aaberge_axiomatic

8/7/2019 Aaberge_Axiomatic

1/18

Journal of Economic Theory 101, 115132 (2001)

Axiomatic Characterization of the Gini Coefficient and

Lorenz Curve Orderings1

Rolf Aaberge

Research Department, Statistics Norway, P.O. Box 8131 Dep., N-0033 Oslo, Norwayrolf.aabergessb.no

Received June 10, 2000; final version received August 22, 2000;published online April 26, 2001

This paper is concerned with distributions of income and the ordering of relatedLorenz curves. By introducing appropriate preference relations on the set of Lorenzcurves, two alternative axiomatic characterizations of Lorenz curve orderings areproposed. Moreover, the Gini coefficient is recognized to be rationalizable underboth axiom sets; as a result, a complete axiomatic characterization of the Gini coef-ficient is obtained. Furthermore, axiomatic characterizations of the extended Gini

family and an alternative ``generalized'' Gini family of inequality measures areproposed. Journal of Economic Literature Classification Numbers: C43, D31,D63. 2001 Elsevier Science

Key Words: Lorenz curve orderings; axiomatic characterization; measures ofinequality; the Gini coefficient.

1. INTRODUCTION

Analyses of income distribution focus on both the level of income andthe relative income differences, i.e., on the size and the division of the cake.In applied work the standard approach is to separate these two dimensionsand use the Lorenz curve as a basis for analysing the relative income dif-ferences. By displaying the deviation of each individual income share fromthe income share that corresponds to perfect equality, the Lorenz curve

captures the essential descriptive features of the concept of inequality. Thenormative aspects of Lorenz curve orderings have been discussed by Kolm[1618], Sen [22], and Atkinson [3] who demonstrated that Lorenzcurve orderings of distributions with equal means may correspond to socialwelfare orderings. The assumption of equal means, however, limits theapplicability of their results. Real world interventions that alter the income

doi:10.1006jeth.2000.2749, available online at http:www.idealibrary.com on

1150022-053101 35.00

2001 Elsevier ScienceAll rights reserved.

1

The author thanks John K. Dagsvik for useful comments on a previous version of thispaper and Anne Skoglund for typing and editing the paper. Parts of this paper were writtenwhile the author was visiting ICER in Torino. ICER is gratefully acknowledged for providingfinancial support and excellent working conditions.


2/18

distribution are usually not mean preserving changes; taxes and transferprograms, for example, are interventions that decrease and increase themean level of income. If, in such situations, concern about inequality isover relative rather than absolute income differences,2 the condition of

scale invariance has to be introduced. The condition of scale invarianceimplies that inequality is compatible with the representation given by theLorenz curve. Thus, adopting the Lorenz curve as a basis for judgingbetween income distributions means that we are only concerned about thedistributional aspects independent of the level of mean income. This is inline with common practice in applied economics where the Lorenz curveand related summary measures of inequality are used to compare

inequality in distributions with different mean incomes.3 This practicedemonstrates that the ordering of Lorenz curves in cases of variable meanincome is of interest in its own right, irrespective of how we judge apossible conflict between the level of mean income and the degree of(in)equality and its implications for social welfare.

In theories of social welfare it has long been considered very importantto decompose social welfare with respect to mean income and inequality.

The standard approach is to derive this decomposition and the relatedmeasures of inequality from specified social welfare functions; see e.g. Kolm[16], Atkinson [3], Sen [22], and Blackorby and Donaldson [6], whoalso proposed a method for deriving social welfare functions from givenmeasures of inequality. For a critical discussion of the KolmAtkinsonapproach see Sen [25] and Ebert [12], who also introduced an alternativeapproach by explicitly taking into account value judgments of the trade-off

between mean and (in)equality in deriving social welfare functions. As afirst step, Ebert [12] introduced a mean-independent ordering of incomedistributions in terms of inequality. Then, to deal with the mean-equalitytrade-off an ordering was defined on pairs of mean income and degrees ofinequality. Moreover, the second ordering was required to be consistentwith the first ordering in the sense that the distribution with lowestinequality is preferred in comparisons of distributions with equal means.

Ebert [12] demonstrated that by combining these two orderings a socialwelfare ordering is obtained and furthermore that the related social welfarefunctions allow a mean-inequality split-up.4

116 ROLF AABERGE

2 The importance of focusing on relative incomes has been acknowledged since ancienttimes and was e.g. discussed by Plato, who proposed that the ratio of the top income to thebottom should be less than four to one (see Cowell [8]). See also Sen's [26] discussion ofrelative deprivation in the space of income and Smith's [28] discussion of necessities.

3

See e.g. Atkinson [3], Coder et al. [7], and Atkinson et al. [4] who make intercountrycomparisons of Lorenz curves, allowing for differences between countries in level of income.4 See also Ben Porath and Gilboa [5] who proposed an alternative approach for dealing

with the mean-equality approach.


3/18

Now, referring to the standard practice of separately comparing themeans and Lorenz curves of income distributions, it appears attractive torepresent the distribution by the mean income and the Lorenz curve. Thus,following Ebert's two-step approach, orderings defined on Lorenz curves

can be used as a basis for deriving social welfare orderings and relatedwelfare functions. The starting point is to introduce a preference orderingon the set of Lorenz curves as a basis for assessing the degree of inequality.Next, combining this ordering with a second ordering defined on pairs ofmean income and degrees of inequality, a social welfare ordering isobtained.

The purpose of this paper is to provide an axiomatic basis for Lorenz

curve orderings. Judgments concerning trade-off between mean and(in)equality are, however, beyond the scope of the paper. Section 2 presentstwo alternative sets of assumptions concerning a person's preferences overLorenz curves and gives convenient representations of the correspondingpreference relations. Furthermore, complete axiomatizations of the Ginicoefficient, the extended Gini family and a new ``generalized'' Gini family ofinequality measures are proposed. Section 3 introduces an alternative

characterization of first-degree Lorenz dominance as a criterion forinequality aversion to those provided by Atkinson [3] and Yaari [32].

2. REPRESENTATION RESULTS

In this section we shall demonstrate that the problem of ranking Lorenzcurves can, formally, be viewed as analogous to the problem of choiceunder uncertainty. In theories of choice under uncertainty, preferenceorderings over probability distributions are introduced as basis for derivingutility indexes. In the present context the corresponding point of departureis to assume appropriate preference relations on the set of Lorenz curves.

The Lorenz curve L for a cumulative income distribution F with mean+ is defined by

L(u)=1

+ |F(x)u x dF(x), 0u1. (1)

L is an increasing convex function with range [0,1]. Thus, L can be con-sidered analogous to a convex distribution function on [0,1].

Now, let L denote the family of Lorenz curves. Note that a convex com-

bination of Lorenz curves is a Lorenz curve and hence a member of L.A person's ranking of elements from L may be represented by a preferencerelation o

, which will be assumed to satisfy the following basic axioms.

117CHARACTERIZATION OF LORENZ ORDERINGS


4/18

Axiom 1 (Order). o

is a transitive and complete ordering on L.

Axiom 2 (Dominance). Let L1, L2 # L. If L1 (u)L2 (u) for all u #[0, 1] then L1o

L2 .

Axiom 3 (Continuity). For each L #L, the sets [L* # L : Lo L*] and[L* # L : L*o

L] are closed (w.r.t. L1-norm).

Then it follows by Debreu [9] that o

can be represented by a con-tinuous and increasing preference functional V from L to R. Hence, Vdefines an ordering on the set of all Lorenz curves. In order to give theorder relation o

an empirical content it is necessary to impose further

restrictions on V. We can obtain convenient and testable representations ofo

by introducing appropriate independence conditions. Specifically, we

shall consider the following two axioms, where L&1 is the left inverse ofL,L&1(0)=0, and L&1(1)=1.

Axiom 4 (Independence). Let L1 , L2 , and L3 be members ofL and let: # [0, 1]. Then L1o

L2 implies :L1+(1&:) L3o

:L2+(1&:) L3 .

Axiom 5 (Dual Independence). Let L1 , L2 , and L3 be members ofLand let : # [0, 1]. Then L1o

L2 implies (:L&11 +(1&:) L&13 )&1o

(:L&12 +(1&:) L&13 )

&1.

Axioms 4 and 5 correspond to the independence axioms of the expectedutility theory and Yaari's dual theory of choice under uncertainty, respec-tively (see Yaari [31]), and require that the ordering is invariant withrespect to certain changes in the Lorenz curves being compared. If L1 is

weakly preferred to L2 , then Axiom 4 states that any mixture on L1 isweakly preferred to the corresponding mixture on L2 . The intuition is thatidentical mixing interventions on the Lorenz curves being compared do notaffect the ranking of Lorenz curves; the ranking depends solely on how thedifferences between the mixed Lorenz curves are judged.

In order to clarify the interpretation of Axiom 4, consider an examplewith a taxtransfer intervention that alters the shape of the income distribu-tions and leaves the mean incomes unchanged: Let

F1and

F2be income

distributions with means +1 and +2 and Lorenz curves L1 and L2 . Nowsuppose that these distributions are affected by the following taxtransferreform. First, a proportional tax with tax rate 1& : is introduced. Second,the collected taxes are in both cases redistributed according to appropriatescale transformations of some distribution function F3 with mean +3 . Thismeans that the two sets of collected taxes are redistributed according to thesame Lorenz curve. It is understood that this redistribution is carried out

so as to give equal-sized transfers or transfers that are less progressive thana set of equal-sized transfers. Specifically, this means that (1&:)(+i+3)F&123 (t) is the transfer received by the t-quantile unit of the income

118 ROLF AABERGE


5/18

distribution Fi. At the extreme, when F3 is a degenerate distribution func-tion, the transfers are equal to the average tax (1&:) +i. At the otherextreme, F3 will give all the collected tax to the best well-off unit. After thistaxtransfer intervention, the inverses of the two income distributions are

given by

:F&1i (t)+(1&:) + iF&13 (t)

+3, i=1, 2, (2)

where F&1i (t) is the left inverse of Fi. Now, it follows readily from (2) thatthis intervention leaves the mean incomes, +1 and +2 , unchanged.

Moreover, (2) implies that the Lorenz curve for Fi after the interventionhas changed from L i to

1

+i |u

0 _:F&1i (t)+(1&:) + i

F&13 (t)+3 & dt

=:Li(u)+(1&:) L3 (u), i=1, 2. (3)

Hence, if L1 is weakly preferred to L2 , then Axiom 4 states that thechanges in L1 and L2 that follow from the above intervention will not affectthe ranking of Lorenz curves.

Axiom 5 postulates a similar invariance property on the inverse Lorenzcurves to that postulated by Axiom 4 on the Lorenz curves. The essentialdifference between the two axioms is that Axiom 5 deals with the rela-tionship between given income shares and weighted averages of corre-

sponding population shares, while Axiom 4 deals with the relationshipbetween given population shares and weighted averages of correspondingincome shares. Thus, Axiom 5 requires the ordering relation o

to be

invariant with respect to aggregation of subpopulations across cumulativeincome shares. That is, if for a specific population the Lorenz curve L1 isweakly preferred to the Lorenz curve L2 , then mixing this population withany other population with respect to the distributions of their incomeshares does not affect the ranking of Lorenz curves. As an illustration, con-sider a population divided into a group of poor and a group of rich whereeach unit's income is equal to the corresponding group mean. In judgingbetween two-point distributions, a person who approves Axiom 4 and dis-approves Axiom 5 will be more concerned about the number of poor ratherthan about how poor they are. In contrast, a person who approves Axiom5 and disapproves Axiom 4 will emphasize the size of the poor's incomeshare rather than how many they are.

By restricting to distributions with equal means we see that Axiom 4 canbe interpreted as a weaker version of Yaari's dual independence axiom.This means that a person who approves Yaari's dual independence axiom



6/18

will always approve Axiom 4. Thus, the result in Theorem 1 can be con-sidered as an alternative (and slightly different) version of the representa-tion result of Yaari [31, 32].

Theorem 1. A preference relation o

on L satisfies Axioms 14 if andonly if there exists a continuous and nonincreasing real function p( } ) definedon the unit interval, such that for all L1 , L2 # L

L1o

L2 |l

0

p(u) dL1 (u)|l

0

p(u) dL2 (u). (4)

Moreover, p is unique up to a positive affine transformation.

Proof. Assume that there exists a continuous and nonincreasing realfunction p(}) such that (4) is true for all L1 , L2 # L. Then by noting that

|p(t) d(L1(t)&L2 (t))=&|(L1 (t)&L2 (t)) dp(t),

it follows by straightforward verification that o

satisfies Axioms 14.To prove sufficiency, note that L is a subfamily of distribution functions.

Furthermore, it follows from Axioms 14 that the conditions of Theorem3 of Fishburn [13] are satisfied and thus that there exists a continuousfunction p(}) satisfying (4) where p(}) is unique up to a positive affine trans-formation. It follows from the monotonicity property of Axiom 2 that p(})

is nonincreasing. KNow, let VP be a functional, Vp : L [0, 1], defined by

Vp (L)=|l

0

P$(u) dL(u), (5)

where P$ is the derivative of P, which is a continuously differentiable andconcave distribution function defined on the unit interval. Theorem 1demonstrates that a person who supports Axioms 14 will rank Lorenzcurves according to the criterion VP. For convenience, and with no loss ofgenerality, we assume P$(1)=0. This is a normalization condition whichensures that VP has the unit interval as its range, taking the maximumvalue 1 if incomes are equally distributed and the minimum value 0 if oneunit holds all income. Thus, JP, defined by

Jp (L)=1&|1

0

P$(u) dL(u), (6)

120 ROLF AABERGE


7/18

measures the extent of inequality in an income distribution with Lorenzcurve L, when P is the chosen preference or weight function.5 By choosingP(t)=2t&t2 it follows directly from (6) and Theorem 1 that the Gini coef-ficient is rationalizable under Axioms 14. Note that +VP corresponds to

the utility representation of the ``dual theory'' of choice under risk proposedby Yaari [31, 32], who also demonstrated that the absolute Gini differencewas rationalizable under the ``dual theory.'' Moreover, by choosingappropriate preference functions we can derive attractive alternatives to theGini coefficient which are consistent with Theorem 1. For example, bychoosing the following family of P-functions,

Pk(t)=1&(1&t)k+1

, k0, (7)

we obtain the following family of measures of inequality:

Gk(L)=1&k(k+1) |1

0

(1&u)k&1 L(u) du, k0. (8)

The family [Gk] is the extended Gini family of inequality measures, intro-duced by Donaldson and Weymark [10] and by Kakwani [15] as anextension of a poverty measure proposed by Sen [24]. For a discussion ofthe extended Gini family, we refer the reader to Donaldson and Weymark[10, 11] and Yitzhaki [33]. By exploiting the fact that the Lorenz curvecan be considered analogous to a cumulative distribution function,Aaberge [1] demonstrated that the subfamily of the extended Gini family

formed by the integer values of k in (8) uniquely determines the Lorenzcurve.6 This means that no information is lost when we restrict attentionto the integer subscript subfamily of the extended Gini family of inequalitymeasures. Further discussion of the relationship between the Lorenz curveand the integer subscript subfamily of the extended Gini family is left forthe next section.

As indicated above, Axiom 4 is closely related to Yaari's dual inde-

pendence axiom and can thus be considered to be an alternative of theindependence axiom that forms the basis of the expected utility theory forchoice under uncertainty. Another alternative is provided by Axiom 5which postulates independence in terms of the inverse Lorenz curve. While


5 Mehran [20] introduced an alternative version of (6) based on descriptive arguments. Foralternative normative motivations of the JP family and various subfamilies of the JP familywe refer the reader to Donaldson and Weymark [10, 11], Weymark [30], and Ben Porath

and Gilboa [5].6 Actually, Aaberge [1] demonstrated that the integer subscript subfamily of the extendedGini family is uniquely determined by a family of inequality measures which is formed by themoments of the Lorenz curve and called the Lorenz family of inequality measures.


8/18

the Lorenz curve only can be considered analogous to a distribution func-tion, its inverse actually proves to be a distribution function. This factfollows from the following expression

L&1

(u)=F(K&1

(u)), 0u1, (9)where K defined by

K(y)=1

+ |y

0

x dF(x) (10)

is the first-moment distribution. Thus, L&1 is the cumulative distribution ofK(Y) where Y is a random variable with cumulative distribution functionF. By aggregating the ordered incomes, starting with the lowest, up toK&1(u), 100u percent of the total income is included. Hence, L&1(u) can beinterpreted as the probability that a randomly drawn income from F isequal to or lower than K&1(u). Expression (9) of L&1 demonstrates thatAxiom 5 postulates a similar invariance condition on the transformedincome distributions as the conventional independence axiom of the

expected utility theory postulates on the income distributions. The essentialdifference between these axioms emerges in how they treat the introductionof a tax reform that exclusively concerns a certain region of a nation.Whilst the conventional independence axiom requires the judgment of thisreform to be independent of the tax structure and distribution of incomesin the remaining part of the country, Axiom 5 assumes that the judgmentof the reform depends on the income structure outside the region as well

as on the distributional consequences of the reform.Now, by replacing Axiom 4 with Axiom 5 in Theorem 1, we obtain the

following alternative representation result.


on L satisfies Axioms 13 andAxiom 5 if and only if there exists a continuous and nondecreasing real func-tion q(}) defined on the unit interval, such that for all L1 , L2 # L,

L1o

L2 |1

0

q(L1 (u)) du|1

0

q(L2 (u)) du. (11)

Moreover, q is unique up to a positive affine transformation.

Proof. It follows from (1) that there is a one-to-one correspondencebetween the Lorenz curve and its inverse. Hence, the ordering relation o

defined on the set of inverse Lorenz curves is equivalent to the orderingrelation defined on L. Note that L&11 (u)L

&12 (u) for all u # [0, 1] if and

only if L1 (u)L2 (u) for all u # [0, 1]. Then, by replacing Axiom 4 with

122 ROLF AABERGE


9/18

Axiom 5, Theorem 2 follows directly from Theorem 1 where the orderingrepresentation is given by

|

1

0 q(t)

d L

&1(t)=

|

1

0 q(L

(u

))du

. K

Now, let V*Q be a functional, V*Q : L [0, 1], defined by

V*Q (L)=|1

0

Q$(L(u)) du=|1

0

Q$(t) dL&1(t), (12)

where Q$ is the derivative of Q, a continuous and convex distribution func-tion defined on the unit interval. It follows from Theorem 2 that V*Qrepresents preferences that satisfy Axioms 13 and 5. The implication isthat a person whose preferences satisfy Axioms 13 and 5 will chooseamong Lorenz curves so as to maximize V*Q . Further restrictions on thepreferences can be introduced through the preference function Q. For nor-malization purposes we impose the condition Q$(0)=0. This condition

implies that V*Q has the unit interval as its range, taking the maximumvalue 1 if incomes are equally distributed and the minimum value 0 if oneunit holds all income. Thus, J*Q defined by

J*Q(L)=1&|1

0

Q$(L(u)) du (13)

measures the extent of inequality in an income distribution with Lorenzcurve L when social preferences are consistent with Axioms 13 and 5. Bychoosing Q(t)=t2 in (13) it follows that J*Q coincides with the Gini coef-ficient. Surprisingly, there seem to be no proposals of alternatives to theGini coefficient that are consistent with Theorem 2. However, by specifyingappropriate preference functions in (13) we can derive measures ofinequality which are consistent with Theorem 2. For example, by introduc-ing the family of preference functions

Qk(t)=tk+1, k0, (14)

we obtain the related family of inequality measures

G*k(L)=1&(k+1) |1

0

Lk(u) du, k0, (15)

where G*i is the Gini coefficient.As mentioned above, L&1 can be considered as a distribution function

defined on [0, 1]. Thus, the inverse Lorenz curve is uniquely determined



10/18

by its moments, which means that the Lorenz curve can be recovered fromthe knowledge of[G*k(L): k=1, 2, ...], which we will use to denote the dualLorenz family of inequality measures. The inequality aversion properties ofthe J*Q-measures will be examined in the next section.

Note, however, that J*Q-measures can be viewed as a sum of weightedpopulation shares, where the weights depend on the functional form of theLorenz curve in question and thereby on the magnitude of the incomeshares. By contrast, JP measures can be viewed as a sum of weightedincome shares, where the weights depend on population shares rather thanincome shares. Hence, these weights do not depend on the magnitudes ofthe income shares, but merely on the rankings of income shares. Now, let

us return to the above discussion of two-points distributions in the contextof Axioms 4 and 5. Note that the effect on JP measures of increasing theincome share of the poor depends solely on the relative number of poorirrespective of their share of income, while a similar effect on J*Q measuresdepends on the poor's share of the population as well as on their incomes.In contrast, the effect on J*Q measures of an increase in the relative numberof poor depends merely on the poor's share of the incomes, while a similar

effect on JP measures depends both on the poor's share of the populationand the income.Postulating further conditions on the ordering relations, in addition to

Axioms 14 and Axioms 13 and 5, respectively, will allow us todemonstrate that it is possible to obtain axiomatizations of the extendedGini family defined by (8) and the alternative ``generalized'' Gini familydefined by (15). To this end it will be convenient to introduce the Lorenz

curves La, b , defined by

0, u


11/18


on L satisfies Axioms 14 for allL # L and Axiom 5 for all L # L if and only if there exists a positive realnumber k such that for all L1 , L2 # L,

L1o

L2 |1

0

(1&u)kdL1 (u)|1

0

(1&u)kd L2 (u).

Moreover, the representation is unique up to a positive affine transformation.


on L satisfies Axioms 13 and 5for all L # L and Axiom 4 for all L # L if and only if there exists a positive

real number k such that for all L1 , L2 # L,

L1o

L2 |1

0

(L1 (u))kdu|

1

0

(L2(u))kdu.

Moreover, the representation is unique up to a positive affine transformation.

Proof of Theorems3

and4. The necessary parts of Theorems 3 and 4

follow by straightforward verification. To prove sufficiency, note thatTheorems 1 and 2 imply that there exists a continuous and nonincreasingreal function p(}), a continuous and nondecreasing real function q(}), anda monotone real function (}), such that

|1

0

p(u) dL(u)= \|1

0

q(L(u)) du+ for all L # L . (17)

Hence, inserting for L # L, Eq. (17) equals

\(1&a)Q(b)

b +=b1&P(a)

1&afor all a # [0, 1) and b # [0, 1], (18)

where Q(u)=u0 q(t) dt and P(u)=u0 p(t) dt. Without loss of generality we

introduce the suitable normalization P(0)=Q(0)=(0)=0, P(1)=Q(1)=(1)=1, and p(1)=q(0)=0. Then, (18) yields

(1&a)=1&P(a)

1&afor all a # [0, 1) (19)

and

\Q(b)

b +=b for all b # [0, 1]. (20)



12/18

Now, inserting (19) and (20) into (18) we obtain the equation

\(1&a)Q(b)

b +=(1&a) \Q(b)

b + for all a, b # [0, 1]. (21)Note that (21) is equivalent to the functional equation

(xy)=(x) (y) for all x, y # [0, 1], (22)

which has the solution (see, e.g., Aczel [2]) #0 or 1, or there exists r>0such that

(x)=xr for all x # [0, 1]. (23)

Inserting (23) into (19) and (20), respectively, the results of Theorems 3and 4 are obtained. K

In addition to providing a rationale for the extended Gini family G andthe alternative ``generalized'' Gini family G* of measures of inequality,

Theorems 3 and 4 demonstrate that G and G* are ordinally equivalenton L .Although several authors have discussed rationales for the absolute Gini

difference (see Sen [23], Hey and Lambert [14], Weymark [30],Donaldson and Weymark [11], and Yaari [31, 32]), for the Gini coef-ficient (Thon [29]) and for the absolute Gini differences as well as for theGini coefficient (Ben Porath and Gilboa [5]), no one has established

a rationale for the Gini coefficient as a preference ordering on Lorenzcurves. Moreover, a complete axiomatic characterization of the absoluteGini difference andor the Gini coefficient has not been provided. However,as was demonstrated above, the Gini coefficient is rationalizable underAxioms 14 as well as under Axioms 13 and 5. Thus, we may conjecturethat the Gini coefficient represents the preference relation which satisfiesAxioms 15.


on L satisfies Axioms 15 if andonly ifo

can be represented by the Gini coefficient.

Proof. The necessary part of the theorem follows by straightforwardverification. To prove the sufficiency part we observe from the proof ofTheorem 3 and 4 that o

satisfies Axioms 15 for all L #L if and only if

(r+1) |1

0(1&u)r dL(u)=

_\1+1r+ |

1

0L1r(u) du

&r

, r>0,

for all L #L , (24)

126 ROLF AABERGE


13/18

where Eq. (24) is established by inserting (16) into (17) and is given by(23). Now, as (24) has to hold for all L #L we obtain the followingequation by inserting L(u)=u2 in (24):

2r+2=\r+1r+2+

r

, r>0. (25)

By applying Lemma 1 in the Appendix, it follows that Eq. (25) has aunique solution for r=1. K

Theorem 5 provides a complete axiomatization for the Gini coefficient.Thus, application of the Gini coefficient means that both independence

axioms are supported jointly. Hence, the preferences of a person whoseethical norms coincide with the Gini coefficient are invariant with respectto certain types of taxtransfer interventions and with respect to aggrega-tion of subpopulations across income shares.

Remark. By restricting the comparison of Lorenz curves to distribu-tions with equal means Axioms 15 can be considered to be defined on theset of generalized Lorenz curves rather on the set of Lorenz curves.7 Thus,in this case the above representation results are valid for generalizedLorenz curves as well as for Lorenz curves.

When the preference relations in Theorems 1 and 2 are defined on the setof distribution functions rather than on the set of Lorenz curves therepresentation results in Theorems 1 and 2 coincide with the conventionalexpected utility theory and Yaari's rank-dependent utility theory for choice

under uncertainty. Thus, it follows from Theorem 5 that risk-neutralbehavior is completely characterized by Axioms 15, provided that theseaxioms are defined on the set of distribution functions F.

Corollary 1. A preference relation o

on F satisfies Axioms 15 ifand only ifo

can be represented by the mean.

3. INEQUALITY AVERSION

In expected utility theory it is standard to impose restrictions on theutility function applying various types of stochastic dominance rules. Forexample, ``risk aversion'' is equivalent to second-degree stochasticdominance and imposes strict concavity on the utility function. Analogous

to the standard theory of choice under uncertainty Atkinson [3] defined


7 The generalized Lorenz curve, defined as a mean scaled-up version of the Lorenz curve,was introduced by Shorrocks [27].


14/18

inequality aversion as being equivalent to risk aversion. This wasmotivated by the fact that the PigouDalton transfer principle is identicalto the principle of meanpreserving spread introduced by Rothschild andStiglitz [21] which is equivalent to the condition of dominating noninter-

secting Lorenz curves when attention is restricted to distributions withequal means. These principles may even be used as a basis for discussingand interpreting cross-country comparisons of Lorenz curves. To performinequality comparisons with Lorenz curves we can deal with distributionsof relative incomes or alternatively simply abandon the assumption ofequal means. The latter approach normally forms the basis of empiricalstudies and is also employed in this paper.

An interesting question is what kind of restrictions does inequality aver-sion (dominating nonintersecting Lorenz curves) place on the preferencefunctions P and Q? Yaari [32] demonstrated that JP defined by (6) sup-ports the criterion of dominating nonintersecting Lorenz curves if and onlyif P is strictly concave. An alternative characterization of the criterion ofnonintersecting Lorenz curves is provided by Theorem 6.

Let Q1 be a class of preference functions related to J*Q and defined by

Q1=[Q: Q$ and Q" are continuous on [0, 1], Q$(t)>0 and Q"(t)>0 fort # (0, 1) , and Q$(0)=0].

Theorem 6. Let L1 and L2 be members of L. Then the followingstatements are equivalent;

(i) L1(u)L2(u) for a u # [0, 1], and the inequality holds strictly forat least one u # (0, 1) .

(ii) J*Q (L1)


15/18

An interesting question is whether the characterization of Lorenzdominance can be achieved for smaller classes of preference functions thanP1 and Q1 . As referred to above, Aaberge [1] demonstrated that the sub-family [Gk: k=1, 2, ...] of the extended Gini family uniquely determines

the Lorenz curve. Thus, to characterize Lorenz dominance we may restrictour attention to the family ofP-functions defined by (7). Similarly, we mayrestrict attention to the Q-functions defined by (14) since the dual Lorenzfamily of inequality measures [G*k: k=1, 2, ...] uniquely determines theinverse Lorenz curve and consequently the Lorenz curve. These results aresummarized in Theorem 7.

Theorem 7. Let L1 and L2 be members of L and let Gk and G*k bedefined by (8) and (15), respectively. Then the following statements areequivalent:

(i) L1 (u)L2 (u) for all u # [0, 1] and the inequality holds strictlyfor at least one u # (0, 1).

(ii) Gk(L1)


16/18

Inserting (28) in (13) yields

J*Qa (L)=1&+

F&1(1), (29)

where F&1(1) is the largest income. Thus, J*Qa is the upper limit ofinequality aversion for members ofQ1 and is in this respect ``dual'' to theRawlsian relative maximin criterion. Thus, we may denote it the relativeminimax criterion. In contrast to the Rawlsian relative maximin criterion,the relative minimax criterion focuses on the relative income of the best-offunit. If decisions are based on the relative minimax criterion, the income

distributions for which the largest relative income is smaller is preferred,regardless of all other differences. The only transfers which decreaseinequality are transfers from the richest unit to anyone else.

The two axiomatic based approaches for measuring inequality differnotably with respect to their descriptions of the most inequality aversebehavior. When evaluation of inequality is based on the JP-measures, rais-ing the emphasis on transfers occurring lower down in the Lorenz curve

attains the most inequality averse behavior. In contrast, if inequality isassessed in terms of J*Q-measures, raising the emphasis on transfers occur-ring higher up in the Lorenz curve attains the most-inequality-aversebehavior. Note that this difference in inequality aversion originates fromthe difference between Axioms 4 and 5.

APPENDIX

Lemma 1. Let # be a function on [0, ) defined by

#(x)=x log \x+1x+2++log (x+2)&log 2.

Then x=1 is a unique root of #(x)=0 in (0, ) .

Proof. By straightforward first- and second-order differentiation of#(x)we get

#$(x)=log \x+1x+2++

xx+1

&x&1x+2

and

#"(x)= &x2&x&3

(x+1)2 (x+2)2.

130 ROLF AABERGE


17/18

Hence, #"(x)=0 ifx=x1#12(1+-13) and

#"(x){>0

0, #(x) can have at most oneroot in (0, x1) and at most one root in (x1 , ) . However, observing that#(x)>0 for x>x1>1, we have that #(x)>0 for all x # [x1 , ) and thusthat x=1 is the only root in (0, ) . K

Lemma

2.

Let H be the family of bounded, continuous, and nonnegativefunctions on [0, 1] which are positive on ( 0, 1) and let g be an arbitrarybounded and continuous function on [0, 1]. Then

|g(t) h(t) dt>0 for all h # H

implies

g(t)0 for all t # [0, 1],

and the inequality holds strictly for at least one t # (0, 1) .

The proof of Lemma 2 is known from mathematical textbooks.

Proof of Theorem 6. To prove the equivalence of (i) and (ii) we use that

L

&1

1(u

)L

&1

2(u

) for allu

# [0, 1] if and only ifL1

(u

)L2

(u

) for allu # [0, 1]. Next, from the definition (13) of J*Q and using integration byparts it follows that

J*Q (L2)&J*Q(L1)=|1

0

Q"(t)(L&12 (t)&L&11 (t)) dt.

Hence, if (i) holds then J*Q (L2)&J*Q (L1)>0 for all Q # Q1 .

The converse statement follows by straightforward application ofLemma 2. K

REFERENCES

1. R. Aaberge, Characterizations of Lorenz curves and income distributions, Soc. Choice

Welfare 17 (2000), 639653.2. J. Acze l, ``Lectures on Functional Equations and their Applications,'' Academic Press,

New York, 1966.3. A. B. Atkinson, On the measurement of inequality, J. Econ. Theory 2 (1970), 244263.



18/18

4. A. B. Atkinson, L. Rainwater, and T. Smeeding, `Ìncome Distribution in OECD Coun-tries: The Evidence from the Luxembourg Income Study (LIS),'' Social Policy Studies,Vol. 18, OECD, Paris, 1995.

5. E. Ben Porath and I. Gilboa, Linear measures, the Gini index, and the income-equalitytrade-off, J. Econ. Theory 64 (1994), 443467.

6. C. Blackorby and D. Donaldson, Measures of relative inequality and their meaning interms of social welfare, J. Econ. Theory 18 (1978), 5980.

7. J. Coder, L. Rainwater, and T. Smeeding, Inequality among children and elderly in tenmodern nations: The United States in an international context, Amer. Econ. Rev. Proc. 79(1989), 320324.

8. F. A. Cowell, ``Measuring Inequality,'' Philip Allan, Deddington, 1977.9. G. Debreu, Continuity properties of Paretian utility, Int. Econ. Rev. 5 (1964), 285293.

10. D. Donaldson and J. A. Weymark, A single parameter generalization of the Gini indices

of inequality, J. Econ. Theory22

(1980), 6786.11. D. Donaldson and J. A. Weymark, Ethically flexible indices for income distributions in thecontinuum, J. Econ. Theory 29 (1983), 353358.

12. U. Ebert, Size and distribution of incomes as determinants of social welfare, J. Econ.Theory 41 (1987), 2333.

13. P. C. Fishburn, ``The Foundation of Expected Utility,'' Reidel, Dordrecht, 1982.14. J. D. Hey and P. J. Lambert, Relative depreviation and the Gini coefficient: comment,

Quart. J. Econ. 94 (1980), 567573.15. N. C. Kakwani, On a class of poverty measures, Econometrica 48 (1980), 437446.

16. S. CH. Kolm, The optimal production of social justice, in ``Public Economics'' (J. Margolisand H. Guitton, Eds.), Macmillan, New YorkLondon, 1969.

17. S. CH. Kolm, Unequal inequalities, I, J. Econ. Theory 12 (1976), 416442.18. S. CH. Kolm, Unequal inequalities, II, J. Econ. Theory 13 (1976), 82111.19. M. O. Lorenz, Method for measuring concentration of wealth, JASA 9 (1905), 209219.20. F. Mehran, Linear measures of inequality, Econometrica 44 (1976), 805809.21. M. Rothschild and J. E. Stiglitz, Increasing risk: a definition, J. Econ. Theory 2 (1970),

225243.22. A. Sen, `Òn Economic Inequality,'' Clarendon Press, Oxford, 1973.23. A. Sen, Informational bases of alternative welfare approaches, J. Public Econ. 3 (1974),

387403.24. A. Sen, Poverty: an ordinal approach to measurement, Econometrica 44 (1976), 219231.25. A. Sen, Ethical measurement of inequality: Some difficulties, in ``Personal Income Dis-

tribution'' (W. Krelle and A. F. Shorrocks, Eds.), North-Holland, Amsterdam, 1978.26. A. Sen, `Ìnequality Reexamined,'' Clarendon, Oxford, 1992.27. A. F. Shorrochs, Ranking income distributions, Economica 50 (1983), 317.28. A. Smith, `Àn Inquiry into the Nature and Causes of the Wealth of Nations,'' Clarendon,

Oxford, 1979.29. D. Thon, An axiomatization of the Gini coefficient, Math. Soc. Sci. 2 (1982), 131143.30. J. A. Weymark, Generalized Gini inequality indices, Math. Soc. Sci. 1 (1981), 409430.31. M. E. Yaari, The dual theory of choice under risk, Econometrica 55 (1987), 95115.32. M. E. Yaari, A controversial proposal concerning inequality measurement, J. Econ. Theory

44 (1988), 381397.33. S. Yitzhaki, On an extension of the Gini inequality index, Int. Econ. Rev. 24 (1983),

617628.

132 ROLF AABERGE

aaberge_axiomatic

Documents