WPS3687
Re-Interpreting Sub-Group Inequality Decompositions
Chris Elbers, Peter Lanjouw, Johan A. Mistiaen, and Berk Özler1
Abstract
We propose a modification to the conventional approach of decomposing income
inequality by population sub-groups. Specifically, we propose a measure that evaluates
observed between-group inequality against a benchmark of maximum between-group
inequality that can be attained when the number and relative sizes of groups under
examination are fixed. We argue that such a modification can provide a complementary
perspective on the question of whether a particular population breakdown is salient to an
assessment of inequality in a country. As our measure normalizes between-group
inequality by the number and relative sizes of groups, it is also less subject to problems of
comparability across different settings. We show that for a large set of countries our
assessment of the importance of group differences typically increases substantially on the
basis of this approach. The ranking of countries (or different population groups) can also
differ from that obtained using traditional decomposition methods. Finally, we observe
an interesting pattern of higher levels of overall inequality in countries where our
measure finds higher between-group contributions.
World Bank Policy Research Working Paper 3687, August 2005
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
1Elbers is at Vrije (Free) University of Amsterdam. Lanjouw, Mistiaen and Özler are at the World Bank. We are grateful to
Tony Atkinson, Francois Bourguignon, Sam Bowles, Valentino Dardanoni, Jean-Yves Duclos, Francisco Ferreira, Gary
Fields, Ravi Kanbur, Jenny Lanjouw, Branko Milanovic, Adam Przeworski, Martin Ravallion, Tony Shorrocks, and Jacques
Silber for comments and/or helpful discussions. We are particularly indebted to Marta Menéndez for numerous contributions
to this project. Correspondence: planjouw@worldbank.org, jmistiaen@worldbank.org, or bozler@worldbank.org.
2
I. Introduction
The significance of group differences in wellbeing is often at the center of the
study of inequality. Roemer (1998) suggests that inequality of opportunity occurs when
the ability of people to pursue lives of their own choosing depends on predetermined
characteristics, such as gender, race, social group, or family background.2 This
perspective implies that it can be instructive to disentangle inequalities due to differences
between groups, defined in terms of such predetermined characteristics, from those due
to, say, individual differences in effort, talent, or luck. Given two countries with the
same overall income inequality, one might worry more about social stability and
prospects for inclusive long-term prosperity in the country with higher inequality
between groups.
Statistical methods are often used to `decompose' economic inequality into
constituent parts. Sub-group decomposable measures of inequality can be written as the
sum of inequality that is attributable to differences in mean outcomes across population
sub-groups and that which is due to inequality within those sub-groups.3 Many have used
such decompositions to `understand' economic inequality and guide the design of
economic policy. Indeed, Cowell (2000) argues: "It is almost essential to attempt to
`account for' the level of, or trend in, inequality by components of the population."
Although decompositions of inequality, as described above, have long been the
workhorse in this literature, empirical implementation has tended to find little evidence of
significant between group differences. For example, in a classic reference, Anand (1983)
2World Development Report 2006, entitled "Equity and Development," adopts a notion of equity that combines the concept of
equality of opportunities with the avoidance of absolute deprivation a Rawlsian form of inequality aversion in the space of
outcomes.
3See Bourguignon (1979), Shorrocks (1980, 1984) and Cowell (1980). Cowell (2000) provides a recent survey of methods of
inequality measurement, including a discussion of the various approaches to sub-group decomposition.
3
showed that inequality between ethnic groups in Malaysia accounted for only 15% of
total inequality in the 1970s. This led to his recommendation that government strategy
should focus on inequality within ethnic groups rather than that between them. Cowell
and Jenkins (1995), who find that most income inequality remains unexplained even after
taking into account the age, sex, race and earner status of the household head in the U.S.,
argue that the real story of inequality is to be found within these population groups and
point to the importance of chance.
Not everyone is comfortable with such interpretations, however. Kanbur (2000)
states that the use of such decompositions "...assists the easy slide into a neglect of inter-
group inequality in the current literature." He argues that finding a relatively small share
of inequality between groups does not mean that the mean differences between them are
less important than inequalities within such groupings. In particular, he argues that social
stability and racial harmony can break down once the average differences between groups
go beyond a certain threshold, with the threshold varying from country to country.4
There are also difficulties with comparisons of such decompositions across
settings (e.g among countries or over time). This is because underlying population
structures often vary. Consider three countries where the issue of racial differences in
income features prominently in public discourse: the United States, Brazil and South
Africa. The shares of income inequality attributable to differences between racial groups
4Foster and Sen (1997) point to the `separatist' view implicit in these sub-group consistent measures, which they claim
ignores potentially relevant information when making inequality comparisons. For example, should a change in inequality
within a certain group (while the means and population shares remain unchanged) when that group is richer than a second
group affect inequality in exactly the same manner as in the presence of a much wealthier second group? Sub-group
consistency requires this to be true. Kanbur (2000) builds on this argument and suggests that invoking such separatist axioms
"...go[es] against basic intuition and considerable evidence which suggest that individuals do indeed pay special attention to
outcomes for their particular racial, ethnic, or regional group."
4
in these countries are 8%, 16%, and 38%, respectively.5 Do these numbers provide a
good yardstick with which to judge the relevance of race to an understanding of
inequality in these countries? Should South African and Brazilian policy-makers worry
much more about racial differences in incomes than do their American counterparts?
Does the small percentage of income inequality attributable to race in the U.S. mean that
racial inequality is not a pertinent economic and social issue?
Conventionally, between-group inequality is calculated as a function of two
arguments: differences among groups in mean incomes and the relative size and number
of the groups. The figures above are based on four population groups for Brazil, three for
South Africa, and five for the U.S., but the population shares of the white groups versus
non-white groups differ tremendously.6 In each country, the mean income of the non-
white groups is much below that of the white group, but the non-white groups form the
majority in South Africa (80%), half of the population in Brazil (50%), and a minority in
the U.S. (28%). The difference in between-group inequality observed between these
three countries could in fact be due largely to the difference in population shares of the
racial groups instead of the differences in relative mean incomes of these groups.7
The conventional between-group share is calculated by taking the ratio of
observed between-group inequality to total inequality. Total inequality, however, can be
viewed as the between-group inequality that would be observed if every household in the
5These figures have been calculated by the authors using data from PNAD (2001) for Brazil, IES(2000) for South Africa, and
LIS(2000) for the U.S.
6The racial groups used in our analysis are "White", "Black", "Pardo", and "other" in Brazil, "White", "African", and "other"
(combining Coloreds, Asians/Indians, and others) in South Africa, and "White", "Black", "Hispanic", "Asian", and "American
Indian" in the U.S.
7The observed differences in between-group inequality may also depend on the number of groups under consideration,
making the specific definition of groups a non-negligible issue. For example, the share of between-group inequality
attributable to caste in India when one groups people simply into "high", `medium", or "low" caste groupings, can be quite
different from that which emerges when the partitions are finer, i.e. when one makes distinctions between castes within each
broad category.
5
population constituted a separate group. Thus, the conventional practice is equivalent to
comparing observed between-group inequality (across a few groups) against a benchmark
(across perhaps millions of groups) that is quite extreme. It is not surprising that one
rarely observes a high share of between-group inequality.8 In this paper, we propose an
alternative measure to assess between-group inequality. Specifically, we suggest
replacing total inequality in the denominator of the conventional ratio with the maximum
between-group inequality that could be obtained if the number of groups and their sizes
were restricted to be the same as for the numerator. Because our proposed measure
normalizes by the number of groups and their relative sizes in a country, decompositions
can be better compared across settings where the number of groups (or the population
shares for those groups) is very different.
We also argue that our measure is better suited to capture the salience of a
specific population breakdown to the assessment of inequality of opportunities. As
indicated earlier, inequality of opportunity is concerned with systematic differences
among groups who differ only in skin color, caste, gender, etc. predetermined
characteristics that are arguably "morally irrelevant". Consider the following example
(illustrated in figure 1). Imagine a country with two population groups of equal size:
serfs and landlords. Mean income of the serfs is low while landlords enjoy a high mean
income. In period 1, there is no variation in the incomes of individuals who belong to the
same group, i.e. one's social group determines his or her income perfectly. In period 2,
some random noise, i, is added to each income. One can think of i as pure luck or,
alternatively, measurement error. Hence, in the second period, there is income inequality
within each group, although the respective means are still the same as in period 1.
8See Anand (1983), Cowell and Jenkins (1995), Elbers et al. (2004).
6
Suppose that the resulting two distributions do not overlap. Has inequality of opportunity
changed from period 1 to period 2?9
It is not clear why the presence of some random variation around these group
means should be indicative of any change in underlying opportunities. After all, even the
luckiest serf is still far poorer than the poorest landlord. However, if the question above
were to be assessed on the basis of the between-group share that derives from traditional
inequality decomposition, one would conclude that inequality of opportunity had fallen
from period 1 to period 2. This is because within-group inequality in period 2 increased
while between-group inequality remained unchanged (as mean incomes and population
shares for the two groups were unchanged), implying that the share of between-group
inequality fell between periods 1 and 2. As will be elaborated further in section 2, our
proposed measure would remain unchanged suggesting that inequality of opportunity
did not fall between periods 1 and 2.10
9Some readers will note that this example is somewhat similar to Example 1 in Esteban and Ray (2004), where their Figure 1b
is analogous to period 1 in our example, while Figure 1a is akin to period 2. However, there is one important difference: the
groups in Esteban and Ray are defined by incomes. Whether blacks and whites, or serfs and landlords fill the income
distribution is of no consequence in their example. We are interested in income differences between groups defined by
another characteristic, such as race, class, gender, parent's education, etc. In general, in an example such as this, the Esteban
and Ray polarization index would register a decline in polarization as we move from period 1 to period 2. However, it is
interesting to note that in the specific case of the two groups having identical sizes, the Esteban and Ray polarization index
would in fact register an increase in polarization. For our purposes, the point to emphasize is that the polarization index, like
between-group inequality, would not remain unchanged in moving from period 1 to period 2.
10Our alternative measure would also start declining in period 2 if the two income distributions started to overlap, but at a rate
much slower than the traditional between-group inequality share (see section 2).
7
Figure 1: Inequality Between Serfs and Landlords
The rest of the paper is organized as follows. Section 2 briefly reviews the
theoretical inequality decomposition literature and introduces our alternative approach.
Section 3 draws on a newly compiled database of inequality and sub-group contributions
for just under 100 developed and developing countries to demonstrate that qualitative
assessments of the importance of between-group differences can, but need not, be
markedly higher when based on our alternative approach. This section also discusses a
thought-provoking finding of a strong positive correlation across countries between
overall inequality and our proposed measure of inequality between groups. Section 4
concludes.
8
2. Methodology
The standard approach to decomposing inequality by population sub-groups
breaks overall inequality into a between-group and a within-group component. The first
component indicates how much of overall inequality would remain if incomes were
equalized within each population group, i.e. each member of a particular group being
given the group's average per capita income. The within-group component captures the
amount of inequality that would remain if differences between groups in terms of their
average incomes were eliminated and only within-group differences remained.
Not all summary measures of inequality can be neatly decomposed into these two
components. The most commonly decomposed measures in the literature come from the
General Entropy class. These take the following form:
GE = 1 yi c
for c 0,1
c(c -1)
i fi -1
= - fi log yi for c=0
i
= fi log
yi yi for c=1
i
where fi is the population share of household i, yi is per capita consumption of household i,
is average per capita consumption, and c is a parameter that is to be selected by the user.11
This class of inequality measures can be decomposed into a between and within-group
component as follows:
11Lower values of c are associated with greater sensitivity to inequality amongst the poor, and higher values of c place more
weight to inequality among the rich. A c value of 1 yields the well known Theil entropy measure, a value of 0 provides the
Theil L or mean log deviation, and a value of 2 is ordinally equivalent to the squared coefficient of variation.
9
GE = 1 j c j c for c 0,1
c(c -1) 1- gj + GEjgj
j j
= gj log
+ GEjgj
for c=0
j j j
= log + GEjg j j
g for c=1
j
j j j
j
where j refers to the sub-group, gj refers to the population share of group j and GEj refers to
inequality in group j. The between-group component of inequality is captured by the first
term: the level of inequality if everyone within each group j had consumption level j. The
second term gives within-group inequality.
Table 1 decomposes inequality on the basis of "social group" in eight countries.
Social group is defined differently across countries, but refers loosely to the racial, ethnic,
or caste breakdown that is relevant to each country. For example, the breakdown for the
United States corresponds to five racial groups: Whites, Blacks, American Indians,
Asians and Hispanics. In India the three groups comprise Scheduled Caste households,
Scheduled Tribes, and Others. The number of groups and their respective sizes are
clearly not the same in all countries. Inequality is estimated on the basis of per-capita
consumption for each country and we have chosen to measure it using a General Entropy
Class measure with parameter value zero. This is often referred to as the Theil L measure
or the mean log deviation, and compared with other General Entropy Class measures
places a good deal more weight on inequalities amongst the poor.
Based on the standard approach to decomposing inequality, as described above,
between group inequality in each country in our list is rather low (Table 1, column III).
10
Only South Africa stands out with a between-race share of 38%. Even here, however, it
is striking to note that nearly two-thirds of total inequality in South Africa can be
attributed to differences within racial groups as opposed to differences across groups.
The generally low between-group shares we observe in Table 1 are typical in the
literature, even for other population breakdowns commonly encountered.12 But how
should this finding of a low between-group share be interpreted? Does it mean that the
population breakdown along social dimensions is not terribly relevant to thinking about
inequality in these countries?
In what follows, we propose an alternative perspective on the between-group
share of inequality. While South Africa's between race inequality share is "only" 38%,
we show that observed between race inequality accounts for more than 50% of the
`maximum possible' between-race inequality in South Africa given its current income
distribution, the number of racial groups, their sizes, and their ranking in terms of average
income (see Table 1, column IV). A similar point can be made when comparing Brazil
and Panama. Based on the standard decomposition by race/ethnic group, the between-
group share of inequality in both countries is only about 16%. This calculation would
conventionally be interpreted as suggesting that race or ethnicity is of limited relevance
to an understanding of inequality in these two countries. However, in Panama, observed
between-race inequality accounts for well over a third of `maximum possible' between-
race inequality, while in Brazil the conclusion based on our measure is only slightly
different from that which obtained from the standard calculation (see Table 1).
12For example, Elbers et al. (2004) demonstrate that the share of inequality attributable to differences between the 1248
communities into which Madagascar can be subdivided is approximately 25%.
11
2.1 Maximum Between-Group Inequality
In studies of inequality one often encounters statements of the following type:
"between-group income inequality accounts for only 20% of total inequality". Such
statements, however, should not be taken to mean that 100% of total inequality would
have been a realistic possibility. A between-group share of 100% would be possible only
under two, rather unlikely, scenarios. First, if each household constitutes a separate
"group" then total inequality is clearly also equal to between-group inequality. Second, a
between group share of 100% would occur if there were fewer groups than households,
but somehow all the households within each of these groups genuinely happened to have
identical per capita incomes. Rather than having a bell-shape, the density function of
income in this latter case would consist of a series of spikes occurring at the average per
capita income level for different groups (as in the first period in figure 1 above). It is
difficult to imagine a realistic setting in which this would occur: for virtually any
empirically relevant income distribution and a limited number of groups, the share of
maximum between-group inequality that can be attained is strictly below 1.
Hence, not all possible groupings of the population are equally relevant in
assessing the salience of inequality between certain groups. But, what groupings are
most relevant? In this section, we propose a measure that evaluates observed between-
group inequality (BGI) against a benchmark of maximum between-group inequality that
can be attained when the number and relative sizes of groups under examination are
fixed:13
13
A different approach could be based on statistical testing. One could ask how (un)likely it is that a particular value of
between-group inequality is the result of pure chance given the number of groups and their relative sizes, i.e., test the null
hypothesis that it is the result of a random allocation of incomes over households in society (keeping the overall income
distribution constant). However, in practice, this exercise always leads to the rejection of that null hypothesis, i.e. observed
12
Rb = BGI = Rb total inequality
maximum BGI maximum BGI
Since BGI can never exceed total inequality, it follows that Rb' cannot be smaller
than Rb.14 It is also clear from the formula above that if maximum BGI attainable is
close to total inequality, then Rb and Rb' will also be close to each other. Put differently,
if there is a way of reordering the population into a given number of groups with fixed
sizes such that the inequality between the resulting groups is almost equal to total
inequality, then Rb' will not differ significantly from Rb. This is true, for example, in the
case of inequality between social groups in Brazil, where our alternative measure is only
slightly higher at 20% than the conventional between-group share of 16%, but not in
Panama, where the analogous figures are 36% and 17%, respectively (see Table 1).
To see how the maximum attainable BGI can differ from context to context, take
the rectangular and triangular distributions depicted in figure 2 below. In both cases,
assume that there are two groups, each containing half the population. A necessary
condition for BGI to be at its maximum is that these two groups occupy non-overlapping
partitions of the distribution of income: if {y} is an income distribution for which BGI is
maximized, and g and h are different groups then either all incomes in g are higher than
all incomes in h, or vice versa. (See Shorrocks and Wan, 2004, section 3.). Hence, for
each of the distributions in figure 2, the maximum BGI is attained when one group
between-group inequalities are almost always above zero in a statistically significant manner. While we could use these
significance levels to rank levels of between-group inequality, it is not so clear how to interpret a difference between
significance levels of, say, 0.999 and 0.9995.
14We follow the notation in Cowell and Jenkins (1995).
13
occupies the bottom half of the income distribution, and the other the top.15 In this
particular example, it can be readily verified that maximum BGI as measured by GE(0) is
0.14 in the uniform case and 0.06 for the triangular distribution. In this hypothetical
society, an observed between-group inequality of, say, 0.05 is arguably much more
extreme (or salient) in the case of the triangular income distribution than the uniform one.
Figure 2: Uniform and triangular densities
In order to calculate Rb' we need to know BGI, which can be calculated in the
usual way, and maximum BGI, which is slightly more difficult to compute. A "brute-
force" approach to calculating maximum BGI uses the property (mentioned above) that
under a BGI-maximizing distribution, group incomes occupy non-overlapping intervals.
In the case of n groups, the following approach can be followed: take a particular
permutation of groups {g(1),..., g(n)}, allocate the lowest incomes to group g(1), then to
g(2), etc., and calculate the corresponding BGI. Repeat this for all possible n!
permutations. The highest resulting BGI is the maximum sought. This approach is
obviously easier when the number of groups under examination is small. In the
15Because the sizes of the groups are identical, it does not matter which group occupies which part of the distribution.
14
appendix, we describe some alternative approaches to solving the problem of maximizing
between-group inequality for the Gini coefficient.16 The same appendix also shows that
without restrictions on the income distribution, no group order can be a priori excluded as
a candidate for the BGI maximizing one.
In general, maximum BGI need not increase if the number of groups grows.17
However, BGI cannot decline if more groups are obtained via proper sub-divisions of the
existing groups.18 In the limit, when every individual constitutes her own group,
maximum BGI equals total inequality, and consequently Rb=Rb'.
A possibly more appealing benchmark against which to evaluate between-group
inequality can be obtained by introducing one more restriction. In addition to fixing the
number of groups and their relative sizes, we can also arrange the groups under
examination according to their observed mean incomes, keeping their `pecking order'
unchanged.19 In many cases, there is a well-understood hierarchy of population groups in
terms of their mean incomes. Comparing actual between-group inequality to a
counterfactual maximum BGI which preserves the actual, observed, rank ordering of
groups is conceivably of greater interest than a counterfactual which allows for random
16The choice of Gini is for computational convenience and is not uncommon in the literature that precedes us, such as Davies
and Shorrocks (1989).
17In the case of the Gini coefficient, the difference between total inequality and maximum BGI can be crudely bounded and
the bounds are functions of the population share of the largest group, and not by the number of groups (see the appendix for a
proof). Maximum BGI will stay bounded away from total inequality unless the maximum group size becomes sufficiently
small. This implies that although the expected value of between group inequality might increase as the number of groups
increases (Shorrocks and Wan (2004), proposition 3), this value may be well below total inequality if one of the groups
remains very large. For example, with a lognormal (0,1) distribution and one group occupying 70% of the population while
every other individual constitutes a separate group (implying effectively an infinite number of groups), the maximum possible
between group inequality (measured by GE(0)) would be 0.373 well below the inequality level of 0.5 in this distribution.
For related results, see Davies and Shorrocks (1989) and Shorrocks and Wan (2004).
18 Starting from a BGI-maximizing group order, split-up a group in such a way that mean incomes in the new groups are equal
to the old parent group. BGI among these groups is equal to the old BGI. Obviously, if the incomes of the two new groups
are non-overlapping, then BGI will increase. Hence a refinement of a particular grouping will generally lead to an increase in
BGI. See also Shorrocks and Wan (2004), proposition 2.
19Ordering population groups by their mean incomes using, say, a household survey would introduce a possible difficulty due
to sampling variability. In other words, our ability to order groups by mean income (or consumption) could be limited by the
fact that some of the group means are statistically indistinguishable from each other. For the time being, we ignore the
standard errors associated with the observed group means.
15
re-ordering of groups. For example, when decomposing inequality by race in Brazil,
South Africa, or the U.S. (see the example in Section 1), the ordering of racial groups in
terms of mean incomes is well-documented, and it is not obvious to what extent a
counterfactual of say, average income of blacks exceeding that of whites would be
realistic and of any inherent interest. This approach is also appealing for practical
reasons as it involves just one, rather than n! calculations of BGI.
Obtaining the maximum possible BGI given the current income distribution,
relative group sizes, and their rankings by mean incomes is trivial: allocate the lowest
incomes to members of the group with lowest mean income, the lowest remaining
incomes to the group with the 2nd lowest mean income, etc. In the rest of this paper, Rb'
will refer to our index of BGI normalized by the maximum possible BGI given the
current income distribution, relative group sizes, and their "pecking order."
Salience of grouping for income inequality
In the preceding sections we have introduced Rb' in an effort to assess whether
group sub-divisions that should not be relevant in a moral sense, in fact display
significant between group inequality. We can compare one way of sub-dividing the
population to another. We could have asked an alternative question: how well does a
person's income predict his or her group membership in one kind of population grouping
relative to another? It is clear that in the case where an income distribution is divided
into non-overlapping groups, a person's group can be perfectly predicted once her
income is known. In general, if income is a good predictor of group membership then it
would seem reasonable to regard that particular grouping as "salient" to the analysis of
16
inequality, especially inequality of opportunity. Thus we are led to ask if Rb' indicates
salience in the above sense. We are able to show that, compared to Rb, it is indeed more
sensitive to overlap in the support of the groups' income distributions and is less sensitive
to inequality within those groups.20 An illustration may clarify.
Consider again a population consisting of two groups. Each group's (weighted)
density of the income distribution is graphed in figure 3. Suppose we introduce a series
of progressive transfers amongst the population represented by density f1. Specifically,
we transfer incomes from those individuals with income below b but above a, to those
individuals with income below a. We continue with these transfers until all individuals
with incomes below b have the same income a. Clearly, redistributing incomes in this
specific way has not affected our ability predict membership of either of the two groups
based on knowledge only of observed incomes. Because group means, BGI, and
maximum BGI are unaffected, Rb' is also unchanged. However, within-group inequality
and total inequality decrease (because of the progressive transfers within group 1) and so
Rb will go up. This change in Rb reflects a drop in total inequality that is not correlated
with one's success in the `salience game' described above.21
20Whether Rb' can be interpreted as an average success rate of guessing a person's group membership on the basis of income
information alone, is a question we leave for future research.
21Pyatt (1976) also invokes the concept of a "game" when exploring the feasibility of sub-group decomposition of the Gini
coefficient. He highlights the significance of the degree of "overlap" between groups in this procedure.
17
Figure 3
f2
f1
a b
The perspective on `salience' of group definitions offered by Rb´ may be of
interest also in settings other than the analysis of inequality of opportunity. While the
latter exercise starts from the position that certain predetermined group definitions are
judged to be "morally irrelevant" (Roemer, 1998) and then seeks to ascertain to what
extent these groups are relevant to an understanding of inequality, there may also be
situations where group definition does not precede the analysis, but is determined ex-
post. For example, a politician may be interested in tailoring his economic policies and
messages to specific groups in the population and would like to know which group
definitions are most `salient' in the sense we are considering here.22 In this case, he (or
she) might consider performing a search across different group definitions until the most
`salient' definition is identified. Basing this "search" on Rb´ would appear to be
22Kanbur (2005) gives a similar example while discussing the policy implications of using conventional inequality
decompositions.
18
particularly appealing as it is more readily comparable across different group definitions
as a result of the normalization by maximum BGI.23
3. Evidence
Expressing observed between-group inequality as a fraction of the maximum
possible BGI can provide additional insight in the analysis of inequality. Figures 4-6
decompose inequality on the basis of three different sub-group definitions for a number
of developed and developing countries. Our data come from nationally representative
household surveys from each of the countries and all refer to a year during the 1990s.
We consider three ways of breaking down the population in each country: by social group
membership, rural-urban location of residence, and education of household head (not all
countries' data permit a breakdown along all three dimensions, however). Of these
groups, only social group membership (which loosely refers to racial, ethnic, or caste
breakdown relevant to each country) can be truly considered as a `circumstance' or a pre-
determined characteristic, and as such consistent with measuring inequality of
opportunity. In the rest of our empirical work, we would have ideally used place of birth
instead of rural-urban residence, and parents' education in place of education of
household head. However, many of the household surveys in our database do not permit
us to break the population down by these circumstance variables, and hence the variables
we use instead can be viewed at best as crude proxies for an individual's circumstances.24
For each country and for each sub-group definition, the between-group share is calculated
23We are grateful to Sam Bowles for suggesting to us this interpretation of Rb´.
24Such data on individual circumstances, of course, exists for an increasing number of countries. For example, drawing on the
distinction between `circumstance' and `effort' variables in Roemer's work on equality of opportunity, Bourguignon, Ferreira,
and Menéndez (2003) uses a different method to decompose earnings inequality in Brazil into a component due to unequal
opportunities and a residual term.
19
in both the conventional manner, with total inequality as denominator, as well as on the
basis of the Rb' calculation outlined above.
The data are not strictly comparable as inequality is typically measured differently
across countries based sometimes on a consumption measure of welfare and sometimes
on an income measure. Even where the welfare indicators are based on the same
concept, the precise definition is almost never the same across countries. We have
explored the sensitivity of decomposition analyses to alternative welfare indicators for a
sub-set of 14 countries in which we have both income and consumption data. We have
found that while overall measured inequality typically varies markedly across welfare
indicators (with measured inequality based on an per-capita income measure usually
being higher than inequality based on a per-capita consumption measure), decomposition
results tend to vary only slightly. This finding that inequality "profiles" are less sensitive
to different underlying welfare definitions than direct inequality comparisons echoes a
similar finding in the poverty literature that poverty profiles are often quite robust to
varying underlying welfare definitions and poverty lines (see Lanjouw and Lanjouw,
2001). Thus, while the data examined here are far from comparable in terms of overall
measured inequality we contend that comparisons of decomposition results are much less
problematic.
Moreover, as we have emphasized above, one of the attractions of the Rb'
measure is that it normalizes by the observed number and relative size of observed
groups, within each distribution of income that is being considered. We have already
described above how, for example, decomposition by social group involves quite
20
different group definitions and sizes in different countries. Working with Rb' rather than
Rb is thus less subject to comparability concerns from this perspective as well.
Figure 4 decomposes inequality on the basis of a rural/urban breakdown in 85
countries. Countries are grouped by region and ranked within each region by
conventionally calculated Rb. In each country both Rb and Rb' are reported. Several
observations can be offered. First, in most countries the conventionally calculated
between-group share is generally well below the Rb' calculation. Indeed, in Senegal,
Guinea, Burundi, Kenya, Guatemala, Panama and Bolivia, the between-group
contribution based on Rb' rises above 40%, suggesting that in these countries inequality is
strongly colored by these spatial issues. In only two out of 85 countries is conventionally
calculated Rb as high as 30%, but the number increases eight-fold to 16 out of 85 on the
basis of Rb'. The between-sector inequality contribution is generally lowest for the most
developed countries in our sample, as well as for a number of the Eastern Europe and
Central Asian countries, irrespective of the manner in which this contribution is
calculated (see further below).
Figure 5 returns to the breakdown of inequality by social groupings described in
Table 1, now for a total of 35 countries. Again, the definition of social group (and
number of groups) differs across countries, but is generally based on some criterion
related to ethnicity, races, or religion. The evidence in Figure 5 suggests that social
grouping is a particularly important dimension of the inequality profile in South Africa,
Paraguay, Guatemala and Panama. In these countries, Rb' is above 30%, and indeed in
South Africa it reaches to nearly 60%.
21
While Rb' is always higher than conventionally measured Rb, the degree to which
these two statistics differ varies considerably across countries. In Nepal and Madagascar,
for example, one's assessment of the salience of social groups does not much vary across
the two approaches, while in South Africa, Paraguay, Vietnam, France, Panama, and
Peru, they yield very different conclusions. It is interesting to note the rank reversals
between U.S., Germany, and France when our alternative approach is employed.
Figure 6 decomposes inequality on the basis of roughly five education groups in
each of 91 countries.25 Education level of household head is a particularly salient
dimension of inequality in many Latin American countries, as well as in several African
countries and Thailand. In general, although Rb' is naturally higher than Rb, the
difference between the two statistics is not as large for decompositions by education as in
the previous two population breakdowns. Indeed, the ranking of countries on the basis of
Rb' is not much different from that on the basis of Rb.
Overall, we can see from these illustrative calculations that employment of the
Rb' calculation has the general effect of significantly raising one's assessment of the
importance of group differences in an examination of inequality. To the extent that this
approach is viewed to contribute a meaningful perspective on the importance of group
differences, the qualitative conclusions that have tended to be drawn in the conventional
literature may merit reconsideration.
Correlating Total Inequality and Between-Group Inequality
25The five broad education categories correspond to levels achieved by the household head, and refer to: no education, up to
primary only, above primary but below secondary completion, secondary completion, post-secondary education. This
definition of education groupings could not be applied in an exactly identical manner in all countries and is therefore only
broadly comparable across countries.
22
As mentioned in Section 1, Kanbur (2000) has cautioned against concluding that
simply because (conventionally calculated) between-group contributions to inequality are
generally low, this should be taken to imply that between group differences are of only
limited importance to an overall assessment of inequality. In the spirit of probing further
this concern we ask here whether, across our set of countries, there is any statistical
relationship between overall inequality and the percentage contribution that is attributable
to between-group differences. We regress overall inequality in each country separately
on the between-group contribution (based on Rb') attributable to four population
breakdowns: rural-urban location of residence, social group, occupation of household
head, and education of household head. As we have noted, our data are far from
comparable in terms of overall measured inequality due to different definitions of welfare
being employed in different countries. To accommodate this concern, albeit only
partially, we include in our regression a set of regional dummy variables as well as a
dummy indicating whether a particular country's inequality is measured on the basis of
per-capita consumption or income. Regression results have also been screened for the
influence of outliers and influential observations.26
Figure 7 presents our results. There is strong evidence of a positive correlation
between overall inequality and the between-group contribution, irrespective of the
specific group definition. It is important to realize that there is nothing inherent in the
mechanics of the decomposition calculation that ensures that there should be a positive
relationship between the overall level of inequality and the percentage contribution that
26We do not report regressions results based on a model of overall inequality on Rb. Our qualitative findings are similar, but
as described in the text there are grounds for doubting the comparability across countries of these measures of between-group
inequality contribution.
23
can be attributed to between group differences.27 In Figure 7, we can see that in all cases
considered here there is great sensitivity of overall inequality to between-group
differences and this is strongly significant for all group decompositions.
These correlations are suggestive but, of course, far from conclusive.
Nevertheless they are consistent with an argument that has been articulated most recently
in the World Bank's 2006 World Development Report, namely that overall inequality in
the developing world tends to be high and to persist over long periods of time in those
countries in which inequalities of opportunity across population groups are accentuated.28
The Report argues that the level and persistence of such inequalities of opportunity act as
a brake on economic growth and dampen prospects for rapid poverty reduction. For this
reason policy makers have an important instrumental reason for concentrating on
reducing group differences alongside the more conventionally acknowledged intrinsic
objections to inequality.
4. Concluding Remarks
In this paper, we propose a modification to the conventional approach of
decomposing income inequality by population sub-groups. We note that the conventional
practice of calculating the share of between-group inequality is equivalent to comparing
observed between-group inequality (across a few groups) against a benchmark (across
perhaps millions of groups) that is quite extreme. Specifically, we propose a measure
that evaluates observed between-group inequality against a benchmark of maximum
27Indeed, if there were concerns about noise in the data, high inequality countries would likely be countries in which there
was more noise. Pure noise would result in smaller between-group shares (because of greater overlap across groups). As a
result, if anything one might expect a negative relationship.
28As mentioned earlier in this section, only differences between `social groups' in these countries can strictly be interpreted as
inequality of opportunity in the Roemer sense. The income/consumption differences between other groups, such as rural-
urban, education, and occupation are likely due, at least in part, to choices people have made.
24
between-group inequality that can be attained when the number and relative sizes of
groups under examination are fixed. As our measure normalizes between group
inequality by the number and relative size of groups under examination, it is also less
subject to problems of comparability across different settings. We argue that our
modification can provide a complementary perspective on the question of whether a
particular population breakdown is salient to an assessment of inequality in a country.
It is important to note that our measure is not the result of a statistical
decomposition of any inequality measure of a certain class. Rb' is concerned with
evaluating between-group inequality against a proper benchmark and as such places less
emphasis on inequality within groups. It is our contention that if one is interested in
assessing inequality of opportunity between certain groups, using the traditional
contribution of between-group inequality to overall inequality may unduly color that
assessment.
Our measure is simple to calculate, particularly when we preserve the "pecking
order" of the groups under examination. We find that for a large set of countries our
assessment of the importance of group differences typically increases substantially on the
basis of this alternative approach. The ranking of countries (or different population
groups) can also differ from that obtained using traditional decomposition methods.
Finally, we observe an interesting pattern of higher levels of overall inequality in
countries where our measure finds higher shares of between-group contributions.
25
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26
Lanjouw, J. O. and Lanjouw, P. (2001). `How to Compare Apples and Oranges: Poverty
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27
Appendix: Maximum Between Group Inequality: Analysis and Programming
In this appendix we will be mainly concerned with the Gini coefficient for measuring
inequality. For reasons of exposition we will assume that the income distribution is
absolutely continuous with density f(y), CDF F(y) and Lorenz curve L(p). We define
between-group inequality as the total inequality one would obtain if all incomes within
groups were equal. As mentioned in Foster and Shneyerov (2000), between-group
inequality can be defined in several ways that lead to different outcomes for a between-
group Gini if groups have overlapping income ranges. However, the same ambiguity does
not exist for maximum between-group inequality, since the maximum implies non-
overlapping income ranges.
Figure A.1 depicts the Lorenz curve: L(p), p[0,1], as well as a Lorenz curve based
on the same distribution, but for groups j=1,...,n, with non-overlapping incomes, and
equal incomes within groups: Lg(p) . The share of each group in the population is wj.
The following are easily verified:
1. If the groups have non-overlapping income they can be mapped along the
horizontal axis of the graph according to increasing per capita income, as adjacent
intervals of width wj.
2. L(p) = Lg (p) at boundery points p of the intervals.
3. Lg is a piecewise-linear approximation to L. The approximation is better, the
smaller are the population shares wj.
4. L(p) Lg (p). Hence the Gini (and all Lorenz-consistent inequality indices) of Lg
is smaller than that of L.
28
The between-group Gini for groups with non-overlapping income ranges is
unambiguously given by the Gini coefficient corresponding to Lg(p) . Since non-
overlapping income ranges are a necessary condition for achieving maximum between-
group inequality we have proved
Theorem
Under the conditions of this section, maximizing the between-group Gini is equivalent to
ordering the groups j=1...,n along the horizontal axis in such a way that the area under
the resulting Lorenz curve Lg is minimal.
In principle, one can solve the max-BGI problem by trying out all n! group orders and
calculate the BGI index (Gini or other) for the resulting piecewise-linear Lorenz curves.
Obviously, this is not a viable strategy if the number of groups is big.29
The max-BGI problem can be formalized as an integer programming problem as
follows. Define integer variables aij to equal 1 if group i has strictly lower expenditure
than group j, and zero otherwise. (So group i comes before j in the income distribution.)
The aij must satisfy the following conditions:
aij{0,1}
aii=0
ijaij+aji=1
i,j,k:aikaij+ajk-1
29One may verify that if the Lorenz curve is quadratic, the between-group Gini does not
depend on the order of groups. Further, it should be clear that one cannot exclude a
particular group order a priori. Take any group order and give equal incomes to members
from the same group (and incomes increasing with group rank). Then that particular order
maximizes BGI for the resulting income distribution.
29
The last condition is in fact a `linear version' of the transitivity condition
i, j,k :aik aijajk which states that if i is poorer than j and j is poorer than k, then i is
poorer than k. The location of group j's interval on the horizontal axis can now be
expressed as (lj,uj], where
lj= wiaij
i
uj=lj+wj,
while group j's income share is L(uj)-L(lj) . Obviously, u j = l . Geometrically,
j+1
from the total size of all groups poorer than j, i.e. lj, and group j's size wj, the location of
the group's chord on the Lorenz curve L(p) can be determined. See again Figure A.1.
Group j's contribution Sj to the area under the linearized Lorenz curve is
S j = wj L(l ) + L(u j )
j
2
and so the programming problem becomes
N
min wj (L(lj)+L(uj)) with respect to aij ,
j=1
subject to
aij{0,1}
aii=0
ijaij+aji=1
aikaij+ajk-1
lj= wiaij
i
uj=lj+wj,
where i,j,k=1,...,N.
30
For other inequality measures the max-BGI problem can be formulated using the
same constraints but a different objective function. Many inequality measures (such as all
those in the GE class and the Atkinson inequality measures) are defined as functions of
the groups' relative mean incomes j/, where denotes overall average income and j
average income in group j. Such inequality measures can be expressed in terms of the
above notation by noting that with non-overlapping income ranges
j/ = (L(uj)-L(lj))/wj.
Although all the constraints in the program are linear, the objective function is not,
and the problem turns out to be a highly `non-convex' integer program. Easily available
solvers for this type of problem do not work.30 Fortunately, some further analytical
results can be obtained for the Gini. Moreover, as mentioned in the text, a maximum
BGI that preserves the income ranking of groups represents a practical, if not entirely
exact, alternative approach.
The Gini coefficient of BGI: further analysis and computation
In this sub-section we present some further analytical results on the problem of
calculating maximum BGI. These results are for the Gini coefficient and as such of
limited value for the analysis of between-group inequality. On the other hand, a between-
group Gini maximizing group order is very likely to have a close-to-maximum between-
group inequality index for other measures as well.
30We tried the GAMS solvers DICOPT and SBB.
31
We use the following extension to Jensen's inequality.
Lemma
Let g(x) be a (not necessarily strictly) convex function on a convex set X. Then for t>0
and (x-t,x+t)X:
g(x + t) + g(x - t) 1 t g(x + u)du g(x).
2 2t
-t
Proof
Obvious. The theorem essentially states that the effect of a mean-preserving spread gets
stronger as probability mass is shifted outward, the mean being equal to x here. Further
below we apply the lemma with the first derivative of the Lorenz curve (i.e. F-1(p) ) in
the role of g(x).
Corollary
Let G the an anti-derivative of g defined by G(x + t) = x+t
g(s)ds , then
x
H(t) = G(x + t) - G(x - t)
is increasing in | t |.
2t
This can be seen by taking the derivative of the above expression and noting that for t > 0
H'(t) = 1 g(x + t) + g(x - t)- 1 [G(x +t) -G(x -t)]
t 2 2t2
= 1 g(x + t) + g(x - t) - 1 t
t 2 2t
-t g(x + u)du
32
from which H'(t)0 follows by applying the lemma. The corollary can be used to proof
the following
Theorem
Let groups A and B be adjacent in a group order that maximizes the between-group Gini
coefficient, with A having the lower incomes. Let F -1(p) be convex on the interval
(lA,uB]. Then wA wB .
Remark
Since F(y) is nondecreasing, it is concave where F -1(p) is convex and vice versa. This
corresponds to an income distribution which has monotonically decreasing density.31
According to the theorem a BGI (Gini) maximizing group order must have decreasing
group sizes if the density is decreasing over the relevant income range.
Proof
Let be average income, then the Lorenz curve satisfies
L(p) = F-1(s)ds.
p
0
Recall that the contribution of groups A and B to the between-group Gini is minus the
surface SAB under the chords connecting the points (l A, L(l A)) , (uA,L(uA)) = (lB,L(lB))
and (uB, L(uB )). If the group order A,B is part of a Gini-maximizing order, the surface
under the chords must not be higher than with groups A and B in reverse order.
31Convex functions are differentiable almost everywhere.
33
Let A and B be located between percentages s = l A and s + wA + wB = uB . If A is before
B the contribution of the two groups is
2SAB = wA(L(s) + L(s + wA)) + wB (L(s + wA) + L(s + wA + wB )).
Alternatively, if B is before A the contribution is
2SBA = wB (L(s) + L(s + wB )) + wA(L(s + wB ) + L(s + wA + wB )).
The difference can be written as
SAB - SBA = (wB - wA)(wA + wB )
1 L(s + wA + wB ) - L(s) - .
2 wA + wB L(s + wB ) - L(s + wA)
wA - wB
The difference can be negative only if wA > wB . To see this set x = s + (wA + wB )/ 2,
tAB=(wA+wB)/2 and tBA = (wB - wA)/ 2. Since tAB > tBA it follows from the lemma's
corollary that
L(s + wA + wB ) - L(s) = L(x + tAB ) - L(x - tAB )
wA + wB 2tAB
L(x + tBA) - L(x - tBA) = L(s + wB ) - L(s + wA) .
2tBA wA - wB
QED
Note that the same method of proof can be used mutatis mutandis to show that groups
must be in increasing size order if the income distribution has increasing density. Thus
we have the following corollary.
34
Corollary
If the income distribution is unimodal, then a between-group Gini maximizing group
order has groups arranged in increasing size where the density is increasing and in
decreasing size where the density is decreasing.32
This last result is useful for practical computation. It implies that for the common case of
a unimodal distribution the largest group is located near the mode of the distribution with
increasing group sizes for incomes lower than the mode, and decreasing group sizes for
higher incomes. When the fraction of the population with incomes below the mode is of
the same order of magnitude of (or smaller than) group sizes, only n or n2 of the n! group
orders will have to be checked to find the order maximizing BGI. Note further that if a
particular group order maximizes the between-group Gini, it is likely also to lead to
values for other inequality measures, close to the maximum.
Bounds on the maximum between-group Gini
The difference between the maximum between-group Gini and the overall Gini originates
from the fact that Lg is an imperfect approximation to L. As mentioned above, the
difference is smaller the smaller are the group sizes. In fact, the difference can be crudely
bounded by maximum group size wM=maxc({wc}) . First note that uj=lj and define
+1
ln =un=1. Then the difference in Ginis is
+1
32See also Davies and Shorrocks (1989) and Shorrocks and Wan (2004).
35
2 Lg (p)dp - 2 L(p)dp
1 1
0 0
n
2w 1[ n
j L(l ) + L(l )]- 2
j+1 j w jL(l ) =
j
j=1 2 j=1
w n
j[L(l ) - L(l )]
j+1 j
j=1
n
wM [ L(l ) - L(l )] =
j+1 j
j=1
wM .
The difference is bounded by a weighted average of group sizes, the weights being
Lorenz-curve increments. If the maximum group size is less than 10% of the population,
the BGI for an arbitrary group order is at most 10% (points) below the overall Gini and
the difference will be even smaller for the BGI maximizing group order. With group sizes
smaller than 10% the gain of using maximum BGI as a benchmark instead of total
inequality is therefore limited.
To derive a lower bound for the difference, note that the between-group Gini does not
decrease (and will typically increase) if groups are refined into smaller groups. It follows
that for w wM we have
minLg 2 Lg (p)dp - 2 L(p)dp
1 1
0 0
s+w
minsw(L(s) + L(s + w)) - 2
s L( p)dp
mins (w(L(s) + L(s + w)) +
- 2[14 w(L(s)+ L(s + w))+ w(L(s + w)+ L(s + w))])=
1 1 1
2 4 2
wmins L(s)+ L(s + w) - L(s + w).
1
2 2
36
Putting w = wM and a(w) = mins[(L(s) + L(s + w))/ 2 - L(s + w/ 2)], it follows33 that
minLg 2 Lg (p)dp - 2 L(p)dp a(wM )wM .
1 1
0 0
Figure A.1: Lorenz curve L(p) and piecewise linear approximation Lg(p). There are
three groups, with w1 = 0.4,w2 = 0.45,w3 = 0.15.
33g(w) is a measure of the convexity of L(p) over an interval of width w.
37
Table 1: Decomposing Inequality by "Social" Group in 8 countries.
Country No of "social" GE(0) Between-Group Rb'
groups Contribution
(%) (%)
India 3 0.136 5.1 10.1
Bangladesh 4 0.181 20.3 28.7
Kazakhstan 3 0.217 9.0 14.7
Nepal 10 0.220 23.3 23.7
United States 5 0.295 8.4 14.7
Panama 10 0.423 16.7 36.4
Brazil 4 0.442 16.2 20.0
South Africa 3 0.563 38.0 55.0
Note: data for India refer to rural areas only.
38
Figure 4: Between-group inequality decompositions: urban-rural
Proportion
0 0.1 0.2 0.3 0.4 0.5
Senegal
Guinea
Cameroon
Burundi
Mali
Kenya
Sub-Saharan Africa South Africa
Uganda
Mauritania
Benin
Tanzania
Madagascar
Burkina Faso
Cote d'Ivoire
Niger
Ethiopia
Thailand
Vietnam
East Asia and Pacific Philippines
Indonesia
East Timor
Laos
Romania
Lithuania
Kazakhstan
Serbia
Poland
Kyrgyzstan
Russia
Turkey
Armenia
Europe and Central Asia Hungary
Ukraine
Moldova
Tajikistan
Albania
Bosnia & Herzegovina
Georgia
Bulgaria
Macedonia
Estonia
Belarus
Azerbaijan
Taiwan
France
Australia
Norway
United States
Finland
High Income Countries Canada
Sweden
Italy
Germany
Denmark
United Kingdom
Switzerland
Guatemala
Panama
Bolivia
Paraguay
Nicaragua
Haiti
Jamaica
El Salvador
Latin America and the Mexico
Ecuador
Caribbean Honduras
Colombia
Costa Rica
Brazil
Dominican Republic
Guyana
Chile
Venezuela
Sta. Lucia
Suriname
Trinidad Tobago
Morocco
Middle East and North Yemen
Africa Jordan
Israel
Bangladesh
Nepal
South Asia Pakistan
Sri Lanka
Rb Rb'
Source: Authors' calculations from household survey data.
39
Figure 5: Between-group inequality decompositions: social group of the household
head
Proportion
0 0.15 0.3 0.45 0.6
South Africa
Madagascar
Benin
Sub-Saharan Africa Cote d'Ivoire
Niger
Guinea
East Asia and Pacific Vietnam
Europe and Central Asia Kyrgyzstan
Romania
United States
Germany
France
Luxembourg
United Kingdom
High Income Countries Canada
Belgium
Switzerland
Australia
Ireland
Norway
Sweden
Austria
Finland
Paraguay
Guatemala
Bolivia
Panama
Latin America and the
Peru
Caribbean
Brazil
Guyana
Nicaragua
Sta. Lucia
Middle East and North Israel
Africa Jordan
Nepal
South Asia
Sri Lanka
Bangladesh
Rb Rb'
Source: Authors' calculations from household survey data.
40
Figure 6: Between-group inequality decompositions: education of the household head
Proportion
0 0.15 0.3 0.45
Cameroon
Madagascar
Senegal
Burundi
Uganda
Cote d'Ivoire
Sub-Saharan Africa Benin
Mali
Guinea
Kenya
Tanzania
Burkina Faso
Nigeria
Mauritania
Ethiopia
Niger
Thailand
Indonesia
Papua New Guinea
East Asia and Pacific East Timor
Philippines
Laos
Vietnam
Lithuania
Romania
Serbia
Poland
Turkey
Hungary
Albania
Macedonia
Estonia
Europe and Central Asia Kyrgyzstan
Georgia
Moldova
Bosnia &
Russia
Kazakhstan
Ukraine
Armenia
Tajikistan
Azerbaijan
Luxembourg
France
Taiwan
United States
Ireland
United Kingdom
Italy
High Income Countries Germany
Finland
Netherlands
Switzerland
Sweden
Norway
Canada
Austria
Australia
Belgium
Guatemala
Brazil
Panama
Peru
Nicaragua
Argentina
Chile
Paraguay
Mexico
Colombia
Latin America and the Ecuador
Caribbean Jamaica
Haiti
Bolivia
Honduras
Costa Rica
Uruguay
Dominican Republic
El Salvador
Venezuela
Trinidad Tobago
Sta. Lucia
Guyana
Suriname
Middle East and North Morocco
Jordan
Africa Israel
Yemen
Bangladesh
South Asia Sri Lanka
Pakistan
Nepal
Rb Rb'
Source: Authors' calculations from household survey data.
41
Figure 7:
Regressions of total inequality on shares of between-group inequality of different
household characteristics
(based on Rb')
Urban/Rural Social Group of the HH Head
.6 .3
) .4 ) .2
X|0eg .2 X|0eg .1
0
e( 0 e( 1-.
2-. 2-.
-.2 0 .2 .4 -.2 0 .2 .4
e( sbge0ur_bge0mx | X ) e( sbge0so_bge0mx | X )
coef = .27642359, se = .08964296, t = 3.08 coef = .39394927, se = .09971086, t = 3.95
Occupation of HH Head Education of HH Head
.3 .6
) .2 ) .4
X|0eg .1
0 X|0eg .2
e( 1-. e( 0
2-. 2-.
-.2 -.1 0 .1 .2 .3 -.2 -.1 0 .1 .2
e( sbge0oc_bge0mx | X ) e( sbge0ed_bge0mx | X )
coef = .36113935, se = .16698371, t = 2.16 coef = .32095749, se = .11905169, t = 2.7
42