International Bank for Reconstruction and Development Development Research Center Discussion Papers No: 12 APPLICATIONS OF LOREN2 CURVES IN ECONOMIC ANALYSIS N.C. Kakwani August 1975 SOTE: Discussion Pnpcrs are prelin~inarymaterials circulated to stimulnte diecuosion and crirical-comment. References in publication to Discussion Papers 5ould be cleared with the author(s) to protect the tentative character of these The Lorenz curve r e l a t e s t h e cumulative p r o p o r t i o n of i c c m e u n i t 6 t o t h e c u n u l a t i v e proportion of income received when u n i t s a r e arranged i n ascending order of t h e i r income. I n t h e p a s t t h e curve h a s been mainly used a s a convenient g r a p h i c a l d e v i c e t o r e p r e s e n t t h e s i z e d i s t r i b u t i o n of incone and wealth. The i n t e r e s t i n t h e Lorenz curve technique has been r e c e n t l y revived by Atlcinson [ 1 ] who provided a theorem r e l a t i n g t h e s o c i r l w e l f a r e f u n c t i o n and t h e Lorenz curve. He showed t h a t t h e ranking of income d i s t r i b u t i o n s according t o t h a Lorenz curve c r i t e r i o n is i d e n t i c a l w i t h t h e ranking implied by aggregate economic w e l f a r e r e g a r d l e s s of t h e form of t h e w e l f a r e f u n c t i o n of t h e i n d i v i d u a l s (except t h a t it be i n c r e a s i n g and concave) provided t h e Lorenz curves do n o t i n t e r s e c t . Hou- e v e r , i f t h e Lorenz curve do i n t e r s e c t , one can always f i n d two functions t h a t w i l l rank them d i f f e r e n t l y . Das Gupta, Sen and S t a r r e t t [ 2 ] have shown t h a t t h i s r e s u l t is i n f a c t more g e n e r a l and does n o t depend on t h e assumption t h a t t h e w e l f a r e f u n c t i o n s should n e c e s s a r i l y be a d d i t i i e . I n the'present paper t h e Lorenz Curve technique is used a s a t s o l t o introduce d i s t r i b u t i o n a l c o n s i d e r a t i o n s i n economic a n a l y s i s . The G n c e p t o f Lorenz curve h a s been extended and generalized t o study t h e '9 L r_elationships among t h e d i s t r i b u t i o n s of d i f f e r e n t economic v a r i a b l e s . D.e * S n e r a l i z e d Ll~renzcurves a r e c a l l e d c o n c e n t r a t i o n curves and the Lorenz.I! curve is only n s p e c i a l c a s e of such c u r v e s , v i z , , t h e concentration curve f o r income.-1 / -I / Professor Mahalonobis [ 6 ] used concentratiari curves to describe the consumption p a t t e r n f o r d i f f e r e n t commodities based on the National sample Survey Data. See a l s o Roy, Chakravnrty and Laha [ 7 1 Section 2 gives the derivation of the Lorenz curve. Some theorezs r e l a t i v e the concentration curve of a function and its e l a s t i c i t y a r e provided i n Section 3. These theorems provide the basis t o study relation- ships amon; the dietributione of different economic variables. Applications of the theorem a r e discussed i n Section 4. 2. THE WRENZ CURVE Suppose t h a t income X of a family is a raildam variable with probability density function f ( X ) . Then the distribution function F(x) is defined as: and t h i s function can be interpreted ae the proportion of families having income l e s a than or equal t o x. - If it is us& that the mean E(X) N of the dietribution exists and X > 0 - , then t h e f i r s t moment distribution function of X is dcf incd as: ; . The Lormz curve i a the relationehipbetveen - P(x) and Fl(x). The * grap%gf the curve i e repreeented i n a unit square. The equatioxr of the -. l i n e P1 F l a called t h e egalitarian l i n e and i f the Lorenz curie c o i n c i d e s with thim l i n e i t implies that each family receive8 the same income. Tlie most widely used measure of inequality is Gini's Index which is equal to twice the area between the Lorenz curve and the egalitarian line. It can be written as: - a0 and i t v a r i e s from zero t o one. 3. THE CGNCENTRATIGN CURVES Let g(X) be a continuous function of X such t h a t its f i r s t derivative e x i s t 8 and g(X) 2 0 f o r X > 0 - . I f E [g(X)1 e x i s t s , then one can define: so t h a t F [g(x)] is monotonic increasing and F1 [g(o) 1 0 and PI [g(m) 1 11. 1 The rel.ationehip between F1 [g(x)] and F(x) w i l l be called the concenkra- .? - - tfon curve of th~efunction g(x). & . It can be se)n that the Lorenz curve of income - x is a special* - I case of the concentration curve f o r the function g(x) when g(x) x. The above generalization of the Lorenz curve was suggested by Profesrsor P.C. 4lnhalanobis t o describe the consuroer behaviour pattern v i t h respect t o d i f f e r e n t commodities. The relationship between 5 [g(x)] and P (x) w i l l be called the 1 r e l a t i v e concentration curve of g(x) with respect t o x. Similarly. l e t * g (x) be another continuous function of x, then the graph of F1 [g(x)] * F1 [g (x)] will1 be called the r e l a t i v e concentration curve oL g(x) v i t h * respect t o g (x) . Let 0 (x) be the e l a s t i c i t y of g(x) v i t h respect co 8 x , then: where g l ( x ) ,is the f i r s t derivative of g(x) . * Similarly denote (x) a s the e l a s t i c i t y of g (x) with respect 8* We can now s t a t e t h e following theorem: Tli60REV I : - The concentration curve f o r the function g(x) w i l l l.ie above * (below) the concentration curve f o r the function g (x) i f 0 (x) is lees (great5r) than 0 (x) f o r a l l x > 0 - : B 8* Proof of the Tkeorw 1 Using the equation (3.1) we obtain: - 5 - vhich g i v e t h e slope of t h e r e l a t i v e concentration curxVeof g(x) with * respect LO g ( x ) a s : The equation (3.6) implies t h a t t h e r e l a t i v e concentration curve is monotonic increasing. Since the curve must pass through ( 0 , O ) and ( 1 , l ) i t f o l l w s t h a t a s u f f i c i e n t condition f o r El [ g ( x ) ] t o be g r e a t e r ( l e s s ) than * F1 [ g ( x ) ] is that the curve be convex (concave) from above. To e s t a b l i s h curvature we o b t a i n t h e second d e r i v a t i v e of Fl [g(x) ] with r e s p e c t t o the sign of the second d e r i v a t i v e is given by t h e s i g n of n (x) - n (x) . g g* Thus the second d e r i v a t i v e is p o s i t i v e (negative) i f n is greater (less) R then f o r a l l x . & * Hence t h e concentration curve f o r g(x) is above * (below) t h e concentration curve f o r g (x) i f rl (x) is less (greater) g . . P than (x) for a l l x 70 i3 -S * - ' Let g (x) = constant f o r a l l x > 0, then t h e e l a s t i c i t y ngit (x)=O L * . an* F1[g (x) = F(x) which is t h e equation of the e g n l i t a r i n n l i n e . Thus* m we have the following c o r o l l a r j . .';.~G~L.t.?? I : z h e c o n c c n t r a t i a n curve i c r :hc ~ J ~ I C L : ~ ? g ( x ) v i l l 'l:e z b c . ~ ~ . (below) t h e e g a l i t a r i a n l i n e if (x) is l e s s (greater) thz.-. g -- zero. -. The proof of Corollary 1 !s a l s o d i v e 0 by ?.;.;, ,rlzXravarti acd * i a h a [ 7 1. Next we assume t h a t g (x) = x s o ti-.at - x ) = 1 and the * * ,4 c o n c e n t r a t i o n curve f o r g (x) i s ~.c-+~ the Lzrens Tcr t h e d l s t r i b u t i o ~ x. I c follows from t h e Corollary i t!:clt tk.- Lcrenz carve f o r x lies belcr. t h e e g a l i i a r i a n l i n e and t h e r e f o r e thz cT:~;c I s ccncnve fro2 abme. F u r t h k r , from Theorem 1 we have t h e following C o r o l l a r j . CO?CLLARY 2: The concentration curve f o r the f ~ : i c " . -- --- g(x) lies above -7- (below) t h e L o r e n ~curve f o r t h e d i s t r i b u t i o n sf x i-f I ~ ~ ( x ) is l e s s (preater; thnn tinicy for a= x > 0 - . I f t h e f u n c t i o n g(x) h a s t h e u n i t e l a s t i c i t y f o r a l l x - 0 > , t':e second d e r i v a t i v e f o r t h e r e l a t i o n c o n c e n t r a t i c n of g(x) with r e s p e c t t o x w i l l be zero which implies t h a t s l o p e of t h e r c l a t i . : ~concentratlion curve w i l l h e constant For a l l v a l u e s of x . Since t h e curve s : ~ s tpnss through (0,Oj and ( 1 , l ) it means that the relntive zoncentri?:!~~ g(x) with clt reepect t o x , c o i n c i d e s with the l i n e ( 2 , G ) and (I,! i . l i ~ n c e - F1 [g(x)] F ( x ) f o r a l l x ; k-hjch pr-jvco ti?c follob-fng: 1 -- .a CCROLLARY 3: The concentration curvc f o r g ( x ) ccir.cidcs with the Lore-. - - curve f o r i f r? (x) i o r A ;I ;. ; of X . e- ---- g It should be pointed out that i!le ~ c n r r n i r a t ? r i .For g(x) ? s :;.: the same thing a s t h c Lorenz cur-Jc i i t r 7 ':,, . 1 7 -. : i.c.- lio,;t~ :kc v ( condition under which both a r e i d c n : i c ~ ~ . let y =: g(x) be a random variable with prokability density * * f~r.c:ion f (y) and the distribution function F ( y ) , and if rean of v exists, the first moment distribution funstior. cf y I s given by: - * 0 * then [ F (y) , F (y) ] is a point on the Lcrenz curve f c r g(x). The 1 following theorem gives the conditions uader which: * * F (y) = F(x) and F1(y) a Fl [ 6 ( x ) 1 (3.9) fcr all values of x. TI!E(?PE?':' 2: - &strictly If g(x) ~onotonicand has a continuous derivative g' 0:) > 0 for all x, then the concentration curve for - g(x) -coincides with the Lorenz curve for the distribution of g(x). t>oo f of the Theorem 2 Under the oeeumption that g(x) is strictly conotonic and has a cor,tinuouenon-vanishing derivative in t1:c region : , :he probo~bility density function of y is given by II * f (Y) 1 (3. ; 3 ) E: f [ h ( y ) j h f ( y ) ( - - -- where x h ( y ) is the solution of y g ( x ) . 'J - L *, - Let US now consider the gl-apt1of F (x) -JC F f g ( > r j ]which has ti:< 8 . i C slope vhich ion using (3.10) becomes one if h l ( y j > 0 . h ' ( y j is obviously g r e a t e r than zero f o r a l l y. Further s i n c e g l ( x j > 0 and t h e curve must * pass through (0.d) and ( 1 , l ) i t implies that the curve F [ g(x) ] .vs F(x) vhich h a s c c n s t a n t s l o p e one must coincide wich t h e l i n e passing * - through (0,O) and ( 1 , l ) . Hence F [ g(x) ] = F(x). * S i m i l a r l y it can b e proved t h a t t h e graph of F, [g(x)] v s F1 [ g ( x ) ] 1 has s l o p e one i f h t ( y ) > 0 . Since t h e curve pasaes through (0.0) and ( i , i ) , i t must coincide with t h e straight l i n e joining (C,O) and ( 1 , l ) * which implies F1 [g(x) 1 = F1 [g(x)] . This proves t h e theorea. I I T 1: The f u n c t i o n g(x) is said t o be Lorens superior (inferior) * t o another f u n c t i o n g (x) i f t h e Lorenz curve f o r g(x) * l i e s above (below) t h e Lorenz curve f o r g (x) f o r a l l It follows from the definition of Gini-Index that the distribution generated from f u n c t i o n g(x) w i l l have lower (higher) v a l u e of Gini- Index * thnn the d i s t r i b u t i o n generated from g (x) i f g(x) l a Lorenz euperior * ( i n f e r i o r ) t o g (x). * C I f t h e f u n c t i o n s g(x) nnd g ( x ) a r e s t r i c t l y monotcnic and have continuous d e r i v a t i v e s s t r i c t l y g r e a t e r thnn z ~ r o ,then from Theorem -- 2 i t followe t h a t t h e i r c o n c e n t r a t i o n curves co.incide with t h e i r r e s p e c t i v e J -Q Lorenz curves. Then using Thebrem 1 we o b t a i n t h e fol!\wing Corollary. E * Js I 1 -- fcFc.-!*lf?Y 5: I f t h e f u n c t i o n s g(x) a$ g (x) ore e t r i r t l y ~ o n o t o n i c -- and hnve continuous d e r i ~ n t i . ~ , e~st r i c t l vg r e a t e r than zero, R it re^^ go:) is r i r ( i f 2 (x,) . ' -i f q (x) is l e s s (grcnter) than ----- *( 9 X I f c r a?: S * ; \ ~ a i :if we p u t ~ g (x) = x * X S O t h a t q ) 1 rke 1 Ccrcll.:r:; g - 5 l e a d s t o t h e i-ollowlng C o r o l l a r y . :;,L.,;LL/:?)' - - 6: I f g ( x ) is s t r i c t l y nosotonic and i:zs a c3nt!~uous ceriva- - t i v e g ' ( x ) > 0 f o r a l l :i: t h e n g(x) Is ;,c:enz s u s e r l u r ( i n f e r i o r ) t o x - iip(x) i f is l e s s (2ri-;iir t:ioo r, ( x ) g -- f o r a l l x > O - . C 2: The c o n c e n t r a t i o n i n d e x f o r gix) l e f l n e d a3 9ne zin3.i~ t w i c e t h e a r e a u n d e r t h e c o n c e n t r a t i o n cErve fcr_ g ( x ) . I n ollr n o t a t i o n , t h e c o n c e n t r a t i o n i n d e x f o r g(x) is given by: ' 1) C g = * - 2 I F1 [g(r)] f ( x ) dn. 1t 1!1 co h c not.ed t11nt j f g ( x ) = corlr;t.nnt., t h e c o ~ ~ r ~ ~ ni :c ~r n, crirvD coi-.c5,.t2. ~ : - w i t h t h e cp,altl:arlan l i n e s o t h a t C 0. If ( x ) : -I i s dny c o n s t a n t , t h e n t h e c o n c e p t r a t i o n is e q u a l t o t h e Gini-Inr!t.x of x. F\:r'&her, i f g ( x ) 0 f o r a l l x , t h e n C is n l v z y s p o s i t l : . ~ ' 8ind v l l l bc: cq:inl r; R t h c GlnL-Index of t h e E u n c t l o r ~ g ( x ) . F i n a l l y i f b ( x ) C) for a l l X, t h ~ nt h c c-oncerrtration c u r v e f o r g ( x ) is above the er,,+l!t ~ r ; a ni:r.c - <=r.d C wlll Se equal t o minus t i m e s F - k :c -,.--..-, 1 ( ' X I ] -.zL:L.-L,4 3: ~f g(x) = gi(x) s o t h a t E !a(:.:)] t 5 i i=1 1: 1 where E is the expected value ope=r, :ten: I Y C ~ the Tnecrem 3 f of k 3 1 S u b s t i t u t i n g g(x) = gi(x) ia (3.1) g'.vtJ:-- %=I N o w Pi [gi(x)] is given by: t h i c h on subetitu!ting i n (2.13) g i v e s t h e r e s u l t s t a t e d i n T h e o r a 3. Let g(x) = n+bx EO t h a t E [ g ( x ) ] = a+bu, ~ i . - - - - . E x ; the;. g(x) cnn b e t r e a t e d RB t h e Bum of two f u n c t i o n s , v i z , a and bx. Hence f r o 2 TIleorem 3 ve o b t a i n : Because ttt! conccntrntion curve f o r a constant funce on c o i n c t d e s with the - egalitarl& l i n e . The e q ~ a t i o n(3.16) con a l s o be writttn ;ri: I -- 31 The interchange of eumriatiorr s i g n 2nd k is f i n i t e . S i n c e F ( x ) .- F1(x) > f o r all. x i t i ~ ~ p l i ethat t h e c o l : c e r . t r c ~ i . ncur-;? s i c r R 1 i n c a r function (a i- l i e s above ('telow) the Lorc:;z curve f o r bx) u i f n is g r e a t e r ( l e s s ) than zern. F x r t h e r i f b>O , t h e function g < x ) = a - bx is a monotonic increasing function of x, frox Theor& 2 5~ f u l l - ~ sthz: t h e ~ o n c e n t r ~ a t i ocurve f o r n (a + bx) c o i r c i d e s w i t h the Sorenz CLL:-ve of f u n c t i o n (a + bx). Thus w e have t h e f o l l o c i n g corollar;J. TChOI.Ur7Y 7: Tf , b > 0, then t h e l i n e a r f u n c t i o n (a 4 b u j --15- -Lorer.z s u p e r i o r ( i n f e r i o r ) t o x if a is greater (lessj t h a ~ . zero. -- - 1 titi,,ich on interchanging the s-tion and i n t e g r a l ~ i g ; : f i 5rrcL z ~ 9 : a w L U k .. I 2 5 - - " 1 - - - 3 . 1 E[p.i(~p:)j 1 - i f (u):;:< E Ig(x)! i . L > fiow C is defined ofi: 81 -12- L u [g, (x)] f (x)cix C g l - l - 2 \ 1 F ( 3 . 2 ~ ; 0 k - SttSstituting (3.24) i n (3.23) and u s i n g t h e f a c t that E[g(xjj = 1 E [ zi(x,: i=1 gives t h e r e s u l t (2.3Gj. This proves the theorex. - Let us again assur;le t h a t g(x) = a+bx so that E[g(x)] a+Ep . I f b > 0, g(x) 13 a monotonic i c c r e ~ s i c gfcnct:cn, t h e r e f c r e the c o n c a - t t a t i o n index f o r g(x) w i l l be saze as t h e Gini - index of the flalnctim. N w using t h e f a c t the Gini- Index of a constant is z c - o , and t h e Gini- Index of bx is same as the Gici-index of x, i t foilcws from T h e o r c * where G i s the Giri-Index of x arld G is the Gini-Index of the l i n e a r function x(a + bx). We have the follcding corollary. -- ---- L'CR(?LIARY 8: - I f G is the Gini-Index of n random variable x, then ihe * Gini- Index G of a l i n e a r functicn ( a 5x) -- f o r b > 0 . c is given by: - where - E(x) P , - * s * * 'e In the above Corollary i f a 0 , G = G w h f c h !.npifei? r:-,st if all incc-,__. arr n r l l t i p l i r d by a anme c o n s t a n t , then t h e * Further, G is iess ( g r e a t e r ) than G If n I n this s e c t i o n w e s h a l l c o s i d e r s o ~ ?:lf i h e a::;lic,;tic!zs of :he 4 / t?i~crczqg i v e n i n t h e i n s t s e c t ion.- -,then , - . . . If g(x) is t h e equation of Er;gal Ci;rvt'. 91 z .-r~..;nc; t-'. < r fo!lor;s f r o n C o r o l l a r y 1 and 2 thz: i f i t s concentrsririn zclrve lies above t h e e g a l i t a r i a n l i n e , i t is a n i n f e r i o r coi-l-okJlt>, iL : ocrtntratiun curve l i e s between t h e t o r e n z curve of x an3 r;!c ksnl3tiir:nn i i n e , 1: is a n e c e s s a r y commodity and i f t h e conc-.ntration curlre lit:; r e i o 1 t h e Lorenz c u r v e , t h e commodity is l u x u r y . 4 . 2 Ccnsmption and .Saving Filnctio~ro I n t h e Keynesian c a s e t h e consumption is r e l z c e d t o incoxe either l i n e a r l y o r c u r v i l i n c ~ r l y . L e t u s f i r s t a s s n n e t h a t t h e rtls:!on be iinea;: * . vhere 3 i s t h e rlarg'nal p r o p e n s l i y t o cansuz:* f:::ci - i :!spor.-~F.le ;- :!-:c. jnconr a n d c is the conGumption expenditure of an inr'.i:.i:;;:n-. S i n c e n and r a r e :renter than z e r o , i t f o l l o u s f r o n Corol2.zr-y 7 ~!i:>th e t -- ; E T : - c : . ~ - 'i ccrisur~ptione y e n d i t u r e 1s moLe e q u a l l y d i s t r ! ' 8.:t.i : t -.- . r c - r t o r , ~ ? " - L d i s p o s a b l e income. - - !+/ .'.!any nore a p p l i c a t i o n s of tho ti;carLl:a:; w!; l i-.;: .-:li ;-;:,-: ' . . I ' ,. , 3 lc~::::-;~:<:L.:.. :.onoqraph which is ;:nc!e:- prep~r;\:t:;i?. ~+..f..Lchg a i n f r o n Corollary 7 implies t h ? t t h e persGr.e7 sr;vir,zs -dill be c o r e a uEequally d i s t r i b u t e d than t h e personal d i s p o s a b l e income provzded t h e a a r g i n a l propensity is l e s s than oxe. Let UE now i n t r o d u c e t h e r a t e of i n t e r e s t a s a n ac!di:ional v a r i a b i ~ i n the savings function ( 4 . 2 . 2 ) : & e r e r is the r a t e of interest. I f 8 < 1, then fron Ccrollary 8 w e obtain: vhere G and G are Gini-Indices of dispoeable income and r,avings, respec- 8 t i v e l y . i n t h e mean d i s p o s a b l e incoine and us is the cean saving x.iiick. ?a given by: . i h Dl fret e n t i o t i n g (4.2.4) w i t h r e s p e c t t o r gives: C vh c h l e o d r t o the conclcsion t h a t higher t h c intc:. - i:,?Tc, xsre cr,';ai r >- 5e t h e d i s t r i b u t i o n of savings. This concltlo!on i:. of cp.arrr. 5:iseZ CI; :':,, ~ s s : m p t i o n t h a t th2 i n c r e a s e i n t h e lntcrc,st ro:c -.4- ' ter "--re ::L,c. L I C d i a t r i b u t i o n of the disposable incane. I -.16- i I 1 i . . J-ZZXIILZ i?!un TniI.~tim"aryi ' c s z ~ , ~ ~ I ? . I I 1 Consider an econcny i n which p r i c e s and prodcc~i-:i:y ar2 r i s i n g 2: dnnclal r a t e of 100 p and 100 s percent. Sus-,ose t h e ~ ~ I Z P T C S of a l l inccae ,.:nits a r e i ~ c r e a s i n gi n t h e same proportion. Then ir.coc.c i?f a u n i t a f t e r t where x is the i n i t i a l income. Let t h e t a x function 1;s: then the t a x c o l l e c t e d a t time t from an incorre unit ult!: i n i t i a i income ;; w i l l b e : l a thc mean tnx paid at time zero, then t h e ccnn tax p o i 2 n : r:me - t v i i l be bhich g i v e s the average tax r a t e z t tic.e L;,CX c ;:(Y) = ,. y-[r.(t) ] = , ~ ( t ) . T'llus, i f tile t a x e s tire ;rogrcssive : > 1, t!le s-~:ragc t a x r a t e w t l l ir!crcl:!se (ciecretise) c7,.cr ti.-* i f ? , s a r p g r e a t e r ( l e s s ) t h a n z e r o b u t l e s s t h a n one i n a b s o l u t e v a i u e . The d i s p o s a b l e income a t t i n e t of a u n i t havir2g i a i t l z i . iaco~iz x is x(t) - T { x ( t ) ] a n d , t h e r e f o r e , a p p l y i n g Theoi-er 3 xc. o:,tain: where $ 1 q(x) = a x6 f (x) dx is t h e proportion of tax paid by incone u n i t s having incczz l e s s than or equai * L C J x a t time z e r o and Ft (x) is the proportion of the * t h e dispos;lblc income of t h e same incone u n i t s a t Lime e . , (r) is t l ~ e ne3n d i s p o s a b l e income a t t i m e t: * t , ? t p ( t ) = { ( l f p ) ( l + s j } p - {(l.ll1) ( l L : j j - (4.L.G) (: - Tl:e ecluat i n n (4.4. 7) simplifies to: I f t h e t a x f u n c t i o n is p r o g r e s s i v e , i . e . 5 > L, then f r o n C o r o l l a r y 2 , F, (x) > q (x) .. f o r a l l x u h i c h from (1.6.10) i z p l i s s that the c o n c e n t r a t i o n c u r v e f o r t h e d i s p o s a b l e incor-e ar: ti-ze : Is higher - than ?he Lcrenz c u r v e f o r i n c o ~ e . F u r t h e r , i f ::?e zarg::,.?l t c x rate :Ls less than one, t h e d i s p o e a b l e i n c o a e is a nonotonic i n c r e s s i c g f u n c t i o n of x v h i c h f r o n Theoren 2 i n p l i e s that t h e 8ccr.cectrat?o?. C ' I N ? f o r :he d i s p o s a b l e income a t time t coincides with its Lorenz z:rve. Thus f c r n p r o g r e s s i v e t a x system t h e a f t e r t a x i n c o ~ eat t i r e r ? s m r c e q u a l l y d i s t r i b u t e d t h a n t h e b e f o r e t a x i n c o r e . Dif f e r e ~ n t i a t i n g(4.4.10) w i t h r e s p e c t t o p gi3;es: Again, i f t h e t a x system is p r o g r e s s i v e 6 > 1 and i (x) > q(x) 1 which i m p l i e s t h e right - hand s i d e o f (4.4.11) is p~sit!-.~e~nci,therefore, ,is p i n c r e a s e s t h e Lorenz c u r v e f o r a f t e r - t a x income d i s t r i l u ~ i o nv i l i s h i f t cpvard. S i m i l a r l y , i f t h e t a x system i n r e g r e s s i v e , 6 1 fin] qkx) t h e right- hand s i d e o f (4.4.11) is again pouitive. The !.%crr,i?z curve ~ i h its f L upvnrJ no p i n c r e a s e s . Thus we can c o n c l u d e t h a t t h e i n f l n e l o n decrease^ t h e a f t e r t a x income- inequality f o r b o t h p r o g r e s s i t c and :r;;r:.r:3ivc t a x systerno provided the b e f o r e t a x d i s t r i b u t i o n is n o t affected by i n f i a r i o n . - The above c o n c l u s i o n i o v a l i d ably i f t h e 'isx;.~ ace c a t 3djus;ea t o i n f l a t i o n . Su,ppose w e change t h e t a x r a t e s every yeAr kj- b c s p i a g t c c n e t n n t b u t change t h e parameter t . "i.-ne t a x f ~ m c ~ ' l atit. - -2r-c i <:an then be * n i t t e n ae: - v h e r e a 3 a t t = 0. Then t h e mean t a x a t t i n e t will be: t a t ~ ( t )= [ (1 + p) (1 + 9)j 6 t Q (4. h . l j ) and, t h e r e f o r e , t h e average t a x r a t e becmes: Suppose we a d j u s t at every year such a way that t!.e r a t i o of t o x t o income rer0,aine c o n s t a n t . Then from (4.4.14) i t can be seen t h a t vhich means a is t o be reduced every year i f t h e tax f ~ x . c t i o nis t progreesive and f o r a regreabive t a x f u n c t i o n a should he increoseci. t Nuw ueing (4.4.15) I n (4.4.10) gives: - - * which implieu t h a t d Pt(x) / dx . Thus we conclude: thnt tf t h e : n x I 0 3 - function is a d j u ~ t e devery y e a r such n \jay t h a t t h e tax-lncrtr,.e r a t 2 0 i s * * concirsnt rvery y e a r , then t h e i n f l a t i o n w i l l n o t ctiiinge t k r nf:er ?ax incs:, I di!ltrib.dtion f o r any tax system progrensive o r rcgressivc. L e t : * a Cini- index of t h e a f t e r - t a x d i s t r i b u t i c n 3t Ct t i n e t. C = Gini-index of before- tax incone a t t=0 . - Concentration index of t a x e s paid a t t i n e zero. from Theorem 4 we o b t a i n : vhich g i v e s t h e e l a s t i c i t y of t h e Gini-Index of t h e a f t e r t a x d i s t r i b u t i o n with r e s p e c t t o i n f l a t i o n r a t e a s : We can now compute t h e Gini- index and the e l o c t i c i t y of t h e Gini- index v i t h r e s p e c t t o i n f l a t i o n r a t e . The source of d a t a use6 f o r t h i s purpose is t h e A u s t r a l i a n Taxatlolf S t a t i e t i c s f o r t h e asscssrrsnt year 1971-;2 (Income t n x yenr 1970-71). lT.e data nre svoilnble i n gronped form. The incoae coneidere2 i a t h e a c t u a l income f o r i n d i v i d u a l t n x pnycrs less t h e ' ? - expenditure incurrzd i n g a i n i n g t h a t income. - ~ i n e - - i n & of b e f o r e t a x i n c ~ m evat3 conprlteC t o b e .3456 and f o r the t a x paid t h e c o n c e n t r a t i o n index w a s . 5 4 1 9 . The t a x funcl.ion was e a t i ~ m t e dt o be:- 5 / - 5 / The weighted regreseion method was uaed t o e s t l r i t c t h e ta:: function. l o g T = -6.2064 +1.583 l o g x ( L . 4 . i > ; -.h,!re .,: rt,srcsc-:~tsi n c o ~ ran2 T t~3:ics. 'The squsre.? currelntior, bc.:.;c_en ~ ~ t : ; r . j t and a c t u a l values of ~ d T ~ 3 conputed t o be .99. s ~ t b l cI j r e s e n t s t h e C i n i - i n d e xof t h e a f t e r - t a x incose and i ~ s e l a s t i c i t y v i t h r e s p e c t t o t h e raLe cE i n f l a t i o n . It is t o 52 noted the Gin[-index is q u l t e s e n s i t i v e t o t h e i n f l a t i o n asd t h e s e n s i r i v i t y i n c r e a s s s with t h e r a t e of i n f l a t i o n an2 a l e o over t i z e . Table 1: GIXI-LWEX OF THE kT@iTAX LNCOME AND ITS ELASTICITY WITH RESPECT TO IWPLirTiOH RATE * u 9 Rate a t - 1970 - 1971 1971 - 1972 1972- 1973 1973 1974 I n f l a t i o n Glni-Index l l a s t i c i t y C i n i - I n d u E l a s t i c i t y Gini-lnden E l a s t i c i t y Gini- Index E l a s t i c i ~ y - I I 1 .0143 .0201 -10 .3124 0.00 .3141 .0076 .?I 57 .0109 .3124 0.00 .3i29 .0038 .3134 .0074 .3138 .0067 .1124 -312G 1 .0022 .3124 .0045 .31?5 i I 0 .3124 O.oO j .)I17 0.0000 .31iO 0.0000 .31G3 0.0000 1I I .,I24 0.00 .3110 -,0021 .3096 1 -.0047 .JOB1 -.0071 1 I I I ,3121; .3105 .3086 -.3080 . 'Q65 -.012 3 1I I ra -.0074 I 1 -.0:<7Y 10 ! . 3 1 2 4 0.00 .309k .30hl -.0165 .7021+ r d ! I 1 I [ 1 -.0110 -. 0 -.01+55, i 5 i .3124 0.00 .3083 . 3075 2 1 , .7Qiil I i 1 I 1 i I _ -- - - .-. -i I-------- ---a S u ~ p o s et I ~ et o t a l family inzone x is i i r i ~ c e nas :he sdn, of n f z c t r s i ~ l c o r n ~ sx, , Y.~,....xn' then from Thecrem 4 , ~ - 2o b t a i n A C i is t h e concentration index of t h e i - t h f & z f o r i n c c z : ~ceni3onent b-hich b.2; :can income . 111s equation expresses the Gini-index of t i e t o t a l family Income aa t h e weighted average of tlrc concclltration i n d i c e s of each, f a c t o r income component, the weight8 being p r o p o r t i o n a l t o tile zezn ixcone of each lke equation (4.5.1) czn be used t o analyze -:, c o n t r i S u t i o n of i n s j t z ' i i t y of each f a c t o r income t o t h e t o t a l I n e q u a l i t y . / Tc i i l u s t r e t e thitr n m r r i c a i 2 . a =e . , ~ i l i ~t hee d a t a obLained from t h e A u s t r a l i a n Survey of C o n s u ~ e rExpenditure 2nd Finance, 1967-68.L/ The r e s u l t 8 a r e preaented i n Table 2. It is seen fro5 :he :able t h a t the income from employment, i . e . , wages and ssiaries c o n t r i b u t e 92.687 t o the t o t a l inequality. Unincorporated business i n c o ~ eii; sccond con t r i b c t -np L l . i8X nlid t h e property income, i . e . i n t e r e s t , dividend nxd rant c o n t r i b u t e ~ 7 r r I v 3 . 2 4 1 t o the t o t a l i n e q u a l i t y . -- ---- - ----- . . *Iihi* srohlernhas also been considered by, ' i a n i a - z 0 ! 2 j 2r.c ?-.- i i 5 1. .7'- See Prydder and Kakvani i 8 1 . L ' T ? . t . dt.lvirlJ r q u a t i o n s of t h e l i n e a r c x p e n d i t u r c ~ y s t t - n(LES) a r e gi-;:: 5y vi plyi 7- bi(v - a) ( L . 6 . l ) - w!~erc vi piqi is the per capira expenditcre f o r the L-th c o r z c d i t p , pi n Is its p r i c e and q i is the per cnpita quantity derar.cle.!. v = 1 piqi ie i=l n t o t a l p e r c a p i t a e x p e n d i t u r e and a - 'L p i y i is the sdbsistence expenditcre. I=1 E i 1s i n t e r p r e t e d as t h e marginal budget s h a r e of t h d i t ! comodity. The above system of demand e q u a t i o n s is derived by rnximizing t h e b Klein and Rubln [ 4 ] form of t h e u t i l i t y f u n c t i o n . n u = B i l o g ( q t - yi) (L. 6.2) i-1 n In r h l c h t h e 13's and y ' s are parametel8 with 0 < E l 1, B j = 1 , yl 2 0 i=1 and q i - y i > 0. Let b e t h e Gini-index f o r t h e d i s t r i b u t i o n of t h e e x p e n d i t u r e sc G i t h e i-ttl conin~odityand G* be the (;!~il-index f o r t h e t o t d l e r p c n d i t u r e , their u s l n p C o r o l l a r y 6 nn t h e e q u a t i o n (4.6.1) we o b t a i n * w t ~ c r c u 1s t h e m a n t o t a l e x p e n d i t u r e and - u i i~thc rneen expenditure ;.: ' i t h c i - t h cnrrmodit?. Tliis e q u a t i o n car1 a l s o be w r i t t e n ds t:xper!r!it UILP cl.tstjc i t y of t h e i - t h c c ~ n o dtly d t t h e mc3a-1t s i ~ c n d ! t u r e s t~ t-i;;uzl t o c t ~ rr a i i o of t h e Ginl-indices of tile d j s t r i b ~ l t i o n sni tLlc I-t:? c o m c d i ~ : : expenditures and t h e t o t a l expenditure r e s p e c t i v e l y . If the elastici.ty is g r e a t e r ( l e s s ) chan one, t h e expenditure on t h e i - t h c o m o d i t y is more ( l e s s ) r ~ n e l u ~ l ldvi s t r i b u t e d t'!ian t h e t o t a l expenditure. 4 . 5 . 1 Tnconc I n e q u a l i t y and Pr- We now c o n s i d e r t h e e f f e c t of p r i c e changes on t h e income i n e q u a l i t y of t h e r e a l income. S u b s t i t u t i n g (4.6.1) i n t o (4.6.2), we o b t a i n t h e i n d i r e c t u t i l i t y function a s *, I Suppose t h e p r i c e s pi change t o pi and t h e t o t a l expenditure -J chenges t o v*, then t h e r e e u l t i n g change i n t h e u t i l i t y w i l l be h - n * where s 1 p i y i. I f t h e change i n u t i l i t y is a e t t o r e m , we o b t a i n t h e i-1 . t o t a l per c a p i t a expenditure v* i n order that t h e family maintains t h e same . - - - - v4 w i l l be t h e - r e e l expenditure. Let GR b e t h 8 ~ i n i - i n d e xof t h e r e a l e-rn- - d l t u r e , then apply Corollary 8 on t h i e equation gives s . * --.'ere is the Gini-indsx of the ncney expenditure i n t h e base year. Tr is obvious from the equation ( 4 . 6 . 8 ) t h a t i f a l l t h e p r l c e s c;.ange - i n :he same proportion GR G* i . e . , t h e i n e q u a l i t y of t h e d i s t r i b u t i o n of t h e mozey expsnditure i n t h e base year is szme a s t h e i n 2 q u a l i t y of t h e r e a l expenditure. * The r a t i o l-isthetruecostoflivingindex.?j It converts the v money expenditure i n t o real expenditure. In t h e s p i r i t of t r u e c o ~ s tof l i v i r g - c.R index, we propoee t o u s e t h e r a t i o u s an index of t h e iccom~ei n e q u a l i t y G* t o t a k e i n t o account t h e e f f e c t s of r e l a t i v e p r i c e changes. T i i ~index converts t h e i n e q u a l i t y of t h e money liousehold expenditure d i s t r i b u t i o n t o t h e i n e q u a l i t y of :he r e a l household expenditure. I f t h i o index is l e s s than one, it implies t h a t t h e r e l a t i v e p r i c e changes a r e making t h e e x p e n a i t u r e d i s t r i b u t i o n more iaequnl. The numerical r e s u l t s on t h e index of ir.corne i n e q u a l i t y a r e presented i n Table 3. The U-K d a t a was used f o r t h i s purpose.?-/ It i s eecn from the t a b l e t h a t t h e r e l a t i v e p r i c e changee from 1964 t o 1972 have t h e e f f e c t of L i n c r e a s i n g income i n e q u a l i t y . The 1971-72 change f s p a r t i c u l a r l y aarkcd. 4 . 6 . 2 Zrlcone Inequnlity and P r i c e s: An A l t e r n a t i v e Approach -- - ' Z - Suppose t h e p r i c e of j-tt commodity changes by clj p e r c e n t , therf * - the dcmnnd f o r t h e i t commodity w i l l change by n i j a j percent:, v h e r e ) I rlij is the price e l a s t i c i t y of the i - t h c o m o d i t y with r e s p e c t t o j - t h p r l c e . -The r e s u l t i n g demand f o r t h e i - t h c o m o d i t y b e c o ~ e s '/ See Kleln and Rubin [ 4 1. See Xaellbauer [ 5 1 f o r t h e d e t a i l e d d e s c r i p t i o n of :b,c dtira. Table 3: INDEX OF INCOME INEQUALITY IN U.K. 1964-72 .-,- - - - --- - ' - T-- ,, r:hanpc I n Gin!-'Ii-,'~ ' --- - - Food 1 . 2 2 1 C l o t h i n g .037 I I!ousing --. 148 ! Durables -.l b 0 1 t I Others - 1 . 5 2 L I I I - ---- ----- -- ; ICihle6 g i v e s the p e r c e n t a g e change 111the Gini-j::?~:: 5: t>.t ril;,l P X ; ) ~ ~ Z - d l t * rt. ~I'cl: thc p r i c e of e a c h comodity 11as Incrc;,stn4 t*. -' ...- iit~~:e. 1s !7cc.r. : l 4 . 3 tt h e p r i c e i n c r e a s e of foot- and clothirrg i n c r c j t t . , r : , t incruaiity of r{:,il e x r c n c i i t \ ~ r ew h i l e the i n c r e a s e i n p r i c e of three oti,c.r gocds ce; r a s e :kt The expenditure on t h e I-th c o m d i t y a t base y e a r p r i c e s w i l l be The t o t a l expenditure is then obtained a s : n where t h e u s e h a s been made o f t h e r e e t r i c t i o n 1 Si - 1 . i-1 Ad Let C;R b e t h e Gini-index of t h e r e a l expenditure , then a p p l y h g Corollary on t h e e q u a t i o n (4.6.12) g i v e s The eirpreeeion (4.6.13) pro.~ides t h e percentnge chnngc i n t h e Gini-ir-dex . of t h e r e a l e x p e n d i t u r e w h e n G h e p r i c e o f t h e - j - t h corvlaodity chsmgea by a + Z, 4 '2 - o t h e r p r i c e s :remaining c o n s t a m . For t h e numerical i l l G t r a t l o n we used t h e d a t a obtained from t h e !-lexica I Household Survey conducted by t h e Bank o f Kexico in 1368. The f a a i l i e ~con- ~ i d e r e dwcre urban enterpreneurs. The parameters of the l i n e a r expenditure :;ye- tern were s e t h t e d using individual observations, REFERENCES (11 Atkinaon, A.B. "On the Measurener't of InequalityH, JomaZ of Ecorm~c Theory, Vol. 2 , 1970. i [2] DasGupte~, P., A.K. Sen and S t a r r e t t , D.,"Notes on the Measurement of i Inequality" , JournaZ of Economic l%eoq, Vol. 5, 1973. [3] Pie, John C.H., Gustav Ranis and Shirley W. Kuo, "Grovth and the Family Distribution of Income by Factor Components: The Case of ~ a i v z i " , Economic Growth Center, Yale University, March 1975 (mineo). (41 Klein, 1L.R. and H. Rubin, "A Conatant U t i l i t y Index of the b s t of Living", Review of Economic Studies, .W (1947-1948), 84-87. [51 Muellbauer, J. " Prices and Inequality: The United Kingdom I3perienceW, , The Economio Journal, Vol. 84,'March 1974, 32-55. [6] Mahalan~obis,P.C., "A Method of Fractile Graphical Analysis," Ecmetrica, 28, 1960, pp. 325-351. [7] Roy, J., I.M. Chakravarti and R.G. Lana, "A Study of Concentration Curves ae Description of Consumer Pattern", Studies m Commer Behmn'oxr, Indian S t a t i s t i c a l I n s t i t u t e : Calcutta, 1959. [8] Podder, N. and N.C. Kakwani, " Distribution and Redistribution of Household Income i n Australia" , The University of New South Wales (mimeo) Pebr-ry 1974. [ 9 ] Pyatt, Graham, "On the Interpretation nnd Diflaggregation of Gini Coeffi- cients" , Development Research Center, The World Bank (mimeo), February 1975. [ I ] Atkinson, A.B. "On t h e Measurement of InequalityH, Jouznai! of EcormLc Theory., Vol. 2, i970. [2] DasGupta, P., A.K. Sen and S t a r r e t t , D.,"Notes on the Xeasurement of Inequality" , JournaZ of Economic -Theo~j,Vol. 5, 1973. (31 Pie, John C.H., Gustav Ranis and Shirley W. Kuo, "Grovth and the Family Distribution of Income by Factor Components: The Carse of Taiv-", Economic Growth Center, Yale University, March 1975 (mimeo). [ 4 ] Klein, L.R. end H. Rubin, "A Constant U t i l i t y Index of the Cost of ~ i v ~ n g " , Review of Economic Stzdies, XV (1947-1948), 84-87. [5] Muellbauer, J., " Prices and Inequality: The United Kingdom Experience", The Economic J m Z , Vol. 84,. March 1974, 32-55. [ 6 ] Mehalanobis, P. C., "A Method of Fractile Graphical Analysis," Econmetrica, 28, 19160, pp. 325-351. (71 Roy, J., I.M. Chakravarti and R.G. Laha, "A Study of Concentration Curves ae Deecription of Consumer Pattern" , Studies on Commer BeFdollr, Indian S t a t i a ~ t i c a lI n s t i t u t e : Calcutta, 1959. (81 Podder, N. and N.C. Kakwani, " Distribution and Redistribu~tionof 3ousehold Income! In Australia" , The University of New Soutll Wales (mfmeo) Fkbruary 1974. [9] Pyatt:, Graham, "On the Interpretation and Disaggregation of Gin1 Coeffi- c iento", Development Research Center, The World Bnnk (rnioe!~),February 1975.