Decomposition of the Gini coefficient-香川大学学術情報リポジトリ

(1)

Vo1.67， N 0..2， October1994， 343-354

*

F L 4 L

d

m

n.

コ

・

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白

・

1 e

c J V

、

J

m A

。

叩

. m

門 } ・唱 EA

m

G

D

K

ゐ_L

Y

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s

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t

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u

Y

okoyama

1 Introduction

The Gini coe伍cienthas been extensively used to measure income inequal -ityれItsmany properties are made plain by use of graph and algebra.. Especially the relation between the Lorenz curve and the Gini coefficient is popular and intui'tively clear.. In spite of such desirable characters

，

the Gini coe伍cientisn't always used to express subgroup decomposition of

income inequality. Thele are some di伍cultiesin decomposition.. To avoid trouble

，

the altemative way is to use decomposable inequality measures，

The entropy measure is useful in this case.. But since the Gini coe伍cientis the most popula.rmeasure

，

it is important to consideI the decomposition，

This subject deserves mOle than a passing notice，.

The Gini coe伍cientcan be decomposed into three sllbparts : between・ groups term

，

within-groups term札ndresidual ternL The implication of

residual term is di伍cultto understand. Numerous works have been made to explain the meaning of the terlll..Mookl悶 jeeand Shorrocks

[

6 )

argued

the qm、stionof decompositioll of illequality measures. Thei1 conclusion of resiclual term in tはh民eGinic<ω伍仁

d

ω

Je仙川lu凶L凶tおspe悶s邸叩s幻叩1凶1I

its tenn depended on income retanking，. Lm由。rtaば Aro凶 on[5) showed meaning of the resiclual telm f10m a graphical point of view，.

"'The earliel'vCl"sion of this papcl'was wIitten in ln) gl'みdllateschool days at KYllShu U niversity 1 am

(2)

344- Kagawa Universiか EconomicReview 578 III this paper

，

w巳 cOllsiderdecompositioll of the Gini coe伍cientand

its implicatioll To begill with we consider decomposition of the Gini coeffici日ntinto two sul】groupsぅillwhich we llSe matrix expression. 8uch

an operation makes it possible to illterpret the tenn.. Then仏1

，

we a凱n凶sw九νr位erthe

q

午伊u問le回S“凶tionra悶 dbyMo

∞

ok1悶 Jeea凶 8ho山刈王s

例

[

6

司

J.刊 T引l山 a1凶材ly戸Slおsma北.k(王es res司idua札1te引rm【cle況aL Finally

，

we give decomposition in the general case The form will give us one implicatioll of the residual terlll..

2 Decomposition i

n

the Two Cases

First

，

we consider that there are n members in a groupぅmemberi receives

Xi where

Xl :::; X2

:

;

ミ

Xn-l:::; .x"

Total group consists of two subgroups.. One is subgroup SI which has nl

members and small mean income μ1ぅandthe other is subgroup S2which

has rl2 members and large mean incomeμ2ぅ.ie μ1:::;μ2 In addition to

this

，

an overlap exists.. In other words

，

the richest member of subgroup 1 is wealthier than the poorest member of subgroup 2..Accordingly we clefine the over lapping setS

，

S = {XI

I

X

I

ε

1

5

2

2 (

1

2 )

ぅ

!

;

緊

(J:j)]}

alld we take SI as the llumber of members in SI alld S

，

simil礼r1y勺川 inS2alldS..Therefore we can clivide the total members illtO four parts

，

SlnScうSI

n

S

，

Sz

n

S

，

and Sz

n

SC.. The numbers of them aren1 -~h SI

，

82

，

and n2 - S2

，

respectively Clearly

，

the relatiolln = nl

+

rl2 llOlds 8econclly

，

w♂define new vect.orstlalldt2to express members of sub

-group in total group.. The rlX 1 vectort1 has Olle in ith position

，

if ith

incomel;i belongs to the subgroup 1

，

and zeros elsewhere..Itis the Sa1nE、

with

t

z

(3)

The matrix A is defilled出 min(Xl

，

Xl) 凶n(XhX2) min(白川 ) 凶n(X2'X2) lnin(xl

，

:Cn) mill(X2ヲ.Xl1) A = min(xn

，

Xl)min(xn

，

X2) . m n

z

z and vector in as 九 =[1

，

1

，

1

，

れ

，

1

，

1

，

1

'

1

:nxl

Now we can express the Gini coefficient by matrix and vectoIS. η2μ(1 -G) = i~Ain because of

)

噌_l & ( n n dμ(1 -

G

)

=

L

min(x;

，

xi) Next

，

we derive the between-group Gini coefficiellt Gゎillwhich ea( h of t

.he members in subgroup 1 has subgroup 1 meanμ1 and e(11出llveach of the members in subgroup 2 ge.ts subgroup 2 meanμ2 That is to白川口1 members getsμ1

，

and n2 members takeμz

7

1.μ (21 -

G

b)

=

(71.

f

+

2川口2)/1¥

+

Tl~f-L2 Then we can derive Go easily

，

η2μG_o= nln2(μ2ーμ

J

)

Next we cOllsider partitioll of matrix and vectors.. I .tn1-s.1 I 111 - 81 IOWS tl=lg¥

I

S1 +S2!OW8 L 0町一 S2J n2 - S2 rows I 0"1-8¥ I 111 - 81 IOWS t2=

I

g2

I

81 + 82 !OW8 L ，Zn2-s.2 I n2 - 82 l'OWS by definition the relations exists in the following 九 =t1

+

t2 (2)

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-346- Kagawa University Economic Rωzeω 580 and ~s l+ s2 = gl

+

g2 Likewise vectors

，

we consider 3 x 3 partition of matrix A12 A =

I

A21 A22 A23

I

81+ 82 IOW8 A31 A32 A33

I

n2 - 8

，

IOW8 111 - 81 81+ 82 112 - 82

columns column8 columns

By using this partition

，

the Gini coefficients of subgroup 1 and subgroup 2 are written as nbtl(l-GI)=t;Atl n，h2(1-G2)=tiAt2 (3) (4) Alternatively

，

w h U

一μ

n

一一

今 A 円ヤム凶 Q υ

一一

g Q L A れU Q リ μ ' n 叫，_L

“

υ 内 4 n

一一

Z 円ヤ臼凶内， b Q υ 内₄ n

一一

・命 z u

。。

A 内 ' e 。， h ・命 @ g'lA23in2-s2= (n2 -82)

L

Xi= (n2

一

句

)

ー

1戸 (7) x

，

ES

，

nS where戸iis the mean in品川ぅ附I吋 μ 士丸山c;:ri We consider the Gini coe伍cientinS asGin..Next between-Gini coe

伍-cientGc inS

，

namely each member inS

n

SIぅgetsequally income戸1

，

and

each member inS

n

S2gets equal income戸2 And the Gini coe担cientsδl

and G2 measure the income inequalities inS

n

SIand S

n

S2

，

respectively We can express these as

s

?

戸1(1-

Gd

=

g~ A22g1 4F2(l-52)=giA2292 (8) (9) Now consider the cross effect，. This is the counterpart to the between四 Gini coe伍cientin subgroups，. Suppose 81 members have each income TI_l_>

(5)

and 82 members have each income

i

I

2 If

i

I

1

三戸

2ぅ (81 + 82)(81

i

I

1 + 82戸2)(1-Gc) = (8i + 28182)戸1+siF2 else if戸1<戸2ラ (81十82)(81戸1+ 82戸2)(1-Gc) = (S~ + 2.'1182)戸2十81戸1 by a simple calculatioll (ぢ 1+ ち 2K~1戸1+ 82戸2)G

，

= 土 81~'2(Ti1 一戸2) (10) where + part of the土sighis used if戸1~戸2 alld -part is used if戸1<戸2 Next

，

the interactioll effect is defilled拙 (81 + 82)(81]71 + 82戸2)(1-Gin) i~I +S2A22isl+s2 = (91 + 92)'A22(91 + 9

Z

)

91A2291

+

29~A刈2

+

92A2292 and we solve for29~ A2292 by usi時 (8)and (9) 29~A2292 = (81十82)(81戸1+ 82戸2)(1-Gin) -sh(1-51)-d戸2(1-G2)

=一

(81+ 82)( 81

i

I

l + 82戸2)Gin +8182(戸1+戸2)+shul+si戸'2G2 (11) using (5)

，

(6)

，

(7) and (11)

，

so from (10)

2t~At2 = 2(i'AI291 + i'Al3i + 9~A2292 + 9~A23i) = 282(nlμ1 -81戸1)+ 2(n2 -82)(n1μ1 -81戸1)

+29~A2292 + 2(n2 -1>'2)81戸1

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-348ー Kagawa Univers#y EconomκRevieω = 2(η1r!2μ1 -SIS2"Jl1)ー(.'>1+ 82)(81戸1十82戸2)Gin +8182(戸1+戸2)+ 8i

i

1

jG1十siF2U2 = 2r!jn2μ1_一(81+82)(宮1戸_{1+ 8}2戸2)G_i_l₁ -5jS2(戸l一九)+shul+91戸2

G

2 2川η2μl一(81十勺)(δl戸1十82戸2)Gi7! 平(δl 十o92)(.~ 1戸1+ S2戸2)Gc+shul+83戸2G2 2nl向μl 一 ('~1

+

乃

)(.5j戸1+九戸2)(Gi1!土Gc) +イ戸jGj十品戸₂G2 P川 (3)alld (4)_うweωnget n2Jl(1 -G) i;

，

Ai" = (tl

₊

t2)' A(tj

₊

t2) = t'lAtj

+

2t;At2

+

叫

At2 =nh(1-GI)+niμ2(1 -G2) + 2ηIn2μl 582 (12) 一(SI十S2)(9ol戸1+ S2

i

1

2)( Gi " 土Gc)十8i戸l否1+ s~

戸

2

G2 Thell the transfonnation is η2μ_G₌_nip₁_GI_{+ n~þL2G2 +}_n_l_n₂₍_μ_I_十_μ₂_)-₂_n_l_r_!₂_f_-_l₁ +(81 + 82)(81戸1十82戸2)

(

G

in土

G

c)-si戸lEl-siF252 niμIGI + _n~þL2G2+ r!1r!2(μ2一μd +( 81 + 82)(宮市j+ 82戸2)(Gin土Gc)-8i戸151-d戸2G2 sゆstituting(2) in these =niμlGI+niμ2G2 + n2μG" + (81 + 82)(SI]11 + 82戸2)(土Gc+ Gin) -si戸252-si戸jGI Finally we get ncw decomposition of the Gilli coefficicnt n~ Ll I η2tlq G = 一千二GJ

+

一千三Gz+G" n"μ n"μ (13)

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(81

+

092)(81_~I-")戸)+匂_T_"_'₂

7

_/

I

_<2₂)₎

₍

_土

_G c

+

Gi1l) 11"μ

一

年

G)

一年百

2 (14) rI"μ Tl"μ more simply weιau write whele and

G=

α

jG)+

α

2 G

2 +

G

b

+

R

n?μz α

，

-

ー

τ一一 η&μ

R=

土

bG

c

+

b

G

in -

C

)

G

)

-C

2 G

2

b = (8)十82)(行声)

+

82戸2) 一 ηーμ .-cg=szμt -11"μ Although the 'dif五cultto interpretation termう Ris alwaγs regarded as iuteraction e丘ects

パ

heimplication of such e丘ectsis often obscure.. In this paper

，

we get more useful decomposition

，

which consists of four parts.. Two of them are the Giui coe侃cielltsof interactioll area of subgroup 1 and subgroup 2.. The term Ginexpresses 'pure' diverse illteraction.. The

te1m Gc explaills the mean di丘erenceeffects Itgives positive effect if

戸l三戸2or negative effect if戸)<戸2' And

G

)

alld

G

2 show the illCOme

illequalities ill overlapping intervaL They give the negative effe< tsぅifthe

IIIωme inequalities inι;1 ease in each overlapping area

¥V'e はu give one implication to the ‘ troublesome E'xample うp戸ropoω〉泌se伐吋dby Mo

∞

ok王.<恥hh民1erωe引lr巾 :

&ω11以吋cl(仁d、':1比tethe example to illustrate itι“ It has two subgroups

，

subgroup 1

{3

，

4

，

14} and subgro叩 2: {8

，

11}

，

出

ellG) = 0..349

，

G2 = 0.079

，

仙

dthe overall Gini coefficient is 0.29山Ifa redistribution of subgroup 1 illcomes resr山sin the ω w set ofincomes {1

，

7

，

13}

，

G) increasesω0.381

，

alld G₂is unchanged.. But the overall Gini coe伍cientdeclines to 0..27..Although this

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-350- Kagawa University Economic Review 584 example is difficult to understand intuitively

，

we Call disaggligate the e妊ect to some factors.. In case 1

，

S = {8

，

11ぅ14}alld ill case 2

，

S = {8ぅ11

，

13}

，

we show the vallles of terms in table Coefficiellt G Gj G₂ Gb Gc Gin Gj G2 Case 1 0..29 0..349 0079 0..075 0..091 0.121 0..000 0..079 Case 2 0.27 0“381 0079 0.075 0..073 0.104 0..000 0079 alld their weights are Coefficient α1 α2 b C1 C2 Cas.e 1 0..270 0.190 0495 0070 0..190 Case 2 0270 0.190 0480 0.065 0..190 Thell

，

we call express the overall Gini coe伍cientsby factors.. The case1 Gj = 0..270 x 0..349 + 0..190 x 0079 + 0.075 +0495 x 0091 + 0495 x 0..121 -0..070 x 0.000 -0190 x 0079 0.09423 + 0..01501 + 0.075 +0045045 + 0..059895 -0..00000 -0..01501 The casぞ2 G2 = 0270 x 0..381 + 0190 x 0..079 + 0..075 +0480 x 0073 + 0480 x 0.104 -0..070 x 0000 -0190 x 0.079 0..10287 + 0.01501 + 0075 +0.03504 + 0..04992 -0..00000ー 0.01501 This (:a1<ulatioll helps to ac(()unt for the result.The ex札mplecited hy them shows the overlappillg ratio is fairly large.. Therefore the ill.telaction Gini eoefficients and the cross Gini coe伍cientsin the area have a grea.tdeal of illfiu旬以e011 the.total Gini coefficient..In.this case

，

.the redistribution gives a gain in Gj

，

but it gives losses inGc and Gin..This change can be

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the decrease of illequality in over1apping area

3 Decomposition i

n

the General Case

1n the previous seιtiOllぅwederived new derompositioll of one-ovedapping

case引 Weconsider mOle general multi-overlapping cωe ill this sectiOll.. To expand the clecomposition， we use a di任erentnotation. There arek:(

<

n) subgroups

，

we denote ith subgroups 5; and consider overlappillg set of subgroup i and subgroup j.We express this as5;j

，

that is to saぅr

5ij= {Xl IXI

ε[

!ll}I，l(XI)

，

ln~l(xl)]} ヒ':>j ，XIヒ.)， whellμ12Eμ;， The llumber in the5i

n

5ijis expressed by t he 8がうallclthe mean in this area is clenoted by戸i.iG ijis the Gini coefficient ill

5

i

n

5ij Gc[iiJis the Gini coefficiellt whぼl早川 membersget each

i

1

ijillCOme alld S)i members get each戸jiincome.G川[ijJis the Gini coefficient ill5i_i 1 Similmly (12)

，

2 t

;

A

t

j

can beは pres;;eclas 2t;

At

i when i

<

j

，

and 2n;r1j min(μHμj) ー(o9iJ+ 5 ji)(o9i(Jlij十8ji戸ii)(土Gζ[iiJ+ Gin[iJJ)

+イ九百

ii+

ふ

i

1 ;

;

G

li

t

;

A

t

;

= r

山

内

(1-

G

;

)

By usi時 (15)<lllcl (16) η2μ(1-G) i'Ai

(

t

1 +t

2+ +

tdA(t1

十

t

2 ，

.

+

t

k

)

一

乞

t

;

A

t

;

+

L

2 t

;

A

t

j

i=l i<j (15) (16) lHe肘 al'esリ#勺;，ji'j#戸laj.ndGij:fi万'jiThe subs<riptijeXl沼田sest hc set丸n九剖，，1lt悶 l' Sj nヲリ AJtelllatively.G，[，jJandG叩

(10)

586 、、， a ' ' - 7 a 句 E ム

(

Kagawa University Ewnomic Review

L

，

n7μi(l - Gi)

+

乞

2ninjmin(μi

，

μj) ;=1 くy

一

ε

5 二(

仏

s

釘zりj+S勺

1 μ

i

)

い

(

8

釘8iけjjM丸戸gりj+8匂J'戸Mjμρ;)(土G c[ド初川行川jJ

+Gi

叫? '<J

+

2 :

8

3 F

i

1 U

Z

J

+

Z

8

7

3

1 Z

5

1

2

Zく '<J

-

L

，

η?μiGi

+

，

L

ninj min(μi

，

J-lj)

-L

，

(Sij

+

8ji)(8ij戸ij+8ji戸ji)(土GC[ijJ

+

Gin[ijJ) '<.J

+

，

L

8 ;

凡

Gij十

E

J

s

?

i

戸

1

1 E

1 t

'<.J '<.J The between-Gini coe伍ιientis expressed as -352-η2μ(1 -Gb) =

，

L

7?j7?j mi

叫

μiぅμj) (18) From (17) and (18)

L

，

71;μjGi+712μGb

+乞

(Sij

+

Sji)(8;jMij

+

8ji戸ji)(土

Gc

[

i

j

J

+

Gin[ijJ) '<.J

-乞

sLR1U27-Zs?

ん

Gji '<.J z<.J

G

μ ' の L n ( 19)

(

2

0 )

The general form is G = 2

乞

:

α向_iGi+Gb+2

ε

=

b

ん_'_J_バ_(土G_c_[_ドη2_川_似_川1けJ + Gi州 ? i=1 i<i if.j αi=-

今

t η&μ where b;;.-_ー (8ij

+

Sji)(8ijMij

+

8ji戸'ji) 71"μ

S

J

;

_μ" Cii ~ _ηρμ The residual term Now we get the decomposition of the Gini coefficient

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is largely e狂ectedby the condition in overlapping area川 Oneof them is the product of population share and income share in overlapping ar伺 " The others are the cross e妊ect

，

the illteractioll effects and diversity in overlapping area 1n the k deviation case

，

the number of Gc

，

Gilland G are k(k -1)/2

，

k(k-1)/2 and k(k -1)ヲrespectIvely.. Th児ぼ悶e哩refo侃rE'the位reaωω出r 2k(k一1

吋

)

ef妊fectsin residual term

4 Conclusion

We have derivecl a llew clecomposition of the Gini coe伍cient.. Although

孔 largellllluber of studies have heen made on the illtelpretation ()f the

Ie1:;iclual telm

，

most of studies giw weight to income lerallking. While thele

lSλfairly gelleral agreement that int位以:tiOlleffect increases the value of the residual termぅnostudy has ever tried to express the interaction effect

COllcretely.. 1ll this paper we clerivecl new clecomposition from allother point of view

，

ancl got the form of the resiclual term.

The resiclual term is infiuencecl by the overlapping Iatio ancl the clegree of the cliversity of income. Such a shape is composed of two overlapping ωmbinations.. However the residual telms have a fine form

，

it is too com-plicated to state the economic implication clirectly. The overlapping is more limited in practice

，

but we cannot avoicl the complexities whell the number of groups increases“ Because if the cleviation increases

，

the num-ber of interaction effects increases geometrically The resiclual term has an original evaluatioll to the overlapping area.. lts evaluation is based on the combination of two subsets Although the Gini coe伍cielltis not so strange to consider income inequalityうtheresiclual term has a complica

-tion to conside1 effects intuitivel

y

.

In particular

，

when there are many subsets and overlapping p紅白ヲwecannot neglect that the Gini coefficient

(12)

-354ー Kagaωa U

，

仰 ers#yEconomic Review 588

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，

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ヘ

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。

fEC0110111icTlleory 2

，

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，

(1967)

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，

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，

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，

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，

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，

1 J

t

.

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，

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，

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，

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，

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，

(1982)

“

A Decomposition anal -ysis of trencl in U K Income Illequality'¥The EconolllicJoulna1 92

，

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“

The illterpretation ancl clisaggregation of Gini Co-e

伍

.

c

i

ellts"

，

The Ecollomic Jou111a186

，

pp.. 243-255 [8] Silber J.

，

(1989)

“

Factor Componellts

，

Population Subgroups allcl the Computation of the Gini Inclex of Inequality"The Reviev;r of Eco-llomicsand Statistics71

，

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