Chapter III. The bivariate Gini-index

Chapter III The bivariate Gini-index 3. Introduction The need f including me than one attribute in the analysis of economic inequality is emphasized in the wks. of Atkinson and Bourguignon (982,989), Kolm (977), Maasoumi (986), Maasoumi and Nickelsbu rg (988), Mosler (994a), Rietveld (990) and Slottje (987). Arnold (987) has given a definition f the Lenz curve in the bivariate setup. The problem of extending the Gini-index to higher dimensions was also considered by Koshevoy and Mosler (996,997). The utility of the truncated fm of the Gini-index in the univariate setup is being mentioned in section (.4). In the present chapter, we propose a measure of income inequality f the truncated variable in the bivariate setup and look into the problem of characterizing certain bivariate probability distributions using this measure. 3.2 Bivariate Gini-index As pointed out in section (.2), in the reliability context, the failure rate f a two dimensional random vect is defined in two ways. [(.2.3) and (.2.4)]. Analogous to the vect valued failure rate, we propose-a definition f the Gini-index in the bivariate setup and look into the problem of characterization of probability distributions by the fm of the bivariate Gini-index. Let X= (X;,X 2 ) represent a bivariate random vect, where X, and X 2 represents two attributes of measuring income in a population. The random variable ~ = X;I X 2 > 2 cresponds to the distribution of X; subject to the condition that X 2 is greater than an amount equal to

52 t 2 Using the terminology used by Ord, Patil and Taillie (983), quoted in section (.4), one can define the Gini-index f the random variable Y, as 00 G,(t"t 2 ) = 2 f F(x"t,,/ 2 ) df,(x"t,,/ 2 ) - 4 (3.2. ) where F(xl'/ l,/ 2 ) is the distribution function of Y, namely Similarly f the random variable ~ = x 2 X, > I, the Gini-index turns out to be 00 ~(t" 2 ) = 2 f F(x 2, " 2 ) d~(x2' " 2 ) - Iz where F(x 2,t,,/ 2 ) is the distribution function of ~ defined by (3.2.2) and ~(x2,/'/2) is the first moment distribution of ~ given by Definition 3. F a non-negative random vect X = (Xl' X 2 ) admitting an absolutely continuous distribution function, we define the bivariate Gini index f the truncated distribution as the vect (3.2.3)

53 where G,(~, t 2 ) and ~(~, t 2 ) are defined as in (3.2.) and (3.2.2) respectively. Let X = (X;, X 2 ) represent two attributes of income, say income from the land and income from the employment. Suppose among the population of individuals whose income in one of the components, say X 2 ' exceeds a certain threshold value say t 2, then G,(~,t2) measures the disparity of income in source one. Similarly, ~(~,t2) measures the disparity of income from the second source subject to the condition that the income from the other source exceeded a threshold value say ~. Hence G(~,t2) can be viewed as a measure of inequality when the two facts are taken into consideration simultaneously. 3.3 Characterization Theems In this section, we discuss characterization theems associated with some bivariate models based on the functional fm of the bivariate truncated Gini-index. We first establish a relationship between the Gini-index defined by (3.2.3) and the vitality function defined by (.2.39), which is useful f the calculation of bivariate Giniindex f particular distributions as well as f establishing characterization theems, in the sequel. Theem 3. Let X = (X;, X 2 ) be a non-negative, non-degenerate random vect admitting an absolutely continuous distribution function. If ml(~,t2)' i=,2 represents the components of the bivariate vitality function defined' by (.2.40) and ~(~,t2), i=,2 represents the components of the bivariate Gini index defined by (3.2.3), then the following relationship holds.

54 Proof From the definition (3.2.). we get - ~(,./2) <Of x ~F(x.t) dx = 2 "'f x, [F(/,.!3)-F(X,./2)) - ax;f(x,./ 2 ) dx, F(/,./ 2 ) 't ' ax,', 2' ~ F(/,./ 2 ) F(/,./ 2 ) a - The above equation can be written as (3.3.2). This gives (- ~(I,. 2)) ( I, + j F(X,./2) dx,) = I, + F j F (x,. 2) dx,. F(I,. 2) ~ (,./ 2 ) 't The above equation can be written as "'- ( - ~ (I,. 2) ) ( I, + r, (I,. 2)) = I, + F f F (X,. 2) dx,. (,./2) 't (3.3.) with i = is immediate from the above equation. The proof f ;=2 is similar. In the following theem we look into the property of the bivariate Gini-index from the point of view of truncation invariance.

55 Theem 3.2 Let X = (X;, X 2 ) be a non-negative random vect admitting an absolutely continuous distribution with respect to Lebesgue measure in the suppt of (a"oo)x(~,oo). The relation G,(t"t2) = a, + b, log tj i,j =,2, i * j (3.3.3) where a"b, are constants, holds f all real t,,t2 ~ 0 if and only if X is distributed as the bivariate Pareto type- I distribution with survival function specified by Proof When (3.3.3) holds, with i =, using equation (3.3.2) we get Differentiating with respect to t, and rearranging the terms we get (3.3.5) Differentiating (3.3.5) with respect to tl we get This gives ( + ) ( F(t" t 2) + tl ~ F(t" t2)) = a, + ~ log t2 at t, af(t"t2) ( hi) 0 =--'-- -:...:...:.;.. + + a, + ' ogt2 =. F(tl,t 2 ) at, The solution of the above partial differential equation is F(t" t 2 ) = r - S, - lit Iogt. C (t2) (3.3.6) where C (t2) is independent t,. Proceeding on similar lines with i=2 in (3.3.3) one can also obtain PI(I. I.) I. - - Bz - ~ Ioglt c. (I.) '2 = 2 2 (3.3.7)

56 where C 2 (t) is independent of 2, When t, =, in (3.3.6) we get C(t2) = F2(t2) Thus, (3.3.6) becomes F(t,,/ 2 ) = F2(t2) (- a, - ~ log~ (3.3.8) In a similar manner, when 2 =, (3.3.7) reads as so that C2(t) = F(t,) F(t,,/ 2 ) = F(t) 2 - - a, -l2 log/, (3.3.9) When 2 =, in (3.3.8) we get and from (3.3.9) we have F(t,) = t, -- a, F(t" 2 ) = r -a, 2 - -a,-l2 l og/, (3.3.0) Similarly when = in (3.3.0) we have and from (3.3.8) we get F2(t2) = 2 - - a, A(t t) - t - - a, t- - a, - ~ Jog/z '2-2 From (3.3.0) and (3.3.) we get (3.3.) t --a, t - - a, - l2 Jog/, - t - - a, t- - a, - ~ Jog~ 2-2 Taking logarithm on both sides and rearranging the terms in the above equation, we get b, = 4 = b (say) From (3.3.) we get the desired fo rm fo r F(t" 2 ) Conversely, when the distribution of X is specified by (3.3.4), by direct calculations we get ().i.j~.2.i~j. t. 2a,- + 20 log -L 8 so that the conditions of the theem are satisfied.

57 In the following theem, we look into the situation where (-G,(tl,/ 2 )) m i (t,,/ 2 ) is linear in li,i=,2. Theem 3.3 Let X = (Xl' X 2 ) be a non-negative, non-degenerate random vect admitting an absolutely continuous distribution function. The relationship ( - Gi(/l' 2 ) ) mi(t,.t 2 ) = A~ + Bi(lj),i, j =,2,i '" j (3.3.2) where Bi(tj) are non-negative functions of Ij holds f all t,.t 2 ~ 0 if and only if X follows (i) the Gumbels bivariate exponential distribution with survival function if A = F(t,,/ 2 ) = e-j.,tl-a.z~-(jtl~,a,,~>0,/'/2>0,0 ~ e ~ A,~ (3.3.3) (ii) the bivariate Pareto type-i! distribution with survival function if A > and F(t"2) = ( + S, t, + B..! 2 + bt, 2 ) - c,/,/2 > 0,s"B..!,c > 0,0 ~ b ~ (c+) s,b..! (3.3.4) (iii) the bivariate finite-range distribution with survival function F(t" 2 ) = ( - P t, - P 2 / 2 + qt, 2 ) d, 0< t, < -.!..,O < 2 < -Plt, Pl P2 - qt, if A <. Proof When (3.3.2) holds with i =, using (3.3.) we have

58 j F(xl' 2) dx, = ( (A-), + 8,(2) ) F(l'/2) I, Differentiating with respect to " we get - F(I,,/ 2 ) = (A-) F(I,,/ 2 ) + ( (A-) I, + 8,(/ 2 ) ) 2F(I,,/2) ~F(/'/2) al, The above equation simplifies to af(i"2) A =--- -~::..:... = ------ F(I" 2 ) al, 2(A-) I, + 8,(/2) Denoting by h = (,(,,/ 2 ),hz(i,,/ 2 ) ), the vect valued failure rate discussed in Johnson and Kotz (975), using (.2.5) the above equation gives, ( " 2 ) =...",.2--:(--:A---::)----:=2-B=-:-"( -,...). - 2 A I, + A Proceeding on similar lines with i = 2, we also get hz(i,,/ 2 ) = 2 (A-) 2 ~(~)' A 2 + A The above expressions f h j (I,,/ 2 ),j =, 2 are reciprocal linear in If" The rest of the proof follows from Roy (989) reviewed in section.2. The if part of the theem follows from the expressions f ( '::( )) ( ). b t (2C ) t ( + ~/2) d -...,,,/2 ~,,/2 given y + 2(A, + Ill), 2c- + (2c-)(a, + b 2 ) an (2!~), + (2d~~~/~qI2) respectively f distributions specified by (3.3.3), (3.3.4) and (3.3.5) with similar expression f (-~(,,/2)) 77:!(,,/ 2 ). Hence the condition of the theem holds. Collary 3. When () = 0 in (3.3.3), we have F(I" 2) = exp ( - A,I, - ~/2 ) (3.3.6)

59 so that Xl and X 2 this set up the relation takes the fm which is characteristic to (3.3.6). are independent and exponentially distributed. In = t. + -, I 2,,, i=,2 Collary 3.2 When b= 0 in (3.3.4), we have F(i,,/ 2 ) = ( + a, / + ~/2rc,i,,/ 2 >O,c>O (3.3.7) which is the model obtained by Lindley and Singpurwalla (986) under a different set of conditions. In this case, the property 2c ) ( + all).... ( 2c- a,(2c-) (-G,(tl,/ 2 )) m i (tl,/ 2 ) = -- I, +,,=,2,*. is characteristic to (3.3.7). It may be noted that the right-hand side of the above equation is a linear function of / and 2, The following theem provides a characterization f the three distributions considered in the above theem based on the relationship between the bivariate Gini-index defined in (3.2.3) and the bivariate mean residual life function defined in (.2.25). Theem 3.4 relationship F the random vect Xconsidered in theem 3.3 the (-G,(i,,/ 2 )) m,(tl,/ 2 ) = I, + k,,(tl,/ 2 ) J=,2 J*j,k>O. (3.3.8) holds f all /,/ 2 ~ 0 if and only if X follows anyone of the three distributions specified by (3.3.3), (3.3.4) and (3.3.5) respectively accding as k=- k<- and k>-. 2' 2 2 Proof When (3.3.8) holds with i=, using (3.3.), we have

60.., f F(X,t2) dx, = I, k r,(i"t2) F(I"t2) Differentiating with respect to I, we get 2 k r,(i"t 2 ) af(t"t2) + k ar,(t"t2) = -. F(I" t 2 ) al, al, Using the relationships (.2.5) and (.2.30), the above equation can be written as -k ~(I" t2) r,(i" t2) = T Proceeding on similar lines with i = 2, one can also get -k ~(I"t2) '2(I"t2) = T The rest of the proof follows from Roy (989), reviewed in section.2. The if part of the theem follows from the expressions f ( -~(I"t2) ) m,(i"t2) and r,(i"t2) given below with similar expressions f (-~(I"t2)) n;(i"t2) and '2(I"t2). Distribution (-~(tl't2)) m,(i"t2) r,(i" t2) (i) exponential t ( ) +'2 A, +tl} A, + Bt2 (ii) Pareto t c- ( I, (+~t2)) + 2c- c- + (a, + bt2)(c-) (iii) finite-range t d + ( -I, (- P2t2) ) + 2d + d + + (A - qt 2 )(d + ) I, + ~t2 -+ c- (a, + bt2)(c-) -I, - P2 t2 --+ d + (Pl - qt2)( d + )

6 Instead of the mean residual life function if we consider the bivariate failure rate, we get a characterization f the three distributions considered in theem 3.4, which we state as theem 3.5 below. Theem 3.5 relationship F the random vect X considered in theem 3.4, the = I, + k,i =,2 h,(t" 2 ) (3.3.9) holds f all,,/ 2 ~ 0 if and only if X follows anyone of the three distributions specified by (3.3.3),(3.3.4) and (3.3.5) respectively accding as k = -, k > - and k < -. 2 2 2 Proof When (3.3.9) holds using (3.3.) we have I, + F j F (x" 2 ) dx, = (,,/ 2 ) I, j F (x" 2 ) dx, = k F (t" 2 ) z(t,,/ 2 ) I, Differentiating with respect to t" we get = k F(t t) az(t" 2 ) + 2 k A(t t) af(t,,/ 2 ) z(t" t ) 2 " 2 at, " 2 at,. - = k az(t,,/2 ) + 2 k z(t t) af(t,,/2 ) at, "2 F(t" 2 ) at, Using (.2.5), the above equation gives

62 az(t,,/ 2 ) 2k- ----:...:...::..:.. = -- at, k Solving the above partial differential equation, we get z(t,,/2) = (-k- 2k-) t, + c,(/2) where c,(/ 2 ) is independent I,. This gives Proceeding on similar lines with i = 2, we get h.z(t,,/ 2 ) = (2k-). -k- 2 + c2(t,) The rest of the proof follows from Roy (989), mentioned in section.2. The if part of the theem follows from the expressions f ( -6,(t" 2) ) m,(t,,/2) and h,(t,,/2) given below with similar expressions f ( -~(/,,/2) ) m.z(t,,/2) and h.z(/,,/2). Distribution ( - 6,(t" 2) ) m, (I" 2) h,(/,,/2) (i) exponential t ( ), +"2 A,+I/) A, + 02 (ii) Pareto t, + ( + a,t, + ~/2 + bt, 2) c (a, + b2) (2c-) (a, + b 2 ) ( + a,t, + ~/2 + bt,2) (iii) finite-range t, + - p,l, - P2 / 2 + qt,2 d (p, - q2) ( + 2d) (p, - q2) - p,t, - P2 / 2 + qt,2

63 The following theem provides a characterization result f the bivariate Pareto type- distribution using a possible relationship between the Gini-index and the vitality function in the bivariate setup. Theem 3.6 relationship F the random vect X considered in theem 3.5, the G( ) m,(,,2) - ~,. 2,,2 -,I -, m,(i" (2) + I, (3.3.20) holds f all " 2 ~ 0 if and only if X follow the bivariate Pareto distribution with survival function (3.3.4). Proof When (3.3.20) holds with i=, using (3.3.), we get 2 I, m, (I" (2 ) <DJ ~( ) d = I, + ~ J- x,,2 X, I, + m, (I" (2 ) J- (" ( 2 ) ' (3.3.2 ) 2 I, m, (I" ( 2 ) ~ (" ( 2 ) = I, (I, + m, (" ( 2 )) ~ (" ( 2 ) + (I, + m, (I" (2)) j ~ (x" (2) dx,. 4 Differentiating with respect to I, and rearranging the terms we get 2 t (t t) af(i,,2) + t am,(,,2) + (t t) = F(I"(2),,, m, l' 2 at' at m, " 2 2,2 af(i,,2) +2,-(m,(I,, 2 )+,)+(+ am,(i,,~))_. jf(x,, 2 )dx,. F(I"(2) al, al, ~(,,2)'l Using the relationships (.2.5) and (3.3.2), the above equation can be written as -2 I, m, (I" ( 2 ) 7, (,, 2 )( I, + m, (" ( 2 )) + I, (I, + m,(i" ( 2 )) am, (,, 2 ) + m, (,, 2 )( I, + m, (" ( 2 )) al, = -2,2 7,(" ( 2 ) (I, + m, (" ( 2 )) + I, (I, + m, (I" (2)) - m, (" ( 2 ) (I, + m, (I" ( 2 )) + ( + a~~'(2)) + (,m,(,,2)-,2).

64-2 I, m,2(tl,/2),(,, 2 ) + 2,2 ~,(,,/ 2 ) + 2,2(,,/ 2 ) = -2,3,(,,/2). al,. Using (.2.44) and simplifying, we get -2,m2 (I" 2)~ 77,(,,/ 2 ) + (2,2 ~ 77,(,,/ 2 ) + 277,2(" 2 ))( m (I" 2 ) -,) = -2,3 ~ m(i" 2 ) ~ ~ ~ -2 I, ~,(,,/2) + 2,2 ~Iog 77,(" 2) + 2,(,,/2) - 2, = o. al, al, On solving the above partial differential equation, we get m,(tl,/2) = C(t2) I, where C (t2) is a function of 2 alone. This gives '(,,/ 2 ) = ( c (/ 2 )- ),c (/ 2 ) is a function of 2 Proceeding on similar lines with i = 2, in (3.3.20), we get '2(,,/2) = ( c 2 (,)- ) 2,c 2 (,) is a function of,. Using the pair of identities f the survival function of X=(X"X 2 ) in terms of the components of the bivariate MRLF, specified in (.2.28), (.2.29) and inserting the values of '(,,/ 2 ) and '2(,,/ 2 ), we get and -A( t, t ) = -a,~ exp {- I og ( -I, ) - I og ( -2 )} 2,/2 (Cl(~) -) a, (c 2 (,) -) ~ (3.3.22) -A( ) a,~ { I ( I, ) I ( 2 )},,2 =,/2 exp - (C (t2) -) og a; - (c 2 (a,) -) og ~ (3.3.23) Equating (3.3.22) and (3.3.23), we get I (I,) I exp (/2)} { - (Cl(~) -) og a, - (c 2 (,) -) og ~ - exp{ - 0g(.i) _ log (ll)} (3.3.24) (C,(t2) -) a, (c2(a,) -) ~ Setting a, =,i=,2,i:l:-j, (3.3.24) can be wr.itten as c,(a j ) -

65 exp {- a log (.i) - log (!L)} = exp {- log (.i) - a 2 log (!L)} ~. (c 2 (/ )-) 8z (C (t2)-). ~ 8z (3.3.25) ( -a) 0g(.i) - ( -a ) 0g(!L) (C (t2) -) ~ - (c 2 (f,) -) 2 8z' Dividing both sides by 0g( ~) 0g(!) we get ( ) ( ( )J _ ( ( )J- _ a 0!L = ( _ a J lo.i (C (/ ) 2 -) g 8z (c 2 (f,) -) 2 g ~ This means that each quantity should be a constant, say (), independent of and 2, Thus, and = a + () log(;) (c (/ 2 ) -) -z = a 2 + () log ( ~ ). ( c 2 ( f,) - ) -, Inserting values of and in (3.3.24), the required (C (t2)-) (c2(f,)-) distribution is obtained. The if part of the theem follows from the expressions f G,(t,,/2) and ~(/' 2) given by ( ) and 2a - + 2(}log! a + (}IOg( ~) a + () log ( ~ ) - respectively,wit~ similar expressions f ~(/'/2) and 7lz(/,/2).

66 The following theem provides a characterization result f the three distributions considered in theem 3.6 based on a functional relationship between the bivaraite Gini-index defined in (3.2.3) and the bivariate vitality function defined in (.2.39). Theem 3.7 F the random vect X considered in theem 3.6, the relation holds f all " G,(I,,/ 2 ) = k (- I, ), ;=,2, k>o (3.3.26) m,(i,,/ 2 ) 2 ~ 0 if and only if X follows anyone of the three distributions specified by (3.3.3), (3.3.4) and (3.3.5) respectively accding as k=..:!.,k>..:!. and k<..:!.. 2 2 2 Proof When (3.3.26) holds with ;= using (3.3.) we have ( - k (- m,(;', )) J m,(i,,/2 ) = I, + F j f!(x"2 ) dx,. 2 - (I" ) 2 ~ Using the relation (.2.43), we get (-k) r,(i,,/ 2 ) f!(i" 2 ) Differentiating with respect to " 2(-k) r.(t t) A(t t) af(i,,/ 2 ) we get = j f!(x" 2 ) dx,. " + (-k) ar,(i,,/ 2 ) f!(t t) = - f!(t t). l' 2 l' 2 at at l' 2 l' 2 Using the equations (.2.30) and (.2.5), the above equation takes the fm

67 This gives. k h.,(~.t2) r,(~.t2) = -. -k Similar expression can be obtained f ;(~.t2) '2(~.t2). The rest of the proof is analogous to that of theem 3.4. The if part of the theem follows from the expressions f G,(~.t2) and ( - ~ J given below. ~(~. t 2 ) Distribution ~(~. t2) ( ~) ~(~. t2) (i) exponential 2( + ~A, + ~ tl}) A, + Bt2 (ii) Pareto c ( + a,~ + ~t2 + b~t2) (2c-) ( + ~t2 + ca,~ + bc~t2) ( + a,tl +~t2 +b~t2) ( + ~t2 + ca,~ + bc~t2) (iii) finite-range d (- A~ - P2t2 + q~t2) (- A~ - P2t2 + q~t2) ( + 2d) ( + dpltl - P2t2 - dq~t2) ( + dpl~ - P2t2 - dq~t2)