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
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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.
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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
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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.
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