I n t e r n a t i o n a l Bank f o r Reconetruct ion and Development

Development Research Center

Diecuesion Papers

No. 10

EFFICIENT ESTIMTi3N OF THE LORENZ CURVE .WD

ASSOCIATED INEQUALITY MEASURES FROM GROUPED OBSET;'?ATIONS

N.C. Kakwani and N . Podder

October 1974

NOTE: Discussion Papers a r e pre l iminary m a t e r i a l s c i r c u l a t e d t o s t i m u l a t e discuaeion and c r i t i c a l comment. References i n publ ic&t ion t o D i s - cusaion Papere should be c leared wi th t h e au thor (e ) t o p ro t ec t t h e t e n t a t i v e cha rac t e r of t he se papers .

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

Pub

lic D

iscl

osur

e A

utho

rized

EFFICIENT ESTLXATION OF THE LORENZ CURVE AM) ASSOCIATED INEQUALITY MEASURES FROM

GROUPED OBSERVATIONS

-

by N.C. Kalmani and N. Podder Development Research Center, The World Sank

The Universi ty of New South Wales

I. INTRODUCTION

The Lorenz curve is widely used t o represent and analyee the o i z e

d i s t r i b u t i o n of income and wealth. The curve r e l a t e s the crmnrlativla propor-

t i o n of lncome ur,ite t o the cumulative proport ion of income received when

u n i t e a r e arranged i n ascending order of t h e i r income.

The equat ion of t he Lorenz cu:rva can be derived from the dens i ty

funcrion of the inccme d i s t r i b u t i o n . I n p r a c t i c e , t h e d e m i t y funct ion i e

not h am and eo f a r the approach Fa8 been t o f i t some re11 known denai tp

func t ion , f o r example, t h e Pareto o r t h e Lognormal. The shortcoming of

such nn approach is t h a t t he well-known d e n s i t y funct ion hard ly givks a

reasonably good f i t t o a c t u a l d a t a . An a l t e r n a t i v e approach i e t o f ind a n

equation o t the Lorenz curve which would f i t a c t u a l da t a reasonably well. ' f

The Lorenz-curve has a number of p rope r t i e s which can be e f f e c t i v e l y - - *

uc i l i zed t e s p e c i f y such an equation. S m -

' R e purpose of t h i s paper i e t o i n t roduce a new coo rd ina t e system

For rlle I.orenz curve. P a r t i c u l a r a t t e n t i o n ie paid t o a s p e c i a l c a s e of

wide emp i r i c a l v a l i d i t y . Four a l t e r n a t i v e methods have been used t o

e s t i m a t e t h e proposed Lorenz cu rve from t h e grouped obse rva t i ons . The wel l -

'mown i n e q u 3 l i t y measures a r e ob ta ined as t h e f u n c t i o n of t h e e s t i n a t e d

paraxr.eters of t h e Lorenz curve . The procedure of e s t ima t i ng t h e asympto t ic

s t anda rd e r r o r s of t h e i n e q u a l i t y neaeures i s aLso provided. I n a d d i t i o n

the frequency d i s r r i b u t i o n is de r ived from t h e equa t ion of t h e Lorenz curve .

X new r e p r e s e n t a t i o n of t h e Lorenz cu rve is in t roduced i n t h e

next s e c t i o n . Sec t i on 3 prov ides t h e r e l a t i o n s h i p between t h i s represen ta -

t i c n of the Lorenz curve and a number of convent iona l measures of income

i n e q u a l i t y . S e c t i o n 4 d e s c r i b e s a number of e s t ima t i on methoda. The l a a t

s e c t i o n r e p o r t s some e m p i r i c a l r e s u l t s based on t he d a t a from t h e A u s t r a l i a n

survey of Conaumer Expendi ture and Finances (1967-68) .

2. A NEW CO-ORDINATE SYSTEM FQR,THE LOkENZ CURVE

I Suppose t h a t income X of a fami ly i s a random v a r i a b l e w i th

p r o b a b i l i t y d i s t r i b u t i o n func t i on F(x). F u r t h e r , I f lit i s assurcsd that * aeon v of t h e d i s t r i b u t i o n e x i r t o and X is def ined on ly f o r p o e i t i v e

value&', t he f i r s t moment d i s t r i b u t i o q f u n c c i o n of X is then given by

where g ( X ) is t h e d e n s i t y f unc t i on .

11 The i n c o c e X can be nega t i ve f o r some f a m i l i e s but i s assuned t o be always p o s i t i v e h e r e because of n o t a t i o n a l convenience.

- 3 -

The Lorenz cu rve is t h e r e l a t i o n e h i p betveen F ( x ) and Fl(x).

The cu rve i e shovn i n F igu re 1. The eque t ion of t h e l i n e F1 - F ia

c a l l e d t h e e g a l i t a r i a n l i n e , which i n the diagram, is t h e diagonal th rough

t h e o r i g i n of t h e u n i t equare .

L , t

Let P be any p o i n t on t h e cu rve w i t h co -o rd ina t e s (F,FL).

1 '9 (F + F1) and n - fi (F - F1). = - 4 - - *

# * .. then rl w i l i be t h e l e n g t h of t h e o r d i n a t e from P on t h e e g a l i t a r i a n

l i n e and will be t h e d i e t a n c e of the o r d i n a t e from t h e o r i g i n a long

t he egalitarian l i n e . S ince t h e Lorenz cu rves l i e s below t h e

< F which impl ies n 0 Fur the r , i f income is e g a l i t a r i a n l i n e , Fl - always poeic ive t he equa t ion (2.2) w i l l imply n t o be l e s s than o r equal

The equa t ion of t h e Lorenz curve i n terms of 7 and n can now

be w r i t t e n as:

where a v a r i e s from zero t o fi .

I f g(X) i s cont inuous, t he d e r i v a t i v e s of F(x) and F ~ ( X )

e x i s t ; d~ I dF1 g(x) and - :i x dx

, Using these va lues i n (2.2) U

e ivcs t h e d ~ r i v a t i v e s of n wi th r e s p e c t t o a as :

Thus q w i l l b e maximum a t x - v .

2 1 I f t he Lorenz curve represented by t he equat ion (2 .3 ) is symmetric,-

. . thc vnlue of a t n and ( - 7) should be equal Eor a l l va lues of ?, - which impl ies I . 2 1 The s ~ e t r i c i t y of t he Lorenz curve is def ined with r e s p e c t -

t o the diagonal drawn perpendicular t o t he e g a l i t a r i a n l i n e .

- 5 -

f ( n ) - f ( f i - n ) f o r a l l n

The cu rve w i l l be skeved toward8 (1, 1 ) i f £ (a ) > f ( f i - n) f o r

1 < fi and i t w i l l b e skewed tcrlards (0, 0 ) i f f (a) < (fi - a ) f o r

1 ;r < fi . f o r i n s t a n c e , assume t h a t t h e equa t ion of t h e curve is

n = a n a ( f i - a)', a > 0 , a > 0 and 0 > 0.

The r c ~ t r i c t i o n a > 0 impl ies t h a t n > 0 i.e. t h e Lorenz c u r v e - l i e s below the e g a l i t a r i a n l i n e . Fu r the r , a > 0 and 0 > 0 mean t h a t s

assuaes v a l u e ze ro when a - 0 o r r - fi . Using (2.6) i t is seen t h a t t h e

curve i s symmetric i f a = B , skewed towards (1 , 1 ) i f 0 > a and skewed

tounrds (0 , 0 ) o therwise . Fu r the r r e s t r i c t i o n s on t h e c o e f f i c i e n t s of (2.7)

cnn be imposed on t h e b a s i s of equa t ions (2.4) and (2.5). I f f l ( n ) s t a n d s

For t he f i r s t d e r i v a t i v e of f ( r ) w i th r e s p e c t t o r, t h e equa t ion (2.4)

i t . p l i e s t h a t Cot X > 0 , [ l - £ ' ( a ) ] and [ l + f ' ( n ) ] should be of t h e eame - s i g n $0 t h a t t h e i r r a t i o is always p o s i t i v e . The equa t ion (2.5) means t h a t

f o r a l l v ~ l u e s of X , t h e second d e r i v a t i v e £"(a) should be nega t ive .

F o r c h i equa t i on (2.7). t h e s e t h i e e q u a n t i t i e s a r e ob ta ined a s

and

s 1 F ) + 1 -

+ - I

1 + ( T I ) = an (2.9)

(fi - a) ?r

where uea has been made of (2.2). It is t h u s obvious t h a t t h e s u f f i c i e n t I

I

condi t iona f o r equa t ione (2.4) and (2.5) t o b e s a t i s f i e d a r e 0 < a 51'

and 0 < 6 5 1. These s u f f i c i e n t cond i t i ons r u l e ou t p o s s i b i l i t y of p o i n t s

of i n f l e x i o n on t h e cu rve which a r e , of cou r se , no t p e d s s a b l e i n t h e ,

Lorenz curve.

Aa a l t e r n a t i v e c l a a s o f equa t ione of t h e Lorenz cu rve which ~ look s i m i l a r t o t h e well-known CES produc t ion func t i on proposed by &r&

Chenery, Minhaa and Solow [ I ] i a g iven b;*

vhcre t h e parameters a , 6, p and v a r e a l l g r e a t e r t han ze ro . ~ e a r r a d ~ i n ~ I

t h e equa t i on (2.11) w e o b t a i n

I

which =] .ear ly ohowa t h a t n a ~ ~ s u m e e va lue ze ro when n - 0 and n = I

h e curve i s s p m e t r i c i f 6 -+, skewed towards (1. 1 ) i f 6 > 1 and ~ 2 I

skewed towards (0, 0 ) i f 6 < f: k u r t h e r , t h e l i m i t of ( 2 . l i ) a s p ~ I

approaches ze ro becomes

- I

e is t h e same c h s s of equa t ione a s (2 .7) wi th a = 6v and 8 - v ( l - 6 ) .

1

F i n a l l y , the s u f f i c i e n t c o n d i t i o n s t h a t t h e equa t i ons (2 .4 ) and (2.5) a r e

always s a t i s f i e d f o r t h i a c l a a s of equa t ione 1 . e . (2.11) a r e 0 < 6 < 1 and

The income d e n s i t y f unc t i on u n d r r l y i n g t h e Lorenz cun r r (2.3)

is ob t a ined a s

xhere use h a s been made of equa t i ons (2.2) and (2.4).

The equa t i on (2.4) w r i t t e n he

v - x f ' ( n ) - U - x , (2.15)

g ives t he r e l a t i o n e h i p between n and x. Under t h e e u f f i c i e n t condi-

t i o n s diacueaed above, f " (n ) < 0 , which impl iee t h a t f ' ( n ) is a mno-

t o n i c a l l y dec r ea s ing f u n c t i o n of n and, t h e r e f o r e , t h e equa t i on (2.16)

can alwaye be so lved f o r n i n terms of x. S u b s t i t u t i n g t h e va lue o f n

f o r n given va lue of x i n (2.2) g i v e s t h e d i s t r i b u t i o n func t i ons P(x)

and Fy(x) . D i f f e r e n t i a t i n g (2.15) w i th r e s p e c t t o n g i v e s

* I t

d a which impl iee t h a t > 0 i . e . n i n c r ea see ae x inc reasee . Uaing

dn the va lue of. n so lved from (2.15) i n t o (2.16) ; we o b t a i n t h e v a l u e of - . . 'J

i dx

' in terms of x, which on s u b s t i t u t i n g i n ( 2 . 1 4 t g i v e s the d e n e i t y f u n c t i m

b * . g ( x ) Thue, i f t h e cond i t i on t h a t f " (7) < 0 3s s a t i t l f i e d f o r t h e given

e q u a t i o n for t h e Lorenz curve i t is alwaye p o s s i b l e t o d e r i v e t h e income

d e n s i t y f u n c t i o n under ly ing tyie equa t ion of t h e Lorenz curve.

3. INEQUALITY MEASURES AND THEIR DERIVATION

Anor,g a l l t h e measures, t h e w e t wide iy used is G i n i ' s concen t ra t ion

r a t i o which is e q u a l t o twice t h s a r e a between t h e Lorenz curve and t h e ega-

l i t a r i a n l i n e . Thus i f t h e ~ o r e n z curve is fom.ulated i n terms of IT and 0 ,

t h e concen t r a t i on r a t i o becomes

K C R d 3 1 f ( n ) d ~ (3.1)

0

which f o r t h e s p e c i f i c curve (2.7) is

3 / where B(l+a, 1+B) is t h e Beta f unc t i on which ha s been widely tabulated.-

The p a r t i a l d e r i v a t i v e of CR w i th r e s p e c t t o a , a and 0 are eva lua ted a s

a (CR) CR I, - (3.3)

aa a

= [ l u g If-+ Y(1 + 0) - Y(?+ ca + B)](CR) 3 0

where Y(l + a ) is t h e E n t e r ' s p s i func t ion which can ?-s numer ica l ly comp6ted ' C L

4 / by making uqe of t he fo l lowing r e l a t i onsh ip .- - C i

S e e Pearoon and Johnson [9 1. 4' In o r d e r t o f i n d t h e d e r i v a t i v e s and , ve r e q u i r e t h e

aa as p a r t i a l d e r i v a t i v e a of B(l+a,l+B) w i th respec t t o a and 0 . Formula 4-2531 of Gradshteyn and Ryekik [ 4 ] i s used t o e v a l u a t e t h e i n t e g r a l ob ta ined a f t e r d i f f e r e n t i a t i n g p a r t i a l l y t h e Beta func t ion .

Using these p a r t i a l de r iva t ive , t h e aeymptotic var iance of CR can

nov be obtained from t h e est imated var iances and covari.ances or' t h e pata-

C 51 ~ e t e r estizzatee a, a and 0.-

Another important measure of i nequa l i ty vhich is we l l knovn in t h e -

l i t e r a t u r e is r e l a t i v e mean d e v b t i o n . This meaoure is defined a8

whore xi l a t h e income of t h e i t h family.

It can b e a h d l t h a t T 1. aqua1 Lo the -bun discrepancy bet-

vcen F ( x ) and El (x), which is a l s o equal t o & t h e e the m u W value of s.

I n order t o ob ta in t h e maximum value of rl, equat ion (2.3) is t o be d i f fe ren-

t i a t e d with reepect t o n and equated t o zero. Than so lv ing f o r n, t h e maxiram

value of n can be obtained from the equat ion of t he Lorenz curva. For in s t auce ,

if the equat ion of t h e Lorenz cr3rve i r (2.7). equating its de r iva t ive t o zero ,

rm ob ta in

3 IL a a ~ - l ( ~ - n)' - n6na(h - nlB-' - o d r c

(3- 8)

.dhich givae ,n - 4% a + 6 and, tharafora , t h e r e l a t i v e mean dev ia t ion vill be

5 -' See Xakwani and Poddar 161.

61 c. f . Gestwlrch [ 3 ] .

Again iC t h e v a r i a n c e s and c o v a r i a n c e s o f 8 6 and 0 a r e known, i t

19 p o s e i b l e t o compute t h e asympto t ic v a i r o n c c of T.

E l t e t o and Fr igyee [2 ] have r e c e n t l y proposed a e e t o f t h r e e new

F n e q c a l i t y measures which can b e e a s i l y cornputea from t h e e q u a t i o n of t h e -

Lorenz c u r v e (2.3) bv c s i n g t h e v a l u e of n a t which q i s maximum. It is

thuo obvious t h a t t h e d e r i v a t i o n o f E l t e t o and F r i g y e s 121 measures a r e

y i n i l a r t o t h e r e l a t i v e mean d e v i a t i o n . Recen t ly , Kondor [ 7 ] has shown

t h a t thecie measures do n o t corlvey much more in format ion t h a n t h e r e l a t i v e

ncnn cic?vi.ition and it is , t h e r e f o r e , unnecessary t o d i s c u s s t h e i r d e r i v a t i o n

here . However, t h e numerical v a l u e e o f t h e s e measures d o n 3 w i t h t h e i r asymp-

t o t i c s t a n d a r d e r r o r s have t e e n computed u s i n g A u s t r a l i a n d a t a i n S e c t i o n 5 .

F u r t h e r , t h e e s t i m a t e d Lorenz curve (2-3) can be used t o o b t a i n zny

p e r c e n t i l e of t h e d i s t r i b u t i o n . To i l l u e t r e t e t h i s p o i n t t h e e s t i m a t e d

s h a r e s o f income going t o t h e p o o r e s t and r i c h e s t 5 and 15% have been compu-

t e d i n S e c t i o n 5.

4 . ON THE ESTIMATION OF 'IXE LOREN2 CURVE

The e s t i m a t i o n o f t h e Lorenz curve from grouped o b s e r v a t i o n s i s con-

. ; idered h e r e . Suppose t h e r e a r e N f a m i l i e s which have been grouped i n t o 3 I *

(T+1) income c l a s s e s , v i z . , (0 t o x l ) , (xl t o x2) ,... , (xT t o x ~ + ~ ) . Let n t

be t h e number of f a m i l i e s e a r n i n g income i n t h e i n t e r v a l x ~ , ~ - and x t , then

'9 n - -& i s t h e relative frequency f t - 5 is e c o n s i s t e n t e s t i m a t o r of t h e ,t N t

- ~ o b a b i l i t y 4 t of a fami ly belonging t o t h e t - t h income g r o u p . l l * .

I f x: is t h e sample mean f o r t h e t - t h incone group, then t h e cons fs -

t e n t e s t i m a t e s o f F(x ) and Fl(x ) a r e t t

W L I respec t ive ly . vhe rc t-1.2,. . . .T and Q - 1 $f i a the nesn +come of

Y-1 Y Y I

B, + Q Pp - 4,

t I and y t

4- d ' r B

a l l t h e f ami l i e s . Now using t h e equat ion (2,2), t h e consistent

of nt and qt a r e obtained a s

reepec t ive ly . (rt and y d i f f e r from nt snd qt by soma random d s turbance t P

es t ima to t a

t e r n . ) Then t h e equat ion of t h e Loranz curve (2.7) i n term8 of he observa-

t i o n s on rt and yt can ba w r i t t e n oo I e / of order i n probabi l i ty . -

In vhat fo l love , it w i l l be u r e f u l t o .nite t h e above

vec tor and m t r h n o t a t i o n s a s . L

l o g yt = a ' + a l o g rt + Blog (6- r t ) + wlt . where a ' - log a and wit i a the random dis turbance which can be

vherc Y1 is a Txl vedtor of T obeervatione on l o g yl, Xl i s a '3 L

T abserva t ione on t h e r ighthand a ide v a r i a b l e s of (4.3). wl i a *

r vector od T obeervat ibns on the d is turbance term and 6

c o m J ~ s t i n g of =he t h r ee elements a ' , a and 0. Then t h e

I (4 3) ~

rhovn t o be

8 - See Kakvani and Podder ( 61 ,

vhich w i l l be r e f e red to a s ?lethod I i n subsequect d i s c ~ s s i o n s .

Following Kakwitni and Podder 161 i t can be shown t h a t 6 is a con-

s i s t e n t e s t ima to r of 6 and its asymptotic vairance - covariance matrix i s

given by

va r (2 ) ( X i X i rill X (X' X )-I

where

is t h e var iance and covariance matr ix of vl. However, t he a e m p t o t i c more

e f f i c i e n t es t imator o f 6 is

which con a l s o be shown t o be cons i s t en t and i t s asymptotic var iance -

covariance m ~ t r i x would be

This genera l ized leas t- squares m ~ + . h o b w i l l bz r e f e r r ed t o a s Method 11. 'a L

The information on incone ranges is a v a i l a b l e f o r most income d i s t r i b u t i o n s - - *

which can be e f f e c t i v e l y u t i l i z e d t o improveqhe p rec i s ion of the es t imates . I)

Tc show t h i s , we cons ide r the equation (2.4) which f o r t he Lorenz curve (2.7)

con be wr ic ten a s

Substituting t h e ee t ima t e s of fi t , n t and u, t h e above equa t i on

becomes

where w is t h e random e r r o r which can a g a i n b e shown t o b e of o rder T ~ / ~ , 2 t

i n p r o b a b i l i t y .

Write (4.11) i n v e c t o r and mritrix n o t a t i o n s a8

where Y2 is a column v e c t o r of T o b s e ~ a t i o n s on t h e dependent v a r i a b l e i n

t h e equa t i on (4.11), X2 i s a Tx3 matrix, t h e f i r s t column of which c o n e i s t s

o f T obse rva t i on on t h e exp lana tory v a r i a b l e s ( - r ) and -rt of the e q u a t i o n

(4.111, and w2 is t h e v e c t o r o f s t o c h a s t i c d i s t u rbances .

The equa t i ons (4.4) and (4.12) c a n n w b e combined t oge the r ae

where

- w fa now t h e v e c t o r o f 2T d i s tu rbances w i th ze ro mean and covariances'*%atrix

The c o e f f i c i e n t v e c t o r 6 can now be e s t ima t ed from (4.13) by t h e

d i r e c t l e ae t - squa re s method which w i l l b e r e f e r r e d t o a s Hethod 111. How-

e v e r , t h e aeymptot ic more e f f i c i e n t e s t i m a t o r of 6 w i l l be

w i t h i ts asympto t i c v a r i a n c e - c o v a r i a n c e m a t r i x

v a r (i*) - (x' h1 ~ 1 - l (4.17)

- 9 / 111e e s t i m a t o r $ w i l l b e r e f e r r e d t o as Method N.-

The above p rocedure o f e s t i m a t i n g t h e pa ramete r s by combining two

s t o c h a s t i c e q u a t i o n s h a s been e a r l i e r used by T h e i l [ I ] i n connec t ion w i t h

t h e n ixed e s t i m a t i o n and by Z e l l n e r [12] i n connec t ion w i t h t h e aeeu4ngly

u n r e l a t e d r e g r e s s i o n s . It can b e demonstra ted t h a t t h e est!mntors of t h e

c o e f f i c i e n t v e c t o r 6 o b t a i n e d from t h e combined e q u a t i o n would b e more e f f i -

c i e n c t l ~ n n from t h e i n d i v i d u a l e q u a t i o n (4.4).

5. SOME EMPIRICAL RESULTS

l < e s u l t s of t h e e s t i m a t i o n o f t h e Lorenz curve and a s s o c i a t e d inequa-

l i t y measurea a r e p r e s e n t e d in t h i s s e c t i o n . I h e s o u r c e o f d a t a used f o r

t h i s purpose is t h e A u s t r a l i a n Survey o f Consumer F inances and Expend i tu res ,

1967-68 c a r r i e d o u t by t h e Hacquar ie U n i v e r s i t y and t h e U n i v e r s i t y of Queens-

Land. The n a t u r e of t h e Survey h a s been e x t e n s i v e l y d i s c u s s e d e l sewhere [ 1 3 ] .

The d a t a were s u p p l i e d t o u s i n t h e form of a set of i n d i v i d u a l o b s e i v a t i o n s .

':'he Lnc&me considered h e r e i? n e t of t a x e s b u t doe8 n o t include&fmput,ed r e n t

frvm umer occupied housee. I n d i v i d u a l income f i g u r e s made i t p o s s i b l e LO

compute t h e a c t u a l v a l u e s o f t h e c o n c e n t r a t i o n r a t i o and o t h e r measuies which

were u s e f u l i n judg ing t h e d e g r e e of a c c u r a c y ?f t h e methods d i s c u s s e d i n t h i s

paper . The grouped d a t a a r e p r e s e n t e d f n ~ a b l ! I.

9/ TI-.! d e r i v ~ t i o n of t h e c o n s i s t e n t e s t i m a t o r of t h e c o v a r i a n c e m a t r f x 2 f o l l o w s a p a t t e r n s i m i l a r t o t h a t g iven i n t h e e a r l i e r work of t h e a u t h o r s [ 6 ] and, t h e r e f o r e , h a s been o m i t t e d h e r e .

TABLE I

Income D i e t r i b u t i o n

Table I1 presen t6 t h e es t imated parametere of t h e Lorenz curve (2.7)

a long w i t h t h e d i f f e r e n t i n e q u a l i t y measures. The equa t i on h a s been es t imated

us ing f o u r a l t e r n a t i v e methods o f e s t i m a t i o n discueaed i n Sec t i on 4.

In t he t a l b e u ' , v' and w ' r ep reene t t h e t h r e e new i n e q u a l i t y measures

r

.046849

.I43297

.354601

.639493

.878343 1.061248 1.165696 1.246493 1.297084 1.346030 1.414213

proposed by E l t e t o and F r igyes (21. The latat row of t h e t a b l e p r e s e n t s t h e Q

* t

Y I 1

.03373

.080799

.I31280

.I56525

.I47471

.I22633 ,099813 .076730 .059415 .039588 - 0

P

.056?8

.I5846

.34357

.56287

.72536

.a3713

.a9485

.93566

.95919

.97978 1.00000

Mean Incame ($1

674.39 1426.10 2545.79 3469.35 4470.33 5446.60 6460.93 7459.14 8456.66 9788.38

15617.6909

Incoma range ($1

i el ow 1000 1000 - 1999 2000 - 2999 3000 - 3999 4000 - 4999 5000 - 5999 6000 - 6999 7000 - 7999 8000 - 8999 9000 -10999

11030 and over

a c t u a l v a l u e s of t h e i n e q u a l i t y measures computed on t h e b a s i s of t h e I n d i v i d u a l

q

.009274

.044193

.I57912

.341510

.516805

.663701

.753693

.a27147

.a75164

.923794 l.000000

Number of f a m i l i e e

310 552

1007 1193

884 608 314 222 128 112 110

~ J e e r v a t i o n e . I t is observed t h a t t h e i n e q u a l i t y measures computed by a l l f o u r - s t h o d s are very c l o s e t o t h e a c t u a l va luee and t h e i r a tandard e r r o r s a r e gene- - g l l y low. Method 4 g ivee t h e bee t rcmlt i n t h e s ense t h a t t h e a t a n d a d e r r o r e !e ..

a r e t h e lowes t and t h e ee t imated i n e q u a l i t y measures are c l o e e s t t o t h e i r a c t u a l

values .

TABLE I1

Results of the Different Methods of ~ s t i r : a t i o n *

Inequality Measures Hethod of Coef f ic ient

* Figures i n brackets are the asymptotic standard errors.

It should be pointed out t h a t various approximeions have been used

i n the pas t t o e r t ima te the concentrat ion r a t i o from grouped da ta . The most

common is the l l n e a r approximation which assunea t h a t wi th in each income

range the inequa l i tv i e z e r o . E 1 Therefore t h i s approximation,provides only

the lower l i m i t of the CR which i n t h e present case is ,3134. Gastwirth [3]

has r ecen t ly suggested t h e method of obta in ing the upper l i m i t of the CR which

has been computed a s .3223 i n our case. The CR computed i n Table I11 l ies w i t h i n

the lower and upper l i m i t f o r a l l four methods est imation.

The equation of t h e curve (4.3) was a l s o f i t t e d t o 20 income groups

e a r l i e r considered by t h e au thor s ( 6 ] using Method I of est imation. The

CR and T have been computed a s .3201 and .2241 respec t ive ly . Thus increasing

the number of income groups from 11 t o 20 iaprovee the accuracy of t he technique.

Table 111 presen t s t h e a c t u a l and est imated values of y using d i f f e r e n t

methoda of eet imation. It is obviour t h a t t he patimated values of y by a l l f o u r

methods a r e very c l o s e t o t h e a c t u a l values. Since the graphica l r e p r e s e n t a t i o n s

of the est imated Lorenz curve8 could not be v i s u a l l y d is t inguished from t h e a c t u a l

one, i t is superf luous t o present them here.

- lo' See Morgan [ B ] .

- 1 8 -

TABLE 111

I n o r d e r t o o b t a i n t h e es t imated f requence d i e t r i b u t i o n of t h

income we need t o s o l v e f o r n in tenne o f x from t h e f o l l o w i n g n o n - l i

equa t ion . I ~

where t h e estimates of a, a and B a r e given i n Table 11. The Newton- L

method vae uaed t o compute n f o r g iven v a l u e e of x . G / The e s t i m a t e d

quency d i a t r i b u t i u n f o r t h e family income.eo ob ta ined is g iven i n Tab

k " I t can be concluded from t h i s t a b l e t h a t t h e d e n s i t y f u n c t i o n u n d e r l y ng t h e

Lorenr c u r v e ( 2 . 7 ) p r o v i b s a reasonab ly good f i t t o t h e whole range o f t h 4 I I -

I

observed income d-iatr ibution. I

I

- "/ See H e n r i c i [ S ] .

TABLE I V

Ac tua l and Est imated Frequency D i s t r i b u t i o n of Family Income

Income Range R e l a t i v e Frequency

A c t u a l Est imated I

10% a r e p reuen ted i n Tab le V. I t is c l e a r from t h e t a b l e t h a t t h e e s t i m a t e d

Mean Iqcome

A c t u a l Es t ima ted I Under 1000 1000 - 1999 2000 - 2999 3000 - 3999 4000 - 4999 5000 - 5999 6000 - 6999 7000 - 7999 8000 - 8999 9000 - 10999

11000 +

incone a h a r e s a r e q u i t e c l o s e t o t h e a c t u a l based on t h e i n d i v i d u a l o b s e n - ? t i o n s .

The twenty income groups a g a i n p rov ide more a c c u r a t e r e s u l t s t h a n t e n groups .

The e s t i m a t e d s h a r e s of incomes going t o t h e p o o r e s t and r i c h e s t 5 a n d

TABLE V

S h a r e s of Incomee: The Poores t and Riches t 5 and 10% . . t *

.0569

. l o 1 5

. I 8 5 1

.2193

. I 6 2 5

. I118 ,0577 ,0408 .0235 .0206 .0203

SHARES OF INCOME

674.39 1426.10 2545.79 3469.35 4470.33 5446.60 6460.93 7459.14 8456.66 9788.38

15617.69

.0403

. I188

.2089

.2029

. I527

. l o 3 3 ,0661 .0406 .0245 .0236 .0183

380.47 1583'. 66 2515.38 3482.29 4469.68 5463.85 6451.13 74 71.49 8440.27 9808.39

15964.67 i

P o o r e s t 5%

Es t ima ted from 11 groups

Poores t 10% I 2.260 I 2.31A

-

Est imated from 20 groups

Riches t 5% I 14.200 I 14.380

Ac tua l from Individual

Observa t ions

Riches t 10% 1 23.600 23.780

- 20 -

The above procedure can be used to o b t a i n t h e r e l a t i v e frequency

and the mean income of income rangee which could be made as emall a s one

wiehas. By d iv id ing t h a whole income range i n t o a l n r g e number of incone

groups i t i a thue pos s ib l e t o compute a c c u r a t e l y a number of Inequa l i t y mea-

eurea which could no t be othe-se obtained from group observations. I n

a d d i t i o n , t h e d e n s i t y functioal can be u s e f u l f o r o t h e r purpoeee which need

not be mentioned here.

REFERENCES

f 1 1 Arrow, K.J., M.B. Chenery, B. S. Minhas and R.M. Solow, "Capital- Lnbour Sube t i tu t ion and Economic Eff ic iency , Reviev o f Economice and S t a t i s t i c s , X X X X I I I (1961). ~

:21 El te to , 0. , and Frigyes , "New Income Inequa l i ty Measurea ae E f f i c i e n t Tools f o r Causal Analysis and Planning", Eco ometrica, (1968), 383-396. -+- Gastv i r th , Joseph L., "The Estimation of t h e Lorenz Cu e and G I N 1 Index", The Review of Economics and S t a t i s t i c s , LIV, 1 72, 306-316.

Gradaheteyn, I.S. and Ryshik, I.M., Tables of I n t e g r a l S e r i e s and Products, (New York, Academic Press , 1965).

1 i

Henrici , P., Elements of Numerical Analysis, (New ~ o r k b John Wiley C Sons, 1967). I

I

Kakwani, N.C. s,:d N. Podder. "On t he Estimation of Curvee

Kondor, Y. "An Old-New Measure of Income Inequal i ty", hconometrica XXXIX, 1971, 1041-1042. ~ from Grouped Observations", In t e rna t iona l Economic Rev:.ew, (June, 1973), 276~291.

Morgan, Jameq, "The Anatomy of Income Dis t r ibut ion" , view of Economics and S t a t i s t i c e , XXXXIV (August, 1962), 270-2 +- 3.

X I V ,

Pearson, E.S., and N.I. Johnson, Tables of t h e Incornplire Beta- Function, (Cambrid8e: The Cambridge Univers i ty Press , $968).

Podder, N., " Diet r ibut ion of Household Income i n Austr lie", The Economic Record, XXXXVIII, (June, 1972). 181-201. 1

I Theil , H., "On t h e Use of Incomplete P r io r Information i n Regres- sion Analysis", Journa l of t h e American S t a t i s t i c a l As o c i a t i o n , Vol. LVIII, June, 1963, pp. 401-414.

I r I a

Zel lner , A., "An Eff i c i en t Method of Regressions and Teete f o r Aggregation S t a t i a t i c n l Associat ion, Vol. LVII, -

-? - -