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On the axiomatization of the weakly decomposable inequality indices
Mathematical Social Sciences 83 (2016) 71–78
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On the axiomatization of the weakly decomposable inequality indices Pauline Mornet ∗ Université Montpellier, UMR5474 LAMETA, F-34000 Montpellier, France Faculté d’Économie, Av. Raymond Dugrand, Site de Richter C.S. 79606, 34960 Montpellier Cedex 2, France
highlights • • • •
A new axiomatization of the weak decomposition property is provided. Standard weighting functions depending on n and µ are defined. A class of weakly decomposable inequality indices is characterized. No specific invariance value judgment is assumed through the characterization.
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info
Article history: Received 13 August 2015 Received in revised form 20 July 2016 Accepted 24 July 2016 Available online 1 August 2016
abstract In this paper, we focus on a decomposition property recently introduced in the inequality literature and known as the weak decomposition. Such a property provides interesting analyses by allowing one to separate the within-group contribution to total inequality from the between-group contribution. A limitation of the current method of decomposition is that, depending on the structure – absolute, relative, compromise – of the inequality index, specific weights have to be used. To avoid such a problem, we propose a unique decomposition property where the weighting functions depend on the size of the population and the mean income. This allows us to characterize a large family of weakly decomposable inequality indices without any recourse to implicit invariance value judgments. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Since the 1970s, particular attention has been paid to the axiomatic foundations of inequality measures. As pointed out by Kolm (1976a), these researches aimed at justifying the structure of the inequality measures. ‘‘[. . .] To really see what these measures imply, it is necessary to build an axiomatic of them, i.e., to find for each one a set of properties which are equivalent to its adoption. If these properties are small in number and as intuitive as possible, they will display the implicit assumptions made by this choice.’’ [Kolm (1976a, p. 426)] In order to get an inequality measure as representative as possible of the economic reality, specific requirements have to be imposed. Beyond basic requirements such as continuity, normalization, symmetry, or population replication invariance, particular
∗ Correspondence to: Université Montpellier, UMR5474 LAMETA, F-34000 Montpellier, France. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.mathsocsci.2016.07.004 0165-4896/© 2016 Elsevier B.V. All rights reserved.
emphasis is placed on subgroup decomposability. Different sets of axioms have been introduced in order to lay the groundwork for a thorough evaluation of the income inequalities when the population consists of exclusive and exhaustive groups of individuals. Subgroup decomposability is particularly useful for empirical studies when one is interested in the contribution to overall inequality of particular groups. For instance, when the population is partitioned into subgroups reflecting socioeconomic criteria (e.g., genders or occupations), a more accurate analysis of inequality is provided when the discrepancies are measured within and between subgroups. In this paper, we are interested in the characterization of the weakly decomposable inequality indices investigated by Ebert (2010). The concept of weak decomposability contrasts with that of strong decomposability introduced by Bourguignon (1979) and Shorrocks (1980, 1984). In both approaches, within-group inequality is computed as the weighted sum of the inequalities in the different groups. It is the definition of the between-group term that makes the difference. While in the strong decomposition process the computation of the between-group term is based on the representative income of the subgroups (generally the mean income), in the weak decomposition process it is based on the
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P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
pairwise comparisons of individuals’ income.1 To make clear the difference between both approaches, let us take an example. Let x be an income distribution partitioned into two subgroups, such that x = (x1 , x2 ) with x1 = (0, 1, 4, 15) and x2 = (2, 3, 5, 10). Note that the average incomes and the population sizes of both subgroups are identical. In order to measure income inequality, we appeal to the square of the coefficient of variation, which, as a member of the generalized entropy family satisfies the strong decomposition property. Application of this property to the income distribution x reveals that the overall inequality is only due to within-subgroups inequality since the between-group term (based on average incomes) is zero. Now consider the income distribution x¯ = (¯x1 , x¯ 2 ) such that x¯ 1 = (0, 0, 0, 20) and x¯ 2 = (5, 5, 5, 5). Both subgroups represent extreme cases. In subgroup 1 all of the income belongs to one individual while in subgroup 2 the same amount of income is held by each individual. Note that average income in each subgroup is still equal to 5. As expected the computation of the total inequality in both distributions highlights that the distribution x¯ is more unequal than the distribution x. However the between-group inequality is the same for both distributions. Indeed, with the strong decomposition process, between-group inequality is computed by smoothing income within groups by using the mean or the equally distributed equivalent income (EDEI). Since all the income distributions have the same mean incomes, the nullity of the between-group terms follows directly. So, the application of the strong decomposition process to the income distributions x and x¯ suggests that inequalities occur only within the groups. In particular, in the case of the distribution x¯ , inequalities are only within the subgroup 1. This simple example illustrates the limit of the measurement of inequality based on representative income of the subgroups such as the average income. The strong decomposition is not the only decomposition process that fits with the structure of the square of the coefficient of variation. Indeed, the weak decomposition of Ebert (2010) also applies to such an index but then the interpretation of the within- and between-group inequality terms, is different. For both partitioning (x1 , x2 ) and (x¯ 1 , x¯ 2 ), two non-zero terms are obtained. Besides, the contribution to overall inequality of the within-group and the between-group terms is identical in both partitions. Imagine now that one is interested in absolute inequality as suggested by Kolm (1976a,b). In such a case, the recourse to the variance makes sense. If the variance can be strongly decomposed in the same way as the square of the coefficient of variation, the weak decomposition cannot be performed on this indicator unless resorting to an alternative formulation of this property. To this aim, Ebert (2010) distinguishes two axiomatic formulations to decompose inequality. A first property (denoted (WD1)) is dedicated to the decomposition of absolute and relative inequality indices, while a second one (WD2) is applied to ‘‘compromise2 ’’ inequality indices. Because of this double axiomatization, all the inequality indices cannot be decomposed according to exactly the same scheme. More specifically, the choice of the decomposition property implicitly imposes that the inequality index verifies a particular invariance property. Invariance conditions (such as scale invariance or translation invariance) allow one to describe the sensitivity of inequality
1 As pointed out by Ebert (2010), individuals belonging to different subgroups are only considered. 2 The terms compromise indices refer to inequality indices of which the structure depends directly on the average income (e.g., the standard Gini index or the squared coefficient of variation), see Blackorby and Donaldson (1980) or Ebert (1988) for more details.
indices to particular economic changes. It is well known that when an inequality index remains insensitive to a proportional increase in individuals’ incomes, it is scale invariant (relative inequality). On the other hand, if the inequality remains unchanged when the same amount of income is added to each individual in the population, then the index is said to be invariant to translations (absolute inequality). Thus, it may seem restrictive to limit the appreciation of inequalities in a population to only one value judgment implicitly embodied in a decomposition property. The aim of this paper is to propose a reformulation of the weak decomposition property that does not depend on a particular invariance condition. While Ebert’s (2010) approach consists in mixing the properties of decomposability and invariance, it is shown that by introducing weighting functions for the within- and between-group components which take into account not only the population sizes but also the mean income, in the same manner as Shorrocks (1980, 1984), the application of both properties can be disentangled. On the basis of such a decomposition property the characterization of the large family of weakly decomposable indices is first provided. Then, by invoking invariance conditions, extensions of the Gini’s mean difference, the standard Gini coefficient, the variance of logarithms and also the Krtscha index are presented as subfamilies of the weakly decomposable indices. The paper is organized as follows. Section 2 introduces the notations and the axioms underlying decomposable inequality indices. Our general formulation of the weak decomposition is also presented in this section. In Section 3 the class of weakly decomposable indices consistent with our decomposition property is axiomatically characterized. Concluding remarks are presented in Section 4. The proofs of the theorems are relegated to Appendix. 2. Notation and axioms Let N := {1, 2, . . . , n} (n > 1) be fixed population of individuals. An income distribution for N is a list x := (x1 , x2 , . . . , xn ), where xi ∈ R+ is the income of individual i ∈ N , and R+ is the non-negative part of the real line. The arithmetic mean of x ∈ Rn+ is indicated by µ(x) := i∈N xi /n, or simply by µ when no ambiguity arises, and the total income by Y := nµ. A partition of N is a list (N1 , . . . , Ng ) such that ng = #Ng > 1, ∀g ∈ {1, . . . , G} (G > 1) and
G
g =1 Ng = N . Then x = n (x1 , x2 , . . . , xG ) such that for each subgroup g , xg ∈ R+g is the income distribution, µg the mean income and Yg := ng µg the total income. We denote by µ := (µ1 , . . . , µG ) the list of the subgroup means, and 1n := (1, . . . , 1) a list where 1 is repeated n times.
Finally, an inequality index for a population of n individuals is a function I (·; n) : Rn+ −→ R+ . We first introduce four standard axioms, usually employed in the literature on income inequality measurement. The first one is a regularity condition imposed on the inequality index. Continuity (CN). For all x ∈ Rn+ , I (x; n) is continuous. The next axiom requires that incomes among the permutation of the individuals do not affect the value of the inequality index. This is captured by the following axiom also referred to an anonymity condition. Symmetry (SM). For all x ∈ Rn+ , I (x; n) is symmetric in x. Another standard axiom requires that the inequality index is always non-negative, with equality to zero if and only if the income distribution is perfectly equal. Normalization (NM). For all x ∈ Rn+ , I (x; n) > 0 and I (x; n) = 0, if and only, if there exists λ ∈ R+ such that x = λ1n . The last standard axiom is the Dalton’s (1920) population replication invariance, according to which a m-fold replication of the distribution has no impact on inequality. This property is useful for making comparisons across populations of different sizes.
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
Population replication invariance (PP). For all x ∈ Rn+ and all m > 2, I (x[m] ; mn) = I (x; n), where x[m] := (x, . . . , x) ∈ Rmn + is a m-fold replication of x. We now consider the different decomposability approaches that have been proposed so far in the literature. According to Shorrocks (1980), an inequality index is strongly subgroup decomposable if total inequality can be rewritten as the sum of the weighted averages of the inequalities within groups and the inequality between the group mean incomes. Strong Decomposition (SD). The inequality index is strongly decomposable if I (x; n) =
G
ωgG (µ, n)I (xg ; ng ) + I (µ1 1n1 , . . . , µG 1nG ; n),
(SD)
g =1
∀x ∈ Rn+ , ∀n = (n1 , . . . , nG ) with G > 2 and where ωgG is the weight attached to subgroup g (see Shorrocks (1980), p. 614). This decomposition is useful for empirical investigations, and it is at the heart of the class of the generalized entropy indices (see Shorrocks (1980)). However, this decomposition approach captures very specific views of the between-group component. More precisely, if one considers again the income distributions mentioned in the introduction such that x = (x1 , x2 ) = ((0, 1, 4, 15), (2, 3, 5, 10)) and x¯ = (¯x1 , x¯ 2 ) = ((0, 0, 0, 20), (5, 5, 5, 5)), it clearly appears that x is obtained from x¯ by a sequence of progressive transfers. Hence, for all inequality indices Schur-convex, it follows that I (x; 4) 6 I (¯x; 4). If the inequality index is strongly subgroup decomposable, then, for both income distributions, the total inequality is equal to the within-group inequality only, the between-group inequality being zero. Now observe that: x1 is obtained from x¯ 1 by a sequence of progressive transfers, x2 is obtained from x1 by a sequence of progressive transfers, x¯ 2 is obtained from x2 by a sequence of progressive transfers, then, one deduces that there is more inequality between-groups in x¯ than in x, i.e., IB (x; 4) 6 IB (¯x; 4). Thus, assuming strong decomposability results in an underestimation of inequality, if one subscribes to the argument above.3 To address this issue, Ebert (2010) introduced a weak decomposition axiom. Assuming that the population is partitioned into two subgroups, an inequality index is weakly decomposable if total inequality can be expressed as the sum of a weighted average of inequality within the groups and of the between-group term based on all possible pairwise comparisons where one individual belongs to one group and the other to another distinct group. Ebert (2010) proposes two different formulations for this (new) decomposition approach, denoted in this paper by WD1 and WD2. Ebert’s Weak Decomposition. The inequality index is weakly decomposable if I (x; n) = α1 (n)I (x1 ; n1 ) + α2 (n)I (x2 ; n2 )
+ β(n)
I (xi , xj ; 2),
(WD1)
i∈N1 j∈N2
∀x ∈ Rn+ , ∀n := (n1 , n2 ), where αg (n) and β(n) denote the weights, alternatively,
3 Moreover, this approach is sometimes considered as a too demanding property. One criticism is related to the loss of information about the variance and the asymmetry of distributions due to the consideration of the representative incomes (mean incomes) for the calculation of the between-group component (see Dagum (1997) for further details).
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ε
ε
µ(x2 ) µ(x1 ) I (x1 ; n1 ) + α2 (n) I (x2 ; n2 ) µ(x)ε µ(x)ε ε µ(xi , xj ) I (xi , xj ; 2). + β(n) µ(x)ε i∈N j∈N
I (x; n) = α1 (n)
1
(WD2)
2
∀x ∈ Rn++ , ∀n := (n1 , n2 ) and for every ε > 0. The weak decomposition is reminiscent of Kolm’s (1999) pairbased inequality indices – including the Gini index – which capture envy between individuals,4 but also with some indices usually considered as additively separable such as the square of the coefficient of variation (belonging to the family of the generalized entropy indices) or the variance. In this paper we propose to generalize the two previous formulations of the weak decomposition axiom, by making the different weights depend on the population shares and also on the income shares. Weak decomposition (WD*). The inequality index is weakly decomposable if I (x; n) =
G
α(ng , n, Yg , Y )I (xg ; ng )
g =1
+
G h −1 h=2 k=1
β(2, n, xi + xj , Y )I (xi , xj ; 2) ,
i∈Nk j∈Nh
(WD*)
∀x ∈ R+ , ∀n = (n1 , . . . , nG ) with G > 2, and where α(ng , n, Yg , Y ) and β(2, n, xi + xj , Y ) denote the weights. n
This axiom considers a decomposition of the population into G > 2 subgroups. In the case of Ebert’s (2010) approach, where the whole population is divided into two subgroups such that N = N1 N2 , condition WD* reduces to I (x; n) = α(n1 , n, Y1 , Y )I (x1 ; n1 ) + α(n2 , n, Y2 , Y )I (x2 ; n2 )
+
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈N1 j∈N2
In the following section we characterize the class of inequality indices that are consistent with this new decomposition property. 3. Characterization of weakly decomposable inequality indices We present below the main result of the paper. Theorem 3.1. Let n > 2. An inequality index satisfies CN, NM, SM and WD*, if and only if, there exists θ : {2, 3 . . . , n}× R++ −→ R++ , such that: I (x; n) =
1
θ 2, µij I (xi , xj ; 2),
2 θ (n, µ) i∈N j∈N
∀x ∈ Rn+ ,
(1)
where I (·; 2) : R2+ −→ R+ is continuous, symmetric and such that I (λ, λ; 2) = 0, for all λ ∈ R+ . Theorem 3.1 provides the basic structure of the weakly decomposable inequality indices. It is interesting to note that only four axioms are necessary to obtain such a result. The introduction of additional requirements will allow us to put more structure on the weakly decomposable inequality indices. A property usually required in the measurement of inequality is the population replication invariance (PP). This property allows one to refine the structure of the weighting functions as shown in Theorem 3.2.
4 See also Bosmans and Öztürk (2013) for the decomposition of envy indices within and between subgroups.
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P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
Theorem 3.2. Let n > 2. An inequality index satisfies CN, NM, SM, WD* and PP, if and only if, there exists a strictly positive and continuous real-valued function c : R++ −→ R++ , such that: I (x; n) =
2
n2 c (µ) i∈N j∈N
c (µij )I (xi , xj ; 2),
∀x ∈ Rn+ ,
where I (·; 2) : R2+ −→ R+ is continuous, symmetric and such that I (λ, λ; 2) = 0, for all λ ∈ R+ . The structure of the weakly decomposable inequality indices still remains quite general. In particular, it imposes no specific restriction on I (·; 2). From the expression proposed in Theorem 3.2, numerous indicators can be obtained. Clearly, all the families of indices pointed out by Ebert (2010) are included into our class of weakly decomposable inequality indices. For example, assume that the inequality index is translation invariant, that is, inequality is not affected by the addition of the same amount to each income. Choosing I (xi , xj ; 2) := |xi − xj |ε (ε > 0), c (µ) = 4 and c (µij ) = c, where c is a constant ∀i, j ∈ N , one retrieves the extensions of the Gini’s mean difference (including the variance). c |xi − xj |ε . I (x; n) = 2n2 i∈N j∈N For ε = 1 and c = 2, we obtain the Gini’s Mean Difference 1 |xi − xj | = GMD(x; n) I (x; n) = 2 n i∈N j∈N and, for ε = 2 and c = 1, we get the variance 1 2 I (x; n) = xi − µ2 = Var (x; n). n i∈N Assume next that the inequality index I (·; 2) is scale invariant, i.e., inequality remains unchanged when a proportional change applies to each income. Choosing I (xi , xj ; 2) := | ln(xi /xj )|ε (ε > 0), c (µ) = 2 and c (µij ) = 1, ∀i, j ∈ N , one gets Ebert’s extension of the variance of logarithms: 1 | ln(xi /xj )|ε , I (x; n) = 2 n i∈N j∈N which reduces to the variance of logarithms when ε = 2 1 I (x; n) = 2 | ln(xi /xj )|2 = VL(x; n). n i∈N j∈N Indices considered as compromise can also be obtained. Dividing the extensions of the Gini’s mean difference by the mean income to a power ε , one gets the last family of indices characterized by Ebert which includes the standard Gini coefficient and the square of the coefficient of variation. c I (x; n) = |xi − xj |ε . 2n2 µε (x) i∈N j∈N For ε = 1 and c = 1, we obtain the standard Gini coefficient 1 I (x; n) = |xi − xj | = G(x; n) 2 2n µ(x) i∈N j∈N and, for ε = 2 and c = 1, we get the square of the coefficient of variation 1 I (x; n) = |xi − xj |2 = CV 2 (x; n). 2 2 2n µ (x) i∈N j∈N Imagine now another kind of compromise indices such that they satisfy Bossert and Pfingsten’s (1990) compromise property.5
5 Compromise property (CP): for all n > 2 and x ∈ R , I (x; n) < I (λx; n) for all ++ λ > 1, and I (x; n) > I (x + ε 1n ; n) for all ε > 0 (see Bossert and Pfingsten (1990, p. 123) for details).
Choosing I (xi , xj ; 2) := |xi − xj |ε /µij (ε > 0), c (µ) = 4µ and c (µij ) = µij , ∀i, j ∈ N , one obtains extensions of Krtscha’s index (Krtscha, 1994) which are not included into Ebert’s families of indices: I (x; n) =
2n2
1
µ(x)
i∈N j∈N
|xi − xj |ε ,
which reduces to the Krtscha’s index when ε = 2 I (x; n) =
2n2
1
µ(x)
i∈N j∈N
|xi − xj |2 .
Notice that, for ε = 1, one retrieves a coefficient corresponding to the standard Gini. Since WD* is not directly related to an invariance property, contrary to WD1 or WD2, one gets more flexibility in the choice of the structure of inequality indices. As suggested by Ebert (2010), the value of ε may be chosen to reflect a decision maker’s preferences towards inequality. Thanks to WD* a wider range of value judgments can be now embodied into the structure of the weakly decomposable inequality indices. 4. Concluding remarks In this paper, we have shown that the family of weakly decomposable inequality indices comprises more than just the extensions of three well-known indicators such as the Gini’s mean difference, the Gini coefficient and the variance of logarithms, as suggested by Ebert (2010). The family of weakly decomposable inequality indices can be extended provided that one reformulates the weak decomposition property. By introducing weighting functions depending both on the mean income and on the size of the population, we have obtained a generalization of the decomposition property compatible with any invariance condition. Implicitly, the weak decomposition axioms of Ebert put restrictions on the structure of the inequality index that have to do with the invariance conditions. The new version of the weak decomposition (WD*) does not place such restrictions and the results are more flexible families of inequality indices. Each of the families of inequality indices highlighted in this paper embodied into their structure a parameter ε which can be interpreted as reflecting the decision’s maker preferences towards inequality. Additional requirements, such as the Schur-convexity of the inequality index I (·; n) may also be imposed. The Schurconvexity condition, which is equivalent to impose the respect of the principle of transfers according to which the inequality diminishes when ‘‘rich-to-poor’’ transfers are implemented, will have a direct impact on the choice of the family of indices retained for the analysis of inequality. Indeed, as pointed out by Foster and Ok (1999) the variance of logarithms is not Schur-convex. Finally, the possibility to weight differently the inequality within groups and the inequality arising between groups could also be investigated in further research. Acknowledgments This paper benefited from valuable suggestions and helpful comments provided by Casilda Lasso de la Vega, Brice Magdalou, Stéphane Mussard and Claudio Zoli to whom I am greatly indebted. I am very grateful to the two anonymous referees for their very helpful comments and suggestions which highly contributed to the improvement of this paper. Participants at the 2013 ASSET and 5th ECINEQ meetings are also acknowledged. I am responsible for the remaining shortcomings.
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
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Appendix. Proofs of Theorems 3.1 and 3.2
From NM, we have I (xh , nh ) = 0 for all h ̸= g. The previous expression becomes:
The proofs of Theorems 3.1 and 3.2 consist of several intermediate results which are detailed in this section. The first lemma investigates the simple case where n = 2. We obtain a particular function β(·).
I (x; n) = α(ng , n, Yg , Y )I (xg ; ng )
+
for all Y ∈ R++ .
(A.1)
Proof. Consider an income distribution x := (x1 , x2 ) where x1 , x2 ∈ R+ , x1 ̸= x2 , such that each individual is considered as a subgroup, that is ng = #Ng = 1 for g = 1, 2. According to WD*, the inequality index can be written as:
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N−g j∈N−g j>i
+
Lemma 1. If an inequality index satisfies NM and WD*, then the function β(·), as defined in the WD* axiom, is such that:
β(2, 2, Y , Y ) = 1,
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.5)
i∈Ng j∈N−g
Now, we propose to decompose the subgroup Ng into ng subgroups of only one individual. By applying WD*, it follows that: I (xg ; ng ) =
α(1, n, xi , Yg )I (xi ; 1) (within Ng )
i∈Ng
+
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈Ng j>i
I (x1 , x2 ; 2) = α(1, 2, x1 , x1 + x2 )I (x1 ; 1)
(between {i} and {j} in Ng ).
+ α(1, 2, x2 , x1 + x2 )I (x2 ; 1) (within N ) + β(2, 2, x1 + x2 , x1 + x2 )I (x1 , x2 ; 2) (between {1} and {2} in N )
Using again NM, we have I (xi ; 1) = 0 for all i ∈ Ng . Thus: (A.2)
from NM, I (x1 , 1) = I (x2 , 1) = 0, and (A.2) becomes: I (x1 , x2 ; 2) = β(2, 2, x1 + x2 , x1 + x2 )I (x1 , x2 ; 2)
I (xg ; ng ) =
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.6)
i∈Ng j∈Ng j>i
(A.3)
since x1 ̸= x2 , I (x1 , x2 ; 2) > 0 from NM. It follows that β(2, 2, x1 + x2 , x1 + x2 ) = 1, for all (x1 + x2 ) ∈ R++ .
By reintroducing (A.6) into (A.5), we have: I (x; n) = α(ng , n, Yg , Y )
+
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N−g j∈N−g j>i
+
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.7)
i∈Ng j∈N−g
Now, consider another partitioning of the same distribution x such that x = (x1 , . . . , xG ) with ng = #Ng = 1 for all g = 1, . . . , G. Then, according to WD*: I (x; n) =
α(ng , n, Yg , Y )β(2, n, xi + xj , Y )
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈Ng j>i
Now we consider a population of n > 2 individuals. The following three lemmas are devoted to establish a functional relationship between the functions α(·) and β(·), whatever the admissible values of all the parameters. Lemma 2. If an inequality index satisfies NM and WD*, then for all x ∈ Rn+ with n > 2 and every partition x = (x1 , . . . , xG ) with G > 2, there exist strictly positive weighting real-valued functions α(ng , n, Yg , Y ) and β(2, n, xi + xj , Y ) such that, for all possible subgroups Ng , we have:
α(1, n, xi , Y )I (xi ; 1) (within N )
i∈N
i∈Ng j∈Ng j>i
+
− β(2, n, xi + xj , Y ) I (xi , xj ; 2) = 0,
Proof. Consider an income distribution x ∈ R+ with n > 2, which can be partitioned into G > 2 subgroups, and assume NM and WD*. First, let x := (x1 , . . . , xG ) be a partition such that ng = #Ng > 2 and nh = #Nh = 1 for all h ̸= g. We define the whole population, but subgroup Ng , by N−g := N \ Ng . Application of WD* gives:
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N j∈N j>i
(A.4)
where 2 6 ng < n and Yg 6 Y . Moreover, by construction, xi +xj 6 Yg for all i, j ∈ Ng .
(between {i} and {j} in N ). Since I (xi ; 1) = 0 for all i ∈ N by NM, this expression reduces to:
n
I (x; n) =
G
α(nh , n, Yh , Y )I (xh ; nh ) (within N )
I (x; n) =
Subtracting (A.8) from (A.7), we finally get: 0 = α(ng , n, Yg , Y )
β(2, n, xi + xj , Y )I (xi , xj ; 2)
+
+
+
(between {i} in Ng and {j} in N−g ).
β(2, n, xi + xj , Y )I (xi , xj ; 2)
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈N−g
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈Ng j∈N−g
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N−g j∈N−g j>i
i∈N−g j∈N−g j>i
(between {i} and {j} in N−g )
i∈Ng j∈Ng j>i
h=1
+
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈N j∈N j>i
−
i∈N j∈N j>i
β(2, n, xi + xj , Y )I (xi , xj ; 2),
(A.8)
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k > 1 times, x′ repeated k′ > 1 times and 0 < x < x′ . The size of subgroup Nh is nh = k + k′ + 1 > 2 and its total income is Yh = kx + k′ x′ . According to Lemma 2 and Eq. (A.4), we have:
which can be simplified as follows: 0=
α(ng , n, Yg , Y )β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈Ng j>i
−
0 = k[α(nh , n, Yh , Y )β(2, nh , x, Yh ) − β(2, n, x, Y )]I (0, x; 2)
β(2, n, xi + xj , Y )I (xi , xj ; 2).
+ k′ [α(nh , n, Yh , Y )β(2, nh , x′ , Yh )
i∈Ng j∈Ng j>i
− β(2, n, x′ , Y )]I (0, x′ ; 2)
We finally obtain the expression in Eq. (A.4).
+ kk′ [α(nh , n, Yh , Y )β(2, nh , x + x′ , Yh )
In the following lemma, a functional equation based on the weighting functions α(·) and β(·) is defined. The proof relies on simple cases demonstrating that the result holds for any distribution x ∈ R+ when n > 2. Lemma 3. Let n > 2. If an inequality index satisfies CN, NM and WD*, then the functions α(·) and β(·) in the WD* axiom are such that:
α(ng , n, Yg , Y )β(2, ng , X , Yg ) − β(2, n, X , Y ) = 0,
(A.9)
for all ng , n ∈ N and all X , Yg , Y ∈ R+ such that, respectively, 2 6 ng < n and 0 < X 6 Yg 6 Y . Proof. The proof is decomposed into two independent steps. In the first step, we show that the lemma is true under the restriction that Yg 6 (ng − 1)X . We establish in the second step that this result is also true under the restriction that Yg > (ng − 1)X . Step 1. Consider a distribution x ∈ Rn+ and a partition x = (x1 , . . . , xG ) with G > 2. The income distribution of subgroup Ng is such that xg := (x, . . . , x, x′ , . . . , x′ ) with x repeated k > 1 times, x′ repeated k′ > 1 times and 0 6 x < x′ . The size of subgroup Ng is ng = k + k′ and its total income is Yg = kx + k′ x′ . According to Lemma 2 and Eq. (A.4), we have: kk′ α(ng , n, Yg , Y )β(2, ng , X , Yg )
− β(2, n, X , Y ) I (x, x′ ; 2) = 0,
(A.10)
where we let X := x + x . By definition, β(·) is a strictly positive function (see WD*) and, according to NM, we have I (x, x′ , 2) > 0 (recalling that 0 6 x < x′ ). Hence we deduce that, in this case: ′
α(ng , n, Yg , Y )β(2, ng , X , Yg ) − β(2, n, X , Y ) = 0.
(A.11)
We obtain a functional equation. By construction of the distribution x, we have enough flexibility to consider any admissible value for ng , n, X and Y (by definition, 2 6 ng < n and 0 < X 6 Yg 6 Y ), by choosing appropriately k, k′ , x and x′ but, nevertheless, not enough to consider also any admissible Yg (which by definition is not lower than X ). Indeed X and Yg are not independent. Clearly, by choosing appropriately k, k′ , x and x′ , we can also obtain any Yg , but under the restriction that Yg 6 (ng − 1)X . Because X = x + x′ with 0 6 x < x′ , and ng = k + k′ with k, k′ > 1, the upper bound of Yg = kx + k′ x′ is immediately obtained, for a given X and a given ng , by letting k = 1, k′ = ng − 1, x = 0 and x′ = X : Yg 6 (ng − 1)X
(upper bound of Yg ).
To sum up we deduce that, for any admissible value of the parameters ng , n, X , Yg and Y (such that, 2 6 ng < n and 0 < X 6 Yg 6 Y ) but under the restriction that Yg 6 (ng − 1)X , we can find a distribution x as defined above, and choose k, k′ , x and x′ such that Eq. (A.11) is true. Before continuing, we have to discuss the special case where ng = 2. We emphasize that, in this case, we necessarily have X = Yg , and thus Yg 6 (ng − 1)X . Hence when ng = 2, whatever the value of the other parameters, Eq. (A.11) is necessarily true. In the following step, we restrict attention to the case where the subgroup size is strictly greater than 2. Step 2. Consider a distribution x ∈ Rn+ and a partition x = (x1 , . . . , xG ) with G > 2, such that the income distribution of subgroup Nh is such that xh := (0, x, . . . , x, x′ , . . . , x′ ) with x repeated
− β(2, n, x + x′ , Y )]I (x, x′ ; 2).
(A.12)
First notice that, whatever the values chosen for k, k , x and x′ , because k + k′ = nh − 1 and 0 < x < x′ , we have: ′
Yh = kx + k′ x′ < (nh − 1)x′ < (nh − 1)(x + x′ ).
(A.13)
Now consider the third term in brackets in Eq. (A.12). By using Step 1, and recalling that Yh < (nh − 1)(x + x′ ), we have:
α(nh , n, Yh , Y )β(2, nh , x + x′ , Yh ) − β(2, n, x + x′ , Y ) = 0.
(A.14)
The same reasoning applies to the second term in brackets in Eq. (A.12). By using Step 1, and recalling that Yh < (nh − 1)x′ , we have:
α(nh , n, Yh , Y )β(2, nh , x′ , Yh ) − β(2, n, x′ , Y ) = 0.
(A.15)
We know that k, k > 0 and I (a, b; 2) > 0 whenever a ̸= b. Hence, by reintroducing Eqs. (A.14) and (A.15) into (A.12), we deduce that: ′
α(nh , n, Yh , Y )β(2, nh , x, Yh ) − β(2, n, x, Y ) = 0.
(A.16)
By construction of the chosen distribution x, we can obtain any admissible values for nh , n, x and Y (by definition, 2 < nh < n and 0 < x 6 Yh 6 Y ), if and only if, for given nh , n, x and Y , there exist k, k′ , x and x′ such that nh = k + k′ + 1, Yh = kx + k′ x′ and x < x′ . Let k = nh − k′ − 1 and, by rewriting Yh = kx + k′ x′ , we have: x x′ = ′ k
Yh x
− (nh − k′ − 1) .
(A.17)
Hence x′ > x if and only if Yh /x − (nh − k′ − 1) /k′ > 1, or equivalently Yh > (nh − 1)x. Thus, for any admissible value of the parameters nh , n, x, Yh and Y (by definition, 2 < nh < n and 0 < x 6 Yh 6 Y ) but under the restriction that Yh > (nh − 1)x, we can find a distribution x as defined above, and choose k, k′ and x′ such that Eq. (A.16) is true. The proof of the lemma is obtained by combining the results of the two steps.
Lemma 4. Let n > 2. If an inequality index satisfies NM and WD*, then the functions α(·) and β(·) in the WD* axiom are such that:
β(2, n, X , Y ) = α(2, n, X , Y ), for all X , Y ∈ R+ such that 0 < X 6 Y . Proof. Consider any distribution x := (x1 , . . . , xn ), such that each individual is considered as a subgroup. By applying WD*, one obtains: I (x; n) =
α(1, n, xi , Y )I (xi ; 1) (within N )
i∈N
+
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N j∈N j>i
(between {i} and {j} in N ) (NM)
=
i∈N j∈N j>i
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.18)
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
Then, consider a new partition of x into G subgroups, such that #Ng = 2 and #Nh = 1 for all h ̸= g. Moreover, let xg := (xg1 , xg2 ) with xg1 ̸= xg2 , and denote the whole population, but subgroup Ng , by N−g := N \ Ng . Application of WD* gives: I (x; n) =
α(1, n, xi , Y )I (xi ; 1)
β(2, n, xi + xj , Y )I (xi , xj ; 2)
(NM)
= α(2, n, xg1 + xg2 , Y )I (xg ; 2) β(2, n, xi + xj , Y )I (xi , xj ; 2) +
(A.23)
(R4)
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈Ng j∈N−g
Applying again WD* to the subgroup Ng such that each individual constitutes one subgroup, it follows:
× I (xg1 , xg2 ; 2) + β(2, n, xi + xj , Y )I (xi , xj ; 2) (A.19)
i∈Ng j∈N−g
From Lemma 1, β(2, 2, xg1 + xg2 , xg1 + xg2 ) = 1, thus, subtracting (A.19) from (A.18) yields:
Recalling that xg1 ̸= xg2 , we have I (xg1 , xg2 ; 2) > 0. We deduce from (A.20) that:
Notice that, because xg1 , xg2 xg1 + xg2 > 0.
I (x; n) =
(R3)
I (x; n) =
β(2, n, xi + xj , Y )I (xi , xj ; 2).
1 θ (2, µij ) 2 i∈N
+
Lemma 5. Let n > 2. If an inequality index satisfies CN, NM and WD*, then the functions α(·) and β(·) in the WD* axiom can be written as:
j∈N j>i
θ (n, µ)
I (xi , xj ; 2)
1 θ (2, µji ) 2 i∈N
j∈N i>j
θ (n, µ)
I (xj , xi ; 2),
by definition µij = µji , then θ (2, µij ) = θ (2, µji ). Recalling that from NM, I (xi , xi ; 2) = I (xj , xj ; 2) = 0, we finally obtain:
and
I (x; n) =
where θ : {2, 3 . . . , n} × R++ −→ R++ is a strictly positive and continuous real-valued function. Proof. Recall that µij = (xi + xj )/2 is the mean income of two individuals, and µg is the mean income of subgroup Ng . Setting n =: n˜ and Y =: Y˜ as constant and letting X = xi + xj , we can write from Lemmas 3 and 4:
α(2, n˜ , 2µij , Y˜ ) α(2, ng , 2µij , Yg ) = . α(ng , n˜ , ng µg , Y˜ )
∀x ∈ Rn+ ,
Replacing β(·) by its expression defined in (R4) and considering I (·; 2) is continuous and symmetric such that I (xi , xj ; 2) = 1 (I (xi , xj ; 2) + I (xj , xi ; 2)), the previous equation becomes: 2
∈ R+ and xg1 ̸= xg2 , we have
θ (ng , µg ) , θ (n, µ) θ (2, µij ) β(2, n, 2µij , Y ) = , θ (n, µ)
2 θ (n, µ) i∈N j∈N
i∈N j∈N j>i
(A.20)
β(2, n, xg1 + xg2 , Y ) = α(2, n, xg1 + xg2 , Y ).
θ 2, µij I (xi , xj ; 2),
Proof. Let x be an income distribution partitioned into G > 2 subgroups such that x = (x1 , . . . , xG ) with ng = #Ng = 1 for all g = 1, . . . , G, and assume NM and WD*. Then, from Eq. (A.8):
β(2, n, xg1 + xg2 , Y ) − α(2, n, xg1 + xg2 , Y ) × I (xg1 , xg2 ; 2) = 0.
1
where I (·; 2) : R2+ −→ R+ is continuous, symmetric, and such that I (λ, λ; 2) = 0, for all λ ∈ R+ .
i∈N−g j∈N−g j>i
β(2, n, xi + xj , Y )I (xi , xj ; 2).
Theorem 3.1. Let n > 2. An inequality index satisfies CN, NM, SM and WD*, if and only if there exists θ : {2, 3 . . . , n} × R++ −→ R++ , such that: I (x; n) =
(NM)
I (x; n) = α(2, n, xg1 + xg2 , Y )β(2, 2, xg1 + xg2 , xg1 + xg2 )
α(ng , n, ng µg , Y ) =
θ (2, µij ) , θ (n, µ)
Finally, by invoking the continuity (CN) of I (x; n), we deduce that θ (n, µ) is continuous for all x ∈ Rn+ .
i∈N−g j∈N−g j>i
(A.22)
α(ng , n, ng µg , Y ) =
(between {i} in Ng and {j} in N−g )
+
θ (2, µij ) , θ (ng , µg )
θ (ng , µg ) and θ (n, µ) θ (2, µij ) β(2, n, 2µij , Y ) = . θ (n, µ)
i∈Ng j∈N−g
α(ng , n˜ , ng µg , Y˜ ), Eq. (A.21) can be
with θ (n, µ) ̸= 0. Substituting back expressions (A.22) and (A.23) into (A.21), yields in general:
(between {i} and {j} in N−g )
+
α(2, ng , 2µij , Yg ) =
α(2, n, 2µij , Y ) =
i∈N−g j∈N−g j>i
+
Defining θ (ng , µg ) := expressed as follows:
where θ (ng , µg ) ̸= 0 is a real-valued function by construction. Thus:
i∈N−g
+ α(2, n, xg1 + xg2 , Y )I (xg ; 2) (within N ) β(2, n, xi + xj , Y )I (xi , xj ; 2) +
77
(A.21)
1
θ 2, µij I (xi , xj ; 2),
(R5)
2 θ (n, µ) i∈N j∈N
such that I (·; n) is symmetric and continuous, for n > 2.
Theorem 3.1. Let n > 2. An inequality index satisfies CN, NM, SM, WD* and PP, if and only if there exists a strictly positive and continuous real-valued function c : R++ −→ R++ , such that: I (x; n) =
2 n2
c (µ) i∈N j∈N
c (µij )I (xi , xj ; 2),
∀x ∈ Rn+ ,
78
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
where I (·; 2) : R2+ −→ R+ is continuous, symmetric and such that I (λ, λ; 2) = 0, for all λ ∈ R+ . Proof. Consider an income distribution x ∈ Rn+ replicated m times and let M denote the corresponding population in which each individual is also replicated m times, such that #M = m · n, with m > 2, n = #N and x[m] = (x, . . . , x) ∈ Rmn + . According to (R5):
By applying the same structure to θ (2, µ˜ij ), it follows:
θ (2, µ˜ij ) = 22 · c (µ˜ij ). Finally, we obtain: I (x; n) =
1
m times
I (x
[m]
1
; mn) =
=
θ 2, µij I (x[i m] , x[j m] ; 2)
2 θ (mn, µ) i∈M j∈M m2
=
2 θ (mn, µ) i∈N j∈N
I (x
; mn) = I (x; n) m2 θ 2, µij I (xi , xj ; 2) 2 θ (mn, µ) i∈N j∈N θ 2, µij I (xi , xj ; 2),
2 θ (n, µ) i∈N j∈N
which is equivalent to: m2
2 θ(mn, µ)
−
1 2 θ (n, µ)
θ 2, µij I (xi , xj ; 2)
i∈N j∈N
= 0.
(A.24)
Because θ (·) is defined as a strictly positive and continuous realvalued function, according to NM, that: it follows, i∈N j∈N θ 2, µij I (xi , xj ; 2) > 0, for all xi ̸= xj . Focusing on the terms in brackets in Eq. (A.24), we deduce that: m2 2 θ(mn, µ)
=
1 2 θ (n, µ)
⇐⇒ m2 θ (n, µ) = θ (mn, µ).
Under PP, µ remains constant, that is, µ ˜ := µ; thus: m2 θ(n, µ) ˜ = θ (mn, µ), ˜ or analogously: n2 θ(m, µ) ˜ = θ (mn, µ). ˜ Mimicking Ebert’s (2010) reasoning (Ebert, 2010, pp. 99-100), we get:
θ(n, µ) ˜ n2
= c (µ). ˜
(µ)
c (µij )I (xi , xj ; 2),
(A.25)
i∈N j∈N
References
[m]
1
2 n2 c
22 · c (µij )I (xi , xj ; 2)
such that I (·; 2) : R2+ −→ R+ is continuous, symmetric and I (λ, λ, 2) = 0, for all λ ∈ R+ .
θ 2, µij I (xi , xj ; 2).
From PP, it follows:
=
2 n2 c (µ) i∈N j∈N
Blackorby, C., Donaldson, D., 1980. A theoretical treatment of indices of absolute inequality. Internat. Econom. Rev. 21, 107–136. http://dx.doi.org/10.2307/ 2526243. Bosmans, K., Öztürk, Z.E., 2013. An Axiomatic Approach to the Measurement of Envy. Maastricht University School of Business and Economics, Graduate School of Business and Economics (GSBE), File url. Bossert, W., Pfingsten, A., 1990. Intermediate inequality: concepts, indices, and welfare implications. Math. Social Sci. 19 (2), 117–134. http://dx.doi.org/10. 1016/0165-4896(90)90055-C. Bourguignon, F., 1979. Decomposable inequality measures. Econometrica 47, 901–920. http://dx.doi.org/10.2307/1913126. Dagum, C., 1997. A new approach to the decomposition of the gini income inequality ratio. Empir. Econom. 22 (4), 515–531. http://dx.doi.org/10.1007/978-3-64251073-1_4. Dalton, H., 1920. The measurement of the inequality of incomes. Econ. J. 30, 48–361. http://dx.doi.org/10.2307/2223525. Ebert, U., 1988. On the decomposition of inequality: Partitions into non-overlapping subgroups. In: Eichorn, W. (Ed.), Measurement In Economics. Physica-Verlag, New-York, pp. 399–412. http://dx.doi.org/10.1007/978-3-642-52481-3_30. Ebert, U., 2010. The decomposition of inequality reconsidered: Weakly decomposable measures. Math. Social Sci. 60 (2), 94–103. http://dx.doi.org/10.1016/ j.mathsocsci.2010.05.001. Foster, J.E., Ok, E.A., 1999. Lorenz dominance and the variance of logarithms. Econometrica 67 (4), 901–907. http://dx.doi.org/10.1111/1468-0262.00057. Kolm, S.-C., 1976a. Unequal inequalities I. J. Econom. Theory 12, 416–442. http://dx.doi.org/10.1016/0022-0531(76)90037-5. Kolm, S.-C., 1976b. Unequal inequalities II. J. Econom. Theory 13, 82–111. http://dx.doi.org/10.1016/0022-0531(76)90068-5. Kolm, S.-C., 1999. The rational foundation of income inequality measurement. In: Silber, J. (Ed.), Handbook of Income Inequality Measurement. Kluwer Academic Publishers, pp. 19–94. http://dx.doi.org/10.1007/978-94-011-44131_2. Krtscha, M., 1994. A new compromise measure of inequality. In: Eichhorn, W. (Ed.), Models and Measurement of Welfare and Inequality. Springer, Heidelberg, pp. 111–120. http://dx.doi.org/10.1007/978-3-642-79037-9_6. Shorrocks, A.F., 1980. The class of additively decomposable inequality measures. Econometrica 48, 613–625. http://dx.doi.org/10.2307/1913126. Shorrocks, A.F., 1984. Inequality decomposition by population subgroups. Econometrica 52, 1369–1386. http://dx.doi.org/10.2307/1913511.
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
On the axiomatization of the weakly decomposable inequality indices Pauline Mornet ∗ Université Montpellier, UMR5474 LAMETA, F-34000 Montpellier, France Faculté d’Économie, Av. Raymond Dugrand, Site de Richter C.S. 79606, 34960 Montpellier Cedex 2, France
highlights • • • •
A new axiomatization of the weak decomposition property is provided. Standard weighting functions depending on n and µ are defined. A class of weakly decomposable inequality indices is characterized. No specific invariance value judgment is assumed through the characterization.
article
info
Article history: Received 13 August 2015 Received in revised form 20 July 2016 Accepted 24 July 2016 Available online 1 August 2016
abstract In this paper, we focus on a decomposition property recently introduced in the inequality literature and known as the weak decomposition. Such a property provides interesting analyses by allowing one to separate the within-group contribution to total inequality from the between-group contribution. A limitation of the current method of decomposition is that, depending on the structure – absolute, relative, compromise – of the inequality index, specific weights have to be used. To avoid such a problem, we propose a unique decomposition property where the weighting functions depend on the size of the population and the mean income. This allows us to characterize a large family of weakly decomposable inequality indices without any recourse to implicit invariance value judgments. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Since the 1970s, particular attention has been paid to the axiomatic foundations of inequality measures. As pointed out by Kolm (1976a), these researches aimed at justifying the structure of the inequality measures. ‘‘[. . .] To really see what these measures imply, it is necessary to build an axiomatic of them, i.e., to find for each one a set of properties which are equivalent to its adoption. If these properties are small in number and as intuitive as possible, they will display the implicit assumptions made by this choice.’’ [Kolm (1976a, p. 426)] In order to get an inequality measure as representative as possible of the economic reality, specific requirements have to be imposed. Beyond basic requirements such as continuity, normalization, symmetry, or population replication invariance, particular
∗ Correspondence to: Université Montpellier, UMR5474 LAMETA, F-34000 Montpellier, France. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.mathsocsci.2016.07.004 0165-4896/© 2016 Elsevier B.V. All rights reserved.
emphasis is placed on subgroup decomposability. Different sets of axioms have been introduced in order to lay the groundwork for a thorough evaluation of the income inequalities when the population consists of exclusive and exhaustive groups of individuals. Subgroup decomposability is particularly useful for empirical studies when one is interested in the contribution to overall inequality of particular groups. For instance, when the population is partitioned into subgroups reflecting socioeconomic criteria (e.g., genders or occupations), a more accurate analysis of inequality is provided when the discrepancies are measured within and between subgroups. In this paper, we are interested in the characterization of the weakly decomposable inequality indices investigated by Ebert (2010). The concept of weak decomposability contrasts with that of strong decomposability introduced by Bourguignon (1979) and Shorrocks (1980, 1984). In both approaches, within-group inequality is computed as the weighted sum of the inequalities in the different groups. It is the definition of the between-group term that makes the difference. While in the strong decomposition process the computation of the between-group term is based on the representative income of the subgroups (generally the mean income), in the weak decomposition process it is based on the
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P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
pairwise comparisons of individuals’ income.1 To make clear the difference between both approaches, let us take an example. Let x be an income distribution partitioned into two subgroups, such that x = (x1 , x2 ) with x1 = (0, 1, 4, 15) and x2 = (2, 3, 5, 10). Note that the average incomes and the population sizes of both subgroups are identical. In order to measure income inequality, we appeal to the square of the coefficient of variation, which, as a member of the generalized entropy family satisfies the strong decomposition property. Application of this property to the income distribution x reveals that the overall inequality is only due to within-subgroups inequality since the between-group term (based on average incomes) is zero. Now consider the income distribution x¯ = (¯x1 , x¯ 2 ) such that x¯ 1 = (0, 0, 0, 20) and x¯ 2 = (5, 5, 5, 5). Both subgroups represent extreme cases. In subgroup 1 all of the income belongs to one individual while in subgroup 2 the same amount of income is held by each individual. Note that average income in each subgroup is still equal to 5. As expected the computation of the total inequality in both distributions highlights that the distribution x¯ is more unequal than the distribution x. However the between-group inequality is the same for both distributions. Indeed, with the strong decomposition process, between-group inequality is computed by smoothing income within groups by using the mean or the equally distributed equivalent income (EDEI). Since all the income distributions have the same mean incomes, the nullity of the between-group terms follows directly. So, the application of the strong decomposition process to the income distributions x and x¯ suggests that inequalities occur only within the groups. In particular, in the case of the distribution x¯ , inequalities are only within the subgroup 1. This simple example illustrates the limit of the measurement of inequality based on representative income of the subgroups such as the average income. The strong decomposition is not the only decomposition process that fits with the structure of the square of the coefficient of variation. Indeed, the weak decomposition of Ebert (2010) also applies to such an index but then the interpretation of the within- and between-group inequality terms, is different. For both partitioning (x1 , x2 ) and (x¯ 1 , x¯ 2 ), two non-zero terms are obtained. Besides, the contribution to overall inequality of the within-group and the between-group terms is identical in both partitions. Imagine now that one is interested in absolute inequality as suggested by Kolm (1976a,b). In such a case, the recourse to the variance makes sense. If the variance can be strongly decomposed in the same way as the square of the coefficient of variation, the weak decomposition cannot be performed on this indicator unless resorting to an alternative formulation of this property. To this aim, Ebert (2010) distinguishes two axiomatic formulations to decompose inequality. A first property (denoted (WD1)) is dedicated to the decomposition of absolute and relative inequality indices, while a second one (WD2) is applied to ‘‘compromise2 ’’ inequality indices. Because of this double axiomatization, all the inequality indices cannot be decomposed according to exactly the same scheme. More specifically, the choice of the decomposition property implicitly imposes that the inequality index verifies a particular invariance property. Invariance conditions (such as scale invariance or translation invariance) allow one to describe the sensitivity of inequality
1 As pointed out by Ebert (2010), individuals belonging to different subgroups are only considered. 2 The terms compromise indices refer to inequality indices of which the structure depends directly on the average income (e.g., the standard Gini index or the squared coefficient of variation), see Blackorby and Donaldson (1980) or Ebert (1988) for more details.
indices to particular economic changes. It is well known that when an inequality index remains insensitive to a proportional increase in individuals’ incomes, it is scale invariant (relative inequality). On the other hand, if the inequality remains unchanged when the same amount of income is added to each individual in the population, then the index is said to be invariant to translations (absolute inequality). Thus, it may seem restrictive to limit the appreciation of inequalities in a population to only one value judgment implicitly embodied in a decomposition property. The aim of this paper is to propose a reformulation of the weak decomposition property that does not depend on a particular invariance condition. While Ebert’s (2010) approach consists in mixing the properties of decomposability and invariance, it is shown that by introducing weighting functions for the within- and between-group components which take into account not only the population sizes but also the mean income, in the same manner as Shorrocks (1980, 1984), the application of both properties can be disentangled. On the basis of such a decomposition property the characterization of the large family of weakly decomposable indices is first provided. Then, by invoking invariance conditions, extensions of the Gini’s mean difference, the standard Gini coefficient, the variance of logarithms and also the Krtscha index are presented as subfamilies of the weakly decomposable indices. The paper is organized as follows. Section 2 introduces the notations and the axioms underlying decomposable inequality indices. Our general formulation of the weak decomposition is also presented in this section. In Section 3 the class of weakly decomposable indices consistent with our decomposition property is axiomatically characterized. Concluding remarks are presented in Section 4. The proofs of the theorems are relegated to Appendix. 2. Notation and axioms Let N := {1, 2, . . . , n} (n > 1) be fixed population of individuals. An income distribution for N is a list x := (x1 , x2 , . . . , xn ), where xi ∈ R+ is the income of individual i ∈ N , and R+ is the non-negative part of the real line. The arithmetic mean of x ∈ Rn+ is indicated by µ(x) := i∈N xi /n, or simply by µ when no ambiguity arises, and the total income by Y := nµ. A partition of N is a list (N1 , . . . , Ng ) such that ng = #Ng > 1, ∀g ∈ {1, . . . , G} (G > 1) and
G
g =1 Ng = N . Then x = n (x1 , x2 , . . . , xG ) such that for each subgroup g , xg ∈ R+g is the income distribution, µg the mean income and Yg := ng µg the total income. We denote by µ := (µ1 , . . . , µG ) the list of the subgroup means, and 1n := (1, . . . , 1) a list where 1 is repeated n times.
Finally, an inequality index for a population of n individuals is a function I (·; n) : Rn+ −→ R+ . We first introduce four standard axioms, usually employed in the literature on income inequality measurement. The first one is a regularity condition imposed on the inequality index. Continuity (CN). For all x ∈ Rn+ , I (x; n) is continuous. The next axiom requires that incomes among the permutation of the individuals do not affect the value of the inequality index. This is captured by the following axiom also referred to an anonymity condition. Symmetry (SM). For all x ∈ Rn+ , I (x; n) is symmetric in x. Another standard axiom requires that the inequality index is always non-negative, with equality to zero if and only if the income distribution is perfectly equal. Normalization (NM). For all x ∈ Rn+ , I (x; n) > 0 and I (x; n) = 0, if and only, if there exists λ ∈ R+ such that x = λ1n . The last standard axiom is the Dalton’s (1920) population replication invariance, according to which a m-fold replication of the distribution has no impact on inequality. This property is useful for making comparisons across populations of different sizes.
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
Population replication invariance (PP). For all x ∈ Rn+ and all m > 2, I (x[m] ; mn) = I (x; n), where x[m] := (x, . . . , x) ∈ Rmn + is a m-fold replication of x. We now consider the different decomposability approaches that have been proposed so far in the literature. According to Shorrocks (1980), an inequality index is strongly subgroup decomposable if total inequality can be rewritten as the sum of the weighted averages of the inequalities within groups and the inequality between the group mean incomes. Strong Decomposition (SD). The inequality index is strongly decomposable if I (x; n) =
G
ωgG (µ, n)I (xg ; ng ) + I (µ1 1n1 , . . . , µG 1nG ; n),
(SD)
g =1
∀x ∈ Rn+ , ∀n = (n1 , . . . , nG ) with G > 2 and where ωgG is the weight attached to subgroup g (see Shorrocks (1980), p. 614). This decomposition is useful for empirical investigations, and it is at the heart of the class of the generalized entropy indices (see Shorrocks (1980)). However, this decomposition approach captures very specific views of the between-group component. More precisely, if one considers again the income distributions mentioned in the introduction such that x = (x1 , x2 ) = ((0, 1, 4, 15), (2, 3, 5, 10)) and x¯ = (¯x1 , x¯ 2 ) = ((0, 0, 0, 20), (5, 5, 5, 5)), it clearly appears that x is obtained from x¯ by a sequence of progressive transfers. Hence, for all inequality indices Schur-convex, it follows that I (x; 4) 6 I (¯x; 4). If the inequality index is strongly subgroup decomposable, then, for both income distributions, the total inequality is equal to the within-group inequality only, the between-group inequality being zero. Now observe that: x1 is obtained from x¯ 1 by a sequence of progressive transfers, x2 is obtained from x1 by a sequence of progressive transfers, x¯ 2 is obtained from x2 by a sequence of progressive transfers, then, one deduces that there is more inequality between-groups in x¯ than in x, i.e., IB (x; 4) 6 IB (¯x; 4). Thus, assuming strong decomposability results in an underestimation of inequality, if one subscribes to the argument above.3 To address this issue, Ebert (2010) introduced a weak decomposition axiom. Assuming that the population is partitioned into two subgroups, an inequality index is weakly decomposable if total inequality can be expressed as the sum of a weighted average of inequality within the groups and of the between-group term based on all possible pairwise comparisons where one individual belongs to one group and the other to another distinct group. Ebert (2010) proposes two different formulations for this (new) decomposition approach, denoted in this paper by WD1 and WD2. Ebert’s Weak Decomposition. The inequality index is weakly decomposable if I (x; n) = α1 (n)I (x1 ; n1 ) + α2 (n)I (x2 ; n2 )
+ β(n)
I (xi , xj ; 2),
(WD1)
i∈N1 j∈N2
∀x ∈ Rn+ , ∀n := (n1 , n2 ), where αg (n) and β(n) denote the weights, alternatively,
3 Moreover, this approach is sometimes considered as a too demanding property. One criticism is related to the loss of information about the variance and the asymmetry of distributions due to the consideration of the representative incomes (mean incomes) for the calculation of the between-group component (see Dagum (1997) for further details).
73
ε
ε
µ(x2 ) µ(x1 ) I (x1 ; n1 ) + α2 (n) I (x2 ; n2 ) µ(x)ε µ(x)ε ε µ(xi , xj ) I (xi , xj ; 2). + β(n) µ(x)ε i∈N j∈N
I (x; n) = α1 (n)
1
(WD2)
2
∀x ∈ Rn++ , ∀n := (n1 , n2 ) and for every ε > 0. The weak decomposition is reminiscent of Kolm’s (1999) pairbased inequality indices – including the Gini index – which capture envy between individuals,4 but also with some indices usually considered as additively separable such as the square of the coefficient of variation (belonging to the family of the generalized entropy indices) or the variance. In this paper we propose to generalize the two previous formulations of the weak decomposition axiom, by making the different weights depend on the population shares and also on the income shares. Weak decomposition (WD*). The inequality index is weakly decomposable if I (x; n) =
G
α(ng , n, Yg , Y )I (xg ; ng )
g =1
+
G h −1 h=2 k=1
β(2, n, xi + xj , Y )I (xi , xj ; 2) ,
i∈Nk j∈Nh
(WD*)
∀x ∈ R+ , ∀n = (n1 , . . . , nG ) with G > 2, and where α(ng , n, Yg , Y ) and β(2, n, xi + xj , Y ) denote the weights. n
This axiom considers a decomposition of the population into G > 2 subgroups. In the case of Ebert’s (2010) approach, where the whole population is divided into two subgroups such that N = N1 N2 , condition WD* reduces to I (x; n) = α(n1 , n, Y1 , Y )I (x1 ; n1 ) + α(n2 , n, Y2 , Y )I (x2 ; n2 )
+
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈N1 j∈N2
In the following section we characterize the class of inequality indices that are consistent with this new decomposition property. 3. Characterization of weakly decomposable inequality indices We present below the main result of the paper. Theorem 3.1. Let n > 2. An inequality index satisfies CN, NM, SM and WD*, if and only if, there exists θ : {2, 3 . . . , n}× R++ −→ R++ , such that: I (x; n) =
1
θ 2, µij I (xi , xj ; 2),
2 θ (n, µ) i∈N j∈N
∀x ∈ Rn+ ,
(1)
where I (·; 2) : R2+ −→ R+ is continuous, symmetric and such that I (λ, λ; 2) = 0, for all λ ∈ R+ . Theorem 3.1 provides the basic structure of the weakly decomposable inequality indices. It is interesting to note that only four axioms are necessary to obtain such a result. The introduction of additional requirements will allow us to put more structure on the weakly decomposable inequality indices. A property usually required in the measurement of inequality is the population replication invariance (PP). This property allows one to refine the structure of the weighting functions as shown in Theorem 3.2.
4 See also Bosmans and Öztürk (2013) for the decomposition of envy indices within and between subgroups.
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P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
Theorem 3.2. Let n > 2. An inequality index satisfies CN, NM, SM, WD* and PP, if and only if, there exists a strictly positive and continuous real-valued function c : R++ −→ R++ , such that: I (x; n) =
2
n2 c (µ) i∈N j∈N
c (µij )I (xi , xj ; 2),
∀x ∈ Rn+ ,
where I (·; 2) : R2+ −→ R+ is continuous, symmetric and such that I (λ, λ; 2) = 0, for all λ ∈ R+ . The structure of the weakly decomposable inequality indices still remains quite general. In particular, it imposes no specific restriction on I (·; 2). From the expression proposed in Theorem 3.2, numerous indicators can be obtained. Clearly, all the families of indices pointed out by Ebert (2010) are included into our class of weakly decomposable inequality indices. For example, assume that the inequality index is translation invariant, that is, inequality is not affected by the addition of the same amount to each income. Choosing I (xi , xj ; 2) := |xi − xj |ε (ε > 0), c (µ) = 4 and c (µij ) = c, where c is a constant ∀i, j ∈ N , one retrieves the extensions of the Gini’s mean difference (including the variance). c |xi − xj |ε . I (x; n) = 2n2 i∈N j∈N For ε = 1 and c = 2, we obtain the Gini’s Mean Difference 1 |xi − xj | = GMD(x; n) I (x; n) = 2 n i∈N j∈N and, for ε = 2 and c = 1, we get the variance 1 2 I (x; n) = xi − µ2 = Var (x; n). n i∈N Assume next that the inequality index I (·; 2) is scale invariant, i.e., inequality remains unchanged when a proportional change applies to each income. Choosing I (xi , xj ; 2) := | ln(xi /xj )|ε (ε > 0), c (µ) = 2 and c (µij ) = 1, ∀i, j ∈ N , one gets Ebert’s extension of the variance of logarithms: 1 | ln(xi /xj )|ε , I (x; n) = 2 n i∈N j∈N which reduces to the variance of logarithms when ε = 2 1 I (x; n) = 2 | ln(xi /xj )|2 = VL(x; n). n i∈N j∈N Indices considered as compromise can also be obtained. Dividing the extensions of the Gini’s mean difference by the mean income to a power ε , one gets the last family of indices characterized by Ebert which includes the standard Gini coefficient and the square of the coefficient of variation. c I (x; n) = |xi − xj |ε . 2n2 µε (x) i∈N j∈N For ε = 1 and c = 1, we obtain the standard Gini coefficient 1 I (x; n) = |xi − xj | = G(x; n) 2 2n µ(x) i∈N j∈N and, for ε = 2 and c = 1, we get the square of the coefficient of variation 1 I (x; n) = |xi − xj |2 = CV 2 (x; n). 2 2 2n µ (x) i∈N j∈N Imagine now another kind of compromise indices such that they satisfy Bossert and Pfingsten’s (1990) compromise property.5
5 Compromise property (CP): for all n > 2 and x ∈ R , I (x; n) < I (λx; n) for all ++ λ > 1, and I (x; n) > I (x + ε 1n ; n) for all ε > 0 (see Bossert and Pfingsten (1990, p. 123) for details).
Choosing I (xi , xj ; 2) := |xi − xj |ε /µij (ε > 0), c (µ) = 4µ and c (µij ) = µij , ∀i, j ∈ N , one obtains extensions of Krtscha’s index (Krtscha, 1994) which are not included into Ebert’s families of indices: I (x; n) =
2n2
1
µ(x)
i∈N j∈N
|xi − xj |ε ,
which reduces to the Krtscha’s index when ε = 2 I (x; n) =
2n2
1
µ(x)
i∈N j∈N
|xi − xj |2 .
Notice that, for ε = 1, one retrieves a coefficient corresponding to the standard Gini. Since WD* is not directly related to an invariance property, contrary to WD1 or WD2, one gets more flexibility in the choice of the structure of inequality indices. As suggested by Ebert (2010), the value of ε may be chosen to reflect a decision maker’s preferences towards inequality. Thanks to WD* a wider range of value judgments can be now embodied into the structure of the weakly decomposable inequality indices. 4. Concluding remarks In this paper, we have shown that the family of weakly decomposable inequality indices comprises more than just the extensions of three well-known indicators such as the Gini’s mean difference, the Gini coefficient and the variance of logarithms, as suggested by Ebert (2010). The family of weakly decomposable inequality indices can be extended provided that one reformulates the weak decomposition property. By introducing weighting functions depending both on the mean income and on the size of the population, we have obtained a generalization of the decomposition property compatible with any invariance condition. Implicitly, the weak decomposition axioms of Ebert put restrictions on the structure of the inequality index that have to do with the invariance conditions. The new version of the weak decomposition (WD*) does not place such restrictions and the results are more flexible families of inequality indices. Each of the families of inequality indices highlighted in this paper embodied into their structure a parameter ε which can be interpreted as reflecting the decision’s maker preferences towards inequality. Additional requirements, such as the Schur-convexity of the inequality index I (·; n) may also be imposed. The Schurconvexity condition, which is equivalent to impose the respect of the principle of transfers according to which the inequality diminishes when ‘‘rich-to-poor’’ transfers are implemented, will have a direct impact on the choice of the family of indices retained for the analysis of inequality. Indeed, as pointed out by Foster and Ok (1999) the variance of logarithms is not Schur-convex. Finally, the possibility to weight differently the inequality within groups and the inequality arising between groups could also be investigated in further research. Acknowledgments This paper benefited from valuable suggestions and helpful comments provided by Casilda Lasso de la Vega, Brice Magdalou, Stéphane Mussard and Claudio Zoli to whom I am greatly indebted. I am very grateful to the two anonymous referees for their very helpful comments and suggestions which highly contributed to the improvement of this paper. Participants at the 2013 ASSET and 5th ECINEQ meetings are also acknowledged. I am responsible for the remaining shortcomings.
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
75
Appendix. Proofs of Theorems 3.1 and 3.2
From NM, we have I (xh , nh ) = 0 for all h ̸= g. The previous expression becomes:
The proofs of Theorems 3.1 and 3.2 consist of several intermediate results which are detailed in this section. The first lemma investigates the simple case where n = 2. We obtain a particular function β(·).
I (x; n) = α(ng , n, Yg , Y )I (xg ; ng )
+
for all Y ∈ R++ .
(A.1)
Proof. Consider an income distribution x := (x1 , x2 ) where x1 , x2 ∈ R+ , x1 ̸= x2 , such that each individual is considered as a subgroup, that is ng = #Ng = 1 for g = 1, 2. According to WD*, the inequality index can be written as:
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N−g j∈N−g j>i
+
Lemma 1. If an inequality index satisfies NM and WD*, then the function β(·), as defined in the WD* axiom, is such that:
β(2, 2, Y , Y ) = 1,
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.5)
i∈Ng j∈N−g
Now, we propose to decompose the subgroup Ng into ng subgroups of only one individual. By applying WD*, it follows that: I (xg ; ng ) =
α(1, n, xi , Yg )I (xi ; 1) (within Ng )
i∈Ng
+
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈Ng j>i
I (x1 , x2 ; 2) = α(1, 2, x1 , x1 + x2 )I (x1 ; 1)
(between {i} and {j} in Ng ).
+ α(1, 2, x2 , x1 + x2 )I (x2 ; 1) (within N ) + β(2, 2, x1 + x2 , x1 + x2 )I (x1 , x2 ; 2) (between {1} and {2} in N )
Using again NM, we have I (xi ; 1) = 0 for all i ∈ Ng . Thus: (A.2)
from NM, I (x1 , 1) = I (x2 , 1) = 0, and (A.2) becomes: I (x1 , x2 ; 2) = β(2, 2, x1 + x2 , x1 + x2 )I (x1 , x2 ; 2)
I (xg ; ng ) =
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.6)
i∈Ng j∈Ng j>i
(A.3)
since x1 ̸= x2 , I (x1 , x2 ; 2) > 0 from NM. It follows that β(2, 2, x1 + x2 , x1 + x2 ) = 1, for all (x1 + x2 ) ∈ R++ .
By reintroducing (A.6) into (A.5), we have: I (x; n) = α(ng , n, Yg , Y )
+
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N−g j∈N−g j>i
+
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.7)
i∈Ng j∈N−g
Now, consider another partitioning of the same distribution x such that x = (x1 , . . . , xG ) with ng = #Ng = 1 for all g = 1, . . . , G. Then, according to WD*: I (x; n) =
α(ng , n, Yg , Y )β(2, n, xi + xj , Y )
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈Ng j>i
Now we consider a population of n > 2 individuals. The following three lemmas are devoted to establish a functional relationship between the functions α(·) and β(·), whatever the admissible values of all the parameters. Lemma 2. If an inequality index satisfies NM and WD*, then for all x ∈ Rn+ with n > 2 and every partition x = (x1 , . . . , xG ) with G > 2, there exist strictly positive weighting real-valued functions α(ng , n, Yg , Y ) and β(2, n, xi + xj , Y ) such that, for all possible subgroups Ng , we have:
α(1, n, xi , Y )I (xi ; 1) (within N )
i∈N
i∈Ng j∈Ng j>i
+
− β(2, n, xi + xj , Y ) I (xi , xj ; 2) = 0,
Proof. Consider an income distribution x ∈ R+ with n > 2, which can be partitioned into G > 2 subgroups, and assume NM and WD*. First, let x := (x1 , . . . , xG ) be a partition such that ng = #Ng > 2 and nh = #Nh = 1 for all h ̸= g. We define the whole population, but subgroup Ng , by N−g := N \ Ng . Application of WD* gives:
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N j∈N j>i
(A.4)
where 2 6 ng < n and Yg 6 Y . Moreover, by construction, xi +xj 6 Yg for all i, j ∈ Ng .
(between {i} and {j} in N ). Since I (xi ; 1) = 0 for all i ∈ N by NM, this expression reduces to:
n
I (x; n) =
G
α(nh , n, Yh , Y )I (xh ; nh ) (within N )
I (x; n) =
Subtracting (A.8) from (A.7), we finally get: 0 = α(ng , n, Yg , Y )
β(2, n, xi + xj , Y )I (xi , xj ; 2)
+
+
+
(between {i} in Ng and {j} in N−g ).
β(2, n, xi + xj , Y )I (xi , xj ; 2)
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈N−g
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈Ng j∈N−g
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N−g j∈N−g j>i
i∈N−g j∈N−g j>i
(between {i} and {j} in N−g )
i∈Ng j∈Ng j>i
h=1
+
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈N j∈N j>i
−
i∈N j∈N j>i
β(2, n, xi + xj , Y )I (xi , xj ; 2),
(A.8)
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P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
k > 1 times, x′ repeated k′ > 1 times and 0 < x < x′ . The size of subgroup Nh is nh = k + k′ + 1 > 2 and its total income is Yh = kx + k′ x′ . According to Lemma 2 and Eq. (A.4), we have:
which can be simplified as follows: 0=
α(ng , n, Yg , Y )β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈Ng j∈Ng j>i
−
0 = k[α(nh , n, Yh , Y )β(2, nh , x, Yh ) − β(2, n, x, Y )]I (0, x; 2)
β(2, n, xi + xj , Y )I (xi , xj ; 2).
+ k′ [α(nh , n, Yh , Y )β(2, nh , x′ , Yh )
i∈Ng j∈Ng j>i
− β(2, n, x′ , Y )]I (0, x′ ; 2)
We finally obtain the expression in Eq. (A.4).
+ kk′ [α(nh , n, Yh , Y )β(2, nh , x + x′ , Yh )
In the following lemma, a functional equation based on the weighting functions α(·) and β(·) is defined. The proof relies on simple cases demonstrating that the result holds for any distribution x ∈ R+ when n > 2. Lemma 3. Let n > 2. If an inequality index satisfies CN, NM and WD*, then the functions α(·) and β(·) in the WD* axiom are such that:
α(ng , n, Yg , Y )β(2, ng , X , Yg ) − β(2, n, X , Y ) = 0,
(A.9)
for all ng , n ∈ N and all X , Yg , Y ∈ R+ such that, respectively, 2 6 ng < n and 0 < X 6 Yg 6 Y . Proof. The proof is decomposed into two independent steps. In the first step, we show that the lemma is true under the restriction that Yg 6 (ng − 1)X . We establish in the second step that this result is also true under the restriction that Yg > (ng − 1)X . Step 1. Consider a distribution x ∈ Rn+ and a partition x = (x1 , . . . , xG ) with G > 2. The income distribution of subgroup Ng is such that xg := (x, . . . , x, x′ , . . . , x′ ) with x repeated k > 1 times, x′ repeated k′ > 1 times and 0 6 x < x′ . The size of subgroup Ng is ng = k + k′ and its total income is Yg = kx + k′ x′ . According to Lemma 2 and Eq. (A.4), we have: kk′ α(ng , n, Yg , Y )β(2, ng , X , Yg )
− β(2, n, X , Y ) I (x, x′ ; 2) = 0,
(A.10)
where we let X := x + x . By definition, β(·) is a strictly positive function (see WD*) and, according to NM, we have I (x, x′ , 2) > 0 (recalling that 0 6 x < x′ ). Hence we deduce that, in this case: ′
α(ng , n, Yg , Y )β(2, ng , X , Yg ) − β(2, n, X , Y ) = 0.
(A.11)
We obtain a functional equation. By construction of the distribution x, we have enough flexibility to consider any admissible value for ng , n, X and Y (by definition, 2 6 ng < n and 0 < X 6 Yg 6 Y ), by choosing appropriately k, k′ , x and x′ but, nevertheless, not enough to consider also any admissible Yg (which by definition is not lower than X ). Indeed X and Yg are not independent. Clearly, by choosing appropriately k, k′ , x and x′ , we can also obtain any Yg , but under the restriction that Yg 6 (ng − 1)X . Because X = x + x′ with 0 6 x < x′ , and ng = k + k′ with k, k′ > 1, the upper bound of Yg = kx + k′ x′ is immediately obtained, for a given X and a given ng , by letting k = 1, k′ = ng − 1, x = 0 and x′ = X : Yg 6 (ng − 1)X
(upper bound of Yg ).
To sum up we deduce that, for any admissible value of the parameters ng , n, X , Yg and Y (such that, 2 6 ng < n and 0 < X 6 Yg 6 Y ) but under the restriction that Yg 6 (ng − 1)X , we can find a distribution x as defined above, and choose k, k′ , x and x′ such that Eq. (A.11) is true. Before continuing, we have to discuss the special case where ng = 2. We emphasize that, in this case, we necessarily have X = Yg , and thus Yg 6 (ng − 1)X . Hence when ng = 2, whatever the value of the other parameters, Eq. (A.11) is necessarily true. In the following step, we restrict attention to the case where the subgroup size is strictly greater than 2. Step 2. Consider a distribution x ∈ Rn+ and a partition x = (x1 , . . . , xG ) with G > 2, such that the income distribution of subgroup Nh is such that xh := (0, x, . . . , x, x′ , . . . , x′ ) with x repeated
− β(2, n, x + x′ , Y )]I (x, x′ ; 2).
(A.12)
First notice that, whatever the values chosen for k, k , x and x′ , because k + k′ = nh − 1 and 0 < x < x′ , we have: ′
Yh = kx + k′ x′ < (nh − 1)x′ < (nh − 1)(x + x′ ).
(A.13)
Now consider the third term in brackets in Eq. (A.12). By using Step 1, and recalling that Yh < (nh − 1)(x + x′ ), we have:
α(nh , n, Yh , Y )β(2, nh , x + x′ , Yh ) − β(2, n, x + x′ , Y ) = 0.
(A.14)
The same reasoning applies to the second term in brackets in Eq. (A.12). By using Step 1, and recalling that Yh < (nh − 1)x′ , we have:
α(nh , n, Yh , Y )β(2, nh , x′ , Yh ) − β(2, n, x′ , Y ) = 0.
(A.15)
We know that k, k > 0 and I (a, b; 2) > 0 whenever a ̸= b. Hence, by reintroducing Eqs. (A.14) and (A.15) into (A.12), we deduce that: ′
α(nh , n, Yh , Y )β(2, nh , x, Yh ) − β(2, n, x, Y ) = 0.
(A.16)
By construction of the chosen distribution x, we can obtain any admissible values for nh , n, x and Y (by definition, 2 < nh < n and 0 < x 6 Yh 6 Y ), if and only if, for given nh , n, x and Y , there exist k, k′ , x and x′ such that nh = k + k′ + 1, Yh = kx + k′ x′ and x < x′ . Let k = nh − k′ − 1 and, by rewriting Yh = kx + k′ x′ , we have: x x′ = ′ k
Yh x
− (nh − k′ − 1) .
(A.17)
Hence x′ > x if and only if Yh /x − (nh − k′ − 1) /k′ > 1, or equivalently Yh > (nh − 1)x. Thus, for any admissible value of the parameters nh , n, x, Yh and Y (by definition, 2 < nh < n and 0 < x 6 Yh 6 Y ) but under the restriction that Yh > (nh − 1)x, we can find a distribution x as defined above, and choose k, k′ and x′ such that Eq. (A.16) is true. The proof of the lemma is obtained by combining the results of the two steps.
Lemma 4. Let n > 2. If an inequality index satisfies NM and WD*, then the functions α(·) and β(·) in the WD* axiom are such that:
β(2, n, X , Y ) = α(2, n, X , Y ), for all X , Y ∈ R+ such that 0 < X 6 Y . Proof. Consider any distribution x := (x1 , . . . , xn ), such that each individual is considered as a subgroup. By applying WD*, one obtains: I (x; n) =
α(1, n, xi , Y )I (xi ; 1) (within N )
i∈N
+
β(2, n, xi + xj , Y )I (xi , xj ; 2)
i∈N j∈N j>i
(between {i} and {j} in N ) (NM)
=
i∈N j∈N j>i
β(2, n, xi + xj , Y )I (xi , xj ; 2).
(A.18)
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
Then, consider a new partition of x into G subgroups, such that #Ng = 2 and #Nh = 1 for all h ̸= g. Moreover, let xg := (xg1 , xg2 ) with xg1 ̸= xg2 , and denote the whole population, but subgroup Ng , by N−g := N \ Ng . Application of WD* gives: I (x; n) =
α(1, n, xi , Y )I (xi ; 1)
β(2, n, xi + xj , Y )I (xi , xj ; 2)
(NM)
= α(2, n, xg1 + xg2 , Y )I (xg ; 2) β(2, n, xi + xj , Y )I (xi , xj ; 2) +
(A.23)
(R4)
β(2, n, xi + xj , Y )I (xi , xj ; 2).
i∈Ng j∈N−g
Applying again WD* to the subgroup Ng such that each individual constitutes one subgroup, it follows:
× I (xg1 , xg2 ; 2) + β(2, n, xi + xj , Y )I (xi , xj ; 2) (A.19)
i∈Ng j∈N−g
From Lemma 1, β(2, 2, xg1 + xg2 , xg1 + xg2 ) = 1, thus, subtracting (A.19) from (A.18) yields:
Recalling that xg1 ̸= xg2 , we have I (xg1 , xg2 ; 2) > 0. We deduce from (A.20) that:
Notice that, because xg1 , xg2 xg1 + xg2 > 0.
I (x; n) =
(R3)
I (x; n) =
β(2, n, xi + xj , Y )I (xi , xj ; 2).
1 θ (2, µij ) 2 i∈N
+
Lemma 5. Let n > 2. If an inequality index satisfies CN, NM and WD*, then the functions α(·) and β(·) in the WD* axiom can be written as:
j∈N j>i
θ (n, µ)
I (xi , xj ; 2)
1 θ (2, µji ) 2 i∈N
j∈N i>j
θ (n, µ)
I (xj , xi ; 2),
by definition µij = µji , then θ (2, µij ) = θ (2, µji ). Recalling that from NM, I (xi , xi ; 2) = I (xj , xj ; 2) = 0, we finally obtain:
and
I (x; n) =
where θ : {2, 3 . . . , n} × R++ −→ R++ is a strictly positive and continuous real-valued function. Proof. Recall that µij = (xi + xj )/2 is the mean income of two individuals, and µg is the mean income of subgroup Ng . Setting n =: n˜ and Y =: Y˜ as constant and letting X = xi + xj , we can write from Lemmas 3 and 4:
α(2, n˜ , 2µij , Y˜ ) α(2, ng , 2µij , Yg ) = . α(ng , n˜ , ng µg , Y˜ )
∀x ∈ Rn+ ,
Replacing β(·) by its expression defined in (R4) and considering I (·; 2) is continuous and symmetric such that I (xi , xj ; 2) = 1 (I (xi , xj ; 2) + I (xj , xi ; 2)), the previous equation becomes: 2
∈ R+ and xg1 ̸= xg2 , we have
θ (ng , µg ) , θ (n, µ) θ (2, µij ) β(2, n, 2µij , Y ) = , θ (n, µ)
2 θ (n, µ) i∈N j∈N
i∈N j∈N j>i
(A.20)
β(2, n, xg1 + xg2 , Y ) = α(2, n, xg1 + xg2 , Y ).
θ 2, µij I (xi , xj ; 2),
Proof. Let x be an income distribution partitioned into G > 2 subgroups such that x = (x1 , . . . , xG ) with ng = #Ng = 1 for all g = 1, . . . , G, and assume NM and WD*. Then, from Eq. (A.8):
β(2, n, xg1 + xg2 , Y ) − α(2, n, xg1 + xg2 , Y ) × I (xg1 , xg2 ; 2) = 0.
1
where I (·; 2) : R2+ −→ R+ is continuous, symmetric, and such that I (λ, λ; 2) = 0, for all λ ∈ R+ .
i∈N−g j∈N−g j>i
β(2, n, xi + xj , Y )I (xi , xj ; 2).
Theorem 3.1. Let n > 2. An inequality index satisfies CN, NM, SM and WD*, if and only if there exists θ : {2, 3 . . . , n} × R++ −→ R++ , such that: I (x; n) =
(NM)
I (x; n) = α(2, n, xg1 + xg2 , Y )β(2, 2, xg1 + xg2 , xg1 + xg2 )
α(ng , n, ng µg , Y ) =
θ (2, µij ) , θ (n, µ)
Finally, by invoking the continuity (CN) of I (x; n), we deduce that θ (n, µ) is continuous for all x ∈ Rn+ .
i∈N−g j∈N−g j>i
(A.22)
α(ng , n, ng µg , Y ) =
(between {i} in Ng and {j} in N−g )
+
θ (2, µij ) , θ (ng , µg )
θ (ng , µg ) and θ (n, µ) θ (2, µij ) β(2, n, 2µij , Y ) = . θ (n, µ)
i∈Ng j∈N−g
α(ng , n˜ , ng µg , Y˜ ), Eq. (A.21) can be
with θ (n, µ) ̸= 0. Substituting back expressions (A.22) and (A.23) into (A.21), yields in general:
(between {i} and {j} in N−g )
+
α(2, ng , 2µij , Yg ) =
α(2, n, 2µij , Y ) =
i∈N−g j∈N−g j>i
+
Defining θ (ng , µg ) := expressed as follows:
where θ (ng , µg ) ̸= 0 is a real-valued function by construction. Thus:
i∈N−g
+ α(2, n, xg1 + xg2 , Y )I (xg ; 2) (within N ) β(2, n, xi + xj , Y )I (xi , xj ; 2) +
77
(A.21)
1
θ 2, µij I (xi , xj ; 2),
(R5)
2 θ (n, µ) i∈N j∈N
such that I (·; n) is symmetric and continuous, for n > 2.
Theorem 3.1. Let n > 2. An inequality index satisfies CN, NM, SM, WD* and PP, if and only if there exists a strictly positive and continuous real-valued function c : R++ −→ R++ , such that: I (x; n) =
2 n2
c (µ) i∈N j∈N
c (µij )I (xi , xj ; 2),
∀x ∈ Rn+ ,
78
P. Mornet / Mathematical Social Sciences 83 (2016) 71–78
where I (·; 2) : R2+ −→ R+ is continuous, symmetric and such that I (λ, λ; 2) = 0, for all λ ∈ R+ . Proof. Consider an income distribution x ∈ Rn+ replicated m times and let M denote the corresponding population in which each individual is also replicated m times, such that #M = m · n, with m > 2, n = #N and x[m] = (x, . . . , x) ∈ Rmn + . According to (R5):
By applying the same structure to θ (2, µ˜ij ), it follows:
θ (2, µ˜ij ) = 22 · c (µ˜ij ). Finally, we obtain: I (x; n) =
1
m times
I (x
[m]
1
; mn) =
=
θ 2, µij I (x[i m] , x[j m] ; 2)
2 θ (mn, µ) i∈M j∈M m2
=
2 θ (mn, µ) i∈N j∈N
I (x
; mn) = I (x; n) m2 θ 2, µij I (xi , xj ; 2) 2 θ (mn, µ) i∈N j∈N θ 2, µij I (xi , xj ; 2),
2 θ (n, µ) i∈N j∈N
which is equivalent to: m2
2 θ(mn, µ)
−
1 2 θ (n, µ)
θ 2, µij I (xi , xj ; 2)
i∈N j∈N
= 0.
(A.24)
Because θ (·) is defined as a strictly positive and continuous realvalued function, according to NM, that: it follows, i∈N j∈N θ 2, µij I (xi , xj ; 2) > 0, for all xi ̸= xj . Focusing on the terms in brackets in Eq. (A.24), we deduce that: m2 2 θ(mn, µ)
=
1 2 θ (n, µ)
⇐⇒ m2 θ (n, µ) = θ (mn, µ).
Under PP, µ remains constant, that is, µ ˜ := µ; thus: m2 θ(n, µ) ˜ = θ (mn, µ), ˜ or analogously: n2 θ(m, µ) ˜ = θ (mn, µ). ˜ Mimicking Ebert’s (2010) reasoning (Ebert, 2010, pp. 99-100), we get:
θ(n, µ) ˜ n2
= c (µ). ˜
(µ)
c (µij )I (xi , xj ; 2),
(A.25)
i∈N j∈N
References
[m]
1
2 n2 c
22 · c (µij )I (xi , xj ; 2)
such that I (·; 2) : R2+ −→ R+ is continuous, symmetric and I (λ, λ, 2) = 0, for all λ ∈ R+ .
θ 2, µij I (xi , xj ; 2).
From PP, it follows:
=
2 n2 c (µ) i∈N j∈N
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