Cornell University/London School of Economics
Inequality: New Directions
Ithaca, New York, USA
September 12-13, 2009
A wide class of inequality measures based on the Bonferroni curve.1
Luis José Imedio Olmedo
Elena Bárcena Martín
Encarnación M. Parrado Gallardo
Dpto. Estadística y Econometría
Facultad de Ciencias Económicas y Empresariales
Universidad de Málaga
Correspondent author:
Elena Bárcena Martín
Dpto. Estadística y Econometría (68)
Facultad de Ciencias Económicas y Empresariales
Campus El Ejido s/n. Universidad de Málaga
29013. Málaga
Telf. +34 952 137 188. e-mail: [email protected]
Abstract
This paper introduces and analyses, both normatively and statistically, a class of inequality
measures. This class generalizes and comprises different well-known families of inequality
measures as particular cases. The elements of this new class are obtained by weighting local
inequality evaluated through the Bonferroni curve. The weights are the density functions of the
beta distributions over [0,1]. As a consequence of the different weighting schemes attached to
the indexes, the elements of the class introduce very dissimilar value judgements in the
measurement of inequality and welfare. Moreover, in this class of inequality measures it is
possible to select elements that focus on a particular part of the income distribution.
1
The authors gratefully acknowledge the financial support from the Spanish Institute for Fiscal Studies.
1
1. Introduction.
If inequality is assessed using a single inequality measure, a number of important
dimensions of the change in inequality due to a certain policy will not be picked up. Each
inequality measure incorporates assumptions about the way in which income differences in
different parts of the distribution are summarized. It is therefore desirable to calculate a wide
range of inequality indexes incorporating different assumptions, but at the same time having a
common theoretical foundation in order to thoroughly evaluate a redistribution policy. Although
indexes with distinct theoretical foundations are occasionally employed together2, this
procedure can make it difficult to evaluate their capacity as complementary measures of
inequality. It is therefore convenient to have homogeneous families of indexes that provide
sufficient information about the distribution and at the same time differ from and complement
one another in terms of their normative aspects. It is even more desirable to estimate indexes
that pay special attention to income differences in a certain part of the income distribution at
which the policy is aimed, and sometimes this part of the distribution may not be the extremes.
This paper introduces a class of inequality measures whose elements can be expressed
as a weighted average of the local inequality in each income percentile. The weights are the
density functions of the beta distributions over [0,1], which depend on two positive real
parameters. When both parameters change, a set of weights is obtained. These weights represent
different attitudes in terms of how inequality is evaluated throughout the income distribution.
Consequently, we obtain a family of indexes that have common properties and a clear formal
analogy, but at the same time differ and complement each other in the normative aspect.
The Gini (1912) and Bonferroni (1930) indexes belong to this class of inequality
measures, as well as families of inequality measures previously described in the literature such
as the generalized Gini measures of inequality (Kakwani, 1980; Yitzhaki, 1983) and the most
recent proposals of Aaberge (2000, 2007) and Imedio et al. (2008). In the indexes of these
families, the weights attached to local inequality behave monotonically throughout the income
distribution. The greatest weight is assigned to one of the extremes. In the class of inequality
measures that we propose, the weights are not necessarily monotonic. They can achieve the
maximum or minimum value in any percentile. This allows us to choose the inequality measures
that are more or less sensitive to changes that could take place in any part of the distribution.
The possibility of choosing the index that focuses on a specific percentile, and not necessarily
on the extremes of the distribution, is one of the advantages of our proposal.
2
It is common practice to combine the Gini index (1912) with the Atkinson indexes and/or entropy
indexes. Newbery (1970) highlights the differences between the theoretical foundations of the Gini index
and the Atkinson indexes. The entropy indexes derive from information theory.
2
Normative aspects are addressed following Yaari’s approach (1987, 1988) based on
social preference distributions. This approach for relating inequality and welfare is more general
than the classic AKS (Atkinson, 1970; Kolm, 1976; Sen, 1973), making it an easy task to
compare the level of inequality aversion3, or preference for equality, that the indexes introduce.
In some cases, it also allows indexes to be ordered in terms of this criterion. Families of
inequality measures that show increasing inequality aversion (respectively, decreasing)
approaching the maximum aversion, or Rawlsian leximin4 belong to this class of inequality
measures. Social preference distributions also make it easy to analyse the behaviour of the
indexes with respect to principles that are more demanding than the Pigou-Dalton Principle of
Transfers (PT), namely the Principle of Positional Transfer Sensitivity (PPTS) and the Principle
of Diminishing Transfers (PDT). Both the PPTS and the PDT analyse the particular sensitivity
of the indexes to progressive transfers that may occur in different parts of the distribution.
The behaviour of the indexes and their corresponding welfare functions are illustrated
empirically using the Spanish Living Conditions Survey for the year 2006. By doing so, we are
able to illustrate the sensitivity of the indexes and their corresponding welfare functions to
changes in the distribution depending on the part of the income distribution where the change
has a greater effect.
2. Analytical framework. Lorenz and Bonferroni curves.
Let us assume that the income distribution of a population is represented by the random
variable X, whose domain is the semi-straight positive real, R+ = [0, ∞), where F is its
∞
distribution function5, and μ = E (X ) = ∫ xdF( x ) < ∞ its mean income.
0
The associated Lorenz curve, L(p), p=F(x), is defined by:
L : [0, 1] → [0, 1] , L(p) =
1x
1 p −1
sdF
(
s
)
=
F ( t )dt , 0 ≤ p ≤ 1 ,
μ ∫0
μ ∫0
where F −1 ( t ) = inf{s: F(s)≥t}, 0≤t≤1, is the inverse to the left of F, F-1(0)=0. For each p=F(x),
L(p) is the proportion of total income volume accumulated by the set of units with an income
3
An index shows inequality aversion if it satisfies the Pigou-Dalton Principle of Transfers. This principle
states that an income transfer from a richer to a poorer individual, whose ranks in the income distribution
remain unchanged (progressive transfer), reduces income inequality. When the level of inequality
aversion increases, the effect of this type of transfer on the inequality index is greater.
4
The Rawlsian leximin focuses on the poorest individual of the population. Between two distributions,
the distribution with the greater minimum income is preferred or, in the event of equality, the distribution
in which the minimum income is less frequent. This approach is derived from the theory of social justice
defined by Rawls (1971).
3
lower than or equal to x. It is clear that for 0≤p≤1 it is L(p)≤p, and L(p)=p in the case of perfect
equality and L(p)=0 for 0≤p<1, L(1)=1 if the concentration is maximum. For any distribution,
X, the Lorenz curve is increasing and convex and given the mean income, the density function
of X is obtained from the curvature of L(p).
A simple transformation of the Lorenz curve allows the information contained in the
curve to be interpreted in an alternative manner. Bonferroni (1930) defines his index through the
curve6
⎧ L( p)
, 0 < p ≤ 1,
⎪
B : [0, 1] → [0, 1] , B(p) = ⎨ p
⎪0 , p = 0.
⎩
It satisfies B(p)≤1, 0≤p≤1. For an egalitarian distribution the curve is B(p)=1, 0<p≤1, whereas if
the concentration is maximum, the curve is B(p)=0, 0≤p<1 and B(1)=1. In the literature, B(p) is
known as the Bonferroni curve or the scaled conditional mean curve, given that:
B(p) =
E (X / X ≤ F −1 (p))
, 0 < p = F(x) ≤ 1 , B(0) = 0.
μ
That is, if p=F(x) is the proportion of units whose income is lower than or equal to x, B(p) is the
ratio between the mean income of this group and the mean income of the population.
Although from a formal standpoint the Bonferroni curve represents inequality in an
equivalent manner to the Lorenz curve and both curves are determined mutually, the
information they yield is different. The values of L(p) are fractions of the total income, while
the values of B(p) refer to relative income levels.
The shape of B(p) depends on the shape of the underlying distribution, F. It is verified:
B' (p) =
1 p t 2 f ' (F −1 ( t ))dt
1 p tdt
=
−
>
0
,
B
'
'
(
p
)
,
∫
∫
μp 3 0 (f ( F −1 ( t ))) 3
μp 2 0 f (F −1 ( t ))
provided that lim (p 2 / f (F −1 (p))) = 0 . Therefore, B(p) is increasing but the concavity/convexity
p →0 +
of the Bonferroni curve depends on the concavity/convexity of the associated distribution.
When each of these curves is compared in a certain percentile x=F-1(p), 0≤p≤1 with its
corresponding curve in case of equidistribution, we obtain the inequality accumulated up to this
percentile. If we use the Lorenz curve, then:
D L (p) = p − L(p) , 0≤p=F(x)≤1
[1]
5
Sometimes F is assumed to be continuous in order to obtain theoretical results in a simpler manner. In
such a case, f(x)=F´(x) is the density function of the distribution.
6
In the following equality, if the minimum income is x0>0, then B(0) = lim (L(p) / p) = L' (0 + ) = x / μ
p →0 +
4
0
is the difference between the share in the total amount of income of individuals whose income is
lower than or equal to x in the case of equidistribution and the real share in total income in the
distribution under consideration.
If we consider the Bonferroni curve, the function
D B (p) = 1 − B(p) =
μ − E(X X ≤ x )
μ
, 0≤p=F(x)≤1
[2]
measures the relative difference between the mean income of the population and the mean
income of individuals whose income is lower than or equal to x.
Although both curves are determined mutually, we introduce value judgements when
choosing one of them to measure local inequality because we attach greater or lesser importance
to inequality located in certain parts of the distribution. DL(p) and DB(p) assign more weight to
local inequality in the middle and on the left-hand side of the distribution, respectively.
In the next section we introduce a class of inequality measures that is one of the
contributions of this paper. In this class, local inequality is measured by means of the function
D B ( p) = D L (p) / p and the density functions of the beta distribution in [0,1] are used as
weights.
3. The β class of inequality measures.
Let us assume D : [0, 1] → R is a function such that for each p ∈ [0,1] , D(p) measures
inequality accumulated up to percentile p and ω : [0, 1] → R is a non-negative weight function
1
∞
0
0
such that ∫ ω(p)dp = ∫ ω(F( x ))dF( x ) = 1 . It is clear that the real number
1
I D ,ω = ∫ D( p)ω(p)dp
0
measures inequality in the distribution F. Its value depends on the functions D and ω, which
respectively introduce a way to evaluate cumulative local inequality and a criterion to weight
this inequality along the income distribution. This procedure to generate inequality indexes
underlies (sometimes in an implicit manner) the papers of Amato (1948), Mehran (1976),
Yitzhaki (1983) and Piccolo (1991), among others.
In the class of indexes that we propose, local inequality is measured by means of the
function D B ( p) = D L (p) / p defined in [2], and the density functions of the beta distribution in
[0 , 1] are used as weights. That is:
5
ω(s ,t ) : [0, 1] → R + , ω(s,t) (p) = (B(s, t )) −1 p s−1 (1 − p) t −1 , s > 0, t > 0 ,
[3]
where B(s,t) is the Euler beta function.
The above is set out in the following definitions.
Definition 1. For each (s, t ) ∈ R + × R + , the index I(s,t) is given by:
1
1
I(s, t ) = ∫ D B (p)ω(s ,t ) (p)dp = (B(s, t )) −1 ∫ (1 − B(p))p s−1 (1 − p) t −1 dp .
0
[4]
0
It is immediate that I(s, t ) (respectively μI(s, t ) ) is a relative (respectively absolute)
measure of inequality, with I(s,t)=0 in case of equidistribution and I(s,t)=1 in case of maximum
concentration. That is, I(s, t ) is a compromise index.
Definition 2. We denote the biparametric set β= {I(s, t )}s ,t >0 as the beta class of inequality
measures.
The elements of β are consistent with the ordering of the distribution induced by the
7
Bonferroni curve, and for s ≥ 2 with those induced by the Lorenz curve. Therefore, the
elements of β satisfy the Pigou-Dalton Transfer Principle: progressive transfers decrease
income inequality.
The β class adds a broad set of judgements relative to the weight that the social
evaluator attaches to the local inequality accumulated in different parts of the distribution. These
judgements are derived from the shape of the function ω(s,t)(⋅). So, we have:
(i)
If 0 < s < 1, 0 < t < 1, then ω(s,t)(p) is U shaped, is symmetric for s=t, and reaches its
minimum value for p = (s − 1) /(s + t − 2) .
7
(ii)
If 0 < s < 1, t ≥ 1, ω(s,t)(p) is decreasing and convex.
(iii)
If s ≥ 1, 0 < t < 1, ω(s,t)(p) is increasing and convex.
If X and Y are two income distribution functions ( B X (p) ≥ B Y (p) , 0 ≤ p ≤ 1)⇒ ( I X (s, t ) ≤ I Y (s, t ) ).
Analogously, the Lorenz consistency: ( L X (p) ≥ L Y (p) ) ⇒ ( I X (s, t ) ≤ I Y (s, t ) ).
6
If s = 1 (respectively t = 1), t ≥ 1 (respectively s ≥ 1), ω(s,t)(p) is decreasing
(iv)
(respectively increasing), and ω(1,1)(p) = 1.
If s >1, t >1, ω(s,t)(p) is bell shaped, if s = t it is symmetric and reaches its maximum
(v)
for p = (s − 1) /(s + t − 2) .
Therefore, in (i) (respectively (v)) less (more) weight is attached to local inequality in
the middle incomes and more (less) weight to the tails. These weights are more focused in the
middle incomes as s and t are greater and closer to each other. In the rest of the cases, except for
s = t = 1, greater weight is attached to the local inequality in one of the tails of the distribution.
Figure 1 shows the functions ω(s,t) for different values of its parameters.
Figure 1. ω(s,t) functions.
When
local
inequality
is
measured
through
the
Lorenz
differences,
DL(p)=p−L(p)=pDB(p), an equivalent expression for the indexes is:
1
I(s, t ) = (B(s, t )) −1 ∫ (p − L(p))p s−2 (1 − p) t −1 dp .
[5]
0
Nevertheless, I(s,t) is not really a weighted mean of DL(p) because the weights used are the
functions:
π (s ,t ) : [0, 1] → R + , π (s,t) (p) =
ω( s , t ) ( p )
p
= (B(s, t )) −1 p s−2 (1 − p) t −1 , s > 0, t > 0 ,
[6]
1
and ∫ π (s ,t ) (p)dp = (s + t − 1) /(s − 1) ≠ 1.
0
Expression [5] proves that the elements of β are linear measures of Mehran (1976).
They can also be expressed in terms of the income differences F −1 (p) − μ , 0 ≤ p ≤ 1, as
1
1
I(s, t ) = ∫ ( F −1 ( p) − μ)Π (s ,t ) (p)dp ,
μ0
where
1
Π (' s ,t ) (p) = π (s.t ) (p) , 0<p<1,
∫ Π (s,t ) (p)dp = 0 .
0
The above results are summed up in the following proposition.
7
Proposition 1. The β class is a subset of the linear measures of Mehran. Their elements are
compromise indexes and consistent with the ordering of the distribution induced by the
Bonferroni curve, and for s ≥ 2 with those induced by the Lorenz curve.
3.1. Identification of well-known indexes and families.
The β class comprises not only known indexes, but also families of inequality measures
frequently used in the literature. For (s, t ) = (1,1) and (s, t ) = ( 2,1) we obtain the Bonferroni, B,
and Gini, G, coefficients, respectively:
1
I(1,1) = ∫ (1 − B( p))dp = B ,
[7]
0
1
I(2,1) = 2∫ (p − L(p))dp = G .
[8]
0
For some particular values of the parameters of β, we obtain families that generalize the G and
B indexes. For s = 2 we obtain the family of the generalized Gini indexes, γ = {I(2,t)}t>0, where
1
1
0
0
I(2, t ) = t ( t + 1) ∫ (1 − B(p))p(1 − p) t −1 dp = t ( t + 1) ∫ (p − L(p))(1 − p) t −1 dp =
1
= 1 − t ( t + 1) ∫ (1 − p) t −1 L(p)dp , t > 0.
[9]
0
If t = 1 and s ∈ N = {1,2,...} is a positive integer, we obtain the countable family
α={I(s,1)}s∈N defined in Aaberge8 (2007).
1
1
1
0
0
0
I(s,1) = s ∫ (1 − B(p))p s−1dp = s ∫ (p − L(p))p s−2 dp = 1 − s ∫ p s−2 L(p)dp , s>0.
[10]
B and G belong to this family.
When s =1, another interesting family is obtained. The elements of this family are:
1
1
1
0
0
0
I(1, t ) = t ∫ (1 − B(p))(1 − p) t −1 dp = t ∫ (p − L( p))p −1 (1 − p) t −1 dp =1 − t ∫ (1 − p) t −1 B(p)dp, t > 0 . [11]
B also belongs to this family. Imedio et al. (2008) introduce and analyse the countable family,
δ={I(1,t)}t∈N, and compare it, in normative terms, with α and γN={I(2,t)}t∈N.
Aaberge (2000) introduces a countable family, λ, with λ=α−{B}, from the moments with respect to the
origin of the Lorenz curve. That is, λ contains the elements of α except the Bonferroni index.
8
8
In the indexes belonging to the families α and δ, the differences D B (p) and D L (p) are
weighted monotonically along the distribution. The same occurs when obtaining the elements of
γ from the Lorenz differences. However, we can choose indexes in class β that are more
sensitive in certain parts of the distribution, or even in a specific percentile. If policymakers are
interested in both tails of the distribution and attach less weight to the inequality accumulated by
middle incomes, they must select an index I(s, t ) with (s, t ) ∈ (0, 1) × (0, 1) . In this case, when t
increases (s), given s (t), the minimum value of the weight is reached for values of p that
approach 1 (0). Particularly, if s = t, we obtain the minimum of the weight function for p = 0.5.
On the contrary, if policymakers want to use an index that is more sensitive to changes taking
place in the middle of the distribution, they must consider an index in which
(s, t ) ∈ (1, + ∞) × (1, + ∞) . Given s (t), the maximum weight is attached to incomes with values of
p that approach 0 (1) as t (s) increases. In particular, if s=t, the weight function is bell-shaped
and symmetric with respect to p=0.5, where it reaches its maximum. In short, the parameters t
and s introduce different value judgements in measuring inequality.
3.2. The βN subclass.
When the parameters s and t are both positive integers (s, t ) ∈ N × N , we obtain a set of
indexes that we will call the subclass βN, βN ⊂ β. The cardinal of βN is less9 than the cardinal of
β, but the set of value judgements introduced by the elements of βN in the evaluation of
inequality is still wide. Among its elements we find classic indexes such as B and G, families
that generalize these indexes and uncommon indexes that are interesting by themselves. βN is
represented in the following triangular diagram.
I(1,1)
I(1,2)
I(1,3)
I(1,4)
......
I(1,s+t-1)
.......
I(2,2)
I(3,1)
I(2,3)
........
I(2,s+t-2)
I(2,1)
I(3,2)
........
.........
.....
.....
é
Focus on low incomes
9
I(4,1)
.......
I(s+t-2,2)
I(s+t-1,1)
......
é
Focus on high incomes
We go from β (whose cardinal is that of the set of real numbers) to βNwhich is countable.
9
The vertex of the triangle is the Bonferroni index, B=I(1,1). Family α={I(s,1)}s∈N is located on
the right-hand side. On the left side of the triangle, whose vertex is the Gini index G=I(2,1), we
find the indexes of the family γN={I(2,t)}t∈N. The elements of δ={I(1,t)}t∈N are on the left side
of the triangle.
The γN, α, and δ families share an interesting property when considering the curves L(p) and
B(p) as distribution functions.10
Proposition 211. Given the mean income, each of the families γN, α and δ characterizes the
income distribution.
Other interesting families that belong to βN are those whose elements are the indexes in
each row of the triangular diagram. The sum of the parameters of the indexes of each of these
families (finite) is a constant. In row n − 1, n ≥ 2, we have n−1 indexes, such that {I(s, t )}s+ t =n :
I(1, n − 1), I(2, n − 2), ......, I(n − 2, 2), I(n − 1, 1) .
The weight these indexes attach to the accumulated local inequality, as assessed by D B (p) ,
ω( s , n −s ) ( p ) =
(n − 1)!
p s−1 (1 − p) n −s−1 ,
(s − 1)!( n − s − 1)!
Is strictly decreasing for s=1, bell-shaped, and takes it maximum for p = (s − 1) / n − 2 ,
2 ≤ s ≤ n − 2 ; and is strictly increasing for s = n−1. Therefore, given s+t, when s increases, a
lower weight is attached to inequality on the left-hand side of the distribution, while more
weight is attached to inequality in middle and right-hand side incomes. Figure 2 shows the
weights of the four indexes {I(s, t )}s+ t =5 .
Figure 2. Weights for {I(s, t )}s+ t =5
The meaning of the Bonferroni index within the family βN is reinforced by the
following property.
10
Both curves are cumulative and increasing with values in the interval [0,1].
See Imedio and Bárcena (2007) and Imedio et al. (2008) for proof of this proposition and proposition 9.
Proof for the rest of the proposition is available upon request from the authors.
11
10
Proposition 3. For each n ∈ N , n≥2, B is the arithmetic mean of the indexes {I(s, t )}s+ t =n . That
is, B is the arithmetic mean of each row of the triangular diagram:
n −1
∑ I(s, n − s) =(n − 1)B .
s =1
The Gini index, G=I(2,1), satisfies a property that is similar to the previous property
when the index corresponding to s=1 is eliminated in each row of the triangular diagram.
Proposition 4. For each n ∈ N , n≥3, G is the weighted mean of the indexes {I(s, t )}s≥ 2,s + t = n :
I(2,n−2), I(3, n−3), ....., I(n−1,1). It is satisfied:
n −1
2
∑ (s − 1)I(s, n − s) = G ,
(n − 1)(n − 2) s =1
n −1
2
∑ (s − 1) = 1 .
(n − 1)(n − 2) s =1
In this weighted mean the weights increase as s increases, hence it attaches more weight
to the indexes that pay more attention to higher incomes.
The previous propositions show the algebraic relationship among the elements of βN.
From any element of the families α, δ or γN, we can obtain the element of the rest of the
families.
Proposition 5. The elements of α, δ and γN satisfy:
t
⎛t ⎞ i
(1 − I(i + 1, 1) ) , t∈N.
I(2, t ) = 1 + ( t + 1)∑ (−1) i ⎜⎜ ⎟⎟
i =1
⎝ i⎠ i +1
∞
∞
i=t
i=t
I(1, t ) = t ∑ I(2, i) / i(i + 1) , t ∑ 1 / i(i + 1) = 1 , t∈N.
In particular, the second expression above shows that the Bonferroni index, I(1,1) is a
weighted mean of all the generalized Gini indexes with a positive integer parameter.
In the following section we analyse some normative aspects of the elements of β.
4. Normative foundation. Preference distributions.
In order to establish the relationship between inequality and social welfare we follow
the Yaari approach (1987, 1988). If F is the income distribution and φ:[0, 1]→R is a distribution
11
function12 that represents social preferences, the Yaari social welfare function (YSWF) is given
by
1
Wφ (F) =
1
'
−1
−1
∫ xdφ(F(x )) = ∫ F (p)dφ(p) = ∫ φ (p)F (p)dp .
R+
0
[12]
0
Thus, Wφ is additive and linear in the incomes and weights them according to the
rankings attached to the individuals in the distribution13. The weight attached to the income of
an individual with rank p, 0<p<1, is φ ' (p) ≥ 0 . Yaari (1988) shows that Wφ (F) presents an
aversion to inequality if, and only if, φ ' (p) is decreasing, which is equivalent to the concavity
of φ.
If μ is the mean of F and L(p) is its Lorenz curve, YSWF can be expressed as a social
welfare function associated to a linear measure of inequality of the type defined in Mehran
(1976). Then,
[
]
Wφ (F) = μ 1 − I φ (F) ,
[13]
where
1
I φ ( F) = ∫ (p − L(p))π φ (p)dp , π φ (p) = −φ '' (p) ,
[14]
0
which yields an explicit relationship between the preference distribution and the weighting
scheme of the Lorenz differences.
[
According to the Blackorby and Donaldson approach (1978), the expression
]
μ 1 − I φ (F) is the equally distributed equivalent income14, in which case μIφ(F) measures the
loss of social welfare due to inequality.
If ℑβ = {φ(s ,t ) }s ,t >0 is the family of preference distributions associated to the elements of
β, from [6] and [14] we obtain:
φ'('s ,t ) (p) = −π (s ,t ) (p) = −(B(s, t )) −1 p s−2 (1 − p) t −1 , s > 0, t > 0 .
12
[15]
We assume the distribution function to be a class C2 function, which is twice continuously derivable.
When necessary, we will admit the existence of higher order derivatives in later results.
13
Ben Porath and Gilboa (1994) axiomatize the YSWF for discrete distributions. Zoli (1999) shows that
the YSWF dominance is equivalent to the inverse stochastic dominance of Muliere and Scarsini (1989).
14
This refers to a level of income such that if it is equally attached to all the individuals of the population,
it will provide an identical level of social welfare, according to the specified SWF, to that of the existing
distribution. This concept is the basis of the AKS approach (Atkinson, 1970; Kolm, 1976; Sen, 1973) for
relating social welfare and inequality.
12
The functions of ℑβ are strictly concave. Therefore, in all the indexes of the β class and in their
corresponding YSWFs, there is an underlying preference (aversion) for equality (inequality).
This common attitude, however, presents different degrees of intensity depending on the index.
Although it is not always necessary to know the preference distributions in order to
prove some normative properties of the indexes, it is convenient to know the preference
distributions to study certain aspects. For example, the possible ranking of the elements of a
family of inequality measures according to their inequality aversion is equivalent to rank their
respective preference functions depending on their concavity.
4.1. Preference functions of the elements of βN.
The expressions of the functions φ(s,t) when (s, t ) ∈ N × N are given in the following
proposition
Proposition 6. The preference functions of the indexes of βN are:
⎧p − p ln(p) , t = 1
⎪
t −1
φ (1,t ) (p) = ⎨ ⎡
( −1) i
t
p
p
ln(
p
)
−
+
∑ i
⎪⎢
i =1
⎩ ⎣⎢
t −1
φ (s ,t ) (p) = (B(s, t )) −1 ∑
i =0
⎛ t − 1⎞⎛
p i+1 ⎞⎤
⎟⎥ , t ≥ 2 .
⎜⎜
⎟⎟⎜⎜ p −
i + 1 ⎟⎠⎦⎥
⎝ i ⎠⎝
[16]
(−1) i ⎛ t − 1⎞⎛
p s +i ⎞
⎟ , s≥2, t≥1.
⎜⎜
⎟⎟⎜⎜ p −
s + i − 1 ⎝ i ⎠⎝
s + i ⎟⎠
If we apply the previous result to the families γN, α and δ, the function of their
respective sets of preference distributions, ℑγ N = {φ ( 2,t ) }t∈N , ℑα = {φ (s ,1) }s∈N and ℑδ = {φ (1,t ) }t∈N
are 15:
φ ( 2,t ) (p) = 1 − (1 − p) t +1 , t∈N.
⎧p − p ln(p) , s = 1
⎪
φ (s ,1) (p) = ⎨ 1
s
⎪⎩ s − 1 sp − p , s ≥ 2 ,
(
)
and the expression [16] for φ(1,t)(p). Particularly, the preference distributions associated to the
Bonferroni and Gini index are:
15
An equivalent expression of φ(1,t), t∈N, is:
∞
φ (1,t ) (p) = 1 − t ∑
i=t
t −1
(1 − p) i +1
⎞
⎛
=1 + t ⎜ p − p ln(p) − 1 + ∑ (1 − p) i +1 / i(i + 1) ⎟ .
i(i + 1)
i =1
⎠
⎝
13
φ B (p) = φ (1,1) (p) = p − p ln(p), 0<p≤1, φ(1,1)(0)=0.
φ G (p) = φ ( 2,1) (p) = 2p − p 2 , 0≤p≤1.
Both functions are strictly increasing and strictly concave in the interval [0,1], but φB is more
concave than φG. Thus, B shows more inequality aversion than G, which influences the
normative properties of both indexes. For example, the different behaviour when considering
principles of transfers that are more demanding than the Pigou-Dalton Principle.
As the following proposition illustrates, the relationship between B and G in terms of
preference for equality can be extended to the families γN, α and δ.
Proposition 7. The functions of families ℑγ N = {φ( 2,t ) }t∈N , ℑα = {φ (s ,1) }s∈N and ℑδ = {φ (1,t ) }t∈N
are ranked according to their degree of concavity. As the corresponding parameter increases, the
concavity of the functions of ℑ γ N and ℑδ increases, while the opposite occurs for the functions
of the family ℑα .
According to the above result, the indexes of γN and δ show an increasing aversion to
inequality, while the elements of α show the opposite behaviour. As the Bonferroni index
I(1,1)=B is a common element of the families δ and α, any function of ℑδ is more concave than
any function of ℑα , except φ(1,1) ≡ φB.
If λ=α−{B} is the family obtained when we eliminate the Bonferroni index from α, any
function of ℑ γ N is more concave than any function of ℑλ , except for the function
corresponding to the Gini index, I(2,1)=G, which is a common element of λ and γN.
In sum, in the subclass βN we can consider two pairs of families, δ and α, δ∩α={B},
together with γN and λ, γN∩λ={G}, which reveal similar behaviour in terms of the degree of
preference for equality that underlies their elements. Moreover, both union δ∪α and union
γN∪λ cover the total range of inequality aversion from maximum aversion to indifference. This
is a consequence of the following proposition.
Proposition 8. (i) The functions of ℑδ and ℑ γ N converge to the function of maximum
concavity in the interval [0,1], which is constant and equal to the unit, except for p=0. That is:
14
⎧0 , p = 0 ,
lim φ (1,t ) ( p) = lim φ ( 2,t ) (p) = ⎨
t →+∞
t →+∞
⎩1 , 0 < p ≤ 1 .
(ii) The functions of ℑα converge to the identity in the interval [0, 1] :
lim φ (s ,1) ( p) = p , 0 ≤ p ≤ 1 ,
s→+∞
with null concavity.
The following figures show the preference functions of the indexes I(1,2), I(1,1)=B,
I(2,1)=G and I(3,1), and the preference functions of I(2,3), I(2,2), I(2,1)=G and I(3,1). The
extreme case functions, maximum concavity and linear function are included in both figures.
Figure 3. Preference functions.
As a consequence of Proposition 8, the behaviour of the YSWFs and their associated
indexes in the extreme cases is given by
lim Wφ(1,t ) (F) = lim Wφ( 2,t ) (F) = x 0 , lim I(1, t ) = lim I(2, t ) = 1 − x 0 / μ ,
t →+∞
t →+∞
t →+∞
t →+∞
1
lim Wφ(s,1) ( F) = ∫ F −1 (p)dp = μ , lim I(s,1) = 0 .
s→+∞
0
s→+∞
In the families δ and γN, when t→+∞, the preference distributions converge to the
maximum concavity in such a way that the only relevant income transfers are those directed at
the poorest individual of the population (Rawlsian leximin). Social welfare is identified by the
minimum income in the distribution and the value of the inequality16 is 1 − x 0 / μ or the unit if
the minimum income is null. The opposite occurs for family α. When s→+∞, the preference
distribution tends to the identity and its concavity is null. Hence, progressive transfers have no
effect on social welfare or inequality. In this case, social welfare associated to any distribution is
identified by its mean income and the associated inequality index is null. This occurs not
because the distribution is egalitarian, but due to its indifference to inequality.
The elements of the families of absolute indexes δA={μI(1,t)}t∈N, γNA={μI(2,t)}t∈N and
αA={μI(s,1)}s∈N represent the cost of inequality in terms of social welfare. Based on the above,
it is clear that the cost of inequality is null in the case of indifference and maximum when
inequality aversion is also maximum, being equal to the difference between the mean and
minimum income.
16
If the range of the income variable is [0,∞), welfare is null and the inequality index is the unit.
15
In the triangular diagram that represents βN, the vertex is the Bonferroni index, I(1,1).
When we move from I(1,1)over the right-hand side of the triangle, family α, the indexes show a
decreasing aversion to inequality and attach less weight to low incomes. The opposite occurs
when we move from the vertex (from I(2,1)) over the left-hand side of the triangle, family δ
(family γN). In this case the indexes show an increasing aversion to inequality and pay more
attention to the incomes in the right-hand side of the distribution.
The sum of the parameters of the indexes in the same row of the triangle is a constant, s+t=n,
n≥2. If n≥3, when we move over any row from left to right, we start with an element of δ and
finish with an element of α. The degree of aversion to inequality decreases in each row when
we move from left to right. As an example, Figure 4 shows the four preference functions of the
indexes of the fourth row of the triangle, {I(s, t )}s+ t =5 .
Figure 4. Preference distribution of the indexes {I(s, t )}s+ t =5 .
As shown in Figure 4, the preference distribution of I(1,4) has the greatest degree of concavity.
The degree of concavity, together with the preference for equality, decrease successively as we
go from I(2,3 to I(3,2) and to I(4,1). The same behaviour can be observed in the indexes of any
row of the triangle, but as s+t increases, the number of indexes also increases. Hence the degree
of inequality aversion (or concavity of their corresponding preference functions) covers a
broader spectrum and there is a smoother change in the concavity between consecutive indexes.
5. Principles of Transfer.
The indexes of β satisfy the Pigou-Dalton Principle of Transfers (PDPT) given the
concavity of their respective preference distributions. However, when studying this type of
measures it is common to analyse if they satisfy more demanding redistributive criteria. An
obvious step is to consider principles by which the effect of a transfer is greater when it takes
place in the lower part of the distribution. Kolm (1976) and Mehran (1976) propose two
alternative versions of a principle of this type. According to the Principle of Diminishing
Transfers (PDT), a progressive transfer between two individuals with a given difference in
income implies that the lower the income of these individuals, the greater the reduction
(increase) in the index (social welfare). A different version of the PDT is given by the Principle
of Positional Transfer Sensitivity (PPTS). According to the PPTS, when there is a given
difference in ranks among the individuals for whom the transfer takes place, the effect of the
transfer is greater when it occurs among individuals in the lower part of the distribution.
Although both principles are analogous with regard to the transfers, the income difference
16
between the donor and the recipient is relevant for the PDT, while the proportion of individuals
located between both is relevant for the PPTS.
To provide a formal definition of the PDPT, let I be an inequality measure, and
ΔI p (δ, s) the change in I resulting from a transfer δ from a person with income F−1 (p) toward a
person with income F −1 (p − s) , where the transfer is assumed to be rank-preserving. Thus,
according to the PDPT, ΔI p ( δ, s) <0. Furthermore, let ΔI q ,p ( δ, s) be defined by
ΔI q , p (δ, s) = ΔI p (δ, s) − ΔI q (δ, s) .
Definition 3a. The inequality measure I is said to satisfy the PPTS if, and only if,
(q < p) ⇒ Δ q , p (δ, s) > 0 .
Analogously, let ΔI x (δ, z) denote the change in I resulting from a transfer δ from a
person with income x to a person with income x−z, ΔI x (δ, z) <0 if the PDPT is satisfied. Let
ΔI y ,x ( δ, z ) be defined by
ΔI y ,x ( δ, z ) = ΔI x ( δ, z ) − ΔI y ( δ, z ) .
Definition 3b. The inequality measure I is said to satisfy the PDT if, and only if,
( y < x ) ⇒ Δ y , x (δ, z) > 0 .
The following result shows how both principles are satisfied.
Proposition 9. Let F be an income distribution with mean μ and Iφ(F) an inequality index whose
preference distribution, φ, is concave. Then
(i) (Mehran, 1976; Zoli, 1999) Index Iφ(F) satisfies the PPTS if, and only if, φ ''' (p) > 0 .
(ii) (Aaberge, 2000) Index Iφ(F) satisfies the PDT if, and only if, φ '' (F( x ))F ' ( x ) is strictly
increasing for x>0. This is equivalent to the condition
−
φ ''' (F( x ))
F '' ( x )
>
, x >0.
φ '' (F( x )) (F ' ( x )) 2
[17]
17
The above proposition proves that an inequality measure satisfies, or does not satisfy,
the PPTS depending on the properties of its preference distribution, φ, irrespective of the
income distribution to which it is applied. It is, therefore, a characteristic of the index. However,
the same does not occur with the PDT. That Iφ(F) satisfies the PDT does not only depend on the
properties of its preference distribution, but also on the shape of the income distribution.
Expression [17] gives the relationship that must be satisfied by both distributions. That is, given
φ, index Iφ(F) verifies the PDT only for a given class of income distributions whose extension
depends on the degree of inequality aversion of φ.
By applying the above result to the preference distributions associated to the elements
of β, we obtain the behaviour of these indexes with regard to both principles.
Corollary 1.
a) The elements of β= {I(s, t )}s,t >0 satisfy the PPTS if, and only if, the following is satisfied17:
A(s, t , p) = [(s + t − 3)p − s + 2] > 0 , 0<p<1.
[18]
b) Let F be the distribution function and I(s,t) the inequality index associated to F, the PDT is
satisfied if, and only if:
(s + t − 3)F( x ) − s + 2
F '' ( x )
> '
, x >0.
F( x )(1 − F( x ))
(F ( x )) 2
[19]
According to the above, the Bonferroni index, B=I(1,1), satisfies the PPTS. However,
this is not true for the Gini index, G=I(2,1), because A(2,1,p)=0, 0<p<1. In the case of a fixed
difference in ranks, the Gini coefficient attaches an equal weight to a given transfer irrespective
of where it takes place in the income distribution Other indexes show the opposite behaviour
with respect to the PPTS. In the case of a fixed difference in ranks, they assign more weight to
transfers at the upper than at the central and the lower parts of the distribution. This is the case
of I(3,1) because A(3,1,p)=p−1<0, 0<p<1. There are other indexes which behave in a nonuniform manner with respect to the PPTS. For example, for I(3,3) A(3,3, p) = 3p − 1 , therefore,
it satisfies the PPTS if p>1/3, but does not if 0<p<1/3.
We obtain the following result for the families γN, α and δ from the expression [18]
17
The sign of the third derivative of the preference distribution is the same as the sign of A(s,t,p). It is
obvious that the sign can or cannot be constant in the interval [0,1] depending on the values of s and t.
18
Proposition 11. (i) The elements of γN, except the Gini index, and all the elements of δ satisfy
the PPTS. (ii) The indexes of α, except the Bonferroni index, show a behaviour that is opposite
to the behaviour implied by this principle.
In the triangular diagram that represents βN, the elements in any of the rows,
{I(s, n − s)}1≤s≤n −1 , n≥2, can show different behaviours with respect to the PPTS. When we move
to the right in a row, the indexes go from satisfying the PPTS for I(1,n−1)∈δ, I(2,n−2)∈γN, to
only satisfying the principle in a subinterval of [0,1] and, finally, to satisfying the contrary to
what the PPTS implies in the case of I(n−1,1)∈α.
With respect to the PDT, we should observe that if an index shows inequality aversion
( φ'' (p) < 0 ) and the third derivative of the preference function is non-negative ( φ ''' (p) ≥ 0 ), then
it will satisfy the PDT for all the concave income distributions ( F'' (p) < 0 ) because the
condition [17] is then satisfied. Thus, all the elements of γN and δ satisfy the PDT when the
income distributions are concave. In these cases, the concavity of F is a sufficient condition.
Particularly, the Gini index satisfies the PDT when the income distribution is strictly
concave. For the Bonferroni index, B=I(1,1), expression [17], or equivalently [19], is
(1 / F( x )) > (F '' ( x )) /(F ' ( x )) 2 , x > 0 , which is equivalent to the strict concavity of ln(F(x)); a less
demanding condition than the concavity of F(x). Therefore, the set of distributions for which B
satisfies PDT strictly contains the set of distributions for which G satisfies this principle. It is,
let Ω I be the set of distributions for which the index I satisfies the PDT, the relation of
inclusion is Ω G ⊂ Ω B .
The behaviour of the indexes of γN, α and δ with respect to PDT is shown in the
following proposition.
Proposition 12. It is satisfied:
(i) I(2,t)∈γN satisfies the PDT for all the distributions, F, such that (1 − F) t is strictly convex.
(ii) If I(1,t)∈δ satisfies the PDT, (1 − ln(F)) t is strictly convex.
(iii) If s>2, I(s,1)∈α satisfies the PDT for all the distributions such that Fs is strictly concave.
19
The relation Ω G ⊂ Ω B is extended to the set of distributions for which the indexes
(ordered with respect to their inequality aversion) satisfy the PDT. Proposition 13 includes some
results in this regard.
Proposition 13. If Δ I is the set of distributions for which the inequality index I satisfies the
PDT, the relations of inclusion are:
(i) For I(1,t)∈δ and I(2,t)∈γN, Ω I ( 2,t ) ⊂ Ω I (1,t ) , t∈N.
(ii) The elements of δ ∪ α satisfy:
⋅ ⋅⋅ ⊂ Ω I (1, t +1) ⊂ Ω I (1, t ) ⊂ ⋅ ⋅ ⋅ ⊂ Ω I (1,1) = Ω B ⊂ ⋅ ⋅ ⋅ ⊂ Ω I (s ,1) ⊂ Ω I (s +1,1) ⊂ ⋅ ⋅ ⋅
(iii) Similarly, the indexes γN∪λ satisfy:
⋅ ⋅⋅ ⊂ Ω I ( 2, t +1) ⊂ Ω I ( 2, t ) ⊂ ⋅ ⋅ ⋅ ⊂ Ω I ( 2,1) = Ω G ⊂ ⋅ ⋅ ⋅ ⊂ Ω I (s ,1) ⊂ Ω I (s +1,1) ⊂ ⋅ ⋅ ⋅
(iv) For n≥2, the set of n-1 indexes, {I(s, n − s)}1≤s≤n −1 , in any row of the triangle that represents
the subclass βN verifies:
Ω I (1, n −1) ⊃ ⋅ ⋅ ⋅ ⊃ Ω I (s , n − s ) ⊃ ⋅ ⋅ ⋅ ⊃ Ω I ( n −1,1) .
According to the previous proposition, as the degree of inequality aversion of the
indexes of a family increases (decreases), the set of distributions for which the indexes satisfy
the PDT is larger (smaller). (i) shows that the relationship between the indexes of γN and δ, with
respect to the PDT, is analogous to the relationship that exists between them when their degree
of inequality aversion is compared. According to section (ii) of proposition 13, as the parameter
increases in family δ, the indexes show a greater preference for equality and the set of
distributions which satisfy the PDT is larger. The opposite occurs in family α. When
considering the union of both families, the sets ΔI, I∈δ∪α, are nested. In this sense, the joint
behaviour of γN and λ is similar to the behaviour of families δ and α. Section (iv) of the
previous proposition is a consequence of the descending order of the inequality aversion of the
elements of the set {I(s, n − s)}1≤s≤n −1 .
6. Illustration.
This section empirically illustrates the different degree of inequality aversion of the
indexes of the β family and their corresponding SWF. That is, they show different sensitivity to
20
changes in the income distribution depending on the part of the distribution where the change
takes place.
We use data from the European Union Statistics on Income and Living Conditions18
(EU-SILC) for the year 2006 on Spain. We consider three variables:
-
Total annual disposable household income (X1).
-
Total annual disposable household income before social transfers other than old-age and
survivor's benefits (disposable household income before transfers, X2).
-
Total annual disposable household income plus tax on income and social contributions
(income before taxes, X3).
Household incomes are adjusted (‘equivalised’) to take account of the differences
between them in terms of size and composition. We use the modified OECD equivalence
scale19. Hence, if Xi is the household income, the equivalent income, Yi, is defined as
Yi=Xi/m(ni), where m(ni) is the number of equivalent members.
Social transfers are skewed towards the left tail of the distribution. On the other hand,
tax on income and social contributions are skewed towards the right tail of the distribution.
Figure 5 shows the distribution of both income components by deciles.
Figure 5. Distribution (in percentages) of social transfers other than old-age and
survivor's benefits, and tax on income and social contributions, by deciles.
We evaluate the effect of both income components on the indexes {I(s, t )}2 ≤ s + t ≤ 6 and
their corresponding SWF, {Wφ(s , t ) }2 ≤ s + t ≤ 6 . They are the elements of the first five rows of the
triangle that represents βN.
Table 1 shows the inequality indexes for X1, X2 and X3, respectively. Under each of the
indexes obtain the percentage of variation when moving from X2 to X1 and from X3 to X1 in that
order.
18
EU-SILC is the reference source for comparative statistics on income distribution, living conditions and
social exclusion at the European level.
19
The OECD equivalence scale attaches a value of 1 to the first adult member of the household, 0.5 to the
remaining adult members and 0.3 to each member under 14 years of age.
21
Table 1. Inequality indexes for X1, X2 and X3.
I(1,1)
0.422, 0.448, 0.443
-5.6%, -4.5%
I(1,2)
I(2,1)
0.545, 0.579, 0.565
0.3000,0.316, 0.320
-5.8%, -3.5%
-5.2%, -6.4%
I(1,3)
I(2,2)
I(3,1)
0.610, 0.648, 0.627
0.416, 0.440, 0.439
0.241, 0.254, 0.261
-6.0%, -2.8%
-5.5%, -5.3%
-5.0%, -7.4%
I(1,4)
I(2,3)
I(3,2)
I(4,1)
0.651, 0.693, 0.667
0.484, 0.513, 0.507
0.348, 0.368, 0.372
0.206, 0.216, 0.224
-6.0%, -2.4%
-5.6%, -4.6%
-5.3%, -6.4%
-4.9%, -7.9%
I(1,5)
I(2,4)
I(3,3)
I(4,2)
I(5,1)
0.682, 0.726, 0.696
0.530, 0.563, 0.553
0.414, 0.438, 0.439
0.304, 0.321, 0.327
0.181, 0.190, 0.198
-6.1%, -2.1%
-5.7%, -4.0%
-5.4%, -5.6%
-5.2%, -7.0%
-4.7%, -8.3%
22
Social transfers other than old-age and survivor's benefits, as well as tax on income and
social contributions reduce inequality. The percentage reductions in the indexes differ
depending on the effect these components have on the income distribution. Social transfers
(mainly received by people with low incomes) produce a greater reduction in the indexes
focused on the left tail of the distribution. When we move diagonally from the top to the bottom
right corner of the triangle, the indexes attach more weight to inequality in the upper part of the
distribution and the reduction in the value of the indexes is therefore smaller. The same thing
occurs for the same reason when we move from left to right in a given row.
Tax collection is mainly concentrated in the right tail of the distribution. The
redistributive effect of the tax is more pronounced when we use indexes that show more
sensitivity to changes in higher incomes. In this case, the percentage reductions in the indexes
are opposite the previous case (transfers) when moving in the triangle.
The above demonstrates that the same transfer can be evaluated as having a greater or
lesser impact on inequality depending on the part of the distribution that the social evaluator
pays more attention to. If policymakers focus on low incomes and choose I(1,5) to measure
inequality, then transfers will reduce inequality more than taxes (6.1% vs. 2.1%). On the other
hand, if policymakers focus on higher incomes and evaluate the impact of both components of
income with I(5,1), they will point to taxes as being more effective for reducing inequality
(8.3% vs. 4.7%). Social transfers (respectively, taxes) reduce inequality more when we measure
inequality with indexes that are sensitive to changes in the low part of the income distribution
(respectively high part of the income distribution). In short, when measuring inequality it is
important to bear in mind the characteristics of the indexes being used.
These considerations are applied to the SWF. Table 2 shows the values of
{Wφ(s , t ) }2 ≤ s + t ≤ 6 for X1, X2 and X3. Under these values we show the percentage of variation in
welfare when we go from X2 to X1 and from X3 to X1. Mean incomes are given in euros20.
20
μ X1 = 13452 , μ X2 = 12843 , μ X3 = 15904 .
23
Table 2. Welfare corresponding to the indexes of βN for distributions X1, X2 and X3.
Wφ(1,1)
7770, 7095, 8866
9.5%, -12.4%
Wφ(1,2)
Wφ(2,1)
6119, 5408, 6923
9421, 8782, 10810
13.1%, -11.6%
7.3%, -12.9%
Wφ(1,3)
Wφ(2,2)
Wφ(3,1)
5251, 4518, 5927
7855, 7190, 8914
10204, 9579, 11758
16.2%, -11.4%
9.3%, -11.9%
6.5%, -13.2%
Wφ(1,4)
Wφ(2,3)
Wφ(3,2)
Wφ(4,1)
4688, 3938, 5290
6941, 6257, 7838
8768, 8122, 9991
10683, 10064, 12348
19.0%, -11.4%
10.9%, -11.4%
8.0%, -12.2%
6.1%, -13.5%
Wφ(1,5)
Wφ(2,4)
Wφ(3,3)
Wφ(4,2)
Wφ(5,1))
4281, 3518, 4833
6317, 5617, 7115
7877, 7216, 8922
9362, 8726, 10703
11013, 10399, 12759
21.7%, -11.4%
12.5%, -11.2%
9.2%, -11.7%
7.3%, -12.5%
5.9%, -13.7%
24
Social transfers increase welfare because they increase mean income. The greatest
increment in percentage terms occurs when the SWF attaches more weight to low incomes.
However, the increment in percentage terms is lower when the SWF shows less inequality
aversion. The opposite effect occurs with taxes. Welfare is reduced as a consequence of the
decrement in mean income. The greatest reduction occurs for the SWF corresponding to indexes
that attach more weight to inequality in higher incomes because taxes have a greater effect on
these incomes.
7. Conclusions.
The indexes of the β class introduce different criteria in the measurement of inequality.
At the same time, they share a common set of properties and have a clear formal analogy.
Because of this we are able to treat different well-known families of inequality measures
uniformly. The main advantage of this class of inequality measures is that it is possible to
choose the indexes that focus on a particular percentile of the income distribution.
Each element of β weights cumulative local inequality in a different way. The
characteristics of the weighting schemes define the properties of the indexes: i.e. income
percentiles where the index attaches more importance, level of inequality aversion and the effect
of transfers depending on rank or income differences between the donor and the receiver, and
depending on the part of the distribution where the receiver and the donor are located.
The subclass βN is obtained when the two parameters of the indexes of β are positive
integers and the subclass contains the families γN, α and δ. These families cover a wide
spectrum of inequality aversion that can be introduced by an index. The families introduce
indexes with different behaviours with respect to the PPTS or to the PDT and contain infinite
and countable indexes. But the indexes in the same row of the triangle that represent βN are
finite families of the kind {I(s, n − s)}1≤s≤n −1 , n≥2. The degree of inequality aversion in these
families changes, but does not reach the extreme, while the responses to the principles of
transfers also change. When we move from left to right in each row, the sensitivity of the
indexes to changes in different parts of the income distribution also changes.
It is obvious that the real effectiveness of an economic policy to affect inequality or
welfare in a certain part of the distribution does not depend on the index used to assess the
impact. Even so, it is interesting to be able to choose indexes that are more sensitive to changes
in the part of the distribution we are interested in focusing our attention on. All of this allows us
to quantitatively assess the policy to be implemented in a better manner.
25
In practice, the choice of a small set of elements of β allows inequality to be measured
according to different distributive criteria, the preferences of the social evaluator and the
particular nature of each empirical case. If we attempt to rank a set of income distributions with
these indexes, it is likely that different rankings will be obtained depending on the index.
Bearing in mind the characteristics of each measure, a result of this type would be highly
revealing, such as the case of robust rankings.
8. References.
Aaberge, R. (2000), “Characterizations of Lorenz curves and income distributions”, Social
Choice and Welfare 17, pp. 639-653.
Aaberge, R. (2007), “Gini’s nuclear family”, Journal of Economic Inequality, 5-3, pp. 305-322.
Amato, V. (1948), “Sulla misura della concentrazione dei redditi”, Rivista Italiana di Economia,
Demografia e Statistica, vol. 2, pp. 509-529.
Atkinson, A. B. (1970), “On the measurement of inequality”, Journal of Economic Theory 2,
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Ben Porath, E. and I. Gilboa (1994), “Linear measures, the Gini index, and the income equality
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Blackorby, C. and D. Donaldson (1978), “Measures of relative equality and their meaning in
terms of social welfare”, Journal of Economic Theory 18, pp. 59-80.
Bonferroni, C. E. (1930): Elementi di statistica generale, Libreria Seber, Firenze.
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February 2008.
Gini, C. (1912), “Variabilità e mutabilità”, Studi Economico-giuridici, Università di Cagliari,
vol 3-2, pp. 1-158.
Imedio Olmedo, L. J. and E. Bárcena Martín (2007): “Dos familias numerables de medidas de
desigualdad”, Investigaciones Económicas XXX(1), pp. 191-217.
Imedio Olmedo, L. J., Bárcena Martín, E. and E. M., Parrado Gallardo, (2008): “Familias de
índices de desigualdad que caracterizan la distribución de la renta”, XV Encuentro de
Economía Pública. Salamanca, 2008.
Kakwani, N. C. (1980), “On a class of poverty measures”, Econometrica 48, pp. 437-446.
Kolm, S. C. (1976), “Unequal inequalities, I, II”, Journal of Economic Theory 12, pp. 416-442;
13, pp. 82-111.
26
Mehran, F. (1976), “Linear measures of inequality”, Econometrica 44, pp. 805-809.
Muliere, P. and M. Scarsini (1989), “A note on stochastic dominance and inequality measures”,
Journal of Economic Theory 49, pp. 314-323.
Newbery, D. M. (1970), “A theorem of the measurement of inequality”, Journal of Economic
Theory 2, pp. 264-266.
Piccolo, D. (1991), “Risultati asintotici per misure di variabilità sull’insieme dei numeri interi”,
Quaderni di Statistica ed Econometria, vol. XIII, pp. 91-107.
Rawls, J. (1971), A theory of justice, Harvard University Press, Cambridge.
Sen, A. K. (1973), On economic inequality, Clarendon Press, Oxford.
Yaari, M. E. (1987), “The dual theory of choice under risk”, Econometrica 55, pp. 99-115.
Yaari, M. E. (1988), “A controversial proposal concerning inequality measurement”, Journal of
Economic Theory 44, pp. 381-397.
Yitzhaky, S. (1983), “On an extension of the Gini index”, International Economic Review 24,
pp. 617-628.
Zoli, C. (1999), “Intersecting generalized Lorenz curves and the Gini index”, Social Choice and
Welfare 16, pp. 183-196.
27
Figure 1. ω(s,t) functions.
w(0,5;2)
w(0,5; 0,5)
8
4
6
3
4
2
2
1
0
w(0.7; 0.4)
6
5
4
3
2
1
0
0
0
0.25
0.5
0.75
1
0
0.25
w(2; 0,5)
0.5
0.75
1
6
4
2
0
0.5
0.75
1
0.5
0.75
1
w(2; 2)
3
2.5
2
1.5
1
0.5
0
0.25
0.25
w(2; 6)
8
0
0
2
1.5
1
0.5
0
0
0.25
0.5
28
0.75
1
0
0.25
0.5
0.75
1
Figure 2. Weights for {I(s, t )}s+ t =5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
s=1; t=4
s=2; t=3
s=3; t=2
s=4; t=1
29
1
Figure 3. Preference functions.
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
s=1; t=2
s=1; t=1
s=2; t=3
s=2; t=2
s=2; t=1
s=3; t=1
s=2; t=1
s=3; t=1
lineal
concav. Máx.
lineal
concav. Máx.
30
Figure 4. Preference distribution of the indexes {I(s, t )}s+ t =5 .
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
s=1; t=4
s=2; t=3
s=3; t=2
s=4; t=1
31
1
Figure 5. Distribution (in percentages) of social transfers other than old-age and survivor's
benefits, and tax on income and social contributions, by deciles
35%
30%
25%
20%
15%
10%
5%
0%
D1 D2
D3 D4
D5 D6
D7 D8
Social transfers
D9 D10
Tax on income and social contributions
32