We establish a general relationship between the standard form of the individualistic social-welfare function and the ?reduced-form? version that is expressed in terms of inequality and mean income. This shows the relationship between the property of monotonicity and the slope of the equity-efficiency trade-off. Particularly simple results are available for a large class of inequality measures that includes the Gini. These results do not require differentiability of the social welfare function.Inequality, social welfare monotonicity

Frank A Cowell

Yoram Amiel

English

Research Papers in Economics

Inequality, Welfare and Monotonicity 
 
by  
 
Yoram Amiel 
Ruppin Institute, Israel 
 
and 
 
Frank Cowell 
STICERD, London School of Economics and Political Science 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The Toyota Centre 
     Suntory  and  Toyota  International  Centres  for 
     Economics  and  Related  Disciplines 
          London School of Economics and Political Science 
Discussion Paper      Houghton Street 
No.DARP/29     London  WC2A  2AE 
February 1997      Tel.: 020-7955 6678 
 
 
This research was partially supported by an ESRC Grant No.R000235725. We are grateful 
to Leonardo Gasparini and Peter Lambert for comments.  
Abstract 
 
We stablish a general relationship between the standard form of the individualistic 
social-welfare function and the "reduced-form" version that is expressed in terms of 
inequality and mean income. This shows the relationship between the property of 
monotonicity and the slope of the equity-efficient trade-off. Particularly simple results 
are available for a large class of inequality measures that includes the Gini. These 
results do not require differentiability of the social-welfare function. 
 
Keywords: Inequality, social welfare, monotonicity. 
JEL Nos.: C13, D63. 
 
 
 
 
 
 
 
 
©  by the authors. 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. 
 
 
 
 
 
 
 
 
Contact address: Contact address: Professor Frank Cowell, STICERD, London School 
of Economics and Political Science, Houghton Street, London WC2A 2AE, UK. Email: 
f.cowell@lse.ac.uk 
 
 
 
 1 Introduction
Inequality indices embody explicitly or implicitly social values on income distrib-
ution.So it is common for those who are concerned with comparing distributional
changes over time or between economies to draw conclusions from comparisons
of estimates of average income and inequality. However the relationship between
intuitive comparisons in terms of say, mean income and the Gini coe±cient, and
formal principles of social welfare is not always made clear. The present paper
focuses upon this relationship by characterising the shape of the equity-e±ciency
trade-o®..
This trade-o® is at the heart of much theoretical and applied research. For
example, it is become standard practice to use the obvious two-way relationship
between certain classes of inequality indices and their associated social-welfare
functions to infer propositions about distributional rankings and the ethical mean-
ing of inequality measures.1 In addition there is an interest in constructing social-
welfare functions for inequality measures with principally pragmatic appeal: for
example, there have been a number of attempts to formalise a social-welfare func-
tion on the basis that the Gini coe±cient is the appropriate index of inequality.2
The precise method of doing this formalisation does not concern us here, except
as illustrations of our approach. We want to examine in a general way what may
1See for example Atkinson (1970), Blackorby and Donaldson (1978, 1980), Kolm (1976a,
1976b).
2See for example Chipman (1974), Katz (1972), Kondor (1975), Lambert (1985), Newbery
(1970), Sheshinski (1972).
1be said in terms of standard welfare criteria if we use well-known practical indices
as basic tools of distributional analysis.
2 The Approach
First, some notation and de¯nitions.
De¯nition 1 The set of income distributions X is the set of all ordered ¯nite-
dimensioned non-negative vectors, excluding the zero vector.
In implementing this de¯nition we adopt the convention that for any x 2 X:
x1 · x2 · x3 · :::. Also, for any x 2 X, denote the number of persons in
the distribution as n(x) and the arithmetic mean of the distribution as ¹(x).
Social welfare and inequality are given, respectively, by functions W : X ! <,
I : X ! <. Economic meaning is given to W and I by a number of assumptions
about their mathematical properties, many of which have come to be accepted
as standard. Foremost among these are
De¯nition 2 The social welfare function is monotonic if, for any x;x0 2 X,
x0 > x implies W(x0) > W(x).3
De¯nition 3 The social welfare function (resp. an inequality measure) satis¯es
the principle of transfers if, for any x;x0 2 X and ± > 0 such that x0
i = xi + ±,
3\ x0 > x " means \x0 ¸ x, x0 6= x".
2x0
j = xj ¡ ±, x0
k = xk;k 6= i;j and xi < x0
i · x0
j < xj W(x0) ¸ W(x) (resp.
I(x0) · I(x)).
The property of monotonicity is of particular interest in view of its relationship
to the Pareto principle - see Amiel and Cowell (1994b) - although it does not
enjoy universal support as a welfare principle - see Amiel and Cowell (1994a),
McClelland and Rohrbaugh (1978).
De¯nition 4 The class W1 of social welfare functions consists of functions W :
X ! < that satisfy the principles of monotonicity and transfers. If, for some
x;x0 2 X, it is the case that W(x) > W(x0) for all W 2 W1 we shall say that
distribution x is W1-preferable to x0.
We shall establish results for a fairly broad class of inequality indices. To
introduce this we need the following two de¯nitions:
De¯nition 5 For any x 2 X, the income share of person i is:
si(x) :=
xi
n(x)¹(x)
(1)
De¯nition 6 An additive distributional-shares inequality index can be written
as
n(x) X
i=1
wiÁ(si(x)) (2)
32.1 The Reduced-form SWF
It is common to ¯nd the social welfare function written in a \reduced form" -
as a function of mean income and inequality.6 To formalise this approach let us
introduce the concept of a distributional pro¯le z := (¹;I), a mean-income and
inequality pair. In this interpretation the admissible set of pro¯les Z µ <2
+ may
be written
Z :=
©
(¹;I) : ¹ ¸ 0;0 · I · ¹ I
ª
(3)
where ¹ I is the least upper bound on I. Then the reduced-form SWF H(W) : Z !
< is given by
W(x) = H
(W) (¹(x);I(x)) (4)
where H(W) is non-decreasing in its ¯rst argument and non-increasing in its
second argument.
Where there is no problem of ambiguity we will drop the superscript on H.
Notice that the adoption of this approach implies that two distributions with
same z-pro¯le will be regarded as equivalent in welfare terms, and that the trade-
o® implicit in the use of H is only well-de¯ned for a particular cardinalisation
of the inequality index I. We shall assume that the issue of the appropriate
cardinalisation has been settled by criteria which are not of concern to the present
study.
6This is the term used by Champernowne and Cowell (1997). See also the term \abbreviated
social-welfare function" used in Lambert (1993), Chapter 5.
53 Di®erentiable Social-welfare Functions
The ADSI class of measures contains a number of well known indices and yields
some readily interpretable results. To see this let us endow I with the elementary
cardinalisation of (2). Substituting from (2) into (4) we have
W(x) = H
0
@¹(x);
n(x) X
i=1
wiÁ(si(x))
1
A (5)
Assume further that W is di®erentiable. Then di®erentiating (5) with respect to
xi we have:
n(x)
@W(x)
@xi
= H¹ + HI
"
wiÁs(si(x))
¹(x)
¡
Pn(x)
j=1 wjsj(x)Ás(sj(x))
¹(x)
#
(6)
where the relevant partial derivatives are denoted by subscripts. From (6) the
following result is immediate
Theorem 1 Given a di®erentiable reduced form social-welfare function H the
underlying individualistic welfare function satis¯es monotonicity if and only if
the following condition on the slope of the welfare-contour is satis¯ed:
¡
H¹
HI
> max
(
wiÁs (si(x)) ¡
Pn(x)
j=1 wjsj(x)Ás(sj(x))
¹(x)
)
(7)
63.1 Applications
The condition (7) can be easily interpreted in the case of the Generalised Entropy
index:
¡
H¹
HI
> max
[n(x)si(x)]®¡1¡1
®¡1 ¡ ®I(x)
n(x)¹(x)
(8)
and the relative mean deviation:
¡
H¹
HI
>
2m(x)+1
n(x) ¡ I(x)
n(x)¹(x)
(9)
where m(x) = #fxi · ¹(x)g. The interpretation of these results will be discussed
below.
3.2 The Gini Index
The Gini index G(x) can also be obtained as a special case of (7) by rewriting.
it as:7
G(x) =
n(x) X
i=1
wisi (10)
where
wi =
2i ¡ n(x) ¡ 1
n(x)
(11)
We will call the extended Gini subclass those inequality measures of the form (10)
but with the restriction (11) replaced by the mild condition that wi be increasing
7See, for example Cowell (1995), Sen (1973).
7with i.8 Notice that for the extended Gini and the true Gini Ás = 1 wherever the
di®erential is de¯ned. Also note that for the true Gini maxwi =
n(x)¡1
n(x) which
for large populations becomes 1; in this limiting case the right-hand side of (7)
becomes:
1 ¡ G(x)
¹(x)
(12)
However in the case of the Gini index the di®erentiability assumption used in
Theorem 1 will be inappropriate. For this reason and for the reason that -
apart from mathematical tractability - di®erentiability has little to recommend
it in social-welfare analysis, we seek a more general characterisation of the main
point.
4 A General Result
To avoid the over-simpli¯cation associated with the di®erentiability assumption
consider the general shape properties of the reduced-form social-welfare function
at any arbitrary point in the set of all pro¯les Z.
For any distributional pro¯le z0 2 Z it is clear that any point in
A(z0) := fz 2 Z : ¹ ¸ ¹0;I · I0; z 6= z0g (13)
8See for example the contributions by Donaldson and Weymark (1980, 1983), Weymark
(1981), , Yitzhaki (1983). However note that Chakravarty's extended Gini does not ¯t within
this category (Chakravarty 1988).
8will be regarded as W1-preferable to z0. However, we can be more precise about
the set of W1-preferable pro¯les. First de¯ne the set of income distributions
consistent with any pro¯le z0 2 Z:
X(z0) := fx : x 2 X;¹(x) = ¹0;I (x) = I0g (14)
and a conditional upper-bound value of inequality, of a given population size
n = n(x):
I
¤(¹;z0) := max
x2X(z0)
I (x + ¶nn[¹ ¡ ¹0]) (15)
where ¶n := (0;0;:::;0;1). These constructs then permit the de¯nition of the
following subset of A(z0):
P (z0) := fz 2 Z : ¹ ¸ ¹0;I · I
¤ (¹;z0); z 6= z0g (16)
Theorem 2 8W 2 W1;8z1 2 P (z0) :
H
(W)(z1) > H
(W)(z0): (17)
Proof. For some z0 2 Z and some z1 := (¹1;I1) 2 P (z0), given that ¹0 < ¹1
by construction, if I0 ¸ I1 then it is trivial to show that (17) holds. So take the
9and by the principle of transfers I¤(¹;z0) = I (x2) ¸ I1 ¸ I (x3) = I0. By
continuity there must be some ® 2 [0;1] such that9
I (®x2 + [1 ¡ ®]x3) = I1 (22)
in which case it is immediate from (20) that (17) holds.
Notice that, by construction, P (z0) is the largest set of pro¯les which are
regarded as W1-preferable to z0, and that it is bounded by the graph of I¤:
the set of points of the form ¹ P(z0) := fz : z = (¹;I¤(¹;z0));¹ ¸ ¹0g provides a
boundary conditional on a particular distributional pro¯le z0. For the ADSI class
we have
I
¤(¹;z0) :=
n¡1 X
i=1
wi
·
Á
µ
xi
n¹
¶¸
+ wn
·
Á
µ
xn + n[¹ ¡ ¹0]
n¹
¶¸
(23)
which, in the case of the subclass of extended Gini measures, reduces to
I
¤(¹;z0) := ¹ I ¡
£¹ I ¡ I0
¤ ¹0
¹
(24)
Notice that through each point (pro¯le) z0 2 Z, there is a frontier ¹ P and a
contour of the reduced-form social welfare function H(W). We may then state:
Theorem 3 For any inequality-averse monotonic SWF and any z0 2 Z, when-
9See the pro¯le z1 in Figure 2. Notice that I must be continuous in view of the convexity
of Á.
11Corollary 4 If W satis¯es monotonicity, the limit of the slope of I¤ (¹;z0) as ¹ #
¹0 must be less than the limit of the slope of the contour of H(W)(z) = H(W)(z0)
as ¹ # ¹0.
Substitution from (23) or from (24) will con¯rm the result for the general
ADSI class and for the special case of the Gini index respectively. In the latter
case we ¯nd that the condition for monotonicity amounts to:
¡
H
(W)
¹
H
(W)
I
>
1 ¡ I
¹
(25)
where the slope is to be interpreted in the manner of Corollary 4. This can be re-
expressed as a requirement on the reduced-form SWF in terms of the associated
equality index 1 ¡ I: it should have the property that the elasticity of equality
with respect to mean income be less than unity. The condition (25) can then
be used to check the properties of reduced-form Gini-SWFs presented in the
literature. Examples of measures satisfying the elasticity condition include n2[1¡
I]¹ (Sheshinski 1972)11, log¹¡I (Katz 1972) and
¹
1+I (Kakwani 1986). However
1¡I
1+I¹ (Chipman 1974, Dagum 1990) does not satisfy the condition and it is easy
to see that this index violates monotonicity: Table 1 shows an example where
distribution B has a higher mean income and higher inequality than A. B also
¯rst-order dominates A. However the Chipman-Dagum social welfare function
11A generalisation of this form is to be found in Lambert (1985). See also Ben Porath and
Gilboa (1994)
13A B
x1 4500 4600
x2 4000 4000
x3 3500 3500
¹(x) 4000 4033
I(x) 0.056 0.061
W(x) = 1¡I
1+I¹ 3579 3572
Table 1: The Gini coe±cient I and the Chipman-Dagum Welfare Index W
indicates that society is worse-o® in B than in A.
Similar interpretations can be provided in terms of other members of the ADSI
class (2).
5 Conclusions
The use of ad hoc inequality measures begs the question of whether the empirical
judgements made using them will be in accordance with conventional welfare
properties. The reduced-form SWF is the appropriate tool to clarify this issue.
The transfer principle is inherited automatically by the SWF from an inequality
measure; now we have shown that the condition on the reduced-form SWF that
ensures consistency with the monotonicity criterion is surprisingly simple. In the
case of the Gini index this condition is particularly appealing, and does not rely
on arbitrary assumptions of di®erentiability of the SWF.
Finally, a word of caution. Cardinalisations of inequality measures are some-
times viewed as irrelevant, but in the analysis of the so-called \equity-e±ciency
14trade-o®" this is not quite so. As the discussion of the elasticity of the H(W)
-contours makes clear, the results we have obtained are speci¯cally for the \natu-
ral" cardinalisation of the index (2). A modi¯ed version of Theorem 1 and related
results will be applicable under continuous monotonic transformations of I.
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19

Inequality, Welfare and Monotonicity

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Inequality, Welfare and Monotonicity

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