In Tax Public Finance (2018) 25:64–98
h ps://doi.o g/10.1007/s10797-017-9443-6
Op imal axa ion and public p o ision o po e y
educ ion
Ra i Kanbu 1·Tuuli Paukke i2,3·
Jukka Pi ilä4,5·Ma i Tuomala5
Published online: 27 Ap il 2017
© The Au ho (s) 2017. This a icle is an open access publica ion
Abs ac The exis ing li e a u e on op imal axa ion ypically assumes he e exis s a
capaci y o implemen complex ax schemes, which is no necessa ily he case o many
de eloping coun ies. We examine he de e minan s o op imal edis ibu i e policies
in he con ex o a de eloping coun y ha can only implemen linea ax policies due
o adminis a i e easons. Fu he , he educ ion o po e y is ypically he exp essed
goal o such coun ies, and his ea u e is also aken in o accoun in ou model. We
de i e he op imali y condi ions o linea income axa ion, commodi y axa ion, and
public p o ision o p i a e and public goods o he po e y minimiza ion case and
compa e he esul s o hose de i ed unde a gene al wel a is objec i e unc ion. We
also s udy he implica ions o in o mali y on op imal edis ibu i e policies o such
coun ies. The exe cise e eals non- i ial di e ences in op imal ax ules unde he
di e en assump ions.
BTuuli Paukke i
uuli.paukke i@ a . i
Ra i Kanbu
[email p o ec ed]
Jukka Pi ilä
jukka@wide .unu.edu
Ma i Tuomala
[email p o ec ed]
1Co nell Uni e si y, I haca, NY, USA
2VATT Ins i u e o Economic Resea ch, Helsinki, Finland
3Aal o Uni e si y, Helsinki, Finland
4UNU-WIDER, Helsinki, Finland
5Uni e si y o Tampe e, Tampe e, Finland
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Op imal axa ion and public p o ision o po e y educ ion 65
Keywo ds Redis ibu ion ·Income axa ion ·Commodi y axa ion ·Public good
p o ision ·Po e y
JEL Classi ica ion H21 ·H40 ·O12
1 In oduc ion
High le els o wi hin-coun y inequali y in many o he wise success ul de eloping
coun ies ha e become a key policy conce n in global de elopmen deba e. While
some coun ies ha e e y unequal inhe en dis ibu ions (e.g., due o his o ical land
owne ship a angemen s), in o he s he ui s o economic g ow h ha e been unequally
sha ed. No ma e wha he unde lying eason o he high inequali y, o en he only
di ec way o go e nmen s o a ec he dis ibu ion o income is ia edis ibu i e ax
and ans e sys ems. Clea ly, public spending on social se ices also has an impac on
he dis ibu ion o well-being, al hough some o he e ec s (such as skill-enhancing
impac s om educa ional in es men ) only ma e ialize o e a longe ime ho izon.
Re lec ing he desi e o educe po e y and inequali y, edis ibu i e ans e sys-
ems ha e, indeed, p oli e a ed in many de eloping coun ies. S a ing om La in
Ame ica, hey a e now sp eading o low-income coun ies, including hose in Sub-
Saha an A ica.1In low-income coun ies, in pa icula , edis ibu i e a angemen s
ia ans e s a e s ill a an ea ly s age, and hey o en consis o isola ed, dono -d i en
p og ams. The e is an u gen and well- ecognized need o mo e away om sca e ed
p og ams o mo e comp ehensi e ax-bene i sys ems.
This pape examines he op imal design o cash ans e s, commodi y axes (o sub-
sidies), he p o ision o public and p i a e goods (such as educa ion and housing), and
inancing hem by a linea income ax. The pape also includes an analysis o op imal
income axa ion in he p esence o an in o mal sec o . The pape he e o e p o ides
an o e iew o many o he mos ele an ins umen s o edis ibu i e policies ha
a e needed o a sys em-wide analysis o social p o ec ion. We build on he op imal
income ax app oach, which is ex ensi ely used in he de eloped coun y con ex 2,
bu much less applied o he design o edis ibu i e sys ems in de eloping coun-
y ci cums ances. This app oach, ini ia ed by Mi lees (1971), allows o a igo ous
ea men o e iciency conce ns (e.g., he po en ially ha m ul e ec o dis o iona y
axa ion on employmen ) and edis ibu i e objec i es. Achie ing he go e nmen ’s
edis ibu i e objec i es is cons ained by limi ed in o ma ion: he social planne can-
no di ec ly obse e indi iduals’ income-ea ning capaci y, and he e o e i needs o
base i s ax and ans e policies on obse able a iables, such as g oss income. The
mos gene al o mula ions o op imal ax models apply nonlinea ax schedules, bu in a
de eloping coun y con ex , using ully nonlinea axes is a ely easible. In his pape ,
we he e o e limi he analysis o edis ibu i e linea income axes, which combine a
1Fo a ecen ea men and su ey, see Ba ien os (2013).
2See IFS and Mi lees (2011) o an in luen ial applica ion o op imal ax heo y o policy analysis o ich
coun ies.
123
66 R. Kanbu e al.
lump-sum ans e wi h a p opo ional income ax, and which can be implemen ed by
wi hholding a sou ce i necessa y.
Linea income axes a e no e y common in p ac ice: less han 30 coun ies had la
ax a es o pe sonal income in 2012, wi h some concen a ion in ex-So ie Eas e n
Eu ope (Peichl 2014). I is no ewo hy ha e en hough la axes a e no pa icula ly
common in low-income coun ies, in many ins ances in such coun ies he p og essi e
income ax eaches only a small sha e o he popula ion. This would indica e ha
despi e he exis ence o a p og essi e income ax, hese coun ies do no ye possess
enough ax capaci y o implemen well- unc ioning p og essi e income axes. This is
one mo i a ion o ou in e es o modeling op imal linea axes. Peichl (2014) sugges s
ha simpli ica ion bene i s can be especially ele an o de eloping coun ies.3
In con en ional op imal axa ion models, he go e nmen ’s objec i e unc ion is
modeled as a social wel a e unc ion, which depends di ec ly on indi idual u ili ies.
We depa om his wel a is app oach by p esen ing gene al non-wel a is ax ules, as
in Kanbu e al. (2006), and, in pa icula , op imal ax and public good p o ision ules
when he go e nmen is assumed o minimize po e y. We ha e chosen his app oach
as i esembles well he one o much o he policy discussion in de eloping coun-
ies, including he Millennium De elopmen Goals (MDGs) and he new Sus ainable
De elopmen Goals (SDGs), whe e he objec i e is explici ly o educe po e y a he
han maximize well-being.4Simila ly, he discussion ega ding cash ans e sys ems
is o en couched especially in e ms o po e y alle ia ion. While we do no necessa ily
wan o ad oca e po e y minimiza ion o e o he social objec i es, we ega d exam-
ining i s implica ions, and con as ing hem wi h adi ional wel a is ic app oaches,
use ul. Using non-wel a is objec i es is, as such, no hing new in economics. In ac ,
as Sen (1985) has a gued, one can be c i ical o u ili a ianism o many easons. No e
also ha he objec i e o po e y minimiza ion is no a odds wi h he es ic ion o a
linea ax scheme ha we impose: a la ax egime oge he wi h a lump-sum income
ans e componen can achie e simila amoun s o edis ibu ion owa d he poo as a
p og essi e ax sys em, i speci ied sui ably (Keen e al. 2008;Peichl 2014). In all ou
analysis, we i s p esen wel a is ax ules (which a e mos ly al eady a ailable in he
li e a u e) o p o ide a benchma k o examine how applying po e y minimiza ion as
an objec i e changes he op imal ax and public se ice p o ision ules.
We also deal wi h some ex ensions o exis ing models, which a e mo i a ed by
he de eloping coun y con ex , such as he case whe e public p o ision a ec s he
indi iduals’ income-ea ning capaci y, hus cap u ing (albei in a e y s ylized way)
possibili ies o a ec hei capabili ies. An impo an ea u e o ake in o accoun in
ax analysis o de eloping coun ies is he p esence o a la ge in o mal sec o , and we
also examine he implica ions o his o op imal edis ibu i e policies.
Ou pape is ela ed o a ious s ands o ea lie li e a u e. Fi s , Kanbu e al. (1994)
and Pi ilä and Tuomala (2004) s udy op imal income ax and commodi y ax ules,
espec i ely, om he po e y alle ia ion poin o iew, bu hei pape s build on he
nonlinea ax app oach which is no well sui ed o de eloping coun ies. Kanbu and
3No e ha i migh be easonable o some coun ies o mo e o a p og essi e income ax sys em as hei
ax capaci y inc eases wi h de elopmen ; he s udy o such dynamics is beyond he scope o his pape .
4In ac , he i s SDG is simply “End po e y in all o i s o ms e e ywhe e.”
123
Op imal axa ion and public p o ision o po e y educ ion 67
Keen (1989) do conside linea income axa ion oge he wi h po e y minimiza ion,
bu hey do no p oduce op imal ax ules bu ocus on a ax e o m pe spec i e, and
p o ide ax a e simula ions. O he s ha e conside ed di e en depa u es om he
wel a is s anda d. Fo example, Fleu baey and Manique (2007) conside ai ness
as an objec i e o he ax- ans e sys em and i s implica ions on op imal axa ion.
Roeme e al. (2003) employ a maximin ype o social goal and cha ac e ize how
well ax and ans e sys ems achie e he goal o equali y o oppo uni y. Second, ou
wo k is ela ed o new con ibu ions in beha io al public inance, which add ess he
si ua ion whe e he beha io al biases o he indi iduals lead he social planne o adop
a di e en objec i e unc ion han he indi iduals ha e; see Che y (2015), Ge i sen
(2016), Fa hi and Gabaix (2015). A hi d s and o li e a u e conside s axa ion and
de elopmen mo e gene ally, such as Go don and Li (2009), Keen (2009,2012), Bi d
and Gend on (2007) and Besley and Pe sson (2013).5This ield, while clea ly e y
ele an , has no concen a ed much on he design o op imal edis ibu i e sys ems.
Finally, op imal linea income axa ion has been s udied om he s anda d wel a is
pe spec i e. We desc ibe hese models in Sec . 2.1. The mos ecen desc ip ion o
linea income ax models can be ound in Pike y and Saez (2013). They also emphasize
how linea ax ules, while analy ically mo e easible, p o ide he same in ui ion as
he mo e complica ed nonlinea models. The linea ax ules, hey a gue, a e obus o
al e na i e speci ica ions6, and examining his o ms pa o ou mo i a ion: we s udy
op imal linea ax policies, in ou unde s anding o he i s ime, om he po e y
minimiza ion pe spec i e.
The pape p oceeds as ollows. Sec ion 2examines op imal linea income axa-
ion, while Sec . 3 u ns o op imal p o ision ules o publicly p o ided p i a e and
public goods ha a e inanced by such a linea income ax. Sec ion 4analyzes he
combina ion o op imal linea income axes and commodi y axa ion and asks unde
which condi ions one should use di e en ia ed commodi y axa ion i he go e nmen
is in e es ed in po e y minimiza ion and also has op imal cash ans e s a i s disposal.
The ques ion o how op imal po e y-minimizing income ax policies a e al e ed in
he p esence o an in o mal sec o is examined in Sec . 5, whe eas Sec . 6p esen s a
nume ical illus a ion o op imal income axa ion o po e y minimiza ion. Finally,
conclusions a e p o ided in Sec . 7.
2 Linea income axa ion
2.1 Op imal linea income axa ion unde he wel a is objec i e
In his sec ion, we gi e an o e iew o some o he models and esul s o op imal
linea income axa ion as hey ha e been p esen ed in he li e a u e. Many o mulae o
op imal axa ion we e de eloped in he 1970s and 1980s (see Dixi and Sandmo 1977;
5Besley and Pe sson (2013) use a model wi h g oups ha can di e in hei income-ea ning abili ies. Thei
analysis ocuses, howe e , on explaining how economic de elopmen and ax capaci y a e in e ela ed, and
no on edis ibu ion be ween indi iduals.
6They also desc ibe some implica ions o depa u es om he wel a is s anda d in he op imal nonlinea
ax model.
123
68 R. Kanbu e al.
Tuomala 1985 and he su ey by Tuomala 1990), and hey a e s ill being used, whe eas
Pike y and Saez (2013) o e esh exp essions o he ax ules. Ou exposi ion mainly
ollows ha o Tuomala (1985), bu Appendix 1shows how he esul s ela e o hose
in Pike y and Saez (2013).
The go e nmen collec s a linea income ax τ, which i uses o inance a lump-
sum ans e b, along wi h o he exogenous public spending R. The indi iduals di e
in hei income-ea ning capaci y (wi), and zideno es indi idual labo income (wiLi,
whe e Li ep esen s hou s wo ked). Consump ion equals ci=(1−τ)zi+b, whe e he
supe sc ip -i e e s o indi iduals.7The e is a disc e e dis ibu ion o Nindi iduals,
whose he e ogeneous p e e ences o e consump ion and labo a e cap u ed by he
u ili y unc ion ui(ci,zi). The maximized (subjec o he indi idual budge cons ain )
alue o his u ili y unc ion is cap u ed by he indi ec u ili y unc ion, which is deno ed
by Vi(1−τ,b), and we e e o he ne -o - ax a e as 1 −τ=a. To simpli y no a ion,
subsc ip -a e e s o he de i a i e wi h espec o he ne -o - ax a e.
The go e nmen has edis ibu i e objec i es ep esen ed by a Be gson–Samuelson
unc ion WV1,...,VNwi h W>0, W <0. The go e nmen ’s p oblem is o
choose he ax a e τand ans e bso as o maximize he social wel a e unc ion
WVi(a,b)unde he budge cons ain (1−a)zi=Nb+R.8We deno e he
social ma ginal u ili y o income by βi=WVVi
b.
All he ma hema ical de ails a e p esen ed in Appendix 1. The e i is shown ha
he op imal ax ule is gi en by
τ∗
1−τ∗=1
ε1−z(β)
¯z,(1)
whe e ε=d¯z
d(1−τ)
(1−τ)
¯zis he elas ici y o o al income wi h espec o he ne -o - ax
a e, ¯zis a e age income and z(β) =βizi
βiwel a e-weigh ed a e age income. De ine
Ω=z(β)
¯z, so ha I=1−Ωis a no ma i e measu e o inequali y o , equi alen ly,
o he ela i e dis o ion a ising om he second-bes ax sys em. Clea ly Ωshould
a y be ween ze o and uni y. One would expec i o be a dec easing unc ion o τ
(gi en he pe capi a e enue equi emen g=R
N). The e is a minimum easible le el
o τ o any gi en posi i e g, and o cou se gmus no be oo la ge, o no equilib ium
is possible. Hence any solu ion mus also sa is y τ>τ
min i he ax sys em is o be
p og essi e. Tha is, i he ax does no aise su icien e enue o inance he non-
ans e expendi u e, R, he sho all mus be made up by imposing a poll ax (b<0)
on each indi idual. One would also expec he elas ici y o labo supply wi h espec
o he ne -o - ax a e o be an inc easing unc ion o τ(i need no be).
We can ew i e (1)asτ∗=1−Ω
1−Ω+ε o illus a e he basic p ope ies o he op imal
ax a e. Because ε≥0 and 0 ≤Ω<1, bo h he nume a o and denomina o a e
nonnega i e. The op imal ax a e is hus be ween ze o and one. The o mula cap u es
7We conside “income” he e as he labo income o indi iduals, bu conside ing ha ou model is in ended
especially o he poo e coun ies, ag icul u al income could as well be included in he concep o income.
In Sec . 5we discuss he implica ions o un axed home consump ion in ag icul u al p oduc ion.
8Summa ion is always o e all indi iduals i, which is supp essed o simpli ica ion.
123

Op imal axa ion and public p o ision o po e y educ ion 69
nea ly he e iciency-equi y ade-o . τdec eases wi h εand Ω, and we ha e he
ollowing gene al esul s: (1) In he ex eme case whe e Ω=1, i.e., he go e nmen
does no alue edis ibu ion a all, τ=0 is op imal. We can call his case libe a ian.
Acco ding o he libe a ian iew, he le el o disposable income is i ele an ( uling
ou bo h basic income b, and o he public expendi u es, g, unded by he go e nmen ).
(2) I he e is no inequali y, hen again Ω=1 and τ=0. The e is no in e en ion by he
go e nmen . The inhe en inequali y will be ully e lec ed in he disposable income.
Fu he mo e, lump-sum axa ion is op imal; b=−go T=−b. (3) We can call he
case whe e Ω=0 as “Rawlsian” o maximin p e e ences. The go e nmen maximizes
ax e enue (op imal τ=1
1+ε) as i maximizes he basic income b(assuming he wo s
o indi idual has ze o labo income). In ac , maximizing bcan be ega ded as a non-
wel a is case, which is he ocus in he nex subsec ion.
2.2 Op imal linea income axa ion unde non-wel a is objec i es
A non-wel a is go e nmen is one ha ollows a di e en se o p e e ences han hose
employed by indi iduals hemsel es (Kanbu e al. 2006). Thus, ins ead o maximizing
a unc ion o indi idual u ili ies, he go e nmen has o he , pa e nalis ic objec i es ha
go beyond u ili ies. A special case aken up in mo e de ail below is he objec i e o
minimizing po e y in he socie y. To be as gene al as possible, le us de ine a “social
e alua ion unc ion” (as in, e.g., Kanbu e al. 2006)asS=F(ci,zi), which he
go e nmen maximizes ins ead o he social wel a e unc ion. F(ci,zi)measu es he
social alue o consump ion ci o a pe son wi h income ziand can be ela ed o
u(ci,zi)bu is no es ic ed o i . Following Tuomala’s model as abo e, gi en he
ins umen s a ailable, linea income ax τ, lump-sum g an band o he expendi u e
R he go e nmen hus maximizes F(azi+b,zi)subjec o he budge cons ain
(1−a)zi−Nb =R. De ine
Fc(zi+azi
a)+Fzzi
a
Fc(1+azi
b)+Fzzi
b≡˜
F,(2)
which e lec s he ela i e impac o axes and ans e s on he social e alua ion unc-
ion. Using his de ini ion, and ollowing he same s eps as in he p e ious sec ion (see
Appendix), he op imal ax a e becomes:
τ∗
1−τ∗=1
ε1−
˜
F
¯z.(3)
The esul esembles he wel a is ax ule in (1). In addi ion o labo supply conside a-
ions ia he e m 1
ε, hey bo h en ail a e m ha measu es he ela i e bene i s o axes
and ans e s, in he wel a is case ia wel a e-weigh ed income, in he non-wel a is
case ia ˜
F, he ela i e impac on he social e alua ion unc ion. No e ha since unde
non-wel a ism indi iduals a e no necessa ily a hei u ili y op imum, he en elope
condi ion does no apply and hus he beha io al esponses zi
aand zi
ba e no cancelled
123
70 R. Kanbu e al.
ou in ˜
F. Tha is, he impac s o ax changes on labo supply a e no i ial unde
non-wel a ism. The e ms zi
a(Fca+Fz)in he nume a o and zi
b(Fca+Fz)
in he denomina o o (2) cap u e hese e ec s on he social e alua ion unc ion. I
axa ion had no beha io al impac s (zi
a=zi
b=0), i would a ec he alue o he
social e alua ion unc ion only by mechanically al e ing indi idual a e - ax income.
No e ha in his case, ˜
F=Fczi
Fcwould be a mo e di ec equi alen o z(β) =βizi
βi.
The same equi alence would be achie ed also when Fca+Fz=0, ha is, he social
ma ginal a e o subs i u ion be ween income and consump ion equals he p i a e a e:
−Fz
Fc=a=−
ui
z
ui
c
( he la e is ob ained om he indi idual’s i s -o de condi ion).
In hese cases, ˜
Fwould be a pu ely edis ibu i e e m, albei a non-wel a is ic one.
Pa e nalis ic conce ns addi ionally en e he op imal ax ule ia labo supply changes,
cap u ed by he esponse o z.In his way, he ax ule in (3) can be decomposed, and
his decomposi ion is simila in spi i o he co ec i e pa s o he ax o mulae in he
new op imal ax li e a u e wi h beha io al agen s, such as Fa hi and Gabaix (2015)
and Ge i sen (2016).
The signs and magni udes o Fcand Fzand hus o ˜
Fdepend on he speci ic
objec i e o he go e nmen , ha is, on he shape o F. Le us conside he speci ic
case o po e y minimiza ion below.
2.2.1 Special case: po e y minimiza ion
Now le us de i e he op imal linea ax esul s o a go e nmen whose objec i e is o
minimize po e y in socie y. The ins umen s a ailable o he go e nmen a e he same,
τand b, and o he exogenous expendi u e is R. No e i s ha he e enue-maximizing
ax a e is in ac equi alen o he ax a e ob ained om a maximin objec i e unc ion,
since when he go e nmen only ca es abou he po e y (consump ion) o he poo es
indi idual, i s only goal is o maximize edis ibu ion o his indi idual, i.e., maximize
ax e enue.
Le us i s de ine he objec i e unc ion o he go e nmen explici ly. Po e y is
de ined as dep i a ion o indi idual consump ion ci ela i e o some desi ed le el ¯c
and measu ed wi h a dep i a ion index Dci,¯c, such ha D>0∀c∈[0,¯c)and
D=0 o he wise, and Dc<0,Dcc ≥0∀c∈[0,¯c),asinPi ilä and Tuomala
(2004). A ypical example o such an index would be he Pα amily o Fos e –G ee –
Tho becke (FGT) po e y indices. We discuss he applica ion o FGT indices in ou
model in Appendix 2. No e, howe e , ha he choice o po e y index depends on he
p e e ences o he go e nmen , whe he hey wish o minimize he o al amoun o
dep i a ion in he socie y, o a e o ins ance conce ned especially abou he incomes
o he poo es o he poo . The social e alua ion unc ion F(ci,zi)becomes Dci,¯c
and he objec i e unc ion is min P=Dci,¯c.NowFc=Dcand Fz=0, so
˜
F=˜
D=Dczi+azi
a
Dc1+azi
b,(4)
and he op imal ax ule becomes:
123
Op imal axa ion and public p o ision o po e y educ ion 71
τ∗
1−τ∗=1
ε1−
˜
D
¯z.(5)
Since now Fz=0, he esul is close o (1) han (3) was, al hough pa o he labo
supply impac s s ill emain. He e ˜
Ddesc ibes he ela i e e iciency o axes and
ans e s in educing dep i a ion. Bo h he nume a o and denomina o o ˜
Ddepend
on Dc, so he di e ence in he ela i e e iciency o he wo depends on zi
aand zi
b.The
mo e people eac o axes ( ela i e o ans e s) by ea ning less, he highe is ˜
Dand
he lowe should he ax a e be. In (1), he highe is he social alue o income, he
highe is z(β) and he lowe should he ax a e be.
Since he o m o he esul is simila in he wel a is and he po e y minimiza ion
cases, he analysis could be also seen as a special case o he a gumen in Saez and
S an che a (2016), who de i ed gene alized social wel a e weigh s and exp ess he
ax o mulae in e ms o hose.9He e, he gene alized social wel a e weigh would
hus be de i ed om a po e y minimiza ion objec i e. I could be close o a sui ably
de ined wel a is c i e ion, and clea ly i would be exac ly he same only i he wel a is
c i e ion would co espond o he chosen po e y minimiza ion objec i e.
We can also ew i e ˜
D,usinga=1−τ,as:Dczi+(1−τ) ∂zi
∂(1−τ)
Dc1+(1−τ)zi
b=
Dc1+(1−τ)
zi
∂zi
∂(1−τ)zi
Dc1+(1−τ)zi
b=Dc1+εizi
Dc1+(1−τ)zi
b. Thus he ˜
Din he op imal ax esul (5)
en ails a u he conside a ion ha depends on labo supply esponses. I com-
bines pa e nalis ic p e e ences—how much po e y is educed—wi h he beha io al
esponses o a ax sys em—how much labo income inc eases when he ake-home pay
goes up. The la e e ec ends o lowe he op imal ax a e o induce he poo o wo k
mo e. Kanbu e al. (1994) ind a simila esul in hei nonlinea po e y-minimizing
ax model. He e, howe e , we a e es ic ed o lowe he ax on e e yone ins ead o
only he poo es indi iduals.
To summa ize, he non-wel a is ax ules di e om he wel a is ones, depending
on he de ini ion o non-wel a ism in ques ion ( he Fcand Fz e ms). Howe e , when we
ake po e y minimiza ion as he speci ic case o non-wel a ism, he ax ules a e qui e
simila o wel a is ones. The basic di e ence is ha equi y is no conside ed in wel a e
e ms bu in e ms o po e y educ ion e ec i eness. A mo e no able di e ence a ises
om e iciency conside a ions. Wi h linea axa ion, aking in o accoun labo supply
esponses means ha e e ybody’s ax a e is a ec ed, ins ead o jus he a ge g oup’s.
I we wan o induce he poo o wo k mo e o educe hei po e y, we need o lowe
e e yone’s ax a e. The wel a is linea ax ule does no ake his in o accoun . I is no ,
howe e , possible o s a e ha unde po e y minimiza ion ax a es a e op imally lowe
han unde wel a e maximiza ion, since we canno di ec ly compa e he wel a e and
dep i a ion e ms. Howe e , he e is an addi ional e iciency conside a ion in ol ed
unde po e y minimiza ion. Nonlinea ax ules o cou se make i possible o a ge
lowe ax a es on he poo e indi iduals, bu in a de eloping coun y con ex wi h
9We a e g a e ul o a e e ee o his poin .
123
72 R. Kanbu e al.
lowe adminis a i e capaci y his is no necessa ily possible, and such conside a ions
a ec e e yone’s ax a e.
3 Public good p o ision wi h linea income axes
3.1 Op imal public p o ision unde he wel a is objec i e
Le us i s ex end he wel a is model o linea axa ion o include he p o ision o pu e
public goods. The go e nmen o e s a uni e sal pu e public good G, which en e s
indi idual u ili ies in addi ion o he consump ion o p i a e goods. The go e nmen ’s
objec i e unc ion is now WVi(a,b,G), whe eas he budge cons ain becomes
(1−a)zi−Nb −NπG=Rwhe e πis he p oduce p ice o he public good.
The consume p ice o p i a e consump ion is no malized o 1. Le us now de ine
he ma ginal willingness o pay o he public good by he exp ession σ=VG
Vband
σ∗=βiσi
βias he wel a e-weigh ed a e age ma ginal a e o subs i u ion be ween
public good and income o indi idual i. The ule o public p o ision can hen be
w i en as
π=σ∗−τσ∗¯zb−¯zG.(6)
This public good p o ision ule is a e sion o a modi ied Samuelson ule. I equa es
he ela i e cos o p o iding he public good o he wel a e-weigh ed sum o ma ginal
a es o subs i u ion (MRS). I also includes a e enue e m, which akes in o accoun
he impac s o public good p o ision and income ans e s on labo supply and hus
ax e enue.
Conside i s he case when labo supply does no depend on public good p o ision
and he e a e no income e ec s, i.e., ¯zG=¯zb=0. Then we a e le wi h a mo e
amilia ule ha wel a e-weigh ed agg ega e MRS mus equal he cos o he public
good. When we add income e ec s so ha ¯zb<0, and since σ∗is posi i e, hen
because o he second e m in (6), he inancing cos s o he public good a e educed.
Likewise, i labo supply and public p o ision a e posi i ely ela ed, he inancing
cos s o he public good a e educed.
3.2 Op imal p o ision o public goods unde po e y minimiza ion
Now conside a non-wel a is go e nmen in e es ed in minimizing po e y. The public
good Gwhich i o e s en e s he dep i a ion index sepa a ely om o he , p i a e
consump ion x:Dx,G,¯x,¯
G. The go e nmen s ill o e s a lump-sum cash ans e
bas well and inances i s expenses wi h he linea income ax τ.
Again al e na i e o mula ions o he public good p o ision ule can be w i en.
The i s is
π=D∗−τD∗¯zb−¯zG,(7)
which can be compa ed wi h Eq. (6). He e, D∗=DG+Dxazi
G
Dx1+azi
bcap u es he
e iciency o he public good in educing dep i a ion ela i e o he income ans-
123
Op imal axa ion and public p o ision o po e y educ ion 79
whe eas, unde he assump ion ha he e a e no income e ec s in e asion, he i s -
o de condi ion wi h espec o bs ays he same. F om he e, we can de i e a ule o
he op imal ax ollowing he same s eps as in Sec . 2.2:
τ∗
1−τ∗=1
εe1−
˜
De
¯ze,(17)
whe e now εeis a ax elas ici y o he ne -o -e asion ax base ¯ze=¯z−¯eand ˜
De
ep esen s he ela i e impac o axes and ans e s on he dep i a ion index (see
Appendix 1 o u he de ail). The ule ep esen s a ade-o be ween po e y educ-
ion and e iciency, bo h o which a e now al e ed by e asion. The e is a p essu e
owa d lowe ax a es, as now dis o ions o axa ion a e inc eased by e asion beha -
io , so εe>ε. Con a y o his e ec , ˜
Deis educed compa ed o ˜
Dbecause educing
axes (inc easing a) is now a less use ul ins umen o po e y educ ion, as pa o
he axes ha e been e aded. As ∂e
∂a<0, people pay mo e axes when ax a es a e
educed, and he e o e po e y in ac inc eases. ˜
De hus wo ks o inc ease ax a es.
The e o e, an in e es ing ade-o a ises: in o mali y inc eases he cos o aising
axes, bu i also means ha highe axes a e less ha m ul as hose in he in o mal
sec o do no need o pay hem (and hey a e s ill en i led o he lump-sum ans e ).14
These coun e ailing o ces ha e no been no ed by he li e a u e be o e. The p esence
o in o mali y he e o e seems o gi e ise o ax policy ules ha a e a om i ial.
Fu u e wo k could also look mo e deeply in o he issue o he ax mix in he p es-
ence o in o mali y. I income ax is mo e easily e aded han commodi y axa ion, as
Boadway e al. (1994) sugges , his could gi e ise o policies ha ocus axa ion and
edis ibu ion on commodi y axes and subsidies, ins ead o income axes and lump-
sum ans e s. Slem od and Gilli ze (2014) ha e also sugges ed ocusing on a “ ax
sys ems app oach” and including, among o he hings, e asion beha io in o op imal
axa ion analysis o ob ain mo e use ul p esc ip ions o ac ual ax policy. This opic
ce ainly dese es a mo e de ailed analysis.
6 A nume ical illus a ion
To u he illus a e he di e ences o ax a es unde po e y minimiza ion and wel-
a ism, we p o ide a simple nume ical simula ion. He e we concen a e on he special
case whe e he e a e no income e ec s on labo supply and he elas ici y o labo sup-
ply wi h espec o he ne -o - ax wage a e is cons an . I εdeno es his elas ici y, he
quasi-linea indi ec u ili y unc ion is gi en by (w(1−τ),b)=b+[w(1−τ)]1+ε
1+ε,so
ha εis cons an . Like mos wo k on op imal nonlinea and linea income axa ion, we
use he logno mal dis ibu ion ln(n,mσ2) o desc ibe he dis ibu ion o p oduc i i ies
wi h suppo [0,∞)and pa ame e s mand σ(see Ai chison and B own 1957). The
i s pa ame e , m, is he log o he median wage. The second pa ame e , he a iance
14 The idea ha hose in he in o mal sec o can s ill ecei e ans e s ma ches well wi h eali y: many o
he cash ans e sys ems each hose wi h li le o no connec ion o he o mal sec o .
123

80 R. Kanbu e al.
o log wage σ2, is i sel an inequali y measu e. As is well known, he logno mal dis i-
bu ion i s easonably well o e a la ge pa o he income ange bu di e ges ma kedly
a bo h ails. The Pa e o dis ibu ion in u n i s well a he uppe ail. We also use
he wo-pa ame e e sion o he Champe nowne dis ibu ion (known also as he Fisk
dis ibu ion). This dis ibu ion app oaches asymp o ically a o m o Pa e o dis ibu ion
o la ge alues o wages bu i also has an in e io maximum. In ou simula ions, he
e enue equi emen is se o ze o; hus, he sys em is pu ely edis ibu i e.
To illus a e he po e y-minimizing ax o mula in (3), we also need o speci y
a measu e o po e y. Typically, po e y indices consis o compu ing some a e age
measu e o dep i a ion by se ing indi idual needs as de ined abo e a he ag eed upon
po e y line ¯c. Fo his pu pose, we ake a po e y index o he o m de eloped by
Fos e e al. (1984). They ha e p oposed de ining a po e y index as he a e age o
hese po e y gaps ac oss indi iduals aised o some powe α. When α=1, i is jus
he p opo ion o uni s below he po e y line mul iplied by he a e age po e y gap.
(See Appendix 2 o mo e de ails.) We conside he cases whe e ei he 30 o 40% o
he popula ion lie below he po e y line.
The esul s om he simula ion o he op imal ax when he go e nmen minimizes
he po e y gap o he logno mal case a e p esen ed in Table 1. Resul s a e shown o
wo di e en alues o labo supply elas ici y ε, wo di e en alues ega ding income
dispe sion σ, and wo alues o he sha e o popula ion below he po e y line F(¯w).
The ax a es a e high, abo e 60%, o all he combina ions o pa ame e alues.15
Compa ing hese esul s o he wel a is case is no s aigh o wa d, as hose depend
on he chosen wel a e unc ion. We adop a cons an ela i e inequali y a e sion o m
o he wel a e unc ion: he con ibu ion o social wel a e o he i h indi idual is
w1−η
i
1−η, whe e ηis he cons an ela i e inequali y a e sion coe icien . Hence, he social
ma ginal alue o income o an indi idual wi h wage a e wis p opo ional o w−η.
Using he p ope y o he logno mal dis ibu ion ln(E(ws)) =sm +s2σ2
2, we can
calcula e he op imal ax a e om he ollowing o mula: τ
1−τ=1
ε[1−e−η(1+ε)σ2].
O , using he p ope y o he logno mal dis ibu ion ha ln(1+c 2)=σ2, whe e c
is he coe icien o a ia ion, we can ew i e τ=1
1+ε/[1+c 2]−η(1+ε) .
A wide ange o alues o he inequali y a e sion pa ame e ηha e been employed
in he li e a u e, a ying ypically om 0.5 o 2. No e ha , as discussed in Sec . 2.1,as
η→∞, social p e e ences app oach “maximin” p e e ences, whe e he op imal ax
a e is he same as he e enue-maximizing ax a e, τ=1
1+ε, which does no depend
on he o iginal income dis ibu ion. Na u ally, i he e is no ega d o inequali y in
he socie y, η=0 and τ=0. Table 2displays he wel a is ic ax simula ion esul s
o wo di e en alues o labo supply elas ici y ε, o wo di e en alues o income
dispe sion σ, and o i e di e en alues o inequali y a e sion η.
The simula ion esul s illus a e clea ly ha a con en ional inequali y a e sion
le els, op imal wel a is ic ax a es lie well below he po e y-minimizing a es. Only
as inequali y a e sion becomes ex emely high do he wel a is ic a es app oach he
15 The esul s a e e y simila using he Champe nowne dis ibu ion (wi h income dispe sion pa ame e s
chosen so ha inequali y is simila in bo h cases), which is no e y su p ising as he dis ibu ions only
di e a he op o he income schedule. These esul s a e a ailable upon eques .
123
Op imal axa ion and public p o ision o po e y educ ion 81
Table 1 Simula ed ax a es o po e y minimiza ion unde di e en alues o ε,σ,andF(¯w)
εσ=0.7σ=1.0
F(¯w) =0.3F(¯w) =0.4F(¯w) =0.3F(¯w) =0.4
0.25 79 77 79 78
0.5 65 63 66 64
Table 2 Simula ed ax a es in
he wel a is ic case unde
di e en alues o ε,σ,andη
εσ=0.7
η=0.5η=1η=2η→∞
0.25 43 58 69 80
0.5 31 44 56 67
εσ=1.0
η=0.5η=1η=2η→∞
0.25 52 65 74 80
0.5 38 51 61 67
po e y-minimizing ones. Wi h po e y minimiza ion as he social objec i e, op imal
ax a es a e close o he e enue-maximizing “maximin” a e.
Ano he poin o compa ison could be he wel a is ic linea ax simula ions o S e n
(1976). His calcula ions di e om ou s as he inco po a es income e ec s and a non-
cons an elas ici y o labo supply wi h espec o he ax a e.16 Wi h he elas ici y o
subs i u ion be ween consump ion and leisu e a 0.5 and income dispe sion desc ibed
by σ=0.39, as conce n o inequali y ises om low o medium and high, he inds ax
a es ising om 19 o 43 and 48%. The ex eme “maximin” esul is 80%. These ax
a es a e also clea ly lowe han he po e y-minimizing a es, excep a e y ex eme
alues o inequali y a e sion.
These nume ical examples and S e n’s (1976) esul s end o sugges ha he ax
a es o he po e y minimiza ion case a e likely o be highe han o many wel a is
examples. The esul s compa e o Kanbu e al. (1994), who also ound ha he (non-
linea ) ma ginal ax a es on he poo a e ai ly high unde he po e y minimiza ion
objec i e. Bo h hei and ou esul s a e in e es ing om he poin o iew ha he ana-
ly ical o mulae o he op imal ax a e include a e m ha , ce e is pa ibus, encou ages
labo supply, bu in compu a ional esul s i s in luence is o se , mos likely, by he
need o minimize he po e y gap. The highe he po e y a e, he highe he lump-
sum g an inanced by hese axes needs o be, in o de o aise mo e people ou o
po e y.
16 Ou simpli ying assump ions allow us o p o ide ax a es wi h espec o he h ee pa ame e s in Table
2.
123
82 R. Kanbu e al.
7 Conclusion
This pape examined op imal linea income axa ion, public p o ision o public and
p i a e goods and he op imal combina ion o linea income ax and commodi y axes
when he go e nmen ’s aim is o minimize po e y. The linea ax en i onmen was
chosen because such axes a e mo e easily implemen able in a de eloping coun y
con ex and since op imal linea ax ules a e seen o p o ide simila in ui ion as he
mo e complex nonlinea ax o mulas.
The esul s show ha he linea income ax includes addi ional componen s ha
wo k owa d lowe ing he ma ginal ax a e. This esul a ises om he goal o
boos ea nings o educe income po e y. Unlike in he op imal nonlinea income ax
amewo k, his lowe ma ginal ax a ec s all axpaye s in he socie y. Howe e , he
nume ical simula ions o e ed sugges ha his mechanism is o se by he dis ibu i e
conce ns and in p ac ice he op imal ax a es o po e y minimiza ion appea high.
Public good p o ision in he op imal ax amewo k unde po e y minimiza ion was
shown o depend on he ela i e e iciency o public p o ision e sus income ans-
e s in gene a ing po e y educ ions. One pa icula a enue whe e public p o ision is
use ul is ia i s po en ially bene icial impac on indi iduals’ ea nings capaci y. Thus,
public p o ision can be desi able e en i i s di ec wel a e e ec s we e non-exis en .
Pe haps mo e impo an ly, po e y minimiza ion as an objec i e changes com-
ple ely he condi ions unde which uni o m commodi y axa ion is op imal. When he
go e nmen ’s objec i e is o minimize po e y ha depends on disposable income,
uni o m commodi y axa ion is unlikely o be e e op imal: his is because he com-
modi y ax changes ha e i s -o de e ec s on consume s’ budge ia he di ec impac
on he cos o li ing, and his di ec e ec depends on he ela i e impo ance o di -
e en goods in he o e all consump ion bundle. Sepa abili y in demand coupled wi h
linea Engel cu es is no su icien o gua an ee op imali y o uni o m commodi y
axes. In eali y, he adminis a i e di icul ies o implemen ing commodi y axa ion
wi h many ax a es mus , o cou se, be aken in o accoun , as well.
We also examined he implica ions o he p esence o an in o mal sec o o op imal
ax and ans e policies. The esul s e ealed ha when he go e nmen is conce ned
abou income po e y, he p esence o he in o mal sec o is, on he one hand, use ul,
as i educes he po e y-inc easing e ec o highe axes bu , on he o he hand, i is
also cos ly since i is likely o inc ease he elas ici y o he ax base. Examining he
implica ions o in o mali y on he ole o o he ins umen s o go e nmen policies is
an impo an a enue o u u e wo k.
Ano he s and o ollow-up wo k should add ess he ques ion o complemen-
a y policies o edis ibu ion, such as minimum wages. I should be bo ne in mind
ha di e en policies impose di e en equi emen s on adminis a i e capaci y,17 and
17 Fo example, Lee and Saez (2012) show how a minimum wage policy can use ully complemen an
op imal nonlinea income ax and ans e policy unde wel a is objec i es. Howe e , imposing minimum
wage egula ion implies ha he go e nmen needs o be ei he able o obse e indi idual wage a es, o has
su icien ins i u ional s eng h o ely on whis leblowe s o denounce non-complying employe s, in o de
o en o ce he legisla ion.
123
Op imal axa ion and public p o ision o po e y educ ion 83
examining which po e y educ ion ins umen s become a ailable only as he socie ies
ad ance on hei de elopmen pa h is an in e es ing a enue o u he wo k.
Acknowledgemen s We a e g a e ul o wo anonymous e e ees and semina audiences a he UNU-
WIDER Con e ence on ‘Inequali y: Measu emen , T ends, Impac s and Policies’, he Annual Mee ing o
he Finnish Economic Associa ion, he HECER De elopmen Economics semina and he IIPF con e ence
in Dublin o use ul commen s. This esea ch o igina es om he UNU-WIDER p ojec The economics
and poli ics o axa ion and social p o ec ion. Funding om he Academy o Finland (G an No. 268082)
is g a e ully acknowledged.
Compliance wi h e hical s anda ds
Con lic s o in e es The au ho s decla e ha hey ha e no con lic s o in e es .
Open Access This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na-
ional License (h p://c ea i ecommons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided you gi e app op ia e c edi o he o iginal au ho (s) and he
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Appendix 1: Ma hema ical appendix
Linea income axa ion
Wel a ism
Conside i s he wel a is case. Using λ o deno e he mul iplie associa ed wi h
he budge cons ain , he go e nmen ’s Lag angian is L=WVi(a,b)+
λ(1−a)zi−Nb−R. The social ma ginal u ili y o income is βi=WVVi
b.
Using Roy’s heo em, Vi
a=Vi
bzi,weha eWVVi
a=βizi. The i s -o de condi ions
wi h espec o aand b, espec i ely, a e hen:
βizi=λzi−(1−a)zi
a(18)
βi=λN−(1−a)zi
b.(19)
Di ide (18)by(19) o ge :
βizi
βi=zi−(1−a)zi
a
N−(1−a)zi
b
.(20)
Deno e a e age income ¯z=zi
Nand wel a e-weigh ed a e age income z(β) =βizi
βi
o ge :
z(β) =¯z−(1−a)¯za
1−(1−a)¯zb
.(21)
Mul iply he go e nmen ’s e enue cons ain by 1
Nand de ine g=R
N o ge (1−
a)¯z−b=g, and o ally di e en ia e, keeping gcons an :
123
84 R. Kanbu e al.
db
da|gcons =¯z−(1−a)¯za
−1+(1−a)¯zb
=−z(β). (22)
The ac ha z(β) =−
db
da|gcons ells us ha wel a e-weigh ed labo supply should be
equal o he cons an - e enue e ec o ax a e changes in b.
By o ally di e en ia ing a e age labo income ¯zand using (22), we ha e
d¯z
da|gcons =¯za+¯zb
db
da|gcons =¯za−¯zbz(β). (23)
When we impose gas a cons an we ha e o gi e up one o ou deg ees o eedom.
Now he in e p e a ion o d¯z
da|gcons is hen he e ec on labo supply when ais changed,
as is b, in o de o keep ax e enue cons an . Using (23) we can w i e (21):
z(β) −¯z=−(1−a)d¯z
da|gcons =−τd¯z(1−τ)¯z
d(1−τ)(1−τ)¯z,(24)
om which we ge he op imal ax a e o Eq. (1).
We now de i e he esul s in he o m o he Pike y and Saez (2013) model. In hei
model, he e is a con inuum o indi iduals, whose dis ibu ion is ν(i)(popula ion size
is no malized o one). Indi iduals maximize hei u ili y ui((1−τ)zi+b,zi), and
hei FOC implici ly de ines he Ma shallian ea nings unc ion zi
u(1−τ,b). Using his,
agg ega e ea nings a e Zu(1−τ,b). The go e nmen ’s budge cons ain b+R=
τZu(1−τ,b)implici ly de ines bas a unc ion o τ, and consequen ly Zucan also
be de ined solely as a unc ion o τ:Z(1−τ) =Zu(1−τ,b(τ)).Zhas elas ici y
ε=1−τ
Z
dZ
d(1−τ).
To s a , no e ha i he go e nmen only ca ed abou maximizing ax e enue
τZ(1−τ), i would se τsuch ha ∂(τZ(1−τ))
∂τ =0: Z(1−τ)−τdZ
d(1−τ) =0. Using
τ
Z
dZ
d(1−τ) =τ
1−τε, his gi es
τ∗
1−τ∗=1
ε
⇔τ∗=1
1+ε.(25)
When he go e nmen is conce ned abou social wel a e, i s p oblem is o
max SWF =ωiW(ui((1−τ)zi+τZ(1−τ)−R,zi)) dν(i), whe e use has been
made o he indi idual consump ion ci=(1−τ)zi+b=(1−τ)zi+τZ(1−τ)−R.
He e ωis a Pa e o weigh and Wis an inc easing and conca e ans o ma ion o
u ili ies. The FOC ∂SWF
∂τ =0is:
ωiWuui
c−zi+(1−τ)∂zi
∂τ +Z+τdZ
dτ+ui
z
∂zi
∂τ dν(i)=0,
which, using he indi idual’s en elope condi ion, becomes:
ωiWuui
c−zi+Z−τdZ
d(1−τ)dν(i)=0.
123

Op imal axa ion and public p o ision o po e y educ ion 85
Taking Z−τdZ
d(1−τ) ou o he in eg and and lea ing i o he le -hand side, we
ha e on he igh -hand side ωiWuui
czidν(i)
ωiWuui
cdν(i). Pike y and Saez de ine βi=ωiWuui
c
ωiWuui
cdν(i)
as a no malized social ma ginal wel a e weigh o indi idual i, so ha he e m can
be simpli ied o:
Z−τdZ
d(1−τ) =βizidν(i).
Using he de ini ion o agg ega e elas ici y o ea nings and de ining ¯
β=βizidν(i)
Zas
he a e age no malized social ma ginal wel a e weigh , weigh ed by labo incomes zi
(i can also be in e p e ed as he a io o he a e age income weigh ed by indi idual
wel a e weigh s βi o he a e age income Z), we can ew i e his as 1 −τ
1−τε=¯
β,
which gi es he op imal social wel a e-maximizing ax a e:
τ∗
1−τ∗=1
ε1−¯
β.(26)
Acco ding o Pike y and Saez, ¯
β“measu es whe e social wel a e weigh s a e concen-
a ed on a e age o e he dis ibu ion o ea nings.” The wel a e-maximizing ax a e
is hus dec easing in bo h he a e age ma ginal wel a e weigh and he ax elas ici y
o agg ega e ea nings. A highe ¯
β e lec s a lowe as e o edis ibu ion, and hus a
lowe desi e o ax o edis ibu i e easons.
Pike y and Saez also no e ha (26) can be w i en in he o m o τ∗=−co βi,zi
Z
−co βi,zi
Z+ε.
I highe incomes a e alued less (lowe β), hen he co a iances a e nega i e and he
ax a e is posi i e. This is a simila o mula ion as in Dixi and Sandmo (1977), Eq.
(20), whe e τ∗=−
1
λ
−co zi,μi
∂¯z
∂(1−τ)|comp.
(he e λ ep esen s he go e nmen ’s budge cons ain
Lag ange mul iplie and μi he indi idual’s ma ginal u ili y o income, s. . uc=μi).
He e he nume a o e lec s he equi y elemen and he denomina o he e iciency
componen , simila as in (26).
Non-wel a ism
In he non-wel a is case, he Lag angian unc ion is L=Fazi+b,zi+λ((1−
a)zi−Nb−R). The i s -o de condi ions wi h espec o aand ba e:
Fc(zi+azi
a)+Fzzi
a=λzi−(1−a)zi
a(27)
Fc(1+azi
b)+Fzzi
b=λN−(1−a)zi
b.(28)
123
86 R. Kanbu e al.
Di iding he i s equa ion wi h he second and di iding h ough he igh -hand side
wi h N, we ge :
Fc(zi+azi
a)+Fzzi
a
Fc(1+azi
b)+Fzzi
b=¯z−(1−a)¯za
1−(1−a)¯zb
,(29)
which gi es Eq. (3). Minimizing a dep i a ion index Dis a special case o his, such
ha Fc=Dcand Fz=0. O he wise he de i a ion o (5) is analogous o he abo e.
Le us nex de i e he po e y-minimizing ax ule ollowing he o mula ion o
Pike y and Saez. Gi en he go e nmen ’s ins umen s, consump ion is ci=(1−
τ)zi+b=(1−τ)zi+τZ(1−τ)−R. The po e y minimiza ion objec i e in he
con inuous case hus eads:
min P=Dci,¯cdν(i)
=D(1−τ)zi+τZ(1−τ)−R,¯cdν(i). (30)
The op imal ax a e is ound om he go e nmen ’s FOC, ∂P
∂τ =0:
Dc−zi+(1−τ)∂zi
∂τ +Z+τdZ
dτdν(i)=0
⇔Dc−zi−(1−τ) ∂zi
∂(1−τ) +Z−τdZ
d(1−τ)dν(i)=0.(31)
De ine a “no malized ma ginal dep i a ion weigh ” as βi=Dc
Dcdν(j).Using
his de ini ion, Z−τdZ
d(1−τ)Dcdν(i)=Dczi+(1−τ) ∂zi
∂(1−τ)dν(i)can
be w i en as:
Z−τdZ
d(1−τ) =βizi+(1−τ) ∂zi
∂(1−τ)dν(i). (32)
Using he de ini ion o he elas ici y o indi idual labo ea nings εi=1−τ
zi
∂zi
∂(1−τ),we
ha e (1−τ) ∂zi
∂(1−τ) =ziεiand using elas ici y o agg ega e ea nings ε=1−τ
Z
dZ
d(1−τ),
we ha e Z−τdZ
d(1−τ) =1−τ
1−τεand we can ew i e he abo e as:
Z1−τ
1−τε=βizi+ziεidν(i). (33)
This leads o he po e y-minimizing ule o
τ∗
1−τ∗=1
ε1−¯
β−¯
βε,(34)
123
Op imal axa ion and public p o ision o po e y educ ion 87
whe e analogously o Pike y–Saez, ¯
β=βizidν(i)
Z=Dczidν(i)
ZDcdν(j)is an a e age
no malized dep i a ion weigh , weigh ed by labo incomes (o , analogously, a e age
labo income weigh ed by indi idual dep i a ion weigh s). In addi ion, we ha e de ined
¯
βε=βiziεidν(i)
Z=Dcziεidν(i)
ZDcdν(j), which desc ibes a e age labo incomes weigh ed
by hei co esponding indi idual elas ici ies and dep i a ion weigh s. This can be
in e p e ed as a combined dep i a ion and e iciency e ec .
As in he wel a is se ing, he mo e elas ic a e age ea nings a e o axa ion, he lowe
is he op imal ax a e (a egula e iciency e ec ). The op imal po e y-minimizing
ax a e is dec easing in he a e age dep i a ion weigh ¯
β, as a highe as e o edis-
ibu ion owa d he ma e ially dep i ed implies a lowe ¯
βand hus highe axa ion
o edis ibu i e pu poses. The e ec is analogous o he wel a is ax a e, o cou se
wi h sligh ly di e en de ini ions o ¯
β.
The new e m ¯
βεcan be in e p e ed as a combined dep i a ion weigh and e iciency
e ec . The elas ici y e m implici in ¯
βε akes in o accoun he incen i e e ec s o
axa ion on wo king and wo ks o educe τ∗. To a oid discou aging he poo om
wo king, hei ax a es should be lowe . Bu because he ax ins umen is o ced o
be linea , ax a es a e hen lowe ed o e e yone, as we ound in he Tuomala model
in Eq. (5). The alue o ¯
βεdepends on he ela ionship o he indi idual ea nings
elas ici ies and income: i he elas ici y is he same ac oss income le els, he e is jus
a le el e ec mo ing om ¯
β o ¯
βε; howe e , i he elas ici y we e highe o mo e
dep i ed indi iduals, o example, ¯
βεwould mos likely be highe han unde a la
elas ici y. This wo ks owa d a lowe ax a e in o de o a oid discou aging he poo es
om wo king. Howe e , whe he ¯
βεis high o low does no depend only on he shape
o he elas ici y bu also on he shape o he dep i a ion weigh s, which also a ec ¯
β.
Finally, he hi d way o exp essing he op imal ax ule in he case o po e y
minimiza ion is one ollowing he Dixi and Sandmo (1977) o mula ion and i can be
w i en as
τ∗=−1
λ
co Dc,zi+1
NDca˜zi
a+co Dcazi
b,zi
1
N˜zi
a
.(35)
In his exp ession, he denomina o is he same as in Eq. (20) o Dixi and Sandmo
(1977) p esen ed be o e, ha is, he a e age de i a i e o compensa ed labo supply
wi h espec o he ne -o - ax a e. In he nume a o , he i s e m measu es he s eng h
o he associa ion be ween income and po e y impac : when he associa ion be ween
o e all po e y and small income is s ong ( his would be he case wi h he squa ed
po e y gap), he ax should be high so ha i will inance a sizable lump-sum ans e .
I he associa ion is weake (as wi h he headcoun a e), he ax a e is op imally
smalle . The second and he hi d e ms in he nume a o a e new. They measu e he
indi ec e ec s om changes in he ax a e on labo supply. He e ˜zis he compensa ed
(Hicksian) labo supply. The g ea e is he educ ion in he labo supply ollowing an
inc ease in he ax a e (i is he compensa ed change as he ax inc ease is linked wi h
a simul aneous inc ease in he lump-sum ans e ), he smalle should he ax a e be
in o de o a oid inc eases in dep i a ion a ising om lowe ea ned income. The las
wo e ms in he nume a o a e closely linked wi h a o mula ion Dc(1−τ)∂z
∂q|comp,
123
88 R. Kanbu e al.
whe e he idea is ha he las co a iance e m se es as a co ec i e de ice o he
mean impac o axes on labo supply (simila ly as in he denomina o in he o iginal
Dixi –Sandmo o mula ion).
Public good p o ision
Wel a ism
The Lag angian is L=WVi(a,b,G)+λ(1−a)zi−Nb−NπG−R.
Maximizing he Lag angian wi h espec o band Ggi es:
βi=λN−(1−a)zi
b(36)
WVVi
G=λNπ−(1−a)zi
G.(37)
Di iding (37)by(36) we ob ain
βiσi
βi=π−(1−a)¯zG
1−(1−a)¯zb
,(38)
whe e we de ine σ∗=βiσi
βi o be he wel a e-weigh ed a e age ma ginal a e o
subs i u ion be ween public good and income o indi idual i. Rew i ing his ule gi es
Eq. (6) in he main ex .
Ex ending he Pike y and Saez app oach o include public p o ision, he go e n-
men ’s goal unc ion is
SWF =ωiWui(1−τ)zi+τZ((1−τ),G)−R−πG,G,zidν(i).
The FOC o τis as be o e, and he FOC o public good p o ision Gis
ωiWuui
G+ui
x(1−τ)∂zi
∂G+τdZ
dG−πdν(i)=0,
which p oduces he ollowing public good p o ision ule:
ωiWuui
G+ui
x(1−τ)∂zi
∂Gdν(i)
ωiWuui
xdν(i)=π−τdZ
dG.(39)
The le -hand side ela es he wel a e gains o public good p o ision (a di ec (uG)
and indi ec e ec (ux(1−τ)∂zi
∂G ia labo supply eac ions)) o he wel a e gains o
di ec ly inc easing consump ion (cash ans e s) and he igh -hand side ela es he
123
Op imal axa ion and public p o ision o po e y educ ion 95
βi=Dc
¯c
0Dcdν(i)
=
−α
¯c¯c−ci
¯cα−1
¯c
0−α
¯c¯c−ci
¯cα−1
dν(i)
=¯c−ci
¯cα−1
¯c
0¯c−ci
¯cα−1
dν(i)
and consequen ly ¯
β=¯c
0βizidν(i)
Zand ¯
βε=¯c
0βiziεidν(i)
Z, as be o e. E e y hing
else s ays exac ly he same as in he calcula ions o Appendix 1. Also in he case o
Tuomala’s and Dixi and Sandmo’s models, he esul s s ay he same, and we can plug
in he explici de ini ion o Dc, he de i a i e o he po e y measu e wi h espec o
disposable income, in o he esul s.
Po e y measu emen in he con ex o public good p o ision
Employing he FGT po e y measu e in he con ex o public good p o ision
o po e y educ ion is mo e complica ed han in he case o jus disposable
income. In Sec . 3.2 he go e nmen ’s objec i e unc ion was de ined as min P=
Dxi,G,¯x,¯
Gdν(i), ha is, dep i a ion was measu ed bo h as dep i a ion in p i-
a e consump ion (i.e., disposable income) as well as wi h espec o he public good.
Bu he FGT index is a uni-dimensional measu e, measu ing dep i a ion wi h espec
o one dimension only (e.g., disposable income). I one wan s o conside publicly
o e ed goods such as educa ion as sepa a e om p i a e consump ion, a mul idi-
mensional FGT measu e is needed. Mul idimensionali y, howe e , en ails a di icul
ques ion o de e mining when a pe son should be de e mined as dep i ed.
The e a e se e al app oaches o mul idimensionali y o FGT- ype po e y mea-
su es.19 Fo example, Besley and Kanbu (1988), who conside he po e y impac s
o ood subsidies, employ he uni-dimensional FGT measu e bu de ine dep i a ion
in e ms o equi alen income: Pα=z
0zE−yE
zEα
(y)d(y), whe e yEis equi a-
len income, de ined implici ly om V(p,yE)=V(q,y), and zEis he po e y line
co esponding o equi alen income. Bu gi en ou aim o de ining op imal policy in
e ms o po e y educ ion, i espec i e o indi idual wel a e, he use o equi alen
income is p oblema ic as i o ces he solu ion o be such ha , by de ini ion, indi idu-
als a e kep as well o as be o e. Pi ilä and Tuomala (2004) employ shadow p ices
in a po e y-minimizing con ex o allow o se e al goods in he po e y measu e.
Fo hem, dep i a ion is measu ed as D(z,y(q,w
)) whe e zh=sxx∗−sh
LL∗and
yh(q,wh)=sxx(q,wh)−sh
LL(q,wh). This app oach equi es de e mining shadow
p ices sx,sL o consump ion and leisu e in o de o cons uc a e e ence bundle
espec i e o which dep i a ion can be measu ed, bu he e is no clea guideline o he
choice o he shadow p ices.
The app oach in Bou guignon and Chak a a y (2003) is mo e sui able o ou
pu poses. They p o ide a mul idimensional ex ension o he FGT measu e, acco ding
o which a pe son is poo i she is dep i ed in a leas one dimension. A simple example
o such an ex ension o he FGT is
19 See Fos e e al. (2010, pp. 504–5) o a b ie o e iew o mul idimensional FGT ex ensions ha allow
he inclusion o dimensions such as heal h, educa ion, and nu i ion in addi ion o o he consump ion.
123

96 R. Kanbu e al.
Pθ=1
n
m

j=1
i∈Sj
ajzj−xij
zjθj
,
whe e θjand aja e weigh s gi en o dimension j, and Sjis he g oup o
people who a e poo in dimension j.Alki e and Fos e (2011) o hei pa p o-
ide a simila measu e which uses a weigh ed coun o dimensions in which he
pe son is dep i ed o de e mine whe he she is poo . An aspec o his is also
whe he he goods unde conside a ion a e complemen s o subs i u es. Following
he Bou guignon–Chak a a y app oach and de ining xi1=xias p i a e consump-
ion, z1=¯x,xi2=Gas he amoun o public good, and z2=¯
Gwould
gi e us Pθ=1
ni∈Sja1¯x−xi
¯xθ1+a2¯
G−G
¯
Gθ2. Using his measu e, Dx=
−θ1a1
¯x¯x−xi
¯xθ1−1
and DG=−
θ2a2
¯
G¯
G−G
¯
Gθ2−1
. These can hen be inse ed o he
public p o ision ules. Fo example, (9) becomes
θ2a2
¯
G¯
G−G
¯
Gθ2−1+θ1a1
¯x¯x−xi
¯xθ1−1
(1−τ)∂zi
∂Gdν(i)
θ1a1
¯x¯x−xi
¯xθ1−1
dν(i)
=p−τdZ
dG
and (40) becomes
θ2a2
¯
G¯
G−G
¯
Gθ2−1+θ1a1
¯x¯x−xi
¯xθ1−1
azi
G
θ1a1
¯x¯x−xi
¯xθ1−11+azi
b
=p−(1−a)zi
G
1−(1−a)zi
b
,
om whe e i can be seen ha he ela i e e iciency o he public good e sus cash
ans e s on educing po e y can be di ec ly aced back o he magni udes o θ1and θ2.
Po e y measu emen in he con ex o commodi y axa ion
In he case o commodi y axes, we un in o he same issues ega ding dep i a ion
measu emen as wi h public goods. Howe e , in Sec . 4.2 dep i a ion was measu ed
only in e ms o disposable income, c. We hus escape he mul idimensionali y issue
and employing he FGT po e y measu e is hus as simple as in he linea income ax
case: we simply need o de ine D=Pαand hus Dc=−
α
¯c¯c−ci
¯cα−1
in Eq. (15).
Po en ially he go e nmen migh also conside weigh ing di e en goods acco ding
o hei impo ance o measu ed po e y.
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