We provide a parsimonious axiomatisation of the complete class of absolute nequalityindices. Our approach uses only a weak form of decomposability and does not require a priori that the measures be differentiable.inequality measures, decomposability, translation invariance

Frank A Cowell

Kristof Bosmans

English

Research Papers in Economics

 
 
 
 
 
 
The Class of Absolute Decomposable 
Inequality Measures 
 
Kristof Bosmans
∗ and Frank A. Cowell
† 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
DARP 99       The Toyota Centre 
February  2009         Suntory  and  Toyota  International   
       Centres  for  Economics  and  Related 
       D i s c i p l i n e s  
       London  School  of  Economics 
       H o u g h t o n   S t r e e t  
       L o n d o n   W C 2 A   2 A  
 
       (+44  020)  7955  6674)
                                                 
∗ Maastricht University.  Correspondence: Kristof Bosmans, Department of Economics, Maastricht 
University, Tongersestraat 53, 6211 LM Maastricht, The Netherlands.   
e-mail: k.bosmans@algec.unimaas.nl 
† London School of Economics   
Abstract 
 
We provide a parsimonious axiomatisation of the complete class of absolute nequality 
indices. Our approach uses only a weak form of decomposability and does not require 
a priori that the measures be differentiable. 
 
Keywords:   inequality measures, decomposability, translation invariance 
 
JEL codes:    D63, H20, H21 
 
  
Distributional Analysis Research Programme 
 
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© Authors: Kristof Bosmans and Frank A. Cowell.   All rights reserved. Short 
sections of text, not to exceed two paragraphs, may be quoted without explicit 
permission provided that full credit, including © notice, is given to the source. 1 Introduction
Absolute inequality indices have the property that, for any income distribution, if any
given number of dollars is added to (or subtracted from) every income, inequality remains
unchanged. This means that such indices have the property that they are deﬁned for
negative incomes: this is particularly important in empirical applications where one is
interested in applying inequality indices to the distribution of wealth (net worth may
often be substantially negative) or to speciﬁc components of income (for example business
income). Decomposability of inequality indices requires that, for an arbitrary subgroup
of the population, if inequality in the subgroup increases then, ceteris paribus, inequality
overall increases. Previous descriptions and characterizations of the class of indices that
has these properties have invoked additional assumptions about structure — see section 4
below. Our approach (in sections 2 and 3) is deliberately minimalist in that it uses just
the axioms required to deﬁne (a) an inequality measure (b) the “absolutist” property and
(c) the population structure.
2 Setting
The income of individual i is a real number xi, and the income distribution for a population
of n individuals is a vector x := (x1,x2,...,xn) in Rn. The set of income distributions for
m or more individuals is Xm :=
S∞
n=m Rn. For an income distribution x, we write n(x) for
the dimension and µ(x) for the mean. A vector of dimension n of which all components
are equal to 1 is denoted by 1n.
An inequality measure is a continuous function I : X1 → R with the property that
I(x) = 0 if x is an equal income distribution. The higher the value of the function I, the
higher income inequality. To give meaning to I it is standard to assume the following two
properties:
Anonymity. For every x ∈X1, I(x) = I(x0) if x0 is obtained from x by rearrangement of
components.
Transfer principle. For every x ∈X2 and any positive real number δ, we have that if xi <
xi+δ ≤ xj−δ < xj, then I(x1,...,xi,...,xj,...,xn) > I(x1,...,xi+δ,...,xj−δ,...,xn).
The essential property for an abolute inequality measure is this:
Translation invariance. For every x ∈X1 and any real number δ,
I(x) = I(x + δ1n). (1)
Finally, the following two axioms describe the relationship between population struc-
ture and inequality:
Decomposability. There exists a function A such that, for all x,y ∈ X2,
I(x,y) = A(I(x), I(y), µ(x), µ(y), n(x), n(y)),
where A is continuous and strictly increasing in its ﬁrst two arguments.
Replication invariance. For every x ∈X1, I(x,x) = I(x).
13 Result
Our result characterizes the class of inequality measures satisfying the ﬁve properties listed
in section 2.
Theorem. An inequality measure I satisﬁes anonymity, the transfer principle, replication
invariance, decomposability and translation invariance if and only if there exists a real
number c and a continuous and strictly increasing function f : R → R, with f(0) = 0,
such that, for all x ∈X1,
f(I(x)) =

      
      
1
n(x)
n(x) X
i=1
{ec[xi−µ(x)] − 1} if c 6= 0,
1
n(x)
n(x) X
i=1
[xi − µ(x)]2 if c = 0.
(2)
Proof. The inequality measures given in (2) satisfy anonymity, the transfer principle, repli-
cation invariance, decomposability and translation invariance. We focus on the reverse
implication.
If I satisﬁes anonymity, the transfer principle, replication invariance and decompos-
ability, then (Shorrocks, 1984, Theorem 4) there exist functions F and φ such that
F(I(x),µ(x),n(x)) =
n(x) X
i=1
[φ(xi) − φ(µ(x))], (3)
where φ is continuous and strictly convex, and where F is continuous in I and µ, strictly
increasing in I, additive in n, with F(0,µ,n) = 0 . Choose any x1 and x2 in X2, and write
x12 := (x1,x2), Ik := I(xk), nk := n(xk) and µk := µ(xk). We may write
I(x12) = H(F(I1,µ1,n1) + F(I2,µ2,n2), µ1, µ2, n1, n2) (4)
where H is a function implicitly deﬁned by
F(H(z, µ1, µ2, n1, n2)), µ1, µ2, n1, n2) = z + n1φ(µ1) + n2φ(µ2) − n12φ(µ12).
Deﬁne λ := eδ and θk := eµk. Then λθk = eµk+δ, and (4) and (1) imply
H(F(I1,ln(θ1),n1) + F(I2,ln(θ2),n2), ln(θ1), ln(θ2), n1, n2)
= H(F(I1,ln(λθ1),n1) + F(I2,ln(λθ2)), ln(λθ1), ln(λθ2), n1, n2),
or, with appropriate deﬁnitions of G from F and of J from H,
J(G(I1,θ1,n1) + G(I2,θ2,n2), θ1, θ2, n1, n2)
= J(G(I1,λθ1,n1) + G(I2,λθ2,n2), λθ1, λθ2, n1, n2).
From Lemma 1 in Shorrocks (1984):
G(I1,λθ1,n1) = cG(I1,θ1,n1),
G(I2,λθ2,n2) = cG(I2,θ2,n2),
2from which it follows that
c =
G(I,λθ,n)
G(I,θ,n)
= c(λ),
independent of I, µ and n. Therefore
G(I,λθ,n) = c(λ)G(I,θ,n)
= c(λ)c(θ)G(I,1,n)
= c(λθ)G(I,1,n).
Because c(λθ) = c(λ)c(θ) for all positive λ and θ, we have c(θ) = θc (Acz´ el, 2006, Theorem
4, p. 144). Consequently
G(I,θ,n) = θc G(I,1,n),
F(I,µ,n) = eµcF(I,0,n),
and (3) yields
F(I(x),0,n) = e−µc
n X
i=1
[φ(xi) − φ(µ)]. (5)
In the case of two individuals,
0 = F(I(x + δ12),µ(x) + δ,2) − ecδF(I(x),µ(x),2),
and writing x := (u,v) this becomes
0 =

φ(u + δ) + φ(v + δ) − 2φ

u + v
2
+ δ

− ecδ

φ(u) + φ(v) − 2φ

u + v
2

. (6)
If we deﬁne
ψ(u,δ) := φ(u + δ) − ecδφ(u), (7)
then (6) becomes
0 = ψ(u,δ) + ψ(v,δ) − 2ψ

u + v
2
,δ

.
This is a Pexider equation with solution (Acz´ el, 2006, p. 142)
ψ(u,δ) = a(δ)u + b(δ). (8)
From (7),
ψ(u,µ + δ) = φ(u + µ + δ) − ec[µ+δ]φ(u)
= ψ(u + µ,δ) + ecδφ(u + µ) − ec[µ+δ]φ(u)
= ψ(u + µ,δ) + ecδψ(u,µ). (9)
Substitute from (8) into (9) to get
a(µ + δ)u + b(µ + δ) = a(δ)(u + µ) + b(δ) + ecδ[a(µ)u + b(µ)],
3which implies
a(µ + δ) = a(δ) + ecδa(µ) = a(µ) + ecµa(δ), (10)
b(µ + δ) = a(δ)µ + b(δ) + ecδb(µ) = a(µ)δ + b(µ) + ecµb(δ). (11)
There are two cases to consider.
Case 1 (c 6= 0). In this case (10) gives
a(δ)(ecµ − 1) = a(µ)

ecδ − 1

,
a(µ) =
a(δ)
ecδ − 1
(ecµ − 1),
so that, for any given value of δ and all µ,
a(µ) =
k1
k2
(ecµ − 1),
where k1 := a(δ) and k2 := ecδ −1 can be taken as constants (conditional on the chosen δ
and the arbitrary value of c). So the only solution is
a(µ) = α(ecµ − 1).
Therefore, from (11)
b(µ + δ) = µα(ecδ − 1) + b(δ) + ecδb(µ) = δα(ecµ − 1) + b(µ) + ecµb(δ),
so that
µα(ecδ − 1) + b(µ)(ecδ − 1) = δα(ecµ − 1) + b(δ)(ecµ − 1),
b(µ)(ecδ − 1) = [δα + b(δ)](ecµ − 1) − µα(ecδ − 1),
b(µ) =
δα + b(δ)
ecδ − 1
(ecµ − 1) − µα.
Since this must be true for arbitrary δ and c, the solution is
b(µ) = β(ecµ − 1) − αµ. (12)
Case 2 (c = 0). Here (10) becomes
a(µ + δ) = a(µ) + a(δ),
and the solution to this Cauchy equation is a(µ) = κµ (Acz´ el, 2006, Theorem 2, pp. 35-36).
Therefore, from (11)
b(µ + δ) = κµδ + b(δ) + b(µ), (13)
Using that b(0) = 0 and letting δ = −µ and κ = 2γ, we get
b(µ) + b(−µ) = 2γµ2,
4to which the solution is
b(µ) = γµ2 + p(µ),
where p(µ) := Π(µ,−µ) with the property p(µ) = −p(−µ) (Polyanin and Manzhirov,
1998). Plugging this into (13) gives
p(µ + δ) = p(µ) + p(δ),
to which the solution is p(µ) = ηµ. Therefore
b(µ) = γµ2 + ηµ. (14)
From (8), we have ψ(0,x) = b(x); so from (12) and (14) we ﬁnd
ψ(0,x) =
(
β(ecx − 1) − αx if c 6= 0,
γx2 + ηx if c = 0.
(15)
Note that, from (7), we have:
φ(x) = ψ(0,x) + ecxφ(0). (16)
However the constant φ(0) is arbitrary. Setting φ(0) = 0 and substituting from (15) and
(16) into (5) produces (2).
4 Concluding remarks
While it is well known that the resulting measures in (2) belong to the class of absolute
decomposable measures (Chakravarty and Tyagarupananda, 1998, 2009), it has not previ-
ously been demonstrated that only these measures belong to the class: previous treatments
have introduced a priori additional restrictions such as diﬀerentiability and have used a
stronger decomposability property.
Letting c = 1, we obtain the variance and, letting c < 0, the family (2) is ordinally
equivalent to the Kolm (1976) family of measures. As the parameter c decreases, the
corresponding measures are more sensitive to transfers between incomes at the bottom of
the income distribution. Note that the Kolm measures are the only members of the class
of absolute decomposable measures satisfying transfer sensitivity (Shorrocks and Foster,
1987), which is the requirement that a rich-to-poor transfer combined with a simultaneous
poor-to-rich transfer at a higher income level (such that there is no eﬀect on the variance)
decreases inequality.
References
Acz´ el J., 2006. Lectures on Functional Equations and Their Applications. Mineola: Dover Publications,
Inc.
Chakravarty S.R. and S. Tyagarupananda, 1998. The subgroup decomposable absolute indices of inequality.
In: S.R. Chakravarty, D. Coondoo and R. Mukherjee (eds) Quantitative Economics: Theory and
Practice, Essays in Honor of Professor N. Bhattacharya. New Delhi: Allied Publishers Limited,
247-257.
5Chakravarty S.R. and S. Tyagarupananda, 2009. The subgroup decomposable intermediate indices of
inequality. Forthcoming in Spanish Economic Review.
Kolm S.-C., 1976. Unequal inequalities I. Journal of Economic Theory 12, 416-442.
Polyanin A.D. and A.V. Manzhirov, 1998. Handbook of Integral Equations: Exact Solutions. Moscow:
Faktorial.
Shorrocks A.F., 1984. Inequality decomposition by population subgroups. Econometrica 52, 1369-1385.
Shorrocks A.F. and J.E. Foster, 1987. Transfer sensitive inequality measures. Review of Economic Studies
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6

Handbook of Integral Equations: Exact Solutions.

Inequality decomposition by population subgroups.

Lectures on Functional Equations and Their Applications.

The subgroup decomposable absolute indices of inequality. In:

The subgroup decomposable intermediate indices of inequality. Forthcoming in Spanish Economic Review.

Transfer sensitive inequality measures.

Unequal inequalities I.

The Class of Absolute Decomposable Inequality Measures

http://sticerd.lse.ac.uk/dps/darp/darp99.pdf

The Class of Absolute Decomposable Inequality Measures

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