INEQUALITY OF WELL-BEING:
A MULTIDIMENSIONAL APPROACH
KOEN DECANCQ AND MARÍA ANA LUGO
Abstract. An important aspect of the multidimensional well-being distribution of a society is the correlation between the different dimensions of well-being.
We propose two indices for measuring inequality of multidimensional well-being,
derived from two underlying social evaluation functions. These functions aggregate both across the dimensions of well-being and across individuals. The two
social evaluation functions differ only with respect to the sequencing of aggregations. We believe that aggregating first across dimensions is more attractive
since it allows the inequality index to depend on the correlation between the dimensions. We illustrate both indices using Russian household data between 1995
and 2005 for four dimensions of well-being: expenditure, health, schooling and
housing quality. This illustrates the empirical impact of correlation-sensitivity
on the obtained results.
Keywords: correlation sensitivity, multidimensional inequality, Russia, single
parameter Gini index, two-step aggregation function.
JEL Classification: D63, I31, O52.
Koen Decancq: Center for Economic Studies, Katholieke Universiteit Leuven, Naamsestraat
69, B-3000 Leuven, Belgium; CORE, Université Catholique de Louvain, B-1348 Louvain-la-Neuve,
Belgium; Herman Deleeck Centre for Social Policy, University of Antwerp, B-2000 Antwerp, Belgium E-mail: [email protected].
Marı́a Ana Lugo: Department of Economics, University of Oxford, Manor Road Building,
Manor Road, OX1 3UQ, Oxford, UK and Office of the Chief Economist, Latin America and
the Caribbean Region, The World Bank, 1818 H ST. NW, Washington DC, 20433, USA. E-mail:
[email protected].
1
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
2
Introduction
In their report on the measurement of economic performance and social progress,
Stiglitz et al. (2009, p.14) argue that individual well-being is a multidimensional
notion.1 Individuals care about many different aspects of their lives, including
their material standard of living, health and schooling. These non-monetary dimensions are neither freely tradable nor perfectly correlated with income. If we
want to take the multidimensionality of individual well-being seriously, it follows,
we need to incorporate the various dimensions explicitly into the analysis of inequality. Yet, most analyses of inequality have confined themselves to the analysis
of a sole dimension, that is, income (or expenditure).
A straightforward approach to take the multidimensionality of well-being into
account is to consider each dimension separately, that is to study the evolution
of inequality dimension by dimension (examples of this approach are by Atkinson
et al., 2002; World Bank, 2005; Fahey et al., 2005)2 This method clearly goes
beyond a sole focus on incomes and may provide additional insights. Unfortunately,
a dimension-by-dimension approach leads us to ignore the interrelationships and
possible correlations between the dimensions of well-being. A society where one
individual is top-ranked in all dimensions, another second-ranked, and so on, is
arguably more unequal than a society with the same distributional profiles in each
dimension but where some individuals are top-ranked in some dimensions, and
other individuals in other dimensions. To neglect these interrelationships would
be to abandon one of the primary motivations for a multidimensional approach to
inequality.3
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
3
The concerns about multidimensionality of well-being and the correlation between its dimensions are not merely academic. In this paper we use Russian household data from the RLMS-HSE to show that these issues lie at the heart of the
evolution of inequality during the recent transition from a centrally planned to a
free market economy. Figures 1 and 2 summarize two stylized facts about that
transition in the period between 1995 and 2005. The first figure presents the
evolution of inequality in four dimensions of well-being: expenditures, health, education and housing quality (for every dimension, inequality is measured by the
Gini coefficient and normalized such that 1995=100). As can be seen, inequality
in each dimension behaved quite differently throughout the period. Expenditure
inequality peaked during the financial crisis of 1998, returning thereafter quickly
to its pre-crisis level. Over the same period, health inequality increased slightly,
educational inequality decreased considerably and housing inequality remained relatively stable. In other words, the evolution of inequality in the four dimensions
differs substantially throughout the period. For the second stylized fact, figure 2
presents the correlation structure between the same four dimensions of well-being.
The correlation structure over the considered period fluctuated considerably (Decancq, 2009). Most notably, there is a sharp drop in (rank) correlation between
expenditures and housing in the crisis year 1998. This is consistent with the findings that the crisis hit the hardest Russians living in urban areas who have, on
average, better housing quality (Gorodnichenko et al., 2010). Other pairwise correlations also fluctuated throughout the period, though they are generally much
more stable and in line with the findings of the literature.4 In such a turbulent
context, by focusing on a sole dimension or analyzing each dimension separately,
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
4
Figure 1: Percentage change in Gini coefficient of household expenditure, health,
education and housing quality in Russia between 1995-2005.
[Figure 1 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
Figure 2: Pairwise rank-correlation coefficients between household expenditure,
health, education and housing quality in Russia between 1995-2005.
[Figure 2 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
one risks overlooking important aspects of the changes in the multidimensional
distribution of well-being.
The goal of this paper is to find a multidimensional generalization of the popular Gini coefficient that combines the information on inequality of the different
dimensions and incorporates that on the correlation between them. The Gini coefficient is particularly interesting for several reasons. First, the one-dimensional
Gini coefficient is probably the best known and most used inequality index in
economics and yet, no convincing extension to multiple dimensions has been proposed.5 Second, the Gini coefficient is attractive in part due to its interpretation
in terms of the underlying rank-dependent social evaluation function. Such a function attaches welfare-weights to individuals that depend on their position in the
total distribution. This is in line with the recent evidence that people care not
only about their levels of attainment but also about their relative position (see,
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
5
for instance, Ferrer-i-Carbonell 2005, Luttmer 2005). Finally, the class of rankdependent social evaluation functions is a large one, which can encompass many
of the non rank-dependent ones and hence can serve as a starting point for a wide
range of measures (Ebert, 1988).
In the present paper, we use a normative approach to derive the multidimensional Gini coefficient from its underlying rank-dependent social evaluation
function.6 The multidimensional social evaluation function used is derived in two
steps combining two distinct aggregations in an explicit way: one aggregation
across dimensions and one across individuals. Two-step social evaluation functions received some recent attention in the literature after the work of Dutta et al.
(2003), and have their pedigree in the early work of Kolm (1977).7 The aggregation across dimensions of well-being leads to an index of well-being and depends on
the preferences of the society (or ethical observer) over the relative importance of
the different dimensions.8 An essential and novel feature of the aggregation across
individuals in this paper is that we allow for rank-dependence.
The sequencing of both aggregations leads to two alternative procedures. In
the first procedure, one first aggregates across all individuals in each dimension,
resulting in a vector of dimension-specific summary statistics, which are then aggregated in the second step. The second procedure is its mirror-image and aggregates
first across the dimensions to come to a well-being index for each individual, which
are in a second step aggregated over all individuals. An example of the first procedure is the multidimensional Gini index by Gajdos and Weymark (2005). We will
argue that the second procedure is more attractive, since it does not exclude the
resulting measure to comply with a multidimensional concern about the correlation
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
6
between the dimensions. Besides the normative approach, two other broad strategies have been followed to generalize the Gini coefficient into multiple dimensions,
each extending an alternative one-dimensional definition of the Gini coefficient.9
Koshevoy and Mosler (1996) introduced the Lorenz zonoid as an m-dimensional
generalization of the standard Lorenz curve. From the volume of the Lorenz zonoid,
a multidimensional Gini coefficient can be derived naturally (Koshevoy and Mosler,
1997). An alternative strategy is followed by Arnold (1987), Koshevoy and Mosler
(1997) and Anderson (2004), who extend the definition based on the sum of all
distances between pairs of individuals. In particular, they propose a multidimensional distance measure to measure the pair-wise distances between the vectors of
outcomes. Both approaches represent a mathematical or geometrical extension of
the one-dimensional Gini coefficient. As such, they lack normative content in the
sense that inequality cannot be readily interpreted in terms of welfare losses. In
addition, the properties satisfied by these indices are not made explicit and are
sometimes hard to unravel. We thus regard them as less attractive to measure
well-being inequality.
The rest of the paper is structured as follows. Section I surveys some attractive properties for the aggregation across individuals and dimensions. Depending
on the sequencing of the aggregation two multidimensional Gini social evaluation
functions are derived. Multidimensional distributional concerns are introduced in
section II, paying special attention to a multidimensional generalization of the onedimensional Pigou-Dalton transfer principle and the effect of changes in correlation
between dimensions. Both multidimensional Gini social evaluation functions are
compared with respect to their compliance to these distributional concerns. From
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
7
the resulting social evaluation functions two multidimensional single parameter
Gini inequality indices are derived in section III. Section IV illustrates the use of
these indices based on Russian household data. Section V concludes the paper.
I. Two multidimensional S-Gini social evaluation functions
We assume that there are m relevant dimensions of well-being (e.g. expenditures, health, schooling, housing, . . . ) for a population of n individuals.10 Each
distribution matrix X in Rn×m
++ represents a particular distribution of the outcomes
for the n individuals in the m dimensions. An element of the distribution matrix
xij denotes the outcome of individual i in dimension j. A row of matrix X refers to
the outcomes of one individual and a column refers to the outcomes in one dimension. Distribution matrices can be compared by making use of a social evaluation
function Wn×m that maps a positive n × m distribution matrix to the positive
real line. As mentioned above, the social evaluation function carries out a double aggregation. The aggregation over the m dimensions (columns) of well-being
will be performed by aggregation function Wm . It can be interpreted as an index
of multidimensional well-being. The other aggregation, carried out by function
Wn , is across the n individuals (rows) and can be interpreted as a standard onedimensional social evaluation function. The sequencing of both aggregations will
turn out to play a crucial role in the following.
Let us describe two procedures. In the first procedure, we first aggregate
across the different individuals (arranged in the columns of the matrix) by making
use of Wn in each dimension. This step obtains for each dimension a summary
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
8
statistic and generates a single m-dimensional vector. Then, this vector is aggregated using Wm , across dimensions. Kolm (1977) calls this first procedure a
specific one, Pattanaik et al. (2008) refers to it as the column-first two step aggregation procedure. In the second procedure, the order of aggregation is reversed:
the first step attaches to each individual a level of well-being and generates an ndimensional vector; in the second step this vector is aggregated -across individualsusing Wn . Following Kolm (1977) this second procedure will be referred to as an
individualistic one, or a row-first aggregation procedure according to Pattanaik
et al. (2008). The following diagram summarizes the social evaluation functions
obtained by both procedures:
1
1
Wn×m
: Rn×m
++ → R++ : X 7−→ Wn×m (X) = Wm [Wn (x1 ), . . . , Wn (xm )] ,
2
2
1
n
Wn×m
: Rn×m
++ → R++ : X 7−→ Wn×m (X) = Wn Wm (x ), . . . , Wm (x ) .
Both procedures split the complex multidimensional aggregation into two (easier)
one-dimensional aggregations, which have been studied extensively in the literature
before, see Ebert (1988) for an overview. We present and discuss a list of interesting
properties for a generic one-dimensional aggregation function Wk : Rk++ → R++
that maps a positive vector x = (x1 , . . . , xk ) on the positive real line. The aggregation functions Wm and Wn are examples of such aggregation functions. The
properties crystallize different value judgements on how the aggregation should
be done. We do not claim that the set of properties laid down here is the only
one possible, au-contraire, but we suggest that it represents an attractive set for
the problem of measuring societal well-being. We restrict ourselves to continuous
aggregation functions, so that the result is not overly sensitive to small changes
in one of its entries, for instance caused by measurement errors. Let the vector
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
9
inequality y > x denote that yl ≥ xl holds for all its entries l = 1, . . . , k and yl > xl
for at least one entry l and let 1k denote a k-dimensional vector with all entries
equal to 1.
Property 1. Monotonicity (MON) For all x, y in Rk++ : if y > x, then
Wk (y) > Wk (x).
Property 2. Symmetry (SYM) For all x in Rk++ , and all k × k permutation
matrices P : Wk (P x) = Wk (x).
Property 3. Normalization (NORM) For all λ > 0 : Wk (λ1k ) = λ.
Monotonicity captures the intuition that all entries of x are desirable. If a
vector is obtained by increasing at least one entry of another vector, it should be
preferred to the initial one. Monotonicity is an attractive property for aggregation
functions across dimensions as well as across individuals. Symmetry states that
any information other than the quantities stated in the entries of x are unimportant in the aggregation. Symmetry is an attractive property in the aggregation
across individuals since it assures an impartial treatment of all individuals. In the
aggregation across dimensions, however, one might want to treat the dimensions
differently to give priority to certain dimensions. Therefore we will not impose
symmetry in the aggregation across dimensions. Normalization makes sure that
whenever all entries of x are equal to λ, the result should be λ as well.
Let K be the set of all k entries and let L be a subset of K. The next property
we introduce is separability of the subset L.
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
10
Property 4. Separability (SEP) For all x, x0 , y, y 0 in Rk++ : if there is an L ⊂ K
such that for all l in L xl = yl and x0l = yl0 whereas for all k in K\L, xk = x0k and
yk = yk0 , then Wk (x) > Wk (y) ⇔ Wk (x0 ) > Wk (y 0 ).
Separability is a practical property. It imposes that, in the comparison of two
vectors, the magnitude of the ‘unconcerned’ entries in L should not matter. An
example helps to clarify. Let Wm aggregate across three dimensions of well-being
-income, health and schooling-. Suppose there are two individuals who have the
same outcome in the income dimension and different outcome levels in health and
schooling. Separability asserts that the exact level of income is not important to
order the individuals with respect to their well-being. Separability implies, for
instance, that the marginal rate of substitution between health and schooling is
independent of the income level of the individuals. It is a commonly made, but
arguably strong, property to aggregate across dimensions of well-being (see Deaton
and Muellbauer, 1980, for a discussion). We will impose this property when aggregating across dimensions. However, we will prefer a weaker version for the
aggregation across individuals. This is because separability excludes all considerations about the position or rank of the entries in the total vector. Yet, recent work
on self-reported well-being and happiness has documented that individuals do not
only care about the levels of their outcomes, but also about the relative position
vis-à-vis other individuals in the distribution (Ferrer-i-Carbonell, 2005; Luttmer,
2005).
To allow considerations about the positions to play a role in the aggregation
across individuals, we use a weakening of the separability property, which states
that the comparison of two vectors is not affected by the magnitude of common
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
11
entries in both vectors as long as the initial ranking is maintained. This property
allows us to take both the level and position in the distribution into account.
e k . The first entry x1 of
Let us define therefore the set of ordered vectors R
++
an ordered vector has the highest level, the second entry the second highest; and
so forth.
Property 5. Rank-dependent Separability (RSEP) For all x, x0 , y, y 0 in
e k : if there is an L ⊂ K such that for all l in L xl = yl and x0 = y 0 whereas for
R
++
l
l
all k in K\L, xk = x0k and yk = yk0 , then Wk (x) > Wk (y) ⇔ Wk (x0 ) > Wk (y 0 ).
Next, we impose three invariance properties. These properties specify which
transformations or standardization procedures of the data will leave the ordering
of two vectors x and y unaffected. In any multidimensional analysis the transformation and standardization of the data is an essential step to make the potentially
very different dimensions comparable.11
Property 6. Weak ratio-scale invariance (WSI) For all x, y in Rk++ and all
positive λ : Wk (x) > Wk (y) if and only if Wk (λx) > Wk (λy).
Property 7. Strong ratio-scale invariance (SSI) For all x, y in Rk++ and all
positive diagonal matrices Λ : Wk (x) > Wk (y) if and only if Wk (Λx) > Wk (Λy).
Property 8. Weak translation invariance (WTI) For all x, y in Rk++ and
all κ : Wk (x) > Wk (y) if and only if Wk (x + κ1k ) > Wk (y + κ1k ).
Property 6, weak ratio-scale invariance, states that a rescaling of all entries
of the two vectors x and y with the same positive number does not affect their
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
12
ordering. Doubling all outcomes, for instance, should not lead to a reordering of
both vectors. This property is standard in the aggregation across individuals and
ensures that, for instance, Wn is unaffected by a general inflation. In the aggregation
across dimensions, it is an appealing property when the units of measurement of the
entries are the same, for example, when the entries are different sources of income
or incomes at different points in time. We will, at first, impose this property to
the aggregation functions both across dimensions Wm and across individuals Wn .
Given that in the empirical application we will deal with dimensions that have
distinct units of measurements, property 6 becomes less appealing. We will thus
impose a much stronger version of this property. Property 7, strong ratio-scale
invariance, requires that a rescaling of all entries of x and y should not lead to a
reordering. This rescaling factor is allowed to differ across the entries of vectors,
which makes it particularly attractive in the aggregation across dimensions when
the variables are expressed in very different units of measurement, such as income
in dollars and schooling in years. The property allows the entries of both vectors
to be standardized by an entry-specific rescaling such as a division by their respective mean, that is, individual income divided by the mean income, and individual
schooling by the mean schooling level of the distribution.12 Unfortunately, this
property will turn out to be fairly restrictive in terms of the functions satisfying it.
Property 8, weak translation invariance, imposes that the ordering of two
vectors by Wk is not affected if a common amount is added to all entries. We will
impose weak translation invariance together with weak ratio-scale invariance to
the aggregation across individuals to come to a parsimonious aggregation function
which is invariant to common linear transformations of the entries.
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
13
The final two properties will allow us to compare k-dimensional vectors with
a variable size k. Since we assume the number of relevant dimensions m to be fixed
throughout the analysis, these properties will only be imposed on the aggregation
function across individuals Wn . These properties permit the comparisons of distributions of different population sizes. We say that z is a replication of x if z is
obtained by cloning x l-times, so that z = (x, x, . . . , x)(l-times).
Property 9. Replication invariance (REP) For all x in Rk++ and all z in
Rlk
++ which is a replication of x : Wlk (z) = Wk (x).
ek :
Property 10. Restricted aggregation (RA) For all x in R
++
Wk (x) = Wk (Wl (x1 , . . . , xl ) , . . . , Wl (x1 , . . . , xl ) , xl+1 , . . . , xk ).
If we impose replication invariance to Wn , the aggregation takes place on a
per-capita basis. Replication invariance has been introduced in the literature on
one-dimensional inequality measurement by Dalton (1920). Restricted aggregation
asserts that the aggregation of the total vector is equivalent to an aggregation
in which the outcomes of the better-off subgroup containing l entries are first
aggregated into one aggregate. This property has been studied by Donaldson and
Weymark (1980) and is appealing because it implies a social evaluation function
that depends on a single parameter representing the distributional judgments.
The result below summarizes the properties imposed to the aggregation function across dimensions Wm and derives the sole class of functions satisfying them
all.
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
14
Proposition 1. A continuous aggregation function Wm : Rm
++ → R++ satisfies
(a) MON, NORM, SEP and WSI, if and only if for each x in Rm
++ we have
(1)
Wm (x) =
m
X
!(1/β)
wj (xj )β
,
j=1
where wj > 0 for all j and
Pm
j=1
wj = 1,
(b) MON, NORM and SSI, if and only if for each x in Rm
++ we have
(2)
Wm (x) =
m
Y
xj wj ,
j=1
where wj > 0 for all j and
Pm
j=1
wj = 1.
Proof. See for (a) Blackorby and Donaldson (1982, theorem 2) and for (b) Tsui
and Weymark (1997, theorem 4)
Result (a) defines the aggregation across dimensions to be a Constant Elasticity of Substitution (CES) function, where parameter β reflects the degree of
substitutability between the dimensions of well-being. In particular, β is related
to the elasticity of substitution between the dimensions σ and equals 1 − 1/σ.
When β = 1, the dimensions of well-being are seen as perfect substitutes. As β
tends to −∞, the dimensions tend to perfect complementarity; at the extreme,
individuals are judged by their worst outcomes.13 Result (b), itself a limiting case
of the previous one for β = 0, is in some respects disappointing and reveals how
restrictive the requirement of strong ratio-scale invariance can be. In the presence
of the other properties, strong ratio-scale invariance is only satisfied by a so-called
Cobb-Douglas well-being function which has unit elasticity of substitution, that
is, σ = 1. Other elasticities between the dimensions cannot be obtained without
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
15
relaxing this property. One has to make a painful trade-off here: it is impossible
to obtain results that are robust to alternative rescaling procedures of the different
dimensions if one desires more flexibility in the functional form. In other words,
the functional flexibility of Wm can only be obtained when one can rely on a theoretically sensible and ethically justifiable standardization procedure of the original
dimensions.
The weighting scheme w = (w1 , . . . , wm ) consists of the weights wj , which are
all positive and sum to one, and reflects the relative importance of the different
dimensions. In interplay with parameter β and the standardization chosen, the
weights determine the marginal rates of substitution or trade-offs between the
dimensions (Decancq and Lugo, forthcoming).
The well-being functions characterized by proposition 1 are popular measures
of well-being in the literature. The Human Development Index advocated by the
UNDP, for instance, is a special case of expression (1) with β = 1, and weights wj
equal to 1/3.14 Other examples may be found in the literature on multidimensional
inequality measurement: Maasoumi (1986), for instance, derives a CES well-being
function based on different considerations rooted in information theory.
An analogous result can be obtained for the properties imposed on Wn , the
aggregation across individuals.
Proposition 2. A continuous aggregation function Wn : Rn++ → R++ satisfies
MON, SYM, NORM, RSEP, WSI, WTI, REP and RA if and only if for each x
in Rn++ we have
(3)
"
#
n
r δ r − 1 δ
X
i
i
Wn (x) =
−
xi ,
n
n
i=1
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
16
where δ > 0 and ri is a shorthand for ri (x), that is the rank of individual i on the
basis of the levels of vector x, ordered from the richest to the poorest person.
Proof. See Ebert (1988, proposition 10).
The class of aggregation functions obtained in proposition 2 is a convenient
one, since it consists of a weighted average where the welfare weights, and the
associated value judgements, are captured by a single parameter δ, which represents
the bottom sensitivity of the aggregation function.15 If δ equals 1, the aggregation
function becomes an unweighted average in the utilitarian tradition. The higher δ,
the more weight is given to the bottom of the distribution, with as limiting case
δ = +∞, which leads to a Rawlsian aggregation function where only the worse-off
individual is counted for the social evaluation.16 For values of δ between 0 and 1
the best-off individuals are given more weight. The standard Gini social evaluation
function is obtained by setting δ = 2.
By substituting the obtained one-dimensional aggregation functions (1) and
(3) into the two initial two-step procedures we obtain two multidimensional social
evaluation functions. The specific (column-first) social welfare function is
(4)

m
X
1

Wn×m (X) =
wj
j=1
where δ > 0, all weights wj > 0,

" δ δ # !β (1/β)
n
i
i
X
rj
rj − 1
−
xij 
,
n
n
i=1
Pm
j=1
wj = 1, and rji is a shorthand for rji (xj ) , that
is the rank of individual i in dimension j. Similarly,the individualistic (row-first)
social welfare function is
(5)
" δ !(1/β)
δ # X
n
m
i
i
X
r
r
−
1
2
Wn×m
(X) =
−
wj (xij )β
,
n
n
i=1
j=1
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
where δ > 0, all weights wj > 0,
Pm
j=1
17
wj = 1, and ri is a shorthand for ri (s) ,
the rank of individual i on the basis of s, the vector of well-being levels, with
(1/β)
P
m
1
i β
is a special case of the
)
. The first alternative, Wn×m
si =
w
(x
j
j=1 j
multidimensional generalized Gini social evaluation functions proposed by Gajdos
2
and Weymark (2005). The second social evaluation function, Wn×m
has, to the best
of our knowledge, not yet been introduced in the literature. In the following section
we will compare both aggregation procedures with respect to their sensitivity to
two specific multidimensional distributional concerns and investigate the empirical
differences based on a Russian data set.
At this point two concerns might be brought to attention. The first is that
both social evaluation functions impose the same degree of substitution between all
pairs of dimensions. However, one could reasonably argue that this should not be
the case. A possible solution is to adopt a nested approach which first aggregates
subsets of dimensions using expression (1) – each with a different parameter β –
and then aggregates these subsets using again the same expression. The second
concern is that the social evaluation functions derived make use of the same level
of bottom sensitivity for all dimensions. Yet, as Anand (2002, p.485) argues,
given that health has both intrinsic and instrumental value “we should be more
averse to, or less tolerant of, inequalities in health than inequalities in income”.
A solution, albeit an imperfect one, is to transform the original dimensions before
the aggregation is done (for instance by a logarithmic transformation of the health
variable).
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
18
II. Multidimensional distributional concerns
The reader will note that so far we have not introduced any property that
captures distributional concerns. In the standard one-dimensional analyses, distributional sensitivity is obtained by imposing some form of the Pigou-Dalton transfer
principle. The principle states that a transfer of income from a poorer to a richer
individual leads to a social inferior situation. Some proposals have been made
to generalize the one-dimensional Pigou-Dalton principle to the multidimensional
setting. In this section we focus on two popular distributional concerns crystallizing the recommendations of Stiglitz et al. (2009) about multidimensionality
and correlation-sensitivity. We investigate their effect within the multidimensional
two-step rank dependent framework of the previous section. The two distributional
concerns affect both aggregation functions Wn and Wm at the same time and are
therefore defined as properties of the multidimensional two-step social evaluation
function Wn×m . In view of the empirical analysis in the following section, we are
especially interested in the parameter-restrictions imposed by the distributional
concerns on the parameters β and δ in expressions (4) and (5) .
The first distributional concern asserts that if a uniform mean-preserving averaging is carried out in all dimensions, the resulting distribution matrix is socially
preferred to the original one. In the literature, the concern is referred to as uniform majorization principle (Kolm, 1977; Marshall and Olkin, 1979; Tsui, 1995;
Weymark, 2006) and its formalization is rooted in multidimensional majorization
theory. A uniform mean-preserving averaging can be obtained by applying the
same bistochastic transform to all dimensions.17
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
19
Property 11. Uniform Majorization Principle (UM) For all distribution
matrices X and Y in Rn×m
: if Y = BX for some n × n bistochastic matrix
++
B, Y 6= X and Y is not a permutation of X, then Wn×m (Y ) > Wn×m (X).
An example can clarify the uniform majorization principle. Consider the
following matrices that summarize the outcomes of three individuals (rows) for
two dimensions of well-being (columns),






 0.75 0.25 0 
 50 80 
 60 65 












B =  0.25 0.75 0  ; X =  90 20  and Y =  80 35  ,












0
0 1
10 50
10 50
where indeed Y = BX so that Y is obtained from X by a bistochastic transform.
The uniform majorization principle imposes that distribution matrix Y should be
socially preferred to X. Note that the uniform majorization principle is indeed a
rather weak distributional concern, since it imposes only that a mean-preserving
averaging performed uniformly in all dimensions leads to an increase in social
welfare.
Atkinson and Bourguignon (1982) identify another multidimensional distributional concern. They argue that a multidimensional social evaluation should be
sensitive to the correlation between the dimensions of well-being. We use again
a fairly weak version of correlation sensitivity, by requiring that the initial distribution matrix is preferred to one in which the distributional profiles in all dimension are unaltered, but where the dimensions are perfectly (rank)correlated.
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
20
This version of correlation sensitivity has been introduced in the literature on multidimensional inequality by Dardanoni (1996) and will be referred as the unfair
rearrangement principle.
Property 12. Unfair Rearrangement Principle (UR) For all distribution
matrices X and Z in Rn×m
++ : if Z is obtained from X by the sequence of dimensionwize permutations which make one individual in Z top-ranked in all dimensions,
another individual second ranked in all dimensions and so forth and Z 6= X, then
Wn×m (X) > Wn×m (Z).
Again, an example can clarify this principle,




 10 20 
 50 80 








X =  90 20  and Z =  90 80  .








50 50
10 50
In distribution matrix Z the first individual is bottom-ranked in all dimensions, the second individual is top-ranked in all dimensions and the third one is
middle-ranked, so that the dimensions are perfectly (rank)correlated. Since the
property only implies social preference of the initial distribution matrix to the
distribution matrix with perfectly (rank)correlated dimensions, this property in
indeed fairly weak.18 However, in terms of implied parameter-restrictions the property will turn out to be remarkably strong.
We investigate the impact of introducing both distributional concerns to the
two multidimensional S-Gini social evaluation functions derived in the previous
section summarized in expression (4) and (5). We start by the social evaluation
1
function resulting from the column-first (specific) procedure Wn×m
.
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
21
n×m
1
: R++
→ R++ ,
Proposition 3. A continuous double aggregation function Wn×m
where Wn satisfies MON, SYM, NORM, RSEP, WSI, WTI, REP and RA; and
Wm satisfies MON, NORM, SEP and WSI,
1
i) satisfies UM if and only if Wn×m
satisfies equation (4) with δ > 1,
ii) cannot satisfy UR.
Proof. See appendix.
The intuition for the first part of this result is the following: the bistochastic
transform leads to more equally distributed dimensions, so that any aggregation
across individuals with a preference for equality (that is, δ > 1) leads to larger
summary statistic than the one corresponding to the initial distribution. This is
because one is giving a larger weight to the lower part of the distribution which,
after the transfer, has a larger mass. Monotonicity of the aggregation across dimensions ensures that the distribution matrix after the mean preserving averaging is
preferred. The impossibility result in the second part of the proposition, involving
the unfair rearrangement principle, crystallizes the intuition that an aggregation
of a dimension-by-dimension approach cannot be sensitive to the interrelations between the dimensions. Similar impossibility results have been obtained by Gajdos
and Weymark (2005, proposition 10) and Pattanaik et al. (2008).
2
Imposing both distributional concerns on the social evaluation function Wn×m
obtained by the row-first (individualistic) procedure, leads to the following restrictions on the parameter-space.
n×m
2
Proposition 4. A continuous double aggregation function Wn×m
: R++
→ R++ ,
where Wm satisfies MON, NORM, SEP and WSI; and Wn satisfies MON, SYM,
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
22
NORM, RSEP, WSI, WTI, REP and RA,
2
i) satisfies UM if and only if Wn×m
satisfies equation (5) with β < 1 and δ > 1,
2
satisfies equation (5) with δ > δ 0 , where δ 0 is
ii) satisfies UR if and only if Wn×m
a threshold depending on X, w and β.
Proof. See appendix.
For both distributional concerns to be satisfied, the degree of substitutability
β should be smaller than one and the bottom-sensitivity of the aggregation across
individuals Wn should be ‘large enough’, that is, larger than a lower-bound δ 0 ,
which depends on the initial matrix X, the weighting scheme w and the degree of
substitutability β.
The difference between both aggregation procedures concerning their compliance with both distributional concerns is essential. Concerning uniform majorization, the column-first procedure imposes no restrictions on the degree of substitutability in the aggregation across dimensions, whereas the row-first procedure
does. More importantly, aggregating according to the specific column-first procedure excludes a-priori any compliance with correlation sensitivity, even when formulated in the very weak form of the unfair rearrangement principle. Instead, the
aggregation according to the row-first individualistic procedure allows the unfair
rearrangement principle to be satisfied for a specific subset of the parameter-space
above lower-bound δ 0 . Unfortunately, this subset depends on X, the distribution
matrix at hand, which is rather inconvenient from an applied perspective. In the
empirical section, we will compute the minimal bottom sensitivity δ 0 such that for
a given distribution matrix X, for a given weighting scheme w and a degree of substitutability β, an unfair rearrangement leads to a decrease in societal well-being.
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
23
III. Two multidimensional S-Gini inequality indices
In the one-dimensional normative approach, a relative inequality index is derived from its underlying social evaluation function as the fraction of total welfare
wasted due to inequality (Atkinson, 1970; Kolm, 1969; Sen, 1973). In a seminal
article, Kolm (1977) generalizes the one-dimensional definition to the multidimensional setting by defining multidimensional inequality to be the fraction of the
aggregate amount of each dimension of a given distribution matrix that could be
destroyed if every dimension of the matrix is equalized while keeping the resulting
matrix socially indifferent to the original matrix (see also Weymark 2006). Formally, a multidimensional relative inequality index I(X) is defined as the scalar
that solves
(6)
Wn×m ((1 − I(X)) Xµ ) = Wn×m (X) ,
where Xµ is the equalized distribution matrix defined such that all the elements in
the j−th column of the matrix are the dimension-wise mean µ(xj ). By substituting
1
Wn×m
obtained in (4) in expression (6), the following (specific) multidimensional
S-Gini inequality index I 1 can be obtained.
"
#(1/β)
Pm
Pn rji δ rji −1 δ i β
− n
xj
j=1 wj
i=1
n
(7)
I 1 (X) = 1 −
P
m
j=1
wj µ(xj
)β
1/β
,
which belongs to the class of multidimensional generalized Gini indices proposed by
Gajdos and Weymark (2005). This index is derived from the column-first approach
to multidimensional welfare presented in the previous section.
2
Based on the row-first social evaluation function Wn×m
summarized in ex-
pression (5) and the definition of a relative inequality index in expression (6),
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
24
the individualistic multidimensional S-Gini inequality index can be expressed as
follows:
Pn
i=1
(8)
2
I (X) = 1 −
δ
i
r
n
−
P
ri −1
n
m
j=1
δ P
wj µ(xj )β
m
j=1
1/β
wj (xij )β
1/β
.
Both multidimensional inequality indices are generalizations of the S-Gini
inequality index and lead to the one-dimensional S-Gini index when m equal to one
(Donaldson and Weymark, 1980; Kakwani, 1980; Yitzhaki, 1983).19 By inspection
of expressions (7) and (8) it is clear that their difference arises from two elements.
First, the sequence of the summations across individuals and dimensions differs and
second, the weights attached to each individual are different. The inequality index
I 1 uses individual weights depending on rji , that is their rank in the distribution of
each dimension j. In contrast, in the inequality index I 2 , the individuals’ weight
depends on ri , their rank in the overall distribution of well-being. Both elements
may lead to very different empirical results, when I 1 and I 2 are computed for the
same distribution matrix X. A priori it is even hard to predict which of both
indices will show more inequality.20 The comparison between both indices depends
on the inequality within each dimension and the correlation structure between the
dimensions as will be illustrated in the next section based on a Russian data set.
IV. Empirical illustration: Russian well-being inequality between
1995 and 2005
In this section we illustrate that the selection of the appropriate properties for
the multidimensional generalization of the Gini coefficient is not only important
from a theoretical perspective, but that, indeed, alternative choices may lead to
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
25
empirically different answers. We will analyze the evolution of well-being inequality
in Russia between 1995 and 2005. In this period, the Russian Federation underwent
a fundamental transition to a market economy and, in August 1998, it was hit by
a severe financial crisis. As can be expected, these dramatic changes have affected
the distribution of household expenditures among Russian citizens. In addition
to household expenditure data, we include in the analysis information on health,
education and housing quality to obtain a more complete understanding of the
changes observed throughout the period. As shown in figures 1 and 2 in the
introduction, the changes in the economy have affected the distributions of these
four dimensions of well-being in different ways and, furthermore, they have led to
a change in the correlation structure between them.
Data used in this section come from the Russian Longitudinal Monitoring
Survey (RLMS-HSE), a series of nearly annual, nationally representative surveys
designed to monitor the effects of Russian reforms on health and economic welfare of households and individuals.21 We use four indicators for the respective
dimensions of well-being: equivalent real household expenditures, a constructed
composite health indicator, years of schooling attained and a constructed composite indicator of housing quality. Equivalent real household expenditures are widely
used in the literature as an indicator of material standard of living.22 We adjust
household expenditure by household composition using the square root of household size as equivalence scale. Secondly, we use a composite indicator of health
status. The RLMS-HSE is particularly rich in objective health indicators, with
information of health problems, life-style habits, and use of health services in the
recent period. We aggregate eight of these indicators to obtain the index of health
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
26
status. The weights attached to each indicator are derived from an ordinal logit
regression of self-assessed health status on the objective health indicators.23 By
this procedure, the constructed measure is as close as possible to the self-reported
health status of the individual, while ensuring that individuals with the same objective characteristics obtain the same health measure. Years of schooling attained
is constructed by combining the information on the highest grade attained and
the number of years in post-secondary education. Finally, a composite indicator of
housing quality captures usable equivalized living space and a set of quality indicators such as the presence of hot water, central sewerage and a phone. The weights
attached to each indicator are derived, similarly as for the health indicator, from
a hedonistic regression using self-reported housing value as dependent variable.24
We restrict the analysis to adult individuals with complete information in all
four indicators, leaving us with sample sizes between 5,500 and 8,500 individuals in
each wave. Figure 3 shows the evolution of the S-Gini inequality index of the four
dimensions separately for a bottom sensitivity parameter δ of 2 and 5 (panel a and
b respectively). The inequality indices are normalized such that 1995 equals 100
to compare changes in inequality more easily.25 Expenditure inequality increases
between ten and fifteen percent from 1995 to 1998 -for instance, the estimated
Gini coefficient (δ = 2) went from 0.39 to 0.45. These findings are in line with
the literature (Gorodnichenko et al. 2010). After the financial crisis, inequality
in expenditures returned relatively quick to its 1995 values. By contrast, educational inequality drops sharply over the considered period and health inequality
increases slightly. Inequality in housing quality remains relatively stable. In short,
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
27
Figure 3: Percentage change in S-Gini index (for δ = 2 and 5) of household expenditure, health, education and housing quality in Russia between 1995 and 2005.
[Figure 3 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
.
all four dimensions experience rather distinct patterns, highlighting the need for a
multidimensional approach to the analysis of the distribution of well-being.
In order to compute both multidimensional Gini indices presented in the preceding section, we are inevitably forced to make four crucial decisions over the
value of the parameters. These are, first, the appropriate standardization for each
dimension; second, the degree of substitutability β; third, the weighting scheme
w = (w1 , . . . , wm ) and, fourth, the bottom sensitivity parameter δ. In the present
illustration we standardize every dimension by dividing the outcomes by its respective mean in 2000. We set β equal to 0, so that the aggregation across dimensions
follows expression (2). The reason for this choice is to make results robust to
alternative standardization procedures involving a dimension-wise rescaling. As
already explained, this is an attractive property when the indicators used are of
very different nature as it is the case here. Third, we use equal dimension weights
(wj = 1/4) for simplicity.26 Finally, we compare two alternative values of the bottom sensitivity parameter δ, the first corresponds to the standard Gini coefficient
(δ = 2) and the second gives higher weight to individuals at the bottom of the distribution (δ = 5). The selection of parameters values is an essential and difficult
step in the computation of any multidimensional index of inequality. Inevitably,
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
28
Table 1: Implied marginal rates of substitution between the dimensions of wellbeing.
expenditures
health education
health
-8405.97
education
-496.211 -0.05903
house (in 1,000)
-27.594 -0.00328
-0.05561
Source: Russian Longitudinal Monitoring Survey, 2005.
the parameters imply value judgements about the nature of well-being and the
contribution of each dimension to it. It is important to make these choices in an
explicit and clear way so that they can be open to public scrutiny. To clarify what
our parameter-choices mean in terms of the well-being function, we present the
implied marginal rates of substitution in Table 1. The average Russian in 2005,
who spends 5,632 (equivalized) rubles, has a health indicator of 0.67 (on a scale
of 0 to 1), attends 11.35 years of schooling and has a housing quality of 204,000
is willing to pay about 841 rubles for an increase of 0.1 on the health scale (which
means, for instance, roughly 577 rubles for not being hospitalized) or 496 rubles
for an extra year of schooling and 28 rubles for an increase of 1000 for the housing
quality measure.27
Before we compute the evolution of well-being inequality, we first check whether
the selected parameters lead to compliance of the indices with the distributional
concerns introduced in section II. Following the conditions obtained in proposition 3, index I 1 satisfies uniform majorization for the parameters chosen and it
is by construction not sensitive to the correlation between the dimensions. Index
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
29
Figure 4: Compliance with the unfair rearrangement principle of the RLMS-HSE
data set between 1995-2005.
Source: Authors’ calculations from RLMS-HSE.
[Figure 4 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
I 2 satisfies uniform majorization for the parameters selected. We check the unfair
rearrangement principle for a wide range of parameters and summarize the results
in figure 4. This figure depicts a part of the normative space given by the degree of
substitutability β and bottom sensitivity δ. In every point of the normative space,
multidimensional inequality of the distribution matrix at hand X is compared to
the corresponding Z, the distribution matrix obtained by an unfair rearrangement.
The curves connect the points where I 2 (X) equals I 2 (Z) for a given year. Above
the curve, the unfair rearrangement principle is satisfied and below the curve it is
not. Hence, for a degree of substitutability β of 0 and a bottom sensitivity parameter δ equal to 2 or 5, it is clear that the unfair rearrangement principle is satisfied
for all years, although the measure is more responsive to correlation in case when
δ equals 5.
Welfare losses due to well-being inequality are quite high all throughout the
period: about 30% for δ equal to 2 and 50% for δ equal to 5.28 The values of I 1 are
significantly higher than of I 2 (at a 95% confidence level) for δ = 5, but do not differ
significantly for δ = 2. In order to focus on the changes of the two measures, rather
than the levels, figure 5 depicts the evolution of the normalized I 1 and I 2 , such
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
30
Figure 5: Percentage change in S-Gini multidimensional indices I 1 and I 2 (for
δ = 2 and 5) of household expenditure, health, education and housing quality in
Russia between 1995-2005.
[Figure 5 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
that both indices equal 100 in 1995. For most combinations of parameter values,
well-being inequality increases up to the crisis year 1998 and afterwards declines
almost monotonically. However, for δ equal to 5, the indices disagree about when
the peak in well-being inequality is reached. According to I 1 inequality is largest
in 1998, whereas I 2 reaches its peak already in 1996. In addition, for both bottom
sensitivity parameters, multidimensional inequality measured by I 2 shows a less
pronounced relative increase than I 1 over the first waves, and a stronger decrease
afterwards.
To obtain more insights into these differences between the indices, we construct four counterfactual distribution matrices, and present the evolution of wellbeing inequality using both inequality indices I 1 -figure 6- and I 2 -figure 7-, and
for bottom-sensitivity parameter δ equal to 2 -left panels- and 5 -right panels. The
counterfactual distributions will keep the position of each individual in each dimension constant throughout the period -i.e. as it was in 1995- while introduce changes
in levels for each indicator one dimension at a time. In this way, we will be able
to ‘decompose’ the observed changes in multidimensional inequality into changes
in the inequality for each dimension, while leaving the rank correlation untouched
from the initial period.29 The difference between the last counterfactual and the
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
31
observed data, can be interpreted as the effect on multidimensional inequality of
the changing correlation structure.
The first counterfactual distribution keeps the distributional profile (levels
and positions) of the health, schooling and housing dimension as in the initial year,
1995, while incorporating the change in levels in expenditure, though keeping the
position in the expenditure distribution constant. That is, if in 1995 a person was
the poorest individual in terms of expenditure, the 5th poorest in health, the 20th
poorest person in education and the 40th poorest in housing quality, we use for
the year 2000 the value of health, education and housing that she had in 1995,
but assign her the expenditure level that the poorest individual had in the year
2000. The multidimensional inequality according to measures I 1 and I 2 of this
first counterfactual are given by the dashed light grey line in figure 6 and 7. The
second counterfactual distribution incorporates the profile of the health dimension,
while keeping the values of educational achievements and housing quality (and all
positions) as they were in 1995. The evolution of well-being inequality according
to this counterfactual is given by the dotted darker grey line. As can be seen,
incorporating the changing distributional profile of the health dimension increases
multidimensional inequality further. The third counterfactual incorporates the
observed distributional profile of the educational dimension. The evolution of this
counterfactual is given by the grey dash-dotted line and leads to a decrease in
well-being inequality. The decrease in inequality due to the inclusion of education
offsets the increase due to the health for I 2 but not for I 1 . Finally, the fourth
counterfactual adjusts the distributional profile of the housing quality which leads
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
32
Figure 6: Counterfactual S-Gini multidimensional indices I 1 (for δ = 2 and 5) of
household expenditure, health, education and housing quality in Russia between
1995-2005.
[Figure 6 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
Figure 7: Counterfactual S-Gini multidimensional indices I 2 (for δ = 2 and 5) of
household expenditure, health, education and housing quality in Russia between
1995-2005.
[Figure 7 here]
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
to an increase of multidimensional inequality. This counterfactual is given by the
solid dark grey line.
The most interesting insight from this counterfactual analysis stems from the
comparison of this last counterfactual with the actual distribution -depicted by
the solid black line-. This difference can be completely attributed to the change
in correlation structure. Given that index I 1 is not sensitive to changes in correlation, the solid black and and the solid grey lines in both panels of figure 6 indeed
coincide. Instead, in figure 7, the index I 2 based on the observed data diverges
from the fourth counterfactual, and quite markedly in the right panel. When one
gives considerable weight to bottom of the well-being distribution, the decrease
in correlation between dimensions in 1998 offsets the increase in inequality within
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
33
dimensions of well-being completely. In contrast, the increase in correlation between dimensions after the crises reduces the otherwise favorable performance of
inequality within each dimension.
This important change in structure of dimensions of well-being would have
been completely missed if one were to use only the column-first inequality index
I 1 . This example highlights the empirical importance of the theoretical concerns
about correlation-sensitivity in multidimensional inequality measurement.
V. Conclusion
Russian inequality showed a different pattern across the different dimensions of
well-being (expenditure, health, education) between 1995 and 2005. Furthermore,
during this period the correlation structure between these dimensions changed significantly. To make an overall assessment of the evolution of well-being inequality,
we have proposed two indices of well-being inequality that summarize the multidimensional inequality into a single value. These indices are multivariate generalizations of the popular Gini coefficient, in which both the levels and the position
of individuals in the overall distribution matter for the social evaluation. We followed a normative approach to derive the inequality indices from their underlying
rank-dependent multidimensional social evaluation functions. These functions are,
in turn, conceived as explicit two-step aggregation functions, one across attributes
and another across individuals. In both steps, we imposed a set of properties to
come to a single class of functions. We think that the set of properties suggested
in this paper entails an acceptable compromise between functional flexibility and
INEQUALITY OF WELL-BEING: A MULTIDIMENSIONAL APPROACH
34
the parsimony needed to come to an applicable and practically manageable index
of well-being inequality.
The sequencing of both aggregations turns out to be essential in terms of
the underlying principles. Aggregating first across individuals and then across
dimensions leads to an index which is a-priori insensitive to the correlation between
the dimensions. In our view this is a serious drawback. The second procedure, in
which we first aggregate across dimensions and then across individuals, leads to a
index that can be sensitive to correlation between the dimensions for specific choices
of parameter-values. We show that researchers who want to obtain a correlationsensitive rank-dependent inequality index, have to be willing to give a (potentially)
large weight to the bottom of the distribution.
The illustration based on Russian household data highlights the empirical difference between both multidimensional Gini inequality indices. Although they are
both derived from a two-step social evaluation function respecting similar properties in each step, the sequencing of both steps makes them very different with
respect to their sensitivity to correlation between the dimensions. The indices differ
most saliently in their evaluation of the evolution of multidimensional inequality
during the years of the financial crisis. In 1998, within dimension inequality increased, but this increase in overall inequality is compensated to a larger or lesser
extend by the decreasing correlation between the dimensions (depending on the
selected bottom-sensitivity parameter). The trade-off between both forces (increasing well-being inequality due to increasing within dimension inequality and
decreasing well-being inequality due to decreasing correlation between dimensions)
is a matter of value judgements on the notion of multidimensional inequality.
NOTES
35
To apply the multidimensional indices in a satisfactory way to real-world
data, hard choices have to be made about the appropriate parameter values (on
standardization, weighting, substitutability between the dimensions and bottomsensitivity). Theoretical guidelines on how these parameter choices can and should
be made, are required. Furthermore, there is a pressing need to collect more and
richer individual data including non-monetary dimensions of well-being together
with monetary ones, so that well-being inequality can be increasingly analyzed in
a truly multidimensional way.
Acknowledgements
We thank Rolf Aaberge, Tony Atkinson, Conchita d’Ambrosio, André Decoster, Stefan Dercon, Jean-Yves Duclos, James Foster, Luc Lauwers, Erwin Ooghe,
Erik Schokkaert, John Weymark, the editor of the journal and two anonymous referees for very helpful comments and suggestions to this or an earlier version of the
paper entitled “Measuring inequality of well-being with a correlation-sensitive multidimensional Gini index”. Remaining errors are all ours. We use the “Russia Longitudinal Monitoring survey, RLMS-HSE”, conducted by HSE and ZAO “Demoscope” together with Carolina Population Center, University of North Carolina at
Chapel Hill and the Institute of Sociology RAS. (RLMS-HSE sites: http://www.cpc.unc.edu/projects
hse, http://www.hse.ru/org/hse/rlms). Funding from the UK Economic and Social
Research Council are gratefully acknowledged.
Notes
1
For earlier references on the multidimensionality of well-being see, for instance, Rawls (1971);
Sen (1985); Streeten (1994).
NOTES
2
36
In other words, the approach is to analyze inequalities within each dimension separately.
Inequalities in health, for instance, refer to the differences between people in health status, and
not to the ‘income gradient’ (the covariation of health and income) as in Wilkinson (1996) or
Lokshin and Ravallion (2008).
3
The idea that a multidimensional inequality or welfare measure should be sensitive to de-
pendence has been brought to the fore by Atkinson and Bourguinon (1982), Rietveld (1990),
Dardanoni (1996), Tsui (1999) and Pattanaik et al. (2008) and appears also in the work of political philosophers such as Michael Walzer (1983) and Thomas Pogge (1989, 2002). Finally, in the
cited report, Stiglitz et al. (2009) highlight the importance of incorporating “information from
the ‘joint distribution’ of the most salient features of quality of life [...] across all people” (Stiglitz
et al., 2009, p. 56). Indeed, they argue that an attractive measure of well-being inequality should
at least be sensitive to the interrelationships between dimensions, since “the various inequalities
might strengthen each other” (Stiglitz et al., 2009, p. 205).
4
Note however the increased gradient between health and expenditures (as documented by
Lokshin and Ravallion (2008) and Blam and Kovalev (2006)) and the increasing correlation
between schooling and expenditures, reflecting an increased skill premium (Gorodnichenko and
Peter, 2005).
5
The Gini coefficient is the measure of choice for various international organizations datasets
such as The World Bank ‘Inequality around the world’ dataset, the UNU-WIDER World Income
Inequality Database, the EurLIFE database, and the UNDP annual report.
6
For a recent survey on the normative approach to derive multidimensional inequality mea-
sures, the reader is referred to Weymark (2006). An alternative approach, based on multidimensional stochastic dominance allows for some disagreement about the value judgements, but
leads inevitably to an incomplete ordering, so that some societies can be ordered with respect to
their well-being inequality, but others can not. This indecisiveness may be informative in its own
right, but for policy purposes a complete ordering is more convenient. Trannoy (2006) provides
a recent survey of the multidimensional dominance literature.
NOTES
7
37
See, for instance Pattanaik et al. (2008), Lasso de la Vega et al. (2009) and Seth (2009) for
recent two-step procedures in multidimensional welfare, poverty and inequality measurement.
8
The well-being function is thus common to all members of society and permits well-being
judgements that are not purely subjective but interpersonally justifiable and comparable (Gaspart, 1998). A potential concern with the ‘objective’ well-being approach is that it is overly
paternalistic or perfectionist, since it represents the preferences of the external observer about
what constitutes a good life for the individuals, and not the preferences of the individuals themselves (Fleurbaey, 2005). On the impossibility of combining differences in individual preferences
with multidimensional egalitarianism, see Fleurbaey and Trannoy (2003).
9
Many alternative definitions and interpretations of the one-dimensional Gini coefficient exist
in the literature. See Sen (1973), Anand (1983) and Yitzhaki (1998). Arnold (2005) gives a
survey of the recent developments on multidimensional generalizations of the Gini coefficient.
See also List (1999).
10
Throughout the analysis, the number of dimensions m is assumed to be fixed, whereas we
allow for variable population size n. We do not discuss which dimensions of well-being should be
included, rather we assume that these are either obtained by a democratic process or given by
philosophical reasoning like the primary goods defined by (Rawls, 1971), the list of ‘functionings’
proposed by (Nussbaum, 2000), or the basic needs approach advocated by (Streeten, 1994).
11
On the issue of making meaningful comparisons of multidimensional outcome vectors, see
Ebert and Welsch (2004). Alternatively, we could assume the data to be standardized from
the beginning to leave the invariance to the standardization outside the characterization of the
measure. We prefer to incorporate the standardization into the characterization given its potential
effects on the final result, see e.g. Decancq et al. (2009).
12
This property does not restrict the standardization to using the same central tendency mea-
sure for each dimension. One could, for instance, choose to standardize cardinal variables (such
as income) by their respective mean and ordinal variables (such as self-reported health status)
by their median, as suggested by Allison and Foster (2004).
NOTES
13
38
This extreme case is excluded by the monotonicity property and should be considered as a
limiting case.
14
Original variables (GDP per capita, life expectancy, school enrolment and literacy rate) are
standardized by a linear rescaling so that their minimum value is 0 and the maximum is 1.
Additionally, GDP per capita is first transformed by a logarithmic transformation, to allow the
marginal rates of substitutions between life expectancy and GDP per capita, and educational
outcomes and GDP per capita to vary over the range of GDP per capita values.
15
Ebert (1988) shows that dropping either WSI or WTI from the list of axioms, leads to
respectively the Gini-Kolm and Gini-Atkinson welfare indices proposed by Berrebi and Silber
(1981).
16
Again, this extreme case is excluded by the monotonicity property and should be considered
as a limiting case.
17
A bistochastic transform of a distribution matrix X involves a premultiplication of the dis-
tribution matrix by a bistochastic matrix, which is a square nonnegative matrix with all row and
column sums equal to 1.
18
Logically stronger principles of correlation-sensitivity are based on the notion of a correlation
increasing transfer, see Epstein and Tanny (1980) and Tsui (1999).
19
The recently proposed multidimensional Gini index by Banerjee (2010) belongs to the class
of indices given by expression (8) . In particular, it is obtained when setting β equal to 1; δ equal
to 2 and the weighting scheme w by the first normalized principal component of distribution
matrix X.
20
However, from proposition 4 it follows that I 2 (X) < I 2 (Z) whenever δ > δ 0 (i.e. when
the unfair rearrangement principle holds) where Z is the distribution matrix introduced in the
previous section obtained from X by an unfair rearrangement. Since I 1 is not sensitive to changes
in correlation, it always holds that I 1 (X) = I 1 (Z). When all dimensions are perfectly correlated,
the ranks rji equal ri for all j dimensions. Moreover, if β equals 1 both aggregations across
individuals and dimensions are weighted averages so that the order can be switched without
NOTES
39
affecting the results. Hence, I 1 (Z) = I 2 (Z) whenever β = 1. In sum, in the specific case for
δ > δ 0 and β = 1, I 2 shows always less inequality than I 1 .
21
The RLMS-HSE data are obtainable from: http://www.hse.ru/org/hse/rlms. The survey
(phase II) has been carried out annually since 1994 (with the exception of 1997 and 1999) and is
still ongoing.
22
We use the original household expenditure variable, where nominal values are corrected for
yearly inflation using June 1992 as the reference year. No other adjustment is made. For a detailed
discussion on possible adjustments, see Decoster (2005) and Gorodnichenko et al. (2010).
23
More precisely, for every individual, the composite index of health is the predicted value of
a pooled ordered logit health regression with the following explanatory variables: indicators of
diabetes, heart attack, anaemia or other health problems; a life-style indicator such as smoking
and indicators of a recent medical check-up, hospitalization or operation in the last three months.
All variables are highly significant and have the expected sign. The predicted values from this
regression are linearly transformed such that the most unhealthy individual over the considered
period obtains slightly more than 0 and the healthiest almost 1. Results are available from the
authors on request. A similar procedure has been used by van Doorslaer and Jones (2003) and
Nilsson (2010). See also Lokshin and Ravallion (2008) for a similar procedure applied to the
RLMS-HSE data using an OLS regression.
24
The composite index of housing quality is the predicted value of a pooled OLS hedonistic
regression with the following explanatory variables: (logarithm of) equivalent usable living space
and indicators of the presence of central heating, hot water, metered gas, central sewerage, a
phone and a video. Time and regional variation of the housing value are corrected for by time
dumies and a rural indicator. All variables are highly significant and have the expected sign.
Results are available from the authors on request.
25
Bootstrapped standard errors are available from the authors on request.
26
On the issue of setting weights in multidimensional measures of well-being, see Decancq and
Lugo (forthcoming).
NOTES
27
40
For reference, the exchange rate in June 1992 (base year of expenditure data) was 1 dollar
= 138 rubles (IFS/IMF); that is, the average Russian spent approximately 40.811-dollars per
month, and is willing to pay about 3.6 dollars for an extra year of education (per month) or 4.2
dollars for not being hospitalized.
28
Results are available from the authors on request.
29
The counterfactuals are constructed based on the balanced subsample which consists of 1601
individuals to keep the sample size fixed. We correct all counterfactuals for the underestimation
of inequality due to attrition (in the balanced panel inequality is about 4 percentage points lower
than the total sample) by a year-specific multiplicative attrition-correction.
Appendix: Proofs of Propositions
n×m
1
Proposition 3. A continuous double aggregation function Wn×m
: R++
→
R++ , where Wn satisfies MON, SYM, NORM, RSEP, WSI, WTI, REP and RA;
and Wm satisfies MON, NORM, SEP and WSI,
1
satisfies equation (4) with δ > 1,
i) satisfies UM if and only if Wn×m
ii) cannot satisfy UR.
Proof. For part i) concerning uniform majorization (UM), let Y = BX, so that
1
1
1
Wn×m
satisfies UM if and only if Wn×m
(Y ) > Wn×m
(X) . By construction, in every
dimension j it holds that yj = Bxj , so that for all strictly Schur concave aggregation
functions across individuals it holds that Wn (yj ) > Wn (xj ) (Marshall and Olkin,
1979, Chapter 3). Strict Schur concavity of Wn is obtained by restricting δ to be
larger than 1 (Ebert, 1988, proposition 6). If in all dimensions j, Wn (yj ) > Wn (xj )
holds, by monotonicity of Wm it is the case that Wm (Wn (y1 ), . . . , Wn (ym )) >
Wm (Wn (x1 ), . . . , Wn (xm )) .
NOTES
41
Part ii) concerning the unfair rearrangement principle (UR), follows from the
fact that an unfair rearrangement is obtained from by a dimension-wize permutation, so that by SYM of Wn , [Wn (x1 ), . . . , Wn (xm )] = [Wn (z1 ), . . . , Wn (zm )] and
1
1
hence Wn×m
(X) = Wn×m
(Z).
n×m
2
Proposition 4. A continuous double aggregation function Wn×m
: R++
→
R++ , where Wm satisfies MON, NORM, SEP and WSI; and Wn satisfies MON,
SYM, NORM, RSEP, WSI, WTI, REP and RA,
2
satisfies equation (5) with β < 1 and δ > 1,
i) satisfies UM if and only if Wn×m
2
satisfies equation (5) with δ > δ 0 , where δ 0 is
ii) satisfies UR if and only if Wn×m
a threshold depending on X, w and β.
2
Proof. A double aggregation function Wn×m
satisfies the required properties on
Wm and Wn if and only if it can be written as:
2
Wn×m
(9)
i
with Wm (x ) =
hP
m
j=1
" δ δ #
n
X
ri
ri − 1
(X) =
−
Wm (xi ),
n
n
i=1
wj (xij )β
i(1/β)
for all i. Note that ri is the rank of individual
i based on the vector of well-being levels. In other words, ri = 1, if individual i is
the best off; ri = 2 if the individual is the second best off; and so forth.
For part i) concerning the uniform majorization principle (UM), Kolm (1977,
2
theorem 6) shows that UM holds if and only if Wn×m
satisfies the following three
conditions: (a) Wm is strictly concave, (b) Wn is increasing and (c) Wn is strictly
Schur concave.
Note that (a) is fulfilled if and only if β < 1. Condition (b) is fulfilled by the
monotonicity of Wn and condition (c) is fulfilled if and only if δ > 1. (Ebert, 1988,
proposition 6)
NOTES
42
For part ii) concerning the unfair rearrangement principle (UR), let Z be
obtained from X by the sequence of dimension-wize permutations which make one
individual in Z top-ranked in all dimensions, another individual second ranked in
2
all dimensions and so forth and Z 6= X, so that Wn×m
satisfies (UR) if and only if
2
2
(Z) > 0.
(X) − Wn×m
Wn×m
(10)
2
it follows that
From the definition of Wn×m
" δ δ #
n
i
i
X
r
r
−
1
2
2
(11) Wn×m
(X) − Wn×m
(Z) =
−
Wm (xi ) − Wm (z i ) .
n
n
i=1
By monotonicity of Wm , it follows that for the worst-off individual i0 , for
0
0
0
0
which Wm (xi ) 6= Wm (z i ), we have that Wm (xi ) > Wm (z i ), since the unfair
rearrangement makes individual i0 worst-off in all dimensions (whereas in the initial
distribution, she was not worst-off in at least some dimensions). Similarly, for i00 ,
00
00
00
the best-off individual for which Wm (xi ) 6= Wm (z i ), we have that Wm (xi ) <
00
Wm (z i ). What happens to the intermediate individuals is unclear and depends on
2
2
(Z) > 0, we have:
(X) − Wn×m
the parameters X, w and β. To reassure that Wn×m
" δ i0 δ i0
δ
δ #
n
X
r
r −1
ri
ri − 1
Wm (xi ) − Wm (z i )
−
>
,
−
n
n
n
n
Wm (xi0 ) − Wm (z i0 )
i=1,i6=i0
which can be solved for δ, and leads to δ > δ 0 where δ 0 is a threshold depending
on X, w and β.
NOTES
43
References
Allison, R. A. and Foster, J. E. (2004). Measuring health inequality using
qualitative data. Journal of Health Economics, 23 (3), 505–524.
Anand, S. (1983). Inequality and Poverty in Malaysia: Measurement and Decomposition. A World Bank research publication, New York: Oxford University
Press.
— (2002). The concern for equity in health. Journal of Epidemiology and Community Health, 56 (7), 485–487.
Anderson, G. (2004). Indices and tests for multidimensional inequality: multivariate generalizations of the Gini coefficient and Kolmogorov-Smirnov twosample test. In 28th General Conference of the International Association for
Research in Income and Wealth, Cork, Ireland.
Arnold, B. C. (1987). Majorization and the Lorenz order: A Brief Introduction.
Berlin: Springer.
— (2005). Inequality measures for multivariate distributions. Metron, 63 (3), 317–
327.
Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic
Theory, 2 (3), 244–263.
— and Bourguignon, F. (1982). The comparison of multi-dimensioned distributions of economic status. The Review of Economic Studies, 49 (2), 183–201.
—, Cantillon, B., Marlier, E. and Nolan, B. (2002). Social Indicators: The
EU and Social Exclusion. Oxford: Oxford University Press.
NOTES
44
Banerjee, A. K. (2010). A multidimensional gini index. Mathematical Social
Sciences, 60 (2), 87–93.
Berrebi, Z. M. and Silber, J. (1981). Weighting income ranks and levels : A
multiple-parameter generalization for absolute and relative inequality indices.
Economics Letters, 7 (4), 391 – 397.
Blackorby, C. and Donaldson, D. (1982). Ratio-scale and translation-scale
full interpersonal comparability without domain restrictions: Admissible socialevaluation functions. International Economic Review, 23 (2), 249–68.
Blam, I. and Kovalev, S. (2006). Spontaneous commercialisation, inequality
and the contradictions of compulsory medical insurance in transitional russia.
Journal of International Development, 18 (3), 407–423.
Dalton, H. (1920). The measurement of the inequality of incomes. The Economic
Journal, 30 (119), 25–49.
Dardanoni, V. (1996). On multidimensional inequality measurement. In
C. Dagum and A. Lemmi (eds.), Research on Economic Inequality, vol. 6, London: JAI Press, pp. 201–205.
Deaton, A. and Muellbauer, J. (1980). Economics and Consumer Behavior.
Cambridge University Press.
Decancq, K. (2009). Copula-based measurement of dependence between dimensions of well-being, hEDG Working Paper 09/32.
—, Decoster, A. and Schokkaert, E. (2009). The evolution of world inequality in well-being. World Development, 30 (1), 11–25.
NOTES
45
— and Lugo, M. A. (forthcoming). Weights in multidimensional indices of wellbeing: An overview. Econometric Reviews.
Decoster, A. (2005). How progressive are indirect taxes in russia? Economics
of Transition, 13 (4), 705–729.
Donaldson, D. and Weymark, J. A. (1980). A single-parameter generalization
of the Gini indices of inequality. Journal of Economic Theory, 22 (1), 67–86.
Dutta, I., Pattanaik, P. K. and Xu, Y. (2003). On measuring deprivation and
the standard of living in a multidimensional framework on the basis of aggregate
data. Economica, 70 (278), 197–221.
Ebert, U. (1988). Measurement of inequality: an attempt at unification and
generalization. Social Choice and Welfare, 5 (2-3), 147–169.
— and Welsch, H. (2004). Meaningful environmental indices: A social choice
approach. Journal of Environmental Economics and Management, 47 (2), 270–
283.
Epstein, L. G. and Tanny, S. M. (1980). Increasing generalized correlation: a
definition and some economic consequences. The Canadian Journal of Economics
/ Revue canadienne d’Economique, 13 (1), 16–34.
Fahey, T., Whelan, C. T. and Maı̂tre, B. (2005). First European Quality of
Life Survey: Income Inequalities and Deprivation. European Foundation for the
Improvement of Living and Working Conditions, Luxembourg: Office for Official
Publications of the European Communities.
NOTES
46
Ferrer-i-Carbonell, A. (2005). Income and well-being: an empirical analysis
of the comparison income effect. Journal of Public Economics, 89 (5-6), 997–
1019.
Fleurbaey, M. (2005). Equality of functionings, le Centre de recherche, sens,
éthique, société, CNRS-Université Paris V, Mimeo.
— and Trannoy, A. (2003). The impossibility of a paretian egalitarian. Social
Choice and Welfare, 21 (2), 243–263.
Gajdos, T. and Weymark, J. A. (2005). Multidimensional generalized Gini
indices. Economic Theory, 26 (3), 471–496.
Gaspart, F. (1998). Objective measures of well-being and the cooperative production problem. Social Choice and Welfare, 15 (1), 95–112.
Gorodnichenko, Y. and Peter, K. S. (2005). Returns to schooling in russia
and ukraine: A semiparametric approach to cross-country comparative analysis.
Journal of Comparative Economics, 33 (2), 324 – 350, returns to Schooling in
Translation: The Chinese, European, and Russian Experiences.
—, — and Stolyarov, D. (2010). Inequality and volatility moderation in russia:
Evidence from micro-level panel data on consumption and income. Review of
Economic Dynamics, 13 (1), 209 – 237.
Kakwani, N. (1980). On a class of poverty measures. Econometrica, 48 (2), 437–
446.
Kolm, S.-C. (1969). The optimal production of social justice. In J. Margolis and
H. Guitton (eds.), Public Economics, London: Macmillan, pp. 145–200.
NOTES
47
— (1977). Multidimensional egalitarianisms. The Quarterly Journal of Economics,
91 (1), 1–13.
Koshevoy, G. and Mosler, K. (1996). The Lorenz zonoid of a multivariate
distribution. Journal of American Statististical Association, 91 (434), 873–882.
— and — (1997). Multivariate Gini indices. Journal of Multivariate Analysis,
60 (2), 252–276.
Lasso de la Vega, C., Urrutia, A. and Diez, H. (2009). The Bourguignon
and Chakravarty multidimensional poverty family: A characterization. Working
Papers 109, ECINEQ, Society for the Study of Economic Inequality.
List, C. (1999). Multidimensional inequality measurement: a proposal, nuffield
College.
Lokshin, M. and Ravallion, M. (2008). Testing for an economic gradient in
health status using subjective data. Health Economics, 17 (11), 1237–1259.
Luttmer, E. F. P. (2005). Neighbors as negatives: relative earnings and wellbeing. Quarterly Journal of Economics, 120 (3), 963–1002.
Maasoumi, E. (1986). The measurement and decomposition of multi-dimensional
inequality. Econometrica, 54 (4), 991–997.
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization
and its Applications. New York: Academic Press.
Nilsson, T. (2010). Health, wealth and wisdom: Exploring multidimensionsl inequality in a developing country. Social Indicators Research, 95 (2), 299–323.
NOTES
48
Nussbaum, M. C. (2000). Women and Human Development. Cambridge: Cambridge University Press.
Pattanaik, P., Reddy, S. and Xu, Y. (2008). On procedures for measuring
deprivation and living standards of societies in a multi-attribute framework.
Working paper 08-02 Andrew Young School of Policy studies. Georgia State
University.
Pogge, T. (1989). Realizing Rawls. Cornell University Press.
— (2002). Can the capability approach be justified? Philosophical Topics, 30 (2),
167–228.
Rawls, J. (1971). A Theory of Justice. Cambridge, Mass.: Belknap Press of
Harvard University Press.
Rietveld, P. (1990). Multidimensional inequality comparisons: on aggravation
and mitigation of inequalities. Economics Letters, 32 (2), 187–192.
Sen, A. K. (1973). On Economic Inequality. Oxford: Clarendon Press.
— (1985). Commodities and Capabilities. Amsterdam and Oxford: North-Holland.
Seth, S. (2009). Inequality, interactions, and human development. Journal of
Human Development and Capabilities, 10 (3), 375–396.
Stiglitz, J. E., Sen, A. and Fitoussi, J.-P. (2009). Report by the commission
on the measurement of economic performance and social progress. Commission
on the Measurement of Economic Performance and Social Progress, mimeo.
Streeten, P. (1994). Human development: means and ends. The American Economic Review, 84 (2), 232–37.
NOTES
49
Trannoy, A. (2006). Multidimensional egalitarianism and the dominance approach: a lost paradise. In F. Farina and E. Savaglio (eds.), Inequality and
Economic Integration, London: Routledge, pp. 284–302.
Tsui, K. Y. (1995). Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson-Kolm-Sen approach. Journal of Economic
Theory, 67 (1), 251–265.
— (1999). Multidimensional inequality and multidimensional generalized entropy
measures: An axiomatic derivation. Social Choice and Welfare, 16 (1), 145–57.
— and Weymark, J. A. (1997). Social welfare orderings for ratio-scale measurable utilities. Economic Theory, 10 (2), 241–256.
van Doorslaer, E. and Jones, A. (2003). Inequalities in self-reported health:
validation of a new approach to measurement. Journal of Health Economics,
22 (1), 61–87.
Walzer, M. (1983). Spheres of Justice: A Defense of Pluralism and Equality.
New York: Basic Books.
Weymark, J. A. (2006). The normative approach to the measurement of multidimensional inequality. In F. Farina and E. Savaglio (eds.), Inequality and economic
integration, London: Routledge.
Wilkinson, R. (1996). Unhealthy Societies. The Afflictions of Inequality. London:
Routledge.
World Bank (2005). World Development Report 2006: Equity and Development.
Washington D.C.: World Bank Publications.
NOTES
50
Yitzhaki, S. (1983). On an extension of the Gini inequality index. International
Economic Review, 24 (3), 617–628.
— (1998). More than a dozen alternative ways of spelling Gini. Research on Economic Inequality, vol. 8, London: JAI Press, pp. 13–30.
Figure 1: Percentage change in Gini coefficient of household expenditure, health, education and housing quality in Russia between 19952005.
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
Figure 2: Pairwise rank-correlation coefficients between household expenditure, health, education and housing quality in Russia between
1995-2005.
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
1
Figure 3: Percentage change in S-Gini index (for δ = 2 and 5) of
household expenditure, health, education and housing quality in Russia
between 1995 and 2005.
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
.
Figure 4: Compliance with the unfair rearrangement principle of the
RLMS-HSE data set between 1995-2005.
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
Figure 5: Percentage change in S-Gini multidimensional indices I 1 and
I 2 (for δ = 2 and 5) of household expenditure, health, education and
housing quality in Russia between 1995-2005.
Source: Authors’ calculations from Russian Longitudinal Monitoring Survey-HSE.
INEQUALITY OF WELL-BEING:
A MULTIDIMENSIONAL APPROACH
SUPPLEMENTARY MATERIAL
KOEN DECANCQ AND MARÍA ANA LUGO
Table 1: Health equation based on the pooled sample (Russia 1995-2005).
Self-assessed Health
diabetes
-0.686∗∗∗
heart attack
-0.951∗∗∗
anemia
-0.565∗∗∗
smokes
-0.228∗∗∗
health problem -1.641∗∗∗
hospitalized
-0.779∗∗∗
check up
-0.171∗∗∗
operation
-0.255∗∗∗
age 20
-0.315∗∗∗
age 30
-0.927∗∗∗
age 40
-1.539∗∗∗
age 50
-2.047∗∗∗
age 60
-2.658∗∗∗
age 70
-3.399∗∗∗
age 80
-4.181∗∗∗
age 90
-4.685∗∗∗
male
-0.545∗∗∗
N
86,464
2
pseudo − R
0.225
(0.0370)
(0.0446)
(0.0385)
(0.0178)
(0.0175)
(0.0341)
(0.0192)
(0.0393)
(0.0271)
(0.0278)
(0.0278)
(0.0307)
(0.0322)
(0.0361)
(0.0535)
(0.135)
(0.0167)
Standard errors in parentheses
∗p
< 0.05,
∗∗ p
< 0.01,
∗∗∗ p
< 0.001
Source: Authors’ calculations from RLMS-HSE.
1
Table 2: House equation based on the pooled sample (Russia, 2000-2005).
logarithm of market value
log. equival. space
central heating
hot water
metered gas
cental sewerage
phone
video
rural
2000
2002
2003
2004
2005
const
N
R2
0.466∗∗∗
0.0848∗∗∗
0.344∗∗∗
0.276∗∗∗
0.303∗∗∗
0.328∗∗∗
0.291∗∗∗
-0.767∗∗∗
-0.235∗∗∗
0.229∗∗∗
0.415∗∗∗
0.638∗∗∗
0.868∗∗∗
9.823∗∗∗
35,279
0.568
(0.0105)
(0.0172)
(0.0123)
(0.0101)
(0.0175)
(0.00945)
(0.00853)
(0.0119)
(0.0149)
(0.0140)
(0.0142)
(0.0143)
(0.0144)
(0.0354)
Standard errors in parentheses
∗p
< 0.05,
∗∗ p
< 0.01,
∗∗∗ p
< 0.001
Source: Authors’ calculations from RLMS-HSE.
Table 3: Summary Statistics. Russia from 1995 to 2005.
year
mean
1995 (N=6,844)
expenditures
health
schooling
house
1996 (N=6,572)
expenditures
health
schooling
house
1998 (N=5,417)
expenditures
health
schooling
house
2000 (N=7,193)
expenditures
health
schooling
house
2001 (N=7,891)
expenditures
health
schooling
house
2002 (N=8,512)
expenditures
health
schooling
house
2003 (N=8,509)
expenditures
health
schooling
house
2004 (N=8,398)
expenditures
health
schooling
house
2005 (N=8,197)
expenditures
health
schooling
house
st.dev.
min
max
4,915
3,992
414.9
0.664
0.164
0.0673
10.53
3.606
0
183,883 110,226 17,806
2,6964
0.946
21
727,112
4,600
4,023
244.3
0.663
0.165
0.145
10.61
3.560
0
180,894 107,931 17,806
27,719
0.946
21
551,276
3,912
3,661
231.1
0.652
0.168
0.103
10.64
3.500
0
193,841 119,461 15,297
25,806
0.946
20.50
561,189
3,963
3,375
374.2
0.660
0.167
0.0948
10.88
3.358
0
191,422 119,802 15,297
28,274
0.946
21
824,656
4,591
3,698
403.9
0.654
0.169
0.0108
11.05
3.279
0
205,392 122,043 18,481
26,111
0.946
21
690,253
4,771
3,748
366.4
0.659
0.169
0.0673
11.13
3.211
0
2073,94 125,580 17,282
28,908
0.946
21
967,367
5,231
4,319
452.6
0.661
0.172
0.0794
11.16
3.216
0
204,963 122,220 16,814
33,310
0.946
21
773,531
5,340
4,243
512.3
0.663
0.169
0.119
11.29
3.157
0
207,953 122,610 16,515
31,543
0.946
21
773,531
5,632
4,459
569.9
0.670
0.170
0.119
11.35
3.138
0
204,102 120,402 17,232
32,533
0.946
20
909,199
Source: Authors’ calculations from RLMS-HSE.
Table 4: Dimension-by-dimension inequality (Bootstrapped standard errors)
year
expenditures
δ=2
δ=5
coef std. error
coef
std. error
health
δ=2
δ=5
coef std. error
coef std. error
1995
1996
1998
2000
2001
2002
2003
2004
2005
0.394
0.416
0.448
0.399
0.394
0.385
0.393
0.388
0.383
0.00573
0.00588
0.00586
0.00508
0.00491
0.00496
0.00548
0.00504
0.00520
0.140
0.141
0.146
0.144
0.147
0.146
0.147
0.145
0.143
year
0.638
0.662
0.697
0.636
0.635
0.630
0.634
0.631
0.621
education
δ=2
δ=5
coef std. error
coef
1995
1996
1998
2000
2001
2002
2003
2004
2005
0.190
0.185
0.182
0.170
0.164
0.159
0.159
0.154
0.152
0.00530
0.00559
0.00550
0.00552
0.00479
0.00473
0.00485
0.00481
0.00487
0.00235
0.00251
0.00260
0.00232
0.00198
0.00203
0.00188
0.00168
0.00190
0.4246
0.4153
0.4058
0.3804
0.3634
0.3525
0.3519
0.3419
0.3375
0.00291
0.00296
0.00332
0.00281
0.00267
0.00269
0.00256
0.00254
0.00272
std. error
0.310
0.312
0.320
0.315
0.321
0.320
0.324
0.319
0.319
housing
δ=2
δ=5
coef std. error
coef
0.00581
0.00572
0.00614
0.00496
0.00501
0.00491
0.00451
0.00403
0.00462
0.341
0.340
0.353
0.355
0.341
0.346
0.341
0.336
0.337
0.00491
0.00501
0.00512
0.00490
0.00495
0.00439
0.00474
0.00456
0.00520
Source: Authors’ calculations from RLMS-HSE.
0.00135
0.00132
0.00161
0.00136
0.00137
0.00133
0.00142
0.00130
0.00126
0.00436
0.00409
0.00489
0.00392
0.00398
0.00405
0.00394
0.00384
0.00396
0.648
0.643
0.665
0.664
0.656
0.665
0.654
0.647
0.649
std. error
Table 5: Multidimensional inequality measured by two S-Gini indices for δ = 2 and
δ = 5 (Bootstrapped standard errors)
I1
year
δ=2
coef
1995
1996
1998
2000
2001
2002
2003
2004
2005
0.274
0.280
0.293
0.276
0.269
0.267
0.269
0.264
0.262
I2
std. error
δ=5
coef
0.00218
0.00237
0.00246
0.00226
0.00189
0.00211
0.00213
0.00193
0.00197
0.525
0.531
0.550
0.522
0.517
0.516
0.515
0.509
0.505
std. error
δ=2
coef
0.00316
0.00363
0.00341
0.00286
0.00295
0.00271
0.00282
0.00295
0.00293
0.280
0.286
0.295
0.278
0.270
0.269
0.269
0.263
0.260
std. error
δ=5
coef
std. error
0.00297
0.00322
0.00326
0.00285
0.00275
0.00242
0.00280
0.00252
0.00280
0.463
0.469
0.465
0.455
0.446
0.447
0.445
0.437
0.433
0.00505
0.00546
0.00527
0.00436
0.00403
0.00390
0.00400
0.00394
0.00424
Source: Authors’ calculations from RLMS-HSE.