The distribution of income by factor components

Kcrid Bank Development Research Center I Discussion Papers No. 29 1 THE DIST?.IBLITION O F INCONE BY FACTOR COMPONEN S Graham P y a t t World Bank I Chau-nan Chen The I n s t i t u t e of t h e Three 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 and John F e i Yale U n i v e r s i t y , L 5 - - 'r - SOTE: isc cuss ion Papers a r e preliminary m a t e f l a l s c i r c u l a t e d t o s t i m u l a t e d i s c u s s i o n and c r i t i c a l comment. References ir p u b l i c a t i o n t o D i s c u s s i o n Papers shouJJ be 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 t e n t a t i v e c h a r a c t e r of t h e s e papers. The papers e x p r e s s t h e viewsLof t h e author and should n o t b e i n t e r p r e t e d t o reflest t h o s e of the World Bank.  th authors I Revised, Eeb uary 3 , i979. The Distribution of Income by Factor Comgonents Grahzm Pyatt -1/ and -3/ John Fei L 1/ -- - Developcent Research Center, World Bank ' i i -2 - - . . / The I n s t i t u t e of the a r e e Principles of the People, .kcsden.a Siarca, ~s:ie: t -3/ 'Tale University E t w I ? r s l k i ~ a r ; rzr.2 C c n f l 2 ~ r . t l a l : ':ot fcr yuoria:,c?. - or z:=rikctrlcn -;icr.cct -ri;r cl=ara~.c.-f r o 2 :he s-zhcrs. '.'le-;s exressea are t , c s e a f :he z -.--necessarilj- ..,, - , refizct tile -.-iz~sr t n e Xorld Sar,k. o  I n a recent paper: "Grcwzh 2nd the Fanily D i s t r i b i ~ t by Factor Co~ponents", Joilr. F e i , Gustav Ranis and S h i r l e y Ku have set out a methodology f o r Che s t a t i s t i c a l decomposition of fainily I incomz i n e q u a l i t y i n t e n s of such components of family inco e s a s wages, P income from a g r i c u l t u r a l a c t i v i t i e s , property income, etc;. . 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 decomposition of income ' n e q u a l i t y i n t o 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 the conceptual of economic theory. A methodology which allows 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 attempt t o understand the economic development. The Fei, Ranis and Kuo (F-R-K) paper a numerical i l l u s t r a t i o n of the methodology with rzfert-lce to Taiwan f o r t h e period 1964 t o 1972. i The measure of i n e q u a l i t y used is the Gini c o e f f i c i e .t. Links w i t h n ihe 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 purpose a t hand, and recent c o n t r i b u t i o n s t o the l i t e r a t u r e a t h e G i n i c o e f f i c i e n t have added t o our cnderstanding of its properties1 and charac- s t e r i s t i c s . The presenr paper is d i r e c r e d towards f u r t h e r i n g l h i s understanding i n t h e context of t h e P-X-ii a p p l i c a t i o n , with r t i c u l a r eaphnsis on t h e i s s u e s which a r i s e when data f o r individual r' ,I ~ o atv a i l a b l e and the decomposition of family incone has t o b srouped data of the 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  d .b aspect of the problens raised by grouped data has pre iously been discussed 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) has analyzed t h e e r r o r s involved when a GLni c o e f f i c i e 2 t is 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 observations. The present a n a l y s i s shows t h a t t h i s problen i s only a minor f a c e t of t h e i s s u e s which a r i s e w-len I groupsd d a t a a r e used a s a b a s i s f o r t h e deconposition of t o t a l i n c n e i n t o i a d d i t i v e f a c t o r conporents. The n a r u r z of t h e broader i s s u e s has b en t h e s u b j z c t of exchanges between t h e present authors 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) ha exanined t h e problems usirig some 1967-8 d a t a f o r Colombia. He u;it~/s: "lT'ne evidence] suggests t h a t f o r t h i s p a r t i c u l a r decomposition k d t h t k i s p a r t i c u l a r type of grouped d a t a , t h e o p t i o n of doing a l l r a t h e r than using what i n p e r i z c t d a t a w e have deserves s e r i o u s cbn- s i d e r a t i o n . " This i s a s t r o n g conclusion vhich 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 hhas receritly b e c c ~ ea v a i l a b l e . Zn exposlng t5-e problaas which a r i s e vhen a Gizi z o e f f i c i n: is I decocposzd i n t o aciditive f z c t o r cccpccects, t h e present a n a l y s i s b u r d s oc .1 r e 10 a s t a t i s t i c known a s t?ie concencratlon r a t i o . Sore authors, the s t a c i s c i c a l l i t e r a t u r e , use t h e t - e " concentration racio" s:.vnonl//cously - vich r h a t of a G i n i coegicien:. iio.;ever, Ln the d e f i n i t i c n s f the f n e rb i s e t out ir s e c t i o n 2 b s b w , * t h e concentratior, r a c l o is a r o r e general c m c e p z -.~i:h res;?c: cc vhicn :5e Gizf c c e f f i c i e s : is ;s ~ e c l a lcsse. So rco I s  t5.e pseudo-Giiii c o e f c i c i s n t xhic.h playa such an Paportant t in che F-it-K I I= ~eittodology. Fcveve-r, t r u e Ginis and pieudo-Ginls a r e not t h 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 is c e n t r a l t o the prese t discussion.- me next 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 concentration 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 I 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 ehold. The G L q i c o e f f i c i e n t f o r z i s then defined as the concentration r a t i o of z w i t h respect 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 i h a t b o t h f o r I g r o c ~ a dd a t a and when i q d i v i d u a l family d a t a a r e avail-able, t c02fficien: of i n e q o a l i t y f o r t o t d family incooes ;an be dec$ioposed e x a c t l y a s z sum over a l l types of f a c t o r incomes, of the cro s products 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 concentration 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 with r e s p e c t t o t h a t f o r t o t a l Lxoize. Soreover, when individual family d t a a r e a v a i l - on r a t i o f o r e he . . of households according t o t o t a l income and incone of t h e f a c t o r type; 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 xact lecorpo- s i t i o n inro 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 decomposition is illustrated fcr Taixan (19 - 'f - i n s e c t i a n 7 below. - 13 tk,e absence of individual fm' y data, i t is stilt possibls *- .I m tc abcaiz 2n ziact decocposition of t h e Gini :oefiicienr f o r i y t a l inccre (5ased cn s r o l ~ p e ddata) F n t a a v e i ~ n t e dz\.erage cf t5e ccnce>Z~atlcnr a t L z s Scr  I - 4 - each f a c t o r i x c c e v i t h respect t o t ~ ; r iiz.ccme. Tn t h e F-X-K Fape these c o n c e n t r a ~ i o nr a t i c s v e r e b-terpreced as being (approximtely r eqcal t o t h e factor G k i c o e f f i c i e n t s . Xowever, u s L ~ gt h e bdi.lidu{l family data sow available, nunerical r e s u l t s LI section 6 show chat G i n i c o e f f i c i e n t s and concentration r a t i o s can b e mmrkedly d i f f e r e n t both 1 in l e v e l a d t r e a d s over t h e . A-ccordhgly, our r e s u l t s show t h a t s a ger-era1 r u l e Ft is 2ot s a f e t o i n t e r p r e t t h e concentration r a t i c s factor c c ~ p o c e n t sor' inccze a s i f they were Gini c c e f f i c l e ~ t s factor 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 clude t h a t the deccrpositicn of t h e t o t d irtcome Gini l e g i t d t e , but t h e i r h t e r p r e t a t i o n of its corpczents requires recolsideration. Before presenticg our ~ u z e r i c a lr e s u l t s i n s e c t i c c s 6 and 7 , 2 *; conceptual basis is esta5lished in sectic-s 2 t o 5 . A s previously nqted, , - the concensration r a t i o is leficed i n section 2 and exact decoxposi~;on by * .%b+ e t&&.. - additive fzctor cocponents is e:cplcred f o r the case vhere individual data are avnil+ble, Srceion 3 :ken presents the exact < e c a r ? o s i i i c ~rhlclj i a??ropziace for grau?ed lacs. P r c b l e ~ sof acc9Jracy axd i n z ~ r ~ r e c a c i c :hen discnssed i n seccicn 5. T i L s shob;r: Chat the eccuracy ~ r o b l e 3 i i s c - s s e l II .L.. Ga~suorc5.(1972) is essentiaily irrelevant to the ~ a l nf s s ~ e ,;;nich - IS r5.e zxtent co chich factor concentration r a t i o s can be Lnterpretsa as Gini Z '1 c s e f f i c i e z t s . Section 5 tken coaplttas che t h e o r e t i c a l disccssion. - f i r s t =:: --- shcwLqg t 5 z t :he c c r . c ~ n t r a t i a nr a t i o s aiTcapsucfere-stlnaze the c:rres-3cr.dln; w - - " I f2c:sr SLni';, and tken b y d e r i v i n g necessary 2r.d surrLcier.:  .. fie r e s u l t s cc exacc decozposi:icn uhich z r e cbtaiceliin t h i s pa2er a r e largely a r i c i p a r e d by the wark of Rao (1969) vhich, i n t a m , has I a t e c e d e n t s in the l i t e r a t u r e . It seens, howzver, r5at the ,:a l i e r statistical I l i t e r a t u r e has not focussed on the prol>lams of em?Friczl analy is, and especially those which a r i s e from grouped Cata and the US. of Secondary source 3 s t e r i a l . Insofar a s t h e pioneering work of Fei, fianls being reproduced in cootenporary studies,y t h e analysis and r a r e ?resented here aay cantion other investigators of the li3i a r e inherent izi working with secondary, as opposed t o pribary aca sources.  2 . Gini c o e f f i c i e n ~ sand concentration r a t i o s 3 e f o r e coming t o =he problems which a r i s e vhen only grouped dlata 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 on individlial fainily d a t a , it is useful t o s t a r t our discussion v i t h gene I d e f i n i t i o n s of Gini c o e f f i c i e n t s and concentratj.on r a t i o s which c u l be applied i n t h e various s i t u a ~ i o n sto b e encountered subsequently. The s t a r r i n g point is t o assume t h a t there 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 for each, ( i = I , ? , ... ti i F a n i l i e s a r e L O be ranked according t o t i : the ranking of the i i a ~ l w; i~l l be denoted '(ti) with t h e convention t h a t r = 1