THE GINI mEFFICIENT Al\JD POVERI'Y INDICES:
SOME RECONCILIATIONS*
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1484
Hay
1985
THE Gnu COEE'FICTI:N.I' AND POVFRI'Y UIDlCES
SCNE RECC1'1CILIATICNS
*
By PRAI\JAB KlJMAR SEN
'The Gini coefficient plays a vital role in the fomulation of the
poverty indices. 'ITI'-transfonna.tions are adopted to provide a new interpretation of the Gini index, and the
SarrE
is incorporated in the formu-
lation of a robust version of the Sen poverty index.
1. TIJrRODUCTION
Poverty is usually defined as the extent to which individuals in a cammunity or society fall bela'! a minimal acceptable standard of living. l'n index of
poverty is generally based on tl1e distribution F (y), Y E p+ = [0,00), of the
ineome variable
=
a
(1.1)
and a set poverty line
F (w)
=
w
(> 0), so that
proportion of people 0010;'1 the poverty line
may be taken as a crude index.
'Ihe
w
income gap ratio of the IX'Or people may
tllen L'e defined by
(1.2)
-1
= 1 - w
[3
Generally,
{a
-1.Lll
.'0
ydF(y)} •
l..xJth a and S are taken into consideration in the formulation of a
rreaningful poverty index. Sen (1976) considered the following :
(1. 3)
TI
S
where G ( 0
a
at w
a{S + (l-[3)G } ,
a
=
~
G .::. 1) is tlle Gini coefficient of the incoEe distribution censored
a
(as will be defined later on ). Though this index has been used extensively,
there remains sane issues relating to its nonnative content. '::::'ak.ayarna (1979),
incorporating a sarewhat different set ofaxia'lS, has argued that a proper
~asure
(1.4)
of the poverty is
TI,.,
1
=
G ,the Gini coefficient for
a
a - ~ (x), 0 < x < w.
l'ihereas, Sen (1976) aclvocated that an index should be given by tl1e
~.veighted
agrregate gap of tl1e people belo;,v the poverty line, the 'l'akayaw.a (1979) index
* ~Jork partially supported by the Office of Naval Research, Contract qo. E0001483-1'-0387.
-2-
is based solely on
t.e~
inccr,l8 inequality of the censored incare distribution
tnmcated fran above by w • Blackorby and Donaldson (1980) have suggested another
index
(1.5)
'JT
BU
=
which Joes not involve the Gini coefficient G explicitly..1'm obvious drawback of
ex.
(1.4) is its insensitiveness to
pretations in
SaTE
ex. , and this may lead to ratJ1er misleading inter-
..
cases. For example, if all the people belcw the poverty line
have an equal incane, say, Yo ( 0 .::. Yo ::.. w ), then Gex. = 0, so that TIT = 0, irrespective of the particular y
heavy unerllployrrent (where y
and ex. • A situation of this type may arise due to
o
o
is close to 0) or due to sore social welfare system
providing indirect incorre below the poverty line and therety leading to sorre
clusters in the censored incare distriliution; in both the cases, G (and hence,
ex.
TI'l') may be a rreaningless entity. As we shall see in section 2, the indices
B and
Gex. are interrelated, and hence, all the three indices TIS' TIn:' and TIED are dependent
on the Gini coefficient G
ex.
all of these cases.
~1ote
in sane way. But, t."le role of G is not Hie sarre In
ex.
that
TIS':" TI
BD
but TIS and
TIT ( or TIT ans TIBD ) may
not be canparable in this way.
'Ihe object of the present study is to focus on the role of C;a
111
the for-
mulation of poverty indices. This examination leads us to a More robust variant
of the Sen poverty index , viz.,
(1. 6)
TI*
= ex. ( BI-Gex. ).
In tilis context, the Gini coefficient is critically examined in Section 2. 'I'1le
main results are then considered in section 3 and sane general remarJ<-.s are also
appended there.
2. THE GINI COEFFICIENT : A FRESH LCOK
First,
,'18
consider the Gini coeffici8Ilt G for the uncensored distribution F.
VJe assurre that
(2.1)
).l
~y
w
= f 0 ydF(y)
is finite.
e
~
-3Thus, defining (the survival distribution)
00
(2.2)
Let
= fa
}l
1 - F (x),
X
£
R+,
,,~ have
F(y)dy.
then
-1 (t)
(2.3)
F
(2.4)
set)
t~
F(x) =
_
=
inf { x : F (x)
=
.2
t } , a .::. t .::. 1,
y-l frl(t) ydF(y) ,
a<t<l.
may consider the Lorenz curve :
S (t)
JErl------- LDrenz curve
j.AIlI::~-----~--~t
'!hen, in terms of the shaded area A , we have
(2.5)
G
Analytically,
(2.6)
G
=
the Gini coefficient
2A •
G may also be expressed as
= (2)l)-~1
Y - Y2
l
I,
where Y and Y are independent randan variables, both having the sarre distril
2
bution F. Note that
where a" b and a I\b stand respectively for the rnaxirnurn and I!"inimum of a and b.
Therefore, by (2 • 6) and (2. 7), we obtain that
(2.8)
G
=
(2y)-1{ E(Y VY ) - E(Y "Y ) }
I
2
l
2
y-1E (Y " Y )
2
l
-1 00
1 - 2y fa y F (y) dF (y) •
1 -
=
At this stage, we borrow the concept of total time on test CTTT) fran
reUabi-
Uty theory [ viz., Klefsjo (1983) ], and consider the scaled-TTT transformation:
(2.9)
In
<P(t)
= )l-lf~-l(t)
F(u)du
= f~-l(t)
this context, we may also note that
F(u)du/
f~
F(u)du , a < t < 1 •
-4+
(2.10)
X E R
is called the equiUbriwn renewal distribution corresponding to F. Furtl'.er, by
(2 • 4 ) and (2 • 9) ,
=
<p(t)
(2.11)
l;(t) + lJ-l(l-t)F-l(t) , t
E
[O,lJ.
Let then
(2.12)
<P
=
f~
<p
=
1.1
f~
<p (t) dt =
ri'F (x) dF (x) = scaled-'j'TT mean,
so that
(2.13)
-1
x-
00
fa {fa
F(u)du}dF(x)
=
1.1-1{ ff
=
-1 r=--2
1.1 {J
F (u)du }
F(u)dudF(x)}
a<u<x<oo
o
=-
2 1.1
-1
-1
00
00
=
2 ]J
=
1 - G •
-
-
{fa uF(u)dF(u)}
{fa
-
u 1" (u) dF (u)
'rhus, the scaled-TilT rrean
}
¢ and the Gini coefficient
G
are ccxnplenentary to each
other. 'Ihis representation not only provides additional insight into G, but also
is quite useful in our sUDsequent analysis.
rm extend the result to the censored case, we set
(2.14)
F (x) = a -~ (x) ,
(2.15)
F
(2 .lb'-)
]Ja
a
=1
(x)
a
= fooa
- F (x)
a
....:n. ( )
~~ x
a
< x <
=1
Vx >
a -
(F (x) = 1,
W
- a-
~ (x)
= fW0 x dF a (x ) =
, 0 < x < w,
-
-
F
a
W
)
,
= 0, 'd x
(x)
_>
llJ
..-'n.'. ( )
a -1 fW0 ~
y •
Then, Parallel to (2.10), we have
(2.17)
'IF (x)
a
=
]J-lfX F lu)du
a o a
=
a-\-lfxo{a -F(y)}dy , 0
a
2.
X
_<
W,
so that
(2.18)
<p
a
=
=
1 - G
a
=-
lJ -1 fWO y&2 (y)
a
l
2 lJ- fW yF (y) dF (y)
u a
a
a
=lJ
-1
a
a
E(YlI\Y2)'
a
a
where Y and Y
are indePendent randan variables , both having the ccmnon
a2
al
distribution F
a
• Note that by (2.18),
-5-
= ]J a (l
(2.19 )
(~ .16)
v"hile, by (1. 2) and
B
(2.20)
= 1 - w-l]J
a
,
= 1 -
(l-G)
-1
-1
w
a
~
( y a 1 A Ya 2)'
so that
(2.21)
(I-B) (1 - G
a
Obviously,
<
(2.22)
note that y
Y
< (I-B) V (l-G )
a -
a
=
(w
-1
]J) V qi
a
a
<
1 .
is a scale-free neasure of the inequality of the censorOO inCcr:1e
a
distrilJution Fa. Clearly, when B = 0 (Le., ].la = W ), Fa has the unit
III
,
so that
-
epa
IY1aSS
at
= 1 (Le., Go. = 0 ), and hence, Ya = 1. Similarly, Ya = 01:.vhen
p -- 1 (i.e., ]Ja = 0). In the other extren-e case, where Fa has the unit mass at
-1
sore inte:r.rre(~iate point Yo (0 < yo < w ), Go. = 0 and hence, Y (= 1 - 6= w Yo
a
depends solely on the ratio
index.
~.,€
w-ly
o
• This reflects the desirability of Y as an
a
shall make rrore came.nts on it in the next section. Tn passing, we
may remark that
(2.23)
Y
a
= 2.a-1 J1 u { 1 - a -1r(uW) } dF(uw) ,
0
* stand
so that if there are m people below the poverty line w and if y 1* .::.•••..::. Yill
for the ordered values of their incares, we have
(2.24)
-1 -~ IT,
*
Yo. = 2 w f:l
2:i~l (m-i+l)Yi
(2.25)
]Ja = m
(2 • 26)
~
-1
G
a
'
*
ill
2: i =l Yi '
= ( 2:.m
1
l=
,.,n1
L.
J=
1
Iv.* - V.*1 ) I (2m
l
-J
1
.-JTl
L.
l=
1
*) .
v.
'l
shall find these expressions useful in our subsequent analysis.
3. POVERrY Th"'DlCES : ROBUSTJ.FICATION
'Ib start with, we may note that by (2.10) and (2.16),
(3.1)
]Ja =
=
rl
- a
-1
w -1
Ja xdF (x) = - a {
]Ja-l~,(w) + w - wa- l
llerefore, by (1.2) and (3.1), we have
w M[xF (x) J 0 - J 0 F (x) dx }
-6-
S = a- l { 1 -
(3.2)
llW-1TF(W)} ,
so that by (1.2), (1.3), (2.21) and (3.2), we have
a {S
(3.3)
+ ( 1 - 6)G
a
}
a {l - (1- S ) (1- G )}
a
=
=
a{ 1 -
Y }
=
a{ 1 -
W E(Yal/\Ya2)} .
a
-1
'Ihis shows that in the sen poverty index, both w-llla (= 1-6 ) and cPa (= l-G
a
play an equally irrportant role. 'lms feature is, hCMever, not shared by the other
biO
indices ,
~
a
(j.4)
7f
Note that Y
a
and
T
7f
7f
. 'Ihis also reveals that
ED
> {(l- (l-S»V(l-(1-G » }
a
S
=
a{SVG
=
7f
=
7f
}
a
V (a. 7f )
T
BD
BD V
7f
*
say.
T
is small if either B or G is close to 1 , while if both S and
a
a
7f S sare~vhat
are close to 1, then Y is of second order srrallness. r:;his rray make
a
rrore inflated when both
~;
S and G are close to 1 , a case that may arise when
a
there is a higher degree of poverty. 'Therefore, there seeIt's to be a genuine need
to curb the sensitivity of
7f S
S and Ga are close to 1.
when both
tiJis point, we consider a hypothetical case where B
0.0625.
*
~en
7f~ =
0.75a,
7f
BD = 0.75a while
7f
S
=
G
a
=
'Tb illustrate
O. 75, so that Y
ex
= 0.9375a
and
'Ire insensitivity of 7frr to a makes it sareY-nat less appealing than
while
lla
=
7f
S
and
7f
BD
0 CLe., B
Also, for
S=
=
7f
T
=
= 0.75.
7f~
or
7f
BD
,
differ by as much as 25% • vIe have noticed earlier that for
1 ), Yais equal to 0 , so ti1at
0 , we have G
a
=
0 , so tilat both
7f
7f
s
S
and
and
7f
BD
7f
ED
both equal to a .
are ecmal to O.
Consider a third case where the censored inCCITE distribution Fa has two clusters
at 0 and w with respective probability masses l-p and p. In such a case, we
have II
a (l-p)
a
=
P
= 7fT*
<P
a
• 11ms
=p
, B = 1 - P and Y
a
7f
BD
= 7fT*
= p
2
,so that
7f S
2
= a (l-p ) and
7f ED
and the extent of divergence depends on p .
=
-7-
* is rrore appealing than the Takayama index
It also reveals that
TIT
Before
TIT.
we present a roLust version, we consider the folla.ving:
Theorem I . For every a : 0
a
(3.5)
< 1
< a
>
>
and every censored distribution F ,
a
>
lJhere aU the three indices are equal to
FoY' 0 <]J
a
and
< (;J
TI
o,
>
a or 0 when 1J
a
are different from 0 while
BD
equal to 0 or
1-S
*J.
TI~
(;J.
o.
may be equal to
Proof. By virtue of (3.4), to prove the set of inequalities in (3.5), it suffices
to show that
(3.6)
G ,for every F and 0
a
a
>
< a
< 1.
'I'ovJards tllis, note that by (2.18) and (2.20),
(3.7)
(1- S)/CI-G
)
a
= -1J a2/{wU (;J-2
yF ly)dy}}
O
(;J -
= ( !0
Fa, (y) dy)
so t,.'1at on using the Cauchy-Schwarz
2
(;J
I {(I 0
. (;J-2
dy ) ( ! 0 Fa (y) dy )},
inequality on the nunerator on the right
hand side of (3. 7), we imneiliately obtain that
<
(1- 6) / (1- G
(3.8)
a
1 , with tlle equlality sign holding only in the
case of degenerate F ,
a
and (3.8) ensures (3.6).
TI
TI
:=:
S
BD
=
a
~Vhen
=
Also, for
•
]Ja
]J
=
a
0 , we have B
a
andG
a
a
=
a ,
< lJa<w, we have 0 <
W ,
we may have G
a
a
S
a
=
0 , so tllat
is not properly defined. Conventionally,
*
equal to 1, and tllis will make t.1Len
so that Y
1 and y
0, there is a perfect equality of incOHE of the
p::-!ople below the poverty line, but G
we may let G
a
=
= 1 and ~-ve have
< 1, so that
TIED
TIS =
TI
TI~
= a • For 1J
=
TIT
.1
BD
*
a
=
W, B= 0
= 0 • ~lote that for
> 0 • On the other hand, for 0 <]J
arbitrarily close to 0, rendering
TI rn* also close to
o.
-1
Yo ( > 0)
,while
G
a
Hill be equal to 0 , rendering that
=
TIT* = 0 • Q.E.D.
Coming back to tile inequality in (3.6), it is not difficult to construct
suitable censored distributions F
is particularly true
small, vvhile
[3
~"lhen
F
a
a
, for 'Which
BIG
a
<
For
example, i f Fa has tlle unit mass at the point YO (0 < YO < (;J ), we have S
1 - (;J
a
may be CJUite large. 'l'his
has one or few clusters 'Whid1. would make G
a
cruite
need not be. As such, fran the robustness considerations, we
-8-
* may not 1::Je a gocxi canpetitor to
feel that
TIT
or
TIS
mind, we would like to construct a robust variant of
TI S
is bounded fran above by
a and frau 1::Jela-v by
• With this picture in
BD
TIS.
TI ED
' where the two bounds
S
while, for
B closer to 1, the upper bound is closer. In (3.6), we have further
is equal to 1. For
to 0 ,
moves more ta-vards
TIE
TI
BD
,
B -> Ga • i'loreover, we may rewrite (1. 3) as
.
(3.9)
'Ihus,
is an weighted arithrretic mean of the two bounds a and
TIS
TI ED
' \<\nere
the relative weights are given by the Gini coefficient and its cCInpleI"'el1t. T-lhile
we do not want to change the role of the Gini coefficient in this context, we
t.~e
advocate the use of
gearetric mean instead of the arithmatic mean, and
thereby, propose the folla-ving index:
(3.10)
TI
* = VJeighted
a Ga
=
Note that as G
a
TI
gearetric mean of
1 - GNu.
a
and
TIBD
= a U3 1- Ga )
BD
goes to 0 (or 1),
TI*
converges to
1T
BD
( or a ) • Also, using
the 'I11ell kno.vn inequality 1::Jetween the arithrretic and gearetric means of nonnegative
numbers, we readily obtain frau (3.9) and (3.10) that
(3.11)
TIs':::'
TI*,
where the equality sign holds when
Ga
is equal to 0 or 1, or when
a
=
TIED •
Also, by virtue of the set of inequalities in (3.5),
0.12)
<
a
where the equality signs hold when either G is 1 or 0 , or a
a
fran (3.5), (3.11) and (3.12), we obtain that for every
censored distribution F
a
(3.13)
->
TI
s ->
a
TI*
It is the median ordering of
TIS
or
TI
BD
a :
= TI BD . Therefore,
a
< a
< 1 and every
,
>
TIED
TI*
>
T
>
o.
that makes it more appealing as an index than
• 'Ib illustrate this point,
G and canpare the three bounds.
a
TI*
,-ve
consider
e
Note that by 'Iheore.J11. 1,
agree \vhen
noticed that
S close
TI
SCJIre
typical values of f3
and
-9TABLE 1
Table for the values of a
_.~--'-_.'
-1
TIs'
u
a -1TI* and a
-1
TI
BD
for
typical (8 ,Go.) •
SCIre
.~
B
a -1TIS
G
a
a -1TI*
a
-1
TI
BD
r
1-1.
0.1
0.1
0.05
0.10
0.145
0.190
0.112
0.126
0.100
0.100
0.2
0.2
0.10
0.20
0.280
0.360
0.235
0.276
0.200
0.200
0.3
0.3
0.20
0.25
0.440
0.475
0.382
0.405
0.300
0.300
0.4
0.4
0.20
0.30
0.520
0.580
0.481
0.527
0.400
0.400
0.5
0.5
0.30
0.40
0.650
0.700
0.616
0.660
0.500
0.500
0.6
0.6
0.40
0.50
0.760
0.800
0.736
0.793
0.600
0.600
0.7
0.7
0.50
0.60
0.850
0.880
0.837
0.859
0.700
0.700
0.8
0.8
0.50
0.60
0.900
0.920
0.895
0.915
0.800
0.800
0.9
0.9
0.60
0.70
0.960
0.970
0.959
0.969
0.900
0.900
0.95
0.95
0.70
0.80
0.985
0.990
0.985
0.990
0.950
0.950
clear picture errerges for
range of variation of
as a middle runner between
TI;
B and
G •
a
~ote
have limited ourselves to values of G
a
clear that for small values of Go.
Go. (and S ) close to I ,
aspects of
BD
over the
that by virtue of (3.6), in Table 1, we
less than or equal to B • It is also
(and B ), TI* is closer to
is closer to
TIS
than
TIED. Tb
TI
RD
, while for
stress the robustness
has two clusters at 0 and w with respective probability
masses l-p and p, so tllat
TI*
TI
n* a bit fcore, we consider again the situation vlhere the censored
incoEe distribution Fa
have
TI*
and
TIs
= a(l-p)p
TIc:
= (l_p2) and
TIED
= (l-p).
(~a(l-p2 (1+p(1-p)/2)). 'Ihis soo..vs that
lJy this clustering effect than
TIS'
As
TI*
here I-Go. = p, we
is less affected
though they are generally close to each other.
'rhis aspect rrerits consideration,particu1ar1y when a heavy une.rrployment daninat.es
-10-
the J.X>verty picture. In any case, B , the inc<::lJ1:'e gap ratio is not very sensitive
to the ince>rre inequality , so that
inequality, while G
a
rrakes
lx>th
TI~
.i
TI
BD
is not that sensitive to the incare
is not that sensitive to the inCCFe gap ratio, and this
rather insensitive to the incare gap ratio. Looking from this perspective,
TIS and
TI* take into accoill1t both
fluctuative than
TIC'.
.::>
S and Ga ' and between the two, n* is less
On this groill1d, we reccmrend n* as a proper povertv index.
...
Universi-ty of North Carolina" Chapel Hill
REfERENCES
[1] Blackorby, C. and lXmaldson, D. (1980). Ethical ind.ices for the
m~asurer:'ent
of
poverty. Econometrica 48" 4, 1053-1060.
12J IQefsjo, B. (1983). Se>rre tests against aging based en the total tirre on test
transform. Commun. Statist.- Theor. Meth. l2" 8" 907-927.
[3]
sen, A.K. (1973).
On
Econom1:c Inequality. OXford Universitv Press, lDndon
[4] Sen, A.K. (1976). The rreasurernent of poverty: An axiomatic approad'l.
Econometrica 44" 1" 219-232.
[5J 'Takayarna, H. (1979). Poverty, inccxre inequality and their rreasures: Professor
Sen's axianatic approach reconsidered. Econometrica" 47" 3, 747-760.