A note on Sen’s representation of the Gini coefficient: Revision and repercussions

S a k, Oded
A icle — Published Ve sion
A no e on Sen’s ep esen a ion o he Gini coe icien :
Re ision and epe cussions
The Jou nal o Economic Inequali y
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: S a k, Oded (2024) : A no e on Sen’s ep esen a ion o he Gini coe icien :
Re ision and epe cussions, The Jou nal o Economic Inequali y, ISSN 1573-8701, Sp inge US, New
Yo k, NY, Vol. 22, Iss. 4, pp. 1061-1067,
h ps://doi.o g/10.1007/s10888-024-09623-y
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/315322
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The Jou nal o Economic Inequali y (2024) 22:1061–1067
h ps://doi.o g/10.1007/s10888-024-09623-y
RESEARCH
Ano eonSen’s ep esen a ion o he Gini coefficien :
Re ision and epe cussions
Oded S a k1,2
Recei ed: 24 Augus 2023 / Accep ed: 19 Janua y 2024
© The Au ho 2024
Abs ac
Sen (1973 and 1997) p esen s he Gini coe icien o income inequali y in a popula ion as
ollows. “In any pai -wise compa ison he man wi h he lowe income can be hough o be
su e ing om some dep ession on inding his income o be lowe . Le his dep ession be
p opo ional o he di e ence in income. The sum o al o all such dep essions in all possible
pai -wise compa isons akes us o he Gini coe icien .” (This ci a ion is om Sen 1973,
p. 8.) Sen’s e bal accoun is accompanied by a o mula (Sen 1997, p. 31, eq. 2.8.1), which
is eplica ed in he ex o his no e as equa ion (1). The o mula yields a coe icien bounded
om abo e by a numbe smalle han 1. This c ea es a di icul y, because he “mission” o a
measu e o inequali y de ined on he uni in e al is o acco d 0 o pe ec equali y (maximal
equali y) and 1 o pe ec inequali y (maximal inequali y). In his no e we show ha when he
Gini coe icien is elici ed om a nea measu e o he agg ega e income- ela ed dep ession
o he popula ion ha consis s o he people who expe ience income- ela ed dep ession,
hen he ob ained Gini coe icien is “well beha ed” in he sense ha i is bounded om
abo e by 1. We conjec u e a eason o a d awback o Sen’s de ini ion, and we p esen
epe cussions o he usage o he “well-beha ed” Gini coe icien .
Keywo ds: Sen’s de ini ion o he Gini coe icien o income inequali y; Agg ega e
income- ela ed dep ession o a popula ion; A “well-beha ed” Gini coe icien
JEL classi ica ion: D31; D63; I31
1. In oduc ion
When he dis ibu ion o income in a popula ion is such ha one pe son ecei es all he
income, hen he Gini coe icien , as de ined by Sen (1973 and 1997), egis e s a alue ha
I am indeb ed o he e iewe s o hei sound ad ice and hough ul commen a y.
This no e is dedica ed wi h a ec ion and admi a ion o Ama ya Sen on his 90 h bi hday.
Oded S a k
[email p o ec ed]
1Uni e si y o Bonn, Bonn, Ge many
2Uni e si y o Wa saw, Wa saw, Poland
Oded S a k
is smalle han 1. This c ea es a di icul y, because he “mission” o a measu e o inequali y
de ined on he uni in e al is o acco d 0 o pe ec equali y (maximal equali y) and 1 o
pe ec inequali y (maximal inequali y). This is how Sen (1973, p. 8) p esen s he Gini
coe icien . “In any pai -wise compa ison he man wi h he lowe income can be hough
o be su e ing om some dep ession on inding his income o be lowe . Le his dep ession
be p opo ional o he di e ence in income. The sum o al o all such dep essions in all
possible pai -wise compa isons akes us o he Gini coe icien .” In addi ion, Sen (1973,
p. 6) commen s as ollows. “Undoub edly one appeal o he Gini coe icien , o o he ela i e
mean di e ence, lies in he ac ha i is a e y di ec measu e o income di e ence, aking
no e o di e ences be ween e e y pai o incomes.”
In his no e we sugges an explana ion o a d awback in Sen’s p esen a ion o he
Gini coe icien , and we p opose an amendmen . We show ha when he Gini coe icien
o income inequali y in a popula ion is elici ed om a “clean” measu e o he agg ega e
dep ession o he popula ion ha expe iences income- ela ed dep ession, hen he ob ained
Gini coe icien is “well beha ed:” when he income dis ibu ion is such ha one pe son
ecei es all he income, hen he Gini coe icien egis e s he alue o 1.
The income- ela ed dep ession o an indi idual is he dep ession ha he indi idual
expe iences when he obse es ha his income is lowe han he incomes o o he indi iduals
in his e e ence (compa ison) g oup. D awing on a nea measu e o he sum o he le els
o dep ession o he dep essed indi iduals u ns ou o be a way o ensu ing ha he Gini
coe icien ob ains he alue o 1 when one pe son ecei es all he income.
To begin wi h, we p esen he Gini coe icien in he o m speci ied by Sen (1997,p.31,
eq. 2.8.1). In popula ion N={1,2, ..., n},n≥2, le y=(y1, ..., yn)be he ec o o he
incomes o he indi iduals. Then, G, he Gini coe icien o popula ion N,is
G≡
n

j=1
n

i=1yi−yj
2n2y,(1)
whe e y=(1/n)
n

i=1
yiis he popula ion’s a e age income.
We can eplace he ep esen a ion in (1), which is based on uno de ed incomes, wi h
a ep esen a ion based on o de ed incomes, ha is, we can le he incomes be a anged
in ascending o de : 0 ≤y1≤y2≤... ≤yn.AsinS a kandBudzinski(2021), on no ing
ha
n

j=1
n

i=1yi−yj=2
n−1

i=1
n

j=i+1
(yj−yi), an equi alen ep esen a ion o Gin (1), he eby
elimina ing he need o ope a e wi h absolu e alues, is
G=
n−1

i=1
n

j=i+1
(yj−yi)
n
n

i=1
yi
.(1’)
On he basis o Sen’s ep esen a ion, he sum o he income- ela ed le els o dep ession
o he membe s o popula ion Nis ob ained in he ollowing way. Le IRDideno e he
income- ela ed dep ession o indi idual i,i=1,2, ..., n −1, whose income is yi.IRDiis
1062
Ano eonSen’s ep esen a ion o he Gini coefficien
de ined as
IRDi≡1
n
n

j=i+1
(yj−yi).
This cumula i e measu e collec s he income excesses o which indi idual iis subjec ed
( he “d i e s” o his income- ela ed dep ession) and hen (w ongly, as will be a gued
momen a ily) di ides he sum by he size o he popula ion.1
Le TIRD deno e he sum o he le els o IRDi:
TIRD =1
n
n−1

i=1
n

j=i+1
(yj−yi). (2)
On subs i u ing (2)in o(1’), we ob ain
G=
n−1

i=1
n

j=i+1
(yj−yi)
n
n

i=1
yi
=TIRD
n

i=1
yi
.(3)
Wi h his equi alence, when he income dis ibu ion is y=(0,0, ..., 0,y
n), hen
TIRD =n−1
nyn, and he Gini coe icien as pe (3)is
Gy=(0,0,...,0,yn)=
n−1
nyn
yn
=n−1
n.
To yield G=1, a co ec ion (mul iplica ion) by n
n−1is equi ed.
2. A e ision
This “p edicamen ” is no una oidable, howe e . The sou ce o he sho coming is de ini ion
(1): he denomina o he e should be 2(n−1)ny, no 2n2y. The eason o s a ing his ela es
o he logic ha unde lies he cons uc ion o he TIRD measu e. Speci ically, he “mission”
o TIRD is o collec he le els o IRDicon ibu ed by he membe s o he popula ion.
Suppose ha he op posi ion in he income hie a chy is occupied by a single indi idual.
In he g oup o hose who con ibu e o he “po ” o agg ega e dep ession we need no
include his op-income indi idual because his con ibu ion is ze o: whe eas e e y indi idual
1,2, ..., n −1 “collec s” and “deli e s” dep ession by obse ing a highe -income indi idual
highe up in he income hie a chy, he op-income indi idual has no one highe up in he
income hie a chy. In he ascending income dis ibu ion he e is no one o his igh . Thus,
because he does no compa e himsel o anyone, he is no , so o speak, “in he game.” I
is he case ha n−1 indi iduals con ibu e o he agg ega e “po .” The e o e, when we
1An in ui i e exposi ion o he o ma ion o his measu e and an accoun o i s “emb ace” by economis s a e
p o ided in wo appendices in S a k (2023).
1063
Oded S a k
assemble he con ibu ing compa isons (income di e ences), we collec hem om n−1
indi iduals, which leads o a “cleansed” TIRD ha we deno e by TIRD∗:
TIRD∗=1
n−1
n−1

i=1
n

j=i+1
(yj−yi).
Upon using TIRD∗and ew i ing (3) in e e se, we ge a “well-beha ed” Gini coe icien
G∗:
TIRD∗
n

i=1
yi
=
1
n−1
n−1

i=1
n

j=i+1
(yj−yi)
n

i=1
yi
≡G∗.(4)
When he income dis ibu ion is y=(0,0, ..., 0,y
n), hen each o he indi iduals whose
income is ze o has a le el o income- ela ed dep ession yn, so ha he sum o he “deli e ies”
o he le els o he income- ela ed dep ession is (n −1)yn. Di iding his sum by he numbe
o con ibu o s, which is n−1 ( his yields he nume a o o he middle e m in (4)), and
hen by he popula ion’s agg ega e income, yn, yields G∗=1; G∗is a “well-beha ed” Gini
coe icien .
This cons uc ion p o ocol un a els an asymme y be ween wo dis inc and dispa a e
me hods o no maliza ion: income pe indi idual (income pe capi a), which in i es d awing
on nas a denomina o , and income- ela ed dep ession pe dep essed indi idual, which
in i es d awing on n−1 as a denomina o .
3. Repe cussions
Rema k 1.Whywas he n
n−1de iciency o e looked? I seems ha he eason ela es o
he di e ence be ween he s anda d measu e o he a e age o a phenomenon ha is based
on he en i e membe ship o a popula ion, and a measu e o he a e age o a phenomenon
ha is based on membe s o a popula ion who a e esponsible o “p oducing” ( ha is, o
gi ing ise o) he phenomenon. I in calcula ing he agg ega e income- ela ed dep ession
we coun hose who a e subjec ed o income- ela ed dep ession and hen o mula e he
agg ega e income- ela ed dep ession pe con ibu o , hen he di ision needs o be by n−1.
We canno exclude he high-income indi idual om he calcula ion o income pe capi a
because his indi idual’s con ibu ion is “pi o al,” bu we can (and should) exclude his
same indi idual om he agg ega ion o he le els o dep ession o he indi iduals who a e
subjec ed o income- ela ed dep ession, as he is no one o hese indi iduals.
Rema k 2.Le y=(y1,y
2)be he ec o o he incomes o he indi iduals, and le hese
incomes be o de ed, 0 ≤y1<y
2.Then
G∗=y2−y1
y1+y2
.
1064

Ano eonSen’s ep esen a ion o he Gini coefficien
I y1=0andy2>0, hen G∗=y2
y2
=1. This is wha we expec he Gini coe icien o be.
Fo su e, his is be e - o he same magni udes o y1=0andy2>0 - han ha ing
G=
1
2(y2−y1)
y1+y2
=
1
2y2
y2
=1
2.
Rema k 3.Sen(1973 and 1997;1976;and1982) sough o measu e social wel a e by means
o he unc ion SWF, o mula ed as μ(1−G), namely as he p oduc o income pe capi a,
μ=
n

i=1
yi
n,and1−G,whe eGis he Gini coe icien as de ined in (3). Expanding SWF
while subs i u ing om (3), and hen inco po a ing he case o n=2, we ge
SWF =
n

i=1
yi
n
⎛
⎜
⎜
⎝
1−TIRD
n

i=1
yi
⎞
⎟
⎟
⎠
=y1+y2
2⎛
⎜
⎝1−
1
2(y2−y1)
y1+y2
⎞
⎟
⎠=3y1+y2
4.
I , howe e , we ew i e Sen’s SWF o he case o n=2 wi h G∗ins ead o wi h G, henwe
ob ain
SWF∗=y1+y2
21−y2−y1
y1+y2=2y1
2=y1.
This is an in iguing esul : in he modeled case, when embedded wi h he “well-beha ed”
Gini coe icien , Sen’s social wel a e unc ion coincides wi h he Rawlsian social wel a e
unc ion.2
Rema k 4. Le he e be a popula ion o i e a me s named R, S, T, U, V who a e a anged
in a layou such ha he a m o a me T occupies a high g ound, whe eas each o he o he
ou a ms occupies a sepa a e alley in alleys ha su ound he high g ound. Whe eas
a me T is obse ed by each o he o he ou a me s, hese o he a me s do no obse e
each o he . In a lean c op yea , a me T has c op y, while each o he emainde a me s has
c op 0. In his case, a me s R, S, U, and V expe ience c op- ela ed dep ession a he le el
o yeach and, hus, hey con ibu e 4y o he popula ion’s “po ” o agg ega e dep ession.
Fa me T con ibu es no hing. Di iding 4yby he numbe o a me s who con ibu e o he
popula ion’s agg ega e dep ession, which is 4, gi es y. Di iding by he a me s’ o al c op,
y, yields 1. In his cons ella ion, he Gini coe icien , G∗, is “well beha ed:” he mos unequal
c op dis ibu ion occu s when he en i e c op o he popula ion o a me s is ecei ed by a
single a me , in which case we will, indeed, wan he Gini coe icien o be equal o 1. We
has en o add ha his magni ude o G∗=1 is dis inc om G=
4y
5
y=4
5.
2Consul Rawls (1999), and S a k (2020).
1065
Oded S a k
4. In conclusion
The d awback iden i ied in his no e is no likely o be oo oubling when he Gini
coe icien is applied o la ge popula ions because hen n
n−1≈1. La ge popula ions do
no ypically ha e a Gini coe icien ha is close o 1. This is no he case, howe e , when
he popula ions conce ned a e small. And, a e all, a good measu e o inequali y should
ideally accommoda e popula ions o di e en sizes.
Al hough he need o and applica ion o “an n
n−1co ec ion” we e al eady ea u ed
a while ago in se e al pape s, o example, in pape s by Weine and Solb ig (1984)and
Del as (2003), he e is an ob ious di e ence be ween co ec ion and a oidance. The sou ce
o o igin o he d awback and he co esponding emedial ac ion iden i ied in his no e we e
no singled ou p e iously.
When in 1912 Co ado Gini de eloped a ma hema ical o mula o measu ing dispe sion,
he did so independen ly o social-psychological p inciples and p e e ences, his being a
sociologis (no jus a s a is ician) no wi hs anding. Su ely, Gini had no idea ha his index,
he Gini coe icien , would become “ he mos commonly used measu e o inequali y in
empi ical wo k” (Sen, 1973, p. 149); Ce iani and Ve me (2012) p esen an illumina ing
accoun o he hinking ha led Gini o o mula e his index. As implied by his no e, whe eas
in he cons uc ion o a measu e o dispe sion, symme y is a na u al a ibu e, his does no
ca y o e o he cons uc ion o a measu e o inequali y, whe e asymme y p e ails: in he
la e case - using Sen’s ocabula y - while he indi idual wi h he op income in luences he
dep ession o (in lic s dep ession on) lowe -income indi iduals, ha indi idual is no subjec
o dep ession in lic ed by lowe -income indi iduals. I is he exploi a ion o his asymme y
ha enabled us o de ine a “well-beha ed” Gini coe icien : o any popula ion, when he
income dis ibu ion is such ha one pe son ecei es all he income, his “well-beha ed”
coe icien akes a alue o 1.
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