Kcr id Bank

Development Resea rch C e n t e r

D i s c u s s i o n P a p e r s I No. 29

THE DIST?.IBLITION O F INCONE BY FACTOR COMPONEN S 1 Graham P y a t t

World Bank

Chau-nan Chen The I n s t i t u t e of t h e T h r e e P r i n c i p l e s of t h e Peo l e

Academia S i n i c a , T a i p e i I and

J o h n F e i Y a l e U n i v e r s i t y

, 5

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SOTE: isc cuss ion P a p e r s a r e p r e l i m i n a r y m a t e f l a l s c i r c u l a t e d - d i s c u s s i o n and c r i t i c a l comment. R e f e r e n c e s ir p u b l i c a t i o n P a p e r s s h o u J J b e c l e a r e d w i t h t h e a u t h o r ( s ) t q p r o t e c t t h e of t h e s e p a p e r s . The p a p e r s e x p r e s s t h e viewsLof t h e a u t h o r i n t e r p r e t e d t o ref les t t h o s e of t h e World Bank.

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th authors

Revised, Eeb uary 3 , i979. I The Distribution of Income

by Factor Comgonents

1 / Grahzm Pyatt -

and

John Fei 3 / -

L

1 / Developcent Research Center, World Bank -

2 / The I n s t i t u t e of the a r e e P r i n c i p l e s of t h e People, .kcsden.a - - 3 / 'Tale University - E

I

? r s l k i ~ a r ; r zr.2 C c n f l 2 ~ r . t l a l : ':ot f c r yuoria:,c?. or z:=rikctrlcn - cl=ara~.c.- f r o 2 :he s-zhcrs. '.'le-;s exressea are t , c s e a f :he z -.-- - ,

- - ' i i

- . . Siarca , ~s:ie: t

t w

-;icr.cct -ri;r

..,, necessarilj- refizct tile -.-iz~s o r t n e Xor ld Sar,k.

I n a r e c e n t paper : "Grcwzh 2nd t h e F a n i l y D i s t r i b i ~ t

by F a c t o r C o ~ p o n e n t s " , Joilr. F e i , Gustav Ranis and S h i r l e y Ku

have s e t o u t a methodology f o r Che s t a t i s t i c a l decomposi t ion of fa in i ly

incomz i n e q u a l i t y i n t e n s of such components of f a m i l y inco e s a s wages, I income from a g r i c u l t u r a l a c t i v i t i e s , p r o p e r t y income, e t c ; . . e i r

a n a l y s i s t r a n s l a t e s t h e s t a t i s t i c a l decomposi t ion of income ' n e q u a l i t y i n t o P rceaningful ecsnomfc components c f growth and i n e q u a l i t y t o

?zapping from t h e s t a t i s t i c a l magnitudes i n t o t h e concep tua l

o f economic t h e o r y . A methodology which a l lows t h i s t o be

c l e a r l y of g r e a t i n t e r e s t f o r any a t t e m p t t o under s t and t h e

economic development. The F e i , Ranis and Kuo (F-R-K) paper

The measure of i n e q u a l i t y used is t h e G i n i c o e f f i c i e . t . L inks w i t h i

a numer ica l i l l u s t r a t i o n of t h e methodology w i t h r z f e r t - l c e to

f o r t h e pe r iod 1964 t o 1972.

i h e L 5 r u curve make c h i s c o e f f i c i e n t an a t t r a c t i v e s r a t i s t i t f o r t h e

Taiwan

purpose a t hand, and r e c e n t c o n t r i b u t i o n s t o t h e l i t e r a t u r e a t h e G i n i n c o e f f i c i e n t have added t o our c n d e r s t a n d i n g of i t s p rope r t i e s1 and cha rac-

t e r i s t i c s . The p r e s e n r paper i s d i r e c r e d towards f u r t h e r i n g l h i s

unders tanding i n t h e c o n t e x t o f t h e P-X-ii a p p l i c a t i o n , w i t h r t i c u l a r s , I

e a p h n s i s on t h e i s s u e s which a r i s e when d a t a f o r i n d i v i d u a l r'

~ o t a v a i l a b l e and t h e decomposit ion of fami ly incone has t o b

srouped d a t a of t h e type t h a t a r e r e a d i l y a c c e s s i b l e i n ? u b l i

I .b a s p e c t of t h e p r o b l e n s r a i s e d by grouped d a t a h a s p r e d i o u s l y been

groupsd d a t a a r e used a s a b a s i s f o r t h e d e c o n p o s i t i o n o f t o t a l i n c n e i n t o I

d i s c u s s e d in t h e l i t e r a t u r e . S p e c i f i c a l l y , G a s t u i r t h (1972) h a s

t h e e r r o r s i nvo lved when a GLni c o e f f i c i e 2 t i s c a l c u l a t e d from grouped

d a t a a s opposed t o i n d i v i d u a l o b s e r v a t i o n s . The p r e s e n t a n a l y s i s

t h a t t h i s p r o b l e n i s on ly a minor f a c e t o f t h e i s s u e s which a r i s e

a d d i t i v e f a c t o r c o n p o r e n t s . The n a r u r z of t h e b roade r i s s u e s has b en t h e i

analyzed

shows

w-len

s u b j z c t o f exchanges between t h e p r e s e n t a u t h o r s f o r sone t i n e .

r e s u l c s i n an e a r l y round of t h e s e exchanges , Gary F i e l d s (1977) h a

exanined t h e problems usir ig some 1967-8 d a t a f o r Colombia. He u ; i t ~ / s :

"lT'ne e v i d e n c e ] s u g g e s t s t h a t f o r t h i s p a r t i c u l a r decomposi t ion

k d t h t k i s p a r t i c u l a r t y p e o f grouped d a t a , t h e o p t i o n of do ing

a l l r a t h e r t h a n u s i n g what i n p e r i z c t d a t a w e have d e s e r v e s s e r i o u s cbn-

s i d e r a t i o n . " T h i s i s a s t r o n g conc lus ion v h i c h r - q u i r e s a p e r s p e c t i .e. i.

f z ~ i L i e s ~ n i c h h a s r e c e r i t l y b e c c ~ e a v a i l a b l e .

Zn expos lng t5-e p r o b l a a s w h i c h a r i s e vhen a Gizi z o e f f i c i n: is I decocposzd i n t o ac id i t i ve f z c t o r c c c p c c e c t s , t h e p r e s e n t a n a l y s i s b u r d s o c .1

r e

10 a s t a t i s t i c known a s t?ie c o n c e n c r a t l o n r a t i o . Sore a u t h o r s ,

t h e s t a c i s c i c a l l i t e r a t u r e , u s e t h e t e " c o n c e n t r a t i o n r a c i o " s:.vnonl//cously - - vich r h a t of a G i n i c o e g i c i e n : . iio.;ever, Ln t h e d e f i n i t i c n s f t h e f n e r

i b s e t out ir s e c t i o n 2 b s b w , t h e c o n c e n t r a t i o r , r a c l o i s a r o r e g e n e r a l

* -.~i:h res;?c: cc vhicn :5e Gizf c c e f f i c i e s : is ; s ~ e c l a l csse . So r c o

c m c e p z

I s

I t5.e pseudo-Gii i i c o e f c i c i s n t xhic.h p laya such an Papor t an t t i n che F-it-K

I ~ e i t t o d o l o g y . Fcveve-r, t r u e G i n i s and p ieudo-Gin l s a r e n o t t h I= same t h i n g ,

1 1 / and a c l a r i f i c a t i o n of the d i f f e r e n c e i s c e n t r a l t o the p r e s e t d i s c u s s i o n .-

me n e x t s e c t i o n ( s e c t i c n 2 j of t h i s paper provide a d e f i n i t i o n

I of t h e c o n c e n t r a t i o n r a t i o f o r a v a r i a b l e r v i t b r e s p e c t t o ( s o n e o t h e r

v a r i a b l e , t , when b o t h z 2nd t a r e observed f o r each hou eho ld . The I G L q i c o e f f i c i e n t f o r z i s then d e f i n e d a s t h e c o n c e n t r a t i o n r a t i o o f z

w i t h r e s p e c t t o i t s e l f . Fron t h i s b a s i s we a r e a b l e t u show h a t b o t h f o r i I g r o c ~ a d d a t a and when i q d i v i d u a l f a m i l y d a t a a r e a v a i l -a b l e , t

c02f f i c i en : of i n e q o a l i t y f o r t o t d f a m i l y incooes ;an be dec$ioposed

e x a c t l y a s z sum o v e r a l l t ypes o f f a c t o r incomes, of t h e c r o s p r o d u c t s of 4 two t e r n s , v i z : ( 2 ) t h e s h z r e of each f a c t o r i n t o t a l income 2nd ( i i ) t h e

c o n c e n t r a t i o n r a t i o f o r t h e d i s t r i b u t i o n of each f a c t o r incon w i t h r e s p e c t

t o t h a t f o r t o t a l Lxoize. Soreover , when i n d i v i d u a l fami ly d t a a r e a v a i l - / on r a t i o f o r e he

. .

of households a c c o r d i n g t o t o t a l income and incone of t h e f a c t o r t y p e ;

and (5) t h e G i c i c c e f f i c i e n t f o r t5.e d i s t r i b u t i o n of t h e

f a c t o r incoze . Hence, v i t h i n d i v i z u a l f a n i l y d a t a , ve x a c t l eco rpo-

s i t i o n i n r o f a c t o r s h a r e s , c o r r e l a t i o n e f f e c t s , 2nd t h e e f f i c i e n t f o r

each f a c t o r . This decomposi t ion i s i l l u s t r a t e d f c r Taixan (19 - i n s e c t i a n 7 below. ' f - - - 13 tk,e absence of i n d i v i d u a l fm' y d a t a , i t i s s t i l t p o s s i b l s

I * . m

tc a b c a i z 2n z i ac t d e c o c p o s i t i o n of t h e Gin i : o e f i i c i e n r f o r

(5ased cn s r o l ~ p e d data) F n t a a v e i ~ n t e d z\.erage cf t5e ccnce>Z~atlcn

i y t a l i n c c r e

r a t L z s Scr

I

- 4 -

each f a c t o r i x c c e v i t h r e s p e c t t o t ~ ; r i iz.ccme. Tn t h e F-X-K Fape

t h e s e c o n c e n t r a ~ i o n r a t i c s v e r e b - t e r p r e c e d a s be ing ( a p p r o x i m t e l y r e q c a l t o t h e f a c t o r G k i c o e f f i c i e n t s . Xowever, u s L ~ g t h e bdi . l idu{l

family d a t a sow a v a i l a b l e , n u n e r i c a l r e s u l t s LI s e c t i o n 6 show cha t

G i n i c o e f f i c i e n t s and c o n c e n t r a t i o n r a t i o s can b e mmrkedly d i f f e r e n t b o t h

in l e v e l a d t r e a d s o v e r t h e . A-ccordhgly, our r e s u l t s show t h a t s a 1 ger-era1 r u l e Ft is 2 o t s a f e t o i n t e r p r e t t h e c o n c e n t r a t i o n r a t i c s

f a c t o r c c ~ p o c e n t s or' i n c c z e a s i f t hey were Gini c c e f f i c l e ~ t s

f a c t o r coz?ocents a l s o depend on c o r r e l a t i c n e f f e c t s . We

c lude t h a t t h e d e c c r p o s i t i c n of t h e t o t d irtcome G i n i

a r e a v n i l + b l e , S rce ion 3 :ken p r e s e n t s t h e exac t < e c a r ? o s i i i c ~ rh lc l j i

l e g i t d t e , b u t t h e i r h t e r p r e t a t i o n of i t s corpczen t s r e q u i r e s

Before p r e s e n t i c g our ~ u z e r i c a l r e s u l t s i n s e c t i c c s 6 and

conceptual b a s i s i s e s t a 5 l i s h e d i n s e c t i c - s 2 t o 5 . A s p r e v i o u s l y

t h e concens ra t ion r a t i o is l e f i c e d i n s e c t i o n 2 and exact decoxpos i~ ;on

a d d i t i v e f z c t o r cocponents i s e:cplcred f o r t h e case vhere i n d i v i d u a l

a??ropziace f o r grau?ed lacs. P r c b l e ~ s of acc9Jracy axd i n z ~ r ~ r e c a c i c

r e c o l s i d e r a t i o n .

7 , 2

n q t e d , , *; -

by * . %b+ e

t&&.. - d a t a

I - " f2c:sr SLni ' ; , and tken b y d e r i v i n g n e c e s s a r y 2r.d sur rLc ier . :

:hen d i s c n s s e d i n s e c c i c n 5. T i L s shob;r: Chat t h e eccuracy ~ r o b l e 3 II

. L.. - Ga~suorc5. (1972) is e s s e n t i a i l y irrelevant t o t h e ~ a l n f s s ~ e , ;;nich

r5.e z x t e n t co c h i c h f a c t o r c o n c e n t r a t i o n r a t i o s can be L n t e r p r e t s a a s

c s e f f i c i e z t s . Sec t ion 5 t ken c o a p l t t a s che t h e o r e t i c a l d i s c c s s i o n . - shcwLqg t 5 z t :he c c r . c ~ n t r a t i a n r a t i o s aiTcaps uc fe re - s t lnaze t h e

w

i i s c - s s e l

IS

Gini Z '1 -

f i r s t =:: - - c:rres-3cr.dln; -

f ie r e s u l t s cc exacc decozposi:icn uhich z r e c b t a i c e i n t h i s li a t e c e d e n t s in t h e l i t e r a t u r e . It seens , howzver, r 5 a t t h e ,:a l i e r s t a t i s t i c a l I pa2er a r e l a r g e l y a r i c i p a r e d by t h e wark of Rao (1969) vhich,

l i t e r a t u r e has n o t focussed on t h e prol>lams o f em?Fr icz l analy i s , and I

i n t a m , has

e s p e c i a l l y those which a r i s e from grouped Cata and t h e US. of Secondary

source 3 s t e r i a l . I n s o f a r a s t h e p ioneer ing work of F e i , fianls

being reproduced in cootenporary s t u d i e s , y t h e a n a l y s i s and r

a r e ? resen ted h e r e aay can t ion o t h e r i n v e s t i g a t o r s of t h e li3i

a r e i n h e r e n t izi working wi th secondary, as opposed t o p r i b a r y aca sources . ..

2 . G i n i c o e f f i c i e n ~ s and c o n c e n t r a t i o n r a t i o s

3 e f o r e coming t o =he problems which a r i s e vhen o n l y grouped dlata

i n d i v i d l i a l faini ly d a t a , i t i s u s e f u l t o s t a r t ou r d i s c u s s i o n v i t h gene I

a r e a v a i l a b l e , and s i n c e t h e new r e s u l t s i n s e c t i o n 7 below a r e based

d e f i n i t i o n s of G i n i c o e f f i c i e n t s and concent ra t j .on r a t i o s which c u l b e

on

a p p l i e d i n t h e v a r i o u s s i t u a ~ i o n s t o b e encountered subsequen t ly .

The s t a r r i n g p o i n t is t o assume t h a t t h e r e a r e n f a n i l i e s and

that :vo v a r i a b l e s , zi and t are observed f o r each , ( i = I , ? , ... i

F a n i l i e s a r e L O be ranked accord ing t o t i : t h e r ank ing o f t h e i ti i a ~ l ; ~ w i l l be denoted ' ( t i ) w i t h t h e convent ion t h a t r = 1 £0 i t h e f ami ly f o r which t i s s m a l l e s t , and r ( t . ) = n f o r t h e f ami ly f

i 1

which ti

i s l a r g e s t . If zwo f a m i l i e s have t h e s a c e v a l u e f o r t . 1 '

they 2 re ~ a c h t o be gi-ren t h e average o f t h e ranks t h a t t h s y would g e t

t h e r e i a s an i n f i n i t e s i a a l d i f f e r e n c e between t h m . K i t h t h e s e conventlions

t h e average o i a l l r anks r(.: .) i s given by 1

-. ,ne zverage raxk is theref o r e inceven2ext of the racking czi t z r i c n , t . I '

y~nich i s adopted. I

Sext, it w i l l be assuned t h a t r h e average value of z i s p o s i t i v e I

It can be noted t 5 a t this does not r e q u i r e t h a t 2; b e pos tive f o r sll i . -

Given , w e can a l s o d e f i n e

f o r each household. From ( 2 ) and ( 3 ) i t i s a u p a r s n t :ha

1 1

*

n = 1

i=.h '6

- - The c o n c e n t r a t i o n w t i o of z . r l i t h r e spzc r t 3 t

m I. i

frsc a c o n c e n t r a t i o n cu:..:e. Elis curyre graphs C-LTJ :c~A~ '_ -~E

2qzi?.sZ a c c l l a l . r 2 lues sf r(r , ) in when cczulatl. .-e 3c-s &

' 2 : ~ ~zkiz3 f a z i l i s s i n the o r d e r i n d i c z t e d by r ( ~ ~ ) . -. Li;cr2 .

( 2 )

is o b t a i n e i

.:z?ses of 7 . A

a r e f o r n e l

1 ilL2s----o L - L - - S

I I

a concen t ra t ion curve. Tne concen t ra t ion r a t i o of z vith respec t t o t , I

2enoted a s C (z / t ) , is now def ined a s one inu us t v i c e t h e a r ea . c d e r t he 4 I

concen t ra t ion cunre. It can be noted from thc f i g u r e and t he de ,n,t,on o f 9' ' ' I

t h e curve t h a t t h e l a t t e r need no t be monotonic. Hence a concendration r a t i o

can be negat ive . Equally, a concen t ra t ion r a t i o can have a value g r e a t e r I thvl one i f sone z ' s a r e negat ive .

Ii can be seen from inspec t ion of F i g u r e 1 tha : t he area c n d s r :he ! concen t ra t ion curve is

In t h i s f ranework, a G i a L c c e f f i c i e n t Fs the concen t ra t

a v a r i a b l e w i t h r e s p e c t to itself s h c e in tl:is c a s e d e concen t r

F i g u r a 1 i s now the L o r ~ n z curve . I f yi i s t h e t o t a l income of

i th family, t h e Gin% c o e f f i c i e n t ? o r f a m i l y incomes i s

I n e i - i - K s t u d y is c a n r e n e d r ; th t h e d e r c n p c i i ~ i o n of ;Iy) i n t e n s of t h e f a c t o r components or' Yi . I f Xik i-he c n t r i b u - "I t i o n or' f a c t o r component k (e.g. wages) t o the t o t a l income of the

then

where t o r 2 1 incone ts uade cp of c f a c t o r cocponents ( k = 1 . . . . (m);

f a i l y , yi ,

f o r i = 1, ...., n k=l

and

( 9 )

where 01,

From (10) and (12) i t fol lows t h a t

"

so t h a t r e s u l t (11) expresses t he GFni c o e f f i c i e n t f o r t o t a 1 income, Yi ,

a s t h e weighted average of t h e concen t ra t ion r a t i o s :or fact04 incomes

wi th respec t t o t o t a l incomes.

Result (11) is due t o Rao [1969] . I t provides an e- 1 a c t decon-

pos i t i on of ~ ( y ) i n which t he weights , Qk , a r e the sqares of each I

Factor t y p ~ $n t o t a l incomes. Conditiop ( 2 ) i n p l i e s t ha t th s e weigh ts a r e 4 I

non-negarive. However, t he concen t ra t ion r a t i o s which aTpear i n (11)

- .. can be nega t ive . .r i To examine t he d i f f e r ence be t -~een f a c t o r concen t r a t i * - ) i a c t o r G in i s , i t fol lows from (8) t h a t the l a t t e r a r e given ..

C s i o p (7), t h i s g i v e s

The r a t i o R(y , 5) w i l l be r e f e r r e d t o as a r a n k c o r r e l a t i o n r a t o . It k

' i . e . o n l y i f f a i l i e s h a v e t h e same r a n k i n g w i t h r e s p e c t t o f a c t o r i n one 4 k as t h e y have w i t h r e s p e c t t o t o t a l income.

It c a n a l s o b e shorn t h a t uniey i s a n uppe r bound f o r X(y 1 , %) . This can b e a rgued with r e s p e c t t o c o n c e n t r a t i o n c u r v e s . To I

o b t a i n G(qL) , f a m i l i e s a r e r anked by t h e s i z e o f xik. Hence t h e r t h

p o i c t on tF.e c o n c e n t r a t i o n c u r v e d e p e n d s on t h e r familfes f o r which xik

is smallest. Sone o t h e r o r d e r i z g is i n v o l v e d i n ob t a in ' i ng C(\/y) ,

t h e o r d e r l . ~ g o f E a n i l i e s a c c o r d i n g t~ the s i z e . oS ;ria The sm sf s. L 1

canr-ot b e s-mller f o r the first r f z m i l i e s a c c o r d i n g t c t h i s o t h e r o d e r l z g . i-

..- -- . . . L A - a b l a c;:zo: s:<ceed its Uini coeffLcienc, acc z e c c e t h ~ i

Fience t h e c o n c e n t r a t i o n c u r v e v h i c h y i e l d s C(\/Y) c ~ r i n o t l i e belo.; .* i

- ~ h i c h y i e l d s GCqK) . F e n c e - - %?

C(:L / ) / t(:i) = ?.(:c/ , y j < i - iC L ."

t h a t

( 1 7 )

for all

3. Grouped Data

For p r e s e n t pu rposes t h e d i s c u s s i ~ n o f grouped d a t a can

r e s t r i c t e d t o i n s t a n ~ e s where grouping r e s u l t s in an e q u a l number 01 f a m i l i e s i n each group i r r e s p e c t i v e o f t h e grouping c r i t e r i o n t h a t s used. k T h i s c o v e r s two s p e c i a l c a s e s of i n t e r e s t , v i z . t h e d e c i l e grouping o f

f z m i l i e s as employed i n F-R-R, and t h e c a s e o f o n l y one f a m i l y i n e c h f group, i . e . t h e ungrouped d a t a case .

I f f a m i l i e s a r e grouped by s i z e of income

r e s t r i c t e d t o t h e a v e r a g e i ccone w i t h i n each group.

used f o r t h e average i n c o s e i n t h e group w i t h i n which

is d e f i n e d f o r each f a m i l y 2nd ;an b e used t o d e f i n e

group. Thus, i f t h e r e a r e s 3roups or' f a m i l i e s , t h e c o n c e n t r a t i o n kurve

- ObvFously yi and r(Yi) w i l l be t h e sane f o r a l l f a m i l i e s w i t h i n

w i l l be a connected s e r i e s of s l i n e a r segnenrs .2 ' I t w i l l y i e l d t ,e G i ~ i : c o e f f i c i e n t f o r grouped i?cone d a t a

a g iven

1,' R e c a l l c h a t ;,1 our e a r l i e r d e f i n i t i o n o f r ( t .) c2 adopted t h e 3nventi.cn - t h a t i f t , is t h e s a s e f o r two o r =ore f m i i i e s , t hen f o r each of ~ h e s e

9 I f a m i l i e s -rk;) t a k e s on t h e ave rzge of t h e v a l u e s i t would h ve i f tS.ere was an i n i k i t e s i 7 a l d i f f e r e ~ c e i n t i be tveen t h e f an i l : s . I

2 / T h i s i s e q u i v a l e n t t o s a y i z g t h a t :he c o n c e n t r a t i o n - by j o i n i n g t h e origi i? and s of t h e 3 s u c c e s s i v e c e n t r a t i o n c u m e by straight l i e e s . i n t h e n o t a t i o n s p o i n t s correspond t o t h e a c t u a l ? o i n t s ac which . a , ( s - l ) n / s , 1 .

1 / y e c a n now n o t e t h a t -

cov G, , r t? , ) ) = :ov (Y, . r ( y i ) ) .-

Bence , f rom U8).

i . e . t h e G i ~ i c o e f f i c i e n t c z l c u l a t e d f rom grouped d a t a sus: a ways under-

e s t i m a t e t h e t r u e G i n i . 1 t h a t i t i s of n a r g h a l impor t ance . It i s o b v i c u s l y a n elemen i n tk.e I

The s i z e o f t h i s u n d e r e s t i m a t i o n h a s been e x p l o r e d

a c c u r a c y of the F-X-K s t a t i s t i c s and is a t t r i b u t a b l e t o the u e Fn that k

by G a s t w i r t h

L n v e s t i g a t i o n of g rouped d a t a . Eowever , o u r w i n c o n c e r n s h e e a r e w i t h I

(1972) a o n g o t h e r s . H i s a n a l y s i s and o u r own e m p i r i c a l r e s u ts s u g g e s t P

p r o b l e m t h a t a r i s e f rom t h e i n t e r a c t i o n o f decompos i t i on zet and g roupsd

d a t a . From this p e r s p e c t i v e t h e b i a s i m p l i e d by (21) i s c o t

a s ve w i l l show i7 s e c t i o n 4 .

Lien i n l o m a t i o n i s r e s t r i c t e d t o d a t a f o r f a m i l i e s ' grouped k:;

s i z e o f t o t a l i n c b d , w e nay a l s o 'know t h e aveQqge v a l u e o f t c h f a c t o r f i n c s c e v i t h i n e a c h o f t h e t o t a l iacome g r o u p i c g s . L t I

- i e z o t e t h e ik - 11 .%is f o l l o w s d i r e c t l y f r c n - i - - - - - cov (v; , r ( y i ) ) = cov (yi , r ( y , ) ) + cov (7: -

- 6 A A e E -

a d c o t 2 i ~ :ha: :he second t e n cr: t h e r i gh t - ? l ane s l d e This Ls j e c a u s e t h i s s ecozd t c r z depends or, c o ~ i a r l z n c ~ s r;i:hia g r c c p s - fsr -2hLch cns o f the v z r l z b l e s , - r l z . r ( y 5 ) , i s In f z c t c ? . s t a c t . t

average v a l u e of type X i qcooe f o r households i n t h e t o t a l income group

ave rage wage lncooe o f f a i l i e s which f a l l i n , s a y , t h e t h i r d s l z e gro

accord ing t o t o t a l Incone . Then t h e sum o v e r k of 2 i s t h e ave r ik

v h i c h i z c l u d e s household i . X t y p i c a l e l e n e n t O F 2 i s t h e r e f o r e i k

of =he conbizled incone f r c n a l l s o u r c e s of f a m i l i e s i n a p a r t i c u l a r s i

group f o r t o t a l i ncone , i . e .

m

che

>!aking t h i s s u b s t i t u t i o n i n (19) y i e l d s

Th i s i s an exac t d e c o m ~ o s i t l o n of t h e G l n i c o e f z i c i e n t f o r i n c c r e s , . I

i s a ~ a ? a g c u s t d that ?revlocsLy ob ta ined f o r ungrouped

S ince ( 2 3 ) corresponds :o t h e c a s e where the c n l y data - - by incone l e v e i s , yi , 1: f c l l o u s t h a t such grou+ng, of i t s e l f , does 2 t

& C - 7 r e c l u d e an exac t b e c c ~ p o s i t i c n . - +

8 !B . - Xesu l t ( 2 3 ) i s t 3 e b a s i c d e c o c p o s i t i o n k q c a t i o n ecployed s y :

t

The c o n c e n t r a t l c n r a t i o s C I ) which ~ p p e z r 5 It are r e l s r r e d :o 4

r-zl:,sis as j seudo-GbFs. Ecivever, we p r e i e r :a sse h e r s :ke E i q e r 35-71 z c r e

. . 2 r ~ c i s e nA=e c f " f a c t z r cczceccra: ioc rati-s 53: da ta zroTd;ec 3:: : C ; E ~

- , - .. ii Ces ' 2 i , ?an:s ar,d x . 2 ~ ( 1 9 i E ) , isser?dkc scxaz:,-r. (,?.1.,-I. -

?roble-s o f a c c u r a c y and h t e r p r e t s t i c n

%suit 1 3 3 ) in the p r e v i c u s s e c t i o n p r o v i d e s an exac I. 2 e c o 3 p o s i t i o n of t E e G i n i c o e f f i c i e c t f o r t o t a l i n c c n e when d a t a r e I

x h e r e t h e t e r n in c u r l e d b r a c k e t s on t h e r igh t- hand s i z e of (24' must be 1

grouped i n t o s i z e c l a s s e s d e f i n e d by t h e l e v e l of c o t a l incone.

can be v r i t t e n a s

m I G(Y) = +k c(zk /? + { G(Y) - ~ ( 7 ) 1 i k=: I

p o s i t i v e i n view of c5e i n e q u a l i t y ( 2 1 ) . W e have p r e v i o u s l y su

t h i s term, which is d i s c u s s e d by G a s t w i r t h (1972) , is n o t l a r g e

Th i s

(24)

l b i t e d r e l e v a n c e t o t h e problems which concern u s . The n e x t s ep i n t h e 4 p r e s e n t argcment i s t o j u s t i f y t h i s p o s i t i o n , g iven t h a t ou r ica'n c c l c e r n i is t h e e x t e n t t o which i t i s l e g i t b a t e t o i n t e r p r e t t h e concen r a t i o n r a t i o s

C ) a s G i n i c o e f f i c i e n t s f o r f a c t o r incomes. 5 i

was a v a i l a b l e , we c c u l 2 group f a n i l i e s ( i n t o d s c i l e s , f c r

. Q

e - discarc iea and a Gin2 coef f i c i e n r , G ( x i c ) , could be

grouped d a t a on t h e assumption t h a t eac? f a n i l y had 2n a roun t of

Lnccice equa l t o t h e ave rage p r c p e r f y inccrte anong f z i l i e s xh ich

l ~ i z i d a s e r i e s o f s t z c i s t i c s G ( s , - ) r.

(:I), x-ould :la1:@ rke P roFer ty

p rope rzy - f a l l Z d l t h i n -

c n e s a r e s i z e c l a s s ( d e c i l e ) a c c c r d i n g t o t h e l e v e l o f 7 rcpe rzy 8 I

c a l c u l a t i c n s f o r each t;-7e of f a c c o r i n c o ~ e ! k = 1,1, . . . , 2 )

C - I r ,?c~+ S e p a r a t e

- - v o c l J 9cT.j

G(qK1 G(%) f o r k = 1 , . . , , * -

G (<) f o r k = 1 , 2 , . . . ,n r e q u i r e s a d i s t i n c t r e r a n k i n g s of t h e i

d a t a . C l e a r l y , such r e r a n k i n g s a r e i n p o s s i b l e i f i n d i v i d u a l d a t a

( 2 5 )

It shou ld b e no ted t h t t o c a l c u l a t e each G(< ) r e q u i r e s a

households accord ing co t h e s i z e o f x ik ' and then grouping of t h e

d a t a . These g roup ings w i l l ? q i y t h a t f o r a g iven k t h e r e a r e no

i n t h e s i z e of x between groups . And t o compute t h e set of i k

re-r;.niing of

r acked

o v e r l a p s

s t a r i s t i c s

t i c s

a v a i l a b l e . Hence, a c c e s s t o p r i n a r y f a n i l y d a t a i s 2 e c e s s a r y i f t h e

a r e t o b e

s t a t i s -

computed . S i a p l e u a n i p u l a t i o n of e q u a t i o n ( 14 ) y i e l d s the r e s u l t I

I

.;here I

- = c ( n , / . j ) - G(;~) f o r k = 1 , . . . , 3 "k h (19)

2nd

? e -

a : G C . i ) - S C ; ) ! - ) +k '; G C X ~ ) - G(;<,,) 1 (29) k: 1

Bence, f r o 3 ( 27 ) , :hero_ a r ? two t y ? e s of e r r o r involve'd in assuniag 8 I :he G h i c o e f f i c i a n c f o r Eoca: !.r,co~e can be sxpressed as a i i e i g h c e d

Li

,

t.!at

$122

t o S e d i s c c s s e d i n d e c a i l i.3 :he z e x t s e c t l o n . Tee s2cond typu of e r r o r , e ,

is l e s s h p o r t a n t f o r t h r e e r e a s o n s . I Components of t h e e r r o r , e , a r f s e from usLrg groupe d a t a t o k

c a l c u l a t e G i n i c c e i f i c i e n t s . I f t h e s i r e of a v a r i a b l e , n , is! t h e c r i r e r i o n

csed t o group f a i l i e s , then t h e e r r o r in c a l c u l a t i n g t h e G i n i o e f f i c i e n t 0 l/

f o r z from grouped d a t a can b e expressed as-

P v i s t h e p o p u l a t i o n p ropor t ion i n t h e vth group; is t h e ( s h a r e of

v

where i t is a s s w e d t b t t h e p o p u l a t i o n has been d iv ided i n t o

group v Lr t h e t o t a l of z f o r the whole p o p u l a t i o n ; and gJ i s t h e

s grcups ;

G f n i c o e f f i c i e n t of inequali t ;? f o r the d i s t r i b u t i o n of z w i t i n group v . Hence, i f t h e popu la t ion is grouped i n t o d e c i l e s a c c o r d i l g t o

z , u e have s = 10 and p y = 1/10 f o r a l l v . S i n i l a r l y , i

example, a v i l l 5ave an ave rage v a l u e of 1/10 u h i l e g 1 v V

be tueen 0 and 1 i f z i s a non- negative v a r i a b l e . Hence t h e e r o r t e r n (36) h ~ s t an c r d e r of mgn ' i rcde of 1 /10 . I n :act Gastwirch (1972) is z 5 l i t o i q o s e r u c i +. L , g h ~ e r 2 ~ o u n d s on t h i s e r r o r t e n . Tkese p rov ide t h e f i r s t rea on f o r r e g a r d 5 2

t h e t e n e 5n (29) a s n e g l i g i b l e . , e

The second reason f o r r e g a r d i n g e a s c e g l t g i b l e i s

in c u r l a b r a c k e t s in (29) o u s t be p o s i t i v e , while t h e iua of t i e ; -e ights , - i

d e t 2 m i n a t L o n a s Lr, (20) Fnvolves compensating e r r o r s . Eacn of -

2 . is i i : y . Xer.ce, ii C e c i l e g rocp ing i s a l x a y s e r p l c y e d , t d e s r r o r e -

.C ' - B *- - 1 C

Ls tha 6 zerence b e t - ~ e e n c-*.o 7 .~2-nega t ive ce--- L - s each of tb ssqe crLer o f I

t h e t e r z s

. ' -, See, Ecr e s a z ~ l e , 3ha t t acha rya a ~ . d !!a'r,alacoois ( l G 6 7 ) . -

%cizce i t nay b e q u i t e r e a s o n a b l s t o i g n o r e e i n

n u c e r i c t l work.

X c h i r d r e a s o n f o r ignorj,,g e is t h a t t h e problem re a r e

i n t e r e s t e d -t? can ba e x p l o r e d w!.thout r e f e r e n c e t o i t . Zf a v a i l a b l e d a t a

i s r e s t r i c t e d t o a grouped d i s t r i b u t i o n accordi.ag t o t o t a l f z r n i l ~ i ~ c o r r e ,

then G ( y ) is unknobn azc a n a l y s i s a u s t n e c e s s a r i l y b e r e s t r i c t e d t o t h e

decomposi t ion of G . Hscce, t a k i n g ( 2 3 ) as t h e s t a r r a g p o i n t , i t i s

e a s y t o show t h a t

i . e , a decon?os i t ion of GG) which d o e s n o t i n y o l y e errors o[ t h e t y p e

d i s c u s s e d by Gas twi r th .

Our a n a l y s i s i n s e c t i o n s 5 a r d 6 w i l l b e based on equat fon ( 3 1 ) .

I n t e ~ s of i t , t h e i n t e r p r e t a t i o n p r o b l e n u d n g t h e F-B-K nethodolcgy is

v h e t h e r t h s c o n c e n t r a t i o n r a t i o s C(% / ) a re good a p p r o x i n a t i o n s of

t h e G i n i c o e f f i c i e n t s G(: ) i . e . whether t h e e r r o r terns E a r e rC k

n e g l i g i b l e .

5. T h e ~ r o h l e l n I

I The i n t e r p r e t a t i o n p r o t ~ l e n de f ined in the p r e v i o u s s r c r i o n is

ccncemed vith t h e s 4 2 e of t h e e r r o r terms ck i3 e q u a t i o n 4 8 ) : i f ciL

is z e r o , then t h e f a c t o r c o n c e n t r a t i o n r a t i o CGk /? ) and J t h e f a c t o r

C h i CGk) a r e equa l . In this s e c t i c ~ we vFll show thzt

t o e x p r e s s E~

a s the sun of two terms, b o t h of which are non

Beyond this, n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s a r e ob ta ined o r each

component of ck t o b e ze ro . Hence we o b t a i z c o n d i t i o n s unde which t h e r e

is n o i n t e r p r e t a t i o n problem. I I

The a n a l y s i s t? this s e c t i o n r e q u i r e s a new concept dtsnoted 1 i

GUk) , 'vhich e2n b e c a l l e d a q u a s i- f a c t o r Gini . Given t h i s , c want t o i prove t h a t I

I i . e . that t h e q u a s i- f a c t o r G i n i l i e s between t h e c o n c e n t r a t i o n ! r a t i o

, c(% 1 and t h e Gini c o e f f i c i e n t G ( ; ~ 1. -4nd i f (32) i s v g i d , chen i t

k

f o l l o w s t h a t

where

a n d

3 e . ra r -kbis -s 2 , have previousl:,. b e e x d e f ~ n e C L;1 s e c t i o c 2 . L.

~ C - - - P I ;I.,d f o r eac5 k 5 7 g r a u > i n g ~ . o u s e h o l c s a c c o r a i i i ~ t h e sl:e

. . . i ~ c s c e , 7 . , ;r.C :hen : a l c i l l z r b . 5 t h e l v e r a p c 711ze 0 5 1 I<.. - . ~ L z L - i

! ;-: I

I

e a c h o f ckese g roups . T i e g r g u p i r g s a r e t h e r e f o r e i ndependen t o f k . They are t h e s a e g r c u p h i g o r c l u s t e r i n g o f f m i l i e s a s is involve t i i n

I - c o ~ p a t i r . g :he y i ' s , i . e. t h e a v e r a g e v a l u e s o f t o t a l i n c o n e amon g roups i: of f a d l i e s d e f i n e d w i t h r e s p e c t t o th: s i z e o f t h e i r t o t a l income1 tiow-

e v e r , t h e r e 1 2 no r e a s o n why r ( Z fk) 2nd r(Yi) shou ld be t h e s a e : n

t h e r e i s no r e a s o n , f o r e ~ a m p l e , why a v e r a g e t r a n s f e r income s h o u l

i n c r e a s e ~ o n o t o n i c a l l y a s w e move upwards between groups defi.r.7.d w t h i r e s p e c t t o t h e s i z e o f yi. Hence t h e r a n k i n g s r(:i ) and r(7ii a r e n c t

i k

2 e c e s s a r i l y t h e s a n e , ar.d r e s u l t s ( . 7 ) and (18) a r e s u f f i c i e n t 40 I I

e s t a b l i s h t h e i n e q u a l i t y (34). 1 I

This d i s c u s s i o n o f t h e d i f f e r e n c e be tveen r acd r ( 7 , ) I -

s u g g e s t s t h e f o l l o w i n g c o c d i t i o n :

F i r s t L n t e r p r e t a t i o n c o n d i ~ i o n I I

A. n e c e s s a r y and s u f f i c i e n t c o n d i t i c n f o r E = 0 i s tdPt t h e l k

I

graph of r ( 7 i ) v e r s u s r (1 ) i s monotonic n o n - d e c r e ~ s i n g . ik

- 10 e s t a b l i s h t h i s f i r s t i n t e r p r e t a t i o n c o n d i t i o n we c+n h r g c e I

a s f o l l o i i s . The c o n c e n t r a t i o n cu rve which y i d l d s t i 2 pseudo- Gini , b(2 ) k '

I

r a ~ k s f a n i l i e s a c c o r d i x g t o r(?; ) , i . e . a c c o r d i n g yo t h e ave rage ).q:el of Lk I

4 I :< f c r grougs ~ h i c h r e s u l t f r c n t h e s e r a t i f l c a r t o : ~ of f a ~ i l i e s by1 i n c c a e i k

l e v e l s . As p r e v i o u s l y no:ed, t h i s must Lzply t h e saze c l u s t e r l r g c$ ZroupFrg

- - a s r ( y i j . Tke o n l y d i f f e r e n c e t h a t can a r i s e b e t - ~ e e n C<:7 . /y ) And

G ( 5 ) = C / ) i s t h e r e f o r e i n t h e o r d e r i n g of t h e s e g r c u p s in che L

der i s ra t5cn of rhe c o n c e n t r a t i o n c u r i e . ? ? l e W o r d e r 5 g f o r G ( % , ) iS :he tc

o f t h e a b c i s s a . >-T-,-- c t i e r o r l e r i n g e i z k e r ~ c s : r ~ s c l : Fz 2 I z r z e r v a l c s cf

- ,:.e - or<L.nace cr , , --st have tY.e s a ~ e v a l u e f o r 2: 1z35t t : i ~ Sr3- i3~ . 2 ~ : L h

I

I

i f t h i s l a t c e r p c s s i b i l i t y z i a i ~ t a i n s , ';he_3, f ~ r txo S U C ~ ; g roups , ~ ( 2 . . ) is 1 r C

a l s o the s a e . Hence the c o n d i t i o n is zot v i c l a t e d . L';..i~er t h e f o r z e r I

? o s s i b f l i t y , the c o n d i t i o n is v i o l a t e d 2nd t h e concz r i t r s t ion c c e f f i c i e n t

I / CG /Yi) w i l l b e s t r i c t l y less th- the ? s e o d o - G i n i Gd? ) - . k k

To e s t a b l i s h i n e q u a l i t y (35) w e can note t h a t l o t h G ( l k : and

G a r e G i n i c o e f f i c i e n t s c a l c u l + t e d £ r o c grouped d a t . The d i f f e r e n c e I between t h e n n u s t t h e r e f o r e l'e in t 6 ~ group in^ c r i t e r i o n g iven t h a t w e i assume an e q u a l r.-mber o f groups i n till c a s e s , and an equ 1 number o f house-

h o l d s in e a c h group. For 5tik , t h e grouping c r i t e r i o n i t h e s i z e o f pi ;

. - c r i t e r i o n f o r computing x i s t h e one which n i n h i z e s he o r d i n a t e of t h e ik

- whLie f o r : t h e c r i t e r i o n is t h e s i z e of x i k '

Hence i k

c o n c e n t r a t i o n cu rve C i n e . t h e Lorenz curvr: in t h i s & s t a n e) f o r ezch v a l u e 4

t h e g roup ing

of t h e a b s c i s s a . Hence t h e c o n c e n t r a t i o n CLorenz) cu rve r/hich y i e l d s G ( i k )

can neve r l i e below t h a t which y i e l d s ( ) . Aid from t h i s , m e q u a l i t y (35)

f o l l o w s d i r e c t l y .

Th i s argument l e a d s t o ou r

Second i n t e r ? r e t a t i o n c o n d i t i o n

h n e c e s s a r y and s u f f i c i e n t

- r (3ik) = r ( x i h f o r a l l i .

t

T h i s c o n d i t i c n is z l e a r l y n e c e s s a r y s i n c e , i f t h e r e Ls any I I

- i I 1x

- d i f f e r e n c e ir. :he weak o r d e r i n g of f a n i l i e s given by r ) arid r(:;.. ) .

- then :he c o n c e n t r a t i o n cu rve f o r ~ ( 2 ~ ) e u s t be above t 4 c f o r G ( x ~ ) s r

I

1 f one p o i n t a t l e a s t . S h c e t h o l a t t e r c o n c e n t r a t i o n cu rve i n v a r i a b l : ~ a 1 r 1 - l c u e r Loand f o r t h e former , i t fo l iows t h a t E = O o ~ l y :here Lc no 2 1i

. . - - divergence iz t 5 e s e xeak order5.zgs. 2e;:ofid t?.is, if t>Azre - 0 :::~~re:-=

- 1/ I t can b e x t e d ti-,a: s i n c e :: - tsr l- :es S r c z ik

- 2 - . -;slues of ::i , -, : s -2 .~0ss lb le tc ' na~re :he s a r e

- :-*-a g r cv?s -A-hic.- has.? dLf f e r e n t v z l u e s of r(x;. )

A 5

s;lould b e 23 ? r e s ~ y t f o c t h a t :his Fsplies snall o r csgligi3lc y~al-es f o r

firs: :./;re :or t k i s l z t a s e t , -.ihich re fe r s : o ~ 2 ~ 2 IT;c"u=~s. L. e al;ar?ge -1 fd-~ .jag% L ~ c o r e s , %.L;r , izcrezse i n s t rF= t =or,o.-snic fasi..iqn 2s 7 s =eve

- , - =.,* . T O Fllust=a'e 5 5 s ? o L ? r , so=-=, 1 9 7 5 dz:a f o r 121:~22 have , s

-.j-L:Ir c'ne resul:s s ~ r - o c z L? Tz5le I.

A

7 r z t a c L s n c o c c i t i o n i s fa r f r o n sa:zsfFed. i f i: .<ere

ra:.:dng 1 2 ~ 3 C e c i l e s 3:7 -+-age L;.core 1 e a ; e l , : E : ~ , ant

. -0xl6 - ?Lzce a l l f-ies on zk.2 a a i 2 2lagcazl o:

&een a2aL:xei

9?L our o f 5 2 5 5 ia=l:les (17.6 ? e r c e n i ) r re t o be ion-d :>ere.

-=1 . . .

--,scz xh;. t>.Ls f igcre kts zo 5s 5i';en as a?. u;psr -1=::,

. - 2:<2c= : .cber, i s Lecacse 231 f a l i l i e s iz c n e rlrsr c-..-o

, I

. - :a :he 12.: ; e r ~ 2 n r .cf a l l ?.-.'lies ::?a: 'c.a*;e z e r o *;age 5-ccco-.

Y . - - .. . act =ea-whils : z s rac: :hzz 13. l ?ercen: sf a l l sa3:l:ss ';'.lq:e : r 3 ':age e

I 1 --- 1: l ) IS 'TI~IUlI1 ' ION OF' 1:AHILIE:S AND AVEltAGE WAGE INCOMES 13Y DECI1,ES O F TIlE WAGE LNCOME AN11 TO'I'AL INCOFIII DIS1'RLBU'l 'IONS

T A I W A N : 1974

/ ' I ' o t a l s v a r y bctwcen d e c l l e s b e c a u s e a l l f a m i l i e s w i th t h e same (wage) income have been p l a c e d i n t h e same -

cell 01 tllc t a b l e .

i 1)cc i l e s of t h e d i ~ t r l b u t i o n of wage income x

Average W a g e

1/ Income 6 t h 7 t h 8 t h 9 t h 1 0 t h T o t a l - I 2 / - iw

incornt: I - - - - .-

1s t I Qi

518 21.8 --- j 'I: .-.

J 1- u 2nd , 1 0 86 124 213 1 6 529 35.2 ui b: I ;, .A

-..- ---- .

3 rd 8 6 10 6 0 4 th 109 16 4 5

-

61 119 186 5 35 1 44 129 146

5 th 36 1 40 1 72 157 9 2

0 2 7 30 1 49 1 96 169 19

V1 9 ;: 2 4 2 8 35 1 50 1 147 - 140

24 2 0 16 19 37 1 61 1 207 2 9 1 7 15 16 16 20 30 1 1 2 1 1 194

9 1 11 1 3 1 7 9 1 5 24 40 1 3 0 7

951 526 514 532 531 526 523 527 530 --P

7 . 0 23 .1 39.4 50.9 61.6 71.8 83 .8 103.9 173.5

527 5 24

41.2 44.0

527 513 5 38 5 26 525 529

5256

52.2 57.4 65.3 69.8 87.5

135.8

-

61.1 - --

t x u c t L~com.rubLtlon of Taral Income Ineq11.11lty Ualng F s c l o r Sttareb and Concer~truLion Y b ~ i o u . r&ld YdcLur G t n i Cortllrlenrb C.,r T n l u d n , 1964 to 1S76, Baac.3 on 1)ut.l C ~ U L I ~ L ~ Ly Dcciles -

- . a s := -~2 C5:i~ svac v t e n :here is r o i:rer?:+raricz p r o j l e z a: i r e i t i s ;

:jrTe. I -., Ft fal lc*;s :kt g2:, 3et-;eeo C / and ~ ( ? c ) -,,sr: 5e :he c c l y

i rc I zeascrc o f r e l i a5 f - l l cy of = h e ?-X-K =eckodolcgy 'I? t?,e s e c s e :hat ,

I

z u s t 3 2 sx3jact t 3 s u b s t a z t i a l s r r z r s . Tab le 2 s e ~ i ax: s c z e r + s c i ~ i 5s;

?erl:d :%6i :a 1 9 7 4 cc zke s l z e cf r y e ga?.

. - ! ~ c . e r i c z l l y , t i e r e s l ~ l c s 1- : ~ 3 l = ? a r e no: s z r l c r l : ; cc-p

-' L..csz ?r?se-:ed Ln F21, 38111s a ~ d Kuo ( 1 9 7 3 ) . T?.erz zrc m ~ c re2s

C3,L1~ --? z ~ r - - . - . r - P ~ - - 3 f ~ ~ - - L =-- h u b r v ~ r 3 ~ ~ 3 3 ~ 2 5 , 3 0 - & - f -..=- rescl:; f o r ::12 earl:e yesrs -8 / - I . - - - - '- . . -

a s ; e t : ~ z - - y zr? c p e n =s I c c k : . ~ ~ ~ ~ ~ d l ; ~ , - a - e s.~r. :e: .s .:sz-, _ _ . - . , - - -1-3 .. - - .... ..'- - - - = - . .

r . I

. I - - . - - - -, z z n 3 e n 3 t e d ~ 5 . t ~ 15ese zqm5srs 2 1 z z e : - . - . . . 5elc;- ';ccsuse L X - 2 ~ ~ e - c a t ? a r e t zker , - .. . - r s x - : = s Secause 2a:z f o r r ? . 2 ~ <j

- . I - - . - -3 52:>:5 : = 4 : $ - . ' 5 7 . - I ~ c = < , ~ s l^rs 55 : ' 3 - - 3 - * - C : - -.._ - . - - . - - . - - k- - ,-,- - - L

- - - " - - 2 - m L - c - . . ,--- . --, - - - J - - >b - r J -, . - -, -I 2 .. -

7 --. - . - - .^ -,, ,- ,-, - - & . - - - - ".. + J . _ D .. . . - - . , I f . . ? . y : . 5': :20 .2=:

Yirh c h ~ s e r e s e r v a t i o n s , Tzb le 2 shows :,:_ti i-.t.::e?ci rg -1112 1 . . s ~ g r r l f i c a c r gaps . Slnce t h e c o n s r n t r a t i o ? r a t i o s C: / y ) ~ u l : always

k

be s m a l l e r t han G(;~) ( E ~ I 0) 5t f o l l o v s t h a t

FLi! f i n a l c o l u m s s f Tab le 2 show t h e d i f f e r e n c e between t h e ex ct d e c o n p o s i t i o n .I ( 2 3 ) a n d t h e d e c o r p o s i t i o n ( 3 6 ) which i s impl i ed by i n t e r p r e t i z g c o n c e n t r a t i c n

r a t i o s a3 Gin i c c e f f i c i e n t s . The gap i s s u b s t a n t i a l , s o t b t t h e

f a c t o r incone of a g i v e n type and t o t a l i c t c n e : t h e c o n c e n t r a t i o + r a t l o is

F-X-X i n t e r -

a p p r o p r i a t e .

- T a b l s 2 i l l u s t r a t e s t h e f a c t t h a t c c n c e n t r a t i o n ratio:;

n e g a t i v e . Such an o b s e r v a t i o n i m p l i e s a n i n v e r s e r e l a t i o n s h i p

ze ro f o r a f a c t o r i f a l l income grcups r e c e i v e an c q c n l ancunt o 1 i n c o r e

? r e t a t i o n of c o n c e n t r a t i o n r a t i o s (pseudo-Ginis) a s t r u e G i n i s i h a r d l y s can be

bet :~een a

c f t h e g iven f a c t o r V p e . Table 2 a l s o i l l u s t r a t e s r t e p c s s i b i l / t r t h a t

t r e n d s i n c o n c e n t r a t i o n r a t i o s can be q u i t e d i f f e r e n t frcm t!-.cse f o r Gici

c o e f f i c i e n t s . Tm e x t r e n e e x a p l e s a r e provided by t h e d a t a on Tbages 2nd

a g r i c u l t u r h l incornei f o r a i l f a m i l i e s . !;ith r e s p e c t t o a ' r i c u l t C r . l

i ncone , t h e c o n c e n t r a t i o n r a t i o f a l l s f r o n 0 . 4 5 t o 0 .10 cve r t h e

(wi th a a i ~ o r break i n ~ o n o t o n i c i t y f o r 1972) . I n c o c t r a s t , he

p e r i o d

G i n i

7 . Exact Decoupca i t i on csLnq Ir ,dFvidual F a n i l y Eacz ----- !hen i n d i v i d c a l fami1.y eats a r e a v a i l a b l e t h e r e s u ts i n 4

s e c t i o n 2 above c a n be used t o o b t a i n an e x s c t decomposi t ion . The key I r e s u l t s a r e I

G(~) = 1 $ k C ( x k / ~ ) k

and

Combining t h e s e we o b t a i n

h t o s e p a r a t e cociponents f o r e.ach t y p e of f a c t o r i n c c z e . r o r g i v e n t y p e

of f a c t o r icccme, i t s cuEponent of t h e d e c c c p o s i t i o n is t h e pr d u c t o f - I t h r e e t e r n s : (1) t h e s h a r e of t h e f a c t o r i n t o t a l income; ( i i a rank P c o r r e l a t i o n r a t i o a s d e f i n e d i n e u q a t i o n ( 1 5 ) ; and ( i i i ) t h e G 'n i c o e f f i c i e n t 1 f o r t h e d i s i r i b u t i o n of incone of t h e g iven f a c t o r t ype . ~;tlk 3 s e t s o u r

i . e . an e x a c t d e c o c p o s f t i o n of t h e G i n i c o e f f i c i e n ~ f o r t o t a l

1 1 :his exps t C e c o n j o s i t i o n u s i n g r h e s a n e daca s c u r c e a s T a b l c z~,- (T.zlces f o r

I t I: 4

$k a r e t h e s a c e a s i n Tab le 2 and t h e r e f o r e a r e n o t r epea ted n T z b l e 3 . ) I

income, G(y i ) ,

Apart from t h e n o d i f i c a t i o n s t o concent :a t ioc r a c l o

c o e f f i c i e n t s which a r i s e f r o n t h e u s e of i n d i v i d c a l 2s oppcsed

fanil:: c ' a t a , t h e r,ev i n f o m a t i o n <r, Tab le 3 conce rns t h e rank Lor?elat i l --r .

- _ . A . _ - T '., . --r-3 -,, , Y , ~ ) . T?.ese r a t l c s sh?:< scce 5 - r e r e s - i n g t r s n d .-

- .. . . L;.e c c c p u t a r i c n of a G in i c o e f i i c i e o : i c z a 7:2r222ra ~ n l i h :~'K;S s . 2 2 ~ 2 : i-;e v a l c e s f o r sc:e f a=+ i i e s ,

- 30 - .

t h e d e c l i s e aoong a l l Z a 3 i l i s s and among a g r i c u l t u r a l f a m i l i e s i n cbe r a t i o

f o r t o t a l in:ome v e r s u s a g r i c u l t u r a l incocle. The i n p l i c a t i o n of t h e s e

r e s u l t s is t h a t t b e ranking of f a m i l i e s by a g r i c u l t u r a l incone has declir.ed

i n iz i jor tance u s a d e t e r o i n a n t of ranking by t o t a l income. And t h i s is

perhaps what one would expect given t h e d e c l i n i n g h p o r t a n c e of agr f u l t u r a l

a l s o be noted t h a t t h e G i n i c o e f f i c i e n t f o r a g r i c u l t u r a l income mon

I incone a s a s h a r e o f t o t a l income which can be seen f r c n Table 2. It can

a g r i c u l t u r a l f n n i l i e s i s tending t o i n c r e a s e s i n c e the l a t e ' 6 0 s . T i s t would be c o n s i s t e n t w i t h a g r e a t e r tendency t o move towards otP.er f o b s of

r ~ n u n e r a t i v e a c t i v f t y (e .g . wage e a p l o p e n t ) by rembers of those I

which a r e a t the l o v s r end of t h e a g r i c u l t u r a l ixccne distribution.

Fron t h e poii i t of view of s t a t i s t i c a l decomposit ion, the e s u l t s i i n Table 3 t e l l a complete s t o r y of t h e t r e n d s i n t o t a l izcode G i r d do-

e f f i c i ~ c t s i n t e n s of f a c t o r components. Eowever, t h e u p p i n g from the

s t a t i s t i c a l magnitudes TO t h e concepts of econonic theory is not s t r j i p h t -

f fo rdard . Such a m p p i n g is provided LIL F-R-X, but u ~ d e r t h e assumpt on t h a t i t h e c o r r e l a t i c n e f f e c t s R ( x & , y j a r e s t z b l e over t i x e ar?d

I m i r y . C u r nevd r e s u l ~ s shcu t h a t c h i s is too r e s t r l c t i c e , acd poi2t t o t h e

I . I I

i & E G f o r an i n t e r p r e t a t i c n of the c n r r e i z t i o i ? e f f e c z s in : e n s of ec I

coacep t s . Ke have no such i n t e r 2 r e t a z i o n t o o f f e r a t t h i s junct- re . hr - - S e s t , us can suggest t h a t t h e c o r r + a t i o z e f f e c ~ s a r e ? c ~ e n t i ; i l l : l a s

L

vi:h o r e d 3 a ~ i . c a s p e c t s o f d e v e l q x e n t , z c t l e a s t ;r i tS xigra tLcn c1dt of

e the farm s e c t o r an2 i n t s >:age enp loy ,en t . 3 i s :;oulZ Se consis tzr i t .+i:h 1 t h e ~ obser-~a:ici: t h a t .;ages g r c w i? Iz-,crtar,co a s zr. s : . ~ i a ~ . a t i c n 2 2 (i. t. 12 :?.el=

I - . . . . : c r r e i ~ : i c ~ ~ ~ ~ 5 t h ) t o t a s ::.ccre; z n c ~:t>- E ----- . A d - - - w k ; 3 : z ! : ~ ~::~zTsT.L~~z: ;?a

5-cores j e t xeen f a x zzd z c c - f a n fircil . ies, ~ h i c h ~ead.; t c redcce t he

corre lz ; ion betveen t c z a l i c c c r e a d icccxe f r o 1 ag l - i cu l tu re . Such phsnorssa I

a r e cons i s t en t wi th our obser:ztions acd s e c s i b l e from a o r e t i c a l per-

spec t i ve . I c going beyond theg , i t raay be t h a t a c h ~ s t a t i s -

f o r t he c o r r e l a t i o n e f f e c t s . 2ut such developeents East v a ' t f o r t h e f u t u r e : I t i c a l approach i n t h i s pzpez and t h e i n t e r p r e t a t i o n o f Gini

f o r t h e p resen t we l e ave t h e s e ques t i cn s open.

c o e f f i c i e n t s i n

Fyat; (1976) vould t ake cs f u r t h e r tcwards a sound t h e o r e t i b a l i n t e r p r e t a t i o n

We have been concerned th roughou t thi.i pape r w i t h a n a t t e p t t o b f u r t h e r t h e d i s c d s s i o n o f i n c c n e i n e q u a l i t y decomposi t ion a s s e t o u t

F i r s t , we have s e t o u t t h e Rao decomposi t ion o f t h e G i c i o r i

i n

p r o b l e ~ s which a r i s e i n t h i s c o n t e x t when a v a i l a b l e d a t a a r e r e s t r i c t e d

t o t h e d i s t r i b u t i o n o f f a c t o r incomes be tueen groups of f a m i l i e s

by t h e i r t o t a l inccme l e v e l . We have n o t d i s c u s s e d t h e i s s u e s which

c e r n t h e u s e of f a m i l y income d a t a a s opposed t o i n d i v i d u a l lncones

f a r i l y i n c c n e Ter c 2 p i t a ) f o r t h e p r o b l e ~ s t o hand . These a r e s i d e

o u r n a i n conce rcs h e r e . The l a t t e r f a l l under tvo heads .

t o t a l incame i n t e r n s o f f a c t o r s h a r e s and f a c t o r c o n c e n t r a t i o n ratids. I

!7zo (1969) and F e i , k n i s and Kuo (1978) . The main f o c u s h a s been ob t h e

d e f t n e d

con-

:cr

i s s u e s t o

This i s an e x a c t d e c o r c ~ o s i t i c n which c a n b e used w i t h grcuped o r

d a t a . For t h e l a t t e r c a s e we have s e t ' o c t t he f u r t h e r

c c c c e c t r a t L c n r a t i c s i n r o r ank c o r r e l a t i o n r a r i o s acd f a c z c r

t h e p rocedure we r eco rnend ~ghen i n d i v i d u a l d a t a a r e available and

c o e f f i c l t n t i s chosen as t h e s t a r t i n g p a i a t . 1 I

S t z o n d l y , w e have shcl.,:n t k a t i t I s a i s l e a d i n g t a I c ~ e r 2 r e - Li

cr?rr of x a g z l t u d e of t-*-o i n o c r r e s u l t s znd can show r a r k e d l y l i f f z r n t I c e n t r a t i o n r a t l s s a s G i l l c o e f f i c i e n t s . The cr;o d i f f e r on avera-,& 3y

1

::enis. T r a n s l a t i o n from t h e e x a c t b o deccr,?osi:',on sf ~ F i s 12 ln5 fc;

: o r a l i n c c z e i n t s t h e f raaework of r c o z c n i c t h e o r y i s t k e r z f c r e l e s s - s t r a i g h t f o r ~ a r d than T e i , Xar.is and Kuc have a swed . 1: r e q u l r e s a. - 1

I I ana logcz La scocon ic t h e o r y f o r t h e ~ E R B c o z r s l a t l c n r a z i z s 1~hLch ? r d v i ? e

- - . . :>.e s z a t l s ~ i c a l l I 7 . X 5e:veen :he Cinl c c e r r z c 1 e T - t ~ i s r fzc:3r i-zczed 2-6 I

ac

To p r o v i d e s c c h ; n ana logxe n71st b e thl? n e x t s t e ? i n t h e d e c o n p b s i t i c n

a n a l y s i s of h c o ~ e i n e q u a l i t y &c:ordinz t o f a c t o r conqocents i f

c o e f t i c i e n t Ls t o ba r e t a i n e d a s t h e b a s i c a e a s u r e of i c e q u a l i t

To o b t a h o u r r e s u l t s w e have n e c e s s a r i l y had :o e w

a l t e r n a t i v e c o n c e p t s and medsurenents which a r e p o s s i b l e when i d i v i d u a l C f a n f l y d a t a a r e a v a i l a b l e , and t o r e l a t * t h e s e t o t h e n o r e lini ed set of r c o n c e p t s which can b e c a l i b r a t e d when a v a i l a b l e d a t a a r c t a k e n rclo a £

t o t a l i ncone and n o t v i t h r e s p e c t t o wage income. r'nus, grou?i. g u e r s e L s --

seconda ry s o u r c e which shows t h e c o n t r i b u t i o n s of e a c h t y p e of a c t o r i n c o n e £

sot t h e cause o f t h e problems which have concemzd u s h e r e . 3a h e r , i t i s I

t o t o t a l i n c o ~ e f c r groups of f a m i l i e s d e f i n e d w i t h r e s p e c t t o

t h r i r t o t a l income. Such g roup ing i n p l i e s o n l y minor problems

e s t i n a t i o n of h e q u a l i t y in t o t a l income. 3 u t f o r wages, f o r

g roup ing can h p l y u j o r e r r o r s because i t i s a g rocp ing v i t h

:he l e v e l o f

f o r t h e

e x m p l e , t h e

r e s p e c t t o

t h e f a c t t h a t s econda ry s o u r c e s p r o v i d e an i n e f f i c i e n t gr0upL.g

t o t h e d i s t r i b u t i c n o f i a d i v i d u a l f a c t o r components of t o t a l

l e a d s u s t o conclnde that r e s e a r c h e r s shoulC b e e x t r e r e l y c a u t l ~ u s

- b t e r ? r e t i ? g r e s u l t s based an seconda ry s o u r c e s o f Lncoze i s e q x z l i t g

v i t h r e s F e c t

i n r c c e . T h i s

i n

by

; i ; ; - l = = z ~ z , S . and 3. Ya5alenohi; (1967) : "Xeglonal :%-usehold Consunptior, i n I n d i a" , Jou rna l of - 5 - ~ r i s t i c a l X s s o c i a t i o n , 7 , r ~ l . 62 , So. 317

- _--22 Case of ?a lk i s t an f ' , unpubl i shed d c c ~ o r a l - - - A s r n i v e r s i t y , 1977.

- ---- -+ .., . .-:=.. , h i s , G . , and K C ~ , s .I T . ~ . (1978) : "Grouth and ;he Bzinily

6 i s s e r t a t i o n ,

-=- - G L r r j u t i o n - - of Znccoe by F a c t c r Corcponerts", - - - --- - - - :cor.cnics, X C I I , Xo. 1 ( F e b r u a r j 1978) .

- - - . . . 1 9 ) "Ixcome I n e q u a l i t y i n Urban C o i ~ r b i a : A 2 e c o z p o s i t i c n - d r s i s " , D i s c u s s i o n Paper ?To. 267, Economic Grow h C e s t e r , Ya le .

z . 1 J . . (1972) : "The E s t i m a t i o n of t h e Loresz cur-& an6 G i n i - ---.r'.=", Xeview of E c o ~ o n i c s and S t a t i s t i c s , LIV, = 7 2 ) .

-- ---J- &-- by F m i l y and by Type o f I;lcotceW, ? h i l i ~ o i z e lieview of

~ - i z e s s and Econonics , Vol . XIII, December --?----- 1976

, i-?-- -- - -,- . . - 6 ) "Cn t h e I n t e r ~ r e t a t i c n and 3 i s a g g r e g a t i o 1 . o f Gin5 - - .st?:=iciea;s", Econonic J o u r z a l , Yol. 56 (Jcr.2 1 9 i F ) .

- - . - - , p .. . - - ) : '''30 i e c ~ ~ p o s i t i o i l s o f Concer.tra:ior. ~ i + r k o " , J o u r ~ a l of - ? n ; ~ a l S t a t i s t i c a l S o c i e t y , S e r i e s A , Vol. 1 3 2 , ? a r t 3.