, 1 ] ha de i e u ili y om he consump ion bundle ( ) xxx=,…, iii n1, and disu ili y om labo supply ℓ ℓUVx E E:((,),,) ii i i . The e a e N consump ion ex e nali ies E EE=( ,…, ) N1, wi h E EX=() jj , whe e X is he ec o o he agg ega e consump ion o hose goods ha cause he ex e nali ies: ∫ ∈⊆ {} Xx=…, ,… im mSS n,{1,…,} . 1 Disposable income is gi en by Ry( ) i, wi h he nonlinea ax schedule R only a unc ion o p e ax income yi . We assume cons an e u ns o scale so ha p ices pE( ) and wages w E( ) ido no depend on mic oeconomic ac i i y le els. Indi idual wage w E( ) iis p i a e in o ma ion and p e ax income is equal o ℓywE=() iii . P e ious li e a u e o en conside s ex e nali ies ha only a ec consump ion p e e ences V .To keep as much gene ali y as possible, we assume ha ex e nali ies may in luence consump ion p e e ences, he labo ‐consump ion choice, as well as labo and o al ac o p oduc i i y. Assump ion 1. Le u ili y V xE(, )be such ha he indi ec u ili y unc ion ∣≤νpRE VxE px R(,,)=max{(,) } x and he expendi u e unc ion 1 The no a ion ∫ x i k is a sho ‐hand o ∫ xdi ik 0 1 , and all unc ions indexed by i a e assumed o be Lebesgue in eg able in his pape . The esul s a e also alid o a ini e se o consume s, p o ided i is assumed ha in hei ma ke beha io , consume s neglec he in luence o hei consump ion on he ex e nali y. FLEURBAEY AND KORNEK | 365 ∣≥ep E pxVxE (,,)=min{ (,) } x sa is y he ollowing iden i y ≡νpep E E (, (, , ), ) . Assump ion 1 educes ou se ing o cases whe e he e is duali y be ween u ili y maximiza ion and expendi u e minimiza ion. Duali y holds, o example, i u ili y V xE(, )is con inuous and locally nonsa ia ed in ≫xp,0 and ≠ V E(0, ) (see, e.g., Mas‐Colell e al., 1995, p. 59). Fu he , de ine he Hicksian demand o commodi ies as: ∣≥hp E pxVxE (, , )=a gmin{ (, ) } . x The go e nmen aises unds h ough income axes and axes xy(,) iion goods ha cause ex e nal e ec s. The ax on an indi idual good may be posi i e o nega i e; o he la e he go e nmen subsidizes consump ion o his good. Go e nmen p ices gene ally depend on he consump ion o goods and income o consume s. The budge o he go e nmen is: ∫∑ ∈ GyRy xyx=−()+ (, ) . ii kS kiiik Indi idual i 's budge cons ain can hen ead as ≤pE x y x Ry ( ()+(, )) () iiii . Consume s maximize hei u ili y Uigi en hei budge cons ain . All a iables a he ini ial equilib ium wi h nonlinea p icing a e deno ed wi h a ha , o example, ℓx ˆ,ˆ ii , and E ˆ. Le U ˆi deno e consume i 's u ili y a his equilib ium, and V ˆ i he subu ili y om consump ion. 3|RESULTS 3.1 |Wi h adjus men o income ax schedule This sec ion i s shows ha he economy wi h nonlinea , income‐dependen p icing xy(,) ii and ini ial income ax scheme Ry() is weakly domina ed by an economy wi h linea p ices o goods, deno ed ¯ , and an adjus ed income ax scheme Ry ¯( ) . This esul holds ue unde Assump ion 1and he assump ion ha , a he linea p icing scheme ¯ , ex e nali ies emain a hei o iginal le el E ˆ. Based on his main inding, we p o ide a b ie ex ension showing ha a single, linea ax on C O 2 ‐emissions weakly domina es a sec o ‐speci ic, nonlinea and income‐ dependen ax. Las ly, we p o ide su icien condi ions o he wo assump ions unde lying ou main esul o hold i only one good causes an ex e nali y. The main esul o ou pape is summa ized in he nex p oposi ion. P oposi ion 1. Take a linea p icing scheme ¯ le ied on goods causing ex e nal e ec s (i.e., ∀∉ mS ¯=0 m). Assume ha wi h his linea p icing scheme and a ini ial subu ili ies V ˆ i , o al Hicksian demand o goods wi h ex e nal e ec s emain a hei ini ial le el X ˆ k o all ∈kS . The e is an alloca ion wi h an al e na i e income ax schedule Ry ¯( ) and ex e nali y p icing a ¯ in which each consume keeps ini ial u ili y U ˆiand he go e nmen budge is weakly g ea e han a he ini ial le el. P oo . Since u ili y is sepa able in consump ion and labo , each consume 's op imal consump ion bundle only depends on he income. The ollowing unc ion yE ˆ(,ˆ)is he u ili y de i ed om consump ion when be o e ax income is y : 366 | FLEURBAEY AND KORNEK ∣≤ yE VxE pE xy x Ry ˆ(,ˆ)=max{ ( , ˆ)( ( ˆ)+ ( , )) ( )} . x (1) Wi h yE ˆ(,ˆ)gi en, each consume chooses labo by maximizing ℓℓU wE E E(ˆ(( ˆ), ˆ), , ˆ) iii i . Now, ix he p icing scheme a he linea ¯ . Le ℓxhpE wEEE ¯=(( ˆ)+¯,ˆ(( ˆ)ˆ,ˆ), ˆ) iii .By assump ion o he p oo , agg ega e consump ion emains he same wi h he linea p icing scheme as be o e: ∫ ∫ ∀∈xxXkS ¯=ˆ=ˆ, ikikk . We nex cons uc a di e en income ax schedule Ry ¯( ) ha keeps he unc ion yE ˆ(,ˆ)unchanged, so ha labo choices emain he same wi h linea as wi h nonlinea p icing. Le Ry epE yE E ¯()= ( ( ˆ)+¯,ˆ(,ˆ), ˆ) . The u ili y maximiza ion p oblem de ines ∣≤ yE VxE pE x Ry νpE Ry E ¯(,ˆ)=max { ( , ˆ)( ( ˆ)+¯)¯()}= ( ( ˆ)+¯,¯(),ˆ) x. By Assump ion 1, yE yE ¯(,ˆ)=ˆ(,ˆ) . By de ini ion o Hicksian demand, one has, o all i , ≤pE x pE x ( (ˆ)+¯)¯(( ˆ)+¯)ˆ. ii The e o e, ≤pEx pEx x x(ˆ)¯(ˆ)ˆ+¯(ˆ−¯ ) iiii , and summing up o e he agen s, ∫ ∫∫ ∑ ≤⏟ ∈ () pEx pEx x x(ˆ)¯(ˆ)ˆ+¯ˆ−¯. ii kS kikik =0 (2) The esul hen ollows since: ∫∫∫ ∫∫∫ ℓℓ ℓ ℓ ≤ℓ ℓ ℓ G wE RwE xwE x wE pEx wE pEx wE RwE x G ˆ=( ˆ)ˆ−(( ˆ)ˆ)+ ( ˆ,( ˆ)ˆ)ˆ=( ˆ)ˆ−(ˆ)ˆ (ˆ)ˆ−(ˆ)¯=( ˆ)ˆ−¯(( ˆ)ˆ)+¯¯=¯ ii ii iiii ii i ii i ii ii i □ Nonlinea , income‐dependen p icing o he ex e nali ies is hence supe luous, i espec i e o whe he he income ax schedule is op imal and he ex e nali y is ully in e nalized in he adjus ed economy. Gi en he assump ions o he p oposi ion, i is always possible o mo e o he ax sys em wi h linea p icing while keeping all consume s a hei o iginal u ili ies. The go e nmen expe iences a su plus in ax e enue h ough he ax e o m. The go - e nmen budge is weakly la ge wi h linea p icing o a simila eason ha , in he absence o ex e nali ies, commodi y axa ion is no needed when income axa ion is only es ic ed by incen i e cons ain s (A kinson & S igli z, 1976; Kaplow, 2006; La oque, 2005). Labo axa ion dis o s he economy o good eason: income axes aise e enue and edis ibu e be ween consume s. Commodi y axa ion is needed due o he ex e nal e ec s. Di e en ia ed com- modi y axa ion would only be needed i i we e be e able o aise e enue, edis ibu e o egula e he ex e nali y. Howe e , leisu e is sepa able om he consump ion choice and con- sump ion choices a e homogenous ac oss consume s, making all goods equal subs i u es o complemen s o leisu e. In addi ion, he ex e nali ies a e caused by agg ega e consump ion wi h no pa icula consume con ibu ing mo e o less a he ma gin. Nonlinea and income‐ dependen commodi y axa ion canno p o ide u he bene i s i he income ax can be lexibly adjus ed, and a wo s c ea es u he dis o ions in he consump ion choice. The go e nmen may use he su plus in i s budge o inc ease he u ili y o all consume s. Assume ha u ili y o consume s has mul iplica i e o addi i e sepa abili y be ween subu ili y V x() , labo choices, and ex e nal e ec s. This is he case, o example, i ⋅ℓ⋅UUVxg dE=(()()() ) ii i iii o ℓUUVx g dE=(()+()+() ) ii i iii . A Pa e o‐supe io poin can be eached i he su plus inances a educ ion in he ha m (in case o nega i e ex e nal e ec s) o FLEURBAEY AND KORNEK | 367 an inc ease in he bene i s (in case o posi i e ex e nal e ec s) om he ex e nali ies, o example, change unc ion d E( ) i o some d E ¯( ) i wi h UdE UdE(… ( ) ) < (… ¯() ) ii ii . In he con ex o pollu ion om C O 2 emissions, he addi ional esou ce would inance adap a ion o clima e change, he eby weakly inc easing he u ili y o all consume s. Assuming ha he su plus in he go e nmen 's budge can be edis ibu ed o consume s so ha e e yone is be e o , consume s would also abso b he bene i s o he ax e o m. La oque (2005), Rema k 1, discusses condi ions unde which his is possible in he absence o ex- e nali ies: each consume ecei es he same lump‐sum ans e . In ou se up, edis ibu ion o consume s may change he le el o ex e nali ies, and he eby could make some, o all, con- sume s wo se o . Kaplow (2012) shows ha a Pa e o‐supe io poin can be eached e en when he le els o ex e nali ies change, howe e only unde he assump ion ha he ma ginal ha m o all ex e nali ies is cons an . As in Kaplow (2012) one may jus assume ha “mo e is be - e ,”so ha he go e nmen can and will use he su plus a hand o imp o e he si ua ion o consume s, o example, by p o iding mo e o a public good ha does no ad e sely in e e e wi h he le el o ex e nali ies. In es iga ing mo e p ecise ci cums ances unde which a Pa e o‐ imp o emen is possible in ou se up is beyond he scope o his pape bu should be conside ed in u u e esea ch. Ou p oo is an ex ension o La oque (2005). The key addi ion, o deal wi h he ex e nali y, is he assump ion ha he e is a linea p icing scheme ¯ ha keeps he agg ega e le els o goods ha cause ex e nali ies a hei o iginal le els wi h nonlinea p icing xy(, ) . In ac , any linea p icing scheme wi h his p ope y does he ick. This addi ional assump ion d i es ou possibili y esul . Such a linea p icing scheme should exis in common economic ci cums ances, and we below p o ide su icien condi ions o he case o only one good ha causes he ex e nali y. Rema k 1. P oposi ion 1can be e o mula ed using di e en assump ions abou linea p icing. Assume ha a linea p icing scheme ¯ , ha gene ally allows ≠∀ k ¯0 k, exis s ha (i) a ini ial subu ili ies V ˆ i leads o Hicksian demand which keeps ex e nali ies a hei o iginal le el E ˆ(while agg ega e le els o consump ion may change) (ii) does no inc ease he p oduce alue o consump ion compa ed o he o iginal le el ∫ pEx(ˆ)ˆ i . Again, wi h linea p icing a ¯ and adjus ed income axa ion, gi en by he p oo o P oposi ion 1, he go e nmen budge weakly inc eases. Rema k 2. Following up on Rema k 1, we may ac ually sea ch o he p icing scheme ha minimizes he p oduce alue o consump ion. 2 The p og am ∫ ∀∀ () xpxEEXjVVxEi ¯=min ˆ=() andˆ=(, ˆ) xij j i i de ines he alloca ion wi h maximum go e nmen budge while keeping ex e nali ies ixed and u ili ies a hei o iginal alue (ensu ing ha incen i e cons ain s a e ul illed). Assuming ha e e y unc ion is di e en iable, one can show ha commodi y p ices wi h maximum go e nmen budge a e linea . 3 2 We a e g a e ul o an anonymous e e ee who poin ed ou his app oach. 3 The i s ‐o de ‐condi ions a e ∀∑⋅ ∂ ∂ ∂ ∂ ki p ζλ,: = + kj NEj XkjiVi xik. F om hese we ha e: ∂∕∂ ∂∕∂ ∑∂∕∂ ∑∂∕∂ = Vixik Vixim pkj NEjXk pmj NEjXm − − , showing ha he ma ginal a e o subs i u ion be ween goods is he same o all consume s. They can hence no ace di e en ia ed a e ‐ ax p ices. 368 | FLEURBAEY AND KORNEK Based on P oposi ion 1 u he ex ensions can be de i ed. Rema k 3. Checking he p oo , ou main esul in P oposi ion 1ex ends o ex e nali ies de i ed om p e ax income o consume s ∈ E EX y=(,{} ) ii[0,1] . 4 Fo he case o CO 2 ‐emissions, an addi ional speci ica ion can be made: he p oduc ion o almos all consump ion goods en ails C O 2 ‐emissions ha cause he same ex e nali y—global wa ming. The ollowing esul is a s aigh o wa d ex ension o P oposi ion 1. We show ha a mo e om a nonlinea p icing scheme o a di e en income ax and a linea ax on emissions s (a scala )— he same in all sec o s—is weakly p e e able. Deno e he emission in ensi y o each good k as α k (amoun o CO 2 ‐emissions pe amoun o good). The clima e change ex e nali y is de e mined by he agg ega e amoun o emissions: ∑ E αX= k nk k =1 . We now conside a nonlinea ax on emissions xy(, ) . The ax may di e be ween sec o s o he economy so ha gene ally ≠≠ xy xy m k (, ) (, ), mk . P oposi ion 2. Assume ha i is possible o ind a single linea ax on emissions, s ¯ , such ha , a ini ial subu ili ies V ˆ i , he o al Hicksian demand o all goods is such ha o al emissions a e equal o he ini ial ∑ E αX ˆ=ˆ k nk k =1 . The e is an alloca ion wi h an al e na i e income ax schedule Ry ¯( ) and an emission ax a s ¯ in which each consume keeps ini ial u ili y U ˆiand he go e nmen budge is weakly g ea e han a he ini ial le el. P oo . The p oo is almos exac ly equi alen o ha o P oposi ion 1. We only need o adjus summing o e expendi u es a Equa ion (2) o ∫ ∫∫ ∑ ≤⏟ pEx pEx s αxαx(ˆ)¯(ˆ)ˆ+¯(ˆ−¯) . ii k nkikkik EE =1 =ˆ−¯=0 □ The same ax a e on CO 2 emissions in all sec o s o he economy is weakly supe io o di e en ia ed axa ion. As a po en ially ele an example in which axes a e di e en ia ed, conside he case o EU ca bon p icing on emissions by powe s a ions, which en ails a lowe ma ginal cos o aba emen in he powe sec o han in oad anspo a ion, whe e egula ion is done h ough pe o mance s anda ds and na ional axes (Hein ichs e al., 2014). In oducing he same CO 2 p ice on EU‐wide emissions can gene a e a su plus in he EU's budge , assuming ha income axa ion be ween consume s can be lexibly adjus ed. Howe e , since he EU is a mone a y bu no a iscal union (ye ), such an adjus men migh ac ually no be easible. We e isi his poin in Sec ion 3.2. So a , we had o ely on wo assump ions o show ou main possibili y esul in P oposi ion 1, and i s ex ension in P oposi ion 2. Fi s , he e is a linea p icing scheme ha keeps agg ega e consump ion o goods a hei o iginal le els. Second, we assume duali y be ween u ili y maximiza ion and expendi u e minimiza ion (Assump ion 1). We nex p esen su icien con- di ions o bo h assump ions o hold in he case when only good 1 causes an ex e nali y. Conside he case in which ex e nali ies s em only om o al consump ion o good 1, E X= 1 . In wha ollows we s udy possible a ia ions o demand in a hypo he ical wo ld in which he ex e nali y X ˆ 1 is ixed (e en i o al demand o good 1 does no add up o his le el). 4 We hank an anonymous e e ee o poin ing his ou o us. FLEURBAEY AND KORNEK | 369 F om he u ili y unc ion V xX(, ) 1, one de i es he u ili y V xzX(,, ˆ) R11, whe e ∑≥ zpXx=( ˆ) k nk k 2 1. Since ≥pX k(ˆ), 2 k1a e ixed when X ˆ 1 is ixed, we will ea he ex- pendi u e on hese goods as a composi e commodi y ( he p ice o which is 1). 5 Le ∣≤ ≤χVx X x X=sup{ ( ,0, ˆ)0 ˆ} R1111. Lemma Assume ha V xzX(,, ˆ) R11is con inuous, locally nonsa ia ed and s ic ly quasi‐ conca e in xz ( , ) 1 . Fu he assume ha V xzX χ (,, ˆ)> R11 o all xz, 1 such ha z>0 . Gi en u ili y le els ∀ V xX χi(ˆ,ˆ)> i1,Assump ion1holds ue ( o XX=ˆ 1 1 )and he e exis s a linea p ice o he ex e nali y ¯ 1 so ha he agg ega e Hicksian demand o good 1 is X ˆ 1 . P oo . See Appendix A. □ The Lemma adds a he mild assump ions abou subu ili y VR and he ini ial alloca ion o show ha P oposi ion 1holds ue and Assump ion 1is ul illed. S anda d assump ions abou u ili y VR a e: con inui y, local nonsa ia ion and s ic quasi‐conca i y. In addi ion, we assume ha ini ial subu ili ies V ˆ i a e no a ainable by consuming good 1 only. The assump ions in he Lemma show why we choose o wo k wi h VR ins ead o V . The Lemma can also be o mula ed using he o iginal V xX(, ) 1; howe e one needs s ic quasi‐conca i y o V in x . We belie e i is less es ic i e o assume s ic quasi‐conca i y o VR ins ead o V . 3.2 |Wi h income ax a e ixed Supe io i y o linea p icing in he p esence o ex e nali ies c i ically depends on a well‐ designed adjus men o he income ax sys em. As highligh ed by S igli z (2019), a change in he income ax migh no be a ainable when e o ming commodi y p icing, and di e en ia ed p icing should be used o alle ia e dis ibu ional conce ns. This di icul y p ominen ly ea u es o global wa ming and he axa ion o CO 2 emissions. When a emp ing o equalize in e - na ional ca bon p ices, policy‐make s canno ely on income axa ion o deal wi h dis ibu- ional issues as i does no exis be ween coun ies. To accommoda e such policy‐ ele an se ings, his sec ion illus a es ha i he income ax a e is ixed a a subop imal le el compa ed o he second‐bes op imum, a nonlinea p ice o he ex e nali y, di en ia ed by income, 6 can imp o e social wel a e. To ha end, we es ic a en ion o he model wi h only good 1 causing a consump ion ex e nali y. The analysis is inspi ed by Chichilnisky and Heal (1994), who showed ha in he absence o edis ibu ion, ma ginal cos s o public good p o ision a e gene ally no equalized be ween agen s in he op imum. Sheshinski (2004) and S igli z (2019) discuss simila esul s. He e, we e ain he incen i e cons ain s o labo supply in he p esen op imal axa ion model. The nex p oposi ion shows ha whene e a social imp o emen is possible wi h a di e - en ia ed p ice, i has o be edis ibu i e, ha is, display a nega i e co ela ion be ween he agen s' pe sonal ax e o m impac and hei ma ginal social alue o money, echoing he esul s o 5 The cons uc ion o V R ollows Va ian (1992) p. 148. Deno e p 0 as he p ice ec o o all o he goods bu good 1. Hence we ha e ⋅zpx x=(,…,) n02 . We can cons uc he indi ec u ili y ⋅ p PR Vx s px Pp x x R (,,)=max(),.. + (,…, )= x n 111 0 2 . U ili y V R is V xz pPR s px Pz R (,)=min(,,),.. + = R pP 1 1,111 , wi h which we ha e p PR V x z s px Pz R (,,)=max (,),.. + = xz R1 1, 111 . 6 Since he consump ion o goods depends on income only, due o iden ical consump ion p e e ences in he popula ion, he e is no need in his sec ion o make he p ice o he ex e nali ies depend on consump ion in addi ion o income. The unc ion xy(, ) iiwas in oduced in p e ious sec ions only o allow o ull gene ali y in he ax unc ion. 370 | FLEURBAEY AND KORNEK