WPS6728
Policy Research Working Paper 6728
OpportunitySensitive Poverty Measurement
Paolo Brunori
Francisco Ferreira
Maria Ana Lugo
Vito Peragine
The World Bank
Africa Region
Office of the Chief Economist
December 2013
Policy Research Working Paper 6728
Abstract
This paper offers an axiomatic characterization of two based on a rankdependent aggregation of typespecific
classes of poverty measures that are sensitive to inequality poverty levels, is also introduced. In empirical analysis
of opportunity—one a strict subset of the other. The using household survey data from eighteen European
proposed indices are sensitive not only to income countries in 2005, substantial differences in country
shortfalls from the poverty line, but also to differences rankings based on standard FosterGreerThorbecke
in the opportunities faced by people with different indices and on the new opportunitysensitive indices are
predetermined characteristics, such as race or family found. Crosscountry differences in opportunitysensitive
background. Dominance conditions are established for poverty are decomposed into a level effect, a distribution
each class of measures and a subfamily of scalar indices, effect, and a population composition effect.
This paper is a product of the Office of the Chief Economist, Africa Region. It is part of a larger effort by the World
Bank to provide open access to its research and make a contribution to development policy discussions around the world.
Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted
at fferreira@worldbank.org.
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
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Produced by the Research Support Team
Opportunitysensitive poverty measurement
Paolo Brunori, University of Bari
Francisco Ferreira, World Bank and IZA
Maria Ana Lugo, World Bank
Vito Peragine, University of Bari
December 17, 2013∗
JEL Classiﬁcation: D31, D63, J62
Keywords: Inequality of opportunity, poverty measures, poverty comparisons.
∗ We are grateful to Fran¸cois Bourguignon, James Foster, Sean Higgins and Dirk van de Gaer
for helpful conversations and comments on previous versions of this paper. We would also like
to thank seminar and conference participants at the University of Bari, University of Milan,
the Universidade Federal Fluminense and IPEA, Rio de Janeiro, ECINEQ 2011 in Catania, the
ABCDE 2011 in Paris, LACEA 2013 in Mexico City, the Second World Bank Conference on
Equity in Washington, the IX Young Economists Workshop on Social Economy in Forl` ı, the
Conference “Equality of opportunity: concepts, measures and policy implications” in Rome, and
the IV Worskshop GRASS in Modena. The usual caveat applies: all remaining errors are ours
alone. Ferreira would like to acknowledge support for the Knowledge for Change Program, under
project grant P132865TF012968. The views expressed in this paper are those of the authors,
and should not be attributed to the World Bank, its Executive Directors, or the countries they
represent.
1
1 Introduction
Two foundational contributions to the modern literature on poverty measurement,
Sen (1976) and Foster et al. (1984), were motivated at least in part by a concern to
introduce poverty indices that were sensitive to inequality. Sen’s paper begins by
noting that the headcount measure is ”...completely insensitive to the distribution
of income among the poor” (p.219). Foster et al. introduce a parametric class of
measures which, crucially, ”satisfy the transfer sensitivity axiom” (p.763).
But whereas sensitivity to inequality of outcomes (e.g. income or consumption
expenditures) has been widely incorporated as a desirable property for poverty
indices, we are aware of no poverty measures that are sensitive to inequality of
opportunity. This stands in contrast to the inﬂuence that the theory of inequality
of opportunity, and in particular its formalization by John Roemer (1998), has had
on the measurement of inequality more generally.
Since Rawls’s (1971) A Theory of Justice, a number of moral philosophers and
social choice theorists have argued that not all income diﬀerences are identical from
a normative or ethical perspective. In particular, income diﬀerences due to personal
choices, or otherwise attributable to personal responsibility, may be judged to be
less morally objectionable than diﬀerences due to predetermined attributes  such
as gender, race or family background  over which individuals have no control.
Based on increasingly sophisticated variants of this basic distinction, a number
of authors have argued that egalitarian justice ought to focus less on the space
of outcomes, and more on the space of opportunities. See, for instance, Dworkin
(1981a, b), Arneson (1989), Cohen (1989), Roemer (1998), Fleurbaey (2008), and
Fleurbaey and Maniquet (2011).
Building on that theoretical literature, other authors have suggested ways in
which inequality due to predetermined circumstances can be appropriately quan
tiﬁed, for the purpose of measuring inequality of opportunity in real laborforce or
household survey data. See, for example, Bourguignon et al. (2007), Lefranc et al.
(2009), Checchi and Peragine (2010) and Ferreira and Gignoux (2011).
Given the close proximity between the literatures on inequality and poverty
measurement, it is surprising that the inﬂuence of opportunity egalitarianism on
inequality measurement has not, to our knowledge, extended to the measurement of
poverty. After all, if our assessment of poverty is informed by the extent of inequal
ity in society, and if our judgment of inequality is in turn aﬀected by whether income
diﬀerences arise from personal responsibility or predetermined circumstances, then
why should our judgment of poverty not be similarly aﬀected by that distinction?
This paper asks whether poverty measures that are sensitive to inequality of
opportunity are conceptually meaningful, and empirically relevant.1 Conceptu
ally, sensitivity (or aversion) to inequality of opportunity implies two fundamental
changes to the standard axiomatic approach. First, partitioning the population
into types (circumstancehomogeneous groups) can lead to violations of the general
anonymity axiom. Second, aversion to inequality between types, but not within
them, raises problems for the transfer axiom: transfers within and across types can
be valued diﬀerently, and a potential conﬂict arises between outcomeinequality
aversion and inequalityofopportunity aversion.
We propose logically consistent solutions to these challenges by introducing rel
atively modest modiﬁcations to the standard set of axioms in poverty measurement.
1 The paper is concerned exclusively with unidimensional poverty. An analysis of opportunity
sensitive measures of multidimensional poverty is left for future work.
2
We show that inequalityofopportunity aversion and outcomeinequality aversion
need not be inconsistent; the tension between them can be resolved, as long as one
is given priority over the other. Our axioms allow us to characterize two classes of
opportunitysensitive poverty measures (OSPM)  one a strict subset of the other.
Dominance conditions are established for each class. The ﬁrst condition, based
on the the broad class of OSPM, turns out to be a reinterpretation of an existing
result in the literature on poverty dominance for heterogeneous populations (Atkin
son and Bourguignon 1987, Jenkins and Lambert 1993). This condition involves
a typesequential comparison of the cumulative distribution functions truncated at
the poverty line. The second condition, for the narrower class of OSPM, is a new
sequential dominance procedure, which involves comparisons of the group speﬁcic
headcount ratios and of average incomes among the poor in each type. Moving
from partial to complete orderings, we also propose a speciﬁc parametric family of
OSPM which combines elements from the Sen and Foster et al. frameworks. We
call it the opportunitysensitive Foster, Greer and Thorbecke index, or OSFGT.
We then apply both the partialordering analysis and the OSFGT index to
eighteen countries in the European Union, using the EUSILC 2005 data. When
aversion to inequality of opportunity is incorporated into the assessment of poverty,
we are able to uncover aspects not captured when using standard incomepoverty
methods. The broad OSPM dominance condition yields a large number of unam
biguous rankings: 128 of 153 possible pairwise comparisons. The more demanding
suﬃcient conditions we establish for dominance in the narrow OSPM class are
satisﬁed much more seldom. In addition, income poverty rankings and OSFGT
rankings, although positively correlated, frequently diﬀer: whereas the Nether
lands has less income poverty than Germany, for example, the reverse is true for
opportunitysensitive poverty. There are a number of such rerankings, and we
explore the extent to which they are caused by each of three key factors: country
diﬀerences in overall income poverty; diﬀerences in population shares across types;
and diﬀerences in typespeciﬁc income distributions among the poor.
The paper proceeds as follows: Section 2 describes the basic set up and nota
tion. Section 3 lists the axioms that deﬁne the two classes of opportunitysensitive
poverty measures: the broad OSPM class, and the narrow OSPM class. Section 4
establishes dominance conditions for both classes. Section 5 deﬁnes the OSFGT
family of indices, and discusses some of its properties. Section 6 describes the
empirical application to eighteen European countries. Section 7 concludes.
2 The basic set up
Society consists of a large number of singleindividual households. Each individual
h is completely described by a list of characteristics, which can be divided into two
diﬀerent classes: traits that lie beyond the individual’s responsibility, which are
denoted by a vector of circumstances c, belonging to a ﬁnite set Ω = {c1 , ..., cn };
and factors for which the individual is responsible, which can be summarized by a
scalar variable denoted eﬀort, e ∈ R+ . There is no luck, nor random components in
our model. The advantage variable of interest  which we will refer to as ”income”
for short  is solely determined by circumstances and eﬀorts, thus generated by a
function g : Ω × R+ → R+ .
We partition the population into n types, where a type Ti ∈ T is the set of
individuals whose circumstance vector is ci . The set of types, T , is an exhaus
tive partition of the entire population. T ∈ , where is the set of admissible
3
partitions.
The (equivalized) income of person h in type i is denoted xh , h ∈ Ti , and given
by xh = g (ci , eh ) . Income is distributed according to the typespeciﬁc income
distribution Fi (x), with density function fi (x).2 The maximum income in the
population and in type i are denoted byd xmax and xmax i respectively, and the
F n F
population share of type i is denoted qi . F (x) = i=1 qi Fi (x) is the overall
income distribution of the society, deﬁned as the componentmix distribution of all
the type distributions. F (x) ∈ D, where D is the set of admissible distributions.
The support of each typespeciﬁc income distribution represents the set of out
comes which can be achieved  through the exertion of diﬀerent degrees of eﬀort 
by individuals with the exact same vector of circumstances, ci . That is to say, the
support of the type distribution is an exante representation (that is before eﬀort
levels are realized) of the opportunity set open to any individual endowed with cir
cumstances ci . Furthermore, one could interpret the frequency distribution fi (x)
as an exante indication of the probability attached to each outcome. On this basis,
a number of authors have used moments, or other attributes, of the typespeciﬁc
distribution function Fi (x) to evaluate the type’s opportunity sets. See, for exam
ple, van de Gaer (1993), Bossert (1997) and Ooghe et al. (2007). Most commonly,
the mean income of type i has been used to value opportunity sets, and therefore
to rank types.
For our purposes, it is not essential that types be ranked by their mean incomes.
Indeed, because we are concerned with poverty, rather than general welfare, our
application will use a ranking based on the typespeciﬁc poverty rates instead. But
while we are agnostic about the exact criterion used to rank types, our approach
does require that some ranking is agreed on, so that all types can be ordered in
terms of the value of their opportunity sets. We assume, therefore, that at any
particular point in time there exists a generally agreed ordering on the set of
types T so that Ti+1 Ti for i ∈ {1, ..., n − 1}. The ordering implies that
individuals in Ti face a ”worse” opportunity set than those in Ti+1 .3
Identiﬁcation of the poor is not the primary object of this exercise. In line with
standard practice in unidimensional poverty measurement, we denote a poverty
line z ∈ [0, xmax ] and deﬁne the set of poor individuals in each type as Tz i :=
{h ∈ Ti xh ≤ z }. The poverty line is typeinvariant: any diﬀerences in household
needs across types should be fully accounted for by the equivalence scale used to
construct the income variable x. We denote by µF i the mean income of type i, and
by µ (Fiz ) the mean income of the poor in type i in distribution F.
Our aim is to propose a poverty measure that is based on the poverty condition
of all individuals in the society, while being sensitive to the type to which each
person belongs. In the next section we introduce a set of axioms that are used to
deﬁne two classes of opportunitysensitive poverty measures. As noted earlier, the
narrow OSPM class is a strict subset of the broad OSPM class.
2 Although society consists of discrete individual households, there is a very large number
of them. For simplicity, we use continuous notation for income distributions, without loss of
generality.
3 Whereas the ranking criterion is in principle permanent, types may of course be reranked
over time as their values of the ranking variable change. If, for example, Ti+1 Ti ⇔ Fi (z ) ≥
Fi+1 (z ), then any change in the headcount ranking of types over time would be reﬂected in their
ranking by .
4
3 Two classes of opportunitysensitive poverty mea
sures
Our objective in this section is to deﬁne a poverty measure P : D × R+ × → R+ ,
of the form P (F (x) , z, T ), which is a meaningful opportunitysensitive poverty
measure. To do so, we present a set of desirable axioms to be imposed on the
function, including some that are standard in the literature; a new axiom that
introduces the property of inequality of opportunityaversion; and some suitable
modiﬁcations of other properties. We then use the axioms to deﬁne two classes of
opportunitysensitive poverty measures.
The ﬁrst two axioms are standard in the poverty measurement literature. If an
individual sees her income increase, ceteris paribus, overall poverty cannot increase.
A1 Monotonicity (MON): For all F ∈ D, for all i ∈ {1, ..., n}, P is non
increasing in x :
∂P
∂x ≤ 0, ∀i ∈ {1, ..., n}, ∀x ∈ [0, xmax ]
In addition, the level of poverty in a society is independent of the incomes of
the non poor (Sen, 1976):
A2 Focus (FOC): For all F, G ∈ D, for all i ∈ {1, ..., n},
P (F (x), z, T ) = P (G(x), z, T ) if Fi (x) = Gi (x), ∀x ∈ [0, z ], ∀i ∈ {1, ..., n}
The next property allows us to express aggregate poverty as the sum of indi
vidual poverty functions.
A3 Additivity (ADD). There exist functions pi : R+ × R+ → R+ , for all types
i ∈ {1, ..., n} , assumed to be twice diﬀerentiable (almost everywhere) in x,
such that
n xmax
i
F
P (F (x), z, T ) = qi pi (x, z )fi (x) dx for all F ∈ D.
i=1 0
Here pi (x, z ) is the individual poverty function for a person earning income
x in type i. Next, we introduce a suitably modiﬁed version of the symmetry,
or anonymity, axiom:
A4 Anonymity within types (ANON): For all F, G ∈ D, P (F (x), z, T ) =
P (G(x), z, T ) whenever, for any i ∈ {1, ..., n} , G (x) is obtained from F (x)
by permuting incomes xh and xk , where h, k ∈ Tiz .
ANON is a partial symmetry axiom (see Cowell, 1980). Anonymity within types
implies that, within types, all that matters for the evaluation of poverty is the
income accruing to each individual. In other words, the identity of the individuals
does not matter within each type.
The next two axioms are intended to incorporate speciﬁc versions of the two
fundamental principles of the theory of equality of opportunity  compensation
and reward  into the poverty measure. In the exante approach, inequality of
opportunity is captured by inequality in the value of the opportunity sets people
face. If, as discussed above, all individuals in type Ti share the same opportunity
set, and hence the same value of the opportunity set, then there is no inequality
5
of opportunity within types. All inequality of opportunity is between types, and
the Compensation Principle requires that those with ”worse” circumstances be
compensated for them. Since, as discussed earlier, types are ranked by from
”worst” (i = 1) to ”best” (i = n), the principle of compensation can be incorporated
by the following axiom:
A5 Inequality of opportunity aversion (IOA):
∂P ∂P
h ∈ Tz
i ≤ k ∈ Tz z z
i+1 , ∀i ∈ {1, ..., n − 1}, ∀h ∈ Ti , ∀k ∈ Ti+1
∂xh ∂xk
IOA requires that an opportunitysensitive poverty measure fall by more if an
extra income unit is given to a poor individual in a ”worse” type, than to a poor
individual in a ”better” type. (And, given FOC, any individual in a ”better” type.)
IOA is analogous to a transfer principle between types, but it is deliberately
silent on transfers within types. How should an OSPM measure respond to changes
in inequality within types? One (strict) view is that all inequality within types is
due to eﬀorts, and hence of no ethical concern. The following axiom imposes a
requirement of no aversion to inequality within types.
A6 Inequality neutrality within types (INW)
∂2P z
∂x2 z = 0, ∀i ∈ {1, ..., n}, ∀h ∈ Ti
h h ∈ Ti
This axiom expresses the Natural Reward Principle  speciﬁcally its utilitarian
version  whereby income inequalities within types are considered equitable and
need not be compensated, because they are the result of diﬀerences in eﬀort exerted
(Peragine, 2004; Fleurbaey, 2008).
Alternatively, one might wish to allow for a certain degree of aversion to inequal
ity within types. This might be justiﬁed if a society’s ethical attitudes to inequality
combine inequality aversion in the space of opportunities with some residual aver
sion to outcome inequality, whatever its source. Or it might be motivated by
practical considerations: in any empirical application, the full set of circumstances
ci is unlikely ever to be observed, so that some of the withintype inequality actu
ally reﬂects diﬀerences driven by unobserved circumstances, as well as those caused
by diﬀerences in relative eﬀort. (See Ferreira and Gignoux, 2011). In this spirit,
the next axiom requires that a progressive PigouDalton transfer within a given
type, all else constant, does not increase poverty.
A7 Inequality aversion within types (IAW)
∂2P
≥ 0, ∀i ∈ {1, ..., n}∀h ∈ Tz
i
∂x2
h h ∈ Tz
i
INW is clearly a particular case of IAW, so that two classes of opportunity
sensitive poverty measures can be deﬁned. The ﬁrst is the the narrow OSPM class,
the members of which satisfy A1A6. It is denoted by:
PN := {P : D×R+ × → R+  P satisﬁes MON, FOCUS, ADD, ANON, IOA, INW }
6
The second is the broad OSPM class, whose members satisfy A1A5 and A7. It
is denoted by:
PB := {P : D×R+ × → R+  P satisﬁes MON, FOCUS, ADD, ANON, IOA, IAW }
Clearly, PN ⊂ PB . The combination of these axioms leads immediately to two
remarks about these classes of measures:
n F
Remark 1 P ∈ PN if and only if, for all F ∈ D, P (F (x), z ) = i=1 qi pi (x)fi (x) dx,
where the functions pi : R+ × R+ → R+ satisfy the following conditions:
P1. pi (x, z ) = 0, ∀x > z and ∀i ∈ {1, ..., n}
P2. pi (x, z ) ≥ 0, ∀x ≤ z and ∀i ∈ {1, ..., n}
∂pi (x,z ) ∂pi+1 (x,z )
P3. ∂x ≤ ∂x ≤ 0 ∀x ∈ [0, xmax ], ∀i ∈ {1, ..., n − 1}
∂ 2 pi (x,z )
P4. ∂x2 = 0, ∀x ∈ [0, xmax ], ∀i ∈ {1, ..., n}
n
Remark 2 P ∈ PB if and only if, for all F ∈ D, P (F (x), z ) = i=1 qi
F
pi (x)fi (x) dx,
where the functions pi : R+ × R+ → R+ satisfy conditions P1  P3 and P4 :
∂ 2 pi (x,z )
p4 . ∂x2 ≥ 0, ∀x ∈ [0, xmax ], ∀i ∈ {1, ..., n}
4 Opportunitysensitive poverty dominance
In this section, we identify the poverty dominance conditions corresponding to each
of the two classes. The poverty dominance condition corresponding to the broad
OSPM class, PB , has already been identiﬁed in the literature, although it has a
diﬀerent interpretation in the current context. It is given by Theorem 1.
Theorem 1 (Jenkins and Lambert (1993), Chambaz and Maurin (1998)) For all
distributions F (x), G(x) ∈ D and a poverty line z, P (F (x), z ) ≤ P (G(x), z )) ∀P ∈
P B if and only if the following condition is satisﬁed:
j j
G F
qi Gi (x) ≥ qi Fi (x), ∀x ≤ z, and ∀j ∈ {1, ..., n} (1)
i=1 i=1
Proof See Chambaz and Maurin (1998).
This dominance condition can be rearranged in order to show how diﬀerences
in poverty can be decomposed into diﬀerences in the income distributions and
diﬀerences in the population compositions:
j j
G G F
qi (Gi (x) − Fi (x)) + (qi − qi )Fi (x) ≥ 0 (2)
i=1 i=1
Theorem 1 establishes a partial ordering in D: when equation 1 is satisﬁed, and
only then, we can say that opportunitysensitive poverty is lower in distribution F
than in G for any poverty measure in the broad class P B . Since P N is a strict subset
of P B , this obviously also holds for the OSPM subclass with inequality neutrality
within types, which complies with the utilitarian Natural Reward Principle.
7
Testing the above dominance condition requires comparisons of cumulative dis
tribution functions. For the narrow class of OSPM it is also possible to obtain suf
ﬁcient dominance conditions that make use of typespeciﬁc income means (among
the poor), poverty headcounts and population proportions, as established in The
orem 2 below. The dominance conditions in Theorem 2 are less informationally
demanding, and they render the exercise of checking the conditions computation
ally simpler, albeit at the cost of arriving only at suﬃcient (but not necessary)
conditions.
Theorem 2 For all distributions F(x), G(x) ∈ D and a poverty line z , P (F (x), z ) ≤
P (G(x), z ) ∀P ∈ PN if the following conditions are satisﬁed:
j j
G F
(i) qi Gi (z ) ≥ qi Fi (z ) , ∀j ∈ {1, ..., n} ;
i=1 i=1
j j
F
(ii) qi µ FiZ ≥ G
qi µ(Gz
i ), ∀j ∈ {1, ..., n} .
i=1 i=1
Proof. See appendix.
Theorem 2 states that a suﬃcient condition for declaring poverty in distribution
F lower than poverty in distribution G according to any index in the family P N , is
that the following sequential conditions are satisﬁed: (i) the weighted proportion
of poor (the headcount) is lower in F than in G at each step of the sequential
procedure; (ii)the weighted average income among the poor is higher in F than in
G, at each step of the sequential procedure.
The sequential procedure in Theorem 2  starting from the lowest type, adding
the second and so on  incorporates our concern for equality of opportunity. The
fact that these conditions are not necessary implies that a diﬀerence in means can
be counterbalanced by a diﬀerence in the proportion of poor individuals (given a
poverty line). This is crucial in understanding the empirical results presented below:
although the conditions in Theorem 2 are less informationally demanding, they
are nevertheless quite strong. The (sequential) requirement that both population
weighted headcounts be lower and populationweighted average incomes be higher
is likely to make the conditions for Theorem 2 diﬃcult to observe in practice.
Note that the distributional conditions characterized respectively in Theorems
1 and 2 are independent. In fact, while the condition in Theorem 1 (Equation
1) clearly implies condition (i) in Theorem 2, condition (ii) in Theorem 2 is not
implied by (and does not imply) Equation 1.4 The only case in which Equation 1
implies both conditions (i) and (ii) of Theorem 2 is the case of equal type partitions;
F G
that is when qi = qi for all i.
In Section 6 we apply these two sets of dominance conditions to eighteen EU
countries. We ﬁnd that Theorem 1 allows us to rank distributions in 128 of the 153
possible pairwise comparisons, while Theorem 2 yields only 21 instances of domi
nance. But before turning to the empirical application, we now turn to complete
orderings in D, by deﬁning a speciﬁc family of scalar OSPM indices, within PB .
4 To see this, consider a case in which the condition of theorem 1 is satisﬁed: that is,
j G j F
i=1 qi Gi (x) ≥ i=1 qi Fi (x), for all x ≤ z, and for all j ∈ 1, ..., n. Now focus on the
ﬁrst step of the sequential procedure (that is j = 1) : q1 G G (x) ≥ q F F (x). Consider what hap
1 1 1
pens when q1 F 0 and q1G 1. In this case condition (i) of theorem 2 is satisﬁed; on the other
F
hand, looking at condition (ii) , we have that: q1 µ F1 Z G
0 and q1 µ GZ µ GZ
1 1 . That is,
the condition (ii) of theorem 2 is not satisﬁed.
8
5 From dominance to scalar measures
In this section we focus on a speciﬁc family of opportunitysensitive poverty indices,
within the broad class PB characterized above. First we note that, in general, there
is an obvious tension between axioms IOA and IAW. To see this, consider a simple
example, with four individuals in three types. Individual A is in the highest type
(h), individuals B and C are in the intermediate type (m), while individual D is
in the lowest type (l). Suppose B is poorer than C, and imagine positive transfers
from A to C and B to D. Both transfers are progressive across types, but they
result in an increase in inequality within type m (see Figure 1). This is simply a
manifestation of the potential tension between aversion to inequality of opportunity
and aversion to inequality of outcomes. If this conﬂict were irredeemable, the class
PB would collapse to its narrow subset PN , where the tension is eliminated by
complete neutrality with respect to withintype inequality.
Figure 1: Progressive transfers across types: Conﬂict between IOA and IAW
−γ
type h 
A A
−δ +γ

type m 
B B C C
+δ

type l 
D D
That the complement of PN in PB is not an empty set can be demonstrated by
considering a speciﬁc example of individual poverty measures. For any wi ∈ R+ ,
i ∈ {1, ..., n}, let pi (x, z ) = wi p(x, z ), such that wi − wi+1 > p(x, z ), ∀x. Then,
n z
F
P (F (x), z, T ) = qi wi p(x, z )fi (x) dx,
i=1 0
where p(x, z ) is weakly convex in x, will always satisfy both IOA and IAW.
A special member of this family of indices is obtained by using the FGT formula
x α −r (i)
tion for p(x) = z− z , and selecting inverse ranks as type weights: wi = n+1n ,
where r(i) is an increasing function of the rank of each type, r : {1, ..., n} → R+ ,
based on the agreed ordering Ti+1 Ti for i ∈ {1, ..., n − 1}, introduced in sec
tion 2. In particular, let r(i) = i, for all i ∈ {1, ..., n} whenever Ti+1 Ti , ∃
i ∈ {1, ..., n − 1}, and r (i) = 1, for all i ∈ {1, ..., n} whenever Ti+1 ∼ Ti for all
i ∈ {1, ..., n − 1}. In that case, nwi = n + 1 − i, unless the ranking collapses to
indiﬀerence across all types (in which case all weights collapse to 1).
With that weighting scheme, we obtain the family of opportunitysensitive FGT
(OSFGT) measure PF GT ⊂ PB :
n z α
1 F z−x
PF GT (F, z, T ) = qi (n + 1 − r(i)) fi (x) dx (3)
n i=1 0 z
9
where the parameter α expresses a concern for withintype inequality among the
poor, and α ≥ 0.
PF GT has a number of interesting properties:
PF GT satisﬁes both IOA and IAW. This can be seen by noting that, ∀i ∈
{1, ..., n − 1}, ∀h ∈ Tz z
i , ∀k ∈ Ti+1 ,
∂P ∂P
h ∈ Tz
i ≤ k ∈ Tz
i+1
∂xh ∂xk
α−1
1α F z − xh
− q (n + 1 − i) fi (x) ≤
nz i z
α−1
1α F z − xk
− q (n − i) fi+1 (x)
n z i+1 z
In words, a marginal income transfer from any poor individual h in type i to any
poor individual k of a higher type increases (or at least cannot reduce) poverty. The
povertyincreasing eﬀect of the outward transfer from type i is larger in absolute
value than the povertydecreasing eﬀect of the inward transfer of the same amount
to any higher type, i + 1. This is a consequence of the fact that rank diﬀerences
z −x α−1
are integers, while q F ≤ 1, α
z z ≤ 1, α, z ≥ 0.5
PF GT ∈ [0, 1).
The minimum level is achieved when no individual in the society is considered
poor, that is, no one falls below the poverty line z . The maximum possible value of
any poverty measure in PF GT is achieved when all individuals in the society have
zero income and r(i) = 1, ∀i. This would occur if, for instance, ranks were deﬁned
according to the poverty headcount, or type mean incomes, and were equal in case
of ties.
When there is perfect equality of opportunity, the poverty status is indepen
dent of the group to which the person belongs. Perfect equality of opportunity
attains when Fi (x) = F (x) , ∀i. In this case, PF GT is given by
n
Q F
PF GT (F (x), z, T ) = n+1− qi r(i) ,
n i=1
z z −x α
where Q = 0 z dF (x) is the poverty rate common to all types. Since
Ti+1 ∼ Ti for all i ∈ {1, ..., n − 1}, r(i) = 1, ∀i, and PF GT (F (x), z, T ) = Q.
The opportunitysensitive rankdependent FGT measure collapses to the standard
FGT poverty measure.
If one is strictly neutral with respect to inequality within types, and seeks
indices that belong to the narrow OSPM class PN , then α can be set equal 1 or 0.
The opportunitysensitive poverty gap measure PG (F (x, z, T )) ∈ PN sets α = 1,
so that p (x, z ) = z− x
z for all x < z . Hence,
5 There is also a potential clash between IOA and the standard inequality aversion for the full
distribution. It is perfectly possible that, in the above example, xh h ∈ Tz
i > xk k ∈ Tz i+1 .
There exists a non empty set of possible transfers which are both progressive in the standard
PigouDalton sense, and opportunityregressive in the sense of IOA. The maximum reconciliation
between the PigouDalton transfer axiom (with no regard to types) and IOA is given by the
withintype inequality aversion (IAW) axiom deﬁned above.
10
n z
1 F z−x
PG (F (x), z, T ) = qi (n + 1 − r(i)) fi (x) dx (4)
n i=1 0 z
When α equals zero, p (x, z ) = 1 for all x < z , and we obtain the opportunity
sensitive poverty headcount measure deﬁned as
n z n
1 F 1 F
PH (F, z, T ) = qi (n +1 − r(i)) fi (x) dx = qi (n +1 − r(i))Fi (z ) (5)
n i=1 0 n i=1
Following Roemer (1998), Fi (z ) can be interpreted as the degree of eﬀort  ex
pressed in percentile terms  necessary to escape poverty for a person endowed with
circumstances ci . An extreme and interesting case arises when xmax i < z : in this
case type i is said to be ‘povertytrapped’ in the sense that, given circumstances ci
that deﬁne Ti , no amount of eﬀort is suﬃcient to escape poverty.
6 Poverty, inequality and opportunities in the Eu
ropean Union
Introducing a concern for inequality of opportunity into poverty measurement ac
quires practical importance if the assessment of the evolution of poverty in a coun
try  or poverty comparisons across diﬀerent regions or countries  diﬀer when the
concern is incorporated. This section provides an empirical application of the ap
proach proposed in the paper. Using data for eighteen European Union countries,
we show that when aversion to inequality of opportunity is explicitly incorporated
into the measurement of poverty we are able to uncover some aspects of poverty
not captured when using traditional poverty measures.
We use data from the EUSILC 2005 round (User Database), which includes
a special module on intergenerational transmission of poverty that contains infor
mation on socioeconomic background characteristics that can be used to deﬁne
circumstances.6 Poverty is assessed on the basis of equivalent disposable household
income, expressed in Euros at PPP exchange rates and in 2004 prices (Eurostat,
2008).7 We restrict the sample to household heads  deﬁned as the individual
with the highest gross earning in the family  and their spouses, with nonnegative
disposable household incomes.
We use three circumstance variables to deﬁne types: gender, parental education
and parental occupation when the respondent was between 12 and 16 years old.
Parental education is deﬁned as the highest level attained by either of the two
parents (when both are available), and is categorized in two groups: higher when
at least one parent completed upper secondary, and lower otherwise.8 Parental
occupation status is based on the highest ISCO 88 occupation status of the parents,
grouped into four categories: highly skilled nonmanual (ISCO codes 1134), lower
skilled nonmanual (4152), skilled manual (6183), and elementary occupation (91
93). The total population is thus partitioned into 16 types.
6 The EUSILC project aims at obtaining internationally comparable information across EU
countries. For more information see http://epp.eurostat.ec.europa.eu/portal/page/portal/
microdata/eu_silc
7 Equivalent disposable household income is computed as the disposable household income
divided by the square root of the household size
8 Educational levels are deﬁned according to the International Standard Classiﬁcation of Edu
cation (ISCED).
11
From the original sample of 26 countries we exclude Denmark, Norway, Sweden,
and the UK due to the high proportion of missing values in parental occupation
and parental education variables in those countries (see Nolan, 2012, for a dis
cussion). Iceland, Ireland, Portugal, and Slovenia are also excluded because the
sample frequency of some types is too low (fewer than 10 individuals) to compute
credible typemean disposable income and typemean poverty rates. Therefore,
the subsample used in the following exercise includes 18 countries: Austria (AT),
Belgium (BE), Cyprus (CY), Czech Republic (CZ), Germany (DE), Estonia (EE),
Spain (ES), Finland (FI), France (FR), Greece (GR), Hungary (HU), Italy (IT),
Lithuania (LT), Luxembourg (LU), Latvia (LV) Netherlands (NL), Poland (PL),
and Slovakia (SK). The resulting sample thus includes a mix of Eastern European,
Mediterranean, and NorthWestern European countries, with substantial variation
in income levels and poverty.
As noted in Section 2, our approach requires an agreed ordering on the set
of types T . In the inequality of opportunity literature, types are often ranked
according to some evaluation of the opportunity set of individuals belonging to
the type. A common criterion is to order types by their mean advantage (in this
case, income) level. Since the main focus of this paper is on poverty rather than
welfare, we choose to rank types according to the typespeciﬁc poverty headcount,
z
pi (x) = 0 fi (x) dx = Fi (z ). Alternative criteria could obviously be applied such
as, for example, the typespeciﬁc average income among the poor.
6.1 Partial rankings: Testing dominance conditions
Once a criterion for ranking types within countries has been agreed, we can test
for poverty dominance across all possible pairs of countries, using both Theorems
1 and 2. We begin with Theorem 1, which is more computationally intensive, but
which yields both necessary and suﬃcient conditions, and refers to the broader
OSPM class. Testing for the condition in Equation 1 requires deﬁning a single
poverty line, up to which the dominance condition is checked. We follow Decancq
et al. (2013) and deﬁne an Europewide poverty threshold as 60% of the median
equivalent household income of the European distribution (of countries included in
our sample). Individuals are considered poor if their annual equivalent disposable
income is below 9, 275 euros at PPP exchange rates, in 2004 prices. This threshold
is used for testing the conditions in both Theorems.
Table 1 presents the results from testing the sequential dominance condition
in Theorem 1, Equation 1. In this table, opportunitysensitive poverty dominance
(OSPM dominance) of country F (G) over country G (F ) is indicated by a ”<”
(”>”) sign in cell (F , G). Whenever dominance of G over F exists, opportunity
sensitive poverty is higher in country F than in country G for any poverty measure
in the class P B .
Of the 153 possible pairwise comparisons in our 18country sample, we ﬁnd
OSPM dominance in 128 pairs (though fewer are signiﬁcant at 10, 5, and 1% level).
It is possible to rank some countries unambiguously against most other countries
in the sample. For instance, Latvia is opportunitysensitive poorer than all other
countries, except Lithuania. Lithuania is OS poorer than all other countries, except
Estonia, Spain, Latvia and Greece. At the other extreme, Luxembourg is found to
be less OS poor than all but two countries: Cyprus and France. It is possible to
rank any country in our sample against at least half of the other countries. At a
more general level, this exercise suggests a pattern among three traditional Euro
pean social models: with the exception of the Czech Republic, Eastern European
12
countries are all opportunitysensitive poorer than Mediterranean countries which,
in turn, are poorer than NorthWestern European countries. But within these three
groups of countries, dominance is harder to ﬁnd.
Table 1: Dominance conditions associated with Theorem 1
AT BE CY CZ DE EE ES FI FR GR HU IT LT LU LV NL PL SK
AT . >
BE < .
CY .
CZ > > > .
DE < < .
EE >*** >*** >*** >*** >*** .
ES > >** >*** > > <*** .
FI > < <*** <*** .
FR < < < < <*** <*** .
GR >*** >*** >*** > >*** < > >*** >*** .
HU > > >*** >** > < > >* >*** < .
IT > > >*** > > <*** >*** >* < .
LT > >*** >*** >* >*** >*** >*** > >** .
LU < < < < < <*** <*** < < <*** <* <*** <*** .
LV >*** >*** >*** >*** >*** > >*** >*** >*** > > >*** >*** .
NL < <*** <*** < < <*** <** <*** <*** > <*** .
PL >* >*** >*** >*** >*** > >*** >*** > > < >*** < >*** .
SK > >*** >*** >** >*** < >*** >*** > < >*** < >*** .
Source: Authors’ calculation from EUSILC (2005). ”>” in row i and column j means that
poverty is higher in country i than in country j . The dominance condition for each pair
is obtained checking the sequential conditions in Theorem 1 (equation 1). 90%, 95%, 99%
conﬁdence intervals are based on the quantile distribution of 200 bootstrap resampling with
replacement.
Table 2: Dominance conditions associated with Theorem 2
AT BE CY CZ DE EE ES FI FR GR HU IT LT LU LV NL PL SK
AT .
BE .
CY .
CZ > > .
DE .
EE .
ES .
FI < .
FR .
GR .
HU > > > .
IT .
LT > > > >* .
LU .
LV > .
NL < < < .
PL > > >*** > .
SK > > > .
Source: Authors’ calculation from EUSILC (2005). ”>” in row i and column j means that
poverty is higher in country i than in country j . The dominance condition for each is obtained
checking the sequential conditions in Theorem 2 . 90%, 95%, 99% conﬁdence intervals are
based on the quantile distribution of 200 bootstrap resampling with replacement.
Signiﬁcantly fewer instances of dominance are found when testing the conditions
in Theorem 2, as shown in Table 2. Only 21 of the 153 possible pairwise comparisons
can be ranked by these conditions, and only two of those are signiﬁcant at 1% and
10%: France is less OS poor than Poland, and Lithuania is OS poorer than Italy.
As expected, by relying on strong sequential requirements for both typespeciﬁc
poverty incidence and mean incomes among the poor, Theorem 2 yields instances of
dominance much less frequently than Theorem 1, which relies on weaker, necessary
and suﬃcient conditions.
13
6.2 Complete rankings: The opportunitysensitive FGT for
Europe
Even when poverty dominance is widespread, as is the case in Table 1 above, one
may wish to establish a complete poverty ordering across all pairwise comparisons.
This requires imposing further conditions and deﬁning a scalar poverty measure.
In this section we employ the OSFGT family of indices, introduced in Section 5,
for the eighteen European countries in our sample. We also compare the resulting
ranking of countries with that obtained for the traditional FGT index, which is
insensitive to inequality of opportunity among the poor.
Poverty comparisons in the European Union typically rely on countryspeciﬁc
relative poverty lines, deﬁned as 60% of the median of the equivalent disposable
household income distribution. Whereas the OSPM dominance exercise reported
above required a single poverty line across countries, the same is not true for com
plete orderings, and in this section we choose to adhere to common European
practice, by using 60% of the country median as the poverty line for each country.
These lines vary from Euro 3,051 (in annual equivalized disposable household in
come) in Lithuania, to Euro 18,253 in Luxembourg (see Table 4 in the Statistical
Annex). As noted above, types are ranked according to their poverty headcount,
Fi (z ). As expected, in most countries the most disadvantaged types are those
with parents that had little formal education and were employed in elementary
occupations when the respondent was young.9
Figures 2 and 3 illustrate the main results of the section.10 Figure 2 plots
the country rank (from least poor to poorest) by the opportunitysensitive poverty
headcount measure PH (F, z, T ), on the vertical axis, against the country rank
according to the standard poverty headcount F GT (0) in the horizontal axis. Al
though the correlation between the two is clearly positive, there is also a substantial
number of rerankings. The red line represents the 45 degree line; countries above
(below) the line rank higher (lower) in the opportunitysensitive poverty measure
than when the distribution of opportunities is ignored. For instance, Germany
(DE) is placed sixth in terms of the poverty headcount but second once we in
corporate information about how poverty is distributed among the diﬀerent types.
Conversely, the Netherlands is the least poor country in our sample according to
standard F GT (0), but it is placed fourth according to the PH (F, z, T ). A similar
reranking is found between Mediterranean countries (such as Greece), on the one
hand, and Eastern European countries (such as Estonia), on the other.
Figure 3 plots the actual values of these measures (as opposed to the ranks).11
In addition to the rerankings, and consistent with results from the previous sub
section, this graph is suggestive of the existence of three groups of countries. The
ﬁrst group consists of the richer countries in the sample, at the bottom left of the
graph, with relatively low values of both standard and opportunitysensitive poverty
headcounts. These are predominantly NorthWestern European countries, but also
include the Czech Republic. The second group includes the Mediterranean coun
tries (Greece, Italy and Spain) with substantially larger poverty headcounts and
even higher opportunitysensitive poverty headcounts, relative to other countries
9 Countryspeciﬁc tables with poverty rates by type are available from the authors upon request.
10 To economize on space, we present only the results from the opportunitysensitive poverty
headcount, but similar conclusions are obtained when computing the measure for α = 1, 2. Results
using opportunitysensitive poverty gap and severity measures are available from the authors upon
request.
11 All values of P (F, z, T ) , with standard errors, are in the Statistical Annex, ﬁgure 4.
H
14
Figure 2: Poverty headcount and opportunitysensitive poverty headcount. Rank
ings
20
GR
LT
ES
15
LV
IT
PL
Op. FGT(0) rank
EE
CY
10
FR
HU
LU
BE
SK
FI
5
NL
AT
DE
CZ
0
0 5 10 15 20
FGT(0) rank
Source: Authors’ calculation from EUSILC (2005)
with similar levels of poverty. The third group is composed of Eastern European
and former Soviet Union countries with even higher standard headcount poverty,
but somewhat lower opportunitysensitive headcounts.
Where do these crosscountry diﬀerences in opportunitysensitive poverty come
from? Diﬀerences in opportunitysensitive poverty across distributions arise from
three basic sources: diﬀerences in overall poverty levels in the population (the
level eﬀect); diﬀerences in the distribution of poverty across types (the distribution
eﬀect); and diﬀerences in population shares across types (the population compo
sition eﬀect). For the case of the headcount OSPM, deﬁned in Equation 5 as
1 n F
PH (F, z, T ) = n i=1 qi (n + 1 − r (i))Fi (z ) , the diﬀerence between two countries,
F and G, can be written as follows:
F G F ∗ ∗ ∗∗ ∗∗ G
PH − PH = PH − PH + (PH − PH ) + PH − PH (6)
where the two counterfactual poverty incidence measures are given by:
n
∗ 1 F G (z )
PH = qi (n + 1 − r (i))Fi (z ) (7)
n i=1
F (z )
n
∗∗ 1 F
PH = qi (n + 1 − r (i))Gi (z ) (8)
n i=1
Analogous decompositions can straightforwardly be written for α > 0, by re
z x α
placing Fi (z ), with 0 z−
z fi (x) dx. Notice that the ﬁrst term in brackets on
the righthand side of Equation 6 denotes the level eﬀect, which is obtained by
scaling up poverty incidence across types in country F by a common factor, so
15
Figure 3: Poverty headcount and opportunitysensitive poverty headcount. Levels
r = 0.8859 (p=0.00)
GR
.15
ES LT
IT LV
PL
EE
Op. FGT(0)
.1
CY
FR
LU HU
BE
FI SK
NL
AT
DE
CZ
.05
0.1000 0.1500 0.2000 0.2500
FGT(0)
Source: Authors’ calculation from EUSILC (2005)
as to replicate country G’s standard poverty incidence, while preserving the cross
type population composition and poverty distribution of country F . The second
term corresponds to the distribution eﬀect : with country G’s overall poverty level,
it moves from the distribution of poverty across types observed in country F to
that of country G.12 Finally, the third term comprises the population composition
eﬀect : it incoporates into the counterfactual distribution the population shares of
types in country G.
The decomposition speciﬁed in (6) is exact, but it is also pathdependent: the
value of each of the three eﬀects would vary if the counterfactual indices were
diﬀerently speciﬁed, corresponding to diﬀerent ”paths” for the decomposition. One
might have changed the population shares ﬁrst, for example, then the relative
poverty incidences across types (given the overall poverty level in country F ), and
only shifted to the overall poverty in G in the last step. That such diﬀerent paths
yield diﬀerent results for the decomposition, and that no path is in any sense more
correct than another, is well understood (see, e.g. diNardo, Fortin and Lemieux
1996). So is the solution to the problem, which consists of taking the Shapleyvalue
of each eﬀect across all possible paths of the decomposition (see Shorrocks, 2013).
This ”average of decomposition paths” is now commonly known as the Shapley
Shorrocks decomposition. With three eﬀects, there are six (3!) possible paths.
Table 3 reports the ShapleyShorrocks decomposition of the diﬀerences in the
opportunitysensitive poverty headcount measure for three pairs of countries: The
Netherlands and Lithuania; the Czech Republic and the Netherlands; and Luxem
bourg and France. These pairs were chosen to exemplify countries ranked very far
12 Since types are ranked by Fi (z ), this step includes any necessary reranking of types between
countries F and G: when the distribution of poverty across types that prevails in country G 
Gi (z )  is imported into the counterfactual poverty index, the inverserank weights associated
with each type adjust accordingly.
16
Table 3: Changes in opportunitysensitive headcount
Countries ∆opF GT 0 population composition eﬀect distribution eﬀect level eﬀect
NLLT 0.0716272 0.0160272 0.0157706 0.0713705
CZNL 0.0231285 0.0101637 0.0133965 0.0004316
LUFR 0.0065687 0.0200333 0.0181715 0.0047069
Source: Authors’ calculation from EUSILC (2005). Poverty thresholds are deﬁned as the
countryspeciﬁc relative poverty lines in 2004 PPP Euro.
apart by both F GT (0) and C (NL vs. LT); countries ranked diﬀerently by the two
measures (CZ and NL); and countries ranked similarly by the two indices (FR and
LU). The entries listed in the table are the Shapley values for the column eﬀects
across all possibly paths of the decomposition given by (6).
Results indicate that diﬀerences in opportunitysensitive poverty arise from dif
ferent sources in each case. Diﬀerences between the Netherlands and Lithuania 
ranked at the two extremes of the F GT (0) range in our sample  are unsurpris
ingly driven by the level eﬀect, with the distribution and population composition
eﬀects largely oﬀsetting each other. The Czech Republic is less OS poor than the
Netherlands both because the distribution of poverty across types is more favor
able, and because poorer types are less populous there. The level eﬀect, as implied
by the reverse ranking, goes in the opposite direction. Diﬀerences between France
and Luxembourg are interesting: although the overall diﬀerence in F GT (0) is very
small, this actually reﬂects two relatively large but mutually oﬀsetting eﬀects: The
distribution of poverty is more concentrated in poorer types in Luxembourg, but
the poorest types are less populous there than in France.
7 Conclusion
The last decade and a half has seen growing interest in inequality of opportunity.
A number of diﬀerent approaches to its formal measurement have been proposed,
and applications to both developing and developed countries now abound. From a
normative perspective, there is a coalescing consensus that equality of opportunity
is the appropriate “currency for egalitarian justice” (Cohen, 1989). From a positive
perspective, there is some evidence that inequality of opportunity is more closely
(and negatively) associated with future economic performance than inequality of
outcomes (Marrero and Rodriguez, 2013).
Yet, although a concern with inequality among the poor has been central to
poverty measurement at least since the mid1970s, sensitivity to inequality of op
portunity has hitherto not been introduced into formal poverty measurement (so
far as we are aware). This may reﬂect, at least in part, the fact that measures of
inequality of opportunity explicitly depend on personal characteristics other than
income, thereby clashing with the standard anonymity axiom. Similarly, most per
spectives on inequality of opportunity would treat transfers within and between
types diﬀerently, requiring adjustments to the transfer axiom.
In this paper, we have sought to address these challenges and to axiomatically
derive a class of opportunitysensitive poverty measures (OSPM). A broad OSPM
class was deﬁned, which satisﬁes the standard axioms of monotonicity, focus and
additivity, as well new axioms of withintype anonymity, inequality of opportunity
aversion, and (weak) inequality aversion within types. A narrow OSPM subclass
was also deﬁned, for the case when weak inequality aversion within types is replaced
17
by a more stringent axiom of inequality neutrality within types.
We then identify poverty dominance conditions corresponding to each of the two
classes. For the broad OSPM class we rely on a reinterpretation of the earlier results
by Jenkins and Lambert (1993), and Chambaz and Maurin (1998). A separate,
original, suﬃcient condition is identiﬁed for dominance in the narrow OSPM class.
We also consider complete poverty orderings by proposing a speciﬁc parametric
family of indices, which belongs to the broad OSPM class. This measure is es
sentially a transformation of the seminal Foster et al. (1984) ‘FGT’ class, where
rankdependent weights are attached to diﬀerent types. These inverse rank weights
are in the spirit of Sen (1976), but are applied here to groups (types) rather than
to individuals, and are thus consistent with decomposability. They also allow for
an elegant resolution of the tension between inequality of opportunity aversion and
inequality aversion within types. Like the traditional FGT, this family of indices
ranges between zero and one. Under equality of opportunity, each member of the
family converges to the corresponding standard FGT index.
In an application to poverty comparisons across eighteen European countries,
we ﬁnd that the broadOSPM dominance conditions are satisﬁed rather often: there
are 128 instances of dominance, of 153 possible pairwise comparisons. The more
stringent suﬃcient conditions for narrowOSPM dominance proposed in Theorem
2 hold much less frequently, suggesting that their computational simplicity exacts a
high cost in practice. Broadly, three groups of countries emerge from the dominance
comparisons: Eastern European countries (other than the Czech Republic) tend to
be most opportunitypoor, and are dominated by most other countries. Mediter
ranean countries (such as Greece, Italy and Spain) tend to dominate the Eastern
European countries, but are dominated by the third group, namely NorthWestern
Europe.
The existence of these three broad country groupings is conﬁrmed by comput
ing the scalar OSFGT index. This index is positively correlated with the standard
FGT index, but a large number of rerankings is observed, and some are substan
tial. Germany, for example, ranks as the sixth leastpoor country in terms of the
standard headcount measure (poorer than the Netherlands), but second leastpoor
in terms of the opportunitysenstive headcount (less poor than the Netherlands).
Crosscountry diﬀerences in the OSFGT index are driven by three factors: dif
ferences in overall poverty levels in the population (the level eﬀect); diﬀerences in
the distribution of poverty across types (the distribution eﬀect); and diﬀerences in
population shares across types (the population composition eﬀect).
We hope that both the dominance conditions we have identiﬁed for a broad
class of opportunitysensitive poverty measures and the speciﬁc, rankdependent
FGT index we have proposed may be useful to empirical researchers and practi
tioners interested in characterizing the nature of poverty in diﬀerent settings, or
in monitoring changes in poverty over time. The measures should be of particular
interest to societies averse both to income poverty and to unequal opportunities.
18
8 Statistical Annex
Table 4 contains descriptive statistics for the eighteen country samples used in
Section 6, including EUSILC sample sizes, average incomes, relative poverty lines
(60% of the median of equivalent household income distributions), FGT (0, 1, 2),
the mean logarithmic deviation (E (0)) of incomes, and the betweentype share of
E (0), as a measure of inequality of opportunity.
Figure 4 graphically depicts the 95% bootstrapped conﬁdence intervals around
PH for each country in the sample.
Figure 4: 95% conﬁdence intervals for PH
0.15
Op. FGT(0)
0.10 0.05
CZ DE AT NL FI SK BE LU HU FR CY EE PL IT LV ES LT GR
Source: Authors’ calculation from EUSILC (2005)
19
country sample average income poverty threshold FGT(0) FGT(1) FGT(2) total inequality Ineq of Opp
Austria 5,650 22,380 12,080.47 0.1298 0.0328 0.0150 0.1270 2.75
Belgium 4,697 22,230.25 11,713.56 0.1481 0.0360 0.0144 0.1674 10.23
Cyprus 4,486 20,521.00 10,800.54 0.1577 0.0398 0.0160 0.1468 4.87
Czech Rep. 5,098 10,601.02 5,559.589 0.1175 0.0287 0.0108 0.1260 6.05
Estonia 4,779 7,557.72 3,843.491 0.2185 0.0757 0.0396 0.2192 7.46
Finland 7,241 19,604.28 10,450.91 0.1374 0.0317 0.0122 0.1380 3.81
France 10,830 19,730.18 10,525.69 0.1437 0.0307 0.0110 0.1248 4.58
Germany 13,152 21,003.09 11,091.51 0.1390 0.0376 0.0164 0.1455 1.51
Greece 6,050 15,835.91 8,018.675 0.1827 0.0570 0.0286 0.1880 7.05
Hungary 7,293 7,855.96 4,054.386 0.1461 0.0359 0.0137 0.1512 8.23
Italy 22,328 20,312.67 10,520.81 0.1864 0.0569 0.0287 0.1900 6.83
20
Latvia 3,931 6,593.19 3,239.392 0.2264 0.0847 0.0484 0.2611 9.08
Lithuania 5,185 6,226.57 3,051.889 0.2305 0.0851 0.0471 0.2479 6.96
Luxembourg 4,608 33,996.06 18,253 0.1368 0.0347 0.0137 0.1247 9.57
Netherlands 5,281 21,110.64 11,400.55 0.1167 0.0294 0.0136 0.1210 2.22
Poland 19,127 7,724.96 3,785.902 0.2116 0.0745 0.0401 0.2483 7.35
Slovakia 6,191 7,213.88 3,930.293 0.1463 0.0413 0.0180 0.1295 2.23
Spain 13,968 16,931.32 8,964.699 0.2082 0.0695 0.0365 0.1925 6.54
Table 4: Descriptive Statistics
Source: Authors’ calculation from EUSILC (2005). Poverty thresholds are deﬁned as the countryspeciﬁc relative poverty lines in 2004 PPP Euro. Total
inequality is the mean logarithmic deviation of incomes, IOp is exante inequality of opportunity calculated non parametrically.
Appendix
Proof  Theorem 2
P (F (x), z ) ≥ P (G(x), z ) ⇐⇒
j z j z
F G
∆P = qi pi (x)fi (x)dx − qi pi (x)gi (x)dx ≥ 0
i=1 0 i=1 0
Integrating by parts: vdu = uv − udv : v = pi (x), u = Fi (x) from which:
z z
pi (x)fi (x) = [Fi (x)pi (x)]z
0 − Fi (x)pi (x)dx
0 0
∆P becomes:
j z j z
F
∆P = qi [Fi (x)pi (x)]z
0 − Fi (x)pi (x)dx − G
qi [Gi (x)pi (x)]z
0 − Gi (x)pi (x)dx
i=1 0 i=1 0
If Fi (0) = 0, then [Fi (x)pi (x)]z z
0 = [Gi (x)pi (x)]0 = 0, as pi (z ) = 0. Hence
j z j z
F G
∆P = qi − Fi (x)pi (x)dx − qi − Gi (x)pi (x)dx
i=1 0 i=1 0
j z j z
G F
∆P = qi Gi (x)pi (x)dx − qi Fi (x)pi (x)dx
i=1 0 i=1 0
x
Integrating again by parts: udv = uv − duv , u = pi (x) and v = Fi (y )
j Z z x
G
∆P = qi [pi (x)]Z
0 Gi (x) − pi (x) Gi (y )dy
i=1 0 0
j Z z x
F
− qi [pi (x)]Z
0 Fi (x) − pi (x) Fi (y )dy
i=1 0 0
that is
j Z Z z x
∆P = [pi (x)]Z
0
G
qi F
Gi (x) − qi Fi (x) + pi (x) F
qi G
Fi (y ) − qi Gi (y ) dy
i=1 0 0 0
Assuming pi (x) = 0 the second term disappears.
Z
Now, 0 Fi (x) = zFi (z ) − µ (Fiz ),
where µ FiZ is the mean of the distribution Fi truncated at z .
This comes from the fact that:
z
µ (Fiz ) = xf (x)dx
0
integrating by parts one obtains
21
z z
µ (Fiz ) = [xF (x)]z
0 − F (x)dx = zH − F (x)dx
0 0
.
From which:
j
G
∆P = pi (z ) qi (zGi (z ) − µ (Gz F z
i )) − qi (zFi (z ) − µ (Fi ))
i=1
As pi (x) ≤ pi+1 (x) for all i = 1, ..., n − 1 (see property 3 of Remark 1) we can
apply Abel lemma, thus obtaining that ∆P ≥ 0 if and only if
j
G
qi (zGi (z ) − µ (Gz F z
i )) − qi (zFi (z ) − µ (Fi )) ≤ 0
i=1
This can be written in the following way:
j j
F G G
z qi Fi (z ) − qi Gi (z ) + qi µ (Gz F z
i ) − qi µ (Fi ) ≥ 0 (9)
i=1 i=1
F
Alternatively, adding and subtracting qi µ(Gz F
i ) and zqi G(z ), as
j
F G F F
z qi Fi (z ) − qi Gi (z ) − zqi Gi (z ) + zqi Gi (z ) +
i=1
j
G
qi µ (Gz F z F z F z
i ) − qi µ(Gi ) − qi µ (Fi ) + qi µ(Gi ) ≥ 0
i=1
F G
(qi − qi )(zGi (z ) − µ(Gz F F z z
i ) + zqi (Fi (z ) − Gi (z )) + q (µ(Gi ) − µ(Fi )) ≥ 0 (10)
We obtain a decomposition of the diﬀerence in responsibility sensitive poverty
in three terms: the diﬀerences in population shares, in headcount poverty ratios,
and in average incomes of the poor. The sign of the contribution of each term is
positive as they are multiplied by a positive number:
z
zGi (z ) − µ(Gz
i) = Gxdx ≥ 0
0
From (9) we obtain that suﬃcient conditions for ∆P ≥ 0 are:
j j
F G
(i) qi Fi (z ) ≥ qi Gi (z ) , ∀j ∈ {1, ..., n} .
i=1 i=1
j j
F
(ii) qi µ(Fiz ) ≤ G
qi µ(Gz
i ), ∀j ∈ {1, ..., n} .
i=1 i=1
From (10) we obtain that suﬃcient conditions for ∆P ≥ 0 are:
j j
(i) µ(Fiz ) ≤ µ(Gz
i ), ∀j ∈ {1, ..., n} .
i=1 i=1
j j
(ii) Fi (z ) ≥ Gi (z ) , ∀j ∈ {1, ..., n} .
i=1 i=1
22
j j
F G
(iii) qi ≥ qi , ∀j ∈ {1, ..., n} .
i=1 i=1
Remark 1.2
If Fi (0) = 0
j z j z
F
∆P = qi [Fi (x)pi (x)]z
0 − Fi (x)pi (x)dx − G
qi [Gi (x)pi (x)]z
0 − Gi (x)pi (x)dx
i=1 0 i=1 0
j j z j z
G F G F
∆P = qi Gi (0)pi (0) − qi Fi (0)pi (0) + qi Gi (x)pi (x)dx − qi Fi (x)pi (x)dx
i=1 i=1 0 i=1 0
we know the suﬃcient conditions for the second term to be positive, the ﬁrst
term adds a new condition:
j
G F
qi Gi (0)pi (0) − qi Fi (0)pi (0) ≥ 0
i=1
j
G G F
qi (Gi (0) − Fi (0)) + (qi − qi )Fi (0) ≥ 0
i=1
j j
Gi (0) ≥ Fi (0) (11)
i=1 i=1
Where Fi (0), Gi (0) are the proportions of the individuals in the type i with no
income. The sum of the these proportions at each step i ≤ j = 1, ..., n must be
larger in G than in F .
23
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