Economic Growth and Equality of Opportunity

Vito Peragine1, Flaviana Palmisano, and Paolo Brunori

In this paper, we argue that a better understanding of the relationship between inequali-
ty and economic growth can be obtained by shifting the analysis from the space of ﬁnal
achievements to the space of opportunities. To this end, we introduce a formal frame-
work based on the concept of the Opportunity Growth Incidence Curve. This frame-
work can be used to evaluate the income dynamics of speciﬁc groups of the population
and to infer the role of growth in the evolution of inequality of opportunity over time.
We show the relevance of the introduced framework by providing two empirical analy-
ses, one for Italy and the other for Brazil. These analyses show the distributional impact
of the recent growth experienced by Brazil and the recent crisis suffered by Italy from
both the income inequality and opportunity inequality perspectives. JEL codes: D63,
E24, O15, O40

In recent years, a central topic in the economic development literature has been
the measurement of the distributive impact of growth (see Ferreira 2010). This
literature has provided analytical tools to identify and quantify the effect of
growth on distributional phenomena such as income poverty and income in-
equality. Indices for measuring the pro-poorness of growth have been proposed,2
and the Growth Incidence Curve (GIC), measuring the quantile-speciﬁc rate of
economic growth in a given period of time (Ravallion and Chen 2003; Son
2004), has become a standard tool in evaluating growth from a distributional
viewpoint. The interplay among growth, inequality, and poverty reduction has

` di Bari, Italy; his email address is
1. Vito Peragine (corresponding author) is a professor at Universita
v.peragine@dse.uniba.it. Flaviana Palmisano is posdoctoral fellow at the Universita ` di Bari; her email
address is ﬂaviana.palmisano@gmail.com. Paolo Brunori is assistant professor at Universita ` di Bari; his
mail address is paolo.brunori@uniba.it. The authors thank Francisco Ferreira, Dirk Van de Gaer, the
editors and three anonymous referees for helpful comments on earlier drafts. The authors also wish to
thank Jean-Yves Duclos, Michael Lokshin, and Laura Serlenga. Insightful comments were received at
conferences or seminars at the World Bank ABCDE Conference, the University of Rome Tor Vergata, VI
Academia Belgica-Francqui Foundation Rome Conference and GRASS workshop, and the College
dEtudes Mondiale, Paris. The authors also thank Francisco Ferreira and Maria Ana Lugo for kindly
providing them with access to data.
2. See Essama-Nssah and Lambert (2009) for a comprehensive survey.

THE WORLD BANK ECONOMIC REVIEW, VOL. 28, NO. 2, pp. 247– 281 doi:10.1093/wber/lht030
Advance Access Publication October 14, 2013
# The Author 2013. Published by Oxford University Press on behalf of the International Bank
for Reconstruction and Development / THE WORLD BANK. All rights reserved. For permissions,
please e-mail: journals.permissions@oup.com

247
248 THE WORLD BANK ECONOMIC REVIEW

also been investigated (Bourguignon 2004). All of these tools are now used ex-
tensively in the ﬁeld of development economics to evaluate and compare different
growth processes in terms of social desirability and social welfare (see Atkinson
and Brandolini 2010; Datt and Ravallion 2011).
A common feature of this literature is the focus on individual achievements,
such as (equivalent) income or consumption, as the proper “space” of distribu-
tional assessments.
In contrast, recent literature in the ﬁeld of normative economics has argued
that equity judgments should be based on opportunities rather than on observed
outcomes (see Dworkin 1981a, b; Cohen 1989; Arneson 1989; Roemer 1998;
Fleurbaey 2008). The equal-opportunity framework stresses the link between the
opportunities available to an agent and the initial conditions that are inherited or
beyond the control of this agent. Proponents of equality of opportunity (EOp)
accept the inequality of outcomes that arises from individual choices and effort,
but they do not accept the inequality of outcomes caused by circumstances
beyond individual control. This literature has motivated a rapidly growing
number of empirical applications interested in measuring the degree of inequality
of opportunity (IOp) in a distribution and evaluating public policies in terms of
equality of opportunity (see, among others, Aaberge et al. 2011; Bourguignon
et al. 2007; Checchi and Peragine 2010; LeFranc et al. 2009; Roemer et al.
2003). Book-length collections of empirical analyses of EOp in developing coun-
tries can be found in World Bank (2006) and de Barros et al. (2009).
The growing interest in EOp, in addition to the intrinsic normative justiﬁca-
tions, is motivated by instrumental reasons: it has been convincingly argued (see
World Bank 2006, among others) that the degree of opportunity inequality in an
economy may be related to the potential for future growth. The idea is that when
exogenous circumstances such as gender, race, or parental background play a
strong role in determining individual income and occupation prospects, there is a
suboptimal allocation of resources and lower potential for growth. The existence
of inequality traps, which systematically exclude some groups of the population
from participation in economic activity, is harmful to growth.
We share this view, and we believe that a better understanding of the rela-
tionship between inequality and growth can be obtained by shifting the analy-
sis from the space of ﬁnal achievements to the space of opportunities. If two
growth processes have, say, the same impact in terms of poverty and inequali-
ty reduction, but in the ﬁrst case, all members of a certain ethnic minority - or
all people whose parents are illiterate - experience the lowest growth rate
whereas poverty reduction in another case is uncorrelated with differences in
race or family background, our current arsenal of measures does not readily
allow us to distinguish them. Moreover, although a set of tools has been pro-
posed to explain changes in outcome inequality as the result of differences in
growth for individuals with different initial outcomes, to the best of our
knowledge, the relationship between the change in IOp and growth has never
been investigated.
Peragine, Palmisano, and Brunori 249

Our aim is to address this measurement problem3 by proposing a framework
and a set of simple tools that can be used to investigate the distributional effects
of growth from an opportunity egalitarian viewpoint. In particular, with refer-
ence to a given growth episode, we address the following questions: is growth re-
ducing or increasing the degree of IOp? Are some socio-economic groups
systematically excluded from growth?
To answer these questions, we depart from the concept of the GIC provided
by Ravallion and Chen (2003) and further developed by Son (2004) and
Essama-Nssah (2005), and we extend it to the space of opportunities. Hence, we
introduce the concept of the Opportunity Growth Incidence Curve (OGIC),
which is intended to capture the effect of growth from the EOp perspective. We
distinguish between an individual OGIC and a type OGIC: the former plots the
rate of growth of the (value of the) opportunity set given to individuals in the
same position in the distributions of opportunities. The latter plots the rate of
income growth for each sub-group of the population, where the sub-groups are
deﬁned in terms of initial exogenous circumstances. As shown in the paper, these
tools capture distinct phenomena: the individual OGIC enables us to assess the
pure distributional effect of growth in terms of increasing or reducing aggregate
IOp; the type OGIC, in contrast, allows us to track the evolution of speciﬁc
groups of the population in the growth process to detect the existence of possible
inequality traps. For each of the two, we also provide summary measures of
growth.
These tools can be used as complements to the standard analysis of the pro-
poorness of growth and may provide interesting insights for the design of public
policies. In particular, they may help target speciﬁc groups of the population
and/or identify priorities in redistributive and social policies. Moreover, these
tools can be used for the evaluation of public policies in terms of equality of op-
portunity. In fact, the two-period framework could easily be adapted for the
comparison of pre- and post-public intervention distributions–for instance, if
one is interested in evaluating the distributive impact of a certain ﬁscal reform in
the space of opportunities.
In this paper, we adopt this theoretical framework to analyze the distributional
impact of growth in two different countries, Italy and Brazil, in recent years. These
two countries experienced very different patterns of growth in the last decade. On
the one hand, Italy experienced a period of very limited growth. According to the
Bank of Italy, in the 2002–04 and 2004–06 periods, the average household
income increased by 2 percent and 2.6 percent, respectively, whereas the equiva-
lent disposable income of Italian households was characterized by a long spell of
negative growth during the recent economic crisis: it decreased by 2.6 percent in

3. Hence, we investigate the relationship between growth and inequality of opportunity using a
“micro approach”; an alternative “macro approach” would also be possible by investigating the
relationship between growth and IOp from a cross-country or longitudinal perspective (see Marrero and
Rodriguez 2010).
250 THE WORLD BANK ECONOMIC REVIEW

the 2006–10 period and by 0.6 percent between 2008 and 2010 (Banca d’Italia
2008, 2012). Inequality in the same period increased, but only slightly. On the
other hand, Brazil faced a period of sustained growth (with an average 5 percent
GDP yearly growth in the last decade), and this growth, as shown in the literature,
was markedly progressive. In fact, the Gini index for the entire distribution de-
creased during the period considered from 60.01 in 2001 to 54.7 in 2009 (see con-
tributions by Ferreira et al. 2008, World Bank 2012).
Therefore, it is interesting to examine how the perspective of opportunity in-
equality can add elements of knowledge to the analysis of two markedly different
distributional dynamics.
We use the Bank of Italy’s “Survey on Household Income and Wealth”
(SHIW) to assess the distributional impact of growth in Italy. In particular, we
consider four of the most recent available waves to compare the 2002–06
growth episode with the 2006–10 episode. We use the “Pesquisa Nacional por
Amostra de Domicı ´lios” (PNAD), provided by the Istituto Brazilero de
Geograpia e Estatistica, to analyze growth in Brazil, and we focus on the 2002–05
growth episode against the 2005–08 episode.
As far as Italy is concerned, when we focus on each single growth episode,
some relevant insights arise. For instance, when the 2002–06 growth period is
considered, the standard GIC shows a clear progressive pattern, but this pattern
is reversed when the individual OGIC is adopted. When the 2006–10 period is
considered, the regressive pattern shown by both the individual OGIC and the
type OGIC demonstrates that the burden of the economic crisis has been borne
by the weak groups in the population. Important information can be gained
when we compare the two periods. The ﬁrst period dominates the second accord-
ing to the GIC and the individual OGIC, but this dominance does not hold when
the type OGIC is adopted. We suggest that these results may be interpreted as the
consequence of differences in per capita income growth between regions and
some structural changes introduced in the Italian labor market in the recent past.
With respect to Brazil, it is interesting to note that although the growth experi-
enced by the individual outcome in 2002–05 appears considerable for the whole
distribution (with the exception of the top 15 percent), the growth experienced
in terms of opportunities is less prominent. Indeed, most of the types suffer a re-
duction in the value of the opportunity during the growth process.4 In contrast,
the 2005–08 growth episode appears to be beneﬁcial for the whole population
regardless of the focus of the analysis (whether outcome or opportunity). Our
analysis shows that the 2005–08 growth process is not only generally progressive
but that it also leads to a reduction in the IOp ( progressive individual OGIC).
Furthermore, the initially disadvantaged groups of the population seem to
beneﬁt more from growth than those that were initially advantaged (decreasing
type OGIC). When the two processes are compared, the dominance of the

4. To obtain this conﬂict between type OGIC and GIC, it is necessary that rich individuals
experiencing losses are spread across the majority of socioeconomic groups.
Peragine, Palmisano, and Brunori 251

2002–05 growth episode over the 2005–08 episode is evident for every perspec-
tive adopted.
Hence, we contribute to the literature by showing how it is possible to extend
the existing frameworks proposed for the distributional assessment of growth to
make them consistent with the EOp approach. The empirical analyses conducted
in the paper show that the evaluation of growth may differ if the opportunity
inequality perspective is adopted instead of the standard income inequality
perspective.
The rest of this paper is organized as follows. Section I introduces the models
used in the literature on the distributional effect of growth and in the EOp litera-
ture. It then proposes the opportunity growth incidence curves and summary
indexes to assess the distributional impact of growth in terms of opportunity.
Section II provides the empirical analyses based on Italian and Brazilian data.
Section III concludes.

THE INCIDENCE OF GROWTH IN THE SPACE OF OPPORTUNITIES

A well-developed body of literature has proposed a number of tools that can be
used to evaluate the distributive impact of growth5 in the space of ﬁnal achieve-
ments. After a brief survey of these tools, this section will propose a set of
formal tools that can be used to evaluate the impact of growth in the space of
opportunities.

Growth and Income Inequality
Let F( yt) be the cumulative distribution function of income at time t, with mean
income m( yt), and let yt ( p) be the quantile function of F( yt), representing the
income corresponding to quantile p in F( yt). To evaluate the growth taking place
from t to t þ 1, Ravallion and Chen (2003) deﬁne the Growth Incidence Curve
(GIC) as follows6:

ytþ1 ðpÞ L0 1 ðpÞ
gðpÞ ¼ À 1 ¼ tþ ðg þ 1Þ À 1; forall p [ ½0; 1 ð1Þ
yt ðpÞ L0t ðpÞ

where L0 ( p) is the ﬁrst derivative of the Lorenz curve at percentile p and g ¼
m( ytþ1)/m( yt) 2 1 is the overall mean income growth rate. The GIC plots the
percentile-speciﬁc rate of income growth in a given period of time. Clearly,
g( p) ! 0 ( g( p) , 0) indicates positive (negative) growth at p. A downward-
sloping GIC indicates that growth contributes to equalize the distribution of

5. In what follows, we focus, in particular, on those tools that will be extended to the EOp model in
the next section. For a detailed survey of other existing measures of growth, see Essama-Nsaah and
Lambert (2009) and Ferreira (2010).
6. For a longitudinal perspective on the evaluation of growth, see Bourguignon (2011) and Jenkins
and Van Kerm (2011).
252 THE WORLD BANK ECONOMIC REVIEW

income (i.e., g( p) decreases as p increases), whereas an upward-sloping GIC indi-
cates non-equalizing growth (i.e., g( p) increases as p increases). When the GIC is
a horizontal line, inequality does not change over time, and the rate of growth
experienced by each quantile is equal to the rate of growth in the overall mean
income.
Growth incidence curves are used to detect how a given growth spell affects
the different parts of the distribution. In addition, they are used as criteria to
rank different growth episodes. Ravallion and Chen (2003) apply ﬁrst-order
dominance criteria based on the GIC: ﬁrst-order dominance implies that the GIC
of a growth spell is everywhere above the GIC of another growth spell. Son (2004)
elaborates on this concept by proposing weaker second-order dominance condi-
tions, requiring that the mean growth rate up to the p poorest percentile in a
growth episode - or the “cumulative GIC” - be everywhere
Ðp Ð p in another.
larger than
In this case, the cumulative GIC is given by Gð pÞ ¼ 0 gðqÞyt ðqÞdq= 0 yt ðqÞdq for
all p [ [0,1].
Building on the concept of the GIC, the literature has provided a variety of ag-
gregate measures of growth. We recall, among these, Ð 1 the rate of pro-poor growth
proposed7 by Essama-Nssah (2005): RPPGEN ¼ 0 vð pÞgð pÞdp, where v( p) . 0,
and v0 ( p) 0 is a normalized social weight, decreasing with the rank in the
income distribution. Hence, RPPGEN represents a rank-dependent aggregation of
each point of the GIC and measures the overall extent of growth, giving more im-
portance to the growth experienced by the income of the poorest individuals.8 We
enrich this framework by looking at the literature on EOp measurement.

From Income to Opportunity Inequality
In the EOp model (see Roemer 1998, Van de Gaer 1993, Peragine 2002), the in-
dividual income at a given time, t [ f1,. . .,Tg, yt, is assumed to be a function of
two sets of characteristics: the circumstances, c, belonging to a ﬁnite set v and
the level of effort, et [ Q # Rþ. The individual cannot be held responsible for c,
which is ﬁxed over time; he is, instead, responsible for the effort et that he auton-
omously decides to exert in every period of time. Income is generated by a pro-
duction function g:V Â Q ! Rþ:

yt ¼ gðc; et Þ: ð2Þ

This is a reduced form model in which circumstances and effort are assumed to
be orthogonal, and the function g is assumed to be monotonic in both argu-
ments. Although the monotonicity of g is a fairly reasonable assumption, the or-
thogonality assumption rests on the theoretical argument that it would be hardly

7. In the original paper, RPPGEN is applied to discrete distributions. Here, we use a continuous
version of the same index to be consistent with our notation.
ÐH
8. Ravallion and Chen (2003) also propose the RPPGRC ¼ 0 t gð pÞdp=Ht where Ht is the initial
poverty headcount ratio. RPPGRC measures the proportionate income change of the poorest individuals.
Peragine, Palmisano, and Brunori 253

sustainable to hold people accountable for factor et if it were dependent on exog-
enous circumstances.
In line with this model, a partition of the total population is now introduced.
Each group in this partition is called a type and includes all individuals sharing
the same circumstances. For example, if the only two circumstances were gender
(male or female) and race (black or white), then there would be four types in the
population: white men, black men, white women, and black women. Hence,
considering n types, for all i ¼ 1,. . .,n, the outcome distribution of type i at time
t is represented by a cdf Fi ( yt), with population size mit, population share qit, and
mean mi ( yt).
Given this analytical framework, the focus is on the income prospects of indi-
viduals of the same type, represented by the type-speciﬁc income distribution
Fi ( yt). This distribution is interpreted as the set of opportunities open to each in-
dividual in type i. In other words, the observable actual incomes of all individu-
als in a given type is used to proxy the unobservable ex ante opportunities of all
individuals in that type.
Let us underline here a dual interpretation of the types in the EOp model: on
the one hand, the type is a component of a model that, starting from a multivari-
ate distribution of income and circumstances, allows us to obtain a distribution
of (the value of ) opportunity sets enjoyed by each individual in the population.
On the other hand, given the nature of the circumstances typically observed and
used in empirical application, the partition in types may be of interest per se:
they can often identify well-deﬁned socio-economic groups that may deserve
special attention by the policy makers. As we will see, this dual interpretation of
the types will be exploited in the analysis of the impact of growth on EOp.
A speciﬁc version of the EOp model, which is called “utilitarian”, further
assumes that the value of the opportunity set Fi ( yt) can be summarized by the
mean mi ( yt). This is clearly a strong assumption because it implies neutrality with
respect to the inequality within types. Assuming within-type neutrality, the next
step consists of constructing an artiﬁcial distribution in which each individual
income is substituted with the value of the opportunity set of that individual,
that is, the mean income of the type to which the individual belongs. More for-
mally, by ordering the types on the basis of their mean such that m1( yt) . . .
mj ( yt) . . . mn( yt), the smoothed distribution corresponding to F( yt) is
deﬁned as Ys t t t t
t ¼ (m1,. . .,mj ,. . .,mN). N is the total size of the population, and mj is
the smoothed income, interpreted as the value of the opportunity set, of the indi-
vidual ranked j/N in Ys t . Hence, in this model, measuring opportunity inequality
simply amounts to measuring inequality in the smoothed distribution Ys t.
Some authors have questioned this “utilitarian” approach (see Fleurbaey
2008 for a discussion of the issue). For instance, some authors argue that in addi-
tion to circumstances and effort, an additional factor, luck, plays a role in deter-
mining the individual outcome (see, inter alia, Van de Gaer 1993; LeFranc et al.
2008, 2009). Therefore, they argue, only part of within-type heterogeneity can
be directly attributable to differences in effort. In particular, the unequal
254 THE WORLD BANK ECONOMIC REVIEW

outcomes resulting from “brute” luck should be compensated for.9 Furthermore,
these authors argue, individuals may be risk averse; hence, the within-type in-
equality may have a cost for them. Following this line of reasoning, alternative
models of EOp that consider within-type heterogeneity have been proposed in
the literature.10
The model adopted in this paper, based on the assumption of within-type in-
equality neutrality and the use of the mean income conditional on each type as
the value of the opportunity set, is well grounded on normative reasons and, in
particular, is consistent with a strong version of the reward principle; see
Fleurbaey (2008) and Fleurbaey and Peragine (2013) for a discussion. However,
it is also motivated by practical reasons; accounting for within-type heterogenei-
ty is very demanding in terms of data. It is often the case that the small size of the
samples used makes it difﬁcult to obtain easily comparable within-type distribu-
tions. This approach makes our empirical analysis fully consistent with most of
the analyses performed in the existing literature.11 Nevertheless, although our
theoretical model is built on the assumption of within-type neutrality, we
explore the issue of within-type heterogeneity in the empirical section by looking
at growth within each type. It is shown that the dynamic of inequality within
types can be a source of divergence between the standard approach based on
income inequality and the opportunity egalitarian approach.
A ﬁnal methodological consideration is in order here and concerns the issue of
omitted circumstance variables. We use a pure deterministic model where, given
a set of selected circumstances, any residual variation in individual income is
attributed to personal effort. This amounts to saying that once the vector of cir-
cumstances has been deﬁned, on the basis of normative grounds and observabili-
ty constraints, all other factors are implicitly classiﬁed as within the sphere of
individual responsibility. However, the vector c observed in any particular
dataset is likely to be a sub-vector of the theoretical vector of all possible circum-
stances that determine a person’s outcome. Whenever the dimension of the ob-
served vector c is less than the dimension of the “true” vector, then we obtain
lower-bound estimators of true inequality of opportunity; that is, the inequality

9. The literature distinguishes between brute luck, which is unrelated to individual choices and hence
deserves compensation, and option luck, which is a risk that individuals deliberately assume and does not
call for compensation. See Ramos and Van de Gaer (2012), Fleurbaey (2008), and LeFranc et al. (2009)
for a detailed discussion of the different meanings of luck.
10. For example, LeFranc et al. (2008) and Peragine and Serlenga (2008) use stochastic dominance
conditions to compare the different type distributions. Moreover, LeFranc et al. (2008) measure the
opportunity set as (twice) the surface under the generalized Lorenz curve of the income distribution of the
individual’s type, that is mi (1 2 Gi), where the type mean income mi and (1 2 Gi) represent, respectively,
the return component and the risk component, with Gi denoting the Gini inequality index within type i.
See also O’Neill et al. (2000) and Nilsson (2005) for empirical analyses that attempt to provide alternative
evaluations of opportunity sets using parametric estimates.
11. As discussed in Brunori et al. (2013), the (ex ante) utilitarian approach has been by now adopted
by several authors to assess IOp in about 41 different countries, making an international comparison of
inequality of opportunity estimates across the world possible.
Peragine, Palmisano, and Brunori 255

that would be captured by observing the full vector of circumstances. The impli-
cation is that the empirical estimates obtained using this model should be inter-
preted as lower-bound estimates of IOp.12 Similarly, it is worth underlining that
whenever circumstances are partially unobservable, the change in IOp due to
growth should be interpreted as the change in the lower bound IOp conditioned
to the observable circumstances. An evaluation of change in IOp based on a dif-
ferent set of variables could lead to different conclusions.

The Opportunity Growth Incidence Curve
In this section, we introduce the two versions of the Opportunity Growth
Incidence Curve (OGIC), which can be considered complementary tools to the
GIC, to improve the understanding of the distributional features of growth when
an opportunity egalitarian perspective is adopted. The two versions, the individ-
ual OGIC and the type OGIC, capture two different intuitions about the rela-
tionship between growth and EOp. The ﬁrst focuses on the impact of growth on
the distribution of opportunities. The second focuses on the relationship between
overall economic growth and type-speciﬁc growth.
Given an initial distribution of income Yt and the corresponding smoothed
distribution Yst introduced in the previous section, the individual OGIC can
simply be obtained by applying the GIC proposed by Ravallion and Chen (2003)
to the smoothed distribution. Hence, the individual OGIC can be deﬁned as
follows:

j mt
j
þ1
go
Ys ¼ t À 1; 8j [ f1; : : :; Ng: ð3Þ
N mj

go
Y s ( j/N) measures the proportionate change in the value of opportunities of the in-
dividuals ranked j/N in the smoothed distributions. Obviously, g o o
Y s( j/N) ! 0( g Y s( j/
N) , 0) means that there has been positive (negative) growth in the value of the op-
portunity set given to the individuals ranked j/N respectively in Ys
t and in
13 s
Ytþ1.
The individual OGIC provides information on the impact of growth on IOp.
Consider the Lorenz curve of Ys t:

j
P
mt
k
j k¼1
LYts ¼ N ; 8k [ f1; : : :; N g; 8t [ f1; : : :; T g: ð4Þ
N P t
mk
k¼1

12. For a discussion of this issue with reference to a non deterministic, parametric model of EOp, see
Ferreira and Gignoux (2011) and Luongo (2011).
13. Note that, given the assumption of anonymity implicit in this framework, the individuals ranked
j/N in t can be different from those ranked j/N in t þ 1.
256 THE WORLD BANK ECONOMIC REVIEW

The individual OGIC deﬁned in eq. (3) can be decomposed in such a way that it
becomes a function of the Lorenz curve deﬁned in eq. (4), as follows:

j
DLYtsþ1
j N
go
Ys ¼ ðg þ 1Þ À 1; 8j [ f1; : : :; N g ð5Þ
N j
DLYts
N

where DLYst( j/N) ¼ mt j /m( yt) is the ﬁrst derivative of LYs t
( j/N) with respect to j/N,
and g ¼ m(Ytþ1)/m( yt) 2 1 is the overall mean income growth rate.
Thus, when growth is proportional, it does not have any impact on the level of
IOp: DLYstþ1( j/N)/DLYst(i/N) ¼ 1, and g o Y s( j/N) will just be an horizontal line, with
goY s( j/N) ¼ g for all j. On the contrary, when growth is progressive (regressive) in
terms of opportunity, growth acts by reducing (worsening) IOp: DLYstþ1( j/N)/
DLYts(i/N) = 1, and g o Y s( j/N) will be a decreasing (increasing) curve.
The main aspect that distinguishes the individual OGIC from the standard
GIC is represented by the distributions used to construct that curve. This varia-
tion allows us to establish a link between growth and IOp. Note that the
smoothed distribution at the base of the individual OGIC is the same used by
Checchi and Peragine (2010) and Ferreira and Gignoux (2011) to measure ex
ante IOp. Therefore, our evaluation of growth based on the individual OGIC is,
by construction, consistent with the IOp index they proposed; other things being
equal, an individual OGIC curve that is downward sloping in all of its domain
implies a reduction in IOp.
However, the individual OGIC is unable to track the evolution of each type
during the growth process. In the smoothed distribution, types are ranked ac-
cording to the value of their opportunity set at each point in time. Thus, the
shape of the curve depends not only on the change in the type-speciﬁc mean
income but also on the type-speciﬁc population share and the reranking of types
taking place during the growth process. Now, although these features are desir-
able when one is interested in studying the evolution of IOp over time, the same
characteristics make it impossible to detect the individual OGIC if there are
groups of the population that are systematically excluded from growth.
However, this can provide valuable information for analysts and policy makers.
For example, consider a very small type that suffers a deterioration of its condi-
tion over time. This information could be irrelevant for the evolution of the
overall opportunity inequality, but it would be extremely important for the
design of tailored policy interventions toward that group.
To address this speciﬁc issue and to investigate the relationship between
overall economic growth and type-speciﬁc growth, we introduce a second
version of the OGIC, which we label the type OGIC.
Letting Ymt ¼ (m1( yt),. . .,mn ( yt)) be the distribution of type mean income at
time t, where types are ordered increasingly according to their mean,
Peragine, Palmisano, and Brunori 257

i.e., m1( yt) . . . mn( yt), and Y ˜ mtþ1 ¼ (m ~ n( ytþ1)) is the distribu-
~ 1( ytþ1),. . .,m
tion of type mean income at time t þ 1, where types are ordered according to
their position at time14 t, we deﬁne the type OGIC as follows:

i ~ ðytþ1 Þ À mi ðyt Þ
m
go
~ ¼ i ; 8i [ f1; : : :; ng: ð6Þ
n mi ðyt Þ

The type OGIC plots, against each type, the variation of the opportunity set of
that type. This can be interpreted as the rate of economic development of each
social group in the population, where these groups are deﬁned on the basis of
initial circumstances. g ˜ o(i/n) is horizontal if each type beneﬁts (loses) in the
same measure from growth. It is negatively ( positively) sloped if the initially
disadvantaged types get higher (lower) beneﬁt from growth than those initially
advantaged.15
The type OGIC differs from the standard GIC in two aspects. The ﬁrst is rep-
resented by the distribution used to plot the curve: the GIC is based on the
income distribution, whereas the OGIC is based on the distribution of opportu-
nity sets. The second is represented by the weakening of the anonymity assump-
tion for types. Thus, the type OGIC, tracking the same type over time, provides
information on the temporal evolution of the opportunity set.
The OGIC, in both the individual and the type versions, can be used to rank
different growth episodes. Analogously with the literature on the standard GIC,
we can apply ﬁrst-order dominance criteria based on the OGIC.16 First-order
dominance implies that the OGIC of a growth spell is everywhere above that of
another.
However, the two approaches (individual and type OGIC) are generally not
equivalent, and they can generate a different ranking of growth processes. In
fact, beyond their interpretation and the fact that they can be used to investigate
different aspects of the relationship between economic growth and EOp, the dif-
ferences between the individual and the type OGIC are mainly due to demo-
graphic and reranking issues. The following remark makes this point clear.
Remark 1. Let YA B
t and Yt be two initial distributions of income, and let G
A

and G B be two different growth processes taking place, respectively, on YA t and
YBt and generating, respectively, two ﬁnal distributions of income, YA
tþ1 and YBtþ1.
A B A B
Moreover, let n and n be the number of types, respectively, in Yt and Yt and
mAi and mBi be the number of individuals in each type i ¼ 1,. . .,n, respectively,
in YA B
t and Yt . If (i) nAt ¼ nBt, 8 t ¼ 1,. . .,T, (ii) mAit ¼ mBit 8i [ f1,. . .,ng,8
t ¼ 1,. . .,T, (iii) no reranking of types, then g ˜ Bo(i/n) 8i [ f1,. . .,ng if
˜ Ao(i/n) X g
Ao Bo
and only if g Y s ( j/N) X g Y s ( j/N) 8j [ f1,. . .,Ng.

14. Note that we track the same type but do not track the same individuals.
15. Note that the type OGIC is a generalization of the idea underlying the ﬁrst component of
Roemer’s (2011) index of development, that is, “how well the most disadvantaged type is doing”.
16. For a normative justiﬁcation of these dominance conditions based on a rank-dependent social
welfare function, see the working paper version of the paper: Peragine et al. (2011).
258 THE WORLD BANK ECONOMIC REVIEW

Proof. See appendix.
This remark establishes that when the two distributions have, at each point in
time (i), the same number of types and (ii) the same type-speciﬁc population size,
and when (iii) types keep their relative position in the type mean income distribu-
tion over time, ranking income distributions according to the individual OGIC is
equivalent to ranking income distributions according the type OGIC. Because
conditions (i) and (ii) basically impose restrictions on the types’ demography and
condition (iii) imposes restrictions on the rank of the types, it is clear that possi-
ble differences in the ordering provided by the two OGICs are determined by
variations in the type’s population shares, between the two distributions and the
two periods compared, and by the reranking of types over time.
Although the conditions in Remark 1 may seem demanding, an interesting
case in which they are met is the comparison of growth processes taking place
on the same initial distribution. This is the standard case in the literature on
microsimulation analyses17 and, in general, in the case of an evaluation of policy
interventions.

The Cumulative OGIC
So far, we have focused on ﬁrst-order OGIC dominance, which is a strong condi-
tion that is rarely veriﬁed with real data. A weaker condition is obtained by
second-order dominance. This order of dominance builds on the deﬁnition of the
cumulative18 OGIC.
To obtain the cumulative OGIC, one should look at the proportionate diffe-
rence between the generalized Lorenz curves applied to the smoothed distribu-
tion at time t and t þ 1, which, after rearranging, gives the following expression
for the individual version:

j
0 1
P k j
go
Ys mt L s
j N k B Ytþ1 N C
Go
Ys ¼ k¼1 ¼B
@ ðg þ 1ÞC
A À 1; 8j [ f1; : : :; N g: ð7Þ
N j
P t j
mk LYt s

k ¼1
N

The cumulative individual OGIC plots the mean income growth rate up to the
jth poorest individual in Y s. It can be downward or upward sloping depending
on the pattern of growth among smoothed incomes. Clearly, at j/N ¼ 1, G o
Y s( j/N)
equals the overall mean income growth rate, g.
The above decomposition allows to express the cumulative OGIC as depend-
ing on two components: the overall mean income change and the variation in the

17. See, inter alia, Sutherland et al. (1999).
18. Similar to the OGIC, the derivation of its cumulative version closely follows the methodology
proposed by Son (2004), adequately adapted to be consistent with the EOp theory.
Peragine, Palmisano, and Brunori 259

level of the IOp. In case of proportional growth, the Lorenz curves do not
change, and the cumulative OGIC is equal to overall mean income growth rate.
On the other hand, the cumulative type OGIC is deﬁned as follows19:

P
i j
go
~ m ðyt Þ
o i j¼1 n j
~
GYm ¼ ; 8i [ f1; : : :; ng: ð8Þ
n P
i
mj ðyt Þ
j¼1

The cumulative type OGIC plots the mean income growth rate up to the type
ranked i in the initial type mean distribution against each type in the population.
It can be downward or upward sloping, depending on the pattern of growth
among types. At i ¼ n, G ˜o
Ym(i/n) equals the overall mean growth rate of Ym.

OGIC Indexes
To avoid inconclusive results because of the partiality of the dominance condi-
tions based on the curves presented so far, we propose aggregate measures of
growth that incorporate some basic EOp principles.
From the individual perspective, adopting a rank-dependent approach to the
evaluation of growth, an aggregate measure of growth consistent with the EOp
theory can be expressed as follows:20

P
N j o j
v gY S
1 j¼1 N N
GY S ¼ : ð9Þ
N PN j
v
j¼1 N

Given the assumption of anonymity of the individual OGIC, the weight v( j/N)
depends on the relative position of individuals in the smoothed distribution, re-
spectively, in t and t þ 1. Thus, the same weight is given to the value of the op-
portunity set of individuals ranked the same in the smoothed distribution of the
two periods21. v( j/N) represents the social evaluation of the growth in the oppor-
tunity enjoyed by individuals in the same position in t and t þ 1.
Thus, eq. (9) represents a rank-dependgent aggregation of the information
provided by each single point of the individual OGIC. In particular, imposing
monotonicity, v( j/N) ! 0, 8j [ f1,. . .,Ng, and opportunity inequality aversion,

19. Similar to the cumulative inividual OGIC, the cumulative type OGIC is obtained by rearranging
˜m .
the difference between the Generalized Lorenz curves applied to the type mean distributions Ymt and Y tþ1

20. The approach is close in spirit to Essama-Nssah (2005), reviewed in a previous section. For a
normative justiﬁcation of the rank-dependent approach to IOp analyses, see Peragine (2002), Aaberge
et al. (2011), and Palmisano (2011)
21. See endnote 12.
260 THE WORLD BANK ECONOMIC REVIEW

v( j/N) ! v( j þ 1/N), 8j [ f1,. . .,N 2 1g, we obtain a measure of opportunity-
sensitive growth. This measure is increasing in each individual opportunity
growth and is more sensitive to the growth in the opportunity experienced by
those individuals with the lowest opportunities. Using the speciﬁcation v( j/N) ¼
2(1 2 j/N), we obtain a Gini-type measure of opportunity-sensitive growth.
If, instead, one is interested in assessing the pure progressivity of growth
without concern for the aggregate growth, then the following index can be
adopted:

OGY S ¼ GY S À GY S ð10Þ

1X N
o j
where GY S ¼ gY S . OGY S ¼ 0 if growth is proportional; it is positive
N j¼1 N
(negative) if growth is progressive (regressive).
An alternative expression can be obtained by using a weighted average of the
growth experienced by each type, with weights incorporating a concern for the
initial condition of the types:

P
n i o i
w g
~
1 i ¼1 n n
GYm ¼ : ð11Þ
n Pn i
w
i¼1 n

The function w(i/n) is the social weight associated to type i and depends on the
rank of the type in the initial distribution of income. As before, this index satisﬁes
monotonicity: w(i/n) ! 0, i [ f1,. . .,ng (that is, aggregate growth is not decreas-
ing in each type growth) and opportunity inequality aversion: w(i/n) ! w(i þ 1/n),
i [ f1,. . .,n 2 1g (that is, more weight is given to the income growth experienced
by the most disadvantaged types).
i P i
Following Aaberge et al. (2011) and choosing w ¼1À q jt , a Gini-
n j¼1
type index of opportunity-sensitive growth results.

TH E EM PI R I CA L AN A LY S ES

This section investigates the distributional changes that occurred in Italy and
Brazil in the last decade. These analyses pursue two additional aims: (i) assessing
the main consequences of the actual economic crisis on the Italian distribution of
income according to the EOp perspective and (ii) assessing the distributional im-
plications of the most recent economic development experienced by Brazil in
terms of EOp.
Peragine, Palmisano, and Brunori 261

For both applications, we ﬁrst provide an assessment of growth according to
the equality of outcome perspective. We then move to the analysis of growth ac-
cording to the EOp perspective.22

Opportunity and Growth in Italy: The Data
Italy is the ﬁrst country considered in this section. This analysis is developed
using the Bank of Italy’s “Survey on Household Income and Wealth” (SHIW), a
representative sample of the Italian resident population interviewed every two
years. Three waves of the survey are considered: 2002, 2006, and 2010 (the latest
available).
The unit of observation is the household, deﬁned as all persons sharing the
same dwelling. The individual outcome is, then, measured as the household
equivalent income in 2010 euro.23 Income includes all household earnings, trans-
fers, pensions, and capital incomes, net taxes, and social security contributions.
The richest and poorest 1 percent of the households in each wave are dropped to
avoid the effect of outliers. To identify the types, the distribution is partitioned
into 18 types using information about three characteristics of the head of the
family: the highest educational attainment of her parents (three levels: up to ele-
mentary school, lower secondary, and higher), the highest occupational status of
her parents (two levels: not in the labor force/blue collar and white collar) and
the geographical area of birth (three areas: North, Centre, and South). Note,
however, that those households for which the identiﬁcation of the type is not
possible because of missing information about one or more circumstances are ex-
cluded. The sample sizes of each wave considered are 6,428 in 2002, 6,354 in
2006, and 6,579 in 2010.
The list of types with their respective opportunity proﬁles24 is reported in
Table 1 for each wave. Types are ranked according to their average income.
Rankings are clearly driven by the regional origin of the household head. In par-
ticular, although some reranking takes place for types of other regions, ﬁve of the
six types from the South of Italy are the lowest-ranked at all times.

22. We calculate conﬁdence intervals for the difference between individual OGIC, type OGIC, and
indexes in the two growth processes. The resampling procedure that we use is in line with the approach
proposed by Lokshin (2008) for the GIC. We assume that the income distributions observed at the two
points in time, y t, y tþ1, are independent and identically distributed observations of the unknown
probabilityp distributions F( y t),F( y tþ1). g is the statistic of interest, and its standard
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ error iss(F( y t),
F( y tþ1)) ¼ Varg ^ ðyt ; ytþ1 Þ. Our bootstrap estimate of the standard error is s ^ ¼s F ^ ðyt Þ; F ^ ðytþ1 , where
^ ðyt Þ; F
F ^ ðytþ1 Þ are the empirical distributions observed. The 95 percent conﬁdence interval is obtained by
resampling B ¼ 1,000 ordinary non parametric bootstrap replications of the two distributions y*
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
*
t , ytþ1.
PB 2
The standard
XB error of parameter g
^ is obtained using s
^ B ¼ b¼1 f g
^ Ã ðbÞ À g ^ ð : Þg = ð B À 1 Þ , where
^ ð:Þ ¼
g b¼1
gÃ ðbÞ/B. We know that s ^B ! s ^ when B ! 1, and, under the assumption that g is
approximately normally distributed, we calculate conﬁdence intervals: g ^¼g ^ + z1Àa=2 s ^B . Our estimate
quality relies on strong assumptions. However, as will be clear in the discussion of the results, dominances
appear rather reliable for the illustrative purpose of the exercise.
23. We use the OECD equivalence scale given by the square root of the household size.
24. All standard errors are obtained using the sample weights according to the suggestion in Banca
d’Italia (2012).
262

T A B L E 1 . Italy 2002-2006 –2010: Descriptive Statistics and Partition in Types
Area Education Occupation rank 02 sample 02 q02
i m02
i rank 06 sample 06 q06
i m06
i rank 10 sample 10 q10
i m10
i

South No-edu/Elementary Blue c./not in l.f. 1 1241 0.2174 14065.82 2 1273 0.2291 15279.71 3 1512 0.2385 14974.97
South Lower secondary Blue c./not in l.f. 2 110 0.0214 14386.26 4 124 0.0214 17783.99 1 198 0.0408 13593.33
South Higher Blue c./not in l.f. 3 137 0.0233 15673.90 1 104 0.0150 14800.64 2 126 0.0214 14749.59
South No-edu/Elementary White c. 4 682 0.1130 16949.30 3 604 0.1098 17149.07 4 594 0.0990 17021.24
South Lower secondary White c. 5 213 0.0324 17917.02 6 230 0.0421 20127.67 5 228 0.0372 17903.09
Centre No-edu/Elementary Blue c./not in l.f. 6 657 0.0822 19477.92 7 604 0.0755 21970.48 9 622 0.0729 23528.86
Centre Lower secondary/ Blue c./not in l.f. 7 51 0.0068 20106.76 12 49 0.0082 26077.04 13 60 0.0111 26010.30
Higher
THE WORLD BANK ECONOMIC REVIEW

North Lower secondary Blue c./not in l.f. 8 135 0.0237 20910.44 10 182 0.0301 24799.79 10 162 0.0294 23548.54
North No-edu/Elementary Blue c./not in l.f. 9 1137 0.1623 22095.60 8 1121 0.1591 23292.56 8 1022 0.1465 23063.41
Centre No-edu/Elementary White c. 10 316 0.0384 22579.76 9 287 0.0401 23873.59 14 260 0.0268 26348.91
South Higher White c. 11 270 0.0406 22828.57 13 239 0.0356 26290.72 11 295 0.0375 24052.45
North No-edu/Elementary White c. 12 594 0.0996 23922.43 11 543 0.0839 25240.80 12 474 0.0709 25209.78
Centre Lower secondary White c. 13 107 0.0187 24702.06 16 93 0.0128 30371.49 16 119 0.0202 28257.28
North Higher Blue c./not in l.f. 14 71 0.0094 25625.36 14 94 0.0140 27060.96 7 100 0.0160 22652.13
Centre Higher Blue c./not in l.f. 15 32 0.0039 25664.17 5 45 0.0059 20096.12 6 30 0.0034 21798.12
North Lower secondary White c. 16 253 0.0421 26890.26 15 250 0.0387 27748.28 15 247 0.0471 27114.15
North Higher White c. 17 296 0.0452 29955.46 17 363 0.0519 32143.62 18 343 0.0543 32106.09
Centre Higher White c. 18 126 0.0197 30786.71 18 149 0.0268 33395.35 17 187 0.0268 30670.72

Note: Types are ranked in ascending order according to the average income at the beginning of each growth period.
Source: Authors’ calculations on SHIW (Banca d’Italia).
Peragine, Palmisano, and Brunori 263

To analyze growth, we consider two four-year periods: 2002–06 and 2006–10.
The exercise is appealing because it compares two periods during which Italy
faced two different economic slowdowns. The former was characterized by the
almost total absence of growth in 2002 and 2003. The latter, triggered by the
2008 ﬁnancial crisis, was characterized by a deep fall in the GDP growth rate in
2008 and, after a slight respite between 2009 and 2010, is ongoing.

Opportunity and Growth in Italy: The Results
The GICs for the two periods are reported in Figure 1. These curves are obtained
by partitioning the distribution into percentiles and by plotting against each per-
centile its speciﬁc growth rate, expressed in yearly percentage points.
Two features stand out. First, the GICs for the two periods lie in two different
domains: positive for the ﬁrst period and negative for the second period, with the
exception of the last percentile. This feature is further captured by the mean
income growth rate relative to each period, which is 1.96 percent for 2002–06
and 2 0.66 percent for 2006–10. Second, the two growth processes show very
different and symmetric patterns. The income dynamic is progressive between
2002 and 2006, but it becomes quite regressive between 2006 and 2010. Their
symmetrical shape suggests that the two processes might have an equally
opposed redistributional impact. The sign of the variation over time of their re-
spective aggregate indexes of inequality conﬁrms this supposition: income in-
equality decreases during the ﬁrst period and increases during the second
period25 (see Table 2 in the appendix).
We proceed in our analysis with the assessment of the distributional effects of
growth in the space of “opportunities”. The individual OGIC for the periods
considered are reported in Figure 2.
The individual OGIC of 2002–06 shows that growth acts by increasing the
value of the opportunities for all quantiles of the smoothed distributions.26
However, the growth rate is not stable across quantiles. In particular, the slightly
increasing pattern of the individual OGIC over the whole distribution demon-
strates an opportunity-regressive impact of growth.
The peculiarities of this growth process are conﬁrmed by the value of the syn-
thetic measures of growth (see Table 2 in the appendix). The ﬁrst index, measur-
ing the extent of the opportunity-sensitive growth, is positive, as expected
because the individual OGIC lies above 0. The second index, exclusively captur-
ing the equal opportunity-enhancing effect of growth is negative, demonstrating
that growth might have failed in its role as an instrument to reduce IOp. These
results emphasize the relevance of extending standard analyses of growth to the
space of “opportunity”. For instance, the different shapes characterizing the GIC

25. The results for the second period are consistent with other empirical evidence on the effect of the
last ﬁnancial and economic crisis. See, for example, Jenkins et al. (2013).
26. To make the individual OGIC and the type OGIC graphically comparable, we partitioned the
smoothed distributions into 18 quantiles.
264 THE WORLD BANK ECONOMIC REVIEW

F I G U R E 1. Italy 2002–2006–2010: Growth Incidence Curve

Source: Authors’ calculation from SHIW (Bank of Italy).

T A B L E 2 . Italy: 2002–2006–2010 Dominance Conditions
quantiles/ cum. type individual cum. individual
types rank GIC type OGIC OGIC OGIC OGIC

1 10.5691*** 2.6484*** 2.6180*** 3.9839*** 3.9744***
2 4.6810*** 11.5799*** 7.3428*** 2.6985*** 3.3317***
3 4.1114*** -1.5181 4.3373*** 2.6562*** 3.1058***
4 4.4694*** 0.5413 3.3125*** 2.5996*** 2.9757***
5 3.6610*** 5.9404*** 3.9061*** 3.4201*** 3.0512***
6 3.3625*** 1.3937 3.3944*** 1.6977*** 2.7881***
7 3.2277*** 7.8721** 4.0511*** 2.8942*** 2.8017***
8 2.8174*** 6.0244*** 4.3561*** 5.4506*** 3.1885***
9 2.5479*** 1.6141** 3.9883*** 1.8843*** 2.9947***
10 2.4750*** -1.1908 3.3700*** 2.1158*** 2.8801***
11 2.3956*** 5.7042*** 3.6263*** 1.5239*** 2.7224***
12 2.7012*** 1.4691 3.3977*** 1.3037*** 2.5751***
13 2.8946*** 7.2706** 3.8027*** 2.6333*** 2.5808***
14 2.7802*** 5.3008** 3.9270*** 2.6164*** 2.5835***
15 2.4743*** -7.5717** 3.0613*** 1.8334*** 2.5185***
16 2.9552*** 1.4023 2.9156*** 2.6758*** 2.5292***
17 1.8412*** 1.9006 2.8161*** 3.4850*** 2.6090***
18 0.3548 4.2672** 2.9169*** 2.5781*** 2.6063***

* ¼ 90 percent, ** ¼ 95 percent, *** ¼ 99 percent are signiﬁcance levels for the difference
between curves obtained from 1,000 bootstrap replications of the statistics.
Source: Authors’ calculations on SHIW (Banca d’Italia).
Peragine, Palmisano, and Brunori 265

F I G U R E 2. Italy 2002–2006–2010: Individual Opportunity Growth Incidence
Curve

Source: Authors’ calculation from SHIW (Bank of Italy).

and the individual OGIC explain the diverging trends of inequality of outcome
compared to the trend of IOp: inequality of outcome decreases, whereas IOp
increases.
For the second period, the 2006–10 individual OGIC lies below zero for most
of the distribution, suggesting that growth generates a reduction in the values of
the opportunities enjoyed by individuals. In particular, it appears that the highest
cost of the recession is borne by the individuals in the poorest quantiles of the
smoothed distributions. Furthermore, similar to the previous period, the individ-
ual OGIC for 2006–10 shows an increasing trend, implying that growth might
have acted by worsening opportunity inequality. The severe consequences of the
recession are also captured by the two synthetic measures of growth, which both
take a negative value.
Turning now to the comparison of the two episodes, the results are clear. The
individual OGIC of 2002–06 lies always above the individual OGIC of 2006–10,
and the dominance is statistically signiﬁcant at all points of the curves.27.
Hence, the growth process in 2002–06 dominates the growth process in
2006–10 when both the extent of growth and progressivity components are
considered. However, if we want to focus exclusively on their opportunity-
redistributive impact (that is, on the extent to which these processes act by increas-
ing or reducing IOp), the dominance is not clear because they both show a

27. This dominance is conﬁrmed by the comparison of their cumulative individual OGICs (ﬁgures
and data available upon request).
266 THE WORLD BANK ECONOMIC REVIEW

regressive pattern. It can be helpful, in this case, to compare the values of their re-
spective opportunity-equalizing indexes, which show that 2002–06 is, with statis-
tical signiﬁcance, less regressive than 2006–10.
We can conclude that both of the income dynamics under scrutiny act by in-
creasing IOp. However, whereas this trend is consistent with the change in
outcome inequality in the second period, in the ﬁrst period, the variation of
outcome inequality and the variation of opportunity inequality are in the oppo-
site direction. This result reveals that a conﬂict may arise in the evaluation of
growth when these two different perspectives are adopted for the assessment of
the same growth process.
It is interesting to examine why such a conﬂict arises. If inequality between
types increases while overall outcome inequality declines, the within-type share
of total inequality must necessarily decline.28 From this perspective, it may be
helpful to look at Figure 3, which reports the GICs within types for the nine
poorest and the nine richest types in each process. As expected, growth is pro-
gressive in both the poorest and richest types, with an higher average growth in
the richest type.29 This within-type dynamic explains the divergence between the
income- and opportunity-based distributional assessments.
Turning the focus to the type-speciﬁc growth, the picture changes dramati-
cally. The type OGIC for 2002–06, reported in Figure 4, does not always lie
above zero for the whole distribution; in particular, the types ranked 3 and 15 ex-
perience a loss. Most importantly, the shape of the type OGIC differs signiﬁ-
cantly from the shape of the individual OGIC. According to this perspective,
growth can no longer be classiﬁed as regressive. For the Italian case, this is equiv-
alent to saying that households whose heads were born in the South grow, on
average, less than households with different geographical origins.30
The type population share and the anonymity implicit in the individual OGIC
explain why a regressive individual OGIC is coupled with a non-regressive type
OGIC. The smoothed distribution, constructed to evaluate distributional phe-
nomena from an EOp perspective, ranks the types according to their average
income at each point in time. Hence, growth is evaluated by comparing the
average of different types whenever there is a reranking of types over time. In
contrast, the type OGIC tracks types over time. Hence, types are ranked accord-
ing to their average income at the initial period of time. Whenever there is a
reranking of types over time, some GIC-OGIC divergence may emerge.
For the second growth process, the 2006–10 type OGIC shows some similari-
ty to the individual OGIC of the same period. In particular, most of the types ex-
perience a reduction in the value of their opportunity set, and this reduction is
higher for the disadvantaged types. In sum, both the individual and the type

28. Note that in these empirical applications, the inequality measure used is additively decomposable
for within and between groups.
29. We aggregate types to have sufﬁcient observations in each quantile of the within-type GIC.
30. As reported in Table 1, the circumstance “head born in the South” appears in the ﬁve poorest
types in 2002 and 2010 and in the four poorest types in 2006.
Peragine, Palmisano, and Brunori 267

F I G U R E 3. Italy 2002–2006–2010: Within-Types Growth Incidence Curve

Source: Authors’ calculation from SHIW (Bank of Italy).

F I G U R E 4. Italy 2002–2006–2010: Type Opportunity Growth Incidence Curve

Source: Authors’ calculation from SHIW (Bank of Italy).
268 THE WORLD BANK ECONOMIC REVIEW

OGIC conﬁrm the negative impact of the crisis in terms of the extent of opportu-
nity and the distribution of opportunity.
Interestingly, the only three types that demonstrate positive growth in this
period share the circumstance of coming from central Italy. This ﬁnding is consis-
tent with the reduction of between-region inequality in Italy due to their different
rates of income decline during the recent economic recession. Whereas the
North-South gap remained stable, the recession narrowed the gap between the
North and the Centre. Among the reasons that may explain this trend is the nega-
tive performance of incomes in the North during the recent slowdown, which is
generally attributed to the decline of the car industry and other manufacturing
sectors, largely developed in Piedmont and Friuli-Venezia-Giulia (Istat, 2012).
A severe crisis in the agricultural sector and a growing service industry (especially
in the health care sector) may explain, at least in part, the diverging trend of the
Southern and Central regions.
The comparison of the two growth episodes is less clear because they have a
specular shape: types that beneﬁt most from growth during the ﬁrst process are
those that lose more during the second. The two type OGICs intersect more than
once; hence, it is not possible to establish a ranking between the two growth pro-
cesses.31 It is possible to obtain an unambiguous ordering by weakening the
dominance conditions and comparing the cumulative type OGICs. We ﬁnd that
the ﬁrst process dominates the second and that this dominance is always statisti-
cally signiﬁcant. This result is also supported by the comparison of the synthetic
measures of growth between the two periods. The index evaluating the extent of
growth, with concern for the growth experienced by the initially disadvantaged
types, is positive for the ﬁrst period and negative for the second, and their diffe-
rence is statistically signiﬁcant (see Table 3).
It is not an easy task to understand the driving forces of these transformations.
Given that, by deﬁnition, the rank of types and income are correlated, it is ex-
tremely difﬁcult to disentangle the changes that may have affected, in opposite
directions, the distribution of outcome and the distribution of opportunities.32
However, the trend of the North-South divide and labor market reforms may be
considered among the determinants of redistribution since 2002. First, the differ-
ent reforms realized in the recent past to reduce the gap in the opportunities ac-
cessible to different individuals have not been able to fulﬁll the desired goal. In
particular, as shown by Pavolini (2011), among others, different public services,
particularly different measures and interventions of the welfare state, are still
suffering from territorial divergences with consequences in terms of an increase
in IOp over time, as witnessed by the lower growth rates experienced by the
Southern types.

31. Although the ﬁrst process is better than the second and the dominance is statistically signiﬁcant for
most of the types, for type 15, the second process is preferred to the ﬁrst one with statistical signiﬁcance.
32. This may be a challenging question for future research.
T A B L E 3 . Italy: 2002–2006–2008 Complete Rankings and Inequality
2002 2006 2010

m( y) eq. 20116.82 (4735.42) 21692.12 (5275.08) 21117.34 (5445.91)
mld (all) 0.1422 (0.0026) 0.1301 (0.0021) 0.1437 (0.0027)
mld (between) 0.0256 (0.0006) 0.0274 (0.0001) 0.0313 (0.0007)
‘02-‘06 ‘06-‘10
GYs 1.821 (0.0145) 2 0.9532 (0.0155)
OGYs 2 0.0946 (0.0080) 2 2.869 ( 0.0244)
GYm 2 0.2340 (0.3707) 2 1.2618 (0.0197)

Note: mld ¼ mean logarithmic deviation or generalized entropy index with parameter 0, GYs ¼ EOp consistent aggregate measures of growth (eq. 9),
OGYs ¼ EOp consistent aggregate measures of growth progressivity (eq. 10), GYm ¼ Aggregate measure of between-type inequality of growth (eq. 11); 95
percent bootstrapped standard errors are reported in parenthesis.
Source: Authors’ calculations on SHIW (Banca d’Italia).
Peragine, Palmisano, and Brunori
269
270 THE WORLD BANK ECONOMIC REVIEW

Second, the labor market reforms introduced in 1998 and extended in 2000
and 2003, which mainly aimed to reduce the labor protection legislation ( partic-
ularly for temporary workers), have increased wage ﬂexibility and job turnover,
increasing the “instability” in the opportunity faced by individuals (Jappelli and
Pistaferri, 2009). This instability may explain why growth appears more oppor-
tunity regressive in the second period, a period of crisis. Boeri and Garibaldi
(2007) suggest that although job ﬂexibility generates instability, it may provide
more job opportunities during periods of positive growth. This is not the case
during recessions because these workers, in all categories of atypical job con-
tracts, are more likely to be ﬁred and are often excluded from social security ben-
eﬁts. We suggest that such an effect has been stronger in the southern regions,
thereby explaining the territorial gradient in the diverging trends of different
types.

Opportunity and Growth in Brazil: The Data
Our theoretical framework may be of particular interest in the analysis of devel-
oping and emerging economies that experience lively growth processes with a dra-
matic impact on poverty and redistribution. For this reason, the second country
considered in this paper is Brazil. To perform this analysis, the 2002, 2005, and
2008 waves of the Brazilian Pesquisa Nacional por Amostra de Domicı ´lios
(PNAD), a representative survey of the Brazilian population, are used.
The unit of observation is the household, and the individual outcome is mea-
sured as the monthly household equivalent income, expressed in 2008 Brazilian
real.33 Household income is computed as the sum of all household members’ in-
dividual incomes, including earnings from all jobs, and all other reported
income, including income from assets, pensions, and transfers.
The population is partitioned into 15 types using the information on two cir-
cumstances: region of birth and race. Region of birth is coded in ﬁve categories
(North, Northeast, Southeast, South, Center-west), and race is coded in three cat-
egories (white/east Asian, black/mixed race, and indigenous). Individuals who
were born abroad and those classiﬁed as “other” for the variable race are exclud-
ed because the number of observations is too low to make appropriate inference.
Hence, the sample sizes of each wave considered in this analysis are as follows:
366,388 households in 2002, 390,046 in 2005, and 372,581 in 2008.34
The full opportunity proﬁles for the three waves are reported in Table 4 in the
appendix.35 In this table, it is clear that race is the main determinant of the dis-
parity in opportunities. Consistent with a number of contributions on socio-
economic inequality in Brazil, racial relationships appear to be the major source
of outcome and opportunity inequality in Brazil (Telles 2004; Bourguignon et al.
2007; among others).

33. Equivalent income is obtained by dividing total income by the square root of the household size.
34. Again, the richest and poorest 1 percent of the household distribution in each wave are dropped.
35. All estimates are based on the sample weights according to Silva et al. (2002).
T A B L E 4 . Brazil: 2002–2005–2008 Descriptive Statistics and Partition in Types
Region Race rank 02 sample 02 q02
i m02
i rank 05 sample 05 q05
i m05
i rank 08 sample 08 q08
i m08
i

Northeast black-mixed 1 91118 0.2227 516.73 2 97846 0.2229 550.09 1 93547 0.2272 695.64
Northeast indigenous 2 299 0.0007 576.47 6 309 0.0006 702.42 2 398 0.0010 715.49
North black-mixed 3 25874 0.0381 631.47 3 35053 0.0542 604.64 3 33200 0.0556 769.59
South black-mixed 4 10121 0.0270 683.06 7 11549 0.0292 748.19 6 12006 0.0319 937.98
Southeast black-mixed 5 42007 0.1448 768.61 9 48800 0.1606 806.90 8 47725 0.1633 969.41
Center-west black-mixed 6 16052 0.0300 777.33 8 17223 0.0306 799.66 10 17472 0.0321 1006.28
Center-west indigenous 7 154 0.0003 806.05 1 136 0.0002 444.41 4 175 0.0003 859.83
Northeast white-east asian 8 42720 0.1094 821.07 10 42911 0.1017 823.36 9 40880 0.1018 975.68
South indigenous 9 119 0.0002 866.19 5 128 0.0003 628.87 7 183 0.0005 940.65
North indigenous 10 98 0.0002 879.60 4 206 0.0002 622.59 5 236 0.0003 861.65
North white-east asian 11 9916 0.0146 970.79 11 11088 0.0167 903.47 11 9942 0.0164 1102.10
Southeast indigenous 12 117 0.0004 1082.98 12 105 0.0004 1011.33 12 153 0.0005 1192.87
South white-east asian 13 49021 0.1311 1169.46 14 49133 0.1244 1229.42 14 44957 0.1198 1456.16
Center-west white-east asian 14 12717 0.0244 1179.96 13 13147 0.0238 1176.54 13 12642 0.0236 1433.06
Southeast white-east asian 15 66055 0.2561 1385.93 15 62412 0.2341 1387.16 15 59065 0.2255 1613.84

Note: Types are ranked in ascending order according to the average income at the beginning of each growth period.
´stica).
Source: Authors’ calculations on PNAD (Instituto Brasileiro de Geograﬁa e Estatı
Peragine, Palmisano, and Brunori
271
272 THE WORLD BANK ECONOMIC REVIEW

To analyze the distributional impact of growth in Brazil according to the EOp
perspective, two three-year period growth processes are considered: 2002–05
and 2005–08. The choice of these particular periods is driven by the observation
that during these years, Brazil experienced quite diverging economic trends. The
former was a period of economic slowdown; the PNAD data record an increase
in the overall mean income of only 0.26 percent. In contrast, the latter period
was a period of pronounced growth, with an overall mean income growth of ap-
proximately 6.36 percent.

Opportunity and Growth in Brazil: The Results
As in the ﬁrst illustration, we begin this analysis with the assessment of growth
according to the equality of outcome perspective. The GICs for the two periods
considered are reported in Figure 5.
Although both curves lie almost always above zero, growth is outstanding in
the second period. In fact, it is possible to unambiguously order the two
growth processes because the difference between the GIC coordinates in the
two periods is always statistically signiﬁcant (see Table 5 in the appendix). The
redistributive impact of the two processes is very similar. The respective curves
are both neatly decreasing, demonstrating that growth acts by alleviating
outcome inequality.
We now proceed in the evaluation of the Brazilian growth by endorsing an
opportunity-egalitarian perspective. The individual OGICs for the two growth
episodes are reported in Figure 6.
One feature stands out. For the 2002–05 growth episode, although the GIC
lies almost always above zero, the individual OGIC is positive only for half of the
smoothed distribution. This conﬂict indicates that although the majority of
households experience positive growth, the extent of the losses borne by the
richest 15 percent is substantial in determining the change in the value of the op-
portunity sets. This effect is plausible whenever the richest households are not
concentrated only in the richest type; that is, income quantiles and types are not
perfectly correlated, as for the case of Brazil during 2002–05.
This does not happen during 2005–08, when the individual OGIC lies above
zero, implying that growth plays a positive role in determining an improvement
of the opportunities faced by the entire population. As a result, the second
process also dominates the ﬁrst when an opportunity-egalitarian perspective is
adopted, and the dominance is statistically signiﬁcant (see Table 5 in the appen-
dix). The sign of the dominance is also conﬁrmed by the plot of the cumulative
individual OGIC. The progressivity of the two growth episodes is clariﬁed by the
decreasing shape of the two curves. These results are further supported by the es-
timation of the synthetic measures of growth. The index capturing the
opportunity-sensitive extent of growth is positive for both the 2002–05 and
2005–08 processes, but it is higher for 2005–08. In the same way, the value of
the index capturing the progressivity of growth, in terms of equality of
Peragine, Palmisano, and Brunori 273

F I G U R E 5. Brazil: 2002–2005–2008 Growth Incidence Curve

´stica).
Source: Authors’ calculation from PNAD (Instituto Brasileiro de Geograﬁa e Estatı

T A B L E 5 . Brazil: 2002–2005–2008 Dominance Conditions
quantiles/ cum. type individual cum. individual
types rank GIC type OGIC OGIC OGIC OGIC

1 5.9040*** 29.4150*** 29.4150*** 9.7517*** 9.7517***
2 6.3070*** 0.9296 13.9522*** 5.3240*** 7.3602***
3 6.5042*** 10.7992*** 12.6996*** 5.4259*** 6.6773***
4 6.6490*** 8.7660** 11.5471*** 6.5167*** 6.6375***
5 6.7888*** 18.3645*** 12.9442*** 8.3194*** 7.0596***
6 6.9937*** 2 1.1505 10.0393*** 6.6153*** 6.9932***
7 6.6517*** 23.7151*** 12.5330*** 6.6142*** 6.9419***
8 6.6463*** 9.3222*** 12.0397*** 6.6130*** 6.9018***
9 6.6600*** 16.2690*** 12.5658*** 6.6119*** 6.8700***
10 6.3123*** 15.8542*** 12.9423*** 6.8518*** 6.8707***
11 6.2099*** 8.8956*** 12.4650*** 6.6791*** 6.8519***
12 6.0596*** 5.1427 11.5463*** 5.3881*** 6.7002***
13 6.0661*** 5.6651*** 10.8799*** 5.3695*** 6.5675***
14 6.2796*** 6.6171*** 10.4224*** 5.2087*** 6.4396***
15 5.7342*** 5.4429*** 9.8781*** 5.2070*** 6.3336***

* ¼ 90 percent, ** ¼ 95 percent, *** ¼ 99 percent are signiﬁcance levels for the difference
between curves obtained by 1,000 bootstrap replications of the statistics.
´stica).
Source: Authors’ calculations on PNAD (Instituto Brasileiro de Geograﬁa e Estatı

opportunity, is positive for both processes. This means that during the two
periods, growth acts by alleviating the disparities in opportunities, but this effect
is stronger for the 2005–08 process (see Table 6 in the appendix).
274 THE WORLD BANK ECONOMIC REVIEW

F I G U R E 6. Brazil: 2002–2005–2008 Individual Opportunity Growth Incidence
Curve

´stica).
Source: Authors’ calculation from PNAD (Instituto Brasileiro de Geograﬁa e Estatı

Similar features characterize the assessment of growth when the focus is on
the type-speciﬁc growth. Figure 7 reports the type OGICs for the 2002–05 and
2005–08 periods.
Regarding the ﬁrst period, it is possible to observe that, consistent with the in-
dividual OGIC, most of the types experience a reduction in the value of their op-
portunity set. These types particularly include households with an indigenous
head.36 However, the curve does not appear to show a clear pattern; it is progres-
sive for the lowest part of the distribution up to type 7 and then takes a clear re-
gressive shape. The unstable trend is conﬁrmed by the negative value of the
opportunity-sensitive growth measure. It can thus be inferred that the negative
growth experienced by certain types more than compensates for the positive
growth experienced by the poorest types.
For the second period, the positive distributional implications of the growth
process are again conﬁrmed by the type-speciﬁc OGIC. All types experience an
increase in the values of their opportunity set with a quite progressive trend.
These results are supported by the positive value of the index measuring the
extent of opportunity-sensitive growth (see Table 6 in the appendix). Thus, we
can conclude that this growth process is beneﬁcial in terms of opportunity when
both size and distributional aspects are considered.

36. However, recall that this curve does not take into account the relative size of types. In this speciﬁc
case, in fact, the types that experience an increase in the value of their opportunity set represent over 90
percent of the population.
T A B L E 6 . Brazil: 2002–2005–2008 Complete Rankings and Inequality
2002 2005 2008

N 366,388 390,046 372,581
m( y) eq. 934.66 (333.55) 937.057 (324.13) 1113.48 (355.26)
mld (all) 0.4738 (0.0014) 0.4327 (0.00131) 0.3922 (0.0010)
mld (between) 0 0.0672 (0.0004) 0.0618 (0.0004) 0.0512 (0.0003)
‘02-‘05 ‘05-‘08
avg. growth. 0.26% 6.36%
GYs 0.7547 (0.0910) 7.4937 (0.1169)
OGYs 0.4384 (0.0559) 0.7891 (0.0566)
GY 2 1.0213 (0.6120) 9.5304 (1.0758)

Note: mld ¼ mean logarithmic deviation or generalized entropy index with parameter 0, GYs ¼ EOp consistent aggregate measures of growth (eq. 9),
OGYs ¼ EOp consistent aggregate measures of growth progressivity (eq. 10), GYm ¼ Aggregate measure of between type inequality of growth (eq. 11), 95
percent bootstrapped standard errors are reported in parenthesis.
´stica).
Source: Authors’ calculations on PNAD (Instituto Brasileiro de Geograﬁa e Estatı
Peragine, Palmisano, and Brunori
275
276 THE WORLD BANK ECONOMIC REVIEW

F I G U R E 7. Brazil: 2002–2005–2008 Type Opportunity Growth Incidence
Curve

´stica).
Source: Authors’ calculation from PNAD (Instituto Brasileiro de Geograﬁa e Estatı

As is reasonable to expect, the comparison of the two processes highlights an
unambiguous dominance of the second period growth over the ﬁrst. The diffe-
rence in the OGIC coordinates is statistically signiﬁcant for almost all types, as
shown in Table 5 in the appendix. For robustness purposes, we also test the diffe-
rence of the respective cumulative type OGICs coordinates, which is clearly stat-
istically signiﬁcant for all types, and the difference, which is again signiﬁcant, of
their aggregate index of growth (see Table 6 in the appendix).
Finally, Figure 8, reporting the within-type GICs, explains how the progressive
growth of Brazil between 2002 and 2005 is the joint effect of a reduction of
between- and-within type inequality. The four within-type GICs are downward
sloping, and the average growth rate in the poorest seven types is higher in both
cases than the same rate in the eight richest types.
This considerable change in the overall inequality for the time span considered
is well known in the literature. Ferreira et al. (2008) suggest a number of determi-
nants of this change: the decline in inequality between educational subgroups, a
reduction in the urban-rural gap, a reduction of inequalities between racial
groups, a dramatic increase in the minimum wage, and improvements in social
protection programs. Clearly, these variables have a direct impact on inequality
of outcome and on the distribution of opportunities. Moreover, our analysis
shows that these growth processes have been beneﬁcial in terms of improving op-
portunities and that Brazil has experienced an impressive increase in the degree
of EOp, particularly during the 2002–05 period. Our conclusions complement
Peragine, Palmisano, and Brunori 277

F I G U R E 8. Brazil: 2002–2005–2008 Within-Types Growth Incidence Curve

´stica).
Source: Authors’ calculation from PNAD (Instituto Brasileiro de Geograﬁa e Estatı

the ﬁndings of Molinas et al. (2011), who look at the development of IOp in
Brazil with a speciﬁc focus on the opportunities of children.

CONCLUSIONS

In this paper, we have argued that a better understanding of the relationship
between inequality and growth can be obtained by shifting the analysis from
ﬁnal achievements to opportunities.
To this end, we have introduced the individual OGIC and the type OGIC. The
former can be used to infer the role of growth in the evolution of IOp over time. The
latter can be used to evaluate the income dynamics of speciﬁc groups of the popula-
tion. For both versions of the OGIC, we have also proposed aggregate indices that
can be used to measure the distributional impact of growth from the EOp perspec-
tive when it is not possible to rank growth episodes through the use of curves. We
have shown that possible divergences in the rankings obtained through the use of
the individual OGIC and the type OGIC are mostly due to demographic issues.
We have provided two empirical applications, for Italy and for Brazil. These
analyses show that the measurement framework we have introduced can be used
to complement existing tools for the evaluation of the distributional implications
of growth. Moreover, our tools appear to be potentially relevant for the
278 THE WORLD BANK ECONOMIC REVIEW

understanding of the joint dynamic of income inequality and inequality of op-
portunity. Another ﬁeld of application of our framework is the analysis of tax-
beneﬁt systems of reforms. Typically, the distributional aspects of these reforms
are analyzed through microsimulation techniques and are evaluated in terms of
income inequality reduction. Comparing reforms with the help of the tools devel-
oped in this paper, which allow the evaluation of the IOp reduction, seems a
promising path for future research.

APPENDIX

Proof of Remark 1 We start by showing the sufﬁciency that the individual
OGIC implies the type OGIC dominance. Let the two type OGICs be deﬁned
as follows: gAo ði/nA Þ ¼ m
~ ~A A
i ðytþ1 Þ/mi ðyt Þ À 1 8i [ f1,. . .,nAg and
Bo B B
g ði/nB Þ ¼ m
~ ~ i ðytþ1 Þ/mi ðyt Þ À 1 8i [ f1,. . .,nBg. If (i) holds and there is type
OGIC dominance between the two growth processes G A and G B, we will have
the following:

i i ~ B ðytþ1 Þ
~ A ðytþ1 Þ m
m
gAo
~ gBo
!~ , iA ! iB ; 8i [ f1; : : :; ng: ð12Þ
n n mi ðyt Þ mi ðyt Þ

If (iii) holds, the type OGIC dominance of the growth processes G A over G B
will be

i i mA ðytþ1 Þ mB ðytþ1 Þ
gAo
~ gBo
!~ , iA ! iB ; 8i [ f1; : : :; ng ð13Þ
n n mi ðyt Þ mi ðyt Þ

where mi ( ytþ1) is the mean income of the type ranked i in the ﬁnal distribution
of the types’ mean income, which, under (iii), corresponds to m ~ i ( ytþ1). Now, let
the two individual OGICs be deﬁned as follows: g Y s ( j/NA) ¼ mAt
Ao
j
þ1
/mj At 2 1
8j ¼ 1,. . .,NA and g Bo Btþ1
Y s ( j / NB ) ¼ m j /mj Bt 2 1 8j ¼ 1,. . .,NB. (i) and (ii)
implies NA ¼ NB. Hence, if there is individual OGIC dominance of the growth
process G A over G B, we will have the following:

j j mAt
j
þ1
mBt
j
þ1
gAo
Ys
Bo
! gY s , At ! Bt 8j [ f1; : : :; N g: ð14Þ
N N mj mj

Now, for the individuals j belonging to type i, given (ii) and because we use
smoothed income, we can write eq. (13) in terms of (14):

Aoi Bo i
m
X it þ1
mAt
j
þ1 m
X itþ1
mBt
j
þ1
g
~ g
!~ , ! 8i [ f1; : : :; ng: ð15Þ
n n j ¼1
mAt
j j¼1
mBt
j
Peragine, Palmisano, and Brunori 279

If eq. (14) holds, than it must be the case that the dominance in their type aggre-
gation holds, providing the dominance in eq. (15). Hence we have proved the
sufﬁciency of the remark.
We now prove the necessary condition by contradiction.
Suppose that eq. (15) holds. Now, pick a type i [ f1,. . .,ng. Assume that for
that type 9kf1,. . .,mig such that mAt
k /mk , mBt
þ1 At
k
þ1 Bt
/mk , then because all individ-
Pmi mAtþ1 Xmi mBtþ1
j j
uals in the same type are given the same mean income, À ,0
j¼1 mj
At
j¼1
mBtj
for a given type i, contradicting eq. (15). QED

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